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Diocesan Response to the Sexual Abuse Crisis
Home → Diocesan Response to the Sexual Abuse Crisis
https://www.fallriverdiocese.org/wp-content/uploads/2020/12/Video-Intro-to-List.mp4
Dear Friends in Christ,
As we begin this New Year, I pray that God will bless us all with His grace and bring peace to those in our community who have suffered greatly during the pandemic and the many other challenges we have all faced in 2020.
The scourge of clergy sexual abuse has deeply wounded so many people in our Church. It has touched every diocese worldwide and continues to affect us all – laity and clergy – in significant ways. Today, it is with a contrite heart and commitment to the healing process that I have published a list of clergy, diocesan and religious related to the Diocese of Fall River who have been credibly or publicly accused of committing sexual abuse of a minor.
The review of Diocesan records, some going back 70 years, was incredibly arduous and time consuming. While this review has taken longer than first anticipated, it was crucial that we took the time needed to do it right.
To all survivors of child sexual abuse, I am deeply sorry. We as a Church failed when you were most vulnerable. I firmly resolve to do all we can to help you heal, make certain we are accountable, and protect our children so that no other child suffers as you have.
It is part of our sacred mission to remain vigilant in efforts to protect children and vulnerable adults in our parishes, schools, programs and ministries. As we conducted this review of Diocesan records, our Diocese also revamped our Safe Environment program to strengthen our child protection efforts, as well as our protocols for handling allegations and responding to those impacted by abuse.
We require all personnel to submit to a background check, agree to a code of conduct, and complete a Safe Environment Training Program. Our Diocesan Policy for Protecting the Faithful has detailed instructions for Diocesan personnel who receive allegations of abuse. In addition, many Diocesan personnel are mandated reporters. We have a dedicated Victim Assistance Coordinator, a licensed social worker who is the first point of contact when an allegation is received and works closely with me to ensure that we are adequately addressing the rights and needs of any victims.
If you, or anyone you know, have suffered abuse by a priest or someone affiliated with the Diocese of Fall River, I urge you to call our Victim Assistance Coordinator at 508-985-6508 or visit our website. You have my assurance that your voice will be heard.
It is my prayerful and deepest hope that the publication of this list will help in the ongoing healing and care of survivors of clergy sexual abuse. They are of paramount concern to me and remain always in my prayers.
To every one of you who has suffered, I know that finding the ability to trust again is a slow and difficult journey. Together, may we find hope in Jesus Christ, may the Blessed Virgin Mary be a mother to us all, and may God grant us peace.
Sincerely yours in Christ,
Bishop Edgar M. da Cunha, S.D.V.
Bishop of Fall River
View/Download the Bishop's Letter in Portuguese
View/Download the Bishop's Letter in Spanish
Priests Accused of Sexual Abuse of a Minor
In 2019, in a public letter, Bishop da Cunha announced the intention of the Diocese of Fall River to publish a list of clergy against whom credible allegations of sexual abuse of a minor have been made.
The list published here contains the names of clergy who have been incardinated or ministered in the Diocese of Fall River and have been credibly or publicly accused of sexual abuse of a minor. Most of the allegations pertain to abuse alleged to have taken place many decades ago; approximately half of those on this list are deceased.
It is for the survivors of clergy sexual abuse that we publish this list. We know there has been grave damage done and that this is long overdue. It is the Diocese's hope that publishing this list will help bring healing to the survivors and their families who have been so grievously harmed. The publication of this list is also an expression of our sincere commitment to transparency and accountability.
Compiling the List
The Diocese underwent a lengthy process to compile and maintain this list. The first steps involved the collection of 70 plus years of data. This was not clear cut given that files were kept in various locations, were neither digitized nor organized, and included everything from academic records, to vacation requests, to other forms of correspondence. Early on, it was quite clear that, historically, the record-keeping was inadequate and incomplete and that we would not be able to complete our review as quickly as we had hoped. While this review, conducted with the assistance of experts in the field, took longer than first thought, it was essential that we took the time needed to do it right.
An initial evaluation of the files was performed by former Federal Bureau of Investigation (FBI) Assistant Director William Gavin of The Gavin Group. After an initial review of the files, the Diocese engaged Kinsale Management Consulting, under the leadership of Kathleen McChesney, Ph.D., former Executive Assistant Director for the FBI and former head of the Office of Child Protection of the U.S. Conference of Catholic Bishops. Among other things, Kinsale provides expertise on prevention and response to allegations of abuse of minors and vulnerable adults, and in this case conducted a detailed file review to identify allegations against individual priests.
The Diocese then engaged the law firm Ropes & Gray LLP to assist it in creating and publishing this list. Ropes & Gray conducted a further investigation and review of Diocesan records and legal files and, at the conclusion of its review, provided a summary of its findings to a core advisory group from the Diocese.
The Core Advisory Group consisted of:
Fr. Richard Wilson, Vicar General
Fr. John Murray, Moderator of the Curia
Mr. Kevin Kiley, Chancellor
Mrs. Lorraine Levy, Director of Professional Standards and Oversight
Mr. Michael Carroll, General Counsel and Chief Legal Officer
After careful review, the Core Advisory Group presented its findings to The Edward Davis Company to conduct a third-party review (see The Edward Davis Company Review). Led by former Boston Police Commissioner Ed Davis, The Edward Davis Company is comprised of a team of former law enforcement officials with deep knowledge of investigations and experience in protecting victims of crime. The Edward Davis Company was given access to clergy files and other relevant information, in order to advise Bishop da Cunha on whether the process (described above) was thorough, fair, and reasonable, to review the placement and category for each name considered for publication, and to present recommendations to Bishop da Cunha. After review and consideration by Bishop da Cunha, the published list reflects those recommendations.
Though the Diocese has taken great care to prepare this list, we understand that this information may still be imperfect. We appreciated the gravity of this effort and did all we could to make the best judgments based on the information available. We are committed to maintaining this list and updating it if and when new information becomes available.
Criteria and Standard of Proof
The Diocese prepared a standard of review to guide its decisions in preparing this list. It adopted the definitions of "minor" and "sexual abuse of a child" from the Diocesan Policy for Protecting the Faithful.
While the Diocese already had a review board in place, in 2020, it updated its policies and reconstituted the review board as the Ministerial Review Board (MRB). The purpose of the MRB is to review allegations made against clergy and make recommendations to the Bishop. The MRB includes a former Chief Justice of the Massachusetts Appeals Court; a retired Senior Executive of the FBI; a Lieutenant Colonel Judge Advocate (Retired) with the United States Marine Corps with experience in sexual abuse matters; a psychologist specializing in child, adolescent and adult trauma and treatment of youth with sexual behavior problems; a licensed social worker; a registered nurse; and others. A complete list of MRB members and their titles can be found here.
When making determinations about the evidence presented to the MRB, the Ministerial Review Board Policy provides:
6.5 The MRB will be required to assess the credibility of the allegations, specifically whether the allegation has a "semblance of truth" under canon law, using a standard of proof approximating probable cause, and whether the allegations conform to the definition of sexual abuse of a minor as outlined in the policy developed and adopted by the Diocese of Fall River.
The Core Advisory Group adopted this same standard of proof in evaluating the credibility of the allegations made against a priest. It found allegations credible when they had "a 'semblance of truth' under canon law, using a standard of proof approximating probable cause[.]"
The Diocese has published a "Frequently Asked Questions" document that addresses many of the topics raised, including:
The commitments of the Diocese pertaining to eradicating and addressing clergy sexual abuse;
Steps the Diocese takes when it receives an allegation of sexual abuse of a minor or vulnerable adult;
Review of Diocesan policies and procedures;
Prevention of child sexual abuse in Diocesan parishes, schools, and ministries;
Understanding the Ministerial Review Board and how it functions; and
Removal of credibly accused priests.
The list includes the following three main categories. To access each category, please click the category name or links below.
Credibly Accused:
Includes priests, seminarians, and other religious against whom an allegation of sexual abuse of a minor was made that meets the semblance of truth standard defined above and who were incardinated in the Diocese of Fall River and/or assigned to public ministry in the Diocese of Fall River. This category is divided into three subcategories: (A) Incardinated in the Diocese of Fall River; (B) Other Diocese; and (C) Religious Order. For those in sub-categories B and C, respectively, the priest was incardinated in another diocese or was a member of a religious order but exercised public ministry in the Diocese of Fall River. CLICK TO VIEW LIST OF CREDIBLY ACCUSED
Publicly Accused:
Includes priests, seminarians, or other religious who were either incardinated in the Diocese of Fall River or who were assigned to public ministry in the Diocese of Fall River, and have been publicly named in media reports, third party lists prepared by advocacy groups and/or other dioceses and religious orders, as having allegations against them involving sexual abuse of a minor. The individuals included are clergy or other religious against whom a publicized accusation was made, but for whom the Diocese lacks sufficient information at this time to make its own determination whether the accusation was credible. In some cases, a religious order or other diocese may have determined that the priest was credibly accused after investigation or upon admission of the priest.
It is important to note that just because a name appears on the publicly accused list, that does not necessarily mean that the individual was accused of abusing a child in the Diocese of Fall River. For example, a priest may have been publicly accused of abuse in another diocese or may appear on a credibly accused list for a religious order for allegations unrelated to the Diocese of Fall River. If we were able to confirm that the priest also worked in the Diocese of Fall River at some point in his career, we have included the priest on the publicly accused list even if the Diocese did not itself receive an allegation regarding that priest. CLICK TO VIEW LIST OF PUBLICLY ACCUSED
Cases in Process:
Includes living priests of the Diocese of Fall River who have been publicly accused of sexually abusing a minor, but the canonical and/or civil proceedings involving these priests have not yet been resolved. There has not been a determination either under canon (Church) or civil law regarding whether the allegations against clergy listed in this category are credible. Consistent with the principles of the American justice system and Church law, these individuals are afforded a presumption of innocence. The priests in this section are prohibited from engaging in public ministry while their cases are in process. CLICK TO VIEW LIST OF CASES IN PROCESS
Note: This list is provided based on information available at the time it is published. If the need arises to add names to this list based on the criteria explained above, the Diocese will do so.
Lists of Clergy Accused of Sexual Abuse of a Minor
– Credibly Accused
– Publicly Accused
– Cases in Process
Glossary of Terms and FAQs
The Edward Davis Company Review
Report an Incident of Abuse
Clergy List News Release | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,160 |
/// Copyright 2019 Pinterest Inc.
///
/// Licensed under the Apache License, Version 2.0 (the "License");
/// you may not use this file except in compliance with the License.
/// You may obtain a copy of the License at
///
/// http://www.apache.org/licenses/LICENSE-2.0
/// Unless required by applicable law or agreed to in writing, software
/// distributed under the License is distributed on an "AS IS" BASIS,
/// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
/// See the License for the specific language governing permissions and
/// limitations under the License.
#include <algorithm>
#include <string>
#include <tuple>
#include <unordered_set>
#include <vector>
#include "gtest/gtest.h"
#include "librdkafka/rdkafkacpp.h"
#include "common/kafka/tests/mock_kafka_cluster.h"
#include "common/kafka/tests/mock_kafka_consumer.h"
namespace kafka {
class KafkaConsumerTest : public ::testing::Test {
protected:
void SetUp() override {
mock_kafka_cluster_ = std::make_shared<MockKafkaCluster>();
topic_names_ = {"topic0", "topic1", "topic2", "topic3"};
partition_ids_ = {1, 2, 3, 4, 5, 10};
records_ = {{"a", 2}, {"b", 4}, {"c", 8}, {"d", 16}, {"e", 32}, {"f", 64}, {"g", 128}};
for (auto& topic_name : topic_names_) {
for (auto partition_id : partition_ids_) {
for (auto& record : records_) {
mock_kafka_cluster_->AddRecord(
topic_name, partition_id, record.payload, record.timestamp_ms);
}
}
}
}
std::shared_ptr<MockKafkaCluster> mock_kafka_cluster_;
std::vector<std::string> topic_names_;
std::vector<int32_t> partition_ids_;
std::vector<MockKafkaCluster::Record> records_;
};
TEST_F(KafkaConsumerTest, TestSingleTopic) {
const int32_t partition_id = 4;
const std::string topic_name = "topic0";
KafkaConsumer kafka_consumer(std::make_shared<RdKafka::MockKafkaConsumer>(mock_kafka_cluster_),
std::unordered_set<uint32_t>({partition_id}),
std::unordered_set<std::string>({topic_name}),
"UnitTestKafkaConsumer");
ASSERT_TRUE(kafka_consumer.Seek(topic_name, 16 /* timestamp_ms */));
int offset = 3;
RdKafka::Message* message;
for (size_t completed_topic_partitions = 0; completed_topic_partitions < 1;) {
message = kafka_consumer.Consume(-1 /* timeout_ms */);
if (message->err() == RdKafka::ERR__PARTITION_EOF) {
completed_topic_partitions++;
continue;
}
// Since there's only one topic all consumed messages should be in the same order as the records
EXPECT_EQ(RdKafka::ERR_NO_ERROR, message->err());
EXPECT_EQ(records_[offset].payload, std::string(static_cast<char*>(message->payload())));
EXPECT_EQ(RdKafka::MessageTimestamp::MSG_TIMESTAMP_CREATE_TIME, message->timestamp().type);
EXPECT_EQ(records_[offset].timestamp_ms, message->timestamp().timestamp);
EXPECT_EQ(offset, message->offset());
EXPECT_EQ(topic_name, message->topic_name());
EXPECT_EQ(partition_id, message->partition());
offset++;
}
EXPECT_EQ(records_.size(), offset);
}
TEST_F(KafkaConsumerTest, TestMultipleTopicPartitions) {
const std::unordered_set<uint32_t> partition_ids{1, 2, 10};
const std::unordered_set<std::string> topic_names({"topic0", "topic2"});
KafkaConsumer kafka_consumer(std::make_shared<RdKafka::MockKafkaConsumer>(mock_kafka_cluster_),
partition_ids,
topic_names,
"UnitTestKafkaConsumer");
// Seek to timestamp
ASSERT_TRUE(kafka_consumer.Seek(63 /* timestamp_ms */));
std::vector<std::tuple<int32_t, std::string, std::string, int64_t>> expected_results{
std::make_tuple(1, "topic0", "f", 64),
std::make_tuple(1, "topic0", "g", 128),
std::make_tuple(1, "topic2", "f", 64),
std::make_tuple(1, "topic2", "g", 128),
std::make_tuple(2, "topic0", "f", 64),
std::make_tuple(2, "topic0", "g", 128),
std::make_tuple(2, "topic2", "f", 64),
std::make_tuple(2, "topic2", "g", 128),
std::make_tuple(10, "topic0", "f", 64),
std::make_tuple(10, "topic0", "g", 128),
std::make_tuple(10, "topic2", "f", 64),
std::make_tuple(10, "topic2", "g", 128)};
size_t num_records = 0;
RdKafka::Message* message;
for (size_t completed_topic_partitions = 0; completed_topic_partitions < 6;) {
message = kafka_consumer.Consume(-1 /* timeout_ms */);
if (message->err() == RdKafka::ERR__PARTITION_EOF) {
completed_topic_partitions++;
continue;
}
EXPECT_EQ(RdKafka::MessageTimestamp::MSG_TIMESTAMP_CREATE_TIME, message->timestamp().type);
std::string str_payload = std::string(static_cast<char*>(message->payload()));
std::tuple<int32_t, std::string, std::string, int64_t> result = std::make_tuple(
message->partition(), message->topic_name(), str_payload, message->timestamp().timestamp);
EXPECT_TRUE(std::find(expected_results.begin(), expected_results.end(), result) !=
expected_results.end());
num_records++;
}
EXPECT_EQ(expected_results.size(), num_records);
}
TEST_F(KafkaConsumerTest, TestMultipleTopicPartitionsDiffSeekTimes) {
const std::unordered_set<uint32_t> partition_ids{1, 2, 10};
const std::unordered_set<std::string> topic_names({"topic0", "topic2"});
KafkaConsumer kafka_consumer(std::make_shared<RdKafka::MockKafkaConsumer>(mock_kafka_cluster_),
partition_ids,
topic_names,
"UnitTestKafkaConsumer");
// Seek to timestamp
ASSERT_TRUE(kafka_consumer.Seek("topic0", 32 /* timestamp_ms */));
ASSERT_TRUE(kafka_consumer.Seek("topic2", 64 /* timestamp_ms */));
{
std::vector<std::tuple<int32_t, std::string, std::string, int64_t>> expected_results{
std::make_tuple(1, "topic0", "e", 32),
std::make_tuple(1, "topic0", "f", 64),
std::make_tuple(1, "topic0", "g", 128),
std::make_tuple(1, "topic2", "f", 64),
std::make_tuple(1, "topic2", "g", 128),
std::make_tuple(2, "topic0", "e", 32),
std::make_tuple(2, "topic0", "f", 64),
std::make_tuple(2, "topic0", "g", 128),
std::make_tuple(2, "topic2", "f", 64),
std::make_tuple(2, "topic2", "g", 128),
std::make_tuple(10, "topic0", "e", 32),
std::make_tuple(10, "topic0", "f", 64),
std::make_tuple(10, "topic0", "g", 128),
std::make_tuple(10, "topic2", "f", 64),
std::make_tuple(10, "topic2", "g", 128)};
size_t num_records = 0;
RdKafka::Message* message;
for (size_t completed_topic_partitions = 0; completed_topic_partitions < 6;) {
message = kafka_consumer.Consume(-1 /* timeout_ms */);
if (message->err() == RdKafka::ERR__PARTITION_EOF) {
completed_topic_partitions++;
continue;
}
EXPECT_EQ(RdKafka::MessageTimestamp::MSG_TIMESTAMP_CREATE_TIME, message->timestamp().type);
std::string str_payload = std::string(static_cast<char*>(message->payload()));
std::tuple<int32_t, std::string, std::string, int64_t> result = std::make_tuple(
message->partition(), message->topic_name(), str_payload, message->timestamp().timestamp);
EXPECT_TRUE(std::find(expected_results.begin(), expected_results.end(), result) !=
expected_results.end());
num_records++;
}
EXPECT_EQ(expected_results.size(), num_records);
}
// Seek again after consume to check that reseeking to an earlier position works
ASSERT_TRUE(kafka_consumer.Seek("topic0", 16 /* timestamp_ms */));
{
std::vector<std::tuple<int32_t, std::string, std::string, int64_t>> expected_results{
std::make_tuple(1, "topic0", "d", 16),
std::make_tuple(1, "topic0", "e", 32),
std::make_tuple(1, "topic0", "f", 64),
std::make_tuple(1, "topic0", "g", 128),
std::make_tuple(2, "topic0", "d", 16),
std::make_tuple(2, "topic0", "e", 32),
std::make_tuple(2, "topic0", "f", 64),
std::make_tuple(2, "topic0", "g", 128),
std::make_tuple(10, "topic0", "d", 16),
std::make_tuple(10, "topic0", "e", 32),
std::make_tuple(10, "topic0", "f", 64),
std::make_tuple(10, "topic0", "g", 128)};
size_t num_records = 0;
RdKafka::Message* message;
for (size_t completed_topic_partitions = 0; completed_topic_partitions < 3;) {
message = kafka_consumer.Consume(-1 /* timeout_ms */);
if (message->err() == RdKafka::ERR__PARTITION_EOF) {
completed_topic_partitions++;
continue;
}
EXPECT_EQ(RdKafka::MessageTimestamp::MSG_TIMESTAMP_CREATE_TIME, message->timestamp().type);
std::string str_payload = std::string(static_cast<char*>(message->payload()));
std::tuple<int32_t, std::string, std::string, int64_t> result = std::make_tuple(
message->partition(), message->topic_name(), str_payload, message->timestamp().timestamp);
EXPECT_TRUE(std::find(expected_results.begin(), expected_results.end(), result) !=
expected_results.end());
num_records++;
}
EXPECT_EQ(expected_results.size(), num_records);
}
}
TEST_F(KafkaConsumerTest, TestMultipleTopicPartitionsSeekOffset) {
const std::unordered_set<uint32_t> partition_ids{1, 2, 10};
const std::unordered_set<std::string> topic_names({"topic0", "topic2"});
KafkaConsumer kafka_consumer(std::make_shared<RdKafka::MockKafkaConsumer>(mock_kafka_cluster_),
partition_ids,
topic_names,
"UnitTestKafkaConsumer");
// Seek to offset
std::map<std::string, std::map<int32_t, int64_t>> last_offsets{
{"topic0", {{1, 3}, {2, 4}, {10, 5}}},
{"topic2", {{1, 4}, {2, 4}, {10, 4}}},
};
kafka_consumer.Seek(last_offsets);
std::vector<std::tuple<int32_t, std::string, std::string, int64_t>> expected_results{
std::make_tuple(1, "topic0", "e", 32),
std::make_tuple(1, "topic0", "f", 64),
std::make_tuple(1, "topic0", "g", 128),
std::make_tuple(2, "topic0", "f", 64),
std::make_tuple(2, "topic0", "g", 128),
std::make_tuple(10, "topic0", "g", 128),
std::make_tuple(1, "topic2", "f", 64),
std::make_tuple(1, "topic2", "g", 128),
std::make_tuple(2, "topic2", "f", 64),
std::make_tuple(2, "topic2", "g", 128),
std::make_tuple(10, "topic2", "f", 64),
std::make_tuple(10, "topic2", "g", 128)};
size_t num_records = 0;
RdKafka::Message* message;
for (size_t completed_topic_partitions = 0; completed_topic_partitions < 6;) {
message = kafka_consumer.Consume(-1 /* timeout_ms */);
if (message->err() == RdKafka::ERR__PARTITION_EOF) {
completed_topic_partitions++;
continue;
}
EXPECT_EQ(RdKafka::MessageTimestamp::MSG_TIMESTAMP_CREATE_TIME, message->timestamp().type);
std::string str_payload = std::string(static_cast<char*>(message->payload()));
std::tuple<int32_t, std::string, std::string, int64_t> result = std::make_tuple(
message->partition(), message->topic_name(), str_payload, message->timestamp().timestamp);
EXPECT_TRUE(std::find(expected_results.begin(), expected_results.end(), result) !=
expected_results.end());
num_records++;
}
EXPECT_EQ(expected_results.size(), num_records);
}
TEST_F(KafkaConsumerTest, TestSeekInvalidTopic) {
const std::unordered_set<uint32_t> partition_ids{1, 2, 10};
const std::unordered_set<std::string> topic_names({"topic0", "topic2"});
KafkaConsumer kafka_consumer(std::make_shared<RdKafka::MockKafkaConsumer>(mock_kafka_cluster_),
partition_ids,
topic_names,
"UnitTestKafkaConsumer");
EXPECT_FALSE(kafka_consumer.Seek("topic1", 2));
}
} // namespace kafka
int main(int argc, char** argv) {
testing::InitGoogleTest(&argc, argv);
return RUN_ALL_TESTS();
}
// Created by Timothy Koh on 4/15/19.
//
| {
"redpajama_set_name": "RedPajamaGithub"
} | 795 |
The publisher gratefully acknowledges the generous support of the Fletcher Jones Foundation Humanities Endowment Fund of the University of California Press Foundation.
# Funnybooks
# Funnybooks
## The Improbable Glories of the Best American Comic Books
Michael Barrier
UNIVERSITY OF CALIFORNIA PRESS
University of California Press, one of the most distinguished university presses in the United States, enriches lives around the world by advancing scholarship in the humanities, social sciences, and natural sciences. Its activities are supported by the UC Press Foundation and by philanthropic contributions from individuals and institutions. For more information, visit www.ucpress.edu.
University of California Press
Oakland, California
© 2015 by Michael Barrier
Library of Congress Cataloging-in-Publication Data
Barrier, J. Michael.
Funnybooks : the improbable glories of the best American comic books / Michael Barrier.
pages cm
Includes bibliographical references and index.
ISBN 978-0-520-24118-3 (cloth : alk. paper)—ISBN 978-0-520-28390-9 (pbk. : alk. paper)—ISBN 978-0-520-96002-2 (ebook)
1. Comic books, strips, etc.—United States—History and criticism. I. Title.
PN6725.B37225 2015
741.5'973—dc232014031037
Manufactured in the United States of America
24 23 22 21 20 19 18 17 16 15
10 9 8 7 6 5 4 3 2 1
In keeping with a commitment to support environmentally responsible and sustainable printing practices, UC Press has printed this book on Natures Natural, a fiber that contains 30 percent postconsumer waste and meets the minimum requirements of ANSI/NISO Z39.48-1992 (R 1997) (Permanence of Paper).
To Ida and Vernon, and Michael Sporn
# Contents
List of Illustrations
Preface
Acknowledgments
Introduction: "The Very Good Ones"
1 Mickey in a Magazine
2 Oskar Lebeck Meets Walt Kelly
3 Whitman, K.K., and Dell
4 Learning on the Job in L.A.
5 A Feel for Walt Kelly's Stuff
6 Animal Magnetism
7 Cartoon Conundrums
8 Carl Barks Makes His Break
9 Barks Becomes the Duck Man
10 The Workman: Gaylord DuBois
11 The Observer: John Stanley
12 "I Am a Backwoods Bumpkin"
13 "Pure Corn" at Disney's
14 Special Talents
15 Barks Masters His Medium
16 An Arena for All the Passions
17 Animal Kingdoms
18 Walt Kelly Branches Out
19 Strong-Handed Friends
20 Carl Barks: The Virtuoso
21 Walt Kelly Escapes
22 Oskar Lebeck in Exile
23 Manifest Destiny
24 Uncle Scrooge: Play Money
25 Carl Barks in Purgatory
26 The Slow Fade
27 Disasters
Epilogue: Can These Bones Live?
Abbreviations
Notes
Index
# Illustrations
From Uncle Pogo So-So Stories (1953), Walt Kelly's first original trade paperback
Oskar Lebeck in 1943, with Western Printing colleagues and others
The front cover of Crackajack Funnies no. 39, September 1941
Ruth and Oskar Lebeck in 1930
An early Walt Kelly page for the November 1935 issue of St. Nicholas: The Magazine of Youth
The front cover of Looney Tunes and Merrie Melodies Comics no. 1 (1941), the first comic book produced by Western Printing's Los Angeles office
From "Porky Pig" in Looney Tunes and Merrie Melodies Comics no. 2, November 1941, by Roger Armstrong
Roger Armstrong in late 1941 or early 1942, at work on "Sniffles and Mary Jane"
From "Bugs Bunny and the Goofy Goose," by Carl Buettner, in Bugs Bunny Large Feature Comic no. 8 (1942)
From "Seaman Sy Wheeler," by Walt Kelly, in Camp Comics no. 2, March 1942
From "Little Black Sambo," by Walt Kelly, in Fairy Tale Parade no. 1 (1942)
From Walt Kelly's inaugural story for the "Our Gang" feature in the comic book of that name (1942)
Pogo Possum and Albert Alligator making their debut in Walt Kelly's lead story for the first issue of Animal Comics (1942)
From "Albert the Alligator," by Walt Kelly, in Animal Comics no. 8, April–May 1944
From "Superkatt," by Dan Gordon, in Giggle Comics no. 50, February 1948
Carl Barks in the 1940s
From the "Donald Duck" story in Walt Disney's Comics & Stories no. 32, May 1943, the first comic-book story that Carl Barks both wrote and illustrated
From "The Terror of the River," by Carl Barks, in Donald Duck Four Color no. 108 (1946)
Gaylord and Mary DuBois in the mid-1940s
From "Tom and Jerry" in Our Gang Comics no. 6, July–August 1943, written by Gaylord DuBois and illustrated by John Stanley
From "Andy Panda" in New Funnies no. 92, October 1944, written and illustrated by John Stanley
From "Oswald the Rabbit" in New Funnies no. 119, January 1947, written by John Stanley and illustrated by Dan Gormley
John Stanley playing the guitar at a mid-1940s pool party at the Oskar Lebeck home in Croton-on-Hudson
John Stanley and Oskar Lebeck around 1950
The front cover of the first Little Lulu comic book, Four Color no. 74 (1945), which was written and drawn by John Stanley
Carl Barks in San Francisco in 1919
Carl Barks's self-caricature for the August 1930 Calgary Eye-Opener
Carl Barks in Minneapolis in 1934
Carl Barks as a Disney story man in 1937, playing a supporting role to his writing partner Harry Reeves
From "The Ghost of the Grotto," by Carl Barks, in Donald Duck Four Color no. 159 (1947)
From "Luck of the North," by Carl Barks, in Donald Duck Four Color no. 256 (1949)
From "Letter to Santa," by Carl Barks, in Walt Disney's Christmas Parade no. 1 (1949)
From "Albert and the Barbecue," by Walt Kelly, in Albert the Alligator and Pogo Possum Four Color no. 105 (1946)
From "Mountain Climbers" in Little Lulu no. 1, January–February 1948, written by John Stanley and illustrated by Charles Hedinger and Irving Tripp
From "The Kid Who Came to Dinner" in Little Lulu Four Color no. 146 (1947), written by John Stanley and illustrated by Charles Hedinger and Irving Tripp
From "Mr. Owl and the Atomic Bomb" in Albert the Alligator and Pogo Possum Four Color no. 148 (1947)
The front cover of Santa Claus Funnies Four Color no. 128 (1946), by Moe Gollub
From "Chuckwagon Charley's Tales" in Animal Comics no. 29, October–November 1947, written by Gaylord DuBois and illustrated by Moe Gollub
From "The Men of Greed" in Tarzan no. 5, September–October 1948, written by Gaylord DuBois and illustrated by Jesse Marsh
From "The Men of A-lur" in Tarzan no. 9, January–February 1949, written by Gaylord DuBois and illustrated by Jesse Marsh
From "Albert and Pogo," by Walt Kelly, in Animal Comics no. 28, August–September 1947
The front cover of Easter with Mother Goose, Four Color no. 140 (1947), by Walt Kelly
From Our Gang Comics no. 40, November 1947, Walt Kelly's pioneering depiction of black and white adolescents as equals
A Kelly editorial cartoon from the New York Star, October 2, 1948
A Kelly editorial cartoon from the New York Star, January 13, 1949
From "Donald Duck," by Carl Barks, in Walt Disney's Comics & Stories no. 138, March 1952
From "Donald Duck" by Carl Barks, in Walt Disney's Comics & Stories no. 145, October 1952
Walt Kelly, in publicity photos taken around 1951
From "Feelin' Mighty Hale, and Farewell," by Walt Kelly, in Pogo Possum no. 3, August–October 1950
From "Cinderella and the Three Bears," by Walt Kelly, in Pogo Possum no. 8, January–March 1952
Oskar Lebeck with members of his comic-book staff in a photo taken around 1950
One of the dozen Surprise Books—this one illustrated by Dan Noonan—that Oskar Lebeck shepherded into print in the fall of 1950
Oskar Lebeck in the early 1950s, around the time his Twin Earths comic strip was launched
From "The Little Rich Boy" in Little Lulu no. 40, October 1951, written by John Stanley and illustrated by Irving Tripp
From Western Marshal Four Color no. 534 (1953), illustrated by Everett Raymond Kinstler
Moe Gollub's wraparound painting for Zane Grey's Sunset Pass, Four Color no. 230 (1949)
From "Only a Poor Old Man," by Carl Barks, in the first issue of Uncle Scrooge, Four Color no. 386 (1952)
From the third Uncle Scrooge one-shot by Carl Barks, Four Color no. 495 (1953)
Carl Barks in 1956, with his friend Bob Harmon and Harmon's wife, Eileen
John Stanley in 1976
Chase Craig in 1969
Carl Barks at his easel in 1974
# Preface
I am sure there were people in mid-twentieth-century America who began reading comic books after they reached adulthood, but there cannot have been many such people compared with the millions for whom comics were among their earliest reading experiences. I was one such child, many years ago; I "read" aloud Walt Kelly stories in Animal Comics to my stuffed animals before I could make out the words. My childhood attachment to comic books was unusually strong. I dreamed of being a cartoonist, and I can remember clearly when and where I first saw many of my comic books—on a newsstand or in a variety store or at a friend's home—even though my memories of my teachers and classmates have dimmed almost to the point of vanishing.
Memories like mine are at once so commonplace and so particular to the person doing the remembering that there can be no point in devoting much attention to them here. What really matters about comic books, especially old comics like the ones from the 1940s and 1950s that are the principal subjects of this book, is whether they repay reading today, and not just by elderly people who want to bathe in nostalgia.
Funnybooks: The Improbable Glories of the Best American Comic Books is my answer to that question, and my answer is, of course, yes. A qualified yes, to be sure, since most comic books, from any period, have very little to recommend them. At two times, separated by about thirty years, I devoted hundreds of hours to reading and rereading old comics, trying to sift out the best of them. The first time was when the late Martin Williams and I were choosing stories to include in A Smithsonian Book of Comic-Book Comics (1982). I made my second and more intensive survey when I was writing this book. In both cases, nostalgia wound up playing no role in choosing stories—either to reprint, in the Smithsonian book, or to write about, in this one.
When I was a boy, I read every kind of comic book, as most children did, but the comics that attracted me most strongly, and that I read and reread, were produced by Western Printing & Lithographing Company and published under the Dell label. "Dell Comics Are Good Comics" was the company's slogan in the 1950s. Not every Dell comic was good, by any means, and certainly there were comic books from other publishers that repaid multiple readings; but, in work on this book, as when I was a child, I became aware of how distinct the Dell comics were from those of every other publisher, and how much better the best Dell comics were than almost all other comic books.
My initial plan was to cast my net wider, but eventually Funnybooks became a history of the Dell comic books, concentrating on the years before comics of all kinds fell under the censor's axe and with only a nod to great cartoonists like Harvey Kurtzman and Will Eisner, whose work was for other publishers. Kurtzman and Eisner, and other artists like them, have already been the subjects of books—in some cases, many books—but there has been no book like this one. At that, my book is only a partial history of Dell and Western Printing, so there are names missing from the index that many devotees of the Dell titles will expect to find, or to find mentioned more often. But although Dell published the work of many writers and artists who deserve to be admired, it published only a few whose work demands to be read—Carl Barks (Donald Duck), John Stanley (Little Lulu), and Walt Kelly chief among them.
Dell never did more than dabble in superheroes, the genre that for many people has long defined what is meant by the term comic book. The absence of superheroes was a large part of Dell's appeal for me. When I was a boy I never cared for any comics of that kind, except for a brief infatuation with the lighthearted Captain Marvel titles. More recently, I have come to appreciate the tongue-in-cheek quality of many of Will Eisner's "Spirit" stories and Stan Lee's early-1960s stories with the Marvel superheroes. But superhero comic books in general, and especially those with the more serious superheroes, like Superman and Batman, have always seemed to me hopelessly inferior to the best comics with "funny animals" like Donald Duck. I found a clue as to why I have felt that way in what a respected science-fiction writer has written of Superman: "He is our universal longing for perfection, for wisdom and power used in service of the human race."
That is true, surely. But in the twentieth century, that longing for perfection was expressed not just in a benign form through Superman and the superheroes that followed him, each of them sharing a larger or smaller piece of Superman's perfection, but also in odious totalitarian ideologies that pursued perfection through mass murder. The longing for perfection is a deeply suspect longing, even when it comes cloaked in the innocent wish fulfillment that the superheroes have always offered.
I have always strongly preferred comic books with characters of a different kind—funny characters most of them, cartoon animals many of them, who on the rare occasions when they aspire to wisdom and power invariably reveal, with comical flourishes, their hopeless imperfectibility. Characters, that is, very much like their readers.
Michael Barrier
Little Rock, Arkansas
February 2014
A postscript: A book of this kind will inevitably contain at least a few errors. As they surface, I will post corrections on my website, www.michaelbarrier.com.
# Acknowledgments
Like several of my earlier books, this book makes abundant use of the research that the veteran animator Milton Gray conducted with me and for me over many years, since I first began writing about the Hollywood animation studios and their comic-book offshoots in the 1960s. Milt, whose enthusiasm for classic hand-drawn animation has not flagged after decades as a highly regarded professional, also read the manuscript for this book, caught some mistakes, and, as always, provided encouragement and support when it was most needed.
Geoffrey Blum, the preeminent student of the work of Carl Barks and an editor of surpassing skills, read the manuscript twice and made innumerable helpful suggestions.
John Kimball; his wife, Virginia; and his sister Kelly generously shared with me Walt Kelly's correspondence with Ward Kimball, John and Kelly's father, as well as drawings Walt Kelly made when he worked at the Walt Disney studio. I owe thanks, too, to Amid Amidi, proprietor of the Cartoon Brew website and Ward Kimball's biographer, for paving the way for me with the Kimballs. I can only hope that his important book, becalmed because of copyright issues, will eventually be published in some form.
Robert R. Barrett shared with me some of the fruits of his many years of exploring the work of Edgar Rice Burroughs and the comic-book and comic-strip incarnations of Burroughs's characters. He lent me his copies of almost thirty years of the correspondence between Edgar Rice Burroughs Inc. and its licensee Western Printing & Lithographing Company, an invaluable window into Western's dealings with its licensors.
Donald Draganski, a friend since my student days in Chicago, lent me copies of some early and rare Disney comic books that eluded me when I was actively collecting decades ago and that I would probably never have seen otherwise.
Steve Schneider, whose outstanding collection of artwork from the Warner Bros. cartoons has been recognized in museum exhibitions throughout the United States and abroad, shared with me correspondence from Warner cartoonists who were working on the Dell comic books in the early 1940s—rare contemporaneous documentation of those early comics.
Hames Ware, whose eye for artists' styles has no equal in my experience, shared his expertise repeatedly and read the manuscript, saving me from a number of mistakes.
Oskar Lebeck's daughter, Letty Edes, shared not just memories of her parents, through telephone interviews and letters, but also rare photos of them and of some of her father's colleagues at Western Printing.
David Saunders, an indefatigable and ingenious researcher into the lives and work of the artists for the pulp fiction magazines of the first half of the twentieth century, was tremendously helpful in tracking down elusive information, about Oskar Lebeck in particular.
It is one of my greatest regrets that I was never able to meet Walt Kelly and tell him how much he meant to me, but I did have the pleasure of meeting his son Peter and chatting with Pete about his father over lunch at a Cracker Barrel restaurant in Virginia. Not a venue Walt would have chosen, probably, but I am sure he would approve the gentlemanly manner in which Pete superintends his father's legacy.
Another son of an illustrious father, John Stanley's son, James, generously shared memories of his father and copies of his few surviving papers through email. I look forward to meeting Jim, too, in person one of these days.
I received invaluable assistance from staff members at any number of research libraries and archives, some of which house what are from all appearances the only significant surviving records of Western Printing's dealings with its licensors. There is no "Western Printing archive" accessible to researchers, and it seems likely that most of Western's own records, and those of its publishing partner Dell Publishing Company, have long since been destroyed, since they would have no continuing legal or commercial usefulness for the current owners of the companies' assets.
I am indebted to David R. Smith, Rebecca Cline, and Ed Ovalle of the Walt Disney Archives, Burbank, California. Dave Smith long ago shared with me the archives' holdings of Carl Barks's correspondence, and he and Becky and Ed have since provided me with important information from such sources as Walt Disney's correspondence files and the Disney studio's "main files," which hold the records of Disney's dealings with Western Printing and are off-limits to most researchers. Shortly before he retired in 2010, Dave Smith provided me with the number of copies of some of the Disney comic books on which Western paid royalties to Walt Disney Productions, information that turned out to be extremely useful.
Thanks to Jenny Robb and Susan Liberator of the Billy Ireland Cartoon Library and Museum, The Ohio State University, Columbus, I was able to spend several enjoyable days, on two visits, examining Walt Kelly's voluminous and endlessly fascinating papers, as well as those of other people with ties to Western. The agent Tony Mendez's papers, which include revealing correspondence with her client Oskar Lebeck, were especially helpful.
Another exceptionally important collection is the "Marge" collection—the papers of Little Lulu's creator, Marjorie Henderson Buell, including a full record of her dealings with Western Printing—at the Schlesinger Library, Radcliffe Institute, Harvard University, Cambridge, Massachusetts. I enjoyed the help of Laurie Ellis and other members of the library's staff.
Marva Felchin, director, libraries and archives, Autry National Center, Los Angeles, made available to me the surviving records (many were lost to fire long ago) of Gene Autry's long affiliation with Western and Dell.
I am also grateful to David Sigler of Special Collections, California State University at Northridge, where Chase Craig deposited letters from Carl Barks and items from his own long career in comics, and to the following librarians and archivists: Martin Gedra of the National Archives and Records Administration, College Park, Maryland; Paul E. Schlotthauer, librarian and archivist, Pratt Institute Library, Brooklyn, New York: Kevin Hallaran, archivist, Riverside Metropolitan Museum, Riverside, California; Jonathon Auxier and Sandra Joy Lee of the Warner Bros. Archives, University of Southern California, Los Angeles; Mary Robison, reference librarian, General Theological Seminary, New York; Eileen L. Fay of the Department of Rare Books and Special Collections, University of Rochester Library; Andy Needham of the Oregon State Archives; Molly Bruce, archives and research assistant, and Susanne Belovari, archivist for reference and collections, digital collections and archives, Tisch Library, Tufts University, Medford, Massachusetts; Ian Stade of Special Collections, Minneapolis Central Library; Connie Von Der Heide, director of reference and outreach services, Wisconsin State Law Library, Madison; Lee Spilberg of the New York Public Library; Allan Raney of the New York State Library; Carol Thunem, archives and periodicals assistant, Carleton College Library, Northfield, Minnesota; and Jan Emberton, Leland Razer, and the members of the interlibrary loan staff of the Central Arkansas Library System.
I am also grateful, for assistance of many kinds, to David Dunn, civil service director, City of Bridgeport, Connecticut; Maxine Hansen, executive assistant to Mrs. Gene Autry, Gene Autry Entertainment, Studio City, California; Shannon Fifer of Warner Bros.; James Sullos and Cathy Wilbanks of Edgar Rice Burroughs Inc.; Stephanie Cassidy of the Art Students League of New York; Colin D. Riley of Boston University; John Ellis of the Milton Caniff estate; Margaret Adamic of the Walt Disney Company; and W. Christopher Barrier, Esq., Mark Kausler, Dana Gabbard, Thomas Andrae, Gunnar Andreassen, Steve Thompson, Mark Evanier, Donald Ault, Andrew Barnes, Didier Ghez, the late Michael Sporn, Art Spiegelman, Frank Young, John Benson, E. B. Boatner, Leonard Marcus, Donald Phelps, Ron Wolfe, Susan Orleans, Thad Komorowski, John Canemaker, and Gary Brown and Alan Hutchinson, whose privately published but wholly professional index to the Dell Four Color Comics was repeatedly useful.
My longtime friend Patrick Garabedian shared with me the tape recording of his 1971 interview with Carl Barks. In addition to the Garabedian interview and many published interviews, this book draws on interviews that Milt Gray and I recorded—most of them long before this book was contemplated—with a number of people who worked on the comic books, worked for Dell Publishing in other capacities, or had connections of other kinds with the Dell comics and the people who made them. Regrettably, most of the people we interviewed are now deceased. I am grateful for those interviews to Roger Armstrong, Carl Barks, Jack Bradbury, Don Christensen, Ross Claiborne, Letty Lebeck Edes, Morris Gollub, Richard Hall, Jack Hannah, David Hilberman, Lynn Karp, Hank Ketcham, Ward Kimball, Harvey Kurtzman, Richard "Sparky" Moore, George Nicholson, Dan Noonan, Milt Schaffer, Gordon Sheehan, Frank Tashlin, Lloyd Turner, and Clair Weeks. I also appreciate the responses I received by mail over the years from Bob Burkert, Robert S. Callender, Del Connell, Chase Craig, Wendall Mohler, Veve Risto, George Sherman, and Bill Spicer. Thanks, too, to my editors and their colleagues at University of California Press: Mary Francis, Kim Hogeland, Rose Vekony, Aimée Goggins, and Carl Walesa.
It remains only to thank my wife, Phyllis Barrier, for putting up with yet another book.
## INTRODUCTION
# "The Very Good Ones"
In 1949, a writer for the Catholic magazine Commonweal interviewed a man named Harry Wildenberg, who was then a "scholarly cigar merchant" in Key West, Florida, but years before had been the sales manager for Eastern Color Printing Company in New York City. The article's author, John R. Vosburgh, wrote of Wildenberg that he "invented the comic book back in 1932," when his job was "to concoct ideas that would sell color printing for Eastern, which . . . printed the comic sections for a score of newspapers along the Atlantic Seaboard."
Wildenberg hit upon the idea for the comic book, Vosburgh wrote, as "he was idly folding a newspaper in halves, then in quarters. . . . As he looked at the twice-folded paper it occurred to him that it was a convenient book size, about seven by ten inches.
"'Why not a comic book?' he reflected. 'It would have 32 or 64 pages and make a fine item for concerns which distribute premiums.'"
That idea gave birth in 1933 to a few premium comic books. They were made up of reprints of Sunday comics pages, one quarter the original size, and could be had by mailing a coupon. The next step was recounted in 1947 by Coulton Waugh, himself a cartoonist and the first serious chronicler of comic strips and comic books:
During the premium book period, M. C. Gaines had become connected with Harry Wildenberg, as salesman for premium books. . . . He had been impressed, he says, with the public reaction to the comic books, and he, for one, did not see why they should not be sold directly over the news stands. Determined to test public reaction, he took several dozen of the premium books labeled Famous Funnies, pasted a sticker which read Ten Cents on them, and induced a couple of news stands to carry them over the week end. Every one was sold out when he visited the stands Monday morning.
Despite such early and unequivocal signs of success, it was months longer before Eastern Color Printing got into the comic-book business by publishing what Waugh called "not the first comic book, but the first American comic magazine in modern format"—that is, roughly ten inches tall by seven and a half inches wide, stapled along the spine, and printed in four colors—"to be placed on newsstands for sale, independently of newspaper or premium connections." That was Famous Funnies no. 1, dated July 1934 and put on sale, according to all accounts, in May 1934.
Eastern Color wound up publishing Famous Funnies because a leading publisher of popular magazines, George Delacorte Jr., decided against exercising the option he held to the title and the format. According to Waugh, advertisers and the most important magazine distributor, American News Company, were cool to comic books' coarse paper and garish color, and to the reprinting of comic strips that millions of people had already seen in newspapers. Their skepticism persuaded Delacorte to bow out. There was a hint here of the hostility that comic books, simply by virtue of being comic books, would encounter repeatedly in the years ahead.
Over the next four years the comic book slowly gained traction, as Famous Funnies became profitable and was joined on the newsstands by a small but growing number of titles from other publishers—including, through a change of heart, George Delacorte's Dell Publishing Company. As Waugh wrote: "He was awake to the new trend now, and in the fall of 1935 he arranged with the American News Company for distribution of a new magazine, Popular Comics. This was produced for Delacorte to publish by M. C. Gaines, who by this time was connected with the McClure Syndicate."
Delacorte had pioneered in comic books, starting in 1929 with an unsuccessful weekly effort: original comics in a tabloid format, called The Funnies. That setback may have discouraged him from picking up his option with Eastern Color Printing. Besides acting to remedy that mistake by publishing Popular Comics, Delacorte also resumed publication of The Funnies in October 1936, in the soon-to-be-standard quarto size. Like Famous Funnies, both Dell titles were anthologies selling for ten cents and made up mostly of reprinted newspaper comics.
In 1935, original material began turning up in some comic books alongside the reprinted newspaper comics. Then, in the spring of 1938, came the introduction of Superman in the first issue of Action Comics. The popularity of Superman and the other costumed superheroes that followed in his wake fueled tremendous growth in both the number of titles and the number of copies sold. By 1949, according to Vosburgh, comic books were selling sixty million copies a month. Such comprehensive sales figures were necessarily inexact, but there was no question but that comics were selling in the tens of millions. M. C. Gaines died in a boating accident in 1947, so by the time Vosburgh interviewed Wildenberg there was no one else who could claim credit for inventing the comic book. Or, as Wildenberg would have it, accept the blame.
"I don't feel proud that I started the comic books," he said. "If I had had an inkling of the harm they would do, I would never have gone through with the idea." But there was, of course, the lure of profit: "You must remember that in the beginning I gave little thought to the social aspects of the matter. In business a man seldom thinks beyond profits. The social impact of an idea, an invention, is secondary, if he contemplates it at all. It was my business to sell comics and it did not even occur to me to weigh the effects they would have."
When comic books began to appear in quantity in the late 1930s, they quickly dominated a niche that other mass-market publications had never filled so completely. The superhero comics were a particularly gaudy manifestation of what was by then a well-established subliterature: magazines that children sought out and embraced despite the misgivings or outright disapproval of their parents. That subliterature originated in the nineteenth century with dime novels, continued in the pulp magazines of the early years of the twentieth century, and flowered in the comic books of the thirties and forties. Comics were even easier to read than their predecessors, and just as cheap.
Some of the principal ingredients of the superhero comics, as of their predecessors, were continuing characters, sharply defined heroes and villains, a high level of violence (or, at least, furious physical activity), and directness and even crudeness in their appeal to young, mostly male readers. Comic books provided those ingredients so efficiently that they became very popular very quickly.
By the time Vosburgh interviewed Wildenberg, and for years afterward, the strongest criticism of comic books was directed at their graphic depictions of violence—not so much in the superhero comics, which by the late 1940s were on the wane, but in crime comic books like Crime Does Not Pay and then in horror comics like Tales from the Crypt—and how such depictions supposedly encouraged juvenile delinquency. But all comic books, even the most innocuous, were the targets of widespread disapproval. The complaints went not just to what comic-book stories were like, but also to what comic books were in their essence.
The most vocal and effective of comics' critics, like the psychiatrist Fredric Wertham, were not distracted by any pleas on behalf of "good" comic books, because in their eyes all comics were inherently bad. In his most famous book, Seduction of the Innocent (1954), Wertham condemned the medium itself as hopeless, for this reason among others:
An important area where comic books do specific harm is the acquisition of fluent left-to-right eye movements, which is so indispensable for good reading. The eyes have to form the habit of going from left to right on the printed line, then returning quickly to the left at a point slightly lower. Reversal tendencies and confusions are common among children at the age of six. As better reading habits are acquired, including the all-important left-to-right movements, reversals and other errors gradually diminish and may automatically disappear. It is different with the comic-book reader who acquires the habit of reading irregular bits of printing here and there in balloons instead of complete lines from left to right.
Wertham had a point, but not exactly the point he thought he was making. Although comic books were vulnerable to criticism on many grounds, the problem was not that the medium itself was fatally deficient. It was that comic-book stories were easy to make, given a bare minimum of drawing skill, but they were surprisingly difficult to make well. Most comic-book editors, writers, and artists had no idea how to make good comics, and no idea why they would want to try.
Even though the comic-book story grew out of the comic strip, the comic-book story's challenges were different, subtler, and more severe. In particular, because such stories were typically much longer than the three or four panels of a daily strip (or the ten or twelve on Sunday), they demanded a more sophisticated handling of time. A panel in a comic-book story usually did not represent an instant, as a frame of film did. The words spoken in the dialogue balloons would take more time than that. Comic-book panels were thus a little like the shots in a movie, which could last for seconds or even minutes—but not really, because movement was absent in the comics. The cartoonist's task was thus to compose somehow a drawing that was satisfying in itself but that also suggested the passage of time within the panel.
Just as important, making good comics required aligning dialogue and drawings so that there could be no doubt that a character that was supposed to be speaking was actually saying the words in the balloon over his head. Not only did the dialogue need to be distinct and individual, but the face beneath it had to mirror those words, even when, or especially when, that required distorting the speaker's features, to the point of flirting with the grotesque.
Good comics also required sensitivity to how each panel's elements were composed, so that drawings and dialogue balloons were in balance, and reading the dialogue in the balloons was as natural and easy as reading type on a printed page. Panels had to lead one to another in what felt like a natural order, and with a rhythmic subtlety that made a page as a whole, and then a story as a whole, come alive as it was read.
Such conditions were almost never met. It was much more common for a story to be clotted with dialogue, its balloons packed with words when words were not conspicuously absent, and for its crude drawings to lurch up and down and across the page, sometimes squeezed into as many as a dozen claustrophobic panels. Such stories required of the reader constant adjustments of the sort Wertham decried. They also relied almost entirely on the power of the medium even at its crudest.
Some people working in comic books in the 1940s and 1950s understood their medium's challenges and relished meeting them, but many more of their colleagues did not. They were in a sort of unspoken conspiracy with educated readers, who were repelled not just by comics' crude printing but also by their pervasive shallowness. A comic-book story's essentials could be absorbed in a single hasty reading, even when the story's clumsiness threw up obstacles in the reader's path. It seems never to have occurred to anyone that the best comic-book stories might reveal themselves fully only through many rereadings, and that they might be—in that respect, at least—more like paintings or music than prose fiction.
In 1963, Lloyd E. Smith, a dedicated bibliophile of sophisticated tastes and for thirty-five years a principal editor and executive with one of the largest publishers, responded skeptically to a letter that praised some of his company's comics: "It is true that comic books generally are not preserved and probably there is a question whether they should be preserved." He softened that remark subsequently, suggesting to his student correspondent that if some university library decided to "build up a collection of comic books" that would be available for research, that might not be a bad thing. Assuming, that is, there was space available and the comics would be in the charge of "a sympathetic librarian." But clearly he could not bring himself to attach any importance to such a collection.
The comic books that Smith supervised, like almost all midcentury comics, were conceived as narrowly commercial enterprises. When a comic book from any publisher became the venue for artistry of any kind, it was invariably because of very unusual circumstances, and such an opportunity invariably vanished within a few years at most.
There was, for example, Will Eisner, who as a very young man in the late 1930s was the coproprietor of a shop that produced comic books for a number of publishers, some of them on the fringes of respectable business. Free of any literary pretensions, the early comic-book publishers were interested only in turning out lots of comics quickly and at minimal cost, and Eisner's factory, as he called it, met that need. But then the nature of his work changed.
Early in 1940, Eisner began producing a sixteen-page syndicated comic book that newspapers could distribute along with their Sunday comics sections. He wrote and illustrated (with help) the eight-page lead feature, "The Spirit," about a vigilante hero. Even though the Spirit was born in the first flush of costumed superheroes' popularity, Eisner insisted on presenting his character in street clothes, the only costume a mask and gloves. His syndicate made almost no demands on him, except that he meet deadlines; he produced his stories for an audience—newspaper readers—that was composed more of adults than was typical for comic books. He thus enjoyed a greater freedom of action than most comic-book creators, and he soon began to take advantage of it.
What really set "The Spirit" apart, after its first year or so, was Eisner's increasingly inventive use of his medium. His explorations were interrupted by military service in World War II—he left "The Spirit" in the hands of lesser artists while he was gone—but after he took charge of "The Spirit" again he became preeminently a short-story writer who wrote mostly with pictures instead of words. He worked in what John Benson has called "an effortless visual narrative style, which is so unlabored that most readers are unaware of how much more his pictures tell us than do the pages of most comic books."
The clumsiness so common in prewar comic books, the simple ignorance of what a good story required, persisted in many if not most comics after the war, but Eisner's best postwar stories are strikingly different. Their panels expand and compress time in unpredictable ways, and the characters are drawn in styles ranging from straight illustration to highly exaggerated cartooning—not just on the same page, but the same character, and not in any arbitrary fashion, but according to what will enhance the story.
The richest of Eisner's stories present small urban dramas, some fanciful, others grim, in which the Spirit himself is not much more than a bystander. The Spirit was at bottom a conventional comic-book hero, and Eisner often had to shoehorn him into stories that had no need of him. But a "Spirit" comic-book section without the Spirit was not feasible at a time when comic books, even one as unique as the "Spirit" section, were defined very narrowly. As a result, the overwhelming sense from the postwar "Spirit" stories is of playfulness. Eisner enjoyed exploring his medium's possibilities, but without asking it to do very much serious work.
The "Spirit" comic-book section shrank from sixteen pages to eight before it died in 1952. By then Eisner had already made a decisive turn. He had left the "Spirit" section in other hands and begun devoting his time to producing educational materials: a monthly magazine on preventive maintenance for the army. He was always a very practical artist, and when there was no longer room in comic books for stories of the kind he had mastered, he left the field.
Eisner's departure occurred when another significant publisher was under siege. This was EC, whose initials on comic-book covers stood at first for Educational Comics. It was founded in the mid-1940s by the pioneering M. C. Gaines, who published comic-book versions of the Bible and American history. After Gaines's death, his son, William, shifted the company's focus toward more popular kinds of comics, and by 1950 EC was publishing three unvarnished horror titles. The horror comics sold well—and spawned a host of imitators—but Gaines also published less popular and more ambitious science-fiction comics.
Al Feldstein, the editor, principal writer, and occasional artist for the science-fiction titles (and the horror titles, too), relied so heavily on captions that the panels beneath his captions were in effect illustrations for stories told in prose. Harvey Kurtzman, EC's other principal editor, was more purely a comic-book creator, and his comics were less indebted to literary models. The titles of Kurtzman's comic books could have been those of any publisher's war comics—Frontline Combat and Two-Fisted Tales—but the content, particularly the harshly realistic stories set during the Korean conflict, could not have been more different. Like Feldstein and everyone else connected with EC, Kurtzman worked in a short-story form, but he wrote his stories as pages of rough sketches that guided other artists, and he illustrated some of his stories himself, with vigorous brushwork that fitted his subject matter perfectly.
In 1952, Kurtzman's resistance to the prevalent comic-book fantasies—especially the fantasies of wish fulfillment in the superhero comics—found perfect expression not in the war comic books but in the satirical Mad, a comic book just as serious in its own way as Frontline Combat, and, needless to say, much funnier. In Mad, Kurtzman lampooned movies and comic strips, but he was at his most penetrating when he was savaging superhero comics and "kiddie" television shows like Howdy Doody, coldly commercial fare that patronized a vulnerable audience.
Mad sold well, but other kinds of success eluded it. The esteemed critic Robert Warshow, writing in Commentary, could come no closer to praise than acknowledging the "irritated pleasure" he took in Mad, which his young son brought into his home along with other comics. He called it a "wild, undisciplined machine-gun attack on American popular culture. . . . The tendency of the humor, in its insistent violence, is to reduce all culture to indiscriminate anarchy." Kurtzman's work was anything but undisciplined, but the discipline he observed was one that Warshow—and many people like him—could not recognize.
Whether a creator took comic-book stories seriously by using them to explore ideas, as Kurtzman did, or to explore the form itself, as Eisner did, there was no shelter from the intense hostility that confronted all comic-book publishers in the early 1950s. EC's Bill Gaines responded to the attacks on his comics by making them even more serious, wiping away the horror and crime titles and replacing them with "New Direction" comics, among them one called, remarkably, Psychoanalysis. When the New Direction comics failed, Gaines shut down his entire comic-book line and converted Mad into what turned out to be a highly successful magazine, distinct in its larger format and lack of interior color from comic books and thus relatively immune from attack.
What Gaines tried to do with the New Direction titles, unsuccessfully, was free comic books from a prison created by their own success. Even at its most popular, the comic book was dominated by formulaic genres, each monthly or bimonthly title as easily categorized as a Hollywood B movie. Successful comics that fell outside existing genres tended to establish new genres, almost instantaneously. That was most famously true of the superheroes, but the pattern was repeated with the crime and horror comics after World War II and with comics of other kinds as well. Harvey Kurtzman eviscerated one comic-book genre after another in EC's Mad, but such mockery quickly mutated into a genre of its own, as the newsstands filled with clumsy copycat titles that included one from EC itself, Panic. On the rare occasions when publishers tried to escape the genre trap, as with EC's New Direction line, their efforts failed, quickly and decisively.
In many genres, the best comics were not the most popular. In the teenage comedy genre, for example, the little-known and short-lived 1950s title Henry Aldrich compares very favorably with the much more popular Archie titles. Other genres, like the romance comics that flourished in the late 1940s, resisted producing any material of value. Even more than the teenage comics, which also appealed to a largely female audience, the romance comics pandered to their readers, stoking their self-pity and insecurity.
It turned out, though, that a few comic-book genres could support stories worthy of a place alongside intelligent fiction of other kinds—novels and short stories and movies. Those comic-book stories depicted characters and situations that were essentially realistic, however far removed from reality the characters and situations might at first seem to be. Read without preconceptions, such stories did not require special pleading; neither was it necessary to pigeonhole them as experiments or as vehicles for ideas larger than the stories themselves. There were only a few such comic-book stories, compared with the many that were mediocre or worse, but there were enough to put the lie to any general condemnation of comics.
Of the comic books that transcended the limitations of their genres, the best came from Western Printing & Lithographing Company, a Racine, Wisconsin, concern that was in the 1940s and 1950s a highly successful publisher of not just comics, but also children's books and printed products of many other kinds. Western was the company of which Lloyd E. Smith was an executive. It made its comics for sale by Dell Publishing Company, under the Dell label.
When Helen Meyer, Dell's vice president, defended her company's comic books—all of them "designed and produced" by Western Printing—before a U.S. Senate subcommittee in 1954, she testified that the average Dell title sold eight hundred thousand copies and that the best-selling titles, like Donald Duck, ranked among the best-selling newsstand publications in the country. Around that time, roughly one-third of the comic books sold in the United States carried the Dell label, and the percentage rose as the comic-book industry as a whole contracted later in the 1950s.
No one at Dell or Western Printing had any elevated conception of what comic books could be, but circumstances combined to give a few gifted cartoonists who worked for Western the openings they needed to realize some of their medium's potential. The best stories for the Dell comics did not succeed as Eisner's and Kurtzman's best stories did. There is in them no dazzling critical exploration of the comic-book form, no ridicule of the falsehoods infesting comic books and other low forms of popular culture. There is mastery of the comic book's graphic language, to be sure, and often sly satirical commentary, but what is central in the best Dell stories is character, as revealed through narrative.
Such priorities can be glimpsed even in some of the Dell comic-book westerns and in the Dell incarnation of Tarzan of the Apes. But the very best Dell comics belong to genres of other kinds: the comic books with animated characters—"funny animals," talking animals like Donald Duck—and precocious child characters like Little Lulu. Those genres were largely vehicles for the most puerile sort of comedy, but it turned out that they could also accommodate greater achievements than might have seemed possible for anyone working in comic books.
As for what gave a few cartoonists the openings they needed, there was, for one thing, length. The Dell comics published stories longer than those of almost any other publisher—as many as thirty or forty pages or sometimes even more. The stories' length was often even greater than might at first appear, because from the mid-1940s on, the standard Dell page for the titles with cartoon characters had eight panels, not the six that prevailed for years afterward at other publishers. The stories in the comics from other publishers were mostly short, or broken up into short chapters, which allowed room for advertisements. For more than a decade, Dell, unlike most other publishers, included no advertising in its comics, except to solicit subscriptions.
Greater length was not always necessary or desirable—or even available, when a comic book was a monthly anthology title with a fixed number of pages for each feature. Another check on story length was a widespread editorial impulse that favored greater variety in a comic book's features. But in many of the Dell comics, especially those devoted to a single character, length was there for the storyteller who needed it. With more pages and panels to work with, a creative cartoonist—in particular, one who was both writing and drawing his stories—could present his characters fully and let a story grow out of those characters, instead of simply rushing them through a truncated plot.
Western's characters typically came to its writers and artists from other media, as with the Disney characters and other creatures of film. Sometimes those writers and artists fumbled with the characters they had inherited, but in other cases such characters gave Western's people a valuable head start. The characters, continuing from one story to another (but usually not in a single overarching narrative), could become richer and more interesting than the borrowed originals.
It was in the career of one cartoonist, Carl Barks, that the virtues—and ultimately the limitations—of Western Printing's approach to comic-book publishing were most clearly visible. One title, Walt Disney's Comics & Stories, became a monthly showcase for Barks, who wrote and drew a ten-page lead story about Donald Duck and his three nephews, Huey, Dewey, and Louie, for almost every issue between 1943 and 1965. In the best of those stories, and in the best of his stories for other Disney comics (most of them considerably longer), Barks demonstrated more effectively than any of his contemporaries that the comic-book story was a valid literary and artistic form. It was a form whose demands were all too easily ignored, but one that could offer unique rewards, especially as a vehicle for comedy, when those demands were respected by someone who was able to meet them.
Barks received very little stimulation or encouragement from his readers and editors, and the same was true of the creators of the Dell comic books in general, some of whom produced work that rivaled in quality Barks's stories in the Disney titles. They produced that work for magazines that were, emphatically and unmistakably, intended for an audience made up mostly of children—but that was not necessarily a handicap, as the newspaper columnist Murray Kempton observed in 1953. He was reviewing a book called Uncle Pogo So-So Stories, by Walt Kelly, who had become a syndicated newspaper cartoonist in 1949 after spending most of the decade writing and drawing Dell comics. Kelly's comic strip, Pogo, was born in one of those comic books, Animal Comics, and at the time of Kempton's review Dell was still publishing a quarterly Pogo Possum comic book.
Uncle Pogo So-So Stories was Kelly's third paperback book but his first original paperback, the first two having been made up of reprinted comic strips. The new book's publisher was not Dell but the more prestigious Simon & Schuster, it was printed in black and white, and it sold for a dollar, ten times as much as most comic books in 1953. But a comic book it was, with panels and dialogue balloons and compact stories like those in the ten-cent comics produced on high-speed four-color presses.
Uncle Pogo So-So Stories (1953), Walt Kelly's first original trade paperback, was a comic book in all but name and price—a dollar rather than a dime. © Okefenokee Glee & Perloo, Inc. Used by permission.
For that matter, the Pogo comic strip itself, with its cast of talking animals, stood revealed as a comic book in disguise when it was collected in the first two paperbacks, so that its stories could be read in a single sitting instead of as daily four-panel installments. Something similar might be said of other comic strips with continuing stories, but few if any other comic strips were ever so rich in drawing, language, and incident as Pogo. When Kelly edited his stories for the paperbacks, culling the weakest material and smoothing out the transitions, what remained were first-rate comic-book stories, memorable not just for verbal and physical pratfalls but also for the dozens of characters that made up Kelly's cast, all of them richly comic (except when menace was the point) and distinctly individual.
In June 1953, when Murray Kempton wrote his review, Kelly had just made his strongest claim for adult acceptance—of which he already had a great deal—by introducing into his comic strip a menacing cat named Simple J. Malarkey, a pointed caricature of Senator Joseph McCarthy. But Kempton was not fooled. Kelly, he wrote, "is an idol among the eggheads, which is odd when you consider that he is a simple man . . . and quite obviously peddles his wares for children, bright children, but children." Dismissive as that might sound, Kempton was headed elsewhere: "Which is, of course, the only worth-while audience there is for the very good ones, childhood being almost the only period when the subject is exposed to consistently good writing; how many grown-ups read Mark Twain, who is obviously Kelly's master[?]"
That was an overstatement, but with more than a little truth in it. All of the great comic-book creators wrote and drew for children, even if some, like Harvey Kurtzman and Will Eisner, wrote and drew for audiences a few years older than the typical readers of Walt Kelly's stories in Animal Comics. What distinguished the "very good ones" from their peers was not so much the nature of their audience as it was the skill with which they managed to respond simultaneously to its requirements and their own hearts' imperatives.
In the best of the Dell comic books, like those by Walt Kelly and Carl Barks—and John Stanley, the man responsible for the excellence of Little Lulu—there was remarkably rich comedy, in stories told for children but by an unmistakably adult storyteller. Those stories were told through words and drawings executed not just with a high level of skill but also with a clear understanding of what the comic-book form demanded. And in the best stories, strong threads of words and drawings were wound together inextricably, so that, as in Carl Barks's case, separating Barks the writer from Barks the illustrator is impossible. Few other comic books demand rereading so urgently or reward it so fully.
For Barks, Kelly, Stanley, and a few other Dell cartoonists, writing and drawing stories for children was more liberating than confining. They were foreclosed from dealing directly with some kinds of adult behavior, but they could explore a great many adult concerns under the cover of characters that were children or talking animals. Barks's best stories, especially, boiled with adult passions—greed, pride, jealousy, joy, sorrow, contempt, disgust, and many more, enough to fit out a contemporary novel, but in as few as ten pages. And because the principal characters were talking ducks, the drawing style so cool and precise, and the stories' basic shape reassuringly familiar, as farce or fable or mystery or something else of the kind, there was almost never in those stories anything that might set off alarm bells in anxious adult minds.
Walt Kelly was allowed to sign some of his comic-book work, and his editor, Oskar Lebeck, passed along to him dozens of admiring letters, many from obviously intelligent and educated adults who had discovered his stories by reading them to their children. Such letters could only have encouraged Kelly to pursue newspaper syndication. Others of Western's cartoonists were not so fortunate. Carl Barks's stories were never signed, and John Stanley's almost never were; moreover, Stanley's drawings were usually cloaked by the ink lines of a lesser cartoonist. Readers' letters never made their way to Barks when he was doing his best work—except for a few inane complaints—and the same was probably true of Stanley. Even so, many children knew that Little Lulu was a special comic book, and they recognized Barks's distinctive drawings as those of "the good artist." They knew that Barks was, like Kelly, one of the very good ones.
How those cartoonists came to be such, and to have the company of a few other very good ones, is the most important part of the story of the Dell comics and of Western Printing & Lithographing, the most successful and unusual of the many companies whose comic books filled the newsstands of mid-twentieth-century America. Western Printing, despite its impressive record as a publisher, never acknowledged the quality of its best comics, asking only that they be accepted as "wholesome" and harmless. When Western's long run ended, it was the comic books that survived, to be collected, admired, reprinted, and written about, even as their publisher vanished as completely as water poured onto desert sand.
## 1
# Mickey in a Magazine
The story of Western Printing's comic books, and thus of the unlikely triumphs of Carl Barks, Walt Kelly, John Stanley, and a few other cartoonists, began with a man of a very different sort: Hal Horne, a peripatetic publicist who from offices on Fifth Avenue in New York City published the first nine issues of Mickey Mouse Magazine.
That particular Mickey Mouse Magazine was actually the third publication to bear the title. The first two, in 1933 and then in 1933–35, were monthly promotional pamphlets, sixteen pages in digest size, about half the size of a standard comic book. The first issue of the third version—the first Disney newsstand periodical—was published in May 1935 and identified on its cover as a "summer quarterly." It was exceptionally large for such a magazine, more than ten inches wide and thirteen inches high, and it cost twenty-five cents, an imposing figure for a children's magazine in that Depression year.
The price fell to a dime with the second issue, dated October 1935, the first on a monthly schedule, and the dimensions shrank, too, by almost two inches on each side. With the March 1936 issue the page count fell from forty-four to thirty-six, including covers, and the trim size shrank a little further. Times were tough, and Horne's print orders for the first three issues of the new Mickey Mouse Magazine had turned out to be far too ambitious. He ordered three hundred thousand copies of each issue but sold fewer than half. Horne scaled back subsequent print orders, but sales continued to decline. The problem was certainly not the very popular Disney characters, but rather the magazine's non-Disney content.
One author of that content was Irving Brecher, who in 1935 was a young gag man for what he called "the cheapest form of human life, small-time vaudevillians." He then became one of Horne's two "associate editors" for Mickey Mouse Magazine and wrote what he described many years later as "stories with wit in them that were amusing only to grown-ups." He invoked college humor magazines in describing his work, no doubt having in mind stories like "Frank Verywell in College by Horatio Algebra" in the May 1936 issue.
Mickey Mouse Magazine's level of inspiration was as low as that title suggests. The magazine offered a conventional mixture of short stories, poems, puzzles, and drawings, but the drawings, including those of the Disney characters, were often weak and even amateurish, the jokes lame, the stories pedestrian. Mickey Mouse Magazine was peculiar competition for the leading children's magazine of the day, the sober and literary St. Nicholas: The Magazine of Youth.
The strongest echoes throughout the magazine were of Hal Horne's gag file. Over the years, Horne had accumulated a huge file of around six million jokes on three-by-five-inch cards (along with magazine cartoons on larger cards), housed in several rooms in a New York office building. Horne rented selections from his gag file to a range of clients identified as "comic strip artists, stage, screen, and radio comedians, playwrights, columnists, governors, senators, house organs and advertising agencies." As to the nature of the gags in the file, there is a clue in Hap Lee's Radio Joke Book: Famous Gags of Radio Stars (1935), since "Hap Lee" was a Hal Horne pseudonym. A sample (setting aside the all too abundant racist and misogynist material):
"Young man, take your hands off my daughter's knee!"
"Excuse me, sir, I was just going to say what a nice joint you have here!"
Everything in the book is generally similar.
Horne had connected with Walt Disney himself as director of advertising and publicity for United Artists (UA), the movie company that began distributing the Disney animated cartoons in 1932. In that role, he was identified as "editor" of the two giveaway versions of Mickey Mouse Magazine. On July 24, 1935, as the New York Times reported, Horne announced his resignation from UA to "organize and head a new advertising and publicity company in New York." By then, though, a new company called Hal Horne Inc. had been in existence for some time. Its name was on the first issue of the new Mickey Mouse Magazine, which actually appeared a couple of months before Horne resigned from UA.
As subsequent events were to show, Hal Horne Inc.'s financial foundations were fragile, and Disney, fearful of that, may have shopped the new version of the magazine to more established publishers before deciding to leave it with Horne. Years later, Ned L. Pines, who published a line of pulp magazines before becoming a publisher of comic books, said that "Walt Disney's magazine was offered to us for publication" in 1934. Pines turned it down.
Hal Horne was from all appearances an exceedingly restless and intense man, someone who, in Motion Picture Daily's words, "burned up the track as an exploiter and theatre operator" before he joined UA in 1931 and began "attacking his new job . . . with characteristic vigor." His intensity did not translate into success as publisher of a Disney magazine. In a December 27, 1935, letter, Horne lamented to Roy O. Disney, Walt's brother and business manager, that the magazine "to date . . . has cost me a terrific amount of heartaches and exactly $50,000, all of which seems such a crime when you consider the magazine has been loved by those who have read it." Roy was sympathetic. In February 1936 he wrote to Horne that he was "more concerned now with saving you from a loss than with trying to get any revenue from the magazine." Accordingly, he authorized Horne to publish the magazine on "a non-royalty basis" for the rest of the year.
Horne published Mickey Mouse Magazine for only a few more months, through the June 1936 issue, with sales declining almost every month. In early June, as the July 1936 issue went to press, Horne surrendered the magazine to a new publisher. He then became a producer for RKO Radio Pictures, which Walt and Roy Disney had chosen as their cartoons' new distributor three months earlier. In August 1936 Horne sold his "gag library" to Walt Disney for twenty thousand dollars, for use by the writers of the Disney animated cartoons and comic strips. They received it with a predictable lack of enthusiasm. Buying the library was probably, at least in part, Disney's way of compensating Horne for his losses on Mickey Mouse Magazine.
Herman "Kay" Kamen, a former Kansas City, Missouri, advertising man, succeeded Horne as Mickey Mouse Magazine's publisher. Kamen, in charge of Disney's licensing efforts since July 1932, was an energetic and resourceful businessman who had talked himself into a deal in which he and Disney split the proceeds from licensing the Disney cartoon characters to other companies. Kamen delivered on that deal, spectacularly, by traveling incessantly and licensing Mickey Mouse and other characters to hundreds of manufacturers. The official total, when Mickey Mouse Magazine began publication in May 1935, was 230.
Kamen had been based in New York since 1932, so there were no geographic obstacles to his taking charge of Mickey Mouse Magazine. Moreover, by 1935 there were in Kamen's offices, as the New York Times reported, "workrooms where Disney artists, trained in the technique of the Hollywood studio, draw the countless pictures used in Mickey's commercial undertakings." Whatever that meant, exactly—there is scant evidence in Mickey Mouse Magazine of 1936–37 of successful training in how to draw the Disney characters—Kamen at least had plenty of people available to fill the magazine's pages. But despite his success in licensing the Disney characters, Kamen was no more successful as a magazine publisher than Horne had been. Success for a Disney magazine would come only when that magazine became a comic book, but in 1937 that was not a conclusion that anyone connected with Mickey Mouse Magazine had yet reached.
Disney comics of a sort, self-contained gag pages with dialogue balloons, appeared sporadically starting with Mickey Mouse Magazine's first issue, but when the magazine began reprinting full-color Mickey Mouse and Silly Symphonies Sunday pages, as of the July 1937 issue, they stood apart from the rest of the contents by virtue of their crisp professionalism. That issue was also the first since the very first one, in 1935, to have half its pages in full color, and it was the first to be published not by Kay Kamen Ltd. but by a new corporation that borrowed Kamen's initials: K.K. Publications.
K.K.'s address was the same as that of the Kamen firm, 1270 Sixth Avenue in Manhattan, but there was a signal of major change in the fine print that identified the magazine's owners and its status with the post office. There was now "additional entry at Poughkeepsie, N.Y."—that is, authorization to mail the magazine from that city eighty-five miles north of Manhattan in the Hudson Valley. Poughkeepsie had become the site in October 1934 of Western Printing & Lithographing Company's first plant outside its home base of Racine, Wisconsin, on Lake Michigan. Western was by then already a leading publisher of children's books, and in 1937 it was on the verge of also becoming a publisher of children's magazines—comic books included.
Western traced its origins to September 1907, when Edward Henry Wadewitz, a son of German immigrants and the twenty-nine-year-old bookkeeper for a Racine ship chandlery, bought the West Side Printing Company, a struggling basement print shop. The purchase price almost equaled what the print shop owed Wadewitz in fees for the bookkeeping services he had provided in his spare time. Wadewitz knew nothing about printing, but he wanted a business of his own, and within a year he took on a partner, Roy A. Spencer, who was an experienced printer. The company—incorporated as Western Printing & Lithographing in August 1910, after the addition of lithographic presses—grew steadily until by 1914 it occupied a six-story building in Racine.
Western entered book publishing on February 9, 1916. As the chief creditor of a failed Chicago publisher named Hamming-Whitman, it acquired all of Hamming-Whitman's assets, including its unsold books and the Whitman name. Later that month, Western set up Whitman Publishing Company as a subsidiary and began manufacturing and selling its own juvenile books.
By the 1930s, Western was producing not only children's books under the Whitman name, but also dozens of titles for other publishers. It was, besides, a leading producer of playing cards, games, and greeting cards, as well as being a large-volume commercial printer. In 1934, as a company publication said, Western was "looking for a site in the East, preferably within 100 miles of New York City, primarily to better and more economically serve the very substantial eastern markets." Western found what it wanted in Poughkeepsie: an empty 125,000-square-foot building on the Albany Post Road. The new plant had been built twenty-five years earlier to make Fiat automobiles when that Italian company was trying to establish itself in the American market.
When K.K. Publications came into existence in 1937, there was already a strong link between Western and Disney. Western was one of the earliest licensees for Disney books—an association that had its beginnings on April 19, 1933, when Samuel E. Lowe of Whitman wrote to Walt Disney. He explained that Whitman had just launched the Big Little Books, chubby little books that fit in the palm of a child's hand and told their stories through text and drawings on facing pages. Lowe sent Disney the first two, which were based on the popular comic-strip characters Dick Tracy and Little Orphan Annie. (The drawings were taken from the comic strips and the text adapted from the dialogue balloons.) He wrote: "Dick Tracy came out about two or three weeks before Christmas [1932], and we have printed over six hundred thousand of these books. Orphan Annie has been out about five weeks and we have printed over six hundred thousand. In the case of Dick Tracy we are already publishing a second book and we are planning to do the same with Orphan Annie. . . . We wonder if it is possible to get the right to Mickey Mouse in a book of this kind, which is different than anything already published."
Roy Disney's response was positive—with sales figures like those in Lowe's letter, it was unlikely to be otherwise—and later in 1933 Whitman published its first two Disney books—a Mickey Mouse coloring book and a Big Little Book, titled simply Mickey Mouse, that reworked a 1931 episode, "Mickey Mouse and the Gypsies," from the Mickey Mouse comic strip. Before long, Western was producing all the Disney-licensed books, whether they were published by Whitman or some other company. In 1937, two and a half years after the opening of Western's Poughkeepsie plant, its close relationship with Disney found additional expression in K.K. Publications and Mickey Mouse Magazine.
In a 1979 memorandum, Howard Anderson, who was working for Kay Kamen in 1937 and later became Western Printing's executive vice president and chief financial officer, recalled K.K. Publications' genesis:
K.K. Publications, Inc., came into existence in April 1937 for the purpose of taking over the publication of the Mickey Mouse Magazine. . . . [I]t was owned 60 percent by Kay Kamen, Ltd., and 40 percent by E. H. Wadewitz. . . .
When K.K. Publications, Inc., was formed Western took over the printing of the magazine. In mid-1938 the subscription promotion and fulfillment function was moved from Kay Kamen's office in New York to Western's plant at Poughkeepsie.
The Mickey Mouse Magazine (circulation 45,000) continued to be published by K.K. Publications, Inc., until the fall of 1940. It was not a profitable venture and it was decided that a change in format was necessary. In September 1940 it was decided that Western, through K.K. Publications, Inc., would take over the entire ownership. The 60 percent of the capital stock in K.K. Publications, Inc., which had been owned by Kay Kamen, Ltd., was transferred to two other individual Westerners, R. S. Callender and F. J. Leyerle, thus putting 100 percent ownership within Western. At the same time it was decided to discontinue the Mickey Mouse Magazine and in its place a new publication, namely Walt Disney's Comics and Stories, was born.
Assuming Anderson's circulation figure is correct, Mickey Mouse Magazine's sales had continued to slide since Hal Horne's departure—a slide that may have been all the more galling because a new and competing children's magazine, Curtis Publishing's Jack and Jill, was proving to be far more successful. In June 1939, eight months after Jack and Jill's launch, Curtis claimed that monthly circulation had risen to 125,000, at a cover price of twenty-five cents, the same as the fading St. Nicholas, versus Mickey Mouse Magazine's dime. In light of such a marketplace defeat, converting Mickey Mouse Magazine to a comic book might have seemed like an obvious and appealing response. But by mid-1939, two years after reprinted Disney comic strips began appearing in the magazine, they had never established much more than a toehold, usually taking up only five pages. Despite the occasional ballyhooing of the comic strips on the cover of the magazine, there was always the sense that its editors begrudged the pages given to them.
Mickey Mouse was listed facetiously as editor on the magazine's masthead, but the real editor by the fall of 1938, as identified in an annual statement required by the U.S. Post Office, was Lily Duplaix. She was the wife of Georges Duplaix, a French artist who had worked for the Artists & Writers Guild, a whimsically named Western Printing subsidiary, from around the time it opened its doors in Manhattan in 1935. The Artists & Writers Guild—which was in no sense a true guild—packaged children's books for other publishers, thereby generating printing work for Western. By the time Georges Duplaix became its director in 1940, the Guild was making a serious effort to drive down the prices of such books, to increase their sales and take advantage of the Poughkeepsie printing plant's economies of scale. Western's price cutting bore fruit most spectacularly in the Little Golden Books, priced at twenty-five cents each when Simon & Schuster first offered them in 1942. In Leonard Marcus's words, "It soon became clear that, at twenty-five cents, millions of parents would take a chance . . . by purchasing Little Golden Books not just one at a time but by the handful."
Such mass-market books were, along with the children's books that Western published under its own Whitman label, regarded with distaste or worse by most other trade publishers, by traditional booksellers, and especially by librarians. Those people had even less use for comic books. Mickey Mouse Magazine's gingerly handling of its comic-strip content amounted to a sort of confession that giving the comics more prominence really would doom the magazine, and thus the people associated with it, to permanent pariah status.
Jack and Jill made scarcely a bow toward the comics, apart from a couple of one-page "picture stories" in each issue, and it was, in its prevailing sweetness and gentleness, the antithesis of most early comic books. Mickey Mouse Magazine in its Hal Horne phase proclaimed itself "A Fun-Book for Boys and Girls to Read to Grown-ups," but Jack and Jill was much more the sort of insistently wholesome magazine that parents were likely to buy for their children. Jack and Jill's one true comics page, "Peggy and Her Pinto Pony," complete with dialogue balloons, vanished after two early-1939 appearances. By late in the 1930s, though, comic books were clearly ascendant: they were the publications that children eagerly bought and read even when their parents hesitated.
It had taken the comic book a surprisingly long time to assume its definitive shape. Comic books of a sort, reprinting comic strips (often in hard covers), had appeared early in the twentieth century, soon after the first true comic strips appeared in newspapers, but what now seems like a short and natural step—to original comic-book narratives extending over multiple pages—took a surprisingly long time, awaiting an inspiration like the one that Harry Wildenberg described to John Vosburgh. It was not until Superman's debut that a feature appeared whose leading character seemed to be perfectly suited to the new format: a generic mythological figure, as coarse and obvious as the color and the paper. But even that was deceptive. Superman's writer and artist—Jerry Siegel and Joe Shuster, respectively—envisioned their creation not as a comic-book feature, but as a newspaper comic strip, and they accepted a meager offer from a comic-book publisher only when the syndicates that distributed comic strips turned them down.
In a publishing environment changed dramatically by the popularity of Superman and the superpowered characters that followed him, Mickey Mouse Magazine underwent what now seems like an inevitable transformation into a comic book. As of the September 1939 issue, its entire contents were printed in full color, and its covers on slick paper. With the June 1940 issue, its dimensions shrank slightly, to those of a standard ten-cent comic book. Then, with its last issue, dated September 1940, Mickey Mouse Magazine doubled in page count, to sixty-four—the usual number of pages in a contemporaneous comic book—with reprinted comics filling thirty-four of those pages. It had become a comic book in all but name, anticipating the introduction of the new Walt Disney's Comics & Stories the next month.
The October 1940 Walt Disney's Comics & Stories, vol. 1, no. 1, was a real comic book with more than forty pages of reprinted newspaper comic strips—the culmination of the halting progress that had begun with the July 1937 issue of Mickey Mouse Magazine. It had been beaten to the newsstands by two other Western Printing comic books, as Howard Anderson explained: "In addition to publishing Walt Disney's Comics and Stories K.K. Publications, Inc., took over the publication of Super Comics and Crackerjack [sic] Funnies, which prior to that time had been published by Western under an operation designated as 'Whitman Newsstands' . . . . These three publications made up Western's entry into the comic magazine business."
Western introduced both Super Comics and Crackajack Funnies early in 1938, less than a year after taking charge of Mickey Mouse Magazine. Like many other early comic books, Super and Crackajack offered mostly reprinted newspaper comics—a few pages each of a wide range of Sunday pages, straight adventure alongside slapstick comedy. Even though comic books with new material had multiplied thanks to Superman's success, Walt Disney's Comics & Stories, when it made its debut in 1940, was still very much a mainstream comic book in its reliance on reprinted comic strips. Some of those comic strips were self-contained gags (Donald Duck), and others were continuing adventure stories (Mickey Mouse).
Many of the reprinted strips in Walt Disney's Comics were a surprisingly awkward source for the content of a children's magazine. The comic strips, products of the Depression years, portrayed a harsh world that invited correspondingly harsh laughter when they were supposed to be funny at all. Mickey Mouse, especially in the continuing stories of the daily strip, was an adventure hero whose life was threatened, fiercely and sometimes for days at a time, by murderous thugs. Donald Duck, in that gag-a-day strip, was often portrayed as a malicious delinquent whose misdeeds were repaid by savage beatings. Walt Disney's Comics was, moreover, anachronistic in its reliance on text pieces like those that dominated Mickey Mouse Magazine until 1940. When the Disney animated feature Dumbo was adapted for Walt Disney's Comics, it was not as comics but as text, spread over four issues, like the adaptations of Snow White and the Seven Dwarfs and Pinocchio in Mickey Mouse Magazine before it. But such was the public's appetite for comic books in the early 1940s, and such the popularity of the Disney characters, that the circulation of Walt Disney's Comics immediately outstripped that of Mickey Mouse Magazine.
Western paid a tiny royalty—one quarter of a cent—to Disney and other copyright holders on each copy of a ten-cent comic book it printed, with the royalty payable upon completion of the print run. As E. H. Wadewitz explained to one of his licensors, the comic-book royalty was only half the royalty on children's books sold through chain stores because "the distributor must have a larger margin in order to allow for returns of unsold copies." But there was a trade-off: the licensor got paid for every comic book printed, even though 40 percent or more of those copies might not be sold. It was a trade-off that could become more attractive as sales and print runs rose. Western paid Disney royalties on 252,000 copies of the first issue of Walt Disney's Comics & Stories. Two years later, as of no. 24, the September 1942 issue, Western for the first time paid Disney a royalty on a million copies of Walt Disney's Comics, and sales were continuing to rise.
By 1942, Western Printing had long since opened a Los Angeles office, the better to be close to Disney and the other movie studios whose cartoon characters, in particular, were becoming increasingly important to it. Western had hired as a liaison with the studios Eleanor Packer, a widow in her early forties who had written Whitman books for years and had worked for a major studio besides. That office would become a vital adjunct to Western's original comic-book operation based in New York City.
## 2
# Oskar Lebeck Meets Walt Kelly
Between 1935 and 1939, a German immigrant named Oskar Lebeck illustrated at least three coloring books for Western Printing & Lithographing's consumer arm, Whitman Publishing. He also illustrated a Whitman softcover potpourri of fiction and fact for children called Chatterbox, drawing for all of those books in a spare, cleanly inked style. In addition, he wrote and illustrated three hardcover books and illustrated one more, an adaptation of L. Frank Baum's The Wizard of Oz, for Grosset & Dunlap, which was, like Whitman, a producer of mass-market children's books, most notably the series with the juvenile heroes Tom Swift and Nancy Drew.
Lebeck intended his own books for children younger than Tom Swift's readers, and he may have taken a little too seriously the responsibility of entertaining an audience about the same age as his own daughter (and only child), who was born in 1932. "His humor was a little German, a little heavy," said Morris "Moe" Gollub, who drew comic books for Western Printing after World War II. Indeed, both The Diary of Terwilliger Jellico (1935) and Stop Go: The Story of Automobile City (1936) are hopeless literary plodders. Only Lebeck's last Grosset book, Clementina the Flying Pig (1939), has real imaginative flair, but that quality turned up in his work at just the right time, when he was hiring artists and writers some of whom turned out to have talents greater than his own. Because he recognized and nurtured those talents, Lebeck became one of the very few comic-book editors of real importance, and his comic books among the very few that published stories of lasting value.
Grosset & Dunlap's kinship with Western Printing was closer than mere resemblance to Whitman. In 1935, Grosset was the first customer for Western's new Artists & Writers Guild, the subsidiary that packaged children's books for other publishers. There is nothing in Lebeck's Grosset books to confirm that they originated with the Artists & Writers Guild, but since he was working for Whitman at the same time, they may well have. His association with both Grosset and Whitman probably gave Lebeck a leg up when Western was choosing an editor for its new Whitman comic-book line in 1938, but he had more in his favor than mere visibility. Not only was Lebeck a working artist; he was also, in the words of John Stanley, one of the gifted cartoonists who later wrote and drew for him, "an impressive, imposing man"—he was six feet tall—"who could sell himself anywhere." Lebeck was, said his daughter, Letty, "very gregarious, and people liked it," even though a German accent was no social advantage in the 1930s and 1940s. "He was easy to make friends with," she said.
Lebeck became a Whitman employee no later than March 1938, and he probably edited the first two monthly Whitman comic books, Crackajack Funnies and Super Comics, from their inception in the spring of that year. By early in 1939 he was editing not just those comics but two more monthly titles, Popular Comics and The Funnies, after Western began producing them for Dell Publishing Company. With four such titles, Western could have a new monthly comic book on sale almost every week, in addition to the occasional special or one-shot comic. Lebeck's office was at 200 Fifth Avenue in Manhattan, diagonally across Fifth from the Flatiron Building. The sixteen-story office building was called the Toy Center because its tenants were overwhelmingly companies in the toy industry, good neighbors for a company producing children's books. Western opened sales and display offices at the Toy Center sometime in the early 1930s.
By 1939, as comic books with original material became increasingly popular, Western and Lebeck were responding by adding a raft of new features, to Popular Comics and The Funnies especially. Those features were for the most part continuing stories in a boy's-adventure mode. "Shark Egan" (in Popular Comics) was described extravagantly but accurately by the comic-book historian Ron Goulart as "a South Seas adventure opus full of sailing ships, seaplanes, pretty ladies in distress, fabulous caches of pearls, and mutinous crews." The hero of another new feature, "Speed Martin" (in The Funnies), was, as Goulart wrote, "an ace newsreel cameraman who roamed the globe," another figure readily imaginable in series fiction.
This photo, showing Oskar Lebeck at work in 1943, was taken in New York at what a handwritten note on the back of the photo identifies as the Martinique, presumably the hotel at 32nd Street and Broadway in Manhattan. Around the table: clockwise from the left: Ruth Lebeck, Oskar Lebeck's wife; Leon Schlesinger, the proprietor of the Warner Bros. cartoon studio; Mary DuBois, wife of Lebeck's premier comic-book writer, Gaylord DuBois; Harold Spencer, a Western Printing & Lithographing Company vice president, general manager of Western's Poughkeepsie, New York, plant, and the brother of Roy Spencer, one of Western's founding fathers; Spencer's wife, Todd; Gaylord DuBois; Oskar Lebeck; and a woman who may be either Schlesinger's wife, Bernice, or Helen Meyer, Dell Publishing Company's vice president. Courtesy of Letty Lebeck Edes.
One of the first original features in the comic books Lebeck edited was "John Carter of Mars," based on novels by Edgar Rice Burroughs, creator of Tarzan of the Apes. Whitman and Burroughs had by 1939 been doing business for half a dozen years, first through the licensing agent Stephen Slesinger and then directly; Whitman published its first Tarzan Big Little Book in 1933. Adding a Burroughs property to The Funnies, once Western began producing that comic book, must have seemed like a natural step.
The first four "John Carter" installments, beginning with The Funnies no. 30, May 1939, were drawn by Jim Gary, but almost immediately John Coleman Burroughs, Edgar Rice's son, wrote to the Western executive Robert S. Callender to say that he wanted to illustrate "John Carter" himself. Lebeck replied on May 4, 1939, that he was "of course, delighted," but the surviving correspondence from the next two years, until "John Carter" disappeared from The Funnies, reflects one worry after another. Lebeck was unfailingly diplomatic, but it is clear that editing a comic book when the format itself was still unsettled could be surprisingly difficult.
The Funnies, like other early anthology comic books, cast its net for potential readers of many kinds, and so it was packed with features that each took up only a few pages—four pages, at first, in the case of "John Carter," and later eight. To get more story into that limited space, Lebeck prodded Burroughs to draw pages of eight panels, not five or six. Then he resisted paying Burroughs for adapting his father's first Mars novel to the comics. He mentioned that Western was paying "an exceptionally high royalty" to the senior Burroughs, but there was a more general policy involved: "In all cases where we supply the artist with either book character, or radio or movie script, the artist is handling the writing of the strip with his art work and does not get paid extra for this." That policy would change soon, as the quantity of original material in the comic books grew and the writing and illustrating of comic-book stories was increasingly in separate hands. Lebeck anticipated the change when he told Burroughs that if "rewriting the strip" was too much work, "let me know and I will do this myself and then send you the script each time."
Even the lettering for "John Carter's" dialogue balloons and captions was a headache. When he got one batch of eight pages, Lebeck complained that the lettering "comes down too small, please have it about twice as large on the next set. We have had quite a number of complaints from our readers that many features in our magazines are hard to read, the lettering being too small." That was a problem for early comic books generally, since so many of their features were reprinted Sunday comic pages that had to be drastically reduced in size to fit onto the smaller comic-book pages. Dialogue balloons were difficult to read in the reduced size, so they were sometimes simplified and relettered to make them legible.
The "John Carter" drawings themselves were too small, Lebeck told Burroughs a few months later, after Burroughs sent him a second batch of pages in "reduced size": "Working in this smaller size and where there are many characters, it is impossible to get a good result, and I hope that your next set will again be done in the larger size." There was even a problem with the way that Burroughs erased his pencil drawings after he drew over them in ink: "In many spots, the inked work has been very gray and to such an extent, that the camera was unable to pick up the line." Later that year, Lebeck was urging Burroughs not to use the "drybrush" technique, which was "very difficult for us to reproduce . . . on our rotary press plates. We have to shoot for line only, and a lot of the drybrush effect drops out completely where it is too gray and in other cases, it becomes solid black."
"John Carter" finally disappeared from The Funnies, in the middle of a continued story, as of no. 56, June 1941. By that time comic books of all kinds, but especially the titles Lebeck oversaw, were going through major changes, and his own role had changed accordingly. When a federal census taker questioned Lebeck on April 18, 1940, he gave his occupation not as "editor" but as "illustrator" for an unidentified publishing company. As such things were viewed at Western, the "editorial" function was exercised by those executives at Racine and Poughkeepsie who dealt directly with the copyright holders that licensed use of their characters. Lloyd E. Smith at Racine negotiated contracts, and Robert S. Callender at Poughkeepsie was the editor of Western's comic books in a general supervisory sense. Lebeck's job was to assemble the actual contents—the illustrations—of the comic books. As original material filled more and more pages in the comics, that became unquestionably an editor's job in every important respect.
As to how Lebeck went about that work, there is the recollection of a cartoonist named Frank Thomas (not to be confused with the Disney animator of the same name). Lebeck hired Thomas early in 1940 to write and draw the adventures of a masked hero called The Owl starting with Crackajack Funnies no. 26, August 1940. Another cartoonist had drawn the first episode, in the June issue, with The Owl as a shoddy imitation of radio's Shadow, and Lebeck hired Thomas to transform The Owl into an athletic hero more like Batman.
New costumed heroes were proliferating then, and starting late in 1939, all four of Lebeck's monthlies began publishing stories with new characters best described as clumsy responses to other publishers' superheroes, most of which were already clumsy enough. Martan the Marvel Man, an alien who regarded the human race with benign condescension, first appeared in Popular Comics no. 46, December 1939. The vaguely Tarzan-like Magic Morro followed in Super Comics no. 21, February 1940, and the gigantic, transparent Phantasmo, "master of the world"—he can, a caption assures the reader, "pick up a mountain and throw it into the sea"—in the July 1940 Funnies, suspiciously soon after a generally similar superhero, The Spectre, debuted in a competitor's More Fun Comics.
Crackajack Funnies no. 39, September 1941, cover featuring The Owl, a halfhearted effort by Oskar Lebeck to feed on the popularity of costumed heroes like Batman.
Thomas had already originated a handful of superhero features for other early comic-book publishers. He had been sent to Lebeck by Helen Meyer, vice president of Dell Publishing, after he showed her his portfolio; Dell and Western were then in the early stages of their comic-book-publishing collaboration, which would last more than twenty years. The editors and publishers of other early comic books "seemed to be promoters and business men rather than creative types," Thomas wrote in a 1965 letter. But under Lebeck,
the procedure was different than I had before experienced. Oskar was a writer and artist in his own right and recruited a good staff of free-lancers of which I was proud to be one. . . . Oskar usually sparked the original feature idea, then called in the artist-writer he felt was best suited for it, the idea was talked over and enlarged upon, model sheets drawn up, a few pages executed and edited, then full steam ahead. After the first few episodes Oskar left the artist-author to go pretty much on his own, for he respected and encouraged individual talent and tendencies. And most were artist-authors—I recall very few features on which a team of artist and author were employed—that came later.
As Lebeck's subsequent career showed, costumed heroes were not his forte, but he did not burden his freelancers with his misgivings.
Oskar Lebeck was born in Mannheim, Germany, on August 30, 1903, and immigrated to the United States in March 1927. "Dad was pretty independent," his daughter recalled, "and I remember [my mother and father] always laughing about it, [that] he left to get away from his mother. . . . Dad came over first, and when he was making some money, so he could get an apartment, my mother [Ruth Seelig] came over [in December 1927], and they got married on Ellis Island" In April 1930 a census taker found the Lebecks living in the Forest Hills section of Queens in New York City, with Oskar's occupation listed as "artist."
According to biographical information that he must have provided, Lebeck had worked in Europe as a stage designer for Max Reinhardt. After he came to the United States, he did the same sort of work for Florenz Ziegfeld and Earl Carroll on Broadway, and he was also employed as an "industrial designer." Probably, like millions of other people in the Depression years, he scratched for work of almost any kind, and so for him, as for many other artists, comic books were a godsend.
Ruth and Oskar Lebeck in 1930, a few years after they immigrated to the United States. Courtesy of Letty Lebeck Edes.
By the spring of 1939, the Lebecks had moved at least twice since leaving Queens—first to Staten Island, and then north to Croton-on-Hudson in Westchester County, in the Hudson River valley. Croton is roughly midway between Manhattan, where Whitman and Dell had their editorial offices, and Poughkeepsie, where Whitman's parent, Western Printing, had its East Coast plant, so the convenience of the location may have recommended it.
By 1941 new stories dominated the Whitman and Dell comic books produced under Lebeck's aegis—Crackajack, Super, Popular, and The Funnies, plus the occasional color or black-and-white one-shot—even though there were still many pages of reprinted newspaper comic strips. The original material in the Lebeck-edited comic books was drawn by a cluster of young men in their twenties, some of them already veteran illustrators of pulp fiction. One of them, William Ely, told Ron Goulart that his family home in White Plains, New York, north of Manhattan, became what was, in Goulart's words, "a sort of informal studio for Dell comic book artists." Lebeck drove to the Ely home to pick up finished pages and deliver scripts.
This increased activity yielded new features that were for the most part respectably drawn but also strongly reminiscent of newspaper strips or other publishers' comic books. Lebeck's young artists especially liked what they saw in Milton Caniff's Terry and the Pirates comic strip—the fluid brushwork, the rich shadows, the stalwart heroes, the glamorous women, the exotic settings. But there was ultimately no confusing their drawings with Caniff's. The closest to an exception was Ken Ernst, who drew "Magic Morro"; he lived in Chicago and submitted his work not to Lebeck but to Don Black, an editor at Western's Racine headquarters, just seventy-five miles north on Lake Michigan. Ernst ultimately became a syndicated cartoonist himself, drawing the Mary Worth comic strip for many years.
The superhero fever was ebbing in the Dell comic books as early as the summer of 1941, when Captain Midnight, a radio hero who lacked superpowers, took Phantasmo's place as the featured character on the cover of The Funnies. At the same time, the emphasis in Super Comics shifted back to comic-strip reprints. Change was in the offing in other respects. Most of Lebeck's artists were of draft age, and as the military depleted his pool of freelancers, he had to fill his comic books with the work of cartoonists who were either too old for the draft or exempt for other reasons.
That was a daunting task, but one of those draft-exempt cartoonists more than took the place of the cartoonists who had left. The big change in the Whitman/Dell titles, the first step toward comic books with an identity distinct from anything else in newspapers or on the newsstands, came a few months before Pearl Harbor, in the fall of 1941, when Walt Kelly connected with Lebeck. Kelly was still in his twenties then, but he was a veteran member of Walt Disney's animation staff. He was already a competent draftsman when he was in his teens, and his natural bent toward cuteness and comedy fitted him for work on the Disney cartoons. He spent more than five years at that studio, under the tutelage of experienced animators who drew extremely well. As his high school classmate Ray Dirgo said, "That's where Kelly learned how to make his drawings look effortless."
Kelly's Disney credentials undoubtedly won him easy entrée to Lebeck's office. As it happened, though, Kelly drew almost no Disney comics in his first couple of years as a Western freelancer. When he began drawing for Western's comic books, it was the polish and warmth of his best drawings—not any specific Disney content—that made them so unusual and attractive, and that made his comics work stand out from the beginning.
Walter Crawford Kelly Jr. was born in Philadelphia on August 25, 1913, the son of Walter C. Kelly Sr. and Genevieve MacAnnulla Kelly, both native Pennsylvanians; he had one older sister, Bernice. The Kellys moved to Bridgeport, Connecticut, on Long Island Sound sixty miles northeast of New York City, when Walter Jr. was two years old. As he wrote in 1959, "The First World War brought to Bridgeport many strangers to work in the factories." Kelly's father was one of them; he was a foreman first in an immense Remington Arms munitions plant on Boston Avenue and then for General Electric, after it leased that plant in 1920 to make civilian products. The Kelly home at 478 East Avenue on Bridgeport's East Side was less than a mile from the plant.
Bridgeport's population burgeoned from just over 100,000 in 1910 to about 175,000 during the war, then fell to 150,000 in the early 1920s and hovered around that figure for the next few decades. Despite the strains and labor strife that accompanied its rapid industrial growth, Bridgeport remained, as Kelly described it, an accommodating sort of town for a child. "We had our differences," he wrote of himself and his schoolmates, "fighting over the important things, such as erasers, marbles, candy, but we didn't quarrel about race or name. . . . It would be nice, Manhattan, if everything outside New York were Bridgeport . . . which was more flower pot than melting pot, more by-way than highway, maybe even more end than beginning."
In 1969, speaking to other cartoonists in blunter language than he had used in print a decade earlier, he called Bridgeport a "good town," even though lacking in important respects. "Such sophistication as it had was brought back by kids from college and so on. And it hated New Yorkers, and it hated Negroes, and it hated Jews, hated anybody that was outspoken or needed something done for them. So it was a very crab-ass town, but a great place to grow up until you get to be about 14 years of age."
It seems likely that Kelly's memories of Bridgeport, and then his proximity to it when he was drawing comic books and living elsewhere in Connecticut, shaped at least a few of his stories, particularly those in Our Gang Comics. Not that Kelly and his friends ever had to deal with exotic threats like those that the children in the "Our Gang" stories encountered repeatedly, but the stories took place in mundane, Bridgeport-like surroundings where children could roam freely without worrying their parents.
"In my early teens," Kelly wrote many years later, "I started work as a high-school reporter for the Bridgeport Post. This came about because while still somewhat of a recluse [because of a prolonged illness], I had submitted some drawings to the Sunday Post for its Junior Page section. Out of this, I became the high-school reporter." Kelly also drew cartoons for the student newspaper and yearbook at Warren Harding High School. Ray Dirgo told Bill Crouch Jr. that art instruction at Harding, like the city of Bridgeport itself, had a distinctly utilitarian cast: "[Y]ou had to take mechanical drawing. This was because some people would be eventually getting jobs in the local factories as draftsmen. Then you were able to get into fine art. Mostly it boiled down to a lot of drawing of a vase with a flower stuck in it and stuff like that. However, if you did art like Kelly and I did"—that is, drawings in imitation of popular cartoonists of the time, like Percy Crosby (Skippy) and Roy Crane (Wash Tubbs)—"you were able to work on a poster or cartoon during regular class time."
Dirgo continued: "I'd had a course on lettering from a guy from Yale who was a friend of the art teacher. . . . Because of this I could letter better than Kelly in high school. Normally we'd get out of school about 2 P.M. and Kelly would head off to the newspaper. However, he wanted to pick up some of this stuff I'd learned and we would go up to the art department and work on lettering. He got good at it"—a statement buttressed by the expressive character of the comic-book and comic-strip lettering that is recognizably Kelly's own, notably in the Pogo comic strip of the early 1950s.
After Kelly graduated in 1930 there followed what he called "a brief period of factory work," a few weeks or months of the sort of patchy employment awaiting many people in the early years of the Great Depression. According to a biography distributed by the Post-Hall Syndicate in the early months of Pogo's national syndication (a biography that gives every sign of having been written by Kelly himself), that factory work consisted first of "wrapping scrap cloth in a factory that manufactured undergarments for ladies," and then, when that job ended, "smashing faulty switches" in an electrical appliance plant (almost certainly General Electric, his father's employer).
But then, Kelly wrote, "the Post called me back and wondered if I would lend my highly imaginative style to general assignments. By sheer drift I was next assigned to the art department—indeed, for a long spell, I was it. During this time, fascinated by the angry, funny political work of J. N. (Ding) Darling, I experimented with political cartooning, because Bridgeport was face to face with the Depression and a change was in the air." Kelly drew what he described as his first political cartoons for the Post, including some supporting Jasper McLevy, the Socialist Party candidate for mayor in the 1931 and 1933 city elections. McLevy was victorious in November 1933, but without any help from Kelly cartoons; the newspaper published no editorial cartoons in the weeks before the election except for one syndicated cartoon each Sunday by Ding Darling. Kelly seems to have made his most significant artistic contribution to the Post for seven months starting in July 1931, when he illustrated four panels a day of "P. T. Barnum's Life in Pictures and Prose." Barnum, the nineteenth-century showman and circus magnate, was at one time Bridgeport's mayor and is still remembered as the city's most famous resident.
Not long after McLevy was elected mayor, Kelly left the Post to take a job as an investigator for Bridgeport's welfare department. When he left (or lost) that job, probably by early in 1935, there followed what the syndicate's biography called "a short engagement as a clerk in an art store, most of which time was spent hunting rats in the cellar." Next was a stint as an unsuccessful freelance artist in New York City. That period was vague in Kelly's telling, except for one episode recounted vividly in a 1952 Collier's profile; it took place just before Christmas. Kelly found work with a window-display company but was tricked into decorating two plate-glass windows with drawings of Santa Claus for the price of one—or, as it turned out, for nothing, since his deceitful employer refused to pay him. Kelly, still in his early twenties, "retreated to my attic. . . . I got back to Bridgeport just in time for Christmas dinner."
The "educational history" section of Kelly's Disney personnel record showed he spent a year studying commercial art at night school, location unspecified, as well as five weeks studying "illustration and life" with Franklin Booth, a famed illustrator of the 1920s. Booth taught at the Phoenix Art Institute, a New York school he cofounded in 1925. Kelly told an interviewer: "I was sort of a monitor and swept up the place, this gave me something to do in the afternoons." Booth and Kelly were an odd fit: Booth's drawings were so grand and meticulously rendered in pen lines that they resembled steel engravings. That was not Kelly's style, but occasionally some trace of Booth would surface in his later work when grandeur was wanted.
By late 1935, Walt Disney was expanding his staff in anticipation of the demands that would be imposed by work on Snow White and the Seven Dwarfs, the studio's first animated feature. When Kelly wrote to Disney about a job he got a positive reply, and he left for California. "I borrowed some money from my father, who had to hock everything but the chimney," Kelly told Murray Robinson of Collier's, "and I set out for the West by bus." He became a probationary Disney employee on Monday, January 6, 1936.
In 1935, before he moved west, Kelly drew and probably wrote a few pages for some of the earliest comic books: More Fun, The Comics Magazine, and New Comics. These were simple one- or two-page features, not really comics at all. "The Little People, Irish Tales by Walt Kelly" from More Fun no. 8, February 1936, comes closest—twelve small panels on one page, with captions and no dialogue balloons—but a full page like "Down by the Old Mill Stream," from More Fun no. 7, January 1936, filled with "little folk" that strongly resemble Palmer Cox's Brownies, would have fit just as well in a traditional children's magazine. Kelly drew at least one page for the most venerable of such magazines, St. Nicholas, illustrating a poem by Dorothy Brown Thompson, "Ballad of a Hunter of Renown," for the November 1935 issue. The hunter is a vainglorious boy, and the page anticipates the illustrated poems Kelly contributed to the Dell comic books ten years later.
Kelly's move west and into animation may have been stimulated not so much by his desire to become a full-time cartoonist as by his infatuation with Helen De Lacy, a Bridgeport woman almost seven years his senior who had just moved to Oakland, California, to take a job as an executive with the Girl Scouts. Kelly met her at choir practice at the Methodist church they both attended. They were engaged by June 1937; she resigned then from her job with the Girl Scouts, effective in September 1937, the month she married Kelly.
Clair Weeks, who eventually became a Disney animator, remembered that out of a "class" of perhaps three dozen men, only he and Kelly and one other candidate survived the probationary period to become permanent Disney employees. He said:
Early Walt Kelly efforts like this page for the November 1935 issue of St. Nicholas: The Magazine of Youth anticipated the illustrations he would contribute to many Dell comic books for very young children in the 1940s.
Walt Kelly and I used to go up on the hill behind the annex [the building that housed the pool of "inbetweeners," or animation apprentices] at noontime and eat our little sack lunch and contemplate our futures—wondering what we were doing and whether we were going to make it or not. It was a very uncertain time of life for us. He was a very slim, thin little fellow, and he'd been working on a Connecticut newspaper, as a cartoonist. He came out here, with what little money he had, and he decided to take the gamble.
Weeks said that Kelly "was taken up by the story department . . . because he could draw little kids very nicely"—a judgment consistent with Kelly's drawings for the early comic books, before he moved west. "He used to draw these cute little children while he was practicing inbetweening. . . . He was one of the first of us to sort of graduate and go across the street [to the main Disney plant]. In those days, crossing the street was like crossing the Red Sea, or the Rubicon, or something; that meant you had arrived."
Ward Kimball, who joined the Disney staff in 1934 and was by the late 1930s one of the studio's premier animators, remembered that Kelly "made very funny drawings, and when it was decided that we needed extra animators, he left the story department and took a crack at animation." In 1939 Kelly began working as a junior animator under Fred Moore, a leading Disney animator, and also, less often, under Kimball. Kelly eventually animated dozens of relatively simple scenes in four early Disney features (Pinocchio, Fantasia, The Reluctant Dragon, and Dumbo) as well as parts of a couple of Mickey Mouse shorts.
"We loved the way he drew Mickey Mouse," Kimball said. "His proportions were very subtly different from the model sheet, and even the accredited authority for Mickey Mouse, Fred Moore, would always laugh at Kelly's drawings. They were just basically funny. In fact, everything he drew was funny."
Kelly in Kimball's recollection, as in Clair Weeks's, resembled a cartoon character himself: the stock editorial-cartoon image of "John Q. Public," "the thin guy with the moustache and the bow tie . . . stoop-shouldered, very thin. He always drew himself exactly like John Q. Public" in the caricatures of one another that the Disney animators exchanged constantly. "If anything slightly crazy would happen," Kimball said, "we'd all draw a gag about it."
Kelly was slender—he weighed around 160 pounds when he registered for the military draft in October 1940, although at six feet tall, he was not the "little fellow" Clair Weeks remembered. Despite his meek appearance, Kelly plunged enthusiastically into the rowdy clowning that was typical of the Disney animators, who were overwhelmingly young and energetic as well as talented. Kimball spoke of "playing football in the hall, Kelly and Fred Moore on one side, Bud Swift [Kimball's assistant, David Swift] and me on the other. We'd kick off in the narrow hallway and the boys in the rooms would have to close their doors, because the football would ricochet and go flying in every direction."
Hank Ketcham, who joined the Disney staff in October 1939 and in the 1950s became a famous newspaper cartoonist (Dennis the Menace), remembered Kelly as "a cigar-smoking Irishman who just loved his waking hours. He laughed loudly, and he was a great dramatist; he could act things out in a very funny way. He was a Charlie Chaplin freak and liked to caricature guys with canes and top hats. . . . Kelly was pretty much of a realist, and quite a cynic; a lot of things he didn't care for." That posture may have provoked this note in Kelly's personnel record: "For a while was sort of a sore head griper but got over this at time of leaving."
Kelly's departure from the studio on May 27, 1941—Ward Kimball noted in his journal that Kelly caught a 5:15 train that day, headed east—preceded by less than a day a traumatic strike by several hundred Disney artists. The strike began at 6 A.M. on May 28 and lasted two months, with flare-ups after that. Kimball, who did not join the strike, said that Kelly "had friends on strike and he had friends not on strike. He had to make a choice, and he didn't want to face it." Dave Hilberman, a leader of the striking employees, said many years later that Kelly "left before the strike, he wasn't going to get involved." Although Kelly told his friends that he was only taking a vacation and would be back, Kimball thought that he was leaving Disney's for good, to work for a newspaper.
Kelly's prolonged absence from the studio was in fact with his superiors' blessing. A note in his Disney personnel record said of Kelly that he "was not layed [sic] off but took extended leave of absence because of illness in family." Kelly said many years later: "Someone was needed to take charge." It was his sister, Bernice, two years his senior, who was suffering from some illness that has never been specified, and Kelly wrote from Connecticut to Kimball of "neurotic acrobatics in the family."
Kelly's own life had already been shaped by serious illness. He wrote in a memoir of his childhood in Bridgeport that as a boy he "contracted some still mysterious ailment which paralyzed my left side"—possibly, according to a profile of Kelly by the magazine writer Bob Abel, a reaction to a diphtheria antitoxin injection he received in his back. That condition kept Kelly out of school for two years, and it was probably what kept him out of military service in World War II. He was rejected when he was called up for induction in the fall of 1943.
In June, after Kelly returned to Connecticut, he wrote disparagingly about the strikers and their union in a letter to Kimball and Moore: "I noticed that a good many men in newspaper photos of the picket line were amongst those sterling gentlemen who were canned a month or so ago—after living on charity for a good many years." He at first seemed in his letters to be eager to get back to California and animation, but when he returned to Los Angeles in July 1941, he told Kimball he had decided to remain in Bridgeport.
Kelly crossed the picket line on July 25, a week after his conversation with Kimball, to meet with Walt Disney. According to Helen Kelly, he went to seek Disney's help in finding work in the East. That help was forthcoming on August 11, the day Disney left on an extended trip to South America. Three letters over his signature went out: to Kay Kamen, to Leo Samuels in Disney's New York office, and to Marshall "Mike" McClintock, a writer and editor of the books that Whitman produced through its Artists & Writers Guild subsidiary. Disney's letters, describing Kelly as a "former employee," said that he had "a complete understanding of the handling of any and all of our characters" and urged the recipients to get in touch with Kelly at his Bridgeport address.
A couple of weeks later, Samuels wrote Disney that McClintock was following through: "Mike advised that he is getting in touch with Kelly to have him come in to New York with a view towards giving him some free lance work." It was undoubtedly through McClintock that Kelly met Oskar Lebeck and began drawing comic books for him around the time of his formal separation from Disney on September 12, 1941.
According to Bill Crouch Jr., in fragmentary notes based on Helen Kelly's memories, the Kellys settled first in Nichols, a town northeast of Bridgeport. Presumably Bernice's illness was no longer an impediment to Walt's leaving his parents' home for a home of his own; neither was it needed as a reason for staying away from his job at Disney. Kelly drew comic books at home and commuted by train from Bridgeport to New York City three days a week to deliver finished work and pick up new assignments. The Kellys moved seven times in two years, alighting briefly in New York City and at Oskar Lebeck's home in Croton.
Kelly wrote to Walt Disney around November 1 to thank his former employer for his help: "Your letters to your friends in New York were invaluable—and through them I've managed to land enough free lance work to keep out of the bread lines. As a matter of fact everybody has been quite friendly and generous toward me—and this attitude was due to the warmth of your letters."
A note on Kelly's "value" in his personnel record said that he "was beginning to click on comedy action" just before he left the studio, but, even so, if he had returned to animation after the strike he could have been vulnerable to a layoff. He had skipped the "cleanup" step when he became an animator—the step above inbetweening, and preceding animation itself—and he was thus "very weak and costly" on animation's purely technical requirements. "Also," his personnel record noted in its telegraphic style, "work often so rough that cleanup and inbetween work on scenes very costly." Expensive animators were a luxury Disney could not afford in the straitened environment that followed the strike. For Kelly, comic-book work in New York, so close to Bridgeport, was an obvious alternative to animation: he had already done such work, and the comic-book industry in 1941 was far more robust than in 1935.
## 3
# Whitman, K.K., and Dell
In September 1929, Whitman published A Story of Our Gang, a children's book based on the Hal Roach movie shorts. It was the first Whitman book written by Eleanor Lewis Packer, who had arrived in Los Angeles in July 1928, six months before her thirtieth birthday. Eleanor and her husband were both from Columbus, Ohio, but by early in 1928 they were living in Chicago, where George L. Packer was a sales manager for a stove company. He died in Chicago in April of that year, at the age of thirty-two, of what his death certificate called heart disease.
"Fortified with introduction cards from an old Ohio State [University] friend," the university's alumni magazine later reported, the newly widowed Eleanor moved to Los Angeles, where she found work as a publicist, first for Douglas Fairbanks and then, six months later, for Hal Roach. While she was with Roach she wrote the Whitman book, which had, she told the alumni magazine, "a circulation of two million and a half copies." After four months with Roach, she became a publicist for Metro-Goldwyn-Mayer (MGM), the distributor of the Roach comedies; she served as the studio's liaison with writers for fan magazines. According to the 1930 federal census, she was living in Beverly Hills with her two young sons and her mother.
Packer worked as a publicist for MGM until 1933, when she quit and devoted herself to freelance writing. By then, she had already begun writing more books with movie tie-ins for Whitman, and she did so throughout the 1930s, mostly Big Little Books based on movies like MGM's David Copperfield, Treasure Island, and Sequoia. She had become a Western Printing employee by December 1936. Ohio State's alumni magazine reported in 1939: "As Hollywood representative for the Whitman Publishing Company . . . Mrs. Packer . . . handles all of the contacts of the company with the motion picture stars and writes most of the company's other books based on Hollywood pictures." It was undoubtedly thanks to her Whitman experience that she was chosen to oversee the comic-book operation on the West Coast.
Oskar Lebeck, Packer's counterpart in Whitman's New York office (and at least nominally her superior), was also an author of mass-market children's books, and as it turned out, that background subtly shaped the comic books Packer and Lebeck edited, and set them apart from most of their competition. The earliest superhero titles of the late 1930s and early 1940s often seem to have been written and drawn by children—more specifically, by young boys reveling in action of the most extreme and ridiculous kind—and, in fact, many of those cartoonists were no more than a few years removed from high school. The comic books that were emerging from Whitman/Western by 1941 were, by contrast, written and drawn for children, like the Whitman books that preceded them, and their editors and writers and cartoonists were unmistakably adult. That adult status was, to say the least, no guarantee of quality, but the violent children's fantasies that shaped the superhero comic books were far more confining than the narrative formulas that typically shaped Western's comic books.
By early in 1941, the new Walt Disney's Comics & Stories, produced in Poughkeepsie, was demonstrating the sales potential of comic books based on animated-cartoon characters. Looney Tunes and Merrie Melodies Comics no. 1, the first comic book produced in Packer's shop, was published in the summer of 1941. (The official publication date, according to its copyright registration, was August 1.) Its characters—Porky Pig, Bugs Bunny, Elmer Fudd—were the stars of the animated cartoons produced by the Leon Schlesinger studio and distributed by Warner Bros.
When Western, using its Whitman name, first got into the comic-book business on the West Coast, its offices were in the Quinby Building at 650 South Grand Avenue, in downtown Los Angeles. Roger Armstrong's detailed memories of that period were stimulated when he saw, for the first time in more than twenty years, comic books whose stories he had drawn. He remembered "a fellow named [A. L.] Zerbe who had rented the office; he was the West Coast salesman for Whitman for their novelties," the nonbook items. After Whitman put Packer in charge of the comic books, Armstrong said, "she sort of pre-empted one corner, and bit by bit poor Zerbe got pushed out into the hall." Within a year or so, Packer had moved the Whitman office to the Brighton-Bedford Building at 9629 Brighton Way in Beverly Hills. "It turned out that Eleanor lived in Beverly Hills," Armstrong said, "so it was more convenient for everybody that we have the office in Beverly Hills."
Looney Tunes and Merrie Melodies Comics no. 1 (1941) was the first comic book produced by Western Printing's Los Angeles office and also the first based on the Warner Bros. cartoon characters.
Packer probably met Chase Craig, the first freelance contributor to the comic books she edited, early in the fall of 1939. That was when Craig began assisting another cartoonist, Carl Buettner, with the drawing of a comic strip based on Edgar Bergen's ventriloquist's dummies, Charlie McCarthy and Mortimer Snerd. The comic strip got off to a rough start in July 1939, as a daily gag drawn by yet another cartoonist. Bergen and the McNaught Syndicate then decided it should be what the Ohio State magazine called "a juvenile adventure strip with a continuous story." Packer had written several children's books for Whitman about Bergen and Charlie McCarthy, and Bergen summoned her to write the comic strip. Although the comic strip expired in mid-1940, Packer evidently kept Craig in mind, because he began working for her at Whitman a few months later.
Craig said that he was actually hired not by Packer but by Oskar Lebeck. They probably met when Lebeck visited Los Angeles in March 1941, on what was almost certainly his first trip to the West Coast as Western's comic-book editor. Lebeck himself described his position, in 1943, as "art director and art editor of all the books and magazines we do of animated cartoon characters, for which my company holds the license from the various studios." It was, however, Packer who was Craig's boss; he described her as "virtually a one-woman type of operation. She represented the company in many ways as well as editorially."
Roger Armstrong, then twenty-three, was working on the final assembly line at the Lockheed aircraft plant in Burbank when Craig, his senior by seven years—Craig was born in 1910, Armstrong in 1917—invited him to draw a story for the second issue of the new Looney Tunes and Merrie Melodies Comics. As was so often the case with early comic books, Armstrong's hiring was fortuitous: he and Craig had met just once, probably in 1940, when both were drawing comic strips written by a third cartoonist, Fred Fox. Armstrong was filling in temporarily on a strip called Ella Cinders, and Craig was developing a strip, not yet syndicated, called Odd Bodkins. Armstrong said:
Roger Armstrong believed that his "Porky Pig," in Looney Tunes and Merrie Melodies Comics no. 2, November 1941, "must be the worst comic strip ever done in the history of cartooning." Actually, it had lots of competitors in 1941.
Chase and a fellow named Win Smith had already done the first issue [of Looney Tunes], which included not only the Warner Bros. characters—at that time, the Schlesinger characters—but also an original thing by Win Smith, called "Pat, Patsy and Pete." It just so happened that the timing was right on the button, and I had my first vacation from Lockheed. I picked up a script, and I spent the two weeks of my vacation doing my first Porky Pig strip. I've often thought that I would dearly love to see that monster, because it must be the worst comic strip ever done in the history of cartooning. I had never seen the character Porky Pig, I didn't have the remotest idea what he looked like. But they said, "Can you do it?" and I said, "Hell, yes, I can do it." The only model sheet I had was a strip that Chase had done, and his Porky Pig was entirely different from anyone else's. My Porky Pig was entirely different from any that has been seen before or since. It was so dreadful that I shudder at the recollection. But they hired me.
Looney Tunes and Merrie Melodies Comics was published not by Western, as either Whitman or K.K. Publications, but by Dell, an entirely independent company that had been in the comic-book business a lot longer than Western. George Delacorte, who had been "general manager" of pulp fiction magazines called Snappy Stories and Live Stories, started Dell around the end of 1921, publishing at first a single biweekly magazine called I Confess ("A Magazine of Personal Experiences") with a monthly circulation, the company said many years later, of ninety thousand copies. Over the next two decades, the number of Dell titles and their sales expanded tremendously until by 1940, the company later claimed, thirty-nine titles were selling 3.7 million copies a month. Those Dell magazines, which included Modern Screen, Modern Romances, and Inside Detective, were aimed at a broad popular audience that bought them almost entirely through newsstands. Comic books were a natural adjunct.
When Roger Armstrong was photographed in late 1941 or early 1942, he was at work on "Sniffles and Mary Jane" for Looney Tunes and Merrie Melodies Comics no. 8, June 1942. Courtesy of Roger Armstrong.
George Delacorte was removed, in taste if not entirely in background, from the bulk of his readers. He was born in 1893 to George and Sadie Tonkonogy, Russian Jewish immigrants. His ambitious father had sold newspapers and flowers in Chicago and then worked in a freight yard in Philadelphia while he studied English at night school. In New York, according to one early account, Tonkonogy "took up the work of tutoring young men who wanted to enter college but were deficient in mathematics and Latin." He also studied law and became a lawyer in Brooklyn by early in the twentieth century. George Jr. grew up as a public school student in Brooklyn, but he then attended Harvard College before graduating from Columbia University in 1913. He remained George Tonkonogy Jr. for a few more years but in December 1917 changed his name to the less ethnic Delacorte in a Monmouth County, New Jersey, court.
Delacorte was notably unsentimental about not just the family name but also his company's publications and their audience. He said in 1943: "First, we publish a group of magazines in the motion picture and in the romance field, which are read by more or less literate people. Then we [publish] a group of pulp paper magazines which are read by less literate people between the ages of approximately 10 and 25. Third, we publish a group of magazines called comics for children between the ages of about three and eight directed to children who have not learned to read as yet, and to those who perhaps mumble with their lips as they read."
Dell and Western probably started doing business with each other sometime in 1936, less than two years after Western opened its Poughkeepsie plant. Delacorte said in 1943 that he had no written contract with Western: "It has been an oral agreement for the last seven years." Whatever the nature of that oral agreement, Dell and Western had become tightly connected by 1939. Until then, the McClure Syndicate had been producing Dell's flagship monthlies—Popular Comics and The Funnies—but early in that year Western assumed that role. Original material copyrighted in the name of the Western executive Robert Callender began appearing in both titles, along with the telltale words about additional postal entry at Poughkeepsie. By late in 1939, Dell was also publishing color one-shot comic books that Western produced.
In March 1940 Dell published the first true Disney comic book, a Donald Duck one-shot that reprinted newspaper comic strips. The Donald Duck one-shot appeared a full six months before Western converted Mickey Mouse Magazine into Walt Disney's Comics & Stories, and possibly that one-shot served as a trial run for the monthly comic book. It is not clear, though, what kind of lesson Western drew from the one-shot's sales: Disney received royalties on 335,000 copies of the one-shot, a print run more than 80,000 copies larger than that for Walt Disney's Comics no. 1, so sales of the one-shot may have fallen short of expectations. Or Western may simply have been cautious.
Dell published several more Disney one-shot comic books in 1941, after Walt Disney's Comics was well under way. Western prepared the content of those comic books, printed their covers, and bound them. (It farmed out the printing of the pulp-paper insides to Eastern Color Printing.) Western sold the comic books to Dell, which distributed them mostly through American News Company, the dominant periodical wholesaler, but also sold them directly to chain stores like Woolworth and Kress, and to mail subscribers. Dell assumed the risk that some retailers might not pay, and since retailers could return unsold copies, it also assumed the cost of those returns. How well a comic book sold for Dell would normally determine how frequently a particular title was published, or whether it was published at all.
Oskar Lebeck spoke of submitting rough pencil sketches and color proofs of the covers for approval by George Delacorte or Helen Meyer, Dell's vice president. Meyer said many years later: "Every year I used to go out to Hollywood and see all the new films and cartoons, and then we would work out comics from them." Even so, Dell's editorial role seems always to have been limited. There are few traces in Western's surviving correspondence with licensors of deference to Dell in editorial matters. Similarly, Delacorte said in 1943 that it was Dell, rather than Western, that paid royalties to the owners of the licensed characters, citing exact figures for New Funnies—for example, a total of $3,008.97 in December 1942—but that was not true in later years; or Dell may have been only a conduit for payments that originated with Western.
The comic books themselves were identified as Dell publications, except for the few, notably Walt Disney's Comics, that were published by K.K. In 1948, the K.K. titles began to carry the Dell label on their covers, and in the same year the Dell comic books first bore, in small type on the inside front cover, the credit line "Designed and produced by Western Printing & Lithographing Co." Until then, Western was completely invisible to the readers of its comic books. It was an unusual arrangement, but one with a precedent of sorts in the books that Western produced for other publishers through its subsidiary the Artists & Writers Guild.
In the early years of its collaboration with Dell, Western had what was probably a similar arrangement with at least two other comic-book publishers. Red Ryder Comics was published at first by Hawley Publications, a Stephen Slesinger company, starting in 1940, but Western produced the comic book for Slesinger. Western's own K.K. then took over, starting with the sixth issue, in 1942; many Red Ryder features were copyrighted in the name of Robert Callender. Also starting in 1942, Fawcett published the first ten issues of Gene Autry Comics, but, again, there is abundant evidence—for example, a publication address, North Road in Poughkeepsie, the same as that of K.K. Publications—that Western produced most if not all of those comic books for Fawcett. Western began producing Gene Autry Comics for Dell in 1944. Western and Dell were by then working together too closely for a third publisher's involvement to have been less than awkward.
The two companies may not have signed their first formal, written agreement until November 29, 1944, when Western agreed to manufacture comic books for Dell; in later years that was the earliest such agreement in Western's files. The 1944 agreement acknowledged that the two companies had by then been doing business with each other for years. John C. Worrell, a Western vice president, wrote in 1979 that "formal agreements were not nearly as popular in those days as they are today and E. H. Wadewitz, the founder of Western, much preferred to work on a handshake. As a result of this, many of our records are quite incomplete." The library of its comic books that Western maintained at the Racine headquarters dated back only to 1944.
Even after its affiliation with Dell, Western continued to publish Walt Disney's Comics, Super Comics, and Crackajack Funnies under the K.K. Publications label. The idea, Howard Anderson explained, was "to always have firsthand knowledge of what was going on in the comic magazine field." But far more of Western's comic books, like Looney Tunes and Merrie Melodies Comics, were published by Dell. That company continued for a time to produce a few comic books of its own; they bore the name of George Delacorte's son, Albert, as their editor. Albert, who graduated from Princeton University in 1935, also edited some of Dell's magazines for adults. But by sometime in the early 1940s, the transition was complete, and Western was producing all of Dell's comic books.
By then, a few years after Superman's debut, costumed superheroes were at the center of the comic-book world, and everything else—including the Dell titles—was on the periphery. Most of the Dell characters originated elsewhere, in other media, and were adapted to comic books; the reverse was true of the superheroes. Dell dropped its few superheroes in the early 1940s, and in subsequent years the Dell titles were conspicuously removed from many other such fashions that swept through the comic-book world. Crime, war, horror, romance, science fiction—there were almost no Dell comic books that could be assigned to any of those categories.
Separation from the mainstream bred in the Dell artists and writers a certain detachment that ultimately worked in their favor. They entered the field not because they were attracted to the superheroes, as so many other cartoonists did, but because the work was in some way similar to work they had already been doing. The best of them found challenges in the new industry that they never expected.
## 4
# Learning on the Job in L.A.
Roger Armstrong's first story for Looney Tunes and Merrie Melodies, the "monster" he remembered with a shudder, "was a thing about Porky and a character called Dirty Dog, who I think was a Chase Craig invention. I'd never done anything like that before; I'd never worked with animated-type characters." When Eleanor Packer patched together a Los Angeles–based staff, it was obvious from the first few issues of Looney Tunes that no one involved with it had any comic-book experience, or, for that matter, much experience with comics of any kind, or with the animated characters on which the comic book was based. Armstrong's drawing style had been influenced by Clifford McBride (who drew the Napoleon comic strip about a large dog) and, he said, "the pen and ink stylists of Punch magazine as well as Charles Dana Gibson and Frank Godwin"—all of them artists who worked in a feathery, freely brushed style that had nothing in common with the sharply defined lines of typical animation drawings.
The field was open for cartoonists better prepared to work in comic books. Before long, Craig recruited Carl Buettner, his former collaborator on the Charlie McCarthy comic strip, to draw part of the first Bugs Bunny comic book—Dell Large Feature Comic no. 8, a black-and-white comic book published in July 1942, which was in its dimensions (more than an inch larger than a standard comic book vertically and horizontally) and its lack of color an indicator of publishers' lingering uncertainty about exactly what readers would prefer as the comic-book format. It was the first one-shot of any kind produced in Western's Los Angeles office. Craig wrote the comic book's two stories and drew the first; Buettner drew the second.
"Bugs Bunny and the Goofy Goose," in Bugs Bunny Large Feature Comic no. 8 (1942), was the first story illustrated by Carl Buettner, the dominant cartoonist in Western Printing's Los Angeles office in the 1940s.
Roger Armstrong met Buettner for the first time the day that Buettner delivered the completed artwork to Whitman's office. Armstrong himself was delivering a completed story with Sniffles the mouse, another Schlesinger character, and, he said, "I'll never forget the condescension with which Buettner looked at it . . . and I was absolutely bowled over by the incredible magnificence (to me at 21–22 years of age) of his material."
Buettner was born in Minnesota in 1903, so he was approaching middle age when he drew his first comic-book story, "Bugs Bunny and the Goofy Goose." He had been working as a cartoonist for about twenty years, including, in the late 1920s, a few years as an instructor for the Minneapolis-based Federal Schools correspondence courses. "The Goofy Goose" is a feeble contrivance, in which Bugs Bunny and other Schlesinger characters encounter the goose that laid the golden eggs, but Buettner's brushwork, even if lacking in individuality, is strikingly free and assured compared with the labored drawings in Chase Craig's companion story, "North Woods Mystery," and with comic-book artwork in general.
There was a great deal of room for improvement in the writing as well as the drawing of the comic books. "In those days," Armstrong wrote in 1967, "[New York] writers were hired to write a lot of the animation characters' stories and they didn't know from Tuesday about characterization or anything. . . . So one day, I had one of these dreadful scripts about Porky Pig and I hit a place where it became asinine and un-drawable, so I called Eleanor . . . and explained my problem. She did understand story . . . so she said, OK, if you don't like the material, go ahead and re-write it so it makes sense. . . . I rewrote the story more in keeping with the character of the pig and from then on, did a great deal of writing for myself and for other cartoonists." By the mid-1940s, Armstrong was writing the Looney Tunes stories with Sniffles, along with some of the "Barney Bear and Benny Burro" stories for Our Gang Comics.
As Armstrong's remarks suggest, there was no hard and fast line between Western's New York and Los Angeles comic-book operations, and some of the comic books based on Hollywood cartoon characters were produced on the East Coast. The Dell comic book called The Funnies metamorphosed in the spring of 1942 into New Funnies, made up mostly of stories with the Walter Lantz cartoon characters (Andy Panda, Oswald the Rabbit). Although the Lantz studio was in Hollywood, New Funnies was produced in New York by cartoonists working under Oskar Lebeck. Likewise, when Western and Dell launched Our Gang Comics in 1942, its characters were the stars of MGM's live-action shorts and cartoons, but the stories were written and drawn by New York–based cartoonists.
Lebeck was actually "the first art editor we had" in the Los Angeles office, Roger Armstrong said, even though he was based in New York. "He was a fantastic guy, because he was the kind of editor who, when he came out here, would look at your stuff and say he didn't like this and he didn't like that, 'but we'll run it. But next time . . .' What Oskar would do was give you suggestions on how he felt it could be improved, but he didn't lay down rigid rules." Armstrong wrote many years later that he never met Lebeck, but knew him only through his "kindly and helpful letters."
Armstrong remembered early efforts to bring to the Looney Tunes comic book some of the flavor of the Schlesinger cartoons, which were by the early 1940s very popular with theater audiences. "We used to get the word to be at the Warner Bros. studio at such-and-such a time on such-and-such a night. It was a big gala evening, and we'd go there and sit through maybe fifteen Looney Tunes and Merrie Melodies cartoons, which were supposed to supply us with ideas. I used to live up on Gramercy Place, four or five blocks from the studio, and in the afternoon, if I'd get bored drawing and feel like being entertained, I'd just wander down to the studio and commandeer a projectionist and a projection room and say, 'I want to see ten Bugs Bunny cartoons.' Any ten. I just sat there watching them, all by myself, munching popcorn in absolute lone grandeur."
Armstrong wrote of going "to Schlesinger's in the early 1940s, to get Leon's O.K. on my Porkys, Bugs & Sniffles pages," but it is open to question just how seriously any of the cartoon studios regarded the comic books, especially since they seem not to have been disturbed that the comics so often diverged from the screen versions of their characters. Another of Packer's cartoonists, Veve Risto, wrote of going to the Schlesinger studio to pick up model sheets of the characters, but not for approval of his work.
Looney Tunes and Merrie Melodies Comics in its first few years looked different from the animated cartoons—naturally enough. At the start no one drawing the stories had worked for the Schlesinger studio or had any significant animation experience at all, except for Chase Craig. A native of Texas who studied cartooning in Chicago, Craig moved to California in 1935. He worked for Lantz and Schlesinger briefly, and then tried out for animation at the Disney studio early in 1939, apparently without success. He next entered newspaper comics with a short-lived Los Angeles Daily News comic strip, Hollywood Hams. Carl Buettner, according to one biography, "worked in the animation departments of several of the major studios" after moving west, but he left no traces there.
The comic book also differed from the cartoons in more peculiar ways. Bugs Bunny, who soared in popularity in 1940 and 1941, was a secondary character in the early issues of the comic book; "Porky Pig" was the lead feature in most issues. Daffy Duck, who first appeared in a cartoon in 1937, appeared in the comic book only in the tiniest of bit parts, and in one he spoke with a stereotypical "black" accent. Even though so many Schlesinger characters were being neglected, Packer filled out the comic book with features of Western's own.
Those stories were copyrighted not by Western but by Robert Callender, the Western executive who was one of the three co-owners of K.K. Publications; he subsequently assigned his copyrights to Western, under its Whitman name. Similarly, Oskar Lebeck copyrighted in his own name the comic books and features that he originated on the East Coast, later assigning the copyrights to Western. This odd procedure may have grown out of a misapplication of patent law, where the applicant was supposed to be the "actual inventor," to copyright. In any case, the copyrights helped to camouflage Western's involvement, since it was not acknowledged otherwise in the comic books in the years when Callender and Lebeck were claiming copyright. Western's annual reports to stockholders for 1943–45—possibly the earliest surviving reports, and certainly the earliest that are accessible—do not even mention comic books, and the only bow toward comic books in the 1946 annual report is a photo of a fanned-out assortment of Dell and K.K. Publications titles.
## 5
# A Feel for Walt Kelly's Stuff
One of Walt Kelly's first stories in a Dell or K.K. comic book appeared in the first issue, dated February 1942, of the very unusual Camp Comics. The "camp" in the title referred to the military facilities where thousands of new draftees were learning to be soldiers. Every page of the comic book was pitched directly to draft-age men, from the photo of a pretty girl on the front cover to the cigarette ad on the back. Western's collaboration with Dell was still taking shape in in early 1942, and Camp Comics was published not by Dell but by Western itself, first in its Whitman guise and then as K.K. Publications. It was an odd stablemate for Western's most important K.K. title, Walt Disney's Comics & Stories.
Camp Comics lasted only three monthly issues, but Kelly drew a slapstick feature called "Seaman Sy Wheeler" for all three, initialing the first story "WK" in its last panel. In the second installment, Kelly caricatured himself behind a hotel counter. He is wearing glasses and a moustache. Standing beside him is a taller, more dapper figure with a receding hairline: Oskar Lebeck. This may have been the only time Lebeck appeared, drawn or otherwise, in one of the comic books he edited. Kelly caricatured himself again in the third installment of "Sy Wheeler," as a railroad ticket agent, and he caricatured Ward Kimball as an obnoxious child passenger. More caricatures of his colleagues at Disney and Western would follow in other comic books.
"Seaman Sy Wheeler," by Walt Kelly, appeared in Camp Comics no. 2, March 1942, a comic book intended for young men newly inducted into the armed services. The two men standing together behind the desk are Kelly's caricatures of Oskar Lebeck, on the left, and Kelly himself.
Kelly also drew a two-page feature, "Elmer and Bugs Bunny," for the first issue of Camp Comics. He thus drew comics with the Warner Bros. cartoon characters before he drew any Disney comics.
Dan Noonan knew Kelly at Disney and wrote and drew comic books for Lebeck after World War II. He told Bill Spicer and Vince Davis: "The thing that struck me about Lebeck more than anything else was his feel for what was 'right'—what was good, and what was genuinely funny. He had this feel for Walt Kelly's stuff long before anyone else had it."
It was presumably thanks to Lebeck's benign involvement that early issues of Looney Tunes and Merrie Melodies Comics, starting with the third, dated January 1942, included a feature called "Kandi the Cave Kid," drawn and almost certainly written by Kelly as some of his first work for Lebeck. Kelly's stories about a juvenile caveman were strikingly simple, direct, and even funny, especially compared with the clumsy stories surrounding them. For Looney Tunes no. 20, June 1943, Kelly wrote (apparently) and illustrated a story with a cast made up of stick figures; it is a much higher grade of nonsense than anything else in the comic book. Then he picked up and revived briefly the comatose feature, also owned by Western, called "Pat Patsy & Pete," about two children, their penguin companion, and an inept pirate. In Kelly's hands, those stories became the rowdiest sort of slapstick.
Most likely those Kelly stories never passed through Eleanor Packer's hands but were added to the Looney Tunes comic book when the rest of the artwork for an issue arrived in New York on its way to Poughkeepsie. Other New York–based writers and artists contributed, too—notably Gaylord DuBois, the most important of Western's writers, who evidently wrote at least a few of the stories that Kelly drew, as well as backup features with lesser Schlesinger characters. Throughout the early and mid-1940s, Kelly stories like the Looney Tunes features appeared in comic books that originated in Los Angeles but were almost certainly added in New York at Oskar Lebeck's direction. Walt Disney's Comics & Stories was a special case: Kelly drew the "Gremlins" stories in 1943 issues and the front covers starting that year, but Walt Disney's Comics was edited from Western's Poughkeepsie offices until late in the 1940s, even though the Los Angeles office produced more and more of its content.
As Noonan said, Kelly was fortunate that he worked in New York for an editor who appreciated his work. If he had remained in Los Angeles and entered comic books through Western Printing's office there, the results might have been different. Roger Armstrong remembered the skepticism about Kelly: "[Carl] Buettner never understood about Walt Kelly. Never. The original Pogo stories"—the genesis of Kelly's enormously popular newspaper comic strip of the 1950s and 1960s—"appeared in Animal Comics, and I told Buettner one time, 'This is an absolutely fantastic thing. Why isn't something being done about syndicating it?' He shook his head and said, 'Roger, you don't understand. This kind of stuff, it's too "far out" for the public.'"
Lebeck clearly saw Kelly as his star cartoonist. In March 1942, Dell published Kelly's first important comic-book work, and Oskar Lebeck's first important comic book: the inaugural issue of Fairy Tale Parade, sixty-eight pages (including covers), all of it drawn by Kelly. Even though Kelly was always a fast and productive worker, drawing so many pages must have required a month or longer—a major investment of time and effort not just by him but by his editor. Although Lebeck had in his first few years as a comic-book editor followed a conventional path, by relying on reprinted comic strips and salting them with new stories that imitated successful genres, there is visible for the first time in Fairy Tale Parade a different kind of ambition: to achieve a level of quality and respectability approaching that of the best traditional children's books.
Western and Dell were beginning to use more and more licensed characters from companies like Disney and Warner Bros.—in contrast to leading comic-book competitors like Detective Comics Inc. (DC) and Fawcett, whose characters were mostly their own properties—but not only did Fairy Tale Parade have no stories with licensed characters; it had no continuing characters at all. It was the first comic book Lebeck produced that bore his own copyright. Even though that copyright had no legal significance—Lebeck assigned it to Western a few weeks later, in what became the standard procedure for his copyrights—the comic books and features bearing his copyright became measures of his taste and ambitions, which differed markedly from those of most comic-book editors.
When Fairy Tale Parade no. 1 appeared, the great success of the superhero comic books had already generated a backlash against comic books in general, most importantly in the writings of the newspaper columnist Sterling North. His attacks on comic books started in 1940—that is, just two years after Superman was introduced in Action Comics—and immediately found a sympathetic response from other writers, parents, and educators. Hostility toward comic books bubbled up in magazines and newspapers throughout the early 1940s. Lebeck, in a short piece on the inside front cover of Fairy Tale Parade no. 1 (it is unsigned but could hardly be by anyone else), took pains to separate his new comic book from the herd, describing it as "an attempt to bring to young and old a series of picture books of folk tales and stories of many lands—not as a shortcut to reading but in the hope of instilling the desire to read and re-read the fairy tales, legends and myths of bygone days. Often we have longed for more pictures in our favorite fairy tale book. Now Walt Kelly, the artist who drew all the wonderful pictures in this book, makes our wish come true."
The immediate stimulus for the new comic book may have been the debut in October 1941 of Classic Comics, adaptations of famous books (The Three Musketeers was the first) in the comic-book format. Classic Comics was certainly the inspiration for another Lebeck-edited title, Famous Stories; the first issue, adapting Treasure Island, was published in February 1942, a few weeks before Fairy Tale Parade no. 1. Famous Stories was not successful—it expired after its second issue, based on The Adventures of Tom Sawyer—but Fairy Tale Parade survived.
Walt Kelly drew five stories for the first issue, as well as pictures and decorative borders for the covers. He signed only the back cover's elaborate latticework, which is swarming with birds, small animals, and elfin creatures. The antique lettering there and on the other covers is recognizably Kelly's, but much of the lettering inside the comic book is not. The dialogue balloons have been lettered in upper and lower case, a departure from the standard comic-book and comic-strip practice of using only capital letters. However, the problem is not that, but rather that the dialogue was almost certainly lettered by someone other than Kelly after he finished his drawings in ink. The fit between balloon and words is frequently awkward, the words squeezed in too tightly or floating in too much space.
Such misfit dialogue balloons were to be a nagging deficiency in Lebeck-edited comic books throughout the 1940s. Some cartoonists did not want to do their own lettering. Moe Gollub recalled that he was paid thirty-five dollars a page by Western in the last half of the 1940s for drawing each page in pencil and finishing it in ink (a surprisingly high figure for the time), "but I didn't letter. I always hated lettering." Lebeck may have wanted to conserve other cartoonists' time, especially when the military draft was scooping up artists in the early and mid-1940s. Publishers organized on more of an assembly-line basis typically dealt with such issues by having artists submit penciled artwork to an editor who after approving it or ordering changes gave it to a letterer before it went back to the original artist or, at least as often, to yet another artist, for inking. But Lebeck's cartoonists were more nearly in charge of their own stories. Thanks to that greater freedom, there was the opportunity for more individual work but also a greater risk of messiness—or excessive tidiness, when mechanical lettering was used.
The fairy tales in Fairy Tale Parade no. 1 are a mixed lot. "Thumbelisa," adapted from Hans Christian Andersen, is clouded by the morbidity that suffuses that author's stories, and "Hansel and Gretel," from the Grimms, does not flinch in depicting the gruesome fate of the wicked stepmother and the witch. Two stories, billed as "an old English fairy tale" and "an Irish folk story," appear to be modern inventions, and rather thin ones at that. But the fifth story, "Little Black Sambo," is something else again. The human characters are not Asian, as in the original story, but American blacks (whose speech is mercifully dialect free). The tigers that threaten to eat Sambo and that ultimately chase one another until they melt into butter are a pack of vaudeville hams, hopelessly pleased with themselves as they strut in the clothes they have bullied from the boy. The tigers never seem to notice that they could eat Sambo and keep his clothes, instead of accepting them as bribes not to eat him. Kelly's drawings make their smug stupidity comically believable.
What Kelly brought to "Little Black Sambo" that distinguished him most strongly from Lebeck's other artists was not just that he could draw better—although he certainly could—but also that his drawings got better the more strongly his comic sensibility manifested itself. His superiority was clearly visible in the second issue of Fairy Tale Parade, whose pages he shared with other artists. The first issue was sufficiently successful that Dell began publishing Fairy Tale Parade on a bimonthly schedule in the summer of 1942, and several other artists—Arthur E. Jameson, George F. Kerr, Jon Small—began illustrating stories in addition to Kelly's. There is nothing really wrong with their work, Jameson's in particular. He was a veteran illustrator, a much older man than most comic-book artists of the time, born in England in 1872, and his stories have an antique flavor that suits fairy tales. The drawings seem slightly indistinct, lending them romantic distance. But it is Kelly's stories that combine charm with the emotional openness and immediate appeal that Walt Disney's animators sought and that was such a lively novelty in early comic books.
As good as his stories were from the start, it took Kelly a while to hit his stride. "Little Black Sambo" apart, his drawings for the first issue of Fairy Tale Parade look a little worked over, almost fussy—similar in fact to the drawings he made for very early comic books in 1935. There is some of that same carefulness in other early Kelly stories. Soon, though, as Lebeck piled assignments on him, his drawings for Fairy Tale Parade and other titles looked more spontaneous, more like the work of a cartoonist who was confident of his skills.
Kelly did not mock the fairy tales he illustrated, but he found a great deal of fun in them. That was especially true in stories whose writing was recognizably his, wholly or in part, as when he embraced the comic possibilities in a giant with two quarrelsome heads. There were such giants in two different stories in early issues of Fairy Tale Parade. It was, however, when his characters were animals that he seemed most like himself. Unlike many other cartoonists, he was always comfortable drawing anthropomorphic animals. "[Y]ou can do more with animals," he told an interviewer many years later. "They don't hurt as easily, and it[']s possible to make them more believable in an exaggerated pose, than it is the human."
"Little Black Sambo," by Walt Kelly, in Fairy Tale Parade no. 1 (1942), was a bright spot in that sixty-eight-page comic book illustrated entirely by Kelly.
Kelly knew and admired the work of the best comic illustrators of the early twentieth century, such as A. B. Frost and T. S. Sullivant, both of whom excelled at bringing anthropomorphic animals to comic life on the page. His own drawings echoed theirs in how he marshaled shading, cross-hatching, and other such techniques to give his comic-book panels a rich, worked-up look that benefited especially from his skill with a brush, but his drawings' solid, three-dimensional quality—the sense that his characters were moving freely in real space—owed most to his Disney training. Even the best of Frost's and Sullivant's drawings can seem static compared with Kelly's best. There was in his work a combination of directness and nuance that was extraordinarily rare in cartooning of any kind, but especially so in comic books.
His next major assignment, as he continued to draw for Fairy Tale Parade, did not call for him to draw animals except as occasional incidental characters. Lebeck assigned him to the lead feature in the new Our Gang Comics, whose first issue was published in midsummer 1942. The comic-book feature was based on the short subjects originally produced by Hal Roach and then, starting in 1938, by MGM, about the mostly comic adventures of a "gang" of prepubescent boys and girls. Our Gang Comics was another of Western's growing number of licensed properties; most of the comic book was filled with stories about characters from other MGM movies, notably the cartoon characters Tom and Jerry and Barney Bear. There was not much room for comic animals in the "Our Gang" stories, with their more realistic characters and drawings, but the gang's pet goat Julip gradually became more expressive, for comic purposes, than the stories would at first glance seem to have permitted.
The earliest "Our Gang" comic-book stories, from when the film series was still in production—the last short was released in April 1944—tend to be stiff and awkward, as if everyone was a little self-conscious about translating live actors into cartoon characters, with obvious reliance on publicity stills. Kelly drew the first "Our Gang" story from a script by Gaylord DuBois, Lebeck's principal writer, and DuBois wrote at least two more installments, but as early as the third issue "Our Gang" was starting to seem like something Kelly wrote as well as drew. The first two stories—DuBois may have written the second as well as the first—have Our Gang involved in conflicts, the first with a rival gang that ends when a misunderstanding is cleared up, very much a DuBois marker, and then one with criminals. The third story, though, is slapstick comedy of the kind that became a Kelly specialty.
Walt Kelly's inaugural story for the "Our Gang" feature in the comic book of that name (1942) depicted unexaggerated versions of the young actors who made up the screen "Gang."
There is no obvious reason that the subsequent "Our Gang" stories could not have been conceived and drawn as movie-style comedy, but soon there was instead a pronounced tilt away from comedy and toward adventure. In the eighth and ninth bimonthly issues of Our Gang Comics, the kids are anomalously presented as actors (in the Our Gang series, naturally) on a film set. Then the action segues out to sea—on a sailing ship that is a movie prop—and into a continuing adventure, written at least in part by DuBois. On a Pacific island, the gang and their one adult companion, the ship's old caretaker, are imperiled by a handful of Japanese soldiers.
This was the period when Kelly had his only brush with the military. In 1943 and 1944, as a biographical press release from around 1955 put it, he "fooled around with the Foreign Language Unit of the Army." That fooling around consisted of illustrating at least two of the two dozen or so pocket-size "language guides" that the War Department published and distributed to soldiers and sailors. All of the humorous illustrations in the Dutch language guide are recognizably Kelly's work, as are many of the illustrations in the Japanese language guide. He was credited, as "Walter C. Kelly, Jr.," for his illustrations in two "self-teaching guides"—The Mechanics of English and Building Good Sentences—published in 1944 by the U.S. Armed Forces Institute. Kelly made such drawings as a freelance cartoonist. He was never a federal employee, and although he visited Washington, D.C., for days at a time, he never lived there.
George Kerr drew the "Our Gang" stories in the seventh and ninth issues of Our Gang Comics, probably because Kelly's workload had increased with the number of Lebeck's Dell titles, but otherwise Kelly's identification with "Our Gang," as artist and increasingly as writer, grew steadily stronger. There was no sense in the "Our Gang" stories that the gang members led lives like those of ordinary children: school was rarely a concern. Instead, the stories increasingly resembled not just juvenile series fiction but also adventure comic strips like Little Orphan Annie, Terry and the Pirates, and Wash Tubbs, whose protagonists were children or childlike. The common thread was that these children or almost-children enjoyed a freedom of action, without the constraints of parents or school or money, that their child readers could enjoy only vicariously, through fictional characters.
## 6
# Animal Magnetism
In September 1942, Lebeck originated another comic book bearing his own copyright, Animal Comics—a title that, like Fairy Tale Parade, spoke of a desire to reach a young audience familiar with traditional storybooks. It was the third Dell comic book launched that year for which Walt Kelly drew the lead story. Like the first issues of Fairy Tale Parade and Our Gang Comics, the first issue of Animal Comics was a trial issue, dated only with the year of its publication; but also like those other two comic books, it very quickly began appearing on a bimonthly schedule.
Kelly's lead story for Animal Comics no. 1 was titled "Albert Takes the Cake." The cast was made up of an alligator, Albert; an opossum, Pogo; and a black boy, Bumbazine—all of them residents of a southern swamp. The other six stories in the first issue of Animal Comics (one of them illustrated by Kelly) were, characteristically for the Dell anthology comic books of the time, a mixture of light and serious. "Katonka Flies North," in the second position, is a somber animal story in the Ernest Thompson Seton vein, with a goose as its protagonist.
"Albert Takes the Cake" was the seed from which Kelly's Pogo comic strip grew, although both of the animals, in keeping with the general carefulness of Kelly's early comic-book stories, look much more like real creatures (clothed, in Pogo's case) than the versions that would emerge in the comic strip six years later. Albert threatens to eat Pogo and Bumbazine, but that threat, unlike Albert himself, has no teeth, because the alligator is as easily manipulated as Sambo's tigers. "Albert Takes the Cake" is easily imaginable as a children's picture book, so much so that it is open to question how much of that first story is Walt Kelly's and how much Oskar Lebeck's. "Albert Takes the Cake" barely resembles the stories with Albert and Pogo that followed, all of which show every sign of being entirely Kelly's work. Frank Thomas spoke of how Lebeck "usually sparked the original feature idea" before turning it over to one of his artist-writers; Gaylord DuBois, who wrote for Lebeck but did not draw, spoke of such a procedure when he and Lebeck collaborated on juvenile fiction. The genesis of "Albert Takes the Cake" may have been similar.
Pogo Possum and Albert Alligator make their debut in "Albert Takes the Cake," Walt Kelly's lead story for the first issue of Animal Comics (1942).
Kelly kept a scrapbook filled with almost all of his Albert and Pogo stories, but "Albert Takes the Cake" was conspicuously missing. When he wrote about that story seventeen years after it was published, in Ten Ever-Lovin' Blue-Eyed Years with Pogo, his 1959 survey of the comic strip's first ten years of national syndication, his description was far wide of the mark: "Pogo Possum started active duty in a comic book in 1943 as a spear carrier in a feature called 'Bumbazine and Albert the Alligator.' Bumbazine was a little boy who lived in the Okefenokee swamp and had learned to talk to the animals. The early material was fairly frightening. Albert kept eating things. As time went on, his manners improved, and Pogo stepped forward as a sort of Jeff to Albert's Mutt. Bumbazine was dropped because, being human, he was not as believable as the animals."
Just about everything in that paragraph is wrong (Pogo's first appearance was in 1942, not 1943; the Okefenokee was not designated as Pogo's home swamp until years later; and so on). On that page in his book, Kelly offered a drawing from memory, a version of his original possum that bore scant resemblance to any of the highly variable Pogos that appeared in the early issues of Animal Comics. Neither is there any hint in the story that Bumbazine is a sort of backwoods Christopher Robin who has "learned to talk to the animals"—although that is exactly the sort of storybook connection that Oskar Lebeck could have had in mind.
There are echoes in "Albert Takes the Cake" of the much earlier Uncle Remus stories by Joel Chandler Harris, a source that both Kelly and Lebeck would have known, but neither Bumbazine nor Pogo qualifies as a mischievous trickster of the Brer Rabbit kind. Pogo and Bumbazine try to bargain their way out of being eaten, but they are successful not because they are clever but because their adversary is stupid. Bumbazine has baked a cake for Pogo's birthday, and he and Pogo persuade Albert that he should eat the cake before he eats them. It turns out the cake is so heavy that after Albert eats it and dives into the water in pursuit of Bumbazine and Pogo, he sinks instantly to the bottom and stays there for a week. A trickster sort of outcome, it might seem, but not really, because Bumbazine had no intention of baking so leaden a cake for his friend.
Kelly's work, although instantly recognizable as his, appeared anonymously in the early issues of both Our Gang and Animal Comics. Cartoonists like Kelly were credited only sporadically in the Dell titles of the early 1940s, perhaps because so many Dell features were owned by licensors that were in one sense the "authors" of those features. The potency of those licensed characters was evident in the second issue of Animal Comics, dated February–March 1943; it was the first issue in which Howard R. Garis's rabbit character Uncle Wiggily appeared. The "Uncle Wiggily" stories were drawn by Hubbell R. McBride, who was, like his Western colleagues Arthur Jameson and George Kerr, a veteran illustrator—in McBride's case, of Liberty magazine's covers. Uncle Wiggily and his supporting cast usurped not just the lead position but the front and back covers of Animal Comics no. 2, and Uncle Wiggily monopolized the front cover for the next year and a half.
Considering that "Albert Takes the Cake" was the lead feature in the first issue of Animal Comics and there was an alligator on the cover, Lebeck probably planned sequels from the start, even under different titles for each story. But once "Uncle Wiggily" began appearing, stories set in the swamp dropped in and out of Animal Comics. One even turned up in the sixth issue of Our Gang Comics, in 1943. In the first few stories—all of which assigned star billing to either Albert or Bumbazine—there is no more than a trace of a southern dialect, and not much of a southern setting, either, except what is necessary to make an alligator's presence acceptable. These are playground stories, essentially, with Albert as a smug, greedy bully who gets his comeuppance at the hands of meeker creatures.
But then the stories changed, quite abruptly, as Kelly began to use them as a vehicle for what his friend Ward Kimball remembered as a fascination with the South, a place he had never visited. Kelly "loved anything about the South," Kimball said. "When he was at Disney's, if Edna Ferber's Showboat would come to town, or any show that had that old hokey southern stuff, he'd go to see it two or three times. Then he'd draw gags about his friends based on his memories of the show." In one such drawing, Kelly drew himself as an elderly black retainer of the Uncle Tom or Uncle Remus sort. "He never talked in southern dialect," Kimball said, "only wrote it, but he would remember all the lyrics from Showboat and use excerpts when he drew up a gag for us."
A profile in the Canadian magazine Maclean's, published early in 1950 in the first flush of the Pogo comic strip's popularity, said of Kelly: "He greatly admires southern folk culture, to him the most imaginative and beautiful lore of America, but he picked it up from books and from listening to southerners like his father"—an odd statement, since the senior Kelly was actually born in Philadelphia, the son of native Pennsylvanians. The writer may have misconstrued something Kelly said about what his father read to him, possibly the Brer Rabbit stories, when he was a boy. Kelly's third wife and widow, Selby Daley Kelly, said in an interview published ten years after Walt Kelly's death that "his father used to read him things like Uncle Remus, and he picked up a lot of the Southern accent and the 'fun talk' from his dad."
Kelly's father has also been described as an artistic influence on his son because he supposedly painted theatrical scenery, but any such activity must have been incidental in a working life defined mainly by factory jobs. Kelly himself wrote that it was his father "who first placed a pencil in my hand," but he also wrote: "I thought vaguely that I could be an artist, but experiments with my father at my elbow (he was a pretty fair painter) convinced me that the work was hard." The elder Kelly has been described as a "Sunday painter who . . . enjoyed painting seascapes in oil."
The Kelly family had a radio by 1923, and Walt undoubtedly encountered "southern folk culture" on the radio as well as in books. Throughout the 1930s, the radio historian Arthur Frank Wertheim has written, "the airwaves were full of banjo music, sentimental Southern ballads, and blackface routines." But one radio show with a southern flavor towered above the others, and there is reason to believe that Kelly listened to it attentively.
In the late 1920s and early 1930s, when Kelly was a teenager, he lived in an America that was madly in love with the Amos 'n' Andy radio show, in which two white performers, Freeman Gosden and Charles Correll, pretended to be southern-born blacks. "Amos Jones" and "Andrew H. Brown" had moved from Atlanta to Chicago and then to New York, but they spoke not just with a strong southern accent but also in what was supposed to be a Negro dialect. As they struggled with Standard English, its grammar and pronunciation, they twisted the language into comic knots (most famously, "I'se regusted," but also "repression" for "depression," "incorpulated" for "incorporated," and so on). Blackface comedy, minstrel shows, and "coon acts" had relied on such dialect in the decades before Amos 'n' Andy, but as the first radio show with a truly nationwide audience—roughly one-third of the total population was listening at the daily program's peak—it was successful on a much larger scale. Amos 'n' Andy was heard in Bridgeport over a New York City station, WJZ.
Growing up in Bridgeport, Kelly had little contact with real black people. In 1920, when he turned seven, the city's African-American population was tiny, less than 2 percent, and it grew very little over the following decade. Most of those people worked in low-status jobs as laborers and domestics and so were all but invisible to many whites. Kelly wrote in 1959 that he attended school with black children—two of them—for the first time only when he got to high school: "We saw our first Negro children in class there, and believe it or not, none of us was impressed one way or another, which is as it should be. Jimmy Thomas became a good friend and the young lady was pretty enough to remember even today." That may have been an enhanced memory. No Bridgeport blacks of any age make an appearance in the extended memoir of his childhood and adolescence that Kelly wrote for publication in 1962, and there is only a cameo by a young black man from Philadelphia.
Amos 'n' Andy would have been for Kelly, as for millions of other white Americans, a window onto a version of everyday black life that was mostly imaginary but not necessarily toxic. For all of its roots in stereotypes, Amos 'n' Andy was free of the mean-spiritedness that dominated the portrayals of blacks in other media. Its principal characters were variously sympathetic or simply funny, but Gosden and Correll did not present their characters as objects of contempt.
In one of Kelly's first comic-book stories with Albert and Pogo, "Bumbazine and the Popinjay," in Animal Comics no. 3, June–July 1943, a bird character called the Popinjay, never seen again, speaks in Amos 'n' Andy–like dialect for a few panels. Then there is a small-scale eruption of such talk in the "Albert the Alligator" story in Animal Comics no. 5, October–November 1943, when Albert goes to a train station and, as a "talkin' alligator," throws the black humans there into turmoil. They squabble over ownership of this phenomenal creature. "Jes' a minute!" one cries. "I did sign-tiffic research on him! He's my 'gator!" (There is also a glimpse of a couple of moonshine-swigging white hillbillies whose speech, what little there is of it, has a different but comparably stereotypical ring, perhaps originating in the Li'l Abner comic strip by the former Bridgeport resident Al Capp.) Bumbazine and the animals come closest to speaking Standard English; a few outbursts by Albert aside, they don't do much more than drop their g's and speak of children as "chillun." They speak with what is supposed to be a southern accent, spiced with traces of dialect.
But in the eighth issue of Animal Comics, dated April–May 1944, the animals and Bumbazine alike are suddenly speaking as if they were delivering lines from an overripe Amos 'n' Andy script—and talking so much that the dialogue balloons crowd the panels. "Whilst Bumbazine is gone," Albert says, "Ah'll jes' 'vestigate certain chawklit cake smells what has been permeatin' and percolatin' thru this here vicinity! Hee-hee! Ah sho' is a mean thing!" When he finds the cake—which happens to have Pogo inside it—Albert says: "Mought jes' as well test out de flavor as a favor! Bumbazine cain't not mind if Ah tastes at it a little!" What follows is the first of many crises born of Albert's accidental consumption of one or more of his fellow swamp dwellers.
Murray Robinson, in a 1952 Collier's profile, wrote that Kelly's work on the likes of the Japanese and Dutch language guides was the impetus for this explosion of dialect: "The study of language interested him, and on his own hook he did a little haphazard research on American dialects. He became enamored of the Georgia accent—and stored it up for use in his comic strip." Another article published around the same time described the Pogo characters' speech as "a mixture of Georgia cracker and pure hokum [that] comes from Kelly's interest in speech patterns of the Atlantic seaboard."
Such references to Kelly's "research" were invariably vague, but Kelly himself, writing in the third person early in 1952, was more precise:
[H]e had taken an interest in phonetics and phonemics. This came about because it was necessary to translate foreign phrases into American English sounds. To do this an average level of American speech had to be found and army experts spent tedious hours finding out exactly how people in every part of the country would pronounce the word, "THROUGH" for example. Kelly followed their findings with great interest as far as Georgia and dropped the pursuit right there. He was fascinated. So, what many have declared to be authentic, Georgia sounds started being translated into Kelly's project in the comic books.
Given the timing, it is certainly possible that there was a connection between Kelly's work on the language guides—which were based on phonetic equivalents of foreign words—and the first appearance of full-blown dialect in Animal Comics. But the echoes of Amos 'n' Andy, whose principal characters were supposed to be from Georgia, are much stronger in the Animal Comics dialogue than the echoes of any other credible source.
Albert lunges into overripe (and overly plentiful, as witness the crowded dialogue balloons) southern dialect in "Albert the Alligator" by Walt Kelly, in Animal Comics no. 8, April-May 1944.
In particular—to pursue another oft-cited Georgia connection—it is hard to find any linguistic similarity between Kelly's comic-book stories and the Uncle Remus stories that Kelly supposedly remembered his father reading to him when he was a child. Those stories were written by the native Georgian Joel Chandler Harris in a thorny dialect that Harris offered as an authentic transcription of black speech—"wholly different . . . from the intolerable misrepresentations of the minstrel stage"—and that bears almost no resemblance to the speech of Kelly's characters.
The world of the Remus stories is, moreover, comic in a harsh and often brutal manner befitting the grim world of the slave. In the story called "The Awful Fate of Mr. Wolf," from Harris's first Remus book, Uncle Remus: His Songs and His Sayings, Brer Rabbit scalds his enemy Brer Wolf to death, with his snickering children as an audience. In "Old Grinny Granny Wolf," from the second collection, Nights with Uncle Remus, Brer Rabbit murders the blind and crippled old lady wolf of the title—there is no other way to put it—and then tricks Brer Wolf into eating the stew he makes from the dead body of Grinny Granny, the wolf's own grandmother.
Other stories are just as gruesome, and just as far removed from anything that happened in Kelly's swamp. The notion of Albert as a carnivorous bully was essentially gone from Animal Comics as of its second issue, in the first Albert story that seems to be entirely Kelly's. It resurfaced in very mild form only once or twice before disappearing completely. With it disappeared any material resemblance to the Remus stories.
There were in fact comic books in the 1940s that may have owed something to Joel Chandler Harris. The story called "Brother Rabbit Gets Brother Fox's Dinner," in Nights with Uncle Remus, could have been the template for a series in DC's Real Screen Comics. In Harris's story, Brer Rabbit is helping Brer Fox shingle his house when he nails the fox's tail to the roof—accidentally, of course—and then descends to the ground and helps himself to the fox's dinner pail. That situation recurred with many, many variations in the "Fox and the Crow" stories in Real Screen, with the crow assuming Brer Rabbit's trickster role. Likewise, the "Li'l Bad Wolf" stories in Walt Disney's Comics & Stories take place, like the Remus stories, in a world in which talking animals—some of them Disney versions of Harris's characters—are neighbors who just might kill and eat one another if the opportunity were to present itself (as it never quite does in the comic book).
There are no such echoes of Harris in Kelly's stories. His characters, in Animal Comics and later in his comic strip, live in a different world, one much more like that of The Wind in the Willows, where no one seems to work but cupboards are full of good things to eat, deadly peril is absent except on those rare occasions when the resident carnivores forget their manners, and the urgent question is how to fill the hours of the day most pleasurably. Kelly's luxuriant swamp looks nothing like the scraggly post–Civil War Georgia countryside drawn by several artists, most notably A. B. Frost, for Harris's stories. When Kelly spoke of a literary model for his comic strip, he invoked not Harris but A. A. Milne, creator of the much sweeter-tempered Winnie-the-Pooh stories, and from all appearances he was not speaking facetiously.
But it was Amos 'n' Andy that offered the most to work with. That show, after a fifteen-year run as a daily serial, went off the air in February 1943, reappearing in October of that year as a half-hour weekly situation comedy that quickly became one of the most popular on radio. The weekly Amos 'n' Andy was more heavily weighted toward comedy, and populated with more extravagantly comic characters, than the serial, which was as much drama as comedy. The new show thus lent itself readily to a comic-book cartoonist's purposes. For example, in the second of the new shows Andy, a genial oaf, has just gotten his diploma from a correspondence school for piano playing—without ever touching a keyboard—and is opening a school of his own, with the raffish George "Kingfish" Stevens as his partner. "I'll learn 'em the white keys and you learn 'em the black keys," Andy says to the Kingfish, in dialogue easily imaginable coming from Albert in an Animal Comics story.
The explosion of dialect in Kelly's Animal Comics stories occurred less than six months after the new weekly Amos 'n' Andy show went on the air; that is, Kelly was writing and drawing those stories around the time the show's first episodes were broadcast. The timing suggests that Kelly's "haphazard research" consisted largely of listening to Amos 'n' Andy and reshaping its version of a "Georgia accent"—and almost everything else about it—to meet his bimonthly needs.
As a mock dialect became increasingly prominent in Kelly's Animal Comics stories, the eventual absence of any human characters in those stories obscured their connection with black stereotypes. It did take a while for black human characters to disappear completely. After that handful of humans in the fifth issue of Animal Comics, one more turned up in "Albert Holds That Tigah," in no. 10, August–September 1944, this time conversing with the animals as peers. Otherwise, Bumbazine was the only representative of humankind in Kelly's Animal Comics stories until he, too, disappeared after the twelfth issue, published in the fall of 1944. From then on, the cast was composed entirely of animals, none of them resembling stereotypical blacks.
There was one curious exception in early 1946: in "Albert and the Barbecue," in the first of two one-shots titled Albert the Alligator and Pogo Possum, Albert and Pogo spend a page and a half in casual conversation with a black human, a locomotive engineer who departs declaring, Amos 'n' Andy style, "You annymiles is the craziest people Ah knows!" There is no accounting for that anomaly, which appeared almost two years after Bumbazine's departure—except that stereotypical black humans were still appearing in other Kelly stories, outside Animal Comics and its offshoots, around the same time.
By the mid-1940s Kelly's Animal Comics stories were increasingly circumscribed, and not just by the departure of humans from their cast. Everything had to take place in the swamp, a decidedly rural environment that did not offer much in the way of props or stimulating backdrops. Perilous adventures were out of the question, and since Albert's desire to eat his neighbors had been suppressed, it had to be his accidental (and always temporary) ingestion of smaller animals that was a major engine of what passed for plots. But other artists had found self-imposed limitations a stimulus to creativity, and that was happening with Kelly, as the confines of his Animal Comics stories pushed him toward finding ways to make his characters funny through their day-to-day collisions.
Albert's role remained central as Kelly transformed him from the inept menace of "Albert Takes the Cake" and the swaggering bully of the next few stories until, as the stories thickened with dialect, he emerged as a scheming, finagling, but ultimately ingratiating blowhard, one who could rely on the much more subdued Pogo as a friend. He almost always had a cigar in his mouth, as if to underline his resemblance to the boastful, cigar-smoking Andy half of the radio team. Kelly was a cigar smoker, too, usually photographed when he became famous in the early 1950s with a cigar in hand or mouth, and Albert was in that way, as in others, a funhouse mirror's reflection of his creator.
## 7
# Cartoon Conundrums
After a year, everyone involved with the Looney Tunes comic book was struggling to write and draw stories that did not invite invidious comparisons with the cartoons. The task at hand, which it is unlikely anyone articulated at the time, was to find some way to bring animated-cartoon characters to life on the printed page without the support they enjoyed on the screen: music and movement and idiosyncratic voices. On paper, the characters were more nearly naked, existing mainly as distinct designs. Almost all of them were, moreover, talking animals, and it was in comic books that talking animals seemed most inescapably juvenile.
The great authors of talking-animal fiction in the late nineteenth and early twentieth centuries—Joel Chandler Harris, Beatrix Potter (The Tale of Peter Rabbit), and Kenneth Grahame (The Wind in the Willows)—depicted characters whose natures were shimmering and shifting mixtures of the animal and the human, a duality captured in Potter's illustrations for her own books, and in A. B. Frost's illustrations for Harris's. Even though such books were inevitably pigeonholed as children's books, their subtlety and sophistication were widely recognized. Their literary quality encouraged superficially similar efforts by writers and artists working in harsher environments—newspaper columnists like Howard R. Garis, author of the Uncle Wiggily stories, and comic-strip artists like George Herriman, creator of Krazy Kat.
Krazy Kat made the transition to animated cartoons in 1916, and Felix the Cat was a cartoon star for much of the 1920s, but most of the leading animated characters, like Max Fleischer's KoKo the clown and Paul Terry's Farmer Al Falfa, were human until the success of Walt Disney's Mickey Mouse, starting in 1928, spawned a host of new animal characters. The Disney animals—Donald Duck most prominent among them—led the field until the early 1940s, but animal characters dominated the rolls at the other cartoon studios, too. The behavior of some of those characters, like Disney's Pluto and MGM's Tom and Jerry, bore a slight resemblance to the behavior of real animals. Other characters, Bugs Bunny in particular, straddled the line between the animal and the human, even engaging in conversation with human hunters who regarded them as wild game and intended to kill them. Most often the cartoon animals were indistinguishable in their actions from cartoon humans, but what audiences learned to accept in the cartoons could seem considerably odder on the printed page.
Walt Kelly, testifying in a 1943 lawsuit brought by Dell against another publisher, Nedor, spoke of the difference between animal characters designed for animation and those designed for print. An animation animal, he said, "is usually based on circles with a fluid construction in the body, with rather peculiar but definite markings on the face and on the body, so that these things will appear in good shape on the screen. An ordinary cartoon"—that is, a cartoon animal intended solely for print, and not for the screen—"is not constructed that way."
Kin Platt, a cartoonist who drew for Nedor, testified in the same case. He said that in the design of an animal for animation, "certain features are sublimated and eliminated to provide for an easy manner of projecting the character on the screen, which gives a certain degree of the flexibility to the character, which the ordinary animal itself could not have. For example, a dog's legs are very awkward to draw in actual animated form. So they simplify that by making that in a rubbery fashion. . . . [T]he lines are all curved and circular as opposed to angular lines on the animals itself."
When animals designed for animation appeared in a comic book, the result was, in the words of the attorney questioning Kelly, that "the magazine reproduces the character but not [the] animation." But without the animation, and especially without the quickness and elasticity that distinguished the best animation by the early 1940s, characters might look like their film versions but still be fatally deficient. Many of the conventions that had emerged in movie cartoons by the 1940s—for example, that characters escaped serious harm even when they were the victims of what looked like lethal violence—had no equivalents in comic books. They could not, because those conventions depended on sophisticated animation.
Characters whose design was more than adequate for films could thus be shallow and uninteresting on the page—fit fare indeed for children who, as George Delacorte said, moved their lips as they read. But if those characters were drawn more realistically—that is, more like the animals depicted by Ernest H. Shepard for The Wind in the Willows or by Kurt Wiese for Walter R. Brooks's Freddy series, with its cast made up of talking farm animals—they would not be recognizable as the screen characters that attracted readers to the comic books in the first place. The design of many animated characters became subtler and less formulaic over the course of the 1930s, but there remained a gulf between such characters and even the more caricatured of the animal characters drawn solely for print.
There were occasional signs of life in the new animated-character comic books, of some alternative to clumsily aping the cartoons. For example, by mid-1942, Bugs Bunny and Porky Pig costarred in the lead story in almost every issue of Looney Tunes and Merrie Melodies Comics. They almost never appeared together in the cartoons, and for good reason: they were both relatively understated characters who were naturally cast against hot-tempered bullies. As reworked for the comic books, though, they made a rough sort of sense as an oddly matched team like those already familiar from countless movies and comic strips—Laurel and Hardy, Wash Tubbs and Captain Easy—with Bugs brash and foolhardy and Porky much more cautious.
The mouse Sniffles was a sweet-tempered, very Disney-like character in the cartoons, and so a guarantee of boredom in a comic-book story. "They were looking for a premise on which to hang some kind of story thing with this mouse named Sniffles," Roger Armstrong said in 1975. "They wanted somebody to play opposite him, so they got the concept of this little girl." Mary Jane, a character who never appeared in the animated cartoons, was paired with Sniffles in the first issue of Looney Tunes and Merrie Melodies Comics, got her name in the second issue, and very quickly became the senior partner, as the character with whom very young children might identify. With the help of "magic sand," she shrank to Sniffles's size and joined him—in dreams, at first—in juvenile adventures.
Eleanor Packer hired Carl Buettner, the best cartoonist of those freelancing for Whitman at the time, as the art editor for the comic books by January 1943. "But at the beginning," Armstrong said, "Chase and Carl and I were all artists. And there was a fellow named Ed Volke, a nice little man. He was not a very good cartoonist, but Eleanor Packer thought he was the world's greatest. She used to hold up Ed Volke's work as the shining example—she would say, 'See, Roger, see what nice work Ed Volke can do.'"
Volke signed some of his work in 1943 (as did Armstrong). His drawing is crudely amateurish, and the staging in his panels is clumsy and crowded. But Packer must have liked him: he drew the lead story, with Bugs Bunny and Porky Pig, in most issues from late in 1942 until Carl Buettner began drawing those stories (now titled "Bugs Bunny") with no. 21, July 1943. Packer, like other editors in comic books' early days, seems to have had no trouble accepting poorly drawn work if it met her needs in other respects. Volke probably never missed a deadline.
The Buettner stories were the same sort of puerile stuff that Volke had illustrated, but now they looked much better. Lynn Karp, a former Disney animator who drew stories in the 1940s for Western and other publishers, remembered Buettner as "a picky son of a gun; just as picky as they come," but Buettner at least measured up to the standards he imposed on others. "Carl Buettner was a martinet," Roger Armstrong said. "He was an absolutely superb draftsman, but he expected everyone else to be, too. If you didn't come up to Carl's standards, you redid it. He never accepted you on your terms; it was always on his terms. He was very German."
The first true Schlesinger veteran began drawing for Packer as of Looney Tunes and Merrie Melodies Comics no. 15, January 1943. That was Veve Risto, who started drawing for comic books in 1942 after animating for two Schlesinger directors, Bob Clampett and Norman McCabe. Risto's drawings were hard and stiff, but so were Craig's and Armstrong's at the time, and Risto was unquestionably the superior draftsman. His versions of the characters—he specialized in Elmer Fudd at first—looked like their animated-cartoon counterparts, even as the scripts diverged increasingly from the cartoons.
Risto both wrote and drew some of his early work for Western before concluding that there was more money to be made from drawing, in both pencil and ink, from other people's scripts. The standard rate in the early 1940s was ten dollars for a penciled and inked page, as both Carl Barks and Roger Armstrong remembered. When Risto wrote in December 1942 to John Carey, his friend from Schlesinger's Bob Clampett unit (Carey served in the navy during World War II), he said that he was making more money "than I ever did before anywhere." In that month, he was drawing not just for Looney Tunes but also for Our Gang and even for a new Dell title, Gene Autry Comics, about the fictional adventures of a real movie cowboy; it was the first comic book produced by Packer's shop that was not based on animated cartoons.
The nature of the work, Risto told Carey in a later letter, was deceptive, its requirements seemingly so simple, and yet difficult to master if they were not approached with humility: "I've seen some very extra good artists not make the grade. Seems the utter plain simplicity of comic books of this type brings up a feeling that the new artist is better than the stuff. I guess it's called 'sincerity' that is needed at the outset. The ability to be interested enough to do a good job on a story that you know you could write better yourself; if you had time (and would rather do your own stuff throughout), than make more money by working on stuff already written for you."
Roger Armstrong spoke more bluntly about the difficulty animators in particular might have with comic-book work: "When I was [at the Walter Lantz studio, where Armstrong worked in 1944–45], the guys were very fascinated by the work I was doing for Whitman, and one by one they tried it, and one by one they fell right spang on their noses. For one thing, they didn't know how to do any inking, [and] they had a hell of a time composing or staging a panel—their staging was almost amateurish. . . . They were so used to thinking in terms of movement that they did not think in static terms; and you had to be able to think in both ways. If you're going to be a good comic-book artist, you have to think kinetically, but you also have to put a kinetic concept into a static form."
There are hints in the work even of experienced animation people who became comic-book regulars that making the adjustment could be difficult. From all appearances, the Warner Bros. cartoonist Tom McKimson drew his earliest stories for Looney Tunes and Merrie Melodies Comics without allowing enough space for the dialogue balloons. As a result, those balloons—most likely lettered and inked by someone else over McKimson's pencil drawings—overlap onto the drawings, often obscuring the characters' faces.
For animators who accepted the need for "sincerity" and approached the work with the humility Risto thought was required, the sheer novelty of comic-book work could be refreshing, as Risto told Carey in that 1944 letter: "There's one of the former Disney animators doing the same work I'm doing but for another company here. He said he'd animated everything they could think of so many times it got tiresome."
As cartoonists at the Hollywood animation studios began to take on freelance work on comic books in the early 1940s, most of them gravitated not to Western, the company licensed by their employers to publish comic books with the characters they drew every day, but to a New York company that called itself Cinema Comics but had no cartoon-studio connections. Those cartoonists were recruited not by an editor like Packer, who had no animation experience, but by a fellow animator, Jim Davis.
Davis was a Californian, but he got into comics—probably in 1942—when he was working in Miami for what had been the Max Fleischer studio but had been renamed Famous Studios after Fleischer's departure in 1941. As Davis told Will Friedwald, "I started working, down in Florida, on comics, for Jay Morton," a writer for the Superman cartoons that Fleischer/Famous made under a license from Superman's owner, DC. Morton "went up to New York once, and tramped around and made connections with a fellow named [Benjamin W.] Sangor. So, he was supplying Sangor with comics done by various guys down there in Florida. I don't know who all did them, but I was one of them. I didn't do many, but I did some, and then I wrote to Sangor before I left Florida and told him I was coming back [to California]."
Morton probably met Sangor through Sangor's friend Harry Donenfeld, the publisher of the Superman comic book, or someone else at DC. Sangor was a more than typical comic-book entrepreneur of the early 1940s, a man whose checkered and sometimes criminal past made for an odd contrast with comic books of the kind Davis produced for him: talking-animal fare aimed at a very young audience. That Sangor plunged so heavily into such comics (he also produced other kinds) invited comparisons with Western Printing, but that company's midwestern uprightness never came into question throughout its several decades of comic-book publishing.
Sangor emigrated from Russia as a teenager. He was born in 1889, arrived in the United States in 1904, and was naturalized in 1914. When he registered for the draft in 1917, he was already a self-employed lawyer in Milwaukee. While he was living there, Sangor took part in a scheme to defraud a Milwaukee insurance company, a scheme that resulted in two rulings against him in the Wisconsin Supreme Court. By 1920, his wife, Sophie, had died, and he was living alone in Chicago, where his daughter, Jacquelyn, was a student at a parochial school. By 1922, B. W. Sangor was listed in a legal directory as a Chicago attorney and was advertising real-estate auctions in the Chicago Tribune. Sangor moved to New York by the mid-1920s, and by 1935 he was a failed real-estate developer in Toms River, New Jersey. That year, he and a Toms River banker were convicted of embezzlement, in a scheme that left a widow destitute, and they were sentenced to prison. In February 1938, his appeals exhausted, Sangor was on the verge of entering prison for a term of two to three years. He apparently served much less than that, although no records have survived showing how long he was imprisoned, or even if he was jailed at all. In any case, he was free by June 1940, when he returned to New York by ship from Mexico.
By then, Sangor had entered publishing with Cinema Comics, which was incorporated in September 1939. For a few years, probably starting in 1941, that company produced Cinema Comics Herald, a four-page giveaway that promoted current movies in a comic-book format. It also packaged comic books for other publishers, starting with Nedor Publishing Company. Nedor was owned by Ned L. Pines, who had married Jacquelyn Sangor by 1938. It was probably that family connection that brought Sangor into publishing in the first place (and provided him with the necessary financing). Pines was a leading publisher of pulp fiction that appealed to the same sort of mass-market audience as George Delacorte's magazines. As with Delacorte, Pines needed to take only a short step to enter the comic-book business.
Sangor testified in 1943 that Pines called him in July or August 1941—that is, around the time that Looney Tunes and Merrie Melodies Comics made its debut—"and told me that he would like . . . to get a magazine with animals in the order of the Fairy Tales, talking animals, and we discussed it, and I started on it about that time." Early in 1942 Sangor delivered the contents of the first Nedor talking-animal comic book, Coo Coo Comics no. 1, for publication in August. Coo Coo featured a superpowered rodent, Supermouse, thereby hitching itself simultaneously to two popular genres: talking animals and superheroes.
The connections between Sangor and the Famous Studios animators grew much stronger after Famous moved to New York from Florida early in 1943. The studio was returning home: its predecessor, the Fleischer studio, had moved from Manhattan to Miami in 1938 after a bitter strike by many of the studio's cartoonists. When Famous returned to New York, the animator Gordon Sheehan said, "comic books were coming into their own, and practically all of the animators at Famous were doing comic-book stuff in their spare time, including myself. The outfit that published the comics [Sangor's Cinema Comics] was right next to the building where we worked. We were at 25 West Forty-fifth Street, and they were at 45 West Forty-fifth Street. . . . I remember Mr. Sangor well, a distinguished-looking, pleasant man. In fact, for a while there I think he was interested in taking over Famous animated cartoon studios."
Dell and Western responded quickly to Nedor's Sangor-produced comic books, but not through the Dell comic books themselves. In February 1943, Dell sued Ned Pines; his widowed mother, Dora Pines; and Nedor Publishing Company for unfair competition. Dell complained that three Nedor comic books—Coo Coo Comics, Real Funnies, and Funny Funnies—too closely aped Dell's New Funnies in their depiction of animal characters on their covers.
New Funnies, then still titled The Funnies, began cover-featuring the Walter Lantz animated characters with no. 64, May 1942, the second issue published on what turned out to be a very brief bimonthly schedule. The name changed to New Funnies with no. 65, June–July 1942, and monthly publication resumed with no. 66. Dell said that before The Funnies began featuring animated characters and added New to its name, its average monthly circulation—the period was not specified—was 130,000 copies. In 1942, the complaint said, "the circulation increased to an average of 350,000 copies per month . . . and in January 1943 to 550,000 copies per month." By comparison, the pioneering Famous Funnies sold well in the 1930s, peaking in 1937 with an average net paid circulation per monthly issue of 462,303. By the first half of 1942, though, per-issue sales had dropped to 180,421.
In June 1943 an appellate court affirmed an injunction barring Nedor from mimicking New Funnies too closely. But although Dell's suit was successful, its victory was essentially meaningless. The market for talking-animal comic books was growing too fast, the sales figures were too large, to be much affected by fine distinctions like those embodied in Dell's complaint.
As Gordon Sheehan noticed, Ben Sangor "was a good friend of Harry Donenfeld's." That friendship was of special value when Sangor began publishing his own comic books—the first two titles were Ha Ha Comics and Giggle Comics—in mid-1943. Not only was Donenfeld the publisher of Superman and other popular superhero comic books, but he owned an important distributor, Independent News. Sangor was thus assured of a place on many newsstands.
Sheehan contributed an eight-page feature, "Bow-Wow Beagle Dog Detective," to the first issue of Ha Ha, and he was joined by a half dozen other animators from Famous and a second New York studio, Terrytoons. James Tyer, who animated for Terrytoons in a strikingly bizarre style, drew two stories for the first Ha Ha that look like what might have resulted if a German expressionist painter of the 1920s had somehow wound up drawing comic books with talking animals.
The content of the Sangor comic books would soon take on a much stronger Hollywood flavor. After Jim Davis moved back to Los Angeles, he ran the animation department of Raphael G. Wolff Studios, one of a number of small Hollywood companies that produced industrial films, training films for the armed forces, and other specialized projects. Davis recruited a crew of Hollywood animators, one of whom, Jack Bradbury, had animated at the Disney studio and then on Leon Schlesinger's cartoons for Warner Bros. "As a sideline," Bradbury said in a 1986 interview with Dave Bennett, "Davis had a few guys working for him part time doing comic book work" for the Sangor titles. "A lot of guys were supplementing salaries this way. They would go home after animating and do a little comic book work at night or on weekends. Davis was paying $15.00 a page for writing a story, drawing, and inking. So I thought, what the heck, and started doing some comic book work, too."
Bradbury remembered producing three or four stories as a freelancer before Davis asked him to go into comic books full-time. Although the Disney characters had been appearing in comic books since 1940, and the Schlesinger characters since 1941, Bradbury "had no idea" what comic books were like before he started drawing for them: "The only comic book job that I had seen beforehand was one that Gil Turner [also a Warner Bros. animator at the time] had drawn."
Davis's was one of a number of shops that produced early comic books of various kinds, middlemen who packaged whole comic books for publishers. In Davis's case, there were two layers, himself and Sangor, since Sangor sold to other publishers many of the stories that Davis produced. When his Los Angeles operation was running full blast, Davis said, "there were 65 of us working on those things at one time, and we used to send back sheets of Strathmore [drawing paper] two feet high every month. I just mailed [Sangor] the original art. It finally got to a point where I represented Sangor, and I wrote the checks here, on a local account, for the services involved."
After Davis recruited Hollywood animators to work on the Sangor comic books, many of the stories in those comic books took on a very general resemblance to the Hollywood studios' cartoons. Their animal characters tended to be cuter and more rounded than the often grotesquely proportioned characters drawn by cartoonists from the studios in the East. As a rule, the Sangor characters were not obvious imitations of Hollywood originals, echoing them instead in their general appearance and sometimes in their alliterative names. Like superheroes, "funny animals" could be generated with apparent ease in almost infinite variety.
The Sangor comic books looked different in another respect. Although the earliest of Dell's comic books with talking animals typically were drawn with six panels to the page, by early in 1944 the standard page had become eight panels, four rows of two panels each. This was because the comic books' page count had shrunk from sixty-four to forty-eight, plus covers, under the impact of wartime paper shortages. It was always the cost and availability of paper, not the cost of producing the stories that filled them, that had the greatest impact on comic books' page count and frequency of publication.
The number of pages in the Sangor-produced comic books shrank, too, but those comic books held to a six-panel format with a more "open" appearance than the eight-panel Dell pages. Jim Davis explained: "With Sangor, we used a half-sheet of Strathmore for a page of artwork. It was easier and allowed more spontaneity for both artists and writers." Davis recalled that Sangor matched what Western was paying at the time, the fifteen dollars per page cited by Jack Bradbury. "But Western was also very persnickety," Davis said, "you had to draw on a huge sheet and they always loaded it with stuff"—that is, called for more elaborate drawings than the Sangor norm. "With ours, nobody was telling us what to do or how to do it. We had a lot more freedom, and a lot more fun in working with our stuff."
What Davis remembered as spontaneity expressed itself most often in drawings with fluency but no depth. There is in the Sangor comic books a pervasive shallowness, even by low comic-book standards, so that the characters may look appealing at first glance but are almost never more than puppets. As Davis himself pointed out, "neither I nor anyone else [was] doing any kind of regular characters for Sangor. The . . . only one who had any kind of so-called regular characters was [the former Disney animator Ken Hultgren], because he did his own characters and he did repeats on all of them. But on the other stuff, that was pretty much hit or miss." There was continuity with a few other characters, notably "Superkatt," a Giggle feature written and drawn by Dan Gordon, a former Fleischer and MGM writer, but Gordon worked in New York.
Far more so than Western's, the Davis roster was packed with veterans of the Hollywood animation studios—not just animators like Bob Wickersham but writers like Mike Maltese, Warren Foster, and Cal Howard. Lloyd Turner, like Maltese and Foster a writer for the Warner cartoons, remembered writing stories with Foster: "On a Saturday, we'd get together with a bunch of beer, go up to his apartment, we'd sit and talk, he'd get an idea, I'd get an idea, we'd do 'em, take 'em over to Jim, and get paid. I think they paid, in those days, ten dollars a page. So you do an eight-page story, that's eighty bucks, sitting there drinking beer with Warren."
The domestic animals in "Superkatt," by Dan Gordon, conversed freely with human characters, as in this example from Giggle Comics no. 50, February 1948.
The stories were usually that short, eight pages or less, with an improvised-on-the-spot quality. There was, however, very little of the nimbleness and wit that might have accompanied such improvisation, just a pervasive lack of any sustained effort. Only the Dan Gordon "Superkatt" stories rose a little above the norm. Superkatt, who fashioned his costume from a baby's diaper and bonnet, lacked superpowers, and his stories took place in an ingratiatingly odd world where cats and dogs conversed freely with human characters but were still domestic animals. Typically, though, Gordon's stories sputtered and stalled well before the concluding page.
"We used to get criticism on them, on each month's magazines," Jack Bradbury said. "We'd get a letter back; a fellow who worked for Sangor [the editor Richard Hughes] would say what he thought of the different ones, but they were pretty lenient, because there was a big market at the time, and they wanted lots of stuff, quantity rather than quality." So robust was the market that Davis's shop wound up producing stories with licensed cartoon characters. "I had this group of guys going," Davis said, "and then Whit Ellsworth [an editor for Donenfeld's DC] came out looking for a cartoon subject. The only thing available was Columbia, but I had half the guys there working on Sangor's stuff already. . . . And Sangor was pretty solid with [Donenfeld], and told him that he was interfering with his operation out here. So, the whole thing was swung over to him."
The first issue of Real Screen Comics, with the Fox and the Crow and other Columbia characters—all of them decidedly minor players—appeared early in 1945. Other DC comics with talking animals followed, most of them not based on the Columbia cartoons. All were in the Sangor vein, but marginally more polished. Although the stories were short, as in Sangor's comic books, the drawings were more nearly uniform—DC, more than most publishers, cultivated a house style for each of its different kinds of comic books—and the writing not so careless. But many of the stories were like half-remembered routines from vaudeville or burlesque. As easy as it was to conjure up "funny animals," it was much harder to find interesting things for them to do.
## 8
# Carl Barks Makes His Break
As Sangor and other publishers fed the appetites stimulated first by the animated cartoons and then by the earliest comic books based on them—the Disney titles, Looney Tunes, and New Funnies—Dell and Western were slower than their competitors to take advantage of the demand that they had themselves done so much to create.
The first new comic-book stories with Disney characters did not appear until mid-1941, in Walt Disney's Reluctant Dragon, a one-shot Dell comic book that took its title from a Disney feature. The feature was assembled from animated shorts and a live-action tour of the studio, but the comic book offered only rough approximations of the shorts. A few months later, Dell published two one-shot comic books based on the 1941 Disney animated feature Dumbo, one in color and one a black-and-white "comic paint book." Both were crude looking, and the color Dumbo was filled out awkwardly with reprinted Mickey Mouse comic strips. Like Walt Disney's Comics & Stories, the new one-shot comic books were produced by people who worked at Western's Poughkeepsie plant.
Irving "Bud" Tripp, a nineteen-year-old Poughkeepsie native, started with Western in September 1940—that is, just as the first issue of Walt Disney's Comics was being published. As he told Bruce Hamilton in 1984, he was put to work immediately, turning Disney comic strips into Disney comic books: "A lot came in as newspaper repros [proof sheets] and press sheets and we cut them up. The daily and Sunday newspaper strips we cut up and put into comic form to make a 48-page book. Sometimes you'd have to add on to the edges, so there was a little art work involved. If a panel ran short you'd have to add on the backgrounds. . . . This was what I actually started doing."
The Reluctant Dragon and Dumbo one-shots could not be assembled in that fashion, though, since comic-strip versions did not exist, and so, according to Tripp, Oskar Lebeck came up with a solution: trace the frames of the actual Disney film while using the film-editing machine called a Moviola to project the film up onto a piece of frosted glass inserted in a drawing board: "The projector of the editing machine would shine onto the frosted glass [from] underneath . . . and when we'd see [a frame] we wanted to use [as a panel in the comic book], we just stopped the machine, put in a single piece of Strathmore or tracing paper over the top of it and [made] a rough sketch. Then we'd go to the next panel. Then we'd ink them in and add balloons, put them in comic-page order and make a 48-page comic book out of it." A stenographer transcribed the dialogue so that Tripp and his colleagues knew which words to put in the dialogue balloons.
This curious expedient was a variant on a familiar animation procedure called rotoscoping, in which animators tried to produce lifelike movement by tracing frames of live-action film—the most important difference being that Western's young cartoonists had to trace relatively few frames of film to produce the panels of the comic books, whereas full-scale rotoscoping could involve tracing many hundreds of frames, and so was far more tedious. Tripp recalled his work on those early comic books as "a lot of fun"—a statement no animator would make about rotoscoping. Tripp spoke of making tracings from not just other Disney cartoons but also a couple of MGM cartoons, The Milky Way and The First Swallow, for the first two issues of Our Gang Comics in 1942. As odd as it was, comic-book rotoscoping evidently persisted for a year or two, perhaps until Tripp entered the army in September 1942.
In addition to the Reluctant Dragon and Dumbo comic books, a comic book devoted to Mickey Mouse also appeared late in 1941, but it was made up not of new material but of reprinted daily comic strips—a complete story titled "Mickey Mouse Outwits the Phantom Blot." Reprinted newspaper comics continued to dominate the pages of Walt Disney's Comics & Stories throughout 1942—a successful strategy, as the sales figures argued—although one new story, drawn on the East Coast by Walt Kelly, somehow slipped into no. 23, August 1942. It was a comic-book version of "The Laughing Gauchito," a segment planned for the Disney feature Saludos Amigos but scrapped months before the comic-book story was published.
In the meantime, Eleanor Packer had produced two Disney comic books that were the first to be composed of new material that was not based on a current animated feature. One, a black-and-white comic book in the Large Feature series, no. 7, called Pluto Saves the Ship, was published in July 1942. It was followed in August by a color comic book called Donald Duck Finds Pirate Gold, an early entry, no. 9, in what Dell called its Four Color Comic series of one-shots.
The two comic books, each of which took its title from the story that filled its pages, included the first work for Packer by a veteran Disney cartoonist, Carl Barks. By the time those comic books were published he was forty-one years old and had been a member of the Disney staff for close to seven years, mostly as a writer—or "story man," as the Disney writers were called—for the short cartoons starring Donald Duck.
Fittingly, Barks's first comic-book work was more as a writer than a cartoonist. He shared the writing of Pluto Saves the Ship with two other Disney cartoon writers, Jack Hannah and Nick George. Barks wrote in 1981 that the three men wrote the story
in 1942 in our evening hours. It was not an adaptation of a cartoon story. Eleanor Packer . . . may have dressed up the basic plot. It was only a one-shot special designed to take advantage of the wartime jitters. Anyway, we three did the final draft in rough sketch form in my den room in North Hollywood. The post–Pearl Harbor blackouts were in effect, and we had all window blinds closed and taped shut. It was hot and stuffy, and we consumed many beers. The story shows the effects. One of Disney's layout men with a flair for drawing panel after panel of shipyard scaffolding did the artwork. I can't recall his name.
Because . . . we were only writing action gags to flesh out someone else's story line, none of us felt we deserved any claim to fame. I certainly forgot the whole business very quickly. As for payment, I doubt that we received more than a dollar a page.
Work on the Donald Duck comic book involved some of the same people, but in a different combination. This time Barks was drawing instead of writing. Bob Karp, who wrote the gags for the Donald Duck newspaper strip, also wrote "Donald Duck Finds Pirate Gold," basing the comic book on story sketches for an unproduced feature cartoon. That cartoon, which was to have starred Mickey Mouse, Donald Duck, and Goofy, would have been called Morgan's Ghost. It would have been a modestly budgeted cartoon about an hour long, one of several such features—Mickey and the Beanstalk and The Wind in the Willows were others—that Walt Disney was making or contemplating in 1941, when his studio was in deep financial trouble and about to be wracked by a divisive strike. By 1942, all those features had been shelved, temporarily or permanently.
Carl Barks posed with a feathered friend in the 1940s. Author's collection.
Barks said on at least two occasions that the idea for the comic book originated with Oskar Lebeck when he "was out there at the studio looking for material that could be adapted for comic books." That was probably in February 1942. There is, however, no indication in those interviews that Barks had any personal knowledge of Lebeck's involvement. Like Barks, Jack Hannah remembered the "Pirate Gold" assignment as originating not directly from Lebeck, but elsewhere.
"A fellow named John Rose worked upstairs," Hannah said (Rose managed the Disney story department),
and he had something to do with Whitman Publishing Company. [He was] a sort of go-between with Eleanor Packer and the studio. She wanted someone to draw this book, and Carl Barks and I were picked out; we were asked if we wanted to do it at night, at home, and we said great. I never knew that [the story was taken from the unmade feature Morgan's Ghost]; I never saw a story sketch that came from it. Carl and I were given a typed copy of the story—I think it was broken into panels. That's all that I worked from. Carl and I took it to our respective homes; we took thirty-two pages apiece, and got together on hookups. We'd just take sections; we didn't even split it down the middle. To my recollection, I never saw a sketch of any kind on the thing.
Barks likewise remembered
a typewritten script, broken down by panels. It would describe a ship or a dock or a room in some detail, to show what atmosphere should be developed in that panel. . . . When we began, we were only given four or five pages of script, while Bob [Karp] was working on other pages. I took a couple of pages, and Jack took a couple, and by the time we got those done, there was much more of the script done, and we had a little talk together as to which we enjoyed or felt either of us could draw better than the other. We decided that I would take most of the outdoor scenes, and he would take the indoor ones.
In both comic books, the story has been broken down into too many drawings, recalling not just the sketches on a storyboard but also the layout drawings that cartoon directors used to guide the work of their animators. Drawings of both kinds were not just typically more plentiful, in proportion to the action depicted, than the panels in a comic-book story; they were also never intended to stand on their own. They required a story man's narration or an animator's hand to bring them to life. Especially in the Pluto story, the action has been dissected so thoroughly that each panel looks like a frozen moment, and the drawings lie flat on the page. In addition, as Geoffrey Blum has noted, Barks and his colleagues borrowed heavily from several Disney cartoons that Barks had helped to write in the preceding four years.
Both comic books were, however, striking departures from the comic-book norm in important respects. Their stories were exceptionally long—"Pluto Saves the Ship" fills all but one of that book's fifty-two pages, covers included, and "Pirate Gold" fills sixty-four pages of sixty-eight, again counting the covers as pages. Both stories' prevailing tone is serious. "Pluto Saves the Ship" is a wartime story with Nazi saboteurs (one of them a traitorous dog—that is, a canine) as villains. As for "Pirate Gold," Walt Disney spoke in 1941 of making Morgan's Ghost "a good burlesque of Treasure Island," with the eponymous ghost a comic presence on the screen, but the comic-book story resists the "burlesque" label and demands to be regarded as an imitation of the Stevenson novel, with Mickey Mouse's longtime nemesis Black Pete in the Long John Silver role. The length of the stories, and their serious tone, may have reflected strong sales of the 1941 Mickey Mouse "Phantom Blot" one-shot. That comic book was also devoted to a single story starring a Disney character engaged in conflict with a deadly enemy. Disney received royalties on 427,057 copies—but when Dell next published a Mickey Mouse one-shot, in 1943, Western paid Disney royalties on more than a million copies.
If, as seems likely, Oskar Lebeck was at Disney early in 1942 looking for a suitable source for a full-length Donald Duck adventure story comparable to "Mickey Mouse Outwits the Phantom Blot," that would have been because there was no existing material suitable for reprinting; the Donald Duck newspaper comics offered not stories but a daily joke. Dell had published three Donald Duck one-shots at that point, two in black and white and one in color, and all assembled from daily strips or Sunday pages. Cobbling together such newspaper comics to tell a real story of any length would have been impossible. Morgan's Ghost met that need, once Mickey Mouse and Goofy had been replaced in the cast by Donald's three nephews.
Both Lebeck and Eleanor Packer would have been at ease working with longer stories in any case—Packer especially, because she had written so many of them for Whitman in the 1930s. A standard sort of story in Whitman's juvenile books—the ones published for children who could read for themselves—was an adventure whose heroes were likable characters familiar from comics and the movies. Comic books like Donald Duck Finds Pirate Gold and Pluto Saves the Ship were awkward new variants of what had already become a Whitman formula.
The two Disney comic books appeared within weeks of the first Bugs Bunny comic book, the black-and-white issue with Carl Buettner's first work for Packer. The stories in the Bugs Bunny comic book were not as long, at twenty-four pages each, but were still long by prevailing comic-book standards. A sixty-four-page adaptation of Bambi (the film was released in the summer of 1942) followed soon after Pirate Gold, drawn by the Disney animator Ken Hultgren in the period before he began drawing in quantity for Benjamin Sangor. The lead story in the first Porky Pig one-shot, published late in 1942, was also long, at thirty-nine pages. The stories in Dell anthology comic books like Walt Disney's Comics and Looney Tunes were shorter, ten pages or less, but those stories were longer than many stories in comparable titles from other publishers. As crude as the stories in the early Dell titles were, there was in their length the potential for the development of both plot and character.
Kelly's "Laughing Gauchito" aside, the first original comics did not appear in Walt Disney's Comics & Stories until no. 27, the December 1942 issue, in the form of a twelve-page story, drawn by Carl Buettner, about Joe Carioca, the parrot star of Saludos Amigos. By then, not only were comic books with new material dominating the market, but the well of reprintable Disney comic strips was starting to run dry. New comics based on characters from Bambi, and drawn again by Ken Hultgren, appeared in three issues of Walt Disney's Comics in early 1943, breaking the pattern of text adaptations from the animated features. Those stories were, however, more conspicuously juvenile—simpler in both writing and drawing—than the reprinted comic strips they displaced.
As it turned out, another original story was more significant. This was the ten-page story with Donald Duck and his nephews in Walt Disney's Comics no. 31, April 1943—the first such story to be drawn by Carl Barks, and his first comic-book work since his collaboration on "Pirate Gold."
By the time Barks drew that ten-page story, late in 1942, he had left the studio, without telling Jack Hannah, his partner in the story department, that he was leaving. "I took a trip east for a few days [on family business]," Hannah said, "and when I came back, Carl had quit the studio to go into comic books." Barks had left his Disney job without giving notice. Instead, he wrote to Hal Adelquist, the Disney personnel director, on November 9, 1942, to tell him that he had decided to leave the staff as of the previous Friday, November 6:
I tried to see you last Friday to tell you that I have decided to leave the studio and try farming at my San Jacinto estate (five acres of Russian thistles). I had not planned to leave so suddenly, in fact, I might have stuck around indefinitely had not gasoline rationing forced me to move while it is still possible to do so. . . .
I have become tired of working for wages and have decided to make one reckless effort to survive on my own. . . . I feel that with more time to develop my long-neglected knack for drawing human figures I may be able to break into the comic magazine field. . . .
Certain of the boys have the mistaken idea that the little job of drawing that I did on a duck one-shot last summer gave me comic-strip delusions and accounts chiefly for my itchy feet; such however is not the case. I was working nights to develop a comic-strip technique long before I even heard of the Whitman Publishing Co. I have no promises of work from the Whitman people in the future, and I doubt very much that they would offer me any lest the studio feel that they had in some degree lured me from the fold.
He had discovered in 1941, Barks wrote fifty-five years later, "that the hot sun of the semi desert area east of L.A. made me feel good." He had been suffering from sinus trouble, which he blamed on the studio's air conditioning, to the point that he had undergone an operation of some kind to relieve its symptoms. In San Jacinto, though,
I could breathe through my nose, and my itchy giant hives simply burned away in the heat. The great depression was still raging. Real estate was cheap. I bought five acres with a livable house [on April 17, 1941] and planned to use it as a weekend "recovery" place . . . .
The wartime restrictions that the studio had to impose upon us hired help would have ended my weekend recoveries. I had to make the break, hence my departure and my letter to Hal Adelquist. . . . I had no way of knowing if I could make a living at freelance cartooning or at raising chickens. Luckily I was saved by the decision of Whitman Publishing Co. to try printing original Donald Duck stories in [Walt Disney's Comics].
Barks had been looking at comic books with interest for a few years before he drew one, and he said in 1983 that he found in working on "Pirate Gold" that he liked drawing comic-book stories:
[Y]ou could get in and do some drawing, and you could do it with a fine-pointed pen. Lord, those old blue and red pencils we used to rub out [story sketches] with—they were broad and soft and smudgy—they took all the fun out of art; it just became a type of shorthand. It was fun to be able to draw things in sharp detail for once. I looked in the [National] Geographic and got some of the English architecture to use for the Bucket o' Blood saloon. And I would have used the Geographic for reference on the ship's rigging.
In 1973, Barks remembered getting in touch with Whitman—that is, with Eleanor Packer—soon after retreating to his chicken farm: "I wrote to Western Publishing [sic], and told them that I had left the studio, and that I was available in case they had anything they wanted done." Packer may have had a script in hand or, at least as likely, may have written one herself in short order. "They sent me a ready-written script to draw for story #1 several weeks after I left the studio. . . . The script was the Crow story," in which Donald defends his Victory Garden against a gang of voracious crows. That story was published in the April 1943 Walt Disney's Comics after Barks had not just drawn it but rewritten it. "I noticed some bad errors in the plot," he said in a 1967 letter, "and wrote the editor for permission to change things around."
He trod lightly when he reworked the crow story, he said a few years later: "I believe I sent them a sort of a draft of some change, or suggested some change. I was a little bit new with these people, they hardly knew who I was, so I wasn't going to make myself an obnoxious character right away." He was successful: "The result was they invited me to see if I could do a whole original story myself. The story of the kids and their 'rabbit foot' [in the May 1943 Walt Disney's Comics, no. 32] was my answer to the opportunity." Barks's "Donald Duck" story in the April Walt Disney's Comics began on the fifth page of the magazine, but for May his story moved to the lead position.
It is possible, though unlikely, that the script for the first "Donald Duck" story came to Barks not from Packer but from Dorothy Strebe, who worked for more than twenty years in the publications department at Walt Disney Productions. Her undated memo to Barks, now in the Walt Disney Archives, says: "Here is a 10-page story for Donald Duck. Hope that you like it . . . you are to stage it, of course . . . and if you see that it can be strengthened, or that it deviates from Donald either in narration or action, please make the improvements." She asked for delivery in two weeks—"by the 23rd," probably of December. Barks was to be paid $12.50 per page, an increase from the $10 per page he remembered receiving for his work on "Pirate Gold." Strebe added: "Golly, I didn't know you had left [the studio]."
In 1974, Barks said of the script that accompanied Strebe's memo: "That would have been about the second one, I think. The first [script] came to me directly from Eleanor Packer." But he also recalled getting a script for only the first story, the one in the April 1943 Walt Disney's Comics. Did Barks receive a script for his second story from Strebe but not use it? He definitely submitted his second story to Whitman on December 23, 1942—a date that is consistent with Strebe's memo—but he always said he wrote as well as drew that story.
The question of when Barks began both writing and drawing his stories is of more than incidental interest because the "Donald Duck" stories were for many years such a congenial artistic platform for Barks, and because so much of the merit of those stories was owing to their harmonious marriage of script and drawings. There is obvious attraction in being able to say of one of Barks's comic-book stories, this is where it all began. But any definitive statement of that sort will always be elusive.
The "Donald Duck" story in Walt Disney's Comics & Stories no. 32, May 1943, was the first comic-book story that Carl Barks both wrote and illustrated. © 1943 Disney.
When Barks began writing and drawing stories for Western, he had no comic-book experience. There were precious few models he might have emulated, whether in Western's other comic books or in those of other publishers. Clumsy writing and weak drawing prevailed. As the Looney Tunes cartoonists Veve Risto and Roger Armstrong believed, the comic book imposed new demands that many cartoonists were slow to understand, much less obey. Where Barks stood apart—and what gave him the breathing space he needed to develop skills uniquely suited to comic books—was in his experience at Disney.
Barks was more familiar with animated characters than many of his comic-book colleagues. He knew how to draw them, and, most important, he knew how to write stories for them, even if those stories were better suited to animated cartoons than to comic books. The Donald Duck cartoon stories Barks wrote with Jack Hannah and others while he was at Disney invite criticism—the comedy is often labored and Donald Duck himself noisy and pugnacious, defects magnified by Jack King's cautious direction—but they are clear and coherent, and they had passed muster with Walt Disney himself. That was a credential that no one else writing comic books for Western in 1943 could boast.
Barks's earliest stories for Walt Disney's Comics & Stories, like his pages for "Pirate Gold," speak strongly of the storyboards he had been making for years: the sparse and functional drawings, with their stock expressions, seem to be waiting for an animator's enlivening touch. The connection between storyboard and comics page was one that Barks himself made. He said in 1971 that in his early years of comic-book work "I practically considered the formation of a story like I was working on a storyboard," even though he first wrote a script in longhand.
I bought a sheet of Celotex [fiberboard], four by eight, and I put that up in front of my drawing board, and when I'd get a half sheet or a comic page done in pencil, I'd stick it up there, and then the next one, and the next one, and after I got about five of them done, I would sit back and look at the display and read the continuity. Sometimes I would take down two or three sheets and do a lot of erasing and changing. I was able to visualize my story progress much better that way.
By May 1943, Barks had written as well as drawn three ten-page stories for Walt Disney's Comics. He then surrendered that feature to another cartoonist for one issue while he wrote and drew the three stories in Four Color no. 29, a Donald Duck one-shot published in the summer of 1943, the first such one-shot since Donald Duck Finds Pirate Gold. As in "Pirate Gold," comedy is notably lacking in that comic book's lead story, "The Mummy's Ring." Barks said of it:
I submitted a sort of script in advance there. I don't remember how detailed a thing it was, but I believe I kind of roughed out the drawings and sent the sheets in, or took the sheets and left them there. Eleanor Packer . . . read through what I had planned . . . and I remember that she suggested some big changes, and she asked me to draw up these changes. Well, I did. I drew up the changes, and I think I sent the roughs of those changes back. She looked them over, and sent back word to go ahead with my original version, my original version was better. So after that, I never had any trouble.
In "The Mummy's Ring" there is again a prevailing seriousness (and a heavy reliance on National Geographic for the Egyptian settings). There are threats of death that Donald and his nephews must take seriously and a torrent of dialogue-heavy balloons once the story's mystery is solved. As in other early comic-book stories from Western's Los Angeles office, the sense is of clumsy imitation of comic strips that more successfully combined comedy and adventure, like Roy Crane's Wash Tubbs and Floyd Gottfredson's Mickey Mouse.
Within the next few months, though, Barks began to master the peculiar requirements of comic books. The story for the February 1944 Walt Disney's Comics no. 41, titled "The Duck in the Iron Pants," is readily imaginable translated to the screen—it echoes a 1942 Donald Duck cartoon called Donald's Snow Fight, for which Barks was a writer—but it works as a comic-book story because its panels are not static, like storyboard sketches, or snapshots, or frames of film. For the first time in a Barks story, the individual panels throughout the story clearly embrace varying amounts of time; they "breathe." Dialogue balloons always nudge a cartoonist in that direction—in particular, when two characters are speaking to each other in a panel, the elapsed time within that panel is necessarily more than a split second—and Barks was by this time assigning more weight to dialogue. "In the early stories," he said in 1971, "I carried the progression with action a great deal more, and then in later stories, I was allowing the dialogue to carry a great deal of the story progress."
Dialogue is not just a storytelling tool, though. It imposes demands on a good cartoonist, especially for greater subtlety in how the characters in a panel seem to be responding to one another's presence. On a larger scale, it is an important element in how panels and pages are related to one another. Barks spoke of that aspect of his work at the Boston Newcon comics convention in 1976. "I tried to end each page, especially toward the latter part of the story, with a little zinger that would carry the reader forward. I'd write the story as I went along, and when I was stuck, I'd skip ahead and write the ending, then go back and do the middle of the book, moving the story along to the already-written conclusion."
Within Barks's intensely practical description of his working methods there is visible a concern for what might be called phrasing. Very early in his comic-book career, he began to think like an actor who is delivering a major speech or, perhaps even more, a pianist who is shaping a performance of a sonata, or a conductor a symphony: he began to make of a comic-book story an organic whole, one whose panels' apparent duration expands and contracts in a syncopated pattern grounded in an acute sensitivity to time, and whose pages move in longer and deeper rhythms. Other cartoonists, notably Will Eisner in his Spirit stories, mastered comic-book time as fully as Barks, but none so naturally and unobtrusively. There is almost never in Barks the flamboyance and theatricality that Eisner so much enjoyed, but instead something that makes Barks's stories seem more real: a temporal flow that mimics how people actually experience time.
## 9
# Barks Becomes the Duck Man
From the beginning of Carl Barks's comic-book career he was attentive to the form's demands, as many of his peers were not. Jack Hannah, for one, drew a few undistinguished comic-book stories after his work with Barks on "Pirate Gold," but, he told Jim Korkis, "I remember very little about the work, I'm afraid, because it just didn't seem significant to me. . . . I can't remember doing the stories but it's obvious I did them. I really had no contact with the people at Whitman. . . . I was probably just given some typewritten scripts and drew them up."
There was, however, not much more than a hint of artistic sophistication in the first few years of Barks's Donald Duck stories, and the stories suffered from debilities that extended beyond his tentative grasp of comic-book aesthetics. In the early stories, Donald often battled ferociously intense opponents—notably his surly, beetle-browed next-door neighbor Mr. Jones, but Mr. Jones was never more than a simplified version of a grumpy movie comedian like Edgar Kennedy. Conflict between Donald and his nephews recurred frequently but was rarely true parent–child conflict. Donald was instead like an older kid, a bully rather than a surrogate father. Other stories suffered from basic structural problems. Comic catastrophes accumulated, sometimes with strong echoes of silent two-reel comedies like Buster Keaton's—films of the kind Barks saw as a young man—but there was no culminating disaster, only a contrived happy ending.
Barks was, however, increasingly aware of the possibilities in his new medium, and of the limitations of the animated cartoons that originally served as models for his stories. He said of gags like those in the Donald Duck cartoons and in his early stories that "sight gags are quite limited. You know, there are only so many things you can do with a human body or a duck body and then you start repeating yourself; otherwise you'd kill him." The animated Donald Duck was, in Geoffrey Blum's pithy description, "little more than a feathered temper whose peculiar voice makes conversation limited at best. Donald in the comics is capable of both speech and thought—rudimentary thought when the comics began in the early 1940s, but that was enough for Barks to work with."
Barks said of the gradual transition in his stories toward a greater emphasis on psychological comedy:
You could draw just so much violent action in a comic book before it began to get tiresome. And I think Floyd Gottfredson [who plotted and drew the Mickey Mouse newspaper comic strip] put his finger on it one time when I was talking to him; he says, "In the strip, the reader can hold it up, and he looks at it for a long, long time, but when it's on the screen, he sees it for one twenty-fifth of a second, or something like that, and it's gone." There's no chance for him to start looking at it too long. I remembered what he had told me, and I toned down my action a little bit after having talked with him. . . . I think it was sometime in the 1940s. I'd gone to the studio for something.
Circumstances also pushed his work in a more promising direction. Like Western's other cartoonists, Barks worked at first on stories made up of six-panel pages, but as wartime paper shortages shrank Walt Disney's Comics from sixty-four pages to forty-eight (and then, briefly, thirty-two), plus covers, he wound up working with pages made up of eight panels. Barks said that he preferred working with six panels, so long as the sheets of drawing paper were of a manageable size. When he was working with six panels on large sheets, he said, "[i]n order to draw the stuff up here at the top, I had to fold the bottom part of the sheet of drawing paper back under the drawing board, and it made it difficult to ink." With an eight-panel page, he could cut the paper in half. "A lot of the guys, though, just drew on full size. I used to see them around the office whenever I'd take in stories, there would be other guys' work lying there, great big sheets, and I used to think, good Lord, they must have arms on them six feet long." His work looked better in the tighter format, though, because it fit the growing precision of his drawings.
Barks was still a chicken farmer as well as a cartoonist, but as Western began to "load me up with this comics stuff," he realized that "I could make more money drawing comics than I ever could with those chickens." From about 1944 on, he simply fed his egg-laying hens "and let them live on a pension."
In the mid-1940s, when the Donald Duck one-shots were appearing only once a year, Barks was called upon to draw, and often write, stories with other characters—"Barney Bear and Benny Burro" in Our Gang Comics, single issues of Porky Pig, and Mickey Mouse, and even one story, at the very beginning of his work for Whitman, with Andy Panda, for New Funnies. "They weren't characters that interested me a great deal," Barks said of the "Barney Bear and Benny Burro" series. The two characters had been paired in Our Gang Comics no. 11, May–June 1944, after earlier stories had made clear that neither was strong enough to sustain a series on his own. "[T]hat burro, to try to figure out how to do much with that confounded burro—he's got no hands," only hooves. The stories "were harder to write, because I had to have very special business for those two guys. With the ducks, I could use human business, because a duck could do anything that a human could do." That was because Donald Duck, as designed for the screen, had arms and hands rather than wings.
In Walt Disney's Comics & Stories, new stories with other characters filled in behind Barks's "Donald Duck stories," alongside the continuing comic-strip reprints. Then, starting with no. 39, December 1943, "Bucky Bug" became a regular backup feature. "Bucky Bug" had already appeared for two years in the Silly Symphonies Sunday page, a page that began with "Bucky" in January 1932, almost a year before the release of a Silly Symphony with similar characters called Bugs in Love. "Bucky Bug" had then lain dormant for almost ten years. There is no way to retrieve the reasons for singling out such characters for continuing features, but their potential to support a series of stories must have been paramount, and "Junkville," miniature insect homes built out of humans' trash, may have been an appealing setting.
As it happened, the "Bucky Bug" stories in Walt Disney's Comics were from the beginning slanted toward a very young audience. The stories were extremely simple, drawn in a much broader cartoon style than Barks's, and, as in the 1932–34 Sunday pages, the dialogue was all in rhyme. A side effect was that Barks's stories seemed more adult set beside them. Walt Disney's Comics was not unique among Western's comic books in that respect—in Looney Tunes and Merrie Melodies Comics, the "Sniffles and Mary Jane" stories tilted younger than the "Bugs Bunny" stories—but the spread in the likely audiences' ages was larger in the Disney comic book.
Curiously, the "Bucky Bug" stories very quickly became crime stories, almost without exception (the exceptions usually being stories about insect warfare—also a subject when "Bucky Bug" occupied the Sunday Silly Symphonies page). Crime comic books enjoyed a great burst of popularity in the middle 1940s, and it may be that the writers of "Bucky Bug" were taking note of that; or, at least as likely, such stories were simply easier to write than other kinds. New Funnies, produced in New York, had its own insect feature, "Billy and Bonny Bee," starting with no. 67, September 1942—a feature, drawn most often by Frank Thomas, that Western owned—and it was in such superficially similar stories that the Dell comics from the Los Angeles and New York offices were actually most different. In their whimsical tone if not in their appearance, the New York comic books produced under Oskar Lebeck resembled traditional illustrated children's books, but "Bucky Bug," with its stories dominated by vigorous conflict, was imaginable only as a comic strip or in a comic book.
As Walt Disney's Comics became a true anthology comic, with reprinted newspaper strips playing only a supporting role, there was the possibility that other cartoonists might become as closely identified with their characters as Barks was with the ducks, and develop their stories in the same way—but nothing like that happened. Instead, a character like the Li'l Bad Wolf, whose stories began with no. 52, January 1945, passed from hand to hand, the stories rarely rising above formula (the Big Bad Wolf tries to catch and eat the Three Little Pigs and the Li'l Bad Wolf thwarts him, usually without the Big Bad Wolf's being aware of it) in both writing and drawing. The comic-book stories were not hamstrung by the animated cartoons; Li'l Bad Wolf in no way resembled the young wolves in two Disney shorts with the Three Little Pigs. It was just that, from all appearances, no one was as willing as Barks to take real pains with his work, or else was able to make such pains pay off for more than a story or two.
Barks himself could in the middle 1940s lapse into a disregard for plausibility that was all too typical of comic books generally. Both of the stories in his third Donald Duck one-shot—Donald Duck in Frozen Gold, Four Color no. 62, published late in 1944—suffer in that way, combining the ludicrous (how does Donald know how to fly an airplane?) with a pulpish seriousness (Donald is again under the threat of death). In the second story, called "Mystery of the Swamp," the ducks confront a particularly silly menace, the Gneezles, murderous hillbilly trolls who have been cut off from the outside world for centuries but somehow speak English.
Such longer stories were inevitably more problematic than the ten-page stories. The shorter stories could be, and almost always were, entirely comic, even though they differed in their kind of comedy. The longer stories, serious in tone from "Pirate Gold" on, could never escape the requirement for some strand of adventure, some element of peril, and making the ducks' peril believable without sacrificing the comedy was a continuing challenge.
Despite the deficiencies of "Mystery of the Swamp," its pacing—the rhythm of the transitions from panel to panel—and the comic incidents suggest for the first time that Barks was on the verge of mastery of a craft, the fabricating of the literate comic-book story, that still barely existed. It was in stories published soon afterward, early in 1945, that Barks began to show just what that mastery would look like when he finally achieved it.
The story in Walt Disney's Comics no. 53, February 1945—in which Donald buys a tramp steamer and heads for Acapulco with a load of bubble-bath soap—is a comic adventure, and, even more to the point, a comic-book adventure, like none before it. There are no echoes of film, no suggestion that Barks's panels are awaiting conversion into a more appropriate medium. This was the first ten-page story that took the ducks out of the country, from a starting point in what could only be California, and their environment embraces not just their little ship but tempestuous seas, a rambunctious whale, and a Mexican coastal village. Everything is more persuasively detailed and concrete than in earlier stories.
The story in the next issue—no. 54, March 1945, in which Donald and the nephews race down a frozen river the three miles to Pumpkinburg on ice skates—also benefits from the expertise Barks had gradually acquired in timing and staging. What was lacking now for this kind of story was an emotional focus that would make the ducks seem like rival family members as well as rival racers.
Later stories in 1945 for Walt Disney's Comics were not as good. For the July through October issues, when the page count had shrunk again, Barks had to squeeze as many as twelve panels onto a page. But the end of the year saw another burst of stories distinguished by a new subtlety and self-assurance. In no. 62, November 1945, Donald enters water-ski races but must operate his tow boat by remote control after the nephews eat themselves sick on popcorn and candy. The situation sounds hopelessly contrived, but not only does Barks make it look possible, mechanically; he also makes it feel possible, through his emphasis on exactly those elements—Donald's vainglory and desperation, the nephews' smugness—essential to that result. Two months later, in no. 64, January 1946, the conflict between the nephews and Donald has become unmistakably a conflict between children and parent, with a corresponding increase in psychological verisimilitude.
In "The Terror of the River," in Donald Duck Four Color no. 108 (1946), Carl Barks took the ducks into uncommonly serious territory. © 1946 Disney.
Things could still go wrong, especially in the longer stories. The nephews had to behave with maturity in those stories, because otherwise they would be burdens on Donald, but in "The Terror of the River," in the Donald Duck one-shot for 1946, Four Color no. 108, Barks boxed himself in. Too much of the story's load falls on the nephews' shoulders, to the point that one of them rents a plane and pilot, and another rents a tugboat—adult activities impossible for very young children. The story as a whole has an ominous tone—the villain is a sadistic, murderous psychopath—that is hard to reconcile with its comic machinery.
As embryonic as Barks's stories still were three years after he began writing and drawing them, he was already putting distance between himself and his colleagues. They were improving, too, and a 1946 comic book like Bugs Bunny's Dangerous Venture, Four Color no. 123, with a thirty-page lead story of that title drawn by Tom McKimson, not only looks much better than earlier comic books with the Looney Tunes characters, but also invites comparison with what Barks was doing. A well-plotted story puts Bugs Bunny and Porky Pig in genuine peril, in an exotic Tibetan setting whose appearance suggests that McKimson, or someone, did a little homework. There is even a comic but entirely appropriate name (Mr. Omi-Akin-Bak). The unidentified author of the story punts on the language question—everyone in Tibet speaks English, it seems—but no more egregiously than Barks had done in the Gneezles story the year before.
Still, the "Dangerous Venture" dialogue lacks Barks's economy and pungency—it is burdened especially by Porky Pig's stutter, which more astute writers learned to minimize—and the panel layouts are frequently awkward, with dialogue balloons shoehorned into whatever space the letterer could find. The intriguingly named Omi-Akin-Bak remains offstage throughout the entire story, and the Tibetan settings, however authentic they may look, are lacking in atmosphere. And then there is the lead character himself, Bugs Bunny, who is a little too much the bully, just as he was in some of his early 1940s animated cartoons.
"Bugs Bunny's Dangerous Venture" is a perfectly presentable comic-book story for its period, more attractive than most, but lacking refinements of exactly the kind that would over the next few years invigorate Barks's stories so greatly. Already, by 1946, the range of both visual and verbal expression in Barks's stories was wider—and much more precise—than in most other comic books, especially those with animated characters. Page through a story drawn by almost any of Barks's contemporaries, even a story like "Dangerous Venture," and the sense very quickly is that a limited repertoire of gestures and expressions is being called upon to do a great deal of work. The dialogue is functional at best. Barks's stories are not vulnerable to such complaints.
Barks's colleagues recognized the clear superiority of his stories to most of what was going into Western's comic books. Roger Armstrong remembered that he and Carl Buettner talked about Barks, who was, "even then [in the mid-1940s], the cartoon genius of the group—and we concurred on the fact that Barks was strictly a 'Duck' man. Given any other character . . . he wouldn't have come through on such a tremendously high level. . . . He had a peculiar affinity with those damn Ducks and he really made them as far as comic books go."
His colleagues' admiration found its strongest expression in the ritual called "the showing." Roger Armstrong remembered that ritual as
almost too hideous to get into. We used to bring our work to the office, and . . . the receptionist would call out, "Here comes Roger, with his Porky Pig." Eleanor [Packer] would say, "Goody, goody, we'll have the showing now." We'd all go into this room with a big table, and Eleanor would sit right in front, in a big overstuffed chair. The drawings were placed in front of her, and everybody who was there—the artists or anybody else—would come look at them. Carl [Buettner] would sit at the table, too, and in a very, very amused voice he would read the stories. We were all delighted when it was a Carl Barks story, we would all laugh and roll on the floor. If it was a story by one of the rest of us, we would die a million deaths. Eleanor would peruse the story as Carl Buettner read it aloud. With expression! He would have a blue pencil in his hand; Eleanor would say, "There needs to be a comma there," and he would reach over and mark it with his blue pencil. Or Carl would say, "I think we could stage this a little better, don't you, Roger? Heh, heh, heh." . . . That was the showing. When it was all over, Eleanor would heave herself up out of her chair and we would all go our ways, and the unfortunate whose work had been on display would pick up the pieces, and take his brush, and white paint, and paint out or paste up or whatever the hell had to be done. You did the work right there.
Armstrong recalled the occasion when "they tore one of my 'Sniffles and Mary Jane' stories to pieces"; he identified the story as the one in Looney Tunes no. 38, December 1944. "This was in the afternoon, around three o'clock, and the redrawing and pasting up I had to do on that thing took me until four-thirty the next morning. There was no transportation available, so I had to walk back to my little place on Gramercy Place. It's a long haul from Beverly Hills to Hollywood, even when the streets are deserted."
The rationale for the showings, Barks believed, was that "Eleanor Packer and Buettner and those people were new at their business of editing comic books, and they felt that the best way to see how comic-book stories were put together was to read them out loud. If they would read well, as you read them out loud, they would read well to the children. The parents reading these stories to their children would be able to read them and get the sense out of them."
Armstrong remembered that "Carl Barks was so modest, so quiet during the showings. He'd stand quiet over on one side, and he'd look out the window. He's always been the most modest, self-effacing man in the world. It never bothered him that people were going over his stuff; it may have, but he didn't show it. He probably turned his hearing aid off."
Chase Craig took issue with Armstrong on that point. "We all went into hysterics of laughter" when Buettner was reading a Barks story aloud, he said, "and such appreciation of his work certainly could do nothing but give him a lift. Carl loves recognition of his talent as much as anyone I've ever met, and that's exactly what he got every time he ever came to the office. In fact, it was a great day when Carl came in once a month to deliver his work. We all had lunch together and sort of celebrated the occasion. I repeat, he was not embarrassed by his reception. He loved it."
Barks may or may not have disliked the showings, but they were an element in his artistic growth; he was, after all, getting applause from his peers. He said as much in 1978: "When I got appreciation—when people said, 'That's a great story,' and laughed—it made me feel good. If they had read the stuff through and just deadpanned it, I would have felt disappointed." But he was glad when the showings stopped, by sometime in the late 1940s.
Barks's superiority as a cartoonist and a writer began to emerge clearly in the mid-1940s, just at a time when his editors were likely to be most responsive. The market for comic books of the general Disney type was expanding rapidly, many people entering the field were struggling to master its requirements, and the military was snatching up cartoonists. Chase Craig enlisted in the navy in 1942, and Roger Armstrong entered the army on February 15, 1945. Armstrong had worked in animation for Walter Lantz for about a year before that, while continuing his comic-book work, in an ultimately futile effort to keep from being drafted. (The Lantz studio was making films for the military as well as Woody Woodpecker cartoons.)
"Eleanor was getting panicky because so many people were being drafted," Armstrong said. "She said, 'Roger, you're the only hope we've got, because you've got kids [from a youthful marriage that ended in divorce], but we're going to have to shore this up to make sure we can keep you through the war. If we lose you, I don't know what's going to happen.' She went to Walter Lantz, and she kind of blackmailed him into taking me."
In such circumstances, Barks—an older man, not subject to the draft because of his age and his defective hearing, reliably productive, and clearly a better writer and artist than most of his peers—was a treasure, and he was treated accordingly. Not with higher pay, to be sure, but with an unusual artistic freedom to explore the potential of the kind of story he had been hired to write and draw.
Said Roger Armstrong:
I'm quite sure that Carl was given a great deal more latitude with his Donald Duck stories than the rest of us were with our stories. I've often wondered why his things turned out the way they did, and I realize in hindsight that the reason must have been that he was given complete carte blanche, he could do exactly as he pleased. I don't think any editorial rein was exercised on Carl. As a consequence, he gave free flight to his imagination. I can remember going in there with story ideas and being hooted down—Eleanor would say, "Oh, Roger, that's just too fey"—whereas Carl would just go ahead and do the thing, he wouldn't discuss it with them. As a result, his stuff caught on—Carl Buettner liked it very much, we were all mad about it.
What was most distinctive about Barks's best work, though, was not that he gave "free flight to his imagination" but that he shaped his stories in accordance with a view of life that was essentially pessimistic. In story after story, Barks revealed his understanding of how people's minds and hearts really work—and, typically, lead them astray. As with the best stories by Harvey Kurtzman and John Stanley, many of the best Barks stories invite labeling him a comic-book Ambrose Bierce—an author he never mentioned and probably never read—except that Barks rarely succumbed to the bitterness that is always lying in wait for the cynic and the satirist. Perhaps it was awareness of his readers' youth that steered him away from darkness. However bleak his stories' underlying message might be, it was always delivered with perfect comic timing and robust comic action. He took seriously his obligation to entertain the child reader.
Not only had Barks entered middle age by the time he began writing as well as drawing comic-book stories with the ducks; he had led a life that differed greatly from the lives of most of his colleagues—the young city boys who made up the bulk of the staff at Disney and Whitman. He was conscious of the difference:
On the farms and the cold, snowy ranch in eastern Oregon where I was raised, all the people lived hard, bitter lives. When I think of them now, I think, my God, every day of their lives was just a hell on earth. . . . It was just hard work and suffering and loneliness. . . . I think I found it natural to satirize yearnings and pomposities and frustrations in the ducks because of my earlier contacts with people in woeful ways of life. Those people had the ability to laugh at the most awesome miseries. If they hadn't had humor in their lives, they would have gone crazy.
Barks's life had been more physically demanding than what his colleagues had known, beyond doubt, but also drearier, more tedious, and even a little sleazy, since for about six years he made his living as a cartoonist and editor for what was, by the standards of the time, a smutty joke magazine. Few autobiographical elements made their way into his stories, he said in a 1971 interview, because "I had so few personal experiences, other than just hard work."
His earlier life was in many ways disconnected from what he later accomplished. The very act of writing and drawing comic-book stories tapped abilities barely visible in all of his previous work. He came to comic books after forty mostly grueling years, but he also came to them as a stubborn autodidact, and his stories with the Disney ducks permitted him to use his rough learning, from magazines and encyclopedias and old textbooks, to explore the surprisingly rich meaning to be found even in relatively narrow experiences like his own.
There is in Barks's stories some of the Depression-bred coldness of the Donald Duck newspaper strip. While he worked at Disney he contributed a few gags for that strip, which was drawn by Al Taliaferro. Cynicism surfaces only rarely in Barks's stories, though, as does its twin, sentimentality. Donald Duck's foolish choices are the engine of Barks's plots, but there is never the sense that Barks believes that either he, as author, or his readers could be capable of much better. It was this cool realism that made his stories seem "adult," compared with most comic-book stories, even in the mid-1940s, and that made his best stories so extraordinary when he had fully mastered the demands of his art.
## 10
# The Workman: Gaylord DuBois
Oskar Lebeck's most important writer in the mid-1940s, as for the preceding few years, was Gaylord DuBois. He was also one of the most important figures in the early comic book. That was not because his work was of high quality in a literary or artistic sense, although admirers believe that some of it deserves such praise. Rather, it was because he worked so well in tandem with Lebeck, contributing materially to the creation of an environment in which gifted artists and writers could thrive as it was not possible to thrive then at any other publisher. Walt Kelly and John Stanley, the two best of Lebeck's other artists and writers, illustrated DuBois scripts before they began writing all or almost all of their own stories.
DuBois's scripts set the tone for the whole Dell line, which was free of almost everything that was lurid and morbid and generally excessive in competitors' comic books. DuBois may deserve the greatest credit for that departure from the prevailing norms, or it may be that he simply absorbed better than anyone else the priorities that Oskar Lebeck observed. Either way, because DuBois's scripts were more archetypally "Dell" than anyone else's—because they established a baseline—they opened the way for other creators who preferred to work in the same vein. There was in the Dell comic books the opportunity to make much better stories than the comic-book industry usually permitted.
Gaylord McIlvaine DuBois became a comic-book writer when he was in his midthirties—he was born in upstate New York on August 24, 1899—and almost by accident. In 1935, when he was suffering from brucellosis and casting about for work that he could do while confined to bed, he wrote to ask his college friend Lloyd E. Smith for help. Smith and DuBois had both attended Trinity College in Hartford, Connecticut, in 1921–22. DuBois left Trinity, attended Carleton College in Northfield, Minnesota, briefly, and then held a series of odd jobs before winding up at Boston University in 1925–27; he never received a college degree. His friend Smith remained at Trinity until he graduated in 1923. Smith worked as a freelance writer, then as an English instructor at Trinity, and finally for three years as an editor for Ely Culbertson, a great popularizer of contract bridge, before joining Whitman's staff in Racine, Wisconsin, as a writer and editor in 1934.
Gaylord and Mary DuBois, mid-1940s. Courtesy of Letty Lebeck Edes.
In the meantime, DuBois worked as a necktie salesman and then as a social worker before enrolling in General Theological Seminary, an Episcopal institution in New York City. (He was from a family of Episcopal priests.) After he graduated from the seminary in 1933 he took a job with the federal government's Works Progress Administration, managing shelters for what were then called "the transient unemployed" and would later be called "the homeless." One six-month assignment took him to Wyoming—far from his home ground in New York—and gave him an exposure to western ways that he would soon put to use in books.
DuBois described to an interviewer, Lou Mougin, how Lloyd Smith's response to his plea for work eventually led him into the comic-book business: "I was given a copy of [a] Lone Ranger radio script with the following instructions: 'Write a 60,000 word novel based on this script, and if we accept it, you will have more assignments.' I thought the script was pretty corny, and not very good material to base a novel on, but I managed to use a little of it as the base for a plot." Although DuBois was identified as the author of The Lone Ranger on early printings, he was then displaced on the book's cover by Fran Striker, creator of the Lone Ranger radio series. "I've never had any complaints about that," DuBois said, "because I sold all the rights to the novel to Whitman."
As DuBois's biographer Irvin H. Ziemann has written, "Following the success of DuBois' novel, Lloyd Smith began to send him other work, mainly the writing of Big Little Books—DuBois wrote 30. . . . Gaylord's editors would give him the name of the principal character for a book and a handful of assorted clippings from newspaper comics—out of sequence—based on that character. DuBois' task was to create a plot that would link these illustrations together, with one page of corresponding text opposite each picture." In 1939, recently married and seeking more income, "DuBois set off for New York City to find adequate work, and again Lloyd Smith came to his aid" by recommending DuBois to the licensing agent Stephen Slesinger. Whitman published most of the books, Big Little Books especially, that bore Slesinger's copyright.
Slesinger was cut from the same cloth as Hal Horne and Kay Kamen, all of them furiously competitive. When Slesinger licensed Whitman to publish Tarzan Big Little Books in 1933, his contract with Edgar Rice Burroughs Inc. probably did not permit him to make such a deal. Only after Burroughs had sent several stiff letters to Whitman, complaining that Whitman had infringed on its copyright by publishing a Tarzan Big Little Book, did the two companies enter upon a long and profitable relationship. Burroughs did not forgive Slesinger so easily, parting company with him in 1938.
Slesinger's greatest coup was buying the merchandising rights to Winnie-the-Pooh. Once Slesinger had satisfied himself that the rights were available, he went to London and signed an agreement with A. A. Milne, author of the Pooh books, on January 6, 1930—two and a half years before Kay Kamen, his fellow licensing pioneer, signed a comparable agreement with Walt Disney. Slesinger subsequently acquired the merchandising rights to other properties. In addition to Burroughs's Tarzan, he controlled the licensing of Zane Grey's western novels and some popular newspaper comic strips, including Wash Tubbs, Alley Oop, and Tailspin Tommy. Whitman liked doing business with Slesinger because, in the words of Whitman's president, Samuel E. Lowe, "it is much more difficult to do business with a single organization than it is to do business with an organization which represents a number of items that we are using. We have turned down a good many things simply because we dreaded doing business with the individual." Slesinger and Whitman had the same overriding goal—to sell as much as possible—whereas the creator of a literary property might find other considerations at least as compelling.
Slesinger himself produced two comic strips: Red Ryder, drawn by Fred Harman; and King of the Royal Mounted, a Zane Grey property drawn by Jim Gary. Slesinger needed a writer for both comic strips, and Gaylord DuBois, after writing dozens of Big Little Books with western characters, was a perfect choice.
DuBois had already written what he described as scripts for early comic books. "Back then," as he said, "the whole idea of a comic magazine was experimental. Some of them had twelve panels to a page. For each twelve-panel page I wrote, I was paid fifty cents"—a low figure even then, but there may have been a reason. He was probably describing the very odd "Tom Mix" feature that appeared in Dell's Popular Comics, starting with the May 1936 issue. There were two pages usually, of twelve panels each, with captions rather than dialogue balloons, and each page had its own heading so that it resembled a reprinted Sunday comics page. Whitman was already producing Tom Mix Big Little Books, and the "Tom Mix" pages in Popular Comics were adapted from the illustrations in Big Little Books, reversing the usual procedure in which comics were reworked for the books. It was probably for that adaptation, and not for any original writing, that DuBois was paid fifty cents a page. "Tom Mix" was copyrighted by Slesinger, who owned the licensing rights to the movie cowboy.
Fred Harman had joined the Slesinger staff through yet another Whitman connection: Samuel Lowe liked the illustrations that Harman drew for a 1938 Big Little Book called Cowboy Lingo: Boy's Book of Western Facts and recommended him to Slesinger. Harman had been drawing a cowboy comic strip called Bronc Peeler for about four years, at first syndicating it himself, with only limited success. Almost certainly at Slesinger's behest, Harman canceled his comic strip and transformed the rough-hewn Bronc Peeler into a virtually identical but somewhat slicker red-headed cowboy, Red Ryder. He had already replaced Bronc Peeler's crusty sidekick Coyote Pete with a more readily adorable Indian boy named Little Beaver. Harman began drawing a Red Ryder Sunday page late in 1938, under a contract Slesinger negotiated with a newspaper syndicate, and then a daily comic strip early in 1939.
Harman was usually identified as the creator of the Red Ryder comic strip, which achieved much greater popularity than Bronc Peeler, but Red Ryder bore Slesinger's copyright, and it was always unclear who actually owned the character. It seems highly likely, though, that Slesinger as the copyright owner reaped by far the greatest rewards. The same was true in other, more famous instances. In the early 1940s Superman's creators, Siegel and Shuster, were paid just ten dollars a page for the stories with their character that they wrote and drew for comic books published by Harry Donenfeld's Detective Comics Inc. By then the character was no longer theirs. DC had bought all the rights for $130—the equivalent of $10 a page for the first thirteen-page story—on March 1, 1938, a few weeks before Superman's debut in Action Comics no. 1.
The television commentator Andy Rooney, who knew both Slesinger and Harman, wrote of their relationship: "Steve was a merchandizing [sic] genius. He went to the Daisy Air Rifle Company and sold them a deal to make a Red Ryder BB gun. In Hollywood, Republic Pictures agreed to pay Steve for the rights to make ten B movies based on Red Ryder. Fred Harman, meanwhile, was cranking out the strip in Steve's back room for $60 a week." That sum may have been more, it must be said, than what Harman could have been earning by himself with Bronc Peeler.
Once DuBois began writing comic strips directly for Slesinger, his pay was, in Ziemann's words, "significantly better" than what he had earned in 1936, "enabling DuBois to rent an apartment in Brooklyn and send for his family." According to Ziemann, DuBois met Oskar Lebeck while on "an errand" from Slesinger's office, probably in 1939. "A friendship developed, and Lebeck offered DuBois full-time work as a freelancer for Whitman."
At first that freelance work involved more than comic books. Three juvenile novels, all published by Whitman in February 1941, bore the names of Lebeck and DuBois as coauthors. "Lebeck originated the characters," according to Ziemann, "but DuBois composed the plot and did the actual writing." As with DuBois's reworking of a Tom Mix Big Little Book, that "actual writing" involved adapting existing material, since new stories with the characters from all three books—Rex King of the Deep, Stratosphere Jim and His Flying Fortress, and The Hurricane Kids on the Lost Islands—had already appeared in Dell or K.K. comic books by late in 1939. Rex appeared in The Funnies, Stratosphere Jim in Crackajack Funnies, and the Hurricane Kids in Popular Comics. There is no reason to believe that DuBois wrote any of those comic-book features, except possibly some of the later "Hurricane Kids" installments.
If Lebeck and DuBois hoped that their collaboration would lead to bigger things, they were disappointed. The books had no sequels, and the comic-book features all died within a few years. The Hurricane Kids on the Lost Islands, like the other two books—and, for that matter, like DuBois's very first novel, The Lone Ranger—is pure pulp. Its college-freshman twin heroes, indistinguishable except by hair color, encounter dinosaurs, cavemen, pirates, sea monsters, bloodthirsty giant gorillas, and "Zulu" warriors in a little more than two hundred pages heavily illustrated by Bill Ely, who drew "The Hurricane Kids" for Popular Comics. There is lots of action, much of it involving gunfire and violent death, but nothing that invites emotional investment by anyone more than twelve years old.
Although DuBois wrote a total of eight juvenile novels published in 1940–42, the increasingly popular comic books demanded more and more of his time. Thanks to Lebeck, DuBois told a correspondent around 1962,
I began writing comic book scripts full time. Oskar was a man of immense drive and had a way of developing the best ability and the fervent loyalty of the artists and writers who worked under him. My work load grew so heavy that I had to dictate scripts to my wife while she typed. Rush orders were often telephoned late in the day from Whitman Co.'s office in New York to our Brooklyn apartment. One night I worked with my wife Mary till 4:00 A.M., and rolled onto the bed fully dressed while she went out to the subway with the finished script to mail it in Grand Central Station so it would be delivered at the office that morning!"
DuBois was a paradigmatic comic-book writer in that he turned out huge numbers of scripts with great speed, rarely received published credit for his work, and had no control at all over how his scripts were illustrated, often not even knowing the name of the artist involved. Late in his career, he described his working methods:
Well, a workman like me gets his assignment—a single script or a whole book—from his editors. The characters are already established and so is the background or setting in a general way. I think up the best plot (or plots) I can, and write the plot details out in longhand. Usually I let the thing "settle" overnight, if it's a 12 or 15 page story. In the morning I go over it, visualizing each picture that is to be drawn by the artist. I mark off my longhand rough story in pages of the comic book. Then I put an original and two carbons in my typewriter, and start writing. Each panel is handled the same way; that is, I first describe the picture the artist is to draw, in detail, which includes color, action, expression, background, angle of view, etc. Then I write out the dialogue for [balloons], and finally I write the caption or narrative line.
I have nothing to do with choosing the artist, as he is chosen by the Art Editor. The artist is free to use or not to use my instructions for each panel. Sometimes he changes the picture I described—to suit his own idea or that of the Art Editor. Usually, though, the art department follows my script fairly close. The Script Editor reviews and approves or changes the dialogue and the captions I have written; usually the changes are few and minor.
DuBois kept meticulous records of his work. His account books dated back to the early 1940s, but he had begun writing comic-book stories before the earliest entries. DuBois was able to identify some earlier stories that he wrote, copyright registrations identify him as the author of others, and in still other instances the tone and shape of a story make it recognizable as his. It is clear from a list based on the account books, and from the stories that predate the stories on the list, that Lebeck did indeed work DuBois very hard—and not as any sort of specialist in cowboys, because western stories played a small part in the Dell comic books of the early forties.
Lebeck valued DuBois not just because he was productive and versatile—most of his earliest stories were in the talking-animal vein, in contrast to the adventure stories he had written before—but also because he was highly professional in a new business that was heavily burdened with clumsy amateurs. There is rarely a hint, in the stories based on DuBois scripts, that his panel breakdowns—that is, how the action was divided into individual drawings—hogtied his illustrators by cramming too much action into some panels and leaving others effectively empty. Likewise, words and pictures are usually in balance within each panel. There is almost never an overload of dialogue, whereas awkwardness of that sort is all too frequently visible in stories written for Whitman's titles by other, mostly anonymous writers, not to mention the stories in other publishers' comic books. (Roger Armstrong may have had such deficient scripts in mind when he complained of getting "un-drawable" scripts by New York writers.) DuBois's talents were of a kind that everyone working regularly in comic books in the early 1940s needed but few had.
DuBois contributed to Animal Comics from the second issue on, by writing all of the "Uncle Wiggily" stories; he wrote a few stories for Fairy Tale Parade, also starting with the second issue; he wrote several backup features for Looney Tunes and Merrie Melodies Comics, even though most of that title's content originated on the West Coast; and he wrote extensively for early issues of Our Gang Comics and New Funnies. DuBois's versatility served him especially well when he was writing for Our Gang, since he was called upon to write not just talking-animal stories (the first six "Tom and Jerry" stories, "Barney Bear" in the first issue), but also "King," a deadly serious dog story that calls up thoughts of Jack London.
"King" was copyrighted by Oskar Lebeck, as was "Flip and Dip," for which DuBois wrote the first seven installments. The "Flip and Dip" stories, completely unlike "King," were raucously and even brutally comic, as if The Katzenjammer Kids or some other early twentieth-century comic strip had been superimposed on an ape family in the jungle. "Flip and Dip" survived for many years, but "King" expired after only five issues, tilting Our Gang's tone toward the uniformly comic—with the notable exception of the title feature.
Many early comic books resembled Sunday comics sections not just in their heavy use of reprinted Sunday pages, but also in their mixture of different types of comics—adventure, humor, domestic drama—that appealed to children of different ages and to both boys and girls. New Funnies was a particularly striking example, even after its focus shifted to talking animals with the title change from The Funnies. In the last half of 1942, the title's regular features included—besides stories with the Walter Lantz characters Oswald the Rabbit and Andy Panda, and Johnny Gruelle's talking dolls Raggedy Ann and Andy—"Young Hawk," about Indian boys; and "Keeto the Jungle Boy," a knockoff of Tarzan and Mowgli inherited from the recently deceased Crackajack Funnies. Over time, and especially as original material displaced the reprinted comic strips, most of these anthology titles became more uniform in the kind of comics they included. On the evidence of the comic books themselves, however, assembling a satisfactory lineup was never easy.
Our Gang's early issues, like the early issues of other comic books with talking animals, revealed just how tricky it could be to present animated-cartoon characters in the new format. There could be no satisfactory equivalent on the comic-book page for the strenuous violence in the Tom and Jerry cartoons. There was instead, in the comic book, mortal conflict, grim in outline and almost as grim on the page: the mouse seeking food, the cat trying to kill and eat the mouse.
DuBois found part of a solution in the very first story, by giving Jerry, the mouse, a companion called Tuffy, who wore a diaper so that the two mice could be distinguished easily. Such a character was added to the animated cartoons years later, under the name Nibbles. In DuBois's sixth and last "Tom and Jerry" story, illustrated by John Stanley for Our Gang no. 6, July–August 1943, the mice are in a bathroom, and the story presents them not as a cat's intended prey, but as two curious children. Among the bathroom's accessories is a box of "Kelly's Liver Pills." A toilet is on plain view—a great rarity in cartoons and comic books—and much of the story's comedy centers on its tank.
For New Funnies, DuBois wrote the "Andy Panda" and "Oswald the Rabbit" stories for the first couple of years (that is, assuming he wrote the stories preceding the earliest entries in his account books). He also wrote "Raggedy Ann & Andy" for New Funnies until that feature was spun off into its own monthly comic book in 1946, with DuBois continuing to write it.
New Funnies was, if anything, an even more problematic title than Our Gang. Not only did it present the usual difficulties involved in translating animated characters to the comic-book page, but the principal Lantz characters, Andy Panda and Oswald—the two Lantz characters for whom DuBois wrote scripts—were much less distinct in appearance and mannerisms than competing studios' characters, like Bugs Bunny and Donald Duck. Although Andy Panda and Oswald appeared side by side in New Funnies, Andy had actually succeeded Oswald as Lantz's cartoon star. When New Funnies got under way in 1942, Lantz had made no Oswald cartoons since 1938.
In "Tom and Jerry," in Our Gang Comics no. 6, July-August 1943, written by Gaylord DuBois and illustrated by John Stanley, a toilet was used as a prop, most unusually for a comic-book story.
Andy Panda was originally, in a few cartoons released in 1939–42, a bratty child modeled on Fanny Brice's radio character Baby Snooks, and living with his parents in the wild. The character's appeal depended almost entirely on giant pandas' novelty—the first live panda had been brought to the United States only in 1936—and their easy adorability. In 1942 Lantz abandoned the original version in favor of an older, bourgeois Andy. That Andy appeared in occasional cartoons throughout the 1940s, but he was always amorphous. DuBois struggled with the first version of Andy, writing a multi-issue continuity that took a freakish "talking panda" from the Asian jungle to life as an American movie star, complete with a Hollywood apartment. Andy was portrayed as a bright child, not a comic figure at all.
The "Andy Panda" and "Raggedy Ann and Andy" stories that DuBois wrote for New Funnies were illustrated by George Kerr, who was—like Arthur Jameson of Fairy Tale Parade—a much older man than most of his comic-book contemporaries. Kerr was born in 1870, and so in 1942 was already in his early seventies. He was for many years a newspaper artist, for the Hearst Sunday supplement American Weekly, and he also illustrated children's books by L. Frank Baum and Thornton W. Burgess early in the twentieth century. There is no comedy in Kerr's drawings for DuBois's scripts (and for stories in other Lebeck-edited titles). His drawings are instead airy and light. Many of his characters seem to be dancing on the page, their poses highly stylized, their proportions and sizes changing in a manner that seems more dreamlike than arbitrary. The famed newspaper cartoonist Rube Goldberg, who thought of Kerr as a "great, great artist," said of him: "The thing that I marvel at is he was such a big he-man, and he was doing these beautiful little pixies, delicate stuff."
The "Raggedy Ann" stories, like the Johnny Gruelle books on which they were based—and, for that matter, like the books that Kerr illustrated early in the century and the books that Oskar Lebeck wrote and illustrated in the 1930s—were intended for an audience of very young children whose parents would most likely be reading the stories to them aloud. It was in the "Raggedy Ann" stories, and perhaps most of all in an Oswald the Rabbit one-shot published early in 1943 as a sort of Easter special—a comic book also intended for a very young audience—that DuBois first showed what would make him so distinctive as a comic-book writer.
In that sixty-four-page Oswald story, illustrated by Ken Hultgren, a rabbit community surmounts a cascade of crises after first taking refuge from a flood in a curious "church" that seems to have no religious purpose. The rabbits' overriding fear is not that they will be drowned or left homeless, but that there will be no Easter baskets ready for the young bunnies. With Oswald in the lead, the rabbits enlist the aid of a host of other gentle animals and birds—ducks, beavers, pack rats—and all ends well on Easter morning.
There is in this story, as in other stories drawn from DuBois's scripts, none of the feverish, arbitrary activity that dominates the original material in other early comic books. Instead, his characters calmly work through practical problems, including some created by villains who themselves behave like naughty children. There are passages in DuBois's juvenile novels, such as The Hurricane Kids, that are generally comparable—passages that sometimes abound in technical terms that would be unfamiliar to most children—but those are interludes in otherwise purplish stories with villains who are mortal threats. The emotional temperature is much lower in the stories based on DuBois's scripts for younger children.
It is tempting to read moral lessons into those stories, especially since DuBois graduated from an Episcopal seminary and in later years served as a lay pastor for small churches without an ordained minister, but there is unforced sweetness rather than preaching in his stories. The stories' challenges are of course those that might confront sentient dolls or talking animals—that is, wholly fantastic characters—and so any connection to reality had to arise not from those challenges but from how the characters responded to them. DuBois was at pains to make what happened in his stories plausible on the stories' own terms. In that way, if in no other, he resembled Carl Barks.
For Barks, plausibility was only the grounding of his stories, the solid foundation that freed him to do much more through his dialogue and his drawings, and especially by knitting those elements together to form an artistic whole that was uniquely "comic book." For DuBois, as a script writer who had no control over the realization of his ideas in drawings, plausibility could only be an end in itself. But the kind of plausibility he established, in which sympathetic characters support one another as they master the difficulties confronting them, would prove to be highly adaptable to stories intended for an older audience—stories of the kind that DuBois began writing for the new Roy Rogers Comics by 1944. That was when he left behind most of the work he had been doing for New Funnies and Our Gang Comics and returned to writing western and adventure stories. Many more stories for older readers followed.
## 11
# The Observer: John Stanley
Gaylord DuBois had abandoned the unworkable movie-star conceit for Andy Panda when he wrote the stories for the mid-1943 issues of New Funnies, but the new installments presented Andy as an adorable wanderer and were hopelessly sticky. Finally, in New Funnies no. 79, September 1943, Andy became a middle-class homeowner generally similar to the animated-cartoon version, and he acquired a foil in the form of a belligerent chick, named Charlie, who in the next issue became a full-grown rooster. Oswald the Rabbit had already acquired a companion of his own, a bear—originally a talking stuffed toy—named Toby.
The talking-animal stars of New Funnies, because they were so indistinct, desperately needed such foils, and as a result they soon had domestic arrangements unlike anything in the animated cartoons or in most other comic books. Andy shared a house with Charlie Chicken, Oswald with Toby. None of those characters ever held jobs, except for the convenience of a particular story. Their ages were variable, too—presented as fully adult in one story, they might be children of seven or eight (that is, the same age as their typical readers) in another. Their circumstances were far more fluid than those of Disney characters like Donald Duck and Mickey Mouse, who in the comic books were always adults. In that respect Andy and Oswald echoed Joel Chandler Harris's Uncle Remus stories far more than Walt Kelly's "Albert and Pogo" stories ever did. Brer Rabbit and the other animals in Uncle Remus's tales were in effect furred people in some stories and talking animals in others; they were even less anchored to a particular identity than the Lantz characters.
The "Andy Panda" stories that DuBois wrote (according to his account books) and that John Stanley illustrated (if the resemblance to his later work can be trusted) feel like hybrids, with quirky touches that are not typical of DuBois's scripts and that Stanley may have added with Lebeck's encouragement. So rough-and-ready were the Lebeck-edited comic books in the pinched war years—when manpower and supplies of every kind, paper especially, were in short supply—that the coloring is often crude beyond excuse and the printing beneath even low comic-book standards. Several hands may be visible in the drawings for a single story, and since the stories were rarely signed, identifying authorship can be a little like a burlesque equivalent of assigning anonymous work to Italian Renaissance artists through telltale details like the shape of an ear. But taking as a guide stories that are unquestionably all Stanley's in writing and drawing, for the early issues of Little Lulu (1945–46), and working forward in New Funnies and Our Gang from 1943, certain stories stand out as his, and reveal a sensibility that was strikingly different from the comic-book norm.
By 1944 Walt Kelly was no longer an anomaly among Lebeck's artists, a draft-age cartoonist surrounded by older men, some of them gifted but much older, like Arthur Jameson and George Kerr, and some of them lesser talents that had been ground down by years in the pulps and other low-prestige venues. In Stanley, Lebeck found another cartoonist who drew as well as Kelly did, if in a very different style, and he would soon put Stanley to work on comic-book features that fitted him as well as Kelly's features fitted Kelly.
John Patrick Stanley was born on March 22, 1914, the second of the five children of James Stanley and Anna Ahern Stanley, Irish immigrants who lived in New York City's Harlem neighborhood. James was a conductor on the New York subway and then, by 1930, a ticket agent, by which time the family had moved to 2907 Heath Avenue in the Bronx. James Stanley's occupation may have inspired three of his son's stories for the Lebeck-edited comic books New Funnies and Little Lulu. In one New Funnies story, Woody Woodpecker is an apprentice trolley conductor, learning the ropes from a cheerfully sadistic instructor; in another, Li'l Eight Ball rides a subway train that is inexplicably luxurious. In Little Lulu, Lulu tells a story in which she travels around the world—by trolley.
Stanley attended Textile High School in the Chelsea neighborhood in Manhattan, a vocational school that embraced classes in design as well as manufacturing. A classmate, Gill Fox, himself to become a comic-book artist, remembered Stanley as extraordinarily talented: "You wouldn't believe how good his work was at 16—as good as most professionals today." In June 1932, Stanley received the Saint-Gaudens Medal, given to the graduating student from each New York City high school who had completed an elective art course with the greatest distinction.
In a 1976 profile published for Stanley's first appearance at a comic-book convention, Donald Phelps wrote of him: "His high school artistry was good enough for him to win a two year scholarship to the New York School of Art." That was probably the New York School of Fine and Applied Arts, which offered courses in commercial illustration; in the 1930s it occupied a five-story building on the southwest corner of 80th and Broadway. The school, now known as Parsons The New School for Design, has no record of Stanley's enrollment, but early records are incomplete. "He felt very uncomfortable there," Phelps wrote, "as most of his classmates were well-to-do and, being a city kid, he did not really blend in." After two years, Stanley found a job with the Max Fleischer studio, which was producing animated cartoons at 1600 Broadway, about thirty blocks south of the New York School.
"I worked for Max Fleischer for about a year," Stanley said in 1976 at the Newcon comics convention in Boston (where he and Carl Barks appeared together for the first and only time). That would have been in 1934–35. "I started out as an opaquer and worked from opaques to inks to being an in-betweener." At the Fleischer studio, as at other animation studios of the 1930s, "inkers" traced all the pencil drawings made by the animators and their assistants onto sheets of celluloid. An opaquer, in animation parlance, was the person who filled in the inked tracings with paint. The celluloids were then photographed, a frame or two at a time, to produce the illusion of movement when the film was projected.
At the Fleischer studio, impatient young men who tried to climb the promotion ladder too rapidly risked collisions with resentful superiors, but Stanley's exceptional talent seems to have spared him from reprisals. His promotion to inbetweener, after just a year or so as an opaquer and then an inker, was announced in the August 1935 issue of the Fleischer studio's mimeographed in-house magazine, Fleischer's Animated News. Three months earlier, in the May issue, Stanley had contributed a page of staff caricatures titled "But for the Grace of God (What They Might Have Been)." The caricatures are less than flattering: the animator Dave Tendlar is a peddler of overripe fish (and his name is misspelled "Tendler"), the camera operator Frank Paiker is selling naughty postcards, and the animator Tom Johnson is a piano mover who fobs off the heavy lifting on someone else.
A promotion to assistant animator would have been Stanley's next step, but Hal Horne had begun publishing the third version of Mickey Mouse Magazine by then, and Stanley left the Fleischer studio to join Horne's staff by sometime early in 1936. "Besides drawing characters and illustrations for the magazines," Donald Phelps wrote, "Stanley also drew some covers." The job with Horne could not have lasted more than six to eight months, since Horne surrendered the magazine to Kay Kamen with the July 1936 issue and Stanley left Mickey Mouse Magazine when Horne did. Stanley said, though, that he was on the staff long enough that he "shook hands with Roy Disney once." Stanley next worked for Kamen until late in 1936 if not longer, drawing characters for Disney merchandise items that included one of the Mickey Mouse watches. "After that," Stanley said in 1976, "I free-lanced for a while. I didn't want to be a cartoonist or a writer: I sort of backed into it. Those were hard times and I couldn't find the kind of work I wanted as a graphic artist."
From September to December 1937, Stanley attended evening classes on etching at the Art Students League. "By his own admission," Phelps wrote, "he didn't learn much about [etching] but he did meet a group of 'good time guys' who believed in revelry and a whole lot of tipping the elbow . . . it was a more personally satisfying experience than his stint at the New York School of Art had been." It may have been while he was at the Art Students League—which was, like the New York School of Fine and Applied Arts, famous and prestigious—that Stanley became an alcoholic or something close to it, and, as a corollary, adopted a contemptuously casual attitude toward his work. That was a defensive gesture, surely, since from the beginning, with the Fleischer cartoons and Mickey Mouse Magazine, he worked on unambiguously commercial products that enjoyed none of the prestige of the paintings, drawings, and prints made by his schools' many famous alumni.
Stanley was still living at home with his parents on Heath Avenue, at the age of twenty-three, and he would continue to live there for the next several years. It seems likely that it was the steady income provided by comic books—the least prestigious work he had done, work he did not want to do—that finally permitted him to move to a home of his own. Stanley said in a 1965 newspaper interview that his comic-book career began more or less by accident: "I just drifted in: I was a commercial artist and letterer in New York when a friend who was in the comic book business asked for some help. I did the art work, but was unhappy with the story and suggested I write my own." He completed that account at the 1976 convention: "So they [presumably Oskar Lebeck] said, 'All right . . . try your hand at it.' They let you do anything at Western in those days."
Stanley later claimed to have saved none of his comic-book work. He told Glenn Bray in an undated letter in the early 1970s: "Somehow I never managed to hold on to a single copy of Little Lulu—or any other comic I've done." He even pretended not to remember the titles of comic books he had originated, although he could be much more specific when his authorship was questioned. Neither was he indifferent to praise, at least when it came from his fellow cartoonists: "I wanted guys like Kelly to say, 'Hey you're great[,] man!" But for the most part he maintained a shell of indifference toward the work to which he devoted most of his adult life.
Stanley's younger brother, James, was a second lieutenant in the army air force and died in combat in 1944, but, like Walt Kelly, Stanley escaped the draft. He was rejected for military service in 1941. "I believe he suffered from a slight touch of tuberculosis, which kept him out of the service during the war," his colleague Dan Noonan said—although Noonan's knowledge was secondhand, since he didn't meet Stanley until after the war—"and it was during that time that he got a job with Lebeck doing artwork. He began doing some writing, too, and he submitted a couple of ideas. I think Lebeck sensed his talent instantly." In keeping with his temperament, Stanley himself spoke of Lebeck with a certain cool respect, saying that he was a "good editor in that he gave a free rein to the artists."
Nothing in the Dell comic books suggests Stanley's involvement before 1943, although considering the lead time typically involved he may have started working for Lebeck by late in 1942. The earliest stories that can be identified with confidence as illustrated by Stanley, in late-1943 issues of New Funnies ("Andy Panda") and Our Gang Comics ("Tom and Jerry"), are much better drawn than was the norm for the Dell talking-animal titles that originated in New York, Kelly's work aside. The typical New Funnies story of 1942–44, when the rising demand for comic books was colliding with the loss of cartoonists to the military draft, is not just simply but carelessly drawn. Stanley's drawings are not more detailed—they are lean in much the way that Oskar Lebeck's were—but the draftsmanship is vastly superior.
It was when he began writing his own stories that Stanley became a unique comic-book presence. Speaking at the 1976 Boston convention, Stanley contrasted his methods with those of Carl Barks: "I made no synopsis. I started from the very first panel on the page without having the faintest notion of how it would turn out. And I just hoped for the best, that's all. Generally, it worked out." Many of Stanley's stories for New Funnies read like highly spontaneous sketches, the sort of thing he tossed off quickly and wrapped up however he could. He was indifferent to neat resolutions. Other stories in New Funnies are illuminating by contrast: the "Homer Pigeon" stories, starring another Lantz character as middle-class as Andy Panda, are irreproachably tidy, as narrative and in drawings (most often by the West Coast's Veve Risto), and hopelessly dull.
"Andy Panda" in New Funnies no. 92, October 1944, is one of the first stories that seem to be totally Stanley's, script and drawings, in a way that anticipates his stories for Little Lulu. Andy in this story pursues a goal—to bake a prizewinning cake—with the blinkered obsessiveness that great comedians have always found a fertile resource. A cookbook cannot be found? Very well, Andy will use a recipe he finds in 1001 Chemistry Problems: "Good thing we got the putty!" Does that recipe call for baking his "cake" for thirty minutes? "We'll leave it in for an hour and it'll be twice as good!" So focused is Andy on completing the cake that when he realizes it lacks icing, he concerns himself only with how the cake should look, and so he ices the cake with a can of paint.
The quintessential Stanley stories may be those, like this early example, in which one or more characters pursue their own ruin—or the incidental ruin of bystanders—with a mad single-mindedness. Sometimes things turn out well for his characters, as in this story, but it is always obvious that disaster was at least as likely, and not infrequently disaster arrives for those bystanders. Andy's victory in the baking contest comes at the price of crushing the hopes of all the legitimate contestants. The story falters on a question of simple plausibility—victory depends on Andy's cake weighing a great deal, weight that is evident only sporadically when Andy and Charlie Chicken are handling it—but Stanley soon managed such details better.
By the time he produced that "Andy Panda" story, Stanley was writing and drawing "Woody Woodpecker" stories for New Funnies, too. That character had been, in a starring role in the animated cartoons and a supporting role in the comic books, a violently unhinged maniac. In Stanley's stories he was much calmer. He was easily imaginable in a role like the one assigned to Andy Panda in the cake-baking contest—that is, as the kind of character who does unreasonable things in a perfectly reasonable manner. He was also well suited to comedy of a sort that would blossom in the Lulu comic books, as when Woody, in New Funnies no. 88, June 1944, must struggle with a character even more unreasonable than he, and vicious to boot. Worse, the vicious character is a baby and so cannot be repelled by any normal defense.
"Andy Panda," in New Funnies no. 92, October 1944, was one of the first comic-book stories written and illustrated by John Stanley.
Stanley's drawings for the stories he wrote are simpler than Barks's and Kelly's—not less important, exactly, but they serve unmistakably as a superstructure for the words and ideas that give the stories their forward thrust. Barks's and Kelly's stories are more organically comic-book stories than Stanley's, with dialogue and drawings that more strongly resist being pulled apart. And yet, because Stanley could draw so well, he could communicate his intentions clearly even through very rough drawings. He was thus exceptionally well qualified to be a comic-book writer who did not tell his story in words, like Gaylord DuBois, but sketched each story in rough form, so that he controlled the staging within each panel and the general appearance and the attitudes of the characters.
In any case, Stanley's characters were not planted as firmly at the center of his interest as the characters in Barks's and Kelly's stories were at the center of theirs. Stanley was concerned with broad reaches of foolish and destructive human behavior; he was more purely a satirist than his comic-book colleagues. Barks and Kelly brought nuances to the drawings of their characters, making them more distinct as individuals, that would have detracted from the peculiar strengths of Stanley's stories.
Stanley demonstrated in the "Woody Woodpecker" stories his rapidly increasing technical security, as he dispensed with dialogue for a page or more and on rare occasions for most of a story—the comic-book equivalent, given the usual importance of knitting drawings and dialogue into an indivisible unit, of working without a net. There are strong echoes, on some of these pages, of the New Yorker's dry, understated humor, and Stanley did sell one multidrawing gag to that magazine; it was published in the March 15, 1947, issue. By then Stanley was also writing many comic-book stories he did not illustrate, his work unsigned but always distinguishable by its detached, ironic tone. There is, for instance, "Oswald the Rabbit" in New Funnies no. 119, January 1947, in which Oswald, walking on stilts that lift him far above the street, has his pocket picked. The gag is executed not just with a necessary respect for plausibility but also with sangfroid worthy of Buster Keaton. First Oswald's trust is abused by the pickpocket, and then a policeman is more than skeptical when the indignant rabbit reports the theft.
That script and many other Stanley scripts were illustrated by Dan Gormley, a Whitman standby whose relaxed, cheerful drawings warmed up stories that might have seemed a little chilly otherwise. Stanley borrowed Gormley's name for a taxidermist, an art gallery's director, a pet-shop owner, and a postman, among other characters. (Walt Kelly, for his part, dubbed an island Gormley's Folly in a couple of "Our Gang" stories.) Gormley, in Dan Noonan's recollection, made only the pencil drawings, which were inked and lettered by a woman named Anahid Dinkjian: "She was a very good inker, and she inked almost precisely what you had laid down there." This was an early example at Western of a multipart division of labor that was to become more common there, as at other publishers.
In "Oswald the Rabbit," in New Funnies no. 119, January 1947, written by John Stanley and illustrated by Dan Gormley, Oswald learns that even being on stilts is no protection against a pickpocket.
By the time the Oswald-on-stilts story appeared, Stanley had been firmly attached for a couple of years to a character that would be almost entirely his responsibility for another dozen. "That's the way it went for about a year," he said of his initial freelance work for Western in 1943–44 on Our Gang Comics and New Funnies, "then Lulu came along and the rest is history."
When Oskar Lebeck assigned him to the first Little Lulu comic book, "I'm sure it was due to no special form of brilliance that he thought I'd lend to it," Stanley told Donald Phelps. "It could have been handed to Dan Noonan [actually not, because Noonan had not begun working for Western yet], Kelly or anyone else. I just happened to be available at the time." And yet the fit between character and cartoonist is so tight that it is hard to believe that such perfect casting was accidental.
The character Lulu was born early in 1935. The Saturday Evening Post was losing Carl Anderson's popular weekly panel cartoon, "Henry," starring a mute, hairless boy, to King Features Syndicate. An editor at the Post asked a regular contributor to the magazine, Marjorie Henderson Buell—she signed her cartoons "Marge"—for help in coming up with a replacement for "Henry." The Post wanted another child character, but a girl this time. "The editors noodled out the name," Buell told an interviewer more than thirty years later. "I was too busy thinking about how she'd look and what she'd do." The name the editors "noodled out" was "Little Lulu." For a design, Buell essentially added a skirt and corkscrew curls to Henry's shoe-button eyes, upturned nose, and babyish jawline.
The new character was a devilish child, heiress to a long line of such characters that in newspaper comics extended back to the Yellow Kid, Buster Brown, and the Katzenjammer Kids. Like theirs, her mischief, as displayed in a weekly panel, often bordered on the malicious. Lulu first appeared in the Post for February 23, 1935, and ran there until the issue of December 30, 1944. By that time, she was also appearing in animated cartoons for Paramount and in advertisements for Kleenex tissues. "The Post didn't want me to go into large-scale business with Lulu," Buell said, "but I wanted to see what she could do in all forms. We parted amicably."
John Stanley, Oskar Lebeck's neighbor, plays the guitar at a mid-1940s pool party at the Lebeck home in Croton-on-Hudson, in the Hudson Valley. To Stanley's right is Anne DeStefano, Lebeck's secretary. The woman behind Stanley is Jane Werner (later Jane Werner Watson), author and editor of many of Western's children's books. Courtesy of Letty Lebeck Edes.
Buell's first contract with Western Printing & Lithographing Company covered inexpensive books (ten to fifteen cents retail) but not comic books specifically. It was dated March 15, 1944, and provided for a royalty of a half cent on each copy sold. Seven months later, in a contract dated October 23, 1944—a contract that perhaps reflected the growing popularity of comic books—Buell gave Western "the right to publish in printed form, in color or in black-and-white, comics magazines or comics [sic] books known as one-shots" featuring Little Lulu. In other words, the Lulu comic books were limited to the Dell Four Color series, with no commitment on Western's part to publish on a regular schedule. Buell was to receive five hundred dollars as a royalty on the first three hundred thousand copies printed, and a higher royalty of a quarter cent on each copy for print runs above that figure. She got an advance on her royalties of five thousand dollars—a significant vote of confidence in Little Lulu, who had appeared in animated cartoons for the first time only the preceding January—and was to get an advance of the same size in the contract's second year.
John Stanley (foreground) and Oskar Lebeck around 1950. Courtesy of Letty Lebeck Edes.
The world of comic-character licensing was a small one: William C. Erskine, who negotiated Buell's contracts with Western and managed the licensing of Little Lulu, had worked in the 1930s for Kay Kamen, Disney's licensing maestro—he was by 1938 vice president of Kay Kamen Ltd.—and had thus received what was the best schooling in character merchandising that anyone could get, except possibly under Stephen Slesinger. Erskine also licensed other famous characters—Raggedy Ann and Andy and Uncle Wiggily—for products that included Dell comic books.
Buell maintained a proprietary interest in her character as Erskine signed up more licensees. Gordon Sheehan, who animated on the Little Lulu cartoons for Famous Studios—the successor to the Fleischer studio, where John Stanley worked in the mid-1930s—remembered that Buell came to the studio in Manhattan occasionally; "She seemed to get quite a big kick out of seeing her little character animated. . . . She might have had a few little friendly words of advice, but I don't think she tried to change things very much." After she signed her contract with Western, and Oskar Lebeck chose Stanley to write and draw the new comic book, Stanley went to Philadelphia to meet her.
The official publication date of the first Little Lulu comic book, Four Color no. 74, was May 14, 1945, almost five months after Lulu's last appearance in the Post. That first issue was written and drawn entirely by Stanley, who also wrote and illustrated the second and third issues. The new comic book was an immediate success: Marge's Little Lulu began appearing on a de facto bimonthly schedule in 1946, even though it did not become officially a bimonthly for two more years, under an amended agreement.
As of the fourth issue (Four Color no. 115, 1946), Stanley began to share the drawing of the feature with two other cartoonists, Irving Tripp and Tripp's friend Charles Hedinger, both of whom had just returned to the art staff at Western's Poughkeepsie plant after military service. Tripp remembered that they "wanted something a little different to do, and we went to Oskar Lebeck . . . and he said he'd see what he could come up with."
Western sent Buell proofs for each issue of the comic book, and she had fifteen days to disapprove any of the contents. She noticed the change in drawing style when she saw the proofs for the fourth issue. She wrote to Erskine, who read part of her letter to Oskar Lebeck over the phone. Erskine wrote to Buell on May 14, 1946, reporting on a conversation that is rare specific evidence of Lebeck's diplomatic skills:
He told me that in order to keep pace with the schedule two artists were added on this job to assist and work under the supervision of John Stanley whom you met in Philadelphia one day. Mr. Lebeck said that outside of members of his own art department no one but you had noticed any change, and he said that all of your comments were to the point and the three men are now working efficiently and effectively, and he is sure that you are going to be pleased with the result. . . . Lebeck did not advise me of the change in advance because he did not want to prejudice our comments, but he is really pleased that you noticed the differences however slight they may be.
Hedinger made the pencil drawings for the Little Lulu stories for the next few years, and Tripp finished them in ink. Tripp eventually assumed both penciling and inking duties, with help on the lettering and the backgrounds from two other artists. Stanley drew only the covers of Little Lulu and no stories, with rare exceptions, but, as he told Maggie Thompson, he remained in control: "For the 13 or 14 years I did them, all the stories, gags, etc., good and bad, were written entirely by me. The stories were done in storyboard form . . . and sent to the Poughkeepsie plant to be copied on large sheets of Bristol board, inked and lettered."
The first Little Lulu comic book, Four Color no. 74 (1945), was written and drawn completely by John Stanley. © 1945, 1973, Marjorie Henderson Buell.
Those "storyboards" were rough sketches broken down into the panels for each page, drawn on what Tripp called "regular typewriter paper." Stanley's stories for Little Lulu always had identifying characteristics that pointed toward his authorship, even though his work was almost never signed. Maggie Thompson listed a few of them: "the character screaming 'YOW!' in fury or fright, frequent use of overweight characters, assorted sound effects peculiar to his work, the slapstick situation of disaster in the wake of an unnoticing causative figure."
Tripp spoke of Stanley's scripts warmly, saying they were "beautiful to work with. . . . I mean he had expressions! Everything was so clear and precise, you didn't have to go over it with a magnifying glass or ask questions. . . . You knew." Tripp remembered seeing Stanley at the Poughkeepsie plant only "three or four times," the first time soon after Tripp and Hedinger began drawing the Lulu stories: "I remember John saying, 'the inker can make or break a story,' so it was kind of encouraging. He complimented me on what I was doing and encouraged me to keep with it. He came to Poughkeepsie with Oscar [sic] Lebeck on occasion, but mostly they were just having a good time."
Stanley said in later years, however, that he disliked the published drawings made by the other cartoonists. He told Donald Phelps: "I complained constantly trying to get a change of artist but to no avail. It was too static for me. I would rather have had faster movement." In a letter written about ten years after he spoke to Phelps, Stanley was a little more charitable, but also more revealing of the pride he took in his work. He remembered meeting Tripp—whom he dismissed as a "copiest [sic]"—"maybe two or three times at the Poughkeepsie plant. . . . I never saw Tripp in the New York office. . . . He was sent my storyboards and enlarged and finished them. He did a good job, but he never added, subtracted or changed anything whatever. I wouldn't allow any tampering with my work, and LeBeck [sic], the boss, was totally supportive of this."
That was not a typical comic-book situation—either in Stanley's determination to resist tampering or in Lebeck's support for his artist. Its atypicality was a major reason that Little Lulu became an extraordinary comic book.
## 12
# "I Am a Backwoods Bumpkin"
"I seldom reminisce," Carl Barks wrote in 1975 to an admirer of his stories with the Disney ducks. "Ninety percent of my life has been spent in such drudgery that I studiously avoid recalling anything earlier than ten minutes ago." That was comic hyperbole, and Barks could at times slip into extended riffs in the same vein—self-deprecation with overtones of resentment and just a little self-pity, as in this 1968 letter to the magazine writer Dick Blackburn:
Frankly, I was incapable of writing much in the way of hidden "messages" into my stories. In personal background I am a backwoods bumpkin. My education consists of eight grades in a one-room Oregon schoolhouse. I've traveled nowhere, seen few movies or plays, read few books (e.g. Zane Greys and Perry Masons). The background research evident in my comic book stories was laboriously dug out of my shelves of National Geographics and Encyclopedia Brittanicas [sic]. I've long been afflicted with partial deafness, so I've missed any oral exchanges with people whose ideas might have broadened my intellect. With such limited knowledge of expression I had to stick to writing my stuff straight.
On the physical plane, however, my experiences during my early years when I worked as a farmhand, mule skinner, psuedo [sic] cowboy, and steel worker, always terribly inefficiently, gave me practical knowledge for complicating many of Donald Duck's problems. The perversity of beasts, machines, and nature I knew by heart.
There was mere exaggeration in such statements, nothing more. When Barks spoke in 1973 about his early years in Oregon, he described a boyhood removed by only a few years from the brute harshness of nineteenth-century pioneer life:
My folks had a little wheat ranch in southern Oregon, about two miles over the California line. My dad was a homesteader and moved up there in the 1880s. He'd been a blacksmith and a teamster and so on, on those big grain ranches around Stockton, in the San Joaquin Valley, quite a number of years, and he had come there out of Missouri, as an orphan at the age of about fourteen. He had had a little apprenticeship as a blacksmith, and so he went to work on those big ranches. He worked there for many years, and when he heard about the homestead land guys could pick up in Oregon, why he went up there and took up a homestead of a hundred and sixty acres. He was very industrious at it, cleared the sagebrush off, and he got enough income out of his hundred and sixty to buy other hundred and sixties around him, so by the time I came along, he had about a square mile of land, all in wheat, and doing all right.
His parents, William Barks and Arminta Johnson, had met as schoolchildren in Missouri. After William made his way west—"I remember him telling about riding on top of the boxcars coming out to California," Carl Barks said—he and Arminta "corresponded with each other a little bit over the years." When her parents died, after what were probably long illnesses, Arminta and William married on November 1, 1897—not in Oregon, for some reason, but in Yreka, in northern California about twenty miles south of the Oregon state line. Both William and Arminta were almost forty years old when they married. Their first son, named Clyde, was born in 1899. Carl Barks was born on March 27, 1901, near the small town of Merrill, Oregon, in a two-room cabin. The cabin became a bunkhouse and laundry room around 1905, after Barks's father built a proper ranch house with a long ell to accommodate a dining room that could seat a dozen or more hired hands at harvest time.
The Barks ranch, as all such western farms are called, was only a couple of miles north of the California line, about five miles west of Merrill and fifteen miles south of Midland. Merrill itself was brand new, laid out in the spring of 1894—the first building was a flour mill—and not incorporated until May 1903, when Carl Barks was two years old. The nearest town of any size, Klamath Falls, was hours away from the Barks farm, behind a team of horses. In Merrill, Barks attended the one-room Lone Pine schoolhouse near the ranch with a dozen or so other students. Even in his nineties, Barks remembered life on the ranch as emotionally parched: "The ranch house was a lonely place with no close neighbors. My parents had little patience with the yearnings of a small boy, both being old enough to be my grandparents, and my slightly older brother had little patience with my 'sissy' fascination with drawing and reading so, other than the farm animals, I had little companionship."
Barks's father tired of wheat ranching, and when Carl was about seven years old the family moved from the ranch to Midland, where William Barks
put up some big corrals and feed lots, and then we used to feed the cattle and bed down the shipping cars whenever these big trail herds would come in from eastern Oregon. . . . Real cowboys would come in with those outfits. You'd see the string of cattle coming through the gap in the hills off to the east, and boy, it was a couple of hours before the tailers came through. . . . Those cowboys were tough fellows. They were used to having one little old blanket roll, rolled in stiff canvas on the back of their saddles, and they'd just lay that out on the barn floor and sleep in that. . . . My brother and I we just worshipped those fellows! And oh, what vulgar-talking men they were
After about two years in Midland, William Barks rented the ranch and the feed lot and moved his family to Santa Rosa, California, north of San Francisco. In Santa Rosa, Barks for the first time attended school with boys his own age. He remembered seeing his first movie there. But the move to Santa Rosa did not work out well. William Barks
was so sure that he had to have some solid income, so he went and spent money to buy a prune orchard, in Santa Rosa. Right away, the price of prunes went down to nothing, and right away the rains stopped up in Oregon, and the wheat ranch didn't pay off . . . . [T]he only income that was coming in was rent from the feed lot, and so hard times began to settle in on us, and my dad, he had a nervous breakdown. . . . In the meantime, my mother had had an operation for cancer . . . she knew she had only a few more years to live. She wanted to get us all back up to Oregon, and back on the ranch, if possible. Well, the lease on the ranch had a couple of more years to run, so we went back to the feed-lot business. For two years we were there, then the ranch lease ran out, and we went back to the ranch, and there my mother died.
Arminta died on November 7, 1916. Barks was fifteen. His schooling ended around the same time. "I got through the eighth grade. . . . I would have loved to have gone on to high school, but it was five miles to the nearest high school, and my hearing at that time had begun to deteriorate a little bit to where it was kind of difficult for me to pick up things in class." Barks traced his hearing loss to a childhood case of the measles.
When he finished school, "I helped my dad there on the ranch for . . . a little more than a year. The war was on, World War I, and kids my age could get five dollars a day, which was fantastic wages, working for the farmers, pitching hay, helping to harvest, just about anything . . . . I worked for wages any time I could get time off from working on the home ranch. When I was seventeen, the war ended, and at that time I was eager to go to San Francisco, to get where I could maybe pursue a line of cartooning."
He had first become seriously interested in cartooning
way back when I was going to school in Santa Rosa. There was a kid sat across the aisle from me in the school . . . and he was a little older than I was, but he had evidently had some correspondence-school training in cartooning, or his parents or somebody had, because he used to amaze me with little old drawings he'd make of Woodrow Wilson or Theodore Roosevelt or somebody like that, little cartoons. I would look at them, and see how he constructed them, and oh, I thought that was the most wonderful thing in the world to draw like that kid could draw.
A few years later, soon after his mother's death, Barks tried to harness his continuing interest in cartooning to formal instruction: "I think I hadn't quite turned sixteen yet when I talked my dad into letting me subscribe" to a mail-order course offered by the Cleveland-based Landon School. He spoke almost sixty years later as if his failure to follow through with the Landon course still made him uneasy: "I got about four of the lessons under my belt when the war work began to get so insistent, I didn't have time to work out those problems. . . . But those [lessons] did help me. And I was always able to look at cartoons in the newspapers, the comic strips, and the feature pages and so on, and get something out of looking at how other guys did their cartoons. So, I had developed a fairly good style; I could draw well enough to get by with quite a lot of things."
Of the comic strips that appealed to him when he was a boy, Barks cited only one in a 1976 interview with Donald Phelps: Winsor McCay's lavish Sunday page Little Nemo in Slumberland. "It came out in the San Francisco Examiner. We'd get that through the mail on our home ranch up in Oregon." Barks always expressed admiration for comic strips, Nemo being the oldest, that were drawn in a more realistic style than that of the Disney short cartoons or comic books. His taste for such drawing turned out to be a critical element in his development as a cartoonist: he was not an exceptionally gifted draftsman capable of a Little Nemo or a later Sunday page like Harold Foster's Prince Valiant, but in drawing comic-book stories he increasingly brought to them the same attention to realistic settings that characterized comic strips of the kind he admired.
In 1978, an inspection of his "morgue" showed that it included panels from Prince Valiant (many of those panels depicted small ships on rough seas), Steve Canyon by Milton Caniff, Jungle Jim by Paul Norris, Captain Easy and Buz Sawyer by Roy Crane, Tarzan by Burne Hogarth, The Lone Ranger by Charles Flanders, and Flash Gordon by Mac Raboy. The work of Crane and Foster was particularly useful, he said, "and any other guy who drew fairly accurately, but at the same time with a lot of freedom." He didn't copy individual panels: "I would just take the general appearance of it." Barks carefully distinguished between the kinds of influence he had felt from other cartoonists. E. C. Segar of Thimble Theatre (starring Popeye the Sailor) was unquestionably a powerful influence in shaping Barks's comedy—the kind of comedy that presents the ridiculous with a straight face—but, as he wrote in 1984, "[n]o doubt my drawing style was more affected by [Roy] Crane's staging and compositions."
Barks arrived in San Francisco in December 1918. He wrote more than fifty years later: "I lived there nearly 2 years (1918–20) and I credit the city with most of what I absorbed in culture and feeling for adventure." He got a job as an errand boy for a print shop, but he could not interest the San Francisco newspapers in hiring him as a cartoonist, and after a year and a half he returned to Oregon. He spent the summer working on the ranch, and then returned to San Francisco to study show-card writing at night school. "I soon found that the lettering came so hard for me, that that wasn't one of my natural talents. I used up all of my money, and went back up to Oregon, and worked on the ranch again," until, he said, "I foolishly got married" to Pearl Turner, who was sixteen or seventeen at the time, on October 1, 1921.
Barks was a bust as a rancher: "[A]s soon as I tried to raise wheat and potatoes and things there on the ranch, the rains stopped. Nothing would grow, so I had to work at something else. My father-in-law [William A. Turner] was an old sawmill man, and he had sold out his own sawmill [which was on Stukel Mountain, roughly midway between Merrill and Klamath Falls], but he had a logging contract with a logging outfit. So, with my young bride, I went up to the logging camp, and I spent the summer [of 1923] with the loggers." When the logging operation closed for the winter late that year, leaving Barks at loose ends, "I remembered people that my folks had known in Midland who were in the oil fields, down at Coalinga [in California's Central Valley], I believe."
The Barkses drove to Coalinga in their secondhand Ford, "a real rattletrap" with doors that wouldn't close. Presumably they brought their infant daughter, Peggy, with them, although Barks never mentioned it. Near the end of his life he spoke of his first marriage and his two daughters—a second daughter, Dorothy, was born in 1924—as if they were all the products of mistakes: "[N]either of us should have gotten married. We did and we had no intention of having any children, but accidentally we got one and then we got another one a little later on."
Carl Barks in San Francisco in 1919. Courtesy of Carl Barks.
Once in California, the Barkses found that the family friends had moved away. They drove to Roseville, "where there was a couple that my wife knew. This guy worked for the railroad, in the car shops. He said, 'Oh, hell, I can get you on at the car shops.' . . . So I went over, and right away I got put on in the car shops. And there I was for over six years, in those darned car shops.
Barks was tall (six feet one and a half inches) and slender (he weighed 180 pounds when he registered for the military draft in February 1942, and probably less in earlier years), and he had worked for most of his life in physically demanding jobs. Now, in the car shops, he had another one: "I started out just swinging a sledge hammer, and common labor, and got put on to heating rivets on a riveting gang, and that was piecework. I was on that for five and a half years or so. God, I was getting sick of that." In the meantime, he was continuing to draw: "By the time 1928, '29 rolled around, I had begun to sell some stuff to [the Calgary Eye-Opener], and also sold a couple of cartoons to Judge magazine. That boosted my ego tremendously." Barks's first and possibly only Judge cartoon appeared in the issue dated November 22, 1930.
Barks worked in Roseville for Pacific Fruit Express, a refrigerated-car system owned jointly by Union Pacific and Southern Pacific, from 1923 to 1929, according to his employment record at the Disney studio (which showed him earning $225 a month in that job). "Work was in the repairing of banged up cars," he wrote in 1980. "I worked in the heavy steel section, which mostly repaired the steel underframes and wheel 'trucks.'"
Pearl "resented the long evenings and weekends I spent at trying to draw cartoons and would have preferred that I lived like the other less ambitious husbands around us," Barks said. The marriage deteriorated: "I was always trying to figure out a comic strip or something I could do. That's what used to irritate my wife at that time, she was perfectly satisfied just to be the wife of a laborer on the railroad, that's all she wanted out of life. I was using our evenings, and all of our spare time, working at this darned stuff, and she would rather have been socializing, and so we gradually got to fighting all the time."
The Barkses separated early in 1930, and, with "fifty dollars and what I could carry on my back," he returned to Oregon for the first time in years. His in-laws ("the salt of the earth, wonderful people") invited him to live with them. His father-in-law got him a job at a factory making wooden boxes, but "the Depression was just going down," and he soon lost that job. He began devoting all his time to drawing cartoons, and soon "I was [selling cartoons and] holding up my end on the groceries and everything there. In the meantime, my mother-in-law and I had gone back down to Roseville on the train and picked up my two kids. . . . My wife had just sort of become the town tramp, so I just figured the kids were better off with my in-laws, and they did, too." According to the 1930 U.S. census, the girls were living with their father and their grandparents in Merrill by April of that year.
Barks later asked that the "town tramp" remark be excised from the transcript of the interview in which he criticized his former wife—she was still alive then—and his accounts of both of his unsuccessful marriages were of course one-sided, but there is no reason to doubt that he spoke honestly about how his marital conflicts appeared to him.
After a year or so of living with his in-laws,
I began thinking, well, this isn't the right thing, I shouldn't be doing this. I'm imposing on these people's hospitality a little too much. And my checks coming in were enough that I felt I could break away. I took the two kids and went over and set up in the little town of Medford, in Oregon, where the climate was a little warmer.
I was getting along fine, [but] one day they didn't come from school . . . . So I went along to see what happened to them. I met one of the neighbors that lived down the line, and she said she saw a car pull up and a man and a woman got out of the car and talked to the girls, and the girls got in. . . . It was my wife and her boyfriend, [they] had come up there and picked them up. She took them back over and left them with my in-laws, I found out afterwards. But I spent some pretty hard days there . . . .
After that, I didn't feel I owed anybody a hell of a lot, because they let me be hurt like I was hurt in those weeks of uncertainty. I've always had a lot of respect and love for my in-laws, but that was a hard thing.
Barks and his second wife eventually took the girls, who by then were teenagers, with them on a road trip in 1938. They drove from California as far east as Louisiana, probably in the last half of August, when Disney cut back operations so its employees could take vacations. But he would otherwise see very little of his daughters and later his grandchildren for the next few decades. When he was drawing the lead story in the most popular children's magazine in the country, there were no children in his life except for the occasional inquisitive neighbor, and Barks did not encourage such visits. Children were distractions.
In 1931, though, his most pressing concern was his own survival:
I took my overcoat down to try to hock it, and nobody wanted an overcoat at the time, it was too warm. But I had a few pieces of stainless steel around that I had bought for the kids and I to eat with, and I got fifty cents for what I had accumulated of that stuff. I went and got a haircut, a hamburger, and a package of cigarettes, for fifty cents. And not only that, but I had some left over to buy some postage stamps. I wrote back to the Eye-Opener and asked them why in hell they hadn't sent me my last check.
## 13
# "Pure Corn" at Disney's
The Calgary Eye-Opener took its name from a satirical Canadian newspaper published early in the twentieth century by a footloose Scottish immigrant named Bob Edwards. He died in Calgary, Alberta, in 1922, and the paper evidently expired the next year. By 1925 the Bob Edwards Publishing Company and the Eye-Opener—its name, at least—had been bought by Harvey Fawcett, one of three brothers who had built a publishing empire in Minneapolis. The Fawcett flagship was Captain Billy's Whiz Bang, described accurately by Time as a "magazine of washroom humor." It took its name from Wilford "Captain Billy" Fawcett, who had launched the Whiz Bang after serving as an army captain in World War I. Harvey Fawcett, after leaving the family company in disgrace—he supposedly took kickbacks from paper suppliers—launched the new Eye-Opener as a magazine of the same kind.
According to the Barks scholar Geoffrey Blum, Barks's first signed cartoon appeared in the Eye-Opener's June 1928 issue. Soon after Barks began contributing to it, the magazine changed owners. Harvey Fawcett died in February 1929 (although his name lingered on the masthead for months afterward), and the Eye-Opener was bought by a contractor named Henry L. Meyers. It was he who summoned Barks to work in Minneapolis.
I spent the summer [of 1931] in Medford, free-lancing, and my checks kept getting a little bigger and a little bigger. I was selling more and more stuff to them, and finally . . . the editor they had in charge back there, he was drinking a little too much, and he got so careless about when he would come to work, and what time the magazine's dummy got pasted together and taken over to the presses, and the assistant editor was also hitting the bottle pretty much. . . . Henry Meyers, he was enough of a businessman [that] he could see things weren't being run right around there, there was too much drinking and playing around and not enough production. So he looked over the list of gag men and decided that hell, I was a hard-working son of a gun, so he sent a telegram to me, asked if I would come back there. I had enough money to send a telegram saying I didn't have enough money to get back there. He sent me money to come back there, and I closed up my affairs very rapidly and gave away the big stack of joke magazines I had. What I could carry in a valise, I carried with me. . . .
I got into Minneapolis in November of 1931. I went to work there with the Eye-Opener. Phil Rolfsen, the editor, had been fired by then, and Ed Sumner was editor. Ed Sumner and I worked together. I was the gag man and illustrator, and he was a wonderful poet, and gag writer himself, but he did have that one fault of drinking a little too much.
Carl Barks caricatured himself for the August 1930 Calgary Eye-Opener, published the year before he left Oregon to join the magazine's staff in Minneapolis. Courtesy of Geoffrey Blum.
Barks's Eye-Opener cartoons, his earliest published drawings (besides his single contribution to Judge), could best be described as "careful"—in their execution more than their subject matter, although the cartoons' sexual content is mild by modern standards. Barks's drawings echo the work of more accomplished cartoonists, their black-and-white patterning in particular recalling John Held Jr.'s highly stylized drawings of Jazz Age libertines. And although Barks never cited the cartoonists for the youthful New Yorker as an influence, it is hard to believe that he never paid any attention to the work of Peter Arno, Gluyas Williams, and Rea Irvin. Where the Eye-Opener is most likely to startle is in the crudity of its racism. Barks's depictions of blacks and Jews, among others, resist acceptance except as specimens of attitudes repellent now but extremely widespread in the 1920s.
In 1932 Henry Meyers sold the Eye-Opener to Antoinette "Annette" Fawcett, who was freshly divorced from Wilford Fawcett. Since, as Time said, apart from the names on the cover "there is little to distinguish Eye Opener from Whiz Bang," Annette Fawcett was in direct competition with her former husband. She fired Ed Sumner as editor of the Eye-Opener, Barks said.
She wanted to put on her own type of heavy drinkers. I never did fit into it, because I just couldn't drink a hell of a lot. . . . I just didn't fit into the social scheme of things, but I fitted into the workhorse part of it. Boy, I was there for four years, doing that work. It just got to where I was doing all of it. . . . Toward the last there, I was practically writing the whole thing. We bought a few gags, at a dollar a gag, and two dollars a gag, and so on. Very little of that.
At first identified as the "art director" for the Bob Edwards Publishing Company in the Minneapolis city directory's listings for 1932 and 1933, Barks was the Eye-Opener's editor in the listings for 1934 and 1935. He was an island of stability in the turmoil that accompanied Annette Fawcett, who referred to herself in print as the "Henna-Haired Hurricane of Joy and Laughter." Annette was "a swinger," Barks said. "She was the theatrical type. Among her friends were burlesque queens and so on. She was not the genteel sort."
Barks remembered his Eye-Opener salary as a hundred dollars a month, although he also said, in a 1961 letter to a former contributor, that he was paid only ninety dollars "for writing and drawing more than half the book, editing it, and composing stalling letters to you contribs to gloss over the fact that no money was in the bank to pay for your stuff." His Disney employment record showed him being paid two hundred dollars a month—perhaps an error on the studio's part, perhaps an exaggeration on Barks's; or his nominal salary may have been twice as much as he was actually being paid. Quite often, he said, "there wasn't enough left after Annette got her fingers into the incoming checks. She would entertain these visiting stars that came to town. She had sort of expensive tastes; she lived in the Radisson Hotel in a very expensive suite, and she entertained very lavishly whenever there [were] any visiting celebrities from Hollywood in town. She would just take everything that came in, and spend it."
Carl Barks in Minneapolis, circa 1934. Courtesy of Carl Barks.
Barks had applied for work at Disney, sending as samples of his work drawings of Mickey and Minnie Mouse and his own idea of Snow White and the Seven Dwarfs. That first Disney feature film was still being written in 1935, but the studio was beginning to expand its staff. "Disney's sent for me to come to work, and so I told the head of the printers over there that I was going to leave, and right away, the printers called the linotypers, and they called up the engravers, and said, 'Barks is gonna leave, he's gonna go out and work for Disney, what are we gonna do about it? Well, we'll raise his salary, we'll give him a hundred and sixty dollars a month to stay on.' It was tempting because, oh, a hundred and sixty looked like a fortune."
Disney was going to pay him only about half as much, but, he said, "I thought there was a future, if I went the route to Disney's, and there wasn't any at Minneapolis." When he left for California he was still married to Pearl Turner, but he was accompanied on the move by Clara Ovidia Balken, a woman almost three years his senior who had been the switchboard operator at his apartment hotel. Pearl divorced Barks in Reno, Nevada, on September 10, 1937, and Barks probably married Clara soon after.
Barks became a Disney employee on November 4, 1935, two months before Walt Kelly. He remembered that there were seven members of his November "class." "You worked one month, and you went to their art class, and you looked at their animation, and you studied and figured out how you were going to be able to use your talents to help them out. . . . At the end of one month, they picked out the ones that were likely to make the grade, and those that weren't were given their walking papers. Of the seven that went to work with my bunch . . . I believe there were four of us" who were hired permanently.
Not for the last time, Barks was again just scraping by; he had no money for a return ticket to Minneapolis if he got his "walking papers" from Disney. "I had barely enough money for rent [of] an apartment for a month, and before I got my first paycheck I was just reduced to eating fig bars. That was my breakfast, lunch, and dinner." He was being paid fifteen dollars a week during his trial period, a salary that rose to twenty dollars after he was hired permanently. Clara Balken, meanwhile, found "a job in a printing plant for wages about the same as Disney paid me."
Although Barks had begun dressing up his Eye-Opener drawings through cross-hatching, stipple effects, and the like, as opposed to a more open style, at Disney "it was just roughed-out drawings . . . there was no opportunity, in other words, to use those shading styles that I had developed." The only Disney artists who could work in their own styles were in the layout department, whose artists designed the settings for the cartoons. Otherwise, while there was room for distinct kinds of animation—that is, of movement—the goal was for greater uniformity in drawing styles.
Barks had entered the studio knowing nothing about animation. He worked first as an inbetweener, the lowest rung on Disney animation's career ladder, and, he said,
I wasn't making out too well at it. I began turning in scripts, or gags for the comic strip, and selling those to the comic-strip department, as just a sort of a sideline. I was attracting a little attention that way. The story department would send over a little outline of a story, just an idea of a story that they were going to try to make into a seven-minute short subject, and ask the guys for gags. . . . I was beginning to turn in some pretty good gags, and finally I turned in the gag of the barber chair that was made into a movie, the first of the [short cartoons starring Donald Duck], which was called Modern Inventions. Walt paid me fifty dollars for the gag. He seemed to get the idea that I should be working in the story department, rather than over in animation.
That gag was a "fanny gag," in which a robotic barber chair traps Donald upside down in its seat, trimming and combing the feathers on his rump (and polishing his beak with black shoe polish). It was the sort of gag, with echoes of the barnyard, that Walt Disney—like Barks, a farm boy—always found congenial, and that Barks had produced in quantity at the Eye-Opener.
Jack Hannah, who began working alongside Barks a couple of years later, said of him: "Carl had come from Oregon, and he had a little bit of the sticks in him; his early work was raw but very refreshing. I never worked with a guy who came out with so many new ideas. . . . Some of it was too far out sometimes; you'd have to watch him on continuity." That was what Hannah said in an edited transcript. He had actually said in the interview that Barks's early work was "pure corn," and that Barks's ideas were not "far out" but "very corny."
From 1936 to 1942, Barks worked on almost nothing but the stories for Donald Duck cartoons, among them Donald's Nephews (1938), the first cartoon with Huey, Dewey, and Louie. Disney paired Barks first with Harry Reeves, a veteran story man who had worked on silent Felix the Cat cartoons in New York years before, and they were subsequently joined by another writer, Chuck Couch. After that, Barks worked as a team with Jack Hannah. "During the years when Harry Reeves, Chuck Couch, and I were a story crew," Barks wrote in 1990, "we usually shaped stories from the earliest idea germs through to the finished story boards. Jack Hannah and I worked that way, too, except on a few gagged up preliminary plots that were wished onto us by Harry Reeves or Walt. We preferred to concoct our own story lines."
As a Disney story man in 1937, Carl Barks played a supporting role to his more flamboyant writing partner, Harry Reeves. The storyboard is for a Donald Duck cartoon called Good Scouts (1938). © Disney.
Barks saw Walt Disney only at story conferences:
We'd have a "showing," and when we had these showings, everybody was nervous and worried for days in advance. Walt would come down, and one of the story men would get up and talk the storyboard through for Walt. He would sit there with one eyebrow cocked up high, and when you got through there was dead silence for about five minutes, then he would say something. Usually, he would start in and say something fairly pleasant—you know, "Well, that's a good gag up there . . . pretty good stuff . . . but this down here, I think I would change that." He was very helpful. Very seldom did he ever say a real hurtful thing to any of the story men, something that would cause him [the story man] great discouragement. If he turned down a story completely, he would do it as gently as he could. As he walked out of the door, he would say, "Well, I think the best thing to do with that is just to shelve it for a while." So you knew that was the end.
Barks said of Walt Disney:
I had a great respect for him. His ideas were always good, his analysis of things was always so darned keen. He had a tremendous intellect, as far as that kind of stuff went. He could look at a gag, and like Chaplin, he could tell whether a gag was funny or not . . . he could see it moving in his mind. Now, that's one of the things that handicapped me. I don't think I visualized stuff in action like I should have. I was more of a plot man. I could work out the plots, the timing on gags, but all the individual action that was to depict that on the screen was something that was a little over my head, I didn't see it. . . . Walt could see the animation possibilities, and I couldn't. I could see a gag, and it looked funny in maybe three or four still pictures, but just how many feet of animation it was going to take to put that over, was over my head. . . . It wasn't a serious handicap; it's just that I could never have become a director, or somebody who had to translate those gags into the final product.
Barks repeatedly drew a distinction between what he called "plot gags" and "animation gags." He illustrated the distinction by referring to the 1936 Mickey Mouse cartoon Moving Day, in which Donald Duck struggles to free himself from a plumber's friend stuck on his rump: "Now that was an animation gag. It went on for practically six minutes of the seven [actually, much less than that]. A plot gag would have been merely the planning that went into this situation, by which he happened to get where this plumber's friend was and the method that was used to get the plumber's friend stuck on his fanny. Then, it immediately came to the animation gag, and it just went—all the funny animation problems he could get into with this plumber's friend."
A plot gag was, in other words, merely what was necessary to set up what really mattered, the animated elaboration of a conflict between Donald and some other character or a prop—in this case, a bathroom plunger. At Disney, the greatest value was always attached to those animators who could exploit the possibilities of animation gags. (Fred Spencer animated Donald's warfare with the plunger.) Barks left no doubt as to where his own preferences lay: "I was a PLOT man," he wrote in 1977. "Animation gags were long and tiresome to me. I wanted to see movement from one situation to another rather than movement revolving endlessly within one situation. Thank the lord I had editors at Western who let me ravel out my plots as I saw fit."
Barks's discontent vanished when he worked for perhaps six weeks with Chuck Couch on the story for Bambi. Ken Hultgren, who would draw a comic-book version of Bambi and then become a mainstay of the Sangor talking-animal titles, worked with Barks and Couch, making finished drawings from their rough sketches. Barks loved his Bambi assignment because "you could work for a week and never produce anything and still get your paycheck." He resisted returning to the Donald Duck shorts: "I just couldn't see myself getting back over there and working at honest labor." There was comic exaggeration in Barks's comments, but not a lot: during work on Bambi, in particular, some of the Disney writers were notoriously unproductive. Given how thoroughly Barks's life was defined by work, it is difficult to imagine his being satisfied for long with such a sinecure, but, in any case, once he was back on the Donald Duck cartoons there was no cushion for his discontent—especially after Disney moved from Hyperion Avenue in Hollywood to shiny new quarters in Burbank.
He wrote in 1975,
The physical layout of the Hyperion studio was very informal, and for that reason was a more pleasant place to work. We duck and Pluto crews got moved every few weeks into quarters that were still being hammered together by carpenters. At Burbank we were catalogued and classified and packaged like so many guinea pigs in quarters that seemed as friendly as hospital four-bed wards. . . . From the standpoint of working conditions, the studio was different things to different people. I found its mass milking of minds and talents very discouraging. Being a loner in my creative thinking, at least, I felt ill at ease among groups of thinkers, all trying to get one up on the next guy.
Whatever resentment he might have felt, Barks used his years at Disney—in the period when the Disney cartoons were changing most rapidly, and imposing ever stronger demands on the people making them—to become a much more skilled writer than he had been at Minneapolis. When he started at Disney in 1935, he could get by with gags not too far removed from those in the Eye-Opener: just as corny but cleaner. By the time he left the studio in 1942 there were still people on the staff who were doing that kind of work, but as Barks told Patrick Garabedian in 1971, he had educated himself for something better:
I spent many a night, sitting with an English book, learning how to put sentences together. That was after I'd gone to work at Disney's. I'd already been a magazine editor, but after I'd gone to work at Disney's, I decided [that] if I was ever going to learn the English language, I'd just have to get a bunch of old English books—grammars, you know, school instruction books—and I did, I got them at second-hand bookstores, and I would sit at night and study those darn things, and make up sentence structures, and learn all about adverbs and adjectives and pronouns. . . . That helped me a lot when I went to working out scripts for comic books. In making up my dialogue—I don't think many people ever noticed it—I would condense my dialogue down, to get the idea across in the fewest words possible. I would read those things over and over and over again. Finally I would be breaking it down to the number of syllables. Even after getting it down to a certain number of words, I would choose a word that had the proper number of syllables, so that I had a kind of meter in my writing. Writing would have to flow smoothly for me. I would read it over and over again.
In Barks's dogged self-education, as in his determination to become a cartoonist during his unhappy first marriage, there is an irresistible echo of Thomas Hardy's Jude the Obscure (1895). Fortunately for him, though, Barks did not aim as high as that novel's Jude Fawley. Hardy's character, a poor stonemason, yearns for a classical education at a university modeled on Oxford but suffers crushing disappointment. Barks wanted no more than to learn how to write well for cartoons and then comic books, and he succeeded.
The difference his studies made was visible not just in his comic-book stories but in his letters and interviews. There was in Barks's voice and delivery in his later years still a rustic tinge; his comfortable drawl was punctuated by verbal crutches like "little old." But he often wrote drafts of his letters before sending them, and the finished versions are typically as tart and direct as the speech of his comic-book characters.
Barks's coolness toward the studio as an organization did not prevent him from making good use of what he called his "Disney training." He learned, he said, how to integrate contrasting elements into a coherent story: "I would say the key was, you have to have a reason for everything. And if you could find a reason for something, you could drag anything in. I think that was what I got out of my Disney training more than anything, was to analyze whether anything was necessary to a story." Likewise, he said, "[w]e tried to get a certain amount of logic in what we did with the Duck. . . . If the Duck was going to pick up something, a very heavy weight, for instance, we had to make it look like that darned weight was heavy."
Barks learned at Disney to ask of his readers only a single, overarching suspension of disbelief. There is always the inescapable suggestion in his stories that the ducks are a distinctly different species, half the height of the semihuman characters around them. They are, as John Benson put it, "squat little ducks that walked around half-naked in a world otherwise populated by fully clothed full-sized people with vague animal attributes." The bargain Barks offered was this: accept such a world, in which ducks can talk (and share the streets with "humans" who usually have noses like dogs'), and everything else will follow naturally—with natural being defined differently in the ducks' world than in our own.
That was the same implicit bargain that other talking-animal comic books offered their readers, but most of them either failed to deliver or else, like the bulk of Western's talking-animal titles, gave their readers stories that hung together, barely, but had little else to recommend them. Barks's stories were, by contrast, dense with comic business presented with, as he said, "a certain amount of logic." There was also, in his longer stories especially, an echo of what Walt Disney said in 1941, at a meeting on Morgan's Ghost, the unmade feature cartoon that became the basis for Barks's first Donald Duck comic book: "Suspense is a good thing to remember in these things. If you can get good comic suspense it's swell. If you just have gags it's not exciting."
In 1946–47, as Barks's mastery of a comic-book story's demands deepened, he was able to put to increasingly good use what he had learned at Disney. And as in earlier years, the superiority of his work insulated him from some of the more disagreeable changes occurring in Western Printing's Beverly Hills office.
## 14
# Special Talents
Roger Armstrong returned to Western after he was discharged from the army late in 1945. He remembered drawing a premium comic book of Disney's Seven Dwarfs, one-quarter the size of a regular comic, some months later. It was copyrighted on January 29, 1947, so an incident he recalled must have taken place in 1946, probably in the fall:
As I recall, I banged that thing out in ten days and delivered it to Carl Buettner at his home. The only time I ever paid a "social call" in all the years he and I were associated. Marcie, his wife, put crackers with cheese on them in the oven to toast and I recall how irritated he was because she had to use two separate brands of cheese and one of them didn't work as readily as the other . . . anyway, over crackers and cheese and a couple of drinks (scotch, I think we were drinking) he confided in me that he and a fellow in the Eastern office (Poughkeepsie) by name of Lloyd Smith had just put the knife in old Eleanor [Packer] and she wouldn't be with us much longer. [Buettner] had just returned (as art editor) from a three week trip to the East and among other things, he had helped put the skids under old Eleanor. And he was right.
Actually, Lloyd E. Smith was based in Racine, at Western's corporate headquarters. He did travel to the West Coast occasionally, and he visited there in early 1947, around the time Packer left the company. Since joining Whitman Publishing in 1934 and, according to an official history, "help[ing] to organize its first editorial department"—it was in that capacity that he brought his friend Gaylord DuBois into the Whitman fold—Smith had become the head of Western Printing's rights and royalties department, or, as it was sometimes identified, the editorial and legal department. (He was also "assistant secretary" of the company.) He was in charge of licensing copyrighted characters and negotiating royalties for their use. It was in managing those licensed properties and dealing with their owners that Smith performed his editorial function.
Packer's dismissal was probably a matter of opening a position for an executive who would more aggressively exploit and expand Western's character licenses, and who had a close family connection with Western's principal owners besides. Here is Roger Armstrong again: "Shortly after [his conversation with Buettner], we got the . . . you should excuse the expression . . . boss's son-in-law: Bob Callender arrived on scene, regimentation was instituted, and the whole operation started its slow, gradual disintegration."
Robert Stevens Callender, who was born in Racine in 1913 and grew up there, began his career with Western Printing at Poughkeepsie in 1935. In 1937, in Racine, he married Wynnefred Audrey Wadewitz, the daughter of Western Printing's founder, E. H. Wadewitz. He entered the navy as a lieutenant in 1943, served twenty-eight months and then, late in 1946 or early in 1947, transferred from Poughkeepsie to take charge of Western's Beverly Hills office.
At Poughkeepsie, Callender was not just one of the three Western executives who owned K.K. Publications; he also handled the Dell Publishing account. He was involved with comic books from the earliest days that Western produced them—thus his name as the copyright holder on many of the features in those comic books. As Chase Craig said, "Callender and Alice [Nielsen] had put together the original Disney magazine which had been done in N.Y. just prior to the comic boom"—that is, they had transformed Mickey Mouse Magazine into Walt Disney's Comics & Stories.
Starting in 1941, Nielsen was credited in the annual circulation statements required by the post office as the editor, at Poughkeepsie, of Walt Disney's Comics. She had been a Western employee since 1932, working first in the Racine bindery during high school vacations and then joining the company full-time as a proofreader. She moved to Poughkeepsie after Western opened its plant there. She evidently continued as a proofreader after she began editing Walt Disney's Comics, since that job—especially in the early years, when it was a cut-and-paste operation—would not have demanded all of her time.
After Callender moved to Beverly Hills, Nielsen followed him, by sometime in 1948 at the latest, to edit Walt Disney's Comics and other comic books. The circulation statement published in the December 1947 issue showed Nielsen still editing Walt Disney's Comics & Stories at Poughkeepsie as of September 9, 1947. A year later, according to the notice published in the December 1948 issue, Nielsen had married Jack Cobb, becoming Alice Nielsen Cobb, and she was based at Beverly Hills. By then Eleanor Packer had long since lost her job, leaving behind no trace in Western Printing's official accounts of its comic-book operations.
Said Roger Armstrong: "Things tightened up at Western in the late forties. . . . Bob Callender ran things with a much tighter hand. I'm sure it was all company [policy], but Bob implemented it." One change, Armstrong said, "was a very deliberate policy . . . to keep the various artists from knowing each other . . . everything was very secretive . . . they had a sliding scale of pay and they never wanted artists to become too intimate for fear of comparing page rates. They scheduled our appearances in the office so there wouldn't be overlaps with consequent fraternizing . . . hence, I never knew many of the other cartoonists." That may have been the case, although in the comic-book industry generally, as more and more cartoonists worked as freelancers at home and only rarely delivered their drawings in person, the opportunities for encountering their peers inevitably diminished.
In the postwar years, even modest departures from a straightforward storytelling style were increasingly rare. Armstrong reveled in memories of one such opportunity, in the "Bugs Bunny" story in Looney Tunes and Merrie Melodies Comics no. 73, November 1947. "I had been looking at a lot of Herriman's Krazy Kat strips and I got carried away with borders and tricky ideas of presentation; it was a real experimental job and I loved every bit of it. I'll never know what came over old stodgy German Carl Buettner that he let me have my hand to this degree, but I had a ball drawing this one." Armstrong's "experiments" are, however, more of a garnish than a recasting of the usual structure of a Western-produced comic book, and it was probably for that reason that Buettner raised no objections—if he even noticed.
Carl Barks was by then so well established, and so highly regarded, that the changes Armstrong lamented seem not to have affected him. Barks said that if he had not met Callender on one of his visits to Western's offices, "I wouldn't have known that there had been any changes made. They never corresponded with me." Essentially, he delivered finished pages to Western and the company sent him checks. "I used to like to drive in with a story, finished artwork, because it was an outing for my wife and I. We'd go home by way of the Farmers Market, and stop and nibble the rare cheeses and eat exotic foods, and maybe go by Chinatown and have a big Chinese dinner, then drive back home again."
Barks was isolated, and, as he wrote in 1966, soon after he retired, he did not regret his very limited contact with other comic-book artists. He had a generally low opinion of other comic books, including those in the talking-animal vein:
I usually thought their stories were weak or monotonously formulated or both. . . . I read very few comic books of any kind. I was afraid the formats might be catching to the extent that my own stuff would start following the same patterns. . . . I knew only a few of the artists that worked or now work for Western. I don't know which ones did which books, and never asked. San Jacinto [where Barks was living in the 1960s] is a day's drive round trip from the Western Pub. [sic] offices and I went in no oftener than once a month if I could avoid it. The chances of meeting another free-lancing artist in the few minutes I'd be in the office were dim indeed.
He was, if anything, even more skeptical about the superheroes: "What amazed me about Superman was that these guys could continually keep the public's interest going with the same rehashed plot week after week and year after year."
Barks enjoyed in the late 1940s a position that was all but ideal for someone of his talents and temperament. He had established himself with Western's editors as a superior artist years before, and the comic books for which he provided the principal features were extraordinarily successful. By 1947, Western Printing was paying Walt Disney Productions royalties on more than two million copies of Walt Disney's Comics every month. Most remarkably, hundreds of thousands of those comic books were going to mail subscribers who sent K.K. Publications a dollar for twelve monthly issues.
"At its peak," the Western executive Howard Anderson wrote, "we had over 400,000 paid subscribers to Disney Comics, ran a direct mail campaign every fall using a self-mailer with only full cash payment up front and felt that any mailing not generating a 3 percent pull"—about twice a typical return rate for direct mail—"was unsuccessful!" According to Anderson, "K.K. Publications, Inc., officially became a wholly owned subsidiary of Western in 1949 when all the stock was transferred to Western" by the three executives—Wadewitz, Callender, and F. J. Leyerle—who were its owners.
Starting in the early 1940s, Dell also aggressively sought mail subscriptions for its comic books published on a monthly schedule, with considerable success. By 1950, the Dell titles that accepted subscriptions (Looney Tunes, New Funnies, and so on) accounted for about five hundred thousand copies a month, in addition to the four hundred thousand subscribers to Walt Disney's Comics & Stories. The subscription rate now seems remarkably low—a dollar for twelve issues—especially considering that fulfilling the subscriptions required that the women working at a conveyor belt in Poughkeepsie give prestamped wrappers an edging of paste before rolling them around individual comic books. But postal rates for magazines were very low, too, and mail subscriptions were a stable base that made print runs and ultimate sales more predictable; and Western Printing & Lithographing was, above all, a printer.
Western was in that respect as in others—like being based in a small Wisconsin city far from New York—a very different kind of company from the many comic-book publishers, like DC and Fawcett, whose roots were in pulp magazines of various kinds. For Western, whose publishing roots were in children's books sold in dime stores, its publishing partner Dell was a sort of bridge to the harshly competitive world of the newsstands.
Western differed from many of its rivals most curiously in that it was, compared with the other leading comic-book publishers, a distinctly Gentile company from the top down, the Wadewitzes being German Christians. It is hard to find Jewish representation among Western's comic-book people, whether they worked for the New York or Los Angeles office, or, for that matter, at Racine or Poughkeepsie. By contrast, not only were the owners and managers of the New York–based comic-book publishers overwhelmingly Jewish, but so were a great many of the artists, writers, and editors. Western was, as a Gentile redoubt in its industry, analogous to Disney in the movie industry. That studio, and Walt Disney himself, were the targets of unfounded charges of bias, but there was rarely if ever any complaint that Western did not want to give work to Jews or looked down on other publishers because they were Jewish. Western's affiliation with Dell may have served as a shield against any such accusations.
The flowering of Walt Disney's Comics and Western's Dell titles in the late 1940s took place in the first years of the baby boom, when the potential audience for comic books intended for young children was growing rapidly. Television was not yet ubiquitous, and the periodic waves of indignation about the crime and horror stories in other publishers' comics served mainly to give Western and Dell an opportunity to emphasize how different their titles were. The editors in Western's New York and Los Angeles offices liked good work, even if they were willing to settle for less. As a result, there was for a few years an unusually broad opportunity for the most creative of Western's cartoonists, of whom Carl Barks was the best, to respond to the challenges posed by the comic-book form.
Barks told Malcolm Willits he did not enjoy working at Disney because "I didn't like the pressure and the fact there were so many straw bosses looking over your shoulder to see how you were doing, criticizing your work all the time. I can't take criticism." Barks was by 1946 and 1947 still some distance away from complete mastery, but most of the time the only "straw boss" looking over his shoulder and criticizing his work was Barks himself.
Dell had published only one Donald Duck one-shot each year from 1942 through 1946, but for 1947 there would be three. The print runs, which had hovered around a million copies of each issue, began to rise as the nagging postwar paper shortages finally began to ease: for the first 1947 issue, Four Color no. 147, Western Printing paid royalties to Disney on more than 1.2 million copies. To his relief, Barks shed the last of his non-Disney comic-book features, "Barney Bear and Benny Burro," in January of that year, when he submitted the story that would be published in Our Gang Comics no. 36, July 1947. Now he would write and draw only stories with the ducks.
The temptation always for any writer of comic-book stories was to take shortcuts that looked exactly like what they were, trusting to the audience's lack of sophistication to make them acceptable, and Barks was not immune. In "Volcano Valley," the lead story in Donald Duck Four Color no. 147, a crucial plot device—Donald must become a national hero before he can leave the wretched Latin American country of Volcanovia—seems especially arbitrary because the story's opening pages are so cheerfully cynical, all but inviting the reader to scorn the contrived comedy that follows.
"Adventure Down Under," the second story in the next 1947 issue of Donald Duck, Four Color no. 159, is, by contrast, simply too serious, with the ducks in danger for too much of its length. The ducks were intrinsically comic figures, at home in an environment where characters could be named Wyndham Blowhard and Argus Gimleteye (as they are in this story's opening pages); so, when comic action disappears, as it does in "Adventure Down Under" for pages at a time after Donald is captured and threatened with death by realistically drawn aborigines, the strain is felt. The apparent reality of the settings, not just in Australia but in the ducks' hometown in what is unmistakably California, mostly intensifies the strain.
In "The Ghost of the Grotto," in Donald Duck Four Color no. 159 (1947), Carl Barks for the first time expanded a panel to a full half page. © 1947 Disney.
"The Ghost of the Grotto," the lead story in that issue, is a different matter, with comedy and adventure in perfect balance for the first time in one of Barks's longer stories. "Ghost" fills twenty-six pages, but it originated, Barks recalled, in an idea for a ten-page story—that is, a wholly comic story—for Walt Disney's Comics. The menaces—a huge octopus, a mysterious man in armor—are real, as are the settings—the ducks are gathering kelp in a credibly depicted West Indies—but the reader is never asked to believe that the ducks are in deadly danger, or to laugh at gags present for their own sake. What the story offers, instead, are moments like the spectacular half-page panel—the first such very large panel in any of Barks's stories—in which the octopus, tricked into eating a roll of meat stuffed with chili pepper, rises stunned into the air, shattering the ancient sailing ship it has used for shelter.
In 1946 and 1947, Barks's best efforts were going into his longer stories, and the ten-page stories suffered in comparison. Even so, they were usually well constructed: a story's events were plausible even when they were predictable. In the story in Walt Disney's Comics no. 76, January 1947, for example, Donald insists on giving shelter to a stray cat that rewards his kindness by keeping the ducks awake all night. There is never the slightest doubt that by the end of the story Donald will be pursuing the cat with homicidal intent, but Barks makes sure that his story arrives at that point with no arbitrary twists or turns. For one thing, the cat's behavior is always recognizably feline, and recognizably exasperating.
By 1947, Barks had achieved a level of craftsmanship in his best stories, in both Donald Duck and Walt Disney's Comics, that had no equal in other comic books of the talking-animal kind, and he had few if any peers in other kinds of comic books. But it was not until the fall of that year, five years after he entered comic-book work, that Barks became wholly himself as a creator of comic-book stories. It was then that he wrote and drew the ten-page stories that were published in the early-1948 issues of Walt Disney's Comics.
The story in the February 1948 issue, no. 89, was one of the first in which Barks spoke clearly in his own distinctive voice. It begins as Donald stalks down his front walk, thinking to himself: "I've got to get a job, that's all there is to it! The rent is due, and the kids need new toothbrushes!" There is, to begin with, that absurd disproportion—rent and new toothbrushes—but, besides that, these are toothbrushes for ducks, characters who lack teeth except on those rare occasions when teeth can be gritted to show anger or determination. (The ducks display teeth nowhere in this story.) Barks does not poke his readers in the ribs to make sure they get the joke, because that would distract their attention from Donald; and Donald, it is clear well before the bottom of the first page, is serious about finding a job, serious about those toothbrushes, and, above all, serious about himself: "But I'll not work at just any job!" he thinks in the second panel. "I want something that suits my special talents!"
The sly toughness always lurking in Barks's earlier stories was here out in the open: when the ducks show up at a warehouse for Donald's new job as a night watchman, the coldly suspicious guard on duty edges his pistol out of its holster. Later, when bandits come to steal the silk Donald is guarding, one thug says he will find the night watchman and "bump him off"—that is, kill him.
Donald's three nephews—so often his opponents in broadly comic conflicts in earlier issues—had by 1947 become more subtly depicted characters. Huey, Dewey, and Louie were always interchangeable triplets, and Barks did not abandon triplet comedy, as when he had each nephew speak one part of a complete sentence, but the effect was not gimmicky, as it was when it was overused in stories by other writers and cartoonists. It was instead as if, being triplets, the nephews thought alike and each picked up naturally from what the others were saying. The divided dialogue was that of minds working in concert. One nephew might occasionally stand apart from the others, but there was never any sense that Huey, say, was an individual distinct from Louie and Dewey. It was in how the nephews related to Donald, and vice versa, that the stories were changing.
The story in the October 1947 issue, no. 85, for instance, is more psychologically complex (and believable) than its predecessors. The nephews and Donald are at odds—not over any childish mischief, but because the nephews are determined to learn to play one chord on their stringed instruments and thus earn the frog-hunting trip to Mud Lake that Grandma Duck has promised them. It is the resulting racket (FWANG! RASP! BLOONK!) that drives Donald wild, not misbehavior, of which there is none once the nephews have signed on to Grandma's offer.
When Barks's stories really began to soar a few months later, the weight within the ducks' parent-child relationship shifted from story to story, and within each story, as Donald and the nephews traded roles. In the February 1948 night-watchman story, Donald is at the start the parent in command, seeking work and then ordering the nephews to bed early so that he can rest before going to his new job at midnight. Donald finally falls asleep just as it is time to leave for work, and the nephews, like grimly determined surrogate parents, doggedly follow him on his rounds, keeping him awake until Donald exhausts even their patience by falling asleep on a fakir's bed of nails. Eventually Donald routs the murderous burglars by shooting his pistol while he dreams, and the nephews read about his triumph in the morning paper. They respond again like weary parents, shaking their heads as their ne'er-do-well child escapes the humiliation he so richly deserves. One nephew says: "Well, gents, that proves what I've always said! . . . That when the horseshoes were being handed out, Unca Donald was there with the cavalry!"
In the next story, in Walt Disney's Comics no. 90, March 1948, the ducks are more like competitive siblings—they are rival telegraph messenger boys—than parent and children. In the May 1948 issue, no. 92, the wheel turns again, and this time Donald is not just a father but a martinet, ranting at the nephews and waving a switch when they ride hoes as "horses" instead of weeding his garden. But then Donald falls under the spell of Professor Pulpheart Clabberhead, "the friend of all children" and Barks's top-hatted riposte to Benjamin Spock and his Common Sense Book of Baby and Child Care, published two years earlier. Here was more evidence of the new sharpness and energy in Barks's stories: he had done nothing comparably satirical in earlier stories, even when he was working with the same basic situation. (In Walt Disney's Comics no. 64, January 1946, the nephews run wild after Donald vows to hold his temper.)
The professor, like some latter-day Rousseau, persuades Donald that he should let the nephews do whatever they please—"Only by that way can they ever learn what they are fitted for in later life!"—and the nephews, anything but sober miniature parents this time, seize the reins from Donald with cool, sadistic glee. They steadily escalate their demands on Donald and his bankroll ("Hustle out and buy us some frock coats and high silk hats!") until, finally, Donald identifies the weak spot in the professor's argument. It seems that—as the professor himself puts it while pursuing the nephews across the countryside, switch in hand—"[b]lowing Professor Pulpheart Clabberhead skyhigh with an atomic bomb is strictly against the rules!"
There is tremendous energy in Barks's writing and drawing in his stories from the late 1940s and early 1950s, energy almost always under tight control but constantly molding the stories in large ways and small—in extreme characters like the furious old crone Angina Arthritis (in the March 1948 story), and in the profusion of details like the mildly annoyed birds that observe Donald from their birdbath as he erupts into his backyard and then frets about how to master the chaos that Professor Clabberhead has unleashed. And then there are the names, pouring out now in a gusher. Memorable comic names turned up occasionally in the midforties (for example, Dr. Carver Beakoff), but now they were present in abundance: Gladstone Gander, Prunella Prunepuss, Señor Mañana N. De Patio, J. Morganbilt Giltwhiskers, Rimfire Remington, Blacksnake McQuirt, Trigger Trueshot, and on and on.
Barks's dialogue and captions, too, were, from 1948 on, more pointed and concrete, persuasive evidence of how intensively he had studied his English textbooks while he was on the Disney staff. Thus: "Unca' Donald couldn't catch a fox with a barrel of squabs!" In that fox-hunt story, in Walt Disney's Comics no. 98, November 1948, Donald is calling out to a tame fox named Red Herring that the nephews have planted to help him, but the fox responds only to his name, and Donald cannot remember it. He summons up every fish name he can think of, or perhaps anyone could think of: "Here, finnan haddie! Here, smoked barracuda! . . . Shad roe, where are you? Kippered sprats! Marinated mackerel! Filet of sole! Sardines in olive oil! . . . Columbia River smelt, come to papa! Grilled halibut, ol' boy, it's me! . . . Dogfish! Catfish! Sawfish! Swordfish! Goldfish! Whitefish! Chub! Carp! Minnows! Guppies! . . . Calico bass! Bluegills! . . . Red snapper! Jelly fish!" The fox is too quick for him (and visibly contemptuous of his pursuer): "That needle-nosed nuisance is faster than a boarder's reach!"
Donald has been thrown by his horse, and in desperation he rents a plow horse from a farmer, riding away in such a hurry that he will not take the time to let the farmer remove the horse's harness. And so, as Barks writes in a caption, Donald rides "with trace chains flying"—trace chains that trigger a disaster that is entirely believable because we can see what is happening when the chains catch on a wire fence. Barks's ducks, unlike most other talking animals, live in a world full of real things—things with names, and, especially, dangerous things that must be treated with respect. As Barks told Paul Ciotti in 1972: "I understand enough about machinery from having been a farm boy. I knew what machines should look like, and gear teeth and sprocket wheels and things were just part of my growing up. I've worked on printing presses and threshing machines and engines and so on, so I know a little bit about the principles of mechanics."
By 1948, Barks was beginning to exploit the comic possibilities even in bystanders: in Walt Disney's Comics no. 95, August 1948, as Donald quarrels with his cousin Gladstone Gander, a fat man passes deep in the background with an ostrich on a leash. Animals—that is, "real" animals, like Red Herring the elusive fox, as opposed to the voting population of the ducks' home town, Duckburg—could be mocking commentators on the main action. In Walt Disney's Comics no. 101, February 1949, when Donald suffers nightmares about being pursued by ravening sharks and wolves, those predators are enjoying themselves entirely too much.
Usually if not quite always, Barks was by the late 1940s producing short stories that were simultaneously marvels of narrative construction and psychological acuity. (A mild qualifier is needed because he was, after all, writing and drawing roughly a page a day of finished artwork.) It is, finally, not so much what happens that commands our attention as the ducks themselves, the distinction between "plot" and "character" dissolving as it so often does in the most enduring fiction. Even when, as was frequently the case, a story was by definition a farce, it was a farce whose improbabilities were marshaled so skillfully that the comedy was much richer than it had any right to be.
Barks himself provided a sort of commentary on his art when he worked from a script provided by Western for the story in Walt Disney's Comics no. 99, December 1948, in which the ducks are contestants on a radio quiz show. Barks always rewrote such scripts, and the story's expert construction makes it feel like his; but other characters could have filled the ducks' roles (as was true with the interchangeable talking animals in many other comic books). In most of the stories Barks wrote himself when he was at his peak, any such substitutions were unthinkable.
By 1948, Western was paying Barks twenty-five dollars a page for writing and drawing his stories; he was thus paid $800 for the thirty-two pages of interior art in "The Old Castle's Secret," in Donald Duck Four Color no. 189, and $250 for the ten-page "Donald Duck" story in the May 1948 Walt Disney's Comics. "I know there were a number of guys who were getting higher rates than I was," he said in 1978. "But they were more under the thumb of the editors. They had to make more corrections, and the work they were doing was not so interesting as what I had. I had that freedom. It was worth ten dollars a page to me, at least, to have the freedom to write whatever I wanted to write. If they didn't like it, they paid me for it anyway. I got along well enough."
The longer stories varied in form even more than the ten-page stories, but by this time Barks could move from one type of story to another without ever losing his footing. And so the twenty-page "Christmas on Bear Mountain," in Donald Duck Four Color no. 178, the last 1947 issue—and the first story with Donald's Uncle Scrooge McDuck, a character of Barks's invention—is cheerful farce. Barks submitted his next long story, "Darkest Africa," twenty-two pages for a 1948 issue of the Boys' and Girls' March of Comics giveaway, a few days after he submitted the night-watchman story for the February 1948 Walt Disney's Comics. Like that story, "Darkest Africa" pits the ducks against an adversary, Professor Argus McFiendy, who is prepared to kill to get what he wants—in his case, the world's rarest butterfly. Poison intended for the ducks dispatches a crocodile instead. "The Old Castle's Secret," in the next Donald Duck one-shot, Four Color no. 189, the first in 1948, is very different from its predecessors—a perfectly paced haunted-house story.
"Secret" is also a comic-book story to its core, simple and far-fetched, even though it shows the nephews more believably mature and resourceful than ever before. The same could be said of "Sheriff of Bullet Valley," in Donald Duck Four Color no. 199, the next 1948 issue—a thirty-two-page story in which the nephews disrupt the plans of rustlers who rely on a highly improbable ray gun to change the brands on cattle. "Bullet Valley" is, however, richer in every way than "Old Castle's Secret," filled as it is with comically distorted echoes of the cowboy movies and Zane Grey and B. M. Bower novels that Barks favored. Barks's reading, as he described it, was almost entirely restricted to pulp magazines and western novels of the Grey variety. He also remembered occasionally reading the fiction in popular weeklies like the Saturday Evening Post and Collier's, and he spoke fondly of P. G. Wodehouse. "The [nonfiction] articles [in the popular weeklies], I don't know, never appealed to me much, because I never seemed to agree with the guys who wrote the articles. I'd get mad at what they were writing about."
In "Bullet Valley," the rustlers' leader, Blacksnake McQuirt, is a murderous villain, and in one of the most extraordinary scenes in any of Barks's stories, he actually seems to shoot Donald—to kill him, that is, although Barks reveals quickly that Donald's enormous sheriff's badge has deflected all the bullets. Here, though, in contrast to earlier stories like "Adventure Down Under," the atmosphere is so thoroughly saturated in comedy—even the horses have distinct comic personalities—that it can absorb the most apparently lethal violence.
Barks's next important long story was the thirty-two-page "Lost in the Andes," in Four Color no. 223, the first of the four 1949 issues of Donald Duck. That story is even richer in comic detail than its predecessors. A simple transfer of orders down the chain of command on a museum expedition's ship becomes a satirical commentary on class distinctions: stiff and dignified exchanges at the top give way to the brusque and peremptory as the orders descend through the ranks, until finally Donald is bullying the nephews.
"Lost in the Andes," perhaps the best loved of all of Barks's stories and the one that he more than once singled out as his best ("I never did anything before or after that would come up to that"), centers on the square eggs that Donald, in a real if menial job as a museum guard, accidentally discovers. Ultimately, and plausibly—Barks had by now scrubbed away virtually all traces of the arbitrariness that disfigured the work of so many of his fellow comic-book writers—it is Donald and the nephews who must enter the Andes to find the source of the mysterious eggs. Their search leads them into the ominous mists from which, an aged vicuña hunter tells them, another American, raving and near death, emerged long ago with the square eggs. Blundering through the mists, the ducks find themselves in a primitive city hewn from rock. It is home to a blocky populace that speaks, as Donald puts it, "straight cornpone," a deep-fried southern version of English. The inhabitants have learned it from an American professor from Birmingham, Alabama (or, as they call him, "th' professah frum Bummin'ham"), who dubbed their home Plain Awful. They eat nothing but the square eggs.
There are echoes in Plain Awful of the ducks' encounter with the Gneezles, the swamp goblins in the 1944 story "Mystery of the Swamp," but the difference is enormous. The Gneezles, supposedly isolated from human contact for hundreds of years, nevertheless speak perfectly intelligible English. The Plain Awfultonians speak English, too, but there is no mystery about how they learned it. The sense that Plain Awful is a real society, with complications and dangers lurking beneath its seemingly simple surface, is much stronger in the later story. That sense of reality is enhanced, as before, by settings based on Barks's careful study of National Geographic—in this case, an article in the August 1942 issue called "The Pith of Peru," about the remote Inca fortress Machu Picchu. "Barks was no longer copying whole images into his comic as he had in 'The Mummy's Ring,'" Geoffrey Blum has observed of "Lost in the Andes," "but we can pick out the details he borrowed"—including the "terraced and angular" appearance of Plain Awful itself.
The one very small problem with "Lost in the Andes" is that Barks has a little trouble figuring out what to do with the ducks once they have arrived in Plain Awful. So, echoing "Volcano Valley" of two years earlier, he manufactures a crisis: the round bubbles the nephews blow with their bubble gum are, it seems, sacrilegious in Plain Awful, although there is scarcely a hint of what kind of religion the Plain Awfultonians might observe. The nephews save themselves by seeming to blow square bubbles—which are actually blown, somehow, by concealed square chicks that have hatched from square eggs.
That relatively strained episode, inconsequential in a story otherwise so rich in comic invention, has no counterpart in the thirty-two-page stories that immediately followed "Lost in the Andes." The pacing of "Voodoo Hoodoo," in Donald Duck Four Color no. 238, is impeccable: Barks deposits the ducks in Africa as smoothly and surely as he deposits them in Plain Awful, but once they are there he finds plenty for them to do. In Africa, Donald is under a threat that is both serious and ridiculous on the comic book's own terms, in contrast to the awkward juxtaposition a few years earlier in "Frozen Gold." He has been pursued by a zombie—an emissary of the vengeful witch doctor Foola Zoola—who has mistaken Donald for Uncle Scrooge. Even after Donald clears up that misunderstanding, he remains in jeopardy, as Scrooge's nearest living relative. In Scrooge's absence the witch doctor intends to shrink Donald to the size of a mouse, as punishment for Scrooge's imperialist misdeeds seventy years before.
Before the ducks come face to face with Foola Zoola, there emerges from a clump of grass a tiny man, Professor Cornelius McCobb, dean of mystic lore at the University of Ypsilanti, who has in fact been shrunk by the witch doctor. But there is no terror in this dramatic entrance, nothing to disrupt the story's comic equilibrium. The professor is heard before he is seen, speaking in what are clearly supposed to be unruffled tones. (Barks discreetly sets aside any question that might be raised about the volume of this tiny man's voice.) He enthusiastically endorses his diminished size, which, after all, brings with it greatly reduced clothing and grocery bills: "A peanut feeds me for a week!" Donald is not persuaded.
What is most striking about "Voodoo Hoodoo," and especially the story that immediately followed it, "Luck of the North," in Donald Duck Four Color no. 256, is how masterful Barks had become in his command not just of comic-book grammar, but also of what might be called comic-book rhetoric—the marshaling of a comic-book story's elements to make the strongest possible impression.
In the years he had been working in comic books he had demonstrated better than any of his colleagues how important it was to compress and expand the amount of time a story's panels seemed to take up, through a constantly shifting balance of dialogue and action. In the 1947 "Ghost of the Grotto," for example, when the huge octopus surges up from the wrecked galleon, stunned by the chili pepper in the baited meat, there is counterpoint to that spectacular half-page drawing—which could only represent an instant—in a bit of dialogue from one of the nephews: "Uh oh! I musta used a sprinkle too much pepper!"
Now, in "Luck of the North," Barks almost seemed to move backward, by isolating moments of time in a sequence of panels as Donald gradually, painfully realizes that a satisfying practical joke may have deadly consequences for his obnoxious cousin Gladstone Gander. Barks was, however, not reverting to the storyboard-like drawings of his early stories but was instead dissecting Donald's psychological state with a subtlety that was unique in comic books. There were echoes of Barks's animation work in such panels, but echoes completely different from those in the early stories, as Barks explained in 1971: "Back in the days when I was working there at the studio, the thing was to hold a character for as long as you could. Let the public see him think. And his actions were studied, so that whenever he did pull a fast gag, it was a contrast to the slow action up to that time." Because Barks shows so clearly the workings of Donald's mind, what follows in "Luck of the North"—Donald's frantic effort to catch up with Gladstone and save him from perishing in the Arctic—is entirely believable.
Somewhere in this period, when Barks was starting to share the duck stories with other cartoonists, he prepared a model sheet titled "Magazine Comics Duck" that seems to have been intended to help those cartoonists draw Donald in something resembling Barks's style. Among the instructions: "Avoid excessive distortion of beak and brows. Tilting eyes is key to most expressions." Barks illustrated that point with nineteen drawings showing Donald's face or just his eyes, and added this note: "Use this eye tilting cautiously. It's awfully easy to tilt the eyes too far."
There is no reason to believe, on the evidence of their drawings in the comic books, that the other cartoonists tried to follow Barks's advice, but if they had, they might well have felt boxed in by such restrictions—especially since Barks himself had transcended them. He was by the time he drew that model sheet so completely in command of his medium that when he depicted Donald's gathering anxiety in "Luck of the North" he eschewed not only "distortion of beak and brows" but even "eye tilting." The changes in Donald's face are all but indiscernible, but the movement of his thought, conveyed through his posture and even the position of the pupils of his eyes, is distinct, and culminates in a striking image: Donald is weighed down, his head seemingly squashed by the drawing above it, as he imagines that his joke has led to a polar bear's eating Gladstone. Words—what Donald says or, more often, thinks—are sparse in these panels, but there are just enough. They simultaneously give needed voice to Donald's disquiet, so that the drawings do not have to do too much work, and subdue any danger that the page might seem fragmented into shards of time.
Barks was by the late 1940s increasingly unusual because he both wrote and illustrated his stories. By then, it was much more common for one person to write a story and one or more cartoonists to illustrate it (one making the pencil drawings, another finishing them in ink), with the work passing through the hands of an editor at each stage—an arrangement that lent itself to much greater editorial control and diminished the opportunities for a strong artistic personality to assert itself. John Stanley, whose Little Lulu stories in particular were unmistakably his own even when drawn by Charles Hedinger or Irving Tripp, was the rare comic-book writer who surmounted that obstacle.
"Luck of the North," in Donald Duck Four Color no. 256 (1949), was one of the first stories in which Carl Barks showed Donald's mind at work, with a subtlety that was rare if not unique in comic books. © 1949 Disney.
As for Barks, he not only wrote and drew his stories but submitted them in finished form to his editors. Throughout his best years, there are pages that might have been unacceptable to an editor in script form, whether it was written or roughly sketched like a storyboard. Marvelous panels like those in "Luck of the North" tracing Donald's growing remorse over his misfired joke could easily have been mistaken for padding, but such pages instantly made sense when seen in Barks's inked drawings.
Because Barks was both writer and artist, there was no gap to bridge between the stories he wrote and the drawings he made. Each was always in the service of the other. There is, in particular, complete harmony between words and drawings, harmony of a sort that shames any complaints that the medium itself is hopelessly inadequate. The words or thoughts in the balloons always seem to be emerging from the faces beneath them, and the words, whether dialogue or captions, tell what the drawings do not. Words and pictures are interdependent: neither is superfluous; they complete each other. In most comic books and comic strips, the vital connection between words and drawings is either not made or made only in the most general terms. There is a sort of vacant space between drawings and dialogue—not a literal lacuna, but closer to what happens when a film and its soundtrack are not quite synchronized, and the discrepancy repels acceptance of what is on the screen. In comic books, it is when that gap is bridged—almost invariably by precise comic exaggeration of the sort that Barks and a few other cartoonists mastered—that comics make their strongest claim to be regarded as art.
## 15
# Barks Masters His Medium
As highly regarded as Carl Barks's stories were in the 1940s, they did not invariably pass through his editors' hands unscathed. Western, probably in the person of Eleanor Packer, rejected completely the "Donald Duck" story Barks submitted for Walt Disney's Comics & Stories no. 64, January 1946. That story had Donald's neighbors responding furiously to his Christmas caroling, and it was apparently the juxtaposition of the holiday with exceptionally violent slapstick that prompted the rejection. Christmas was a touchy subject: another holiday-themed story, "The Golden Christmas Tree," for Donald Duck Four Color no. 203, in 1948, underwent surgery to tame it down, at Barks's hands but at the direction of his editors. As he wrote to a fan in 1961, he thought the revisions "took the guts out of the story. I still gag when I read the last two pages. . . . But the rest of the tale was robust enough."
There was occasional tinkering with other stories—changes in both artwork and dialogue. As Barks said, "[T]here are a few words here and there in certain stories that got changed in the office." For instance, in the 1949 Donald Duck story "Voodoo Hoodoo," the word dead has been changed to done for in two instances but—typically for such censorship—left unchanged in a third. Barks wrote in 1966: "Any ending that is sickeningly sweet is almost certain to have been altered from my original product."
Not many endings were changed, though. The most significant such change, apart from "The Golden Christmas Tree," was the substitution of two panels at the end of "The Firebug," a story in the 1946 Donald Duck one-shot, to make Donald's pyromania the stuff of a bad dream. (Barks had him going to jail for burning down a courthouse.) The "pressure from the office"—"the guidance from the editors, just by talking with them"—was apparently light enough otherwise that it was rarely disturbing. In a few instances it may even have been helpful. In 1971, Barks said of his shift toward more comical menaces, and away from characters like the psychopath in "Terror of the River": "That was conscious. . . . I was getting away from the real serious stuff. . . . [The reasons were] not only pressure from the office, but I was beginning to feel that giving villains too much of a character—that is, you build up a respect for them. I don't want people to respect the villains, the bad guys."
There is no sense in the published stories from the late 1940s that Barks felt cramped. He was still developing as a comic-book artist, and the Dell comic books, especially those that Barks wrote and drew, were increasingly popular. There was every incentive for cartoonist and editors to accommodate one another's needs.
By the late 1940s, Barks's drawing had become suppler, free of the residual storyboard stiffness in his earliest stories. He always drew with dip pen and ink, using a flexible nib, and turning to brush for areas of solid black. For him as for other artists, relying on the pen encouraged relatively cool and restrained expression compared with the expansive gestures the brush permitted. This restraint worked to Barks's advantage: now he was drawing broadly conceived cartoon faces and bodies with a new variety and subtlety of expression. The ducks' beaks shrank, as did the pupils of their eyes. "I did deliberately shorten the ducks' beaks around 1949," he said, "when my slow wits finally awoke to the fact that I'd been drawing them too long." He had diminished the size of his characters' most expressive features while simultaneously and paradoxically increasing their capacity for expression, by bringing everything about their faces into better balance.
It was in Barks's stories that the conundrum identified by Walt Kelly and Kin Platt in that 1943 court case—how to adapt characters designed for animation to the printed page—was resolved most satisfactorily. Barks took advantage of both Donald's design—its capacity for human activity and expression—and the opportunity that the comic books provided for dialogue that was much richer than was possible in the cartoons, where Donald's voice was limited by its quacking sound as well as the economy of words that classic animation demanded. Although Barks drew his ducks with poses and expressions whose equivalents could be found in some of the best animation coming then from Hollywood cartoon studios, notably Chuck Jones's Warner Bros. cartoons, those cartoons could not incorporate dialogue as rich as Barks's without sacrificing some of their other virtues. In good animated cartoons of the 1940s the characters often move much faster than is possible in real life, and there was no way to combine such speed with dialogue spoken normally. Barks's ducks were emphatically creatures of print, even though their origins in animation were never in doubt.
The drawings by some of Barks's contemporaries in the Dell talking-animal titles, such as Veve Risto ("Henery Hawk," "Homer Pigeon") and Bill Wright ("Mickey Mouse"), were more rigid than precise, but Barks's drawings never fell into that trap. His writing demanded that his drawings depict emotions and states of mind that were highly specific, as in "Voodoo Hoodoo," when Donald awakens to find a zombie standing at the foot of his bed. Barks shows Donald's awakening and his sudden awareness of his visitor with a panel or two more than most cartoonists would have used, and those panels make a tremendous difference in revealing Donald's state of mind. And then there was "Luck of the North," in which Barks went so much further, devoting most of a page to Donald's dawning realization that he has tricked his insufferable cousin into a treasure hunt that may lead to his death.
Gladstone Gander had first appeared in Walt Disney's Comics no. 88, January 1948, but it was more than a year later, in "Race to the South Seas," a story in a 1949 March of Comics giveaway comic book, that Barks attached to him a characteristic that made him a perfect foil for Donald. Gladstone is, for the first time in "Race," more than a smug dandy: he is also preternaturally lucky. Donald's anger and frustration are coarse-grained compared with how subtly Barks would depict the workings of his character's mind in "Luck of the North," but "Race to the South Seas" is Barks on the verge of his best. The story's broad comedy is supported by the concrete detail of the ducks' perilous voyage across the Pacific to rescue Uncle Scrooge, who is cast away—they think—on a remote island.
Barks wrote and drew "Race to the South Seas" about six months before "Luck of the North," and it is like a preliminary sketch for that later and better story. In "Luck of the North," Donald's fierce resentment of Gladstone's luck is never just a pretext for gags but always the engine of the story: everything flows from it. And there is no doubting that Donald would feel such resentment. In the story's opening four pages, Gladstone (making his first appearance in the Donald Duck series) enjoys one lucky break after another, each unexceptional in itself but overwhelming in cumulative impact, especially since Gladstone crows continuously about his good luck and insists on dragging Donald along to witness it.
Barks's stories by this time were almost always impeccably constructed, but it was his emphasis on how his characters' minds worked, his willingness to take the time—that is, the panels—to make them seem real, that lifted the best of those stories to a higher level than any other comic-book stories of the time. That psychological acuity, too, meant that the ducks, and Donald in particular, could be convincing characters even though their circumstances differed sharply from story to story. In both "Luck of the North" and "Trail of the Unicorn," a story in the next issue of Donald Duck, the ducks, after enduring heartbreak and misery for page after page, triumph over Gladstone and wind up rich—but their wealth is evanescent. There is no trace of it in the stories that follow.
The stories themselves differed more now in their basics than had ever been the case in earlier years, when they tended to fit into broad categories. "Ancient Persia," the lead twenty-four page story in Donald Duck Four Color no. 275, the first 1950 issue, seems at first to be a pulpish sort of story. There is a mad-scientist villain (drawn, like more and more of Barks's characters at the time, with a completely human face rather than a dog's nose) and a plot that hinges on the scientist's efforts to resuscitate ancient royals who were "dried" and turned to dust millennia ago. The kingdom in question is, however, called Itsa Faka, its ruler King Nevvawaza, and his daughter Princess Needa Bara Soapa. Burlesque trumps pulp, but without disrespecting pulp's virtues. For one thing, the ducks (and Barks's readers) can understand what the king is saying in his indignant ravings because, it seems—and this is pure pulp, of the most entertaining kind—the "thought processes" of the dried Persians have risen like perfume from the water in the royal bathtub where they are being revived.
Rather than work around differences in languages, the writers of talking-animal stories usually ignored them, thus affirming their stories' childishness. But Barks disposed of the language problem neatly in "Ancient Persia," and he did the same in "The Magic Hourglass," the twenty-eight page lead story in Four Color no. 291, a later 1950 issue. Uncle Scrooge has fobbed off what he thinks is a worthless hourglass on the nephews, and they try to sell it to a junkman with a pushcart. The junkman sneers at the very idea: "And don't try to tell me it's a magic hourglass just because of what's written on the top of it!" The inscription, it seems, is in ancient Arabic, which the junkman can read; and why shouldn't a junkman have a fascinating history that encompasses knowledge of ancient Arabic? "As long as this glass keeps perfect time, its owner will grow richer hour by hour. . . . What nonsense!" the junkman snorts.
"The Magic Hourglass" is full of such language issues, and Barks resolves the earliest of them so deftly, as with the junkman and then with a camel-riding Arab who learned English from American soldiers during World War II, that he finally can ignore them. The Bedouin-like "raiders of No Issa" speak English without any explanation's being offered—but even here, Barks has left vague whether what we read as English is what the raiders are supposed to be speaking to one another. Only the raiders' leader addresses the ducks directly, in what has to be English. Barks knew that by the time the raiders appeared in his story he had a great deal of leeway, considering how thoroughly he had established the reality of the ducks and their surroundings in other ways.
In many of the comic books published in Barks's heyday, there was the suggestion that the stories were occurring in something like the order in which they were published (as was very much the case where adventure comic strips were concerned, since one story usually segued into another). But in many other comic books, those with talking animals especially, there was only the slimmest connection between stories. The chronological sense in the duck stories was likewise very loose; almost never did a story contain a reference to events in another story. The ducks' external circumstances changed as a story required: sometimes they lived in what had to be southern California; other times they spent the winter up to their necks in snow. Everything was in flux.
Barks spoke as if he always considered his gags to be more important than his characters, a ranking reflected in the mutability of the ducks' surroundings and the ducks themselves. In his prime years, though, he almost never compromised a character for the sake of a laugh. As different as the ducks might seem from one story to the next, they were whole in each story. Barks took great pains to make the events in his stories plausible, to make each story seem real on its own terms, and as a result there was in his stories something that was missing from almost all the superficially similar others: a powerful core of emotional continuity. Most talking animals, whether in comic books or in animated cartoons, were little more than costumes (with perhaps, in the movies, distinctive voices and a few obvious mannerisms), but Barks in his best stories preserved the sense that his characters were the same people, just behaving very differently as their circumstances changed. Those carefully plotted stories never seem mechanical, because so much of what happens seems to originate within the characters themselves.
The work of Barks's best years demands to be read as a suite of distinct but interrelated stories covering hundreds of pages, their nature ranging from farce to domestic comedy to exotic adventure. Only then is it really possible to understand what Barks was doing, and that the mutability of the ducks, and of Donald and his nephews in particular, was not unrestricted but existed within distinct but very wide boundaries. If the ducks change from story to story, Barks says clearly, that is because this is what people are like—never the same from moment to moment, much less day to day. Montaigne could have been writing about Barks's ducks: "Each man bears the entire form of man's estate."
Barks said of Donald: "He was just the everyday sort of a guy. It depended on what mood he had when he got up in the morning whether he was going to be a mean son of a gun all day or whether he was going to be a real good-natured fellow. . . . He was flexible, yes," unlike the Donald of the animated cartoons.
There, he was typed as a guy who had to be always squawking all the time, and making a lot of noise, and he was cranky and belligerent. I got away from it, because that was too hard to follow, to have a character that can only be cranky and belligerent all the time is very difficult. People would get tired of him, I would have thought. . . . As it was, I was able to make him into a sympathetic character at times, and a hero, and a heel. . . . He was just a whole variety of things, and I believe that was one of the reasons that people would pick up the book.
Donald in Barks's stories is variously brave and foolish, generous and spiteful, childish and mature. There are, however, certain things he can be imagined doing only under the most extreme circumstances, like being seriously derelict in his responsibilities as a parent. His mutations thus make him more real, not less, because they make him more like us: like Barks's Donald, we retain some core of identity through what may be tremendous changes in everything about us and around us.
It is difficult to think of comparable characters not just in other comic books but in popular culture outside comic books. Barks's Donald was remarkably complex, and it is for this reason that Barks's stories have always commanded the allegiance of a sizable cluster of adult readers. The stories attracted child readers through their comedy and propulsive narratives, and then held some of them—the ones not persuaded that comic books had to be abandoned with the onset of puberty—as those readers began to recognize themselves in his characters.
Barks was by the late 1940s very self-assured as a comic-book writer and artist, consistently extending himself in ways that other cartoonists never considered. Tom Gill, who began drawing the Lone Ranger comic book for Western in 1950, wrote many years later: "All cartoon illustration must be done from memory when making deadlines. There is no time for involved research. . . . Usually you'll have your picture files handy just to prod your memory, but normally you will fill the drawing area with your visions from your mind." Many of Barks's stories, though, reveal exactly the sort of research that Gill said was impossible.
In 1966, near the end of the most active years of his comic-book career, Barks described his "morgue," whose reference materials extended far beyond the panels clipped from the work of cartoonists he admired: "I have four files [filing cabinets, presumably] full of clippings of every sort of subject and type of drawing. Also have many years of Nat'l Geographics, and an Encyclopedia Britannica. The rock of Gibraltar picture in 'Ancient Persia' and the authentic-looking background props and frescoes are from Nat'l Geo. I simplify such material, naturally."
By the late 1940s, the settings of Barks's stories resembled actual places, especially where he lived at the time or had lived in the past, more than ever before. Any number of his stories are unmistakably set in either Los Angeles or the desert country to the east, and his snowstorms have a distinct Minnesota look to them. References to southern California, both visual (palm trees) and verbal (Burbank), are so plentiful in Barks's stories from the late 1940s and early 1950s that he even mentions a specific Los Angeles street intersection—of Wilshire Boulevard and Vermont Avenue—in "Vacation Time," a long story in the first issue, for summer 1950, of a 128-page "giant" comic book called Walt Disney's Vacation Parade.
Sometimes the sense of reality in one of Barks's stories is so strong that it tilts the story itself in a direction uncharacteristically serious. That is true of "Vacation Time," but what happens in that story is not the same as what happened three years earlier in "Adventure Down Under." Barks's plotting pushed "Adventure Down Under" into overly serious territory, but in "Vacation Time" everything—the settings, the situation (a devastating forest fire), and the characters (especially the villain, the sneering, cigarette-smoking fisherman who starts the fire through his carelessness)—is so insistently real that Barks's plotting, more than satisfactory in itself, is still not quite strong enough to permit him to bring comedy and drama into the most satisfying balance.
The ducks' world, in all of Barks's most successful stories, was realistic but not real. It was instead a world increasingly rich in comic detail—detail that was never so thick as to become overbearing (as could sometimes happen a few years later in the stories drawn by Will Elder for Harvey Kurtzman's Mad comic book) but that established a texture different from that of our own world. Comic-book stories in general, and talking-animal comic-book stories in particular, typically seem to have no "real life" outside the story itself, but in the best stories, especially Barks's best, what happens seems to take place in a current of everyday life. Not everyday life as we know it, but the everyday life of the characters on the page.
The storytelling in most comic books has always been like a crude, reduced version of the kind of storytelling that makes up the bulk of American popular narrative: character is sketchy and undeveloped, plot turns are correspondingly arbitrary, and nothing is allowed to stand in the way of a story's forward movement. A supreme stylist like Will Eisner in his "Spirit" stories could accept those conditions without surrendering any of his essential self, but what Barks did was entirely different. He held the comic-book culture's narrative barbarity at arm's length, acknowledging only a duty to entertain, to give the kids a dime's worth of enjoyment. While always bearing in mind that fundamental obligation, he depicted his characters and worked out his plots with a passionate attention to psychological and emotional exactitude that was worthy of a serious novelist or playwright, and he broke down his pages into panels, and composed each panel, as carefully as any serious filmmaker selected and composed his shots.
The result was an extraordinary truthfulness within the boundaries imposed first by the nature of the intended audience—young children—and then by the nature of the characters and their environment, both of which were always fantastic in important ways. In Barks's case, a sympathetic reader had to accept not only the possibility of real literature's emerging from a cheaply printed pamphlet sold mostly to children or bought for them, but also the legitimacy of continuing characters, which were automatically suspect in some educated readers' eyes no matter how many respectable precedents might be invoked. Moreover, almost all of those characters were talking animals, and Disney animals at that, the Disney association no longer being a positive one in the eyes of many educated readers by the time Barks was doing his best work. A sympathetic reader had to recognize that if stories are truthful, as Barks's best stories are, it makes no difference if they do not—or cannot, for external reasons—explore all the infinite varieties of human behavior. They point toward other, untold stories that are just as true.
What Barks did in comic-book stories could be compared to what Chaplin and Keaton did in films, as they mastered their medium and used it for a comedy that was, within a few years, much richer than anything in their earliest efforts. The difference was that the obstacles to acceptance of Barks's work were much higher. Not only were comic-book stories regarded with suspicion, but for the most part that suspicion was justified, the shallowness of the great bulk of comic-book stories being undeniable (and, for that matter, a major element in their appeal to children, because the stories were so undemanding). Even though the comic-book story itself was not hopelessly suspect as an artistic medium, few readers recognized that, and only a few cartoonists, like Barks, grasped the potential of their medium and tried to realize it, mostly without articulating what they were doing or reaching out to a broader audience. And then there were the publishers, almost none of whom had any respect for their own product.
That Barks enjoyed so much artistic freedom under those circumstances—and moreover, that he took full advantage of it—was all but miraculous, with astonishing results like the story called "Letter to Santa" in the first issue of the first Dell giant comic, Walt Disney's Christmas Parade, from 1949. Unlike most of Barks's longer stories, it is not a mixture of comedy and adventure but is instead completely comic, like most of the ten-page stories in Walt Disney's Comics & Stories. The comedy is sustained without a lapse through twenty-four pages. It is a Christmas story, in a children's comic book, but it is almost entirely sugar free. (After rejecting or manhandling two earlier Christmas stories that were comparably unsentimental, Barks's editors for some reason decided to leave this one alone.) It is also the first full-dress appearance of one of the most memorable comic-book characters, Donald's Uncle Scrooge McDuck—Barks's creation, and a character as fiercely acerbic as any of the great supporting actors in the Hollywood comedies of the 1930s and 1940s.
Scrooge in "Letter to Santa" was making his sixth appearance in one of Barks's stories since he had been introduced in "Christmas on Bear Mountain" two years earlier. When Barks used Scrooge for a second time in "The Old Castle's Secret," six months after "Bear Mountain," it was not because there had been any favorable response to the character either from his editors or from his readers. He heard, he told Malcolm Willits, "not a word, but I kind of liked old Scrooge and he filled a gap. We needed somebody to help Donald out."
In the 1949 story "Voodoo Hoodoo," Scrooge was identified for the first time (in a caption) as "the richest man in the world," but it was in "Letter to Santa" that he was first visibly so rich—as Donald says to himself, the "richest tycoon in the universe." Donald goes to Scrooge to ask for money to buy a steam shovel: he mistakenly believes that the nephews want a full-size steam shovel for Christmas, rather than the toy they actually want, and he has forgotten to mail their letter to Santa Claus asking for one. In Scrooge's office, cash spills around his desk and threatens to topple onto Scrooge himself. "The nerve of you, nephew!" he cries. "If I could afford a steam shovel, I'd have one in here shoveling this money out of the way!"
Scrooge was also more compelling and attractive than before, because he was far more energetic and irascible. When Donald tells him that he wants the money to buy a Christmas gift for the nephews, Scrooge has what looks like a change of heart, giving Donald what he calls a "wad of money! Go out and buy 'em a steam shovel!" Scrooge is indignant, though, when he realizes that the nephews will believe that Santa has brought the steam shovel: "What kind of a deal is that? I furnish the money to buy their present, and Santa Claus gets the credit!" It is his indignation that makes his generous gesture believable: he not only wants to be rich but also wants the world, his relatives especially, to bow to his fortune.
Scrooge quickly concludes that he must buy another steam shovel and deliver it himself. It is inconceivable that anyone else could be at the controls if he is to get the credit he knows he deserves, and Scrooge is clearly the sort of self-made man who is comfortable around heavy machinery. He says to himself as his chauffeur drives him in search of a steam shovel: "What's the use of having eleven octillion dollars if I don't make a big noise about it?"
When the police arrest Scrooge and Donald after they have battled with their steam shovels like two latter-day dinosaurs, Scrooge is as brazenly contemptuous of the law as any nineteenth-century robber baron, dressing down the judge and then, when he is fined a million dollars, tossing two million at the bench ("Put the rest in the kitty—in case we come back"), a remarkably cynical gesture for a children's comic book.
"Letter to Santa," in Walt Disney's Christmas Parade no. 1 (1949), was the first Carl Barks story to present Uncle Scrooge McDuck in all his irascible splendor. © 1949 Disney.
The story's lack of sentiment is underlined rather than contradicted by Scrooge's changes of heart—not one, but two. The second follows the ducks' court appearance, when he tells Donald that he has been an "old brat." Now Scrooge is going to help Donald cover for his failure to come up with a steam shovel by disguising him as Santa, so that Donald can feed the nephews a hard-luck story. Scrooge's acknowledgment of his personal failings turns out to be highly superficial. Within a few pages he is disguised as Santa himself and telling the nephews that if they had only asked Scrooge for a steam shovel, "you'd have gotten some action! Scrooge McDuck is the greatest man in the world! Why, he can give steam shovels easier than Santa Claus can give gumdrops!"
Santa Claus himself shows up as the story draws to an end, and this is a little surprising, because Barks usually avoided characters that were wholly fanciful even on the terms set by the ducks' universe. Barks's Santa is, however, a remarkably plausible personage—Barks has figured out how he can easily navigate up and down chimneys—and he is a very practical fellow, too. He gives the nephews exactly what they want, but only that—there is no flood of unrequested gifts. Most important, Santa by his very existence has generated the turmoil that precedes his entrance: if Santa did not exist, it would not matter that Donald had forgotten to mail the nephews' letter to him.
There is the occasional eccentric detail in "Letter to Santa," like the man who carries Christmas presents on a unicycle, but there are far more traces of a reality closer to our own. They serve simultaneously to leach the sugar out of the central idea and to make what happens in the story more believable. As Donald anxiously stalks the streets after realizing that he has failed to mail the nephews' letter, he passes near a sign: "Flop 25¢." Duckburg, a Disney city, has flophouses. When Donald tries to buy a steam shovel, he does not drive one out of the heavy-machinery lot, as any other comic-book talking animal would, but is told by a gruff overseer: "Listen, bud! You don't buy steam shovels like you do teaspoons! You gotta order 'em from the factory!" How many comparisons might that overseer have made, and how perfect is his "like you do teaspoons," calling up as it does thoughts of a mass-produced item much smaller than a steam shovel, but not pathetically trivial.
The slapstick in "Letter to Santa"—first the battle of the steam shovels, and then Donald's mishaps while he is disguised as Santa, culminating in the explosion that follows when his suit, full of wet beans, expands and bursts in the chimney—is set up and staged with the care that good slapstick always demands but rarely gets. It may be a tad too convenient that Scrooge has a second Santa Claus suit handy that he can wear when he takes Donald's place, and the concluding panels have an obligatory feeling (the story has ended on the previous page, but there has to be a coda of some kind), but if "Letter to Santa" is not perfect, its tiny imperfections are very easy to excuse.
There was, however, one small cloud over this marvelous story, arising from the fact that Scrooge himself was inherently a much more limited character than Donald. As Barks said, "[H]e did always have one characteristic, his desire for money. That was the first thought that would come into his head whenever he was in a dangerous situation, how to save his money rather than himself." And there were nuances in the Scrooge of "Letter to Santa"—in his interplay of greed, vanity, and spasms of self-interested generosity—that would be difficult to sustain in the comic-book environment, with its persistent tropism toward the simple and obvious.
Because he was producing so many long stories for Donald Duck and the new giant comics, Christmas Parade and Vacation Parade, Barks had to cut back on his work for Walt Disney's Comics & Stories. The ten-page "Donald Duck" story in the January 1950 issue was his last for a year, except for the March and June 1950 issues. Carl Buettner had been sidelined for months by a heart attack in the fall of 1949, and so it was his assistant, Tom McKimson, the former Warner Bros. cartoonist and a member of Western's staff since 1947, who wrote to Barks from Whitman's offices (which by then were at 405 North Bedford Drive in Beverly Hills) on August 4, 1950: "Your public is clamoring to see more Donald Duck, a la Barks . . . so we have decided to put your story-art combination back into the monthly magazines." Set aside the one-shot you're working on, Barks was told, and whip up a ten-page story for the February 1951 Walt Disney's Comics as quickly as possible. "When you get the plots worked out" for a couple of ten-pagers, McKimson wrote, "we'd enjoy having a brief synopsis of what you have in mind."
Barks submitted his first new ten-pager very quickly—quickly enough that it was published in no. 124, the January 1951 issue, rather than February. Alice Nielsen Cobb wrote to him on August 21 to tell him that his editors were very happy with it. ("Amen!" Tom McKimson wrote in the margin of the letter.)
But who made up the "public" McKimson mentioned? Were child readers and their parents writing to complain that the quality of the duck stories in Walt Disney's Comics had plummeted (as indeed it had)? It is difficult to read into the royalty figures for 1950 any direct effects of Barks's absence. The number of copies on which Disney received royalties—the number of copies printed—ranged in the course of the year from 2.5 million to 2.875 million, with the high figure for the August 1950 issue, the low for the November 1950 issue. Seasonal variations were almost certainly more important than any response to Barks's absence. Barks's name never appeared on his comic-book work, so anyone complaining that he was gone would have had to refer to him as "the good artist" or something similar. That was in fact the way that many young readers distinguished him from his lesser contemporaries.
The figures for Donald Duck showed a similar pattern, with summer print runs higher. Print runs for that title actually rose in 1951, to well above two million copies per issue, after Barks cut back on his Donald Duck stories so he could resume his monthly appearances in Walt Disney's Comics.
Barks told Donald Ault that he "never got a swelled head over hearing that [Walt Disney's Comics] was selling millions of copies. I knew it wouldn't last. And besides, Alice Cobb . . . told me my work wasn't selling those comics—it was Walt Disney's name over the titles." Cobb was surely right. Even if many children recognized and responded to Barks's superior work, the Disney characters and the comic-book format itself were the strongest lures. That reality would inevitably have some effect on the remarkable artistic freedom Barks enjoyed.
## 16
# An Arena for All the Passions
When colleagues described John Stanley, handsome was the first word they used. He was "strikingly handsome," Dan Noonan said. A "handsome son of a gun," Moe Gollub said. In the first of Walt Kelly's two Albert the Alligator and Pogo Possum one-shots, from 1946, Albert has Pogo in a barber chair when he shows him a wanted poster for "Perty Boy John," a Stanley caricature. "Look!" Albert says. "You kin be as dee-lishus as dish yere feller fo' twenny cents." Pogo responds: "Man—he perty!" Stanley was blue-eyed and prematurely gray, tall and slender—six feet two inches and 170 pounds when he registered for the draft in October 1940, and not much heavier, if at all, in photos taken years later. Cartoonists being what they are, his good looks made him a natural target for Kelly's good-natured abuse, in that story and others. But his colleagues were aware, too, of his exceptional intelligence.
Noonan described Stanley in a 1968 interview as "an omnivorous reader, always. He reads everything he can lay his hands on. I'd say he's an authority on writers like Samuel Pepys and Boswell. He has a very strange, wonderful feel for words; he's similar to S. J. Perelman in that sense. He loves words like 'foot-pad' or 'cut-purse,' or the like; strange-sounding Elizabethan terms appeal to him. He's got a very fine mind, and I think he could have been a serious writer had he chosen to be one." Stanley "used to send ideas to The New Yorker," Noonan said, "and Jim Geraghty, who was the cartoon director there, was so impressed with Stanley he wanted to give him a contract. Stanley wouldn't have any of it; he didn't want to be tied." Stanley's ideas were "very sophisticated gag ideas, all of them," Noonan said.
"Albert and the Barbecue," by Walt Kelly, in Albert the Alligator and Pogo Possum Four Color no. 105 (1946), is notable for its caricatures of Western Printing insiders. "Perty Boy John" on a wanted poster is John Stanley. On the second wanted poster, "Danny the Dip" in front view is Walt Kelly; the "side view" may be of Richard Small, a Western salesman and later an executive at the Poughkeepsie plant.
There seems to be no record in the New Yorker's archives of any such relationship with Stanley, or of anything he contributed to the magazine other than that one 1947 cartoon. Among Stanley's papers was a single letter from Geraghty, dated only November 11, no year, rejecting the ideas he had submitted the previous week. There is, however, every reason to believe that Stanley was a considerably more sophisticated man than the comic-book norm. The discrepancy between his distinguished appearance and literary pursuits on the one hand, and what was unquestionably low-status work in comic books on the other, surely fed his dismissive attitude toward his own work. And it may have made alcohol more attractive, too.
Moe Gollub said of Stanley: "He drank a little bit; he wasn't a lush or anything, but every weekend he'd hit a few, mostly because he couldn't sleep nights. His Irish background, with all the restraints that his religious parents put on him, seemed to have had some negative effects that he could never really consciously offset, although he was not anything like religious."
Like Marjorie Henderson Buell's panel cartoons, Stanley's early Little Lulu stories had a strong flavor of the 1930s, although in his case that flavor lingered throughout the 1950s. Most obviously, the children, especially the boys, dressed in styles of early in the twentieth century—in knickers, short pants, sweatshirts, and cloth caps. Tubby, the fat boy who was Lulu's costar (called "Joe" in the panel cartoons on the rare occasions when he was named), always wore a sort of black suit coat and white cravat—and shorts, regardless of the weather, with a tiny sailor hat perched incongruously atop his large round head.
Lulu's hometown was not suburban but looked like a small eastern city of the Depression and World War II years: plain houses with tiny yards or none at all. (Many front doors opened directly onto the sidewalk.) Families were small—Lulu, Tubby, and Lulu's next-door neighbor Alvin, the third major character, had no siblings—and parents usually looked middle-aged or close to it, as if they had postponed having a child until defense spending lifted the economy after ten years of the Depression. Lulu's town could have been a small city in the Hudson valley, where Stanley himself lived and worked after moving to Croton-on-Hudson in Westchester County—the same town where Oskar Lebeck lived—sometime in the mid-1940s.
Stanley's Lulu stories remained frozen in time, with only a few concessions—like the very occasional appearance of a television set—to postwar changes. Lulu and the other characters visited no other towns, much less other countries. Stanley said he restricted the settings of the Lulu stories for practical reasons: "Why did all the Little Lulu stories take place within one neighborhood? Because it saved me all kinds of research. Why should I send her to Alaska when all the passions that make a story absorbing are already present within the neighborhood setting? It was laziness."
Marjorie Buell and Stanley differed on just how much raw material she gave Stanley to work with. In handwritten notes dating from 1984, Buell said: "I drew model charts of Little Lulu, Tubby, and Alvin for Western as I did for many licensees, and it was clearly understood that I had the final word on the drawings and stories in the Western Little Lulu comics. Western sent drafts of each comic book for me to OK and I returned the drafts sometimes with suggestions for changes but most of the time without." Lulu's parents and Tubby's parents also made the transition from panel cartoons to comic books, she said. "Many other characters in the Saturday Evening Post Lulu cartoons resembled to some extent other characters in the Western Lulu comic books."
Stanley was fully as expansive, and probably more accurate, in his claims to authorship. For one thing, no character identifiable as Alvin can be found in the last three years of the Post panels or any of the half dozen book collections of the Lulu panels published by Rand McNally and the David McKay Company. "LeBeck [sic] gave me two characters, Lulu and Tubby, nothing more, to do something with in a comic book, and I took it from there," Stanley wrote. "Marge supplied their last names, Moppet and Tompkins. The rest of the characters, all of them, till the last issue done by me, were conceived, drawn and named by me." Some details carried over from the Post panels to the comic books—"No wimmen allowed" on a boys' clubhouse in the Post became "No girls allowed" on the clubhouse in the comic books—and a front-cover drawing by Stanley might echo one of the Post drawings. But that did not happen often.
There was no cheeriness in Stanley's depiction of Lulu and his other child characters, none in the writing or in the drawing, not a trace of the cuteness that came so effortlessly to Walt Kelly. In some of the early one-shot issues of Little Lulu, the characters seem isolated, firmly separated by empty space even in those panels with more than one character. In plain black-and-white reproductions of the inked lines, the drawings—simpler and more uniform than Buell's for the Post—look especially stark. There was in Little Lulu no mediation of the sort that Dan Gormley's drawings had provided for Stanley in New Funnies. As Bud Tripp's hard-edged ink lines came to dominate how the Lulu comic book looked, they did not add any Gormley-style warmth to Stanley's scripts—or storyboards, as Stanley himself called them—but rather made them seem more abstract, adding what was sometimes a welcome distance from characters whose sufferings might otherwise have seemed uncomfortably real.
Compared with Kelly's stories, which were rooted in nineteenth-century illustration (by way of the Disney studio) and Victorian children's fiction, Stanley's early stories could seem strikingly modern. They were far removed from the likes of the Archie stories, which resembled television situation comedies in their combination of the earnest and the shoddy, but they were also far removed from stories of the kind that Oskar Lebeck had found most congenial. There was never a hint, though, in anything Stanley said that Lebeck was other than "totally supportive." As Dan Noonan said, Lebeck was guided by his sense of "what was good, and what was genuinely funny," and he could hardly have found Stanley's best work anything other than good and funny.
For all that Lulu's town evoked a real city in a real part of the world, Stanley never gave it a name that stuck, and there was never any sense that it had a fixed geography, no sense that Lulu and the other characters lived in an imaginary world with constraints like those that the real world imposes. Instead, everything was more fluid, not just geography but also names and physiognomies, so that only a handful of characters at the center of the stories—Lulu, Tubby, Alvin, Lulu's parents, and eventually a few other children—were constants. Lots of things were cloudy: Lulu's father had a job (her mother was a housewife), but what kind of job was it? The senior Moppets and Tompkinses were acquainted, but were they really friends, since they addressed one another as "Mr." and "Mrs."? The stores were small and bare and emphatically old-fashioned, like something out of Communist Eastern Europe. A clergyman appeared in one early story, but otherwise it was hard to find any trace of religion. The children attended school, went to the movies, and so on, but all of those places were props that Stanley wheeled onstage as they were needed.
The children in Little Lulu had remarkably little adult supervision by contemporary standards, and not very much even by the standards that prevailed when the stories were published. Lulu and her friends were presented as elementary school children of seven or eight years old (ages and grade level were changeable and usually vague), but they walked about their town as they pleased, and the same was true even of children who were supposed to be a few years younger, like Lulu's neighbor Alvin. They were all latchkey children, entering and leaving their homes freely—a door or window was almost always unlocked—no matter whether their parents were at home. Lulu, Tubby, and their friends also found it remarkably easy to slip out of their homes and engage in extended nocturnal adventures. And not always nocturnal: in Little Lulu no. 1, January–February 1948, Lulu and Tubby scale a five-story building by using protruding bricks as handholds, stepping through one window into an apartment for a drink of water and reaching through another window to make telephone calls to their mothers ("Hello, mother, guess where I am"). Lulu does take the precaution of keeping her eyes shut throughout the climb.
There was, to be sure, a corollary to this remarkable freedom of action: physical punishment, in the form of spankings not just by parents but also by other adults, was easily provoked and could be severe. But parents, like adults generally, were distractions in Stanley's best stories, where, as he said, "all the passions that make a story absorbing" were concentrated not just in a small neighborhood but almost entirely in a handful of child characters. There is, however, never any suggestion in Stanley's Little Lulu stories that he was observing real children as a source, in the manner later made famous by newspaper cartoonists like Hank Ketcham (Dennis the Menace) and Bil Keane (The Family Circus). In some stories, early and late, the sense of Lulu and Tubby as children is so tenuous that Andy Panda and Charlie Chicken can easily be imagined in their places. Stanley was looking at families and children from outside: he was neither husband nor parent during almost all of his time as Little Lulu's guiding intelligence.
In "Mountain Climbers," in Little Lulu no. 1, January-February 1948, written by John Stanley and illustrated by Charles Hedinger and Irving Tripp, Tubby leads Lulu up a building using protruding bricks as handholds.
Stanley's version of Lulu herself initially resembled Marge's. In the first story in the first one-shot she is a tomboyish female bully who resists wearing an angel costume to a children's party and commandeers Tubby's beard—he is dressed as a pirate—so that she can at least go as an old angel. But in the second story, "Lulu at the Beach," she is already not so much mischievous as single-minded and self-absorbed, very much like Stanley's New Funnies characters. Like those characters, she has a clear idea of what is important to her, and she is not distracted by irrelevancies, especially by what adults think is important.
The stage was thus set for collisions between adults and children, all of whom were behaving reasonably on their own terms—not that it made any difference to the frequently calamitous results. Within a few years, though, Lulu was in Stanley's stories no less self-possessed but now dealing forthrightly with destructive forces rather than being one. In a 1970 interview, long after Stanley departed Little Lulu, Marjorie Buell contrasted the humor of her Post panels with the comic book. In the comic book, she said, "Lulu was quite childlike and honest."
Lulu could assume that more congenial role for the title character of the comic book because "Joe," her foil for years in the Post, was available to become Tubby, a character very much like the one she had been, only more so. Usually a victim of Lulu's willfulness in the magazine cartoons—where she might maul "Joe" so she could put her first-aid kit to use—Tubby in Stanley's stories blossomed into a sublimely narcissistic monster, not actively malicious but lacking any trace of a conscience. The transition was complete by "The Kid Who Came to Dinner," in the eighth Little Lulu one-shot (Four Color no. 146, 1947). Tubby invites himself to eat dinner with Lulu and her parents, cheerfully insults his hosts without showing the slightest awareness he is doing so, pokes at Mr. Moppet's food while asking if he is going to finish it (Lulu's father transfers the compromised morsel to Tubby's plate), eats so much that he falls ill, and finally accuses Lulu's mother of poisoning him. When the Moppets return to Tubby's sickbed—or sofa, actually—after calling a doctor, he is gone: he has left in a rush so that he can return to his own home in time to eat dinner there.
"The Kid Who Came to Dinner," in Little Lulu Four Color no. 146 (1947), written by John Stanley and illustrated by Charles Hedinger and Irving Tripp, was one of the first stories to showcase Tubby's mad self-absorption. © 1947, 1974 Marjorie Henderson Buell.
What makes Tubby funny rather than alarming is that Stanley presents him as a child; the younger the child, the more tolerable, and even amusing, the behavior that in an adult would invite a diagnosis of, say, narcissistic personality disorder. Stanley's stories were evolving in a way that made it impossible to think of Tubby as being very much like a real child—real children did not wander freely in a stage-set town—but Stanley made sure he resembled one just enough. His characters were never vulnerable to the suggestion so often made about the children in Charles Schulz's comic strip Peanuts—that they were adults masquerading as children. They were instead children whose quarrels and schemes echoed adult life.
Stanley's strategy took full advantage of his characters' severely stylized design—their solid black oval eyes, up-pointing noses, and simple or invisible mouths—and how it discouraged too close an identification of reader and character. Over the first four or five years of Little Lulu, a small number of child characters acquired a standard appearance and consistent names—Willie, Eddie, and Iggy, the "fellers," were Tubby's clubhouse pals, and Annie, a buck-toothed gamine, was Lulu's friend—but although clothing and hair color differed, their faces were identical, Annie's buck teeth aside. Stanley introduced Gloria, a curly-haired blonde, in Four Color no. 131 (1947), and in 1949 he began presenting her as a preening beauty. Many stories thereafter turned on how much better looking she was, compared with the supposedly very plain Lulu. For anyone who paid attention to what the characters actually looked like, the joke was unmistakable.
It was in the first Little Lulu one-shot in 1945 that Stanley introduced the most important of the comic book's characters aside from Lulu and Tubby: Alvin, Lulu's next-door neighbor. Alvin's personality was not far removed from that of the original Post version of Lulu or Stanley's version of Tubby, but because he was presented as several years younger, he was, compared with the other two characters, pure id. He was savagely egotistical, a three- or four-year-old child who not only disregarded all social restraints but attacked them violently. Stanley was attracted to such small horrors—as when he pitted Woody Woodpecker against a demonic baby a year earlier, in New Funnies—but they were inherently limited as foils. Nihilism lacks nuance, and Stanley had to find some way to make Alvin more interesting without diluting his personality.
His solution, starting with the third Little Lulu one-shot (Four Color no. 110, 1946), had two parts: he made Alvin a little older and more articulate than he was in his first appearance; but, more important, he had Lulu subdue Alvin's destructive nature, if only temporarily, by telling mock fairy tales. These fairy tales were, in marked contrast to Alvin himself, cool and ironical. Lulu was the featured player as well as the narrator, and she usually took the part of a cheerful Candide—almost always "the poor little girl"—in a world dominated by greedy and stupid adults.
In the first such story, "Lulu in Distress," the orphan Lulu is the victim of a fantastically cruel stepmother and a capricious stepbrother (named Alvin and indistinguishable from Lulu's neighbor), who insists on celebrating his birthday every day and twice on Sunday. As the ragged and desperately overworked Lulu gazes longingly at the candies in a shop display, the seemingly sympathetic shopkeeper is reduced to tears—but then, when she asks if he has any samples, he yells at her: "NO!! GEDDADDA HERE!!" before he returns to bawling in the next panel. As Lulu explains in a caption, "People were so sorry for me that they just couldn't bear to have me around . . ."
In the fairy tales, Stanley relied heavily on captions like that one to achieve the effects he wanted. (He used captions sparingly in stories of other kinds.) The captions, in Lulu's voice, exude a mock innocence and sentimentality that is completely at odds with what the panels show: a world in which adults exploit children shamelessly. What the reader sees and what Alvin supposedly hears are thus very different, although Stanley did not explore that disjunction directly, since there was never any question about who the real audience was.
Stanley's reliance on captions was, as so often with him, a procedure whose substantial risks were not immediately obvious. Comic-book captions are typically redundant, duplicating in words what is (or should be) clear from the pictures. The most lamentable examples were the celebrated EC crime, horror, and science-fiction comic books of the early 1950s, edited and largely written by Al Feldstein, and often illustrated with great skill by artists like Wallace Wood and Jack Davis. Feldstein's captions accentuated what were too often hackneyed plots, hobbled by a conception of the short story that required an ironic "twist" ending of the sort associated with O. Henry. Stanley's captions, by contrast, worked always in counterpoint to his drawings and their dialogue balloons, the stories much funnier when the captions were read along with the panels. Sometimes Stanley went so far as to fill a whole panel with words; that is, a caption took over completely. His confidence in his words was almost always justified.
He ran the same sort of risk when he dispensed with words altogether, in the stories he told entirely in pictures. He dabbled in such stories in New Funnies, but he went further in Little Lulu, starting with a six-page story in the third one-shot, Four Color no. 110, and pushing all the way out to ten pages in the last Four Color issue, no. 165 (1947). Such stories required a virtuoso's timing—there could not appear to be too much or too little elapsed time within a panel, or the story's rhythm would be fatally disrupted—as well as construction that made the absence of dialogue, so important an element in most comic-book stories, seem natural in this instance. Stanley passed such tests easily.
There was an underlying sense of experiment in Stanley's Little Lulu stories, so formally tame on the surface. Stanley's staging—how he positioned his characters within each panel—was from the beginning much less varied than that of many other comic-book creators, in keeping with the severe simplicity of his character designs. The characters were most often seen full figure, as if at some distance from the reader (although after a few issues he did soften the starkness of his earliest stories by bringing the characters a little more forward on each panel's stage). The occasional close-up was in striking contrast, and Stanley rarely used one except when a character's features were to be distorted for comic emphasis.
Little Lulu made the transition from a de facto bimonthly schedule in 1947 to an official bimonthly schedule in 1948 and finally a monthly schedule in 1949—a growing success for which Stanley was almost entirely responsible. He was by the late 1940s both exceptionally productive and exceptionally facile. Even when there were dry periods, as happened in 1948 and 1949, he could produce stories that were superior by ordinary comic-book standards, for all that they slipped into predictability within a page or two.
What Stanley needed were more templates like the fairy tales—starting points for stories, of a general kind. He needed such templates in the late 1940s more than Carl Barks did, because Barks did not hesitate to send his ducks to exotic locations. Stanley worked within stricter confines—by choice, as he said, but also out of near necessity, since his comic-book children could not have left their homes without arduous and unconvincing preliminaries. Besides, there was always the risk that Stanley would succumb to boredom without the stimulation that a good template could provide. In a fair number of his Lulu stories, as in some of his stories for New Funnies, there is a strong sense that Stanley—the cartoonist who claimed that he typically started a story without any idea where he would wind up—has lost interest and is ending a story as quickly as he can, even though his stories were never disfigured by the insolent carelessness of which other cartoonists could be guilty.
Stanley gradually accumulated an assortment of templates of varying usefulness. When Lulu was pursued by the truant officer after being mistaken for a hooky player, not much could differ from story to story except the sight gags. Stanley sent his characters to the beach fairly often, too, without exceptional results. But the fairy tales were rich with possibilities, as were, only slightly less so, ghost stories. The first important story of the latter kind appeared in Little Lulu no. 15, September 1949, when a ghost—a real one, in comic-book context—haunts Lulu's dollhouse.
The categories sometimes overlapped, as when Lulu told Alvin a ghost story, and in stories of both kinds Stanley sometimes verged on serious grimness, even though it was always concealed behind innocent-looking cartoon drawings. In the fairy tale in Little Lulu no. 17, November 1949, there is violent death, in quantity, shown not in graphic detail but through its consequences: the complete absence of the many hundreds of people who have been devoured by a dragon. The dragon is guarding the castle where Lulu is imprisoned, and a fortune in gold is waiting for whoever can cross the moat where the dragon is lurking. The mob has rushed the castle, gambling that the dragon can kill only a few people before the rest make it inside. This turns out to be a miscalculation.
In the framing story for this fairy tale, Lulu's storytelling is revealed to be more than a means of taming Alvin: she wants to avoid him and his demand for a new story, but then, when she is successful, she misses him. He is a pest but also a challenge to her imagination.
In the ghost stories, there is more than once the danger, presented seriously if sometimes in what turns out to be a dream, that Lulu or Tubby will themselves become ghosts—that is, that they will die. Such stories somehow passed Marjorie Buell's scrutiny, but she rejected a story, scheduled for the August 1950 issue, called "The Bogyman." It was, she said in a handwritten draft for a 1985 letter, "an ugly, tasteless, scary story, entirely out of character and way below the high standards of the Little Lulu comics." Given that many other stories at least as "scary" were acceptable to Buell, the fundamental problem may have been that the Bogyman was actually shown, as a grotesquely elongated and hairy monster with six ears, and not just talked about, like some of the menaces in other stories.
Stanley's most fruitful source of story ideas, apart from the fairy tales, proved to be the rivalry between groups of boys and girls. The first story of that kind was "Little Lulu Fights Back with a Club," in the fourth one-shot (Four Color no. 115, 1946). Stanley spins out a conflict between juveniles across sixteen pages, but that story is, unfortunately, one of those that end awkwardly. (Tubby and his friends decide that they actually like crocheting doilies.) Until then Stanley, like other writers for the Dell comic books, had benefited from the opportunity to write stories longer than many other publishers permitted. In the early issues of Little Lulu a story like "The Hooky Team" (in Four Color no. 139, 1947) might fill as many as twenty pages.
As Stanley elaborated on the rivalry theme in subsequent stories, Lulu herself came into focus as a "good little girl" who outsmarted the boys, instead of triumphing through sheer brass as she did in his earliest stories with the character. She became, more than that, a sly trickster, much more closely resembling Brer Rabbit than Walt Kelly's Pogo ever did. There were no racial complications in Stanley's stories, either; the entire cast was white and only rarely was there a hint of ethnicity. Even the butcher with a German-sounding name, Mr. Kohlkutz, had no accent.
Lulu's trickster identity was established by 1949, but it took a while for Stanley to realize fully the possibilities, as he finally did in "Five Little Babies," in the August 1951 Little Lulu. Deceived and humiliated by Tubby and his friends, Lulu responds through a carefully worked-out scheme that ultimately delivers the boys, clad only in diapers and stacked under a blanket on a coaster wagon, into the heart of a rival gang's neighborhood. Devising and executing such a scheme would certainly have been beyond the capacities of any real-life grade-school child who was not a precocious monster, but by 1951 Stanley was so sure-handed with his characters that Lulu's triumph commands assent as well as laughter.
Even as Stanley became more tightly linked to Little Lulu, he continued both to write and to illustrate features in other comic books, notably "Jigg and Mooch" (originally "Jigger") in the last seven issues of Animal Comics, in 1946–47, and "Peterkin Pottle" in eight 1949 issues of Raggedy Ann & Andy. Mooch of "Jigg and Mooch," the big canine comedian in a big/small pairing, was so willfully obtuse as to be delusional, and Peterkin Pottle was a juvenile Walter Mitty. These were, it turned out, very narrowly focused features that probably had not much more life in them when they ended their short runs.
Peterkin's daydreams would seem to have been as fertile a source of comedy as Lulu's stories for Alvin, but, curiously, such was not the case. Peterkin daydreamed to retreat from the world, whereas Lulu in her stories for Alvin—he was her adversary, after all—engaged the world on her own terms. Unlike Peterkin, she was almost always in charge.
## 17
# Animal Kingdoms
As John Stanley took up work on Little Lulu in the mid-1940s, what was emerging in Walt Kelly's "Albert and Pogo" stories around the same time was constantly percolating ensemble comedy. Built less on real stories than on how eccentric characters bumped up against one another, this was a kind of comedy that was common in radio, on shows like those of Jack Benny, Fred Allen, and Fibber McGee and Molly—not to mention Amos 'n' Andy—but had no parallels in comic books and relatively few in newspaper comic strips, with prominent exceptions like George Herriman's Krazy Kat and Elzie Segar's Thimble Theatre, especially in its Sunday-page incarnation. Those cartoonists understood that the loose, open-ended comic-strip format was highly accommodating to a free flow of invention but was also forgiving when the cartoonist marked time for a few days while waiting for inspiration to return. It was a format made to order for the sort of cartoonist Kelly was becoming.
Kelly was, in the "Albert and Pogo" stories from 1944–47, a sort of comic-strip cartoonist in waiting—literally so, because heavily reworked versions of some of those stories, or parts of them, turned up a few years later in the Pogo strip. The differences were often subtle—matters of staging, gesture, and emphasis—but the cumulative effect was vast improvement. Kelly also improved rapidly as a draftsman after he began drawing the comic strip, and it benefited as well from his own lettering (or lettering that he supervised) in the dialogue balloons. The underlying problem with "Albert and Pogo" in comic books was that the characters were not sufficiently developed—and at first not sufficiently numerous—to sustain the kind of comedy that Kelly seemed to have in mind; and so he fell back on broader and cruder expedients.
In two stories, a year and a half apart, Kelly had Albert become "mean" in response to insults, suddenly looking and behaving more like a real alligator. The effect is startling because Albert, like Pogo, had become more Disney-like since his first appearance—that is, recognizable as the animal he was supposed to be, but with attributes (expressive eyes and mouth, hands that can grasp, upright posture) that permitted that animal to behave like a human being. In the fifth issue of Animal Comics, when Albert is at the train station, he indignantly declares to the humans who are fighting over who will own him: "I isn't no dawg or hoss! I is a reg'lar hooman!"
Transformations like Albert's into a "mean" version of himself were the kind of storytelling shortcut of which comic books were often guilty, and Kelly was guilty of others. In Animal Comics no. 15, June–July 1945, in the first story to elevate Pogo to costarring status under the title "Albert and Pogo," there is a real villain, a medicine-show huckster of wolflike appearance who threatens Pogo with a knife. (This character could be confused with Seminole Sam, the fox con man of the comic strip, but the first such fox actually appeared in Animal Comics no. 11, October–November 1944.) The villain is menacing in a direct manner and is defeated in the same way, by a skunk. There is no room in such hopelessly blunt stories for comedy of character, or subtlety of any kind.
As Kelly's feature evolved, and especially after its title became "Albert and Pogo," it began to be handicapped by a lack of resemblance to Amos 'n' Andy in one crucial respect. The radio show, particularly in its early years, played the sensible, low-key Amos off against the blustering, foolish Andy—the roles eventually assumed in the Pogo comic strip by Pogo himself as the equivalent of Amos, and Albert as a large reptilian version of Andy. Amos 'n' Andy's supporting cast was made up of such uncomfortably memorable black characters as the boastful Kingfish, the shyster lawyer Algonquin J. Calhoun, and the pokey messenger Lightnin'. As those characters came to dominate Amos 'n' Andy, Amos himself all but disappeared from the show, and its comedy became increasingly raucous and predictable.
In the middle 1940s, Kelly was gradually accumulating a cast of animal oddballs that invited comparisons with Amos 'n' Andy's supporting cast. The crackpot scientist Howland Owl and the pirate turtle Churchy LaFemme both first appeared in Animal Comics no. 13, February–March 1945, the first issue without Bumbazine. But Pogo, unlike Amos on the radio, assumed an increasingly important role in the "Albert and Pogo" stories. Along with Albert, he had become a more Disney-like animal since the earliest issues of Animal Comics, as his snout turned up, his body acquired a generally softer and more rounded appearance, and his head grew larger, so that his body had more childlike proportions. But now he seemed to scowl a large part of the time.
The problem was that Pogo had become too much like the other characters, competing with them instead of serving as a sensible counterweight to the foolishness around him. It was as if Amos, instead of retreating offstage, had assumed a more prominent role on the radio show, one that required that he be much more aggressive in dealing with his feckless friends. In "The Catfish Pirates," in the 1947 Albert the Alligator and Pogo Possum one-shot, Pogo's superior intelligence—always implied in these stories—is fully in evidence when he easily outwits Howland Owl and Churchy LaFemme, but they are too foolish for his defeating them to be either surprising or endearing.
Occasionally, as in a story in the same one-shot, "Mr. Owl and the Atomic Bomb"—later reworked for the Pogo comic strip—there is a hint of a sweeter, more benign Pogo, concerned more with protecting weaker creatures (including his weak-minded friends) than with defeating anyone. It was such a Pogo that would provide the balance Kelly's stories needed—not in Animal Comics, as it happened, since the necessary balance was never quite achieved there, but in the Pogo Possum comic book that began publication in 1949, and most successfully in the comic strip that began national syndication in the spring of that year.
To page through the Dell comic books of the early and mid-1940s is to be reminded that even in those years Kelly drew much better than most of his colleagues. His stories, with their Disney-bred draftsmanship, stand out in the midst of work that is more labored than his, or stodgier looking, or both. Oskar Lebeck and Western Printing's Los Angeles editors were desperately short of writing and drawing talent during the war years. Always in the Lebeck-edited comic books through the mid-1940s there is the sense that a thin line of competent artists and writers is holding chaos at bay, with the semicompetent and sometimes the incompetent filling in behind them. In such an environment, it was no wonder that Disney veterans like Dan Noonan and Moe Gollub were so welcome when they left the military and came looking for work.
"Mr. Owl and the Atomic Bomb," in Albert the Alligator and Pogo Possum Four Color no. 148 (1947), later reappeared in a new version in the Pogo daily comic strip.
Noonan recalled,
I was discharged from the service in May of '45, on V-E Day as a matter of fact. Kelly was then at Western Printing, and I'd gone up to visit him at his home in Darien [Connecticut]. He suggested that I come to work for Lebeck, so I went in and talked to Oskar. Oskar was one of the nicest people I ever worked for—a very good man. He gave me a shot at doing a story for [Fairy Tale Parade]. I wrote this thing, and he liked it, and he asked me if I wanted to draw it. I told him I would try, and he liked that, and I was on.
Noonan and Moe Gollub knew each other from Disney, where they had both worked in the Bambi unit, and Gollub also knew Kelly "fairly well." Both took part in the 1941 strike. Gollub was let go on September 12, 1941, soon after the strike ended—the same date on which Kelly's Disney employment ended, although under different circumstances. "I knew the war was going to start, and I didn't even bother looking for a job," Gollub said. "I was in the service"—the navy—"as soon as the war started." Noonan entered the coast guard.
When Gollub left the navy, he assumed he would be blackballed in the animation industry, as a striker and therefore a potential troublemaker. "I figured that if I couldn't work in the 'good' places, as I thought of it, I didn't want to work in the industry. That's the only reason I ever got into comic books." Although he had enlisted in Long Beach, California, Gollub talked the navy into discharging him at another Long Beach—in New York, on Long Island, on October 7, 1945. "Noonan met me at the station when I got out, and we stayed together," Gollub said. "We had Norman Rockwell's old studio [in New Rochelle, north of New York City]; it was also Frederick Remington's old place. An old place, and man, it was drafty. After a while, we had to move out of there; it was too cold to work. I moved down to New York City eventually, and it was a fair life for me, in a simple sort of way." As for paying work, "it was no time at all that Kelly had me up there seeing Oskar Lebeck, and on Kelly's word, and Noonan's word, I just went to work. And they were very good to me."
The advent of Gollub and Noonan made for a dramatic change in Animal Comics in particular because they could draw real animals better than anyone else who had been drawing for Lebeck. Kelly's animals seemed real, certainly, but usually not in the same literal sense; Gollub and Noonan worked in much straighter styles, Gollub's softer and more heavily modeled, Noonan's tighter and wirier. Suddenly there were in no. 18, the December 1945–January 1946 issue, in addition to Kelly's "Albert and Pogo" and H. R. McBride's crabbed "Uncle Wiggily," features by Noonan about a "fire dog" and bear cubs, both drawn in a realistic style far more persuasive than the few earlier efforts of the kind, and making little or no use of traditional dialogue balloons. Gollub's first story for Animal Comics, "Cubby and Tubby," also about bear cubs and also without dialogue balloons, appeared in the next issue.
More such stories by Noonan and Gollub followed, and Noonan's feature "Rover," about a homeless spaniel, eventually displaced "Albert and Pogo" from the front cover that Kelly's feature had seized from "Uncle Wiggily." "Rover," which Noonan wrote as well as drew, was a very serious and occasionally grim serial (the second installment included a murder and an implied lynching of the murderers), an animal story of the classic Black Beauty kind, although not told in the first person.
What Lebeck had found in Gollub and Noonan, as he had earlier in Kelly, were artists who drew so well that they could bring a strong storybook flavor to his comic books. The 1946 Santa Claus Funnies one-shot was drawn entirely by those three men. Although Kelly drew a genuine comic-book story, "A Mouse in the House," Gollub and Noonan drew, in realistic styles, stories with captions rather than dialogue balloons. Lebeck wrote "Santa and the Angel," which Gollub illustrated, and it filled twenty-six of the comic book's forty-eight interior pages. Neither Lebeck nor Gollub was credited in the book itself, but they were credited by name when Santa and the Angel was reprinted in its own comic book just three years later. Such reprinting was itself unusual, but that story was reprinted a second time in a 1954 giant comic book called A Christmas Treasury—several years after Lebeck left Western but with his credit intact on the title page.
In 1947, there was a second Lebeck–Gollub collaboration, "A Letter to Santa." This time both men were credited on the title page (and Lebeck's name was somehow misspelled "Oscar" on the front cover). The story was published in that year's issue of Santa Claus Funnies. There was another Kelly comic-book story, too, again at the back of the book, as if to emphasize that Lebeck had transferred his affection to the illustrations that Moe Gollub and Dan Noonan were now giving him.
Many of Lebeck's comic books resembled traditional children's books in more than just their emphasis on fairy tales and nursery rhymes. If stories like "Santa and the Angel" and "A Letter to Santa" had been published in boards and on better paper than newsprint, they could have fit very comfortably in the children's section of a bookstore. Lebeck did his best to blur the line: in 1948, Raggedy Ann & Andy published "a monthly book feature," illustrated text stories, leading in March with Dan Noonan's "Framingham," about a fox. "Framingham" was evidently never published as a real book, but the April installment was a new version of just such a book, Beatrix Potter's Peter Rabbit, with Moe Gollub's illustrations in place of Potter's. The Potter book was in public domain in the United States, thanks to its publisher's carelessness about registering the copyright.
The front cover of Santa Claus Funnies Four Color no. 128 (1946), by Moe Gollub, illustrated the featured story, "Santa and the Angel," written by Oskar Lebeck and illustrated by Gollub.
Raggedy Ann's May installment, "Michael Finnegan," illustrated by Gollub, had been published as a book two years earlier by Grosset & Dunlap. That book had been packaged for Grosset by Western's Artists & Writers Guild subsidiary, with illustrations by a different artist. The comic-book version did not credit the author, Irene Little—probably the same person who a few years later was finishing comic-book pages in ink for Western's New Funnies. A few more adaptations of stories produced by other Western subdivisions followed in 1948 before the "monthly book feature"—by then no longer identified as such—disappeared late in the year.
The Western-produced storybooks adapted for Raggedy Ann occupied the lower rungs of the children's-literature ladder when they were first published (the original Potter Peter Rabbit excepted, of course). But the fundamental problem with Lebeck's comic books that resembled children's books was that children—and parents—who were attracted to such material had no reason, apart from price, not to move on to real books instead. Successful comic books were not like children's books or comic strips or anything else, but were comic books and nothing else. Given that climate, Lebeck's aspirations had a sort of poignancy. The antic flavor of some of Walt Kelly's stories in comic books like Santa Claus Funnies was a more realistic response to the circumstances.
Moe Gollub in particular was a pure illustrator of a kind—George Kerr and Arthur Jameson were earlier examples—that had always flourished at Western alongside cartoonists like Kelly. "In the early days," Gollub said, "I tried to write stories. I never liked writing stories; I thought it was time away from the drawing board. Noonan liked to write; he could do it better than I did, and he realized it wasn't really such a big thing, for children's books. I enjoyed the drawing." Lebeck accommodated Gollub by providing him with scripts,
and on top of that, he was kind enough to give me a raise over the then prevailing rate. I had no squawks with him, I must say . . . . He was really good to all of us; there wasn't a guy there who had a squawk with Oskar. He was generous, he never was critical. He took the big view of what you did, he never picked at little things. There were any number of editors in the comic-book business who'd make a big thing out of something that nobody would ever even notice. He had a much more intelligent attitude.
"Chuckwagon Charley's Tales," in Animal Comics no. 29, October-November 1947, was written by Gaylord DuBois and illustrated by Moe Gollub.
Richard Hall, who drew for the Walter Lantz comic books from roughly 1946 to 1955—six years in New York, followed by three in Los Angeles—remembered Lebeck as "a great guy." Another cartoonist usually made the inked drawings over Hall's pencils, in keeping with what was becoming general industry practice. If Lebeck questioned anything in his pencil drawings, Hall said, "he would say, 'Look, Dick, next time, don't do it this way, do it a little different way.' He never made me do anything over; he didn't nitpick. But you did it the way he wanted. I used to write stories, and then finally he said, 'Dick, it's easier for you to just draw. And you will make more money.' After about a year of writing and drawing both," Hall said, he was "very glad" to give up the writing.
Lebeck encouraged Gaylord DuBois in a different way. As DuBois's biographer Irvin H. Ziemann has written: "In 1946, while the DuBoises were living in Westport, N.Y., Gaylord was called to Western's Fifth Avenue office for a conference and told by Oskar Lebeck that his western comics were becoming repetitious. When DuBois asked what to do, Lebeck replied, 'Go west, young man, go west! Find the best bargain you can in a used travel trailer and we'll advance you the money.'"
By that time DuBois had written only six to eight issues of Roy Rogers Comics, which were published as one-shots but on a bimonthly schedule, like Little Lulu; the scripts for all but the first issue, in 1944, are his. The stories are much longer and more skillfully told than the comic-book norm, but DuBois's plots turn repeatedly on the villains' efforts to steal ranch land that is worth more than the rightful owners realize because of the oil, gold, or radium hidden in the soil—plots of a sort that were common in cowboy movies. It may have been that sort of repetition that Lebeck had in mind. It is difficult to imagine other comic-book editors complaining about such very general repetition, but even more difficult to imagine an editor volunteering the money to cure the problem.
In Ziemann's words, "With a canvas-covered trailer that measured 14½ × 6 feet, Gaylord and Mary DuBois covered 18,000 miles, writing wherever they stopped. From Westport they went to Brownsville, Texas. On the way, Gaylord stopped to visit and ride with ranchers." Their route after that took them along the Rio Grande to the Big Bend, and from there to New Mexico, into Navajo country—on horseback. They continued into Arizona, Utah, Idaho, Montana, and ultimately Alberta. "Gaylord and Mary's northernmost destination was the Peace River area, where their daughter Miriam and her husband were building a pioneer ranch. After a few weeks there, they returned to Westport, their heads and notebooks crammed with invaluable material for Gaylord's Western comics. The money they had earned by writing during the long trip not only repaid the advance for the travel trailer and all their other expenses but also left them with a good financial cushion."
The DuBoises continued to travel in the years ahead, often driving hundreds of miles in a day. "Gaylord dictated comic-book scripts to Mary as he drove. When they stopped for a day or a week, he typed finished copy to airmail back to Western in New York or perhaps to the company's Los Angeles office. At motels he would work with his typewriter on the bed—so that the sound would not disturb sleepers on the other side of the wall."
DuBois's scripts always felt calmer than the stories for most of his competitors' comic books, particularly when the former Disney artist Jesse Marsh was the illustrator. Other publishers' comic books designed for an audience somewhat older than the Disney comic books' target audience tended to lunge at subject matter that was simply too complex—too emotionally fraught—to fit into a comic-book story's six to ten pages. Not only were DuBois's stories in the 1940s and early 1950s longer than the norm, often much longer, but the characters were likely to be dealing with a concrete, well-defined problem. It might be a serious problem, a matter of life and death, but its dimensions were clear and its emotional implications relatively limited.
DuBois began writing Tarzan stories with a script for Tarzan no. 2, March–April 1948. Marsh was already on board: he began illustrating Tarzan with the first of the two Tarzan one-shots, published in early 1947, but he had been illustrating Gene Autry for Western since 1945. With plentiful work from Western assured, he left his job as a Disney story sketch artist in January 1947.
Marsh worked very fast and illustrated many other comic books for Western in addition to almost every page of Tarzan, even after it became a fifty-two-page monthly. That speed could take its toll at times, as in panels where Tarzan's anatomy looks not quite right—a common failing in comic books of the time—but such flaws are subsumed in pages whose elements are unusually cohesive. There is rarely if ever any reason to be confused about what is happening, or why it is happening, because Marsh was a visual storyteller whose speed and skill mirrored Gaylord DuBois's. He and DuBois were a highly compatible team for many years on Tarzan, even though the two men never met.
"The Men of Greed," in Tarzan no. 5, September-October 1948, was written by Gaylord DuBois and illustrated by Jesse Marsh. On this page, as on others, Marsh's powerful sense of design leads the eye through the page and overrides the occasional lapse in his depiction of Tarzan's anatomy. Trademark TARZAN® and Edgar Rice Burroughs™ Owned by Edgar Rice Burroughs, Inc., and used by permission. © 1948 Edgar Rice Burroughs, Inc. All rights reserved.
In "The Men of A-lur," in Tarzan no. 9, January-February 1949, written by Gaylord DuBois, Jesse Marsh departed from the usual six-panel grid for the sake of more dramatic staging. Trademark TARZAN® and Edgar Rice Burroughs™ Owned by Edgar Rice Burroughs, Inc., and used by permission. © 1949 Edgar Rice Burroughs, Inc. All rights reserved.
Jesse (or, sometimes, Jessie) Marsh was born in Alabama on July 27, 1907. Like Carl Barks, he became a full-time comic-book artist around the time he turned forty, after decades of other kinds of work—most important, his few years with Disney. He was hired by Disney in January 1940, left after the 1941 strike (almost certainly laid off on the same day Walt Kelly and Moe Gollub left the staff), and was rehired in July 1942. After brief military service in 1942–43 he returned to the studio for a final three and a half years in June 1943.
Early in the life of the Tarzan comic book, Marsh wrote to a fan that he and DuBois were "limited" in how closely they could adhere to the Edgar Rice Burroughs novels on which the comic books were ultimately based because the Dell Tarzan was "constantly under the scrutiny of parent-teacher associations, etc." The Burroughs novels were classic pulp adventure stories, violent and preposterous, if also fast moving and exciting, at least until skepticism's demands could no longer be resisted. They were the same sort of thing that DuBois had written in his own prose fiction, in books like the Hurricane Kids on the Lost Islands, if much less successfully, especially in commercial terms.
Many other publishers' "jungle" comic books, such as Sheena, Queen of the Jungle, were like highly colored offshoots of the Burroughs novels. In the Dell Tarzan, though, thanks to Marsh's matter-of-fact illustrations of Tarzan's fundamentally bizarre adventures in an invented Africa—adventures filled with talking apes, lost civilizations, and strange races—it was as if everything excessive had been scraped away. In DuBois's scripts, the Burroughs books' virtues remained, but thanks to Marsh they were clothed now in eminently sane drawings.
## 18
# Walt Kelly Branches Out
Dan Noonan said of working for Western in the last half of the 1940s: "It was really the heyday of the business. You were very well paid for the work, in those days; Western's page rate was good, and of course it was comparably higher in the forties than it is now [because of intervening inflation]. Most of us would receive Christmas bonuses, too; and there were the Christmas parties up in the Penthouse Club." Western's office building, the Toy Center, "was located in a very pleasant part of the city, and the city was very pleasant in those days, too."
It was standard practice for Lebeck to treat his freelancers to drinks and lunch on Friday afternoon. "It was a very nice group of fellows to work with," Noonan said, "and it had a sort of an easy camaraderie about it. We all worked at home, for the most part, and came in on Friday. Oskar paid everybody by check and took us downstairs for drinks and lunch at a place called the Fifth Avenue Bar; or we went upstairs to a place called the Fifth Avenue Club and had lunch up there." After the cartoonists started "belting away a few martinis," Moe Gollub said, they would "begin to rip poor old Oskar—I don't know why, just because it was an easy thing to do."
On those Friday afternoons, Noonan said, the cartoonists would often "stay and talk until late in the afternoon; the bull sessions sometimes lasted almost all day. There was a lot of ego deflating; anybody who'd get to taking themselves too seriously was in for trouble, because laying in the woods were people like John Stanley and Walt Kelly. And even they'd get it once in a while, too. I remember the time when Kelly came in with a huge stack of fan letters he'd received. 'For Christ's sake, Kelly,' yelled Tony Rivera [who was, like Kelly, a former Disney artist], 'why don't you pay those bills?'"
That was probably in the summer of 1947. Pogo asked for mail in the final panel of the "Albert and Pogo" story in Animal Comics no. 28, August–September 1947: "Anybody wants to send us some lettuhs, us'll spell 'em out . . . we jes' rarin' to hear f'um all our friends—children and grown-ups!" Kelly got fifty or more letters from his readers in July and August, letters addressed to him in care of Dell and forwarded to Oskar Lebeck, who passed them along to Kelly. He saved them in a scrapbook. Some letters were written by children, or dictated by them to their parents, but many of the letters were written by adults who had discovered "Albert and Pogo" by reading the stories to their children and had become Kelly fans as a result.
Kelly's byline had first appeared on one of his "Albert and Pogo" stories in Animal Comics six months earlier, in no. 24, December 1946–January 1947. But the first of the two Albert the Alligator and Pogo Possum one-shots had already been published by then, early in 1946, with Kelly identified as the author on the comic book's front cover as well as in a mock introductory letter on the inside front cover. When the second one-shot was published in the spring of 1947, Kelly was again identified prominently as the comic book's author. The one-shots inspired a steady trickle of admiring mail before the surge in the summer of 1947.
Some letters were surprisingly thoughtful and perceptive, considering comic books' general low repute in the 1940s. Lawrence Cole of Newport, Minnesota, who read Animal Comics with his five-year-old daughter, wrote on July 30, 1947, that he recognized Kelly's hand in "Hector the Henpecked Rooster" in Animal Comics no. 16, published two years earlier—"it seems to me you invented Hector and all the hilarious misadventures his wife had with various forms of eggs, did you not?"—and offered cautious praise for other unsigned work by Kelly. Cole told Kelly he "would feel let down if you were to adapt the hard wit I find in almost all other comics, or the brassy terminology which seems to be the stock-in-trade of all but you and Dan Noonan and Gaylord DuBois" (who were credited in Animal Comics for their work on "Rover" and "Chuckwagon Charley's Tales," respectively). Cole asked Kelly to share his praise with Noonan and DuBois, and Kelly may have done so with Noonan, at least, at one of the Friday luncheons.
"Albert and Pogo," by Walt Kelly, in Animal Comics no. 28, August-September 1947, was a story that generated dozens of fan letters to Kelly.
The names and likenesses of Western's cartoonists turned up repeatedly in the comic books, testifying to the prevailing clubby atmosphere through inside jokes that almost no readers could have detected. In the 1946 Oswald the Rabbit one-shot (Four Color no. 102), which was written and drawn by Kelly, when a cannon fires it is not "BANG" or "WHAMMO" but "FRED" that appears as a visual sound effect. No telling whose name that was, but one of the characters, a bulldog pirate captain, remarks of the cannon: "In 1927 she wouldn't say anything but 'John Stanley.'" In Little Lulu no. 1, January–February 1948, a man named Mr. Gripe, fleeing the noise and disruption of Lulu and her friends, moves to a "dump" on Old Post Road—the street in Croton-on-Hudson where Oskar Lebeck lived. A resentful butler in another of John Stanley's Little Lulu stories is named "Noonan." And so on.
As frequent as Kelly's appearances in Animal Comics, Fairy Tale Parade, and Our Gang were, and as richly detailed as much of his work for those titles was, those efforts hardly exhausted his capacity for work in the mid-1940s. He drew and almost certainly wrote the 1945 one-shots based on the Disney feature films The Three Caballeros and Pinocchio. The Three Caballeros one-shot in particular departed widely from the film, although it was in the Pinocchio one-shot that Kelly inserted a reference to "the gleet"—no doubt accepted by his editors as a mere nonsense word, but actually a medical term for chronic inflammation of the urethra. Kelly's inside joke went undetected through reprintings of his adaptation of Pinocchio in 1954 and 1963.
Kelly was extraordinarily prolific, even if sometimes as the illustrator of very ordinary scripts and not as a writer. His drawings for what was unmistakably an outside script, for a non-Disney version of "Three Little Pigs," turned up in the single 1943 issue of Tiny Tots Comics, and in 1944 and 1945 he illustrated the second and third issues of Mother Goose and Nursery Rhyme Comics—not really so much a comic book as a nursery rhyme compilation for very small children. In 1945, with Fairy Tale Parade near the end of its run, Kelly illustrated a new title for Lebeck called Christmas with Mother Goose, which was followed in the spring of 1946 by Easter with Mother Goose. Kelly was credited by name on the covers of both comic books, the Christmas credit coming a full year before his first credit for "Albert and Pogo" in Animal Comics. Both Mother Goose titles continued throughout the 1940s, always with Kelly credited on the cover, until the last Christmas with Mother Goose appeared in the fall of 1949, but he was present mostly as an artist, and only rarely as a writer.
Walt Kelly drew all of Easter with Mother Goose, Four Color no. 140 (1947), as well as the other Christmas and Easter Mother Goose one-shots published between 1946 and 1949.
Kelly's drawings for the Mother Goose titles are far more polished than some of his other comic-book work, and giving him a cover credit was perhaps Lebeck's way of acknowledging that superiority. Kelly was by 1945 a parent himself—he and Helen had the first of their three children, a daughter, Kathleen, in 1942—and that circumstance may have contributed to his interest in such work, and in making a better living from it through work outside the comic books.
Mike McClintock, the editor Kelly met when he was first looking for work in New York, left Whitman in October 1944 and began editing children's books for Julian Messner, another publisher with an extensive juvenile line. Kelly illustrated three books for Messner—all aimed at very young children and priced at $1 or $1.25—under a pseudonym, Tony Maclay, that would have hidden from Lebeck and other people at Western that he was working for a Whitman competitor. "Tony Maclay" both wrote and illustrated the first book, Trouble on the Ark, which was published in the fall of 1945 by a newly acquired Messner subsidiary, Veritas Press. "Maclay" then illustrated two books by other authors, The Downy Duck and Raffy Uses His Head, which Messner published under its own name in October 1946. Kelly may have found a pseudonym inhibiting: his drawings for The Downy Duck are sweet and bland even when measured against his lesser work for Lebeck's Mother Goose titles.
Hiding Kelly's identity would not have been a consideration where children's records were concerned, since Western's first Little Golden Records were not issued until 1948. So, in 1946, Kelly "created" and "designed," according to the credits on the labels, at least sixteen 78 rpm record sides, released in two sets of four records each, for Story Book Record Company. These were picture records—cardboard records six and a half inches in diameter, whose color illustrations by Kelly on each side were covered with transparent plastic into which the record grooves were pressed. Story Book Record was housed at 200 Fifth Avenue, the address of Western Printing's New York office, and it undoubtedly had some connection with Western, however informal.
The first set of four records credits three narrators, including Kelly, but he is actually the sole narrator on all the records, not just telling familiar stories (Little Red Riding Hood, Three Billy Goats Gruff, and so on) but acting out all the parts. He is variously male or female, animal or insect, gruff or falsetto. He barks and roars and growls and quavers. Andrew "Andy" Barnes, who first met Kelly a couple of years after the records were made, when Barnes was in his teens, remembered that Kelly was always "on," and that is exactly how the records sound. They are filled with broad, emphatic vocal acting that is an aural equivalent of Kelly's most boisterous comic-book stories. It is as if Albert Alligator were the narrator. A third set of picture records was planned—the titles for the additional stories were listed on the record sleeves in the second set—but evidently never materialized, so a Kelly career as an equivalent of radio's storyteller "Uncle Don" also never materialized.
Mike McClintock's wife, Inez Bertail, edited Complete Nursery Song Book, a hardcover collection published in the fall of 1947 by a respected publisher of children's books, Lothrop, Lee & Shepard. It was no doubt through McClintock that Kelly was hired to illustrate Bertail's book, although he had by then demonstrated through the Mother Goose comic books how well suited he was for such work. In her foreword, Bertail thanked Kelly for entering "so wholeheartedly into the planning of the book to make the pictures complement the music."
Bertail's book had actually been announced for publication in November 1945, around the time that Kelly's Trouble on the Ark was published under the Maclay pseudonym. If its publication had not been delayed, Complete Nursery Song Book would also have been illustrated by a disguised Kelly. The announced illustrator was "Jan MacAnulla," as in Kelly's mother's maiden name (minus one "n"). Kelly's relationship with Western Printing had changed by 1947, though, becoming more formal through new written agreements between artist and publisher, and presumably he no longer felt it necessary to conceal his identity.
Even as Kelly produced handsome drawings for nursery rhymes, his drawings for "Albert and Pogo"—conspicuously better than any other drawings in Animal Comics in its first few years—began to look a little rougher in the middle 1940s. His stories always varied a great deal in the quality of their finish, so that, for example, his handful of stories with Famous Studios cartoon characters—Cilly Goose, Blackie the Lamb, Hector the Henpecked Rooster—in 1945 issues of Animal Comics are much simpler in drawing and hastier in execution than "Albert and Pogo," as if in response to the generally poor quality of the Famous cartoons. Earlier stories with those characters, some of them illustrated by Famous Studios animators, were based on John Stanley scripts produced as hastily as Kelly's work. But now the "Albert and Pogo" stories began to look a little blunt and almost crude when set beside some of the other comic books Kelly was illustrating at the same time. Even within Animal Comics, some of his last stories for that title—three 1947 stories about a mouse named Nibble—had the same fluency and charm as his work for the Mother Goose comic books, and considerably more of both than "Albert and Pogo."
But what was happening in the "Albert and Pogo" stories was simply more interesting, and held the seeds of work of greater substance, than the "Nibble" stories or anything in the Mother Goose titles. It was as if Kelly were deliberately suppressing any tendency toward cuteness in his "Albert and Pogo" stories so that cuteness of the Mother Goose kind would not be an obstacle to the adult readers that by 1947 he knew he was attracting. Stories like "Nibble" were, for Walt Kelly then, almost too easy.
"Albert and Pogo" was not the only one of Kelly's features that was going through significant changes in the mid- to late 1940s. "Our Gang," his other longest-running feature, was passing through a remarkable evolution of its own. Starting in 1944, the "Our Gang" stories took a strong turn toward crime melodramas. It was inconceivable that Western would produce such nakedly exploitative comic books as Crime Does Not Pay, but the publishing climate in the mid-1940s undoubtedly pushed the Dell titles a little in that direction. Such material was not entirely unknown in the Our Gang movie shorts—in one of the last shorts, Little Miss Pinkerton (1943), criminals murder a kindly janitor and threaten the Gang members themselves with death—but it was certainly unusual, whereas in the comic books homicidal villains turned up several times a year.
The most striking of these villains was The Barrel, swarthy and stocky, thus his name. He was of uncertain ethnicity and spoke in an unclassifiable accent but was given to exclamations ("Sacre!" "Santos!") that suggested variously he might be French or Hispanic or, perhaps most likely, French-Canadian. Kelly would have encountered that ethnic group as a resident of Connecticut in the 1920s, when hundreds of thousands of French-speaking Canadians crossed the border to work in New England's textile and shoe factories. The Barrel appeared first in Our Gang Comics no. 14, November–December 1944, returning frequently thereafter. He and many of the other villains showed themselves to be perfectly willing to murder children if the need and the opportunity coincided, as never quite happened.
"Our Gang" was a feature that differed in almost every way from "Albert and Pogo," as if Kelly were going out of his way to test his abilities. The exact authorship of those stories is open to question, although Kelly certainly shaped them even if he did not write the initial scripts. Sometimes the link between the "Our Gang" stories and Kelly's later newspaper-strip work is unmistakable: Professor Hector Hannibal Horatio Gravy, the flamboyant proprietor of a pocket circus consisting of a lion and a tiger, is an early version of P. T. Bridgeport, the even more flamboyant bear (Professor Gravy is of course human) who was a member of the Pogo cast starting in 1952, and whose name and dialogue balloons, lettered like circus posters, recalled P. T. Barnum, the great nineteenth-century circus man who lived in Kelly's hometown. In the Professor Gravy stories, Kelly seized every opportunity to indulge in comically overripe language, just as he would when P. T. Bridgeport entered his comic strip.
Occasionally words were all too abundant in Kelly's "Our Gang" stories, as when a character's long-winded explanation, perhaps covering several panels, was required to wrap up some messy plotting, or when the criminals indulged in exposition by telling one another what they had done or were going to do. And those stories were problematic in other ways.
As was true of his Animal Comics stories, Kelly's drawings for "Our Gang" in the last half of the 1940s never looked as finished as his work for some of the other Lebeck-edited titles. His "Our Gang" pages often looked more spontaneous, compared with the likes of the Mother Goose titles, but as beautiful as his brushwork usually was, it sometimes looked merely hasty, especially toward the end of the decade. Our Gang Comics changed to a monthly schedule as of the July 1946 issue, so that Kelly's workload on that feature—which now most often filled sixteen pages a month—more than doubled. Since Kelly was still filling about as many pages as before in Animal Comics and other titles, he had every incentive for haste.
Kelly's staging of action sequences in these comic books was frequently awkward. The farces that pop up in the "Our Gang" series amid the much grimmer adventures often seem labored—silly more than funny—because their slapstick has been cut up and parceled out among the panels with greater regard for clarity than for a comic-book equivalent of comedic timing. One could say that Kelly never flinched from the challenges posed by slapstick in his comic-book years, starting with the clowning in his "Pat, Patsy & Pete" stories—or that he kept making the same mistakes (which recurred in some of his Sunday Pogo pages).
Kelly was hardly alone in that. A common problem in comic-book stories of all kinds is that when a relatively complex action is broken up into a series of panels, seen from a uniform point of view or close to it, such an action is likely to seem chopped up rather than coherent and complete, even when it is clear enough what is going on. In this way, as in others, comics differ significantly from the movies. Action that might be encompassed in a movie's single shot, lasting as long as several minutes, can almost never be accommodated in a single panel, and when divided into several panels it often seems fragmented.
Bernard Krigstein, years later in a famous EC story, "Master Race," used such fragmentation to achieve the effects he wanted. In "Master Race," published in a "New Direction" comic book called Impact in 1955, a former concentration-camp inmate has recognized the camp's Nazi commander on a New York subway car. The fragmentation cools the intense events in the panels, so that what could have been merely lurid becomes instead solemn and momentous, almost as if the story were being told in slow motion.
But not many stories lent themselves to such handling. For most cartoonists, the task was different: to minimize the complexity of the actions they were drawing. The best ones found creative ways to do that; it is hard to find an awkwardly staged passage in Carl Barks's stories once he was a few years away from drawing Disney storyboards. In "Letter to Santa," Barks resisted the temptation to show too much of the climactic battle of the steam shovels, relying instead on a few carefully composed panels and, especially, one astonishing half-page panel depicting the full scale of the battle. In the superhero comic books of later years a similar battle might rage across a dozen repetitive pages or more, but Barks confined his warfare to five panels that fill the equivalent of one page.
The appearance of Kelly's "Our Gang" characters could vary a great deal not just from story to story, but also within a story, particularly as the real-life models for the characters receded in importance. The character variously called Janey or Janet—the name of a member of the movie cast, although by the middle 1940s the comic-book Janet bore scant resemblance to the movie Janet—is alternately a pretty girl or a pug-nosed tomboy in Our Gang Comics no. 19, September–October 1945, depending on the panel. Such variability was a virtue, not a weakness, when Kelly was drawing the Pogo comic strip in the early 1950s, because the variations in a character's appearance usually spoke so clearly of what was going on in the character's mind. The "Our Gang" stories, however, skated closer to differences that were merely arbitrary, or even careless.
Kelly rarely spoke publicly about his comic-book work, perhaps because his comic strip achieved great popularity in the early 1950s just as comic books of all kinds were coming under increasingly hostile scrutiny. When he did say anything in later years, he was dismissive, as when he told the television interviewer Edward R. Murrow, on New Year's Day 1954: "I decided I would recreate the whole comic book industry and take this business away from those who were teaching children how to stab their mothers to death. . . . Well, after this effort of mine folded, out of this wreckage I plucked this one spear carrier named Pogo."
In April 1954, when he testified before a U.S. Senate subcommittee investigating the links between comic books and juvenile delinquency, he spoke almost as if he were Oskar Lebeck:
I got into the comic-book business at one time back in 1940 or 1941 and had some experience with its early days as before the 1947 debacle of so many crime magazines and so on.
In those days there was even then a taste on the part of children for things which are a little more rugged than what I drew. So that I was faced with the problem of putting into book form, into comic form, comic-book form, things which I desired to make popular, such as an American fairy story or American folklore type of stories.
I found after a while that this was not particularly acceptable. . . .
I decided I would help clean up the comic-book business at one time, by introducing new features, such as folklore stories and things having to do with little boys and little animals in red and blue pants and that sort of thing.
So when my comic book folded, the one I started doing that with [Animal Comics], I realized there was more to it than met the eye. Perhaps this was the wrong medium for my particular efforts.
He made no mention of his "Our Gang" stories and their murderous villains, or, for that matter, of the parents and children who had written admiring letters about "Albert and Pogo."
In 1965, in a brief, hand-printed autobiography, Kelly dismissed his comic-book career with sarcasm: "Returned to N.Y. and set out to (HA!) improve comic books." He painted with an even broader brush at a National Cartoonists Society banquet in November 1969: "I worked in the comic book industry for quite a while," he told an audience of fellow cartoonists, "and I made a lot of money by slapping the stuff out fast, and some of that stuff was terrible, awful. . . . We want to be good, but actually we don't work at it hard enough, and a lot of the slop stuff I was putting out during the '40s and early '50s reflected that attitude."
Some of his "stuff" was indeed the product of haste, but Kelly always worked with exceptional speed even when he was drawing newspaper comics, and even taking into account that he usually had the help of at least one assistant. A 1952 profile in Collier's said that it took Kelly just an hour to turn out a daily strip, three hours for a Sunday page, and a week for a comic book. That would have been, in 1952, a fifty-two-page issue of Pogo Possum, almost every page of which Kelly wrote and drew in pencil himself.
The awkwardness of a fairly high percentage of Kelly's comic-book stories had another source. Unlike many other comic-book artists, Kelly clearly had difficulty settling into formulas that would carry him through patches when he lacked either the inspiration or the time, or both, to do his best work. In reading his stories (and the Pogo comic strip, especially when it was at its peak in the early 1950s), there is thus a much stronger sense of encountering an artist in the throes of creation than is usual in the comics. It is, finally, this artistic integrity that most distinguishes Kelly's comic-book work, and is most admirable about it, even when it leads him down a blind alley.
Moe Gollub described Kelly as "one of the brightest guys you'd ever want to know, really very sharp, but a little insecure, and I never knew quite why. That was my feeling about him. He was nervous and anxious, and he had a lot of little perverse streaks. He used to call Noonan and me periodically and talk about some big projects that he had. We knew he was lying, and we couldn't figure out what his motivation was."
Kelly could be harsh in his evaluations of his colleagues. Writing in 1952 to a Western Printing executive, Kelly said that "the miserable son of a bitching experience in trying to help other artists back in 1947 will never be forgotten by me. I learned then to never listen to carpers or gripers and never to offer to help any co-worker in his business dealings." He did not identify his targets by name, but his resentment against not just his fellow freelancers but some of Western's own employees fairly boiled: "The misguided and completely stupid assumption that Kelly wanted Oskar LeBeck's [sic] job left me with something less than admiration for the ability of frightened and/or ambitious men to analyze the motives of anybody who is trying to be helpful. . . . My affection for Western Printing was never increased by the vote of somebody who refused me a bonus due that year on the grounds that I had not paid obeisance to a pipsqueak god in the New York office."
Kelly may have been venting his bile against Oskar Lebeck himself, since Lebeck in his daughter's recollection believed that Kelly wanted to replace him. Lebeck could have interpreted any such ambition as a betrayal, since, she wrote, "I do remember Dad was really behind Kelly, encouraging him on his 'Pogo' strip." The details of that episode are probably irretrievable, but when Kelly wrote as an occasional memoirist in later years—in books, and in fugitive material like his 1952 third-person autobiography—he never mentioned Lebeck or his fellow comic-book cartoonists by name. He was in touch occasionally with some of those cartoonists, and he responded sympathetically when one of them, Tom Hickey, was seeking work. He wrote fondly, though, not of those former colleagues but of his former colleagues at Disney and the two newspapers where he was on the staff, the Bridgeport Post and the New York Star.
Kelly's harshness could extend to cartoonists outside Western's circle, as it did in 1948, when he was working at the short-lived Star as its editorial cartoonist, artist and writer of a new comic strip, and, it seems, screener of talent. Harvey Kurtzman, already published in comic books and soon to become the creator and editor of Mad, remembered a deflating encounter: "I went up to the newspaper . . . when Walt Kelly was doing Pogo and he was the one who saw aspiring young cartoonists. He gave me the heave-ho. He wasn't impressed; as a matter of fact, he gave me the bum's rush. I just didn't like him from that moment on."
When Kelly wrote to Kurtzman on November 2, 1948, probably just after giving him "the heave-ho," his rejection letter was not as harsh as Kurtzman's memory suggests, although Kelly in person may have been more abrasive than Kelly on paper: "We believe that your material is still too specialized for syndication. However, it's actually 'funny' and I hope that sometime you will be able to work out a strip that we can use."
## 19
# Strong-Handed Friends
Over the course of Walt Kelly's seven-year run on the "Our Gang" stories, there was an increasingly strong sense of caricature. Many of the characters that slipped in and out of the stories were strikingly lifelike, as if to confirm what Hank Ketcham said about Kelly: "He was a great observer of people and various funny types that you'd see all over the city, and he could put that down in a drawing very nicely." Like just a few other cartoonists, notably Will Eisner ("The Spirit") and Roy Crane (the Wash Tubbs and Captain Easy comic strips), Kelly at his best could combine comic exaggeration with acute observation, to the benefit of both.
Sometimes his caricatures are identifiable as Kelly's colleagues at Western, or Kelly himself. They are like extensions of the caricatures that Kelly and many other cartoonists drew so abundantly at Disney, and that testify to the friendships and admiration for one another's work (tinged with jealousy, of course) that flourished at both places. According to Ward Kimball, "Sometimes he'd use his old friends [at Disney] as characters in his stories and send us the printed comics."
John Stanley said of Kelly: "Well, most of the artists were mainly concerned with their own stuff and there were some petty jealousies going on here and there but when Kelly walked into the Western office with a stack of his art under his arm, everyone would stop what they were doing to read it. Everyone knew that he was something special. He loved to play jokes and, invariably, he would whip up some special drawing involving his intended target. I must admit that Walt and I painted the town many times. He was a very enjoyable guy to be with."
The story in Our Gang Comics no. 27, October 1946, set on a showboat, is particularly rich in caricatures, among them Kelly himself, first as a small-town lawman and then as a cook with a Yiddish accent who hurls garbage out a window onto a blue-eyed salesman introduced a few issues earlier. The salesman demands identification as a caricature, but cementing any name to that caricature may no longer be possible. The likeliest candidate is Richard Small, a Western salesman in the 1940s and later an executive at the Poughkeepsie plant. Likewise "Siwash Susie," a diminutive barroom singer—could she be Lebeck's secretary, Anne DeStefano? A short, fat, pipe-smoking man with a five o'clock shadow, who was caricatured not just by Kelly but by John Stanley, is Moe Gollub. Gollub was almost six feet tall, but his heartless friends translated his stockiness into shortness.
Stanley himself is readily identifiable in that October 1946 story as "Gentleman John," a top-hatted, temperamental piano player who is much shorter than the real Stanley, and Dan Noonan is present as "Aloysius," a trumpet player. Other real people are no doubt tucked away in corners of the story, as are inside jokes like the one in no. 22, March–April 1946, when a piano mover—he looks like a real person, too—for some now obscure reason speaks for one panel in French.
Caricature and stereotype are just a step or two apart, and for the first few years Kelly's depiction of the black boy Buckwheat in the "Our Gang" stories was far more stereotypical than caricatured. But then the character changed.
The cast of the movie series had turned over gradually as the child stars grew out of their roles, and the comic-book stories followed suit, so that George "Spanky" McFarland disappeared from the comic book after the first two issues. With the end of the movie series the cast of the "Our Gang" comic-book stories began to include fictional gang members—that is, children who were not among the actors in the Our Gang short subjects—starting with the unmistakably Irish Red MacDougal in no. 14, November–December 1944. By no. 31, the February 1947 issue, only Billy "Froggy" Laughlin and Billie "Buckwheat" Thomas of the movie cast remained in the comic book, and both of them were gone in the March issue. As Kelly replaced the real-life Our Gang members with fictional children, those children also grew older gradually, from year to year, in a pattern unusual but not unknown, thanks to comic strips like Gasoline Alley, whose characters famously grew older in "real time."
Buckwheat's name left before he did—he began to be called "Bucky" in the November–December 1944 issue—and then his dialect faded, lingering for several issues as speech more informal than that of his friends before disappearing entirely as of no. 24, July 1946. In the meantime, in no. 20, November–December 1945, a white member of a rival gang disguised himself as Bucky, darkening his skin and speaking in minstrel-show dialect.
The transition in Kelly's handling of Bucky probably mirrored a change in his own thinking, since there are suggestions in other comic books from the mid-1940s that he was as casually accepting of stereotypes as most other white Americans. "The Great Egg Hunt," the Kelly story that fills the Oswald the Rabbit one-shot for the 1946 Easter season, is loaded with broadly stereotypical black cannibals who speak in an incongruous dialect. The story is probably Kelly's in every respect—he did not sign it, but the copyright registration lists him as the sole author—and its dialogue, like that of the contemporaneous "Albert and Pogo" stories, echoes the Amos 'n' Andy radio show. The comedy is very broad and feather light, with an off-the-cuff quality like that in some of John Stanley's stories for New Funnies, and as one result the racial stereotypes carry very little comic weight. It is simply too obvious that nothing serious is going on.
That Oswald one-shot was published in March 1946, the same month as the first Albert the Alligator and Pogo Possum one-shot, which included a story with a stereotypical black locomotive engineer. There is a little more of the same in "Goozy," a Kelly filler set in Africa, in Animal Comics no. 23, October–November 1946.
The cultural climate that permitted such stereotypes to pass unremarked was beginning to change, although the stereotypes lingered in a few of the Dell comic books. "Beloved Belindy," the mammy doll in the "Raggedy Ann" stories, was still speaking in what was supposed to be a Negro dialect as late as Raggedy Ann & Andy no. 28, September 1948, and there are even a couple of stereotypical walk-ons in Carl Barks's 1949 Donald Duck story "Voodoo Hoodoo." But stereotypes' most visible survival in the Lebeck-edited comic books—"Li'l Eight Ball" in New Funnies—disappeared after the August 1947 issue, no. 126.
Eight Ball's grossly stereotypical screen original appeared in just three Walter Lantz cartoons, all released in 1939. The character began appearing in The Funnies with the May 1942 issue, just before it became New Funnies, and he spoke in dialect for four and a half years. Then, starting with New Funnies no. 117, November 1946, Eight Ball spoke Standard English, not some white writer's notion of dialect. His race then became incidental to the stories, but his hopelessly stereotypical name and appearance—in keeping with his name, his head was bald and shiny black—made him a natural target for indignation.
The indignation arrived more or less on schedule, and possibly that is why Eight Ball vanished from New Funnies in 1947. The Amsterdam News, a New York African-American newspaper, published a story about Eight Ball on May 10, 1947, under the heading "'Li'l Eight Ball' Killed by Publisher." It quotes a letter from Oskar Lebeck to the eight-year-old schoolchildren who had complained about Eight Ball. Lebeck wrote:
Dear Boys and Girls:
Aren't you a little unfair to imply that our editors discriminated against the colored people in our Li'l Eight Ball stories? I can assure all of you it was not our intention to make fun of the Negroes as you put it in your letter. If you were right, wouldn't we also discriminate against all the white children when we caricature boys and girls, such as in our Little Lulu strips or Henry or many others? Should we leave out the Irish cop or the funny Italian organ-grinder or the fat German delicatessen man, etc., etc.?
However, in order that there should be no doubt in anybody's mind, I have decided to discontinue the Li'l Eight Ball stories effective with the September issue. We certainly do not want, in these troubled times, to add anything which might cause friction and hamper the efforts to build a happy and peaceful world.
Eight Ball was by then a pretty easy call. The more interesting case, one that seems to have left no document trail, is what happened in Our Gang Comics (which became Our Gang with Tom and Jerry as of no. 39, October 1947) just as Eight Ball was making his exit.
Although Bucky disappeared as of the March 1947 issue along with Froggy, the other vestige of the film gang, Bucky reappeared four months later and made a total of four appearances in the last half of 1947. He was now unmistakably an adolescent, a sexually mature and self-possessed young black male and thus inherently difficult for American popular entertainment to accommodate in the mid-1940s. To cite one conspicuous example of how that difficulty manifested itself, the Disney live-action feature Song of the South, released in 1946, has no black characters with speaking parts except the very old and very young. In the "Our Gang" story in no. 40, November 1947, which centers on a cross-country race and some white boys' efforts to cheat Bucky out of victory, there is no mistaking that Kelly is presenting adolescent black and white boys—and white girls—as equals.
In Our Gang Comics no. 40, November 1947, published long after the Our Gang movie shorts had been discontinued, Walt Kelly not only depicted "gang" members older than the movie children but also showed black and white adolescents as equals-a radical departure from cultural norms in 1947. © 1947, Loew's Inc., 1974, Metro-Goldwyn-Mayer, Inc.
Kelly did not draw his black characters with the particularity of his white characters; there is not the same sense that close observation has preceded caricature. The black characters—Bumbazine, Bucky, and incidental characters like the locomotive engineer in the 1946 Albert the Alligator and Pogo Possum one-shot—look like one another as Kelly's white characters do not. But in the "Our Gang" stories the similarity can be accepted as that of members of the same family, which is what the black characters are supposed to be. There is no trace of the old stereotypes.
By presenting Bucky as he did, Kelly had ventured into uncharted territory. In 1947, even the U.S. armed forces were still segregated. It seems hardly accidental that after that November story, the only black characters to be seen in the "Our Gang" stories were very young children. Were there complaints from white parents, especially in the South? Did some executive at Dell or Western get cold feet? Just what role did Kelly play in keeping Bucky in the "Our Gang" stories and then removing him? There is probably no longer any way to know. But certainly there is every reason to believe that Kelly's thinking had changed—that he had acquired a more acute awareness of racial injustice—in the little more than a year that separated his 1946 Oswald the Rabbit one-shot from his 1947 "Our Gang" stories with Bucky.
Animal Comics, and with it "Albert and Pogo," expired with no. 30, December 1947–January 1948, just one month after the stereotype-free Bucky disappeared from Our Gang with Tom and Jerry. Kelly continued to write and draw for Lebeck, but it was around that time, late in 1947, that he made an unsuccessful effort to sell a Pogo comic strip to the Chicago Tribune–New York News Syndicate, which was the home of Dick Tracy and Little Orphan Annie and a particularly unlikely home for a whimsical comic strip with animal characters. Kelly knew, thanks to the fan letters, that "Albert and Pogo" had a core of discerning admirers, but the syndicate was not interested. Murray Robinson of Collier's offered Kelly's account of this humiliating episode:
Determined to make a daily strip of his swamp creatures, Kelly hopefully took a collection of Pogo samples to the office of [the] syndicate and showed them to the boss, who happened to be a lady.
She looked at Pogo and said, "We're not buying any of these duck strips."
"But it isn't a duck strip," Kelly protested.
The lady boss pointed to Pogo in all his pristine possum prettiness. "What, may I ask, do you think that is?" she inquired icily.
There was no persuading the "lady boss" or anyone else that Pogo was ripe for syndication, and so Kelly retreated to comic books. What are from all appearances two sample Sunday pages—both titled "Albert Alligator and Pogo Possum" and drawn in Kelly's relatively astringent mid-1940s style—survived in his papers and were published in 1985.
By mid-June 1948, still working for Lebeck, Kelly was also working for one of the dozen daily newspapers then being published in New York City: the storied left-wing tabloid PM. He was not drawing a comic strip, though, but editorial cartoons. Kelly cartoons mocking the field of Republican presidential candidates appeared in the last two issues of PM, which ended publication on June 22 and was succeeded the next day by a new morning tabloid newspaper, the New York Star.
PM had been sold to new owners about two months earlier, and the change of name, along with other changes, was intended to salvage a chronically money-losing newspaper. By 1948 Kelly had close friends and drinking companions among New York journalists, and it was presumably through such connections that he became part of the Star's makeover—a more important part, actually, than anyone originally had in mind.
The Star had announced in its first two issues, with considerable fanfare, that Edmund Duffy, a Pulitzer Prize–winning editorial cartoonist for the Baltimore Sun, was joining its staff. Kelly was mentioned in an Editor & Publisher piece about the new Star—his name somehow emerging as "Walt Miller"—but only as a "contributing cartoonist." Then Duffy changed his mind, or the Star's owners changed their minds about him. No new cartoon by Duffy ever appeared in the Star. Instead, Kelly was Duffy's emergency replacement. Within a day or two of the Star's birth he became its principal editorial cartoonist. A Kelly editorial cartoon, and sometimes more than one, appeared in almost every issue.
John Horn, as cartoon editor of the men's magazine Argosy from 1946 to 1948, had urged his friend Kelly "to try some gag cartoons" of the sort John Stanley was trying to sell to the New Yorker. "He did," Horn later wrote, "but he confessed his heart was really not in it. . . . Looking ahead, he thought he would like to be a political cartoonist in five years." Edmund Duffy's sudden exit accelerated that timetable dramatically.
Kelly's editorial cartoons, drawn in grease pencil on textured paper, soon caught fire with their depictions of Governor Thomas E. Dewey, the Republican presidential nominee, as a mechanical man. Other cartoonists, notably Bill Mauldin, creator of the famous wartime cartoon "Willie and Joe," also drew for the Star, but Kelly quickly and unmistakably became the newspaper's principal cartoonist. Kelly said more than twenty years later: "I don't think anybody can really do two good jobs at the same time. He has to do one. . . . I wouldn't want to get into the position of doing a special job, three or four times a week even, as an editorial cartoonist, and also try to do a real job on the comic strip. I draw fast but I don't think fast." But in 1948, as he was about to turn thirty-five, he observed no such restrictions.
This Kelly editorial cartoon appeared in the New York Star on October 2, 1948. Kelly's portrayal of Governor Thomas E. Dewey, the Republican candidate for president in 1948, as a "mechanical man" attracted widespread attention. The dog is obviously closely related to Kelly's comic-book animals.
For one thing, he continued to draw "Our Gang." It had been the lead feature in every issue of Our Gang Comics through November 1947, but it lost that position to "Tom and Jerry" the next month, and in no. 45, April 1948, it was shunted to the back of the comic book. (By then "Tom and Jerry" had also usurped the front cover.) As of the May 1948 issue "Our Gang" shrank from its usual sixteen pages each issue to eight, or in one case six. Two exceptionally grim serials, one showing the cold-blooded murder by drowning of an unconscious man, consumed most of those truncated installments until the series finally ended, five years after the demise of the Our Gang films, with no. 57, the April 1949 issue.
A Kelly editorial cartoon for the New York Star on January 13, 1949, targeted a notorious Georgia case in which a black man was killed by members of a white mob, all of whom escaped punishment for their crime.
And just in time. The fundamental problem was not the darkness of the "Our Gang" stories—even the Pogo comic strip of the early 1950s had some unsettling moments—but that by 1949 Kelly's comic-book drawings looked very rough. As he had been signaling for some time, his interest now lay elsewhere. (In the October 1948 issue, one of Our Gang's allies declares as he slugs a homicidal madman, "I've been waiting five pages for this opportunity!"—exactly the sort of self-referential joke that Kelly used occasionally in "Albert and Pogo.") Kelly's role at the Star had expanded while he was drawing "Our Gang," and as the newspaper's de facto art director—he did not use that title in his correspondence with the cartoonists whose work he was evaluating and usually rejecting—he had assigned himself to draw a daily comic strip, Pogo. It first appeared in the Star (and nowhere else) on October 4, 1948.
Kelly seems to have had no legal right to publish such a comic strip. The "Albert and Pogo" feature in Animal Comics, like "Albert Takes the Cake" and the other early installments that bore different titles, was work for hire. Kelly did not own the characters. "Albert and Pogo" was copyrighted by Oskar Lebeck and then, through assignment, by Western Printing & Lithographing, in its Whitman Publishing incarnation. A one-page gag, "Monkey Biz," in the ninth issue of Animal Comics, bore the line "© Walter Kelly," in a marked departure from Western's usual practice, but "Albert and Pogo" was always Western's property.
Kelly's first agreement with Western about "Albert and Pogo" was dated February 1, 1946. It was subsequently amended in 1948 and 1949, after the comic strip's debut, and then renewed on October 17, 1950. No copies of the original 1946 agreement appear to be part of Kelly's papers at Ohio State University, perhaps because that agreement was completely superseded by an agreement dated September 20, 1951.
Lloyd E. Smith, the Western executive who handled the licensing of characters from their copyright owners, summarized the 1946 agreement when he told Kelly in 1951 that "the entire intent of the agreement was to indicate that we own the feature because you made it originally for us for hire and that we were engaging you to continue to produce it for as long as you cared to do so under the terms agreed upon." That agreement was signed shortly before publication of the first Albert the Alligator and Pogo Possum one-shot, which had Kelly's name on the front cover, and it seems likely that the idea was to anticipate any questions about the ownership of the feature—that is, to confirm that "Albert and Pogo" was Western's property. This was, besides, the unsettled period when Kelly was trying to find other sources of income by working for other publishers under a pseudonym and creating a series of children's records. Whatever Western knew of that activity could only have made a written agreement seem more desirable.
Subsequently, Western surrendered some of its rights in Pogo to Kelly, but not all. In particular, it was through the December 17, 1948, amendment—that is, two and a half months after Pogo began appearing in the Star—that Western released to Kelly "all of the newspaper comic strip syndication rights and all other subsidiary rights, exclusive of printed publications." Kelly thus owned the newspaper strip and had the right, if he wished, to license Pogo to an animated-cartoon producer. The timing of that agreement was no doubt dictated by the Star's plan to form its own syndicate and begin selling its home-grown features like Pogo to other newspapers starting January 3, 1949. What had probably been an informal handshake agreement would not suffice when third parties—other newspapers—were to be involved. As it happened, though, the Star's sudden death on January 28, 1949, made syndication rights a moot point.
As to why Western was willing to surrender Pogo to Kelly without asking anything in return, Smith explained in a memorandum to Lucille Ogle of Western's Artists & Writers Guild subsidiary that "the company's attitude in the matter is, stated quite simply, that Pogo would never have become the famous character that it is if we had retained complete control over it. That is to say, Walt went out and by his own almost unaided efforts made it famous only because we conceded that it should belong to him." In other words, Western recognized that Pogo was uniquely Kelly's creation, and not a commercial property that others could exploit, and so, in an enlightened act of generosity, the company returned character to creator.
Although the Star comic strip lasted less than four months, until the newspaper died, that was long enough to establish differences from the Animal Comics stories in a number of ways. From the start, Pogo was unequivocally the leading character—Albert did not even come onstage in the first ten published strips—and the possum's evolution toward a sweeter, kinder, more Amos-like disposition was far advanced. Kelly's stock company was still rather small when measured against the requirements of an eight-page comic-book story or thirty-six-page comic book, but it was big enough to fill four daily panels, and it was starting to expand. The rhythm of a humorous daily comic strip, one that relied less on a punch line in every day's last panel than on a continuous flow of incident and the elaboration of personality, proved immediately to be congenial to Kelly. The lingering scratchiness of the Animal Comics stories gave way to fluid brushwork that soon recalled his most polished drawings for Oskar Lebeck.
The dialogue in Pogo differed markedly from the dialogue in "Albert and Pogo." Here is Albert, from the last Animal Comics story: "Owl, yo' is allus complain 'bout de fishes yo misses. Why, Ah is missed mo' big fish dan yo' is even see." A year later, in the Star for November 3, 1948, when there is a contested election in the swamp, Albert still sounds like his southern blowhard self, but different: "You li'l rascals is claimin' stuff right and left—I gone leave judgement [sic] of who win to a impartial judge." There is still an Amos 'n' Andy flavor to such dialogue, but its excesses, like "Ah" and "mo'" and "de" and "yo," have been purged. This is no longer dialogue that demands to be read aloud, like an Amos 'n' Andy radio script, if it is to be fully intelligible.
Kelly himself, writing in Pogo Parade, a 1953 giant comic book that reprinted eleven of the Animal Comics stories, explained why his characters' speech had changed when they made the move from comic books to newspapers: "It will be noted, to the dismay of latecomers, that a sort of southern fried hash was once spoken here. Strong handed friends hammered that part of the apparatus into junk. We all grow up, fight it though we will, if we just live long enough."
Ward Kimball spoke of how difficult it was to read the dialect: "[I]t was like a foreign language. A lot of people wrote to the editors and complained about this, and as a result he simplified the language." But that obviously did not happen when "Albert and Pogo" was running in Animal Comics, and the dialect had already been simplified when Pogo began appearing in the Star. Some of Kelly's fans wrote in 1946–47 of difficulty with the dialect, but other correspondents liked it. They resisted his tongue-in-cheek suggestion in the August–September 1947 story, through a letter to "de swamplan' critturs" from "Wallet Kelly," that the animals' comportment could be improved. (In Pogo's words, "De man advise us to ack mo' ree-fined so folks would write lettuhs at us.")
Kelly had a number of "strong handed friends" at the Star who remained his friends for years after the paper closed, as evidenced by, among other things, his use of their names in his comic strip and his books. Joseph Barnes, the Star's editor, was one; John Lardner, who started reviewing Broadway shows for the Star in September 1948, and George Y. Wells, editor of the Star's editorial page, were others. But, when writing in the third person about Pogo's entry onto the Star's comic-strip page, Kelly singled out another Star editor, Richard E. Lauterbach, a former Moscow bureau chief for Time and Life:
After much argument pro and con with the then feature editor, Kelly was able to get the POGO strip into shape so that it could be printed daily. Fortunately, the feature editor, Dick Lauterbach, was able to convince Kelly that he should clarify the dialect. "It's funny after you explain it," he told Kelly, "but believe it or not we now have 165,000 readers and you're going to have a hell of a time running around [and explaining the dialect to each reader]."
"Oh, 165,000 . . . well, why didn't you say so before," asked Kelly and he cleaned up the dialect so that it became dialogue.
Thomas Andrae, coauthor of a book about Kelly, has written that the original artwork for the first Pogo that Kelly drew for the Star has survived and that the dialogue bears evidence of corrective surgery: "[T]he editors had Kelly create pasteovers on the word balloons to simplify the dialogue. For example, in the first panel Pogo says, 'H'lo ol' worm. How'd you like a wonderful job?' Under the pasteover he says, "H'lo ol' worm. How yo' likes a won'erful job?'" (That strip, although the first to be drawn, was the second to be published, on October 5, 1948. According to Andrae, "The strip was marked with a '1' then had a '2' blue-penciled over it.")
Lauterbach may have detected the racial condescension that was an inextricable element in dialect comedy even when that comedy was as benign as Kelly's. Some of Kelly's readers were certainly aware of that racial element—and they approved of it. When Kelly asked for mail from his Animal Comics readers in 1947, a few of them took his tongue-in-cheek plea for more "ree-fined" behavior from his animals as evidence that he was being pressed by "one or various colored associations" to temper his stories' language. Robert E. Kiler of Oakland, California, wrote: "The only charm the Negro can lay claim to is when he operates in his natural state, and chatters in his idiotically moronic fashion. Then, and only then, is he really funny." Kiler urged Kelly that he not "change your very delightful strip one iota." Kelly thus had good reason, apart from making his dialogue easier to read, to put greater distance between his characters' speech and dialect of the Amos 'n' Andy kind. Thanks to Lauterbach, any obvious connection was gone from Pogo as it began its Star run. Pogo attracted fan mail that the Star published on its editorial page, but those letters included no complaints about the dialect.
In an admiring article in Editor & Publisher just a couple of months after Pogo debuted in the Star, Doris Willens wrote: "Kelly started the strip using the standard (or what is thought by Northerners to be standard) Southern dialect. He was accused of poking fun at Negroes and poor Whites [sic]. So he switched to straight English. But after experimenting, he turned to a dialect that combines the Elizabethan English still found in the South, the French of New Orleans, the Negro and the Indian. Kelly has studied phonetics, dialects, anthropology as well as social problems of the South."
That account is, when measured against the comic books and the early Star strips, fanciful at best, but Kelly had every reason not just to sweeten the dialect but also to deflect any questions about its origins. His liberal credentials were well established by late in 1948, thanks to his work as the Star's editorial cartoonist. A. J. Liebling, writing in the New Yorker a few weeks after the Star's demise, singled out Kelly for praise: "It seemed to me that a young Star cartoonist named Walt Kelly, who used to be a Walt Disney draftsman, did a wonderful job during the [1948] Presidential campaign. The only bit of caricature I remember from that period is his mechanized Mr. Dewey, with a torso that might have been either a cash register or a slot machine."
Kelly's Dewey cartoons made up only a small part of an output that was, in keeping with the Star's editorial positions, consistently, insistently left of center (although not as much so as PM had been) and that depicted southern whites as degenerate bigots and blacks as their victims. By the late 1940s, no such politically liberal cartoonist would want to acknowledge publicly that Amos 'n' Andy was one source of the comedy in his comic strip. It was much better to speak of haphazard research on American dialects. And it worked: no embarrassing questions about the racial origins of Pogo's dialect seem to have been raised over the next few years.
As the syndicated strip became wildly popular, Kelly let his liberal political views show themselves in episodes like a "trial"—Albert was the defendant—that was richly comic but also evoked the anti-Communist witch hunts of the late 1940s and early 1950s. Even so, Kelly still presented himself as less liberal than he actually was. In late October 1952, when he was traveling around the country promoting Pogo's mock campaign for president, he visited the racially segregated southern states of Texas, Louisiana, Arkansas, and Tennessee. He told a Kiwanis Club audience in New Orleans that "I have no social messages" to deliver through the comic strip, whose purpose, he said, "essentially is to amuse." That mask would drop the next year when Simple J. Malarkey came onstage, bringing with him Kelly's unmistakable engagement with contemporary politics.
Happily, given the evanescence of most political commentary, in drawings and otherwise, there was much more to Pogo in the early 1950s. In particular, by taming his strip's linguistic excesses Kelly not only removed any lingering racial stain but also made it possible to give his characters more interesting things to say. Over time and especially once the comic strip was well under way, Pogo's speech became increasingly baroque, and increasingly removed not just from speech of the Amos 'n' Andy kind but from actual speech of any kind. The language of Pogo at its peak, in the early to mid-1950s, bore no closer relation to everyday human speech than the language of Shakespeare's plays. It was speech that could exist only in a comic strip, and as with the language of The Tempest or A Midsummer Night's Dream, if very differently and of course more modestly, it made a small, confined space seem much larger, and certainly more magical, than the life outside.
## 20
# Carl Barks: The Virtuoso
Carl Barks drew hundreds of cartoon humans for the Calgary Eye-Opener, but he had no occasion to draw human characters, except for his own pleasure, once he joined the Disney staff in 1935. From then on, he was drawing only ducks and other animals. The urge to draw human characters was too powerful to resist, though, and Barks made many such drawings in the late 1940s, "just trying to keep from getting in a rut so that I couldn't draw anything else but ducks." Then such human characters began turning up in Barks's comic-book stories.
The Australian aborigines who slipped into Donald Duck's supporting cast as early as 1947, in "Adventure Down Under," were drawn as human characters, and Barks drew Indians as humans for "Land of the Totem Poles," in Donald Duck Four Color no. 263, published in late 1949. For a year after that, human characters, mostly drawn in a generic illustration style, multiplied in Barks's longer stories, most conspicuously in "Dangerous Disguise," in Donald Duck Four Color no. 308. That 1950 story is, in another departure, a wicked parody of spy fiction, the sort of thing Alfred Hitchcock might have done if he had found himself drawing Donald Duck instead of directing movies. This time, though, Barks went too far for his editors, who wanted him to draw his supporting casts with doglike noses again.
Barks delivered "Dangerous Disguise" to Western's Beverly Hills office late in June 1950. "I don't know why the office never complained about the human characters in Ancient Persia and others," he wrote in 1969. "Maybe the fact I used a girl for the principal villain in 'Dangerous Disguise' caused them to notice the non-duckness of the cast." In any case, he said in 1971, "As soon as I took 'Dangerous Disguise' in and Carl Buettner took a look at it, he said, 'That doesn't go good, having real humans. It takes the ducks out of their own world.'"
Human characters still turned up occasionally in later stories, as in a few panels in Uncle Scrooge no. 6, June–August 1954. "At times they would slip in without me being able to control it," Barks said. "I would draw them and get them off to the post office before I realized I'd drawn a human face." But Barks complied with his editors' wishes for the Donald Duck story called "In Old California," in Four Color no. 328, published early in 1951. Barks conceived its characters as humans, but by the time he delivered the story to Western in November 1950 he had drawn them with dog faces or, in one case, a pig face. "I would have [drawn them as humans], yeah," Barks said, "but at that time I had received my warning."
The story itself—an extended dream in which the ducks spend weeks in Spanish California just before the Gold Rush—is another departure for Barks, sweet tempered and nostalgic in a way that his stories almost never were, but with comic twists. "A Christmas for Shacktown," his next story for Donald Duck, in Four Color no. 367, published late in 1951 (after three intervening issues drawn by other cartoonists), has some of the same tenderness. Mercifully, though, the poor children of Shacktown simply are not pretty enough to be adorable.
Besides expanding his stories' emotional range, Barks for a few years explored more aggressively than he had in the past his stories' formal elements. He occasionally tinkered with the dialogue balloons, so that, as a gag in the parodistic "Dangerous Disguise," one balloon is chopped off by the right-hand border of a page-width panel. But his more significant departures were from the eight-panel grid that was standard for the Dell talking-animal titles. He experimented with the size and shape and arrangement of the panels, and with the composition of each page as a whole. He had grown increasingly assured in his use of oversize panels, especially, since he first used a half-page panel in "The Ghost of the Grotto," and that assurance is immediately visible in the "Donald Duck" lead stories in the first issues of the giant Disney comic books Christmas Parade (1949) and Vacation Parade (1950). The opening page of each story is taken up by a single striking panel.
Panel size received Barks's close attention in "No Such Varmint," in Donald Duck Four Color no. 318, published early in 1951. When a sea serpent—the genuine item, not a rubber phony like the one piloted by the villain in "The Terror of the River"—first surfaces in front of the ducks' tiny boat, Barks conveys its enormous size by drawing it so that it stretches from just above the page's baseline to almost the full height of the page. It occupies the space that would be filled by three stacked panels and even spills up onto a fourth level, at the top of the page. Over the next few pages, whenever the sea serpent is the center of attention, Barks finds a way—as by dividing the top half of a page into two panels, vertically—to make room for it while simultaneously respecting the grid.
Within other stories from around the same time, like the 1950 "Magic Hourglass" and 1951 "Christmas for Shacktown," some panels have five or six sides and fit together like the pieces of a jigsaw puzzle within the "tray" of a page. Almost always there is the unmistakable sense that Barks was seeking new ways to use such formal elements in service to his story. The odd-shaped panels rarely call attention to themselves. It is as if, instead, Barks were borrowing from one panel to give to another so that he could show more—or tell more, when dialogue was the critical element—than in a standard layout.
Barks's mastery was so complete by 1952 that he could start with a conspicuously ridiculous premise, like the one for "The Golden Helmet," in Donald Duck Four Color no. 408, and still emerge with a marvelous story. The idea is that whoever owns an ancient parchment and a golden Viking helmet will somehow have an unbreakable claim on the entire North American continent. Barks signals clearly from the start of his story that his premise is supposed to be ridiculous. One such signal is the presence of the rat-faced Lawyer Sharkey, who spouts bogus Latin ("Hocus, locus, jocus! Which means, 'To the landlord belong the doorknobs'") in the service of his reprehensible client, Azure Blue, a descendant of the Viking who discovered America. But the story works itself out perfectly—and satirically, as one character after another succumbs to the lure of absolute rule over the continent—with never a false step.
Barks himself, reading the story twenty years later, marveled at its construction in a letter to Donald Ault: "I have just read the 'Golden Helmet' story for the first time in many years. The amount of plotting in the script is amazing, and every device and situation seems to develop easily from what has gone before. I can offer no explanation of how I did it other than to say that I wrote and rewrote and re-polished and double checked backward and forward and counted syllables in the dialogue and read the stuff dozens of times for effectiveness and 'flow.'"
In January 1952, a few weeks after Barks delivered "The Golden Helmet" to Western, he delivered "The Gilded Man," for Donald Duck Four Color no. 422. Both comic books were published in the summer of 1952. Barks was about to exit the Donald Duck series. After "Trick or Treat" in Donald Duck no. 26 (the first issue on an official bimonthly schedule, independent of the Four Color series), he would be gone from Donald Duck except for rare encores.
In "The Gilded Man" Barks showed again just how subtle he could be, in the midst of more boisterous pages. A wealthy, addled stamp collector has given Donald a thousand dollars as a reward for returning an album that Donald actually did not return; Gladstone Gander did, with the dismayed Donald alongside him, and the insufferably lucky cousin has already left with his own reward. But the collector thinks Donald is Gladstone and presses a thousand-dollar bill on him. Outside the house, Donald struggles with his conscience for the better part of a page before deciding to keep the money, in a mild departure from conventional comic-book morality. This page is not quite as impressive as the similar page in "Luck of the North"—Barks relies more on thought balloons than he did in the earlier story—but even so, such sustained attention to the workings of a cartoon character's mind is simply unimaginable in most other comic books.
Two of Barks's Donald Duck stories that he submitted in 1950—"Big Top Bedlam," for Four Color no. 300, and "No Such Varmint"—were twenty-eight pages each but still had the modest scale and neat construction of his ten-pagers for Walt Disney's Comics & Stories, the stories he was not producing at the time because Western had chosen to have him concentrate on the longer stories. Once Barks returned to the ten-page stories, starting with the January 1951 issue of Walt Disney's Comics, it was as if they now commanded his attention in the way the longer stories had in the late 1940s: they were the arena in which he scrutinized his characters most closely.
Although he was still producing stories of twenty pages or more for Donald Duck, like "The Golden Helmet" and "The Gilded Man," there were only six such stories altogether in the issues published in 1951 and 1952, and as virtuosic as they were, those stories were a shade less probing than his best ten-pagers from the same two years. One constant in the new ten-page stories was that Donald was unmistakably an adult most of the time—not really mature, and usually with no obvious job, or with a very menial one, but still, especially in relation to the nephews, a responsible person, a parent who took his obligations seriously, and overall an even more complex character than before.
Although Uncle Scrooge and Gladstone Gander had appeared in the ten-page stories of the late 1940s, they had flowered in longer stories like "Letter to Santa" and "Luck of the North." The richer characterizations of those stories now carried over to the ten-pagers. Scrooge and Gladstone brought fresh comic vigor to Barks's stories, very much in the way that Dickens's characters Alfred Jingle and Sam Weller enlivened The Pickwick Papers, when that novel was growing in monthly installments that invite comparison with the comic books published more than a century later. Scrooge, of course, took his name from another Dickens character, but Barks's direct inspiration was Andy Gump's Uncle Bim, an Australian billionaire in Sidney Smith's comic strip The Gumps. Bim's wealth was a catalyst for Smith's stories in the early 1930s, just as Scrooge's wealth would be for Barks twenty years later. The Gumps was domestic drama, though, and Bim a warm and admirable character, unlike Barks's original version of Scrooge.
Both Scrooge and Gladstone were more distinct and more insistently comic characters than Donald Duck himself, and they were comic, moreover, in ways that fitted them perfectly for roles in American comic books. Scrooge had become, two years after his debut late in 1947, incredibly rich, and Gladstone had likewise become incredibly lucky. Wealth and luck: children growing up in postwar America could easily recognize in Scrooge and Gladstone—or, still more to the point, in Donald's responses to Scrooge's wealth and Gladstone's luck—preoccupations like those of adults they knew.
Both characters were in addition excellent vehicles for Barks's cheerful skepticism about human nature. Sometimes that skepticism dominates a whole story, as in the January 1951 Walt Disney's Comics, Barks's first ten-pager after Western decided to restore him to that series. When Scrooge must take leave of his cash temporarily because he has become allergic to it—oh, the irony!—he reluctantly entrusts to Donald the management of his fortune. Donald immediately begins making loans, with a misplaced sympathy for obviously unworthy borrowers (one is named Miss Lily de la Field) that he has found lacking in Scrooge. Donald also makes those loans with a self-regard symbolized by the halo that appears above his head as he doles out the cash. In debt to Scrooge for four dollars at the opening of the story, Donald has by the end squandered Scrooge's fortune and thus incurred a debt, as a nephew remarks, of "seven hundred and eighty-eight billion, four hundred and twenty-three million, seventeen dollars and sixteen cents."
More often, though, Barks's skepticism flavored stories that were straightforwardly comic. The sheer outrageous bulk of Scrooge's fortune, its existence as coins and bills, lent itself to parable. In Walt Disney's Comics no. 126, March 1951, a huge tornado sucks all of Scrooge's cash out of the enormous corncrib where he has hidden it and spreads it evenly across the whole country. Everyone is now rich and so quits working, but with no one working, there is nothing for anyone to buy—except at Scrooge's farm, where the prices are so astronomically high that Scrooge quickly gathers in all the cash he had lost.
Scrooge's fortune lent itself even more readily to farce, and as Barks sensed the possibilities, the scale of both fortune and farce increased rapidly. In Walt Disney's Comics nos. 134 and 135, November and December 1951, Scrooge is the target of a family gang called the Beagle Boys, dog-faced criminals who dress alike—domino masks, sweatshirts inscribed "Beagle Boys, Inc."—and go by prison numbers instead of names. In the second of those stories Scrooge's fortune is for the first time housed in a huge cube identified as a money bin—"ten stories deep and a block square," as Scrooge says. Like the fortune itself, the circumstances of those two stories defy belief. For one thing, in the November story Scrooge and his money are alone in the McDuck Building, with no guards in sight. As always, though, Barks drew the preposterous with a disarming concreteness and told his ridiculous story with a straight face.
Barks very quickly peaked in his presentation of Scrooge's fortune as the pretext for farce. In Walt Disney's Comics no. 138, March 1952, Scrooge is outraged that his status as the world's richest man has been challenged. To prove his bona fides he provokes a contest with the Maharajah of Howduyustan—the other claimant to the title—to erect the largest statue of Cornelius Coot, Duckburg's founder. The climax comes in the unveiling of the two largest of a series of rival statues, identical except for their size. Both statues are insanely huge, looming over Duckburg like sculpted mountains, so gargantuan that they terrify cattle many miles away. The statues would have dwarfed the Colossus of Rhodes; they would have stunned the pharaohs. Nothing like them is even remotely possible—but there they are in Barks's half-page drawing, offered with the solemn authority of photography. (The caption says, "[T]he camera must pull back to an extreme long shot to show the awesome finish of this third round in the battle of the titans.") The Maharajah himself certifies the reality of what we see, his howl of rage emerging from distant Duckburg in a dialogue balloon with an exceptionally long tail: "May ten thousand demons hound that upstart! He has topped me again!" Barks strains the reader's credulity but rewards the effort.
"Donald Duck" in Walt Disney's Comics & Stories no. 138, March 1952, pitted Uncle Scrooge against a rival who seemed to be just as rich. © 1952 Disney.
Although Scrooge was so imperious that he often overshadowed Donald in their stories together, Gladstone never threatened to swallow up the stories he shared with Donald. (There could have been no serious thought of giving Gladstone a comic book of his own.) The size of Scrooge's fortune, and the difference in ages, meant that there could not be any true rivalry between him and Donald. With Gladstone, though, rivalry—romantic rivalry included, since Donald and Gladstone were rivals for Daisy Duck's affections—was the whole point. Gladstone's smugness and hauteur expressed themselves not just in a dandyish appearance, complete with wavy and un-duck-like hair, but also in ornamental expression. No other comic-book artist besides Barks—except maybe Walt Kelly—would have had one of his characters denounce another as a "quacksalving slanderer," as Gladstone denounces Donald in Walt Disney's Comics no. 151, April 1953.
What made Gladstone so intimidating to the other ducks in the best stories with the character was not his appearance, his language, or even his good luck, but how irresistibly that luck unfolded, without a trace of comic-book arbitrariness. When, in Walt Disney's Comics no. 140, May 1952, Gladstone leaves home with a shopping list, the narrative pattern Barks has established in earlier stories points the story powerfully in a single direction: Gladstone will return home with everything on the list, and without spending a dime or lifting a finger. But everything does not merely drop into his basket, it does so with the utmost naturalness. The story insists, gently but firmly and finally irrefutably, that something like this could happen, and it requires only one small step to conclude that since Gladstone is the beneficiary, it must happen.
The problem was, Gladstone's good luck almost invariably translated into bad luck for others, usually Donald—or, if not outright bad luck, the frustration attendant on taking someone else's good luck too seriously. That is what happens in the May 1952 story, when Donald, the nephews, and Uncle Scrooge break into Gladstone's house in a search for the secret of his luck and find in his safe only a dime that he once worked to earn, to his everlasting shame.
To be sure, Barks mined Donald's frustration for comedy. In Walt Disney's Comics no. 131, August 1951, Donald responds to two discouraging pages with Gladstone by locking himself into his broom closet, where he will remain, he tells the nephews, for the rest of his life. There was, however, just so much psychological punishment that an artist working in a children's magazine could inflict on his principal character before it seemed too cruel. As Barks said of Gladstone: "I don't think anybody likes a character who gets by with so little effort in the world. They like to feel that other people have just as much of a struggle as they themselves have, and Gladstone was a fellow who would just go along, skimming all the cream out of life, without ever sweating for it." And so some of the stories with Gladstone end with an arbitrary twist that works against the plausibility that Barks has in the earlier pages so carefully nurtured.
Plausibility is a dominant concern in all the stories from the years when Barks's artistry was at its peak, roughly 1948 to 1955. In one story after another, events fall into place so surely that there is no doubting them, even if a mere summary might invite skepticism. In Walt Disney's Comics no. 133, October 1951, the nephews are determined to play hooky when school resumes, but they cannot evade teachers and schools and truant officers (and Donald), no matter how hard they try. The story's plausibility is rooted not just in how neatly its events interlock, but in the settings—this is a story that takes place in a real-looking southern California, with palm trees and mountains and deserts—and in the way the ducks are portrayed: Donald as a conscientious working-class parent (he drives a delivery truck for Bumblehead's Bucket Works), the nephews as smart kids smugly certain that it is a waste of their time to attend school.
Huey, Dewey, and Louie are more mature in some Barks stories than in others, but they are almost always highly intelligent, and never more so than in those stories—the first appeared in Walt Disney's Comics no. 125, February 1951—in which are they members of the Junior Woodchucks, Barks's parody of the Boy Scouts. The Woodchucks, in Barks's early stories with them, are a rather prissy organization so absorbed with merit badges and with rank that the lowest rank is major. In the second Woodchuck story, in Walt Disney's Comics no. 132, September 1951, the focus is not so much on the scouts themselves (although the Exalted Grand Marshal of Duckburg Burrow No. 13 is imposing in a distinctly military manner) as on the ducks as parent and children. The nephews, who one month later were to play hooky, are sober and industrious in the September issue, refusing to take the shortcuts Donald urges upon them as they make a canoe and a bow and arrow to fulfill the Woodchucks' requirements.
Though mature in the Woodchuck stories, the nephews are much more like real children in Walt Disney's Comics no. 142, July 1952, when Donald determines to keep them out of trouble by spending the summer on a houseboat in Lake Erie. With the most innocent (and believably childlike) of intentions, they generate one disaster after another, until on its last page this expertly machined narrative delivers Donald over Niagara Falls in a barrel. In an interview with Edward Summer, Barks said of the nephews: "I began making them into sort of smart little guys once in a while, and very clumsy little guys at other times, and always, I aimed at surprise in each story so that nobody could pick up a comic book and say, 'Well, the nephews are going to behave thus and so.' They wouldn't know until they read the story just what those little guys were going to be up to in a particular sequence."
The ducks, Donald and the nephews, were by this time extraordinarily mutable characters, the nature of their relationship, as Barks depicted it, shifting from month to month, sometimes radically. It was Donald who was the most mutable of all of Barks's characters, but never to the point that he dwindled into a cipher, like so many other comic-book characters based on animated originals. By 1952, in stories like the one set on the houseboat, and even in the occasional story that might seem, by comparison, just a bit forced, Barks had proved able to place Donald in almost any situation without losing the character. There was now a psychological precision in his portrayal of Donald, in the writing and especially in the subtleties of his drawing, that permitted him to go almost anywhere he wanted.
In Walt Disney's Comics no. 145, October 1952, the plausibility that Barks had for years brought to his stories was fully present not just in the ordering of the comic events but also in his depiction of Donald's kaleidoscopic mental state. For years, Barks had made Donald seem real, but by 1952 the character's reality was richer and more complex than ever before. Donald's mutability within a story made him seem more real, not less.
As the October 1952 story opens, the nephews are playing with a toy gun that they pretend can hypnotize. Donald's expression changes subtly, echoing but surpassing the transformation in "Luck of the North," as he overhears the cries of "Bing! You're hypnotized!" Donald is for the first two pages very much the concerned parent—a foolish and gullible parent, true, but sincerely worried about a hazard to his young charges. But as he is about to pitch the gun into a river, his expression makes yet another transition, in fewer panels this time, from righteous indignation to sly cupidity: might it be, he thinks, that Uncle Scrooge could be hypnotized? Not just hypnotized, but made generous under hypnosis? Donald the stern and righteous parent has become, in the few seconds represented by three comic-book panels, a cunning would-be thief.
It is difficult to imagine a comparable effect being achieved so successfully in any other visual medium. An animated cartoon or even an exceptionally well-made live-action comedy might possibly combine the broad and the subtle so smoothly, but there would not be the opportunity to savor Barks's subtlety that the printed page allows.
Scrooge is of course not hypnotized, but he plays along with the gag—and it turns out that Donald's own belief in the gun's power is so strong that Scrooge can use it to hypnotize him, transforming Donald into an aggressive bill collector and sending him out to collect a dollar from an intimidating deadbeat named Rockjaw Bumrisk. As the story ends, after several pages of high-octane conflict between Donald and Rockjaw, Donald is back on the bridge, carrying a sack of money that Scrooge has given him as a reward for collecting that buck.
Would Scrooge have given Donald a huge reward for collecting so small a debt? The story offers plentiful reasons to believe that he would, starting with Rockjaw's battering of Scrooge himself—Scrooge tried to collect that dollar before sending Donald after it—and ending with Scrooge's alarm at what hypnosis has done to his nephew. And then there is Donald's success at collecting a debt that would otherwise have to be written off, a much blacker mark on Scrooge's books than an oversized reward.
"Donald Duck" in Walt Disney's Comics & Stories no. 145, October 1952, was perhaps the Carl Barks story that presented the title character at his most complex-but with no sacrifice of comedy. © 1952 Disney.
Once again, Donald is about to pitch the toy gun into the river, this time with the nephews as an audience. He switches effortlessly from adolescent smugness, bragging about his success in hypnotizing Scrooge, to—in the last panel, and once again—parental rectitude, solemnly intoning about the gun's hazards to "somebody with a gullible mind." The story is silent as to any lessons the nephews might derive from witnessing this episode. Just as well.
Not every story that followed in Walt Disney's Comics over the next few months invites comparisons with the October 1952 story, but there are echoes even in superficially very different ten-pagers. The story in the next issue, no. 146, November 1952, in which the ducks become poultry farmers to the ruin of a town called Pleasant Valley—it is ultimately renamed Omelet, if you get the picture—is expertly constructed farce, but its comedy is much richer because of how persuasively Barks depicts Donald's pigheaded behavior. (Although Barks had been a poultry farmer himself, he insisted there was nothing in that story based on his personal experience.) The story in no. 150, March 1953, in which Donald is a postman delivering valentines in a snowstorm, including one from Gladstone to Daisy, benefits at every stage from the exactness of its portrayal of Donald's wildly fluctuating states of mind.
Other ten-page stories from the early 1950s are comparably rich in psychological detail, like the nephews' skeptical posture in no. 149, the February 1953 issue, as Donald tries to track down "Professor Batty," a crackpot philosopher who peddles "Flipism"—Flipists make decisions by flipping a coin—and has skinned Donald out of a dollar. Those stories are also rich in detail of other kinds, like the surprisingly realistic settings in which the "Flipism" story takes place. It is winter in Duckburg, and there is snow on the ground, but it is only patches from a snowfall a few days earlier. Everything works toward making what happens on the page seem real.
Donald threatened to emerge as a distinct but correspondingly limited personality in the May and June 1953 issues, nos. 152 and 153. In both stories he believes in cutting corners—in the May story by trying to pass off a phony talking dog as the real thing, in June by using teams of powerful worms to haul in fish. Gyro Gearloose, a gawky chicken inventor whom Barks introduced in 1952, is in both cases the source of inventions that Donald misuses. But then, in no. 156, September 1953, after a couple of stories in which Donald is little more than a bystander, Barks made him an expert, for the first time—an expert rainmaker, a rainmaker so remarkably skillful that he can ream out a section of a cloud so that rain will not fall on a farm wife's drying clothes while he waters the farmer's gooseberry patch. Donald's expertise is funny because it is so extreme, and also because it is so easily undermined by his jealousy of Gladstone and Daisy. His expertise exists only for comic overthrow. Two issues later, in no. 158, November 1953, Donald is again primarily a parent, this time coping with the bees the nephews are raising as a Junior Woodchuck project, and the nephews are again real children, absorbed with their project and oblivious of the havoc it creates.
The several dozen ten-page stories published in Walt Disney's Comics in the early 1950s differ remarkably from one another in the nature of their comedy and in how the ducks are characterized, but one story, in no. 129, June 1951, stands apart from all the others because it is the only story that is unequivocally satirical from start to finish. That satire is, besides, essentially independent of the ducks; Donald is the victim of characters who are themselves Barks's target. In this story, the pessimism that undergirds all of Barks's best stories—pessimism that never descends into despair, that is never sour or spiteful, but is instead insistently comic—had perhaps its clearest expression.
The nephews are absent for almost the whole of the story, away at camp when Donald, weary of his vegetable garden, replaces it with a swimming pool. Instantly he is besieged by neighbors who claim the pool as their own and pelt Donald with ripe tomatoes when he tries to assert his rights. Donald is surrounded by overbearing mothers and hideous children; he is, really by default, the only sympathetic character. The story is, however, not about him but about the Swiftian portrait of humanity (dog noses notwithstanding) that Barks paints around him. When the nephews return from camp, they find the swimming pool—which they have not seen before and do not recognize as such—filled in and transformed into a sunken vegetable garden, complete with what they see not as a diving board, but as a springboard for Donald's assaults on marauding insects.
Although such full-fledged commitment to satire was rare, throughout the early 1950s as in the late 1940s there were passages within Barks stories that passed beyond farce into something more pointed. In the March 1952 Walt Disney's Comics, the Maharajah's contest with Scrooge leaves him penniless; Scrooge, by contrast, has spent only his petty cash. For all that it is broadly comic, the story also has satirical sharpness. Defeated and impoverished, the Maharajah immediately becomes the target of scorn from those who have been fawning over him. It is Duckburg's mayor who demands that he yield up his fancy clothes—the mayor holds a mortgage on them, it seems—so that the Maharajah is left with nothing to wear but a barrel. There is a similar moment in Walt Disney's Comics no. 157, October 1953. When Donald tries to prove to Scrooge that he can make more friends because he is young and charming, he immediately encounters surly hostility from the people he greets warmly on the street. "Brr!" Donald says, "People sure are cranky this morning." Scrooge replies brightly: "They're always cranky! But watch me!" As he pushes a wheelbarrow full of coins and bills down the sidewalk, the cranky onlookers instantly melt.
Scrooge was the Barks character who evolved in what turned out to be the most consequential way. Having exploited the possibilities of farce in three "Donald Duck" stories with Scrooge in Walt Disney's Comics, and no doubt aware of farce's limitations—for one thing, Donald and the nephews do not have a great deal to do in any of those stories, all of them centered on Scrooge and his money—Barks left farce behind. Early in 1952, alongside the duel with the Maharajah of Howduyustan in Walt Disney's Comics, Dell published another comic book, written and drawn entirely by Barks, that was strikingly different in tone. This was the first issue of Uncle Scrooge, Four Color no. 386. Barks's editors had asked him to write and draw a comic book starring Scrooge that would test the market for a regular series.
Barks remembered drawing that comic book in September 1951, in a motel room in Los Angeles, while his second marriage was falling apart. He said of Clara Balken, his second wife:
She had a lot of talent for cooking, sewing, and drinking. I taught her to black in my stuff—that is, put in the solid blacks, with the brush—and she did that for me for several years. But as she became more and more of an alcoholic, she got to where she would get on belligerent spells, and try to tear up a bunch of my drawings. In fact, that first Uncle Scrooge . . . I drew that in a motel down in Los Angeles, where I had taken refuge. She would have torn up my drawings, and probably chopped me up with a meat cleaver or something, on one of her big drunks. So I had taken all of my drawings and drawing paper and skipped out, in the dark of night . . . and got into a motel on the corner of Alvarado and Seventh Street, I believe it was.
This was not the first time that Barks had worked on one of his longer stories under less than ideal conditions. Clara's rages could well have been fueled by something more than alcoholism. In July and August 1950, she was hospitalized for weeks at La Jolla, California, for cancer surgery that resulted in the loss of a leg below the knee. Barks took his drawing board, paper, and ink with him to the hospital and worked there on the Donald Duck story "No Such Varmint." Tom McKimson's letter directing him to return to work on the ten-page stories for Walt Disney's Comics was addressed to him in care of Scripps Memorial Hospital. Barks subsequently built a prosthetic leg for Clara.
It is not clear when Barks left Clara, but he said that they divorced in December 1951, a little more than a year after her surgery. Over that span in 1950–51, Barks did some of his best work—the ten-page stories for all of the 1951 issues of Walt Disney's Comics and most of the 1952 issues, the thirty-two-page Donald Duck stories "In Old California," "Christmas for Shacktown," and "The Golden Helmet," the first issue of Uncle Scrooge—despite the exceptional stress in his personal life. He made this connection: "It seemed like the more difficulties I had, why, the bigger the inspiration that would come when there was just a moment of calmness. When the dishes would stop flying, the bottles breaking, why, I could sit down and the ideas would just flow in on me. I could forget all of the pains and the scratches and so on, and go right to work."
When Barks left Clara, he moved into one of five small apartments in a converted warehouse, with no possessions other than his drawing board, clothing, two blankets, and his issues of National Geographic. The divorce brought neither peace nor an interruption in the flood of ideas, as Barks wrote to Donald Ault, recalling his situation as it was in November 1951:
I had just given everything I owned to my alcoholic wife in exchange for my freedom. Broke and in debt and facing years of stiff alimony at the age of 50 I chose to keep on working, and I can recall one day when all the bad news had struck me and I should have been heading for a bar, and instead I sat like a zombie with a pad of paper and jotted down gags and plots and situations that seemed to pour onto me from somewhere. . . . I recall that three nearly complete story plots were on my pad before the ideas finally began to dry up.
## 21
# Walt Kelly Escapes
By the time the New York Star folded in January 1949, Walt Kelly was working with an assistant, George Ward, who had joined the Star's art department the previous August. Kelly had surrendered a little to the demands of his workload and delegated the lettering of Pogo's dialogue to Ward after the comic strip began its Star run in October 1948. "Walt was on such a tight schedule that we always sweated out his showing up with the daily Pogo and also drawing his editorial cartoon," Ward said, adding that Kelly "never missed a deadline." After the Star's demise they continued to work together, on comic-book stories and the syndicated Pogo comic strip.
Ward told Bill Crouch Jr. that he spent days at a time in Darien, Connecticut, at the home where Kelly lived with his wife Helen and his three children. "After dinner Walt and I would play with the children and maybe by 8 P.M. we'd sit down at the drawing boards. Walt had two regular drawing boards in his living room in front of a large fireplace. . . . I never saw him work from a script. Walt would just start and write and pencil [drawing with a light blue drawing pencil] as he went along. One night on an eight-page story he looked up and said, 'Hey, George, I'm on page 6 and I don't know how this story is going to end.'"
By then Kelly was delegating to Ward more than just the lettering: "He would ink all the main characters and skip where he felt I could ink. Or he'd draw in half a tree or house and leave the remainder for me to complete." If the work had to be delivered the next day, the morning would begin with a mad dash to catch a train to Manhattan.
Walt Kelly posed for these publicity photos around 1951. Author's collection.
It is an open question whether Kelly gave any serious thought to continuing as an editorial cartoonist, or whether that was even possible in New York in 1949. He found newspaper life congenial, certainly, especially the sociable, drinking side of it. Andy Barnes, the son of the Star's editor, Joe Barnes, said many years later: "My child's eye images of Kelly are vivid and positive." But he also remembered that when Kelly visited the Barneses' Connecticut country home in the Litchfield Hills, he began drinking immediately upon his arrival and was not outside for more than fifteen minutes at a time before retreating to the house in search of alcoholic refreshment. He was in that way indistinguishable from countless other newspaper people.
It was, however, comic strips that offered the more promising career path. Kelly was actually about to launch a second one for the Star, called Bobo Larkin, with human characters, when the newspaper died. Two and a half weeks after the Star's death, the newspaper's owners executed a copyright assignment to him of all rights in both Pogo and Bobo Larkin.
Although Bobo Larkin never escaped from limbo, Pogo's fate was different: the comic strip was very quickly picked up for national syndication. It is not clear who initiated contact, Kelly or the syndicate, but George Ward remembered delivering "the first 42 daily Pogo strips [from the Star, presumably] to Bob Hall at the Post-Hall Syndicate in the New York Post Building" in late February 1949. He delivered "a pile of letters—fan mail" from Star readers to Hall in March. By then, Kelly and Post-Hall had already signed a contract, dated March 2, 1949. Post-Hall announced Pogo as one of its features in mid-March 1949, barely six weeks after the Star's death. The first Pogo strip under the new arrangement appeared in the New York Post on Monday, May 16, 1949.
By 1949 the nature of Western Printing's comic books, the home of Kelly's characters since 1942, had changed considerably. There had been a steady retreat from proprietary material of the "Albert and Pogo" kind—features that Western owned within anthology titles like Animal Comics and Popular Comics—and toward more licensed subjects like the Disney, Warner Bros., MGM, and Lantz cartoon characters and movie cowboys like Gene Autry and Roy Rogers. Such licensed material was more expensive because royalties had to be paid, but potential sales were greater because the characters came with public awareness built in. Other publishers also published comic books with licensed characters—among them Fawcett, with the cowboy star Hopalong Cassidy, and DC, with the actors Alan Ladd and Bob Hope—but Western signed up more of the most popular characters.
Always, successfully translating such material into the comic-book format required ingenuity. Although the lead character in the Roy Rogers movies, like the lead character in Roy Rogers Comics, was called "Roy Rogers," Rogers's western musicals were not otherwise much like the comic books. For one thing, there could be no trace in the comic books of supporting regulars like Gabby Hayes and Dale Evans, both of whom had their own licensing deals and their own comic books. Rogers and Gene Autry in their movies could assume different roles even under their own names, but their comic-book equivalents were almost always depicted as wandering cowboys, or maybe ranchers out running errands. Those two stars, like the less stellar cowboys (Bill Elliott, Johnny Mack Brown) who were also published by Dell, were latter-day equivalents of William "Buffalo Bill" Cody, a real man whose fictional adventures filled hundreds of dime novels in the late nineteenth and early twentieth centuries. The difference was that the dime-novel Cody's adventures had a slender basis in fact—his exploits as an army scout on the frontier—whereas the Dell comic-book cowboys were simply actors.
The stories in the Dell western comic books were typically longer and more complex than the short, abruptly told stories in other publishers' comic books, stories that were essentially interchangeable with the ones in the same publishers' crime comic books. The difference between Dell and its competitors was amplified starting in 1948 when Western began adapting Zane Grey's novels, with Gaylord DuBois as the adapter. Those adaptations filled whole comic books—first of thirty-six pages, covers included, and then, for a couple of years, fifty-two—with complete stories whose length naturally made them attractive to older readers.
As western comics in particular grew in popularity, there was a diminution in the number and frequency of comic books intended for a very young audience—that is, comic books of the kind that had been Oskar Lebeck's specialty and that Walt Kelly had often illustrated. After Animal Comics ended in 1947, a few new ones did appear. In 1948 and 1949, Kelly drew the first two one-shot issues of The Brownies, a property that Western had bought from the estate of the Brownies' originator, Palmer Cox, and he also drew stories about those tiny characters for Raggedy Ann & Andy. But Raggedy Ann, still a monthly in 1949, was cut back to the occasional one-shot in the summer of that year. Kelly's Mother Goose annuals were gone after the 1949 Christmas issue.
The nature of Kelly's work for Western was changing as his old venues disappeared, but he was still producing a great deal of comic-book material. Even after Pogo began syndication, he was writing and drawing every issue of the Adventures of Peter Wheat, an eighteen-page monthly giveaway comic for a bakery, made up of fairy tales about the Tom Thumb–like titular character and the other tiny creatures—birds, animals, and insects—living with him in a wheat field. He continued to draw front covers for Walt Disney's Comics & Stories into 1950. Also, around the time Pogo went into national syndication, he wrote and drew the first fifty-two-page issue of a new quarterly comic book called Pogo Possum.
That first issue, dated October–December 1949, was copyrighted by Western Printing & Lithographing, even though Kelly had by then become the Pogo comic strip's owner. The copyright had no legal significance—Kelly's contract with Western was controlling, and it established Kelly's ownership of everything about Pogo—but Western apparently was determined to maintain some connection with Kelly and his popular comic strip. It was, after all, a property that Western had once owned and that gave every sign of becoming increasingly popular and prestigious, even though Kelly resisted commercializing his characters. But despite the apparent goodwill on both sides, the Kelly–Western relationship yielded up difficulties frequently over the next few years.
Kelly divorced Helen on Halloween 1950, in Mexico, and married Stephanie Wagonny soon afterward. She was the daughter of immigrants from central Europe, unskilled laborers whose origin is listed variously in census records as Austrian or Ukrainian, no doubt in a reflection of shifting European borders. It was a familiar situation: Stephanie had been Kelly's secretary—her sister Julia succeeded her in that job—and she was about twenty years younger than Helen, who by the time of the divorce was well into middle age. In the early 1950s Kelly was supporting two households, both with young children; he and Stephanie had a son in 1951. (Stephanie eventually gave birth to five children in all, although two died in infancy.) The Pogo comic strip was increasingly popular but had yet to produce an income that would separate Kelly decisively from his family's blue-collar life in Bridgeport. Everyone involved—Kelly, Western Printing, Dell Publishing—thus had every incentive to make the Pogo Possum comic book a success. But determining how to do that was not easy.
The need for a new agreement between Western and Kelly arose in 1951 when Kelly wanted to publish a paperback collection of his newspaper strips through Simon and Schuster, since in Western's view it had retained the right to publish such Pogo books under its contract with him. In a September 20, 1951, agreement, Western released "any claim that we may have had or may have under said agreement or otherwise, and we acknowledge that the imaginary character know as POGO or POGO POSSUM, and associated imaginary characters, including ALBERT THE ALLIGATOR and others . . . are your property." Western retained only the exclusive right to publish Pogo comic books, and a more limited right to publish Pogo children's books of the kind that made up its huge Whitman line. Kelly was to be paid Western's standard royalty on his comic books of one-fourth cent for every copy printed, as well as "regular rates" for writing and drawing the comic books. He could terminate the agreement if his royalties fell below five thousand dollars a year. Those "regular rates," as reflected in statements sent to Kelly by Western in September 1951 and March 1952, were $32.75 for writing and drawing each comic-book page (plus $2.25 if he did his own lettering), $50 for a front cover, and $25 each for other covers—rates comparable to those paid then to other cartoonists who did not own the properties they worked on.
While the available record of Kelly's dealings with Western in the 1940s is skimpy, there is fuller documentation of what was going on in the early 1950s. Kelly's position—as the owner of his comic book's copyright as well as its cartoonist—was not unique but it was certainly rare, and his unusual status undoubtedly contributed to the irritable tone of many of his exchanges with Western's editors and executives. He had to be concerned with much more than writing and drawing funny stories.
The U.S. Post Office Department was one source of irritation because Western had to meet the post office's often peculiar requirements for mailing at second-class, or periodical, rates. In 1951, postal officials wanted Kelly's name removed from the front cover of the comic book. In 1952, when Pogo Possum skipped a quarterly issue, the post office noticed, and, as Richard Small of Western told Kelly, it warned that "any other departure from its authorized frequency would call for a cancellation of the second class entry permit. As you know, this means several thousands [sic] dollars difference to Dell and would further upset their entire argument about the Dell magazine having a continuing periodicity over many years."
Anne DeStefano, who was Oskar Lebeck's secretary throughout the 1940s but by the early 1950s was an editor in Western's New York office, fretted to Kelly about the need to "have at least 8 pages of art without Albert or Pogo" to meet another postal requirement. His immediate reply was acid:
The majority of sales on the book is of the newsstand variety, and for me it is a great handicap to try to meet a non-important second-class entry requirement.
Pogo is the lead character with whom the Post Office Department is concerned. They should not be concerned with Albert. Pogo does not appear in a number of pages here. In fact, he does not appear at all in at least one story, and I feel that I have met all the requirements which should be placed upon me in delivering this material.
Furthermore, if there is to be any discussion on this, I think that Dell and Western should place in my hands a review or a record of POGO comic book sales so that I can more accurately judge whether such stupid requirements are in any way necessary.
Kelly groused about Dell, and especially its decision to raise the price and page count of Pogo Possum, in a letter to Lloyd E. Smith on May 13, 1952. After four fifty-two-page issues in 1949 and 1950, Pogo Possum, like other Dell comic books, had been cut back to thirty-six pages in 1951. But then, starting with no. 9, April–June 1952, the page count went back up to fifty-two, with an increase in price to fifteen cents. By the time Kelly wrote to Smith, one fifteen-cent issue had been published, with another soon to follow.
The nagging problem, as Kelly knew, was how to distinguish Pogo Possum from all the other comic books on the newsstands and get it into the hands of its natural audience—the older and more educated readers who followed the daily comic strip and had no interest in comic books in general. Dell's solution was to raise the price to fifteen cents for a fifty-two-page comic book, at a time when the company's other titles were priced at ten cents for thirty-six pages (covers included in both cases). Kelly believed, correctly, that such a solution was no solution at all.
I think Dell should either publish or get off the pot. The comic book if correctly distributed will sell according to all accounts we get from colleges and other spots where the book cannot be found. So far as I am concerned I am sorry that I did not put on the record my objections voiced to you and Richard Small on separate occasions before the 15 cent mag came out. If you'll recall I said that even a novice at distribution could plainly foretell what would happen. The extra nickel did not discourage the buyers but it surer than hell made every other dealer hide the bastard-priced comic so that he wouldn't get stuck for an extra nickel every time a kid bought a clutch of comics. I saw not one copy of the book displayed in Grand Central for one place when it retailed for 15 cents.
It should not come as any surprise to anyone that Dell's activities only irk me and I am not happy with the relationship at all. Why they think they have any sort of claim on POGO or an expectation of reward is beyond every concept of fairness that I picked up as an Eagle Scout. How in hell can they even hope to retail a 25¢ or 50¢ or for that matter a dollar book and give me as good a break as [Simon and Schuster]? This POGO thing has gone because I have worked my can off, night and day. The fruits of the effort are not going to [be] shared with any johnny come lately with the editorial judgment of a codfish.
Dell tried at one point to stake a claim of some kind on the paperback books—or so Kelly believed—and it may have been for that reason that he agreed to the 1953 reprinting of Animal Comics stories in the giant comic book Pogo Parade.
As of no. 14, October–December 1953, Dell dropped Pogo Possum's price to ten cents and the page count to thirty-six. Moreover, about one-third of those pages were now taken up by a reprinted story from Animal Comics. The change was not working, as Small, the liaison with Dell at Western's Poughkeepsie plant, told Kelly early in 1954:
Dell had hoped that the change in price from 15¢ to 10¢ on POGO would make a big increase in the sale of the comic magazine. However, the reverse has proved to be so. Apparently although dealers did not like the unusual 15¢ price, it at least made them keep it separate and adults found their way to it more readily than they now do when it may be buried with all the other 10¢ comics. In addition to this, Dell has received quite a few adverse comments about the reprint material.
Dell advises that they are currently losing $4,000 per issue and while we are all pogophiles, it is hard for us to insist that Dell continue to shell out that kind of money in order to keep the magazine as part of their line. So, it would appear we have two choices: either to go back to a 48 page all original material 15 [cent] item or reluctantly to give up.
Kelly drafted but apparently did not send a brusque reply, reminding Small that he had told him the previous May that "96 pages [a year] is the utmost I can deliver and I have trouble doing that. Also, despite royalties and page rates I lose money too on the comic books. Similar material in the other forms pays twice and four times as much." Another, undated draft expressed his frustration in greater detail:
It doesn't seem to me that Dell's comic line is the place for adult material in the first place. This conclusion I've reluctantly reached after 12 years of having Dell Pub. tell me so through Oskar [Lebeck] and others. Therefore I blink a little when told that the way Dell can best deliver POGO to his readers is to have the newsdealer hide a 15¢ item so the adults will ask for it.
This method of distribution is novel but should be tried with somebody else. The sole remaining value of a POGO comic book, to me, is publicity and promotional. If it's not displayed on the stand, I work for nothing.
Kelly finally sent a more conciliatory reply on February 3, 1954: "As you know, producing even [as] much POGO material as Dell currently publishes is a severe tax on my powers of ingenuity. We discussed the problem perhaps a year ago. Since then the situation has worsened. I have no desire to knock myself out. Comic book work pays only a third as much as the other books." Kelly also complained that the Pogo Possum comic book robbed him of ideas that he could have used more profitably (in every sense) in the newspaper strip. As time went on the comic book began to look more and more slapdash, written and drawn in haste. From the beginning, many of the Pogo Possum stories had a free-association quality more pronounced than anything similar in the comic strip. Kelly surely started such stories in John Stanley fashion, without knowing where he was going to wind up, but, like the good Disney man he was, he never completely abandoned comic plausibility, even when he had enlisted his characters in escalating foolishness.
What made the comic strip invariably better was not just its superior execution—the sense that Kelly was taking more time and care with his dialogue and drawings—but the way that the added time and care translated into stronger characters who were doing funnier things. Albert in particular was vulnerable to misuse, as in the story called "Fire Bugs," in Pogo Possum no. 2, April–June 1950, when he lunges out of character by bursting into a recitation of poetry. A couple of years later, in "Mother's Gooseberry Rinds," in Pogo Possum no. 10, July–September 1952, Kelly had Albert recite Lewis Carroll, again at the cost of diluting his comic personality. In the comic books Kelly's characters were also susceptible to speaking in very ordinary language that testified through its ordinariness to just how much trouble Kelly took with the rich, strange dialect his characters spoke in the comic strip. Kelly may have been deliberately suppressing dialect in the comic books for the sake of young readers, but if so, there seems to be no record of that—and he had not shrunk from using an even thicker dialect in Animal Comics.
The lettering of Kelly's Pogo Possum stories was farmed out, as so often had been the case with his earlier stories. Anne DeStefano wrote to Kelly in 1952 to tell him that Raymond Burley, a veteran illustrator for pulp magazines and early comic books, had lettered forty-six pages for no. 11, January–March 1953. As before, the results, compared with lettering done by Kelly himself or under his direct supervision, were limp (the lettering is mostly italic) and all too uniform. George Ward said that Kelly was "very pleased" with Burley's work, even though it seems clear that the expressive possibilities had been suppressed. In the comic strip, by contrast, those possibilities were sometimes realized flamboyantly, in dialogue that resembled a circus poster or black-letter printing, but even more often in subtle variations of size and weight that made Kelly's increasingly intricate language all but audible.
Kelly put even his panel borders to expressive use. The "Albert and Pogo" panels were stacked four deep in almost every issue for two years, in 1944–46, when paper shortages were driving down comic books' page counts. Then, in Animal Comics no. 20, April–May 1946, they were again stacked three deep, as in Kelly's earliest stories with those characters. Kelly's panel borders had been ruled until that time, like most comic-book borders, but starting with no. 20 he drew them freehand, with an attractive but not overbearing irregularity (and he often dispensed with borders altogether). Ruled panel borders disappeared from Kelly's "Our Gang" stories around the same time, as of the July 1946 issue, and freehand borders were from then on a distinguishing mark of Kelly's comic-book and comic-strip work.
"Feelin' Mighty Hale, and Farewell," by Walt Kelly, in Pogo Possum no. 3, August-October 1950, was a successful reworking of an earlier Animal Comics story.
The shortcomings of Burley's lettering aside, the best stories in Pogo Possum were still far advanced over their Animal Comics predecessors, a point that Kelly himself underlined in Pogo Possum no. 3, August–October 1950, when he reworked a story from the 1946 Albert the Alligator and Pogo Possum one-shot. The new version, "Feelin' Mighty Hale, and Farewell," is one of the many stories in which Albert has accidentally ingested another animal or thinks he has, and also one of a number in which Albert is impenetrably certain of his own attractiveness—in this case, when he is disguised as a female (a cigar-smoking female, at that). Hailed as "Uncle Albert" by passing rabbits, Albert concludes, reasonably enough by his standards: "Natural, they mus' of mistooken me fo' a li'l gal rabbit name of 'Uncle Albert.'" Where the old story is tight and awkward in drawing and staging, the new version is open and relaxed, and the characters, Albert especially, have been drawn with a confidence that says Kelly has come to think of them as good friends.
When he was not equipped with a story that saved him the work of thinking up a new one, Kelly could be tempted into cutting corners: simplifying his characters' dialogue or descending into an aimless running in place that merely filled up pages with static panels. The thicker the comic book, the greater the risk, and the stories in the fifty-two-page, fifteen-cent comic books of 1952–53—there were five of them—are often very thin indeed. It was at this low point that George Ward, after several years as a freelancer, began working for Kelly again, inking his pencil drawings for the last few issues of Pogo Possum. Ward remembered: "The early Pogo comic books were something Walt had a lot of fun doing and we turned them out very fast. Walt would write everything and do the pencils and I would letter some of them and do most of the inking." Unfortunately, not much of that fun wound up on the printed page.
Kelly found one happy solution to his comic-book predicament when he turned to fairy tales of the kind he once drew for Oskar Lebeck. Starting with "A New Jag on the Old Beanstalk" in Pogo Possum no. 7, October–December 1951, he began putting his characters into burlesques of familiar stories. Several more such burlesques followed—the most successful was "Cinderella and the Three Bears," in no. 8, January–March 1952—but there was a complication. In June 1953, when Simon and Schuster published its third Pogo paperback, Uncle Pogo So-So Stories, it was made up not of reprints of the newspaper strip but of new material that included a Mickey Spillane parody and comic retellings of classic tales. Fairy-tale burlesques fit perfectly not just in the comic book but also in the new paperback.
Kelly's 1951 agreement with Western was modified in 1953 to make it unmistakably clear that juvenile books, and not a book like Uncle Pogo So-So Stories, were its subject matter. Kelly had good reason to wall off books like Uncle Pogo So-So Stories from his work for Western and Dell: his Simon and Schuster royalties were higher than his comic-book royalties or any other royalties Western might pay him. Three months after So-So Stories was published, Simon and Schuster announced that the book had sold more than 160,000 copies, at a dollar a copy. It was approaching the sales of the first two daily-strip compilations, Pogo and I Go Pogo, which had sold more than a quarter million copies each. Kelly was receiving royalties of five cents a copy on the first 25,000 copies of each book sold and seven and a half cents on most sales after that. Surviving royalty statements in Kelly's papers show that when he next produced a book of original stories—The Pogo Stepmother Goose, published in the summer of 1954—Simon and Schuster sold more than 100,000 copies in the first few weeks, generating a payment to Kelly of more than nine thousand dollars.
"Cinderella and the Three Bears," by Walt Kelly, in Pogo Possum no. 8, January-March 1952, was a burlesque of classic fairy tales like those that Kelly once drew more seriously for Fairy Tale Parade.
Western was prepared to continue with the comic book, but from Kelly's point of view, continuing with it could have made no sense. As he told Richard Small, stories he drew for a new paperback book were likely to earn him several times the royalties he would earn by drawing an equivalent number of pages for comic books. If he had needed comic books as a source of income a few years earlier, before the Pogo paperbacks' great success, such was no longer the case.
Pogo Possum expired with no. 16, April–June 1954. With it ended Walt Kelly's association with the Dell comic books. He was furiously active throughout the rest of the decade, promoting his comic strip and traveling widely, in the United States and abroad, but he produced nothing more in comic-book form, other than paperback books that were really comic books in disguise. He scaled back in the 1960s, dogged increasingly by ill health—his own and his wife's. Stephanie died in 1970, and Kelly himself succumbed to the complications of diabetes on October 18, 1973, at age sixty.
As early as that death might seem, Kelly had lived longer than many of the close friends, New York Star editors and writers in particular, with whom he had raised a glass at his favorite bars: Tim Costello's on Third Avenue, and Bleeck's (more formally known as the Artist and Writers Restaurant) on West Fortieth Street. Joe Barnes, John Lardner, George Y. Wells, Richard Lauterbach, the New Yorker writer John McNulty, the Newsweek editor Niles von Wettberg—all died before Kelly, some at a much younger age. Kelly's closest comic-book colleagues lived much longer, with one conspicuous exception: Oskar Lebeck.
## 22
# Oskar Lebeck in Exile
By the time Walt Kelly's relationship with Western ended, Oskar Lebeck had been gone from Western for several years. He did not leave under the happiest circumstances.
Moe Gollub thought that Lebeck shared some of his own pugnacity: "He insisted on being his own man. And he was right, as far as I'm concerned, most of the time. He had a crazy kind of integrity, and I liked him for it. I was so spoiled by him that any subsequent employer I could have had would never have seemed as good. I can't regret having worked for Lebeck, even though I never made a lot of money. But they were getting incensed at him up in Poughkeepsie, because he wouldn't follow directives; they couldn't move him around. He was not a sycophant, that was all."
Lebeck was still hiring cartoonists in 1950, when he hired Tom Gill to illustrate the Lone Ranger comic book (until then one of the few remaining Dell titles devoted to reprints of a newspaper strip), but his major project that year was the Surprise Books. Those dozen hybrids, published in the fall of 1950, were the clearest expression yet of his desire to bridge the gap between traditional children's books and comic books.
Eight of the Surprise Book stories were traditional (Little Black Sambo, Sleeping Beauty, Alice in Wonderland), and four were modern. Lebeck adapted or wrote all of them, including a new version of his own 1939 book Clementina the Flying Pig. The artists, like Sheilah Beckett and Tony Rivera, were veterans of either Whitman's children books or book-like features in Lebeck's comic books. Dan Noonan, who illustrated two Surprise Books, The Emperor's New Clothes and Teddy B.B., described them with general accuracy:
Oskar Lebeck posed with members of his comic—book staff around 1950, not long before Lebeck's departure from Western Printing. Standing behind Lebeck are, from the left, Mel Crawford, Dan Noonan, John Stanley, and Dan Gormley. The photo was probably taken for an article for Western's house organ, the Westerner, that was planned but never published.
The Surprise Book was exactly half the size of a comic book [half as tall, that is, with sixty-four pages, not counting covers, the equivalent of a thirty-two-page comic book]; it wasn't a true comic, but it did tell the same kind of story, only with fewer pictures and no dialogue balloons—just a running text. [Actually, most of the Surprise Books had dialogue balloons, but they were stylized, with upper- and lower-case lettering, and only generally resembled typical comic-book balloons.] He felt that by making it this size, and by taking it out of the comic book display racks, it would give kids the basics of a modified comic and at the same time would open new areas of publishing for the Western line. . . . His idea was to force it off the newsstands and up onto the counter, where it would amount to a partially parent-purchase. . . .
Oskar always fought against violence in comics; and in 1950, and for a few years afterward, there was quite a bit of it in other publishers' comics. To Oskar, it was all very lurid sensationalism. He would have none of it. It outraged him. He wasn't a prude by any manner or means, but he just felt that this wasn't for the youngsters' market. He wanted to go into the real young market with the Surprise Books, and he felt that a parent-purchase would insure even a much greater sale than the comic book did. And maybe it would have, with stronger promotion. We'll never know.
Teddy B.B. was one of the dozen Surprise Books—this one was illustrated by Dan Noonan—that Oskar Lebeck shepherded into print in the fall of 1950.
Lebeck cared about the Surprise Books, and there was a trace of bitterness in what he wrote about them to his agent, Toni Mendez, in 1954. He said he had written the stories for the Surprise Books "on my own time. Four of them while I was on a leave of absence (without pay) in the Bahamas. Artwork for two additional Surprise Books was completed but the books were never published." In the wake of the Surprise Books' failure and Lebeck's departure, Western wanted to be rid of the Surprise Books completely. In January 1953, the company sent Lebeck the paperwork for a transfer to him of the copyrights.
For some reason, the transfer never took place, and Western ultimately renewed its copyrights on eight of the books, including the new version of Clementina. In 1957, the flying pig returned in another new story, this one illustrated by Mel Crawford, a deft practitioner of various storybook styles who had begun working for Western at the very end of Lebeck's tenure. The new Clementina appeared in the ninth issue of Dell Junior Treasury, a very Lebeckish comic book devoted to classic stories like Gulliver's Travels (and costing, for the first few issues, a premium price of fifteen cents). Dell had already published a sequel to Lebeck's "Santa and the Angel" in another Junior Treasury issue, also illustrated by Crawford. Both comics were copyrighted in Lebeck's name, so he and Western evidently reached a rapprochement of some kind.
The Lebeck era at Western's New York office had ended sometime in 1951. Lebeck was listed as a vice president of K.K. Publications in Western Publishing's annual reports for 1949 and 1950; he was gone from the 1951 report. He was still working for Western in March 1951, when he was involved in the early stages of work on the new Tom Corbett Space Cadet comic book, but he left sometime soon after that. His successor, George Brenner, previously an editor for another publisher, Quality Comics (Plastic Man, Blackhawk), held the job only briefly before his death in March 1952. He was succeeded by Matthew H. Murphy, who had been an artist for King Features Syndicate and then an editor of the Harvey comic books, which resembled the Dell comic books in their reliance on licensed characters. Murphy remained with Western until 1970.
Lebeck moved on. He collaborated on a science-fiction comic strip called Twin Earths with Alden McWilliams, who had illustrated many of the stories in the comic books Lebeck edited, starting in the 1930s. More recently McWilliams had drawn the first few Tom Corbett comic books for Western, which were published early in 1952. In June 1952, just before the launch of Twin Earths by the United Features syndicate, Editor & Publisher reported that Lebeck was "in semi-retirement, though still a consultant to Dell." After Lebeck left Twin Earths, he tried to float projects of various kinds, as reflected in his correspondence with Toni Mendez, but without success. He and his wife ultimately moved to California, to be closer to their daughter. He died there, suddenly and unexpectedly, on December 20, 1966. He was sixty-three years old.
By the time Lebeck died, Western's comic books had changed fundamentally from what they had been in his hands. Formula storytelling and drawing had become increasingly dominant. A holdover like Carl Barks still stood out from the crowd; Jesse Marsh and Gaylord DuBois were two others. Marsh continued to draw the Tarzan comic books, his work coarser and scratchier as his eyesight deteriorated (he suffered from diabetes) but distinguished still by the calm and order, to the point of stoicism, that separated it so firmly from other "jungle" titles in the late 1940s.
Oskar Lebeck in the early 1950s, around the time his Twin Earths comic strip was launched. Courtesy of Letty Lebeck Edes.
Like Carl Barks, Marsh kept his distance from other comic-book artists. That was not because he did not take his work seriously. As Alex Toth, a fellow artist for Western in the late 1950s and a great admirer of Marsh's work, wrote, "Jesse did it all, always: penciled, lettered and inked the lot. . . . Jesse experimented constantly, finding new methods and techniques to do the job—restless, knowledgeable, astute—but unwilling to 'talk shop' with colleagues."
As Toth said, Marsh's was a low-key style, far removed from the "wild and wooly" drawings of many New York-based cartoonists, but with an exceptionally solid underpinning: "[B]efore laying out a page"—usually three panels deep, with six or seven panels to the page—"Jesse would create an overall abstract design on the page blank—once done, he'd block in his panels, [the dialogue and captions], figures, props, backgrounds—within the context of that design. . . . His inking would either accent or diminish that design, by turns, according to his mood of the moment."
Toth had been a New York–based cartoonist before moving west. He encountered Marsh by accident at Western's Beverly Hills office one day and tried thereafter to see him socially and talk about their work, but Marsh would have none of it: "Jesse had a wall up and few of us could penetrate it, but I kept at it," until finally Toth realized the effort was futile. Marsh was a complete professional and valued by Western for that reason, but, like Gaylord DuBois—and, for that matter, Carl Barks—he seemed to understand that anyone working in the comic-book industry courted heartbreak if he let himself think about his work too seriously—that is, as a kind of art and not just as a job that was a little more interesting than many others.
Speaking as the professional he was, DuBois told a correspondent:
The sort of script I have turned out for comics is, so to speak, a factory product. I have never owned one after it left my hands, because I always sold all rights to the company who assigned me the writing of it. Moreover, the writing of the script was not, strictly, a one-man job. In the case of the Tarzan comics, I was given the characters of Tarzan, Jane, and Boy to build my stories around. Sometimes I was told that all three must feature in a certain story; and always my instructions were clear as to editorial policy—and many things were taboo. Among the taboos were ungrammatical speech by the principals; sex emphasis, and everything that might honestly be called horror.
Despite working within such constraints, DuBois spoke of "buying and digesting all the good books on Africa I could get, treasuring National Geographics and even delving into the Encyclopedia Britannica. Oh yes, my hat is permanently off to Sir H. Rider Haggard—in my mind he has done for 19th century Africa comparable to what Zane Grey has done for the Old West."
There were, to be sure, modest compensations for adhering to a modest view of the work. According to his biographer, DuBois for many years received bonus checks "based on the sales over a specified number. More than any of DuBois's other writings, Tarzan brought in consistently high bonus checks."
And then there was John Stanley, another highly productive worker. As his Lulu workload grew in the early 1950s, Stanley drew less, but his writing style is unmistakable not just in Lulu but in some other titles, notably at least a few issues of Henry Aldrich, a comic book based on a popular radio show about teenage characters. Western may have envisioned Henry Aldrich as competition for the increasingly popular Archie titles, but Stanley's scripts, illustrated with considerable energy and flair by Bill Williams, were more sophisticated than the typical Archie story, and that may have been a major reason that Henry Aldrich lasted only about four years, from 1950 to 1954. Stanley spoke of getting some of Dell's new titles started before turning them over to other writers. Henry Aldrich may have been in that category—later issues, and some stories in the early issues, are pedestrian—and the scripts for a number of other Dell comic books of the 1950s, including Howdy Doody, Krazy Kat, and The Little King, have been identified as probably his by the Stanley scholar Frank Young. Those stories, if they are Stanley's, are on a much lower level than his best Lulu work, which continued to mature and grow in interest.
That was especially true of the Little Lulu fairy tales of the early 1950s, many of which have the flavor of children's books written for exceptionally intelligent young readers. (Stanley ultimately wrote one hardcover children's book, It's Nice to Be Little, which was published by Rand McNally in 1965, long after his Lulu days; he did not illustrate it.) "Fairy tales" is perhaps too loose a title for these stories, some of which could just as easily be labeled fables or picaresque adventures. Whatever they are called, their sharp edges are often barely concealed by the poor little girl's smile and rags, and sometimes there is no concealment at all.
In "The Little Rich Boy," in Little Lulu no. 40, October 1951, the fairy-tale Lulu is an ingenue indeed, grateful for the kindness she thinks is being shown to her by the preposterously rich little rich boy—kindness that is, of course, nothing of the kind. Overcome with pity for the poor little girl when he sees her looking longingly at a Ferris wheel—he has happened to pedal by on his six-wheel solid-gold cycle—the little rich boy buys the Ferris wheel, dismantles it (with the unfortunate riders still in their seats), and has it rolled down a hill. "Now," as a caption tells us, "the poor little girl wouldn't have to look longingly at the Ferris wheel . . ." The poor little girl is overcome with love for this unselfish little plutocrat ("Gosh! How c'n anybody be so GOOD?") and pursues him relentlessly, even though he buys the police department and has her jailed. Finally he convinces her of his hatred and gives her a shovel so that she can dig a deep, deep hole and crawl into it. But ah! When she digs she strikes oil. She is rich, richer than the little rich boy, and now she takes revenge, with a chilling thoroughness, on the one who has scorned her love.
"The Little Rich Boy," in Little Lulu no. 40, October 1951, written by John Stanley and illustrated by Irving Tripp, was Stanley at his most savagely satirical. © 1951, 1979 Marjorie Henderson Buell.
Such savage satirical comedy—and there were other stories of the same kind—invites comparison, if only in terms of temperament, with Harvey Kurtzman's EC comic book Mad, which first appeared the next year, in 1952. Kurtzman said in 1990 that he "wasn't a Little Lulu fan, but for some reason I knew it was good. I probably had read some Little Lulu and was impressed, but I didn't read any more." He and Stanley worked very much alike, since Kurtzman, as the editor and writer of titles like Frontline Combat and Mad, made layout drawings for other cartoonists that were directly comparable to Stanley's in purpose (although very different in style and tone, especially when war comics were involved). Like Stanley, Kurtzman told his stories visually, with captions that amplified rather than duplicated what each panel showed.
Where Stanley is concerned, the most fruitful comparison may be to Carl Barks's "Donald Duck" stories of the same period. There is in Stanley's best stories—as in Barks's best, not that either man ever spoke in such terms—an intensity of feeling arising from a wholehearted engagement with the comic-book medium and a corresponding delight in its capacity for expression. Both Barks and Stanley had found a way, at least for a time, to add depth to their stories without losing their child readers or alienating their editors.
In addition to the fairy tales for Alvin, and the stories that pitted Lulu against Tubby and his pals, Stanley found other ideas that could be nursed into comic life and exploited in one story after another. Tubby had very early assumed Lulu's old role as the serenely egocentric child, resenting all adult efforts to deflect him from his course, and he played that role brilliantly in a string of stories that began with "The Case of the Purloined Popover" in the last Four Color issue, no. 165 (1947). In those stories Tubby was a self-styled detective, wreaking all the havoc necessary to solve a case that usually involved Lulu's being spanked unjustly. Lulu's father was invariably the culprit, although he was no culprit at all by adult standards. In Little Lulu no. 55, January 1953, after a few years of such detective work, Tubby began calling himself "the Spider" (because, he explains, the spider spins a web to trap the unwary).
When Little Lulu began regular bimonthly publication in 1948, Western added short "Tubby" stories to each issue to meet postal requirements. (Those same requirements meant that Lulu could not appear in the stories.) In the "Tubby" stories, Stanley occasionally presented his hero as the selfish monster he had been in the 1947 "Kid Who Came to Dinner." The story in Little Lulu no. 5, September–October 1948, titled "The Gourmet," is essentially a reworking of the earlier story into something grander and more fiendish. This time Tubby is in a restaurant, running up a tab so large that the kindly couple who invited him to join them cannot pay it. Tubby is indignant at the thought that anyone would come to a restaurant without enough money to pay for their meal—or, as it has turned out, his.
Such a character, however fascinating in occasional appearances, had to be toned down to be tolerable over the long haul, so the Tubby of the early 1950s was merely self-absorbed—although the line separating mere self-absorption from mania could be fuzzed over profitably, as it was in Little Lulu no. 52, October 1952. In that issue, determined to prove to the beauteous Gloria that he is a gentleman, Tubby brings a sack of corn on the cob to her house in order to impress her with the gentlemanly manner in which he eats it. Remarkably, she is not impressed.
So successful was the character Tubby that Dell began putting him into one-shots in 1952, then a quarterly of his own in 1953. Stanley illustrated as well as wrote the first seven issues before surrendering the drawing to a Western veteran named Lloyd White, who also drew for New Funnies. Stanley spoke of White more warmly than he did of Irving Tripp, calling him "a dear friend [who] followed my storyboards faithfully" and also drew the "cover finishes"—in contrast to Little Lulu, whose covers Stanley always drew. White's drawings have a surface similarity to Stanley's but are looser and coarser, a description that might be applied to the Tubby stories generally.
In the four Tubby one-shots of 1952–53, one story fills thirty or thirty-two pages of each comic book—that is, the complete comic book (minus covers) or all but a couple of pages. Length was important in Stanley stories, but thirty-two pages were too many. At their best his stories are distinguished by relaxed, open storytelling in which certain panels are not strictly necessary to convey information, but their presence enriches the prevailing tone. In the fairy tales, for example, comedy that might seem merely sardonic, as could be the case with the short stories in Mad (typically six to eight pages), is made slyer and funnier through the judicious addition of panels that expand the scope of the poor little girl's naïveté and the "generosity" of her supposed benefactors. "The Little Rich Boy" could have been told in ten pages, but it fills thirteen and benefits immensely from the added length. The full-length Tubby stories, though, are a little padded. For other stories, eight or even six pages are enough. What matters is that any given story was allowed to grow to its best length. For Carl Barks, the monthly ten pages for "Donald Duck" in Walt Disney's Comics & Stories were a comfortable discipline, but Stanley worked very differently, and such constraints were not helpful to him.
The general excellence of Stanley's work on Little Lulu and Tubby was rewarded not by any favorable critical attention (although there is slender anecdotal evidence that Little Lulu, like Uncle Scrooge and Mad, enjoyed a larger literate adult readership than other comic books) but by very healthy sales. In December 1951, Marjorie Henderson Buell and Western agreed to a new contract covering the next six years, a contract that reflected the success of the Little Lulu comic book. Buell was to receive eighteen thousand dollars upon signing the contract—money that was not an advance on royalties but was instead a signing bonus. Western could extend the contract further each year by paying a similar bonus amounting to 10 percent of Buell's royalties for the previous year. She would receive royalties of one-fourth cent on each copy printed, Western's standard royalty rate for its comic-book licenses. As always, Western would pay the royalty upon completion of the print run.
Copyright holders in the early 1950s could expect an advance on royalties of at least a thousand dollars on the first issue of a new title, based on a printing of at least four hundred thousand copies. The actual number of copies printed could be, and usually was, much higher—Western aimed for a minimum of six hundred thousand copies—with a corresponding increase in the royalties paid. Advances could be higher, too, depending on how large a minimum print run for each issue a contract envisioned. An April 20, 1951, contract for the Cisco Kid comic book called for an annual advance of $7,500, or $1,250 for each of six bimonthly issues, based on a minimum print run of a half million copies per issue. Western's liability, in the event it decided to stop publication of a comic book, was limited to the advance it had already paid a licensor.
Western wrote to Buell on November 9, 1953, to ask to extend its contract. Her royalties for 1952 (exclusive of her earlier signing bonus) totaled $46,924.42, and so she was entitled to a new signing bonus of almost five thousand dollars. The Little Lulu comic book and its offshoots were not the sole source of those royalties—Western published a variety of other Lulu items, including a coloring book, a storybook, a puzzle, and a record—but the comic books accounted for the bulk of her earnings.
Even more than was true of Disney and the ducks, Marjorie Buell as the copyright holder reaped the benefits of popularity that was due overwhelmingly to the creative efforts of a cartoonist who had no ownership stake in what he created. But as with Carl Barks, it is hard to argue that John Stanley was being exploited. Marge's Lulu, like Disney's Donald Duck, gave a gifted artist the head start he needed to do work that was far more impressive than anything he did, or may have been capable of doing, completely on his own.
## 23
# Manifest Destiny
It was somewhere around the time his second marriage fell apart, early in the 1950s, that Carl Barks briefly thought about leaving Western.
One of Western's artists contacted me outside the building one day, and said, "Say, why don't you do some comic books for this outfit"—I think they were publishing comic books with Heckle and Jeckle, or some of those fellows [St. John Publishing, which had licensed Terrytoons characters like Mighty Mouse and the magpies Heckle and Jeckle]. He said I'd make more money, and so on. I thought, well, I'll wait a while before I make any changes, I want first to see how permanent this outfit is, and see if I can find out from somebody else if their checks bounce, and all that sort of thing. One thing I respected Western for was that their checks never bounced. So, I guess it was a good thing that I was hesitant about it. . . . It was not too long after that that I was talking with Bob Karp, who does the gags for the [Donald Duck] newspaper strip. His brother Lynn was doing some comic-book work in New York, and he knew of this outfit that the Western artist had tried to get me to go with. "Watch out for them," Bob told me, "because they're just liable to fold up and disappear some night." . . . He said, "Don't make any changes," so I didn't, and I never regretted it, because that outfit did fold up later.
Other significant publishers of comic books with animated characters and talking animals had long since shut their doors or curtailed production by the time the St. John company left the field in 1956. Benjamin Sangor shut down his Los Angeles operation, run by Jim Davis, in the summer of 1948. That ended what some cartoonists remembered as a cozy and collegial working life. Davis, Jack Bradbury, and two other cartoonists, Al Hubbard and Hubie Karp, had set up a little office "and worked together full time," Bradbury said. Ken Hultgren "preferred working at home, but would join us for lunch every day." The pay was good: before the Davis shop closed, the per-page rate for writing, drawing, and inking had risen to twenty-five dollars. "It was most unfortunate, really," Davis said, "that Sangor didn't realize what he had his hands on, or never tried to make use of it. Sangor was after the quick buck, he was selling half the stuff that we produced to his son-in-law [Ned Pines] and a lot of it to [DC]."
Mark Evanier, a writer for and about comic books, has written that Sangor closed the Davis shop because he was cutting back on such work for other publishers. "There is disagreement as to whether this was by choice or because companies like Standard [one of the labels under which Ned Pines published comic books] no longer wished to pay the cost of production plus a profit for Editorial Art Service [Sangor's packaging arm]. Either way, Sangor opted to lay off Davis and his crew."
Jack Bradbury put it succinctly: "The bottom sort of fell out." He was recruited briefly to work for Standard but by 1949 had begun working for Western. When Carl Barks's ten-page "Donald Duck" story disappeared from Walt Disney's Comics & Stories as of the February 1950 issue, it was Bradbury's "Donald Duck" that took its place. "When I first went with Western," Bradbury said, "[comic books] were selling like hotcakes; they couldn't get enough of the stuff. They'd practically tell you, don't spend too much time on it, but do as much as you can. We did some pretty rotten stuff, too. And everybody and his brother was working for them."
For someone who had grown used to the routine in the Davis shop, Western could seem rigorous by comparison. The changes that had started gathering speed in the 1940s, pointing toward greater editorial control over the stories, were now firmly in place. Carl Barks was more of an anomaly than ever. "There was plenty of work over there," Bradbury told Dave Bennett, "but the only trouble was that you couldn't write your own material. By only doing the drawing you couldn't make as much money. Your work couldn't be as fast either, because it all had to be okayed by some editor before you could ink it. Western's comics also had eight panels to the page instead of six. It was a whole different process working at Western."
Whereas Roger Armstrong remembered more isolation for the artists after Robert Callender took charge (Armstrong had stopped working for Western by early in 1950, when he began drawing the Ella Cinders comic strip), Bradbury remembered crossing paths with old colleagues: "I'd go over to Western once in a while . . . and I'd meet more guys I knew. That's the last time I saw Carl Barks, was over there. I'd see a lot of old animators over there, guys from the different studios, who had left the studios and were doing comic-book work because they could work at home."
Bradbury may have seen more of his fellow cartoonists because more of them were working on Western's comic books; there was, as he said, "plenty of work." By 1949 Western's Beverly Hills office had contracts with "almost 30" freelance artists, according to an in-house publication, although only "about 14" were working steadily on its comic books. There is no such figure available for the early 1950s, but the total had undoubtedly grown along with the number of Dell comic books.
The paper shortages that constrained comic-book publishers during World War II had persisted for a few years after the war. When Western signed a contract with Gene Autry on December 7, 1945, it called for Gene Autry Comics to be published monthly "as soon as manufacturing conditions and available paper permit," but to be published quarterly for the time being. As it happened, Dell published Gene Autry Comics bimonthly at first, switching to monthly with the January 1948 issue. Western anticipated just such swelling popularity: its contract with Autry provided that he would receive fifteen thousand dollars as a signing bonus, independent of his royalties on the comic books.
Western took a similar long view when it launched a new series of Dell Tarzan one-shots in 1947. The planned print run of 600,000 copies for the first issue had to be cut to 456,946 because, as Western's Lloyd Smith explained, "we could not secure the necessary paper." Western paid Edgar Rice Burroughs Inc. the full royalty on 600,000 copies anyway, and absorbed the loss, cementing what turned into a comic-book partnership of more than twenty years.
With paper becoming less of a problem, late 1947 brought a flood of new monthly and bimonthly titles (Tarzan was one of the latter) as Western put comic books it had published as one-shots on new regular schedules. The publishing schedules of one-shots could be juggled if paper shortages demanded it, but once a comic book had been put on a regular schedule and had thus qualified for a permit to mail subscription copies at the reduced second-class rate, maintaining that schedule was very important. The post office frowned on publications that claimed to publish on a regular schedule but in fact did not, and subscriptions were, of course, a foundation of Western's business.
Throughout the 1940s, the comic books based on the Disney, Warner Bros., and Lantz cartoon characters were limited mostly to the monthly anthology titles, plus one-shots that were devoted to two of each studio's characters: Donald Duck and Mickey Mouse, Bugs Bunny and Porky Pig, Andy Panda and Oswald the Rabbit. There were occasional one-shots based on Disney movies, too. As paper became more readily available, most of the one-shots with the established characters increased in frequency, from one a year to three or more, and Western also began publishing one-shots based on some of the backup features from its anthology titles. Woody Woodpecker from New Funnies was one of the first, in 1947, and by the early 1950s new one-shot titles were spilling out in growing numbers. Mary Jane and Sniffles and Tweety and Sylvester were one-shots based on features in Looney Tunes and Merrie Melodies Comics; likewise with Li'l Bad Wolf and Little Hiawatha from Walt Disney's Comics & Stories. Some of these new one-shots gave birth to series on a quarterly or bimonthly schedule, others to more one-shots. Still others, like the Mary Jane comic book, died after a trial issue or two. Dell was publishing a growing number of giant comic books, too—a hundred pages for twenty-five cents—with the Disney characters and other licensed cartoon characters.
By the early 1950s, Western was expanding well beyond the core of licensors whose characters dominated its line in the 1940s. There was a flood of new titles, based on comic strips (Beetle Bailey), movies (Gerald McBoing Boing), radio shows (Sergeant Preston of the Yukon), and especially television (Howdy Doody, Beany and Cecil). Dell's monthly cowboy comic-book stars Gene Autry and Roy Rogers were by the early 1950s stars of weekly television shows. George Delacorte wrote to Autry on January 4, 1951, a few months after the Autry show's debut, to tell him that television's power was making itself felt: the comic book's sales were about 5 percent higher in areas where the show was seen. So popular were the Dell cowboys that Dell published quarterly spin-offs starring their horses, Autry's Champion and Rogers's Trigger. The Lone Ranger went them one better, with a bimonthly devoted to the Lone Ranger's Indian companion, Tonto; and a quarterly about his horse, Silver.
This expanded schedule was the work of Robert Callender. He came to the West Coast "to enlarge the office," Chase Craig said, "and that he did . . . building it up from a one-man office to an office of perhaps 30 people." In June 1951, Western moved its growing staff into its own Whitman Building at 9916 Santa Monica Boulevard in Beverly Hills. Although the New York office continued to produce comic books like Little Lulu and Pogo Possum, among many others, the greater part of Western's comic-book output was shifting westward. Comic books that in Oskar Lebeck's day had originated in New York, like New Funnies and Tom and Jerry, became West Coast products. As Chase Craig put it, "[I]t was Callender's idea that all the comic books based on Hollywood's production should be actually produced here, close to the studio contact." Craig, after freelancing as a cartoonist and then as a writer for the comic books in the 1940s, was hired by Callender (in January 1950, Craig said) "to come into the organization as co-editor with Alice Cobb, to work on comics exclusively, as editor and art director." By 1956 Craig was overseeing at least twenty comic books a month, with Cobb by then listed as one of his "chief assistants," along with the former Disney writer Del Connell.
As Western's line expanded, cartoonists like Jack Bradbury began to do most of their work for Western. Other artists who had not worked in comic books before also became regular contributors to the Dell titles. Richard "Sparky" Moore was cleaning swimming pools in Beverly Hills when he saw a magazine advertisement announcing: "[A]rtists wanted to draw cowboys and Indians. . . . So, I went home and bundled up some of my cowboy drawings and mailed them off." A few weeks later—this was probably in 1951—Moore was summoned to meet with Tom McKimson, who by then had succeeded Carl Buettner as the comic books' art editor in the Beverly Hills office. (Buettner was still supervising the comics' covers and the artwork for the children's books that Western produced for Simon and Schuster and its own Whitman line.) Once a story's script had been approved by an editor like Craig, it was forwarded to McKimson for assignment to an artist.
McKimson gave Moore the script for his first drawing job, for the Johnny Mack Brown comic book. "They gave me one picture of [Brown]," Moore said. "That was it, and you were cut adrift. You never had the feeling that you were launching one for the art league. Here's the book, it's thirty-two pages, and get out of here." As was typical of Western, Moore never met any of the writers whose scripts he illustrated, and there was little or no thought given to the compatibility of artist and writer. "In my case," Moore said, "dogs and horses were very strong, and anything Tom would get like that, I'd get it, usually. . . . At that time, the Western office was made up of a number of small offices, and you really had no knowledge of the script you got, who had handled it or what they had done with it. . . . [I]t was pretty much like a factory." Moore remembered starting at twenty dollars a page for his drawings, "and the rate never went higher than thirty dollars. I could never get the thirty dollars. I got up to twenty-nine dollars, but I could never make it [to thirty]." For that figure, Moore penciled and inked each page but left the lettering to a specialist: "You'd do the pencils, and he'd letter it, and then you could build the picture around that."
In addition to the ongoing Zane Grey series, Dell in 1952 began publishing in its Four Color series comics based on the work of other popular western novelists, including Johnston McCulley (Zorro), Max Brand (Silvertip), Ernest Haycox (Western Marshal), and Luke Short. Starting with the first Silvertip (Four Color no. 491, 1953), and for the next few years, all of those comic books, each with a story filling thirty-four pages (including the back covers), were illustrated with great panache by Everett Raymond Kinstler, whose editor was Matt Murphy, Oskar Lebeck's successor in Western's New York office. Kinstler's staging and brushwork were flamboyant as the Dell comics seldom were, his villains seamier and slimier, his action deadlier and often far more violent, especially in early examples like the first Western Marshal (Four Color no. 534, 1953). Kinstler's skills were so imposing that almost every melodramatic panel was self-justifying.
There are echoes of Lebeck in what Kinstler recalled about Murphy. He said that Murphy "gave me two things I really appreciated: longer stories—that let me cut back on the time I spent searching for work—and a degree of freedom to tell the story my way. . . . The instructions I actually received from the scriptwriters were more staid, but Matt trusted me and allowed me to reinterpret the story if I thought I could make it more exciting." Unfortunately, the scripts remained wordy when they got to Kinstler, their frequently overloaded dialogue balloons getting in the way of the dynamic drawings.
In the 1950s, as in the 1940s, the most important cartoonists and writers at Western—people like Barks, Stanley, DuBois, Marsh, and, for a time, Kinstler—were closely associated with that publisher and a few distinct sets of characters. Barks even became a Western employee, he believed in 1953, and so became eligible after five years for what he called "their big package of fringe benefits." When Barks went onto Western's payroll, he received regular advances every two weeks and accumulated a "credit balance" for work beyond that covered by the advances. It was then that he began receiving vacation pay and bonuses twice a year—bonuses that, he said, "averaged around 10 percent of what I'd been earning."
Western Marshal Four Color no. 534 (1953), illustrated by Everett Raymond Kinstler, was uncommonly melodramatic for a Dell comic book.
The company could afford to be generous. As Western's comic books flourished in the late 1940s and early 1950s, Walt Disney's Comics & Stories was easily the most successful, accounting for about 10 percent of all the Dell comics printed. The print run—the number of copies on which Western paid a royalty to Disney—crested at 3,038,000 copies with the September 1953 issue. The figures for the Dell line as a whole were no less impressive: an in-house publication said that year that "we create and manufacture no less than 375,000,000 comic magazines annually, at the current rate of 30,000,000 or more per month, or about 1,400,000 to 1,500,000 every working day."
Starting late in 1949, and coinciding with the expansion in the number of Dell titles, Western made all of its comic books forty-eight pages plus covers and began emblazoning on each front cover: "52 pages all comics!" Most other publishers' comic books carried advertising, much of it tawdry—notably the ads for the novelty seller Johnson Smith and the bodybuilder Charles Atlas. Between 1940 and 1952 advertising was almost completely absent from all Dell and K.K. titles; the few exceptions (like a General Mills cereal ad on the back cover of the November 1946 Walt Disney's Comics) may have been relics of old contracts. In its 1951 contract for the Cisco Kid comic book, Western agreed "that in none of the publications contemplated hereunder will it permit or cause to be permitted any advertising matter to appear, intended to publicize any products other than its own publications." Only subscription ads were permitted. Presumably other contracts contained similar wording.
Unfortunately, Western's campaign to set itself more clearly apart was poorly timed. In mid-1950 the Korean War arrived, with attendant inflation, and in early 1951 Dell scaled back all of its titles to thirty-two pages, except for the monthlies and a rare one-shot like the comic-book version of Walt Disney's animated feature Alice in Wonderland. In the fall of 1951 the monthlies, too, fell to thirty-two pages. The change came so abruptly that the November 1951 Tom and Jerry Comics bore ghostly traces of the "52 Pages" slogan in the lower left-hand corner of its front cover.
Western attached importance to the missing pages. It began to move carefully into advertising, assuring its licensors that they need not fear being associated with ads of the Johnson Smith kind. In December 1951, with ads in the Dell comic books about six months away, Lloyd Smith wrote to William C. Erskine, Marjorie Buell's representative, that Western would give him and Buell ample opportunity to object to any ads contemplated for Little Lulu and would, besides, do its best to see that any ads appearing in Little Lulu also appeared in other Dell comics, so that Little Lulu would have company if there was any negative reaction. In the summer of 1952, Dell's monthly titles returned to forty-eight interior pages and began running advertisements—on the back cover only—for a highly respectable advertiser, General Mills. The rather stiff new slogan on the front cover was "a 52-page comic magazine" (or, in the case of adventure titles like Tarzan and Roy Rogers, "a 52-page magazine").
It is open to question how much weight child readers attached to the page count and the presence or absence of ads. The surviving figures for titles like Walt Disney's Comics and Little Lulu reveal no obvious pattern of sales that rose or fell with the number of pages. But it was undoubtedly important to Western's executives and editors that their comic books be separated from the competition in as many ways as possible.
By their glossier appearance, for one thing. On the covers of the comic books with screen cowboys like Roy Rogers and Gene Autry were photos of the stars—not publicity stills but Kodachrome or Ektachrome transparencies from shoots staged by Kellogg Adams and Polly Harrison of Western's staff, sometimes with elaborate props. And not always in a studio; one photo surviving from the 1950s shows a shoot under way in the desert, with the canine star Rin Tin Tin and Lee Aker, the boy who played the dog's companion in the TV series. The Autry covers were photographed at the Newhall, California, ranch where Autry shot his movies and TV shows.
For other comics, especially westerns, the covers were impressively expert paintings, sometimes in oils or acrylics, usually in gouache. The shift from drawings to paintings for such Dell adventure titles got under way in the late 1940s, when Moe Gollub—who had already been producing covers in brush and ink for The Lone Ranger and Tarzan, and the occasional painted cover for Santa Claus Funnies—began painting covers for the new series of Four Color one-shots based on Zane Grey's novels. Gollub was Western's first specialist in such covers. He devoted an increasing amount of time to them, and less to illustrating comic-book stories, although sometimes he did both, as with Zane Grey's Drift Fence (Four Color no. 270, 1950), whose cover and forty-eight interior pages are all his work.
When Dell began publishing comic books with painted covers in the late 1940s, Moe Gollub made many of the paintings, notably for the Four Color series based on Zane Grey's western novels. His wraparound painting in gouache for Zane Grey's Sunset Pass, Four Color no. 230 (1949), is poised and quiet, as the covers of other publishers' western comic books never were. Gollub's painting invites comparisons not with other comic books but with the work of illustrators from earlier in the twentieth century, like J. C. Leyendecker.
Gollub was joined in the 1950s by a half dozen artists of comparable skills. All these cover artists tended to specialize. For example, Gollub painted Tarzan covers (except when the cover was filled by a photo of a movie Tarzan like Lex Barker), and Sam Savitt painted covers dominated by horses, like those for Gene Autry's Champion and The Lone Ranger's Famous Horse Hi-Yo Silver. The paintings, the products of consultations between the illustrators and Western's New York art editor, Ed Marine, almost never had any connection with the stories inside the comic book. Instead they usually seemed to be illustrations for intense dramas whose origins were unclear and whose outcomes were perilously in doubt. One Dell hero, King of the Royal Mounted, was depicted on a 1953 front cover trying to repel an attack by a ravening wolf while armed only with a knife, and with more wolves on the way. On the next issue's cover, King was bound and underwater, clearly in imminent danger of drowning. Other hazards awaited him on future covers. The movie cowboys on their carefully posed photo covers usually looked much more relaxed.
It was not just adventure titles that enjoyed the prestige of painted covers. In 1953, the multiplying twenty-five-cent giant comic books with cartoon characters began to be set apart from their ten-cent brethren through covers showing versions of Donald Duck, Bugs Bunny, and their like painted in gouache. Paintings were rare on the covers of competing comic books, one notable exception being Ziff-Davis's G.I. Joe, a war comic and thus inconceivable as a Dell title in the 1950s.
In smaller ways, too, the Dell comic books simply looked different from their competitors. Other comic books' covers tended to be heavy with type, hooks for the stories inside, but the Dell comics rarely did more than identify a featured story in a Four Color one-shot through part of a comic book's title, as with Porky Pig in Ever-Never Land or Donald Duck in "Lost in the Andes" or Gene Autry and the Wildcat. Even that very limited sort of promotion ended in the early 1950s. Likewise, it was common for other publishers' humor titles to use dialogue balloons on gag covers, but such balloons almost never appeared on a Dell cover—except in the case of a series based on the Francis the Talking Mule movies, where the balloon affirmed the "talking" part of the character's name. Dell's covers were remarkably chaste, the gags, such as they were, required to stand on their own. It was not until deep into the 1950s that the comics began promoting their featured stories with a line or two on the front cover. Over the years, the most restless element on the covers was the Dell symbol itself, whose design changed repeatedly and frequently.
Typically Western's licenses for animated characters, TV shows, and cowboy stars extended well beyond comic books, to printed merchandise of many other kinds—games, puzzles, books, stamps. Not much survives to show how closely licensors worked with Western in the early 1950s, but from a little later there is the agenda for an April 1, 1958, meeting between "16 associates of Western Printing Co." and three executives of Warner Bros., one each representing the studio's cartoons, theatrical motion pictures, and television series. That morning, each executive outlined production plans for the forthcoming season's releases—twenty cartoons, four western TV series, and about three dozen feature films, all of them potential grist for Western's mill. After lunch, everyone watched a few cartoons.
## 24
# Uncle Scrooge: Play Money
Of the many cartoon characters given their own Dell comic books in the early 1950s, Uncle Scrooge McDuck was the most unusual. He was not the star of a backup feature; he appeared in support of Donald Duck, in stories bearing that character's name, and for all practical purposes only in stories by Carl Barks. Other writers and cartoonists had begun incorporating Scrooge in their stories as early as the December 1950 issue of Walt Disney's Comics & Stories, when he appeared in a "Grandma Duck" installment, but it was only in Barks's stories that he was a character of any substance. Measures of his popularity, as with sales of the comic books in which he appeared, could only have been vague and highly subjective, since he had been seen, barely, on just two front covers, both in 1951. Barks's editors must have hoped that readers would share their own high opinion of his work.
When Barks started work on the new Uncle Scrooge comic book, he immediately had a problem. Scrooge had been a perfect foil for Donald Duck and his nephews. The question now was how to elevate him to a leading role without fatally diluting what had made him attractive as a supporting player. "When he became the hero with his own book," Barks wrote in 1969, "I had to be careful how bad I made him." In "Only a Poor Old Man," the thirty-two-page story that filled the first issue of Uncle Scrooge, Barks softened Scrooge's character a bit, taming his greed. He gave the miser's attachment to his fortune a sentimental basis by emphasizing how he had acquired it. In doing so, he made Scrooge a little more complex than before, without sacrificing a great deal of what had made him funny and appealing. There are dozens of stories in which Barks demonstrated his mastery of his medium, but "Only a Poor Old Man" may be the one where the nature of that mastery—how he reconciled exercising a mature artistry with meeting the demands of a purely commercial publication for children—is most clearly visible.
In earlier stories, Barks had casually identified Scrooge as a "financier" or the owner of a "mortgage business," but now the ornery old duck spoke of prospecting for copper in Montana in 1882—a full seventy years before the publication of "Only a Poor Old Man," making Scrooge perhaps ninety years old—and of taking gold out of the Klondike in 1898. Barks was reaching back to the decades just before his own birth, when his father led a hard but less lucrative life of the sort Scrooge was describing. Scrooge now scorned the idea that he might have made his money through banking or some other sedentary pursuit. "I made it on the seas, and in the mines, and in the cattle wars of the old frontier," he told the nephews. "I made it by being tougher than the toughies, and smarter than the smarties. And I made it square. . . . You'd love your money, too, boys, if you got it the way I did—by thinking a little harder than the other guy—by jumping a little quicker—"
Needless to say, a real billionaire who had made himself rich from Montana copper would not have plunged into the goldfields, alone, sixteen years later. There may be echoes of Andrew Carnegie in Scrooge's Scottish surname, but the real Carnegie, unlike the fictional duck, hired other people to do such work for him as soon as he could. Scrooge was not a true capitalist, but that was beside the point. What Barks was doing, with his young audience in mind and with remarkable thoroughness and ingenuity, was making the reasons for Scrooge's attachment to his wealth as concrete as possible. If Scrooge carried gold out of the Klondike himself, then of course he would care about it.
Barks made Scrooge's fortune concrete, too. For children, money is bills and coins—rare is the child who grasps that money can be something else—and in "Letter to Santa" in 1949, Scrooge's desk was already awash in the stuff. In the story in the December 1951 Walt Disney's Comics, Barks had gone further, so that Scrooge's wealth was not only measured in cubic acres but actually filled his huge money bin. Scrooge had the kind of fantastic wealth a child could understand, and he hoarded that money in what amounted to a giant piggy bank.
A child with a lot of cash might want to spend it on toys, but for Scrooge his money itself was an enormous toy. As early as the "Donald Duck" story in the March 1951 Walt Disney's Comics, published one year before the first issue of Uncle Scrooge, a sense of play had joined the acquisitive urge as one of Scrooge's dominant characteristics. Staring over his three cubic acres, he muses: "Now, me, I know that money isn't worth anything! It's just a lot of paper and metal!" Then he plunges into the cash: "But I love the stuff! I love to dive around in it like a porpoise! And burrow through it like a gopher! And toss it up and let it hit me on the head!"
"Only a Poor Old Man," in the first issue of Uncle Scrooge, Four Color no. 386 (1952), marked a turn by Carl Barks toward a more sympathetic and likable Scrooge than the crusty billionaire of his first few appearances. © 1952 Disney.
At the end of "Only a Poor Old Man," after the ducks have fended off a determined criminal assault, Donald tells Scrooge: "You may not know it, Uncle Scrooge, but your billions are a pain in the neck. You're only a poor old man!" Scrooge stares bleakly after Donald, but then declares defiantly: "Bah! Kid talk! No man is poor who can do what he likes to do once in a while!" Barks had created a bridge between Scrooge's actions and a child's understanding, without fatally compromising the adult nature of Scrooge's greed and entrepreneurial drive.
Scrooge's adversaries in "Only a Poor Old Man" are again the Beagle Boys, this time not silent bit players, as in their first two appearances in Walt Disney's Comics, but instead the anti-Scrooge: bureaucratic criminals who represent all the large organizations—thus their "Inc."—dedicated to imposing uniformity on prickly individualists. The Beagles care about Scrooge's money only as money, not as the source of the pleasure he takes in it, and the money itself exacts a sort of revenge. At the climactic moment, when the Beagles seem to have won, they cannot resist following Scrooge's example and diving as a group into his cash. Unlike Scrooge, they smash themselves unconscious against all that metal.
The number of Beagle Boys would fluctuate from story to story, according to each story's requirements. In "Only a Poor Old Man," there are seven of them; in the fourth issue of Uncle Scrooge, Scrooge says there are thirty. The Beagles are identical in appearance—only their prison numbers vary—and they are sometimes identified as brothers, members of what must be a very large and genetically very unusual family. Rather than remove their domino masks, when the need arises they disguise themselves with sunglasses.
As is almost always true of Barks's stories of the early 1950s, in "Only a Poor Old Man" everything follows without hesitation or strain from the core comic premises—Scrooge's monomaniacal love for his preposterously tangible fortune and the Beagle Boys' matching obsession. That the Beagles have a laboratory that develops "super termites" seems perfectly reasonable, given what Barks has shown of them by the time the termites appear. After the termites do their work, there is a wonderful mock-solemn moment, set in late-afternoon shadows, as Scrooge is about to suffer what looks like defeat. The Beagles will be gleeful in a moment, but Barks has them pause respectfully, like warriors who have finally conquered a tenacious adversary.
As successful as this story was on its own terms, it held the seeds of future trouble. Scrooge was not nearly as mutable a character as Donald. He could not change as much from story to story. Every story in which he was the principal character would have to be not just about him but also about his money. In "Back to the Klondike," in the second Uncle Scrooge one-shot, Four Color no. 456, Scrooge is, however, an even more sympathetic character than in "Only a Poor Old Man." That one-shot was published early in 1953, a year after the first one. Here an editorial hand, as well as Barks's, was shaping the character. "Back to the Klondike" as published was censored, in the worst such mutilation since the alterations in "The Golden Christmas Tree."
Barks submitted "Back to the Klondike" to Western in September 1952. Atypically, he had written and drawn that story on the road, during a late-summer driving trip to the Pacific Northwest, "away from the heat. . . . I was working on the story for at least a month, but not steadily. It was in Seattle that I got the idea of going down to one of the bookstores and finding some books on the Klondike. I found an old book that was practically an eye-witness account." That book, Klondike '98, by Ethel Anderson Becker, was actually published in 1949, just three years before Barks found it, but it is made up of photos from the Gold Rush days of a half century earlier, accompanied by Becker's commentary. The book inspired five pages by Barks that showed a young Scrooge accumulating his fortune in the Klondike at the turn of the twentieth century. Scrooge in those deleted pages is mean and tough, the decisive victor over a mob of miners in a barroom brawl. He forces Glittering Goldie, a dance-hall girl who has drugged and robbed him, to work on his claim for a month.
The half-page panel showing the brawl was one of Barks's very rare miscalculations at this stage of his career. There is a great deal going on in that panel, many simultaneous actions ignited by Scrooge, but no suggestion that the panel is a composite stretching over a brief period of time (as might have been the case if Scrooge himself were depicted more than once, battling the mob at different spots in the bar). The brawl might be more convincing, and seem less contrived, if Barks had presented it in a few more panels, as he did the battle of the steam shovels in "Letter to Santa." Hyperbole was an important element in Barks's stories, but usually with a more solid grounding than in this case.
Barks wrote in 1972 that the deleted pages were "cut out of the story at the editorial office in Beverly Hills in 1952. I was a little skeptical of whether I could get by with such a bar room atmosphere but I did it anyway for fun. . . . The sequence was cut because of violence and dance hall atmosphere." And, he added in 1974, because of the month Scrooge spent alone with Goldie on his claim: "That was kidnapping, he picked her up and carried her out to his claim and made her go to work. . . . It didn't look very much like kidnapping, yet it was."
In both Barks's original version (which was restored and published in 1981) and the published story, the latter-day Scrooge is, beneath the crust, a tenderhearted soul who still cares for Goldie, now a lonely old woman scratching out a living on Scrooge's claim, but in the published story this sympathetic Scrooge more than occupies center stage.
When Western made major cuts in "Back to the Klondike," it was the third time in 1952 that Barks had been at cross-purposes with his editors. In January he had submitted a ten-page "Donald Duck" story that he notated as the "golden apples" story. It was originally scheduled for the September 1952 Walt Disney's Comics, but it was shelved and the next story in line substituted. Although the artwork vanished long ago, Barks remembered the story as a modern retelling of the myth of Atalanta and the golden apples. He said in 1974:
I only recall that I had Daisy quite angry with Donald because he was trying to win the hand, I guess, of this queen of the apple festival. . . . Every time Donald would be about to catch up with this fleeing queen, or princess, or goddess, whatever she was, she would drop her golden apple and he would pick it up and then he had to chase her again. . . . Daisy was so jealous that she was throwing things at Donald, and she was not acting in a ladylike manner. That was the objection. . . . That was the only excuse they ever gave me.
At the end of March, Barks submitted the finished pages for "Trick or Treat" in Donald Duck no. 26, the first issue numbered as part of a standard bimonthly series; that comic book was dated November–December 1952 and scheduled for publication in September. "Trick or Treat" was based on a Donald Duck animated cartoon directed by Jack Hannah, Barks's former partner in the Disney story department. The cartoon was scheduled for release in October, a few weeks after publication of the comic book. It was by then standard procedure for Dell to anticipate the release of a Disney feature film by publishing a comic-book version a few weeks in advance, but such a tie-in with a short cartoon was unprecedented.
Barks wrote in 1971:
Alice Cobb . . . gave me photostats of the story board sketches and asked me to adapt them to a comic book version. I believe it was sort of open end as to length. Anyway, in the part where Hazel [a witch] is trying to bust into Don's closet [where he has locked away the Halloween candy], I departed from the movie script and added some business with an ogre that Hazel summons. [Barks's departures from the film story were actually more extensive than that, but the introduction of the ogre was the major change.] Alice Cobb deleted the extra business and didn't pay me for the unwanted pages. She was that mad.
What Barks had completed as a thirty-two-page story was twenty-three pages as published, the final version adhering closely to the story line of the animated cartoon. The comic book's remaining nine pages were filled with a Halloween story starring the nephews and Gyro Gearloose that Barks submitted to Western in May 1952. Likewise, he came up with a five-page filler story for the second Uncle Scrooge one-shot after Western cut that many pages from "Back to the Klondike." Barks had no record that he was paid for that five-page story, and it seems likely that he was not, as with the deleted pages from "Trick or Treat."
In an interview with Bill Spicer, the editor Del Connell described Barks's status as it appeared to someone who began working for Western almost a decade after Barks. Connell started by writing comic-book stories in 1950. "When Carl stopped doing the Donald Duck book to concentrate on Scrooge," he said in 1983, "I wrote the duck stories and much of the other Disney material for the regular line of Dell comics and Dell annuals." Connell joined Western's staff as an editor in 1956.
"The fact that Carl turned in finished stories and art, without first having his scripts okayed, was established before my time with Western. I presume it came about because he was so good at his craft and because he knew more about the ducks than anyone in editorial. Carl did send in finished art. Other artists had to have pencils okayed before inking. Carl's material was read and then shipped to Poughkeepsie for coloring and printing. I wasn't involved with checking his stories and art, since I was busy with other books."
Other Western artists were also aware that Barks was treated differently. "Sparky" Moore, as an artist for cowboy titles in the 1950s, spoke of being in the office when Barks—whom he remembered as "kind of a formally dressed and elderly fellow" (Barks was in his fifties)—delivered his work: "[Western] had a little office with a display board that you could lay a full page of comics on, and that's where you usually went when you'd bring your work in. Carl didn't do that. He just went to the desk where the receptionist was and laid down his work. They didn't examine his work like they did the rest of us. He thanked everybody and left."
By the fall of 1952, it was clear that Barks would pay a price for the freedom he enjoyed in submitting finished art without first having his scripts and then his pencil drawings approved by his editors, as was required of Western's other writers and artists. In less than six months he had collided twice with his editors over the content of his longer stories, and he had not been paid for fourteen pages of finished art—at least two weeks' work. He was newly divorced and burdened with alimony payments to Clara. He had no choice but to be more cautious.
For the rest of 1952 Barks produced nothing but ten-page stories. Then in February 1953, he submitted all the interior pages for the third Uncle Scrooge one-shot, Four Color no. 495. It had been six months since he submitted the drawings for "Back to the Klondike," and that mutilated story was published just as he finished work on the new issue. This time Barks submitted two stories—a lead story of twenty-two pages and a backup story of ten pages. Four Color no. 495 was published in the summer of 1953.
The villain in the untitled lead story is named Chisel McSue, and he is trying to use the law—which had bruised Barks, as he saw it, in his divorce from Clara—to wrest Scrooge's entire fortune away from him. In sharp contrast to Glittering Goldie, McSue is a vicious and even murderous adversary. But perhaps because Barks saw something of himself in this mistreated Scrooge—or, at least as likely, because he remembered what had happened to "Back to the Klondike"—he made Scrooge wholly admirable. Scrooge even overcomes (with considerable effort) the temptation to free himself from McSue by letting his odious enemy drown. As Barks told Edward Summer, "It made Scrooge much more likable. If I had had the nephews tie Scrooge down so that they could rescue the crook, you wouldn't have had much sympathy for Scrooge, but the fact that he helped to rescue the darn guy and then got kicked in the face again made him sweet." Sweet is a word that does not come readily to mind to describe the earlier Scrooge.
Ultimately, in that pivotal story, it is thanks not to Scrooge himself but to Huey, Dewey, and Louie that McSue is defeated. Scrooge, facing ruin, has not summoned help from a vast corporate apparatus but has chosen to rely instead only on Donald and his nephews. The McSue story established a workable pattern for Barks's stories in Uncle Scrooge, which Dell began publishing on a quarterly schedule with the fourth issue, late in 1953. In those stories, Scrooge is much less a greedy and aggressive entrepreneur than a fundamentally benign skinflint who battles not to enlarge his fortune but to protect it. His money is a prop that permits Barks to send the ducks to exotic locales (the Caribbean, in the McSue story) or drop them into unlikely situations.
Although Scrooge is supposedly old in the Uncle Scrooge stories, he moves without a hint of physical debility (and there is no reason to believe, as in earlier stories, that his passion for money is so intense that it outweighs his age). Despite the resources presumably at his command, there is almost never a suggestion that Scrooge's wealth confers on him any power. For help, he calls upon only Donald and the nephews, and it is almost always Huey, Dewey, and Louie who step forward at a crucial moment to save the day.
In the third Uncle Scrooge one-shot by Carl Barks, Four Color no. 495 (1953), Scrooge must fight the impulse to protect his fortune by letting his mortal enemy Chisel McSue drown. © 1953 Disney.
This was a template for children's adventure stories that recalled not just comic strips like Milton Caniff's Terry and the Pirates and Harold Gray's Little Orphan Annie, but also nineteenth-century novels like those of G. A. Henty, who inserted child protagonists in the midst of great events. Like the young heroes in those books and comic strips, the nephews were the child reader's surrogates, the central nature of their roles camouflaged by Scrooge and his dazzling wealth. Having Scrooge rely only on Donald and the nephews meant that the surrogates could be put into action that much more quickly. There is, besides, scarcely a hint that the nephews attend school, or, for that matter, that Donald has any kind of steady job. Scrooge's fortune was useful as a pretext for adventures, but the price was an erosion of the plausibility Barks had once worked so hard to maintain.
In the first few quarterly issues Barks worked ingenious variations on his new template, notably in no. 6, June–August 1954. There, for once, Scrooge's fortune is a believable burden, a burden so demanding—he is besieged by anxious employees, corrupt foreigners, obscure charities, and even a revolutionary who demands that he contribute a billion dollars to his own destruction—that he seeks refuge by parachuting into Tralla La, an idyllic variant on Shangri-La in an impossibly remote Himalayan valley. The residents have no knowledge of money, but Scrooge inadvertently introduces them to the concept when they begin lusting after the bottle caps on his nerve medicine. Then the Tralla Lallians turn vengeful, and it is—of course—Huey, Dewey, and Louie who persuade them to let the ducks go free.
Despite the flexibility it permitted, the new template was confining compared with the freedom Barks had enjoyed a few years earlier, when there was no predicting the tone and shape of any of his stories. And even the template could not provide a perfect defense in the panicky climate that prevailed at all comic-book publishers, Western included, in the mid-1950s.
## 25
# Carl Barks in Purgatory
For comic-book publishers, the mid-1950s were dark and frustrating years, marked by increasingly widespread condemnation not just of the titles specializing in horror and crime but of all comic books. Sales fell as the criticism increased. Dell weathered the storm better than most, since almost all of its comic books enjoyed the protection that came with characters that had already won wide acceptance in other media: not just animated cartoons, as with the Disney characters, but also radio, television, newspaper comic strips, live-action movies, books, and even, in the case of Bozo the Clown, phonograph records. But there was no escaping such broad-brush attacks as Fredric Wertham's book Seduction of the Innocent, a sensational and highly effective condemnation of comic books that attracted a great deal of favorable attention when it was published in April 1954.
Wertham's publisher, Rinehart & Company, placed an aggressively worded advertisement for Seduction—it promised "[t]he startling truth about the 90,000,000 comic books American children read every month"—on the cover and the first two pages of the March 6, 1954, issue of Publishers' Weekly. Western's Lloyd E. Smith surmised, correctly, that Wertham would make no exception for comic books like Dell's. Whitman Publishing sent a telegram protesting the ad to Publishers' Weekly on March 9, and Smith followed up with a letter that the magazine published in its March 20 issue. "To say that we regard the publication of this book as well as the publication of the advertisement for it as irresponsible is to put it mildly," he wrote. The Dell line, he insisted, "is well known for its wholesomeness."
George Delacorte and a half dozen of Western's licensors sent strongly worded complaints to Publishers' Weekly and Seduction's publisher, Rinehart & Company. Delacorte got right to the point: attacks like Wertham's could cost Dell a lot of money. "This is a tremendous publishing enterprise and one which I must protect to the best of my ability," he wrote to Rinehart, concluding: "You can realize that we can't sit by idly and allow you, or anybody else, to give the general public the impression that all comic books come in the category discussed in Seduction of the Innocent."
For all their bluster, there was never any chance that Wertham's critics would derail publication of his book. Rinehart's lawyers had reviewed the manuscript carefully the previous fall, suggesting changes that would make clear that Wertham's more extreme statements were expressions of opinion and so legally protected. Seduction was not vulnerable to a libel suit.
When the book was published on April 19, 1954, it enjoyed what Publishers' Weekly called "an appearance of complete cooperation from the United States Senate," since a Senate subcommittee opened hearings two days later in New York City on the supposed connection between juvenile delinquency and comic books. The hearings attracted a great deal of press attention, most notably to the testimony of William Gaines, EC's publisher, whose defense of his company's horror comic books was widely ridiculed.
Gaines was lured into defending the "good taste" of a cover drawing that depicted a severed head. The drawing would have been in bad taste, he said, if the murderer had been shown "holding the head a little higher so the neck would show with the blood dripping from it," but it was instead in good taste—for a horror comic book. "Good taste," always a deadening standard when applied to any kind of art, was especially such where comic books were concerned, since they were inherently offensive to so many educated adults.
Dell and Western took Wertham's bait by defending their comic books as uniformly "wholesome" or "harmless." That meant that a mere handful of questionable episodes in a few comic books could be cited as evidence that the whole publishing enterprise (to use George Delacorte's phrase) was rotten. Wertham responded in Publishers' Weekly to Lloyd Smith's letter without mentioning him or Western or Dell by name, but to devastating effect:
Your correspondent [Smith] singles out one comic book publisher [Dell] for the "wholesomeness" of his product. In recent specimens of this firm's comic books a child will find such episodes as these: mugging; a youth held with arms twisted behind his back being hit in the face with a rifle-butt; a child hitting a man over the head from behind ("CRUNCH!") with such force that he knocks out the man and breaks the bottle. What should be considered "appalling," "irresponsible," "most extreme," and "to be deplored" [all quotations from Smith] is that this kind of thing is found "wholesome" by your correspondent.
That such violent episodes might have some artistic rationale, might be justified by some dramatic purpose, was simply inconceivable to the people on either side of the debate, which left the advantage with Wertham and other critics of comic books.
By 1954 one major comic-book publisher, Fawcett, home of the very popular Captain Marvel titles, had already left the business, and over the next year or two it was joined by many smaller publishers. Most of the surviving publishers banded together in September 1954 to form the Comics Magazine Association of America (CMAA) and within it a Comics Code Authority (CCA) headed by a New York City magistrate named Charles Murphy. The CCA was entrusted with reviewing all of its members' comic books so that objectionable material, as it was strictly defined in the new code, could be excised before publication. Insipidity was no longer simply the desired outcome, as had often been the case, but was instead to be relentlessly enforced.
Dell was one of three publishers that refused to adhere to the code, the others being EC and Gilberton, publisher of what had been called Classic Comics but were now the more respectable-sounding Classics Illustrated. Dell's complaint was that the code was not strict enough. On October 1, 1954, Helen Meyer, Dell's vice president, wrote to Milton Caniff, whose Steve Canyon comic strip was licensed to Western for Dell comic books, about Dell's refusal to join the CMAA: "Of course the main point . . . is that Dell does not publish any crime, horror or love and romance comics, and yet we have 40% of the comic book business. Publishing a crime or horror comic doesn't necessarily spell success. In fact our average comic book sale is over 750,000 copies, and I am certain that this average is greater by more than a hundredfold what any crime, horror or love and romance comic has ever been able to attain."
The slogan "Dell Comics Are Good Comics" began appearing at the bottom of the first page of each Dell comic book in the late summer of 1954. Early in 1955, Dell and Western struck back more directly at the Comics Code with "A Pledge to Parents," published in each Dell comic book. It said in part: "The Dell code eliminates entirely, rather than regulates, objectionable material." In the summer of 1955, Dell went further, not just publishing the "Pledge to Parents" on the inside front covers of one month's comic books (as it had already done earlier in the year) but also insisting through a cover blurb headed "Important!" that readers pay attention to it.
Western's president, W. R. Wadewitz, the founder's brother, had warned in March 1955 of declining sales, in Western's annual report to stockholders:
One of our vexing problems in 1954, and for that matter up to this writing, has been the blanket criticism directed towards comic magazines. . . . [I]n the face of publicity linking comic magazines to delinquency, it is a job of no mean proportion to get across to the public that our magazines have always maintained a high editorial standard and are not injurious to anyone. Until our story is put across, we can expect less than the natural market for our comics, even though in the long run we are sure to benefit from a resurgent demand for clean comics.
By 1956, though, comic books accounted for only 19 percent of Western's revenues, down from 30 percent in 1952.
The shift of emphasis in Western's comic-book operations from New York to Los Angeles, under Robert Callender, had already resulted by the early 1950s in a spreading blandness and uniformity, and the comic-book scare made things worse. The atmosphere was one in which even the most creative comics artists found it hard to avoid becoming a little more self-conscious and uncomfortable. Dell's titles were increasingly as uniform, stylistically, as those of its principal competitors, so that most of its animated-character stories in particular shed the traces of individuality and even eccentricity that were common in the comic books of the 1940s. The typical story in a Dell talking-animal comic book of the mid- to late 1950s was drawn simply and sometimes awkwardly, the writing a string of coincidences and forced gags. It was as if a Barks story like the one in the October 1951 Walt Disney's Comics, in which the nephews are repeatedly thwarted in their efforts to play hooky, were stripped of everything except the coincidences that make up its narrative skeleton—that is, stripped of everything that makes the story worth reading. Greater editorial control was resulting in a crudely efficient kind of storytelling.
Carl Barks told Malcolm Willits that when comic books were the subject of national controversy, "[t]hey did tell me from the office to be awfully careful. Don't put in anything that suggests any kind of horror. Don't use the word 'horror.' . . . Oh, I was kind of pinned down there for a while on that." Barks remembered in 1966 that "the rules got real strict in the 1950's. The office never told me what I couldn't draw until I'd made some bad bungle, then I'd get back a page or two of stuff to correct. Luckily, my natural bent in writing was as clean as new-fallen snow." But what constituted a "bad bungle"? And what were "the rules"?
Del Connell remembered a "dos and don'ts list for writers in the early 1950s, but I have been unable to find a copy. I know I haven't sent such a list to writers in over twenty-five years"—that is, since around the time Connell became a Western editor in 1956. One such list is part of a document titled "Hints on Writing for Dell Comics" that is mostly concerned with story structure: "Make sure your story is not just a series of incidents. Avoid counter and sub-plots." It also includes a section titled "Taboos" that advises writers to "avoid showing or mentioning the following items: anything dealing with minority races, politics, religion, labor, suicides, death, afflictions (such as blindness), torture, kidnapping, blackmail, snakes, sex, love, female villains, crooked lawmen or heavies of any race other than the white race. At all times keep the stories absolutely clean and in good taste."
The novelist Charles Beaumont worked briefly as an assistant editor in the Beverly Hills office where Western, under its Whitman Publishing name, was producing most of the Dell comic books. When he wrote about the taboo list in 1955, it sounded even more comprehensive: "A Whitman story must not mention disease, evil, foul odors, blood, death, spiders, snakes, religion, politics, minority groups, atom bombs or sex. In no story may knives appear as weapons. Characters may carry guns, but must not shoot them (except in the Western department, where arm and leg wounds are permissible). Justice positively must triumph and the villain must be punished. . . . The list is rewritten often, as new taboos are thought up." Beaumont quoted Kellogg Adams, who edited cowboy titles from the Beverly Hills offices for a few years in the middle 1950s, as saying that hanging scenes had recently been outlawed because a Canadian boy had tried to kill himself that way; "claimed he picked up the idea from Red Ryder."
What is most deadly for any artist is not being required to observe rules like those on a "dos and don'ts list," but not knowing what the rules are because they are constantly changing, as was the case at Western in the mid-1950s. Barks's publisher was trying to adapt to a hostile environment by anticipating every possible complaint, a hopeless task; thus the constant changes in the taboo list, which from all appearances served more as protection for the editors than as guidance for writers and artists. That was certainly true with Barks, who said in 1983: "Western didn't give me much direction. It was years after I had made a few mistakes that I found out that they had a list of taboos. Alice Cobb got me the sheet one time and showed it to me. You couldn't use the word 'kill' or use a gun in a dangerous way; you couldn't have poison or sickness or crippled people."
Western was an honorable company whose dealings with its writers and artists were exceptionally straightforward, compared with many other comic-book publishers. As Barks said, its checks never bounced, and Lloyd E. Smith, who was in effect editor-in-chief of Western's publications, believed in paying artists and writers quickly. Royalties to licensors were paid monthly. But there was nothing about the company to suggest that it would ever recognize or encourage artistic accomplishment if that accomplishment entailed the slightest risk of controversy or lost sales. Western's editors were classic bureaucrats, and taking risks of any kind was simply alien to them, as it was to the company as a whole.
Although Barks was well regarded in the 1950s, Del Connell said, "there were many good artists and writers working on comics for Western Publishing. They were busy producing, and the editors were trying to get material together to meet deadlines on the numerous monthly books. It left little time to admire one person's efforts for too long. We all knew Carl was exceptionally good, and we left it at that."
But not really. By the late spring of 1955, when Barks submitted to Western a story for Uncle Scrooge no. 12, to be published that fall, the atmosphere had become so sickly that, as Barks wrote, "I almost had to eat those 32 pages of drawings because I'd used some harpies as menaces." The story's harpies are those of Greek mythology, but Western required him to reletter each mention so that the reference is to "larkies" instead. The reason being, Barks said, that harpies "are street walkers in some obscure synonym in somebody's slang dictionary." Perhaps there is such a definition somewhere, but if so, it is more than obscure; there is no definition of the sort in standard dictionaries. There was no way Barks could protect himself against such irrationality without surrendering whatever artistic freedom he had left.
The ducks, wondrously elastic as personalities in the stories of the late 1940s and early 1950s, were by 1955 in Uncle Scrooge much more the same characters from one story to the next. They were also blander and more generalized as both written and drawn. Necessarily so: the Uncle Scrooge stories were almost always well constructed, as Barks's stories had been since the mid-1940s; but the template, because it was a template, could not accommodate the character variations of the earlier stories. Moreover, because it was a template for children's stories, it could rarely accommodate believable adult motivations. Barks even transferred the nephews' resourcefulness from the characters themselves to a book, the Junior Woodchucks' Guidebook. Initially a parody of the Boy Scouts' handbook—it first appeared in the Tralla La story—the Guidebook very soon became something like a crutch. Faced with a dilemma, the nephews were more likely to rely on the Guidebook, seemingly a compendium of all human knowledge, than on their wits.
As for Scrooge himself, there is a measure of how radically he changed in the contrast between the 1951 Donald Duck story "A Christmas for Shacktown" and "Land Beneath the Ground," published in Uncle Scrooge no. 13, March–May 1956. In both stories, Scrooge's hoard vanishes into a hole in the ground, abruptly and with seeming finality. In the 1951 story, Scrooge's dismay first unmans him—Barks depicts his physical collapse in vivid detail, from rubbery muscles to stunned eyes—but then yields in an instant to mad fury as he attacks the handiest target, which happens to be Donald. In the 1956 story, Scrooge simply gives up and limps away. "I've had it!" he says. Granted, he makes that declaration after pages of struggle to preserve his fortune, but any such statement would have been unimaginable a few years earlier. Scrooge would sometimes show a little fire in the stories that followed, but he was overall a far more passive character.
Scrooge's fortune is restored by the end of the 1956 story, but the 1951 story ends with him retrieving his money a bit at a time with the help of a toy train—a process so slow, he groans, that "I'll be here for two hundred and seventy-two years, eleven months, three weeks, and four days!" He was, of course, as rich as ever in the next story. In the early 1950s, Scrooge's occasional loss of wealth was no more permanent than the occasional windfall that Donald and the nephews enjoyed. Stories began (or ended) with whatever was best for the tale Barks wanted to tell. In "The Magic Hourglass," Scrooge owes his fortune to the hourglass, a lucky charm, but there is no hint of such supernatural intervention in other stories from around that time. By the mid-1950s, even though direct references in one story to another story were rare, the sense was building that the stories were bound together in a sort of web that made all but impossible the sharp variations of a few years earlier.
So rigorous was Barks's craftsmanship, not just in Uncle Scrooge but in the ten-page "Donald Duck" stories in Walt Disney's Comics, that this deterioration was at first barely noticeable. In the ten-page stories there was the same concern for plausibility, the same painstaking construction that recalled nothing so much as perfectly assembled two-reel silent comedies. But the stories were not building as the best ones did; now they more often just stopped, with an air of how-do-I-end-this-thing frustration. Sometimes, as in the story in Walt Disney's Comics no. 165, June 1954, plausibility itself came under strain, even though in that story Donald's obsession with becoming a television performer is ultimately worked out in convincing detail. More often, what was leaking away was not fundamental plausibility but, as in Uncle Scrooge, the characters. Increasingly, the ducks were too simple to hold interest. Now the plot and the gags had to carry almost all the weight, and sometimes, when Barks let down his guard and settled for the merely silly, the results were embarrassing.
As Barks's life as an artist became more constrained, his personal life finally offered some compensations. On July 26, 1954, he married for a third time, to Margaret Wynnfred Williams, known as Garé, herself an artist who by then had been assisting Barks for almost two years. "Garé started doing lettering and a little of my inking in about October of 1952," he said in 1973. Her first work on her future husband's stories was in the "Donald Duck" story for the June 1953 Walt Disney's Comics; she inked the scales on the fish in that story. "She kept doing more and more of it, and I was paying her more and more, and finally we decided, hell, what's the use of going on like this, we might as well get married, and pool the money all in one bundle."
During Barks's descent from the peaks, there were still wonderful departures from an increasingly pedestrian norm. In Walt Disney's Comics no. 178, July 1955, Donald moves to a quiet neighborhood to escape the noise that will not let him sleep—and then seeks out noise that he can retaliate against. The story is a marvel, combining as it does perfect construction—no Laurel and Hardy short was ever better in that regard—and much more psychological comedy than was typical of the midfifties stories. "At the time Carl was writing that story," Garé Barks told Klaus Strzyz, "we lived in an apartment house, and had awfully loud neighbors; it was only quiet upstairs." The story in Walt Disney's Comics no. 180, September 1955, in which Donald is an insurance salesman, has some of Barks's satirical fire, but also, even more important, much of the energy of his best stories; Donald's bombastic boss, Mr. Brasshorn, all but leaps off the page.
The passion visible in Brasshorn's exhortations and sobs and furious outbursts was mostly felt through its absence in the stories that came after. Even in the rare story with Donald and the nephews in conflict, the sharp edge of their earlier battles was missing—and so was its counterpart, the ducks' intense attachment to one another. Barks's stories lost their emotional immediacy in the middle 1950s, as comic books of all kinds retreated from anything with a strong flavor, anything that might be seen as threatening to slip out of control.
Gladstone Gander, while continuing to turn up in one or two stories a year, proved to be an especially problematic character because the air of contrivance hung so heavily around his stories. The problem was not that Gladstone's luck was preposterously good; that was the point of the character. To be funny, his luck had to seem preposterous, to the verge of the supernatural. The problem was instead that any victory by Donald over this rival was almost certain to seem forced; and yet such victories were now all but required in the increasingly censorious environment in which Barks was working. It was when Donald took Gladstone's good luck seriously and was undone by his mistake in doing so that the stories with Gladstone came to life; but such stories—giving unearned victory to a reprehensible character—were awkward when the emphasis was on providing what the Dell "Pledge to Parents" called "clean and wholesome juvenile entertainment."
As to how Barks felt about the changes that were being imposed on his stories, directly and indirectly, the best evidence is not in what he later told fans and interviewers—the first such letters and interviews are dated in the late 1950s and early 1960s, after the major changes had already taken place—but in the furious response he drafted to a letter from Alice Cobb. He kept Cobb's letter and his draft reply, uncharacteristically for the time; he said later that he had discarded 90 percent of his correspondence with Western, and almost nothing survives from the 1940s and 1950s. In June 1956, Cobb had sent Barks a letter that Dell received from a woman named Ruth Downing. Cobb wrote: "Enclosed is a letter from a mother, which we feel is a fair criticism. We should all watch for this sort of thing, we think—and avoid it as much as possible. In addition, we usually use 'Quiet!' instead of 'Shut up!' "
There is no knowing the exact nature of Downing's complaint, since Cobb asked Barks to return that letter, but to judge from Barks's handwritten draft for his reply to Cobb (which he believed he mailed in highly similar finished form), Downing objected to the rivalry between Donald and the nephews in the story in Walt Disney's Comics no. 186, March 1956. In that story, the ducks operate competing ice taxis. Barks wrote:
Very well, I will avoid any more stories using conflict or rivalry between Don & the kids. Also no more "shut ups." . . .
In the past few years I have returned from time to time to the early duck plots that I used in the years when Disney comics were moving up. Thinking that the increased sales of the magazine in those times proved that these story-types were sure fire and as I didn't want to stray too far from the beaten track, I switched the plots. The objectionable Ice Boat Taxi plot is a switch on a story in June 1944 Disney Comics called "Rival Boatmen." . . .
When I worked in the Disney story department in the Duck unit the basic theme in a great number of the Donald Duck shorts was rivalry between the kids & Donald. Certainly the main body of the public didn't object to this theme, for during those years Donald and his nephews overtook and passed Mickey Mouse and his nephews in box office popularity.
After starting to work for Whitman I kept to the proven story lines as much as possible. The 10 page Donalds through the first several years were liberally sprinkled with rivalry plots and with no crime stories among the offerings it's possible that the contests between Don & the kids satisfied the reader's desire for triumphs of good over bad. The boastful over the meek, etc. At any rate the magazine didn't seem to go broke.
Now comes a neurotic female with a cramped fault-finding mind and a cry-baby son, and proves that all of those millions (well, dozens at least) of boys and girls who have bought and read Disney Comics over the years were and are sadists, masochists, murderers, lechers and worse! I agree with her.
From now on you will see changed stories coming from this former breeding place of vice. You will see stories that will cause Ruth Downing to write another letter to say that she just loves the Donald Ducks. For every time she reads to her little nose-picking cry-baby, he goes to sleep in the middle of the second page. . . .
P.S. All of this letter except the first paragraph is so much jousting with windmills.
Cobb's persona may have tempted Barks into such a response: not only was she an up-from-the-ranks company loyalist, but she was, in Charles Beaumont's description, "a tall, angular woman with a voice startlingly like Minnie Mouse's." If she replied to Barks, her letter has not survived. But she may have answered him in another way.
Six months earlier, in December 1955, Barks had submitted a ten-page story for the January 1957 issue of Walt Disney's Comics & Stories. It was identified in his records as "bobsled race." It was never published; "Don't know what became of this story," Barks noted on a list of his work. As with some of his other censored stories, the artwork has long since vanished, almost certainly destroyed (like almost all of the artwork published in almost all of the Dell comic books over the years). The Cobb–Barks exchange took place around the time that Cobb would have been assembling the January 1957 issue, and it seems likely that she scrapped the bobsled story because it involved conflict between Donald and the nephews, and replaced it with a safer story that Barks had submitted later in December.
One more of Barks's ten-page stories was completely suppressed after he submitted it in September 1957. In that story, Donald is a milkman besieged by the villainous McSwine, a pig character who covets Donald's job and pulls one nasty trick after another trying to get it. It was shelved, Barks said, "because Donald was too mean to the villain." In this unusual instance, the artwork survived.
Barks was like many another conscientious middle-aged worker who feels the foundations of his working life shifting under him but has no choice but to try to stay on his feet. He was wedded to the ducks, and thus to the Disney comic books and to Western. But his disgust and impatience surfaced in various ways, and his resentment did not cool with the years. In December 1960, he wrote to one of his first fan correspondents: "About Donald being less vital than he was in the old days. I can only point to the fact that tabu after tabu has been imposed upon us scripter's [sic] freedom of material. My early stories were in many instances based on an intense and violent rivalry between Don and the kids. Can I do that now? Ha!"
However much he resented the damage his stories were suffering, Barks was above all a practical artist, one who in 1966 advised an aspiring cartoonist to "keep the number of characters as limited as possible. . . . All introductions of characters take up story panels. Soon you've used so many there is no room left for the action sequences. . . . One or two shots of a complex machine is about all an artist can do in any one story and make a living wage." He wanted to please his editors: "They knew when an artist was padding his stuff, and that's one thing they liked about my stuff, there was seldom any unnecessary padding. My stories were kind of stripped, they were just naked bones in a lot of cases." And, as much as possible, he wanted to please himself. Not infrequently, Barks revised stories that he had already drawn in ink; some of the excised panels have survived from mid-1950s stories. He said in 1974: "It was just second nature, reading through there, to spot those dull spots and change them. . . . I was always a very severe critic of whatever I had done."
There is a measure of his growing frustration in how he dealt with the occasionally troublesome language question: how do the ducks and these exotic foreigners communicate? In 1954, in the Tralla La story, Scrooge speaks the local language. As he explains, "It's the speech of ancient Cathay, which I learned when I was a yak buyer in Tibet"—a funny but entirely believable explanation, in context. But three years later, in "The Mines of King Solomon," in Uncle Scrooge no. 19, September–November 1957, Scrooge speaks Arabic that he learned "selling lawnmowers in the Sahara." In the next issue, in "The City of Golden Roofs," Scrooge speaks "ancient Bengali" that he learned selling "road maps to Marco Polo." Rather than devise an explanation that might get a laugh but also give Scrooge a richer past, Barks threw up his hands and went for the easy joke.
By this time the expansive environment of the late 1940s and early 1950s, when Dell's comic-book sales were rising steadily, had vanished, and comic books were a declining part of Western's business. Even in their physical appearance, the Dell comic books were losing some of their earlier luster. By 1954, all of the monthlies were slipping back to thirty-two pages (Walt Disney's Comics was one of the last to shrink, as of its October 1954 issue), advertising had spread to the inside covers, and some issues looked surprisingly cheap.
By the late 1950s, comic books based on television shows of various kinds were starting to dominate the Dell line, especially the comic books based on animated cartoons. Western acquired the comic-book rights to Hanna-Barbera's Huckleberry Hound in October 1958 and Quick Draw McGraw in May 1959, and then to all the Hanna-Barbera properties under a comprehensive agreement with Screen Gems, the cartoons' distributor, in January 1960. The new comic books echoed the coolly formulaic writing and drawing of the TV cartoons all too closely. A comic book like Uncle Scrooge, whose author had a more elevated conception of storytelling, was an anachronism even in that comic's debilitated state.
On "orders from the office," as Barks said in 1978, the stories in Uncle Scrooge became shorter. "They'd tell me how long I could make a story. Unless they gave me specific orders to cut down the length, I just assumed that they would be the length they had been before. If I had been doing 24-page stories, I just kept on doing 24-page stories until I suddenly received word that they wanted to cut them down to 17, and here I'd be with one already on the drawing board."
The change from one long story to two shorter stories was dictated by the greater advertising content in the Dell comic books. (In addition, story length was constrained by the need for a four-page "Gyro Gearloose" filler in each issue to meet postal requirements for second-class matter.) Whereas ads had once been restricted to the back covers of the monthlies, then to the inside covers as well, they were by 1956 appearing on interior pages, as had been the case for years with almost all competing companies' comic books. In that year, too, ads began appearing for the first time in Dell comic books other than the monthlies—quarterlies like Uncle Scrooge and bimonthlies like Donald Duck. With sales declining, Dell and Western were trying to take up the slack with advertising revenue. The plan was to cut back the comics pages to twenty-five in each issue, the remaining seven interior pages to be filled with advertising; but when Dell increased the price of its comic books from ten cents to fifteen cents, starting with the issues published on December 15, 1960, the planned increase in advertising pages was halted.
After years of producing stories that ranged in length from ten to thirty-two pages, Barks did not adjust well to the new regimen. His Uncle Scrooge stories from this period often end abruptly, as if they had been conceived longer and then forced into fewer pages. Other stories are like sketches for something longer and better. In these years, it was only in the occasional story that took the ducks to outer space that Barks indulged himself in a relaxed sort of fantasy. He spoke in 1982 of "Island in the Sky," in Uncle Scrooge no. 29, March–May 1960, as the story "that I like best now after all these years in looking back over the whole chain of them that I did." Such stories could have been more enjoyable for him only because he felt free in them to ease back on his concern with plausibility, which had fallen under increasing strain. In "Island in the Sky," the implausible and the impossible cascade through pages in which Scrooge flies a rocket ship far into space to hide his money on a barren asteroid that happens to be populated by tiny savages.
By way of compensation for the loss of plausibility, such stories offered not imaginative extravagance but rather endorsement of the characters' good intentions, so that the reader must step outside the stories to find the greatest value in them. The gap between such Barks stories and the painfully earnest stories in some other publishers' comic books—the "Little Archie" stories by Bob Bolling in the Archie comic books, for example—is remarkably small. Barks at his best concerned himself almost exclusively with bringing his characters fully to life. He found certain kinds of characters particularly congenial—that is, more likely to generate good comic ideas—but there is never the sense in his best stories that his view of the world is determining what his characters do, that he is manipulating them in the service of an uplifting message.
Although Western's editors occasionally sent him complaints like the Downing letter, Barks otherwise continued to work in isolation, as he had since the early 1940s. He was not entirely invisible to the outside world, but the occasional mention of his name and his work in local newspapers caused no ripples, and neither, more curiously, did the article that Charles Beaumont wrote for the May 1955 issue of a California magazine called Fortnight. In an article otherwise devoted to how industriously Western's editors scrubbed from their comic books anything that might conceivably offend an anxious parent, Beaumont paused to acknowledge the special status of Carl Barks, whose work had "a freshness and originality which both inspire and depress colleagues." He wrote of Uncle Scrooge's "popularity among the 'intelligentsia,'" and of the "flood of fan mail" that Barks's stories evoked, especially the complaints when other cartoonists filled in on the monthly "Donald Duck" story for Walt Disney's Comics—the story that Beaumont called "the best-loved story in the comic world."
Apparently no one who read Beaumont's article seized the occasion to try to get in touch with Barks. Barks himself never saw such evidence of his stories' popularity, and when he wanted to see his stories in print, he bought his own copies of the comic books at a newsstand. It was another five years before some of his admirers wheedled his name out of Western and began to write and then visit him. When he got the first fan letter, from John Spicer of Aptos, California, in April 1960, Barks was suspicious at first, thinking that the letter might have been concocted by Bob Harmon, a cartoonist friend and a practical joker. He wrote to Spicer:
I might be justified in wondering how you know my stuff from the other guys' work but I have met people before who knew that difference and not all of them were professional comic bookers. I always try to write a story that I wouldn't mind buying myself. Maybe that is what distinguished it from the writing of those who try only to get a story past the editors. . . .
The front office tells me they get many letters, but over the past 17 years they have shown me only three. Two of which were pan letters that left me cringing for weeks. I suppose it's just as well I don't get much mail. Writing and drawing these comics is a full-time job 7 days a week. I would have little time to answer.
One of the first visiting fans, Malcolm Willits, got from Barks what Willits called a "letter of introduction" to Western. When Willits appeared at Western's offices in Beverly Hills, he wrote to Barks in October 1960, "[t]hey were quite surprised, shocked I should say, to have a REAL FAN of yours turn up. They were sure it was a plot for me to try to get a job from them or pawn off some artwork on them. But one man was very good to answer my questions and I spent about 45 minutes there. The man said they give you the most freedom of any artists, and that they do not know whom they will replace you with." Within days of Willits's visit, Western's editors, aware now that the game was up, began sending Barks's address to anyone who asked for it.
When Carl Barks got his first fan letter in 1960, he thought at first it was a joke played by Bob Harmon, a gag writer for the Dennis the Menace newspaper comic, seen here in a 1956 photo with Barks and Harmon's wife, Eileen. Author's collection.
After the price increase, Western began cutting some of Barks's stories after he submitted them, to make room for the pages of Dell contests it was hoped would lure readers who were resisting the higher price. The page rate for Barks's drawings did go up, though, by one dollar. In the early 1960s he was receiving $11.50 per page for writing the stories and $34 per page for drawing them. Barks had always illustrated an occasional story that he did not write, but now a full one-third of his pages were based on other writers' scripts. He was turning out a much larger number of pages—a total of 358 in 1960—in order, as he told one correspondent, to "build up the family bankroll." With work on the stories themselves offering less and less satisfaction, Barks was now seeking satisfaction of another kind.
One of Western's efforts to economize in the late 1950s affected Barks strongly, though, as he explained:
When I was first drawing, back in the years when my stuff was supposedly at its best, they were furnishing me with a very good grade of Strathmore paper, for pen and ink. Then they got to buying some West German paper [that] was coated with kind of a clay surface, and it was very difficult to draw on. They got it much cheaper, they'd buy it in trainload lots, I guess. There were so many complaints came in from the artists on the first batch of that stuff, they got this German company to make a better grade.
Barks blamed the inferior paper for deterioration in the way he drew the ducks, a loss of plasticity as his characters grew tall and spindly: "I've always drawn that duck too tall, and with good paper I would just erase it and redraw him. But since there was a trench already made there by my pencil from making that tall duck, I was always getting my pen line stuck in that trench, and drawing him that way anyway. So I thought, oh heck with this, I'll just draw that duck the way he comes out in the first rough."
More was at issue than inferior paper, though, as Geoffrey Blum has noted: "By 1960, with the new batch of paper on hand, the ducks have become short and bouncy again—but they are so much simpler. Pie cuts disappear from their eyes, line weight is more consistently thick, and gestures have become less extrovert." All complexity, and with it almost all interest, was disappearing from Barks's stories.
## 26
# The Slow Fade
As advertising took up more space in the Dell comic books, and as shorter, simpler stories became the rule in most titles, there were still a few comics—feature-film adaptations— that necessarily offered full-length stories. A few of these, as it happened, were among the most attractive and interesting Dell comics of the mid- to late 1950s, particularly when Alex Toth was the illustrator.
Before he worked for Western from 1956 to 1960, Toth illustrated stories for Harvey Kurtzman's war comics at EC, and Kurtzman could not suppress his skepticism: "[H]e had this technique of lots of black. Which is legitimate; but sometimes I had the feeling that Toth was just covering space with black, and was shortcutting." But even in a Dell potboiler like Gun Glory (1957), based on a western movie starring Stewart Granger, Toth's careful spotting of blacks, combined with his economical drawings, gave his work greater power and elegance than could be found in most other cartoonists' work. The typical comic-book illustrator in the late 1950s, for the Dell titles especially, produced drawings that showed too much, too literally—sins of which Toth was rarely if ever guilty.
Toth was, however, entirely typical of many comic-book illustrators—Everett Raymond Kinstler was a prominent exception—in his impatience with length, with a story's having enough pages to reveal itself fully. "I'd rather have twenty 10-page stories than one 200-page story," he said in 1968. "I found this to be the case when I was freelancing; I could be tired as hell, having just come off a job, when a new script would arrive in the mail and I'd be perked up by it. . . . Even those 34-pagers"—that is, the Dell movie comics—"used to drive me up the wall." Yet the scripts that Toth illustrated for the comic books that Ned Pines published under the Standard label—scripts that crammed far too much violence and emotional turmoil into very few pages—were beyond rescue even by a talented artist stimulated by their very novelty.
As Toth flourished, almost against his will, by illustrating stories longer than he liked, circumstances were conspiring to diminish what was most distinctive and valuable in John Stanley's work, by depriving his Little Lulu stories of length they needed. Stanley's stories had always varied in length—and in quality, given that he was so prolific—but stories that would have benefited from length were now limited to fewer pages than they needed. The plots were Stanley's, but everything that made his stories special was being squeezed out. The plots began to sag, too, as Stanley submitted himself increasingly to the tyranny of that most hackneyed of comic-book writers' tools—the far-fetched coincidence. He had used coincidences before but had made them funny in the context of a tongue-in-cheek story. Now the coincidences were being asked to do too much work.
Stanley's most consistently funny and inventive work, the fairy tales, also suffered, especially after Stanley began using a witch, Hazel, and later her niece, Little Itch, as continuing characters that simply appeared too often. By 1955 the fairy tales' dry, ironic tone had mostly vanished from stories that had become conventional comic-book conflicts. What were useful templates when a comic-book artist was at his best could turn into crutches when it was difficult or impossible for him to be at his best, and by the mid-1950s that was happening to Stanley. There were still bright moments, stories better than anyone might have expected, like the five-page "Two Foots Is Feet" in Little Lulu no. 94, April 1956, in which a typical Stanley cascade of events is triggered by, and mostly consists of, repetition of the single word foot. Sometimes Stanley even seemed to pick up speed for months at a time, as he did in much of 1956. But he was sailing against the tide.
It took a long time before the decline in Little Lulu's quality was reflected in declining sales, and even then the general shrinkage of the comic-book market was undoubtedly more important. In the 1950s Western was printing more than a million copies of most issues of Little Lulu. When Western wrote to Marjorie Henderson Buell on November 22, 1955, her royalties for the year had grown to $81,183.08, and her signing bonus had risen to more than $8,000. Her royalties leveled off after that but stayed above $60,000 throughout the 1950s.
Buell's supervision of the comic book, very loose but still permitting of the occasional crack of the whip, was not unusual. Although Western produced "all of the art work and story at our own expense," as Lloyd E. Smith wrote to one literary agent, the company was prepared to "submit it, when required, for approval before publication." Walt Kelly was of course intensively involved with the Pogo Possum comic book, as its writer and artist as well as its copyright holder and licensor. Other comic-strip creators, like Mort Walker of Beetle Bailey, were likewise active in the production of their comic books, even when the industry as a whole was in a downward spiral and attention to quality could seem quixotic.
Milton Caniff, one of the comic strip's aristocracy thanks to his enormous success with Terry and the Pirates and then Steve Canyon, oversaw the Steve Canyon comic book so carefully, first for Harvey and then for Western and Dell, that at one point he not only was reviewing the penciled comic-book pages by Ray Bailey but was inking what Alfred Harvey called "the Steve Canyon heads." As it happened, Harvey published only one issue of Steve Canyon before notifying Caniff in 1952 that newsstand competition was so severe that "the success of any venture, no matter how well conceived, seems questionable."
After Caniff switched Steve Canyon to Dell in 1953, it never sold well enough to move out of the one-shot category. There was never more than one issue published in a calendar year, and there were ultimately only seven issues altogether. Even so, Caniff remained actively involved in production of the comic. For the first Dell issue, he said, he "paid the artist I called in on the job [William Overgard] in full so that there will be no hitches in the bookkeeping aspects of the contract."
When Western renewed its Steve Canyon contract in January 1955, Chester Weil of King Features Syndicate wrote to Caniff: "The entire comic book business is shot, as you well know, and I am happy to keep this alive with Dell and I hope you are." Caniff evidently was. Even when the Canyon comic book was at the end of its run in 1959, Matt Murphy submitted the plot for a sixteen-page story to Caniff for his approval and asked for some help: "Could you suggest the type two-seater new plane, with dual controls installed for instructional purposes, that we might use in this story. Any material you might have available on it would be very much appreciated."
The last of John Stanley's stories for Little Lulu were also published sometime in 1959. The exact date is difficult to pinpoint because although first-rate Stanley is immediately identifiable as his work, second- or third-rate Stanley of the sort that was turning up in Little Lulu by the mid-1950s bears fewer identifying marks. But 1959 is certainly the latest year when Lulu stories can with any confidence be identified as his. He had stopped writing for Little Lulu by the summer of 1958, when he began writing for the Nancy comic books based on Ernie Bushmiller's strip. They had become Dell titles after many years with another publisher.
The little girl Nancy and her rough-edged friend Sluggo were a combination reminiscent of Lulu and Tubby, but nowhere near as ingratiating, thanks mainly to Sluggo's indistinctness compared with the occasionally magnificent Tubby. There were supporting characters, too, who recalled the Little Lulu cast, notably Oona Goosepimple, a macabre girl who was close kin to Witch Hazel and Little Itch. Western and Dell would have been happy for Stanley to continue with Lulu. In July 1959, Dell's Helen Meyer asked Matt Murphy, as he told Stanley, "to inquire about your interest in doing some Little Lulu stories. The last time we talked I know you were not interested but if you feel there are still some story ideas kicking around, we would be glad to have them." There is no reason to believe that Stanley accepted that invitation.
Stanley's long bachelor life ended just before Christmas 1957, when, at forty-three, he married a German-born woman named Barbara Tikotin Widmer. Soon after they married they moved from the Hudson valley and its small cities like the one where Lulu lived. For a year or so the Stanleys lived in Manhattan, on Bleecker Street in Greenwich Village, until they relocated to Peekskill, on the Hudson north of Croton. Stanley had children of his own—a daughter and then a son—only when Little Lulu was completely in his past.
By the time Dell and Western ended their long comic-book association in 1962, Stanley had already moved over to Dell. He wrote two of that publisher's first new titles, Around the Block with Dunc and Loo, in the general Archie or Henry Aldrich vein; and Linda Lark, a soap-opera comic book about a nurse, drawn in a realistic style (not by Stanley). The first issues of both comic books appeared in the fall of 1961, almost a year before the Dell–Western split was formalized by publication of Western's first comic books under its Gold Key label. For the next few years, Stanley would write and draw many more comic books for Dell, notably Melvin Monster, about the green, sweet-tempered child in a family of horrors. (His "mummy" is exactly that.) The adults in Melvin's world, like the adults in Stanley's Little Lulu fairy tales, are cruel and thoughtless, but at least they have an excuse, since they are literally and not just metaphorically monsters.
Stanley also wrote a few serious horror stories for Dell, for a one-shot called Tales from the Tomb and the first issue of a continuing title, Ghost Stories, both in 1962. Western had never published any horror comics. After the split with Dell it came no closer than Gold Key titles based on the Twilight Zone television show and a short-lived TV series presided over by Boris Karloff; but Dell, which like Western never adhered to the Comics Code, took advantage of its freedom by publishing the most aggressive examples of the genre since the code had squelched other publishers' titles eight years earlier. The stories identified as Stanley's rely on devices so easily imaginable as starting points for Little Lulu stories—a monstrous arm rises from a manhole; a harmless-looking throw rug is actually the entrance to the lair of a hideous man-eating creature—that they reveal just how close some of the Lulu stories came to being more frightening than comic.
Western's editors surely would have rejected some of the Lulu stories if Stanley had submitted them in the censorious climate of the late 1950s. As it was, Matt Murphy did reject a few of Stanley's scripts for the Nancy comic book, asking him to revise a story called "Nancy Meets the Yoyos" "to make the sequence more 'believable' and eliminate the more 'horrible' aspects." The finished twelve-page story was published in Nancy no. 169, August 1959. While not "horrible" in the manner of some of the Little Lulu stories, it still reads like a bad dream that is taking place inside another bad dream, as Nancy is manipulated and pursued by supernatural creatures. It is a story that if drawn in a realistic style instead of a simple, Bushmiller-like cartoon style (by Dan Gormley) could invite nightmares.
As to why Stanley left Western for Dell, the closest thing to an answer is in a letter Lloyd E. Smith wrote in 1966:
John Stanley might still be working for us except for the fact that he made certain demands as to compensation which as a matter of business policy we were unable to meet. He even went to the extent of consulting an attorney in the matter, not so much to determine his rights, which were always clear and limited, but to insist that there was a legal obligation on our part to make a kind of standing arrangement with him. He found on consulting an attorney that this was not so but nevertheless he refused to continue with us and consequently he has not been engaged by us since that time.
Irving Tripp, the longtime Little Lulu artist, remembered that on one of Stanley's visits to Poughkeepsie "John was very disgruntled. I don't think he was happy and he also had these other characters that he wanted to get involved in, too. He had some of his own." Possibly Stanley wanted Western to publish titles of his own that Dell ultimately published—like Melvin Monster and Thirteen Going on Eighteen—but under terms that Western found unacceptable. As it happened, both Melvin and Thirteen lasted only a few years before Dell canceled them, and other Stanley titles had even shorter lives. After his Dell titles expired, Stanley worked again for Western. He wrote the single 1969 issue of a Gold Key comic book called Choo-Choo Charlie, a candy tie-in. His very last comic-book work, published late in 1970, was the first issue of another of Western's Gold Key comic books, O. G. Whiz. In 1976, Stanley said of his comics work, "I haven't put a pencil to paper for seven or eight years."
John Stanley made a rare public appearance in 1976, at the Newcon comics convention in Boston. Photo by E. B. Boatner.
Over the course of his comic-book career Stanley depicted every phase of childhood from the elementary grades through high school, in Little Lulu, Melvin Monster, Nancy, Henry Aldrich, Thirteen Going on Eighteen, and Around the Block with Dunc and Loo, and just beyond with Kookie. The demise in the 1960s of all his own titles just preceded his exit from the comic-book industry and no doubt contributed to the bitterness he expressed toward it. He was truly successful as a writer and cartoonist only when he was enlarging upon characters originated by other people.
After living in Manhattan again for a few years in the 1960s, Stanley and his family moved to Cold Spring, New York, eighteen miles up the Hudson River from Croton-on-Hudson, late in that decade. His marriage, under strain for years, ended in separation and divorce in the 1970s. At Cold Spring he worked in a factory making silk-screened aluminum rulers—very good ones, apparently—for a company called Fairgate Rule. When Fairgate's small plant was demolished early in 2012, a local government official remembered that "Fairgate made the best rulers. . . . I still have one here on my desk and it is terrific!"
Stanley's son James said: "I think at [my father's] core he was an artisan, a perfectionist who wasn't afraid of getting his hands dirty—so it isn't a stretch to understand where he ended up. He used to complain about the cranky old SOB who owned Fairgate Rule, Charles Brody. He did some advertising artwork for him as well." Brody asked Stanley to run the place, James Stanley said, "and he turned that opportunity down. I clearly remember my father telling me this. Later in life, I ran into locals who worked alongside him as young guys in their 20's, and they made it a point to say what a great guy he was. Hard to reconcile with the nasty stuff at home, but I can understand it now."
John Stanley retired sometime in the late 1980s. He died on November 11, 1993, at the age of seventy-nine.
## 27
# Disasters
It was the decline in comic-book sales that undermined and eventually ended the long-standing Dell–Western partnership.
The partnership was very sociable as well as very profitable. Every fall, the annual "Western–Dell Day" brought members of Dell's New York City staff, including George Delacorte, north to Poughkeepsie for golf, tennis, meals, and tours of Western's plant. In 1957, according to a Western Printing prospectus from that year, the Dell comic books and paperback books that Western produced were being distributed by Dell to "more than 100,000 newsstand and supermarket outlets." That arrangement was, however, severable at will by either party. Western reserved "the right to produce and distribute two other comic magazines of its choice"—by 1957 those titles were Walt Disney's Comics & Stories and Red Ryder Ranch Comics, which would soon cease publication—"and these reserved sales are handled through one of its subsidiaries, K.K. Publications, Inc." K.K. also published the Boys' and Girls' March of Comics giveaway, which was distributed mainly through shoe stores.
Three years later, in the words of a 1960 Western prospectus, discussions were under way "which could materially alter this arrangement. Western has proposed the elimination of the exclusive features of the comic book contract with respect to new material which would allow [Western] to create comic-type books for publication and distribution through other channels and permit Dell Publishing to create its own comic books or purchase from other producers." Western was still producing more than 250 million comic books a year, a total that it estimated was 40 percent of the industry's output, but, the company said, "[d]ollar sales of comic books have declined in recent years in the industry overall and for Western." By 1960, comic books accounted for only 7 percent of Western's total sales—less than half the percentage just five years earlier. The company changed its name to Western Publishing in July of that year, retaining the Western Printing & Lithographing name for its "principal operating unit."
Among the major comic-book publishers, Western and Dell were uniquely dependent on publishing revenues. They paid licensors for the rights to use popular characters, but they owned almost no characters of their own. Publishers like DC, on the other hand, owned almost all of the characters in their comic books and thus could license them to toy manufacturers, television networks, and movie studios. In other words, there was a powerful incentive for both Western and Dell to try to increase comic-book revenue, not just go on taking the same slice of a shrinking pie.
By early in 1961, as the price of their comic books rose to fifteen cents, they had agreed on "a new nonexclusive comic book contract covering live and animated characters under licenses held by Western." The two companies were still on friendly terms and still doing business with each other. Around the time they signed the new comic-book contract, they signed a ten-year agreement for Western to print Dell's paperback books. But if a higher price for comic books did not generate more revenue, Western could now publish under its own label; and, of course, so could Dell.
Dell had tested the fifteen-cent price in some unspecified part of the country in 1957, unsuccessfully, and then again in several western states in 1959 with mixed results. But the logic of a price increase carried the day. As the new price went into effect, William F. Callahan Jr., Dell's executive vice president, noted in an article published in Bestsellers, a trade publication for book and magazine retailers, that items whose prices were comparable when comic books first appeared in the mid-1930s—candy bars, hamburgers, cigarettes, movies—had all at least doubled or tripled in price by 1960. "Yet the comic book alone of all these popular items has remained at a dime. This despite the fact that kids' allowances have increased to keep pace with higher prices."
Bestsellers paired Callahan's article with one by Irwin Donenfeld, Harry Donenfeld's son and the publisher of what had become generally known as "Superman DC Comics." Donenfeld scorned the idea of a 50 percent price increase. His company had tried to sell its World's Finest Comics for fifteen cents but had been unsuccessful even with extra pages (and with both Superman and Batman on the cover). When DC cut the price to ten cents—and reduced the page count correspondingly—sales went up instantly and an unprofitable comic became profitable. "Two of our closest competitors tried the fifteen cent price on their entire line," Donenfeld wrote, "and in both cases the tests were discontinued. Youngsters are the greatest shoppers in the world because of the limitations of their allowance, and they definitely resist paying fifteen cents for comics."
As Donenfeld all but predicted, Dell's price increase turned out to be a disastrous mistake. For the October 1960 issue of Walt Disney's Comics, Disney's records showed it receiving royalties on 1,375,000 copies. A year later, after the higher price took effect, the number of copies had fallen to 900,000. For October 1962 the figure was 461,000.
American News Company, Dell's distributor for many years, had been under assault by smaller distributors and the federal government for its supposedly monopolistic practices, until finally in the spring of 1957 it announced it was leaving magazine distribution altogether. By then, Dell had already decided, as Business Week said, that it "would shift to distribution through independent channels," not just of its comic books but also of its many magazines and books of other kinds. Dell's comic books, which had once enjoyed a clear path onto newsstands thanks to powerful American News, now had to compete with many other publications for the attention of wholesalers, even as the higher price drove down sales. Western's affiliation with Dell, and the huge success of the Dell comics, had insulated Western from such grubby realities of distribution, but now that was at an end.
Another problem was that some of the rival publishers' comic books, those starring costumed superheroes, were enjoying a rebirth. Only a few of DC's heroes—notably Superman, Batman, and Wonder Woman—had survived into the 1950s, but late in the decade DC began introducing new versions of such "golden age" characters as the Flash and Green Lantern. Then in 1961 a company that went under several names—Timely, Atlas, and ultimately Marvel—introduced superheroes that were not simply reworked versions of old characters; rather, the new superheroes incorporated characteristics of some of the other comic-book genres that had flourished since the superheroes' first burst of popularity in the early 1940s.
Marvel's comics were as much soap operas as adventure stories, echoing the romance comics that flourished in the postwar years. Young readers who would have recoiled from television soaps were hooked as firmly by the continuing stories in the Marvel books as their parents were by the stories on TV. Superheroes, always the vehicles for fantasies of wish fulfillment, were even more attractive when pubescent anguish was stirred into the mix. Moreover, Marvel's protagonists successfully united the two strains that had dominated superhero comics until then: heroes who were supposed to be taken seriously, like Superman and Batman, and parody superheroes like Captain Marvel and Plastic Man. Marvel's editor, Stan Lee, and his collaborators treated the heroes seriously—the reader was never asked to regard them as jokes—but the heroes themselves mocked who they were and what they were doing. They could laugh or groan and shake their heads at the craziness of it all, or at their own mistakes, but they had earned the right to do that by being heroes in the first place.
This potent mixture was given greater vitality through furiously energetic drawings by Jack Kirby for Fantastic Four and dry, almost sinister drawings by Steve Ditko for The Amazing Spider-Man and Strange Tales. They were far more distinctive artists than the cartoonists confined by DC's homogenized house style. Both Kirby and Ditko were involved heavily in the writing of their stories—to what extent was a matter of ongoing dispute in later years—and their involvement gave the stories a more personal flavor than the superhero stories in the DC comic books.
Increasingly in the 1960s, the Marvel comic books relied not just on continuing stories but also on crossover appearances, so that the featured character from one comic book turned up in another character's comic. There was risk in tying the books together in that fashion—the risk that young readers would be discouraged by the need to keep up with most of the line if they were to understand what was going on—but it was a risk that paid off, because Marvel cultivated a relatively small but devoted fan base.
Continuing stories were not unknown in the Dell comic books, especially Walt Disney's Comics & Stories. A "Mickey Mouse" serial was part of the package from the beginning, at first in reprints and adaptations from the Floyd Gottfredson comic strip and then, starting in 1952, in new stories in the same vein (Mickey as a sort of detective) that were written by Carl Fallberg and drawn by Paul Murry, both Disney animation veterans. But it was Walt Disney's Comics' huge subscriber base that made continuing stories work. There was never much chance that readers would be swept up by the "Mickey Mouse" stories as they were by the Marvel stories.
After feasting on customers' resistance to Dell's higher price for about a year, DC and Marvel raised their prices, too—but only by two cents, to twelve cents a copy, for a standard thirty-two-page comic book. Dell had no choice but to cut its price by three cents a few months later, starting with issues dated June 1962. Western also priced its standard comic books at twelve cents when it began issuing Gold Key comic books through its subsidiary K.K. Publications in July 1962. But the damage turned out to be irreparable. As of the January 1967 issue, Disney received royalties on only 317,000 copies of Walt Disney's Comics—a figure far below what had once been Western's usual base print run of 600,000 copies.
Not until 1961 did the published statements of ownership that the post office required of periodicals mailed at the second-class rate include circulation figures. The first paid-circulation figure for Uncle Scrooge appeared in no. 33, March–May 1961; it showed each issue selling an average of 1,040,543 copies for the period ending October 1, 1960. Number 33 was, however, the first Uncle Scrooge to sell for fifteen cents, and the effects of the increase did not begin to show until the next year's statement, when the figure fell to 835,928. For some reason the 1962 figure covered only subscription copies. When next an accurate paid-circulation figure was published, for 1963, it showed a precipitous decline to 296,255 (from a print run of 425,900). Circulation recovered a bit over the next few years, but only to around 330,000—less than one-third of the figure before the price increase.
In an echo of earlier efforts, like Oskar Lebeck's Surprise Books, to blur the line between comic books and children's books, Western in 1962 transformed some of the former Dell monthlies and bimonthlies—Bugs Bunny, Tom and Jerry, Woody Woodpecker, Little Lulu—into twenty-five-cent Gold Key quarterlies with more pages, thicker-than-usual slick paper for covers, and panel borders that were strips of color, rather than lines, when they were not missing entirely. In a reversion to the early 1950s, the Gold Key line did not carry outside advertising.
The new design was also inflicted on other comic books, like Uncle Scrooge, that continued to be the standard thirty-two pages. Writing to Malcolm Willits in April 1962, Carl Barks lamented a reduction in the size of the drawings: "[T]he old size of 2½ times up gave [the artists] room to operate with big pens or brushes when advantageous. Now the size is 2 times up. This wouldn't be a calamity, except that some bright boy in the East thought that the pages would look 'different' if the dialogue balloons were inset a minimum of ¼ inch from the top and sides of the panels. Naturally, this compresses the drawing area. . . . I think the pages will look different, all right. So different the kids will leave the books lie right on the stands."
Barks was right. Western acknowledged in its annual report for 1963 that it was losing money on its comic books, blaming the loss on "conservatism growing out of initial experience as a comic publisher in issuing comic magazines which, both in subject matter and quantities printed, proved not to be commercially sound." Although Western continued to tinker with formats, within a year titles like Bugs Bunny and Tom and Jerry were back to thirty-two pages, and by 1965 they were filled with reprinted stories from the 1950s.
Of Barks's stories, only the two in the first Gold Key issue of Uncle Scrooge (no. 40, January 1963) suffered the indignities of the new design before someone in authority recognized the mistake. It was, however, too late for any sort of rescue. By the early 1960s Barks's drawings had settled into a persistent blandness and his stories into mechanical patterns disturbingly similar to those of most other Disney comic books. Like his colleagues' work, Barks's stories now contained far too many arbitrary coincidences and far too much strained dialogue. There were other lapses in craft, like the occasional use of ungainly thought balloons as exposition. Barks was aware of the decline in his work. He wrote to Willits in 1963: "As a refresher last night I got out my 1953 Disney Comics [Walt Disney's Comics & Stories] and read all twelve stories. Then I read the first two I did in 1962. The difference is alarming. . . . My stuff certainly has deteriorated. But in the cold light of a new day I can easily see why. It's simply that twenty years is too long to be writing one type of story for one set of characters."
Barks was by nature a serious storyteller, a man with a bent toward his own kind of realism, and the sheer silliness of many of his later stories in the Gold Key issues of Uncle Scrooge is like the exaggerated sigh of a man who is eager to be gone. Toward the end, he was allowed once again to extend his stories to as many as twenty-four pages, and some of them benefit from the greater length; they are more carefully worked out, and they do not end as abruptly. But it was too late for greater length to make much difference. Any lingering sense of Scrooge as a dynamic, irascible capitalist completely disappeared, as Barks continued to drop him into questionable roles. In one story he is a status seeker in pursuit of the world's top status symbol, in another a scorned applicant to the archeologists' club, in still another a teller of tall tales. In many others Barks had him frantically trying to protect his "Old Number One Dime," the first money he earned, from the sorceress Magica de Spell. That dime (which Barks introduced in a backup story in the third issue of Uncle Scrooge, in 1953) became in the 1960s the same sort of overworked prop as the Junior Woodchucks' Guidebook in the 1950s.
"Some Uncle Scrooge fans have complained that in late years I turned the Uncle Scrooge stories into mere adventure strips," Barks wrote in 1968. "They are right. I was afraid to keep repeating money and miser gags over and over, and the lack of contact between me and my readers left me ignorant of the fact that readers liked the money-miser gags."
By 1963 the exceptional freedom Barks had enjoyed, submitting stories in finished form without clearing them with his editors first as scripts or penciled artwork, had been severely curtailed. At Western's request, Barks began submitting outlines for his Uncle Scrooge stories to Chase Craig. Barks told Malcolm Willits that he felt hobbled by the new procedure: "[A]t times I thought, 'There are so many things I'd like to have that duck do—brand-new original stuff.' But I was a little afraid to try it. I was afraid to write a continuity or synopsis and send it down to the office for fear they might not see it the way I saw it in my own mind. I haven't got the vocabulary to write it out."
On July 9, 1965, after receiving the outline for what was eventually published under the title "House of Haunts" (Uncle Scrooge no. 63, May 1966), Craig wrote offering praise that probably made Barks cringe:
Your story synopsis sounds fine! The only thing that did not sound so good to me was your reference to the Beagle Boys learning all kinds of new crime tricks while attending Prison School. The idea of teaching crime in prison does not sound very well [sic] for our law-enforcing friends. You no doubt have something else in mind here, but the way I read it it would put our prisons in a rather bad light. Why not just have the Boys going to Crook School somewhere and Scrooge gets word that they're about to graduate, etc. Other than that, your story is very pleasing and uplifting.
The irony was that Barks's politics were very conservative, and any implicit criticism in the story would have been not so much of the prisons as of the politicians in charge of them. On those rare occasions in the later stories when Barks's politics rose close to the surface, the stories benefited from the additional energy. "The real truth," Barks wrote in 1973, "is that I had a great fear of 'preaching' in scripts and avoided any sort of social or economic or political slanting unless such slanting aided the entertainment value." "House of Haunts" in Barks's finished version is very much like the outline he had sent to Craig a few weeks earlier, although when the Beagles leave Studious Hours Prison "rehabilitated," they have learned not "crime tricks" but legitimate skills that they put to criminal uses. After Craig received the finished art, he wrote to Barks to tell him how much he liked it: "I just wanted you to know how much I enjoyed your commentary on our kooky society."
When Chase Craig was photographed in 1969 at his desk, Western's Los Angeles offices were in a building on Hollywood Boulevard. The framed painting visible on the wall behind Craig is by Moe Gollub, for the front cover of Tarzan of the Apes no. 140, February 1964. Photo by the author.
Craig's letter came from yet another new address: more than twenty years after moving from downtown Los Angeles to Beverly Hills, Western's editorial offices had moved back downtown again, to 1313 West Eighth Street, in December 1963. That move, from Western's own building on Santa Monica Boulevard to half of the third floor in a three-story office building, was one measure of the comic books' diminished stature in their industry. Western's comics made a gradual comeback until by 1965, the company reported, it was producing about four hundred thousand comic books a day: "The demand is firm at this level, and is carried on at a profit." But production was far below the million-plus copies a day of the early 1950s.
By the time Barks retired, he had clearly exhausted, if not his ability to write and draw stories with the ducks, his interest in doing so. Chase Craig wrote to Barks on November 1, 1965, telling him:
In [a] recent contest-survey . . . we discovered this magazine has a great girl-reader interest. In fact, we got more letters from girls than from boys. Many of them were quite insistent that Daisy should be featured and many were indignant that we didn't up-date her dress, makeup, etc.
Anyway, we do feel that if we add Daisy as a regular in the DONALD DUCK [feature in Walt Disney's Comics] we will make a lot of readers happy, so we would like to try it. You might also like to use Daisy's nieces.
Barks's next-to-last story for Walt Disney's Comics, "Donald and Daisy," in no. 308, May 1966, not only elevates Daisy to a costarring role, but also gives her nieces a much larger role than the nephews, who exit on the third page. The story has Donald as, of all unlikely things, a beautician, and it is uncharacteristically cruel—not dry and satirical, but cold and contemptuous—in its depiction of Donald's female customers, like the very old and desperately unattractive Mrs. J. Crowsfoot Dryskin. Daisy herself does not escape; she is fretful and anxious, at first ashamed of Donald's new business and then jealous of his success with other females. She emerges "beautiful" at the end of the story thanks not to Donald but to the brutal ministrations of her nieces; and it is, of course, beauty of a clownish kind.
Daisy had been an incidental character in almost all of her appearances in Barks's stories, beginning with a two-panel walk-on in Walt Disney's Comics no. 36, September 1943. She was usually no more than a pretext for conflict between Donald and Gladstone, and only very rarely was she presented as a clearly defined character. In Walt Disney's Comics no. 101, February 1949, when she learns from the nephews that Donald has been ordered by his doctor to crochet doilies as therapy to cure his nightmares, she leads a phalanx of determined women in search of this rare specimen, so that he might lecture on crocheting at her needlework club. Confronted by a reality more terrifying than any nightmare, Donald takes refuge in the lions' cage at the zoo; his bad dreams are at an end. But it is Donald's hypermasculine recoil from crocheting that is the story's comic engine, not Daisy's frilly excitement that Donald is sharing a traditionally female pastime.
Like the other ducks, Daisy rose with the arc of Barks's stories in the early 1950s, becoming a more substantial and interesting character, quiet and womanly and compassionate in her most important appearance—in "A Christmas for Shacktown." She made only a few indistinct appearances in Barks's stories in the next dozen years or so—not counting his illustrations of other writers' thin scripts for a comic book called Daisy Duck's Diary—before reemerging diminished, like the other ducks, in that May 1966 story.
Barks submitted only one more "Donald and Daisy" story. He officially retired on June 30, 1966, although he made a few more deliveries before the end of the year—of cover drawings and of stories for which he had received advance payments so that they would not count against the earnings limit for Social Security. He knew it was time to go. Writing to Chase Craig on July 4, 1966, Barks said of himself and his wife: "Our sentiments about leaving the duck work are all of great relief. We'd like nothing better than to be able to jump into a new career of painting without a backward glance at the long drudging years we spent in comic books." In retirement he would receive a small pension. "The company was under no legal requirement to pay me anything," he wrote in 1975. "Bob Callender got me $100 a month. This comes from the company treasurer, not from the pension insurance company."
Despite the distaste for comic books that Barks expressed so often in later years, Chase Craig prevailed upon him to provide scripts (in the form of sketches like those John Stanley made for Little Lulu) and occasionally finished artwork for the Disney comic books for another seven years. His last such story was published in 1974. He never expressed much pleasure in the work. His concern for plausibility, so liberating for him when his characters were most real, had by the 1960s become a lead weight. In 1968, writing about his script called "Pawns of the Loup Garou" (a story penciled by Tony Strobl for Donald Duck no. 117, January 1968), Barks complained:
I get anti-nostalgia every time I approach my files of old [Donald Duck] comics. There's so little about my years of comic book work that I care to remember that I shudder when I pick up a stack of comics and become snowed under with recollections. . . .
Always there were decisions to make. . . . Could a witch be a real witch? How far can I stretch the ridiculous without getting in trouble with the office? How far can I push pure fantasy before some sophisticated 5-year-old kid complains that Donald Duck is only a fairy tale character like Hans C. Anderson's [sic] people? I leaned toward logical explanations of [phenomena] in every day terms and mechanics—just to be safe. I hate to go back into those old stories and relive the struggles I had trying to make the explanations interesting and funny and not a dull let-down.
In retirement in 1974, Carl Barks painted the Disney ducks at his home in Goleta, California. The large painting to the left is based on "The Terror of the River" in Donald Duck Four Color no. 108 (1946). Photo by E. B. Boatner.
Barks and his wife moved from San Jacinto to Goleta, near Santa Barbara, and he joined her in painting in oils—landscapes and the like at first, but then, starting in 1971, at the suggestion of a longtime fan, Glenn Bray, subjects of an entirely different kind. Those paintings proved to be tremendously popular, some of them eventually selling for upward of a hundred thousand dollars, even though, considered strictly as art, they were vulnerable to criticism on many grounds. Their subjects were Donald Duck, Uncle Scrooge, the nephews, and the other characters Barks had drawn for comic books. For most comic-book artists there could be no "late period," no Indian summer of distinctive work like that enjoyed by a Titian or a Verdi or other famous painters and composers, but Barks actually had one—of a sort—and it lasted for about as long as his comic-book career. He painted the ducks for almost thirty years (with an interruption) under licenses from Disney.
At first he fulfilled commissions from fans for paintings based mostly on comic-book covers. In 1976, Disney revoked his license when a purchaser made unauthorized prints from one of the paintings, but the license was restored in 1982 at the instigation of an entrepreneur named Bruce Hamilton. The paintings Barks made under Hamilton's aegis were larger and more elaborate than before. They were reproduced as expensive lithographs, and the paintings themselves were sold to wealthy collectors whose interest in Barks's comic books was often negligible.
As he painted, Barks moved first to Temecula in Southern California in 1977, and then, in 1983, to Grants Pass, Oregon, a little more than a hundred miles from his childhood home. He continued to work into his nineties, less in oils than in watercolors and colored pencils, and often under pressure from the managers who succeeded Hamilton—pressure that equaled or exceeded the deadline pressures of his comic-book work. A widower after Garé Barks's death in 1993, he died on August 25, 2000, at the age of ninety-nine.
## EPILOGUE
# Can These Bones Live?
When Western began publishing under the Gold Key label (derived from the names of two of its subsidiaries, Golden Press and K.K. Publications), the emphasis shifted slightly toward titles like Magnus, Robot Fighter and Doctor Solar, Man of the Atom, which Western itself owned. Such characters were, if not exactly superheroes, more directly competitive with the resurgent DC and Marvel characters, and, perhaps just as important in Western's reduced circumstances, using them did not require paying royalties to a licensor. Magnus was science fiction of a sort Western had rarely attempted. It was an exceptionally handsome comic book, illustrated by Russ Manning in a sleek, athletic style that fitted a hero who was, as Manning said, "a Tarzan of the future."
Late in 1965, Manning succeeded Jesse Marsh as the illustrator of the "Tarzan" stories. Marsh died a few months later, in April 1966. Tarzan of the Apes, as the comic book was now called, would follow a trajectory like that of the Gold Key line as a whole. Manning drew it for about three years (with interruptions) and then was succeeded by a lesser artist until Edgar Rice Burroughs Inc. ended its long affiliation with Western in 1972 and licensed Tarzan to DC. By then the comic book's average paid circulation had declined from more than 350,000 per issue in the mid-1960s to less than 250,000. Manning's Magnus, despite its quality, had never cracked a quarter million per issue.
Gaylord DuBois wrote most of the stories for Tarzan of the Apes until the end, although his affiliation with Western ended soon after that, evidently because Gold Key was no longer publishing adventure stories of the kind he preferred to write. He died in Volusia, Florida, on October 23, 1993, at the age of ninety-four.
The Gold Key comic books' mild prosperity in the mid-1960s was short-lived. Western lost other longtime characters—the Hanna-Barbera characters, King Features stalwarts like the Phantom and Popeye the Sailor—as licensors sought better deals in a shrunken comic-book market and even in some cases launched self-publishing efforts. Late in the decade, as Western renewed license agreements for its comic books, licensors agreed to drop the guarantees based on print runs that had been a standard element of its contracts.
Western's J. J. Barta wrote to Marjorie Buell's representative William C. Erskine on December 3, 1970: "Western is not now nor have we for well over a year been making or paying any royalty guarantees to Walt Disney Productions or Warner Bros. Inc. on our Gold Key comicpubs [sic] that use or feature the Disney or Warner Bros. characters or properties. The royalties are payable strictly on our sales, and when we close out and render the final sales on each comicpub." The only exception was Little Lulu, for which Western was still paying the guarantees required by a contract that had taken effect on January 1, 1962, during the transition from Dell to Gold Key. Under that contract, Buell was no longer paid a royalty on each copy printed. She received instead a royalty of 2.5 percent "of the retail cover price on the net sales thereof," but with minimum guarantees of two thousand dollars on twenty-five-cent comic books and a thousand dollars on comic books selling for less.
Barta asked Buell to surrender even that scaled-down guarantee. By 1969 the circulation of Little Lulu had shrunk to just over two hundred thousand copies, little more than half the print run and not enough to cover the guarantee. "Our Periodical Department simply cannot continue to absorb such overpayment of royalties on the comicpubs," Barta wrote. Buell and Erskine agreed to a change, and a new agreement dated December 14, 1970, provided for an advance on royalties but no guarantee.
Two years later, Buell sold all her rights in Little Lulu to Western for ninety-nine thousand dollars. The payments were spread over seven years, no doubt for tax reasons. Officially Marge's Little Lulu from the beginning, the comic book became simply Little Lulu, the name everyone had always known it by. After Western bought Lulu, the company produced a manual to guide artists and writers working on the comic book and presumably merchandise of other kinds. A final section explained the characters and their relationships: "The outlines of some typical stories will point the direction to take in creating new ones." The "typical stories" were all from the mid-1950s and all written by John Stanley.
The Gold Key comic books continued to decline, and by 1980 they were indistinguishable in their general shabbiness from most other comics. In the February 1980 issue of Walt Disney's Comics & Stories, once the crown of the Dell line, almost one-third of the pages were devoted to advertisements, including one for a company selling novelty items like "talking teeth" and "vampire blood." All the stories were reprints except for one weakly illustrated five-page story with the Disney chipmunks, Chip 'n' Dale. The lead story—as was almost always the case—was a Carl Barks "Donald Duck" ten-pager, this one a reprint from January 1963.
The February 1980 Walt Disney's Comics was the last to bear the Gold Key label. The Gold Key line expired after the toy company Mattel, which had acquired Western Publishing the year before, decided to sell Western's comic books under the Whitman label—and to sell them, moreover, not like newsstand magazines but like toys, in bagged sets at variety and department stores, places where the Whitman name was already familiar. Sales were poor, and in May 1984 yet another new owner, the real-estate tycoon Richard A. Bernstein, closed down Western's comic books altogether, less than three months after buying the company from Mattel.
Dell's demise as a comic-book publisher preceded Western's by more than ten years; the last Dell comic books were published in 1973. Eventually both Western and Dell were absorbed into the Bertelsmann media empire, whose most important publishing brand was Random House. Remnants of the long Dell–Western collaboration, including bound volumes of Dell comic books embossed for George Delacorte's personal library, were sold at auction and subsequently bobbed up on Internet auction sites.
The disappearance of Western Publishing's comic books in the 1980s coincided with major changes both in how comic books were sold and in who made up their audience. As comics' traditional retail outlets—newsstands, drugstores, and retail chains—dwindled, the direct market took their place. Whereas traditional retailers had returned unsold comic books (or parts of their covers) for credit, the new retailers that made up the direct market gave up the return privilege in exchange for a larger discount. By the 1980s the direct market was thriving, but with stores and comics devoted more specifically to superheroes. Customers tended to be collectors who were older than the children who made up the core comic-book audience in earlier years. Now comic books were much more expensive than before and sold in smaller quantities—but because they were tailored to the collectors' market and sales were more predictable, they could still be profitable for all concerned.
Some fans, attracted as teenagers by superheroes, wound up as young adults writing and drawing for the titles they had read so avidly a few years earlier. They strained to come up with superhero stories that were somehow more "adult," but superheroes are not just impossible, as Carl Barks's ducks are impossible; they are impossible in ways that seal off pathways for the imagination rather than opening them. They are rooted in wish fulfillment; thus the typically superfluous secret identity that conceals the superhero behind an appearance like that of the comic books' readers. "Serious" stories about the superheroes—like Frank Miller's Dark Knight (1986), devoted to an aging Batman; or Alan Moore's apocalyptic Watchmen series (1986–87), also for DC—could assert their maturity only by becoming ever more grim and bloody and ridiculous.
In a field desperate for new ways to use old heroes, ingenious writers—who did not draw at all, in contrast to the best comics writers of the past—assumed new importance. But the work of even the most intriguing of those writers had a secondhand air. Neil Gaiman, for instance (whose Sandman series for DC appeared between 1989 and 1996), acknowledged his debt to the fantasist James Branch Cabell.
In this increasingly self-contained and ultimately claustrophobic marketplace, there was no room for comic books of the kind Western had published. There were sporadic efforts to revive the Disney comic books from 1986 on, in various formats and under several different publishers, including Disney itself, but ultimately none was successful. The echoes of the Dell comic books were heard not in new comics of the Dell sort, but in reprints, some encompassing long runs of titles like Little Lulu and Uncle Scrooge. Like the far more numerous superhero reprints, they were intended for an audience of older and affluent collectors.
There were also faint echoes of the Dells in comic books of a very different kind.
The "underground" comics of the 1960s and 1970s—black-and-white comic books that were variously experimental and outlandishly vulgar—rose and fell with the counterculture. They bore scant resemblance to comic books as most Americans knew them. Most underground comics were devoted to explicit sex and language, drug taking, radical politics, scatological criticism of American society, and, almost incidentally, aggressive design. The smooth, featureless illustrator's style that was the norm in superhero comics had no place in the underground comics, many of whose cartoonists embraced the grotesque with a gusto that even Will Eisner never approached.
The few distinctive creators who emerged from the underground comics might have enjoyed success in more traditional environments. Gilbert Shelton, whose characters the Fabulous Furry Freak Brothers appeared in a series of comics, could be readily imagined working for Oskar Lebeck on the Dell comics of the forties. There is in Shelton's stories some of the same predilection for rowdy physical activity found in much of Walt Kelly's work. Robert Crumb, a superb draftsman, brought his characters onto the page with a physical presence that was rare if not unique. Although there was plenty of detail in his drawings, more important was a rich, bending line that traced the contours of each character's body (women's bodies especially) with loving precision. In earlier comic books, dominated as they were by the narrowest commercial considerations, even the most personal work by creators like Carl Barks and Will Eisner stood at one or more removes from the artist's life. Crumb, by contrast, went so far as to make himself a character—eventually the dominant character—in his stories, inviting the reader to share in his fantasies and frustrations.
There was an even more powerful autobiographical element in Art Spiegelman's Maus, the most significant single publication by an underground cartoonist. Maus was based on the harrowing experiences of Spiegelman's parents, survivors of the Holocaust, and on Spiegelman's own troubled relationship with his widowed father. What made this story so arresting was that its characters, including Spiegelman himself and his wife, were drawn as talking animals—Jews as mice, Nazis as cats, Poles as pigs—and not in a reassuring Disney-like manner, but simply and directly, almost crudely. Maus was first serialized between 1980 and 1991 in Raw, an adult comic book edited by Spiegelman and his wife, Françoise Mouly. The first half was published as a book in 1986 by the mainstream publisher Pantheon. In 1992, after the second and concluding volume of Maus was published, Spiegelman received a special Pulitzer Prize.
The very term underground had already all but lost its meaning by the time Maus appeared, and the virtually unanimous praise for Spiegelman's book proved to be a tremendous stimulus to serious efforts by other cartoonists and their publishers, especially in the form of what were now called graphic novels. That term was used as early as the 1960s in fan publications by the pioneering critic Richard Kyle, expressing a hope more than a reality. By the 1980s it had been appropriated by the superhero publishers for fancy comic books with stiffer covers, better printing, and higher prices. But as applied to Maus and to other books that followed, it actually meant something: the comic-book form was being used to tell a long story that in its complexity and subtlety rivaled the best prose fiction.
In the most important of these books, like Daniel Clowes's Ghost World (serialized 1993–97) and Chris Ware's Jimmy Corrigan, or The Smartest Kid on Earth (serialized 1995–2000), there is the sense that the cartoonist is marshaling all the elements of a comic-book story—how the panels are broken down, the weight assigned to dialogue and drawing within a panel, the balance within an entire page or a two-page spread—as skillfully as the best contemporary novelists manipulate words alone. The medium's power may have been revealed most impressively in a 2009 book by Robert Crumb, a version of the book of Genesis, sober and faithful to the biblical text but startlingly alive in the comic-book format.
There are reminders in some of these books of the work of cartoonists like Carl Barks and John Stanley. Jimmy Corrigan is particularly Stanley-like in how Chris Ware maintains a steady distance from his character, resisting the sentimental urge to make him at least a little more sympathetic. The appalling drudge Jimmy Corrigan is, the reader learns when he works his way through the book's intricate jacket (as much a part of its content as anything inside the covers), a comic-book collector—a fact whose implications it may pay the comics-fancying reader not to examine too closely.
Ultimately, though, such graphic novels are separated from the Dell comic books of Western Printing's heyday by much more than a few decades. The cartoonist of the mid–twentieth century, if he caught sight of the possibilities in his medium and wanted to realize them, almost always had to work undercover, concealing his ambitions in comic books that appeared to be as trivial and disposable as those from the competition. Most publishers, even a publisher as upright as Western, tolerated nothing else. In recent years, though, artistic ambition has been acknowledged and celebrated not just in real graphic novels but also in superhero comic books, where adolescent grandiosity has reigned cloaked in art's mantle.
Despite the superhero comic books' pretensions, superheroes have now become preeminently movie characters. The movies once had to struggle to reproduce on film the feats superheroes performed effortlessly on paper, but special effects that were necessarily awkward in a 1940s serial are now convincing in movies that are largely the products of computers. The superhero comic books have for some years served mostly as spawning grounds for new movie characters.
So much has happened in so short a time that it is a little as if the centuries separating the composers and painters of the early Renaissance from the comparable artists of our own time had been compressed into a few decades, so that, say, Guillaume de Machaut overlapped with Philip Glass. Those much earlier artists worked within constraints at least as severe as those binding Carl Barks and John Stanley. They worked for patrons who regarded them merely as useful artisans, and the market for their work was limited almost entirely to religious subjects. Opportunities for expression in the modern sense would seem to have been terribly limited; but in the hands of the greatest artists—the Giottos and Machauts—they were surprisingly plentiful. An even better comparison might be with American jazz, regarded early in the twentieth century as a disreputable popular music but by the end of the century accepted as art music worthy of concert halls.
That evolution in jazz's public acceptance did not mirror an evolution in artistic quality, which was as high when Louis Armstrong and Duke Ellington played early in the century as when successors like Miles Davis and John Coltrane played decades later. Opportunities for expression existed for a little while, too, in the Dell comic books, even though Western Printing's editors were ultimately successful in shutting them down. Today, those editors are long gone and comic books in general enjoy a respectability that was never theirs when they were truly popular, but the psychological obstacles to perceiving an artistic triumph in the work of a Carl Barks or John Stanley or Walt Kelly have by no means disappeared. It is easy to believe that as graphic novels multiply and their experiments in storytelling become ever more radical (or they abandon stories altogether), straightforward, unapologetically comic stories like "Lost in the Andes" and "Five Little Babies" and "Feelin' Mighty Hale, and Farewell" will come to seem as exotic and remote to the average reader as a fourteenth-century mass.
Does that matter? Probably not. The best art, art that survives its own time, will always seem a little strange to its new audiences. All that really matters is that the art itself has the chance to live, to find those audiences.
There was a time when nothing could have seemed more ephemeral than a comic book. Many people undoubtedly shared the opinion Lloyd E. Smith expressed in 1963, when he questioned whether comic books—including the ones he published—were deserving of preservation. But the Dell comics, and many others besides, have survived, collected and read by people who find lasting value in them. Often that value seems remote from anything resembling artistic or literary merit, the comics surviving as objects of nostalgia or simply as investments whose initial value to readers, whatever it was, no longer has anything to do with the much greater market price today of, say, Superman no. 1 or Archie Comics no. 1. At the same time, though, the best stories, like those of Carl Barks, have been reprinted not just with increasing frequency but also with increasing attention to just how difficult it is to reproduce a page from an old comic book satisfactorily.
The better such stories look, the likelier that they will be read, or so their admirers hope. In any case, in years to come those stories will have the opportunity to make their case to new readers. With luck those readers will prove to be as perceptive as the midcentury children and parents who recognized and sought out the work of the "very good ones."
# Abbreviations
AC | Author's collection.
---|---
Dell/Nedor Record | Dell Publishing Co., Inc., v. Ned Pines et al., 266 A.D. 837, 42 N.Y.S. 2d 937, N.Y. A.D. 1 Dept. 1943. The case record and appellate briefs are held by the New York State Library, Albany.
ERB | Edgar Rice Burroughs Inc., Tarzana, CA. Burroughs's files of its dealings with Western Printing & Lithographing Company from 1933 to 1960 were made available to the author by Robert R. Barrett, an expert on the work of Edgar Rice Burroughs who was permitted to copy the files for such research purposes by the late Danton Burroughs, Edgar Rice Burroughs's son.
GA | Autry Library, Autry National Center, Los Angeles.
JS | John Stanley papers, courtesy of James Stanley.
Kimball | Ward Kimball papers, courtesy of John Kimball, Virginia Kimball, Kelly Kimball, and Amid Amidi.
Marge | Marge [Marjorie Henderson Buell] Papers, 1856–1994, Schlesinger Library, Radcliffe Institute, Harvard University, Cambridge, Massachusetts
OSU, MAC | Milton Caniff Collection, Billy Ireland Cartoon Library and Museum, Ohio State University, Columbus.
OSU, TM | Toni Mendez Collection, Billy Ireland Cartoon Library and Museum, Ohio State University, Columbus
OSU, WK | Walt Kelly Collection, Billy Ireland Cartoon Library and Museum, Ohio State University, Columbus. The Pogo Collection and the Walt Kelly Collection are separate collections. Because of the size and complexity of the two Kelly collections in particular, all OSU citations include both box and folder numbers.
SS | Steve Schneider collection
WDA | Walt Disney Archives, Burbank, CA
Wertham papers | Fredric Wertham papers, 1818–1986, Manuscript Division, Library of Congress, Washington, DC
# Notes
## A NOTE ON REPRINTS
Many of the comic-book stories discussed in this book are far more accessible now than they were just a few years ago, thanks to extensive reprint programs. In the most important of these programs, Fantagraphics Books of Seattle is reprinting all of Carl Barks's stories for the Disney comic books, starting with his best work from the late 1940s and early 1950s. Fantagraphics is also reprinting Walt Kelly's daily and Sunday Pogo comic strips, and Floyd Gottfredson's daily and Sunday Mickey Mouse, among many other worthy projects. All of Kelly's comic-book stories with Albert and Pogo from 1942–54 are being reprinted in a set of three volumes by Hermes Press. John Stanley's stories for Little Lulu are also available now in a reprint series from Dark Horse, as are many stories from the Jesse Marsh–Gaylord DuBois collaboration on Tarzan.
## PREFACE
1. Harlan Ellison, as quoted in the preface to Superman at Fifty, ed. Dennis Dooley and Gary Engle (Cleveland, 1987), p. 12.
## INTRODUCTION
1. John R. Vosburgh, "How the Comic Book Started," Commonweal, May 20, 1949, pp. 146–47.
2. Coulton Waugh, The Comics (New York, 1947; reprint, Jackson, MS, 1994), p. 339.
3. Ibid., p. 341 (illustration caption).
4. ibid., p. 342.
5. Vosburgh, "How the Comic Book Started," p. 148.
6. For a full account of Wertham's career, which extended far beyond his attacks on comic books, see Bart Beaty, Fredric Wertham and the Critique of Mass Culture (Jackson, MS, 2005).
7. Fredric Wertham, Seduction of the Innocent (New York, 1954; reprint, Laurel, NY, 2004), p. 127.
8. Lloyd E. Smith to the author, February 22, 1963, as quoted in Smith to the author, November 12, 1963. Unfortunately, Smith's earlier correspondence with the author has been lost. Smith, as head of Western Printing & Lithographing Company's rights and royalties department, negotiated with the proprietors of many popular characters for the rights to use them in comic books and other products. He was also the owner of a library of perhaps forty-five thousand books and donor of thousands of rare books to the Library of Congress and college and university libraries. "A Titan in the World of Books," Racine (WI) Journal-Times, December 16, 1971, p. 10A, is a memorial editorial and a fuller account of Smith's career than the obituary that the newspaper published two days earlier.
9. Smith to the author, November 12, 1963.
10. John Benson, preface to "Art & Commerce: An Oral Reminiscence by Will Eisner," Panels no. 1, Summer 1979, p. 3.
11. Robert Warshow, The Immediate Experience: Movies, Comics, Theatre, and Other Aspects of Popular Culture, enl. ed. (Cambridge, MA, 2001), pp. 54–55. The book is a collection of Warshow's essays for Commentary, including "Paul, the Horror Comics, and Dr. Wertham," in which the quotation appears; that essay was published originally in the June 1954 issue.
12. Hearings Before the Subcommittee to Investigate Juvenile Delinquency of the Committee on the Judiciary, U.S. Senate, April 22, 1954, p. 197.
13. Murray Kempton, "Pogo's So-So Stories (So, So Wonderful)," New York Post, June 21, 1953, p. 12M. It seems doubtful that Mark Twain was a major influence on Kelly, although characters like the King and the Duke in Adventures of Huckleberry Finn resemble some of the rascals in Kelly's "Our Gang" stories, in particular.
## CHAPTER 1
1. Irving Brecher as told to Hank Rosenfield, The Wicked Wit of the West (Teaneck, NJ, 2009) p. 61.
2. "Biggest Gag Factory on Earth Has 6,000,000 Jokes on File," Milwaukee Journal Green Sheet, December 11, 1935, p. 1.
3. As evidenced by the book's copyright registration.
4. "Screen Notes," New York Times, July 26, 1935, p. 14.
5. Dell Publishing Co., Inc., v. Ned Pines et al., 266 A.D. 837, 42 N.Y.S. 2d 937, N.Y. A.D. 1 Dept. 1943 (hereafter cited as Dell/Nedor Record), p. 153.
6. "Purely Personal," Motion Picture Daily, July 20, 1931, p. 5.
7. Cecil Munsey, Disneyana: Walt Disney Collectibles (New York, 1974), p. 144. The figures reproduced by Munsey were presumably sent by Horne to the Disneys. In 2010 the Walt Disney Archives could find no such figures for later issues of the magazine.
8. "Kay Kamen to Take Hal Horne Magazine," Motion Picture Daily, June 2, 1936, p. 10.
9. "Hal Horne to Radio; To Become Producer," Motion Picture Daily, June 17, 1936, p. 14.
10. Walt Disney to Roy O. Disney, August 10, 1936. Walt Disney Archives, Burbank, CA (hereafter cited as WDA).
11. "Mickey Mouse Now Editor!" New York Morning Telegraph, May 15, 1935, p. 4.
12. As reflected in 1932 correspondence in the Leo Hart Printing Co. Inc. papers, University of Rochester. Kamen wrote to Hart from New York as well as from Los Angeles, in the latter case writing on Walt Disney Productions stationery.
13. L. H. Robbins, "Mickey Mouse Emerges as Economist," New York Times Magazine, March 10, 1935, p. 8.
14. "The Western Story," Westerner, March 1949, p. 3; Don Black, E.H.: The Life Story of Edward H. Wadewitz (a privately published memorial volume, ca. 1955); "The Story of Western," Westerner, April 1962, p. 10. The dates and other figures in these official histories differ slightly. The figures adopted here, in cases of conflict, are those that seem to have been best supported by documentary evidence.
15. "Happy Anniversary to Western at Poughkeepsie," Westerner, December 1964, p. 4.
16. Lowe's letter is reproduced in Leonard S. Marcus, Golden Legacy (New York, 2007), p. 13. Also see "Western and Disney Extend Agreement," Westerner, February 1961, p. 14.
17. The Westerner's 1961 article incorrectly identifies the first Mickey Mouse Big Little Book as Mickey Mouse Sails for Treasure Island, also published in 1933 and also based on the comic strip.
18. One of those books, published by Whitman in 1936, was titled 40 Big Pages of Mickey Mouse. It was a reprint of the first, oversize issue of Hal Horne's Mickey Mouse Magazine, with a different cover and no advertisements.
19. Howard Anderson to John C. Worrell, memorandum, March 21, 1979. Anderson wrote his memo to Worrell, who was Western Publishing's vice president for product development and planning, in response to the author's letter to Western's president, G. J. Slade, asking for information about the company's history, and Worrell sent a photocopy to the author. Author's collection (hereafter cited as AC ). It is not clear why ownership of K.K. was initially placed in the hands of three individual Western executives instead of Western itself, since K.K. was always a subsidiary of the parent company.
20. Jack and Jill advertisement, Printers' Ink, June 29, 1939, pp. 54–55.
21. Georges Duplaix's byline, as "George" Duplaix, appeared on "Topsy-Turvy Circus," a story with no Disney content in the October 1938 issue of Mickey Mouse Magazine.
22. Leonard S. Marcus, Minders of Make-Believe (New York, 2008), p. 164.
23. Anderson to Worrell, memorandum.
24. E. H. Wadewitz to C. R. Rothmund (secretary of Edgar Rice Burroughs Inc.), January 26, 1940. Edgar Rice Burroughs Inc., Tarzana, CA (hereafter cited as ERB). Burroughs's files of its dealings with Western Printing & Lithographing Company from 1933 to 1960 were made available to the author by Robert R. Barrett, an expert on the work of Edgar Rice Burroughs, who was permitted to copy the files for such research purposes by the late Danton Burroughs, Edgar Rice Burroughs's son.
25. Malcolm Willits, "George Sherman: An Interview with Another One of the 'Men Behind the Mouse,'" Vanguard (a comic-book fan magazine published by Robert Latona), 1968, p. 33. The pages are unnumbered except on the contents page.
26. Ibid., p. 34.
## CHAPTER 2
1. Morris Gollub, interview with the author, Hollywood, CA, November 2, 1976.
2. Donald Phelps, "John Stanley," Newcon 1976 program booklet (Boston), n.p. When Lebeck registered for the draft on February 14, 1942, the registrar's report showed his height as six feet and his weight as 182 pounds. National Personnel Records Center, Saint Louis.
3. Letty Lebeck Edes, telephone interview with the author, January 29, 2011.
4. Lebeck's application for a Social Security number, showing Whitman as his employer in Poughkeepsie, was dated March 17, 1938. Oskar Lebeck, application for Social Security number, Social Security Administration. The first issue of Super Comics was dated May 1938; the first issue of Crackajack Funnies, June 1938.
5. "The Story of Whitman," Westerner, January 1966, p. 7.
6. Ron Goulart, "Before Superman: Part II, Popular Comics," Comics Buyer's Guide, June 17, 1983, p. 93.
7. Ron Goulart, "Before Superman: Part III, The Funnies," Comics Buyer's Guide, July 29, 1983, p. 20.
8. Oskar Lebeck to John Coleman Burroughs, May 4, 1939. ERB.
9. Oskar Lebeck to John Coleman Burroughs, July 18, 1939. ERB.
10. Oskar Lebeck to John Coleman Burroughs, August 11, 1939. ERB.
11. Oskar Lebeck to John Coleman Burroughs, October 23, 1939. ERB.
12. Oskar Lebeck to John Coleman Burroughs, May 20, 1940. ERB.
13. Oskar Lebeck to John Coleman Burroughs, October 31, 1940. ERB.
14. U.S. Federal Census, 1940, Cortlandt, Westchester, New York, roll T627_2802, page 10A, enumeration district 60–18.
15. Frank Thomas to "Jerry" (probably Jerry DeFuccio), August 12, 1965. AC.
16. Letty Lebeck Edes, telephone interview with the author, July 11, 2010. Oskar arrived in New York on March 8, 1927, and Ruth on December 19, 1927. Passenger and Crew Lists of Vessels Arriving at New York, New York, 1897–1957. Microfilm publication T715, 8,892 rolls. Records of the Immigration and Naturalization Service, National Archives, Washington, DC. The "certificate and record of marriage" in the New York City municipal archives shows that the Lebecks were indeed married on December 19, 1927—the date of Ruth's arrival from Germany—but officially, at least, the ceremony took place at the New York City Municipal Building.
17. U.S. Federal Census, 1930, Queens, Queens, New York, roll 1610, page 4A, enumeration district 1554, image 265.0.
18. Erwin Knoll, "UF Adds 'Twin Earths' to Space Fiction Ranks," Editor & Publisher, June 7, 1952, p. 48.
19. U.S. Federal Census, 1940; Lebeck to Burroughs, May 4, 1939.
20. Goulart, "Before Superman, Part II," p. 94.
21. "Ray Dirgo Remembers Walt Kelly," in The Best of Pogo, ed. Mrs. Walt Kelly and Bill Crouch Jr. (New York, 1982), p. 71.
22. Walt Kelly, Ten Ever-Lovin' Blue-Eyed Years with Pogo (New York, 1959), p. 5.
23. On Bridgeport when Walt Kelly grew up there, and specifically on the Remington and General Electric plants where his father worked, see Cecelia Bucki, Bridgeport's Socialist New Deal, 1915–36 (Urbana and Chicago, 2001), pp. 21, 38, 101, 108; and "Huge War Plant Turned to Peace," New York Times, March 18, 1920, p. 1. Remington had two factories in Bridgeport. There was an older plant on Barnum Avenue, the Union Metallic Cartridge Company, which Remington continued to operate until the 1980s, in addition to the Boston Avenue rifle factory where Walter Sr. worked and that was built in 1915. The senior Kelly's occupation and the family's address were recorded in the 1920 census (U.S. Federal Census, 1920, Bridgeport Ward 12, Fairfield, Connecticut, roll T625_177, p. 13B, enumeration district 90, image 106), in city directories in the 1920s and 1930s, and in the 1930 census (U.S. Federal Census, 1930, Bridgeport, Fairfield, Connecticut, roll 254, page 23A, enumeration district 86, image 138.0).
24. Kelly, Ten Ever-Lovin' Blue-Eyed Years, p. 6.
25. Gil Kane, "Walt Kelly Interview," Comics Journal, February 1991, p. 58. According to the Journal, Kane—himself a well-regarded comic-book artist—interviewed Kelly "for an audience at the National Cartoonists Society banquet in November 1969, which was no mean feat, since Kelly had apparently been celebrating and was not initially in a cooperative mood."
26. Martin Levin, ed., Five Boyhoods (Garden City, NY, 1962), p. 112. The book is made up of memoirs by five writers who grew up in the first half of the twentieth century. Kelly's memoir represents the 1920s.
27. "Ray Dirgo Remembers Walt Kelly," p. 70. Kelly, in Ten Ever-Lovin' Blue-Eyed Years with Pogo, p. 44, said of the circus-poster lettering for P. T. Bridgeport's dialogue that "one editor wrote me that the speech was hard to read. I could only reply that it was mighty hard to letter, too. In those days I did it myself, having a set of younger eyebones."
28. "The Land of the Elephant Squash: A Biography of Walt Kelly," Walt Kelly Collection, Billy Ireland Cartoon Library and Museum, Ohio State University, Columbus (hereafter cited as OSU, WK), box 5, folder 16, was published in Pogo Even Better, ed. Mrs. Walt Kelly and Bill Crouch Jr. (New York, 1984), pp. 49–50.
29. Levin, Five Boyhoods, p. 112.
30. Ibid., pp. 113–14. Possibly Kelly was referring to unsigned political advertisements, with unflattering caricatures of McLevy's opponents, that appeared in the Post just before the 1931 election.
31. Confirming the dates of Kelly's employment is probably no longer possible, since the Bridgeport city government's archives include no personnel records more than thirty years old. David Dunn, Bridgeport civil service director, email to the author, February 4, 2013.
32. "The Land of the Elephant Squash," p. 49. This was probably what Kelly's Disney personnel record called, under the heading "employment history," work in the "art department of a department store." Early in 1939, Disney employees were asked to complete questionnaires about their education, employment history, hobbies, and athletic pursuits; the results, along with start dates, salary information, and other personal data, were transferred to sheets to which small photographs of the employees were attached. The sheets were apparently updated with supervisors' comments, as in Kelly's case, at least until late in 1941. The purpose of the questionnaires, the Disney manager Paul Hopkins explained, was so that the studio could make the best use of what its employees had to offer: "First, for your personal benefit and secondly, for the studio's progress." Some employees apparently were skeptical; thus Hopkins's stress "on the immediate need of your cooperation in answering the questionnaire AT ONCE, for our mutual benefit." Hopkins to Homer Brightman, memorandum, April 5, 1939. AC.
33. Kelly was listed as a commercial artist in the Bridgeport city directory for 1935, with his address shown as his parents' home on East Avenue.
34. Murray Robinson, "Pogo's Papa," Collier's, March 8, 1952, pp. 20–21. There is a detailed version of that episode, clearly the source of Robinson's version, in an untitled Kelly biography, written by him in the third person, that survived in carbon copies in his papers. Internal references date the biography to early 1952. The same file includes a draft version of the biography, with emendations in Kelly's handwriting. OSU, WK 5 16.
35. Don Maley, "Walt Kelly Muses on His 20 Years of Playing Possum," Editor & Publisher, April 10, 1969, p. 46. Maley has Kelly studying under Booth in 1931, rather than the more likely 1935, but establishing the correct date is probably not possible, since the Phoenix Art Institute's records have long since vanished. The school merged with other institutions twice, becoming first the New York Phoenix School of Design in 1944 and then a part of the Pratt Institute in 1974, as well as changing locations several times, and any records were lost along the way. Paul E. Schlotthauer, librarian and archivist, Pratt Institute, email to the author, February 5, 2013.
36. Jim Vadeboncoeur Jr., on a Web page devoted to Kelly, has pointed out the unmistakable Booth influence on the inside front and back covers of the Dell comic book Fairy Tale Parade no. 5, February–March 1943. www.bpib.com/kelly.htm.
37. Robinson, "Pogo's Papa," p. 54.
38. Kelly's Disney personnel record.
39. Bill Crouch Jr., "Two Mavericks at the Disney Studio," in Phi Beta Pogo, ed. Mrs. Walt Kelly and Bill Crouch Jr. (New York, 1989), p. 131. Helen De Lacy announced her resignation from the Girl Scouts, effective in September 1937, in June of that year. "Girl Scouts Entertained at Los Gatos," Oakland Tribune, June 24, 1937, p. 15.
40. Clair Weeks, interview with Milton Gray, Los Alamitos, CA, May 13, 1978. The interview was recorded for the author as part of the research for his book Hollywood Cartoons: American Animation in Its Golden Age.
41. Ward Kimball, interview with the author, Burbank, CA, June 6, 1969. A studio memorandum noted that Kelly had been transferred to the inbetween department—that is, moved into animation—effective January 4, 1937, but that was probably only a temporary reassignment prompted by Snow White's demands. John Rose to Herb Lamb, memorandum, December 31, 1936. WDA.
42. Thomas Andrae and Geoffrey Blum, "Ward Kimball Remembers Walt Kelly," in Phi Beta Pogo, ed. Mrs. Walt Kelly and Bill Crouch Jr. (New York, 1989), pp. 134–35. The interview was not dated in either of its published appearances, but according to Blum, he and Andrae recorded their interview with Kimball on September 8, 1981.
43. Kimball, interview, June 6, 1969.
44. Kelly registered for the draft in Los Angeles on October 16 (or possibly October 18; the handwriting is not clear), 1940. Selective Service System records, National Personnel Records Center, Saint Louis. His wife, Helen, was the registrar for his local draft board, no. 227, in Los Angeles; she signed the "Registrar's Report" when he signed up.
45. Ward Kimball, interview with the author, San Gabriel, CA, November 7, 1976.
46. Hank Ketcham, interview with the author, Monterey, CA, June 10, 1991. Rebecca Cline, archivist for the Walt Disney Company, provided the dates of Ketcham's employment (October 9, 1939, to November 24, 1941) in an email to the author, November 26, 2012.
47. Kelly's Disney personnel record.
48. Kimball, in preparation for his 1981 interview with Geoffrey Blum and Thomas Andrae, assembled two typewritten pages headed "From Record Book (Kelly)," consisting of entries from his journal in which Kelly was mentioned. The first page only is reproduced at the end of that interview. The entry for Tuesday, May 27, 1941, reads as follows: "Kelly is leaving on the 5:15 train for Bridgeport Conn. 'On vacation' he says. We doubt this. We guess he is leaving Disney's for good to work on a newspaper." The wording but not the substance is different in the relevant entry in the journal itself; Kimball's biographer, Amid Amidi, sent a scan of that hand-printed journal entry to the author on March 6, 2013. It reads as follows: "Kelly leaving on the 5:15 for Bridgeport Conn 'on vacation' I doubt it—I think he's quitting to go back to work on the newspaper!" Ward Kimball papers, courtesy of John Kimball, Virginia Kimball, and Kelly Kimball, Pasadena, CA (hereafter cited as Kimball).
49. Kimball, interview, November 7, 1976.
50. David Hilberman, interview with the author, Palo Alto, CA, October 24, 1976.
51. Kelly's Disney personnel record.
52. Bob Abel, "The Man from Pogo," True, October 1966, p. 74.
53. Kelly to Kimball and Fred Moore, ca. June 14, 1941 (the postmark date on the envelope). Kimball.
54. Levin, Five Boyhoods, p. 111; Abel, "The Man from Pogo," p. 74.
55. Selective Service System records. National Personnel Records Center, St. Louis.
56. Kelly to Kimball and Moore.
57. Kimball wrote in his journal of seeing Kelly at the studio on Friday, July 18, 1941, when Kimball returned at 3:30 P.M. from an extended lunch break: "Kelly there! Not coming back to work. Has to stay in Conn. with sister! Gave me a rough outline of his book that he's submitting to a publisher. About Hitler and Mussolini. Looks good." Kimball's biographer, Amid Amidi, sent a scan of that hand-printed journal entry to the author on February 27, 2013. That entry seems to be the basis of the entry that is misdated Thursday, July 17, 1941, on the second of Kimball's typewritten pages of journal excerpts. That version, altered in other respects, reads as follows: "Kelly surprised us by showing at the Tam O Shanter [sic] restaurant. He says that he is not coming back to work. Says he has to stay in Conn. with sister. Gave us a rough outline of a book that he is submitting to a publisher. It is about Hitler and Mussolini. It looked good." Kimball. Kelly's address on his draft registration card was changed from Los Angeles to his parents' home in Bridgeport as of August 11, 1941, the day Walt Disney departed for South America.
58. According to Ed Ovalle of the Walt Disney Archives, "Walt's desk diary does have 'Walt Kelly' on the morning of July 25, 1941." Ovalle, email to the author, August 21, 2013.
59. Crouch, "Two Mavericks."
60. Walt Disney to Kay Kamen, Leo F. Samuels, and Mike McClintock (three separate letters), August 11, 1941. WDA.
61. Samuels to Walt Disney, August 25, 1941. WDA.
62. Crouch's notes were published as footnotes to a "work in progress" biography by Don Thompson and Maggie Thompson, "Walt Kelly," Okefenokee Star, vol. 1, no. 2, Late Summer 1977, n.p.
63. Kelly to Walt Disney, undated letter date-stamped as received at the studio on November 6, 1941. WDA. Kelly wrote to Disney again on May 25, 1960, a friendly letter—mostly about fund-raising for epilepsy research—that includes this paragraph: "Just in case I ever forgot to thank you, I'd like you to know that I, for one, have long appreciated the sort of training and atmosphere that you set up back there in the thirties. There were drawbacks as there are to everything, but it was an astounding experiment and experience as I look back on it. Certainly it was the only education I ever received and I hope I'm living up to a few of your hopes for other people." OSU, WK, box 3, folder 44.
## CHAPTER 3
1. The book is titled A Story of Our Gang on its cover, A Day With Our Gang on its title page.
2. "Eleanor Lewis Packer, w'19, Directs National Publicity for Famous Movie Stars of Metro-Goldwyn-Mayer Company," Ohio State University Monthly, May 1930, p. 365.
3. U.S. Federal Census, 1930, Beverly Hills, Los Angeles, California, roll 124, page 3A, enumeration district 829, image 481.0.
4. "Two Publicity Women Opening New Agency," Hollywood Reporter, May 2, 1933, p. 4.
5. When Packer applied for a Social Security number, her application was dated December 2, 1936, and showed Western Printing & Lithographing Company as her employer; the company presumably provided the form to its employees, since its name was stamped on it. Eleanor Lewis, application for Social Security number, Social Security Administration.
6. "McCarthy's Boswell," Ohio State University Monthly, December 1939, p. 9.
7. Roger Armstrong, interview with the author, Laguna Beach, CA, June 5, 1969.
8. Ibid.
9. Chase Craig, undated autobiographical notes written at the request of Mark Evanier. Courtesy of Mark Evanier.
10. "McCarthy's Boswell."
11. C. R. Rothmund to Lloyd E. Smith, May 27, 1941. ERB. Rothmund, who as secretary of Edgar Rice Burroughs Inc. had his office in the Los Angeles suburb of Tarzana, mentions having seen Lebeck in California in March 1941.
12. Dell/Nedor Record, p. 171.
13. Chase Craig to the author, July 25, 1978.
14. Armstrong, interview. Craig's comic strip, Odd Bodkins, was syndicated for about a year, from June 1941 to June 1942.
15. "Snappy Stories Has New Owners," Printers' Ink, January 13, 1921, p. 28; "G. T. Delacorte, Jr., Forms Dell Publishing Co.," Printers' Ink, January 13, 1922, p. 189; George T. Delacorte, "Dell Rings the Bell," American News Trade Journal, March 1952, p. 12.
16. The 1900 census has the Tonkonogy family listed on two separate pages, both dated June 7, 1900 (although one is dated "June 7 or 8"). One shows George's mother, Sadie, as head, and the other George Tonkonogy Sr. as head. Neither do the addresses match. The citation for George Tonkonogy as head: U.S Federal Census, 1900, Brooklyn Ward 26, Kings, New York, roll T623_1064, page 14B, enumeration district 453. For Sadie as head: U.S. Federal Census, 1900, Brooklyn Ward 30, Kings, New York, roll T623_1069, page 7B, enumeration district 559. The address shown for Sadie, 135 Osborn Street in Brooklyn, is the address listed for George Tonkonogy in a 1901 Brooklyn city directory.
17. "Our New York Letter," Jewish Exponent, October 4, 1907, p. 10.
18. George Tonkonogy Jr.'s graduation from Public School no. 84 was reported in the Brooklyn Standard Union, June 27, 1906. Online version: <http://bklyn-genealogy-info.stevemorse.org/Graduate/1906/1906.PS84.June.html>.
19. Order in the Matter of an Application of GEORGE TONKONOGY, JR. for Leave to Assume the Name of GEORGE DELACORTE, JR., Monmouth County Common Pleas Court, December 6, 1917. New Jersey State Archives, Trenton. The name change took effect on January 6, 1918. Although Delacorte was routinely identified in the press and elsewhere as "George T. Delacorte, Jr.," and even signed letters that way, the "T." presumably standing for Tonkonogy, his legal name was "George Delacorte, Jr."
20. Dell/Nedor Record, pp. 105–6.
21. Ibid., p. 85.
22. K.K. Publications had published a sort of Donald Duck comic book two years earlier—reprints of comic strips in black and white—but that book, with its formal title page, was more like a Whitman children's book than a true comic book.
23. Except as noted otherwise, David R. Smith, head of the Walt Disney Archives until 2010, provided the figures for the number of copies of Disney comic books on which Western Printing & Lithographing Company paid royalties to Walt Disney Productions.
24. Dell/Nedor Record, p.176.
25. Max Hastings, "Grandma Keeps an Eye on the Books," London Evening Standard, July 28, 1976, p. 15.
26. Dell/Nedor Record, p. 48. New Funnies may have been a special case, too, since it originated as The Funnies, published by Dell before its affiliation with Western.
27. What would have been Red Ryder no. 2 was published instead as Hi-Spot Comics no. 2 before the title reverted to Red Ryder with no. 3.
28. Dell published Gene Autry Comics nos. 11 and 12 in 1944 and then brought out subsequent issues as part of its Four Color series of one-shots until it began publishing Gene Autry Comics on a bimonthly schedule in 1946.
29. John C. Worrell to the author, March 22, 1979.
30. Mary Spillane, "Western Tells Its Stand on Comics: No Crime, Horror or Romance," Racine (WI) Sunday Bulletin, May 23, 1954, p. 2.
31. Howard Anderson to John C. Worrell, memorandum, March 21, 1979. AC.
## CHAPTER 4
1. Roger Armstrong to the author, October 8, 1967.
2. Dell's black-and-white series lasted roughly five years, from 1937 until late in 1942.
3. Armstrong to the author. Armstrong was actually twenty-four years old at the time.
4. Buettner's Federal Schools employment was recorded in the Minneapolis city directory from 1928 to 1930 (he was identified as an instructor in 1930) and the 1930 federal census. The 1931 city directory showed him as an instructor at a company called the Bureau of Engraving. In 1932 he was listed simply as a cartoonist, and he was absent from the directory from 1933 on. According to an online biography by David Saunders (www.pulpartists.com/BC.html), Buettner contributed cartoons to Wilford Fawcett's Captain Billy's Whiz-Bang and eventually worked for Fawcett in New York (where he was living in 1935, according to the 1940 federal census). Buettner's employment in Minneapolis overlapped Carl Barks's employment there by Wilford's former wife, Antoinette "Annette" Fawcett, who owned a competing magazine, the Calgary Eye-Opener, but if the two men ever became aware of their common roots in such magazines when they were both working for Western Printing, Barks never mentioned it.
5. Armstrong to the author.
6. Roger Armstrong, interview with the author, Laguna Beach, CA, June 5, 1969.
7. Armstrong, letter dated October 2006 and published in The Comics ("an original first-person history" by Robin Snyder), October 2007, p. 75. Armstrong wrote: "I knew absolutely nothing of the [New York] office, other than Oskar Lebeck's kindly and helpful letters. I never met him, but remember him with great affection."
8. Armstrong, interview.
9. Roger Armstrong to the author, September 23, 1970.
10. Veve Risto to John Carey, May 16, 1943. Steve Schneider Collection, New York (hereafter cited as SS).
11. "New Tryout Class," Bulletin (Disney in-house newsletter), January 17, 1939, p. 1.
12. Chase Craig, undated autobiographical notes written at the request of Mark Evanier. Courtesy of Mark Evanier.
13. "We Mourn" (a Buettner obituary), Westerner, April 1965, p. 16. The two studios most commonly cited as Buettner's employers are Disney and Harman-Ising, but Disney has no record of Buettner's employment there (Rebecca Cline, Disney archivist, email to the author, March 3, 2011), and there is no mention of Buettner in a list of Harman-Ising employees that survived in Hugh Harman's personal papers. Photocopy. AC. Buettner's only work in animation may have been for Cartoon Films Ltd., a small Hollywood studio that made advertising films; he is identified as one of the cartoonists in a 1941 photo of the staff. Dana Larrabee, "Ed Benedict on Animation—the Facts of Life," Film Collectors' World no. 14, May 1, 1977, p. 33.
## CHAPTER 5
1. Bill Spicer and Vince Davis, "Interview with Dan Noonan," Graphic Story Magazine, Summer 1968, p. 13.
2. Randall W. Scott, Alphabetical List of Comic Book Stories by Gaylord Du Bois (East Lansing, MI, 1986). Scott's Alphabetical List is identified as a working paper rather than a formal bibliography. DuBois's account books are part of the comic art collection at Michigan State University. The dates from the account books are difficult to align with the published stories, but he evidently wrote a number of "Pat Patsy & Pete," "Chester Turtle," and "Ringy Roonga" installments that were published in 1943–44.
3. Roger Armstrong, interview with the author, Laguna Beach, CA, June 5, 1969.
4. Lebeck copyrighted in his own name the comic books he edited (apart from material copyrighted by licensors like Walter Lantz and Johnny Gruelle) until 1948. He then assigned those copyrights to Western Printing, which, as it turned out, renewed none of them. In 1948, Western began copyrighting in its own name the comic books and features that it originated.
5. The fullest account of the campaigns against comic books is David Hajdu, The Ten-Cent Plague: The Great Comic-Book Scare and How It Changed America (New York, 2008). Sterling North's role is discussed on pp. 39–47.
6. William B. Jones Jr., Classics Illustrated: A Cultural History, with Illustrations (Jefferson, NC, 2002), p. 9.
7. Morris Gollub, interview with the author, Hollywood, CA, November 2, 1976.
8. The fullest account of Jameson's life and career is by David Saunders at the Pulp Artists website: www.pulpartists.com/Jameson.html.
9. "London Calling: Kelly Interviewed by the Sunday Times," in Pluperfect Pogo, ed. Mrs. Walt Kelly and Bill Crouch Jr. (New York, 1987), p. 96. The interview, by Henry Brandon, Washington correspondent for the Sunday Times of London, took place in the fall of 1959. Brandon provided a transcript to Kelly, and the published version incorporates Kelly's revisions.
10. Hall Syndicate, "Walt Kelly Biographical Sketch," a mimeographed press release dated "10/5," probably October 1955. AC.
11. Japanese: A Guide to the Spoken Language, no. TM [Technical Manual] 30–341 (Washington, 1943); and Dutch: A Guide to the Spoken Language, no. TM 30–307 (Washington, 1943). That is how those booklets are identified on their title pages, but each is identified as a "language guide" on the front cover. As the Kelly scholar Steve Thompson has pointed out, Kelly put his work on the Japanese guide to use in his "Our Gang" stories by having one of the enemy soldiers speak phonetic Japanese in the episode published in Our Gang Comics no. 12, July–August 1944. Steve Thompson, "Returning to Our Gang," in Walt Kelly's Our Gang, vol. 2 (Seattle, 2007), p. 6.
12. A I. Spangler, The Mechanics of English: A Self-Teaching Course, illustrated by Walter C. Kelly Jr. (Madison, WI, 1944); and Spangler, Building Good Sentences: A Self-Teaching Course, no. EM [Education Manual] 102, illustrated by Walter C. Kelly Jr. (Madison, WI, 1944).
13. Bill Crouch Jr., footnote to Don Thompson and Maggie Thompson, "Walt Kelly," Okefenokee Star vol. 1, no. 2, Late Summer 1977, n.p. Surviving records for the language guides, which were classified as "technical manuals," are at the U.S. National Archives, College Park, in Record Group 407, Records of the Adjutant General's Office, but they deal only with the printing and distribution of the manuals, and not with how they were prepared or how artists like Kelly were hired and assigned to particular manuals.
## CHAPTER 6
1. Walt Kelly, Ten Ever-Lovin' Blue-Eyed Years with Pogo (New York, 1959), p. 9.
2. Disney had begun preliminary work on an Uncle Remus animated feature by June 24, 1939, the date on a sheet of "suggested models" that includes an opossum character. Kelly was still in the Disney story department then, but the sheet is initialed by John Miller and Campbell Grant, members of the Disney "model department," and there is no reason to believe Kelly had anything to do with it.
3. Thomas Andrae and Geoffrey Blum, "Ward Kimball Remembers Walt Kelly," in Phi Beta Pogo, ed. Mrs. Walt Kelly and Bill Crouch Jr. (New York, 1989), p. 140.
4. Harrison Fisher, "Pogo's Pal Kelly," Maclean's, April 15, 1950, p. 52. The Pennsylvania roots of the senior Kelly and his parents were recorded in every U.S. census from 1900 on.
5. Nancy Beiman, "Walt and Selby Kelly," Cartoonist PROfiles, December 1983, p. 27.
6. Kelly, Ten Ever-Lovin' Blue-Eyed Years, p. 6; Martin Levin, ed., Five Boyhoods (Garden City, NY, 1962), p. 109.
7. Bill Crouch Jr., footnote to Don Thompson and Maggie Thompson, "Walt Kelly," Okefenokee Star, vol. 1, no. 2, late summer 1977, n.p.
8. Levin, Five Boyhoods, p. 107.
9. Arthur Frank Wertheim, Radio Comedy (New York, 1979), p. 28.
10. Ibid., p. 48.
11. Cecelia Bucki, Bridgeport's Socialist New Deal, 1915–36 (Urbana and Chicago, 2001), pp. 20, 106.
12. Kelly, Ten Ever-Lovin' Blue-Eyed Years, p. 6.
13. Levin, Five Boyhoods, p. 93.
14. Murray Robinson, "Pogo's Papa," Collier's, March 8, 1952, p. 65.
15. Julian May, "Aesop Takes to the Swamp," Today, November 1951, p. 8.
16. From the untitled Kelly biography, written by him in the third person, that survives in carbon copies in his papers. OSU, WK, box 5, file 16.
17. Joel Chandler Harris, Uncle Remus: His Songs and His Sayings, rev. ed. (New York, 1896; reprint, 1921), p. viii.
18. Doris Willens, "Walt Kelly's 'Pogo' Ribs Stupidities of Mankind," Editor & Publisher, December 11, 1948, p. 42. Willens quotes Kelly as saying, in regard to his admiration for Milne: "I sometimes think we've lost something of the human element in becoming so slick and modern in our living, in our writing."
## CHAPTER 7
1. Dell/Nedor Record, p. 68.
2. Ibid., p. 154.
3. Roger Armstrong, tape-recorded letter to the author, January 18, 1975. Chase Craig, who wrote the early stories with Mary Jane and Sniffles, named the girl after Mary Jane Green, a fellow tenant in his rooming house on Gramercy Place; they later married.
4. Carl Barks remembered that Buettner, as art editor, required him to redraw a buxom girl duck in the "Donald Duck" story for the June 1943 Walt Disney's Comics, a story that Barks's records showed him delivering to Western on January 29, 1943. Barks spent several hours at Western's Beverly Hills office redrawing those pages to reduce the girl's bust line. Gottfried Helnwein, "Conversation with Carl Barks," in Carl Barks: Conversations, ed. Donald Ault (Jackson, MS, 2003), p. 151. Helnwein interviewed Barks on July 11, 1992. An official company history of the Los Angeles office showed Buettner freelancing for Western in 1940, then becoming a part-time art director in 1941 and a full-time art director in 1942. The two earlier dates are difficult to line up with what is known of Buettner's work otherwise. "The 'West' in Western," Westerner, August 1949, p. 7. Buettner's obituary in Western's house organ said: "Carl's first association with Western was on a free lance basis, but in 1942 he became a permanent member of the Western staff, having been employed as art director for Western products created on the West Coast." "We Mourn," Westerner, April 1965, p. 16.
5. Roger Armstrong, interview with the author, Laguna Beach, CA, June 5, 1969.
6. Lynn Karp, interview with the author, Lancaster, CA, September 25, 1990.
7. Armstrong, interview.
8. From a biographical questionnaire Risto completed for Who's Who of American Comic Books, ed. Jerry Bails and Hames Ware (a privately published four-volume set, published between 1973 and 1976). Courtesy of Hames Ware.
9. Roger Armstrong to the author, August 11, 1968. Armstrong wrote the day after having dinner with Carl Barks, whom he saw for the first time in perhaps twenty years. "I did find out one astounding thing from Carl: he told me last night that he was paid exactly the SAME PAGE RATE as all the rest of us . . . .he got only $10.00 per 8 panel page, inked, lettered and delivered to the Beverly Hills office!"
10. Veve Risto to John Carey, December 14, 1942. SS. There is nothing readily identifiable as Risto's work in any of the Fawcett and early Dell issues of Gene Autry Comics, so whatever Risto drew may not have been published.
11. Veve Risto to John Carey, June 2, 1944. SS.
12. Roger Armstrong, tape-recorded letter to the author, May 9, 1975.
13. Risto to Carey, June 2, 1944.
14. Will Friedwald, "An Interview with Jim Davis," Comics Buyer's Guide, November 2, 1984, pp. 40–42
15. Naturalization record, U.S. National Archives and Records Administration, Washington, DC.
16. Royal Indemnity Co. v. Sangor et al., 166 Wis. 148, 164 N.W. 821, October 23, 1917. Pietsch v. Sangor et al., 173 Wisc. 301, 181 N.W. 312, February 8, 1921.
17. The 1920 U.S. Federal Census showed "Benjamine Sangor" and his parents as born in Wisconsin, rather than Russia, and Sangor himself as several years older than he actually was, but those were probably either the enumerator's errors or—at least as likely given his history—Sangor's misrepresentations. U.S. Federal Census, 1920, Chicago Ward 3, Cook (Chicago), Illinois, roll T625_313, page 9A, enumeration district 180, image 811. Jacquelyn Sangor's presence at the academy was recorded as follows: U.S. Federal Census, 1920 Chicago Ward 6, Cook (Chicago) Illinois, roll T625_309, page 13A, enumeration district 299, image 349.
18. "Public Auction Sale of Real Estate," advertisement, Chicago Daily Tribune, November 26, 1922, p. A18. Sangor was listed as a Chicago attorney in Martindale's American Law Directory for 1922 and 1924–28.
19. "Major E. S. Farrow Dies in Street Here," New York Times, September 10, 1926, p. 21. As noted in that article, Sangor's office was at 1457 Broadway in Manhattan, but B. W. Sangor & Co. was incorporated in New Jersey on August 26, 1924.
20. "Jersey Bankers Guilty," New York Times, November 3, 1935, p. 33.
21. Sangor arrived on June 11, 1940, aboard the S.S. Mexico. Passenger and Crew Lists of Vessels Arriving at New York, New York, 1897–1957 (National Archives Microfilm Publication T715, 8892 rolls); Records of the Immigration and Naturalization Service, National Archives, Washington, DC.
22. Dell/Nedor Record, p. 163. Cinema Comics Inc. certificate of incorporation filed September 15, 1939. New York State Department of State, Division of Corporations.
23. The passenger list for the S.S. Queen of Bermuda, which sailed from Hamilton, Bermuda, to New York on September 10, 1938, arriving on September 12, includes the names of both Ned and Jacqueline [sic] Pines, but a line has been drawn through their names, presumably indicating a change of plans, although there appears to be no record that the Pineses departed from Hamilton on another ship. Passenger and Crew Lists of Vessels Arriving at New York, New York, 1897–1957 (National Archives Microfilm Publication T715, 8892 rolls); Records of the Immigration and Naturalization Service, National Archives, Washington, DC.
24. Dell/Nedor Record, p. 164.
25. Gordon Sheehan, interview with the author, Evanston, IL, April 20, 1973. Jim Davis remembered Sangor rather differently, as "a short man . . . way overweight, and he smoked big, black cigars." Michael Vance, "Forbidden Adventures: The History of the American Comics Group," Alter Ego no. 61, August 2006, p. 15.
26. The title on the cover of no. 65 is The New Funnies, but the official title in the indicia is still The Funnies. Cover title and indicia title were again aligned in no. 66, as New Funnies.
27. Dell/Nedor Record, p. 7.
28. Ibid., p. 218.
29. Sangor incorporated as Creston Publications Corporation in August 1942. His name is nowhere on the certificate of incorporation for either Creston Publications or Cinema Comics, perhaps because of his criminal record, but he was listed with Creston as the owner of, for example, Giggle Comics in the statement of ownership required by the post office in the February 1947 and April 1953 issues. His estate owned all the stock in both companies when they were dissolved after his death.
30. Dave Bennett, "An Interview with Jack Bradbury," Ace Comics Presents no. 2, July 1987, p. 6.
31. Friedwald, "An Interview with Jim Davis," p. 42.
32. Briefly in 1945–46, many of the Dell comic books shrank even further, to thirty-two pages, plus covers. The stories had about as many panels as before, but the panels were smaller, and as many as a dozen were crowded onto a page. The effect was claustrophobic.
33. Vance, "Forbidden Adventures," p. 14.
34. Friedwald, "An Interview with Jim Davis," p. 42.
35. Ibid., p. 44.
36. Lloyd Turner, interview with the author, Shady Cove, OR, May 13, 1989.
37. Jack Bradbury, interview with Milton Gray, Irvine, CA, March 23, 1977. The interview was recorded for the author as part of the research for his book Hollywood Cartoons: American Animation in Its Golden Age.
38. Friedwald, "An Interview with Jim Davis," p. 44.
## CHAPTER 8
1. Bruce Hamilton, "A Tripp Down Memory Lane," in The Little Lulu Library, ed. John Clark, set 6, vol. 16 (Scottsdale, AZ, 1985), p. 13.
2. Ibid.
3. "The Little Field Mouse," in Our Gang Comics no. 3, January–February 1943, appears to be the product of the same sort of tracing.
4. Dell began publishing one-shot color comic books in late 1939, two years after it launched its series of black-and-white one-shots. After a couple of years, Dell began numbering each color one-shot as a "Four Color Comic" on its cover. The series went through no. 25, in mid-1942, then for some reason resumed with no. 1. The "Four Color Comic" designation was dropped after the first hundred issues of the second series, but the numbering of the series continued for twenty years.
5. Carl Barks to the author, May 17, 1981.
6. When Lebeck registered for the draft on February 14, 1942, he listed his residence address in Croton-on-Hudson, but in the separate line for his mailing address he listed the Knickerbocker Hotel in Los Angeles—an address that was then crossed out, probably because he or someone at his draft board realized that it was a permanent mailing address that was wanted.
7. The interviews with Barks were conducted by Donald Ault and Thomas Andrae on August 4, 1975 (a videotaped interview), and by Bruce Hamilton (from questions by Hamilton, Geoffrey Blum, and Andrae) on September 24, 1983. Blum provided the author with the relevant pages from his transcripts of both interviews. A composite account "by Carl Barks" that includes material from the 1983 interview and is titled "Sailing for Pirate Gold" was published in The Carl Barks Library, ed. Bruce Hamilton, set 1, vol. 1 (Scottsdale, AZ, 1984), p. 131.
8. Jack Hannah, interview with the author, Glendale, CA, November 3, 1976.
9. Barks, interview with Hamilton, September 24, 1983.
10. Geoffrey Blum, "It Started with Pluto," in Carl Barks Collection, ed. Geoffrey Blum, vol. 1, p. 34. Carl Barks Collection, a thirty-volume set reprinting all of Barks's Disney stories in color, with extensive commentary by Blum, was translated into the languages of five European countries (Denmark, Sweden, Norway, Germany, and Finland) and published in those languages between 2005 and 2009, but it has not yet been published in English. Blum made available to the author the English originals of his essays for the set.
11. Transcript of story meeting, "Pirate Story, Mickey Feature #11," March 25, 1941. Photocopy originating with Homer Brightman, one of the four Disney writers—the others were Harry Reeves, Roy Williams, and Gilles de Tremaudan—who attended the meeting with Disney. Barks was not present. AC.
12. Hannah, interview.
13. Carl Barks to Hal Adelquist, November 9, 1942. WDA.
14. The date on the title deed is mentioned in endnote 27 on page 32 of volume 1 of Carl Barks Collection.
15. Carl Barks to the author, April 17, 1996. He mentioned his operation in an interview with the author in Goleta, CA, on November 22, 1973.
16. Barks, interview with Hamilton.
17. Barks, interview with the author.
18. Carl Barks to the author, April 17, 1967.
19. Barks, interview with the author.
20. Barks to the author, April 17, 1967.
21. Dorothy Strebe to Carl Barks, n.d. (ca. December 1942). WDA. Strebe likely wrote that script. She wrote stories for other Western Printing publications in the 1940s, including the first "Li'l Bad Wolf" story in Walt Disney's Comics & Stories no. 52, January 1945. That story caught the eye of the Disney executive Hal Adelquist, who wrote to Walt Disney on December 13, 1944, to suggest that it could be the basis for an animated cartoon: "The attached story, 'The Li'l Bad Wolf,' seems to me to be a good possibility for a short subject. The twist on the little wolf who doesn't want to be bad and prefers being a vegetarian seems to suggest swell possibilities. In checking to find out who did the original story, I find that it was done by Dorothy Strebe, edited by Eleanor Packard [sic], and the art work done by Carl Betner [sic]." Hal Adelquist to Walt Disney, memorandum, December 13, 1944. WDA. No such cartoon was ever made.
22. Carl Barks, interview with the author, Atlanta, October 5–6, 1974.
23. Carl Barks, interview with the author, Goleta, CA, May 30, 1971.
24. Barks, interview with the author, November 22, 1973.
25. Geoffrey Blum, "Imagining Egypt," in Carl Barks Collection, ed. Geoffrey Blum, vol. 1, pp. 223–34. For a comprehensive examination of Barks's use of National Geographic for the settings of his stories, see Thomas Andrae and Geoffrey Blum, "The Far Away and Long Ago" in The Carl Barks Library, ed. Bruce Hamilton, set 1, vol. 1 (Scottsdale, AZ, 1986), pp. 229–52.
26. Barks, interview with the author, May 30, 1971.
27. Martin L. Greim, "Crusader Comments," Comics Buyer's Guide, December 17, 1976, p. 16.
## CHAPTER 9
1. Jim Korkis, "An Animated Life: The Story of Jack Hannah," Persistence of Vision no. 8, 1996, p. 32.
2. Malcolm Willits, Don Thompson, and Maggie Thompson, "The Duck Man," in Carl Barks: Conversations, ed. Donald Ault, p. 7. This article is based on an interview Willits recorded with Barks in December 1962 and subsequently edited and expanded for publication in the Thompsons' mimeographed fan magazine Comic Art in 1968.
3. Geoffrey Blum, "Animating Scrooge," in Carl Barks Collection, ed. Geoffrey Blum, vol. 13, p. 83.
4. Carl Barks, interview with the author, Goleta, CA, November 22, 1973.
5. Ibid.
6. Carl Barks, interview with Patrick Garabedian, Goleta, CA, October 30, 1971. Garabedian made the tape recording available to the author.
7. Carl Barks, interview with the author, Goleta, CA, May 30, 1971.
8. Roger Armstrong to the author, July 16, 1967.
9. Roger Armstrong, interview with the author, Laguna Beach, CA, June 5, 1969.
10. Roger Armstrong to the author, August 18, 1967.
11. Carl Barks, interview with the author, Temecula, CA, August 13, 1978.
12. Armstrong, interview.
13. Chase Craig to the author, July 25, 1978.
14. Barks, interview, August 13, 1978.
15. For Chase Craig: undated autobiographical notes written at the request of Mark Evanier. Courtesy of Mark Evanier. For Armstrong: U.S. World War II Enlistment Records, 1938–1945. National Archives and Records Administration, Record Group 64, National Archives at College Park, College Park, MD.
16. Armstrong, interview.
17. Ibid.
18. Carl Barks, Walt Disney's Uncle Scrooge McDuck: His Life and Times (Millbrae, CA, 1981), p. 365. Barks's comments, based on interviews with Edward Summer, accompany recolored versions of a dozen stories from Uncle Scrooge.
19. Barks, interview, October 30, 1971.
## CHAPTER 10
1. Irvin H. Ziemann, "Gaylord DuBois, King of the Comics Writers," Comics Buyer's Guide, October 6, 1989, p. 64.
2. According to Colin D. Riley, Boston University's executive director of media relations: "There is a record of Gaylord DuBois attending BU's College of Liberal Arts . . . from 1925 to 1927. Unfortunately our records do not indicate that he received a degree." Riley, email to the author, February 21, 2012.
3. "A Titan in the World of Books," Racine (WI) Journal-Times, December 16, 1971, p. 10A.
4. Mary Robison, reference librarian, General Theological Seminary, New York, email to the author, January 19, 2012.
5. Lou Mougin, "Gaylord DuBois," Comics Interview no. 17, November 1984, p. 10.
6. Ziemann, "Gaylord DuBois," p. 65.
7. The Burroughs company, through its secretary, C. R. Rothmund, initiated what became a long relationship with Whitman by writing a short, stiff letter to the publisher on December 7, 1933, advising Whitman to deal with the Burroughs company directly on "all negotiations of any nature for the Edgar Rice Burroughs books, or books concerning any of the characters originated by Mr. Burroughs." ERB.
8. St. Clair McKelway, "Onward and Upward with the Arts: The Literary Character in Business & Commerce," New Yorker, October 26, 1935, p. 90. The January 6 date was on the website of Stephen Slesinger Inc.: www.stephenslesinger.com.
9. S. E. Lowe to C. R. Rothmund, July 8, 1935. ERB.
10. The Big Little Books' credited author was Leon Morgan, who was also credited as the author of several other Big Little Books in the mid-1930s.
11. Fred Harman, "New Tracks in Old Trails," True West, October 1968, pp. 59–60; Mario DeMarco, "Fred Harman, Master Draftsman of the Western Figure," Comics Buyer's Guide, April 17, 1987, p. 114.
12. Siegel v. Warner Bros. Entertainment, Inc., 658 F. Supp. 2d 1047.
13. Andy Rooney, Out of My Mind (New York, 2006), p. 288.
14. Ziemann, "Gaylord DuBois," p. 65.
15. Camille Cazedessus Jr., "Gaylord DuBois, Veteran Tarzan Script Writer," ERBdom, October 1962, p. 7.
16. Ibid., p. 9. The distinction DuBois described between "Art Editor" and "Script Editor" reflected the division of responsibilities in Western's Los Angeles office in the 1950s and 1960s. That office was then handling the Tarzan comic book that DuBois wrote for many years.
17. Randall W. Scott, Alphabetical List of Comic Book Stories by Gaylord Du Bois (East Lansing, MI, 1986).
18. "George F. Kerr Dies; Illustrator Was 84," New York Times, October 23, 1953, p. 23. In fact, Kerr was probably eighty-three at the time of his death. The fullest account of Kerr's career, by David Saunders, is at the Pulp Artists website: www.pulpartists.com/Kerr.html. Kerr's career as an illustrator ended a few years before his death when he lost his eyesight, which had been failing for years.
19. Gil Kane, "Walt Kelly Interview," Comics Journal, February 1991, p. 53. Goldberg was a member of the audience for the interview.
20. Cazedessus, "Gaylord DuBois," p. 7.
## CHAPTER 11
1. U.S. Federal Census, 1920, Manhattan Assembly District 13, New York, New York, roll T625_1209, page 5B, enumeration district 972, image 812; U.S. Federal Census, 1930 Bronx, Bronx, New York, roll 1490, page 3B, enumeration district 694, image 240.0.
2. Jim Amash, "Quality Control: A Conversation with Gill Fox," Alter Ego no. 12, January 2002, pp. 5–6.
3. "Pupils Get Awards in Art Tomorrow," New York Times, June 26, 1932, p. 28.
4. Donald Phelps, "John Stanley," Newcon 1976 program booklet (Boston), n.p.
5. "Interview: Carl Barks and John Stanley," Comics Journal, February 2003, p. 159. The "interview" was a panel discussion moderated by Bruce Hamilton, with questions from the audience. The quotations from Barks and Stanley in this transcript, as edited by Milo George, differ in immaterial details from the quotations published by Martin L. Greim in his column "Crusader Comments," Comics Buyer's Guide, December 17, 1976, p. 16.
6. Fleischer's Animated News 1, no. 9, August 1935, n.p.
7. Ibid., no. 6, May 1935, n.p. Stanley was also credited for a caricature of Dave Fleischer, the studio's co-owner, in the first issue of the Animated News, dated December 1934.
8. According to Phelps, one of those covers "may have been the very first large-size Mickey Mouse Magazine cover!" (Phelps, "John Stanley") but that is unlikely, since that first cover was published in May 1935, when Stanley was still on the Fleischer staff. It is possible, though, that Stanley drew the cover of the October 1935 issue—the second issue and the first published on a monthly schedule.
9. "Interview: Carl Barks and John Stanley," p. 162.
10. Phelps, "John Stanley." Stanley was on Kamen's staff when he applied for his Social Security account number on November 26, 1936. John Patrick Stanley, application for Social Security number, Social Security Administration.
11. "Interview: Carl Barks and John Stanley," p. 159.
12. A scan of Stanley's Art Students League registration card was provided to the author by Stephanie Cassidy, editor for the League. Cassidy, email to the author, March 26, 2012.
13. Phelps, "John Stanley." The reference is to lithography in Phelps's article rather than etching, perhaps as the result of a lapse of memory on Stanley's part.
14. U.S. Federal Census, 1940 New York, Bronx, New York, roll T627_2493, page 2A, enumeration district 3–1305.
15. Dorothy Krumeich, "Stanley Comics Help Quell Furor," Peekskill (NY) Evening Star, August 11, 1965.
16. "Interview: Carl Barks and John Stanley," p. 159.
17. Bill Spicer sent a copy of Stanley's letter to the author together with a letter dated March 29, 1971.
18. Martin L. Greim and Bob Cosgrove, "Crusader Comments," Comics Buyer's Guide, March 11, 1977, p. 17.
19. Stanley registered for the draft in the Bronx on October 16, 1940, probably the same day that Walt Kelly registered in Los Angeles. Selective Service System records, National Personnel Records Center, St. Louis.
20. Bill Spicer and Vince Davis, "Interview with Dan Noonan," Graphic Story Magazine, Summer 1968, p. 13.
21. Phelps, "John Stanley."
22. "Interview: Carl Barks and John Stanley," p. 159.
23. Dan Noonan, interview with Milton Gray, Pasadena, CA, December 12, 1977. The interview was recorded for the author as part of the research for his book Hollywood Cartoons: American Animation in Its Golden Age.
24. "Interview: Carl Barks and John Stanley," p. 159.
25. Phelps, "John Stanley."
26. Robert Ingersoll, "Comic Star Is Real Lulu," Philadelphia Inquirer, March 20, 1966, p. NW5.
27. Contract dated March 15, 1944, between Marjorie Henderson Buell and Western Printing & Lithographing Company Inc. Marge [Marjorie Henderson Buell] Papers, 1856–1994, Schlesinger Library, Radcliffe Institute, Harvard University, Cambridge, MA (hereafter cited as Marge).
28. Contract dated October 23, 1944, between Marjorie Henderson Buell and Western Printing & Lithographing Company Inc. Marge.
29. "News and Notes of the Advertising Field," New York Times, September 29, 1938, p. 44; "Lulu of a Visit," Westerner, January 1954, p. 13.
30. Gordon Sheehan, interview with the author, Evanston, IL, April 20, 1973.
31. Bruce Hamilton, "A Tripp Down Memory Lane," in The Little Lulu Library, ed. John Clark, set 6, vol. 16 (Scottsdale, AZ, 1985), p.14.
32. William C. Erskine to Marjorie Henderson Buell, May 14, 1946. Marge.
33. Maggie Thompson, "The Almost-Anonymous Mr. Stanley," Funnyworld no. 16, Winter 1974–75, p. 34.
34. Hamilton, "A Tripp Down Memory Lane," p. 16. Scripts of this kind were common in the comic-book industry by the mid-1940s and thereafter, as with, for example, the scripts for the Sangor comic books, many of whose authors were accustomed to drawing real storyboards for animated cartoons.
35. There was a single exception: Stanley was among those credited—in his case, for the front cover specifically, as well as a more general credit—on the inside front cover of the July 1952 issue. Among the others receiving general credit were Irving Tripp, Al Owens (who lettered the dialogue), and Gordon Rose (who drew backgrounds).
36. Thompson, "The Almost-Anonymous Mr. Stanley," p. 34.
37. Hamilton, "A Tripp Down Memory Lane," p. 16.
38. Ibid., p. 17.
39. Phelps, "John Stanley."
40. John Stanley to Bruce Hamilton, September 12, 1985. Marge.
## CHAPTER 12
1. Carl Barks to the author, May 8, 1975.
2. Carl Barks to Dick Blackburn, March 8, 1968. Courtesy of Dick Blackburn.
3. Carl Barks, interview with the author, Goleta, CA, November 22, 1973.
4. That is the wedding date and place in Arminta's obituary in the Klamath Falls (OR) Evening Herald for November 13, 1916.
5. Stan Turner, The Years of Harvest (Eugene, OR, 2002), pp. 115–16.
6. "Merrill's Beginnings," Lost River (OR) Star, Special Edition, July 1995, p. 5.
7. "Carl Barks, the Early Years," Lost River (OR) Star, Special Edition, July 1995, p. 10.
8. Barks, interview.
9. Carl Barks in conversation with the author, Grants Pass, OR, August 4–5, 1988. Or maybe not. Barks also said, in a letter published in 1995, that he had seen his first movie in Merrill around 1908. "Carl Barks, the Early Years," p. 16.
10. Barks, interview.
11. Barks, interview.
12. Ibid.
13. ibid.
14. Donald Phelps, "Carl Barks," 1976 Newcon program book (Boston), n.p.
15. Carl Barks, interview with the author, Temecula, CA, August 13, 1978.
16. Carl Barks to the author, March 17, 1984. Barks was responding to a question posed to the author by another cartoonist, Alex Toth, in a postcard dated February 5, 1984: "In all your exchanges with him, was no opinion given re the rollicking good fun, pictorialism, drama and economy of Roy's 'Tubbs/Easy' years?"
17. Carl Barks to the author, August 1969.
18. Marge Burleigh, "Cartoonist Merrill Native," Klamath Falls (OR) Herald and News, March 13, 1983, p. 4. At the time this article—probably the only published interview with Barks's older brother, Clyde—appeared, its author, Marge Burleigh, and her husband, Bob, were living in the old Barks family home in Merrill. Thanks to Gunnar Andreassen for a copy of this article.
19. Barks, interview, November 22, 1973.
20. Donald Ault and Lynda Ault, "Carl Barks Remembers 'A Perfect Life,'" in Carl Barks: Conversations, ed. Donald Ault (Jackson, MS, 2003), p. 184. The Aults interviewed Barks on June 13–14, 1997, when he was ninety-six years old.
21. Barks, interview, November 22, 1973.
22. Barks registered for the draft in Los Angeles on February 14, 1942. Selective Service System records, National Personnel Records Center, St. Louis. Barks's Disney employment record, undated but probably from 1939, shows him as a half inch shorter and ten pounds lighter than his draft registration.
23. Barks, interview, November 22, 1973.
24. Carl Barks to the author, July 24, 1980.
25. Barks, interview, November 22, 1973.
26. U.S. Federal Census, 1930, Merrill, Klamath, Oregon, roll 1945, page 1B, enumeration district 44, image 608.0.
27. Barks, interview, November 22, 1973.
28. Ibid.
## CHAPTER 13
1. "Animated Annette," Time, July 4, 1932, p. 21.
2. Carl Barks, interview with the author, Goleta, CA, November 22, 1973. Philip E. Rolfsen, shown as editor in the 1930 and 1931 Minneapolis city directories, was listed again as editor in the 1936 edition. Apparently he persuaded Antoinette "Annette" Fawcett, to whom Henry Meyers had sold the Eye-Opener in 1932 (and who was divorced from Wilford Fawcett) to give him his old job back after Barks left. Rolfsen was a very young man (born in 1908) when he first became editor.
3. Geoffrey Blum, The Unexpurgated Carl Barks (Prescott, AZ, 1997), reproduces dozens of examples (with color added) of Barks's cartoons, mostly unsigned, for the Eye-Opener.
4. Sumner was listed as the Eye-Opener's editor only in the 1932 edition of the Minneapolis city directory.
5. Barks, interview.
6. Ibid.
7. Carl Barks to John Coulthard, October 8, 1961. WDA.
8. Barks, interview.
9. Ibid.
10. Ibid.
11. "Decrees Granted," Reno Evening Gazette, September 10, 1937, p. 16.
12. The Los Angeles city directory for 1938 showed Clara as Barks's spouse; they resided at 4016 Effie Street in Los Angeles. Barks was erroneously identified as "Paul" Barks.
13. Barks, interview.
14. Donald Ault and Lynda Ault, "Carl Barks Remembers 'A Perfect Life,' " in Carl Barks: Conversations, ed. Donald Ault (Jackson, MS, 2003), p. 192.
15. Donald Ault, "Ideas Flowing Like Waterfalls: Some Reflections from Carl Barks at 98," in Carl Barks: Conversations, ed. Donald Ault (Jackson, MS, 2004), p. 219. "Ideas Flowing Like Waterfalls" is based on multiple sources (telephone conversations, faxed correspondence) from 1999.
16. Barks, interview.
17. Ibid.
18. Jack Hannah, interview with the author, Glendale, CA, November 3, 1976.
19. Carl Barks to the author, January 25, 1990.
20. Carl Barks, interview with the author, Goleta, CA, May 30, 1971.
21. Barks, interview, November 22, 1973.
22. Barks, interview, May 30, 1971.
23. Carl Barks to the author, May 19, 1977.
24. Barks, interview, November 22, 1973.
25. Carl Barks to the author, May 8, 1975.
26. Carl Barks, interview with Patrick Garabedian, Goleta, CA, October 30, 1971. This is the rare instance when Barks's statements in different interviews are at least superficially inconsistent. In the June 1997 interview with Donald Ault and Lynda Ault, he spoke of this self-directed education as occurring much earlier, years before he worked at Disney:
Well, I got to digging into that stuff in the 1920s, and I realized if I was going to write jokes and things I needed to be able to write intelligible sentences. So I began studying, got my old school books out and studied those things. So, by the time I got a job back on the Eye-Opener, when I was in my thirties and became editor of the darn thing, I knew enough about sentence structure that I was just as well off as the guys in there with college educations, as far as that little simple bunch of writing was concerned. . . . I would take about an hour or two every evening and look at the old grammar books and see how they phrased a sentence, and I would do it and take a line out of a newspaper or a letter and analyze it and finally I got so I could just look at a sentence and realize which words were the subject words and which were the predicates, which were back and forth. (Ault and Ault, "Carl Barks Remembers," p. 190)
Barks spoke in a 1991 interview of trying to write short stories in the 1920s—this would have been when he was trying to break in as a magazine cartoonist—"but I hadn't enough education to write English fluently. I was always stuck on sentence construction, just like building a fence in front of me. . . . When I got the opportunity to draw comic books, all those things were taken care of for me. I only had to write simple little dialogue balloons." Geoffrey Blum, "A Conversation with Carl Barks," The Carl Barks Library of Walt Disney's Comics and Stories in Color, no. 2 (Prescott, AZ, n.d. [ca. 1992]), n.p.
27. Barks, interview, November 22, 1973.
28. Carl Barks, interview with the author, Atlanta, October 5–6, 1974.
29. John Benson, "Editor's Note," Panels, no. 2, Spring 1981, p. 2.
30. Transcript of story meeting, "Pirate Story, Mickey Feature #11," March 25, 1941. Photocopy originating with Homer Brightman. AC.
## CHAPTER 14
1. Roger Armstrong to the author, January 5, 1968.
2. Lloyd E. Smith to C. R. Rothmund, April 1, 1947. ERB.
3. Armstrong to the author, January 5, 1968.
4. "Elected Vice President of Western Printing Co.," New York Times, December 24, 1955, p. 22.
5. Chase Craig to the author, July 25, 1978.
6. Nielsen was identified as a proofreader in Poughkeepsie city directories as late as 1946.
7. "The 'West' in Western," Westerner, August 1949, p. 4. Nielsen's 1932 start date was reported in "Beverly Hills Twenty Year Club," Westerner, January 1961, p. 24.
8. Roger Armstrong to the author, August 18, 1967.
9. Roger Armstrong to the author, August 16, 1967.
10. Carl Barks, interview with the author, Goleta, CA, November 22, 1973.
11. Carl Barks to the author, December 18, 1966.
12. Bruce Hamilton, "The Mouse Man and the Duck Man: An Interview with Floyd Gottfredson and Carl Barks," in Walt Disney's Mickey Mouse in Color, deluxe ed. (Prescott, AZ, 1988), p. 101 The joint interview took place on December 5, 1982, in Pasadena, CA.
13. Howard Anderson to John C. Worrell, memorandum, March 21, 1979. AC.
14. Harvey Thompson, "The Mail Goes Out," Westerner, February 1951, p. 19. A few years later, the total number of subscription copies was pegged at 787,000 a month, an evident decline of more than 100,000 copies. "Shipping with a Capital W," Westerner, February 1954, p. 2.
15. Malcolm Willits, Don Thompson, and Maggie Thompson, "The Duck Man," in Carl Barks: Conversations, ed. Donald Ault (Jackson, MS, 2003), p. 5.
16. Carl Barks, interview with the author, Atlanta, October 5–6, 1974.
17. Paul Ciotti, "Writing to Please Myself: An Interview with Carl Barks," in Carl Barks: Conversations, ed. Donald Ault (Jackson, MS, 2003), p. 33. The interview was recorded in September 1972.
18. Barks said in 1974 that twenty-six pages a month of story and art was "as much as I could physically do and do it comfortably." This figure is generally consistent with the number of pages his records showed him submitting annually. Barks, interview, October 5–6, 1974.
19. Carl Barks, interview with the author, Temecula, CA, August 13, 1978.
20. Barks, interview, November 22, 1973.
21. Carl Barks, interview with the author, Goleta, CA, May 30, 1971.
22. Geoffrey Blum, "Gracious Living in Plain Awful," in Carl Barks Collection, ed. Geoffrey Blum, vol. 6, pp. 155–60.
23. Barks, interview, May 30, 1971.
24. That model sheet is reproduced in Michael Barrier, Carl Barks and the Art of the Comic Book (New York, 1981), p. 43.
## CHAPTER 15
1. The artwork for this story survived for years in Barks's hands before all but a half page of its ten pages passed into a private collection. The top half of the first page had disappeared many years before, probably given away by Barks. He reworked the story, eliminating its Christmas aspects, as "The Terrible Tourist," a ten-page "Donald Duck" story for Walt Disney's Comics & Stories no. 248, May 1961.
2. Carl Barks to John Verpoorten, March 22, 1961. WDA.
3. Carl Barks to the author, December 18, 1966.
4. Carl Barks, interview with Patrick Garabedian, Goleta, CA, October 30, 1971.
5. Carl Barks to the author, June 9, 1966.
6. Carl Barks, interview with the author, Atlanta, October 5–6, 1974.
7. Michel de Montaigne, Essays, book 3, chap. 2, "Of Repentance," in The Complete Works: Essays, Travel Journal, Letters, trans. Donald Frame (New York, 2003), p. 740.
8. Barks, interview, October 5–6, 1974.
9. Tom Gill with Tim Lasiuta, The Misadventures of a Roving Cartoonist: The Lone Ranger's Secret Sidekick (Chandler, AZ, 2008), p. 39.
10. Barks to the author, June 9, 1966.
11. Malcolm Willits, Don Thompson, and Maggie Thompson, "The Duck Man," in Carl Barks: Conversations, ed. Donald Ault (Jackson, MS, 2003), p. 13.
12. Barks, interview, October 5–6, 1974.
13. "Hair-Raising Episode," Westerner, February 1950, p. 10
14. Tom McKimson to Carl Barks, August 4, 1950. WDA.
15. Alice Cobb to Carl Barks, August 21, 1950. WDA.
16. Donald Ault, "Ideas Flowing Like Waterfalls: Some Reflections from Carl Barks at 98," in Carl Barks: Conversations, ed. Donald Ault (Jackson, MS, 2003), p. 216.
## CHAPTER 16
1. Selective Service System records, National Personnel Records Center, St. Louis.
2. Bill Spicer and Vince Davis, "Interview with Dan Noonan," Graphic Story Magazine, Summer 1968, p. 13.
3. Dan Noonan, interview with Milton Gray, Pasadena, CA, December 12, 1977. This interview was recorded for the author as part of the research for his book Hollywood Cartoons: American Animation in Its Golden Age.
4. Spicer and Davis, "Interview with Dan Noonan," p. 13. Geraghty became New Yorker's art editor in 1939 and held the job for thirty-four years.
5. Lee Spilberg, Manuscripts and Archives Division, New York Public Library, email to the author, January 20, 2011.
6. Morris Gollub, interview with the author, Hollywood, CA, November 2, 1976.
7. Martin L. Greim, "Crusader Comments," Comics Buyer's Guide, December 17, 1976, p. 16.
8. Marjorie Henderson Buell, notes accompanying a letter to John Clark, August 5, 1984. Marge. Clark, as an editor for Another Rainbow Publishing, was involved in the preparation of the multivolume hardcover reprint collection called The Little Lulu Library and had sent Buell questions whose answers could be used in preparing text features for the books. "Life With Lulu: An Interview with Marge," by John Clark and Bruce Hamilton, was published in The Little Lulu Library, ed. John Clark, set 1, vol. 1 (Scottsdale, AZ, 1992), pp. 13–16, and incorporated material similar to that in Buell's notes.
9. John Stanley to Bruce Hamilton, September 12, 1985. Marge.
10. For example, the cover of the February 1953 Little Lulu shows Tubby, Alvin, and Gloria crowding onto a scale as Lulu divides the total poundage the scale shows by three. The same idea, with four children instead of three, is present in one of the Post cartoons reprinted in Little Lulu and Her Pals (Philadelphia, 1939). Another cartoon in the same book, in which Lulu uses a vacuum cleaner to clear a snowy sidewalk, probably inspired the cover of Little Lulu no. 1, January–February 1948.
11. Charlotte Astor, "Tubby, Today," Florida Accent (Sunday magazine of the Tampa Tribune), September 6, 1970, p. 13.
12. Only Bernard Krigstein, in a celebrated story called "Master Race" in the first issue of Impact (1955), one of the ill-fated EC "New Direction" titles, was able to shed completely the weights attached by Feldstein's methods. Although Feldstein's captions came to him already lettered on sheets of illustration board, with the panels drawn in, Krigstein cut those sheets apart and expanded the number of pages and the number of panels.
13. Marjorie Henderson Buell to John Clark, draft, October 30, 1985. Marge.
## CHAPTER 17
1. Dan Noonan, interview with Milton Gray, Pasadena, CA, December 12, 1977.
2. Morris Gollub, interview with the author, Hollywood, CA, November 2, 1976.
3. Rebecca Cline, archivist for the Walt Disney Company, provided the dates of Gollub's employment in an email, November 26, 2012. Disney has a record only of Dan Noonan's brief employment in the 1960s, not of his earlier employment.
4. Gollub, interview. Gollub entered the navy on January 17, 1942. Official Military Personnel File for Morris Gollub, National Personnel Records Center, St. Louis.
5. Gollub, interview.
6. Official Military Personnel File for Morris Gollub.
7. Gollub, interview.
8. In 1950, more than two years after the demise of Animal Comics and "Rover," Moe Gollub illustrated a strikingly similar serial about a dog, written by Gaylord DuBois. In keeping with Western's growing use of licensed characters, the dog was Lassie, the canine MGM star. MGM's Lassie, first published as two one-shots, was published quarterly and then bimonthly well into the 1950s, when it was superseded by a comic-book version of the Lassie television series.
9. In a strange break with Western's usual practice of not crediting its artists and writers, full credits were published on the inside front covers of New Funnies for June and July 1952 as well as Little Lulu for July 1952. Both titles were then edited in New York. Irene Little was credited with inking Richard Hall's pencil drawings for the "Woody Woodpecker" and "Andy Panda" stories in both issues of New Funnies.
10. Gollub, interview.
11. Richard Hall, interview with the author, Alexandria, VA, September 8, 1978.
12. Irvin H. Ziemann, "Gaylord DuBois: Chapter One, Concluded," Comics Buyer's Guide, October 13, 1989, p. 62.
13. Ibid.
14. Ed Ovalle of the Walt Disney Archives, email to the author, August 21, 2013. Marsh's military records were among those destroyed in a 1973 fire at the National Personnel Records Center in St. Louis, but he was most likely discharged early for medical reasons, since he suffered from diabetes.
15. Jesse Marsh to Ronald J. Goulart, October 18, 1948. Courtesy of Robert R. Barrett.
## CHAPTER 18
1. Dan Noonan, interview with Milton Gray, Pasadena, CA, December 12, 1977. Manhattan telephone directories from the 1940s show a Fifth Avenue Restaurant at the 200 Fifth Avenue address, but no establishments corresponding to the names that Noonan remembered. He also referred to the upstairs establishment as the Penthouse Club, but there is no listing for such a place, either. Presumably the Fifth Avenue Restaurant was the "Bar" or "Club" he remembered.
2. Morris Gollub, interview with the author, Hollywood, CA, November 2, 1976.
3. Bill Spicer and Vince Davis, "Interview with Dan Noonan," Graphic Story Magazine, Summer 1968, p. 13. Rivera was a story sketch artist for early Disney features like Pinocchio.
4. Lawrence Cole to Walt Kelly, July 30, 1947. OSU, WK, box 20.
5. "Books—Authors," New York Times, October 11, 1944, p. 19; "You Meet Such Interesting People," Publishers' Weekly, October 21, 1944, p. 1656.
6. "Messners Acquire Veritas Press," Publishers' Weekly, July 28, 1945, p. 318. Messner used Veritas, which before World War II published English translations of German books for adults, as a vehicle for issuing children's picture books priced at a dollar. Wartime paper allocations were no doubt the major factor in the acquisition, and probably the only one: Veritas, since it was publishing before the United States entered the war, would have had a paper allocation that could be used for children's "dollar flats" was well as adult books. Mike McClintock became Veritas's vice president after Messner acquired what Publishers' Weekly called "a controlling interest."
7. Trouble on the Ark was reprinted (in black and white, along with preliminary drawings and supporting material) in two parts in the Kelly fan magazine Fort Mudge Most, April 1996, pp. 10–48, and June 1996, pp. 9–38. The Downy Duck, "Story by Edith Heal," was reprinted (also in black and white, and also with supporting material) in Fort Mudge Most, March 1994, pp. 16–34. The third Maclay title, Raffy Uses His Head, by Rita Kissin, author of an earlier book about the giraffe title character, is more elusive.
8. Andrew Barnes, conversation with the author, Tampa, FL, June 4, 1974. Barnes reviewed and approved notes from that conversation in 2012.
9. "Story Book Records: Walt Kelly Tells More Fairy Tales," Fort Mudge Most, June 1997, pp. 12–33. This article reproduces the labels for all sixteen sides and the covers for the two boxed sets of four records each, and also includes transcriptions of Kelly's narration for all the records. The description here of Kelly's performances is based on digital copies from the original records made for the author by a private collector. More readily accessible examples of Kelly the uninhibited vocal performer are several tracks on Songs of the Pogo, a 1956 record reissued as a compact disc in 2003.
10. Inez Bertail, Complete Nursery Song Book (New York, 1947), p. 1. Bertail's husband had resigned as "general editor" of Julian Messner early in 1946. The couple remained active in publishing for years afterward, but apparently without any connection with Kelly.
11. "Fall Index of Children's Books," Publishers' Weekly, August 25, 1945, p. 795. Kelly spelled his mother's maiden name the same way when he applied for a Social Security number in 1934, but it was more often spelled "MacAnnulla."
12. The copyright registrations for those stories show Stanley as the author.
13. Person to Person, January 1, 1954, CBS Television, transcript.
14. Hearings before the Subcommittee to Investigate Juvenile Delinquency of the Committee on the Judiciary, U.S. Senate, April 21, 1954, p. 110.
15. "Walt Kelly," in Album of the National Cartoonists Society, ed. Mort Walker (New York, 1965), p. 86. Kelly's half-page hand-lettered autobiography was reprinted on p. 84 of the revised edition, titled The National Cartoonists Society Album (ed. Mort Walker [Greenwich, CT, 1972]).
16. Gil Kane, "Walt Kelly Interview," Comics Journal, February 1991, pp. 52, 53.
17. Murray Robinson, "Pogo's Papa," Collier's, March 8, 1952, p. 65.
18. Gollub, interview, November 2, 1976.
19. Walt Kelly to Lloyd E. Smith, December 8, 1951. OSU, WK, box 19, file 5.
20. Letty Lebeck Edes to the author, March 12, 2013.
21. Kelly's correspondence with Hickey is part of the file labeled "Artists' correspondence." OSU, WK, box 3, file 44. Hickey illustrated "Tom and Jerry" in Our Gang Comics in the 1940s, as well as many adventure stories drawn in a straight illustration style.
22. Harvey Kurtzman, interview with the author, Mount Vernon, NY, March 31, 1990.
23. Walt Kelly to Harvey Kurtzman, November 2, 1948. OSU, WK, box 6, file 4.
## CHAPTER 19
1. Hank Ketcham, interview with the author, Monterey, CA, June 10, 1991.
2. Thomas Andrae and Geoffrey Blum, "Ward Kimball Remembers Walt Kelly," in Phi Beta Pogo, ed. Mrs. Walt Kelly and Bill Crouch Jr. (New York, 1989), p. 138.
3. Donald Phelps, "John Stanley," 1976 Newcon program booklet (Boston), n.p.
4. "'Li'l Eight Ball' Killed By Publisher," New York Amsterdam News, May 10, 1947, p. 9.
5. Murray Robinson, "Pogo's Papa," Collier's, March 8, 1952, p. 65. Robinson does not identify the syndicate, which Kelly names in his untitled 1952 carbon-copy biography. A fan's letter, written in reply to one of Kelly's, indicates that by August 1947 Kelly was aware that Animal Comics' days were numbered and he was contemplating syndication for "Albert and Pogo." Betsy Curtis to Walt Kelly, August 12, 1947. OSU, WK, box 20.
6. The two pages were published in Outrageously Pogo, ed. Mrs. Walt Kelly and Bill Crouch Jr. (New York, 1985), pp. 26–27. An accompanying note says of one of the pages, "The page definitely predates national syndication of the strip" but does not make the connection with Kelly's unsuccessful early effort at syndication.
7. Carle Hodge, "PM 'Experiment' Dies and a Star Is Born," Editor & Publisher, June 26, 1948, p. 5.
8. John Horn, "A Memory of Walt Kelly," in The Best of Pogo, ed. Mrs. Walt Kelly and Bill Crouch Jr. (New York, 1982), p. 42.
9. Gil Kane, "Walt Kelly Interview," Comics Journal, February 1991, p. 58.
10. Kelly did identify himself as the Star's "political cartoonist, art director, comic-strip editor, and comic-strip artist" in Ten Ever-Lovin' Blue-Eyed Years with Pogo (New York, 1959), p. 10, and as its "art director, senior editor, pol. Cartoonist and match game champ" in Album of the National Cartoonists Society, ed. Mort Walker (New York, 1965), p. 86.
11. At least as likely, they may have been destroyed in the flooding of the basement of Kelly's New York townhouse a few years after his death. That flooding probably accounts for the absence of other significant documents from the Kelly papers at Ohio State.
12. Lloyd E. Smith to Walt Kelly, June 15, 1951. OSU, WK, box 5, file 19.
13. E. H. Wadewitz and Lloyd E. Smith to Walt Kelly, November 17, 1948. OSU, WK, box 5, file 20.
14. Ogden J. Rochelle, "N.Y. Star Features Become Star Syndicate," Editor & Publisher, December 18, 1948, p. 44.
15. Quoted in Lloyd E. Smith to Walt Kelly, April 13, 1953. OSU, WK, box 5, file 19.
16. A collection of early Amos 'n' Andy radio scripts was published as Here They Are: Amos 'n' Andy, by Charles J. Correll and Freeman F. Gosden (New York, 1931). The scripts are totally in dialect—Correll and Gosden did little or no improvising—and so reading them can be a chore; but when the scripts are read alongside "Albert and Pogo," the resemblance to that feature's dialogue is striking.
17. Pogo Parade was published in the summer of 1953. Kelly's commentary appeared on the inside front cover.
18. Andrae and Blum, "Ward Kimball Remembers Walt Kelly," p. 138.
19. Kelly, untitled 1952 carbon-copy biography. Lauterbach's name is spelled correctly in Kelly's draft but misspelled "Lauterback" in the retyped version. Lauterbach died in 1950 at the age of thirty-six.
20. Thomas Andrae, "Pogo's Politics," in Thomas Andrae and Carsten Laqua, Walt Kelly: The Life and Art of the Creator of Pogo (Neshannock, PA, 2012), p. 106.
21. Robert E. Kiler to Walt Kelly, July 20, 1947. OSU, WK, box 20.
22. Doris Willens, "Walt Kelly's 'Pogo' Ribs Stupidities of Mankind," Editor & Publisher, December 11, 1948, p. 42. That characterization of Pogo's dialect resurfaced in Editor & Publisher a few years later (Erwin Knoll, "Walt Kelly Is Named Cartoonist of Year," April 26, 1952, p. 145), and it may have been the basis for similar characterizations.
23. A. J. Liebling, "The Wayward Press," New Yorker, February 12, 1949, p. 55.
24. "Walt Kelly Insists Comic Strip Aims at Amusement," New Orleans Times-Picayune, October 22, 1952, p. 14.
## CHAPTER 20
1. Carl Barks, interview with the author, Atlanta, October 5–6, 1974.
2. Carl Barks to the author, January 15, 1969.
3. Carl Barks, interview with the author, Goleta, CA, May 30, 1971.
4. Carl Barks to Donald Ault, September 15, 1972, in "Letters from the Duck Man Part Fourteen: Thieves and Other Nuisances," in The Carl Barks Library of Walt Disney's Comics and Stories in Color, ed. Geoffrey Blum, no. 41 (Prescott, AZ, n.d. [ca. 1995]), n.p.
5. Martin L. Greim et al., "Crusader Comments," Comics Buyer's Guide, February 11, 1977, p. 27.
6. Barks first used the term money bin in "You Can't Guess!" in Christmas Parade no. 2, 1950, where Scrooge's cash is housed in actual bins identified variously as "No. 739" and "Double-decker Money Bin Capacity 100 Tons." The capacity of that set of bins is suggested when Scrooge muses, "Just because I've got three cubic acres of money, people think I can afford anything."
7. Barks, interview, October 5–6, 1974.
8. Edward Summer, "Of Ducks and Men: Carl Barks Interviewed," in Carl Barks: Conversations, ed. Donald Ault (Jackson, MS, 2003), p. 83. Summer's interview was recorded in April 1975.
9. Carl Barks, interview with the author, Goleta, CA, November 22, 1973.
10. Carl Barks, interview with the author, Temecula, CA, August 13, 1978.
11. Barks, interview, November 22, 1973.
12. Donald Ault, "Ideas Flowing Like Waterfalls: Some Reflections from Carl Barks at 98," in Carl Barks: Conversations, ed. Donald Ault (Jackson, MS, 2003), p. 217. Also see, in the same book, "Chronology," p. xxxviii.
13. Barks to Ault, September 15, 1972.
## CHAPTER 21
1. Ward's hiring by the Star was announced in "In the Editorial Rooms," Editor & Publisher, December 11, 1948, p. 40, but he said he started there on August 10, 1948. Bill Crouch Jr., "George Ward Talks about Kelly, Pogo and Times Past," The Best of Pogo, ed. Mrs. Walt Kelly and Bill Crouch Jr. (New York, 1982), p. 76.
2. Ibid.
3. Andrew Barnes, conversation with the author, Tampa, June 4, 1974.
4. Kelly dedicated Beau Pogo (1960), one of the paperback compilations of Pogo comic strips, to Andy Barnes's father, but Kelly and Joe Barnes were estranged by the time of Barnes's death in 1970. Kelly never wrote to Andy or his mother after Joe died, but he dedicated his next paperback, Impollutable Pogo (1970), to Joe.
5. Copyright assignment from New York Star, Inc., to Walt Kelly, February 14, 1949, vol. 704, p. 25. Copyright Office, Library of Congress, Washington, DC.
6. Crouch, "George Ward Talks," p. 82.
7. March 2, 1949, is the date of the original agreement as stated in an amended agreement dated April 19, 1955, a draft edited and signed by Kelly and Robert Hall, president of the Post-Hall Syndicate. OSU, WK, box 4, folder 11. The same file includes a carbon copy of an unsigned letter agreement dated December 31, 1951, in which the syndicate relinquished any ownership of Pogo, recognizing it as Kelly's "sole property." The copyright notice in the comic strip began appearing in Kelly's name, rather than the syndicate's, as of January 1, 1952.
8. "Post Syndicate Name Changed; It's Post-Hall," Editor & Publisher, March 19, 1949, p. 47.
9. Cox's legatees conveyed all rights to his Brownies stories to Western Printing & Lithographing Company by an assignment dated February 11, 1946, vol. 592, pp. 139–40. Copyright Office, Library of Congress, Washington, DC.
10. Del Connell began writing Peter Wheat when Kelly left the comic book after the first thirty-three issues. A former Disney artist named Al Hubbard, who drew in a Kelly-like style, succeeded Kelly as the illustrator.
11. The dates of Kelly's divorce from Helen and of Stephanie's death are noted on Kelly's license for his third marriage, to Selby Daley. The license was issued on October 20, 1972, and Kelly and Daley were married at Lenox Hill Hospital on October 23, just before Kelly lost a leg to diabetes.
12. U.S. Federal Census, 1930, Manhattan, New York, New York, roll 1545, page 5A, enumeration district 37, image 582.0. Stephanie is listed in that census as Estelle Wagonny. U.S. Federal Census, 1940 New York, Kings, New York, roll T627_2560, page 8B, enumeration district 24–529.
13. Don Maley, "Walt Kelly Muses on His 20 Years of Playing Possum," Editor & Publisher, April 10, 1969, p. 20. Maley identifies Stephanie as "formerly a secretary at his syndicate," rather than Kelly's personal secretary, although that may be a distinction without a difference if she did most of her work for Kelly.
14. E. H. Wadewitz and Lloyd E. Smith to Walt Kelly, September 20, 1951. OSU, WK, box 5, folder 19.
15. Remittance advice, Western Printing & Lithographing Company to Walter Kelly, September 4, 1951, and March 4, 1952. Kelly was paid a total of $1,692.50 in March 1952 for producing the fifty-two pages in Pogo Possum no. 10, July–September 1952—a figure that did not include payment for the lettering. Billy Ireland Cartoon Library and Museum, Ohio State University, Columbus, Pogo Collection, box 14, folder 3. The Pogo Collection is a separate collection from the Walt Kelly Collection.
16. Richard Small to Walt Kelly, October 2, 1952. OSU, WK, box 5, folder 19.
17. Anne DeStefano to Walt Kelly, September 11, 1952. OSU, WK, box 5, folder 19. DeStefano became office manager after the comic books moved from 200 Fifth Avenue to quarters they shared with other components of Western's "newsstand division" at 415 Madison Avenue. "Western in New York," Westerner, March 1957, p. 3.
18. Walt Kelly to Anne DeStefano, September 11, 1952. OSU, WK, box 5, folder 19.
19. Walt Kelly to Lloyd E. Smith, May 13, 1952. OSU, WK, box 5, folder 19.
20. Richard Small to Walt Kelly, January 27, 1954. OSU, WK, box 5, folder 19.
21. Walt Kelly to Richard Small, January 29, 1954. OSU, WK, box 5, folder 19.
22. Walt Kelly, undated draft letter to Richard Small. OSU, WK, box 5, folder 19.
23. Walt Kelly to Richard Small, February 3, 1953 [sic]. OSU, WK, box 5, folder 19.
24. In Kelly's trade paperback called The Pogo Peek-a-Book (1955), made up of new stories, Albert reads aloud (rather than recites) "Who Killed Cock Robin?" but he does so entirely in character, assuming one can accept an ability to read that is advanced over his virtual illiteracy in earlier stories.
25. DeStefano to Kelly, September 11, 1952. DeStefano refers to him only as "Mr. Burley," but Burley's full name is given in Crouch, "George Ward Talks," p. 76.
26. Crouch, "George Ward Talks," p. 82. Ward's principal duties subsequently were to ink the Sunday Pogo page and to finish whatever work Kelly left on the daily comic strip.
27. Ibid., p. 85.
28. Seymour Turk (Simon and Schuster) to Walt Kelly, February 28, 1955. OSU, WK, box 4, folder 10.
29. Anne DeStefano to Walt Kelly, June 15, 1953. OSU, WK, box 12, folder 3. DeStefano sent Kelly for his approval a list of four stories chosen for reprinting in Pogo Possum nos. 15–18 and due dates for the new material in each issue, including in each case "20 pages of Pogo art" and a "6 page strip without Pogo and Albert." Only the first two of those four issues were published.
## CHAPTER 22
1. Morris Gollub, interview with the author, Hollywood, CA, November 2, 1976.
2. Bill Spicer and Vince Davis, "Interview with Dan Noonan," Graphic Story Magazine, Summer 1968, pp. 12–13.
3. Oskar Lebeck to Toni Mendez, January 7, 1954. Toni Mendez Collection, Billy Ireland Cartoon Library and Museum, Ohio State University, Columbus (hereafter cited as OSU, TM), box P103, folder 142.
4. Richard Small to Oskar Lebeck, January 28, 1953. OSU, TM, box P103, folder 142.
5. Oskar Lebeck to Toni Mendez, March 30, 1951. OSU, TM, box P40, folder 9.
6. Mort Walker, ed., National Cartoonists Society Album, rev. ed. (Greenwich, CT, 1972), p. 117.
7. Erwin Knoll, "UF Adds 'Twin Earth' to Space Fiction Ranks," Editor & Publisher, June 7, 1952, p. 48.
8. Alex Toth, "Jesse Marsh," Panels no. 2, Spring 1981, pp. 21–27.
9. Camille Cazedessus Jr., "Gaylord DuBois, Veteran Tarzan Script Writer," ERBdom, October 1962, p. 7.
10. Ibid., p. 9.
11. Irvin H. Ziemann, "Gaylord DuBois, King of the Comics Writers, Chapter Four," Comics Buyer's Guide, January 12, 1990, p. 46.
12. Harvey Kurtzman, interview with the author, Mount Vernon, NY, March 31, 1990.
13. John Stanley to Bruce Hamilton, September 12, 1985. Marge.
14. Contract dated December 17, 1951, between Marjorie Henderson Buell and Western Printing & Lithographing Company. Marge.
15. Lloyd E. Smith to Joseph Greene, August 17, 1951; Lloyd E. Smith to Toni Mendez, February 9, 1951. OSU, TM, box P40, folder 9.
16. Contract dated April 20, 1951, between Cisco Kid Products Inc. and Western Printing & Lithographing Company. OSU, TM, box 102, folder 93. The terms of the original 1949 contract are similar, but with a smaller advance.
## CHAPTER 23
1. Carl Barks, interview with the author, Goleta, CA, May 30, 1971.
2. Jack Bradbury to Walt Kelly, January 5, 1949. OSU, WK, box 6, folder 4.
3. Dave Bennett, "An Interview with Jack Bradbury," Ace Comics Presents no. 2, July 1987, p. 14.
4. Will Friedwald, "An Interview with Jim Davis," Comics Buyer's Guide, November 2, 1984, p. 44.
5. Mark Evanier, "POV Point of View," Comics Buyer's Guide, May 1, 1998, p. 49.
6. Jack Bradbury, interview with Milton Gray, Irvine, CA, March 23, 1977. The interview was recorded for the author as part of the research for his book Hollywood Cartoons: American Animation in Its Golden Age.
7. Bennett, "An Interview with Jack Bradbury," p. 32.
8. Bradbury, interview, March 23, 1977.
9. Alice Cobb, "The 'West' in Western," Westerner, November 1956, p. 6.
10. Contract dated December 7, 1945, between Gene Autry and Western Printing & Lithographing Company, Inc. Autry Library, Autry National Center, Los Angeles (hereafter cited as GA).
11. Lloyd E. Smith to C. R. Rothmund, April 9, 1947. ERB.
12. George T. Delacorte Jr. to Gene Autry, January 4, 1951. GA.
13. "Go West Young Man," Westerner, December 1951, p. 11.
14. Chase Craig to the author, July 25, 1978. Craig mentioned the January 1950 start date in a typewritten note attached to the copy of the November 1956 Westerner, the Western Printing house organ, that he lent to the author in 1978.
15. Cobb, "The 'West' in Western," p. 4.
16. "Go West Young Man," p. 12.
17. Moore's first drawings for Johnny Mack Brown are in no. 10, September–November 1952. The earlier issues were illustrated by Jesse Marsh, except for the lead story in no. 8, December 1951–January 1952, which Hames Ware identifies as an unusual collaboration between Mike Arens and John Ushler.
18. Richard "Sparky" Moore, telephone interview with the author and Hames Ware, April 3, 2013.
19. Jim Vadeboncoeur Jr. and Everett Raymond Kinstler, Everett Raymond Kinstler: The Artist's Journey through Popular Culture (Nevada City, CA, 2005), p. 103. Kinstler later became a highly sought after portrait painter.
20. Carl Barks to the author, July 14, 1978.
21. Carl Barks, interview with the author, Temecula, CA, August 13, 1978. Other cartoonists were employees in earlier years, at least at the New York office. Dan Noonan spoke of working as a "staff artist" for Western until 1950 or 1951, when he left to work as a freelancer, continuing to sell to Western "while moving into magazine illustration." He ultimately left such work and returned to animation. Bill Spicer and Vince Davis, "Interview with Dan Noonan," Graphic Story Magazine, Summer 1968, p. 17.
22. Malcolm Willits, "George Sherman: An Interview with Another One of the 'Men Behind the Mouse,'" Vanguard (a comic-book fan magazine published by Robert Latona), 1968, p. 34.
23. Those figures are in a "Family Day" program for Western's Poughkeepsie plant, undated but datable through internal evidence to 1953. Thanks to Dana Gabbard for providing photocopies of two such programs, both originating with the Poughkeepsie Public Library.
24. Contract dated April 20, 1951, between Cisco Kid Products, Inc., and Western Printing & Lithographing Co., Inc. OSU, TM, box 102, folder 93.
25. Lloyd E. Smith to William C. Erskine, December 19, 1951. Marge.
26. Cobb, "The 'West' in Western," p. 5.
27. "Gene Autry in Racine," Westerner, March 1954, p. 4.
28. Savitt is the subject of the fullest examination of the work of any of the Dell cover artists. See Leo Pando, "Sam Savitt: Painter, Author, Teacher & Horseman," Illustration, August 2002, p. 2.
29. The agenda was in the papers of John Burton, producer of the Warner Bros. cartoons at the time. Courtesy of Mrs. John Burton.
## CHAPTER 24
1. Carl Barks to the author, December 11, 1969.
2. Barks introduced that routine in the January 1951 Walt Disney's Comics, but without the third variant, "toss it up and let it hit me on the head."
3. Carl Barks, Walt Disney's Uncle Scrooge McDuck: His Life and Times (Millbrae, CA, 1981), p. 62. Comments by Barks, based on interviews with Edward Summer, accompany a version of "Back to the Klondike" with most of the excised panels (Barks had kept the drawings) restored to their original positions in the story and new panels drawn by Barks to take the place of four that had not survived.
4. Carl Barks to the author, February 17, 1972.
5. Carl Barks, interview with the author, Atlanta, October 5–6, 1974.
6. Ibid.
7. Carl Barks to the author, January 25, 1971.
8. Del Connell, interview with Bill Spicer, Los Angeles, November–December 1983. The interview was conducted for inclusion in Carl Barks Library, but it was not published there. Thanks to Geoffrey Blum for providing a copy.
9. Richard "Sparky" Moore, telephone interview with the author, August 23, 2013.
10. Edward Summer, "Fortune Favors the Bold: An Interview with Carl Barks," in Carl Barks: Conversations, ed. Donald Ault (Jackson, MS, 2003), p. 124. The published interview is a composite of material recorded from 1975 to 1981. The quoted passage was also published in Barks, Uncle Scrooge McDuck, p. 90, as part of Barks's commentary on the Chisel McSue story.
## CHAPTER 25
1. Lloyd E. Smith, "Protest against Ad for Wertham Book," Publishers' Weekly, March 20, 1954, p. 1399.
2. George T. Delacorte Jr. to Stanley Rinehart, March 17, 1954. Fredric Wertham papers, 1818–1986, Manuscript Division, Library of Congress, Washington, DC (hereafter cited as Wertham papers).
3. McLaughlin, Stickles & Hayden, counselors at law, to Rinehart & Company, November 16, 1953. Wertham papers.
4. "F.G.M." [Frederick G. Melcher], "Senate Committee Holds Hearing on the Comics," Publishers' Weekly, May 1, 1954, p. 1906.
5. Hearings before the Subcommittee to Investigate Juvenile Delinquency of the Committee on the Judiciary, U.S. Senate, April 21, 1954, p. 103.
6. Fredric Wertham, "Wertham Replies to Criticism of Ad," Publishers' Weekly, May 1, 1954, p. 1889.
7. Amy Kiste Nyberg, Seal of Approval: The History of the Comics Code (Jackson, MS, 1998), pp. 110–11.
8. Helen Meyer to Milton Caniff, October 1, 1954. Milton Caniff Collection, Billy Ireland Cartoon Library and Museum, Ohio State University, Columbus (hereafter cited as OSU, MAC), box 110, folder 5.
9. Western Printing & Lithographing Company, annual report to stockholders for 1954, n.p.
10. A partial copy of a 1957 prospectus was provided to the author by David R. Smith of the Walt Disney Archives. The cited figures are from p. 9.
11. Malcolm Willits, Don Thompson, and Maggie Thompson, "The Duck Man," in Carl Barks: Conversations, ed. Donald Ault (Jackson, MS, 2003), p. 12.
12. Carl Barks to the author, June 9, 1966.
13. Del Connell, interview with Bill Spicer, Los Angeles, November–December 1983. Courtesy of Geoffrey Blum.
14. The list is reproduced in Thomas Andrae, "The Expurgated Barks," in The Carl Barks Library, ed. Bruce Hamilton, set 3, vol. 2 (Scottsdale, AZ, 1984), p. 522.
15. Charles Beaumont, "The Comic World," Fortnight, May 1955, pp. 49–50. Adams's entry in The Who's Who of American Comic Books, ed. Jerry Bails and Hames Ware, vol. 1 (Detroit, 1973), p. 1, based on information he provided, shows him working for Western from 1952 to 1957, initially writing and inking talking-animal titles and then writing and editing such western titles as Roy Rogers, Gene Autry, and the Lone Ranger.
16. Geoffrey Blum, "Wartime Joys and Jitters," in Carl Barks Collection, ed. Geoffrey Blum, vol. 3, p. 13. Barks made that comment during a September 24, 1983, interview with Bruce Hamilton (with additional questions by Blum and Thomas Andrae).
17. Connell, interview, November–December 1983.
18. Barks to the author, June 9, 1966.
19. An embryonic version, the "Junior Woodchucks' Book of Knowledge," is mentioned in the previous issue, Uncle Scrooge no. 5, March–May 1954.
20. Carl Barks to the author, May 17, 1981.
21. Carl Barks, interview with the author, Goleta, CA, November 22, 1973.
22. Klaus Strzyz, "An Interview with Carl and Garé Barks," in Carl Barks: Conversations, ed. Donald Ault (Jackson, MS, 2003), p. 117.
23. Alice Cobb to Carl Barks, June 4, 1956. WDA.
24. Carl Barks to Alice Cobb, undated draft [ca. June 1956]. WDA.
25. Beaumont, "The Comic World," p. 47.
26. Barks prepared that list from records of his payments from Western for the author's use in compiling a bibliography of his comic-book work. That bibliography was published in Michael Barrier, Carl Barks and the Art of the Comic Book (New York, 1981).
27. Carl Barks to R. O. Burnett, December 13, 1960. WDA. Photocopy provided by John Benson.
28. Carl Barks to the author, March 27, 1966.
29. Barks, interview.
30. Carl Barks, interview with the author, Atlanta, October 5–6, 1974.
31. "Meet the Team of Bill Hanna and Joe Barbera," Westerner, October 1963, p. 9.
32. Carl Barks, interview with the author, Temecula, CA, August 13, 1978.
33. Chase Craig to Carl Barks, August 11, 1960. WDA. William F. Callahan Jr., "The Dell Move to 15¢ Means Maximum Profits for the Retailer!" Bestsellers, December 1960, p. 15.
34. Bruce Hamilton, "The Mouse Man and the Duck Man: An Interview with Floyd Gottfredson and Carl Barks," in Walt Disney's Mickey Mouse in Color, deluxe ed. (Prescott, AZ, 1988), p. 108. The joint interview took place on December 5, 1982, in Pasadena, CA.
35. Beaumont, "The Comic World," p. 54.
36. John Spicer to Carl Barks, April 11, 1960; Barks to Spicer, undated draft. WDA.
37. Malcolm Willits to Carl Barks, October 24, 1960. WDA.
38. Chase Craig to Carl Barks, August 11, 1960. WDA.
39. Barks, interview, October 5–6, 1974.
40. Barks to Burnett, December 13, 1960.
41. Barks, interview, October 5–6, 1974.
42. Carl Barks, videotaped interview by Donald Ault and Thomas Andrae, Goleta, CA, August 4, 1975.
43. Geoffrey Blum, "Running on Empty," in Carl Barks Collection, ed. Geoffrey Blum, vol. 21, p. 10.
## CHAPTER 26
1. Harvey Kurtzman, interview with the author, Mount Vernon, NY, March 31, 1990.
2. "Interview with Alex Toth from Graphic Story Magazine," in Setting the Standard: Comics by Alex Toth, 1952–54, ed. Greg Sadowski (Seattle, 2011), pp. 26–27. The interview was conducted in 1968 by Vincent Davis, Richard Kyle, and Bill Spicer and was published originally in Graphic Story Magazine no. 10, Spring 1969.
3. Western Printing & Lithographing Company to Marjorie Henderson Buell, November 9, 1953. Marge.
4. Western Printing & Lithographing Company to Marjorie Henderson Buell, November 22, 1955. Marge.
5. Lloyd E. Smith to Toni Mendez, February 9, 1951. OSU, TM, box P40, folder 9.
6. Alfred Harvey to Milton Caniff, February 15, 1951. OSU, MAC, box P110, folder 1.
7. Alfred Harvey to Milton Caniff, March 24, 1952. OSU, MAC, box P110, folder 4.
8. Milton Caniff to Chester Weil, July 10, 1953. OSU, MAC, box P110, folder 5.
9. Chester Weil to Milton Caniff, January 31, 1955. OSU, MAC, box P110, folder 5.
10. Matthew H. Murphy to Milton Caniff, March 10, 1959. OSU, MAC, box P110, folder 5.
11. Matthew H. Murphy to John Stanley, July 24, 1959. John Stanley papers, courtesy of James Stanley (hereafter cited as JS).
12. Stanley and Widmer took out a marriage license at the Municipal Building in Manhattan on December 19, 1957, and were married there on December 23.
13. Matthew H. Murphy to John Stanley, September 26, 1958. JS.
14. Lloyd E. Smith to the author, February 25, 1966.
15. Bruce Hamilton, "A Tripp Down Memory Lane," in The Little Lulu Library, ed. John Clark, set 6, vol. 16 (Scottsdale, AZ, 1985), p. 17.
16. "Interview: Carl Barks and John Stanley," Comics Journal, February 2003, p. 160.
17. "The Old Fairgate Ruler Factory Is Demolished," Putnam County (New York) News and Recorder (online edition), February 1, 2012.
18. James Stanley, email to the author, October 27, 2012.
## CHAPTER 27
1. "Dell Comes to Play," Westerner, December 1953, p. 14. Similar visits were reported in other issues of the Westerner.
2. A partial copy of the 1957 prospectus was provided to the author by David R. Smith of the Walt Disney Archives. The cited figures are from p. 9.
3. A partial copy of the 1960 prospectus was provided to the author by David R. Smith of the Walt Disney Archives. The quoted passage is from page 8.
4. Western Publishing Company, Inc., annual report to stockholders for 1960, p. 5.
5. Lloyd E. Smith to C. R. Rothmund, April 15, 1958. ERB.
6. William F. Callahan, "The Dell Move," Bestsellers, December 1960, p. 15.
7. Irwin Donenfeld, "D-C Fast Turnover at 10¢ Produces the Biggest Dealer Profit," Bestsellers, December 1960, p. 14.
8. "Newsstand Giant Shrinks Away," Business Week, May 25, 1957, p. 59.
9. Malcolm Willits, "George Sherman: An Interview with Another One of the 'Men Behind the Mouse,'" Vanguard (a comic-book fan magazine published by Robert Latona), 1968, p. 34. The pages are unnumbered except on the contents page.
10. Carl Barks to Malcolm Willits, April 19, 1962, in "Letters from the Duck Man, Part Three: A Sense of Fandom," The Carl Barks Library of Walt Disney's Comics and Stories in Color, no. 8, ed. Geoffrey Blum (Prescott, AZ, n.d. [ca. 1992]), n.p.
11. Western Publishing Company, Inc., annual report to stockholders for 1963, p. 5.
12. Western published both a Walt Disney Comics Digest and a Golden Comics Digest (with other cartoon studios' licensed characters) from the late 1960s until the mid-1970s, in both cases shrinking standard comic pages to a barely readable digest size.
13. Carl Barks to Malcolm Willits, February 16, 1963, in "Letter from the Duck Man, Part Four: From Duckburg to Middle-Earth," in The Carl Barks Library of Walt Disney's Comics and Stories, no. 13, ed. Geoffrey Blum (Prescott, AZ, n.d. [ca. 1993]), n.p.
14. In Uncle Scrooge nos. 41, 44, and 49, respectively.
15. Carl Barks to Dick Blackburn, March 8, 1968. Courtesy of Dick Blackburn.
16. Malcolm Willits, Don Thompson, and Maggie Thompson, "The Duck Man," in Carl Barks: Conversations, ed. Donald Ault (Jackson, MS, 2003), p. 12.
17. Chase Craig to Carl Barks, July 9, 1966. WDA.
18. Carl Barks to the author, September 11, 1973.
19. Chase Craig to Carl Barks, August 31, 1965. WDA.
20. "Western's New Home in the West," Westerner, January 1964, p. 9.
21. "Western Reports Record Year," Westerner, April 1965, p. 6.
22. Chase Craig to Carl Barks, November 1, 1965. WDA.
23. Carl Barks to Chase Craig, July 4, 1966. Carl Barks Correspondence Collection, 1963–84, Special Collections, Oviatt Library, California State University, Northridge.
24. Carl Barks to the author, January 8, 1975.
25. Carl Barks to the author, January 23, 1968.
26. The fullest and most accurate account of Barks's years as a painter is Geoffrey Blum, Carl Barks Paintings and Drawings 1971–1990 (2011), a companion volume to the thirty-volume Carl Barks Collection that, like that set, has not yet been published in English. Blum made available to the author the original English version of his text.
## EPILOGUE
1. Robert R. Barrett, "Tarzan's Third Great Comic Strip Artist: Russell G. Manning (1929–1981)," Burroughs Bulletin, n.s., no. 13, January 1993, p. 17.
2. J. J. Barta to William C. Erskine, December 3, 1970. Marge.
3. Contract dated January 1, 1962, between Marjorie Henderson Buell and Western Printing & Lithographing Company. Marge.
4. "Final Royalty Earnings Report on Gold Key Comics" for Little Lulu no. 194, published September 11, 1969. Marge. Little Lulu by that time was selling for fifteen cents per copy, and so the royalty was calculated at .00375 cents per copy, rather than the .0025 that was standard for ten-cent comic books.
5. Barta to Erskine, December 3, 1970.
6. Raymond C. Butman to Marjorie Henderson Buell, December 14, 1970. Marge.
7. Contract dated December 24, 1971, between Marjorie Henderson Buell, William C. Erskine, and Western Publishing Company, Inc. Marge. The contract was effective January 1, 1972.
8. Excerpts from the manual were published in the Hollywood Eclectern, an informal fan publication devoted to John Stanley and Little Lulu, in its summer 1997 issue, no. 23.
9. "Mattel to Sell Publishing Unit," New York Times, December 22, 1983, p. D4; "Company Briefs," New York Times, February 29, 1984, p. D22; "Western Suspends Its Whitman Comics Line," Comics Buyer's Guide, May 18, 1984, p. 1.
# Index
Page numbers referring to illustrations are in italics. Continuing FEATURES within anthology titles that are discussed in the text are listed under those titles; a number of the individual STORIES discussed in the text are likewise listed under such overall titles.
Abel, Bob,
Action Comics (comic book), , ,
Adams, Kellogg, ,
Adelquist, Hal, , 373n20
Adventures of Peter Wheat (giveaway comic book), , 388n10
advertisements in Dell comic books, 297–98, 322–23
"Albert Alligator and Pogo Possum" (sample Sunday pages),
Albert the Alligator (character), , , ,
Albert the Alligator and Pogo Possum (comic book), , , , , , , , ,
—STORIES: "The Catfish Pirates," ; "Mr. Owl and the Atomic Bomb" (1947), ,
Alice in Wonderland (comic book),
Alley Oop (comic strip),
Alvin (Little Lulu character), 201–2
Amazing Spider-Man, The (comic book),
American News Company (distributor), , ,
Amos 'n' Andy (radio comedy), 72–73, , , , , , ,
Anderson, Carl,
Anderson, Howard, , , , 359n19
Andrae, Thomas,
Andy Panda (character), , , ,
Animal Comics (comic book), , , , , 211–12, , , , , , 245–46, , , , ; introduction of Pogo characters, 68–71, , 73–78,
—FEATURES: "Albert and Pogo," 207–9, , , , 227–28, , , 244–45; "Hector the Henpecked Rooster," ; "Jigg and Mooch," 205–6; "Rover," , ; "Uncle Wiggily," , , ,
—STORY: "Albert Takes the Cake," (1942) 68–71,
Archie comic books, , , , , ,
Armstrong, Roger, , , 55–56, 81–83, , ; on Carl Barks, 111–12, ; on Carl Buettner, , , , 110–11, ; on Robert Callender, 162–63, ; on starting work for Western, 46–47, ,
Around the Block with Dunc and Loo (comic book),
Artists & Writers Guild, , , , , ,
Art Students League,
Ault, Donald, , ,
Autry, Gene, , , ,
Bailey, Ray,
Balken, Clara Ovidia. See Barks, Clara Ovidia Balken
Bambi (comic book), ,
Bambi (Disney animated feature), 157–58,
Barks, Arminta Johnson (Carl Barks's mother), ,
Barks, Carl, , 13–14, , , , , , , , , , , , , , , 295–97, , 379n26; as cartoonist and writer for Calgary Eye-Opener, 150–54, , ; childhood and young adulthood in Oregon and California, 142–49, ; at Walt Disney studio, 154–60, ; earliest comic-book work, 93–103, , ; earliest contact with fans, 324–25, ; early issues of Uncle Scrooge, 301–10, , ; early 1950s comic-book work, 250–64, , ; essential pessimism, 113–14; late 1940s comic-book work, 163–64, 166–78, , ; late 1940s–early 1950s comic-book work, 179–92; mid-1940s comic-book work, 104–14, ; mid- to late 1950s comic-book work, 316–19, 322–23; 1960s comic-book work, 338–43; and publisher's taboos, 314–16, 319–21; work in retirement, 344–45,
Barks, Clara Ovidia Balken (Carl Barks's second wife), , 263–64
Barks, Clyde (Carl Barks's brother),
Barks, Dorothy (Carl Barks's younger daughter),
Barks, Garé (Carl Barks's third wife), , 344–45
Barks, Pearl Turner (Carl Barks's first wife), 146–47, 148–49,
Barks, Peggy (Carl Barks's elder daughter),
Barks, William (Carl Barks's father), ,
Barnes, Andrew ("Andy"), ,
Barnes, Joseph, , ,
Barnum, P. T., ,
Barta, J. J.,
Beagle Boys (characters), ,
Beany and Cecil (comic book),
Beaumont, Charles, , ,
Beckett, Sheilah, 278–79
Beetle Bailey (comic book), ,
Bennett, Dave, ,
Benson, John, ,
Bergen, Edgar,
Bernstein, Richard A.,
Bertail, Inez,
Bertelsmann media empire,
Bierce, Ambrose,
Big Little Books, 19–20, , 43–44, 117–19
Black, Don,
Blackburn, Dick,
Blum, Geoffrey, , , , ,
Bob Edwards Publishing Company, ,
Bobo Larkin (comic strip),
Booth, Franklin, 36–37
Bower, B. M.,
Boys' and Girls' March of Comics (giveaway comic book), ,
—STORY: "Race to the South Seas" (1949),
Bradbury, Jack, , , , 291–92,
Bray, Glenn, ,
Brecher, Irving,
Brenner, George,
Bridgeport, Connecticut, 34–35,
Bridgeport Post, , ,
Bronc Peeler (comic strip),
Brooks, Walter R.,
Brownies, The (comic book),
Buckwheat/Bucky ("Our Gang" character), 235–36, 237–40, 238–39
Buell, Marjorie Henderson, 136–37, , 195–96, , 204–5, , 328–29,
Buettner, Carl, , 53–55, , , , 81–82, , 110–12, , , , , , 366n4
Bugs Bunny (character), , , ,
Bugs Bunny (comic book), 53–54, ,
—STORY: "Bugs Bunny's Dangerous Venture" (1946), .
Bugs in Love (Disney cartoon),
Building Good Sentences,
Bumbazine (character), , , 70–71, 73–74, , ,
Burley, Raymond,
Burroughs, Edgar Rice, ,
Burroughs, John Coleman, 27–29
Cabell, James Branch,
Calgary Eye-Opener, , 150–54, ,
Callahan, William F., Jr.,
Callender, Robert S., , , , , , , 162–63, , , 293–94, ,
Camp Comics (comic book), 58–59
—FEATURE: "Seaman Sy Wheeler," ,
—STORY: "Elmer and Bugs Bunny,"
Caniff, Milton, , , , ,
Captain Midnight (character),
captions, use in comic-book stories, 202–3
Carey, John,
Charlie McCarthy (character), ,
Chatterbox (Oskar Lebeck),
Chicago Tribune–New York News Syndicate,
Choo-Choo Charlie (comic book),
Christmas with Mother Goose (comic book),
Cinema Comics, ,
Cinema Comics Herald,
Ciotti, Paul,
Cisco Kid (comic book), ,
Clampett, Bob,
Classic Comics (comic book), ,
Classics Illustrated (comic book),
Clementina the Flying Pig (Oskar Lebeck), , ,
Clowes, Daniel,
Cobb, Alice Nielsen, 162–63, , , , , , 319–21
Cole, Lawrence,
Collier's, , , , , ,
Comics Code Authority (CCA),
Comics Magazine, The (comic book),
Comics Magazine Association of America (CMAA),
Commentary,
Commonweal,
Complete Nursery Song Book (Inez Bertail),
Connell, Del, , , ,
Coo Coo Comics (comic book), ,
Correll, Charles,
Couch, Chuck,
Cowboy Lingo: Boy's Book of Western Facts (Big Little Book),
Cox, Palmer, ,
Crackajack Funnies (comic book), , , , , , ,
Craig, Chase, , 53–55, , 81–82, , , 293–94, 340–41, , ,
Crane, Roy, , , ,
Crawford, Mel, ,
Crime Does Not Pay (comic book), ,
Crosby, Percy,
Crouch, Bill, Jr., , ,
Crumb, Robert, ,
Curtis Publishing, 20–21
Daffy Duck (character),
Daisy Duck (character), , , , , 342–43
Dark Knight, The (comic book),
Darling, J. N. ("Ding"),
David Copperfield (MGM feature),
Davis, Jack,
Davis, Jim, , , , 89–90, 290–91
Davis, Vince,
DC. See Detective Comics Inc.
Delacorte, Albert,
Delacorte, George, Jr., , , 48–49, , , , ,
De Lacy, Helen. See Kelly, Helen De Lacy
Dell Junior Treasury (comic book),
Dell Publishing Company, , , , , 47–51, , 164–65, 270–72, 313–14, 334–36,
Dennis the Menace (newspaper comic),
DeStefano, Anne, , , ,
Detective Comics Inc. (DC), , , , , , , , , 335–36, , , ,
Diary of Terwilliger Jellico, The (Oskar Lebeck),
Dick Tracy (character), 19–20
Dinkjian, Anahid,
Dirgo, Ray, ,
Disney, Roy O., , ,
Disney, Walt, , , , , , , , , 156–57,
Ditko, Steve,
Doctor Solar, Man of the Atom (comic book),
Donald Duck (character), , , , , 258–59
Donald Duck (comic book), , , , ,
—STORIES: "Adventure Down Under" (1947), 166–67, , ; "Ancient Persia" (1950), ; "Christmas for Shacktown" (1951), , , , 342–43; "Christmas on Bear Mountain" (1947), ; "Dangerous Disguise" (1950), 250–51; "Donald Duck Finds Pirate Gold" (1942), , 95–96, , ; "The Firebug" (1945), ; "The Ghost of the Grotto" (1947), , , , ; "The Gilded Man" (1952), 252–53; "The Golden Christmas Tree" (1948), ; "The Golden Helmet" (1952), ; "In Old California" (1951), ; "Lost in the Andes" (1949), 173–74; "Luck of the North" (1949), 175–76, , 181–82; "The Magic Hourglass" (1950), 182–83, , ; "The Mummy's Ring" (1943), 101–2; "Mystery of the Swamp" (1945), , ; "No Such Varmint" (1951), 251–52, ; "The Old Castle's Secret" (1948), 172–73; "Pawns of the Loup Garou" (1968), ; "Sheriff of Bullet Valley" (1948), ; "The Terror of the River" (1946), , , , ; "Voodoo Hoodoo" (1949), 174–75, , , ; "Trick or Treat" (1952), ; "Volcano Valley" (1947),
Donald Duck (comic strip), , ,
Donald's Nephews (Disney cartoon),
Donald's Snow Fight (Disney cartoon),
Donenfeld, Harry, , , , ,
Donenfeld, Irwin, 335–36
Downy Duck, The (Edith Heal),
DuBois, Gaylord, , , 65–66, , 115–26, , , 127–28, , , 216–18, , , , 346–47
DuBois, Mary, , , , 216–17
Duffy, Edmund,
Dumbo (comic books), 91–92
Dumbo (Disney animated feature), ,
Duplaix, Georges,
Duplaix, Lily,
Eastern Color Printing Company , ,
Easter with Mother Goose (comic book), ,
EC (Educational Comics), 7–9, , , , , , 382n12
Edes, Letty Lebeck, ,
Edgar Rice Burroughs Inc., 117–18, ,
Eisner, Will, 6–7,
Ella Cinders (comic strip), ,
Ellsworth, Whitney,
Ely, William, ,
Ernst, Ken,
Erskine, William C., , , ,
Evanier, Mark,
Fairbanks, Douglas,
Fairgate Rule,
Fairy Tale Parade (comic book), 60–63, , , ,
Fallberg, Carl,
Famous Funnies (comic book), ,
Famous Stories (comic book), 61–62
Famous Studios, , , ,
Fantasia (feature film),
Fantastic Four (comic book),
Farmer Al Falfa (character),
Fawcett (publisher), , , , ,
Fawcett, Antoinette ("Annette"), 152–53
Fawcett, Harvey,
Fawcett, Wilford, ,
Feldstein, Al, , , 382n12
Felix the Cat (character), ,
First Swallow, The (MGM cartoon),
Flanders, Charles,
Fleischer, Max, , , ,
Fleischer's Animated News,
Foster, Harold,
Foster, Warren,
Fox, Fred,
Fox, Gill,
Francis the Talking Mule (character),
Freddy series (Walter R. Brooks),
Friedwald, Will,
Frontline Combat (comic book), , ,
Frost, A. B., , ,
Funnies, The (comic book), , 26–29, 32–33, , , , , ,
Funny Funnies (comic book),
G.I. Joe (comic book),
Gaiman, Neil,
Gaines, M. C., 1–3
Gaines, William, , ,
Garabedian, Patrick,
Garis, Howard R., ,
Gary, Jim, ,
Gene Autry Comics (comic book), , , , ,
Gene Autry's Champion (comic book),
General Electric, ,
George, Nick,
Geraghty, Jim, 193–94
Gerald McBoing Boing (comic book),
Ghost Stories (comic book),
ghost stories, in Little Lulu,
Ghost World (comic book),
Gibson, Charles Dana,
Giggle Comics (comic book),
Gilberton (publisher)
Gill, Tom, ,
Gladstone Gander (character), , , ,
Godwin, Frank,
Goldberg, Rube,
Golden Press,
Gold Key comic books, , , , 338–39, 346–48
Gollub, Morris ("Moe"), , , , 194–95, , 211–16, , , , , , 298–99, , ; in Animal Comics, 211–12, ; in Santa Claus Funnies, ,
Gordon, Dan, 88–89,
Gormley, Dan, 134–36, , , ,
Gosden, Freeman,
Gottfredson, Floyd, ,
Goulart, Ron, 26–27,
Grahame, Kenneth,
graphic novels,
Gray, Harold,
Grey, Zane, , , , , ,
Grosset & Dunlap, 25–26,
Gruelle, Johnny, 122–23,
Gumps, The (comic strip),
Gun Glory (comic book),
Ha Ha Comics (comic book),
Hal Horne Inc., 16–17
Hall, Richard,
Hamilton, Bruce, ,
Hamming-Whitman,
Hanna-Barbera, ,
Hannah, Jack, , 94–95, , , , ,
Hap Lee's Radio Joke Book: Famous Gags of Radio Stars (Hal Horne),
Harman, Fred, 118–19
Harmon, Bob, ,
Harmon, Eileen,
Harris, Joel Chandler, , 76–77, ,
Harrison, Polly,
Harvey, Alfred,
Hawley Publications,
Hedinger, Charles, , , ,
"Henry" (magazine cartoon),
Henry Aldrich (comic book), , 283–84
Henty, G. A.,
Herriman, George, , ,
Hickey, Tom,
Hilberman, Dave,
Hollywood Hams (comic strip),
Horn, John,
Horne, Hal, 15–17,
Howard, Cal,
Howdy Doody (comic book), ,
Howdy Doody (television show),
Hubbard, Al, , 388n10
Huey, Dewey, and Louie (characters), 168–69, , 308–10
Hughes, Richard,
Hultgren, Ken, , , , , ,
Hurricane Kids on the Lost Islands, The (Lebeck and DuBois), , ,
I Confess (magazine),
I Go Pogo (book),
Independent News (distributor),
Inside Detective (magazine),
It's Nice to Be Little (John Stanley),
Jack and Jill (magazine), 20–22
Jameson, Arthur E., ,
Jimmy Corrigan, or The Smartest Kid on Earth (Chris Ware),
Johnny Mack Brown (comic book),
Johnson, Tom,
Jones, Chuck
Judge (magazine),
Julian Messner (publisher),
Junior Woodchucks, ,
K.K. Publications, , 19–20, , 50–51, , , , , , , ,
Kamen, Herman ("Kay"), 17–18, , , , ,
Karp, Bob, , ,
Karp, Hubie,
Karp, Lynn, ,
Kay Kamen Ltd., ,
Kelly, Bernice (Walt Kelly's sister), , ,
Kelly, Genevieve MacAnnulla (Walt Kelly's mother),
Kelly, Helen De Lacy (Walt Kelly's first wife), , , , ,
Kelly, Selby Daley (Walt Kelly's third wife),
Kelly, Stephanie Wagonny (Walt Kelly's second wife), ,
Kelly, Walt (Walter Crawford Kelly, Jr.), 11–13, , , 33–34, , , , , 230–33, , ; in Albert the Alligator and Pogo Possum, , , , , , , , , , ; in Animal Comics, 68–78, , , 207–9, 221–22, , 227–28, ; in Camp Comics and Looney Tunes, 58–60, ; as caricaturist of himself and colleagues, , , , , 234–35; and Complete Nursery Song Book, ; early comic-book work, , , ; early life in Bridgeport, Connecticut, 34–37; in Fairy Tale Parade, 60–65, ; as illustrator for "language guides" and "self-teaching guides," 66–67; and Harvey Kurtzman, ; in Mother Goose titles, 224–26, ; at New York Star, , 241–42, 242–43, 246–48, 265–66; in Our Gang Comics, 65–66, , , 228–230, 234–36, 237–40, 238–39, 242–43; at PM, ; and Pogo comic strip, 240–41, 244–49, ; and Pogo Possum comic book, 231–32, 268–77, , ; and Story Book Records, 226–27; as "Tony Maclay" (pseudonym) ; at Walt Disney studio, , 39–42; in Walt Disney's Comics & Stories, , ,
Kelly, Walter C., Sr. (Walt Kelly's father), ,
Kempton, Murray, ,
Kerr, George F., , , , ,
Ketcham, Hank,
Kimball, Ward, 39–40, , , 71–72, , , 363n48, 364n57
King, Jack,
King Features Syndicate, , , ,
King of the Royal Mounted (character),
King of the Royal Mounted (comic strip),
Kinstler, Raymond Everett, , ,
Kirby, Jack,
Klondike ' (Ethel Anderson Becker),
KoKo (character),
Korkis, Jim,
Krazy Kat (comic book),
Krazy Kat (comic strip), , ,
Krigstein, Bernard, , 382n12
Kurtzman, Harvey, 7–8, , , , , ; and Walt Kelly,
Kyle, Richard,
Landon School,
language guides, , 74–76
Lantz, Walter, , , , , ,
Lardner, John, ,
Lauterbach, Richard E., 246–47,
Lebeck, Letty. See Edes, Letty Lebeck
Lebeck, Oskar, , , , , , , , , 221–22, , , , ; and Animal Comics, ; as author of children's books, 25–26, , ; copyrights in his name, , , ; departure from Western, , ; Disney studio visit, , ; and Gaylord DuBois, , 120–21, 216–17; as editor of early comic books, 26–31, , , 32–33; and Fairy Tale Parade, 60–62; and Moe Gollub, 211–16, , ; and Richard Hall, ; and Walt Kelly, , , 232–33, ; and Dan Noonan, 211–12, , 279–80, ; personal history, 31–32, ; and John Stanley, , , , , , , ; and Surprise Books, 278–81, ; and Frank Thomas, 29–31, ; Twin Earths comic strip, ,
Lebeck, Ruth Seelig, , ,
lettering in comics, , ,
Leyerle, F. J., ,
Liebling, A. J.,
Li'l Bad Wolf (character), , 173n21
Li'l Bad Wolf (comic book),
Li'l Eight Ball (character), , 236–37
Linda Lark (comic book),
Little Golden Books,
Little Hiawatha (comic book),
Little Itch (character),
Little King, The (comic book),
Little Lulu (character), ,
Little Lulu (comic book), , , , , 139–41, , 195–206, , , 284–88, , , , 329–30, ,
—FEATURE: "Tubby," 286–87
—STORIES: "The Bogyman" (1950, unpublished), 204–5; "Five Little Babies" (1951), ; "The Kid Who Came to Dinner" (1946), 199–201, ; "The Little Rich Boy" (1951), , ; "Lulu in Distress" (1946), ; "Two Foots Is Feet" (1956),
"Little Lulu" (magazine cartoon),
Little Nemo in Slumberland (comic strip),
Little Orphan Annie (character), 19–20
Little Orphan Annie (comic strip), ,
Little, Irene,
Live Stories,
Lone Ranger (comic book), , , ,
Lone Ranger, The (comic strip),
Lone Ranger, The (Gaylord DuBois),
Lone Ranger's Famous Horse Hi-Yo Silver, The (comic book),
Looney Tunes and Merrie Melodies (animated cartoons),
Looney Tunes and Merrie Melodies Comics, , , 46–47, , , , , 59–60, , 81–83, , , 106–7, , , ,
—FEATURES: "Bugs Bunny," 54–55, , ; "Kandi the Cave Kid," ; "Pat, Patsy & Pete" , , ; "Sniffles and Mary Jane," ,
Lowe, Samuel E., , ,
Maclay, Tony (Walt Kelly pseudonym),
Maclean's,
Mad, , , , ,
Magazine Comics Duck model sheet,
Magica de Spell,
Magic Morro (character),
Magnus, Robot Fighter,
Maltese, Mike,
Manning, Russ,
Marcus, Leonard,
Marine, Ed,
Marsh, Jesse, , 218–19, , 282–83,
Martan the Marvel Man (character),
Marvel (publisher), 336–37
Mary Jane (character), , 369n3
Mary Jane and Sniffles (comic book),
Mary Worth (comic strip),
Mattel (toy company),
Mauldin, Bill,
Maus,
McBride, Clifford,
McBride, Hubbell R., ,
McCabe, Norman,
McCay, Winsor,
McClintock, Marshall ("Mike"), , ,
McClure Syndicate, ,
McKimson, Tom, , , , ,
McLevy, Jasper,
McNaught Syndicate,
McNulty, John,
McWilliams, Alden,
Mechanics of English, The,
Melvin Monster (comic book), 330–31,
Mendez, Toni, ,
Merrill, Oregon,
Metro-Goldwyn-Mayer (MGM), , ,
Meyer, Helen, , , , , ,
Meyers, Henry L., 150–52
Mickey and the Beanstalk (planned Disney feature),
Mickey Mouse (Big Little Book),
Mickey Mouse (comic book), , ,
—STORY: "Mickey Mouse Outwits the Phantom Blot" (1941), ,
Mickey Mouse (comic strip), , , ,
Mickey Mouse Magazine, 15–18, , 21–22, , , ,
Milky Way, The (MGM cartoon),
Miller, Frank,
Milne, A. A., ,
Modern Inventions (Disney cartoon),
Modern Romances (magazine),
Modern Screen (magazine),
Montaigne, Michel de,
Moore, Alan,
Moore, Fred, , ,
Moore, Richard ("Sparky"), 294–95,
More Fun (comic book),
Morgan's Ghost (unmade Disney feature cartoon), , ,
Mortimer Snerd (character),
Morton, Jay,
Mother Goose and Nursery Rhyme Comics (comic book),
Motion Picture Daily,
Mougin, Lou,
Mouly, Françoise,
Moving Day (Disney cartoon),
Moviola,
Murphy, Charles
Murphy, Matthew H., , , , ,
Murrow, Edward R.,
Murry, Paul,
Nancy (comic book),
Nancy and Sluggo (characters),
Napoleon (comic strip),
National Cartoonists Society,
National Geographic (magazine), , , , , , ,
Nedor Publishing Company, ,
New Comics (comic book),
New Funnies (comic book), , , , , , , , , , , , 236–37, ; John Stanley stories in, , , , , ,
—FEATURES: "Andy Panda," , , , , , ; "Billy and Bonny Bee," ; "Homer Pigeon," ; "Li'l Eight Ball," 236–37; "Oswald the Rabbit," , ; "Raggedy Ann & Andy," , ; "Woody Woodpecker," ,
New Yorker, , 193–94,
New York Post,
New York School of Fine and Applied Arts,
New York Star, , 241–48, , , ,
Nielsen, Alice. See Cobb, Alice Nielsen
Nights with Uncle Remus (Joel Chandler Harris),
Noonan, Dan, , , , , 196–97, , , 221–22, , ; in Animal Comics, 211–12, ; on Surprise Books, 279–80,
North, Sterling,
Odd Bodkins (comic strip),
Ogle, Lucille,
O. G. Whiz (comic book),
Ohio State University, ,
Oona Goosepimple (character),
Oswald the Rabbit (comic book), 125–26, , ,
Our Gang Comics / Our Gang with Tom and Jerry (comic book), , , , , , , , , , , , , , , ,
—FEATURES: "Barney Bear," ; "Barney Bear and Benny Burro," , , ; "Flip and Dip," ; "King," ; "Our Gang," , 65–66, , 228–30, 234–36, 237–40, 242–44; "Tom and Jerry," , , ,
Overgard, William,
Owl, The (character),
Packer, Eleanor Lewis, , 43–44, , , 81–82, , , , , , , , , , , 161–63,
Packer, George L.,
Paiker, Frank,
panels, size, arrangement, and borders of, 251–52,
Panic (comic book),
paper shortages, , ,
Parsons The New School for Design. See New York School of Fine and Applied Arts
Peter Rabbit, The Tale of (Beatrix Potter), ,
Phantasmo (character), ,
Phelps, Donald, , , ,
Phoenix Art Institute,
Pines, Dora,
Pines, Jacquelyn Sangor, ,
Pines, Ned L., , , , ,
Pinocchio (comic book),
Pinocchio (Disney animated feature), ,
Platt, Kin, ,
Pledge to Parents, A (Dell), 313–14
Pluto Saves the Ship (comic book),
—STORY: "Pluto Saves the Ship" (1942), 95–96
PM newspaper,
Pogo (book),
Pogo (comic strip), , , , , 244–49,
Pogo Parade (comic book), ,
Pogo Possum (comic book), , , , , , 270–77, , ,
—STORY: "Feelin' Mighty Hale, and Farewell" (1950), ,
Pogo Stepmother Goose, The,
Popular Comics, , , , , , ,
Porky Pig (character), ,
Porky Pig (comic book), ,
Post-Hall Syndicate, ,
Post Office Department,
Potter, Beatrix, ,
Prince Valiant (comic strip),
Psychoanalysis (comic book),
radio comedy,
Raffy Uses His Head (Rita Kissin),
Raggedy Ann & Andy (comic book), , , ,
—FEATURE: "Peterkin Pottle,"
Random House,
Raphael G. Wolff Studios,
Real Funnies (comic book),
Real Screen Comics (comic book), ,
Red Ryder (character),
Red Ryder (comic strip), ,
Red Ryder Comics (comic book),
Red Ryder Ranch Comics (comic book),
Reeves, Harry, ,
Reluctant Dragon, The (Disney feature film),
Reluctant Dragon, Walt Disney's (comic book), 91–92
Remington Arms,
Rex, King of the Deep (Lebeck and DuBois),
rhetoric, Carl Barks's mastery of comic-book,
Rinehart & Company, 311–12
Risto, Veve, , 82–83, , ,
Rivera, Tony, ,
RKO Radio Pictures,
Roach, Hal, ,
Robinson, Murray, , ,
Rogers, Roy, , ,
Rolfsen, Phil,
romance comics,
Rooney, Andy,
Rose, John,
rotoscoping,
Roy Rogers Comics (comic book), , , ,
Saludos Amigos,
Samuels, Leo,
Sandman (comic book),
Sangor, Benjamin W., 84–90, , 290–91
Sangor, Jacquelyn. See Pines, Jacquelyn Sangor
Santa Claus Funnies, ,
Saturday Evening Post, ,
Savitt, Sam,
Schlesinger, Leon, , , , ,
Seduction of the Innocent (Fredric Wertham), , 311–12
Segar, E. C., ,
Sequoia (MGM feature),
Sergeant Preston of the Yukon (comic book),
Sheehan, Gordon, , ,
Sheena, Queen of the Jungle (comic book),
Shelton, Gilbert,
Shepard, Ernest H.,
Shuster, Joe, ,
Siegel, Jerry, ,
Silly Symphonies (comic strip), ,
Silvertip (comic book),
Simon and Schuster,
Simple J. Malarkey (character), ,
Skippy (comic strip),
Slesinger, Stephen, , , 117–19
Small, Jon,
Small, Richard, , , ,
Smith, Lloyd E., 5–6, , , , , , 244–45, , , 297–98, 311–13, , , , , 358n8
Smith, Win,
Snappy Stories (magazine),
Sniffles (character),
Snow White and the Seven Dwarfs (Disney animated feature),
Song of the South (Disney feature),
Spencer, Harold,
Spencer, Roy A.,
Spencer, Todd,
Spicer, Bill, ,
Spicer, John,
Spiegelman, Art,
Spirit, The (character), 6–7
"Spirit, The" (newspaper section), ,
Standard (publisher), ,
Stanley, Anna Ahern (John Stanley's mother),
Stanley, Barbara (John Stanley's wife). See Widmer, Barbara Tikotin
Stanley, James (John Stanley's brother),
Stanley, James (John Stanley's father),
Stanley, James (John Stanley's son),
Stanley, John, , , , , , , , 193–95, , 205–6, , , , , , , , ; on Walt Kelly, 234–35, 328–30; later life, ; on Oskar Lebeck, , , ; in Little Lulu, , 139–41, , 195–205, , , 283–87, , , 329–30; in Nancy, ; in New Funnies, 128–36, , ; in Our Gang Comics, , ; in Tubby, 287–88; work for Dell Publishing ,330–333
stereotypes (racial), 236–37,
Steve Canyon (comic book),
Steve Canyon (comic strip), , ,
St. Nicholas: The Magazine of Youth, , ,
Stop Go: The Story of Automobile City (Oskar Lebeck),
Story Book Record Company, 226–27
Story of Our Gang, A (Eleanor Packer),
Strange Tales (comic book),
Stratosphere Jim and His Flying Fortress (Lebeck and DuBois),
Strebe, Dorothy, , 373n21
Strobl, Tony,
Sullivant, T. S.,
Sumner, Ed,
Super Comics (comic book), , , , ,
superhero comics, , , , , ,
Superman (character), , , ,
Supermouse (character),
Surprise Books, 278–80,
Swift, David ("Bud"),
taboos in Dell comics, 315–16,
Tailspin Tommy (comic strip),
Tales from the Crypt (comic book),
Tales from the Tomb (comic book),
Taliaferro, Al,
Tarzan (comic book), , 218–19, , , , ; as Tarzan of the Apes, ,
Tendlar, Dave,
Ten Ever-Lovin' Blue-Eyed Years with Pogo (Walt Kelly),
Terry, Paul, and Terrytoons, , 86–87,
Terry and the Pirates (comic strip), , , ,
Textile High School, New York,
Thimble Theatre (comic strip), ,
Thirteen Going on Eighteen (comic book),
Thomas, Frank, 29–31,
Thompson, Maggie,
Three Caballeros, The (comic book),
Tiny Tots Comics (comic book),
Tom and Jerry Comics (comic book), ,
Tom Corbett, Space Cadet (comic book),
Tonkonogy, George and Sadie,
Tonkonogy, George, Jr. See Delacorte, George, Jr.
Toth, Alex, 282–83, 327–28
Tralla La (in Uncle Scrooge), ,
Tripp, Irving ("Bud"), 91–92, , , , , ,
Trouble on the Ark ("Tony Maclay"),
Tubby (character), 199–201, , 286–87
Tubby (comic book), 287–88
Turner, Gil,
Turner, Lloyd,
Tweety and Sylvester (comic book),
Twin Earths (comic strip),
Two-Fisted Tales (comic book),
Tyer, James,
Uncle Pogo So-So Stories (Walt Kelly), , ,
Uncle Remus, , 76–77,
Uncle Remus: His Songs and His Sayings (Joel Chandler Harris),
Uncle Scrooge (comic book), , , , 301–6, , 308–10, 316–17, , , 338–40
—STORIES: "Back to the Klondike" (1953), 304–6; Chisel McSue story in third one-shot, , ; "House of Haunts" (1966), 340–41; "Island in the Sky" (1960), ; "Land Beneath the Ground" (1956), ; "Only a Poor Old Man" (1952), 301–4,
Uncle Scrooge McDuck (character), , 187–88, , , , , 301–6
"underground" comic books, 349–50
United Artists (UA) ,16
unpublished stories, 204–5, , 320–21
Volke, Ed,
Vosburgh, John R., , ,
Wadewitz, Edward Henry, 18–19, , 23–24, , ,
Wadewitz, W. R.,
Wagonny, Stephanie. See Kelly, Stephanie Wagonny
Walker, Mort,
Walt Disney Productions, , ,
Walt Disney's Christmas Parade, 187–91,
—STORY: "Letter to Santa" (1949), 187–88, , 190–91,
Walt Disney's Comics & Stories, , , , , , 162–63, , , , 191–92, , , , , , , , , , , ; earliest issues , 22–24, , , , ; as K.K. publication, , ,
—FEATURES: "Bucky Bug," 106–7; "Donald Duck," 97–101, 102–3, 168–72, 254–64, , , 318–21, , ; "Gremlins," ; "Li'l Bad Wolf," 76–77, 373n21; "Mickey Mouse,"
—STORIES, "Donald Duck," by issue number and date: no. 32 (May 1943), 99–100, ; no. 41 (Feb. 1944), ; no. 53 (Feb. 1945), ; no. 54 (March 1945), ; no. 62 (Nov. 1945), 108–9; no. 76 (Jan. 1947), ; no. 89 (Feb. 1948), 168–69; no. 92 (May 1948), 169–70; no. 98 (Nov. 1948), 170–71; no. 101 (Feb. 1949), ; no. 124 (Jan. 1951), , ; no. 125 (Feb. 1951), ; no. 126 (March 1951), 254–55, ; no. 129 (June 1951), ; no. 131 (Aug. 1951), ; no. 132 (Sept. 1951), ; no. 133 (Oct. 1951), , ; nos. 134–35 (Nov.–Dec. 1951), , , ; no. 138 (March 1952), , , 262–63; no. 140 (May 1952), 256–57; no. 142 (July 1952), ; no. 145 (Oct. 1952), 259–61, ; no. 146 (Nov. 1952), ; no. 149 (Feb. 1953), ; no. 150 (March 1953), ; no. 156 (Sept. 1953), 261–62; no. 157 (Oct. 1953), ; no. 158 (Nov. 1953), ; no. 178 (July 1955), ; no. 180 (Sept. 1955), ; no. 186 (March 1956), 319–20; no. 308 (May 1966; titled "Donald and Daisy"),
—STORIES, other: "The Laughing Gauchito,"
Walt Disney's Vacation Parade 185–86,
—STORY: "Vacation Time," 185–86
Ward, George, 265–67, ,
Ware, Chris
Warner Bros., , , ,
Warren Harding High School, Bridgeport,
Warshow, Robert,
Wash Tubbs (comic strip), , , ,
Watchmen,
Watson, Jane Werner,
Waugh, Coulton, 1–2
Weeks, Clair, 37–38
Weil, Chester,
Wells, George Y., ,
Wertham, Fredric, , 311–13
Wertheim, Arthur Frank,
Western Marshal (comic book), ,
Western Printing & Lithographing Company, , , 18–19, , 22–24, , 49–50, , , , , , , , , , 291–300, , 325–26, 334–35; as Western Publishing Company, , , 341–42,
West Side Printing Company,
Wettberg, Niles von,
White, Lloyd,
Whitman Publishing Company, 19–20, 25–26, , 44–45, , , , , , , , , ,
Wickersham, Bob, 88–89
Widmer, Barbara Tikotin,
Wiese, Kurt,
Wildenberg, Harry, , ,
Williams, Bill,
Williams, Margaret Wynnfred. See Barks, Garé
Willits, Malcolm, , , , 324–25, , ,
Wind in the Willows, The (Kenneth Grahame), , ,
Wind in the Willows, The (planned Disney animated feature),
Witch Hazel (character),
Wood, Wallace,
Woody Woodpecker (comic book),
Worrell, John C.,
Wright, Bill,
Young, Frank,
Zerbe, A. L., ,
Ziemann, Irvin H., , ,
| {
"redpajama_set_name": "RedPajamaBook"
} | 1,691 |
21y/o snubs West Brom for another club despite better contract offered - report
View: Three changes, one full debut - West Brom predicted XI vs Peterborough
'Disaster' - David Prutton predicts result of West Brom clash v Peterborough
By Olly Allen
3rd Dec, 2021 | 2:10pm
Matt Clarke: Mood in West Brom dressing room is down after recent poor form
West Brom defender Matt Clarke has admitted that the mood in the Hawthorns dressing room has dipped following the club's recent poor form.
Albion have won just one of their last six games, scoring only twice in that run, leaving them fourth in the Championship table and eight points behind the top two.
Valerien Ismael's side face a crucial game against Midlands rivals Coventry City, who currently sit one point below them, on Saturday afternoon (4 December) at the Coventry Building Society Arena.
Ahead of the game, Clarke spoke to the Express & Star about how the squad are feeling.
"There's a determination in the dressing room to put things right," he said. "The results haven't been what we wanted them to be, so the mood isn't sky-high.
"But it's as good as it can be, we're not panicking and there's no apprehension.
"We believe in everything we are doing. We want to be winning games. We want to be performing better than we are. We want to be in a better position than we are.
"It's up to us to find that little bit more. Turning chances into goals and turning draws into victories."
Considering the expectations were so high at the start of the season for West Brom, it is no surprise that the squad are struggling to come to terms with a big dip in form that has left them way off the pace in the fight for automatic promotion.
The frustration in the squad has been clear to see by the fact that Albion have received more red cards in the last three games (two) than they have scored goals (zero).
Jake Livermore was sent off against Huddersfield and then Jayson Molumby picked up two bookings in the space of 10 minutes against Nottingham Forest.
West Brom must now show they are a good team by lifting themselves out of this rut, with the senior players in the squad needing to step up.
However captain Livermore is serving his second suspension of the season which does not set a good example, while 34-year-old Robert Snodgrass has been ostracized by Valerien Ismael.
Of course it is also down to Ismael, and if he cannot do so, then change could be needed at the Hawthorns.
In other West Brom news, David Prutton has predicted the outcome of Saturday's game against Coventry City.
Valerien Ismael confirms West Brom team news v Coventry City
Steve Madeley makes West Brom budget suggestion ahead of January | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,155 |
Barog is a hill station, near City of Solan in Solan district in the Indian state of Himachal Pradesh. The station lies on UNESCO World Heritage Site Kalka–Shimla Railway. Set in the mountains Barog is just 60 km from Chandigarh on the Kalka-Shimla highway.
History
Barog was settled in the early 20th century during the building of the narrow gauge Kalka-Shimla Railway. Currently many residents have their long stays in their houses and flats in Barog. It used to be an important stop in the early decades of the century when the Kalka-Shimla toy train stopped here for an hour while the sahibs and memsahibs enjoyed a lavish lunch.
Geography
Barog is located at at a distance of 60 km from Chandigarh. Shimla, the capital city of Himachal Pradesh is another 65 km from Barog.
Until 2003, National Highway 22 connecting Chandigarh with Shimla passed through Barog. On 6 December 2003, the new section of the highway was inaugurated that would connect the village of Kumarhatti directly to Solan, thus bypassing Barog. This was done to avoid the steep incline to Barog from Kumarhatti.
Barog is located at a height of 1560 metres above the mean sea level. Due to its height, temperatures here range between 23 and 10 °C during summers and between 15 and 5 °C during winters. The summers last from April to July. Winters set in during December and typically last up to February.
Economy
The economy of Barog is primarily dependent upon tourists, who come here because of its cool climate and proximity to Chandigarh. Many hotels including Hotel KorInns and a Himachal tourism resort called Pinewood operate in Barog. Barog is also influenced economically by the nearby Lawrence School, Sanawar.
The local economy mainly depends on the agriculture and especially on tomato growing. Until 1975 the local populace was mostly illiterate, which resulted in stalled economic progress.
Barog also serves as a fitness camp for the Indian National Hockey and other athletic teams.
Barog Tunnel
Barog tunnel is the longest of the 103 operational tunnels on the route of the UNESCO heritage Kalka-Shimla railway, which is 1143.61m long. Barog station is immediately after the tunnel. Trains take about 2.5 minutes to cross this tunnel, running at 25 kilometres per hour.
In the news
Barog station was filmed in Himalaya with Michael Palin in 2004 and on CNN in Season 3 Episode 1, "Punjab, India" of Anthony Bourdain: Parts Unknown during Anthony Bourdain's journey in Kalka-Shimla train, which aired on Sunday April 13, 2014 in USA.
References
Cities and towns in Solan district
Hill stations in Himachal Pradesh | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,509 |
package org.webrtc;
import android.support.test.InstrumentationRegistry;
import android.support.test.filters.SmallTest;
import android.support.test.runner.AndroidJUnit4;
import org.junit.Test;
import org.junit.runner.RunWith;
import org.webrtc.PeerConnectionFactory;
// This test is intended to run on ARM and catch LoadLibrary errors when we load the WebRTC
// JNI. It can't really be setting up calls since ARM emulators are too slow, but instantiating
// a peer connection isn't timing-sensitive, so we can at least do that.
@RunWith(AndroidJUnit4.class)
public class WebRtcJniBootTest {
@Test
@SmallTest
public void testJniLoadsWithoutError() throws InterruptedException {
PeerConnectionFactory.initializeAndroidGlobals(InstrumentationRegistry.getTargetContext(),
false /* videoCodecHwAcceleration */);
PeerConnectionFactory.Options options = new PeerConnectionFactory.Options();
new PeerConnectionFactory(options);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 794 |
Boys basketball Team going from 0-3 to 3-3: How the players view the rest of the season
Captains John Shannon and Jason Giambra, going up for the rebound against Sheehan.
Jess Sanders
The boys basketball team had a rough start to the new season, but is finally turning their season around. With the record 0-3, Coach Ian Kirkpatrick and the players knew that they had to do something to change that. Practicing hard and putting in a hundred percent, they finally see the light in turning their season around.
Co-Captains John Shannon and Jason Giambra, has encouraged their teammates to work together and find ways to keep this winning streak going. They still have a competitive schedule ahead of them but they have no doubt that they won't continue to succeed.
Co-Captain Jason Giambra said "After having a rough start, I have no doubt that the rest of the season will go well. If we continue to work hard in practices and games, it will soon payoff. Last year, the team started the season off the same as we did this year, but we worked hard everyday and that work paid off, considering we went to the state tournament."
Co-Captain John Shannon said, "I started varsity my sophomore and junior year, and I have learned a lot. Our team both years continued to get better with practice and effort. We all do our jobs on the court to get open and pass the ball around and find the best opportunity to shoot. Although that the season didn't start off on how we would hope, I believe that our winning streak will continue. We have some very difficult games coming up but I have no doubt that we will put up a great fight and do what has to be done."
Tyler Griffin, junior who has played all 3 years on freshman, junior varsity, and now varsity, states "I've been able to see how every team has worked together. Playing varsity this year is a lot more competitive, the teams you play are more aggressive and you have to make sure that you stay in the game. We have always done well with our seasons in the past and I see us keep winning in the upcoming games."
All the teammates on the freshman, junior varsity, and varsity team would stay that they give it their all and make it back to states this year as in the past. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,580 |
\section{Introduction}
Video inpainting can help numerous video editing and restoration tasks such as undesired object removal, scratch or damage restoration, and retargeting. More importatnly, and apart from its converntional demands, video inpainting can be used in combination with Augmented Reality (AR) for a greater visual experience; Removing existing items gives more opportunities before overlaying new elements in a scene. Therefore, as a Diminished Reality (DR) technology, it opens up new opportunities to be \textit{paired with recent real-time / deep learning-based AR technologies}. Moreover, there are several semi-online streaming scenarios such as automatic content filtering and visual privacy filtering. Only a small wait will lead to a considerable latency, thus making the speed itself an important issue.
Despite tremendous progress on deep learning-based inpainting of a single image, it is still challenging to extend these methods to video domain due to the additional time dimension. The difficulties coming from complex motions and high requirement on temporal consistency make video inpainting a challenging problem. A straightforward way to perform video inpainting is to apply image inpainting on each frame individually. However, this ignores motion regularities coming from the video dynamics, and is thus incapable of estimating non-trivial appearance changes in image-space over time. Moreover, this scheme inevitably brings temporal inconsistencies and causes severe flickering artifacts. The second row in~\figref{fig:teaser} shows an example of directly applying the state-of-the-art feed-forward image inpainting~\cite{yu2018generative} in a frame-by-frame manner.
\begin{figure}[t]
\def1.0{0.5}
\includegraphics[width=1.0\linewidth]{figures/teaser_rallye.jpg}
\caption{Input video with mask boundaries in red (row-1). Video inpainting results by per-frame image inpainting~\cite{yu2018generative} (row-2), optimization-based method~\cite{huang2016temporally} (row-3), and our method (row-4). \textit{Best viewed when zoomed-in}.}
\label{fig:teaser}
\end{figure}
To address the temporal consistency, several methods have been developed to fill in the missing motion fields; using a greedy selection of local spatio-temporal patches~\cite{shiratori2006video}, a per-frame diffusion-based technique~\cite{matsushita2006full}, or an iterative optimization~\cite{huang2016temporally}. However, the first two methods treat flow estimation to be independent of color estimation~\cite{shiratori2006video,matsushita2006full} and the last relies on time-consuming optimization~\cite{huang2016temporally} (3rd row in \figref{fig:teaser}), which is effective but limits their practicality and flexibility in general scenarios.
One might attempt to maintain temporal consistency by applying a post-processing method. Recently, Lai~\etal~\cite{lai2018learning} proposed a deep CNN model that takes both original and per-frame processed videos as input and produces a temporally consistent video. However, their method is only applicable when those two input videos have a pixel-wise correspondences (\eg colorization), which is not the case for video inpainting.
In this paper, we investigate whether a feed-forward deep network can be adapted to the video inpainting task. Specifically, we attempt to train a model with two core functions: 1) temporal feature aggregation and 2) temporal consistency preserving.
For the \textbf{temporal feature aggregation}, we cast the video inpainting task as a sequential multi-to-single frame inpainting problem. In particular, we introduce a novel 3D-2D feed-forward network which is built upon a 2D-based (image based) encoder-decoder model. The network is designed to collect and refine potential hints from neighbor frames and synthesize semantically-coherent video content in space and time. For the \textbf{temporal consistency}, we propose to use a recurrent feedback and a memory layer (\eg~convoutional LSTM~\cite{xingjian2015convolutional}). In addition, we use a flow loss to learn a warping of the previously synthesized frame and a warping loss to enforce both short-term and long-term consistency in results. Finally, we come up with a single, unified deep CNN model called \textbf{VINet}.
We conduct extensive experiments to validate the contributions of our design choices. We show that our multi-to-single frame formulation produces videos that are much more accurate and visually pleasing than the method of~\cite{yu2018generative}. An example result of our method is shown in the last row of~\figref{fig:teaser}. Our model sequentially processes video frames of arbitrary length and requires no optical flow computation at the test time, thus runs at a near real-time rate.
\paragraph{Contribution.} In summary, our contribution is as follow.
\begin{enumerate}[topsep=0pt,itemsep=0pt]
\item We cast video inpainting as a sequential multi-to-single frame inpainting task and present a novel deep 3D-2D encoder-decoder network. Our method effectively gathers features from neighbor frames and synthesizes missing content based on them.
\item We use a recurrent feedback and a memory layer for the temporal stability. Along with the effective network design, we enforce strong temporal consistency via two losses: flow loss and warping loss.
\item Up to our knowledge, it is the first work to provide a single, unified deep network for the general video inpainting task. We conduct extensive subjective and objective evaluations and show its efficacy. Moreover, we apply our method to video retargeting and super-resolution tasks, demonstrating favorable results.
\end{enumerate}
\section{Related Work}
\label{sec:related}
Significant progress has been made on image inpainting~\cite{ballester2001filling,bertalmio2000image,efros1999texture,pathak2016context,yang2017high,yeh2017semantic,iizuka2017globally,yu2018generative,liu2018image,yu2018free}, to a point of where commercial solutions are now available~\cite{barnes2009patchmatch}. However, video inpainting algorithms have been under-investigated. This is due to the additional time dimension which introduces major challenges such as severe viewpoint changes, temporal consistency preserving, and high computational complexity. Most recent methods found in the literature address these issues using either object-based or patch-based approaches.
In object-based methods, a pre-processing is required to split a video into foreground objects and background, and it is followed by an independent reconstruction and merging step at the end of algorithms. Previous efforts which fall under this category are homography-based algorithms that are based on the graph-cut~\cite{granados2012not,granados2012background}. However, the major limitation of these object-based methods is that the synthesized content has to be copied from the visible regions. Therefore, these methods are mostly vulnerable to abrupt appearance changes such as scale variations, \eg~when an object moves away from the camera.
In patch-based methods, the patches from known regions are used to fill in a mask region. For example, Patwardhan~\etal~\cite{patwardhan2005video,patwardhan2007video} extend the well-known texture synthesis technique~\cite{efros1999texture} to video inpainting. However, these methods either assume static cameras~\cite{patwardhan2005video} or constrained camera motion~\cite{patwardhan2007video} and are based on a greedy patch-filling process where the early errors are inevitably propagated, yielding globally inconsistent outputs.
To ensure the global consistency, patch-based algorithms have been cast as a global optimization problem. Wexler~\etal~\cite{wexler2004space} present a method that optimizes a global energy minimization problem for 3D spatio-temporal patches by alternating between patch search and reconstruction steps. Newson~\etal~\cite{newson2014video} extend this by developing a spatio-temporal version of PatchMatch~\cite{barnes2009patchmatch} to strengthen the temporal coherence and speed up the patch matching. Recently, Huang~\etal~\cite{huang2016temporally} modify the energy term of ~\cite{wexler2004space} by adding an optical flow term to enforce temporal consistency. Although these methods are effective, their biggest limitations are high computational complexity and the absolute dependence upon the pre-computed optical flow which cannot be guaranteed to be accurate in complex sequences.
To tackle these issues, we propose a deep learning based method for video inpainting. To better exploit temporal information coming from multiple frames and be highly efficient, we construct a 3D-2D encoder-decoder model, that can provide traceable features revealed from the video dynamics. It takes total 6 frames as input; 5 source frames and 1 reference frame (\ie the frame to be inpainted). We learn the feature flow between frames to deal with both hole-filling and coherence. The still-unknown regions are synthesized in a semantically natural way based on the surrounding context. We argue that our method provides a better prospect than the previous optimization-based techniques in that deep CNNs are excellent at learning spatial semantics and temporal dynamics from an ever-growing vast amount of video data. To our best knowledge, this is the first work that deeply addresses the general video inpainting problem via a deep CNN model.
\begin{figure*}[t]
\begin{tabular}{@{}c@{}}
\includegraphics[width=0.96\linewidth]{./figures/vinet_v4.png} \\
\end{tabular}
\caption{ {\bf The overview of VINet.} Our network takes in multiple frames (${X}_{t-6}, {X}_{t-3}$, ${X}_{t}$, ${X}_{t+3}, {X}_{t+6}$) and the previously generated frame ($\hat{Y}_{t-1}$), and generates the inpainted frame ($\hat{Y}_{t}$) as well as the flow map ($\hat{W}_{t\Rightarrow t-1}$). We employ both flow sub-networks and mask sub-networks at 4 scales (1/8, 1/4, 1/2, and 1) to aggregate and synthesize feature points progressively. For temporal consistency, we use a recurrent feedback and a temporal memory layer (ConvLSTM) along with two losses: flow loss and warp loss. The orange arrows denote the $\times 2$ upsampling for residual flow learning as in~\cite{sun2018pwc} for 5 streams, while the thinner orange arrow is for only the stream from $\hat{Y}_{t-1}$. The mask sub-networks are omitted in the figure for the simplicity.}
\label{fig:architecture}
\end{figure*}
\section{Method}
\label{sec:method}
\subsection{Problem Formulation}
Video inpainting aims to fill in arbitrary missing regions in video frames $X_1^T := \{X_1, X_2, ..., X_T\}$. The reconstructed regions should be either accurate as in the ground-truth frames $Y_1^T := \{Y_1, Y_2, ..., Y_T\}$ and consistent in space and time. We formulate the video inpainting problem as learning a mapping function from $X_1^T$ to the output $\hat{Y}_1^T := \{ \hat{Y}_1, \hat{Y}_2, ...,\hat{Y}_T\}$ such that the conditional distribution $p(\hat{Y}_1^T|X_1^T)$ is identical to $p(Y_1^T|X_1^T)$. Through matching the conditional distributions, the network learns to generate realistic and temporally-consistent output sequences. To simplify the problem, we make a Markov assumption where we factorize the conditional distribution to a product form. In this form, the naive \textit{frame-by-frame} inpainting can be formulated as
\begin{equation}
p(\hat{Y_1^T}|X_1^T) = \prod_{t=1}^{T} p(\hat{Y}_t|X_t).
\end{equation}
However, to obtain visually pleasing video results, we argue that the generation of $t$-th frame $\hat{Y_t}$ should be consistent with 1) spatio-temporal neighbor frames $X_{t-N}^{t+N}$ where $N$ denotes a temporal radius, 2) the previously generated frame $\hat{Y}_{t-1}$ and 3) all previous history encoded in a recurrent memory $M_t$. Thus, we propose to learn the conditional distribution of
\begin{equation}
p(\hat{Y_1^T}|X_1^T) = \prod_{t=1}^{T} p(\hat{Y}_t|X_{t-N}^{t+N}, \hat{Y}_{t-1}, M_t).
\label{eqn:our_formulation}
\end{equation}
In our experiments, we set $N$ to 2, taking two lagging and two leading frames to recover the current frame. We sample frames with a temporal stride 3, such that $X_{t-N}^{t+N} := \{ {X}_{t-6}, {X}_{t-3}, {X}_{t}, {X}_{t+3}, {X}_{t+6}\}$. We want to recover the current frame by both aggregating information from neighbor frames and synthesizing totally blind regions jointly.
At the same time, the output is enforced to be temporally consistent with the past predictions by the recurrent feedback ($\hat{Y}_{t-1}$) and the memory ($M_t$). We train a deep network $D$ to model the conditional distribution $p(\hat{Y}_t|X_{t-N}^{t+N}, \hat{Y}_{t-1}, M_t)$ as $\hat{Y}_t=D(X_{t-N}^{t+N}, \hat{Y}_{t-1}, M_t)$. We obtain the final output $\hat{Y}_1^T$ by applying the function $D$ in an autoregressive manner. Our multi-to-single frame formulation outperforms a single-frame baseline and even produces results comparable with the optimization-based method, as described in \secref{sec:experiments}.
\subsection{Network Design}
Our full model (VINet) jointly learns to inpaint the video and maintain temporal consistency. The overview of VINet is illustrated in ~\figref{fig:architecture}.
\subsubsection{Multi-to-Single Frame Video Inpainting}
In videos, the occluded or removed parts in a frame are often revealed in the past/future frames as the objects move and the viewpoint changes. If such hints exist in the temporal radius, those disclosed content can be borrowed to recover the current frame. Otherwise, the still-unknown regions should be synthesized. To achieve this, we construct our model as an encoder-decoder network that learns such temporal feature aggregation and single-frame inpainting simultaneously. The network is designed to be fully convolutional, which can handle arbitrary size input.
\paragraph{Source and reference encoders.}
The encoder is a multiple-tower network with source and reference streams. The source stream takes past and future frames with the inpainting masks as input. For the reference stream, the current frame and its inpainting mask are provided. We concatenate the image frames and the masks along the channel axis, and feed into the encoder.
In practice, we use a 6-tower encoder: 5 source streams with weight-sharing that take two lagging (${X}_{t-6}, {X}_{t-3}$) and two leading frames (${X}_{t+3}, {X}_{t+6}$), and the previously generated frame ($\hat{Y}_{t-1}$), and 1 reference stream. The source features that are non-overlapping with the reference features can be borrowed to inpaint the missing regions by the following feature flow learning and learnable feature composition.
\paragraph{Feature flow learning.}
Before directly combining the source and reference features, we propose to explicitly align the feature points. This strategy helps our model easily borrow traceable features from the neighbor frames. To achieve this, we insert flow sub-networks to estimate the flows between the source and reference feature maps in four different spatial scales (1/8, 1/4, 1/2, and 1). We adopt the coarse-to-fine structure of PWCNet~\cite{sun2018pwc}. The explicit flow supervision is only given at the finest scale (\ie~1) and \textit{only between} the consecutive two frames, where we extract the pseudo-ground-truth flow ${W}_{t\Rightarrow t-1}$ between ${Y}_{t}$ and ${Y}_{t-1}$ using FlowNet2~\cite{ilg2017flownet}.
\paragraph{Learnable Feature Composition.}
Given the aligned feature maps from the five source streams, they are concatenated along the time dimension and fed into a $5\times3\times3$ (THW) convolution layer that produces a spatio-temporally aggregated feature map $F_{s'}$ with the time dimension of 1. This is designed to dynamically select source feature points across the time axis, by highlighting the features complementary to the reference features and ignoring otherwise. For each 4 scales, we employ a mask sub-network to combine the aggregated feature map $F_{s'}$ with the reference feature map $F_{r}$. The mask sub-network consists of three convolution layers and takes the absolute difference of the two feature maps $|F_{s'} - F_{r}|$ as input and produces single channel composition mask $m$, as suggested in~\cite{chen2017coherent}. By using the mask, we can gradually combine the warped features and the reference features. At the scale of 1/8, the composition is done by
\begin{equation}
F_{c_{1/8}} = (1-m_{1/8}) \odot F_{r_{1/8}} + m_{1/8} \odot F_{s'_{1/8}},
\label{eqn:feature_composition}
\end{equation}
where $\odot$ is the element-wise product operator.
\paragraph{Decoder.}
To pass image details to the decoder, we employ skip connections as in U-net~\cite{ronneberger2015u}. To prevent the concern raised by~\cite{yu2018free} that skip connections contain zero values at the masked region, our skip-connections pass the composite features similarly to \eqnref{eqn:feature_composition}, as
\begin{eqnarray}
F_{c_{1/4}} = (1-m_{1/4}) \odot F_{r_{1/4}} + m_{1/4} \odot F_{s'_{1/4}},\\
F_{c_{1/2}} = (1-m_{1/2}) \odot F_{r_{1/2}} + m_{1/2} \odot F_{s'_{1/2}}.
\end{eqnarray}
At the finest scale, the estimated optical flow $\hat{W}_{t\Rightarrow t-1}$ is used to warp the previous output $\hat{Y}_{t-1}$ to the current raw output $\hat{Y'}_{t}$. We then blend this warped image and the raw output with the composition mask $m_{1}$, to obtain our final output $\hat{Y}_{t}$ as
\begin{equation}
\hat{Y}_{t} = (1-m_{1}) \odot \hat{Y'}_{t} + m_{1} \odot \hat{W}_{t\Rightarrow t-1}(\hat{Y}_{t-1}).
\end{equation}
\subsubsection{Recurrence and Memory}
To strongly enforce the temporal coherence on the video output, we propose to use the recurrent feedback loop ($\hat{Y}_{t-1}$) and the temporal memory layer ($M_{t}$) as formulated in \eqnref{eqn:our_formulation}.
Our formulation encourages the current output to be conditional to the previous output frame. The knowledge from the previous output encourages the traceable features to be kept unchanged, while the untraceable (\eg~occlusion) points to be synthesized. This not only helps the output to be consistent along the motion trajectories but also avoids ghosting artifacts at occlusions or motion discontinuities.
While the recurrent feedback connects the consecutive frames, filling in the large holes requires more long-term (\eg~5 frames) knowledge.
At this point, the temporal memory layer can help to connect internal features from different time steps in the long term. We adopt a convolutional LSTM (ConvLSTM) layer and a warping loss as suggested in~\cite{lai2018learning}. In particular, we feed the composite feature $F_{c}$ at the scale 1/8 to the ConvLSTM at every time step.
\subsection{Losses}
We train our network to minimize the following loss function,
\begin{eqnarray}
\mathcal{L} = \lambda_{R}\mathcal{L}_{R}+\lambda_{F}\mathcal{L}_{F}+\lambda_{W}\mathcal{L}_{W},
\label{eqn:total_loss}
\end{eqnarray}
where $\mathcal{L}_{R}$ is the reconstruction loss, $\mathcal{L}_{F}$ is the flow estimation loss, and $\mathcal{L}_{W}$ is the warping loss. The balancing weights $\lambda_{R}, \lambda_{F}, \lambda_{W}$ are set to 1, 10, 1 respectively throughout the experiments. For the temporal losses $\mathcal{L}_{F}$ and $\mathcal{L}_{W}$, we set the number of recurrences as 5 $(T = 5)$.
$\mathcal{L}_{R}$ consists of two terms, $\mathcal{L}_{1}$ and $\mathcal{L}_{ssim}$,
\begin{eqnarray}
\pazocal{L}_{1} = \left \| \hat{Y_t} - Y_t \right\|_{1}, \\
\pazocal{L}_{ssim} = (\frac{{(2\mu_{\hat{Y_t}} \mu_{Y_t} + c_1 )(2\sigma_{\hat{Y_t} {Y_t}} + c_2)}}
{{(\mu_{\hat{Y_t}}^2+\mu_{Y_t}^2 +c_1)(\sigma_{\hat{Y_t}}^2+\sigma_{Y_t}^2 +c_2)}}), \\
\pazocal{L}_{R} = \pazocal{L}_{1} + \pazocal{L}_{ssim},
\end{eqnarray}
where $\hat{Y_t}, Y_t$ denote the predicted frame and the ground-truth frame respectively. $\mu, \sigma$ denote the average, variance, respectively. $c_1, c_2$ denote the stabilization constants which are respectively set to $0.01^2, 0.03^2$.
The flow loss $\mathcal{L}_{F}$ is defined as
\begin{equation}
\sum\limits_{t=2}^{T} (\left \| {W}_{t\Rightarrow t-1} - \hat{W}_{t\Rightarrow t-1} \right\|_{1} + \left \| {Y}_{t} - \hat{W}_{t\Rightarrow t-1}({Y}_{t-1}) \right\|_{1}),
\label{eqn:flowloss}
\end{equation}
where ${W}_{t\Rightarrow t-1}$ is the pseudo-ground-truth backward flow between the target frames, ${Y}_{t}$ and ${Y}_{t-1}$, extracted by FlowNet2~\cite{ilg2017flownet}. In \eqnref{eqn:flowloss}, the first term is the endpoint error between the groundturth and the estimated flow, and the second is the warping error when the flow is used to warp the previous target frame to the next target frame.
The warping loss $\mathcal{L}_{W}$ includes $\mathcal{L}_{st}$ and $\mathcal{L}_{lt}$ as,
\begin{eqnarray}
\pazocal{L}_{st} = \sum\limits_{t=2}^T M_{t \Rightarrow t-1} \left \| \hat{Y_t} - {W}_{t \Rightarrow t-1}({Y}_{t-1})\right\|_{1}, \\
\pazocal{L}_{lt} = \sum\limits_{t=2}^T M_{t \Rightarrow 1} \left \| \hat{Y_t} - {W}_{t \Rightarrow 1}({Y}_{1})\right\|_{1}, \\
\pazocal{L}_{W} = \pazocal{L}_{st} + \pazocal{L}_{lt}.
\end{eqnarray}
We follow the protocol in~\cite{lai2018learning} that uses FlowNet2~\cite{ilg2017flownet} to obtain $M_{t \Rightarrow t-1}$ and $W_{t-1}$, which respectively denote the binary occlusion mask and the backward optical flow between the target frames ${Y}_{t}$ and ${Y}_{t-1}$. We adopt both short-term and long-term temporal losses. Note that we use ground-truth target frames in the warping operation since the synthesizing ability is imperfect during training.
\subsection{Two-Stage Training}
We employ a two-stage training scheme that gradually learns the core functionalities for video inpainting; 1) We first train the model without the recurrent feedback and memory to focus on learning the temporal feature aggregation. At this stage, we only use the reconstruction loss $\mathcal{L}_{R}$; 2) We then add the recurrent feedback and the ConvLSTM layer, and fine-tune the model using the full loss function (\eqnref{eqn:total_loss}) for temporally coherent predictions. We use videos in the Youtube-VOS dataset~\cite{xu2018youtube} as ground-truth for the training. It is a large-scale dataset for video object segmentation containing 4000+ YouTube videos with 70+ common objects. All video frames are resized to $256\times256$ pixels for training and testing.
\paragraph{Video mask dataset.}
In general video inpainting, the spatio-temporal holes consist in diverse motion and shape changes. To simulate this complexity during training, we create the following four types of video masks.
\begin{enumerate}[topsep=1pt,itemsep=1pt]
\item {Random square}:
We randomly mask a square box in each frame. The visible regions each of input frames are mostly complementary so that the network can clearly learn how to align, copy, and paste neighboring feature points.
\item {Flying square}:
The motion of the inpainting holes is rather regularized than random in real scenarios. To simulate such regularity, we shift a square by a uniform step size in one direction across the input frames.
\item {Arbitrary mask}: To simulate diverse hole shapes and sizes, we use the irregular mask dataset~\cite{liu2018image} which consists of random streaks and holes of arbitrary shapes. During training, we apply random transformations (translation, rotation, scaling, sheering).
\item {Video object mask}: In the context of the video object removal task, masks with the most realistic appearance and motion can be obtained from video object segmentation datasets. We use the foreground segmentation masks of the YouTube-VOS dataset~\cite{xu2018youtube}.
\end{enumerate}
\subsection{Inference}
We assume that the inpainting masks for all video frames are given. To avoid any data overlap between training and testing, we obtain object masks from the DAVIS dataset~\cite{perazzi2016benchmark, pont20172017}, the public benchmark dataset for video object segmentation. It contains dynamic scenes, complex camera movements, motion blur effects, and large occlusions. The inpainting mask is constructed by dilating the ground-truth segmentation mask. Our method processes frames recursively in a sliding window manner.
\begin{figure}[t]
\def1.0{1.0}
\begin{tabular}{c@{}}
\includegraphics[width=1.0\columnwidth]{./figures/visual/people.jpg} \\
(a) \\
\includegraphics[width=1.0\columnwidth]{./figures/visual/hamster.jpg}\\
(b) \\
\end{tabular}
\caption{\textbf{Visualization of the learned feature composition.} Input frames are on the odd rows, and corresponding feature flows referential to the center, and the inpainted frame are on the even rows. Our network successfully aligns and integrates the source features to fill in the large and complex hole in the reference frame.}
\label{fig:vis_feature_comp}
\end{figure}
\subsection{Implementation Details}
Our model is implemented using Pytorch v0.4, CUDNN v7.0, CUDA v9.0. It run on the hardware with Intel(R) Xeon(R) (2.10GHz) CPU and NVIDIA GTX 1080 Ti GPU. The model runs at 12.5 fps on a GPU for frames of $256\times256$ pixels. We use Adam optimizer with $\beta$ = (0.9, 0.999) and a fixed learning rate 1e-4. We train our model from scatch. The first and second training stage takes about 1 day each using four NVIDIA GTX 1080 Ti GPUs.
\section{Experiments}
\label{sec:experiments}
In this section, we conduct experiments to analyze our two major design choices. Specifically, we visualize the learned multi-to-single mechanism and show the impact of the added recurrence and memory. We then evaluate our video results both quantitatively and qualitatively, compared with the state-of-the-art baselines. Finally, we demonstrate the applicability of our framework on video retargeting and video super-resolution tasks.
\paragraph{Baselines.} We compare our approach to two state-of-the-art baselines in the literature by running their test codes with our testing videos and masks.
\begin{itemize}[topsep=1pt,itemsep=1pt]
\item Yu~\etal~\cite{yu2018generative}: A feed-forward CNN based method, which is designed for single image inpainting. We processes videos frame-by-frame without using any temporal information.
\item Huang~\etal~\cite{huang2016temporally}: An optimization-based video completion method, which jointly estimates global flow and color. It requires on-the-fly optical flow computation and is extremely time-consuming.
\end{itemize}
\subsection{Visualization of Learned Feature Composition}
\figref{fig:vis_feature_comp} shows that the proposed model explicitly borrows visible neighbor features to synthesize the missing content.
For the visualization, we take the model of the first training stage and plot the learned feature flow from each of the four source streams to the reference stream, at $128\times128$ pixel resolution.
We observe that even with a large and complex hole in the reference (center) frame, our network is able to align the source feature maps to the reference and integrate them to fill in the hole. Even without an explicit flow supervision, our flow sub-network is able to warp the feature points in visible regions and shrink the unhelpful zero features in masked regions. Moreover, these potential hints are adjusted according to the spatio-temporal semantics, rather than copied-and-pasted in a fixed manner. One example is shown in \figref{fig:vis_feature_comp}-(b) where the eyes of the hamster are synthesized \textit{half-closed}.
\subsection{Improvement on Temporal Consistency}
\label{sec:TC}
We compare the temporal consistencies of our video results before and after adding the recurrent feedback and the convLSTM. To validate the effectiveness of our method, we also compare with the two representative baselines mentioned above~\cite{yu2018generative,huang2016temporally}. Since the Sintel dataset~\cite{butler2012naturalistic} provides ground-truth optical flows, we use it to quantitatively measure the \textit{flow warping errors}\cite{lai2018learning}. We use the object masks in the DAVIS dataset~\cite{perazzi2016benchmark, pont20172017} as our inpainting mask sequences.
We take 32 frames each from 21 videos in Sintel to constitute our inputs and experiment for five trials. For each trial, we randomly select 21 videos of length 32+ from DAVIS to create corresponding mask sequences and keep them unchanged for all the methods.
In \tabref{tab:warping}, we report the flow warping errors averaged over the videos and trials. It shows that our full model outperforms other baselines by large margins. Even the global (heavy) optimization method~\cite{huang2016temporally} performs marginally better than our 1st-stage method and has a much larger error than our full model. Not surprisingly, Yu~\etal's method turns out to be the least temporally consistent. Note that the error of our full model is reduced by a factor of 10 after adding the recurrent feedback and the convLSTM layer, implying that they significantly improve the temporal stability in the short and long term.
\begin{table}
\centering
\setlength{\tabcolsep}{10pt}{
\normalsize
\begin{tabular}{ l|c }
\hline
& DAVIS masks on Sintel frames \\
\hline
Frame-by-frame~\cite{yu2018generative} & 0.0429\\
Optimization~\cite{huang2016temporally} & 0.0343\\
\textbf{VINet} (agg. only) & 0.0383 \\
\textbf{VINet} (agg. + T.C.) & \textbf{0.0015} \\
\hline
\end{tabular}
}
\caption{\textbf{Flow warping errors.} We evaluate the flow warping errors on the Sintel dataset using 21 videos and ground truth flows.}
\label{tab:warping}
\end{table}
\begin{table}
\centering
\setlength{\tabcolsep}{10pt}{
\normalsize
\begin{tabular}{ l|c }
\hline
& DAVIS masks on DAVIS frames \\
\hline
Frame-by-frame~\cite{yu2018generative} & 0.0080 \\
Optimization~\cite{huang2016temporally} & 0.0053\\
\textbf{VINet} (agg. only) & 0.0073 \\
\textbf{VINet} (agg. + T.C.) & \textbf{0.0046} \\
\hline
\end{tabular}
}
\caption{\textbf{FID scores.} We evaluate the FID scores on the DAVIS dataset using 20 videos.}
\label{tab:FID}
\end{table}
\begin{figure*}[t]
\begin{center}
\def1.0{1.0}
\begin{tabular}{@{}c@{\hskip 0.01\linewidth}c@{}}
\includegraphics[width=0.495\linewidth]{./figures/results/roller4.jpg}&
\includegraphics[width=0.495\linewidth]{./figures/results/drift-turn4.jpg}\\
\includegraphics[width=0.495\linewidth]{./figures/results/swing4.jpg} & \includegraphics[width=0.495\linewidth]{./figures/results/breakdance4.jpg}\\
\includegraphics[width=0.495\linewidth]{./figures/results/gray4.jpg} &
\includegraphics[width=0.495\linewidth]{./figures/results/tennis4.jpg}\\
\end{tabular}
\end{center}
\caption{\textbf{Object removal from DAVIS video sequences.} For each input sequence, we show representative frames with mask boundaries in red. We show the inpainted results using our method in even rows.}
\label{fig:obj_removal}
\end{figure*}
\subsection{Spatio-Temporal Video Quality}
Wang~\etal~\cite{wang2018video} proposed a video version of the inception score (FID) to quantitatively evaluate the quality of video generation. We take this metric to evaluate the quality of video inpainting as it measures the spatio-temporal quality in a perceptual level. As in~\cite{wang2018video}, we follow the protocol that uses the I3D network~\cite{carreira2017quo} pretrained on a video recognition task to measure the distance between the spatio-temporal features extracted from the output videos and the ground-truth videos.
For this experiment, we take 20 videos in the DAVIS dataset. For each video, we ensure to choose a different video out of the other 19 videos to make a mask sequence, so that we have the setting where our algorithm is supposed to recover the original videos rather than remove any parts. We use the first 64 frames for both input and mask videos.
We run five trials as in~\secref{sec:TC} and average the FID scores over the videos and trials.
\tabref{tab:FID} summarizes the results. Our method has the smallest FID among the compared methods. This implies that our method achieves both better visual quality and temporal consistency.
\begin{figure*}[t]
\begin{center}
\def1.0{1.0}
\begin{tabular}{@{}c@{\hskip 0.008\linewidth}c@{\hskip 0.008\linewidth}c@{}}
\includegraphics[width=0.193\linewidth]{./figures/retarget/inputs_r.jpg}&
\includegraphics[width=0.397\linewidth]{./figures/retarget/talls_r.jpg}&
\includegraphics[width=0.39\linewidth]{./figures/retarget/fats_r.jpg}\\
{(a) First input frame} & {(b) Horizontally shrunk frames} & {(c) Vertically shrunk frames} \\
\end{tabular}
\end{center}
\caption{\textbf{Extension to video retargeting.} (a) Original first frame. (b) Horizontally shrunk frames. (c) Vertically shrunk frames.}
\label{fig:retarget}
\end{figure*}
\begin{figure}[t]
\begin{center}
\def1.0{1.0}
\begin{tabular}{@{}c@{}}
\includegraphics[width=\linewidth]{./figures/user_study_v3.png}\\
\end{tabular}
\end{center}
\vspace{-6mm}
\caption{\textbf{User study results}.}
\label{fig:user}
\end{figure}
\subsection{User Study on Video Object Removal}
We apply our approach to remove dynamically moving objects in videos. We use 24 videos from the DAVIS dataset~\cite{perazzi2016benchmark,pont20172017} of which the names are listed in \figref{fig:user}. Examples of our results are in \figref{fig:obj_removal}. We perform a human subjective test for evaluating the visual quality of inpainted videos. We compare our method with the strong optimization baseline~\cite{huang2016temporally} which is specifically aimed for the video completion task.
In each testing case, we show the original input video, our removal result and the result of Huang~\etal on the same screen. The order of the two removal video results is shuffled. To ensure that a user has enough time to distinguish the difference and make a careful judge, we play all the video results once at the original speed and then once at $0.5\times$ speed. Also, a user allows seeing videos multiple times. Each participant is asked to choose a preferred result or tie. A total of 30 users participated in this study. We specifically ask each participant to check for both image quality and temporal consistency. The user study results are summarized in \figref{fig:user}. It shows that, while there are different preferences across video samples, our method is preferred more often by the participants.
\subsection{Application to Video Retargeting}
Video retargeting aims to adjust the aspect ratio (or size) of frames to fit the target aspect ratio while maintaining salient content in a video. We propose to solve video retargeting by \textit{removing and then adding}, which is a potential pipeline where our framework would run in combination with other AR (\ie overlaying) technologies. Specifically, we first remove the salient content by inpainting the background, resize the inpainted frames into the target aspect ratio, and then overlay the salient content after the desirable rescaling. To simplify the settings, we target to horizontally or vertically shrink the frames while keeping the original aspect ratio of the moving object. The saliency masks can be automatically estimated, for example, by a feed-forward CNN~\cite{cho2017weakly}, however we assume a more constrained scenario where the saliency masks are given as the object segmentation masks for all frames. Our method yields little warble and jittering over time and produces natural video sequences. \figref{fig:retarget} shows examples of the retargeted frames.
\subsection{Limitation}
We observe color saturation artifacts when there is a large and long occlusion in a video. The discrepancy error of the synthesized color propagates over time, causing inaccurate warping. The regions that have not been revealed in the temporal radius is synthesized blurry. Also, due to the limited memory footprint, we only experimented with $256 \times 256$ px frames.
\section{Conclusion}
In this paper, we propose a novel framework for video inpainting. Based on the multi-to-single encoder-decoder network, our model learns to aggregate and align the feature maps from neighbor frames to inpaint videos. We use the recurrent feedback and the temporal memory to encourage temporally coherent output. Our extensive experiments demonstrate that our method achieves superior visual quality than the state-of-the-art image inpainting solution and performs favorably against an optimization method both qualitatively and quantitatively. Despite some limitations, we argue that a well-posed feed-forward network has a great potential to avoid computation-heavy optimization method and boosts its applicability in many related vision tasks.
\paragraph{Acknowledgements}
Dahun Kim was partially supported by Global Ph.D. Fellowship Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018H1A2A1062075).
{\small
\bibliographystyle{ieee}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,828 |
Leluchów (j. łemkowski Лелюхів) – wieś w Polsce położona w województwie małopolskim, w powiecie nowosądeckim, w gminie Muszyna. W latach 1975–1998 miejscowość administracyjnie należała do województwa nowosądeckiego.
Na jej terenie znajduje się cerkiew pw. św. Dymitra z 1861 z zachowanym wyposażeniem z XIX wieku. Cerkiew jest siedliskiem podkowca małego, nietoperza zagrożonego wyginięciem w Europie (kolonia liczy ok. 135 osobników).
Jest ważnym punktem na szlaku turystycznym wiodącym przez Beskid Niski.
Miejscowość jest również znana z piosenki Starego Dobrego Małżeństwa pt. Leluchów, do słów autorstwa Adama Ziemianina.
Historia
18 sierpnia 1876 przekazano do użytku linię kolejową nr 96 relacji Tarnów – Tuchów – Stróże – Nowy Sącz – Leluchów (dł. 145 km). Uzyskała ona połączenie z węgierską linią preszowską. Linia leluchowska miała charakter strategiczny. Było to połączenie transkarpackie – dla handlu i dla transportu wojsk na wypadek wojny z Rosją.
Demografia
Ludność według spisów powszechnych w 2009 roku:
Szlaki turystyczne
Żegiestów-Zdrój – Pusta Wielka (1061 m) – Runek (1082 m) – Przełęcz Krzyżowa – Krynica-Zdrój – Góra Parkowa (741 m) – Powroźnik – Leluchów
Zobacz też
Leluchów (przystanek kolejowy)
Przejście graniczne Leluchów-Čirč
Przypisy
Linki zewnętrzne
Wsie w powiecie nowosądeckim | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 410 |
{"url":"https:\/\/zxi.mytechroad.com\/blog\/","text":"There are\u00a0n\u00a0cars traveling at different speeds in the same direction along a one-lane road. You are given an array\u00a0cars\u00a0of length\u00a0n, where\u00a0cars[i] = [positioni, speedi]\u00a0represents:\n\n\u2022 positioni\u00a0is the distance between the\u00a0ith\u00a0car and the beginning of the road in meters. It is guaranteed that\u00a0positioni\u00a0< positioni+1.\n\u2022 speedi\u00a0is the initial speed of the\u00a0ith\u00a0car in meters per second.\n\nFor simplicity, cars can be considered as points moving along the number line. Two cars collide when they occupy the same position. Once a car collides with another car, they unite and form a single car fleet. The cars in the formed fleet will have the same position and the same speed, which is the initial speed of the\u00a0slowest\u00a0car in the fleet.\n\nReturn an array\u00a0answer, where\u00a0answer[i]\u00a0is the time, in seconds, at which the\u00a0ith\u00a0car collides with the next car, or\u00a0-1\u00a0if the car does not collide with the next car. Answers within\u00a010-5\u00a0of the actual answers are accepted.\n\nExample 1:\n\nInput: cars = [[1,2],[2,1],[4,3],[7,2]]\nOutput: [1.00000,-1.00000,3.00000,-1.00000]\nExplanation: After exactly one second, the first car will collide with the second car, and form a car fleet with speed 1 m\/s. After exactly 3 seconds, the third car will collide with the fourth car, and form a car fleet with speed 2 m\/s.\n\n\nExample 2:\n\nInput: cars = [[3,4],[5,4],[6,3],[9,1]]\nOutput: [2.00000,1.00000,1.50000,-1.00000]\n\n\nConstraints:\n\n\u2022 1 <= cars.length <= 105\n\u2022 1 <= positioni, speedi\u00a0<= 106\n\u2022 positioni\u00a0< positioni+1\n\n## Solution: Monotonic Stack\n\nKey observation: If my speed is slower than the speed of the previous car, not only mine but also all cars behind me will NEVER be able to catch\/collide with the previous car. Such that we can throw it away.\n\nMaintain a stack that stores the indices of cars with increasing speed.\n\nProcess car from right to left, for each car, pop the stack (throw the fastest car away) if any of the following conditions hold.\n1) speed <= stack.top().speed\n2) There are more than one car before me and it takes more than to collide the fastest car than time the fastest took to collide.\n\nTime complexity: O(n)\nSpace complexity: O(n)\n\n## C++\n\nYou are given two arrays of integers\u00a0nums1\u00a0and\u00a0nums2, possibly of different lengths. The values in the arrays are between\u00a01\u00a0and\u00a06, inclusive.\n\nIn one operation, you can change any integer\u2019s value in\u00a0any\u00a0of the arrays to\u00a0any\u00a0value between\u00a01\u00a0and\u00a06, inclusive.\n\nReturn\u00a0the minimum number of operations required to make the sum of values in\u00a0nums1\u00a0equal to the sum of values in\u00a0nums2.\u00a0Return\u00a0-1\u200b\u200b\u200b\u200b\u200b if it is not possible to make the sum of the two arrays equal.\n\nExample 1:\n\nInput: nums1 = [1,2,3,4,5,6], nums2 = [1,1,2,2,2,2]\nOutput: 3\nExplanation: You can make the sums of nums1 and nums2 equal with 3 operations. All indices are 0-indexed.\n- Change nums2[0] to 6. nums1 = [1,2,3,4,5,6], nums2 = [6,1,2,2,2,2].\n- Change nums1[5] to 1. nums1 = [1,2,3,4,5,1], nums2 = [6,1,2,2,2,2].\n- Change nums1[2] to 2. nums1 = [1,2,2,4,5,1], nums2 = [6,1,2,2,2,2].\n\n\nExample 2:\n\nInput: nums1 = [1,1,1,1,1,1,1], nums2 = [6]\nOutput: -1\nExplanation: There is no way to decrease the sum of nums1 or to increase the sum of nums2 to make them equal.\n\n\nExample 3:\n\nInput: nums1 = [6,6], nums2 = [1]\nOutput: 3\nExplanation: You can make the sums of nums1 and nums2 equal with 3 operations. All indices are 0-indexed.\n- Change nums1[0] to 2. nums1 = [2,6], nums2 = [1].\n- Change nums1[1] to 2. nums1 = [2,2], nums2 = [1].\n- Change nums2[0] to 4. nums1 = [2,2], nums2 = [4].\n\n\nConstraints:\n\n\u2022 1 <= nums1.length, nums2.length <= 105\n\u2022 1 <= nums1[i], nums2[i] <= 6\n\n## Solution: Greedy\n\nAssuming sum(nums1) < sum(nums2),\nsort both arrays\n* scan nums1 from left to right, we need to increase the value form the smallest one.\n* scan nums2 from right to left, we need to decrease the value from the largest one.\nEach time, select the one with the largest delta to change.\n\ne.g. nums1[i] = 2, nums[j] = 4, delta1 = 6 \u2013 2 = 4, delta2 = 4 \u2013 1 = 3, Increase 2 to 6 instead of decreasing 4 to 1.\n\nTime complexity: O(mlogm + nlogn)\nSpace complexity: O(1)\n\n## C++\n\nYou would like to make dessert and are preparing to buy the ingredients. You have\u00a0n\u00a0ice cream base flavors and\u00a0m\u00a0types of toppings to choose from. You must follow these rules when making your dessert:\n\n\u2022 There must be\u00a0exactly one\u00a0ice cream base.\n\u2022 You can add\u00a0one or more\u00a0types of topping or have no toppings at all.\n\u2022 There are\u00a0at most two\u00a0of\u00a0each type\u00a0of topping.\n\nYou are given three inputs:\n\n\u2022 baseCosts, an integer array of length\u00a0n, where each\u00a0baseCosts[i]\u00a0represents the price of the\u00a0ith\u00a0ice cream base flavor.\n\u2022 toppingCosts, an integer array of length\u00a0m, where each\u00a0toppingCosts[i]\u00a0is the price of\u00a0one\u00a0of the\u00a0ith\u00a0topping.\n\u2022 target, an integer representing your target price for dessert.\n\nYou want to make a dessert with a total cost as close to\u00a0target\u00a0as possible.\n\nReturn\u00a0the closest possible cost of the dessert to\u00a0target. If there are multiple, return\u00a0the\u00a0lower\u00a0one.\n\nExample 1:\n\nInput: baseCosts = [1,7], toppingCosts = [3,4], target = 10\nOutput: 10\nExplanation: Consider the following combination (all 0-indexed):\n- Choose base 1: cost 7\n- Take 1 of topping 0: cost 1 x 3 = 3\n- Take 0 of topping 1: cost 0 x 4 = 0\nTotal: 7 + 3 + 0 = 10.\n\n\nExample 2:\n\nInput: baseCosts = [2,3], toppingCosts = [4,5,100], target = 18\nOutput: 17\nExplanation: Consider the following combination (all 0-indexed):\n- Choose base 1: cost 3\n- Take 1 of topping 0: cost 1 x 4 = 4\n- Take 2 of topping 1: cost 2 x 5 = 10\n- Take 0 of topping 2: cost 0 x 100 = 0\nTotal: 3 + 4 + 10 + 0 = 17. You cannot make a dessert with a total cost of 18.\n\n\nExample 3:\n\nInput: baseCosts = [3,10], toppingCosts = [2,5], target = 9\nOutput: 8\nExplanation: It is possible to make desserts with cost 8 and 10. Return 8 as it is the lower cost.\n\n\nExample 4:\n\nInput: baseCosts = [10], toppingCosts = [1], target = 1\nOutput: 10\nExplanation: Notice that you don't have to have any toppings, but you must have exactly one base.\n\nConstraints:\n\n\u2022 n == baseCosts.length\n\u2022 m == toppingCosts.length\n\u2022 1 <= n, m <= 10\n\u2022 1 <= baseCosts[i], toppingCosts[i] <= 104\n\u2022 1 <= target <= 104\n\n## Solution: DP \/ Knapsack\n\nPre-compute the costs of all possible combinations of toppings.\n\nTime complexity: O(sum(toppings) * 2 * (m + n)) ~ O(10^6)\nSpace complexity: O(sum(toppings)) ~ O(10^5)\n\n## Solution 2: DFS\n\nCombination\n\nTime complexity: O(3^m * n)\nSpace complexity: O(m)\n\n## C++\n\nYou are given an array\u00a0items, where each\u00a0items[i] = [typei, colori, namei]\u00a0describes the type, color, and name of the\u00a0ith\u00a0item. You are also given a rule represented by two strings,\u00a0ruleKey\u00a0and\u00a0ruleValue.\n\nThe\u00a0ith\u00a0item is said to match the rule if\u00a0one\u00a0of the following is true:\n\n\u2022 ruleKey == \"type\"\u00a0and\u00a0ruleValue == typei.\n\u2022 ruleKey == \"color\"\u00a0and\u00a0ruleValue == colori.\n\u2022 ruleKey == \"name\"\u00a0and\u00a0ruleValue == namei.\n\nReturn\u00a0the number of items that match the given rule.\n\nExample 1:\n\nInput: items = [[\"phone\",\"blue\",\"pixel\"],[\"computer\",\"silver\",\"lenovo\"],[\"phone\",\"gold\",\"iphone\"]], ruleKey = \"color\", ruleValue = \"silver\"\nOutput: 1\nExplanation: There is only one item matching the given rule, which is [\"computer\",\"silver\",\"lenovo\"].\n\n\nExample 2:\n\nInput: items = [[\"phone\",\"blue\",\"pixel\"],[\"computer\",\"silver\",\"phone\"],[\"phone\",\"gold\",\"iphone\"]], ruleKey = \"type\", ruleValue = \"phone\"\nOutput: 2\nExplanation: There are only two items matching the given rule, which are [\"phone\",\"blue\",\"pixel\"] and [\"phone\",\"gold\",\"iphone\"]. Note that the item [\"computer\",\"silver\",\"phone\"] does not match.\n\nConstraints:\n\n\u2022 1 <= items.length <= 104\n\u2022 1 <= typei.length, colori.length, namei.length, ruleValue.length <= 10\n\u2022 ruleKey\u00a0is equal to either\u00a0\"type\"\"color\", or\u00a0\"name\".\n\u2022 All strings consist only of lowercase letters.\n\n## Solution: Brute Force\n\nTime complexity: O(n)\nSpace complexity: O(1)\n\n## C++\n\nYou are given two strings,\u00a0word1\u00a0and\u00a0word2. You want to construct a string in the following manner:\n\n\u2022 Choose some\u00a0non-empty\u00a0subsequence\u00a0subsequence1\u00a0from\u00a0word1.\n\u2022 Choose some\u00a0non-empty\u00a0subsequence\u00a0subsequence2\u00a0from\u00a0word2.\n\u2022 Concatenate the subsequences:\u00a0subsequence1 + subsequence2, to make the string.\n\nReturn\u00a0the\u00a0length\u00a0of the longest\u00a0palindrome\u00a0that can be constructed in the described manner.\u00a0If no palindromes can be constructed, return\u00a00.\n\nsubsequence\u00a0of a string\u00a0s\u00a0is a string that can be made by deleting some (possibly none) characters from\u00a0s\u00a0without changing the order of the remaining characters.\n\npalindrome\u00a0is a string that reads the same forward\u00a0as well as backward.\n\nExample 1:\n\nInput: word1 = \"cacb\", word2 = \"cbba\"\nOutput: 5\nExplanation: Choose \"ab\" from word1 and \"cba\" from word2 to make \"abcba\", which is a palindrome.\n\nExample 2:\n\nInput: word1 = \"ab\", word2 = \"ab\"\nOutput: 3\nExplanation: Choose \"ab\" from word1 and \"a\" from word2 to make \"aba\", which is a palindrome.\n\nExample 3:\n\nInput: word1 = \"aa\", word2 = \"bb\"\nOutput: 0\nExplanation: You cannot construct a palindrome from the described method, so return 0.\n\nConstraints:\n\n\u2022 1 <= word1.length, word2.length <= 1000\n\u2022 word1\u00a0and\u00a0word2\u00a0consist of lowercase English letters.\n\n## Solution: DP\n\nLet s = word1 + word2, build dp table on s. We just need to make sure there\u2019s at least one char from each string.\n\nTime complexity: O((m+n)^2)\nSpace complexity: O((m+n)^2)\n\n## C++\n\nO(m+n) Space complexity","date":"2021-03-03 06:07:45","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\": 0.3502368628978729, \"perplexity\": 3377.990354642816}, \"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-2021-10\/segments\/1614178365454.63\/warc\/CC-MAIN-20210303042832-20210303072832-00294.warc.gz\"}"} | null | null |
{"url":"https:\/\/amathew.wordpress.com\/tag\/descent-theory\/","text":"One of the really nice pictures in homotopy theory is the \u201cchromatic\u201d one, relating the structure of the stable homotopy category to the geometry of formal groups (or rather, the geometry of the moduli stack of formal groups).\u00a0A while back, I did a series of posts trying to understand a little about the relationship between formal groups and complex cobordism; the main result I was able to get to was Quillen\u2019s theorem on the formal group of $MU$. I didn\u2019t understand too much of the picture then, but I spent the summer engaging with it and think I have a slightly better feel for it now. In this post, I\u2019ll try to give a description of how a natural attack on the homotopy groups of a spectrum via descent leads very naturally to the moduli stack of formal groups and to the Adams-Novikov spectral sequence. (There are other approaches to Adams-type spectral sequences, for instance in these notes\u00a0of Haynes Miller.)\n\n1. Descent\n\nLet\u2019s start with some high-powered generalities that I don\u2019t really understand, and then come back to earth. Consider an ${E_\\infty}$-ring ${R}$; the most important examples will be ${R = H \\mathbb{Z}\/2}$ or ${R = MU}$. There is a map of ${E_\\infty}$-rings ${S \\rightarrow R}$, where ${S}$ is the sphere spectrum.\n\nLet ${X}$ be a plain spectrum. Then, equivalently, ${X}$ is a\u00a0module\u00a0over ${S}$. Tensoring with ${R}$ gives an ${R}$-module spectrum ${ R \\otimes X}$, where the smash product of spectra is written ${\\otimes}$. In fact, we have an adjunction\n\n$\\displaystyle \\mathrm{Mod}(S) \\rightleftarrows \\mathrm{Mod}(R)$\n\nbetween ${R \\otimes }$ and forgetting the ${R}$-module structure. As in ordinary algebra, we might try to apply the methods of flat descent to this adjunction. In other words, given a spectrum ${X}$, we might try to recover ${X}$ from the ${R}$-module ${R \\otimes X}$ together with the \u201cdescent data\u201d on ${X}$. The benefit is that while the homotopy groups ${\\pi_* X}$ may be intractable, those of ${R \\otimes X}$ are likely to be much easier to compute: they are the ${R}$-homology groups of ${X}$.\n\nLet\u2019s recall how this works in algebra. Given a faithfully flat morphism of rings ${A \\rightarrow B}$ and an ${A}$-module ${M}$, then we can recover ${M}$ as the equalizer of\n\n$\\displaystyle M \\otimes_A B \\rightrightarrows M \\otimes_A B \\otimes_A B.$\n\nHow does one imitate this construction in homotopy? One then has a cosimplicial ${E_\\infty}$-ring given by the cobar construction\n\n$\\displaystyle R \\rightrightarrows R \\otimes R \\dots .$\n\nThe ${\\mathrm{Tot}}$ (homotopy limit) of a cosimplicial object is the homotopyish version of the 1-categorical notion of an equalizer. In particular, we might expect that we can recover the spectrum ${X}$ as the homotopy limit of the cosimplicial diagram\n\n$\\displaystyle R \\otimes X \\rightrightarrows R \\otimes R \\otimes X \\dots .$ (more\u2026)\n\nWe continue in the quest towards descent theory. Today, we discuss the fpqc topology and prove the fundamental fact that representable functors are sheaves.\n\nWe now describe another topology on the category of schemes. First, we need the notion of an fpqc morphism.\n\nDefinition 1 A morphism of schemes ${f: X \\rightarrow Y}$ is called fpqc if the following conditions are satisfied:\n\n1. ${f}$ is faithfully flat (i.e., flat and surjective)\n2. ${f}$ is quasi-compact.\n\nIndeed, \u201cfpqc\u201d is an abbreviation for \u201cfidelement plat et quasi-compact.\u201d It is possible to carry out faithfully flat descent with a weaker notion of fpqc morphism, for which I refer you to Vistoli\u2019s part of FGA explained.\n\nAs with many interesting classes of morphisms of schemes, we have a standard list of properties.\n\nProposition 2\n\n1. Fpqc morphisms are closed under base-change and composition.\n2. If ${f: X \\rightarrow Y, g: X' \\rightarrow Y'}$ are fpqc morphisms of ${S}$-schemes, then ${f \\times_S f': X \\times_S X' \\rightarrow Y \\times_S Y'}$ is fpqc.\n\nProof: We shall omit the proof, since the properties of flatness, quasi-compactness, and surjectivity are all (as is well-known) preserved under base-change, composition, and products. This can be looked up in EGA 1 (except for flatness, for which you need to go to EGA 4 or Hartshorne III). $\\Box$\n\nSo we have the notion of fpqc morphism. Next, we use this to define a topology.\n\nDefinition 3 Consider the category ${\\mathfrak{C}}$ of ${S}$-schemes, for ${S}$ a fixed base-scheme. The fpqc topology on ${\\mathfrak{C}}$ is defined as follows: A collection of arrows ${\\left\\{U_i \\rightarrow U\\right\\}}$ is said to be a cover of ${U}$ if the map ${\\coprod U_i \\rightarrow U}$ is an fpqc morphism.\n\nThis implies in particular that each ${U_i \\rightarrow U}$ is a flat morphism. We need now to check that this is indeed a topology.\n\n1. An isomorphism is obviously an fpqc morphism, so an isomorphism is indeed a cover.\n2. If ${\\left\\{U_i \\rightarrow U\\right\\}}$ is a fpqc cover and ${V \\rightarrow U}$, then the morphism ${\\coprod( U_i \\times_U V )\\rightarrow V }$ is equal to the base-change ${(\\coprod U_i) \\times_U V \\rightarrow V}$, hence is fpqc.\n3. Suppose ${\\left\\{U^i_j \\rightarrow U_i\\right\\}}$ is a cover for each ${i}$ and ${\\left\\{U_i \\rightarrow U\\right\\}}$ is a cover, I claim that ${\\left\\{U_j^i \\rightarrow U\\right\\}}$ is a cover. Indeed, we have that$\\displaystyle \\coprod_{i,j} U^{j}_i \\rightarrow U$factors through$\\displaystyle \\coprod_{i,j} U^{i}_j \\rightarrow \\coprod_i {U_i} \\rightarrow U$and we know that each morphism in the composition is flat (since the coproduct of flat morphisms is flat) and quasi-compact (since the coproduct of quasi-compact morphisms is quasi-compact). Similarly for surjectivity. It follows that ${\\left\\{U^i_j \\rightarrow U \\right\\}}$ is an fpqc cover.\n\nSo we have another topology on the category of schemes, which is very fine in that it is finer than many other topologies of interest (e.g. the fppf and etale topologies, which I will discuss at some other point). (more\u2026)\n\nIt is possible to define sheaves on a Grothendieck topology. Before doing so, let us recall the definition of a sheaf of sets on a topological space ${X}$.\n\nDefinition 1 A sheaf of sets ${\\mathcal{F}}$ assigns to each open set ${U \\subset X}$ a set ${\\mathcal{F}(U)}$ (called the set of sections over ${U}$) together with \u201crestriction\u201d maps ${\\mathrm{res}^U_V: \\mathcal{F}(U) \\rightarrow \\mathcal{F}(V)}$ for inclusions ${V \\subset U}$ such that the following conditions are satisfied:\n\n\u2022 ${\\mathrm{res}^U_U = \\mathrm{id}}$ and for a tower ${W \\subset V \\subset U}$, the composite ${\\mathrm{res}^V_W \\circ \\mathrm{res}^U_V }$ equals ${\\mathrm{res}^U_W}$.\n\u2022 If ${\\left\\{U_i\\right\\}}$ is a cover of ${U \\subset X}$, then the map $\\displaystyle \\mathcal{F}(U) \\rightarrow \\prod \\mathcal{F}(U_i)$is injective, and the image consists of those families ${f_i \\in \\mathcal{F}(U_i)}$ such that the restrictions to the intersections are equal $\\displaystyle \\mathrm{res}^{U_i}_{U_i \\cap U_j} f_i = \\mathrm{res}^{U_j}_{U_i \\cap U_j}$\n\nIn particular, this says that if we have a family of elements ${f_i \\in \\mathcal{F}(U_i)}$ that satisfy the above gluing condition, then there is a unique ${f \\in \\mathcal{F}(U)}$ which restricts to each of them.\n\nI\u2019ve been reading a lot about descent theory lately, and I want to explain some of the ideas that I\u2019ve absorbed, partially because I don\u2019t fully understand them yet.\n\nIn algebraic geometry, we often like to glue things. In other words, we define something locally and have to \u201cpatch\u201d the local things. An example is the ${\\mathrm{proj}}$ of a quasi-coherent sheaf of algebras. Let ${\\mathcal{A}}$ be a graded quasi-coherent sheaf of algebras on the scheme ${X}$. Then, for an open affine ${U = \\mathrm{Spec} A}$, ${\\mathcal{A}|_U}$ is the sheaf associated to a graded ${A}$-algebra ${\\Gamma(U, \\mathcal{A})}$. We can define the ${\\mathrm{Proj} }$ of this algebra; it is a scheme ${X_U}$ over ${U}$. When we do this for each ${U}$ open affine and glue the resulting schemes ${X_U}$, we get the ${\\mathrm{Proj}}$ of ${\\mathcal{A}}$, which we can call ${X}$. This is an example of how gluing is useful. Another example is the construction of the ${\\mathrm{Spec}}$ of a quasi-coherent sheaf of algebras. So gluing is ubiquitous.\n\nWe start with a review of the ideas behind gluing. Let\u2019s now take the simplest possible example of how gluing might actually work in detail. Suppose we have a scheme ${X}$ and an open cover ${\\left\\{U_i\\right\\}}$ of ${X}$, and quasi-coherent sheaves ${\\mathcal{F}_i}$ on ${U_i}$ for each ${i}$. We would like to \u201cglue \u201d the ${\\mathcal{F}_i}$ into one quasi-coherent sheaf on ${X}$ that restricts to each of the ${\\mathcal{F}_i}$ on each ${U_i}$. In order to do this, we need isomorphisms\n\n$\\displaystyle \\phi_{ij}: \\mathcal{F}_i|_{U_i \\cap U_j} \\rightarrow \\mathcal{F_j}|_{U_j \\cap U_i}$\n\nthat satisfy the cocycle condition\n\n$\\displaystyle \\phi_{jk } \\circ \\phi_{ij} = \\phi_{ik}: \\mathcal{F_i}|_{U_i \\cap U_j \\cap U_k} \\rightarrow \\mathcal{F_k}|_{U_i \\cap U_j \\cap U_k}.$ (more\u2026)","date":"2017-10-18 01:40:19","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\": 0, \"img_math\": 109, \"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\": 0.9301375150680542, \"perplexity\": 187.83020593165855}, \"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-2017-43\/segments\/1508187822668.19\/warc\/CC-MAIN-20171018013719-20171018033719-00396.warc.gz\"}"} | null | null |
Fatma Samba Diouf Samoura, más conocida como Fatma Samoura (Senegal, 1962), es una diplomática senegalesa. En mayo de 2016 fue elegida secretaria general de la FIFA, convirtiéndose en la primera mujer en asumir esta responsabilidad en la historia.
Es especialista en gestión internacional de Naciones Unidas, donde durante 21 años ocupó responsabilidades en diversos programas de desarrollo. Su último trabajo antes del nombramiento en la FIFA fue el de coordinadora humanitaria de la ONU en Nigeria, responsable del Programa de las Naciones Unidas para el Desarrollo (PNUD) en este país.
Biografía
Es diplomada en español e inglés en la Universidad de Lyon y ha estudiado relaciones internacionales y comercio internacional en el Instituto de Estudios Superiores Especializados (IECS) en Estrasburgo. Habla francés (su lengua materna), italiano, inglés y español.
A los 21 años empezó a trabajar en la ONU donde ha ocupado diferentes cargos.
Inició su carrera en 1995 como oficial de logística de alto nivel en el Programa Mundial de Alimentos en Roma. De 2000 a 2005 trabajó en Yibuti, de 2005 a 2007 en Camerún, de 2009 a 2010 fue Directora del Programa Mundial de Alimentación en Guinea. También se ocupó de situaciones de emergencia en Kosovo, Liberia, Nicaragua, Afganistán, Costa de Marfil, Sierra Leona y Timor Oriental.
De 2010 a 2015 fue coordinadora residente del Programa de las Naciones Unidas para el Desarrollo en Madagascar y posteriormente en Nigeria (2015-2016) donde en su última etapa tuvo el cometido de organizar el presupuesto, recursos humanos y adquisiciones.
El 13 de mayo de 2016 fue elegida secretaria general de la FIFA durante el 66.º Congreso de la federación en México convirtiéndose en la primera mujer en asumir el puesto. Sucedió al francés Jérôme Valcke suspendido durante 12 años tras acusaciones de corrupción en la venta de entradas durante la Copa Mundial de Fútbol de 2014.
Referencias
FIFA
Directivos de la Organización de las Naciones Unidas
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\section{Introduction}
Spectral and temporal properties of galactic and extra-galactic black holes
suggest time dependence of various degrees. Imaging a black hole accretion disk
would therefore not be accurate without inclusion of the time dependence of
various disk components. There is a class of stellar mass black hole
candidates, known as the outbursting sources, which totally changes its disk configurations
in a matter of few days, when it goes through transitions of its spectral states from
hard to hard intermediate to soft intermediate and then soft states during their
rising phase. Recently, through fits of data with Two Component Advective Flow (TCAF) solution
as prescribed by Chakrabarti \& Titarchuk (1995; hereafter, CT95), the accretion rates
of the halo and disk components, the sizes of the hot electron cloud (`Compton cloud') etc.
were all shown to be highly variable. The Compton cloud in this solution is the
inner part of the advective halo, puffed up due to slowing down by the centrifugal barrier,
which not only inverse Comptonizes the soft photons to produce high energy X-rays, but also
produces outflowing matter just as a normal boundary layer. The disk component on the
equatorial plane supplies soft photons. In the absence of the CENBOL
as in the soft state, the flow behaves as a standard thin disk with the inner edge
close to the inner stable circular orbit or ISCO (Shakura \& Sunyaev, 1973) and no jets or outflows
are seen. In class-variable sources, such as GRS 1915+105, temporal variation in
significant count rates and state transitions such as (burst-on and burst-off states)
are seen in a matter of few seconds (Muno et al. 1999). This source also exhibits strong
relativistic radio jets (Fender et al. 1999). Along with the monotonic changes in the CENBOL sizes, the
frequency of the Quasi-Periodic Oscillations (QPOs) is also seen to change
monotonically (Chakrabarti et al. 2005). Theoretical work which forms the basis of TCAF
is in Chakrabarti (1996) and references therein.
The time dependent simulations (Giri, Garain \& Chakrabarti 2013
and references therein) show the formation of two components in the flow
with the Keplerian disk on the equatorial plane and advective halo surrounding it
formed the shock and outflows exactly as envisaged in CT95. Outflows are intrinsically associated with
accretion mechanism (see C90 and Chakrabarti 1996 for further details) in this solution. The explicit
ratio of mass outflow rate to the inflow rate was first computed in Chakrabarti (1999)
and was found to strongly depend on the compression ratio at the shock.
From Fig. 4 of Giri \& Chakrabarti (2012), it is clear that the ratio of the outflow to
the inflow rate increases with decreasing viscosity parameter and the inviscid flow
produces maximum outflows. In outbursts or class variable sources, outflows are found to be
sporadic and highly variable. Thus, it is very important to have the
time dependent spectra and images of TCAF where outflows are self-consistently
generated from the CENBOL itself. Cooling via Comptonization is an essential mechanism by which the
CENBOL reduces its size making the spectrum softer. While the CENBOL lasts,
rough agreement between the cooling and compressional heating timescales causes it
to oscillate and produce the so-called QPOs (Chakrabarti et al. 2015 and references
therein). Detailed hydrodynamic simulations in presence of Compton cooling
are in Ghosh, Garain \& Chakrabarti (2011) and Garain, Ghosh \& Chakrabarti
(2014) where effects of cooling on the spectra, shock location and the formation of
QPOs are described.
Theoretical efforts for imaging of accretion disks started with the pioneering
work by Luminet (1978) who drew the image of a standard Keplerian disk around a
Schwarzschild black hole (Shakura-Sunyaev, 1973). The method of image construction was done from the
observers end. This technique is very useful as photons were not lost while
tracking them from the observer. Following this, Fukue \& Yokohama (1988) published
the colored version of the image. With that, they also studied the occultation
effect on accretion disk light curve. A few years later, Viergutz (1992) generalized
the transfer functions for Kerr geometry. Marck (1996), for the first time, produced
the Keplerian disk image using the actual Ray-Tracing mechanism. Bromley et al. (2001)
added polarization determination. Dexter \& Agol (2009) first published `geokerr', a
public code to compute images in Kerr geometry. `GYOTO' (Vincent et al., 2011),
`YNOGK' (Yang \& Wang, 2013), `GRTRANS' (Dexter, 2016) came into public domains. The
focus of these works was to prepare one to interpret results from the Event
Horizon Telescope (EHT) whose objective is to identify the horizon of the supermassive
black hole at the centre of our Galaxy.
Armitage \& Reynolds (2003), first performed MHD simulation of Keplerian disk and
produced the images where the disk structure remained the same at all the time.
More recently, Broderick \& Loeb (2009) presented results where simulated images
of force free jets (Tchekhovskoy et al. 2008) launched from M87 are shown for various
model parameters and polarization maps of the magnetically driven jets are obtained.
However, under TCAF paradigm, one does not have to introduce magnetic field to generate
outflows. The hydrodynamic inflow produces time varying outflows self-consistently. Magnetic
field may be useful for collimation and acceleration purposed further out.
GRMHD simulations in the context of our Galactic centre were mostly performed
(Ohsuga et al. 2005; Noble et al. 2007; M\`oscibrodzka et al. 2009, 2011; Dexter et al. 2010;
Hilburn et al. 2010; Dexter \& Fragile 2012; Dolence et al. 2012; Shcherbakov, Penna
\& McKinney 2012, S\c adowski et al. 2013, Roelofs et al. 2017) considering a very
low accretion rate as applicable. Cooling term was neglected as appropriate for Sgr A*.
In fact, Dibi et al. (2012) added the cooling term and found the effect to be negligible in case
Sgr A*. Drappeau et al. (2013) constrained the mass accretion rate to be $\sim10^{-9}~M_{\odot}/yr$
with a highly spinning central black hole. This accretion rate is much lower
than that of the earlier result ($\sim10^{-5}~M_{\odot}/yr$) predicted by Coker \& Melia
(1997). A detailed review on the simulations and observational possibilities of the black hole
shadow of Sgr A* was presented by Falcke \& Markoff (2013). But, most of the earlier studies were
based on the RIAF model to simulate spectra and images of Sgr A* only. However, there exists
a large number of SMBHs in Radio Loud Quasars, Seyfert galaxies, where the accretion rate
could be substantially high (Bian \& Zhao, 2003) and thus formation of disk and cooling becomes
important. Another situation which might induce a sudden high mass inflow is from tidal
disruption of neighboring object (Gillessen et al. 2012).
Chatterjee, Chakrabarti \& Ghosh, (2017a, hereafter CCG17a) showed the static images of TCAF
with general relativistic thick disks acting as the Compton cloud and relativistic
Keplerian disk (Page \& Thorne, 1974) acting as the source of seed photons. Images for various disk
rates and inclination angle are shown with their observable spectral counterparts.
The variation of optical depth of the CENBOL medium is achieved by changing the
halo rate ($\dot{m}_h$). They have shown that by increasing the optical depth, the Keplerian
disk from the other side is completely blocked by the CENBOL medium. Also, with increasing
inclination angle, the spectral hardening can be seen in their work. Following the
same geometry, Chatterjee, Chakrabarti \& Ghosh, 2017b, (hereafter CCG17b) simulated
the time lags of various energy bins and compare the simulated results with the
observed counterparts.
In this paper, for the first time, we present time dependent images of TCAF
with Comptonization and outflows. Basic simulation process by hydrodynamic TVD
code is explained \S 2. Thermodynamic properties of CENBOL and Keplerian
disk are presented in \S 3. Section deals with Monte-Carlo simulation of Comptonization
and cooling mechanism. The spectra, time dependent images with and without
cooling effects are presented in the results section. Convolved images of accretion
disk at different spectral states are presented. We discuss our results and conclude
this paper in the final section.
\section{Time Dependent Simulations of TCAF}
Hydrodynamic simulation procedure is similar to what has been earlier reported in Ryu et al. (1997),
Giri et al. (2010), Giri \& Chakrabarti (2012, 2013), Giri, Garain \& Chakrabarti (2015).
Governing equations of TVD are explicitly presented by Ryu et al. 1996; Molteni et al.
1996; Giri \& Chakrabarti 2012. The conservation equation of mass, momentum and energy for an
axisymmetric, inviscid, non-magnetic flow takes the following form
\begin{equation}
\frac{\partial \boldsymbol {q}}{\partial t} + \frac{1}{r} \frac{\partial\boldsymbol {F_1}}{\partial r} +
\frac{\partial\boldsymbol {F_2}}{\partial r} + \frac{\partial\boldsymbol {G}}{\partial z} = \boldsymbol {S},
\end{equation}
where
$\boldsymbol{q}= \left (\begin{array}{c}
\rho\\
\rho v_r\\
\rho v_{\theta}\\
\rho v_{z}\\
E
\end{array} \right ) $ is the state vector, \\
$\boldsymbol{F_1}= \left (\begin{array}{c}
\rho v_{r}\\
\rho v_{r}^{2}\\
\rho v_{r}v_{\theta}\\
\rho v_{r}v_{z}\\
(E+\rho)v_{r}
\end{array} \right ), $
$\boldsymbol{F_2}= \left (\begin{array}{c}
0\\
p\\
0\\
0\\
0
\end{array} \right ), $
$\boldsymbol{F_2}= \left (\begin{array}{c}
\rho v_{z}\\
\rho v_{r}v_{z}\\
\rho v_{\theta}v_{z}\\
\rho v_{z}^2 + p\\
(E+\rho)v_{z}
\end{array} \right )$ are the flux functions.
The source function is defined as\\
$\boldsymbol{S}= \left (\begin{array}{c}
0\\
\frac{\rho v_{\theta}^2}{r}-\frac{\rho r}{2(\sqrt{r^2+z^2}-1)^2\sqrt{r^2+z^2}}\\
-\frac{\rho v_{r}v_{\theta}}{r}\\
-\frac{\rho z}{2(\sqrt{r^2+z^2}-1)^2\sqrt{r^2+z^2}}\\
-\frac{\rho (rv_{r}+zv_{z})}{2(\sqrt{r^2+z^2}-1)^2\sqrt{r^2+z^2}}
\end{array} \right ). $
The energy density $E=p/(\gamma -1)+\rho(v_{r}^2+v_{\theta}^2+v_{z}^2)/2$ is the sum
of thermal and kinetic energy of the infalling matter. The equation corresponding
to energy density can be extracted as
\\
$
\frac{\partial {E}}{\partial t} + \frac{1}{r} \frac{\partial {(E+\rho)v_{r}}}{\partial x} +
+ \frac{\partial {(E+\rho)v_{z}}}{\partial z} =
-\frac{\rho (rv_{r}+zv_{z})}{2(\sqrt{r^2+z^2}-1)^2\sqrt{r^2+z^2}}.
$
\\
During cooling, change in $E$ directly modifies the flow configurations following Eqn (1).
The gravitational field, as far as the hydrodynamics is concerned, is assumed to follow the
Paczy\'nski \& Wiita (1980) pseudo-Newtonian potential which is given by,
\begin{equation}
\phi(r,Z)=-\frac{GM_{bh}}{r-r_g}
\end{equation}
where $G$ is the universal gravitational constant, $M_{bh}$ is the mass of the black hole kept at
$10~M_{\odot}$ throughout the simulation and $r_g=\frac{2GM_{bh}}{c^2}=1$ is the Schwarzschild radius
and $r=\sqrt{r^2+Z^2}$. Velocity of light is denoted by $c$. The flow is considered to be adiabatic
and axisymmetric with the black hole sitting at centre of the cylindrical co-ordinate system
defined by $(r,\theta,Z)$. In our current simulation, we have ignored viscosity to
have the strong and hot shock and consequently highest outflow rate. Thus, the specific
angular momentum $\lambda$ is constant everywhere. Also, from C89 and C90, it is known that for
a steady state situation the specific energy $\varepsilon$ is also constant.
Matter is injected at the outer boundary with specific energy $\varepsilon=0.003$
(in units of $c^2$) and specific angular momentum $\lambda=1.8$ )in units of $c r_g$.
TVD scheme was originally developed by Harten (1983) to solve the hyperbolic equations present in
the hydrodynamic conservation equations. This method works on the second order accuracy by
modifying first order flux function to use the non-oscillatory results in the second order
solutions. The details of the code were presented in Ryu et al. (1995, 1997) where the code is
modified to suit black hole accretion.
Our computational domain is defined as: $0<r<100$ and $0<Z<100$ in $r-Z$ plane with a grid
resolution of $512\times512$. So, each grid size is $\delta r=0.195 r_g$ and $\delta Z=0.195 r_g$.
The black hole absorption boundary is kept at $1.5 r_g$ after which all informations about the
matter as well as photons are lost. Before starting the simulation, entire grid is filled with a low floor density
($10^{-8}$ where the injected density is $1$ in code unit) to avoid any numerical singularities.
At the outer boundary sub-Keplerian
matter is allowed to flow through. During the accretion, the matter follows the inviscid hybrid
accretion mechanism (Chakrabarti 1990) and forms a shock at a specified position.
We assume axisymmetry and perform our hydro-simulation on the first quadrant of the r-Z plane.
The black hole is located at the origin of the r-Z coordinate system.
The initial pressure is chosen such that the sound speed as of the interior material is same as
the incoming matter. Also, inside the computational domain, all the velocity components are set to
zero initially. The simulation result does not depend on the chosen initial condition as the
initial matter is washed away by the incoming matter in a few dynamical time. On the right
radial boundary, we use an inflow boundary condition. Incoming matter enters the computational
domain through this boundary. Since the density of the incoming matter is chosen to be unity, it
has $10^{8}$ times higher pressure and it rushes towards the central black hole practically through
vacuum until it hits the centrifugal barrier.
Total dynamical time scale for this simulation is $4000$ which in physical units ($\frac{2GM_{BH}}{c^3}\times f_{t}$,
where $f_{t}$ contains informations of each time) is about $40$s. In the present context, the hydrodynamical
simulation has been performed for Galactic black hole candidates. However, these results will also hold for
supermassive or intermediate mass black holes.
\section{Density and Temperature Profile of the Electron Cloud}
CENBOL is a region of hot electron cloud where inverse Comptonization of soft photons
originated from the Keplerian disk occurs. The electrons in this region are ultra-relativistic. So, $\gamma=4/3$
is a good assumption for this region. Number density (per $cm^3$) and temperature (in the kilo electron Volt)
of the electrons are supplied for each grid. The pressure and density is related via polytropic equation
of state $p=K\rho^{\gamma}$, where $p$ is the pressure, $K$ the entropy constant and $\rho$ is the density
of the flow. From this, we can write $T\propto \rho^{1/3}$, where $T$ is the temperature of the electron
cloud. In our studies, the maximum number density of electron is $\sim 10^{19}~per~cm^3$ and temperature is
$\sim 250~keV$. Hydrodynamical simulation changes the configuration of CENBOL region for each time stamp. The
variation of electron cloud geometry with number density and temperature as given in the color bar are presented in
Figs. 1.
\begin{figure*}
\centering
\vbox{
\includegraphics[height=5.5cm,width=8.0cm]{image01a.eps}\hskip 0.5cm
\includegraphics[height=5.5cm,width=8.0cm]{image01b.eps}}
\caption{Electron Density (in the units of number per $cm^3$ shown in log scale, left panel)
and temperature (in the units of keV, right panel) distribution in the post-shock region at
(a) $t=14.25s$, (b) $t=14.35s$, (c) $t=14.45s$, (d) $t=14.55$, (e) $t=14.65s$ and (f) $t=14.75s$.}
\end{figure*}
From Fig. 1, the region of outflows can be distinctly seen as a region where temperature and electron
number density drops rapidly. Over the time, the geometry of the CENBOL region changes as hydrodynamical
simulation progresses. One can easily differentiate between Fig. 1a and Fig. 1f as the amount of outflow increased
substantially in the time passed. The inner isobaric contours of the CENBOL region (Figs. 1a-f)
behave similar to thick disks (Abramowicz et al. 1978; Kozlowski et al. 1978; Paczy\'nski \& Wiita 1980; Begelman, Blandford
\& Rees 1982; Chakrabarti, 1985a) as pointed out by Molteni, Lanzafame \& Chakrabarti (1994, hereafter MLC94).
The TVD method simulates a true accretion mechanism. The injected matter washes away the
initial low floor matter and attains a quasi steady state. Inflowing matter undergoes a shock due
to the centrifugal barrier. After the shock, matter starts to flow towards black hole. The mass
absorption rate through the black hole boundary ($1.5~r_g$) is represented in the Fig. 2. It can
be seen that the mass absorption rate after the transient phase saturates at $\sim75\%$ of the
injected mass.
\begin{figure}
\vskip 0.4cm
\centering
\vbox{
\includegraphics[width=1.0\columnwidth]{image02.eps}}
\caption{Fraction of mass absorption (circle-black), outflows (triangle-red) and the ratio of outflowing
matter to that of absorption (square-green) through the inner boundary with time are shown. Injected
halo rate ($\dot{m}_h$) is kept fixed at $0.1$.}
\end{figure}
First simulation of this kind was performed by MLC94 followed by Molteni, Ryu \& Chakrabarti 1996 (MRC96), Giri \&
Chakrabarti 2012, 2013 (GC12, GC13). The outflows are generated out of
inflowing sub-Keplerian matter forming a shock due to centrifugal barrier.
The oscillation of the shock front after attaining a steady state and the consequence
of that on light curves are well studied in MLC94 and Giri et al. (2010) and Garain et al. (2013).
This formation of shock induces a temperature rise in the post-shock region which acts
as the base of the outflows or Jets. Due to the increase of temperature in the post shock
region, force due to gradient of pressure pushes matter outward along the vertical
direction. This creates the outflows. In HD simulations, this is not well collimated.
One may require magnetic effects for collimation (Chakrabarti, 2013).
To show details, we have shown the post-shock region only. The
sub-Keplerian flow of the pre-shock region has much lower density and optical depth as
compared to the post-shock region (GC13). Maximum number of scatterings occur in the
post-shock region and for an observer trying to capture the image of an accretion disk
in presence of the outflows, the last scattering surface of the CENBOL will appear as
the shape of the Compton cloud.
\subsection{Keplerian Disk Acting as the Soft Photon Source}
In order not to include a time dependence in the Keplerian disk, we put it on the equatorial plane, whose
purpose is to supply soft photons as per Page \& Thorne (1974) prescription. We employ the Monte-Carlo simulations
to compute the resulting spectrum which includes the original source photons from this Keplerian disk as well as those
scattered from the CENBOL and the outflows. The Keplerian disk is truncated at the inner edge at the CENBOL surface
and the outer edge is extended up to $100~r_g$ for simulation purpose.
\begin{equation}
\begin{aligned}
F(r) =
\frac{F_c(\dot{m}_d)}{(r-3/2)r^{5/2}} \\
\times \bigg[\sqrt{r}-\sqrt{3/2}+
\frac{\sqrt{3/2}}{2}log\bigg(\frac{(\sqrt{r}+\sqrt{3/2})(\sqrt{3}-\sqrt{3/2})}
{(\sqrt{r}-\sqrt{3/2})(\sqrt{3}+\sqrt{3/2})}\bigg)\bigg]\\
\\ and \\
T(r)= \bigg(\frac{F(r)}{\sigma}\bigg)^{1/4}
\end{aligned}
\end{equation}
where, $F_c(\dot{m}_d)=\frac{3m\dot{m}_d}{8\pi r_{g}^3}$, $\dot{m}_d$ is
the Keplerian disk accretion rate in Eddington unit, $\sigma=\frac{2\pi^5k_{B}^4}{15h^3c^3}$
is the Stefan-Boltzmann constant.
Photon flux emitted from the Keplerian disk surface of radius $r$ to $r+\delta r$ is written as,
\begin{equation}
n_{\gamma}(r)=\bigg[\frac{4\pi}{c^2} \bigg(\frac{k_bT(r)}{h}\bigg)^3 \times \zeta (3)\bigg]~cm^{-2}s^{-1},
\end{equation}
where, $\zeta (3)=\sum^\infty_1{l}^{-3} = 1.202$ is the Riemann zeta function.
So, the rate of photons emitted from the radius $r$ to $r+\delta r$ is given by,
\begin{equation}
dN(r) = 4\pi r\delta rn_{\gamma}(r),
\end{equation}
where the Keplerian disk height $H(r)$ is considered as $0.1$
(since $\frac{H(r)}{r} << 1$ for Keplerian disk region).
The disk is divided into several annuli of radial width $D(r) = 0.1$ and of mean
temperature $T(r)$. The exact number of photons come to be around $\sim 10^{39}$ --$10^{40}$ per
second for a disk rate of $\dot{m}_d=0.1$. With increasing disk rate this number increases
rapidly. Using a weightage factor $f_W=\frac{dN(r)}{N_{comp(r)}}$ where $N_{comp}(r) = 10^9$,
we bundle these photons to save computational time.
Soft photon energy is calculated using Planck distribution law for $T(r)$.
Photon number density ($n_\gamma(E)$) which corresponds to an energy $E$ is given by,
\begin{equation}
n_\gamma(E) = \frac{1}{2 \zeta(3)} b^{3} E^{2}(e^{bE} -1 )^{-1}.
\end{equation}
The process is similar to the earlier works presented by Ghosh, Chakrabarti \&
Laurent (2009); Chatterjee, Chakrabarti \& Ghosh (2017a, 2017b).
\section{Monte-Carlo Simulation of Comptonization}
In Monte-Carlo process, the photons inside the CENBOL region are assumed to follow straight line trajectories
in between two scatterings. This enhances the Computational efficiency without sacrificing science. This has been pointed out by
Laurent \& Titarchuk (1999). The Keplerian disk emits seed photons with six random number associated
with position and velocity. In reality, the Keplerian disk should have the maximum flux along
Z-axis and minimum along the equatorial plane. This has been implemented by assuming
$F_{\nu}=\int I_{\nu}\mathrm{cos}\theta d\Omega$. Inside the CENBOL region,
a critical optical depth ($\tau_c$) is set up using random number corresponding to each
scattering. The next scattering would occur if the optical depth of a particular photon crosses $\tau_c$.
We choose Klein-Nishina scattering cross section $\sigma$ which is given by:
\begin{equation}
\sigma = \frac{2\pi r_{e}^{2}}{x}\left[ \left( 1 - \frac{4}{x} - \frac{8}{x^2} \right)
ln\left( 1 + x \right) + \frac{1}{2} + \frac{8}{x} - \frac{1}{2\left( 1 + x \right)^2} \right],
\end{equation}
where, $x$ is given by,
\begin{equation}
x = \frac{2E}{m c^2} \gamma \left(1 - \mu \frac{v}{c} \right),
\end{equation}
$r_{e} = e^2/mc^2$ is the classical electron radius and $m$ is the mass of the electron. This yields
Thomson scattering cross section ($\sigma_{T}$) in low frequencies and Compton scattering cross
section ($\sigma_{C}$) in higher frequencies like X-rays and $\gamma$-Rays. Between each pair of
scatterings, the gravitational red-shift modifies the energy of the photon. This process
continues until it leaves the CENBOL region or get sucked by the black hole. The process
is similar to that presented in GCL09; Ghosh, Garain, Chakrabarti, Laurent, 2010; Ghosh, Garain,
Giri, Chakrabarti, 2011 (hereafter GGGC11); CCG17a, CCG17b.
\begin{figure}
\vskip 0.7cm
\centering
\vbox{
\includegraphics[width=1.0\columnwidth]{image03.eps}}
\caption{Spectra obtained after Comptonization. (a) The spectra at $t=14.25s$ (Black-Solid),
$t=14.45s$ (Red-Dotted) and $t=14.75s$ (Green-Dot-Dashed)~and~ injected (Blue-Dot-Dash-Dash) spectrum.
(b) Injected spectrum (Black-Solid), composite spectrum (Red-Dashed), Reflected Spectrum
(Blue-Dot-Dash-Dash), Outflow spectrum (Green-Dot-Dashed). (c) Inclination dependence
of reflected spectra. The inclination bins are $0^{\circ}-20^{\circ}$ (Black-Solid), $40^{\circ}-60^{\circ}$
(Red-Dotted), $70^{\circ}-90^{\circ}$ (Green-Dot-Dashed). (d) Outflow spectra at $t=14.25s$
(Black-Solid), $t=14.45s$ (Red-Dotted) and $t=14.75s$ (Green-Dot-Dashed). Note that the outflow spectra
changed significantly due to the increase in the outflow rate.
Disk accretion rate $\dot{m}_d=0.1$ is kept fixed throughout the simulations.}
\end{figure}
After Comptonization, the spectra at various times are presented in Fig. 3.
Fig. 3a shows the Comptonized spectra of TCAF at $t=14.25s$, $t=14.45s$~and~$t=14.75s$.
The temporal variation in the thermodynamic variables of TCAF has been shown
in Fig. 1. In the absence of cooling, the Compton cloud remains hot.
Thus, the variations in spectra are very small. The Blue-dot-dot-dashed curve
in Fig. 3a represents the injected spectra. In (b), various components such as the net spectrum (red-dashed),
injected spectrum (solid-black), reflected spectrum caused by rescattering of Comptonized radiation by the Keplerian
disk before it reaches the observer (blue-dot-dot-dashed) and outflow
spectrum (green-dot-dashed) are shown at $t=14.45s$. It can be clearly seen
from Fig. 3b that the reflected spectrum contributes more to the net spectrum than
the outflows spectrum. Fig. 3c shows the reflected spectrum at three inclinations
($0^\circ-22.5^\circ$ (black-solid), $45^\circ-67.5^\circ$ (red-dashed) and $67.5^\circ-
90^\circ$ (green-dot-dashed)) at time $t=14.45s$. The hard energy contribution in
reflected spectra increases with inclination angles while the soft energy contribution
decreases. This is an expected result for the reflected component as the reflection dominates
at high inclinations. With the
increase of the outflow rate, the spectral contribution of
the outflow also increases. In Fig. 3d, the outflow spectra have been plotted for
$t=14.25s$~(black-solid), $t=14.45s$~(red-dotted) and~$t=14.75s$~(green-dot-dashed).
\subsection{Cooling Process}
During Comptonization, the photons either gain or lose energy. In the context of accretion physics,
the electrons in the CENBOL region is at much higher temperature than that of the photons coming from a Keplerian disk.
So, mostly inverse Compton scattering dominates and the signature of that is prominent in the
emergent spectra. We supply a steady state hydrodynamical configuration obtained from TVD code
where the density and temperature of the cloud is defined in each grid. After the scattering,
the electron from a particular grid may loose or gain $\Delta E$ amount of energy which is
transferred to the photon via Compton mechanism. In a particular state, after all the
scatterings by the injected photons, the energy of the entire grid is modified to generate a new
hydrodynamical state where the lose or gain of energy from each grid has been accounted. By this
process the final temperature of the cloud steadily decreases. The final temperature of the
electron cloud in a particular grid is expressed as,
\begin{equation}
k_BT_{new}(ir,iz) = k_BT_{old}(ir,iz) - \frac{\Delta E}{3dN_e(ir,iz)},
\end{equation}
where $T_{new}$ and $T_{old}$ are updated and old temperature of electron cloud. $dN_e(ir,iz)$ is
the number density of the electron in a particular grid denoted by $ir$ and $iz$ and $k_B$ is
the Boltzmann constant. Details of this process of cooling is earlier presented in Ghosh, Garain,
Giri \& Chakrabarti 2011 (GGGC11); Garain, Ghosh \& Chakrabarti 2012 (GGC12) and Garain, Ghosh
\& Chakrabarti 2014 (GGC14).
\begin{figure}
\centering
\vbox{
\includegraphics[width=1.0\columnwidth]{image04.eps}}
\caption{(a-c) Electron number density (in per $cm^3$, upper panel), and (b-d) temperature (in the units of keV,
lower panel) distribution in the post-shock region at $t=14.25s$. Compton cooling collapses the post-shock
region (a,c): initial stage of cooling, (b,d): after the cooling process is completed. The shock moves
inward due to loss of post-shock pressure. Disk rates $\dot{m}_d=0.1$ and $\dot{m}_h=0.1$ are kept constant
throughout.}
\end{figure}
In Fig. 4, we see that the shock location moves inward as the Compton cooling
starts to affect the post-shock region. Panels (a) and (d) show the CENBOL region
which has almost the same electron density and temperature as Fig. 1 (a). We have
considered this particular data file to see the effect of Compton cooling. Fig 4(a) is presented
when cooling just started to work. Then, after a few iterations cooling starts to reduce the
CENBOL size. Spectra softens as the size of hotter Compton cloud becomes smaller. Finally, the
Compton cloud reduces more which makes the accretion disk spectra to be in soft state.
The corresponding spectra of Panel (b) and (d) suggest the CENBOL region is
so small that a very few numbers of soft photons are intercepted by the cloud. Thus, the spectrum
softens and spectral slope increases. The effect of cooling on the outflows are
also visible from Fig. 4. With the inclusion of cooling, the amount of matter outflow rate
($R_{\dot{m}}$) reduces drastically.
\begin{figure}
\vskip 0.7cm
\centering
\vbox{
\includegraphics[width=1.0\columnwidth]{image05.eps}}
\caption{Variation of spectrum due to Compton cooling. Panel (a) corresponds
to the injected soft spectrum where solid-black is the same spectrum as injected
to the without cooling cases. Dotted-red corresponds to the injected spectra at final
stage (Fig. 4b,d). The corresponding emergent Comptonized spectra are presented
in Panel (b) where solid-black (same as Fig. 3) and dotted-red (for CENBOL represented
in Fig. 4b,d) curves showing the spectral softening due to Compton cooling.}
\end{figure}
Fig. 6 represents the outflows spectra for two different CENBOLs. Black-solid one is same as
the spectra presented for $t=14.25s$ in Fig. 3d. This is the input TVD file on which Compton
cooling started to act. The CENBOL reduces the its size due to cooling. And outflows component
vanishes with cooling. The outflow spectra shrinks more and more and the it softens. Basically, all
photons in the outflows spectra for the to particular case comes from the base of the jet which is
CENBOL itself. In this paper, our goal is to show the variation of images and spectra when
cooling is included. A very detailed study of the effect of cooling on outflows has already
been reported in GGC12.
\begin{figure}
\centering
\vbox{
\includegraphics[width=1.0\columnwidth]{image06.eps}}
\caption{Emergent outflow spectra for corresponding density and temperature
distribution shown in Fig. 4. Black-Solid curve shows the outflows spectrum
directly from the output of the TVD code (same as the black curve shown in
Fig. 3d). Red-dashed represents outflow spectra of the TCAF after final
stage of cooling.
}
\end{figure}
\section{Ray-Tracing Process}
In Schwarzschild geometry, the non vanishing components Christoffel symbol yield geodesic
equations from the field equation,
\noindent
\begin{equation}
\frac{d^2x^{\mu}}{dp^2}+ {\Gamma}_{\nu\lambda}^{\mu}\frac{dx^{\nu}}{dp}\frac{dx^{\lambda}}{dp} = 0,
\end{equation}
where ${\mu} = [0,1,2,3]$; $x^{0} = t$, $x^{1} = r$, $x^{2} = \theta$ and $x^{3} = \phi$,
$p$ is the Affine parameter. The four coupled, second order differential equations for photons
can be reduced to three using energy $P_t=E=(1-\frac{1}{r})\frac{dt}{dp}$ and
angular momentum $P_{\phi}=L=r^{2}\mathrm{sin}^{2}\theta\frac{d\phi}{dp}$ definitions
(Chandrasekhar, 1985). So, three geodesic equations which dictate the trajectory of photons
can be written as
\begin{equation}
\begin{aligned}
\frac{d^2r}{dt^2} + \frac{3}{2r(r-1)}\bigg(\frac{dr}{dt}\bigg)^2 - \\ (r-1)\bigg(\frac{d\theta}{dt}\bigg)^2 -
(r-1)r\mathrm{sin}^2{\theta}\bigg(\frac{d\phi}{dt}\bigg)^2
+ \frac{r-1}{2r^3} = 0,\\
\frac{d^2\theta}{dt^2} + \frac{2r-3}{r(r-1)}\bigg(\frac{d\theta}{dt}\bigg)\bigg(\frac{dr}{dt}\bigg)
- \mathrm{sin}{\theta}\mathrm{cos}{\theta}\bigg(\frac{d\phi}{dt}\bigg)^2 = 0 ~\mathrm{and}\\
\frac{d^2\phi}{dt^2} +\frac{2r-3}{r(r-1)}\bigg(\frac{d\theta}{dt}\bigg)\bigg(\frac{dr}{dt}\bigg)
+ 2\mathrm{cot}{\theta}\bigg(\frac{d\theta}{dt}\bigg)\bigg(\frac{d\phi}{dt}\bigg) = 0.
\end{aligned}
\label{eq:xdef}
\end{equation}
Velocity components are derived using Tetrad formalism (Park 2006) and are given by,
\begin{equation}
\begin{aligned}
v^{\hat{r}}=\frac{d\hat{r}}{dt}=\frac{r}{(r-1)}\frac{dr}{dt},
~v^{\hat{\theta}}=\frac{d\hat{\theta}}{dt}=\frac{r\sqrt{r}}{\sqrt{(r-1)}}\frac{d\theta}{dt}\\
\mathrm{and} ~
v^{\hat{\phi}}=\frac{d\hat{\phi}}{dt}=\frac{r\sqrt{r}\mathrm{sin}{\theta}}{\sqrt{(r-1)}}\frac{d\theta}{dt}.
\end{aligned}
\label{eq:xdef}
\end{equation}
The redshift factor added to connect the source and the observer frame is given by the relation
\begin{equation}
1+z= \frac{E_{em}}{E_{obs}}=\frac{(P_{\alpha}u^{\alpha})^{em}}{(P_{\alpha}u^{\alpha})^{obs}},
\end{equation}
where, $E_{em}$ and $E_{obs}$ are the energy of emitted and observed photons respectively.
The observed and emitted fluxes are related via the fourth power of redshift factor
\begin{equation}
F_{k}^{obs}= \frac{F_{k}^{disk}}{(1+z)^4}.
\end{equation}
This relationship between the observed flux and the observed temperature can be expressed as,
\begin{equation}
T_{k}^{obs} = \bigg(\frac{F_{k}^{obs}}{\sigma}\bigg)^{1/4}.
\end{equation}
Doppler boosting is added during the emitter-observer frame transformation.
The Ray-Tracing process and formation of images are done in the same way as
as reported earlier in CCG17a, CCG17b.
\section{Results}
\subsection{Time Dependent Images without Cooling}
The static images of TCAF in presence of Comptonization were presented in CCG17a. The Compton
cloud was modelled using natural angular momentum description of General Relativistic thick disks given by
Chakrabarti (1985a). The variations of images and spectra were presented for various inclination angles and
disk accretion rates. They show that due to steep variation of density and temperature inside the CENBOL,
the image does not have sharp edges. On the top of that, for an increasing optical depth of the CENBOL region, the photons from
the Keplerian disk region located opposite to the observer are blocked. In the present case, where TVD code
was used to create the CENBOL and the outflow self-consistently, the optical depth of the CENBOL medium allows
a fraction of photons from the Keplerian disk to pass through the Compton cloud without getting scattered.
This can be seen in each panels presented in Fig. 7 \& Fig. 8.
\begin{figure}
\centering
\vbox{
\includegraphics[width=1.00\columnwidth]{image07.eps}}
\caption{Images of TCAF (in $log(T_{obs})$ scale) seen from an inclination angle of $70^\circ$
in presence of outflows at time $t=14.25s$ ~and~$t=14.75s$. Colorbar is in the range
$10^{6.0}-10^{6.5}$ Kelvin.}
\end{figure}
Figure 7 is drawn at (a) $t=14.25s$~and~ (b) $t=14.75s$. In panel (a), the amount of outflow is much less as
compared to the panel (b). The photons emitted by the base of the lower jet are mostly blocked by the optically
thick Keplerian disk. In Fig. 7a, a very low number of photons is visible from the lower
jet. Due to reflection symmetry of the outflows, these photons are also visible in the upper part of
the outflows. However, due to Lorentz boosting in the upper jet, its observed temperature
is enhanced and it is overall brighter. The sonic surface of the outflow is the cap-like curved
bright surface. The flow is subsonic and hotter below this surface.
We include the photons in this plot which are emitted from the regions with optical depth
greater than unity. This creates a special shape for the base of the outflow.
\begin{figure*}
\centering
\vbox{
\includegraphics[width=2.0\columnwidth]{image08.eps}}
\caption{Images of TCAF (in $log(T_{obs})$ scale) in presence of outflows at $t=14.25s$,
$t=14.35s$, $t=14.45s$, $t=14.55s$, $t=14.65s$~and~$t=14.75s$. Colorbar is in the range $10^{6.0}-
10^{6.5}$ Kelvin and inclination of the observer is at $80^\circ$. Asymmetry in the shape
is due to difference in Doppler shifts in upper and lower jet components.}
\end{figure*}
Figure 8 shows the dynamical evolution of outflows at $t=14.25s$, $t=14.35s$, $t=14.45s$,
$t=14.55s$, $t=14.65s$~and~$t=14.75s$. The mass outflow increases with time in this case. The images
with gravitationally bent rays have been drawn from the perspective of an observer placed at an
inclination angle of $80^\circ$. The asymmetry of outflows become prominent in this Figure also. The
energy range is chosen in such a way that the observer temperature variation of the Keplerian
disk can be found easily.
\begin{figure}
\centering
\vbox{
\includegraphics[width=1.0\columnwidth]{image09.eps}}
\caption{Grid averaged images of TCAF (in $log(T_{obs})$ scale) in presence of outflows at time $t=14.65s$.
Colorbar is in the range $10^{5.4}-10^{6.7}$ Kelvin for panel (a) and $10^{6.7}-10^{10.3}$ Kelvin for panel
(b). The observer is placed at $70^\circ$. For higher energy range detectors, the Keplerian disk and
upper part of jet becomes completely invisible to an observer.}
\end{figure}
Figure 9 shows energy dependent images of TCAF. With a higher energy detector which
works best in the energy range of $1.0-100$ keV, the Keplerian disk might not be seen for disk
rates lower than $0.1$. With increasing $\dot{m}_d$, the disk may reappear. So, for
an outbursting source, whose accretion rates change on a daily basis, the images will change for
any particular energy range of detector. These pictorial changes will be due to the physical
variations of accretion disk geometry which will be triggered by the thermodynamical properties
(see, CT95 for details) of the disk.
In Panel (b), we see the top part of the outflows is also missing in
a high energy observation. This is due to the velocity profile of the outflowing electrons.
Most of the photons suffer downscattering during their collision in the outflowing electrons.
In Panel 3d, we see that the Jet spectra extends upto $1000$ keV. But, in reality,
in our model of non-magnetic jets, maximum hard photons in the Jet spectra are contributed
only from the base of the jet. The photons that are coming from below the equatorial plane are mostly
reflected or absorbed by optically thick Keplerian
disk. So, up to $70^\circ$, visibility of the base of a lower jet is screened by the Keplerian
disk itself. But, at $80^\circ$, as in Fig. 8, both sides become visible to the
observer.
\subsection{Observed Spectra}
The source spectrum as presented in the Fig. 3a has a peak
at $0.6$ keV. This has been modified due to photon bending and shifted to a lower value for
an observer at higher inclination angle. In the observed spectrum of Keplerian disk has a
peak at $0.3$ keV for $80^\circ$ inclination angle (Fig. 10). The
corresponding characteristic temperature ranges from $10^{6.0}-10^{6.6}$ Kelvin and this
is visible in Fig. 9a for $70^\circ$. The observed spectral variation of the black body
part with an inclination angle remains almost the same as reported in CCG17a.
\begin{figure}
\centering
\vbox{
\includegraphics[width=0.8\columnwidth]{image10.eps}}
\caption{Observed spectra of TCAF seen from inclination angles $10^\circ$, $50^\circ$
and $80^\circ$ in presence of Outflows. Increase in the hard photon contribution with inclination
angle is visible. However, with the inclusion of outflows, the spectral slope increases with
increasing the inclination angle.}
\end{figure}
From Fig. 10, we see that the contribution of hard photons increases with inclination
angle. But, the spectral slope is less for low inclination angle. This is to be contrasted
with our earlier studies (see CCG17a) where the outflow component has not been added.
Without the outflows, the hard photons in powerlaw component increases and the slope
of the spectrum decreases with increasing inclination angle. With the inclusion of
outflows, the spectral slope increases with increasing inclination angle. But, the
contribution of hard photons increases more significantly when the observer moves to
a higher inclination angle.
\subsection{Images with Cooling}
To demonstrate the effects of Compton cooling, we consider first the
data file from hydrodynamic simulations at $t=14.25s$. The electron number density and
temperature profile of the post-shock region after Compton cooling is presented
in Fig. 4. Corresponding spectrum in Fig. 5 shows the spectral softening due
to cooling. From Fig. 11, we see that the size of the CENBOL region is
substantially reduced due to cooling effect and the optical depth of CENBOL
region is so low that very few soft photons from Keplerian disk are intercepted
in the Compton cloud. This is the reason of spectral softening. With the inclusion of
Compton cooling the outflow drastically reduces its size. This effect can also be visible
in the images of TCAF. As the CENBOL size is reduced, the inner edge of the truncated Keplerian
disk moves radially inwards.
\begin{figure}
\centering
\vbox{
\includegraphics[width=0.45\columnwidth]{image11a.eps}
\includegraphics[width=0.45\columnwidth]{image11b.eps}}
\caption{Images of TCAF (in $log(T_{obs})$ scale) as seen from the inclination angle of $50^\circ$,
$70^\circ$ in presence of Compton cooling. Colorbar is in the range $10^{5.6}-10^{6.6}$ K. If the
outflows are absent, dark central region of black hole just starts to appear to an observer at $50^\circ$ and
becomes clearer if the onlooker moves to a lower inclination angle.}
\end{figure}
\vskip 0.5cm
\subsection{Realistic Observed Image with a finite beamwidth}
Figure 12, panel (a) shows a convolved image of a Keplerian disk with the inner edge
at $3$ and outer edge at $50$. Convolution is over a beam width of (a) $\sim 20 r_g$
and (d) $\sim 5 r_g$. Corresponding spectrum for this image gives a pure soft spectrum
(see CT95 and SS73 for further details).
The intensity variation due to the disk and the black hole can be seen. In Panels (b,e),
the images show an accretion disk with
CENBOL (same as Fig. 9(i) of CCG17a).
In this case, the spectrum is hard and the CENBOL surface completely screens the inner
region close to the black hole. This case was studied in CCG17a. In Panel (c,f)
outflows are strong as in an intermediate states and the convolutions show complex
structure. If the inclination angle is very low, then possibly the black hole horizon
could be seen, but in the present case of high inclination angle, the identification of
horizon would be difficult, especially if integrated over time. The images we just presented are
relevant for our galactic center under a sudden high mass inflow induced by
tidal disruption. Various measurements (Markoff et al. 2007 and
references therein) suggested that the jet from the Sgr A* is around $75^\circ$ away
from our line of sight. Assuming this is along the perpendicular direction with respect to the accretion plane, and
assuming radio intensity roughly tracks the disk/jet as discussed above, the Galactic centre
should also show images similar to the panels Fig. 12(a-f).
Our conclusion is that we may not be able to discern the horizon itself, but some
features as in our Fig. 12(a-f) should be visible. Similarly the presence and absence of accretion disk
with change in accretion rates should be seen.
\begin{figure*}
\centering
\vbox{
\includegraphics[width=2.0\columnwidth]{image12.eps}}
\caption{Convolved images of TCAF at different states. 2D Gaussian distribution is
considered for convolution with spreading diameter (hypothetical beam width of the instrument)
of $20~r_g$ (for upper panel) and $5~r_g$ for lower
panel. Panels (a,d) are for a pure Keplerian disk (hard state with weak outflow) with outer edge
at $50~r_g$. Panels (b,e) show
images of TCAF without outflows with outer edge at $50~r_g$ (Soft state). Panels (c,f) depict images
of TCAF with outflows having outer edges up to $100~r_g$. Inclination angle is $80^\circ$ for all
panels. Intensity variation close to the black hole could be seen only when the beam width is around
$5~r_g$.}
\end{figure*}
The convolved images presented in Fig. 12 suggest a low spatial resolution of about $5~r_g$
will be able to distinguish features in the hole-disk system, unless the object has a very low
inclination. Integrated counts from the central region is plotted in Fig. 13b for the Keplerian disk only.
A spatial resolution of $5~r_g$ shows double-peaked curve in just resolved condition. Figure 13a shows
unresolved count distribution when $20 r_g$ is used instead. Considering the possibilities
of high mass inflow onto Sgr A*, in its flaring state, we expect an X-Ray image similar
to Fig. 8 which is difficult to resolve at a high inclination. Thus,
probing inner region to capture the event horizon can be difficult in this state. Future projects
which aim at capturing the image of an event horizon will not be able detect the difference in the intensity
of disk to that of the hole unless the object is in the soft state. Otherwise, only
low inclination disks would show the even horizon in any spectral states, even in presence of
outflows. Of course, this conclusion is valid only if the radio emission roughly tracks the inflow-outflow features as
observed in our numerical simulations.
\begin{figure}
\centering
\vbox{
\includegraphics[width=1.00\columnwidth]{image13.eps}}
\caption{Difference in counts from central region received by detectors with spatial resolution of
$20~r_g$ (curve (a)) and of $5~r_g$ (curve (b)). This is case is for a pure Keplerian disk
presented in Fig. 12(a,d).}
\end{figure}
\vskip 0.5cm
\section{Discussions and Conclusion}
Recent study of the effects of the photon bending on standard and two component
accretion disk spectra and images was made by CCG17a. Earlier, static images of
Two component advective flows were presented where we discussed how the Keplerian
disk (valid for soft states only) and the two component disks with its intrinsic
Compton cloud at the inner edge would look like from various angles. However, a
realistic advective flow will be time dependent and also will have outflows as
the outflow and inflow solutions are always treated with a common footing (Chakrabarti,
1989). This outflow must come out from the post-shock region or CENBOL which behaves
as the Compton cloud around a black hole or a neutron star. This has also been
shown by numerical simulations through numerous works referred to in the earlier
Sections. The outflow could be time varying and the image of the disk would, in
general, vary with time.
In the present paper, we have employed numerical simulations of hydrodynamic process to
generate time dependent accretion disk configurations. At every time step, Monte-Carlo
simulation was done to compute the effects of Comptonization and to obtain the spectrum.
We produced images of the CENBOL with the jet as well as the composite spectra
when the effects of cooling due to Comptonization was turned off or turned on. We
find that with cooling effects the CENBOL collapses to a smaller size and the outflow
is also reduced as in a soft state. In presence of outflows, we draw images of an accretion
disk where the Compton cloud or CENBOL is showing temporal variation of its physical
shape and internal properties. With these, the images depict the outflows which are
self-consistently produced from the inflowing matter. We showed the combined effects
of the disk and the jet and particularly how they would look like in different
X-ray energies. Doppler shift was seen to break the symmetry as expected, but
the photon bending made it even more asymmetric.
Though our simulations are carried out in regions which emit X-rays, and this
will remain true for higher mass black holes also, but the efforts are on to
observe the event horizon in radio waves which have higher spatial resolutions.
On the other hand, both the radio and the Comptonization track the high energy
electron distribution. So we believe that our simulations will have relevance for
radio observations as well. However, intensity of synchrotron photon in
radio band depends on the magnetic field strength and are expected to be primarily originated
from the post shock region. Thus, our X-ray images without the cooler
Keplerian disk, can be treated as radio images. We showed, using two different spatial resolutions,
that one requires a resolution of $\sim 5 r_g$ in order to separate distinct features
in disk-jet systems when high inclination objects are observed, unless the object
is in a soft state and only has the Keplerian disk. However, the obscuration of the
central region is not done when viewed at low inclination angle and the event horizon
should be observable at any spectral state if the beam width is low enough.
The images are of particular interest where the mass accretion rate onto central
engine is high enough, such as RL Quasars, Seyfert 1 galaxies. The current
findings would hold for LLAGNs, like Sgr A*, for high mass inflow induced by tidal
disruptions. Considering the future possibilities of observing such Galactic black holes and SMBHs, we
present an example of a high inclination system, to impress how the inclination
basically removes the chances of observing the horizon, especially, when the
jets are active. In the absence of jets and outflows, however, the horizon can be
discerned. This result will be of importance for future missions which could exclude
high inclination Seyfert 2 AGNs and GBHs such as GRS 1915+105, H 1743-322, GRO 1655-40.
In the spectral study using Monte Carlo simulations, we see effects of
photon bending in the multi-color soft photon component. The peak was seen at a lower
energy due to redshifts at an inclination of $70^{\circ}$. We also find that the
spectra harden at higher inclination angles. Of course, a major result, though
expected, in our simulation is that when the cooling is included, the CENBOL itself shrinks
and consequently the spectra become softer. Further simulations including the effects
of synchrotron radiation, Compton cooling and viscosity are being carried
out and will be reported elsewhere.
\section{Acknowledgements}
This research was possible in part due to a grant from Ministry of Earth Sciences
with Indian Centre for Space Physics.
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"redpajama_set_name": "RedPajamaC4"
} | 5,505 |
Q: Expected Minimum Distance between draws of rv If $X$ is a random variable, then Chebyshev's inequality states that $$\Pr(|X-E(X)|\ge t)\le\frac{Var(X)}{t^2}.$$ If the "spread"/variance of the distribution $X$ is drawn from is very large, then for small $t$ this inequality gives no useful information. My question is if we get several draws from $X$, can we say something a bit stronger, something like this:
If $X_1,\ldots, X_N, X_{N+1}$ are iid, then can we bound $\Pr(\min_{1\le i\le N} |X_i - X_{N+1}| \ge t)?$ It seems like we should be able to say that this would be small with much higher confidence than in the vanilla Chebyshev, since the $N$ samples we draw before comparing the distance to $X_{N+1}$ intuitively seems to capture the spread of the distribution, and would potentially yield a better concentration than comparing to the mean. Any help is appreciated!
A: Since $X_{N+1}$ is independent of the vector $\left(X_1,\dots,X_N\right)$, it follows (from Fubini's theorem) that
$$
\Pr\left(\min_{1\le i\le N} |X_i - X_{N+1}| \ge t\right)=\int_{\mathbb R}\Pr\left(\min_{1\le i\le N} |X_i - x| \ge t\right)\mathrm{d}{\Pr}_{X_{N+1}}(x).
$$
Using independence of the vector $\left(X_1,\dots,X_N\right)$ and then the fact that all the components have the same distribution, we get that
$$
\Pr\left(\min_{1\le i\le N} |X_i - x| \ge t\right)=\prod_{i=1}^n\Pr\left(\left\lvert X_i-x \right\rvert \geq t\right)=\Pr\left(\left\lvert X_1-x \right\rvert \geq t\right)^N
$$
hence
$$
\Pr\left(\min_{1\le i\le N} |X_i - X_{N+1}| \ge t\right)=\int_{\mathbb R}\Pr\left(\left\lvert X_1-x \right\rvert \geq t\right)^N\mathrm{d}{\Pr}_{X_{N+1}}(x).
$$
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,484 |
Q: How to make the widgets on the popup window widgets stay with the popup window when the root window is resized The widgets on my popup window do not resize when the root window resizes. The popup window and the labels on the popup window stay where they are. Does it have something to do with the size_hint and size of the popup window itself? It seems that the widgets(icons) are independent of the popup window.
main file
from kivy.app import App
from kivy.uix.screenmanager import Screen
from kivy.uix.button import ButtonBehavior
from kivy.uix.image import Image
from kivy.properties import StringProperty, ObjectProperty,NumericProperty
from kivy.uix.popup import Popup
from kivy.uix.floatlayout import FloatLayout
class MainScreen(Screen, FloatLayout):
mantra_text = ObjectProperty(None)
def printMantra(self):
print(self.ids.mantra_text.text)
def icon_popup(self):
popup = Popup(title="Profile Icon", content=Popup_Content(), size_hint=(None, None), size=(300, 200))
popup.open()
class Popup_Content(FloatLayout):
pass
class ImageButton(ButtonBehavior, Image):
pass
class MainApp(App):
def build(self):
return MainScreen()
def set_profile_icon(self, image):
self.root.ids.profile_icon.source = image.source
print(image)
#print(self.root.ids.profile_icon)
MainApp().run()
kivy file
#:import utils kivy.utils
<MainScreen>
Popup_Content:
id: popup_content
FloatLayout:
canvas:
Color:
rgb: utils.get_color_from_hex("#ffbb99")
Rectangle:
pos: self.pos
size: self.size
GridLayout:
cols: 2
pos_hint: {"x":0.6, "top":1}
size_hint: 0.4,0.2
spacing_horizontal: [0.9*root.width]
Label:
text: "Name"
ImageButton:
id: profile_icon
source: "profile_icon"
on_release: root.icon_popup()
Label:
text: mantra_text.text
pos_hint: {"x":0, "top":0.8}
size_hint: 1, 0.2
text_size: self.size
halign: "center"
font_size: 25
TextInput:
id: mantra_text
pos_hint: {"x": 0.15, "top":0.6}
size_hint: 0.7, 0.1
#text_size: self.size
Label:
text: "Time"
pos_hint: {"x":0.3, "top":0.6}
size_hint: 0.4, 0.2
text_size: self.size
halign: "left"
font_size: 30
Button:
text: "Time"
pos_hint: {"x":0.3, "top":0.5}
size_hint: 0.4, 0.2
on_release: root.printMantra()
<Popup_Content>
#profile_icon: profile_icon
FloatLayout:
GridLayout:
cols: 5
pos_hint: {"x":0.95, "y":1.6}
ImageButton:
id: man_01
source: "icons/male_icon_01.png"
on_release: app.set_profile_icon(man_01)
ImageButton:
id: man_02
source: "icons/male_icon_02.png"
on_release: app.set_profile_icon(man_02)
ImageButton:
source: "icons/male_icon_01.png"
on_release: app.set_profile_icon()
ImageButton:
source: "icons/male_icon_01.png"
on_release: app.set_profile_icon()
ImageButton:
source: "icons/male_icon_01.png"
on_release: app.set_profile_icon()
ImageButton:
id: female_01
source: "icons/female_icon_01.png"
on_release: app.set_profile_icon(female_01)
A: If you want your Popup to change size when you resize the App, then use size_hint. Something like:
popup = Popup(title="Profile Icon", content=Popup_Content(), size_hint=(0.5, 0.5))
Using size_hint=(None, None), size=(300, 200) forces the Popup size to (300, 200) regardless of the size of MainScreen.
And to get the Popup content to follow the Popup, you can use RelativeLayout. In the documentation for RelativeLayout, it says:
When a widget with position = (0,0) is added to a RelativeLayout, the
child widget will also move when the position of the RelativeLayout is
changed. The child widgets coordinates remain (0,0) as they are always
relative to the parent layout.
So if you define your Popup_Content as a RelativeLayout, then the GridLayout will follow it. I suggest defining Popup_Content as:
class Popup_Content(RelativeLayout):
pass
Then, in the kv:
<Popup_Content>
#profile_icon: profile_icon
GridLayout:
cols: 5
# pos_hint: {"x":0.95, "y":1.6}
ImageButton:
id: man_01
.
.
.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,080 |
\section{Introduction}
{{\color{black}Femtosecond laser pump-probe techniques are often used to investigate thermal transport in materials such as metals, semi-conductors, insulators, gases, liquids and at solid-liquid-gas interfaces. These techniques include measuring the transient sample temperature rise following the partial absorption of a femtosecond laser pump pulse through monitoring the change in optical reflectivity by an optically time-delayed probe. Analysis of the recorded waveforms is typically done through comparison to models in order to retrieve the thermal characteristics of the sample, such as thermal conduction, diffusivity or even Kapitza interfacial thermal resistance \cite{Cahill2004,Schmidt2008,Schmidt2008b,Cahill2014,Schmidt2009}. Time-domain Brillouin scattering (TDBS) is also based on ultrafast lasers for optical ultrasound excitation and detection in the materials under study. It is a well suited technique for the measurement of elastic, visco-elastic and optical properties of ultrathin transparent solid or liquid samples at ultrasound GHz to THz frequencies \cite{Lin1991}. TDBS has been applied for the examination of diverse phenomena such as non-linear acoustic waves in solids and liquids \cite{Klieber2015,Bojahr2012}, spatial mechanical inhomogeneities in solids \cite{Mechri2009,Nikitin2015}, and even GHz transverse acoustic phonons in viscoelastic liquids \cite{pezeril09,pezeril12,KHP+13,pezeril16}.
We present a tabletop pump-probe method which enables the measurement through TDBS of the local temperature in liquids in contact to an optical transducer. We demonstrate the performances of TDBS as a contactless local temperature probe with liquid glycerol, a well-known and well-characterized prototypical glass-forming liquid. We also study octamethylcyclotetrasiloxane (OMCTS), a confined liquid prototype, with the interest for future experiments related to molecular confinement in ultrathin liquid layers where the understanding of thermal effects remains elusive.}
\section{Experimental technique}
For sample construction, the liquid under study was squeezed in between two flat optical quality substrates with one of them having a metallic thin film deposited on it which was in contact with the liquid, as sketched in Fig. \ref{fig:sample_holder}. The metallic film, which, depending on the experimental configuration, is either a 40~nm chromium film or a 80~nm aluminium film, was acting as a photoacoustic transducer and the liquid thickness was more than 10~microns. As liquids we used glycerol (Acros Organics$^\textrm{\textregistered}$, 99+\% purity) and OMCTS (Fluka$^\textrm{\textregistered}$, 99+\% purity) which were prepared by forcing them through several linked 0.2~$\mu$m teflon millipore filters to remove dust particles. {\color{black}The fully assembled liquid sample cell was installed inside a cold finger cryostat which was temperature controlled through a feedback sensor directly attached to its heat sink (labeled Sensor A in Fig. \ref{fig:sample_holder}). The sample temperature was measured by an additional Peltier temperature sensor (labeled Sensor B in Fig. \ref{fig:sample_holder}) which was attached to the sample holder mount}, about 1~to~2~cm away from the experimental volume, {\color{black}the closest place for a convenient installation. At each temperature, the sample was given sufficient time to equilibrate before data acquisition. This time was estimated upon equilibration of the cryostat temperature sensors A and B and from repeated TDBS measurements until convergence to a reproducible steady state TDBS signal.}
\begin{figure}[t!]
\centering
\includegraphics[width=9cm]{Fig1.eps}
\caption{Illustration of the sample holder used to construct liquid samples of different thicknesses. A copper jig was used to hold two optically transparent substrates with the sample liquid sandwiched in between. The whole construction is attached to a heat sink of a cryostat with one temperature sensor (A) at the heat sink and a second (B) attached to the jig to monitor the sample temperature as closely to the sample as possible.}
\label{fig:sample_holder}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[width=9cm]{Fig2.eps}
\caption[Fig1]{(a) Sketch of sample with a liquid squeezed in between two flat, optically grade substrates. One substrate holds a metallic thin film serving as photoacoustic transducer. Laser irradiation launches the acoustic waves through the metallic transducer film which are transmitted through the transparent liquid where they are detected through TDBS by means of a time-delayed laser probe pulse. For several different pump fluences and at different temperatures of the cryostat we recorded transient reflectivity data for OMCTS (b) and glycerol (c). For a fivefold change in laser fluence, the Brillouin scattering frequency drastically changes in case of OMCTS which indicates melting through local laser thermal heating. For a similar change in pump fluence for the glycerol sample, the Brillouin frequency changes by over 10\%.}
\label{fig1}
\end{figure}
The optical experiment uses the TDBS technique, suitable for the study of the temperature and frequency dependences of ultrafast mechanical dynamics in liquids in the lower GHz frequency range \cite{pezeril09,pezeril12,KHP+13,pezeril16,Kli10,Shelton2005,Maznev11}. Measurements were carried out using an ultrafast optical pump-probe experimental setup as illustrated in Fig. \ref{fig1}(a). The laser pulses were generated by a femtosecond Ti-Sapphire Coherent RegA 9000 regenerative amplifier operating at a repetition rate of 250~kHz outputting 160~fs pulses with a central wavelength of 790~nm. The laser pulses were separated into two beams with a 790~nm pump beam synchronously modulated at 50~kHz frequency by an acousto-optic modulator (AOM) and focused on the surface of a metallic photo-acoustic transducer film with a gaussian spatial beam profile of FWHM $\sim$100~$\mu$m. {\color{black}For an enhanced signal to noise ratio, the pump modulation frequency has been chosen as a submultiple of the laser repetition rate in order to obtain a stable number of pump pulses that are chopped on each cycle.} The second beam, used as the probe, was a much less energetic beam which was frequency doubled by a BBO crystal to 395 nm. It was time-delayed and tightly focused on the sample surface at normal incidence with a spot size smaller than 20 $\mu$m where it spatially overlapped with the pump spot. A photodiode, coupled to a lock-in amplifier synchronized to the 50~kHz pump modulation frequency, recorded the reflected probe beam and hence measured the transient differential reflectivity $\Delta$R(t) as a function of delay between pump and probe beams. Upon transient absorption of the 790~nm pump pulse over the optical skin depth of the metallic thin film, laser excited acoustic pulses were transmitted across the interface into the adjacent transparent liquid. The out-of-plane acoustic propagation of the strain pulses in the transparent liquid medium leads to the occurrence of TDBS oscillations in the transient reflectivity signal, see Fig. \ref{fig1}(b) and (c). As in any Brillouin scattering process, the frequency $\nu$ of these oscillations is related to the ultrasound velocity $v$ of the liquid, to the probe wavelength $\lambda$, to the refractive index $n$ of the medium, and to the back-scattering angle $\theta$ through
\begin{equation}\label{eq:Brillouin1}
\nu = 2 \ n \ v \cos \theta / \lambda.
\end{equation}
The acoustic velocity and the index of refraction of the liquid and, as a consequence, the TDBS itself are influenced by many external conditions, such as the ambient temperature and the laser fluence. Therefore TDBS is sensitive to a local temperature modification of the scattering liquid medium. In fact, the Brillouin scattering frequency detected in the time-domain reflectivity signal monitors any change of the temperature distribution, which appears as a modification of the detected Brillouin oscillation frequency. {\color{black}As in classical interferometric processes, the characteristic TDBS sensitivity length is given by $\lambda/2n$. This sensitivity length is, in most cases, in the range of 100-200 of nanometers.} It means that any change of the overall temperature or temperature distribution in a liquid volume as small as a couple of pico-liters (200~nm$\times$probe spot surface of 100~$\mu$m FWHM diameter) can be detected from TDBS.
The laser pump pulse can cause permanent damage or irreversible sample modification at a given fluence threshold, as in any optical pump-probe experiment. This effect can be experimentally observed once the recorded data become fluence dependent such as the excitation of shock waves at high laser fluences \cite{Klieber2015} which reveals the departure from the linear acoustic regime to the non-linear acoustic regime. It can be a consequence as well of a local temperature rise caused by cumulative heating of the sample from the multiple laser pump pulses which brings the sample into a steady state temperature regime correlated to the laser pump fluence \cite{Cahill2004,Schmidt2008,Schmidt2008b,Cahill2014,Schmidt2009}. Fig. \ref{fig1}{\color{black}(b) and (c)} displays recorded data obtained in OMCTS and glycerol at a temperature of 260~K and 230~K, respectively, as indicated by the Peltier temperature sensor, at different laser pump fluences. As seen in Fig. \ref{fig1}(b), a fivefold change in the laser pump fluence induces a drastic change in the Brillouin oscillations frequency, from 20.5~GHz to 8.8~GHz, and in the attenuation rate. The Brillouin frequency of 20.5~GHz matches the Brillouin frequency in the glassy state whereas the Brillouin frequency of 8.8~GHz matches the Brillouin frequency in the liquid state, see Fig. \ref{fig2}(a), indicating a major modification of the OMCTS sample which experiences melting mediated by the cumulative heating of the multiple laser pump pulses. Similarly, the Brillouin frequency in glycerol changes from 25.1~GHz to 22.8~GHz, for a fivefold modification of the laser pump fluence. Similarly, the attenuation rate evolves with a modification of the laser fluence, however, its pertinence is out of scope of the manuscript which focus mainly on the analysis of the Brillouin frequency versus laser fluence or temperature.
An important aspect to consider is that when the liquid is driven with large enough strain pulses to produce shock wave formation as in \cite{Klieber2015}, the Brillouin frequency will increase with an increase of the laser pump fluence, which is opposite to our current experimental observations. Therefore, we have neglected the effect of non-linear shock waves and solely assumed cumulative laser heating as the main mechanism responsible for the evolution of the Brillouin frequency in our present measurements. In the following, we will describe how to calibrate the measured Brillouin frequency in the studied liquid in order to employ TDBS as a specific temperature sensor.
\section{Results}
\begin{figure}[t!]
\centering
\includegraphics[width=9cm]{Fig31.eps}
\caption[Fig2]{Time derivative of recorded transient reflectivity signals obtained in (a) OMCTS and in (b) glycerol at different temperatures of the cryostat and at a given laser pump fluence of 0.75 mJ.cm$^{-2}$. At such relatively low fluence, the overheating caused by the multiple laser pump pulses is moderate. (c) Temperature dependent Brillouin frequency in glycerol. The temperature calibration curve displayed in (c) can be used to estimate the absolute liquid temperature from the measured Brillouin frequency. {\color{black}The experimental uncertainties in (c) are estimated to lie below 0.1~GHz}.}
\label{fig2}
\end{figure}
The temperature calibration measurement takes advantage of the fact that the Brillouin frequency in a liquid is strongly temperature dependent. Therefore, the Brillouin frequency can be used as a probe to determine the absolute temperature in the experimentally investigated local region of the liquid. Since the laser pump itself can affect the Brillouin frequency at high fluences, the measurement data shown in Fig.~\ref{fig2} were obtained at sufficiently low pump fluence such that the effect of cumulative heating is moderate. Figure~\ref{fig2} shows the derivative of the recorded transient reflectivity change recorded at several temperatures in OMCTS and glycerol, with a sample structure as sketched in Fig.~\ref{fig1}(a), composed of a silicon substrate holding a 40~nm chromium film in contact with the liquid and a transparent cover glass substrate. In both cases, for both liquids, the temperature influences the Brillouin frequency and the attenuation rate, as shown in Fig.~\ref{fig2}(a) and (b). {\color{black}OMCTS displays a sharp crystalline to liquid phase transition at 260~K and the Brillouin frequency variation across the transition is way more abrupt than for glycerol which is an intermediate fragile glass former with a smooth glass transition temperature around T$_g$~=~186~K. A sharp temperature transition is not appropriate for the TDBS temperature determination of a liquid sample over a wide temperature range. For this physical reason, our TDBS technique applied for the temperature determination of samples is more adapted for glass former liquids. In the following, we detail the meticulous TDBS temperature calibration in the case of glycerol.} For each temperature measurement in glycerol, such as the ones displayed in Fig.~\ref{fig2}(b), the frequency $\nu$ of the Brillouin oscillations observed in the transient reflectivity signal were fitted following a sinusoidal damped function in the form
\begin{equation}\label{eq:Brillouin2}
\Delta\textrm{R}(t) \sim \sin (2\pi\nu t + \phi) \ \exp(-\Gamma t),
\end{equation}
$\phi$ being a {\color{black}constant} phase parameter. As indicated in Fig.~\ref{fig2}(c), the relevant Brillouin scattering frequency fit parameter $\nu$ changes significantly as a function of temperature. The experimental {\color{black}TDBS} temperature calibration has been further fitted by a smooth polynomial function in order to extract an even more reliable temperature behavior of the Brillouin scattering frequency of glycerol under our experimental conditions at 395~nm probe wavelength and a normal incidence. {\color{black}From the calibration curve displayed in Fig.~3(c), we can calculate the temperature rise caused by cumulative laser heating corresponding to the data of Fig. 2(c). The Brillouin frequencies extracted from Fig.~2(c) for 2.4~mJ.cm$^2$ and 12~mJ.cm$^2$ fluences respectively are 25.1~GHz and 22.7~GHz, we thus calculate a temperature rise of 3~K and 33~K respectively. The temperature uncertainty of our measurement is linked to the Brillouin frequency uncertainty. The Brillouin frequency uncertainty of a single TDBS scan, that takes about 5 seconds of acquisition time, at a given fixed temperature, is in the range of 0.1~GHz and the corresponding temperature accuracy is estimated in the range of 0.5~K. It means that our TDBS measurements can monitor dynamic temperature changes of 0.5~K at a sampling frequency of 0.1~Hz. In order to further improve the temperature accuracy of our TDBS measurements, it is required to average several TDBS scans for a better signal to noise ratio and a more accurate frequency determination.}
\section{Summary}
{\color{black}Several liquid samples, OMCTS and glycerol, have been analyzed to experimentally investigate the influence of laser-induced cumulative thermal heating effects on the liquid.} Such effects can be efficiently minimized by using a good thermally conducting substrate like sapphire or silicon. An additional decrease of the influence of cumulative thermal heating effects can be achieved with multilayer sample structure where a thermal insulating SiO$_2$ layer is added in order to shield the liquid from the laser heated metallic transducer film \cite{Chaban2017}. Under certain circumstances, alternative sample structure are required in experimental situations where even slight temperature changes have to be avoided.
The extrapolation of our results to confined liquids could shed light on the thermal properties of ultrathin liquid films \cite{Christenson1982,Heuberger2001,Perkin2012}, which is an exciting experimental challenge for the understanding of nanoscale heat transport \cite{Cahill2014,Cahill2003,Volz2016}.
\section*{ACKNOWLEDGEMENTS}
The authors acknowledge financial support from CNRS (Centre National de la Recherche Scientifique) under grant Projet International de Coop\'eration Scientifique. The authors would like to thank Lionel Guilmeau for technical support as well as Mathieu Edely for Chromium deposition.
This work was partially supported by the Department of Energy under grant No.~DE-FG02-00ER15087, National Science Foundation under grants No.~CHE-0616939 and DMR-0414895, Agence Nationale de la Recherche under grant No.~ANR-12-BS09-0031-01.
| {
"redpajama_set_name": "RedPajamaArXiv"
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package hlib
type Wscript_t struct {
Package Package_t
Options Options_t
Configure Configure_t
Build Build_t
}
type Stmt interface {
is_stmt()
}
type Package_t struct {
Name string
Authors []Author
Managers []Manager
Version Version
Deps []Dep_t
}
type Author string
type Manager string
type Version string
type Branches []string
type Dep_t struct {
Name string
Version Version
Type DepType
}
type DepType int
const (
UnknownDep DepType = 1 << iota
PublicDep
PrivateDep
RuntimeDep
)
func (d DepType) HasMask(mask DepType) bool {
return bool((d & mask) != 0)
}
type Visibility int
const (
Local Visibility = 0
Exported Visibility = 1
)
type Options_t struct {
Tools []string
HwafCall []string
Stmts []Stmt
}
type Configure_t struct {
Tools []string
HwafCall []string
Env Env_t
//Tags []Value
Stmts []Stmt
}
type Env_t map[string]Value
type Build_t struct {
Tools []string
HwafCall []string
Targets Targets_t
Stmts []Stmt
Env Env_t
}
type Targets_t []Target_t
type Target_t struct {
Name string
Features []string
Source []Value
Target string
Group string
Use []Value
Defines []Value
CFlags []Value
CxxFlags []Value
LinkFlags []Value
ShlibFlags []Value
StlibFlags []Value
RPath []Value
Includes []Value
ExportIncludes []Value
InstallPath []Value
Env Env_t
KwArgs map[string][]Value
}
// make Targets_t sortable
func (tgts Targets_t) Len() int { return len(tgts) }
func (tgts Targets_t) Less(i, j int) bool { return tgts[i].Name < tgts[j].Name }
func (tgts Targets_t) Swap(i, j int) { tgts[i], tgts[j] = tgts[j], tgts[i] }
type KeyValue struct {
Tag string
Value []string
}
type Value struct {
Name string
Set []KeyValue // first item is the "default"
}
func DefaultValue(name string, value []string) Value {
kv := KeyValue{
Tag: "default",
Value: make([]string, len(value)),
}
copy(kv.Value, value)
return Value{
Name: name,
Set: []KeyValue{kv},
}
}
// EOF
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,337 |
Winter Storm Watch
Winter Storm Warning Issued for Chicago Area on New Year's Day
Heavy snow and gusty winds are possible across the Chicago area Saturday
A winter storm warning has been issued for the Chicago area on New Year's Day, as a storm system could bring between four and eight inches of snow and gusty winds to the region, according to the National Weather Service.
The National Weather Service issued the warning beginning at 9 a.m. Saturday for Cook, Lake, DuPage, DeKalb, Kane, La Salle, Kendall, Grundy, Will, Winnebago, Boone, McHenry, Ogle and Lee counties through midnight. A winter storm watch was previously issued for the Chicago area from 6 a.m. Saturday until Sunday morning.
Forecasters say that steady, blowing snow will be the primary threat from the storm, causing dangerous travel conditions throughout the area.
Snow accumulations will vary widely depending on the track of the storm, but forecasters say that accumulations of greater than six inches are possible, along with northeasterly winds gusting in excess of 35 miles per hour.
Travel will be difficult at times during the storm, with blowing snow expected to dramatically reduce visibility in open areas.
Forecasters say that the steadiest snow rates will likely occur from 2 p.m. through 8 p.m. Saturday.
The storm track could potentially change in the days leading up to Saturday, but the National Weather Service estimates that the city of Chicago, along with areas immediately to the southwest of the city, has about a 70% chance of seeing at least four inches of snow from the storm.
Any change in the track of the storm will impact those predictions, with a move to the north potentially drawing in warmer air and causing mixed precipitation to fall in some areas, or a move to the south potentially causing higher snowfall accumulations in the southern suburbs.
Stay tuned to the NBC 5 Storm Team for all the latest information.
Winter Storm WatchNational Weather Servicechicago snow | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,219 |
{"url":"https:\/\/www.gradesaver.com\/textbooks\/math\/applied-mathematics\/elementary-technical-mathematics\/chapter-2-section-2-3-multiplication-and-division-of-signed-numbers-signed-numbers-drill-3-page-114\/2","text":"## Elementary Technical Mathematics\n\n(-5)(-9) Multiplying two negatives yields a positive answer $5 \\times 9$ = 45","date":"2018-07-21 02:36:40","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\": 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\": 0.28921017050743103, \"perplexity\": 2938.4358593026036}, \"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-2018-30\/segments\/1531676592150.47\/warc\/CC-MAIN-20180721012433-20180721032433-00189.warc.gz\"}"} | null | null |
Q: CRA Service worker not working in production Hello and thanks in advance for any help. I got a service-worker up and running with my react-app, it runs perfectly on local builds and when deployed from Azure pipelines as well. Problem is in production, which is deployed via Cloudflare, the service worker throws this error in console and cannot figure out why: "Uncaught (in promise) bad-precaching-response: bad-precaching-response :: [{"url":"https://example.com/static/media/getFetch.xxxxxxxxxxxxx.cjs","status":404}]"
Now the said file "getFetc.xxxx.....cjs" seems to be there under static/media/ build files
Any suggestions would be welcome!
A: reason being "getFetc.xxxx.....cjs" this is a static file so you have to mention a rule in your web.config file ,so that this particular file is processed.
Add following line in your web.config file :
<staticContent>
<mimeMap fileExtension=".cjs" mimeType="text/javascript" />
</staticContent>
Hope this will help you..!
A: Ok, problem solved, and in our case it was actually an error in the server not serving the file whne requested, so apparently nothing was wrong client-side
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,698 |
\section{Prologue.}\label{s:pro}
At the outset of his landmark monograph \cite{hajek}, Petr H\'ajek writes:
\begin{quote}{\sl There are various systems of fuzzy logics, not just one. We have one basic logic \textup{(BL)} and
three of its most important extensions: {\L}ukasiewicz logic, G\"odel logic, and the product logic.} \cite[p.5]{hajek}.
\end{quote}
\emph{Basic Logic} is, of course, the creation of H\'ajek himself. One of its several virtues is to afford metamathematical
comparison of many-valued logics to an unprecedented degree of clarity. Our paper is intended as a modest contribution to such comparative studies; it will soon transpire that it would have been impossible to
write it, in the possible but unfortunate worlds orphan of \cite{hajek}.
\smallskip We assume familiarity with Basic (propositional) Logic, triangular norms (\emph{t-norms}, for short), and BL-algebras; see \cite{hajek}, and Section \ref{s:pre} for an outline. Note that in this paper `t-norm' means `continuous t-norm', for the sake of brevity. We write
$\Form$ for the set of formul\ae\ over the countable collection of propositional variables $\textup{{\sc Var}}:=\{X_1,X_2,\ldots\}$, with primitive connectives $\to$ (implication), $\&$ (monoidal conjunction), and $\bot$ (\textit{falsum}). As usual, $\&$ is semantically interpreted by a t-norm, $\to$ by its residuum, and $\bot$ by $0$. We adopt the standard abbreviations, $\neg \alpha:=\alpha \to \bot$, $\alpha \wedge \beta:=\alpha \& (\alpha\to\beta) $, $\alpha \vee \beta:=((\alpha\to\beta)\to \beta)\wedge((\beta \to \alpha)\to \alpha)$, and $\alpha \leftrightarrow \beta := (\alpha \to \beta) \& (\beta \to \alpha)$. We write \BL\ to denote Basic Logic, as axiomatised in \cite{hajek, cignolietal_bl}. An \emph{extension} of \BL\ is a collection of formul\ae\ closed under the (syntactic) consequence relation of \BL, and closed under substitutions. If $\Mog$ is an extension of \BL, we always tacitly assume that $\Mog$ is consistent, we refer to $\Mog$ as a \emph{many-valued logic}, and we denote by ${\vdash_{\Mog}}$ its consequence relation.
\emph{{\L}ukasiewicz logic}, denoted \LL, is obtained by extending \BL\ with the axiom schema $\neg\neg\varphi \to \varphi$. \emph{G\"odel logic}, denoted \G, is obtained by adding to \BL\ the schema $\varphi \to (\varphi\&\varphi)$. To obtain \emph{Product logic}, written \PP, one extends \BL\ with $\neg \varphi \vee ((\varphi\to(\varphi \&\psi))\to \psi)$. See \cite[p.63, Definitions 4.2.1 and 4.1.1]{hajek}, and \cite[Chapter I]{handbook1}.
Over the real unit interval $[0,1]\subseteq \R$, consider a BL-algebra $([0,1],*,\to_*,0)$, where $*\colon [0,1]\times [0,1]\to[0,1]$ is a continuous t-norm with residuum $\to_{*}$. By an \emph{algebra of truth values} we shall mean a subalgebra $T_{*}$ of some such BL-algebra $([0,1],*,\to_*,0)$. Note, in particular, that $\{0,1\}$ is a subset of any algebra of truth values. We write $T_{*}\subseteq [0,1]$ for the underlying set of the algebra of truth values, too, \textit{i.e.}\ for the set of truth values itself.
We say that the pair $(\Log, T_*)$ is a \emph{real-valued logic} if $\Log$ is an extension of \BL\ that is complete with respect to valuations $\mu\colon \Form \to T_{*}$ into the given algebra of truth values. Any algebra of truth values $T'_*$ such that $(\Log,T'_*)$ is a real-valued logic is said to \emph{induce} $\Log$. When $T_{*}=[0,1]$, we also say that $\Log$ is induced by the t-norm $*$. (This makes sense, recalling that $\to_{*}$ is uniquely determined by $*$. See Section \ref{s:pre} below.) Distinct algebras of truth values may of course induce the same logic $\Log$, \textit{i.e.}\ the same extension of \BL. When we write that $\Log$ is a real-valued logic, with no reference to $T_*$, we mean that there is at least one algebra of truth values $T_*$ that induces $\Log$.
\smallskip With this machinery in place, we consider two principles that a real-valued logic $\Log$ may or may not satisfy.
\smallskip
\princ{\P{1}}{For every algebra $T_*$ of truth values inducing $\Log$, the following holds. For each $\alpha, \beta \in\Form$, we have $\vdash_{\Log} \alpha \leftrightarrow \beta$ if, and only if,
\begin{align*}
\mu(\alpha) = 1 \quad \Longleftrightarrow \quad \mu(\beta) = 1
\end{align*}
holds for each valuation $\mu \colon \Form \to T_{*}$.\qed\\}
\princ{\P{2}}{For every algebra $T_*$ of truth values inducing $\Log$, the following holds. For each pair of valuations $\mu, \nu \colon \Form \to T_{*}$, if $\mu\neq \nu$ then
there is a formula $\alpha\in \Form$ such that $\mu(\alpha) >0$ while $\nu(\alpha) = 0$.\qed\\}
\noindent Our first two results are that \P{1} and \P{2} are characteristic of $\G$ and $\LL$, respectively, to within extensions.
\begin{theoremI}A real-valued logic $(\Log,T_*)$ satisfies $\P{1}$ if, and only if, $\Log$ is an extension of G\"odel logic.
\end{theoremI}
\begin{theoremII}A real-valued logic $(\Log,T_*)$ satisfies $\P{2}$ if, and only if, $\Log$ is an extension of {\L}ukasiewicz logic.
\end{theoremII}
\begin{remark}
Observe that the two preceding theorems show that in \P{1} and \P{2} one can safely replace the initial universal quantification by an existential one. In other words, the principles \P{1} and \P{2} display robustness with respect to the specific choice of the algebra of truth values, \textit{salva logica} $\Log$.\qed
\end{remark}
\noindent We prove Theorem I in Section \ref{s:tarski}, and Theorem II in Section \ref{s:leibniz}, after some preliminaries in Section \ref{s:pre}.
\smallskip The question arises, can one also characterise Product logic by means of general principles such as \P{1} and \P{2}. We shall show how to answer this question affirmatively, under one additional assumption. Let us say that the real-valued logic $\Log$ is \emph{closed} if there exists an algebra of truth values $T^*$ inducing $\Log$ such that the underlying set of $T^*$ is closed in the Euclidean topology of $[0,1]$.
Product logic is the unique \emph{closed} real-valued logic that \emph{fails} both \P{1} and \P{2} hereditarily with respect to real-valued extensions, in the following sense:
\begin{theoremIII}A closed real-valued logic $\Log$ is Product logic if, and only if, $\Log$ and all of its non-classical, real-valued extensions fail \P{1} and \P{2}.
\end{theoremIII}
\noindent We prove Theorem III in Section \ref{s:product}.
\smallskip The proofs of Theorems I--III are relatively straightforward applications of known facts about extensions of Basic Logic.
The interest of the present contribution, if any, is thus to be sought not so much in the technical depth of the results, as in the significance of the two principles \P{1} and \P{2} in connection with logics of comparative truth. Before turning to the proofs, let us therefore expound on \P{1} and \P{2} a little.
Logics fulfilling \P{1} share with classical logic the feature that each proposition is uniquely determined, up to logical equivalence, by the collection of its true interpretations (that is, models), where `true' in the latter sentence is to be read as `true to degree $1$'. In the classical case this may be conceived as a consequence of the Principle of Bivalence, along with completeness. (Indeed, if in classical logic $\alpha$ and $\beta$ evaluate to $1$ at exactly the same $\mu$'s, then, by bivalence, they evaluate to
the same value at each $\mu$; hence $\alpha \leftrightarrow\beta$ is a tautology, and we therefore have $\vdash \alpha \leftrightarrow \beta$ by completeness.) Theorem I shows that, remarkably, real-valued logics that \emph{fail} the Principle of Bivalence---for instance, G\"{o}del logic---may still satisfy \P{1}.
Logics fulfilling \P{2} share with classical logic the feature that distinct models of the logic can be separated by some formula. In more detail, classical logic has the property that if $\mu$ and $\nu$ are two distinct true interpretations of its axioms, then there is a formula $\alpha$ that can tell apart the two models $\mu$ and $\nu$, in the sense that $\alpha$ is not false (\textit{i.e.}\ true) in $\mu$ but false in $\nu$. A logic failing \P{2}, by contrast, must allow two distinct true interpretations $\mu$ and $\nu$ of its axioms which are indiscernible, in the sense that no proposition is false (\textit{i.e.}\ evaluates to degree $0$) in $\nu$ and not false (\textit{i.e.}\ evaluates to degree $>0$) in $\mu$.\footnote{It should be emphasised that there is some leeway in formulating the separating conditions $\mu(\alpha)>0$ and $\nu(\alpha)=0$ here: see Corollary \ref{cor:equiv} below for equivalent variants.} In this precise sense, the given real-valued semantics of such a logic is redundant, as one could identify $\mu$ and $\nu$ without any logical loss. Theorem II shows that, remarkably, there is just one $[0,1]$-valued logic---namely, {\L}ukasiewicz logic---capable of avoiding that redundancy, by actually telling apart any two distinct real numbers in $[0,1]$.
\section{Preliminary facts about real-valued logics.}\label{s:pre}
We outline the framework of H\'ajek's Basic Logic.
A (\emph{continuous}) {\em t-norm} is a binary operation $* \colon [0,1]^2 \to [0,1]$, continuous with respect to the Euclidean topology of $[0,1]$, that is associative, commutative, has $1$ as neutral element, and is monotonically non-decreasing in each argument:
\begin{align*}
\forall a,b,c \in [0,1]\ : \ b \leq c\ \Longrightarrow \ a * b \leq a * c.
\end{align*}
For $a,b \in [0,1]$, set $a \to_* b := \sup{\{ c \in [0,1] \mid a * c \leq b \}}$.
It is well known \cite[Sec.\ 2.1.3]{hajek} that continuity is sufficient to entail $a \to_* b = \max{\{ c \in [0,1] \mid a * c \leq b \}}$. The operation $\to_*$ is called the {\em residuum} of $*$.
Recall that the residuum determines the underlying order, that is, $a \leq b$ if, and only if, $a \to_* b = 1$. Further recall that the subset of $\Form$ that evaluates to $1$ under every valuation $\mu\colon \Form \to ([0,1],*,\to_{*},\bot)$, is by definition the collection of all tautologies of \BL. It is one of the main achievements of \cite{hajek}, of course, that this set is recursively axiomatisable by schemata, using \textit{modus ponens} as the only deduction rule; see also \cite{cignolietal_bl} for an improved axiomatisation. Moreover, \BL\ is an algebraizable logic, see \cite[p.25 and references therein]{hajek}; the algebras in the corresponding variety are called \emph{BL-algebras}, and schematic extensions of \BL\ are in one-one natural correspondence with subvarieties of BL-algebras. Each t-norm $*\colon [0,1]^{2} \to [0,1]$ induces a BL-algebra $([0,1],*,\to_{*},0)$, and the variety of BL-algebras is generated by the collection of all t-norms. More generally, each algebra of truth values as defined above is a BL-algebra. We occasionally write `BL-chain' for `totally ordered BL-algebra'.
Given algebras of truth-values $T_*, T_{*}' \subseteq [0,1]$, we say that $\sigma \colon T_* \to T_{*}'$ is
an {\em isomorphism} if $\sigma$ is an isomorphism of BL-algebras; equivalently, $\sigma$ is a bijection,
for all $a,b \in T_*$ we have $\sigma( a * b ) = \sigma(a) * \sigma(b)$, and
$a \leq b$ implies $\sigma(a) \leq \sigma(b)$.
Recall the three fundamental t-norms.
\begin{align}
x \odot y &:= \max\{0,x+y-1\}\\
x \min y&:=\min\{x,y\}\label{e:gtnorm}\\
x \times y &:= xy \label{e:prodtnorm}
\end{align}
The associated residua evaluate to $1$ for each $x,y \in [0,1]$ with $x \leq y$; when $x >y$, they are respectively given by:
\begin{align*}
x \to_\odot y &:= \min\{1,1-x+y\} \\
x \to_{\min} y &:=y \\
x \to_\times y &:= \frac{y}{x}
\end{align*}
The algebra of truth values $T_\odot:=([0,1],\odot,\to_{\odot},0)$ is called the \emph{standard MV-algebra};
the \emph{standard G\"odel algebra}, denoted $T_{\min}$, and the \emph{standard Product algebra}, denoted $T_\times$, are defined analogously using (\ref{e:gtnorm}--\ref{e:prodtnorm}) and their residua. The important \emph{completeness theorems} for \LL, \G, and \PP\ will be tacitly assumed throughout: they state that these logics are complete with respect to evaluations into $T_{\odot}$, $T_{\min}$, and $T_{\times}$, respectively. For proofs and references, consult \cite[Theorems 3.2.13, 4.2.17, and 4.1.13, and \textit{passim}]{hajek}.
\smallskip In the remainder of this section we collect technical results needed in the sequel. We begin with a remark that will find frequent application.
\begin{remark}\label{r:autos}
For any real-valued logic $(\Log, T_*)$, let $T_{*}'$ be an algebra of truth values that is isomorphic to $T_{*}$. Then
the logic induced by $T_*'$ is again $\Log$.
This follows immediately from the fact that $\sigma(1) = 1$ and $\sigma^{-1}(1) = 1$ for any isomorphism $\sigma\colon T_{*} \to T_{*}'$.
The converse statement is false in general: it is well known that non-isomorphic t-norms may induce the same real-valued logic.
However, the following hold.
\begin{enumerate}
\item The only t-norm inducing $\G$ is the minimum operator, for it is the only idempotent t-norm. See \cite[Theorem 2.1.16]{hajek}.
\item Each t-norm inducing $\LL$ is isomorphic to $T_\odot$. See \cite[Lemmata 2.1.22.(2) and 2.1.23]{hajek}.
\item Each t-norm inducing $\PP$ is isomorphic to $T_\times$. See \cite[Lemma 2.1.22.(1)]{hajek}. \qed
\end{enumerate}
\end{remark}
\begin{lemma}\label{l:iffsemantics}For any real-valued logic $(\Log,T_*)$, and for any formul\ae\ $\alpha,\beta \in \Form$, we have:
\begin{align*}
\provesL\, \alpha \leftrightarrow \beta \ \ \ \ \Longleftrightarrow \ \ \ \ \ \provesL\, \alpha \to \beta\, \text{ and } \, \provesL\, \beta \to \alpha \ \ \ \ \Longleftrightarrow \ \ \ \ \mu(\alpha)=\mu(\beta) \text{ for all valuations } \mu \colon \Form \to T_*.
\end{align*}
\end{lemma}
\begin{proof}
Indeed, $\provesL\, \alpha \leftrightarrow \beta$ iff, by the completeness of $\Log$ with respect to $T_*$, for all valuations $\mu \colon \Form \to T_*$ we have $\mu(\alpha \leftrightarrow \beta) = 1$
iff, since $1$ is the neutral element for $*$, $\mu(\alpha \to \beta) = \mu(\beta \to \alpha) = 1$ iff, by the completeness of $\Log$ with respect to $T_*$, $\provesL\, \alpha \to \beta\, \text{ and } \, \provesL\, \beta\to \alpha$ iff, since $\mu(\alpha\to\beta)=1$ is equivalent to $\mu(\alpha)\leq \mu(\beta)$ by the definition of residuum, $\mu(\alpha) = \mu(\beta)$.
\end{proof}
BL-algebras are defined over the signature $( *, \to, \bot )$. {\em Basic hoops} are the $\bot$-free subreducts of BL-algebras, the latter considered over the extended signature that includes $\top:=\bot\to\bot$. Conversely, BL-algebras are {\em bounded} basic hoops, that is, basic hoops with a minimum element which interprets the new constant $\bot$. Let now $(I, \leq)$ be a totally ordered set, and let $\{C_i\}_{i \in I}$ be a family of totally ordered basic hoops,
where $C_i := (C_i, *_i, \to_i, 1)$. Assume further that $C_i \cap C_j = \{1\}$ for each $i \neq j \in I$.
Then the {\em ordinal sum} of the family $\{C_i\}_{i \in I}$ is the structure\footnote{Usage of the symbol $\oplus$ to denote ordinal sums seems fairly standard. It is also standard to use $\oplus$ to denote {\L}ukasiewicz's strong disjunction, see \cite{cdm}. This we will do in Section \ref{s:leibniz}, where context should prevent confusion.}
$$\bigoplus_{i \in I} C_i := \left(\, \bigcup_{i \in I} C_i,\, *\,,\to,\, 1\, \right)\,,$$
where
$$
x * y =
\left\{
\begin{array}{ll}
x *_i y & \mbox{ if } x,y \in C_i, \\
y & \mbox{ if } x \in C_i,\, y \in C_j \setminus \{1\},\, i > j, \\
x & \mbox{ otherwise, }
\end{array}
\right.
$$
and
$$
x \to y =
\left\{
\begin{array}{ll}
x \to_i y & \mbox{ if } x,y \in C_i, \\
y & \mbox{ if } x \in C_i,\, y \in C_j,\, i > j, \\
1 & \mbox{ otherwise. }
\end{array}
\right.
$$
Each $C_i$ is called a {\em summand} of the ordinal sum.
\smallskip
\begin{lemma}[The Mostert-Shields Structure Theorem]\label{l:ordinalsumdecomposition} Each algebra of truth values $([0,1],*,\to_{*},0)$
is isomorphic to an ordinal sum of bounded basic hoops, each of which
is isomorphic to one among $T_{\odot}$, $T_{\min}$, $T_{\times}$, and $\{0,1\}$.
\end{lemma}
\begin{proof}
This
is essentially \cite[Theorem B]{mostert_shields}.
\end{proof}
\begin{lemma}\label{l:subalgebraofordinalsum}
Let $A$ be a subalgebra of an ordinal sum $\bigoplus_{i \in I} B_i$.
Then there exists $J \subseteq I$ and algebras $\{C_j \mid j \in J\}$
such that $C_j$ is a subalgebra of $B_j$ for each $j \in J$, and
$A \cong \bigoplus_{j \in J} C_j$.
\end{lemma}
\begin{proof}
Direct inspection of the definition of ordinal sum.
\end{proof}
\emph{MV-algebras} \cite{cdm} are (term equivalent to) BL-algebras satisfying the equation $\neg\neg x =x$, where $\neg x$ is short for $x \to \bot$. \emph{Wajsberg hoops} are the $\bot$-free subreducts of MV-algebras; equivalently,
MV-algebras are exactly the bounded Wajsberg hoops.
\begin{lemma}\label{l:finiteordsum}
Each finite BL-chain splits into an ordinal sum of finitely many finite MV-chains.
\end{lemma}
\begin{proof}
This is \cite[Theorem 3.7]{agliano_montagna}, together with the observation that finite Wajsberg hoops are necessarily bounded.
\end{proof}
\begin{lemma}\label{l:2summands}Suppose the algebra of truth values $T_*$ is not a subalgebra of $T_{\odot}$. Then $T_*$ splits into a non-trivial ordinal sum of at least two
summands.
\end{lemma}
\begin{proof}
If $T_*$ is finite, from Lemma \ref{l:finiteordsum} it follows that $T_*$ is isomorphic to an ordinal sum of finitely many finite MV-chains. Since, by assumption, $T_*$ is not a subalgebra of $T_{\odot}$,
the ordinal sum must contain at least two summands.
If $T_*$ is an infinite subalgebra of $[0,1]$, by Lemmata \ref{l:ordinalsumdecomposition} and
\ref{l:subalgebraofordinalsum}
it follows that $T_*$ is isomorphic to an ordinal sum $\bigoplus_{i \in I} C_i $ where each summand $C_i$ is isomorphic to
a subalgebra of $T_\odot$, $T_{\min}$, or $T_\times$. If the index set $I$ has at least two elements, we are done; otherwise, by the hypotheses $T_*$ is isomorphic
to a subalgebra of $T_{\min}$ or of $T_\times$, and it has more than two elements.
Now, by direct inspection, $T_{\min}$ is isomorphic to $\bigoplus_{r \in [0,1)} \{0,1\}$, while $T_\times$ is isomorphic to $\{0,1\} \oplus \mathcal{C}$, where $\mathcal{C} = (\, (0,1], \times, \to_\times, 1 \,)$
is known as the {\em standard cancellative hoop}.
Any subalgebra of $T_{\min}$ with more than two elements is then a non-trivial ordinal sum of copies of $\{0,1\}$,
while any subalgebra of $T_\times$ distinct from $\{0,1\}$ is of the form $\{0,1\} \oplus \mathcal{C}'$, for $\mathcal{C}'$ a subhoop of $\mathcal{C}$.
In both cases, $T_*$ splits into a non-trivial ordinal sum of at least two
summands.
\end{proof}
\begin{lemma}\label{l:varseval}Suppose the algebra of truth values $T_*$ splits into a non-trivial ordinal sum of at least two summands, say $\bigoplus_{i \in I} C_i$,
where each $C_{i}$ is a totally ordered basic hoop, and $|I|\geq 2$. Then $I$ has a least element, say $i_{0}$. Further, let $S\subseteq T_*$ be the support of
a summand distinct from $C_{i_{0}}$.
For any two valuations $\mu, \nu \colon \Form \to T_*$ such that $\mu(\textup{{\sc Var}}), \nu{(\textup{{\sc Var}})}\subseteq S$, and for any $\alpha \in\Form$, we have:
\begin{align*}
\mu(\alpha)=0 \ \ \ \ \Longleftrightarrow \ \ \ \ \nu(\alpha)=0.
\end{align*}
\end{lemma}
\begin{proof}Since $T_{*}$ is bounded below, the existence of $i_{0}$ follows from inspection of the definition of ordinal sum.
We first prove the following \emph{claim} by induction on the structure of formul\ae:
For any valuation $\mu \colon \Form \to T_*$ such that $\mu{(\textup{{\sc Var}})}\subseteq S$, and for any $\alpha \in\Form$,
we have $\mu(\alpha) \in S \cup \{0\}$.
If $\alpha$ is either $\bot$ or $\alpha \in \textup{{\sc Var}}$, the claim holds trivially.
Suppose $\alpha=\beta \& \gamma$.
By the induction hypothesis, $\mu(\beta),\mu(\gamma) \in S \cup \{0\}$.
If both $\mu(\beta),\mu(\gamma)\in S$ then, by the definition of ordinal sum, $\mu(\beta \& \gamma) \in S$, too.
If at least one among $\beta$ and $\gamma$, say $\beta$, is such that $\mu(\beta) = 0$, then $\mu(\beta \& \gamma) = 0$.
Hence $\mu(\beta \& \gamma) \in S \cup \{0\}$ for all $\mu$ such that $\mu{(\textup{{\sc Var}})}\subseteq S$.
Next suppose $\alpha=\beta \to \gamma$.
If $\mu(\beta) \leq \mu(\gamma)$, then $\mu(\beta \to \gamma) = 1 \in S$.
If $\mu(\beta) > \mu(\gamma) \in S$ then, by the definition of ordinal sum, $\mu(\beta \to \gamma) \in S$, too.
Finally, if $\mu(\beta) \in S$ and $\mu(\gamma) = 0$, then $\mu(\beta \to \gamma) = 0$. In all cases $\mu(\beta \to \gamma) \in S \cup \{0\}$.
This settles the \emph{claim}.
Consider now $\mu, \nu \colon \Form \to T_*$ such that $\mu(\textup{{\sc Var}}), \nu{(\textup{{\sc Var}})}\subseteq S$, and any formula $\alpha \in \Form$.
It suffices to show that $\mu(\alpha) = 0$ implies $\nu(\alpha)= 0$.
By the preceding claim, we have $\mu(\alpha),\nu(\alpha) \in S \cup \{0\}$.
We proceed again by induction on the structure of formul\ae.
The base cases $\alpha=\bot$ or $\alpha \in \textup{{\sc Var}}$ hold trivially.
Let $\alpha=\beta \& \gamma$. The definition of ordinal sum entails that $\mu(\beta \& \gamma) = 0$ can only occur if
at least one of
$\mu(\beta)$ and $\mu(\gamma)$, say $\mu(\beta)$, lies in the first summand $C_{i_{0}}$. By the preceding claim,
$\mu(\beta) = 0$. By induction
$\nu(\beta) = 0$, and therefore $\nu(\beta \& \gamma) = 0$.
Let $\alpha=\beta \to \gamma$.
Assume $\mu(\beta \to \gamma) = 0$. The definition of ordinal sum entails
either
$\mu(\beta) > \mu(\gamma) = 0$,
or both $\mu(\beta),\mu(\gamma) \in C_{i_0}$.
In the latter case, the preceding claim shows $\mu(\beta) = \mu(\gamma) = 0$, and therefore
$\mu(\beta \to \gamma) =1$, which is a contradiction.
In the former case, by induction
$\nu(\beta) > \nu(\gamma) = 0$. By the preceding claim, $\nu(\beta) \in S$.
By the definition of ordinal sum
$\nu(\beta \to \gamma) = 0$.
This completes the proof.
\end{proof}
\section{Logics satisfying \P{1}.}\label{s:tarski}
\begin{lemma}\label{l:mintnorm}For any real-valued logic $(\Log,T_*)$, we have:
\begin{align*}
\Log \text{ extends } \G \ \ \ \ \Longleftrightarrow \ \ \ \ T_* \text{ is a subalgebra of } T_{\rm min}.
\end{align*}
Moreover, we have:
\begin{align*}
\Log \text{ extends } \G \text{ properly \textup{(}\textit{i.e.}\ $\Log\neq \G$\textup{)}} \ \ \ \ \Longleftrightarrow \ \ \ \ T_* \text{ is a finite subalgebra of } T_{\rm min}.
\end{align*}
\end{lemma}
\begin{proof}
$\Log$ extends $\G$ iff $\provesL\, X_1 \leftrightarrow X_1\&X_1$ iff,
by Lemma \ref{l:iffsemantics},
$\mu(X_1) = \mu(X_1) * \mu(X_1)$ for any valuation
$\mu \colon \textup{{\sc Var}} \to T_*$ iff $a = a * a$ for any $a \in T_*$ iff $T_*$ is a subalgebra of $T_{\min}$. (The latter equivalence follows from Remark \ref{r:autos}.1.)
Now, if $\Log$ extends $\G$ properly, then, by Remark \ref{r:autos}.1,
and the fact that each infinite subalgebra of $T_{\rm min}$ induces $\G$ \cite[Theorem 4]{dummett59},
the underlying set of $T_*$ cannot be
an infinite subset of $[0,1]$, hence $T_*$ is a finite subalgebra of $T_{\min}$.
The other direction follows from \cite[Corollary 4.2.15]{hajek}, stating that any two finite subalgebras of $T_{\min}$ of the same cardinality are isomorphic,
and from the axiomatisation of the subvariety of G\"{o}del algebras generated by the $n$-element chain, essentially given in \cite{godel}.
\end{proof}
\begin{lemma}\label{l:maintarski}Any real-valued logic that satisfies \P{1} is an extension of \G.
\end{lemma}
\begin{proof}We prove the contrapositive: a real-valued logic $\Log$ that does not extend \G\ fails \P{1}. Indeed, by the hypothesis we have $\nprovesL\, X_1\leftrightarrow X_1\&X_1$. On the other hand, for any algebra of truth values $T_{*}$ inducing $\Log$, and for any valuation $\mu\colon \Form\to T_*$, we have
\begin{align}
\mu(X_1)=1 \ \ &\Rightarrow \ \ \mu(X_1\&X_1)=1,\label{d:firstimpl}\\
\mu(X_1\&X_1)=1 \ \ &\Rightarrow \ \ \mu(X_1)=1.\label{d:secondimpl}
\end{align}
Indeed, (\ref{d:firstimpl}) holds by the very definition of t-norm, which includes the condition $1*1=1$; and (\ref{d:secondimpl}) holds by the fact that t-norms are non-increasing in both arguments, whence $\mu(X_1\&X_1)\leq \mu(X_1)$. Now (\ref{d:firstimpl}--\ref{d:secondimpl}) show that $\Log$ fails \P{1} for $\alpha=X_1$ and $\beta =X_1\&X_1$.
\end{proof}
For the proof of the next lemma we recall the notion of semantic consequence with respect to an algebra of truth values $T_*$.
Given a set $\Gamma \subseteq \Form$ and $\alpha \in \Form$, we say that $\alpha$ is a \emph{semantic consequence}
of $\Gamma$ with respect to $T_*$, in symbols $\Gamma \vDash_{T_*} \alpha$ if, for any valuation $\mu \colon \textup{{\sc Var}} \to T_*$,
the fact that $\mu(\gamma) = 1$ for each $\gamma \in \Gamma$ implies $\mu(\alpha) = 1$.
\begin{lemma}\label{l:godimpliestarski}Any real-valued logic $\Log$ that is an extension of $\G$ satisfies $\P{1}$.
\end{lemma}
\begin{proof} Let $T_*$ be an algebra of truth values inducing $\Log$.
By Lemma \ref{l:mintnorm} we know that $T_*$ is a subalgebra of $T_{\rm min}$. Let $\alpha, \beta \in\Form$ be such that $\mu(\alpha)=1$ iff $\mu(\beta)=1$, for each valuation $\mu\colon \Form \to T_*$. By the definition of semantic consequence,
we have $\alpha \vDash_{T_*} \beta$ and $\beta \vDash_{T_*} \alpha$.
Recall that $\G$ is strongly complete with respect to $T_{\min}$ (\cite[Theorem 4.2.17.(2)]{hajek}).
By Lemma \ref{l:mintnorm}, each real-valued extension $\Log$ of $\G$ distinct from $\G$ is induced by a finite subalgebra of $T_{\min}$, and it is moreover strongly complete with respect to any such (essentially unique) subalgebra (\cite[Proposition 4.18 and Corollary 4.19]{Distinguished}).
In all cases we therefore infer $\alpha \, \provesL \, \beta$ and $\beta \, \provesL \, \alpha$. The logic \G\ has the Deduction Theorem by \cite[Theorem 4.2.10.(1)]{hajek}, and the same proof shows that each extension of \G\ also has the Deduction Theorem. We thereby obtain
$\provesL\, \beta \to \alpha$ and $\provesL\,\alpha \to \beta$. Hence, by Lemma \ref{l:iffsemantics}, we conclude $\provesL\, \alpha\leftrightarrow\beta$, as was to be shown.
\end{proof}
\paragraph{Proof of Theorem I} Combine Lemmata \ref{l:maintarski} and \ref{l:godimpliestarski}. \qed
\begin{remark}\label{r:godelandMTL}
Theorem I holds even if we relax the notion of real-valued logic considerably.
Recall that \emph{MTL} (\emph{monoidal t-norm-based logic}) is the logic of all left-continuous t-norms and their residua \cite{eg}; write $\Form'$ for the set of well-formed formul\ae\ of MTL. (In contrast to BL, here it is necessary to regard the lattice-theoretic conjunction $\wedge$ as primitive.) The algebraic semantics corresponding to MTL is provided by \emph{MTL-algebras}.
By a \emph{standard MTL-algebra} we mean an MTL-algebra induced by a left-continuous t-norm on $[0,1]$ and its residuum.
Now replace the definition of real-valued logic by the following. The pair $(\Log, T_*)$
is a {\em real-valued logic} if $\Log$ is an extension of MTL that is complete with respect
to valuations $\mu \colon \Form' \to T_*$ into an arbitrary MTL-subalgebra $T_{*}$
of some standard MTL-algebra.
It is well known that Remark \ref{r:autos}.1 holds even
if we consider all left-continuous t-norms instead of the continuous ones only.
And it is possible to show that Lemmata \ref{l:mintnorm}, \ref{l:maintarski}, and \ref{l:godimpliestarski}
continue to hold. Hence Theorem I holds for real-valued logics in the present sense.
\qed
\end{remark}
\section{Logics satisfying \P{2}.}\label{s:leibniz}
\begin{lemma}\label{l:extL}For any real-valued logic $(\Log,T_*)$, we have:
\begin{align*}
\Log \text{ extends } \LL \ \ \ \ \Longleftrightarrow \ \ \ \ T_* \text{ is isomorphic to a subalgebra of } T_{\odot}.
\end{align*}
Moreover, we have:
\begin{align*}
\Log \text{ extends } \LL \text{ properly \textup{(}\textit{i.e.}\ $\Log\neq \LL$\textup{)}}\ \ \ \ \Longleftrightarrow \ \ \ \ T_* \text{ is isomorphic to a finite subalgebra of } T_{\odot}.
\end{align*}
\end{lemma}
\begin{proof}
$\Log \text{ extends } \LL$ iff $\vdash_\Log \neg\neg X_1 \leftrightarrow X_1$ iff (by Lemma \ref{l:iffsemantics}) $\mu(X_1) = \neg\neg\mu(X_1)$ for any valuation $\mu \colon \textup{{\sc Var}} \to T_*$
iff $a = \neg\neg a$ for any $a \in T_*$ iff (by Remark \ref{r:autos}.2) $T_*$ is an MV-algebra with some underlying set $U \subseteq [0,1]$. Now, if $U$ is finite, say of cardinality $n$,
then $T_*$ is isomorphic to the MV-chain $T_{n-1} = \{\frac{0}{n-1},\frac{1}{n-1},\ldots,\frac{n-2}{n-1},\frac{n-1}{n-1}\}$, by \cite[Proposition 3.6.5]{cdm},
and direct inspection shows that $T_{n-1}$ is a subalgebra of $T_\odot$.
Assume then that $U$ is infinite. Observe that $T_*$ cannot be a non-trivial ordinal sum of at least two summands: consider such a sum $B \oplus C$, and take $1 \neq c \in C$.
Then $\neg\neg c = 1 \neq c$, and hence $B \oplus C$ is not an MV-algebra.
By Lemma \ref{l:2summands} and by Remark \ref{r:autos}.2,
$T_*$ is isomorphic to
a subalgebra of
$T_\odot$.
Clearly, if $T_*$ is isomorphic to a subalgebra of
$T_\odot$ then $\Log \text{ extends } \LL$. This proves the first statement.
Each finite MV-chain generates a proper subvariety of the variety of MV-algebras (see \cite[Theorem 8.5.1]{cdm} for axiomatisations). Thus,
if $T_*$ is isomorphic to a finite subalgebra of $T_\odot$ then $\Log \text{ extends } \LL$ properly.
On the other hand, by \cite[Theorem 8.1.1]{cdm}, every infinite subalgebra of $T_\odot$ generates the whole variety of MV-algebras. This fact, together
with the first assertion of the lemma, suffices to complete the proof.
\end{proof}
\begin{lemma}\label{l:p2impliesL} Any real-valued logic $\Log$ that satisfies \P{2} is an extension of $\LL$.
\end{lemma}
\begin{proof}By contraposition, suppose $\Log$ is not an extension of $\LL$. If $T_{*}$ is an algebra of truth values that induces $\Log$, then $T_{*}$ is not a subalgebra of $T_{\odot}$: for, given that $T_{\odot}$ does induce $\LL$ (\textit{cf.}\ Remark \ref{r:autos}), any such subalgebra clearly induces an extension of $\LL$.
Hence, by Lemma \ref{l:2summands}, $T_{*}$ splits into a non-trivial ordinal sum of at least two summands. With the notation therein, there exists a summand $S$ of $T_{*}$ distinct from the first one that is non-trivial, and thus contains two distinct elements $v\neq w$. Let
$\mu_{v}$ be the unique valuation that sends each variable to $v$, and let $\nu_{w}$ be the unique valuation that sends each variable to $w$. Evidently, we have $\mu_{v}\neq \nu_{w}$, so that $\mu_{v}$ and $\nu_{w}$ fail \P{2} by Lemma \ref{l:varseval}.
\end{proof}
\begin{remark}Let $(\Log, T_*)$ be a real-valued logic. In the next lemma we say, somewhat informally, that ``$\Log$ satisfies \P{2} with respect to $T_*$'', to mean that for any two valuations $\mu\neq \nu \colon\Form \to T_*$ there is $\alpha\in\Form$ with $\mu(\alpha)>0$ and $\nu(\alpha)=0$. \qed
\end{remark}
\begin{lemma}\label{l:inv}Let $(\Log, T_{*})$ be a real-valued logic, and let $\sigma \colon T_{*}\to T'_{*'}$ be an isomorphism, where $T'_{*'}$ is an algebra of truth values. The logic induced by $T'_{*'}$ is again $\Log$, by Remark \ref{r:autos}. Then
$\Log$ satisfies \P{2} with respect to $T_{*}$ if, and only if, $\Log$ satisfies \P{2} with respect to $T'_{*'}$.
\end{lemma}
\begin{proof}Since $\sigma^{-1}\colon T'_{*'}\to T_{*}$ is an isomorphism, too, it suffices to show that $\Log$ satisfies \P{2} with respect to $T'_{*'}$ if
$\Log$ satisfies \P{2} with respect to $T_{*}$. Proof by contraposition. Let $\mu\neq\nu\colon \Form\to T'_{*'}$ be valuations that
fail \P{2}. Thus, for all formul\ae\ $\alpha \in \Form$, we have $\mu(\alpha)=0$ if, and only if, $\nu(\alpha)=0$. Write $\mathrm{Free}_{\aleph_{0}}^{\Log}$ for the Lindenbam algebra of the logic $\Log$. As usual, we may identify formul\ae, modulo the logical-equivalence relation induced by $\provesL$, with elements of $\mathrm{Free}_{\aleph_{0}}^{\Log}$; and valuations with homomorphisms from $\mathrm{Free}_{\aleph_{0}}^{\Log}$ to $T_{*}$ (or to $T'_{*'}$, as the case may be). Then the compositions $\sigma^{-1}\circ \mu$ and $\sigma^{-1}\circ\nu$ are valuations into $T_{*}$, see the commutative diagram below.
\[
\begin{tikzpicture}[scale=0.3]
\node(A) at (0,5) {${\mathrm{Free}_{\aleph_{0}}^{\Log}}$};
\node (P1) at (10,5) {$T_{*}$};
\node (P2) at (10,-4) {$T'_{*'}$};
\draw[transform canvas={yshift=0.4ex},<-] (P1) -- (A) node [above, midway] {$\sigma^{-1}\circ\mu$};
\draw[transform canvas={yshift=-0.4ex},<-] (P1) -- (A) node [below, midway] {$\sigma^{-1}\circ\nu$};
\draw[transform canvas={xshift=0.4ex},<-] (P2) -- (A) node [above right, midway] {$\mu$};
\draw[transform canvas={xshift=-0.5ex, yshift=-0.3ex},<-] (P2) -- (A) node [below left, midway] {$\nu$};
\draw [->] (P2) -- (P1) node [right, midway] {$\sigma^{-1}$};
\end{tikzpicture}
\]
It is not the case that $\sigma^{-1}\circ \mu = \sigma^{-1}\circ \nu$: for else $\mu=\nu$ would follow by pre-composing with $\sigma$. Now for any $\alpha\in \Form$ we have:
\begin{align*}
\mu(\alpha)=0 \ \ \ & \text{iff} \ \ \ \ \nu(\alpha)=0 & \text{(by assumption),}\\
\sigma^{-1}(0)=0 \ \ \ & & \text{(homomorphisms preserve $0$),}\\
\sigma^{-1}(\mu(\alpha)))=0 \ \ \ & \text{iff} \ \ \ \ \sigma^{-1}(\nu(\alpha)))=0 & \text{(by composition).}
\end{align*}
Hence $\Log$ fails \P{2} with respect to $T_{*}$, as was to be shown.
\end{proof}
\begin{lemma}\label{l:separation}{\L}ukasiewicz logic \LL\ satisfies \P{2}.
\end{lemma}
\begin{remark}\label{rem:man}A proof of Lemma \ref{l:separation} can be obtained as a consequence of McNaughton's Theorem \cite[9.1]{cdm}; in fact, the proof can be
reduced to the one-variable case \cite[3.2]{cdm}. Here we give a proof that uses a weaker (and simpler) result from \cite{agu}, thus showing that the full strength of McNaughton's Theorem is not needed to fulfill \P{2}.\qed
\end{remark}
\begin{proof}
In light of Remark \ref{r:autos}.2 and Lemma \ref{l:inv},
it suffices to show that \LL\ satisfies \P{2} with respect to the {\L}ukasiewicz t-norm $\odot$ on $[0,1]$. For terms $s$ and $t$ over the binary monoidal operation $\odot$ and the unary operation $\neg$, set $s\oplus t :=\neg(\neg s\odot \neg t)$. Let us write $nt$ as a shorthand for $t\oplus\cdots \oplus t$ ($n-1$ occurrences of $\oplus$), and
$t^{n}$ as a shorthand for $t\odot\cdots \odot t$ ($n-1$ occurrences of $\odot$). We inductively define the set of \emph{basic literals} (in the variables $X_{i}$, $i=1,2,\ldots$) as follows.
\begin{itemize}
\item $X_{i}$ is a basic literal;
\item each term $s$ either of the form $s=nt$ or of the form $s=t^{n}$, for some integer $n>0$, is a basic literal, provided that $t$ is a basic
literal;
\item nothing else is a basic literal.
\end{itemize}
Given integers $n_{1}\geq 1$, and $n_{2},\ldots,n_{u}>1$, we write $(n_{1},n_{2},\ldots,n_{u})X_{i}$ to denote the basic literal $$(\cdots ((n_{i}\cdots ((n_1X_{i})^{n_2} \cdots ))^{n_{i+1}})\cdots ).$$
In this proof, a \emph{term function} is any function $\lambda_{\tau} \colon [0,1]^{n}\to [0,1]$ induced by interpreting over the standard MV-algebra $T_{\odot}=([0,1],\odot,\neg,0)$ a term $\tau$ whose variables are contained in $\{X_1,\ldots,X_{n}\}$. Below we also use the interpretation of the definable lattice connective $\wedge$ as the minimum operator.
\begin{claim}\label{c:separation}
For any integer $n \geq 1$, and for any two points $p\neq q\in[0,1]^{n}$, there is a term $\tau$ whose term function $\lambda_{\tau} \colon [0,1]^{n}\to [0,1]$ takes value $0$ at $q$, and value $>0$ at $p$.
\end{claim}
\begin{proof}
Since $p \neq q$ there exists an integer $i\geq 1$ such that $p(i) \neq q(i)$, that is, $p$ and $q$ differ at one of their coordinates.
If $q(i) < p(i)$ then there are integers $h,k>0$ such that $q(i) < \frac{h}{k} < p(i)$, with $h$ and $k$ coprime.
By \cite[Corollary 2.8]{agu} there is a basic literal $L =(a_1,\ldots, a_{u})X_{i}$ such that
$\lambda_{L}^{-1}(0)$ is the set $[0,\frac{h}{k}] \times [0,1]^{n-1}$, and $\lambda_{L}$ is monotone increasing in the variable $X_i$.
Hence $\lambda_L(p) > 0$ and $\lambda_L(q) = 0$.
If $p(i) \leq q(i)$ for all integers $i \geq 1$, then one can choose $j$ such that $p(j) < q(j)$.
As before there are integers $h,k>0$ such that $p(j) < \frac{h}{k} < q(j)$, with $h$ and $k$ coprime,
and there is a basic literal $R = (b_1,\ldots, b_{w})X_{i}$ such that $\lambda_{R}^{-1}(1)$ is the set $[\frac{h}{k},1] \times [0,1]^{n-1}$, and $\lambda_{R}$ is monotone increasing
in the variable $X_i$.
Hence $\lambda_{\neg R}(p) > 0$ and $\lambda_{\neg R}(q) = 0$.
\end{proof}
The proof is now completed by a routine translation of Claim \ref{c:separation} from terms to formul\ae\ of $\LL$.
\end{proof}
\begin{remark}In connection with Claim \ref{c:separation}, let us observe that term functions in {\L}ukasiewicz logic (even over an arbitrarily large set $I$ of propositional variables) enjoy an even stronger separation property. Recall (see \textit{e.g.}\ \cite[1.5]{engelking}) that a space is \emph{completely regular} if it is $T_{1}$, and points can be separated from closed sets by continuous $[0,1]$-valued functions. Now, \emph{in each product space $[0,1]^{I}$, points can be separated from closed sets by term functions}. Thus the space of standard models $[0,1]^{I}$ may be described as \emph{definably completely regular}. The proof is essentially the same as the one above, \textit{mutatis mutandis}; \textit{cf.}\ \cite[Lemma 3.5]{marraspada}.
\qed
\end{remark}
\paragraph{Proof of Theorem II}In light of Lemmata \ref{l:p2impliesL} and \ref{l:separation}, it remains to show that each real-valued extension of $\LL$ that is not $\LL$ itself satisfies \P{2}. By Lemmata \ref{l:extL} and \ref{l:inv}, we may safely assume that $\Log$ is induced by a finite subalgebra $T_{*}$ of $T_{\odot}$.
By \cite[Proposition 3.6.5]{cdm}, each such subalgebra is isomorphic to $T_m = \left\{\frac{0}{m},\frac{1}{m},\ldots,\frac{m-1}{m},\frac{m}{m}\right\}$, for a uniquely determined integer $m\geq 1$.
Notice now that if $p\neq q$ are in $T_m^{n}$ then the term function $\lambda'_{\tau}$ obtained by restricting to $T_m^{n}$ the function $\lambda_{\tau} \colon [0,1]^{n}\to [0,1]$
provided by Claim \ref{c:separation} is such that $\lambda'_{\tau}(q) = 0$ while $\lambda'_{\tau}(p) > 0$.
Hence $\Log$ satisfies \P{2}, and the proof is complete. \qed\\
To conclude this section, let us discuss two alternative formulations of \P{2}. We consider the following conditions, for every algebra $T_*$ of truth values inducing $\Log$.
\princ{\P{2$'$}}{For each pair of valuations $\mu, \nu \colon \Form \to T_{*}$, if $\mu\neq \nu$ then
there is a formula $\alpha\in \Form$ such that $\mu(\alpha) <1$ while $\nu(\alpha) = 1$.\qed\\}
\princ{\P{2$''$}}{For each pair of valuations $\mu, \nu \colon \Form \to T_{*}$, if $\mu\neq \nu$ then
there is a formula $\alpha\in \Form$ such that $\mu(\alpha) =0$ while $\nu(\alpha) = 1$.\qed\\}
\begin{corollary}\label{cor:equiv}A real-valued logic satisfies \P{2} if, and only if, it satisfies \P{2$'$} if, and only if, it satisfies \P{2$''$}.
\end{corollary}
\begin{proof}
Let $T_*$ be an algebra of truth-values inducing the real-valued logic $\Log$.
It suffices to prove that if $\Log$ is an extension of $\LL$ then it satisfies $\P{2}'$ and $\P{2}''$,
and otherwise it fails both.
Assume first that $\Log$ is an extension of $\LL$.
Given valuations $\mu \neq \nu$ with values in $T_{*}$, by Theorem II there is a formula $\alpha$ be such that $\mu(\alpha) > 0$ and $\nu(\alpha) = 0$.
Then $\mu(\neg\alpha) < 1$ and $\nu(\neg\alpha) = 1$. Hence \P{2$'$} holds. We now show that \P{2$'$} implies \P{2$''$}.
In light of Remark \ref{r:autos}.2 and Lemma \ref{l:inv}, we may safely assume that $T_*$ is a subalgebra of $T_\odot$.
Then,
if $\mu(\alpha) < 1$ and $\nu(\alpha) = 1$, it is clear by the definition of $\odot$ that there exists an integer $k \geq 1$ such that
$\mu(\alpha^k) = 0$ and $\nu(\alpha^k) = 1$, where $\alpha^1 = \alpha$ and $\alpha^n = \alpha \odot \alpha^{n-1}$.
Assume now $T_*$ does not induce an extension of $\LL$.
By Theorem II,
there are distinct valuations $\mu$ and $\nu$ such that
$\nu(\alpha) = 0$ implies $\mu(\alpha) = 0$ for
any formula $\alpha$.
This suffices to show that \P{2}$''$ fails.
For what concerns \P{2}$'$,
recall that, by Lemma \ref{l:extL} and Lemma \ref{l:2summands}, $T_*$ splits into a non-trivial ordinal sum of at least two summands.
Let $\mu$ be the valuation assigning $1$ to every variable.
Then it is easy to check that $\mu(\alpha) \in \{0,1\}$ for each formula $\alpha$.
Let $\nu$ be a valuation such that $\nu(\textup{{\sc Var}})$ is contained in a summand of $T_*$ distinct from the first one.
Then, by Lemma \ref{l:varseval}, for each formula $\alpha$ we have
$\nu(\alpha) = 0$ iff $\mu(\alpha) = 0$, and hence $\nu(\alpha) =1$ implies $\mu(\alpha) = 1$,
that is, \P{2}$'$ fails.
\end{proof}
\section{Product logic.}\label{s:product}
\begin{lemma}\label{l:noext}The only many-valued logic that extends \PP\ properly is classical logic.
\end{lemma}
\begin{proof}
This is essentially \cite[Corollary 2.10]{cignoli_torrens}.
\end{proof}
\begin{lemma}\label{l:fails}Product logic \PP\ fails both \P{1} and \P{2}.
\end{lemma}
\begin{proof} (\P{1}) \, Choose the standard product algebra $T_\times$ to induce \PP. It follows directly from the definition of t-norm that $\mu(X_1)=1$ if, and only if, $\mu(X_1\&X_1)=1$, for any valuation $\mu\colon\Form\to T_\times$. To see that \P{1} fails, it thus suffices to observe that $\not \vdash_{\PP} X_1 \leftrightarrow X_1 \& X_1$: for else, by soundness and Lemma \ref{l:iffsemantics}, we would have $\mu(X_1\&X_1)=\mu(X_1)\mu(X_1)=\mu(X_1)$ whatever $\mu$ is; this is a contradiction.
\smallskip \noindent (\P{2}) \,
By Remark \ref{r:autos}.3 and Lemma \ref{l:inv}, it suffices to argue about the product t-norm $T_\times$. By direct inspection, we have the decomposition $T_\times = \{0,1\} \oplus \mathcal{C}$, where $\mathcal{C}$ is the standard cancellative hoop. The hypotheses of Lemma \ref{l:varseval} are therefore satisfied, and hence
\P{2} fails for any two valuations $\mu\neq \nu \colon \Form \to T_\times$ such that $\mu(\textup{{\sc Var}}),\nu(\textup{{\sc Var}})\subseteq \mathcal{C}$.
\end{proof}
\begin{lemma}\label{l:justproduct}Let $\Log$ be a closed real-valued logic all of whose non-classical, real-valued extensions fail \P{1} and \P{2}. Then $\Log=\PP$.
\end{lemma}
\begin{proof} We know that $\Log$ is not an extension of $\G$ or $\LL$, by Theorems I and II. Let $T_*$ be any algebra of truth values inducing $\Log$. We will show that $T_*$ cannnot be finite, to begin with.
If $T_*$ is finite,
by Lemma \ref{l:finiteordsum} we know that $T_*$ splits into an ordinal sum of finitely many finite MV-chains. If there is just one summand, then $\Log$ is an extension of $\LL$, and this is a contradiction. If there is more than one summand then, by the definition of ordinal sum, and using the fact that each summand is bounded below by $0$, there is
an idempotent element $0,1\neq e \in T_*$. The subset $G_3:=\{0,e,1\}\subseteq T_*$ is closed under the BL-algebraic operations, as is checked easily, and all of its elements
are idempotent. Hence $G_3$ is isomorphic to the three-element G\"odel algebra. Now consider the collection $\mathscr{E}$ of formul\ae\ that evaluate to $1$ under each valuation into $G_3$. Obviously $\mathscr{E}\supseteq \Log$, and $\mathscr{E}$ is closed under substitutions by its very definition. Hence $\mathscr{E}$ is a real-valued extension of $\Log$ which by construction is three-valued G\"odel logic. Theorem I implies that $\mathscr{E}$ satisfies \P{1}, and we have reached a contradiction.
We may therefore suppose that $T_*$ has an infinite closed subset of $[0,1]$ as its support. By definition, $T_{*}$ extends to a BL-algebra $([0,1],*',\to_{*'},0)$.
By Lemmata \ref{l:ordinalsumdecomposition} and \ref{l:subalgebraofordinalsum}, $T_*$ decomposes into an ordinal sum $\bigoplus_{i \in I} C_i$, where each summand $C_i$ is
isomorphic to a subalgebra of one amongst $T_\odot$, $T_{\min}$, and $T_\times$. If the index set $I$ has more than one element, then using again the fact that each summand $C_i$
is bounded below by $0$, we have an idempotent element $0,1\neq e \in T_*$, and hence $\{0,e,1\}$ is a three-element G\"odel subalgebra of $T_*$.
We then reason as above to conclude that $\Log$ has three-valued G\"odel logic as an extension, reaching a contradiction.
Hence $I$ is a singleton, that is, $T_*$ is isomorphic to a subalgebra of $T_\odot$, $T_{\min}$, and $T_\times$. Using Remark \ref{r:autos},
and Theorems I and II, $T_*$ cannot be isomorphic to a subalgebra of $T_\odot$ --- because it fails \P{2} --- nor can it be isomorphic to a subalgebra of $T_{\min}$
--- because it fails \P{1}. Then $T_*$ is isomorphic to an infinite subalgebra of $T_\times$, and hence $\Log=\PP$, by \cite[Corollary 2.9]{cignoli_torrens}.
\end{proof}
\paragraph{Proof of Theorem III} Lemmata \ref{l:noext}, \ref{l:fails} and \ref{l:justproduct}. \qed
\begin{remark}\label{rem:fails}
Theorem III fails if we drop the assumption that $\Log$ be closed. Indeed, consider the logic $\Log$ induced by $\{0,1\} \oplus \mathcal{C}\oplus \mathcal{C}$, where $\mathcal{C}$ is the standard cancellative hoop (see the proof of Lemma \ref{l:2summands}).
Then it can be verified that $\Log$ is not closed, that $\Log$ is not $\PP$, and that all of its non-classical,
real-valued extensions fail \P{1} and \P{2}. \qed
\end{remark}
\section{Epilogue.} Let us return to H\'ajek's Programme, as embodied in \cite{hajek}. According to H\'ajek, a real-valued logic may be considered as a ``{\sl logic of imprecise \textup{(}vague\textup{)} propositions}'' \cite[p.vii]{hajek}, wherein ``{\sl truth \textup{[}\ldots\textup{]} is a matter of degree}'' \cite[p.2]{hajek}. Classical logic may be viewed as a limiting case, where only two degrees of truth, $0$ and $1$, exist. But as soon as a logic is genuinely real-valued, it must renounce at least one of the familiar features \P{1} and \P{2} of the classical world. We record this fact as a formal statement, by way of conclusion.
\begin{corollarynonum}A real-valued logic $\Log$ satisfies $\P{1}$ and $\P{2}$ if, and only if, $\Log$ is classical logic if, and only if, $T_*=\{0,1\}$ is the unique algebra of truth values that induces $\Log$.
\end{corollarynonum}
\begin{proof}That $\Log$ is classical logic just in case $\Log$ satisfies $\P{1}$ and $\P{2}$ follows from Theorems I--II upon observing that the only common extension of $\G$ and $\LL$ is classical logic, by \cite[Theorem 4.3.9.(1)]{hajek}. It thus remains to show that $\Log$ is classical logic if, and only if, $T_*=\{0,1\}$ as soon as $T_*$ induces $\Log$.
By the very definition of t-norm, $T_* = \{0,1\}$ induces classical logic. On the other hand, if there exists $a \in T_* \setminus \{0,1\}$ then $\max{\{a, a \to_{*} 0\}}< 1$. Indeed, $a \to_{*} 0 = 1$ would entail $a * 1 = 0$ for $a > 0$, which is impossible. Any valuation $\mu\colon \Form \to T_{*}$ that sends $X_{1}$ to $a$ is therefore such that $\mu(X_{1}\vee\neg X_{1})<1$, and the logic induced by $T_*$ cannot be classical.
\end{proof}
\section*{Acknowledgements.}
\noindent We are grateful to two anonymous referees for several remarks on an earlier version of this paper that led to improvements in exposition, and to shorter proofs of some of the results given here.
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,355 |
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WHO ARE HAPPIEST?
AND
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by
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in the Clerk's Office of the District Court of the Eastern District of
Pennsylvania.
Stereotyped by L. Johnson & Co.
Philadelphia.
CONTENTS.
PAGE
WHO ARE HAPPIEST? 9
DICK LAWSON, AND THE YOUNG MOCKING-BIRD. 21
THE MEANS OF ENJOYMENT. 60
MAN'S JUDGMENT. 72
WHAT FIVE DOLLARS PAID. 89
LOOK AT T'OTHER SIDE. 97
THIN SHOES. 115
THE UNRULY MEMBER. 131
THE RICH AND THE POOR. 149
INTRODUCTION.
In this volume, the stories are not illustrative of childish
experiences. Most of the actors are men and women,--and the trials and
temptations to which they are subjected, such as are experienced in
mature life. Their object is to fix in the young mind, by familiar
illustrations, principles of action for the future. While several of the
volumes in this series will be addressed to children as children,
others, like this one, will be addressed to them as our future men and
women, toward which estate they are rapidly progressing, and in which
they will need for their guidance all things good and true that can be
stored up in their memories.
WHO ARE HAPPIEST?
"What troubles you, William?" said Mrs. Aiken, speaking in a tone of
kind concern to her husband, who sat silent and moody, with his eyes now
fixed upon the floor, and now following the forms of his plainly-clad
children as they sported, full of health and spirits, about the room.
It was evening, and Mr. Aiken, a man who earned his bread by the sweat
of his brow, had, a little while before, returned from his daily labour.
No answer was made to the wife's question. A few minutes went by, and
then she spoke again:
"Is any thing wrong with you, William?"
"Nothing more than usual," was replied. "There's always something wrong.
The fact is, I'm out of heart."
"William!"
Mrs. Aiken came and stood beside her husband, and laid her hand gently
upon his shoulder.
The evil spirit of envy and discontent was in the poor man's
heart,--this his wife understood right well. She had often before seen
him in this frame of mind.
"I'm as good as Freeman; am I not?"
"Yes, and a great deal better, I hope," replied Mrs. Aiken.
"And yet he is rolling in wealth, while I, though compelled to toil
early and late, can scarcely keep soul and body together."
"Hush, William! Don't talk so. It does you no good. We have a
comfortable home, with food and raiment,--let us therewith be contented
and thankful."
"Thankful for this mean hut! Thankful for hard labour, poor fare, and
coarse clothing!"
"None are so happy as those who labour; none enjoy better health than
they who have only the plainest food. Do you ever go hungry to bed,
William?"
"No, of course not."
"Do you or your children shiver in the cold of winter for lack of warm
clothing?"
"No; but"----
"William! Do not look past your real comforts in envy of the blessings
God has given to others. Depend upon it, we receive all of this world's
goods the kind Father above sees best for us to have. With more, we
might not be so happy as we are."
"I'll take all that risk," said Mr. Aiken. "Give me plenty of money, and
I'll find a way to largely increase the bounds of enjoyment."
"The largest amount of happiness, I believe, is ever to be found in that
condition wherein God had placed us."
"Then every poor man should willingly remain poor!"
"I did not say that, William: I think every man should seek earnestly to
improve his worldly affairs--yet, be contented with his lot at all
times; for, only in contentment is there happiness, and this is a
blessing the poor may share equally with the rich. Indeed, I believe the
poor have this blessing in larger store. You, for instance, are a
happier man than Mr. Freeman."
"I'm not so sure of that."
"I am, then. Look at his face. Doesn't that tell the story? Would you
exchange with him in every respect?"
"No, not in _every_ respect. I would like to have his money."
"Ah, William! William!" Mrs. Aiken shook her head. "You are giving place
in your heart for the entrance of bad spirits. Try to enjoy, fully, what
you have, and you will be a far happier man than Mr. Freeman. Your sleep
is sound at night."
"I know. A man who labours as hard as I do, can't help sleeping
soundly."
"Then labour is a blessing, if for nothing else. I took home, to-day, a
couple of aprons made for Mrs. Freeman. She looked pale and troubled,
and I asked her if she were not well."
"'Not very,' she replied. 'I've lost so much rest of late, that I'm
almost worn out.'
"I did not ask why this was; but, after remaining silent for a few
moments, she said--
"'Mr. Freeman has got himself so excited about business, that he sleeps
scarcely three hours in the twenty-four. He cares neither for eating nor
drinking; and, if I did not watch him, would scarcely appear abroad in
decent apparel. Hardly a day passes that something does not go wrong.
Workmen fail in their contracts, prices fall below what he expected them
to be, and agents prove unfaithful; in fact, a hundred things occur to
interfere with his expectations, and to cloud his mind with
disappointment. We were far happier when we were poor, Mrs. Aiken.
There _was_ a time when we enjoyed this life. Bright days!--how well are
they remembered! Mr. Freeman's income was twelve dollars a week; we
lived in two rooms, and I did all our own work. I had fewer wants then
than I have ever had since, and was far happier than I ever expect to be
again on this side of the grave.'"
Just then a cry was heard in the street.
"Hark!" exclaimed Mr. Aiken.
"Fire! Fire! Fire!" The startling sound rose clear and shrill upon the
air.
Mr. Aiken sprang to the window and threw it open.
"Mr. Freeman's new building, as I live!"
Mr. Aiken dropped the window, and catching up his hat, hurriedly left
the house.
[Illustration: MR. AIKEN'S RETURN FROM THE FIRE.]
It was an hour ere he returned. Meanwhile the fire raged furiously, and
from her window, where she was safe from harm, Mrs. Aiken saw the large
new factory, which the rich man had just erected, entirely consumed by
the fierce, devouring element. All in vain was it that the intrepid
firemen wrought almost miracles of daring, in their efforts to save the
building. Story after story were successively wrapped in flames, until,
at length, over fifty thousand dollars worth of property lay a heap of
black and smouldering ruins.
Wet to the skin, and covered with cinders, was Mr. Aiken when he
returned to his humble abode, after having worked manfully, in his
unselfish efforts to rescue a portion of his neighbour's property from
destruction.
"Poor Freeman! I pity him from my very heart!" was his generous,
sympathising exclamation, as soon as he met his wife.
"He is insured, is he not?" inquired Mrs. Aiken.
"Partially. But even a full insurance would be a poor compensation for
such a loss. In less than two weeks, this new factory, with all its
perfect and beautiful machinery, would have been in operation. The
price of goods is now high, and Mr. Freeman would have cleared a
handsome sum of money on the first season's product of his mill. It is a
terrible disappointment for him. I never saw a man so much disturbed."
"Poor man! His sleep will not be so sound as yours, to-night, William."
"Indeed it will not."
"Nor, rich as he is, will he be as happy as you, to-morrow."
"If I were as rich as he is," said Mr. Aiken, "I would not fret myself
to death for this loss. I would, rather, be thankful for the wealth
still left in my possession."
Mrs. Aiken shook her head.
"No, William, the same spirit that makes you restless and discontented
now, would be with you, no matter how greatly improved might be your
external condition. Mr. Freeman was once as poor as you are. Do you
think him happier for his riches? Does he enjoy life more? Has wealth
brought a greater freedom from care? Has it made his sleep sweeter?
Far, very far from it. Riches have but increased the sources of
discontent."
"This is not a necessary consequence. If Mr. Freeman turn a blessing
into a curse, that is a defect in his particular case."
"And few, in this fallen and evil world, are free from this same defect,
William. If wealth were sought for unselfish ends, then it would make
its possessor happy. But how few so seek riches! It is here, believe me,
that the evil lies."
Mrs. Aiken spoke earnestly, and something of the truth that was in her
mind, shed its beams upon the mind of her husband.
"You remember," said she smiling, "the anecdote of the rich man of New
York, who asked a person who gave utterance to words of envy towards
himself--'Would you,' said he, 'take all the care and anxiety attendant
upon the management of my large estates and extensive business
operations, merely for your victuals and clothes?' 'No, indeed, I would
not,' was the quick answer. '_I get no more_,' said the rich man,
gravely. And it was the truth, William. They who get rich in this world,
pass up through incessant toil and anxiety; and, while they _seem_ to
enjoy all the good things of life, in reality enjoy but little. They get
only their victuals and clothes. I have worked for many rich ladies, and
I do not remember one who appeared to be happier than I am. And I am
mistaken if your experience is not very much like my own."
One evening, a few days after this time, Aiken came home from his work.
As he entered the room where his wife and children sat, the former
looked up to him with a cheerful smile of welcome, and the latter
gathered around him, filling his ears with the music of their happy
voices. The father drew an arm around one and another, and, as he sat in
their midst, his heart swelled in his bosom, and warmed with a glow of
happiness.
Soon the evening meal was served--served by the hands of his wife--the
good angel of his humble home. William Aiken, as he looked around upon
his smiling children, and their true-hearted, even-tempered, cheerful
mother, felt that he had many blessings for which he should be thankful.
"I saw something, a little while ago, that I shall not soon forget,"
said he, when alone with his wife.
"What was that, William?"
"I had occasion to call at the house of Mr. Elder, on some business, as
I came home this evening. Mr. Elder is rich, and I have often envied
him; but I shall do so no more. I found him in his sitting-room, alone,
walking the floor with a troubled look on his face. He glanced at me
with an impatient expression as I entered. I mentioned my business, when
he said abruptly and rudely--
"'I've no time to think of that now.'
"As I was turning away, a door of the room opened, and Mrs. Elder and
two children entered.
"'I wish you would send those children up to the nursery,' he exclaimed,
in a fretful half-angry voice. 'I'm in no humour to be troubled with
them now.'
"The look cast upon their father by those two innocent little children,
as their mother pushed them from the room, I shall not soon forget. I
remembered, as I left the house, that there had been a large failure in
Market street, and that Mr. Elder was said to be the loser by some ten
thousand dollars--less than a twentieth part of what he is worth. I am
happier than he is to-night, Mary."
"And happier you may ever be, William," returned his wife, "if you but
stoop to the humble flowers that spring up along your pathway, and, like
the bee, take the honey they contain. God knows what, in external
things, is best for us; and he will make either poverty or riches,
whichsoever comes, a blessing, if we are humble, patient and
contented."
DICK LAWSON AND THE YOUNG MOCKING-BIRD.
"Dick!"
"Sir."
"I want a young mocking-bird. Can't you get me one?"
"I d'no, sir."
"Don't you think you could try?"
"I d'no, sir. P'r'aps I might."
"Well, see if you can't. I'll give you half a dollar for one."
"Will you? Then I'll try."
And off Dick started for the woods, without stopping for any further
words on the subject.
The two individuals introduced are a good-natured farmer in easy
circumstances, and a bright boy, the son of a poor woman in the
neighbourhood.
As Dick Lawson was hurrying away for the woods, his mind all intent upon
finding a nest of young mocking-birds, and despoiling it, he met a
juvenile companion, named Henry Jones.
"Come, Harry," said he, in an animated voice, "I want you to go with
me."
"Where are you going?" asked the friend.
"I am going to look for a mocking-bird's nest."
"What for?"
"To get a young one. Mr. Acres said he would give me half a dollar for a
young mocking-bird."
"He did?"
"Yes, he did so!" was the animated reply.
"But don't he know that it's wrong to rob bird's nests!"
"If it had been wrong, Harry, Mr. Acres wouldn't have asked me to get
him a bird. He knows what is right and wrong, as well as anybody about
here."
"And so does Mr. Milman, our Sunday-school teacher; and he says that it
is wicked to rob bird's nests. You know he has told us that a good many
times."
"But Mr. Acres knows what is right as well as Mr. Milman, and if it had
been wrong, he'd never have asked me to get him a bird. And then, you
know, he says he will give me half a dollar for a single one."
"I wouldn't touch a bird's nest for ten dollars," rejoined Henry Jones,
warmly.
"I would then," replied Dick, from whose mind the promised reward had,
for the time, completely dispelled every tender impression received both
from his mother, who had been very careful of her child, and his teacher
at the Sunday-school. "But come," he added, "you'll go with me, anyhow."
"Not, if you are going to rob a bird's nest," firmly responded Henry.
"It is wicked to do so."
"Wicked! I don't see any thing so very wicked about it. Mr. Acres is a
good man, so everybody says, and I know he wouldn't tell me to do a
wicked thing."
"I'm sure it is wicked," persevered Henry Jones, "for isn't it taking
the poor little birds from their mother? Don't you think it would be
wicked for some great giant to come and carry your little sister away
off where you could never find her, and shut her up in a cage, and keep
her there all her life?"
"No, but birds are not little children. It's a very different thing. But
you needn't talk, Harry; for it's no use. If you'll go along, you shall
have half the money I get for the bird--if not, why, I'll go myself and
keep the whole of it."
"I wouldn't go with you for a hundred dollars," said Harry
half-indignantly, turning away.
"Then I'll go myself," was Dick Lawson's sneering reply, as he sprang
forward and hurried off to the woods.
He did not, however, feel very easy in mind, although he attempted first
to whistle gayly, and then to sing. The remonstrance of Henry Jones had
its effect in calling back previous better feelings, awakened by the
precepts of a good mother and the instructions of a judicious
Sabbath-school teacher. To oppose these, however, were the direct
sanction of Mr. Acres, towards whom he had always been taught to look
with respect, and the stimulating hope of a liberal reward. These were
powerful incentives--but they could not hush the inward voice of
disapprobation, that seemed to speak in a louder and sterner tone with
every advancing step. Still, this voice, loud as it was, could not make
him pause or hesitate. Onward he pursued his way, and soon entered the
woods and old fields he had fixed in his mind as the scene of his
operations.
An hour's diligent search ended in the discovery of a nest, in which
were two young ones, with the mother bird feeding them. This sight
softened Dick's heart for a moment, but the strong desire, instantly
awakened, to possess the prize for which he had been seeking, caused him
to drive off the old bird, who commenced fluttering about the spot,
uttering cries and showing signs of deep distress. These, although he
could not help feeling them, did not cause him to desist. In a few
moments he had one of the birds safely in his possession, with which he
bounded off in great delight.
"Well, Dick, have you got my bird?" said Mr. Acres, as Dick came puffing
and blowing into his presence.
"Yes, indeed!" returned Dick with a broad smile of pleasure, presenting
the bird he had abstracted from its warm, soft nest.
"You are a fine smart boy, Dick, and will make a man one of these days!"
said Mr. Acres, patting Dick on the head encouragingly. Then, taking the
bird, he toyed with it for a while fondly--fed it, and finally placed it
in a cage. The promised half-dollar, which was promptly paid to the
lad, made him feel rich. As he was about leaving the house of Mr. Acres,
the latter called to him:
"Look here, Dick, my fine fellow, don't you want a dog? Here's Rover,
the very chap for you."
"May I have Rover?" eagerly asked Dick, his eyes glistening with
delight.
"Yes. I've more dogs now than I want."
"He fights well!" ejaculated Dick, surveying the dog proudly. As he did
so, the animal, seeing himself noticed, walked up to Dick, and rubbed
himself against the lad familiarly.
"He'll whip any dog in the neighbourhood," said Mr. Acres.
"And you'll give him to me?"
"Oh, yes. I've got too many dogs now."
"Here, Rover! Here, Rover! Here! Here! Here!" cried Dick in an animated
tone, starting off. The dog followed quickly, and in a few moments both
were out of sight.
"A smart chap that," remarked Mr. Acres to himself, as Dick bounded
away. "He'll make something before he dies, I'll warrant."
The possession of the dog and half-dollar, especially the latter, were
strongly objected to by Dick's mother.
"How could you, my son, think of robbing a poor bird of her little young
ones?" said she seriously and reprovingly.
"But, mother, Mr. Acres wanted me to get him a bird, and of course I
could not say 'no.' What would he have thought of me?"
"You never should do wrong for any one."
"But if it had been so very wrong, Mr. Acres never would have asked me
to do it, I know," urged Dick.
Mrs. Lawson would have compelled her son to take back the money he had
received, if almost any other person in the village but Mr. Acres had
been concerned. But he was well off, and influential; and, moreover, was
her landlord; and, though she was behindhand with her rent, he never
took the trouble to ask for it. The dog, too, would have been sent back
if any one but Mr. Acres had given it to her son. As it was, she
contented herself with merely reprimanding Dick for robbing the bird's
nest, and enjoining on him not to be guilty of so cruel an act again.
About three days after this event, Dick, accompanied by Rover--now his
inseparable companion--met his young friend, Henry Jones, who had with
him his father's large house-dog, Bose.
"Whose dog is that?" asked Henry.
"He's mine," replied Dick.
"Yours!"
"Be sure he is."
"Why that is Mr. Acres's Rover."
"Not now he isn't. Mr. Acres gave him to me."
"What did he give him to you for?"
"For getting him a young mocking-bird."
"I thought he promised you half-a-dollar?"
"So he did; and what is more, gave it to me, and Rover into the
bargain."
"Well, I wouldn't have robbed a bird's nest for a dozen Rovers," said
Henry Jones, warmly.
"Wouldn't you, indeed?" returned Dick, with a sneer.
"No, I would not. It's wicked."
"Oh, you're very pious! But Rover can whip your Bose, anyhow."
"No, he can't, though," replied Henry quickly, who could not bear to
hear his father's faithful and favourite old dog's courage called in
question.
"Yes, but he can, ten times a day. There, Rover! There,
_sck!--sck!--sketch him_!" At the same time pushing Rover against Bose.
Both dogs growled low, and showed their teeth, but that was all.
"Rover's afraid to touch him!" said Henry, a good deal excited.
"No, he is not, though!" returned Dick, his face glowing with interest;
and, lifting up the forefeet of Rover, he threw him full against old
Bose, who received the onset with a deep growl and a strong impression
of his teeth on Rover.
This brought on the battle. Bose was nine or ten years old, and somewhat
worn down by age and hard service, while Rover had numbered but two
years, and was full of fire and vigor. Still the victory was not soon
decided. During the fight, each of the boys entered into the spirit of
the contest almost as much as the dogs. First one would interfere to
secure for his favourite the victory, and then the other, until, at
last, Dick struck Henry; and then they went at it likewise, and fought
nearly as long, and certainly with as much desire to injure each other,
as did the dogs themselves. The result was that both Henry and Bose had
to yield, and then the parties separated, indulging against each other
bitter and angry feelings. But with Dick there was an emotion of cruel
delight at having triumphed over his friend. As he was crossing a
field, on his way home, he met Mr. Acres.
"Why, what's the matter with you and Rover?" the farmer asked.
"Rover's had a fight," replied Dick.
"Ah! Who with?"
"Mr. Jones's Bose."
"Well, which whipped?"
"Rover, of course," replied Dick, with a smile of triumph; "and I can
make him whip any thing."
"You're a keen chap, Dick," said Mr. Acres, patting the boy on the head,
"and are going to make a man one of these days, I see plainly enough. So
Rover whipped. I knew there was prime stuff in him."
"There isn't another such a fellow in these 'ere parts," was Dick's
proud answer.
"But _you_ look a little the worse for wear, as well as Rover. Have you
been fighting, too?"
Dick held down his head for a moment, and then looking up into Mr.
Acres's face, said--
"Yes, sir," in rather a sheepish way.
"Ah! well, who have you been fighting with?"
"With Harry Jones. He didn't want to give Rover fair play; and once,
when he had Bose down, he kicked him."
"And then you kicked him for kicking your dog?"
"Yes, sir."
"That was right. Never permit a friend to be imposed upon. And after
that you had a regular fight?"
"Yes, sir."
"Which whipped?"
"I gave him a bloody nose; and shouldn't wonder if he had a black eye
into the bargain. And what is more, made him cry 'enough.'"
"That was right. Never fight but in a good cause, and then be sure to
whip your man."
"It'll take a smarter boy than Harry Jones to whip me," said Dick
proudly.
"And you think Rover can whip any thing about here?"
"Yes, indeed. And I'm going to make him do it, too."
"You'd better not try him against Markland's old Nero."
"He'll whip him in ten minutes."
"I'm not so sure of that. Nero is a great deal bigger and stronger."
"I don't care if he is. I'm learning Rover a trick that'll make him whip
a dog twice his size."
"What is that?"
Dick called Rover, and the dog came up to him wagging his tail.
"Give us your paw," said the boy, in a tone of authority.
The dog instantly lifted one of his forefeet, which Dick took in his
hand, and began to squeeze gently at first, and then, by degrees, harder
and harder, ejaculating all the while, in a quick distinct tone--"Leg
him! leg him! leg him!" until the dog, from first indicating signs of
pain, began to whine, and then to yell out as if in agony. At this, Dick
dropped the foot, and looked up into the farmer's face.
"Well, Dick, what does all that mean?" asked Mr. Acres.
"I'm learning him to catch hold of the foot," replied the boy.
"The mischief you are!"
"Yes, sir. And when he's fairly up to it, he can whip any dog, if he's
as big as an elephant."
"But can you learn him?"
"I made him catch Jones's Bose by the foot this morning, and it would
have done your heart good to have heard him yell. If he isn't lame for a
month, then I don't know any thing about it."
"There's no fear of you, I see," was Mr. Acres's encouraging reply to
this, again patting Dick on the head.
In about two weeks from that time it was pretty well known through the
neighbourhood that Dick Lawson had given out that he could make his
Rover whip Markland's Nero, a noble animal that had never been matched
by any dog around. Markland's son felt his pride in his dog touched at
this, and challenged Dick to a battle. The time was set, and the place,
a neighbouring field, chosen. Old and young seemed to take an interest
in the matter, and when the time arrived, and Dick appeared on the
ground with his dog, there were assembled, men and boys, at least one
hundred persons, and among the rest, Mr. Acres, who began to feel
somewhat drawn towards his protege Dick.
[Illustration: CRUEL SPORT.]
The two dogs were brought forward by the two lads, whose parents knew
nothing of the affair, and by pushing them against, and throwing them
upon each other, irritated and angered them until they finally went to
work in real earnest, greatly to the delight of the lookers-on. Rover
fought bravely, but he was evidently no match for his larger and
stronger antagonist, who tore him savagely, while he seemed unable to
penetrate Nero's thick yielding skin. The shouts that arose from the
group around were all in favour of Nero, who was a general favourite--as
he was one of those large, peaceable, benevolent fellows, belieing his
name, whom all liked, while there was something of the churl and savage
about Rover, that caused him to have but few friends.
The contest had waged about ten minutes, fiercely, and Rover was
evidently getting "worsted," when Dick, who had been constantly
encouraging his dog, stooped close to his ear, and spoke something in a
low, quick, energetic tone.
Instantly Rover crouched down, and darting forward, seized the forepaw
of Nero in his mouth, and commenced gnawing it eagerly. The noble
animal, thus unexpectedly and basely assailed, found the pain to which
he was suddenly subjected so great as to take away all power of
resistance. He would not utter a cry, but sat down, and permitted the
other dog to gnaw away at his tender foot without a single sign of
suffering. As the cry of pain, the dog's "enough," was to terminate the
battle, the fine fellow was permitted thus to suffer for several
minutes, before the bystanders came forward and pulled Dick Lawson's dog
off. Nero would have died before a sound could have been extorted from
him.
As Nero had not cried "enough," Bob Markland contended afterwards that
his dog had not been whipped, to settle which difference of opinion he
and Dick had several hard battles, in which the latter, like his dog,
always came off the victor. The upshot of all these contests was, the
expulsion of Dick from the Sabbath-school, into which he carried the
bickerings engendered through the week. Another reason for his expulsion
was the frequency with which he played truant, and of his having, in
several instances, enticed other boys away from the school for the same
purpose.
Except Mr. Acres, nearly every man, woman and child in the
neighbourhood sincerely disliked, and some actually hated Dick Lawson,
for there was hardly a family some member of which had not been annoyed
by him in one form or another. But Mr. Acres liked the spirit of the
lad, as well as his thorough independence in regard to the opinion of
others.
This man, who had first thrown temptation into the lad's way, and
encouraged him to persevere in a conduct which nearly all condemned, was
not a wilfully bad man. By most people he was called a good-hearted,
benevolent person. The truth was, he was not a wise man. When young, he
had indulged in such amusements as catching young birds, fighting dogs
and cocks, and attending horse-races, and all the exciting scenes to
which he could get access. But none of these things corrupted him so far
as to make him a decidedly bad man in the community. As he grew up, he
gradually laid aside his boyish follies; saved up his money; bought
himself a small farm, and, in time, became quite a substantial man, so
far as worldly goods were concerned.
Contrasted with himself were several lads whose parents had been
exceedingly strict with them, and who had, as they grew up, shaken off
the trammels of childhood and youth, run into wild extravagances of
conduct, and some into wicked and vicious habits, from which they were
never reclaimed. Comparing his own case with theirs, his short-sighted
conclusion was that boys ought to be allowed as much freedom as
possible, and this was why he encouraged Dick, who was an exceedingly
bright lad, in the course he had been so willing to pursue. He knew
nothing at all of the different hereditary tendencies to evil that exist
in the mind. His observation had never led him to see how two persons,
raised in precisely the same manner, would turn out very
differently--the one proving a good, and the other a bad citizen. His
knowledge of human nature, therefore, never for a moment caused him to
suspect, that in encouraging a feeling of cruelty in Dick Lawson, he
might be only putting blood upon the tongue of a young lion--that there
might be in his mind hereditary tendencies to evil, which encouragement
to rob a bird's nest, or to set two dogs to fighting, by one occupying
his position and influence, might cause to become so active as to
ultimately make him a curse to society.
And such, in a year or two, Dick seemed becoming. He had in that time,
although but fourteen years of age, got almost beyond his mother's
control. His dog and himself were the terror of nearly all the dogs and
boys in the neighbourhood, for both were surly, quarrelsome, and
tyrannical. Even Mr. Acres had found it necessary to forbid him to
appear on his premises. Rover having temporarily lamed, time after time,
every one of his dogs, and Dick having twice beaten two of his black
boys, farm-hands, because of some slight offence. To be revenged on him
for this, he robbed a fine apricot-tree of all its fruit, both green
and ripe, on the very night before Mr. Acres had promised to send a
basket full, the first produced in the neighbourhood that spring, to a
friend who was very much esteemed by him.
Though he strongly suspected Dick, yet he had no proof of the fact, and
so made no attempt to have him punished.
Shortly after, the boy was apprenticed to a tanner and currier, a severe
man, chosen as his master in the hope that his rigid discipline might do
something towards reclaiming him. As the tanner had as many dogs as he
wanted, he objected to the reception into his yard of Dick's ill-natured
cur. But Dick told his mother that, unless Rover were allowed to go with
him, he would not go to the trade selected for him. He was resolute in
this, and at last Mrs. Lawson persuaded Mr. Skivers, the tanner, to take
him, dog and all.
In his new place he did not get along, except for a very short time,
without trouble. At the end of the third month, for neglect of work,
bad language, and insolence, but particularly for cruelties practised
upon a dog that had gotten the mastery over Rover, Mr. Skivers gave him
a most tremendous beating. Dick resisted, and fought with might and
main, but he was but a boy, and in the hands of a strong and determined
man. For a time this cowed Dick, but in the same ratio that his courage
fell when he thought of resisting his master single-handed, rose his
bitter hate against him. Skivers was a man who, if he had reason to
dislike any one about him, could not let his feelings remain quiescent.
He must be doing something all the while to let the victim of his
displeasure feel that he was no favourite. Towards Dick, he therefore
maintained the most offensive demeanour, and was constantly saying or
doing something to chafe the boy's feelings. This was borne as patiently
as possible, for he did not again wish to enter into a contention in
which he must inevitably get severely beaten. Skivers was not long in
perceiving that the way to punish Dick the most severely was to abuse
his dog; and he, therefore, commenced a systematic process of worrying
Rover. This Dick could illy bear. Every time his master would drive
Rover from the yard, or throw sticks or stones at him, the boy would
make a new and more bitter vow of retaliation in some form.
One day, Rover and a large dog belonging to Skivers got into a fight
about something. Dick's interest in his dog brought him at once to the
scene of action. His master, seeing this, ordered him, in a harsh, angry
tone, to clear out and mind his own business. As he did so, he took a
large club, and commenced beating Rover in a most cruel manner. Dick
could not stand this. His blood was up to fever heat in an instant.
Seizing a long, heavy pole, used for turning and adjusting hides in the
vats, he sprang towards Skivers, and giving it a rapid sweep, brought it
with tremendous force against his head, knocking him into a vat
half-full of a strong infusion of astringent bark, to the bottom of
which he instantly sank.
So incensed did the lad feel, that he made not the slightest attempt to
extricate his master from a situation in which death must have
inevitably ensued in a few minutes, but walked away to another part of
the yard. Two or three journeymen, however, who witnessed the whole
affair, were on the spot in a moment, and took out the body of Skivers.
He was completely insensible. There was the bloody mark of a large wound
on his head. A physician was immediately called, who bled him profusely.
This brought him back to consciousness. In a day or two he was out
again, and apparently as well as ever. In the mean time, both Dick and
his inseparable companion, Rover, had disappeared, and gone no one knew
whither. No effort was made to discover the place to which the boy had
fled, as every one was too much rejoiced that he had left the village,
to care about getting him back. About twelve months after, his mother
died--her gray hairs brought down to the grave in sorrow. Year after
year then passed away, and the memory of the lad was gradually effaced
from the minds of all, or retained only among the dim recollections of
the past.
Mr. Acres, who had first placed temptation in the way of Dick Lawson,
continued to prosper in all external things, and to hold his position of
influence and respectability in the neighbourhood. He, perhaps, more
than others, thought about the lad in whom he had once felt a good deal
of pride and interest, as exhibiting a fair promise for the future. But
he never felt exactly easy in mind when he did think of him. Something
whispered that, perhaps, he had been to blame in encouraging his wild
habits. But, then, how could he have dreamed, he would argue, that the
boy had in him so strong a tendency to evil as the result had proved.
He had once been just as fond as Dick had shown himself to be of
bird's-nesting, dog-fighting, &c., but then, as soon as he had sown a
few wild oats, he sobered down into a steady and thrifty farmer of
regular habits. And he of course expected to see Dick Lawson do the
same.
"And who knows but that he has?" he would sometimes say, in an effort at
self-consolation.
It was some five or six years from the time Dick left the village, that
Mr. Acres was awakened one night from sleep by a dream that some one had
opened the door of the chamber where he slept. So distinct was the
impression on his mind that some one had entered, that he lay perfectly
still, with his eyes peering into the darkness around, in order to
detect the presence of any one, should the impression on his mind really
be true. He had lain thus, with every sense acutely active, for only a
moment or two, when a sound, as of a stealthy footstep, came distinctly
upon his ear, and at the same moment, a dark body seemed to move before
his eyes, as if crossing the room towards that part of it where stood a
large secretary, in which was usually contained considerable sums of
money.
Mr. Acres was a brave man, but thus suddenly awakened from sleep to find
himself placed in such an emergency, made him tremble. He continued to
lie very still, straining his eyes upon the dark moving object intently,
until the figure of a man became perfectly distinct. The robber, for
such the intruder evidently was, had now reached the secretary, where he
stood for a few moments, quietly endeavouring to open it. Finding it
locked, he moved off, and passed around the room, feeling every chair
and table that came in his way. This Mr. Acres could now distinctly
perceive, as his eyes had become used to the feeble light reflected from
the starry sky without. At last his hands came in contact with a chair
upon which the farmer had laid his clothes on disrobing himself for bed.
These seemed to be the objects of his search, for he paused with a quick
eager movement, and commenced searching the ample pockets of a large
waistcoat. The slight jingle of the farmer's bunch of keys soon
explained the movement. Before the robber had fairly gotten back to the
secretary, Mr. Acres's courage had returned, and with it no small share
of indignation. He rose up silently, but, unfortunately, as his foot
touched the floor, it came in contact with a chair, which was thrown
over with a loud noise. Before he could reach a large cane, for which he
was making, a heavy blow from the robber laid him senseless.
When again conscious, Mr. Acres found himself still in total darkness.
On attempting to move, there was an instant, almost intolerable pain in
his head, as if from a violent blow. On lifting his hand and placing it
upon the spot where the pain seemed most severe, it came in contact
with a cold, slimy mass of what he at once knew to be blood. His first
effort to rise was accompanied by a feeling of faintness, that caused
him to stretch himself again upon the floor, where he lay for some time
endeavouring to collect his scattered senses. After he had fully
comprehended the meaning of his alarming situation, he made another and
more successful effort to rise. Sitting up in the middle of the room,
and straining his eyes into the darkness, he began to see more and more
distinctly each moment. He was soon satisfied that he was alone. It did
not take long after this to arouse the whole house. An examination
resulted in ascertaining the fact that his secretary had been robbed of
five hundred dollars in gold.
By daylight, the whole neighbourhood was aroused, and some twenty or
thirty men were in hot pursuit of the robber, who was arrested about
twenty miles away from the village and brought back. The money taken
from the secretary of Mr. Acres, was found upon his person, and fully
identified. The man proved to be quite young, seeming to have passed but
recently beyond the limit of minority. But even young as he was, there
was a look of cruel and hardened villany about him, and an expression of
settled defiance of all consequences. He gave his name as Frederick
Hildich. A brief examination resulted in his committal to await the
result of a trial for burglary at the next court.
The day of trial at length came. The action of the court was brief, as
no defence was set up, and the proof of the crime clear and to the
point. During the progress of the trial, the prisoner seemed to take
little interest in what was going on around him, but sat in the bar,
with his head down, seemingly lost in deep abstraction of mind. At the
conclusion of the proceedings, when the court asked what he had to say
why the sentence of the law should not be pronounced upon him, the
prisoner slowly arose to his feet, lifted his head, glanced calmly
around for a few moments, until his eyes rested upon Mr. Acres, whom he
regarded for some time with a fixed, penetrating, and meaning look.
Then, turning to the Bench, he said in a firm, distinct voice:
"YOUR HONOUR--Although I have nothing to urge against the execution of
the laws by which I am condemned, I would yet crave the privilege of
making a few remarks, which may, perhaps, be useful. The principal
witness against me is Mr. Acres,--and upon his testimony, mainly, so far
as positive proof goes, I am convicted of a crime, the commission of
which I have no particular reason for wishing to deny. But, if I have
wronged him, how far more deeply has he wronged me. If I have robbed him
of a few paltry dollars, he has robbed me of that which he can never
restore, either here or hereafter. In a word, your honour, I stand here,
in the presence of this court, and the people of this town, and charge
upon that man (pointing to Acres) the cause of my present condition. My
real name is Richard Lawson!"
As he said this, the prisoner's voice failed him, and he paused for a
few moments, overcome with emotion. A universal exclamation of surprise
passed through the court-room, and there was scarcely an individual
present who did not wonder why he had not discovered this fact for
himself long before. For, sure enough, it was Dick Lawson, and no one
else, who stood there humbled under the iron hand of the law. As for Mr.
Acres, he became instantly pale and agitated--and when the prisoner
again looked up and fixed his eyes upon him, his own fell to the floor,
as if he were conscience-stricken.
"To that man," resumed the individual, at the bar, pointing steadily
toward the farmer, "as I just said, am I indebted for my ruin. A wild,
but innocent boy, he first led me into conscious wrong, by tempting me
with money to rob a bird's nest. The young mocking-bird was procured for
him, but at the expense of a violated conscience; for a voice within me
spoke loudly against the act of cruelty about to be practised upon the
mother-bird and her young. But I stifled that inward monitor, and
stilled the voice that urged me to depart not from the path of
innocence. I saw that the act was a cruel one, and felt that it was a
cruel one--but to be asked to do even a wrong act by a man to whom I
looked up, as I then did to Mr. Acres, was to rob the wrong act of more
than half of its apparent evil--and so I performed the cruel deed, small
as it was, deliberately. From the moment I took the young bird in my
hand, all my scruples were gone, and after that it was one of my
greatest pleasures to rob birds' nests, and to kill the older birds with
stones. My dog Rover, who is no doubt as well remembered as myself, was
given me by Mr. Acres, and I was, moreover, encouraged by that
individual to make Rover fight, and to fight myself, whenever it came in
the way. Had he discouraged this in me; had he told me that fighting was
wrong, his precept for good would have been as powerful as his precept
for evil. He was kind to me, and had gained my entire confidence, and
could have made almost any thing of me. My cruel, tyrannizing temper,
thus encouraged, grew rapidly, until at last I took no delight in any
good. Finally expelled from the Sabbath-school, and persecuted for my
ill-behaviour and annoyance of almost every one, I became reckless, and
finally left this neighbourhood. Five or six years of evil brought me at
last into a strait. I could not gain even a common livelihood. I must
starve or beg. In this state I thought of my corrupter--of the man who
had been the cause of my wretchedness, and I resolved that he should, at
least, pay some small penalty for what he had done. In a word, I
resolved to rob him--and did so. And now I stand here to await the
sentence of the law for this crime."
The prisoner then suffered his head to fall upon his bosom, and sank
slowly into the seat from which he had arisen. A profound and oppressive
silence reigned through the court-room, broken at last by the judge, who
said--
"Richard Lawson, _alias_ Frederick Hildich, stand up, and receive the
sentence of the law."
The prisoner arose, and looked the judge steadily in the face, while a
sentence of imprisonment in the penitentiary for three years was
pronounced upon him in a voice of assumed sternness.
When the unfortunate man was removed by an officer, the crowd slowly
withdrew, conversing in low, subdued voices, and Mr. Acres turned his
step homeward, the unhappiest man of all who had stood that day in the
presence of offended justice.
And here we must leave the parties most concerned in the events of our
brief story--Richard Lawson to fill up the term of his imprisonment in
the penitentiary; and Mr. Acres to muse, in painful abstraction, over
the ruin his thoughtlessness had wrought--the ruin of an immortal
soul--the corruption of a fellow creature, born to become an angel of
heaven, but changed by his agency into a fit subject for the abodes of
evil spirits in hell.
THE MEANS OF ENJOYMENT.
One of the most successful merchants of his day was Mr. Alexander. In
trade he had amassed a large fortune, and now, in the sixtieth year of
his age, he concluded that it was time to cease getting and begin the
work of enjoying. Wealth had always been regarded by him as a means of
happiness; but, so fully had his mind been occupied in business, that,
until the present time, he had never felt himself at leisure to make a
right use of the means in his hands.
So Mr. Alexander retired from business in favour of his son and
son-in-law. And now was to come the reward of his long years of
labour. Now were to come repose, enjoyment, and the calm delights of
which he had so often dreamed. But it so happened, that the current of
thought and affection which had flowed on so long and steadily, was
little disposed to widen into a placid lake. The retired merchant must
yet have some occupation. His had been a life of purposes, and plans for
their accomplishment: and he could not change the nature of this life.
His heart was still the seat of desire, and his thought obeyed,
instinctively, the heart's affection.
So Mr. Alexander used a portion of his wealth in various ways, in order
to satisfy the ever-active desire of his heart for something beyond what
he had in possession. But, it so happened, that the moment an end was
gained--the moment the bright ideal became a fixed and present fact, its
power to delight the mind was gone.
Mr. Alexander had some taste for the arts. Many fine pictures already
hung upon his walls. Knowing this, a certain picture-broker threw
himself in his way, and, by adroit management and skilful flattery,
succeeded in turning the pent-up and struggling current of the old
gentleman's feelings and thoughts in this direction. The picture-dealer
soon found that he had opened a new and profitable mine. Mr. Alexander
had only to see a fine work of art to desire its possession; and to
desire was to have. It was not long before his house was a gallery of
pictures.
Was he any happier? Did these pictures afford him a pure and perennial
source of enjoyment? No; for, in reality, Mr. Alexander's taste for the
arts was not a passion of his mind. He did not love the beautiful for
its own sake. The delight he experienced when he looked upon a fine
painting was mainly the desire of possession; and satiety soon followed
possession.
One morning Mr. Alexander repaired alone to his library, where, on the
day before, had been placed a new painting, recently imported by his
friend the picture-dealer. It was exquisite as a work of art, and the
biddings for it had been high. But he succeeded in securing it for the
sum of two thousand dollars. Before he was certain of getting this
picture, Mr. Alexander would linger before it, and study out its
beauties with a delighted appreciation. Nothing in his collection was
deemed comparable therewith. Strangely enough, after it was hung upon
the walls of his library, he did not stand before it for as long a space
as five minutes; and then his thoughts were not upon its beauties.
During the evening that followed, the mind of Mr. Alexander was less in
repose than usual. After having completed his purchase of the picture,
he had overheard two persons, who were considered good judges of art,
speaking of its defects, which were minutely indicated. They likewise
gave it as their opinion that the painting was not worth a thousand
dollars. This was throwing cold water on his enthusiasm. It seemed as
if a veil had suddenly been drawn from before his eyes. Now, with a
clearer vision, he could see faults, where before every defect was
thrown into shadow by an all-obscuring beauty.
On the next morning, as we have said, Mr. Alexander entered his library,
to take another look at his purchase. He did not feel very happy. Many
thousands of dollars had he spent in order to secure the means of
self-gratification; but the end was not yet gained.
A glance at the new picture sufficed, and then Mr. Alexander turned from
it with an involuntary sigh. Was it to look at other pictures? No. He
crossed his hands behind him, bent his eyes upon the floor, and, for the
period of half an hour, walked slowly backwards and forwards in his
library. There was a pressure on his feelings--he knew not why; a sense
of disappointment and dissatisfaction.
No purpose was in the mind of Mr. Alexander when he turned from his
library, and, drawing on his overcoat, passed forth to the street. It
was a bleak winter morning, and the muffled passengers hurried shivering
on their way.
[Illustration: "OH! I WISH I HAD A DOLLAR."]
"Oh! I wish I had a dollar."
These words, in the voice of a child, and spoken with impressive
earnestness, fell suddenly upon the ears of Mr. Alexander, as he moved
along the pavement. Something in the tone reached the old man's
feelings, and he partly turned himself to look at the speaker. She was a
little girl, not over eleven years of age, and in company with a lad
some year or two older. Both were coarsely clad.
"What would you do with a dollar, sis?" replied the boy.
"I'd buy brother William a pair of nice gloves, and a comforter, and a
pair of rubber shoes. That's what I'd do with it. He has to go away so
early, in the cold, every morning; and he's 'most perished, I know,
sometimes. Last night his feet were soaking with wet. His shoes are not
good; and mother says she hasn't money to buy him a new pair just now.
Oh, I wish I had a dollar!"
Instinctively Mr. Alexander's hand was in his pocket, and a moment
after, a round, bright silver dollar glittered in that of the girl.
But little farther did Mr. Alexander extend his walk. As if by magic,
the hue of his feelings had changed. The pressure on his heart was gone,
and its fuller pulses sent the blood bounding and frolicking along every
expanding artery. He thought not of pictures nor possessions. All else
was obscured by the bright face of the child, as she lifted to his her
innocent eyes, brimming with grateful tears.
One dollar spent unselfishly brought more real pleasure than thousands
parted with in the pursuit of merely selfish gratification. And the
pleasure did not fade with the hour, nor the day. That one truly
benevolent act, impulsive as it had been, touched a sealed spring of
enjoyment, and the waters that gushed instantly forth continued to flow
unceasingly.
Homeward the old man returned, and again he entered his library. Choice
works of art were all around him, purchased as a means of enjoyment.
They had cost thousands,--yet did not afford him a tithe of the pleasure
he had secured by the expenditure of a single dollar. He could turn from
them with a feeling of satiety; not so from the image of the happy child
whose earnestly expressed wish he had gratified.
And not alone on the pleasure of the child did the thoughts of Mr.
Alexander linger. There came before his imagination another picture. He
saw a poorly furnished room, in which were an humble, toiling widow, and
her children. It is keen and frosty without; and her eldest boy has just
come home from his work, shivering with cold. While he is warming
himself by the fire, his little sister presents him with the comforter,
the thick gloves, and the overshoes, which his benevolence had enabled
her to buy. What surprise and pleasure beam in the lad's face! How happy
looks the sister! How full of a subdued and thankful pleasure is the
mother's countenance!
And for weeks and months did Mr. Alexander gaze, at times, upon this
picture, and always with a warmth and lightness of heart unfelt when
other images arose in his mind and obscured it.
And for a single dollar was all this obtained, while thousands and
thousands were spent in the fruitless effort to buy happiness.
Strange as it may seem, Mr. Alexander did not profit by this
lesson--grew no wiser by this experience. The love of self was too
strong for him to seek the good of others--to bless both himself and his
fellows by a wise and generous use of the ample means which Providence
had given into his hands. He still buys pictures and works of art, but
the picture in his imagination, which cost but a single dollar, is
gazed at with a far purer and higher pleasure than he receives from his
entire gallery of paintings and statues.
If Mr. Alexander will not drink from the sweet spring of true delight
that has gushed forth at his feet, and in whose clear waters the sun of
heavenly love is mirrored, we hoped that others, wiser than he, will
bend to its overflowing brim, and take of its treasures freely. Some one
has beautifully said--"We only possess what we have bestowed." Something
of the meaning of this will be understood by such of our young readers
as have perused this story thoughtfully. Benevolent actions ever bring
their own reward. Far more happiness is gained in seeking to bless
others, than ever comes from efforts to secure merely our own good. God,
who is infinitely good and wise, and from whom comes all true happiness,
is ever seeking to bless others. If we would truly enjoy life, we must
be like Him.
MAN'S JUDGMENT.
"I wouldn't give much for his chance of heaven!" was the remark of a
man, whose coarse, well-worn garments contrasted strongly with the dark,
rich broadcloth of the person to whom he referred. In the tones of the
individual who uttered this sentence was a clearly apparent satisfaction
at the thought of his rich neighbour's doubtful chance of admission into
heaven. It was on the Sabbath, and both had just passed forth from the
sacred edifice, to which each had that morning gone up for the avowed
object of worship.
"Why do you say that?" asked the friend to whom the remark was
addressed.
"You know the Scriptures," was the confident answer. "'How hardly shall
they who have riches enter the kingdom of heaven.'"
"You believe, then, that the mere fact of possessing riches will keep a
man out of heaven?"
"No; I wouldn't just like to say that. But, riches harden the heart, and
make men unfit for heaven."
"I doubt if riches harden the heart more than poverty," was replied.
"How can you say so?" was warmly objected. "Isn't the promise everywhere
to the poor? To whom was the gospel sent?"
"The rich and poor spoken of in the word of God," said the friend, "do
not, it is plain, mean simply those in the world who possess natural
riches, or who are in natural poverty. Remember, that the Bible is a
revelation of heavenly truth, for man's eternal salvation; and that its
teachings must have primary regard to what is spiritual, and refer to
man's internal state rather than to his mere worldly condition.
Remember, that the Lord, while on earth, said, _Blessed are the poor in
spirit_, (not the poor in this world's goods,) _for theirs is the
kingdom of heaven_. And we may, without violence to even the letter of
the word, conclude that when He speaks of its being hard for the rich to
enter the kingdom of heaven, that only the proud in spirit, those who
rested self-confident on the riches of their worldly and natural wisdom,
were meant. That it would be easier for a camel to go through the eye of
a needle than for such rich men to enter heaven, is plain from our
Lord's words when he set a child in the midst of his disciples, and told
them that, unless they became as that little child, they could not enter
the kingdom of heaven. Not externally and naturally as that child, for
that was impossible; but poor in spirit, teachable, and innocent as a
child."
The first speaker, whose name was Maxwell, tossed his head, and slightly
curled his lip as he replied--
"I believe just what the Bible says. As for your forced meanings, I
never go to them. A plain matter-of-fact man, I understand what is
written in a plain, matter-of-fact way. The Bible says that they who
have riches shall hardly enter the kingdom of heaven. And I can see how
true the saying is. As for Clinton, of whom I spoke just now, I repeat
that I wouldn't give much for his chance. It is well that there is a
just God in heaven, and that there will come a day of retribution. The
Diveses have their good things in this life; but our turn will come
afterwards. We sha'n't be always poor. Lazarus went, a beggar, from the
rich man's door, and was received into Abraham's bosom."
"What has made you so bitter against Clinton, just now?" inquired the
friend.
"I'm not bitter against him in particular--I speak of rich men as a
class. They are all selfish, unfeeling, and oppressive. Look at the good
Clinton might do, as a steward of God's bounty, if he chose. He might
make our wilderness blossom as the rose. But settlement-day will come,
ere long, and then a sorry account of his stewardship will he have to
render."
"How do you know that the account will not be approved in heaven?" was
asked in a quiet voice.
"Approved? How do I know?" ejaculated Maxwell, impatiently. "Any man can
see that he is an unfaithful, hard-hearted, and oppressive steward."
"Has he oppressed you?"
"Yes."
"Ah! I was not aware of that. I didn't know that you had any claims upon
him as an almoner of heaven."
"My claims are those of common humanity. But you shall know all, and
judge for yourself. I am a poor man"----
"Well"----
"With a wife and four children, whom I love as tenderly as Clinton, or
any other purse-proud oppressor of the poor can possibly love his wife
and children. They are dependent for daily bread upon my daily labour.
With the sweat of my brow, I keep hunger from my door, and cold from
entering therein."
"An independent man," said the other.
"Yes, an independent man; as independent as any nabob in the land."
"Do let the nabobs alone," was smilingly answered to this. "If you are
independent, why care for them? Why permit yourself to be fretted
because others are blessed by Providence with a greater abundance of
worldly goods? There is danger, in this thing, of going beyond the
nabobs, and arraigning the wisdom of Him who setteth up whom he will,
and whose bounty feeds even the young ravens. So go on with your story.
What is the crime that Mr. Clinton has committed against you and
humanity?"
"I am a poor man, as I said."
"I know you are; a hard-working, industrious, but poor man."
"And as such, entitled to some consideration."
"Entitled to a fair return for your labour, in all cases."
"Of course I am; and to some favour, in the distribution of employment,
when I present equal capacity with those who are less needy than
myself."
"What do you mean by that?"
"A plain story makes all plain. Well: you are aware that Mr. Clinton is
about building a new dam for his mills?"
"I am."
"And that he asked for proposals?"
"Yes."
"I tried to get the contract."
"You!" There was more surprise in this ejaculation than the friend had
meant to convey.
"Certainly! Why not?" was petulantly remarked.
"Of course you had a perfect right to do so?"
"Of course I had; and of course my bid, though the lowest, was thrown
out, and the bid of Jackson, who manages to monopolize every thing in
the village, taken. He and Clinton are leagued together, and the offer
for proposals was only a sham."
"That's assuming a good deal, friend Maxwell."
"No, it isn't. It's the truth, and nothing else but the truth. He's the
jackal, and Clinton's the lion."
"You speak without reflection," said the friend, mildly.
"I'm not blind. I see how things are worked."
"You say your bid was lower than Jackson's? How do you know this? I
thought his bid was not publicly known."
"I knew it; and, in fact, knew what it was to be before I sent in my
proposals, and was, therefore, able to go below it. The truth is, I
managed, between you and I, to find out just what every man was going to
bid, and then struck a mark below them all, to make sure of the job. I
wanted a chance, and was determined to have it at all hazards."
"I hardly think your mode of procedure was fair," said the friend; "but
waiving that, could you have made any thing by the job, at your
bidding?"
"Oh, yes, I'd have made something--more, a good deal, than I can make by
day's work. The fact is, I set my heart on that job as a stepping stone
to contract work; and am bitterly disappointed at its loss. Much good
may it do both Jackson and Clinton. I shouldn't be much sorry to see the
new dam swept away by the next freshet."
"Why, Maxwell! This is not the spirit of a Christian man. Envy,
malice--these are what the Bible condemns in the plainest terms; and for
these sins, the poor have quite as much to answer for as the rich--and
perhaps more. If you go from church on the Sabbath with no better
thoughts than these, I fear you are quite as far from the Kingdom of
Heaven as you have supposed Mr. Clinton to be."
"Good day," said Maxwell, turning off abruptly from his friend, and
taking a path that led by a nearer course than the one in which they
were walking, to his home.
A few weeks later, the person with whom Maxwell thus conversed, had
occasion to transact some business with Mr. Clinton. He had rendered him
a bill for work done, and called to receive payment.
"You've made a mistake in your bill, Mr. Lee," said Clinton.
"Ah? Are you certain?"
"You can examine for yourself. I find an error of twenty dollars in the
additions."
"Then you only owe me sixty dollars?" said Lee, with a disappointment in
his tones that he could not conceal.
"Rather say that I owe you a hundred, for the mistake is in your favour.
The first column in the bill adds up fifty, instead of thirty dollars."
"Let me examine it." Lee took the bill, and added up the column three
times before he felt entirely satisfied. Then he said,
"So it does! Well, I should never have been the wiser if you had only
paid me the eighty dollars called for by the bill. You might have
retained your advantage with perfect safety."
Lee said this on the impulse of the moment. He instantly saw a change in
Mr. Clinton's countenance, as if he were slightly offended.
"Oh, no; not with safety," was gravely replied.
"I never should have found it out."
"But there is coming a day, with every man, when the secrets of his
heart will stand revealed. If not now, it would then appear that I had
wronged you out of twenty dollars."
"True! true! But all men don't think of this."
"No one is more fully aware of that than I am. It is for me, however, to
live in the present so as not to burden my future with shame and
repentance. Knowingly, Mr. Lee, I would not wrong any man out of a
single dollar. I may err, and do err, like other men; for, to err is
human."
After the expression of such sentiments, Lee felt curious to know what
Mr. Clinton thought of, and how he felt towards Maxwell. So he said,
after referring to the new mill-dam in the process of erection--
"You didn't take the lowest bid for its construction."
"I took the lowest competent bid."
"Then you do not think Maxwell competent to do the work?"
"I do not think him a man to be trusted, and, therefore, would not have
given him the contract for such a piece of work at any price. You are
aware that the giving way of that dam would almost inevitably involve a
serious loss of life and property among the poor people who live along
the course of the stream below. I must regard their safety before any
pecuniary advantage to myself; and have given Mr. Jackson, who has the
contract, positive instructions to exceed his estimates, if necessary,
in order to put the question of safety beyond a doubt. I know him to be
a man whom I can trust. But I have no confidence in Maxwell."
"A good reason why you declined giving him the job."
"I think so."
"Maxwell was greatly disappointed."
"I know he has spoken very hard against me. But that avails nothing. My
principle of action is to do right, and let others think and say what
they please. No man is my judge. Maxwell is not, probably, aware that I
know him thoroughly, and that I have thrown as much in his way as I
could safely do. He is not, of course, aware, that one of my sons
overheard him, in reference to this very mill-dam, say--'I'm bound to
have that contract whether or no. I have learned the lowest bid, and
have put in a bid still lower.' 'How did you learn this?' was asked of
him. 'No matter,' he answered, 'I have learned it.' 'You can't go lower
and build the dam safely,' was said. To which he replied--'I can build
the dam, and make a good profit. As to the safety, I'll leave that in
the hands of Providence. He'll take care of the poor people below.' Mr.
Lee! I felt an inward shudder when this was repeated to me. I could not
have believed the man so void of common honesty and common humanity. Was
I not right to withhold from him such a contract?"
"You would have been no better than Maxwell, if you had given it to
him," was answered. "And yet, this same man speaks against the rich, and
thinks their chance of heaven a poor one."
"Simply because they are rich."
"Or, it might with more truth be said, because they will not yield to
his covetous and envious spirit. He is not content with the equivalent
society renders back to him for the benefit he confers, but wants to
share what of right belongs to others."
"That spirit I have often seen him manifest," was replied. "Well, if
simple riches are a bar to man's entrance into heaven, how much more so
are discontent, envy, malice, hatred, and a selfish disregard for the
rights and well-being of others. The rich have their temptations, and so
have the poor, and neither will enter heaven, unless they overcome in
temptation, and receive a purified love of their neighbour. This at
least is my doctrine."
"Of the two, I would rather take Clinton's chance of heaven," said Lee
to himself, as he went musing away, "even if he is a rich man."
[Illustration: ANOTHER DEBT PAID.]
WHAT FIVE DOLLARS PAID.
Mr. Herriot was sitting in his office, one day, when a lad entered, and
handed him a small slip of paper. It was a bill for five dollars, due to
his shoemaker, a poor man who lived in the next square.
"Tell Mr. Grant that I will settle this soon. It isn't just convenient
to-day."
The boy retired.
Now, Mr. Herriot had a five-dollar bill in his pocket; but, he felt as
if he couldn't part with it. He didn't like to be entirely out of money.
So, acting from this impulse, he had sent the boy away. Very still sat
Mr. Herriot for the next five minutes; yet his thoughts were busy. He
was not altogether satisfied with himself. The shoemaker was a poor man,
and needed his money as soon as earned--he was not unadvised of this
fact.
"I wish I had sent him the five dollars," said Mr. Herriot, at length,
half-audibly. "He wants it worse than I do."
He mused still further.
"The fact is," he at length exclaimed, starting up, "it is Grant's
money, and not mine; and what is more, he shall have it."
So saying, Herriot took up his hat and left his office.
"Did you get the money, Charles," said Grant, as his boy entered the
shop. There was a good deal of earnestness in the shoemaker's tones.
"No, sir," replied the lad.
"Didn't get the money!"
"No, sir."
"Wasn't Mr. Herriot in?"
"Yes, sir; but he said it wasn't convenient to-day."
"Oh, dear! I'm sorry!" came from the shoemaker, in a depressed voice.
A woman was sitting in Grant's shop when the boy came in; she had now
risen, and was leaning on the counter; a look of disappointment was in
her face.
"It can't be helped, Mrs. Lee," said Grant. "I was sure of getting the
money from him. He never disappointed me before. Call in to-morrow, and
I will try and have it for you."
The woman looked troubled as well as disappointed. Slowly she turned
away and left the shop. A few minutes after her departure, Herriot came
in, and, after some words of apology, paid the bill.
"Run and get this note changed into silver for me," said the shoemaker
to his boy, the moment his customer had departed.
"Now," said he, so soon as the silver was placed in his hands, "take two
dollars to Mrs. Lee, and three to Mr. Weaver across the street. Tell Mr.
Weaver that I am obliged to him for having loaned me the money this
morning, and sorry that I hadn't as much in the house when he sent for
it an hour ago."
"I wish I had it, Mrs. Elder. But, I assure you that I have not," said
Mr. Weaver, the tailor. "I paid out the last dollar just before you came
in. But call in to-morrow, and you shall have the money to a certainty."
"But what I am to do to-day? I haven't a cent to bless myself with; and
I owe so much at the grocer's, where I deal, that he won't trust me for
any thing more."
The tailor looked troubled, and the woman lingered. Just at this moment
the shoemaker's boy entered.
"Here are the three dollars Mr. Grant borrowed of you this morning,"
said the lad. "He says he's sorry he hadn't the money when you sent for
it awhile ago."
How the faces of the tailor and his needlewoman brightened instantly, as
if a gleam of sunshine had penetrated the room.
"Here is just the money I owe you," said the former, in a cheerful
voice, and he handed the woman the three dollars he had received. A
moment after and he was alone, but with the glad face of the poor woman,
whose need he had been able to supply, distinct before him.
Of the three dollars received by the needlewoman two went to the grocer,
on account of her debt to him, half a dollar was paid to an old and
needy coloured woman who had earned it by scrubbing, and who was waiting
for Mrs. Weaver's return from the tailor's to get her due, and thus be
able to provide an evening's and a morning's meal for herself and
children. The other half-dollar was paid to the baker when he called
towards evening to leave the accustomed loaf. Thus the poor needlewoman
had been able to discharge four debts, and, at the same time
re-establish her credit with the grocer and baker, from whom came the
largest portion of the food consumed in her little family.
And now let us follow Mrs. Lee. On her arrival at home empty-handed,
from her visit to the shoemaker, who owed her two dollars for work, she
found a young girl, in whose pale face were many marks of suffering and
care, awaiting her return.
The girl's countenance brightened as she came in; but there was no
answering brightness in the countenance of Mrs. Lee, who immediately
said--
"I'm very sorry, Harriet, but Mr. Grant put me off until to-morrow. He
said he hadn't a dollar in the house."
The girl's disappointment was very great, for the smile she had forced
into life instantly faded, and was succeeded by a look of deep distress.
"Do you want the money very badly?" asked Mrs. Lee, in a low,
half-choked voice, for the sudden change in the girl's manner had
affected her.
"Oh, yes, ma'am, very badly. I left Mary wrapped up in my thick shawl,
and a blanket wound all around her feet to keep them warm; but she was
coughing dreadfully from the cold air of the room."
"Haven't you a fire?" asked Mrs. Lee, in a quick, surprised tone.
"We have no coal. It was to buy coal that I wanted the money."
Mrs. Lee struck her hands together, and an expression of pain was about
passing her lips, when the door of the room opened, and the shoemaker's
boy came in.
"Here are two dollars. Mr. Grant sent them."
"God bless Mr. Grant!" The exclamation from Mrs. Lee was involuntary.
On the part of Harriet, to whom one dollar was due, a gush of silent
tears marked the effect this timely supply of money produced. She
received her portion, and, without trusting her voice with words,
hurried away to supply the pressing want at home.
A few doors from the residence of Mrs. Lee lived a man who, some months
before, had become involved in trouble with an evil-disposed person, and
been forced to defend himself by means of the law. He had employed Mr.
Herriot to do what was requisite in the case, for which service the
charge was five dollars. The bill had been rendered a few days before,
and the man, who was poor, felt very anxious to pay it. He had the money
all made up to within a dollar. That dollar Mrs. Lee owed him, and she
had promised to give it to him during this day. For hours he had waited,
expecting her to come in; but now had nearly given her up. There was
another little bill of three dollars which had been sent in to him, and
he had just concluded to go and pay that, when Mrs. Lee called with the
balance of the money, one dollar, which she had received from the
shoemaker, Grant.
Half an hour later, and the pocket-book of Mr. Herriot was no longer
empty. His client had called and paid his bill. The five dollars had
come back to him.
LOOK AT T'OTHER SIDE.
"I don't like Mr. Monto at all," said Mr. Jones.
"Nor I," replied Mrs. Mayberry.
"Take him for better or worse," added Mr. Lee, "and I think he is the
strangest and most inconsistent man I ever saw."
"Inconsistent!" resumed Mr. Jones. "He is worse than inconsistent.
Inconsistencies may be pardoned, as constitutional defects and
peculiarities of character. But he is worse than inconsistent, as I
said."
"Yes, that he is," chimed in Mrs. Mayberry. "What do you think I heard
of him last week?"
"What?" said Mr. Jones.
"Yes, what did you hear?" asked Mrs. Lee.
"You know Mr. Barker?"
"Yes."
"There isn't a more gentlemanly man living than Mr. Barker."
"Well, what of him?"
"He was in Mr. Monto's store one day last week, and happened to say
something the little man did not like, when he fired up and insulted him
most grossly."
"Indeed!"
"Yes. Mr. Barker told me himself. He said he was never more hurt in his
life."
"He left the store, of course."
"Oh, yes. He turned on his heel and walked out, and says he will never
darken the door of Monto's store again."
"It is too bad, this habit of insulting people which Monto has. I know
several persons who are hot as fire against him."
"If there were nothing worse about him than that," said Mr. Jones, "I
would be glad. His conduct towards the young man he raised was
unpardonable."
"What was that? I never heard about it," remarked Mr. Lee.
"He had a young man whom he had raised from a lad, and who, it is said,
was always faithful to his interests. Toward the last he became wild,
having fallen into bad company. If Monto had been patient and forbearing
toward him, the young man might have been reclaimed from his error; but
his irascibility and impatience with every thing that did not go by
square and rule, caused him to deal harshly with faults that needed a
milder corrective. The young man, of course, grew worse. At last he got
himself into a difficulty, and was arrested. Bail was demanded for his
appearance to stand a trial for misconduct and breach of law. Monto was
sent for to go his bail; but he heartlessly refused, and the poor fellow
was thrown into prison, where he lay four months, and was then, after a
trial, dismissed with a reprimand from the court. Feeling himself
disgraced by confinement in a jail, he enlisted in the army as soon as
he got free, and has gone off to the Indian country in the West. Isn't
it melancholy? The ruin of that young man lies at Monto's door. His
blood is on the skirts of his garments!"
"Dreadful to think of! Isn't it?" said Mrs. Mayberry. "Just imagine my
son or your son thus cruelly dealt by! A fiend in human shape couldn't
have done more!"
"It'll come back upon him one of these days. I believe in retribution.
No man can do such things with impunity," added Mr. Lee. "Mark my words
for it--Monto will repent of this, as well as a good many other acts of
his life, before he dies."
"He's the meanest man I ever saw," said Mr. Jones. "I don't believe he
ever gave a dollar for charitable purposes in his life."
"You may possibly err, there," remarked a fourth in the company, who had
not before spoken.
"I should like to see the man, Mr. Berry, who can point to a benevolent
act of Monto's," returned Mr. Jones in a decided voice.
"Perhaps," said Mr. Berry, "if we were as willing to look at the other
side of men's characters, we should not entertain the poor opinion of
them we do. If we were to look as closely at the good as we do at the
bad, we might find, perhaps, as much to praise as we do to blame. When I
was a boy, I had a penny given to me, and was about buying a large,
seemingly fine apple, when my brother said in a warning voice, 'Look at
t'other side.' I did look, and found it rotten. When I became a man, I
remembered the lesson, and determined that I would not be deceived by
fair appearances of character, but would be careful to look at t'other
side for blemishes. I saw enough of these, even in the best, to sicken
me with mankind. A few years passed, and I was glad to change my habit
of observation. I began to look at the other and brighter side. The
result surprised and pleased me. I found more good in men than I had
supposed. Even in the worst there were some redeeming qualities."
"You will find few in Monto," said Mr. Lee.
"Do you see that man on the other side of the street?" asked Mr. Berry.
"Who? Miller?"
"Yes; that's the one I mean. I'll call him over, if you have no
objection, and ask him a question or two. I think he can say something
bearing on the subject of our present discourse."
The man was called, and he came over and entered the store of Mr. Jones,
where the conversation happened to occur.
"Good morning, Miller! How are you to-day?" said Mr. Berry.
"Good morning! You've quite a party here. All friends, I see."
"We seem to have met by one of those happy accidents that sometimes
occur. How are you getting along now, Miller? You've been through some
pretty tight places, I believe."
"Yes; and, thanks to a good Providence! I am through them with a whole
skin."
"Cause for congratulation, certainly. We meet with some hard rubs in our
journey through life."
"Indeed we do. Adverse circumstances try us severely, and try our
friends also. It has been so in my case. I thought I had a good many
friends, until trouble came; but, as you know, there were few to stand
by me when I most needed support."
"But you met with friends?"
"Yes, friends in need, who are friends indeed."
"And they were among those who had made no professions, and upon whom
you did not feel that you had any claims?"
"Exactly so. This was particularly the case in one instance. Through
losses, mistakes, and from errors on account of which I do not attempt
to excuse myself, my business became embarrassed. What little real
estate I had was thrown into market and sacrificed, but this did not
meet my necessities. In the hope of weathering the storm, I removed from
the handsome store I occupied into one at half the rent, reduced all
expenses both in my business and family, but still I was not able,
without the most untiring exertions, to meet my payments. More than half
my time I was on the street, engaged in temporary expedients to raise
money. I was harassed to death, and in daily dread of failure. In this
unhappy posture of my affairs, I tried to get some permanent assistance
from friends who were able enough to afford it, and who knew me well.
But they were all afraid to risk any thing.
"One day I had been out from nine o'clock until two, using my best
efforts to obtain sufficient money to meet my notes. I had a thousand
dollars to pay, and could only thus far raise five hundred. Everywhere
that I could think of going I went, but no one would help me through my
difficulty. Dispirited and alarmed at the perilous position of my
affairs, I returned to my store, in order to sit down and reflect for a
few minutes. I thought over all my business acquaintance, but there were
none upon whom I had not already called, that I felt free to ask for the
loan of money. Things seemed desperate. Something must be done, or I
would be ruined. Already the finger of time was past the mark of two. In
less than an hour my paper would be dishonoured, unless I could in some
way command the sum of five hundred dollars. I thought, and thought,
until I felt stupid. At last a man whom I had never liked much came up
before my mind. I had some little acquaintance with him, and knew, or
supposed, that he had money. The idea of going to him I would not at
first entertain. But things were desperate. At last I started up,
determined to see this man.
"'He can but refuse me,' I murmured to myself.
"'It is past two o'clock,' said I abruptly, as I met him standing at his
counter, 'and I am still five hundred dollars short. Can you lend me
that sum for a few days?'
"I expected him to say 'no.' What was my surprise then to hear him
reply--
"'I can, and with pleasure.'
"I could hardly believe my ears. But by the assistance of my eyes, when
he put a check for the amount I had asked for into my hands, I was fully
assured that he was in earnest. I don't know that I ever stopped to
thank him, so overjoyed was I at such unexpected and cheerfully tendered
relief. Three or four days afterward I took him the money he had loaned
me.
"'Keep it longer, if you desire to do so. I have no present use for it,'
said he.
"I hardly knew whether to take him at his word or not. But necessity is
an eloquent pleader.
"'If you can spare it as well as not, it will be an accommodation. My
payments are heavy in the next ten days,' I replied.
"'Retain the use of it and welcome,' said he kindly. After a pause, he
inquired how I was getting along, and did it with so much sincerity
that I was tempted to state frankly the position of my affairs, and did
so. He listened with a good deal of interest, and afterward asked many
questions as to the nature and profits of my business. I concealed
nothing from him in favour or against myself as a business-man.
"'You must be sustained, Mr. Miller,' said he. 'I have a few thousand
dollars uninvested, that I will keep free for six months or so. As far
as you need assistance in meeting your payments, I will afford it. Pay
no more exorbitant interests; waste no more time in running about after
money; but put all your thoughts and energies down to your business, and
twelve months from to-day will see you freed from embarrassment.'
"And he was right."
"He was certainly a noble fellow," said Mr. Jones. "Pity there were not
more like him!"
"That it is," remarked Mrs. Mayberry.
"He belongs to another grade of beings than your Montos."
"Who?" Miller spoke quickly.
"We were talking of Monto when I called you," said Mr. Berry. "Our
friends have a very poor opinion of him."
"Of Mr. Monto? Why, it is of him that I just now spoke."
"Of Monto!" ejaculated Lee.
"Certainly. He it was who so generously befriended me."
"Impossible!" ejaculated Mrs. Mayberry.
"Not at all, for it is true. I never was more mistaken in any one in my
life than in Mr. Monto. He has his faults and defects of character, as
all men have. He is irascible and impatient, and makes in consequence a
great many enemies."
"He was certainly kind to you, Mr. Miller," said Mrs. Mayberry. "But
still, I don't believe in him. Look at the way he treated that poor
young man whom he raised from a boy. That stamps his character. That
shows him to be cruel and vindictive."
"There is another side to that story, without doubt," remarked Mr.
Berry.
"That there is," said Miller; "and suppose we look at it. Monto knew
that young man much better than you or I, or any of us. He had borne
with his irregular habits and evil conduct for years, as well as a man
of his peculiar temperament could bear with them."
"A precious kind of forbearance it was, no doubt. It isn't in him to
bear with any one," broke in Mr. Jones.
"Will you censure a man for what he can't help?" asked Mr. Miller.
"I don't know that we should," was replied.
"It is clear that we ought not; for to do so would be for us to ask of
him an impossibility, and censure him for not performing it. Mr. Monto
is a man, as we all know, of exceedingly impatient temper. Keep that in
view. He takes this boy when quite young, and educates him as well as
teaches him his business. Before he is of age he abuses the confidence
reposed in him by his benefactor, neglects his business, associates with
vicious companions, and purloins his money. Still Monto bears with him,
in the hope that he will change. But he grows worse and worse; and at
length, after a long series of peculations at home, gets into a
difficulty, and is sent to jail to await the judgment of the law in his
case. I happened to be in Mr. Monto's store when he was sent for to bail
the young man out.
"'No,' he said firmly to the messenger, 'he is much better in prison
than out.'
"The man went away, and Monto, turning to me, said--
"'That, Mr. Miller, is the most painful thing I have done in my whole
life. But to have acted otherwise would have been wrong. Kind
admonition, stern reproof, angry expostulation, all have failed with
this young man, in whom I cannot help feeling a strong interest. I will
now leave him to the consequences of his own acts, and to the, I hope,
salutary results of his own reflections. If these fail to reform him,
there is no hope.' This was the spirit in which it was done. He did not
attend court when the trial came on, but he had a messenger there, who
kept him constantly advised of the proceedings. The acquittal gave him
great pleasure, and he expected the young man would return to him,
changed and penitent. He was, alas! grievously mistaken. The enlistment
hurt him exceedingly. I could perceive that his voice was unsteady when
he spoke of it. If he erred in his conduct, it was an error of judgment.
He meant to do good. But I do not believe he erred. In my opinion, the
young man is fit only for the grade he now occupies, and he is better
off where he is."
"There is good in every one," said Mr. Berry, when Miller ceased
speaking; "and we will find it, if we look at the other side."
"No truer word than that was ever spoken," returned Mr. Miller. "Yes,
there is good in every one; and more good than evil in Monto, you may
all be assured."
The censurers of Monto approved the words by a marked and half-mortified
silence.
Yes, there is good in every one; there is another side. Let us look for
this good rather than for what is evil, and we will think better of
mankind than we are now disposed to do.
[Illustration: THIN SHOES.]
THIN SHOES.
"Why, Lizzy, dear!" exclaimed Uncle Thomas, to his pretty niece, Miss
Walton, as she stepped upon the pavement from her mother's dwelling, one
morning in midwinter--"You are not going in this trim?"
"In what trim?" said Lizzy, glancing first at her gloves, then upon her
dress, and then placing her hand upon her neck and bosom to feel if all
was right there. "Is any thing wrong with my dress, uncle?"
"Just look at your feet."
"At my feet!" And Lizzy's eyes fell to the ground. "I don't see any
thing the matter with them."
"Why, child, you have nothing on your feet but paper-soled French
lasting boots."
"They have thick soles, uncle."
"Thick! If you call them thick, you will have to find a new term for
thinness. Go right back, and put on your leather boots."
"Leather boots!" Lizzy's voice and countenance showed an undisguised
amazement.
"Yes, leather boots. You certainly wouldn't think of going out on a day
like this without having your feet well protected with leather boots."
"Leather boots! Why, Uncle Thomas!"--and the musical laugh of Miss
Walton echoed on the air--"who ever heard of such a thing?"
Uncle Thomas glanced involuntarily down at his own thick, double-soled,
calfskin understandings.
"Boots like them!" exclaimed the merry girl, laughing again.
"But come along, my good uncle," she added more seriously, drawing her
arm within his, and attempting to move away. "We'll have all the
neighbourhood staring at us. You can't be in earnest, I'm sure, about my
wearing clumsy leather boots. Nancy, the Irish cook, has a pair; but
I"----
"And pray, Lizzy," returned the old gentleman, as he yielded to the
impulse given him by his niece, and moved down the street beside
her--"are you so much heartier than Nancy, so much stouter and stronger,
that you can bear exposure to damp and even wet pavements, in thin
shoes, while she will not venture out unless with feet well protected by
leather boots?"
"My shoes are not thin, uncle," persisted Lizzy. "They have thick
soles."
"Not thin! Thick soles! Look at mine."
Lizzy laughed aloud, as she glanced down at her uncle's heavy boots, at
the thought of having her delicate feet encased in leather.
"Look at mine!" repeated Uncle Thomas. "And am I so much more delicate
than you are?"
But Miss Walton replied to all this serious remonstrance of her uncle
(who was on a visit from a neighbouring town) with laughing evasion.
A week of very severe weather had filled the gutters and blocked the
crossings with ice. To this had succeeded rain, but not of long enough
continuance to free the streets from their icy encumbrance. A clear,
warm day for the season followed; and it was on this day that Miss
Walton and her uncle went out for the purpose of calling on a friend or
two, and then visiting the Art-Union Gallery.
Uncle Thomas Walton was the brother of Lizzy's father. The latter died
some few years before, of pulmonary consumption. Lizzy, both in
appearance and bodily constitution, resembled her father. She was now in
her nineteenth year, her veins full of young life, and her spirits as
buoyant as the opening spring. It was just four years since the last
visit of Uncle Thomas to the city--four years since he had looked upon
the fair face of his beautiful niece. Greatly had she changed in that
time. When last he kissed her blushing cheek, she was a half-grown
school-girl--now she burst upon him a lovely and accomplished young
woman.
But Uncle Thomas did not fail to observe in his niece certain signs,
that he understood too well as indications of a frail and susceptible
constitution. Two lovely sisters, who had grown up by his side, their
charms expanding like summer's sweetest flowers, had, all at once,
drooped, faded, withered, and died. Long years had they been at rest;
but their memory was still green in his heart. When he looked upon the
pure face of his niece, it seemed to Uncle Thomas as if a long-lost
sister were restored to him in the freshness and beauty of her young and
happy life ere the breath of the destroyer was upon her. No wonder that
he felt concern when he thought of the past. No wonder that he made
remonstrance against her exposure, in thin shoes, to cold and damp
pavements. But Lizzy had no fear. She understood not how fatal a
predisposition lurked in her bosom.
The calls were made; the Art-Union Gallery visited, and then Uncle
Thomas and his niece returned home. But the enjoyment of the former had
only been partial; for he could think of little else, and see little
else, besides Lizzy's thin shoes and the damp pavements.
The difficulty of crossing the streets, without stepping into the water,
was very great; and, in spite of every precaution, Lizzy's feet dipped
several times into little pools of ice-water, that instantly penetrated
the light materials of which her shoes were made. In consequence, she
had a slight hoarseness by the time she reached home, and Uncle Thomas
noticed that the colour on her cheeks was very much heightened.
"Now go and change your shoes and stockings, immediately," said he, as
soon as they entered the house. "Your feet must be thoroughly
saturated."
"Oh no, indeed they are not," replied Lizzy. "At the most, they are only
a little damp."
"A little damp!" said the old gentleman, seriously. "The grass waves
over many a fair young girl, who, but for damp feet, would now be a
source of joy to her friends."
"Why, uncle, how strangely you talk!" exclaimed Lizzy, becoming a little
serious in turn. Just then Mrs. Walton came in.
"Do, sister," said the old gentleman, "see that this thoughtless girl of
yours changes her wet stockings and shoes immediately. She smiles at my
concern."
"Why, Lizzy dear," interposed Mrs. Walton, "how can you be so imprudent!
Go and put on dry stockings at once."
Lizzy obeyed, and as she left the room, her uncle said--
"How can you permit that girl to go upon the street, in midwinter, with
shoes almost as thin as paper."
"Her shoes have thick soles," replied Mrs. Walton. "You certainly don't
think that I would let her wear thin shoes on a day like this."
Uncle Thomas was confounded. Thick shoes! French lasting, and soles of
the thickness of half-a-dollar!
"She ought to have leather boots, sister," said the old gentleman
earnestly. "Stout leather boots. Nothing less can be called a protection
for the feet in damp, wintry weather."
"Leather boots!"
Mrs. Walton seemed little less surprised than her daughter had been at
the same suggestion.
"It is a damp, cold day," said Uncle Thomas.
"True, but Lizzy was warmly clad. I am very particular on this point,
knowing the delicacy of her constitution. She never goes out in
winter-time without her furs."
"Furs for the neck and hands, and lasting shoes and thin cotton
stockings for the feet!"
"Thick-soled boots," said Mrs. Walton, quickly.
"There are thick-soled boots."
And the old gentleman thrust out both of his feet, well clad in heavy
calfskin.
Mrs. Walton could not keep from laughing, as the image of her daughter's
feet, thus encased, presented itself to her mind.
"Perhaps," said Uncle Thomas, just a little captiously, "Lizzy has a
stronger constitution than I have, and can bear a great deal more. For
my part, I would almost as lief take a small dose of poison as go out,
on a day like this, with nothing on my feet but thin cotton stockings
and lasting shoes."
"Boots," interposed Mrs. Walton.
"I call them boots," said the old gentleman, glancing down again at his
stout double-soled calfskins.
But it was of no avail that Uncle Thomas entered his protest against
thin shoes, when, in the estimation of city ladies, they were "thick."
And so, in due time, he saw his error and gave up the argument.
When Lizzy came down from her room, her colour was still high--much
higher than usual, and her voice, as she spoke, was a very little
veiled. But she was in fine spirits, and talked away merrily. Uncle
Thomas did not, however, fail to observe that every little while she
cleared her throat with a low _h-h-em_; and he knew that this was
occasioned by an increased secretion of mucus by the lining membrane of
the throat, consequent upon slight inflammation. The cause he attributed
to thin shoes and wet feet; and he was not far wrong. The warm boa and
muff were not sufficient safeguards for the throat when the feet were
exposed to cold and wet.
That evening, at tea-time, Mr. Walton observed that Lizzy eat scarcely
any thing, and that her face was a little pale. He also noted an
expression that indicated either mental or bodily suffering--not
severe, but enough to make itself visible.
"Are you not well?" he asked.
"Oh yes, very well," was the quick reply.
"You are fatigued, then?"
"A little."
"Go early to bed. A night's sleep will restore all."
Mr. Walton said this, rather because he hoped than believed that it
would be so.
"Oh yes. A night's rest is all I want," replied Lizzy.
But she erred in this.
"Where is Lizzy?" asked Mr. Walton, on meeting his sister-in-law at the
breakfast-table on the next morning. The face of the latter wore a sober
expression.
"Not very well, I am sorry to say," was the answer.
"What ails her?"
"She has taken a bad cold; I hardly know how--perhaps from getting her
feet wet yesterday; and is so hoarse this morning that she can scarcely
speak above a whisper."
"I feared as much," was the old gentleman's reply. "Have you sent for
your doctor?"
"Not yet."
"Then do so immediately. A constitution like her's will not bear the
shock of a bad cold, unless it is met instantly by appropriate
remedies."
In due time the family physician came. He looked serious when he saw the
condition of his patient.
"To what are you indebted for this?" he asked.
"To thin shoes," was the prompt reply of the uncle, who was present.
"I have warned you against this more than once," said the doctor, in a
tone of gentle reproof.
"Oh, no; brother is mistaken," spoke up Mrs. Walton. "She wore
thick-soled shoes. But the streets, as you know, were very wet
yesterday, and it was impossible to keep the feet dry."
"If she had worn good, stout, sensible leather boots, as she ought to
have done, the water would never have touched her feet," said Mr.
Walton.
"You had on your gums?" remarked the physician, turning to Lizzy.
"They are so clumsy and unsightly--I never like to wear them," answered
the patient, in a husky whisper, and then she coughed hoarsely.
The doctor made no reply to this, but looked more serious.
Medicine was prescribed and taken; and, for two weeks, the physician was
in daily attendance. The inflammation first attacked Lizzy's
throat--descended and lingered along the bronchial tubes, and finally
fixed itself upon her lungs. From this dangerous place it was not
dislodged, as an acute disease, until certain constitutional
predispositions had been aroused into activity. In fact, the latent
seeds of that fatal disease, known as consumption, were at this time
vivified. Dormant they might have lain for years--perhaps through
life--if all exciting causes had been shunned. Alas! the principle of
vitality was now awakened.
Slowly, very slowly, did strength return to the body of Miss Walton. Not
until the spring opened was she permitted to go forth into the open air.
Then her pale cheek, and slow, feeble steps, showed too plainly the
fearful shock her system had received.
A week or two after his remonstrance with his niece about her thin
shoes, Mr. Walton returned home. Several letters received by him during
the winter advised him of the state of Lizzy's health. In the spring her
mother wrote to him--
"Lizzy is much better. The warm weather, I trust, will completely
restore her."
But the old gentleman knew better. He had been a deeply interested party
in a case like her's before. He _knew_ that summer, with its warm and
fragrant airs, would not bring back the bloom to her cheeks. In July
came another epistle.
"The hot weather is so debilitating for Lizzy, that I am about taking
her to the sea-shore."
Uncle Thomas sighed as he read this, permitted the letter to droop from
before his eyes, and sat for some time gazing upon vacancy. Far back his
thoughts had wandered, and before the eyes of his mind was the frail,
fading form of a beloved sister, who had, years before, left her place
and her mission upon the earth, and passed up higher.
"The doctor says that I must go South with Lizzy," wrote Mrs. Walton
early in December, "and spend the winter. We leave for Charleston next
Tuesday, and may pass over to Havana."
Uncle Thomas sighed as before, and then became lost in a sad reverie. He
had been to Havana with both of his sisters. The warm South had been of
use to them. It prolonged, but did not save their lives.
And so the months passed on--the seasons came and went--but health,
alas! returned not to the veins of the lovely girl.
It was an autumn day, nearly two years after that fatal cold, taken in
consequence of wearing thin shoes, that Mr. Walton received a letter
sealed with a black seal.
"As I feared," he murmured, in a low, sad voice, gazing
half-abstractedly upon the missive. He knew too well its contents. "Dear
child! I saw this from the beginning."
And the old man's eyes became dim with moisture.
He had not erred in his conjecture. Lizzy Walton was dead.
THE UNRULY MEMBER.
"In trouble again, I find! Ah, Flora! That restless little tongue of
yours is a sad transgressor. Why will you not learn to be more careful?
Why do you not place a guard upon your lips, as well as upon your
actions?"
"So I do, aunt, when I think myself in the company of tattlers and
mischief-makers."
"I do not think Mary Lee either a tattler or a mischief-maker," replied
the aunt gravely.
"Then why did she run off to Ellen Gray, and tell her what I had said?"
"She might have done so from far different motives than those you are
inclined to attribute to her," said Mrs. Marion, the aunt of Flora Mere.
"And from my knowledge of her character, I feel very sure that her
conduct in this has been governed by a strict regard to right
principles."
"But what possible end could she have had in view in repeating to Ellen
my thoughtlessly spoken words? It could do her no good."
"There she is at the door now," Mrs. Marion replied, glancing out of the
window. "We will ask the question direct, as soon as Betty has admitted
her."
The blood mounted to Flora's cheeks as her aunt said this, and her own
eyes caught a glimpse of the young lady whose conduct she had been so
strongly condemning. The aunt and her niece sat silent until Mary Lee
entered.
Here we will take the opportunity to mention the cause of the unpleasant
state of affairs between Flora and her young friend. On the day before,
while in company with Mary Lee, and one or two other of her
acquaintances, she very thoughtlessly and not exactly in the right
spirit, repeated some remarks she had heard about Ellen Gray that
reflected upon her rather unfavourably. Mary Lee at once attempted to
vindicate her friend, but Flora maintained that the allegations were
certainly true, for she had them from an undoubted source. Mary asked
that source, but she declined mentioning it, on the ground that she did
not wish to violate the confidence reposed in her by the individual who
related the facts she had repeated.
"It would, perhaps, be better not to mention any thing of this kind,"
said Mary Lee, "unless the author be given, and full liberty, at the
same time, to make the most free inquiries as to the truth of what is
alleged."
"And get up to your ears in hot water," returned Flora, tossing her
head.
"Even that would be better than to let any one suffer from an untrue
statement."
"Ah! But suppose it should be true?"
"Let the guilt rest upon the right head--where it ought to rest. But
save the innocent from unjust allegations. That is my doctrine."
"A very good doctrine, no doubt," Flora returned; "if you can act it
out."
Here the subject was dropped. On the next morning, Mary Lee called in to
see her young friend Ellen Gray. After conversing for a short time she
said--
"I heard, yesterday, Ellen, that at Mrs. Harvey's party, you acted
towards Mr. Evelyn with much discourtesy of manner, besides actually
telling an untruth."
"I am unconscious of having done either the one or the other of these,"
Ellen replied, in a quiet tone.
"I believed you innocent," said Mary, with a brightening countenance.
"But what ground is there for the idle, ill-natured gossip that has got
on the wind?"
"Not much, if any. I declined dancing with Evelyn, as I had a perfect
right to do."
"Did you tell him you were engaged for the next cotillion?"
"No, certainly not, for I had no engagement then."
"It is said that when he asked you to dance, you excused yourself on the
plea that you were already engaged."
"Who says this?"
"Flora Mere."
"How does she know?"
"That I cannot tell. She declined giving her authority."
"Then, of course, I must believe her the author of the fabrication."
"No--that does not certainly follow. I do not believe Flora would be
guilty of such a thing. But, like too many, she is ready to believe
another capable of doing almost any thing that may happen to be alleged.
And like the same class of persons, too ready to repeat what she has
heard, no matter how injuriously it may affect the subject of the
allegation--while a false principle of honour prevents the open
declaration of the source from which the information has been derived."
"Be that as it may, I shall see Flora Mere at once, and ask her for the
authority upon which the statement rests."
"It was to give you an opportunity of doing this, that I have come and
freely told what I heard."
"Thank you, Mary. I wish all the world were as frank and as
conscientious as you are. I shall, of course, mention from whom I
derived my information."
"You are at perfect liberty to do so. I try never to say or do any thing
that requires concealment."
It was, perhaps, an hour afterward, that Flora Mere was surprised by a
visit from Ellen Gray. She had an instinctive consciousness of the cause
of this visit, which made the blood mount to her face, as she took the
hand of her friend. She was not long in doubt.
"Flora," said Ellen, a few minutes after she had entered. "Mary Lee came
in to see me this morning, and mentioned that you had made statements
about me which are not true--as that I refused to dance with Mr. Evelyn
under the plea of a prior engagement, when, in fact, no such engagement
existed."
"I think Mary Lee had very little to do!" Flora returned petulantly, the
colour deepening on her face and brow, "to tattle about what she hears
in company."
"But reflect," said Ellen, mildly, "that the charge against me was one
of falsehood--no light charge--and that Mary had every reason to believe
me incapable of uttering what was not true. And further, remember, that
you declined giving your informant, so as to place it in her power to
ascertain upon what basis the statement rested. Reverse the case.
Suppose I had heard that you had done some wrong act; and, instead of
carefully satisfying myself whether it were really so or not, were to
begin circulating the story wherever I went. Would you not deem her a
true friend, who, instead of joining in the general condemnation, were
to come to you and put into your power to vindicate your character?
Certainly you would. Just in the relation which that true friend would,
under the imagined circumstances, stand to you, now stands Mary Lee to
me. She has put into my power to arrest a report which I find is
circulating to my injury. It is true that I declined dancing with Mr.
Evelyn. But it is not true that I stated to him that I was engaged. I
was not engaged, and to have said that I was, would have been to have
told a deliberate falsehood. May I, then, ask you from what source you
derived your information?"
Flora cast her eyes upon the floor, and sat silent for some time. Her
pride struggled hard with her sense of justice. At length she said,
looking up, and breathing heavily--
"I would rather not mention my informant, Ellen. It will only make
difficulty. You will go to her, and then there will be trouble. I think
you had better let the matter rest where it is. I do not, now, believe
what I heard. The person who told me, was, no doubt, mistaken."
"But, Flora, that would not be right. You have already repeated what you
heard so publicly, that it is possible at least fifty persons now
believe me guilty of having spoken an untruth. You should have reflected
beforehand. Now it is too late to let the matter drop. My character is
at stake, and I am bound to vindicate it. This I shall have to do in
such a manner as to fully clear myself from the charge. The consequence
will be, as you may at once perceive, that upon you will rest the burden
of having originated a false charge against me. Then, if not now, you
will feel it your duty to give the name of your friend. This, you had
much better do at once. No doubt she has been led into a mistake by a
too hasty judgment of my acts, but half understood. She may have
observed Mr. Evelyn ask me to dance, and have naturally inferred that I
declined on the ground of a previous engagement. This being in her mind,
she may have too hastily concluded, when she soon afterwards saw me
accept another offer, that I had not spoken the truth at the time I
refused to dance with Evelyn. All this can easily be explained, and the
matter put to rest."
Flora hesitated for a short time, and then said--
"It was Araminta Thomas who told me."
"Thank you for this information. Will you now go with me to see
Araminta?"
"I would rather not," Flora returned.
"I think it would be better for you to do so, Flora," urged Ellen. But
she could not be persuaded.
"I must then go alone," said Ellen, rising and bidding Flora good
morning.
In a little while she was at the house of Araminta Thomas. Ellen entered
at once upon the business of her visit, by stating what she had heard.
Araminta looked confused, but denied saying that Ellen had actually
told Evelyn she was engaged for the next cotillion.
"Then what did you say?" mildly asked Ellen.
"I said," replied Araminta, "that I saw you decline Evelyn's offer for
your hand."
"But did not say that I told him I was engaged?"
"_Not positively_; I only _inferred_, as was natural, that you declined
on that ground."
"Was your communication to Flora mere inferential?"
"It was."
"But she says you told her that you heard me say I was engaged."
"In that she is mistaken. I inferred that your refusal to dance was for
the reason stated. But I did not _know_ that it was, and, therefore only
gave my own impression."
"Which Flora has taken for the truth, and so repeated."
"On my authority?"
"Yes. After having been pressed by me very closely."
"In that she was wrong. But I suppose I was as wrong in giving an
impression which might not be a true one, as she has been in giving my
impressions as actual facts, and making me responsible for them. But
will you, as matters have taken this serious and unexpected turn, give
me the exact truth. I will then, so far as in my power lies, endeavour
to correct what I have done."
"Most cheerfully. You know as well as I do, that Evelyn has not acted in
some things with that honour and integrity that becomes a gentleman?"
"I do."
"It was on this ground that I declined. He asked me if I was engaged in
the next set? I said no. He then proffered his hand, which I declined.
In a little while after, and while sitting beside you, a gentleman
wished to have me as a partner. I accepted his invitation. This is the
simple truth."
"And so it seems," said Araminta with a sober face, "that while you were
rebuking vice, and standing up with dignified, virtuous firmness in the
cause of our sex, I was misjudging you. And not only that, was so far
influenced by an improper spirit as to impart to others my wrong
impressions to your injury. Alas! poor, weak human nature! I feel
rebuked and humbled. More for what I thought than for what I said, for
out of the heart proceedeth evil thoughts. If I had not had something
wrong here, I would not have been so ready to misjudge you. But all that
I can do to repair the wrong, I am ready to do."
"All I ask is, that you correct Flora, and take some little care, that,
where she has imparted a wrong impression, the true one is given in its
place."
"That I will do with all my heart," Araminta replied. "I will see Flora
this very hour."
"Do so, and you shall have not only my thanks, but my esteem and love.
We are all liable to do wrong. But to confess and repair the wrong we
have done, as far as we can, is noble. In so doing, power is given us to
conquer in all the temptations that may assail us."
As soon as Ellen had retired, Araminta went out and called upon Flora.
She found her troubled and mortified at the turn matters had taken. She
tried to excuse herself for what she had done, and insisted, at first,
that Araminta had actually stated all she had said of Ellen Gray's
conduct. But this point she soon had to give up. Araminta was too
positive, and her own memory a little too clear on the subject. In fact,
when the whole truth came fully to the light, it was very apparent, that
if there were any falsehood in the matter, she was the most guilty.
Certain it was, that Ellen Gray was innocent, in every particular, of
the charge that had been made against her.
Mrs. Marion knew nothing of all this, until the day after Ellen Gray had
called upon Flora. Then her niece, whose troubled looks had not escaped
her notice, gave a relation of what had occurred. It was in reply to
this that the opening remarks of our story were made. When Mary Lee came
in, as the reader has seen, Flora received her coldly. Mrs. Marion, on
the contrary, welcomed her with genuine cordiality.
"I am glad to see you, Mary," she said--"and particularly at this time.
It seems there has been a misunderstanding among you young ladies, and
that Flora is not altogether pleased with the part you have taken."
"It is to see her in regard to that very matter that I am here this
morning," Mary said. "I know she blames me for having told Ellen Lee
what I did. But in that I acted conscientiously. I did to another as I
would have another do to me. I acted towards Ellen as I would act
towards Flora, were I to hear any one making statements that were
calculated to injure her. The result, I think, should satisfy Flora that
I was right in doing what I have done. Ellen, it now appears, was
entirely innocent of the charge made against her--as I knew she must be.
Araminta Thomas, to whom the report has been traced, regrets extremely,
that upon her hasty inferences, so serious a matter has grown up. She
acknowledged that she only _inferred_ that Ellen told an untruth. Flora
took this inference for a direct assertion, and thence came the charge
of falsehood against Ellen Gray. Has not, then, the result proved that
the course I took was the only right one? Does it not show that I would
have been guilty of a great wrong, if, to save the feelings of any one,
I had left an innocent person to bear the imputation of wrong?"
"It certainly does, Mary. And Flora cannot but see it in the same
light."
"And she will, surely, forgive me the pain I have occasioned her,"
resumed Mary, "seeing that I had no selfish end to gain in what I did,
but was moved only by the desire to vindicate injured innocence."
This appeal softened Flora's feelings toward Mary Lee. She saw that she
was wrong, and that Mary was right. Mary had been governed by a
high-minded regard for right. Pride soon yielded.
"Mary," said she, taking her hand, while the tears came into her eyes,
"I confess that I have been wrong, and you right. I shall not soon
forget this lesson. Forgive the unkind thought I have had of you, and
say to Ellen, from me, that I do most sincerely regret the part I have
taken in this matter."
"Will I ever learn to be guarded in my remarks!" Flora said, to her
aunt, after Mary had left them. "This is the third time I have been
called to account for speaking of others, within the last few months."
"Never, I suppose," Mrs. Marion replied, "until you learn to guard your
thoughts as well as your words. If, like Mary Lee, you were less
disposed to give credence to every disparaging report circulated about
others, you would need no guard placed over your tongue. It is from the
abundance of the heart that the mouth speaketh. _A good man, out of the
good treasure of his heart, bringeth forth good things: and an evil man,
out of the evil treasure, bringeth forth evil things._ Try and keep this
in mind. If you are more ready to believe an evil than a good report of
others, be sure that all is not right with you, and more especially, if
you feel an inward pleasure in convicting them of wrong. A truly good
mind is always grieved at improper conduct in others, and ever seeks to
palliate, rather than to judge with severity. It gives but slow credence
to evil reports. Truly regard the good of all around you, and there will
be no need of placing a bridle on your tongue."
THE RICH AND THE POOR.
A hot and sultry summer had passed away, and autumn was verging on
toward its cooler months, with their long and quiet evenings.
Occasionally a colder day than usual made a fire in the grate necessary
and drew closer together the happy family of Mr. Barton in their evening
circle. It was pleasant to all, thus to feel the warm fire again, and to
see its deep glow reflected from loving faces.
"How good the fire feels!" said James, holding up his small hands to
receive its heat, and smiling as he looked upon it.
"I think I love the winter best after all," remarked William. "It is so
pleasant to sit round the fire, and feel its warmth upon our hands and
face. Home feels more like home. Don't you think so, father?"
"The change of season is always pleasant," replied Mr. Barton. "Have you
never noticed that, my son?"
"Oh yes! I always say, when spring comes, 'I am glad that it is spring.'
And in summer-time, when fruit and flowers are so plenty, I say, 'I am
glad it is summer.' And then I am glad again when the doors and windows
can be closed, and we can all gather around the fire as we do now in
autumn. In winter, when the snow begins to fall, I feel that it is
pleasant to see the light flakes flying about gayly in the air."
"But I always think then," said Mary, the gentle, loving-hearted Mary,
"of the poor children who have no warm clothing, nor good fires, as we
have. I wish, sometimes, that it were always warm, for their sakes."
"And yet, my dear, the Lord knows what is best," remarked Mr. Barton,
looking into Mary's sympathizing face. "The Bible says He is good to
all, and kind even to the unthankful."
"I know it does; and it also says, that He pitieth us even as a father
pitieth his children. But, I can't help thinking, sometimes, that there
is a great deal of suffering in the world."
"And so there is, Mary, a great deal of dreadful suffering, the reason
for which we sometimes find it very hard to understand. But one thing we
know, and this is, that it is all from man, and not from God; and that
God permits it for some good purpose--not to punish people; for the Lord
never punishes any one merely for the sake of punishment, but suffers
evil and sin to punish for the sake of reformation. You remember what I
read to you about the Divine Providence on last Sunday evening?"
"Yes, sir."
"What did I say the Divine Providence regarded?"
"Eternal ends," replied Mary.
"Do you remember what I then told you was meant by eternal ends?"
"Whatsoever had reference to man's salvation in heaven."
"Yes, that is what I said. A great many people believe that the Lord's
Providence, which is over us all, even to the smallest things, has
reference to our worldly well-doing. I remember when a boy, hearing a
man pray, regularly, in his family, every day, and a part of his prayer
always was, that the Lord would increase his basket and his store."
"What did he mean by that?" asked James, who was listening very
attentively to his father, and trying to understand all he said.
"Why, that the Lord would make him rich."
"Did the Lord make him rich?" asked Mary.
"No, my daughter, the Lord knew that to make him rich would be the worst
thing for him, for it might be the means of destroying his soul."
"Then it is best for some to be rich and some poor?" said William.
"Undoubtedly it is, or all would be rich in this world's goods, and have
every comfort and luxury that earth could afford them. For the goodness
of the Lord would seek to bless every one in good things for the body as
well as good things for the mind, if the former blessings could be given
without injury to the latter. But where they cannot, they are always
withheld."
"But all rich people are not good people," remarked William. "I think
they are, generally, more unfeeling and selfish than poor people. I have
often heard it said so; and that there was very little chance of rich
people's going to heaven."
"I know this is said, but it is a great mistake. Poor people are, as a
general thing, just as unfeeling and selfish as rich people, and stand
no better chance of heaven. So far as poverty or riches are concerned,
there is an overruling Providence regarding each, and this, as I before
remarked, looks to the salvation of souls in heaven."
"Then it isn't because one man is better than another, that he is
permitted to get rich, or has money left to him?"
"Not by any means, William," replied the father. "No man's state can be
judged of by his external condition: for the external condition that is
good for one, may be very bad for another. Ever bear this in mind, as
you pass through life, and learn, no matter in what external condition
the Lord places you, therewith to be content."
*** | {
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A Niche for Marilyn
Miguel Anxo Fernández
Title: A Niche for Marilyn
Author: Miguel Anxo Fernández
Original in: Galician
Availability: A Niche for Marilyn - US
A Niche for Marilyn - UK
A Niche for Marilyn - Canada
Un nicho para Marilyn - España (Gallego)
Galician title: Un nicho para Marilyn
Translated by Kathleen March
Premio García Barros, 2002
B- : solid, standard PI novel, but too uneven
A Niche for Marilyn is the first in a series of novels by Fernández featuring Los Angeles private detective (with Galician roots) Frank Soutelo, "a detective who doesn't shoot much, doesn't smoke and gets off topic a lot talking about films and cheap literature". There's a touch of the Galician here, but it's pretty much just at the edges, as Fernández opts to go all in in Raymond Chandler-territory -- complete with a case that involves the remains of Marilyn Monroe.
Soutelo recounts much of the story, but some of the chapters have an omniscient narrator, giving readers information and perspectives not available to the PI -- such as that: "The detective was only supposed to discover where Marilyn's body was and the rest would be taken care of by others". It's an interesting approach, separating the book from your standard PI procedural though arguably also revealing too much too easily. It's as if Fernández didn't trust readers to only discover along with Soutelo how he's been hoodwinked into something bigger and stranger than he might have first believed, or, in another instance, to make clear secretary Pat's deeper interest in her detective boss. But Fernández also undermines some of the potential usefulness of the technique by, for example, preparing readers for the fact -- or reassuring us -- that Soutelo is a pro after all and isn't quite as gullible as his client might have hoped:
Nevertheless, Tara Colbert had definitely underestimated Soutelo's abilities when she gauged him to be just a fellow who needed to make a living.
Fernández also opts for a pretty sensational case. Wealthy, wheelchair-bound recluse Tara Colbert hires Soutelo. She claims to have been a friend of Marilyn Monroe, and to have taken it upon herself to look after the movie star's grave; recently: "I got a message informing me that Marilyn's grave was empty". And she offers Soutelo a whole lot of money to find it.
Apparently there are people who are, for various reasons, interested in the remains of the dead, and Soutelo finds himself immersed in the word of necrophilia -- though presumably the ... enjoyment anyone could get out of a decades-old corpse is of a somewhat different nature than that the recently deceased might offer.
Soutelo begins by trying:
to get my head around the world of necrophilia and all the abnormal facets of this activity, the one most closely related to the urge to remove a dead body from its eternal place of rest.
As a former policeman, he still has some contacts in the department, and a friend is able to get him up to speed on the big local players in the necrophilia game -- wealthy folk able to devote a lot of resources to their hobby. Like Tara Colbert.
Soutelo sniffs around -- which, unsurprisingly, the necrophiliac-enthusiasts are at the very least concerned about -- and eventually settles on an elaborate plot to get Colbert what she wants -- sort of.
A Niche for Marilyn follows the basic hardboiled PI novel template, from its protagonist -- a fairly cultured loner -- to the secretary who is in love with him, the creepy client with some ulterior motives she doesn't share with him, the suspicious other parties who want to get in on the action (or scare Soutelo off). Soutelo is tailed much of the time, and there are some unwanted confrontations. And lots of things (and some people) aren't quite what they seem.
Fernández handles most of this reasonably well, but A Niche for Marilyn is still a bit unbalanced. Indulgent scenes of the food Soutelo enjoys (from the old country ...) are fine, but the case has to be the focus, and Fernández juggles this unevenly. While there are some quite raw descriptions of necrophiliac activities, the bizarre basic situation Soutelo is confronted with -- determining the location of Marilyn Monroe's remains -- isn't considered too closely. And while the sensational denouement, revealing Colbert's plans, is certainly spectacular, it is a bit much of a finale to append to the otherwise underdeveloped case. Soutelo's own ambivalence -- he always tries to learn as much background about what's involved in a case as possible, but here dismissively doesn't want to bother (because anyone involved in this sort of thing was obviously; "crazy or a total degenerate, a real loser") -- carries over to the novel as a whole.
A Niche for Marilyn tries a bit too hard to follow the PI-template with its characters and atmosphere and the strangeness of the case at its heart. It goes through a lot of the right motions, but in particular the case -- and the necrophiliacs -- are underdeveloped, and then also too easily dealt with. The presentation is quite good -- it reads well enough (some unpleasant necrophiliac examples aside) -- and while the chapters that aren't told from Soutelo's perspective don't contribute quite enough, they make for interesting side-views of the action.
A Niche for Marilyn does lay a decent foundation for the series. Soutelo is a solid lead, and Fernández does recreate the Chandler-feel of Los Angeles well enough. It would be interesting to see where the series went from here.
- M.A.Orthofer, 13 January 2017
A Niche for Marilyn:
Small Stations Press publicity page
Editorial Galaxia publicity page
Palabra de Gatsby (Galician)
Gabrielle Wittkop's The Necrophiliac
See Index of Spanish and Galician literature
Galician-writing author Miguel Anxo Fernández was born in 1955.
© 2017 the complete review | {
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{"url":"https:\/\/optimization-online.org\/tag\/permanent\/","text":"## Conic Geometric Programming\n\nWe introduce and study conic geometric programs (CGPs), which are convex optimization problems that unify geometric programs (GPs) and conic optimization problems such as linear programs (LPs) and semidefinite programs (SDPs). A CGP consists of a linear objective function that is to be minimized subject to affine constraints, convex conic constraints, and upper bound constraints \u2026 Read more\n\n## Unharnessing the power of Schrijver\u2019s permanental inequality\n\nLet $A \\in \\Omega_n$ be doubly-stochastic $n \\times n$ matrix. Alexander Schrijver proved in 1998 the following remarkable inequality $$\\label{le} per(\\widetilde{A}) \\geq \\prod_{1 \\leq i,j \\leq n} (1- A(i,j)); \\widetilde{A}(i,j) =: A(i,j)(1-A(i,j)), 1 \\leq i,j \\leq n$$ We prove in this paper the following generalization (or just clever reformulation) of (\\ref{le}):\\\\ For all \u2026 Read more\n\n## Hyperbolic Polynomials Approach to Van der Waerden\/Schrijver-Valiant like Conjectures :\\\n\nThe paper describes various combinatorial and algorithmic applications of hyperbolic (multivariate) polynomials . Section 2.2 introduces a new class of polynomials , which include as hyperbolic polynomials as well volume polynomials $Vol(x_1C_1+\u2026+x_nC_n)$ , where $C_i$ are convex compact subsets of $R^n$. This extension leads to randomized poly-time algorithm to approximate $M(C_1,\u2026,C_n)$ (the mixed volume) within \u2026 Read more","date":"2022-12-02 07:19:27","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\": 0.9348980188369751, \"perplexity\": 1245.2298264659016}, \"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\/1669446710898.93\/warc\/CC-MAIN-20221202050510-20221202080510-00012.warc.gz\"}"} | null | null |
\section{Introduction}
The extraction of the thermodynamic properties of QCD is one of the major goals
of relativistic heavy-ion collision experiments.
The phenomenological thermodynamic models are useful in this regard and have long been employed to estimate the temperatures reached in the relativistic heavy-ion collisions~\cite{SOG1981, Molitoris1985,Stoecker1986,Hahn1987}.
The thermal parameters at chemical freeze-out -- the stage of heavy-ion collision when inelastic reactions between hadrons cease -- have been extracted by fitting the rich data on hadron yields in various experiments,
ranging from the low energies at SchwerIonen-Synchrotron (SIS) to the highest energy of
the Large Hadron Collider
(LHC), within the hadron resonance gas (HRG) model~\cite{CleymansSatz,CleymansRedlich1998,CleymansRedlich1999,
Becattini2001,Becattini2004,ABS2006,ABS2009}.
It has been argued~\cite{DMB}, that the
inclusion of all known resonances into the model as free non-interacting (point-like) particles allows to
effectively model the attraction between hadrons.
Such formulation, a multi-component point-particle gas of all known hadrons and resonances, is presently the most commonly used one in the thermal model analysis.
HRG models which take into account the
repulsive interactions between hadrons are also available.
The HRG model with repulsive interactions have been compared with the lattice QCD data~\cite{EV-latt-1,EV-latt-2,EV-latt-3}, and
it has recently been shown in Ref.~\cite{VS2015} that the inclusion of the repulsive interactions into the HRG model in the form of a multi-component eigenvolume procedure can significantly change the values of chemical freeze-out temperature at $\mu_B=0$ while improving the agreement of the fit with the ALICE hadron yield data as compared to the point-particle HRG.
The analysis reveals a surprisingly strong sensitivity of the thermal fits at LHC to the modeling of eigenvolume interactions, leading to a large systematic uncertainty in the determination of chemical freeze-out temperature.
In the present work we perform a similar analysis at the finite (baryo)chemical potential by considering the data on hadron yields in Pb+Pb and Au+Au collisions of NA49 and STAR collaborations.
In order to study the sensitivity of the obtained results we use two different formulations of the multi-component eigenvolume HRG.
\section{Eigenvolume models}
The repulsive interactions between hadrons can be modeled by the eigenvolume correction of the van der Waals type, whereby the volume available for hadrons to move in is reduced by their eigenvolumes. Such a correction was included into the hadronic equation of state first in Refs.~\cite{HagedornEV1,Gorenstein1981,Kapusta1983}, while the thermodynamically consistent procedure for a single-component gas was formulated in Ref.~\cite{Rischke1991}.
In the present work we consider hadrons of different sizes, thus, a multi-component formulation of the eigenvolume HRG model is necessary. In our study we use two different formulations. Since both of these formulations are rigorously derived only for the case of Boltzmann statistics we will neglect the effects of quantum statistics
in the present work.
\subsection{``Diagonal'' eigenvolume model}
The single-component eigenvolume model of Ref.~\cite{Rischke1991} was generalized
to the multi-component case in Ref.~\cite{Yen1997}. It was assumed that the available volume of each of the hadron species is the same, and equals to the total volume minus the sum of the eigenvolumes of all hadrons in the system.
Let us assume that we have $f$ different hadron species.
The pressure as function of the temperature and hadron densities has the following form
\eq{\label{eq:Pex1n}
P(T,n_1,\ldots, n_f) = T \sum_i \frac{n_i}{1 - \sum_j v_j n_j},
}
where the sum goes over all hadrons and resonances included in the model,
and where $v_i$ is the
eigenvolume parameter of hadron species $i$.
The eigenvolume parameter $v_i$ can be identified with
the 2nd virial coefficient of the single-component gas of hard spheres and
is connected to the effective hard-core hadron radius as
$v_i = 4 \cdot 4 \pi r_i^3 / 3$.
In the grand canonical ensemble (GCE) one has to solve the non-linear equation for the pressure, which reads as
\eq{\label{eq:Pex1}
P(T, \mu) = \sum_{i} \, P^{\rm id}_i (T, \mu_i^*),
}
where $P^{\rm id}_i (T, \mu_i^*)$ is the pressure of the ideal (point-like)
gas at the corresponding temperature and chemical potential, and
$\mu_i^* = \mu_i - v_i \, P(T, \mu)$ is the shifted chemical potential
due to the eigenvolume interactions. The $v_i$ is the eigenvolume parameter of the hadron species $i$, and the number density of these species can be calculated as
\eq{\label{eq:nex1}
n_i(T, \mu) = \frac{n_i^{\rm id} (T, \mu_i^*)}{1 + \sum_j v_j n_j^{\rm id} (T, \mu_j^*)}.
}
The multi-component eigenvolume HRG model given by Eqs.~\eqref{eq:Pex1n}-\eqref{eq:nex1} is the most commonly used one in the thermal model analysis.
Since this model does not describe properly the cross-terms in the virial expansion of the
multi-component gas of hard spheres (see details below) we will refer to it in this work as the ``Diagonal'' model.
\subsection{``Crossterms'' eigenvolume model}
The virial expansion of the classical (Boltzmann) multi-component gas of hard spheres up to 2nd order can be written as~\cite{LL}
\eq{\label{eq:virial}
P(T,n_1,\ldots, n_f) = T \sum_i n_i + T \sum_{ij} b_{ij} n_i n_j + \ldots,
}
where
\eq{\label{eq:bij}
b_{ij} = \frac{2 \pi}{3} \, (r_i + r_j)^3
}
are the components of the symmetric matrix of the 2nd virial coefficients.
Comparing Eqs.~\eqref{eq:Pex1n} and \eqref{eq:virial} one can see that the ``Diagonal'' model is not consistent with the virial expansion of the multi-component gas of hard spheres up to 2nd order and corresponds to a different matrix of 2nd virial coefficients, namely $b_{ij} = v_i$.
While we do not require hadrons to be non-deformable spherical objects and expect that the ``Diagonal'' model to capture the essential features of a system of particles with different sizes, the interpretation of $r_i$ as a hard-core hadron radius can be problematic in such model. Therefore, we additionally consider the
multi-component eigenvolume model of Ref.~\cite{GKK}, which is formulated in the grand canonical ensemble (GCE) assuming Boltzmann statistics, and which is consistent with the 2nd order virial expansion in Eq.~\eqref{eq:virial}. The pressure in this model reads as
\eq{\label{eq:Pex2n}
P(T,n_1,\ldots, n_f) = \sum_i P_i = T \sum_i \frac{n_i}{1 - \sum_j \tilde{b}_{ji} n_j},
}
where
\eq{
\tilde{b}_{ij} = \frac{2\,b_{ii}\,b_{ij}}{b_{ii}+b_{jj}}
}
with $b_{ij}$ given by \eqref{eq:bij},
and where the quantities $P_i$ can be regarded as ``partial'' pressures.
This eigenvolume model given by \eqref{eq:Pex2n} is initially formulated in the canonical ensemble.
In Ref.~\cite{GKK} it was transformed to the grand canonical ensemble.
In the GCE formulation one has to solve the following system of non-linear
equations for $P_i$
\eq{\label{eq:Pex2pi}
P_i = P_i^{\rm id} \left(T, \mu_i - \sum_j \tilde{b}_{ij} \, P_j \right), \qquad i = 1, \ldots , f,
}
where $f$ is the total number of the hadronic components in the model.
Hadronic densities $n_i$ can then be recovered by solving the system of linear equations connecting $n_i$ and $P_i$:
\eq{\label{eq:Pex2ni}
T n_i + P_i \sum_j \tilde{b}_{ji} n_j = P_i, \qquad i = 1, \ldots , f~.
}
We refer to the model given by Eqs.~\eqref{eq:Pex2n}-\eqref{eq:Pex2ni} as the ``Crossterms'' eigenvolume model. We note that this ``Crossterms'' model is very similar to the one used in Ref.~\cite{BugaevEV}.
From the technical point of view, the ``Crossterms'' model is more complicated than the ``Diagonal'' model: a set of coupled non-linear equations~\eqref{eq:Pex2pi} needs to be solved, instead of a single equation~\eqref{eq:Pex1} for the total pressure in the ``Diagonal'' model. In practice, the solution to \eqref{eq:Pex2pi} can be obtained by using an appropriate iterative procedure. In our calculations Broyden's method~\cite{Broyden} is employed to obtain the solution of the ``Crossterms'' model, using the corresponding solution of the ``Diagonal'' model as the initial guess.
\section{Calculation results}
\subsection{Some details about model implementation}
In our calculations we include strange and non-strange hadrons listed in the Particle Data Tables~\cite{pdg}, along with their decay branching ratios. This includes mesons up to $f_2(2340)$, (anti)baryons up to $N(2600)$.
We do not include hadrons with charm and bottom degrees of freedom which have a negligible
effect on the fit results, and we also removed the $\sigma$ meson ($f_0(500)$) and
the $\kappa$ meson ($K_0^*(800)$) from the particle list because
of the reasons explained in
Refs.~\cite{Broniowski:2015oha,Pelaez:2015qba}.
We also omit the light nuclei.
The finite width of the resonances is taken into account in the usual way, by adding the additional integration over their Breit-Wigner shapes in the point-particle gas expressions.
The feed-down from the decays of unstable resonances to the total hadron yields is included in the standard way.
There are three conserved charges in the system: baryon charge, electric charge, and strangeness. Therefore, there are three corresponding independent chemical potentials: $\mu_B$, $\mu_Q$, and $\mu_S$. The chemical potential of the $i$th hadron species is thus determined as $\mu_i = B_i \mu_B + Q_i \mu_Q + S_i \mu_S$.
At each fixed temperature $T$ and baryochemical potential $\mu_B$, the $\mu_Q$ and $\mu_S$ are determined in a unique way in order to satisfy two conditions: the electric-to-baryon charge ratio of $Q/B = 0.4$, and the vanishing net strangeness. Assuming no pre-freezeout radiation, both of these conditions are relevant for the system created in the collision of heavy ions.
\subsection{Eigenvolume parametrizations}
As was mentioned before, the inclusion of the eigenvolume interactions is one of the most popular extensions of the standard HRG model.
In most of the analyses dealing with chemical freeze-out, which did include
the eigenvolume corrections~\cite{ABS2006,PHS1999,Cleymans2006}, it was assumed that all the hadrons have the same eigenvolume.
It has been established that, in this case, the eigenvolume corrections can significantly reduce the densities~\cite{Begun2013,mf-2014}, and, thus, increase the total system volume at the freeze-out as compared to the point-particle gas at the same temperature and same chemical potential.
For this parametrization, however, the eigenvolume corrections essentially cancel out in the ratios of yields and, thus, have a negligible effect on the values of the extracted chemical freeze-out temperatures and chemical potentials.
If, however, one considers hadrons with different hard-core radii, then the ratios may change, and the fit quality can be improved~\cite{VS2015,Gorenstein1999,BugaevEV}.
In our paper we use
a bag-model inspired parametrization of hadron eigenvolume. In this case the hadron eigenvolume is proportional to its mass through a bag-like constant, i.e.
\eq{\label{eq:BagEV}
v_i = m_i / \varepsilon_0 .
}
This kind of eigenvolume parametrization had been obtained for the heavy Hagedorn resonances, and was used to describe their thermodynamical properties~\cite{HagedornEV1,Kapusta1983} as well as their
effect on particle yield ratios~\cite{NoronhaHostler2009}.
It was mentioned in the Ref.~\cite{Becattini2006} that such parametrization would lead to an increase of the freeze-out temperature, but that it does not entail an improvement of the fit quality in the ``Diagonal'' EV model, nor does it change other fit parameters.
Note that the eigenvolume for the resonances with the finite width is assumed to be constant for each resonance, and is determined by its pole mass.
\begin{figure*}[!t]
\centering
\includegraphics[width=0.90\textwidth]{chi2-Tdep-unconstrained-oner}
\caption[]{(Color online)
The temperature dependence of $\chi^2 / N_{\rm dof}$ of fits to data of the NA49, STAR, and ALICE collaborations on hadron yields in central Pb+Pb and Au+Au collisions within the point-particle HRG model (solid black curves), the ``Diagonal'' eigenvolume model (thin dotted blue lines),
and the ``Crossterms'' eigenvolume model (thick dotted blue lines).
The constant $\varepsilon_0$ is fixed in order to reproduce the effective hard-core proton radius of
0.5 fm.
}\label{fig:chi2-vs-T}
\end{figure*}
\begin{figure}[!t]
\centering
\includegraphics[width=0.49\textwidth]{NA49-40-chi2-Tdep}
\caption[]{(Color online)
The temperature dependence of $\chi^2 / N_{\rm dof}$ of fits to data of the NA49 Collaboration on hadron yields in central Pb+Pb collisions at $\sqrt{s_{_{\rm NN}}} = 8.8$~GeV within the point-particle HRG model (solid black curve), the ``Diagonal'' eigenvolume model (thin colored lines),
and the ``Crossterms'' eigenvolume model (differently styled colored lines).
The constant $\varepsilon_0$ is fixed in order to reproduce the effective hard-core proton radius of
0.4,
0.5, and 0.6 fm.
}\label{fig:chi2-vs-T-40}
\end{figure}
Presently not much is well-established about the eigenvolumes of different hadron species, and
it is debatable whether
parametrization \eqref{eq:BagEV} is the most realistic one. For instance, it can be argued, that strange hadrons should have a different (smaller) eigenvolume compared to non-strange ones.
In general, the eigenvolumes might also depend on temperature and chemical potential.
In our study we would like to minimize the number
of the additional free parameters due to the introduction of the finite eigenvolumes and we use
\eqref{eq:BagEV} as a simple parametrization of hadron eigenvolumes.
The bag-like constant $\varepsilon_0$ determines the magnitude of the hadron eigenvolumes,
and in our analysis we vary this constant such that the corresponding hard-core radius $r_p$ of protons (nucleons) takes reasonable values.
Values of $r_p = 0.3$-$0.8$~fm have been rather commonly used in the literature~\cite{Yen1997,BugaevEV,Begun2013,PHS1999,Cleymans2006,Gorenstein1999}.
Additionally, the value $r_p \simeq 0.6$~fm was extracted from the ground state properties of nuclear matter within the fermionic van der Waals equation for nucleons~\cite{NM-VDW}. In the present work we vary the effective proton radius in the range $r_p = 0.0$-$0.6$~fm. Note that, within the bag-like parametrization, the radius $r_i$ of any hadron $i$ is related to the chosen value of $r_p$ through the relation $r_i = r_p \, \cdot \, (m_i/m_p)^{1/3}$, where $m_i$ is the mass of the hadron $i$.
\subsection{Experimental data set}
The thermal equilibrium EV HRG model fits are using the hadron yield
data of the NA49, STAR, and ALICE collaborations.
The data of the NA49 collaboration include
$4\pi$ yields of charged pions, charged kaons, $\Xi^-$, $\Xi^+$, $\Lambda$, $\phi$, and, where available, $\Omega$, $\bar{\Omega}$, measured in the 0-7\% most central Pb+Pb collisions at $\sqrt{s_{\rm NN}} = 6.3, 7.6, 8.8, 12.3$, and in the 0-5\% most central Pb+Pb collisions at $\sqrt{s_{\rm NN}} = 17.3$~GeV~\cite{NA49data-1,NA49data-2}.
The feeddown from strong and electromagnetic decays is included in the model.
Additionally, the data on the total number of participants, $N_W$, is identified with total net baryon number and is included in the fit.
The actual tabulated data used in our analysis is available in Ref.~\cite{NA49data-3}.
The STAR data contains the midrapidity yields of charged pions, charged kaons,
(anti)protons, $\Xi^-$, $\Xi^+$, $\Omega$+$\bar{\Omega}$, and $\phi$ in the 0-5\% most central Au+Au collisions at $\sqrt{s_{\rm NN}} = 200$~GeV~\cite{STARdata,BecattiniSTAR}.
The yields of protons also include the feed-down from weak decays of (multi)strange hyperons, this is properly taken into account in the model. All other yields include the feeddown from strong and electromagnetic decays.
We also perform the fit to the ALICE data on
midrapidity yields of hadrons in the 0-5\% most central Pb+Pb collisions at $\sqrt{s_{\rm NN}} = 2.76$~TeV~\cite{ALICEdata}.
This includes yields of charged pions, charged kaons, (anti)protons, $\Xi^-$, $\Xi^+$, $\Lambda$, $\Omega$+$\bar{\Omega}$, $\phi$, and $K^0_S$.
Since the experimental centrality binning for $\Xi$ and $\Omega$ hyperons in the ALICE data is different from that of the other hadrons, we take the midrapidity yields of $\Xi$ and $\Omega$ in the $0-5$\% centrality class from Ref.~\cite{Becattini2014}, where they were obtained using the interpolation procedure.
All these yields include the feeddown from strong decays.
\subsection{Fitting results}
Figure~\ref{fig:chi2-vs-T} shows the temperature dependence of the $\chi^2/N_{\rm dof}$ of the fit to NA49, STAR, and ALICE data within ``Diagonal'' and ``Crossterms'' EV models for the value of the bag-like constant $\varepsilon_0$ fixed to reproduce the proton hard-core radius $r_p = 0.5$~fm.
At each temperature the two remaining free parameters, namely the baryon chemical potential $\mu_B$ and the radius $R$ of the system volume $V = (4/3)\, \pi \, R^3$, are fitted in order to minimize the $\chi^2$ at this temperature.
At this point, we do not enforce
any limitations on the values of $T$ and $\mu_B$ in our fitting procedure. This means, for instance, that we do not require that the eigenvolume HRG model describes the available lattice QCD data, and also that we employ no limitations on the packing fraction $\eta$ -- the fraction of the total volume occupied by hadrons of finite size.
We will come back to these issues later.
The temperature dependence of the $\chi^2 / N_{\rm dof}$ within the point-particle HRG (solid black line in Fig.~\ref{fig:chi2-vs-T}) shows a narrow minimum, and the temperature of this minimum is slightly increasing with the collision energy. These temperature values are consistent with numerous previous analyses within the point-particle formulations of the HRG. The fit quality within the point-particle HRG is not very good. This is especially the case for the NA49 energy of $\sqrt{s_{\rm NN}} = 12.3$~GeV with $\chi^2 / N_{\rm dof} \simeq 13$.
We note that the fit quality in the point-particle HRG can be improved significantly within the chemical non-equilibrium scenario (see e.g. Refs.~\cite{Becattini2006,Letessier2005,MHBQ,VBG2015}), however, we do not consider this option in the present work.
\begin{figure}[!t]
\centering
\includegraphics[width=0.49\textwidth]{chi2-SN-Tmudep-v2}
\caption[]{(Color online)
Regions in the $T$-$\mu_B$ plane where the ``Crossterms'' eigenvolume HRG model with bag-like constant $\varepsilon_0$ fixed to reproduce the hard-core radius of $r_p = 0.5$~fm yields a better fit to the NA49 and STAR data as compared to the point-particle HRG model.
The locations of the local $\chi^2$ minima are depicted by the diamonds for the point-particle HRG case, and by the dots for the eigenvolume HRG.
For RHIC energy the light shaded area additionally depicts the wide $T$-$\mu_B$ region where $\chi^2$ of the fit to the STAR data within the eigenvolume model has a broad minimum.
The solid lines show the isentropic curves for the eigenvolume model, which
go through the global $\chi^2$ minima.
}\label{fig:chi2-Tmu}
\end{figure}
\begin{table*}
\caption{Summary of the fitted parameters within the point-particle HRG, and within the two eigenvolume HRG models with mass-proportional eigenvolumes fixed to $r_p = 0.5$~fm.
No restrictions on possible values of $T$ and $\mu_B$ are applied.
Full chemical equilibrium is assumed.}
\centering
\begin{tabular*}{\textwidth}{c @{\extracolsep{\fill}} ccccc}
\hline
\hline
System & Parameters & Point-particle & ``Diagonal'' EV & ``Crossterms'' EV \\
& & $r_p = 0$ & $r_p = 0.5$~fm & $r_p = 0.5$~fm \\
\hline
$\sqrt{s_{_{\rm NN}}} = 6.3$~GeV & $T$ (MeV)
& $138$ & $340$ & $233$ \\
NA49 & $\mu_B$ (MeV)
& $486$ & $1335$ & $838$ \\
Pb+Pb & $R$ (fm)
& $7.5$ & $7.2$ & $7.5$ \\
0-7\% central
& $S/A$
& $13.2$ & $14.7$ & $14.4$ \\
& $E/N$ (GeV)
& $1.11$ & $1.41$ & $1.25$ \\
& $\chi^2 / N_{\rm dof}$
& $25.6/6$ & $18.3/6$ & $9.1/6$ \\
\hline
$\sqrt{s_{_{\rm NN}}} = 7.6$~GeV & $T$ (MeV)
& $142$ & $275$ & $249$ \\
NA49 & $\mu_B$ (MeV)
& $427$ & $862$ & $767$ \\
Pb+Pb & $R$ (fm)
& $7.8$ & $7.7$ & $7.7$ \\
0-7\% central
& $S/A$
& $15.4$ & $16.6$ & $17.3$ \\
& $E/N$ (GeV)
& $1.09$ & $1.33$ & $1.25$ \\
& $\chi^2 / N_{\rm dof}$
& $31.2/7$ & $23.1/7$ & $6.9/7$ \\
\hline
$\sqrt{s_{_{\rm NN}}} = 8.8$~GeV & $T$ (MeV)
& $140$ & $307$ & $283$ \\
NA49 & $\mu_B$ (MeV)
& $391$ & $903$ & $822$ \\
Pb+Pb & $R$ (fm)
& $8.8$ & $7.9$ & $8.3$ \\
0-7\% central
& $S/A$
& $17.7$ & $19.3$ & $19.8$ \\
& $E/N$ (GeV)
& $1.03$ & $1.35$ & $1.28$ \\
& $\chi^2 / N_{\rm dof}$
& $56.5/8$ & $44.6/8$ & $24.5/8$ \\
\hline
$\sqrt{s_{_{\rm NN}}} = 12.3$~GeV & $T$ (MeV)
& $142$ & $353$ & $322$ \\
NA49 & $\mu_B$ (MeV)
& $320$ & $898$ & $747$ \\
Pb+Pb & $R$ (fm)
& $10.0$ & $8.4$ & $8.3$ \\
0-7\% central
& $S/A$
& $23.1$ & $26.8$ & $27.4$ \\
& $E/N$ (GeV)
& $0.97$ & $1.39$ & $1.32$ \\
& $\chi^2 / N_{\rm dof}$
& $89.1/7$ & $64.2/7$ & $22.8/7$ \\
\hline
$\sqrt{s_{_{\rm NN}}} = 17.3$~GeV & $T$ (MeV)
& $148$ & $293$ & $320$ \\
NA49 & $\mu_B$ (MeV)
& $254$ & $534$ & $578$ \\
Pb+Pb & $R$ (fm)
& $10.4$ & $9.4$ & $9.1$ \\
0-5\% central
& $S/A$
& $29.3$ & $31.7$ & $35.0$ \\
& $E/N$ (GeV)
& $0.98$ & $1.41$ & $1.27$ \\
& $\chi^2 / N_{\rm dof}$
& $44.8/10$ & $35.4/10$ & $7.0/10$ \\
\hline
$\sqrt{s_{_{\rm NN}}} = 200$~GeV & $T$ (MeV)
& $157$ & $239$ & $190$ \\
STAR & $\mu_B$ (MeV)
& $25$ & $38$ & $31$ \\
Au+Au & $R$ (fm)
& $8.1$ & $8.0$ & $8.7$ \\
0-5\% central
& $S/A$
& $300$ & $332$ & $324$ \\
& $E/N$ (GeV)
& $0.97$ & $1.13$ & $1.01$ \\
& $\chi^2 / N_{\rm dof}$
& $9.6/7$ & $7.9/7$ & $8.4/7$ \\
\hline
$\sqrt{s_{_{\rm NN}}} = 2760$~GeV & $T$ (MeV)
& $153$ & $284$ & $330$ \\
ALICE & $\mu_B$ (MeV)
& 0 (fixed) & 0 (fixed) & 0 (fixed) \\
Pb+Pb & $R|_{y=0}$ (fm)
& $11.0$ & $9.8$ & $9.3$ \\
0-5\% central
& $E/N$ (GeV)
& $0.93$ & $1.20$ & $1.28$ \\
& $\chi^2 / N_{\rm dof}$
& $32.9/10$ & $16.6/10$ & $8.8/10$ \\
\hline
\end{tabular*}
\label{tab:FitUnconstrained}
\end{table*}
The fit quality within the two considered here eigenvolume models is considerably better than in the point-particle case at all energies, the best fits yield significantly higher values of the chemical freeze-out temperature. In Fig.~\ref{fig:chi2-vs-T-40} the temperature dependence of the $\chi^2$ for three different values of $r_p$, namely for $r_p = 0.4, \,0.5,\,0.6$~fm is shown for the SPS data at $\sqrt{s_{\rm NN}}=8.8$~GeV. The behavior of the $\chi^2$ for different values of $r_p$ shown in Fig.~\ref{fig:chi2-vs-T-40} is representative for all other collision energies considered in the present work.
For $r_p = 0.4$~fm the temperature dependence of $\chi^2$ shows two distinct local minima: the first one is located close to the minimum for point-particle HRG and the second one is at much higher temperatures and with considerably smaller $\chi^2$.
This trend is continued when even lower values of $r_p$ are considered.
For $r_p = 0.5$ fm and $r_p = 0.6$ fm a wide (double)minimum structure is observed in the temperature dependence of $\chi^2$ and, at all energies, the fit quality is better in EV models than in the point-particle case for a very wide high-temperature range.
We note that the results presented in Fig.~\ref{fig:chi2-vs-T} for the ALICE energy are consistent with
the recent Ref.~\cite{VS2015}.
Small quantitative differences are attributed to the use
of the Boltzmann approximation in the present work.
The two eigenvolume models considered give the same qualitative picture, however there are significant quantitative differences, especially at higher temperatures: the ``Crossterms'' model consistently gives a considerably better description of the data.
In order to study the effects of the eigenvolume interactions on the fit values of the baryon chemical potential, we consider the structure of the $\chi^2$ of the fit in the $T$-$\mu_B$ plane.
For the purpose of clarity we consider just the ``Crossterms'' model with $r_p = 0.5$~fm.
Figure $\ref{fig:chi2-Tmu}$ depicts the regions in the $T$-$\mu_B$ plane where the fit quality
of NA49 data in the EV model is better than in the point-particle HRG.
At all five NA49 energies wide regions of improved $\chi^2$ values are observed at high temperatures and chemical potentials. We note a clear correlation between the fitted values of the chemical freeze-temperature and the chemical potential: the higher values of the temperature correspond to higher values of the baryochemical potential.
The thermal parameters at the global minima are listed in Table~\ref{tab:FitUnconstrained}. The values of the temperature at the global minima monotonically increase with the exception of the STAR energy. The corresponding values of the baryochemical potential monotonically decrease with an exception of a peak at $\sqrt{s_{\rm NN}} = 8.8$~GeV.
Based on the systematic analysis of the hadron yield data within the point-particle-like formulations of the HRG
it has been suggested that the chemical freeze-out is universally characterized by the constant average energy per hadron of $E/N \simeq 1$~GeV~\cite{CleymansRedlich1998}.
It was shown in Ref.~\cite{Cleymans2006} that this criterium is robust with regards
to the eigenvolume corrections in the specific case when all hadrons have the same
hard-core radius.
It is interesting to analyze whether such a criterion holds as well for the bag-like eigenvolume parametrization considered in our work.
The values of $E/N$ at the global minimum for the point-particle, ``Diagonal'' EV, and ``Crossterms'' EV with $r_p = 0.5$~fm are shown in the Table~\ref{tab:FitUnconstrained}.
The values of $E/N$ are notably larger when the eigenvolume corrections are present,
typically $E/N \simeq 1.3-1.4$~GeV.
Similar result, i.e. larger values of $E/N$, is observed for $r_p = 0.4$~fm and $r_p = 0.6$~fm. Therefore, one can conclude that the energy per particle criterium is, in general, \emph{not} robust with regards to the modeling of the eigenvolume interactions if the chemical freeze-out conditions are determined solely from the thermal fits.
On the other hand, the total entropy per baryon $S/A$ is found to be much more robust.
This can be seen by looking at the $S/A$ values in Table~\ref{tab:FitUnconstrained}, on the lines of constant $S/A$ in Fig.~\ref{fig:chi2-Tmu}, and on the
excitation function of $S/A$ at SPS energies in Fig.~\ref{fig:SA}.
It is seen that $S/A$ remains approximately constant across the vast $T$-$\mu_B$ islands of small $\chi^2$, and that $S/A$ increases only by 5-15\% when going from the point-particle HRG to the eigenvolume HRG, in spite of the massive changes in temperature and chemical potential.
It is also quite remarkable that the values of $S/A$ at the global minimum show almost no dependence on the value of $r_p$. This is best seen in Fig.~\ref{fig:SA} where the energy dependence of the $S/A$ values for $r_p = 0.4$, $0.5$, and $0.6$~fm extracted from- fits to the SPS data are shown.
These results hint towards the impossibility of fixing one ``best'' pair of temperature and chemical potential from a fit to the data: many $T-\mu_B$ pairs yield similarly good fits.
An isentropic expansion of matter with hadrons being chemically frozen out across the extended space-time regions is consistent with this finding. Namely, this implies that the chemical freeze-out is not a sharp process which takes place at one freeze-out hypersurface with similar values of the temperature and chemical potential. Rather a continuous process, occurring throughout the whole space-time evolution of the system created in the heavy-ion collisions, and characterized by a wealth of different values of temperatures, energy densities, and other dynamical parameters. Such a picture has been obtained within the transport model simulations of heavy-ion collisions by analyzing the space-time distribution of the chemical ``freeze-out'' points of various hadrons~\cite{ContFrz1,ContFrz2}.
\begin{figure}[!t]
\centering
\includegraphics[width=0.49\textwidth]{SoverA-vs-en}
\caption[]{(Color online)
The collision energy dependence of the entropy per baryon ratio $S/A$ calculated from the HRG model at the global minimum of the $\chi^2$ fit to the SPS hadron yield data. Calculations are done within the point-particle HRG and within the eigenvolume HRG with mass-proportional eigenvolumes. The parameter $\varepsilon_0$ in Eq.~\eqref{eq:BagEV} is fixed to reproduce the effective hard-core radius of proton of 0.4, 0.5, and 0.6 fm.
}\label{fig:SA}
\end{figure}
In the present analysis we do not determine the uncertainties of the temperature and chemical potential extracted from fits to hadron yield data within the bag-like eigenvolume HRG model. These parameters depend strongly on the chosen eigenvolumes of the hadrons, and different parametrizations can give a fit of comparable quality but with very different values of $T$ and $\mu_B$.
\subsection{Fit constrained to low temperatures}
As seen from Figs.~\ref{fig:chi2-vs-T} and \ref{fig:chi2-Tmu} one can fit the data significantly better than in the point-particle case by employing the mass-proportional eigenvolume and considering high values of temperature and chemical potential.
However, the interpretation of these results is
prone to controversy: to describe QCD matter at $T>160$~MeV as an interacting
HRG with EV, i.e. with hadronic degrees of freedom, seems to contradict the consensual paradigm based on the ideal (point-particle) HRG model established in the community for decades.
Moreover, the high temperature $\chi^2$ minima shown in Fig.~\ref{fig:chi2-vs-T} are excellent fits of the model to the experimental data; however, the equation of state of this EV HRG model is plagued by the fact that
the speed of sound
takes values of $c_s^2 \sim 1$ at the $T$-$\mu_B$ values at the global minima for $r_p = 0.4-0.6$~fm.
The superluminal behavior of the speed of sound is a known problem of the EV model. Avoiding it requires modifications to the model.
Secondly, the packing fraction $\eta$ takes high values, typically
$\eta \sim 0.15$
at the best fit locations.
At such high values of $\eta$ the eigenvolume model may deviate significantly from the equation of state of the hard spheres model~(see, e.g., Refs.~\cite{mf-2014,ZalewskiRedlich}). Evidently, this suggests that the global minima
locations may lie outside the range where the model can be applied safely.
In this respect it would be interesting to study the behavior of eigenvolume HRG
at high densities within other EV formulations which take care of these issues.
Such a study is outside the scope of the present paper.
The mass-proportional eigenvolume interactions give a systematic improvement of the fit quality at all considered collision energies.
However, the interpretation of these results is evidently controversial, for the reasons listed above.
Thus, we also perform an additional calculation where we restrict the temperatures from above, in order to get some consistency with the lattice QCD data.
The lattice QCD calculations reveal a crossover-type transition between hadronic and partonic degrees at $\mu_B = 0$, with a pseudocritical temperature $T_{\rm pc} \sim 155$~MeV\cite{Aoki:2006we,lQCD,Bazavov:2014pvz}.
The lattice data, however, does not give a direct information about the constituent composition of matter at a particular temperature. Due to the crossover nature of the transition it is impossible to uniquely define a transition temperature, which would exclude a presence of the hadronic degrees above this temperature. For example, it was illustrated in~\cite{Mukherjee:2015mxc} that the charm hadron-like excitations remain dominant degrees of freedom at temperatures above the pseudocritical one, up to at least $1.2 \, T_{\rm pc}$~(see also \cite{Biro:2014sfa})
while the $T_{\rm pc} \sim 155$~MeV temperature suggests only the onset of hadron melting~\cite{Jakovac:2013iua}.
Such a picture, a \emph{gradual} transition from hadrons to quarks had been advocated before, in particular in the context of heavy-ion collisions~\cite{Stoecker:1980uk}.
The thermodynamics of the HRG with mass-proportional eigenvolumes was considered in Ref.~\cite{EV-latt-3} in the
context of the lattice QCD equation of state, in particular regarding the gradual transition from hadrons to quarks.
In that work a crossover equation of state of the QCD matter was developed. It was obtained by smoothly matching two models, an
eigenvolume HRG equation of state
at low $T$ and/or $\mu_B$ and a perturbative QCD (pQCD) equation of state at high $T$ and/or $\mu_B$.
The total QCD pressure function at vanishing chemical potential in this crossover model reads
\eq{
P(T) = S(T) \, P_{\rm qg} (T) + [1 - S(T)] \, P_{\rm h} (T),
}
where $P_{\rm h} (T)$ is the hadronic pressure modeled by the eigenvolume HRG,
$P_{\rm qg} (T)$ is the pressure of the quark-gluon phase based on the pQCD calculations, and
\eq{
S(T) = \exp\left[-(T_0/T)^{r} \right]
}
is the so-called switching function.
In one of the scenarios considered in Ref.~\cite{EV-latt-3} the hadronic part was modeled by the ``Diagonal'' eigenvolume HRG model with the mass-proportional eigenvolume \eqref{eq:BagEV}. The bag-like constant $\varepsilon_0$ was determined from the fit to the lattice data. The best fit was obtained for $\varepsilon_0 = 797$~MeV/fm$^3$, which in our notation corresponds to the hard-core proton (nucleon) radius of $r_p \simeq 0.41$~fm, and a ``switching'' temperature of $T_0 \sim 175$~MeV.
Using the point-particle HRG gives a worse but still a satisfactory fit to the lattice data at $\mu_B = 0$, thus, it seems that values of hard-core proton radius in the range $r_p = 0.00-0.41$~fm are all generally compatible with the lattice data within the mass-proportional EV model.
\begin{figure}[t]
\centering
\includegraphics[width=0.49\textwidth]{Crossover-PT4-v2}
\caption[]{(Color online)
Temperature dependence of the scaled pressure at zero baryon chemical potential calculated within the crossover equation of state (solid line), and
the ``Crossterms'' eigenvolume HRG with $v_i = m_i / \varepsilon_0$ and $\varepsilon_0 = 704$~MeV/fm$^3$~(dashed line).
The lattice QCD data of the Wuppertal-Budapest collaboration~\cite{lQCD} is shown by the symbols with error bars.
}\label{fig:crossoverEoS}
\end{figure}
\begin{table*}
\caption{Summary of the fitted parameters within the HRG models restricted to temperatures below 175~MeV. Chemical equilibrium is assumed.}
\centering
\begin{tabular*}{\textwidth}{c @{\extracolsep{\fill}} cccc}
\hline
\hline
System & Parameters & Point-particle & ``Crossterms'' EV \\
& & $r = 0$~fm & $r_i \sim m_i^{1/3}$, $r_p = 0.43$~fm \\
\hline
$\sqrt{s_{_{\rm NN}}} = 6.3$~GeV & $T$ (MeV)
& $137.5 \pm 5.3$ & $158.6 \pm 14.1$ \\
NA49 & $\mu_B$ (MeV)
& $485.7 \pm 4.1$ & $555.2 \pm 36.5$ \\
Pb+Pb & $R$ (fm)
& $7.46 \pm 0.70$ & $7.71 \pm 0.59$ \\
0-7\% central & $\chi^2 / N_{\rm dof}$
& $25.6/6$ & $20.2/6$ \\
\hline
$\sqrt{s_{_{\rm NN}}} = 7.6$~GeV & $T$ (MeV)
& $142.4 \pm 3.7$ & $159.4 \pm 5.9$ \\
NA49 & $\mu_B$ (MeV)
& $427.2 \pm 3.7$ & $479.5 \pm 12.9$ \\
Pb+Pb & $R$ (fm)
& $7.82 \pm 0.41$ & $8.29 \pm 0.30$ \\
0-7\% central & $\chi^2 / N_{\rm dof}$
& $31.2/7$ & $23.9/7$ \\
\hline
$\sqrt{s_{_{\rm NN}}} = 8.8$~GeV & $T$ (MeV)
& $140.0 \pm 2.5$ & $151.6 \pm 4.5$ \\
NA49 & $\mu_B$ (MeV)
& $391.2 \pm 5.0$ & $425.1 \pm 9.8$ \\
Pb+Pb & $R$ (fm)
& $8.83 \pm 0.42$ & $9.21 \pm 0.37$ \\
0-7\% central & $\chi^2 / N_{\rm dof}$
& $56.2/8$ & $50.4/8$ \\
\hline
$\sqrt{s_{_{\rm NN}}} = 12.3$~GeV & $T$ (MeV)
& $141.7 \pm 3.1$ & $157.0 \pm 8.0$ \\
NA49 & $\mu_B$ (MeV)
& $320.2 \pm 4.9$ & $347.1 \pm 9.3$ \\
Pb+Pb & $R$ (fm)
& $10.00 \pm 0.54$ & $9.90 \pm 0.64$ \\
0-7\% central & $\chi^2 / N_{\rm dof}$
& $89.1/7$ & $80.4/7$ \\
\hline
$\sqrt{s_{_{\rm NN}}} = 17.3$~GeV & $T$ (MeV)
& $147.9 \pm 2.1$ & $160.2 \pm 3.9$ \\
NA49 & $\mu_B$ (MeV)
& $254.5 \pm 4.7$ & $277.4 \pm 6.6$ \\
Pb+Pb & $R$ (fm)
& $10.44 \pm 0.41$ & $10.89 \pm 0.37$ \\
0-5\% central & $\chi^2 / N_{\rm dof}$
& $44.8/10$ & $38.7/10$ \\
\hline
$\sqrt{s_{_{\rm NN}}} = 200$~GeV & $T$ (MeV)
& $157.4 \pm 2.3$ & $173.4 \pm 5.1$ \\
STAR & $\mu_B$ (MeV)
& $25.3 \pm 11.2$ & $27.9 \pm 12.7$ \\
Au+Au & $R|_{y=0}$ (fm)
& $8.13 \pm 0.33$ & $8.55 \pm 0.32$ \\
0-5\% central & $\chi^2 / N_{\rm dof}$
& $9.6/7$ & $8.6/7$ \\
\hline
$\sqrt{s_{_{\rm NN}}} = 2760$~GeV & $T$ (MeV)
& $152.9 \pm 2.3$ & $165.8 \pm 4.3$ \\
ALICE & $\mu_B$ (MeV)
& 0 (fixed) & 0 (fixed) \\
Pb+Pb & $R|_{y=0}$ (fm)
& $10.98 \pm 0.47$ & $11.35 \pm 0.45$ \\
0-5\% central & $\chi^2 / N_{\rm dof}$
& $32.9/10$ & $27.0/10$ \\
\hline
\end{tabular*}
\label{tab:FitCo}
\end{table*}
Our hadron list is very similar to the one used in Ref.~\cite{EV-latt-3}.
On the other hand, we believe that the ``Crossterms'' model reflects better the features of the multi-component eigenvolume systems. In particular, as previously mentioned, it is consistent with virial expansion of classical hard spheres equation of state, in contrast to ``Diagonal'' model.
As seen from Fig.~\ref{fig:chi2-vs-T} it also gives a systematically better description of the hadron yield data.
We use the ``Crossterms'' EV model and repeat the calculation of Ref.~\cite{EV-latt-3} for the crossover equation of state. For the pQCD part we use exactly the same model with exactly the same parameters. They are listed in the second-last row of Table~1 in Ref.~\cite{EV-latt-3}.
We reproduce the result of~\cite{EV-latt-3}, i.e. a perfect description of the lattice QCD pressure, by using the value
$r_p \simeq 0.43$~fm ($\varepsilon_0 = 704$~MeV/fm$^3$)
for the proton hard-core radius
within the ``Crossterms'' EV model.
The slightly different value of $\varepsilon_0$ compared to~\cite{EV-latt-3} is attributed to differences between the ``Diagonal'' and ``Crossterms'' models.
The resulting temperature dependence of the scaled pressure $P/T^4$ is shown
in Fig.~\ref{fig:crossoverEoS}. A very good agreement is seen in the whole temperature range, in line with results of Ref.~\cite{EV-latt-3}.
The dashed line in Fig.~\ref{fig:crossoverEoS} shows the purely hadronic pressure $P_h$ of the eigenvolume HRG.
The bag-like eigenvolume models with $r_p = 0.4-0.6$~fm show a qualitatively similar temperature dependence, with larger $r_p$ values bringing the dashed curve further down.
Let us now employ the
``Crossterms'' model with $r_i \sim m_i^{1/3}$ and $r_p = 0.43$~fm to fit the NA49, STAR, and ALICE data, and impose an additional restriction $T < T_0 \simeq 175$~MeV. This ensures that only the temperatures where the hadronic part of the QCD equation of state plays a notable role are considered.
Certainly, due to the crossover nature of the hadron-parton transition, this value should not and cannot be regarded as a sharp transition temperature.
Even with the low temperature restriction, the EV model fits leads to a better description of the data at all the considered energies, as seen in Table~\ref{tab:FitCo}. The improvement is modest but systematic.
The inclusion of the finite EV also leads to some changes in the extracted parameters: the chemical freeze-out temperature increases by about 10-15 MeV and
the baryochemical potential increases by about 10-15\%.
These changes are significant as they are larger than the typical
thermal fit uncertainties reported in the literature.
The resulting temperature profiles of the $\chi^2$
are all qualitatively similar to the $r_p = 0.4$~fm curve in Fig.~\ref{fig:chi2-vs-T-40}.
The fit errors of $T$ and $\mu_B$,
obtained from analyses of the second-derivative error matrices at the minima,
increase notably for the cases with finite EV.
The obtained results also indicate that the chemical freeze-out curve in $T$-$\mu_B$ plane has a smaller curvature in the EV models compared to the one obtained within the point-particle HRG~(see Fig.~\ref{fig:FitCo}). A similar result was obtained in \cite{Becattini2013,Becattini:2016xct} but by employing a different mechanism, namely, by considering the distortion of yields due to the post-hadronization cascade phase.
Due to the possibly irregular structure of the $\chi^2$ profiles one should treat the uncertainties for the extracted freeze-out parameters shown in Table~\ref{tab:FitCo} and Fig.~\ref{fig:FitCo} with care,
since they are calculated from the second-derivative error matrix and
are based on an assumed parabolic behavior of $\chi^2$ near the minima.
\begin{figure}[t]
\centering
\includegraphics[width=0.49\textwidth]{T-muB-constrained-nogs}
\caption[]{(Color online)
The extracted freeze-out parameters within the point-particle HRG,
and the case when eigenvolumes of hadrons are proportional to their mass with $\varepsilon_0 = 704$~MeV/fm$^3$, modeled
by the ``Crossterms'' eigenvolume HRG.
The parameterized freeze-out curves from Refs.~\cite{ABS2009}, \cite{Cleymans2006},
obtained within the point-particle-like (including constant eigenvolume for all hadrons) HRG models, are depicted by lines.
}\label{fig:FitCo}
\end{figure}
It is seen that the extraction of the chemical freeze-out parameters is rather sensitive to the modeling of repulsive interactions between hadrons even if only moderate temperatures $T<175$~MeV are considered.
As mentioned before, the lattice data for the pressure does not conclusively exclude any $r_p$ in the range of $0.00-0.43$~fm for the bag-like parametrization of the hadron eigenvolumes. It is also likely that the upper limit of acceptable $r_p$ is even higher. For these reasons the uncertainties in the extraction of the chemical freeze-out parameters remain sizable even when the lattice constraint is used.
\section{Summary}
In summary, the data of the NA49, STAR, and ALICE collaborations on the hadron yields in central Pb+Pb (Au+Au) collisions at $\sqrt{s_{\rm NN}} = 6.3, 7.6, 8.8, 12.3, 17.3$, $200$, and $2760$~GeV are analyzed within two different multi-component eigenvolume HRG models employing
mass-proportional eigenvolumes
for different hadrons.
For a proton hard-core radius of 0.4-0.6~fm, these models describe the data significantly better than the conventional point-particle HRG model in very wide regions in the $T$-$\mu_B$ plane.
These results show that the extraction of the chemical freeze-out parameters is extremely sensitive to the modeling of the short-range repulsion between the hadrons, and they imply that the ideal point-particle HRG values are not unique.
As far as the thermal fits within the EV-HRG model are concerned,
the constant energy per hadron of 1 GeV criterion proposed in the literature is
changing ($\sim$30\%) depending on
the modeling of the eigenvolume interactions.
On the other hand,
the entropy per baryon extracted from the data for the different energies is found to be much more robust: it is nearly independent of the details of modeling of the eigenvolume interactions and of the specific $T-\mu_B$ values obtained.
This, as well as the
large uncertainties in the extracted $T$ and $\mu_B$ values suggested by the eigenvolume HRG models, is consistent with the scenario of a continuous freeze-out, whereby the hadrons are being frozen-out throughout the extended regions of the space-time evolution of the system rather than from the sharp freeze-out hypersurface.
The interpretation of the high values of temperature and baryochemical potential obtained within EV fits is prone to controversy. These values do give a significant and systematic improvement in the fit quality. However, the fits are also plagued by an irregular behavior of the speed of sound, difficulty in reconciliation with lattice QCD results, and high values of packing fraction. Thus, the extracted high values of $T$ and $\mu_B$ should not be interpreted as estimates of chemical freeze-out conditions, but rather as an illustration of the hitherto unexplored sensitivity of thermal fits to the modeling of EV interactions.
Even the fits with a low temperature ($T<175$~MeV) restriction inspired by analysis of the lattice QCD equation of state suggest a strong influence of EV effects on thermal fits. The fits with this temperature restriction result into 10-15~MeV higher values of the chemical freeze-out temperature and a systematically improved $\chi^2$
compared to the point-particle case.
Hence, the modeling of the eigenvolume interactions plays
a crucial role for the thermal fits to the hadron yield data, a fact that had been largely overlooked in the past.
Many possibilities for the parameterization of the hadron eigenvolumes exist.
It is evident that proper restrictions on the eigenvolumes of different hadrons are urgently needed.
Any conclusions based on thermal fits must be based not just on the location of the $\chi^2$ minimum and its magnitude, but rather on the full profile of the $\chi^2$.
The $\chi^2$ may have
non-parabolic structure around the local minima, and thus, the standard statistical-based estimates of the uncertainties of the extracted parameters may become inapplicable.
The analysis is performed within two different multi-component eigenvolume HRG models: the ``Diagonal'' EV model and the ``Crossterms'' EV model. Both models yield qualitatively similar results but they differ quantitatively.
The commonly used ``Diagonal'' EV model is shown to be not consistent with the 2nd order virial expansion for the equation of state of the multi-component system of hard spheres, while technically more complicated ``Crossterms'' EV model is.
The ``Crossterms'' model also opens new possibilities
to model the repulsive interactions: it allows to
independently specify the virial coefficient $b_{\rm ij}$
between any pair of hadron species.
For example, one can treat differently the baryon-baryon, baryon-antibaryon, meson-baryon, and meson-meson repulsive interactions. These effects were recently explored for LHC energies in Ref.~\cite{Satarov:2016peb}.
Thus, the ``Crossterms'' model is suggested to be used
over the ``Diagonal'' one
in the future
analyses employing the EV corrections.
The collision energy range investigated in this work is relevant for the ongoing SPS and RHIC beam energy scan programs, as well as for the experiments at the future FAIR and NICA facilities.
The strong effects of EV interactions should be taken into account in the future analyses and interpretations of the hadron yield data.
\vspace{0.5cm}
\section*{Acknowledgements}
We are grateful
to
P.~Alba,
M.~Ga\'zdzicki,
M.I.~Gorenstein, L.M.~Satarov, and A.~Tawfik
for fruitful comments and discussions.
This work was supported by the Helmholtz International Center for FAIR within the LOEWE program of the State of Hesse.
Some of the numerical calculations were performed at the Prometheus cluster at GSI.
V.V. acknowledges the support from HGS-HIRe for FAIR.
H.St. appreciates the support through the Judah M.~Eisenberg Laureatus Chair
at Goethe University.
| {
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Put Channing Tatum in everything.
Channing Tatum is a riot.
Before the 37- year-old actor got his start as a heartthrob in some fantastic teen movies in the early 2000s, he was earning money as a dancer at a nightclub. He has since graduated to one of Hollywood's best leading men.
His delightful performance in the NASCAR heist movie "Logan Lucky" — directed by Steven Soderbergh, director of the "Ocean's Trilogy" and the Tatum-led "Magic Mike" movies — has earned him critical praise.
You can catch "Logan Lucky" in theaters starting today.
Here's how Tatum went from dancing in nightclubs to being one of Hollywood's funniest leading men.
Channing Tatum dropped out of college after one year and returned to Tampa, Florida, where he worked odd jobs, like framing houses and working at a Dillard's.
Channing Tatum with his now-wife Jenna Dewan Tatum.
After quitting that job, Tatum started working at a Tampa nightclub where he got paid to dance.
Tatum stripped when he was younger.
"I don't miss anything about stripping," he told People. "I stripped in Tampa for like 25 girls, at best. It wasn't glamorous whatsoever, so there's nothing that I miss about stripping. This isn't stripping. This is a show."
Tatum decided he would be a model and walked into a modeling agency. He was signed immediately and began working for clients like Abercrombie & Fitch and Nautica.
Here's Tatum in a "Nautica" ad.
His first film role was in 2005's "Coach Carter."
Tatum showed his dramatic chops in "A Guide to Recognizing Your Saints," which also starred Shia LaBeouf, Rosario Dawson, and Robert Downey Jr.
"A Guide to Recognizing Your Saints."
2006 was a good year for Tatum as he won hearts and showed his knack for humor as Duke Orsino in "She's the Man."
That same year, he danced his way to heartthrob status in "Step Up," where he also met his now-wife, Jenna Dewan Tatum. He returned for a brief cameo in the sequel a few years later.
He used his talents to do some action films like 2008's war film "Stop-Loss."
He also did 2009's "G.I. Joe: The Rise of Cobra" as well as the sequel, "G.I. Joe: Retaliation."
"G.I. Joe: The Rise of the Cobra."
He tackled some crime movies like "Public Enemies."
For a while, Tatum was all about the romantic roles. He starred in Nicholas Sparks' "Dear John" and "The Vow."
He was also in "10 Years," a high school reunion rom-com also starring Rosario Dawson, Chris Pratt, Oscar Isaac, Justin Long, Aubrey Plaza, Max Minghella, and Jenna Dewan Tatum.
But it was his role in "21 Jump Street" that really showed Tatum could go all in to comedy. He was just as hilarious in the sequel, "22 Jump Street."
Tatum danced onscreen once again in "Magic Mike," which was inspired by his own stint as a dancer. He returned once again for the sequel, "Magic Mike XXL."
He astounded as real-life wrestler Mark Schultz in the Oscar-nominated "Foxcatcher."
He's lent his voice to the animated movie "The Book of Life," as well as voiced Superman in "The LEGO Movie" and "The LEGO Batman Movie."
His role in Quentin Tarintino's "The Hateful Eight" was unlike anything he'd ever done before, but he nailed it.
He followed that up with a part in the Coen brother's "Hail, Caesar!" which included a fun little dance number with Tatum dressed as a sailor.
His part in the heist movie "Logan Lucky" proves that Tatum can lead an action film, but still have superb comedic timing.
He's playing another Southern boy in "Kingsman: The Golden Circle," which hits theaters September 22.
And if that wasn't enough, the actor has an even more impressive slate for the upcoming year, which includes "Van Helsing," and is going to be playing Gambit in the "X-Men" spin-off.
Channing Tatum at Comic-Con 2017. | {
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One of my sorority sisters is facing the same thing I once faced but even greater. Her name is Shanai and one thing that inspires me about her is her will to live. She is in college, working, and goes to dialysis daily. A kidney has been found for her but she is in need of help financially. Please donate what you can and forward this along. Love you all! | {
"redpajama_set_name": "RedPajamaC4"
} | 742 |
Nikolai Dmitrievich Sergeevsky (1849–1908) was a Russian law professor and statesman.
Biography
Graduate in law of St. Petersburg University; professor of Demidov Lyceum in Yaroslavl, and of criminal law in St. Petersburg University (1882), lecturer in the Military Law Academy.
Publisher and editor of Iuridicheskaia Letopis, 1890–1892; Head of the Section for the Codification of the Fundamental Laws of Finland (1893). Member of the Consultative Board in the Ministry of Justice and editor of Zhurnal Ministerstva Iustitsii (1894).
State Secretary of the Section for Codification of Laws of the Imperial Chancellery (1895).
Works
Sergeevskii, N. D. Finland : the question of autonomy and fundamental laws (1911)
1849 births
1908 deaths
Saint Petersburg State University alumni
Russian legal scholars
Russian journalists
Members of the Russian Assembly | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,665 |
Rudno (potocznie: Osiedle Głęboka, niem. Kneipab) – osiedle w Gdańsku, w dzielnicy Śródmieście.
Historia
Rudno zostało przyłączone w granice administracyjne miasta w 1814, jako Knipawa. Należy do okręgu historycznego Gdańsk.
Nazwa Rudno została nadana osiedlu w 1951, kiedy to wybudowano tam zespół bloków mieszkalnych. Współcześnie nazwa ta, odnośnie do osiedla między ulicami Głęboką, Siennicką, a Elbląską, jest coraz rzadziej używana.
Zobacz też
Olszynka Wielka
Osiedle Zawodników
Sienna Grobla
Sienna Grobla I
Sienna Grobla II
Przypisy
Jednostki morfogenetyczne okręgu Gdańsk
Podział historyczny Śródmieścia Gdańska | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,937 |
{"url":"http:\/\/hellenicaworld.com\/Science\/Mathematics\/en\/Multiplicativepartition.html","text":"### - Art Gallery -\n\nIn number theory, a multiplicative partition or unordered factorization of an integer n is a way of writing n as a product of integers greater than 1, treating two products as equivalent if they differ only in the ordering of the factors. The number n is itself considered one of these products. Multiplicative partitions closely parallel the study of multipartite partitions, discussed in Andrews (1976), which are additive partitions of finite sequences of positive integers, with the addition made pointwise. Although the study of multiplicative partitions has been ongoing since at least 1923, the name \"multiplicative partition\" appears to have been introduced by Hughes & Shallit (1983). The Latin name \"factorisatio numerorum\" had been used previously. MathWorld uses the term unordered factorization.\n\nExamples\n\nThe number 20 has four multiplicative partitions: 2 \u00d7 2 \u00d7 5, 2 \u00d7 10, 4 \u00d7 5, and 20.\n3 \u00d7 3 \u00d7 3 \u00d7 3, 3 \u00d7 3 \u00d7 9, 3 \u00d7 27, 9 \u00d7 9, and 81 are the five multiplicative partitions of 81 = 34. Because it is the fourth power of a prime, 81 has the same number (five) of multiplicative partitions as 4 does of additive partitions.\nThe number 30 has five multiplicative partitions: 2 \u00d7 3 \u00d7 5 = 2 \u00d7 15 = 6 \u00d7 5 = 3 \u00d7 10 = 30.\nIn general, the number of multiplicative partitions of a squarefree number with i prime factors is the ith Bell number, Bi.\n\nApplication\n\nHughes & Shallit (1983) describe an application of multiplicative partitions in classifying integers with a given number of divisors. For example, the integers with exactly 12 divisors take the forms p11, p\u00d7q5, p2\u00d7q3, and p\u00d7q\u00d7r2, where p, q, and r are distinct prime numbers; these forms correspond to the multiplicative partitions 12, 2\u00d76, 3\u00d74, and 2\u00d72\u00d73 respectively. More generally, for each multiplicative partition\n\n$$k=\\prod t_{i}$$\n\nof the integer k, there corresponds a class of integers having exactly k divisors, of the form\n\n$$} \\prod p_{i}^{{t_{i}-1}},$$\n\nwhere each pi is a distinct prime. This correspondence follows from the multiplicative property of the divisor function.\nBounds on the number of partitions\n\nOppenheim (1926) credits McMahon (1923) with the problem of counting the number of multiplicative partitions of n; this problem has since been studied by others under the Latin name of factorisatio numerorum. If the number of multiplicative partitions of n is an, McMahon and Oppenheim observed that its Dirichlet series generating function f(s) has the product representation\n\n$$f(s)=\\sum _{{n=1}}^{{\\infty }}{\\frac {a_{n}}{n^{s}}}=\\prod _{{k=2}}^{{\\infty }}{\\frac {1}{1-k^{{-s}}}}.$$\n\nThe sequence of numbers an begins\n\n1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, ... (sequence A001055 in the OEIS).\n\nOppenheim also claimed an upper bound on an, of the form\n\n$$a_{n}\\leq n\\left(\\exp {\\frac {\\log n\\log \\log \\log n}{\\log \\log n}}\\right)^{{-2+o(1)}},$$\n\nbut as Canfield, Erd\u0151s & Pomerance (1983) showed, this bound is erroneous and the true bound is\n\n$$a_{n}\\leq n\\left(\\exp {\\frac {\\log n\\log \\log \\log n}{\\log \\log n}}\\right)^{{-1+o(1)}}.$$\n\nBoth of these bounds are not far from linear in n: they are of the form n1\u2212o(1). However, the typical value of an is much smaller: the average value of an, averaged over an interval x \u2264 n \u2264 x+N, is\n\n\\( {\\bar a}=\\exp \\left({\\frac {4{\\sqrt {\\log N}}}{{\\sqrt {2e}}\\log \\log N}}{\\bigl (}1+o(1){\\bigr )}\\right),\n\na bound that is of the form no(1) (Luca, Mukhopadhyay & Srinivas 2008).\n\nCanfield, Erd\u0151s & Pomerance (1983) observe, and Luca, Mukhopadhyay & Srinivas (2008) prove, that most numbers cannot arise as the number an of multiplicative partitions of some n: the number of values less than N which arise in this way is NO(log log log N \/ log log N). Additionally, Luca, Mukhopadhyay & Srinivas (2008) show that most values of n are not multiples of an: the number of values n \u2264 N such that an divides n is O(N \/ log1 + o(1) N).\n\npartition (number theory)\ndivisor\n\nReferences\n\nAndrews, G. (1976), The Theory of Partitions, Addison-Wesley, chapter 12.\nCanfield, E. R.; Erd\u0151s, Paul; Pomerance, Carl (1983), \"On a problem of Oppenheim concerning \"factorisatio numerorum\"\", Journal of Number Theory, 17 (1): 1\u201328, doi:10.1016\/0022-314X(83)90002-1.\nHughes, John F.; Shallit, Jeffrey (1983), \"On the number of multiplicative partitions\", American Mathematical Monthly, 90 (7): 468\u2013471, doi:10.2307\/2975729, JSTOR 2975729.\nKnopfmacher, A.; Mays, M. (2006), \"Ordered and Unordered Factorizations of Integers\", Mathematica Journal, 10: 72\u201389. As cited by MathWorld.\nLuca, Florian; Mukhopadhyay, Anirban; Srinivas, Kotyada (2008), On the Oppenheim's \"factorisatio numerorum\" function, arXiv:0807.0986, Bibcode:2008arXiv0807.0986L.\nMacMahon, P. A. (1923), \"Dirichlet series and the theory of partitions\", Proceedings of the London Mathematical Society, 22: 404\u2013411, doi:10.1112\/plms\/s2-22.1.404.\nOppenheim, A. (1926), \"On an arithmetic function\", Journal of the London Mathematical Society, 1 (4): 205\u2013211, doi:10.1112\/jlms\/s1-1.4.205.\n\nKnopfmacher, A.; Mays, M. E. (2005), \"A survey of factorization counting functions\" (PDF), International Journal of Number Theory, 1 (4): 563\u2013581, doi:10.1142\/S1793042105000315","date":"2021-04-15 14:36:18","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\": 0.8626092076301575, \"perplexity\": 1266.268629624191}, \"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\/1618038085599.55\/warc\/CC-MAIN-20210415125840-20210415155840-00109.warc.gz\"}"} | null | null |
Q: Insert drawn shape over canvas background image In an html document, I have a canvas with a background image, and am drawing a shape and storing it as an object, and then reinserting it as an element of the canvas. As it stands now, the act of reinserting the shape as a canvas element imposes part of the canvas over the background, so that the shape is surrounded by the color of the canvas instead of being drawn over the canvas background. Is there a way to avoid this but still be able to insert the image where I would like? (I know I could save the image as a png and then make its background transparent, but I'd like to do it in the html itself.) I've seen a few things tangential to this, but none that I've found address it directly.
Fiddle: https://jsfiddle.net/ZLevine/h3mo50w8/4/
I used sample code from https://msdn.microsoft.com/en-us/library/gg589516(v=vs.85).aspx to draw the shape.
Code in fiddle:
<html>
<head>
<meta name="viewport" content="width=device-width, initial-scale=1.0"/>
<style>
canvas {
border:1px solid #d3d3d3;
background-image: white;
}
</style>
<head>
<body onload="startGame()">
<script>
var ship = new Image();
var myBackground;
function startGame() {
myBackground = new component(656, 541, "http://wallpapercave.com/wp/PU5vVEd.jpg", 0, 0, "background");
myScore = new component("30px", "Consolas", "white", 280, 40, "text");
myGameArea.start();
makeShip();
}
var myGameArea = {
canvas : document.createElement("canvas"),
start : function() {
this.canvas.width = 480;
this.canvas.height = 540;
this.context = this.canvas.getContext("2d");
document.body.insertBefore(this.canvas, document.body.childNodes[0]);
this.frameNo = 0;
this.interval = setInterval(updateGameArea, 20);
},
clear : function() {
this.context.clearRect(0, 0, this.canvas.width, this.canvas.height);
},
stop : function() {
clearInterval(this.interval);
}
}
function makeShip() {
ctx = myGameArea.context
// Draw saucer bottom.
ctx.beginPath();
ctx.moveTo(28.4, 16.9);
ctx.bezierCurveTo(28.4, 19.7, 22.9, 22.0, 16.0, 22.0);
ctx.bezierCurveTo(9.1, 22.0, 3.6, 19.7, 3.6, 16.9);
ctx.bezierCurveTo(3.6, 14.1, 9.1, 11.8, 16.0, 11.8);
ctx.bezierCurveTo(22.9, 11.8, 28.4, 14.1, 28.4, 16.9);
ctx.closePath();
ctx.fillStyle = "rgb(222, 103, 0)";
ctx.fill();
// Draw saucer top.
ctx.beginPath();
ctx.moveTo(22.3, 12.0);
ctx.bezierCurveTo(22.3, 13.3, 19.4, 14.3, 15.9, 14.3);
ctx.bezierCurveTo(12.4, 14.3, 9.6, 13.3, 9.6, 12.0);
ctx.bezierCurveTo(9.6, 10.8, 12.4, 9.7, 15.9, 9.7);
ctx.bezierCurveTo(19.4, 9.7, 22.3, 10.8, 22.3, 12.0);
ctx.closePath();
ctx.fillStyle = "rgb(51, 190, 0)";
ctx.fill();
// Save ship data.
ship = ctx.getImageData(0, 0, 30, 30);
}
function component(width, height, color, x, y, type) {
this.type = type;
if (type == "image" || type == "background") {
this.image = new Image();
this.image.src = color;
}
this.width = width;
this.height = height;
this.speedX = 0;
this.speedY = 0;
this.x = x;
this.y = y;
this.update = function() {
ctx = myGameArea.context;
if (this.type == "text") {
ctx.font = this.width + " " + this.height;
ctx.fillStyle = color;
ctx.fillText(this.text, this.x, this.y);
} if (type == "image" || type == "background") {
ctx.drawImage(this.image,
this.x,
this.y,
this.width, this.height);
if (type == "background") {
ctx.drawImage(this.image, this.x,
this.y - this.height,
this.width, this.height);
}
}
else {
ctx.fillStyle = color;
ctx.fillRect(this.x, this.y, this.width, this.height);
}
}
this.newPos = function() {
this.x += this.speedX;
this.y += this.speedY;
if (this.type == "background") {
if (this.y == (this.height)) {
this.y = 0;
}
}
}
}
function updateGameArea() {
myGameArea.clear();
myBackground.speedY = 1;
myBackground.newPos();
myBackground.update();
myGameArea.frameNo += 1;
ctx.putImageData(ship, 200, 200);
}
function everyinterval(n) {
if ((myGameArea.frameNo / n) % 1 == 0) {return true;}
return false;
}
</script>
</div>
</body>
</html>
A: Draw the ship on a second in-memory canvas:
var memCanvas=document.createElement('canvas');
... draw your ship on the second canvas
Then use the second canvas to draw your ship on the main canvas:
canvas.drawImage(memCanvas,x,y);
A: The problem is that using putImageData() will draw verbatim what you stored as ImageData when using getImageData(). As the ship is surrounded by transparent pixels also these are copied to the canvas overriding what is there.
As markE mentions in his answer, drawing to a temporary canvas is a better solution. All you need to do is to change a couple of places in the code:
// Save ship data.
//ship = ctx.getImageData(0, 0, 30, 30);
ship = document.createElement("canvas"); // create canvas
ship.width = ship.height = 30; // set size
var shipCtx = ship.getContext("2d"); // temp. context
shipCtx.drawImage(ctx.canvas, 0, 0); // draw ship to canvas
Then when you want to draw it back:
function updateGameArea() {
myGameArea.clear();
myBackground.speedY = 1;
myBackground.newPos();
myBackground.update();
myGameArea.frameNo += 1;
//ctx.putImageData(ship, 200, 200);
ctx.drawImage(ship, 200, 200); // draw ship from canvas
}
Updated fiddle
An alternative approach is to draw the shape directly onto the canvas, but using a intermediate canvas will give performance benefits.
| {
"redpajama_set_name": "RedPajamaStackExchange"
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\section{Introduction}
The task of preparing a quantum state on a qubit register is of fundamental importance in digital quantum computing. To this end, the Variational Quantum Eigensolver (VQE)~\cite{Peruzzo_2014,Moll_2018,cerezo2021variational} algorithm, of widespread use in several quantum application domains, from chemistry and physics~\cite{Kandala_2017,hempel2018quantum,Kokail2019}, machine learning~\cite{havlivcek2019supervised}, combinatorial optimization~\cite{farhi2014quantum}, and finance~\cite{Chakrabarti2021thresholdquantum} aims at variationally preparing the ground state of a Hamiltonian $\hat{H}$~\cite{Kandala_2017,Cao_2019,wecker2015progress,Chakrabarti2021thresholdquantum,cerezo2020variational,lubasch2020,mazzola2021sampling}.
While the VQE algorithm has been implemented already on existing noisy devices, its importance will persist in the future fault-tolerant regime as it is also necessary for more advanced quantum algorithms such as eigenstate projection methods~\cite{abrams1999quantum,Reiher_2017}.
The two essential ingredients of the method are:
(i) a parametrized quantum circuit with variational parameters $\boldsymbol{\theta}$, which produces a wave function $|\psi(\boldsymbol \theta) \rangle$ expressing the ground state of the problem, and
(ii) a \textit{learning} procedure aimed to optimizing the circuit variational parameters $\boldsymbol{\theta}$. This last step involves a feedback loop between the quantum and classical resources.
The VQE is among the most widely used quantum algorithms, and it has been adapted to many particular contexts and models.
However, most of the effort has been spent so far in devising better variational forms, tailored to various application domains~\cite{Bartlettt_2007,wecker2015progress,adaptvqe,uccsd,cerezo2021variational,Kokail2019,Giulia_gauge2021,Seki_2020} or hardware architectures~\cite{Kandala_2017,Kokail2019}.
Significantly less attention has been paid to the optimization part of the algorithm, with the notable exception of works discussing the concept of the barren plateaus, manifested in gradients vanishing exponentially fast with the system size~\cite{barren2018}.
However, this phenomenon was reported only in the context of random circuit variational forms. Recent works have proven an absence of such plateaus in the quantum CNN architecture~\cite{2021_Pesah} and circuits aimed at the study of quantum magnets~\cite{https://doi.org/10.48550/arxiv.2108.08086,Liu_2019}.
This article aims to provide a phenomenological theory underlying the efficiency of the training (i) within several optimization algorithms, (ii) in various regimes of model parameters, and (iii) as a function of optimization hyperparameters such as the learning rate $\eta$ or the number of gradient-estimating samples $N_s$ per optimization step.
We adopt the best-case scenario of a noise-free hardware setting and study the stochastic gradient descent (SGD) and the stochastic reconfiguration (SR) approach~\cite{Sorella_1998}, as implemented on quantum computers in the Quantum Natural Gradient Descent (QNGD) approach~\cite{stokes_quantum_2020}. In such a hardware-free noise setting, the remaining obstacle for the state preparation is the inherent statistical quantum measurement noise in the gradient estimation ($\boldsymbol \nabla \langle \hat H \rangle$) induced by a finite budget $N_s$ of shots, i.\,e. circuit repetitions~\cite{wecker2015progress,Torlai_2020,cerezo2021variational}.
In this scenario, we analyze how the performance of variational ground-state preparation is affected by the sample size $N_s$ per gradient component and the learning rate $\eta$ of a gradient descent step. First, we observe that the resulting fidelity shows a ``critical'' behavior as a function of a suitably-defined stochastic energy fluctuations measure $\epsilon \propto 1 / N_s$. Namely, we show that the fidelity vanishes when this measure is larger than a certain threshold value, $\epsilon > \epsilon^c$ and shows a rapid growth instead at $\epsilon < \epsilon^c$. {Further in the text, we refer to this sharp change in the algorithm performance as an {\it algorithmic phase transition} and, somewhat loosely, call this behavior {\it critical}. We emphasize that these algorithm performance regimes, as well as the transition between them and threshold (critical) $\epsilon_c$, stem primarily from the optimization hyperparameters such as $\eta,\,N_s$, and are not related to conventional phase transitions in a medium.} Notably, this critical behavior can be qualitatively reproduced for a given circuit with a simple distribution over the parameter space given by the Boltzmann distribution $\Pi(\boldsymbol \theta) \propto \exp \left(-E(\boldsymbol \theta) / T \right)$ with $T$ being the effective temperature of the system and $E(\boldsymbol \theta) = \langle \psi(\boldsymbol \theta)|\hat{H}| \psi(\boldsymbol \theta) \rangle$ being the energy expectation. Second, we address estimating the sample size required to reach a certain overlap with the ground state. For sufficiently large samples, we observe that the circuit infidelity behaves as $A \epsilon + \mathcal{I}_0$ with $\mathcal{I}_0$ representing the circuit's inability to exactly express the quantum state.
By considering numerical simulations of different two-dimensional frustrated spin-$1/2$ magnets, we observe that the prefactor $A$ has a universal behavior and, in the case of such systems, grows as $\sim 1 / \Delta^2$ with the spectral gap $\Delta$. Such dependence imposes a constraint on the minimum circuit shots number $N_s$ required to reach certain fidelity and on the class of quantum systems addressable with VQE. Based on this observation, we also discuss symmetry-based strategies to effectively increase $\Delta$, thus, in some cases, significantly reducing the required resources for the algorithm.
This paper is organized as follows. In Section\,\ref{sec:phenomenology}, we outline the phenomenological theory for the observed algorithmic phase transition and residual infidelity scaling. In Section\,\ref{sec:model_circuits}, we introduce a model for two-dimensional frustrated magnets and symmetry-enhanced circuits. Finally, in Section\,\ref{sec:results} we demonstrate numerical evidence for these claims, and we draw conclusion in Section\,\ref{sec:discussion}.
\section{Phenomenology of state preparation}
\label{sec:phenomenology}
\subsection{An effective stochastic temperature and algorithmic phase transition}
In stochastic optimization methods, such as variational Monte Carlo or gradient-based machine learning, it is known that the power spectrum of statistical noise, under certain assumptions, defines an effective temperature~\cite{Becca_2015,https://doi.org/10.48550/arxiv.1711.04623,NEURIPS2019_dc6a7071}. Concretely, we consider a variational circuit parametrized with a vector of parameters $\boldsymbol{\theta}$ and consider the update law representing the Stochastic Gradient Descent (SGD) approach
$
\boldsymbol{\theta}^{i + 1} = \boldsymbol{\theta}^i - \eta \nabla \langle \hat H \rangle
$
with $\eta$ being the learning rate. {In the actual multivariate VQE optimization, noise is governed by a $\boldsymbol \theta$--dependent and non-diagonal covariance matrix. Here, to analytically examine the stochastic VQE optimization dynamics,} we assume that $\nabla \langle \hat H \rangle = \mathcal{N}\left(\boldsymbol{f}, \sigma \right),$ i.\,e. that the forces on the parameters, $\boldsymbol{f}_k$, are distributed normally with a diagonal and uniform variance, $\sigma_k^2 \simeq \mbox{Var}\,\boldsymbol{f}_k / N_s$. As mentioned above, in this work, $N_s$ stands for the number of shots used for the estimation of each gradient component.
The effective parameters pseudodynamics is therefore given by a Langevin equation of the form
\begin{gather}
\label{eq:lang}
\boldsymbol{\theta}^{t + 1}_k = \boldsymbol{\theta}^t_k - \eta \boldsymbol{f}_k + \mathcal{N}(0, \eta \sigma_k),
\end{gather}
where the index $k$ enumerates components of the variational parameters vector of length $N_p$. Besides, we assume that the gradient variances, $\mbox{Var}\,\boldsymbol{f}_k$, are approximately equal for all parameters.\footnote{This assumption is numerically well verified in the case of the Ansatz states considered in the following}. Then, stationary solution of Eq.\,\eqref{eq:lang} is the Boltzmann distribution $\Pi(\boldsymbol{\theta}) \propto \exp \left(-E(\boldsymbol{\theta}) / T\right)$, where we defined
\begin{gather}
\label{eq:Teff}
T = \mbox{Var}\,\boldsymbol{f}_k \eta / N_s,
\end{gather}
as an effective temperature of the system. This definition agrees with the well-known expression for the effective temperature in variational Monte Carlo optimization~\cite{https://doi.org/10.48550/arxiv.1711.04623,NEURIPS2019_dc6a7071}.
Without loss of generality, we assume that
the ground state energy value is $E = 0$, and is also reached at $\boldsymbol \theta = \boldsymbol 0$. We thus write to the second order $E(\boldsymbol \theta) = \boldsymbol{\theta}^T \hat{D}\,\boldsymbol{\theta} / 2 = (1/2) \sum_k D_k \tilde{\boldsymbol{\theta}}_k^2$ where we consider a basis $\tilde{\boldsymbol{\theta}}$ delivering a diagonal form to $\hat{D}$~\cite{https://doi.org/10.48550/arxiv.1907.03215,160202666,https://doi.org/10.48550/arxiv.1710.11029}.
This implies that energy fluctuations are proportional to the number of parameters $N_p$ with $T / 2$ per degree of freedom\footnote{Strictly speaking, the energy fluctuation should be the sum of effective temperatures $\langle E(\boldsymbol \theta) \rangle = (1/2) \sum_k T_k$ associated to the system parameters. In our case, however, for simplicity we assume all temperatures equal.}, namely $\langle E(\boldsymbol \theta) \rangle = (1/2)\,N_p T$.
Below we will provide numerical evidence supporting the existence of an algorithmic phase transition as a function of the energy fluctuations measure $\epsilon = (1/2) N_p T$, separating the $\epsilon > \epsilon^c$ regime where the optimization is impossible from the $\epsilon < \epsilon^c$ regime where the algorithm finds sizable finite overlap with the ground state. Notably, the switch between the two regimes often occurs not through a smooth crossover but is instead characterized by a sharp phase transition with a well-defined $\epsilon^c$, also marked, similar to heat capacity behaviour in second-order phase transitions, by a peak in energy variance $\langle (E - \bar E)^2 \rangle$. We will also show that $\epsilon^c$ does not decrease exponentially with system size, keeping the state preparation feasible while approaching the thermodynamic limit.
We emphasize that our numerical experiment setup is different from the standard thermodynamic case, where both the energy fluctuations and the full problem Hamiltonian energy scale $\Lambda$ are proportional to the number of system degrees of freedom, i.\,e. are both extensive quantities. In our setup, $\Lambda$, being a characteristic of a problem and not of the Ansatz, is independent of $N_p$. In the meantime, energy fluctuations $\epsilon$, depending on the Ansatz, are linear in $N_p$. Intuitively, the ``trainable'' phase should satisfy $\epsilon \ll \Lambda$. Since the latter is independent of $N_p$, the magnitude of energy fluctuations $\epsilon = (1/2) N_p T$, and not temperature $T$, sets a measure detecting the algorithmic phase transition.
{Substituting non-diagonal $\boldsymbol \theta$--dependent covariance matrix with diagonal uniform variance $\sigma^2$ is a major approximation. We show, however, that sampling a ``parametric'' partition function
\begin{gather}
\label{eq:Z_thermal}
\mathcal{Z}_{\boldsymbol \theta} = \int \mbox{d}\boldsymbol{\theta} \exp \left(-E(\boldsymbol \theta) / T \right)
\end{gather}
can reproduce some features of the original stochastic VQE optimization, such as the mentioned parametrical regions with vanishing training fidelity and pronounced separation from the ``trainable phase'' with the transition point marked by a peak of energy variance. Thus, the reported behaviour manifests itself also within sampling the variational circuit parameters from the thermal Boltzmann distribution $\Pi(\boldsymbol \theta) \propto \exp \left(-E(\boldsymbol \theta) / T\right).$}
\subsection{A phenomenological scaling law for the residual infidelity}
In the $\epsilon < \epsilon^c$ regime, when the optimization reaches sizable fidelity, we propose the following empirical scaling law
for the residual infidelity $\mathcal{I} = 1 - |\langle \psi(\boldsymbol{\theta})|\psi_{0} \rangle|^2$:
\begin{gather}
\label{eq:infidelity}
\mathcal{I} = A \epsilon + \mathcal{I}_0 \, ,
\end{gather}
where $\mathcal{I}_0$ describes the `ideal' representational ineffectiveness of a given variational circuit. It has been demonstrated in an ideal exact gradient setup that $\mathcal{I}_0$ can be made arbitrarily small with a suitable choice of the Ansatz and corresponding circuit depth~\cite{https://doi.org/10.48550/arxiv.2108.08086,Choquette_2021,Mineh_2022}.
However, in addition to the limitations associated with the expressivity of the circuit, the measured state infidelity also stems from the finite sample size of gradient estimates per component, $N_s$.
\begin{figure*}[t!]
\centering
\includegraphics[width=\textwidth]{./panel_1.pdf}
\caption{(a-b) Fidelity and energy variance as a function of $N_s$ for various $j_2 / j_1$ measured on the $4 \times 4$ square lattice with OBC-PBC.\; (c) Fidelity as a function $\beta$ within simulation of the $4 \times 4$ square lattice at various $j_2 / j_1$ with direct sampling from thermal partition function Eq.\,\eqref{eq:Z_thermal}. (d) Critical number of samples $N_s^c$ as a function of learning rate $\eta$ on the $4 \times 4$ triangular lattice with PBC. (d, inset): Critical number of samples $N_s^c$ as a function of number of parameters $d$ on the $4 \times 4$ square lattice with PBC at $j_2 / j_1 = 0.4$. (e-f) Circuit fidelity as a function of $\epsilon / j_1$ within the $L \times 4$ setup at $j_2 / j_1 = 0.4$ and inverse critical fluctuation measure $(\epsilon^c / j_1)^{-1}$ as a function of $1 / L$.}
\label{fig:panel1}
\end{figure*}
We argue that this residual infidelity depends on the same effective energy fluctuations measure $\epsilon$ introduced above and on spectral properties of the system of interest, such as the gap to the first excited state $\Delta$. Our numerical experiments suggest a fast scaling with the inverse gap, which to a reasonable degree follows a $A \propto 1 / \Delta^2$ behaviour.
This proportionality implies an increasing optimization complexity for systems with a closing gap.
This dependence is similar to the one of the Adiabatic Theorem, which imposes the relation $\tau \propto 1 / \Delta^2$ on the smallest time extent of quantum annealing~\cite{sarandy2005consistency}. In the following, we also provide a recipe allowing one to ameliorate this problem by employing symmetry projections~\cite{Seki_2020}.
\section{Model and quantum circuits}
\label{sec:model_circuits}
We perform numerical experiments on the $j_1-j_2$ Heisenberg spin--$1/2$ model on a series of two-dimensional lattices.
We place both model and geometries in focus since (i) the ground state of this model under certain conditions realizes a quantum spin liquid, an exotic and long-sought phase of matter~\cite{savary_quantum_2016,zhou_quantum_2017}, (ii) tuning the couplings ratio $j_1/j_2$ allows one, on each of these geometries, to explore different model regimes, and to trigger gap closing, (iii) lattice models allow for a controllable study of the thermodynamic limit.
The spin--$1/2$ Heisenberg model is described by the Hamiltonian
\begin{equation}
\label{eq:hamiltonian}
\hat H = j_1 \sum\limits_{\langle i,j \rangle} \hat{\mathbf{S}}_i \cdot \hat{\mathbf{S}}_j + j_2 \sum\limits_{\langle \langle i,j \rangle \rangle} \hat{\mathbf{S}}_i \cdot \hat{\mathbf{S}}_j,
\end{equation}
with $\langle \ldots \rangle$ representing $j_1$--bonds and $\langle \langle \ldots \rangle \rangle$ representing the $j_2$--bonds. The range of ratios $j_2 / j_1$ is chosen, in each particular case, to interpolate between frustrated and magnetically-ordered regimes. See Appendix~\ref{appendix:geometries} for the definition of the lattice geometries and for the description of the parameter regimes for which gap closing is expected.
For numerical experiments with SGD and SR techniques, we employ a symmetry-enhanced Ansatz~\cite{Seki_2020}. The spin-spin interaction in the Hamiltonian (see Eq.\,\eqref{eq:hamiltonian}) can be replaced by the {\it SWAP operator} $\hat P_{ij}~=~\frac{1}{2} \left( \hat{\mathbf{S}}_i \cdot \hat{\mathbf{S}}_j + \hat{\bbone}\right)$, which induces an exchange of spin states between sites $i$ and $j$. If the state is prepared in the total spin-zero $S = 0$ sector, such as the dimer product state
\begin{equation}
\label{eq:dimerized}
|\psi_D\rangle = \bigotimes_{0 \leqslant i < N / 2} \frac{1}{\sqrt{2}} \left(|\uparrow_{2 i} \downarrow_{2i + 1} \rangle - | \downarrow_{2 i} \uparrow_{2i + 1}\rangle \right),
\end{equation}
action of eSWAP operators $\exp \left(i \theta \hat{P}_{ij}\right)$ preserves the total spin and the wave function reads
\begin{equation}
\label{eq:wavefunction}
|\psi\rangle(\boldsymbol \theta) = \left(\prod_{\alpha} e^{i \theta_{\alpha} \hat{P}_{i_{\alpha} j_{\alpha}}} \right) |\psi_D\rangle,
\end{equation}
where to define the $(i_{\alpha},\,j_{\alpha})$ pairs we employ checkerboard decomposition of the Hamiltonian Eq.\,\eqref{eq:hamiltonian}. Specifically, in the case of a square lattice, the Hamiltonian can -- up to a constant -- be written as a sum of $L \times 4$ SWAP operators acting between pairs of qubits. We split this set of pairs into $8$ layers of full coverings (when each qubit belongs to exactly one pair), and treat these pairs as $(i_{\alpha},\,j_{\alpha})$. Note that all eSWAP operators within one layer can be applied simultaneously. In addition one can also fix the spatial point group symmetry representation of the wave function (for details of such fixation, optimization protocol and the definition $(i_{\alpha},\,j_{\alpha})$ in the triangular and hexagonal cases, see Appendix\,\ref{appendix:symmetric_circuit}). This Ansatz allows one to effectively change $\Delta$ (if symmetry is imposed, effective $\Delta$ is the energy of the first excited state {\it in the respective irreducible representation}) and improves the ability to express the ground state (decreases $\mathcal{I}_0$), keeping $N_p$ relatively low.
Importantly, the introduced Ansatz only shows a mild decay of the gradient norms with system size, signaling the absence of the so-called barrens plateau issue, proven to emerge in a generic quantum circuit setup~\cite{McClean_2018}.
The proposed circuit has a sub-exponential decay, similarly to the case of the transverse field Ising model reported in~\cite{PRXQuantum.1.020319}, as we demonstrate in Appendix~\ref{appendix:absence_of_barren_plateaus}. Other barren plateaus-free circuits aimed at the study of frustrated two-dimensional magnets were also previously reported~\cite{https://doi.org/10.48550/arxiv.2108.08086,Liu_2019}.
\section{Results}
\label{sec:results}
\subsection{Small sample size regime: critical behaviour}
The optimization of the circuit parameters requires the estimation of the energy gradient with respect to gate parameters $\boldsymbol \theta$, for which a circuit is being executed $N_s$ times per gradient component. Larger $N_s$ leads to better estimates for the wave function projections such as the fidelity $\mathcal{F} = |\langle \psi(\boldsymbol \theta) | \psi_0\rangle|^2$ (i.\,e., the squared overlap with the ground state), as well as for expectation values of the form $\langle \psi(\boldsymbol \theta) |\hat O| \psi(\boldsymbol \theta) \rangle$ for any given observable $\hat O$. A naively expected behavior in such a case would be a gradual growth of fidelity with $N_s$. However, Ref.\,\cite{2020_twesterhout}, which studies Neural Quantum States (NQS)\,\cite{2017_gcarleo} (an ansatz class used in \textit{classical} computing) reported a sharply different behavior with nearly zero fidelity for $N_s < N_s^c$ smaller than some critical $N_s^c$ number of samples\footnote{Another prominent example of an algorithmic phase transition occurring in an optimization problem is the so-called {\it jamming transition} reported for an artificial Neural Network training~\cite{Spigler_2019}.}.
\begin{figure*}[t!]
\centering
\includegraphics[width=\textwidth]{./panel_2.pdf}
\caption{Large-$N_s$ regime study of the $4 \times 4$ square lattice. (a) Infidelity as a function of $N_s$ for a set of depths at $j_2 / j_1 = 0.4$. The dashed line show fit with the Ansatz Eq.\,\eqref{eq:infidelity}. (b) Infidelity as a function of $\eta$ for a set of $j_2 / j_1$ at $D = 8$.}
\label{fig:panel2}
\end{figure*}
\subsubsection{Manifestation of the critical behavior}
To test which scenario (a smooth transition or a critical behavior) is realized with the quantum state preparation, we perform optimization with stochastic gradient descent (SGD) in the small--$N_s$ regime\footnote{In this regime of small $N_s$, the metric tensor inversion requires a very large regularization (see Appendix\,\ref{appendix:symmetric_circuit}), effectively turning SR into SGD.}. We plot the circuit fidelity (or, similarly, infidelity) throughout the Results section on the vertical axis. To obtain the circuit fidelity, we employ the following protocol. Starting from a random set of parameters, we optimize a circuit using the SGD or SR approach until the sliding mean of fidelity with a window of several hundred steps stabilizes. After such convergence, we use the mean over the next thousand iteration steps as the circuit fidelity. The procedure is repeated ten times to estimate the error bars. In Fig.\,\ref{fig:panel1}\,(a) we show the state fidelity as a function of $N_s$ for various $j_2 / j_1$ on the $4 \times 4$ square lattice with open boundary conditions (OBC) in the first and periodic (PBC) in the second dimension to avoid geometrical frustrations in the $j_2 / j_1 = 0$ pure nearest-neighbor case. One can see that the fidelity remains vanishingly small for $N_s < N_s^c$, followed by a rapid growth afterwards. Such drastic change of pattern is suggestive of a {\it critical behaviour}. To explore this transition in $N_s$, in Fig.\,\ref{fig:panel1}\,(b) we show $c_V / N_s^2 = \mbox{Var}\,E$ as a function of $N_s$. We observe that the peak of $\mbox{Var}\,E$ coincides precisely with $N_s^c$, defined as the departure point from the zero-fidelity regime.
The effective specific heat $c_V$ also shows a peak in this region. This quantity generalizes the thermal specific heat $c^{\beta}_V = \beta^2\, \mbox{Var}\, E$, where we assume that the inverse number of samples, $1 / N_s$, plays the role of an effective temperature. We observe a similar critical behaviour across other cluster dimensions and geometries under consideration. Notably, the reported algorithmic phase transition is also accompanied by a qualitative change in the distribution of the overlaps with higher excited states $\mathcal{O}_k = |\langle \psi(\boldsymbol{\theta}) | \psi_k \rangle|^2$. Namely, it changes from being peaked at $1 / |\mathcal{H}|$ (with $|\mathcal{H}|$ being the size of the Hilbert space) to be much wider (see Appendix.\,\ref{appendix:distribution_overlaps}). Thus, at small $N_s$, the circuit learns {\it uniform} overlap with all eigenstates, while at $N_s > N_s^c$ the circuit favors only several low-lying excitations. This is consistent with the argument outlined in Section.\,\ref{sec:phenomenology}, based on the thermal partition function.
{To substantiate this, we simulate the ``parametric'' partition function $\mathcal{Z}_{\boldsymbol \theta}$ defined in Eq.\,\eqref{eq:Z_thermal} using a Metropolis algorithm that performs random walks in the parameter space.}
In Fig.\,\ref{fig:panel1}\,(c) we show the average fidelity computed along the generated Markov chain at different temperatures for the $4 \times 4$ square lattice. We keep the number of parameters fixed and thus present the data as a function of $\beta = 1/T$. Even though the growth after the critical value $\beta_c$ is not as sharp as observed in Fig.\,\ref{fig:panel1}\,(a), we notice also in this case a region of nearly zero fidelity for $\beta < \beta_c$. Similarly to the case in Fig.\,\ref{fig:panel1}\,(b), the specific heat $c^{\beta}_V$ has a pronounced peak at $\beta_c$. The histogram of overlaps $\mathcal{O}_k$ shows a similar qualitative change between the two regimes (see Appendix\,\ref{appendix:distribution_overlaps} for details).
{We note that the abruptness of growth at $N_s > N_s^c$ ($\beta > \beta_c$) is clearly dependent on $j_2 / j_1$, as seen from contrasting the $j_2 / j_1 = 0.6$ and $j_2 / j_1 = 0.0$ curves demonstrated in Fig.\,\ref{fig:panel1}\,(a, c). However, clear separation between the two algorithmic regimes marked by a peak of energy variance, remains present.}
\begin{figure*}[t!]
\centering
\includegraphics[width=\textwidth]{./panel_3.pdf}
\caption{(a) Dependence of the prefactor $A$ introduced in Eq.\,\eqref{eq:infidelity} on the system spectral gap. The symbols denote: {\it sq. (tr.), $4^2$} --- square $4^2$ lattice case with only translational symmetry imposed, {\it sq. (pg.), $4^2$} --- same with only point symmetry group imposed, {\it sq. (all), $4^2$} --- same with both translation and point group symmetries imposed, {\it tr.-(1), $4^2$} --- triangular $4^2$ lattice with full symmetry imposed with starting dimerization along the $j_1$--bonds, {\it tr.-(2)} --- same with dimerization along the $j_2$--bonds, and {\it hex., $3^2$} --- $3^2$ hexagonal lattice with full symmetry group imposed. (b) On the $4 \times 4$ square lattice, dependence the system gap $(j_1 / \Delta)^2$ on $j_2 / j_1$ in two cases: (i) with only point group symmetry imposed, (ii) with only translational symmetry imposed. (c): same for the resulting prefactor $A$.}
\label{fig:panel3}
\end{figure*}
To verify the effective temperature expression Eq.\,\eqref{eq:Teff}, in Fig\,\ref{fig:panel1}\,(d), we plot $N_s^c$ as a function of $\eta$ for the $4 \times 4$ triangular lattice and observe a linear dependence on learning rate $\eta$. This supports the linear $\eta$ proportionality in Eq.\,\eqref{eq:Teff}. The inset shows $N_s^c$ as a function of $N_p$. As expected, we observe a linear dependence, while the variance per parameter, $\frac{1}{N_p} \sum_k \mbox{Var}\,\boldsymbol{f}_k$, remains mostly unchanged. This behavior is also observed in other lattice geometries. This shows a pronounced criticality in the energy fluctuations measure $\epsilon = (1/2) T N_p$, rather than in the mere effective temperature $T$.
\subsubsection{Critical behavior in the thermodynamic limit}
We aim to extrapolate the critical energy fluctuations measure $\epsilon^c$ to the thermodynamic limit. To this end, we consider $L \times 4$ square lattices with $L = 3,\,4,\,5$ and $6$. As circuit architecture, we select the 8 checkerboard decomposition layers of the Hamiltonian in Eq.\,\eqref{eq:hamiltonian}. Thus, due to the high computational cost otherwise, the depth of the circuit does not scale with $L$.
Additionally, OBC in the $L$--direction and PBC in the second direction are imposed, ensuring the absence of geometric frustrations. In Fig.\,\ref{fig:panel1}\,(e) we report the state fidelity as a function of $\epsilon$, while in Fig.\,\ref{fig:panel1}\,(f), we show the dependence of $\epsilon^c$ on the lattice dimension $L^{-1}$. We observe a saturation with the increase of $L$, and note that the critical energy fluctuations do now grow exponentially with the system size\footnote{We note that $\epsilon_c / j_1$ varies only mildly with $j_2 / j_1$, while the system gap has a two orders of magnitude difference between $j_2 / j_1 = 0.0$ and $0.6$. This suggests that in the trainable phase definition $\epsilon \ll \Lambda$ given above, $\Lambda$ is not an energy gap of $\hat H$, but rather some other energy scale of the system.}. Note, that the curves in Fig.\,\ref{fig:panel1}\,(e) approach different final asymptotic values $\mathcal{F}_{\infty}$ for $\epsilon < \epsilon^c$, while ``fair'' extrapolation to thermodynamic limit should ensure non-vanishing (preferably same) saturation. {Thus, as $L$ increases, one needs to employ deeper circuits, which would correspondingly increase $\epsilon^c$ at a given $L$. However, due to high computational cost, we refrain from scaling up the circuit depth at large cluster volumes $L = 5,\,6$. Nevertheless, we believe that such an increase in the circuit depth would not change the overall sub-exponential scaling character of $\epsilon^c(L)$. For instance, Ref.\,\cite{bosse2021probing} reports that, in order to reach a certain fidelity $\mathcal{F}$ with the ground state on the frustrated $N$--site kagome lattice, only $\propto \sqrt{N}$ gates are necessary. Thus, with such depth adjustment, saturating behavior of $\epsilon^c(L)$ shown in Fig.\,\ref{fig:panel1}\,(f) can only change to polynomial dependence on $L$, and not to exponential growth.}
The main practical quantity of interest is $N_{\mbox{\footnotesize tot}}(L) = N_{\mbox{\footnotesize SGD}}(N_s^c) \times N_s^c$, the total number of samples required to train a circuit in the course of $N_{\mbox{\footnotesize SGD}}(N_s^c)$ SGD iterations, which is proportional to the required hardware resources. We define $N_{\mbox{\footnotesize SGD}}(N_s)$ as the number of SGD steps needed to reach 90\,\% of the saturated fidelity.
We observe that $N_{\mbox{\footnotesize SGD}}(N_s)$ required to saturate $\mathcal{F}_{\infty}$ (as defined in Fig.\,\ref{fig:panel1}\,(e)) shows no growth with $L$ (see Appendix.\,\ref{appendix:N_SGD} for details).
Thus, the obtained scaling for $\epsilon^c(L)$ results into polynomial scaling of $N_{\mbox{\footnotesize tot}}(L)$. Note that reducing $\eta$ will allow reaching $\epsilon^c$ with a smaller number of samples per SGD iteration. However, such convergence would require proportionally more SGD steps $N_{\mbox{\footnotesize SGD}}$, making $N_{\mbox{\footnotesize tot}}(L)$ a true lower bound on the required total number of samples.
\subsection{Large sample size regime}
In this section, we provide numerical evidence to validate Eq.\,\eqref{eq:infidelity_gap} in the regime of accurate gradients. We employ the symmetrized wave function, and the Stochastic Reconfiguration (SR) approach proposed in~\cite{Sorella_1998}. In this section, $N_s$ has the meaning of the number of shots per gradient component and per symmetry projection (see Appendix\,\ref{appendix:symmetric_circuit} for projections definition).
We consider the $4 \times 4$ square lattice at $j_2 / j_1 = 0.4$ with a variable circuit depth. The circuit is built of repetitive applications of the checkerboard decomposition of the Hamiltonian in Eq.\,\eqref{eq:hamiltonian} (a single decomposition is described in Section\,\ref{sec:model_circuits}), restricting ourselves to a maximum of $N_p$ optimization parameters. The circuit depth is then $D = N_p / 8$. In Fig.\,\ref{fig:panel2}\,(a) we present the infidelity as a function of $N_s$. To verify the proposed functional form for $\mathcal{I}(N_s)$, the data are fitted with the expression in Eq.\,\eqref{eq:infidelity}. To verify the power law, we replace $N_s \to N_s^{\alpha}$ and observe $\alpha \sim 1$ within error bars consistently within the fitting procedure (for details, see Appendix\,\ref{appendix:fit_parameters}). Finally, in Fig.\,\ref{fig:panel2}\,(b) we fit infidelity with a linear function $C \eta$. These fits support the functional form Eq.\,\eqref{eq:infidelity} and provide an empirical way to verify the $1 / N_s$ power law dependence of the residual infidelity.
In the following, we investigate the dependence of the prefactor $A$ on the spectral gap $\Delta$. To this end, in Fig.\,\ref{fig:panel3}\,(a) we present the offset $A$ against $(\Delta / j_1)^{-1}$ for the different two-dimensional lattices considered in this study. {We consider different lattice geometries, different starting dimerization patterns, and symmetry projectors.} The fit with $A_0 \Delta^{\alpha}$ yields $\alpha \sim -2$. The black dashed line shows the $A_0 / \Delta^2$ fit of the data. {We note that in the cases of other spin systems that we considered, such as the $j_1$--$j_2$ Heisenberg chain, we observed a scaling law significantly different from $1 / \Delta^2$ and being highly depth-dependent. We thus emphasize that our claim $\alpha \sim -2$ only concerns two-dimensional frustrated magnets.}
From this observation, it follows that for increasingly smaller gaps, the systems become intractable, in the sense that it becomes harder to converge to the true ground state. However, imposing symmetrization of the Ansatz (see Appendix.\,\ref{appendix:symmetric_circuit}) can alleviate this problem. We illustrate this approach in the case of the $4 \times 4$ square lattice, where the parameter regime $j_2 / j_1 \to 0.6$ leads to a nearly closing gap, accompanied by the emergence of a quantum spin liquid~\cite{Nomura_2021}. Namely, we contrast the cases of (i) only point group symmetry imposed with $8$ terms in the projector against (ii) only translations imposed with $16$ terms in the projector. As seen in Fig.\,\ref{fig:panel3}\,(b) at $j_2 / j_1 \to 0.6$, $(j_1 / \Delta)^2$ diverges for case (ii), while it remains well-behaving for case (i). This results in dramatically different behaviour for $A(j_2 / j_1)$, with the latter being moderate for (i) and exploding for (ii), as seen in Fig.\,\ref{fig:panel3}\,(c). This improvement in the prefactor $A$ magnitude emphasizes the importance of symmetries in the state preparation process of systems with a vanishing gap. In summarizing, the residual state infidelity, which is not due to the lack of representability of the chosen Ansatz, but rather to the optimization process, scales as
\begin{gather}
\label{eq:infidelity_gap}
\mathcal{I}-\mathcal{I}_0 \propto {\epsilon \over \Delta^2}.
\end{gather}
where $\epsilon = (1/2) T N_p$ with $T$ defined in terms of sampling shots $N_s$, and learning rate $\eta$, as in Eq.\,\eqref{eq:Teff}.
\section{Discussion}
\label{sec:discussion}
Variational state preparation is essential for various quantum computing algorithms in near-term and fault-tolerant regimes.
Here we show that, besides the most apparent dependency on the circuit architecture, the fidelity of the prepared state strongly depends on learning hyperparameters and the system-dependent properties.
We layout a phenomenology of state preparation as a function of two significant factors, (i)
the number of samples $N_s$ used to estimate the gradient of the variational parameters in an SGD step, and (ii) the fundamental gap $\Delta$ of the model Hamiltonian.
In particular, we explored the interplay of these two parameters, focusing on challenging two-dimensional spin-$1/2$ frustrated quantum magnets.
We observe that, in the regime of small $N_s$ (noisy gradient) the stochastic optimization shows critical behaviour with near-zero state fidelity for $N_s < N_s^c$ and rapid growth of fidelity at $N_s > N_s^c$. {The pace of fidelity growth is highly geometry- and parameter regime-dependent. However, the zero-fidelity region is always present and is pronouncedly separate from the ``trainable phase''. Besides, the point of transition, $N_s^c$, always features a peak of energy variance $\langle (E - \bar E)^2 \rangle$, resembling heat capacity behaviour in second-order phase transitions.} Together with the notion of effective temperature, this separation allows us to discuss an effective algorithmic phase transition on the energy fluctuations measure $\epsilon = (1/2) N_p T$ axis. In the case of a two-dimensional square lattice, we found evidence that the critical energy fluctuation $\epsilon_c$ scales only polynomially with the system size (see Fig.\,\ref{fig:panel1}\,(f)), providing the basis for possible applications of VQE in the study of larger-size frustrated magnets, inaccessible to classical algorithms.
To support the notion of effective temperature, we show that the observed criticality and energy variance peak can be reproduced within sampling from a simpler ``parametric'' partition function Eq.\,\eqref{eq:Z_thermal} with the classical circuit parameters $\boldsymbol \theta$ distributed according to the Boltzmann weight.
We thus provide a simplified picture explaining the reported algorithmic phase transition in $\epsilon$ and justifying the notion of an effective temperature of VQE optimization.
We have also observed that the approach to the exact ground state in the large--$N_s$ limit depends heavily on the system spectral properties. Namely, in the studies case of two-dimensional frustrated magnets, the $N_s$--dependent contribution to the residual infidelity Eq.\,\eqref{eq:infidelity} scales as $1 / \Delta^2$ with $\Delta$ being the energy gap of the studied system. We emphasize that this empirical dependence is only applicable to two-dimensional magnets. Even though we express $A$ as the function of the gap to the first excited state, $\Delta$, the $1/\Delta^2$ scaling is also related to the growing contributions to the infidelity coming from the higher-excited states. Explanation of the $A = 1/\Delta^2$ prefactor, as well as the study of $A$ in other physically-relevant systems, is subject to future research.
This $1 / \Delta^2$ scaling poses a significant obstacle for the study of systems with a closing gap. To address this problem, we showed how a symmetry-enhanced wave function -- in addition to being not susceptible to the barren plateaus issue -- can, in some cases, mitigate the effects of a closing gap. Namely, imposing symmetry projection can increase the effective system gap $\Delta$ in this symmetry sector, relieving the exploding $A = 1/\Delta^2$ prefactor and providing a significant improvement in the algorithm efficiency, which can be quantified in several orders of magnitude. This development will open up new possibilities for using VQE to simulate complex many-body systems with near-term quantum computers, which could otherwise be intractable due to a closing gap.
\section{Acknowledgements}
We are sincerely grateful to T.\,Westerhout for useful discussions and help with the technical simulation setup. We thank Titus Neupert for helpful comments on our manuscript. Numerical simulations used the high-performance package \texttt{lattice{\_}symmetries}~\cite{westerhout2021latticesymmetries} for quantum state vectors manipulation. N.\,A is funded by the Swiss National Science Foundation, grant number: PP00P2{\_}176877.
\section{Appendix}
\subsection{Two-dimensional geometries}
\label{appendix:geometries}
\begin{figure}[h!]
\centering
\includegraphics[width=0.49\textwidth]{./geometries.pdf}
\caption{Geometries of lattices considered in this work. Solid lines represent $j_1$--bonds, while $j_2$ bonds are represented by dashed lines. (a) Square lattice, (b) triangular lattice, (c) hexagonal lattice.}
\label{fig:geometries}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=0.49\textwidth]{./circuit.pdf}
\caption{Circuit for measurement of $\langle \psi | \hat h_j \hat g_k | \partial_i \psi \rangle$. The Hadamard scheme applied to the ancilla qubit allows to obtain real and (if necessary for connection and metric tensor) imaginary part of $\langle \psi | \hat h_j \hat g_k | \partial_i \psi \rangle$}
\label{fig:circuit}
\end{figure}
In Fig.\,\ref{fig:geometries} we show the geometries considered in this work: square, triangle, and hexagonal lattices. The $j_1$ bonds are marked as solid lines, while the $j_2$ bonds are shown as dashed lines. Each lattice shows a near-vanishing gap to the first excited state in the $S = 0$ sector in some region of $j_2 / j_1$. For instance, in the case of a square lattice, the vicinity of the $j_2 / j_1 \approx 0.55$ fluctuations melt magnetic orders and result in a frustrated phase, possibly gapless QSL~\cite{Nomura_2021}. On the triangular lattice, gap shrinks in the vicinity of $j_2 / j_1 = 1.0$, where the model is most frustrated~\cite{Iqbal_2016} and in the vicinity of $j_2 / j_1 = 0.2$ on the hexagonal lattice~\cite{PhysRevB.88.165138}.
\subsection{Symmetric circuit routines}
\label{appendix:symmetric_circuit}
{\it Symmetrized wave function:} The in the Hamiltonian Eq.\,\eqref{eq:hamiltonian}, spin-spin interaction can be replaced with the {\it SWAP operator} $\hat P_{ij}~=~\frac{1}{2} \left( \hat{\mathbf{S}}_i \cdot \hat{\mathbf{S}}_j + \hat{\bbone}\right)$, exchanging spin states on the site $i$ and $j$. Notably, the SWAP operator commutes with the total spin operator $[\hat P_{ij}, \hat{S}^2] = 0$, allowing for working in the wave function sector with fixed total spin~\cite{Seki_2020}. As the result, the symmetry-enhanced Ansatz for the system of $N$ spins is constructed by (1) first preparing the system in the simple {\it fully-dimerized} state
\begin{equation}
\label{eq:dimerized}
|\psi_D\rangle = \bigotimes_{0 \leqslant i < N / 2} \frac{1}{\sqrt{2}} \left(|\uparrow_{2 i} \downarrow_{2i + 1} \rangle - | \downarrow_{2 i} \uparrow_{2i + 1}\rangle \right),
\end{equation}
being the direct product of $N / 2$ dimer pairs\footnote{The exact dimerization pattern is chosen to maximize overlap with the ground state} and (2) action of the string of parametrized eSWAP operators preserving the total spin $S = 0$\footnote{To construct wave function in another spin sector, one needs to prepare one (or more) electron pairs in spin-triplet state $|\uparrow \uparrow\rangle$.}:
\begin{equation}
\label{eq:wavefunction}
|\psi\rangle(\boldsymbol \theta) = \left(\prod_{\alpha} e^{i \theta_{\alpha} \hat{P}_{i_{\alpha} j_{\alpha}}} \right) |\psi_D\rangle.
\end{equation}
\begin{figure}[t!]
\centering
\includegraphics[width=0.49\textwidth]{./appendix_I0_alpha.pdf}
\caption{Depth dependence of the power $\alpha$ at $j_2 / j_1 = 0.4$ at the $4 \times 4$ square lattice. (Inset): $\mathcal{I}_0$.}
\label{fig:I0_alpha}
\end{figure}
To define $i_{\alpha}$ and $j_{\alpha}$, i.\,e., we employ checkerboard decomposition of the Hamiltonian Eq.\,\eqref{eq:hamiltonian}. Namely, in case of a square lattice, we split $4 \times N$ Hamiltonian terms into $8$ layers of full dimerizations, when all eSWAP operators within one layer can be applied simultaneously\footnote{In the triangle lattice case, to construct the ($i_{\alpha}$, $j_{\alpha}$) pairs, we effectively add the ``missing'' $j_2$ bonds (with zero exchange interaction) to recover the square lattice setup and employ the same decomposition. In the case of the $3 \times 3$ hexagonal lattice, the $j_1$ and $j_2$ bonds groups are split into incomplete dimerization coverings (including no all spins at once) with parallel dimers.}. The spatial symmetry projector operator is defined as
\begin{gather}
\label{eq:symmetrization}
\hat P = \frac{1}{|G|} \sum\limits_{g \in G} \chi_g \hat g,
\end{gather}
where $G$ is the spatial symmetry group, consisting of the elementary unitary permutations $\hat g$ and $\chi_g$ are the characters, depending on the desired projection quantum number. The projected wave function $|\psi_P(\boldsymbol \theta)\rangle = \frac{\hat P}{\mathcal{N}(\boldsymbol \theta)} |\psi(\boldsymbol \theta)\rangle$ is normalized with $\mathcal{N}(\boldsymbol \theta) = \langle \psi(\boldsymbol\theta) | \hat P | \psi(\boldsymbol \theta) \rangle$. The energy gradient reads
\begin{gather}
\label{eq:gradient}
\partial_i \langle E(\boldsymbol \theta) \rangle = 2 \mbox{ Re} \left[ \frac{\langle \psi(\boldsymbol \theta) | \hat H \hat P | \partial_i \psi(\boldsymbol \theta)\rangle}{\mathcal{N}(\boldsymbol\theta)} - \mathcal{A}_i(\boldsymbol \theta) \langle E(\boldsymbol \theta) \rangle \right],
\end{gather}
where $\mathcal{A}_i(\boldsymbol \theta) = \frac{1}{\mathcal{N}(\boldsymbol \theta)} \langle \psi(\boldsymbol \theta) | \hat P | \partial_i \psi(\boldsymbol \theta) \rangle$ is the {\it connection}.
Finally, the metric tensor for natural gradient descent is defined as
\begin{equation}
\label{eq:metric_tensor}
G(\boldsymbol \theta)_{ij} = \frac{\langle \partial_i\psi(\boldsymbol \theta) | \hat P | \partial_j \psi(\boldsymbol \theta)\rangle}{\mathcal{N}(\boldsymbol\theta)} - \mathcal{A}^*_i(\boldsymbol \theta) \mathcal{A}_j(\boldsymbol \theta),
\end{equation}
and used to improve energy gradient $\boldsymbol \theta_{k + 1}=\boldsymbol \theta_k - \eta \sum\limits_j \left(\mbox{Re}\, G(\boldsymbol \theta) \right)^{-1}_{ij} \partial_j \langle E(\boldsymbol \theta) \rangle$ in the spirit of imaginary time evolution within stochastic reconfiguration (SR)~\cite{Sorella_1998}. The metric tensor obtained within sampling is regularized $G_{\mbox{\footnotesize reg}} = \sqrt{G G} + \beta \hat{\bbone}$ as suggested in Ref.\,\cite{gacon2021simultaneous}.
\begin{figure}[t!]
\centering
\includegraphics[width=0.49\textwidth]{./barren_plateaus.pdf}
\caption{The gradient magnitude per parameter and qubit $||\nabla E|| / (j_1 N_p N_q)$ within the $4 \times 4$ setup on the square lattice with $j_2 / j_1 = 0.4$ as a function $N_p$. Inset: $L \times 4$ setup on the square lattice at $j_2 / j_1 = 0.4$ as a function of $L$.}
\label{fig:barren_plateaus}
\end{figure}
{\it Sample quantum circuit:} In the course of optimization, it is required to measure expectation values of the kind $\langle \psi(\boldsymbol \theta) | \hat{h}_j \hat{g}_k | \partial_i | \psi(\boldsymbol \theta)\rangle$ with $\hat{h}_j$ being $j$--th Hamiltonian term (unitary SWAP operator) and $\hat{g}_k$ being $k$--th unitary permutation. The quantum circuit used to measure such quantity is shown in Fig.~\ref{fig:circuit}. Note that any $\hat{g}_k$ permutation can be written as product of note more than $N - 1$ pair SWAP operators.
\subsection{Large--$N_s$ fit parameters}
\label{appendix:fit_parameters}
To further verify the large--$N_s$ behavior in the variable depth scenario, we introduce additional optimization parameter $\alpha$ with $N_s \to N_s^{\alpha}$ representing possible deviation of the power from being strictly $\alpha = 1$. In Fig.\,\ref{fig:I0_alpha} we show, $\alpha$ and $I_0$ (inset) as functions of $D$ for $j_2 / j_1 = 0.4$ at the $4 \times 4$ square lattice. Inability to express the ground state $I_0$, expectedly, decays exponentially with $D$. In the meantime, $\alpha$ remains close to $1$.
\begin{figure}[t!]
\centering
\includegraphics[width=0.49\textwidth]{./appendix_histogram.pdf}
\caption{Histogram of overlaps with excited states $O_k = |\langle \psi_k | \psi(\boldsymbol \theta) \rangle|^2$ for $k$ corresponding to the lowest 100 states in the $S = 0$ sector measured at $j_2 / j_1 = 0.4$ on $4 \times 4$ square lattice with PBC at $N_s = 6$ and $N_s = 8$.}
\label{fig:appendix_histogram}
\end{figure}
\subsection{Absence of barren plateaus}
\label{appendix:absence_of_barren_plateaus}
We investigate the possible presence or absence of barren plateaus in our small--$N_s$ setup. To this end, in Fig.\,\ref{fig:barren_plateaus} we plot the gradient magnitude per parameter and qubit $||\nabla E|| / (j_1 N_p N_q)$ (i) within the $4 \times 4$ square lattice setup at $j_2 / j_1 = 0.4$ as a function of $N_p$ and (ii) within the $L \times 4$ setup on square lattice at $j_2 / j_1 = 0.4$ as a function of $L$. To obtain the gradient magnitude, we individually draw each angle from the uniform distribution $\theta_k \sim \mathcal{U}(-0.0, 0.1)$, while the starting state is fully dimerized. We then average over 100 such random starting points. We observe that in the first setup, the average gradient per parameter increases and saturates since a deep circuit allows one to rapidly change the outcoming quantum state by only small variations of parameters. In the meantime, in the second setup, the gradients decay sub-exponentially, similarly to the transverse field Ising model case reported in~\cite{PRXQuantum.1.020319}, which is in line with other reported non-exponential decay of gradients with system volume in the case of specific circuits~\cite{https://doi.org/10.48550/arxiv.2108.08086,Liu_2019}.
\subsection{Algorithmic phase transition details}
\label{appendix:distribution_overlaps}
\begin{figure}[t!]
\centering
\includegraphics[width=0.49\textwidth]{./thermal_cv.pdf}
\caption{Energy variance $\langle (E - \hat{E})^2 \rangle$ as a function of $\beta$ on the $4 \times 4$ square lattice within sampling from the classical parametrical partition function $\mathcal{Z}_{\boldsymbol \theta}.$ The positions of maxima correspond to transition between the zero-overlap and trainable regimes.}
\label{fig:cv_therm}
\end{figure}
In this Appendix, we provide additional details concerning observed algorithmic phase transition with energy fluctuation $\epsilon$. In Fig.\,\ref{fig:cv_therm} we plot energy variance $\langle (E - \hat{E})^2 \rangle$ obtained on the $4 \times 4$ square lattice within sampling from the classical parametrical partition function $\mathcal{Z}_{\boldsymbol \theta}.$ The maxima of the energy variance coincide precisely with the separation between zero-overlap and and trainable phases observed in Fig\,\ref{fig:panel1}\,(c).
To get further insight into transition with energy fluctuation $\epsilon$, in Fig.\,\ref{fig:appendix_histogram} we show histograms of fidelity of first 100 $S = 0$ low-lying excited states above and below transition at $j_2 / j_1 = 0.4$ on $4 \times 4$ square lattice. We see that below transition, histogram is very narrow with $\sigma = 10^{-3}$ and small average overlap of $\mu = 10^{-3} \approx 1 / |\mathcal{H}|$ with $|\mathcal{H}|$ being the $S = 0$ Hilbert space size of the problem. This indicates that all states that can possibly have overlap with our Ansatz wave function $|\psi(\boldsymbol \theta)\rangle$ are nearly of the same fidelity with small variation. On contrary, after transition, the histogram widens to $\sigma = 0.036$, showing that only few low-energy states dominate fidelity.
\subsection{Number of iterations to saturation}
\label{appendix:N_SGD}
\begin{figure}[t!]
\centering
\includegraphics[width=0.49\textwidth]{./appendix_N_SGD.pdf}
\caption{The number of SGD steps required to reach 90\,\% of the final fidelity $F_{\infty}$ within the $L \times 4$ setup on the square lattice at $j_2 / j_1 = 0.4$ as a function of inverse energy fluctuation $(\epsilon / j_1)^{-1}$.}
\label{fig:N_SGD}
\end{figure}
We investigate in Fig.\,\ref{fig:N_SGD} the number of SGD steps required to reach 90\,\% of the final fidelity $F_{\infty}$ within the $L \times 4$ setup on the square lattice at $j_2 / j_1 = 0.4$ as a function of effective temperature. We observe no pronounced dependence as a function of $L$ at given $\epsilon$.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,592 |
The Agenda: Feds nix drilling off S.C. coast; Chs. schools to boost classroom size; More jobs coming
by Sam Spence March 15, 2016
[image-1]
Even though college board incumbents do not face competition for re-election, at least one Francis Marion University incumbent is facing a challenge from the daughter of Senate Majority Leader Hugh Leatherman. Source: The State
Charleston Schools officials said yesterday that because of inadequate budget oversight over the past few years, the district will increase classroom size for many grade levels and save $7.35 million through teacher cuts. Source: P&C
A Charleston Metro Chamber/C of C economic forecast predicts at least 20,000 more jobs are expected to be created in the Charleston region by the end of 2017. Source: CRBJ
State Superintendent Molly Spearman told a group of athletic administrators yesterday that the state's uniform grading policy will shift to a 10-point scale from its current 7-point grading scale. Source: P&C
Former S.C. state trooper Sean Groubert pleaded guilty to assault and battery yesterday stemming from the incident where he shot an unarmed man during a 2014 traffic stop in St. Andrews. Source: The State
The federal government says it will reverse course and drop a plan to allow offshore drilling for oil and natural gas off the coast of several Atlantic states, including South Carolina. The mayors of Charleston and Beaufort were in D.C. for meetings ahead of the announcement. Source: NYT, The State, P&C
The Cougars' granny-style free-throw shooter Canyon Barry announced yesterday that he'll graduate from CofC this summer and transfer to another school for his final year of NCAA eligibility. Since he'll be a graduate student wherever he transfers, he'll be able to play next season instead of sitting out a year. Source: P&C
In other Cougar news, the CofC baseball team has cracked into Baseball America's top 25, ranking #23 in this week's poll. The Cougs play at Patriots Point this weekend against VCU. Source: P&C, CofCSports
Tagged: The Agenda | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,492 |
\section{Introduction}
Liquid Argon Time Projection Chamber (LArTPC) detectors provide exceptional calorimetric and track reconstruction capabilities compared to various other detector technologies. This technology has already shown great promise to be one of the ideal detector technologies to study neutrino interactions with matter. In a LArTPC, neutrinos interact with argon atoms and produce secondary charged particles (e.g protons, muons etc). These secondary charged particles ionize argon atoms and produce ionization electrons and scintillation light. The ionization electrons drift to the anode wire planes under an external electric field whereas scintillation light is collected by photo multiplier tubes (PMTs) located behind the anode wire planes. Ultimately, one can retrieve the information registered in the PMT system and anode wire planes to reconstruct 3D images of particle tracks and their energies.
One of the critical operational requirements of a LArTPC is ultra pure liquid argon. Electronegative contaminants like H\textsubscript{2}O and O\textsubscript{2} can degrade the liquid argon purity. These contaminants can capture some of the drifting ionization electrons and thus directly impact the reconstruction of particle energies. This capture process is electric field dependent where at higher electric fields ionization electrons have more chance of drifting all the way to the anode wire planes.
Equation ~\ref{eq:attenuatiion} governs how a cloud of ionization electrons gets attenuated due to the presence of electronegative contaminants inside the detector.
\begin{equation}
\frac {n_e(t_{\mathrm{drift}})}{n_e(t_0)} = \exp(\frac{-t_{\mathrm{drift}}}{\tau})\label{eq:attenuatiion}
\end{equation}
Here $n_e(t_0)$ stands for the initial number of electrons whereas $n_e(t_{\mathrm{drift}})$ is the number of electrons after a time $t_{\mathrm{drift}}$. An important parameter in this equation is $\tau$, which stands for the electron lifetime. The electron lifetime contains information about the amount of electronegative contaminants present in the detector where a higher electron lifetime is indicative of low levels of contamination\cite{ICARUS1O2,ICARUS2O2}. If liquid argon is 100$\%$ pure then $\tau$ should ideally be infinite.
Figure~\ref{fig:drift1} shows the fractional loss of ionization electrons as a function of their drift distance for different electron lifetimes. It is obvious that having a higher electron lifetime in the system guarantees that more ionization electrons survive.
\begin{figure}[htb]
\begin{center}
\includegraphics[scale=0.6]{Fig_1.png}
\end{center}
\caption{Fractional loss of ionization electrons as a function of drift distance for an electric field of 0.273 kV/cm. Different colored curves correspond to different electron lifetimes ($\tau$). Graphs are created using Equation 1 for different electron lifetimes. A drift velocity of 0.1114 cm/$\mu$s (MicroBooNE's drift velocity at nominal electric field of 273 V/cm) is used to covert time into drift distance.}
\end{figure}\label{fig:drift1}
\section{The MicroBooNE LArTPC}
The MicroBooNE experiment\cite{uBDetJINST} at Fermilab uses LArTPC technology to study neutrino-argon cross sections in the 1 GeV energy regime. The other major physics goal of MicroBooNE includes addressing the low energy excess observed by the MiniBooNE experiment\cite{MiniBooNE-excess1,MiniBooNE-excess2}. MicroBooNE also serves as a critical R$\&$D step for upcoming large scale experiments like the Deep Underground Neutrino Experiment (DUNE) and the Short Baseline Neutrino (SBN) program showing the feasibility of LArTPC technology. The MicroBooNE TPC has an active volume of 85 tons of liquid argon where the cathode is kept at -70 kV thus attaining a drift electric field of 0.273 kV/cm. The anode consists of 3 wire planes (U, V and Y) where each plane has a wire pitch (separation between two of the neighboring wires, which make up the plane) of 3 mm. The two induction planes (U and V) are inclined +60\textsuperscript{o} and -60\textsuperscript{o} with respect to the vertical collection plane (Y). The maximum drift distance of the experiment is 2.56 m where it takes $\sim$2.3 ms for ionization electrons to drift from cathode to anode across the detector. The light collection system of the experiment consists of 32 8-inch PMTs located behind the anode wire planes.
The state of the art purification system in the experiment maintains the electronegative contaminants at extremely low levels which is vital for the performance of the detector. The purification system consists of two pairs of filters\cite{H20filter,O2filter} which remove H\textsubscript{2}O and O\textsubscript{2}. The initial design goal of the experiment was to maintain the O\textsubscript{2}-equivalent electronegative contaminant levels below 100 ppt to achieve an electron lifetime of 3 ms under an applied electric field of 0.5 kV/cm. The MicroBooNE detector was fully commissioned summer of 2015 and started taking data in August 2015. By the summer of 2017, experiment collected a $\sim$$6\times10\textsuperscript{20}$ POT (protons on target) equivalent of data which contains more than $1\times10\textsuperscript{5}$ neutrino interactions to study.
\begin{figure}[htb]\label{fig:mic_dim}
\begin{minipage}[h]{0.3\textwidth}
\includegraphics[scale=0.45]{Fig_2.png}
\subcaption{\textsl{}}
\end{minipage} \hspace{0.2\textwidth}
\begin{minipage}[h]{0.3\textwidth}
\includegraphics[scale=0.45]{Fig_3.png}
\subcaption{\textsl{}}
\end{minipage}
\caption{(a) Layout of the MicroBooNE LArTPC. The detector is 10.4 m long (Z-direction), 2.3 m in height (Y-direction) and 2.5 m wide (X-direction). (b) Coordinate system of MicroBooNE. Two induction anode wire planes (U and V) are +60\textsuperscript{o} and -60\textsuperscript{o} with respect to the vertical whereas collection plane (Y) is vertical. }
\label{fig:mic_dim}
\end{figure}
\section{Measuring electron lifetime($\tau$) in MicroBooNE}
There are several methods one can use to measure the electron lifetime in MicroBooNE. They are:
\setlength{\parskip}{0em}
\begin{itemize}
\setlength{\parskip}{0em}
\item Gas analyzers
\setlength{\parskip}{0em}
\item Purity monitors\cite{ICARUSpm}
\setlength{\parskip}{0em}
\item Laser tracks\cite{laser1,uBlaser}
\setlength{\parskip}{0em}
\item Long minimum ionizing cosmic ray muon tracks\cite{ICARUScosmic}
\end{itemize}
Since MicroBooNE is located close to the surface, it sees abundant cosmic muons. Quantitatively, in the 4.8 ms wide readout window of MicroBooNE there are approximately 25 comic muons crossing the detector (Figure~\ref{fig:read_out}). Because of this cosmic muons provide good statistics to perform the measurement. In addition cosmic muons are uniformly distributed throughout making the measurement to be sensitive to any local variations of external electric field and electronegative contaminant levels inside the detector. But knowing the correct arrival time (or, t\textsubscript{0}) of these cosmic muons is critical for this measurement as one needs to know how long the charge drifted in the TPC. There are two types of cosmic muon datasets for which t\textsubscript{0} information is known.
\begin{itemize}
\setlength{\parskip}{0em}
\item Cosmic muons tagged by a small external cosmic ray counter\cite{MuCs}
\setlength{\parskip}{0em}
\item TPC anode-cathode crossing tracks (Crossing tracks)
\end{itemize}
Since crossing tracks have wide angular coverage compared to tracks tagged by the small external cosmic-ray counter and cover the full drift distance, these tracks are preferred. The measurement shown in this document uses the TPC anode-cathode crossing tracks.
\begin{figure}[htb]
\begin{center}
\includegraphics[scale=0.7]{Fig_4.png}
\end{center}
\caption{Cosmic muon tracks in the MicroBooNE readout window. The three boxes show the full readout window of the MicroBooNE detector which corresponds to 4.8 ms or equivalently the total effective drift volume for a MicroBooNE readout window. The red highlighted box shows the physical volume of the TPC. The colored lines shown in the boxes are 3D reconstructed tracks. Different colors represent different tracks.}
\end{figure}\label{fig:read_out}
\section{Event Selection}
Several selection cuts are applied in the analysis to select the best quality cosmic tracks.
\begin{itemize}
\item The track projected length in the drift direction (X) must be between 250 cm to 270 cm.
\end{itemize}
Ideally the X projected track length of a crossing track should be 256 cm (drift distance of MicroBooNE). But due to track reconstruction and space charge effects a spread of this quantity is observed (Figure 4).
\begin{itemize}
\item Exclude tracks with the angular orientation 75\textsuperscript{0}$<$$\theta_{XZ}$$<$ 105\textsuperscript{0} or 85\textsuperscript{0}$<$$\theta_{YZ}$$<$ 95\textsuperscript{0}.
\end{itemize}
$\theta_{XZ}$ is the angle defined in X-Z plane with respect to the Z direction whereas $\theta_{YZ}$ is the angle defined in the Y-Z plane with respect to the Z direction. The $\theta_{XZ}$ ($\theta_{YZ}$) cut eliminates tracks that are nearly perpendicular (parallel) to the collection plane wires. These tracks tend to be mis-reconstructed with low quality calorimetric information.
\begin{itemize}
\item[$\bullet$] The track must have a minimum of 100 hits registered in the collection wire plane.
\end{itemize}
This ensures a uniform density of hits along the drift direction and better reconstructed tracks.
\begin{itemize}
\item Exclude hits from tracks which correspond to shorted channel TPC regions.
\end{itemize}
We require that hit Y and Z coordinates are not in the regions, (-100 cm$<$Y$<$ 20 cm) and (250 cm$<$Z$<$675 cm), respectively. In the shorted channel regions of the TPC, the collection wire plane response is altered and properly reconstructed calorimetric information was not available for hits\cite{uBnoiseJINSTpre}.
\begin{figure}[htb]
\begin{center}
\includegraphics[scale=0.6]{Fig_5.png}
\end{center}
\caption{Distribution of X-projected length of tracks in 10000 event sample consisting of cosmic ray data. Colored band shows the region of interest for TPC anode-cathode crossing tracks. In a $\sim$5000 event sample $\sim$2$\%$ of all tracks are crossing tracks.}
\end{figure}\label{fig:crossig_trks}
\section{Analysis Method}
The following method is used to calculate the electron attenuation (electron lifetime) from a set of crossing tracks which satisfy all the selection criterion described in the previous section.
\begin{itemize}
\item The start time (t\textsubscript{0}) of the track is calculated : For TPC crossing tracks either light information or hit information can be used to extract t\textsubscript{0}. At the time of this analysis, light reconstruction wasn't reliable. So hit information is used to get t\textsubscript{0}. Minimum drift coordinate of the crossing track provides the t\textsubscript{0}. Once t\textsubscript{0} is known, each drift coordinate of the track is corrected using Equation~\ref{eq:x_correction}.
\end{itemize}
\begin{equation}
X_{\mathrm{corrected}} = X_{\mathrm{reconstructed}}-X_{\mathrm{minimum}}\label{eq:x_correction}
\end{equation}
\begin{itemize}
\item The full drift time window (2.2 ms) is split into smaller 100 $\mu$s wide time windows (Figure 5(a)).
\end{itemize}
\begin{itemize}
\item For each drift time bin, the charge deposited per unit length (dQ/dx) distribution is produced and fit with a Landau convoluted with a Gaussian function\cite{lgfit} to get the Most Probable Value (MPV) of dQ/dx representing that time bin (Figure 6).
\end{itemize}
\begin{itemize}
\item[$\bullet$] All 22 most probable dQ/dx values are plotted against drift time and this final distribution is fit with a function f(t) to get the final Q\textsubscript{A}/Q\textsubscript{C} charge ratio (Figure 5(b)).
\end{itemize}
The final Q\textsubscript{A}/Q\textsubscript{C} charge ratio is calculated using Equation~\ref{eq:qa_qc} which gives the fractional change in charge due to capture by electronegative contaminants when a cloud of ionization electrons drift from cathode to anode. Ideally f(t) should be exponential. But the presence of space charge and other effects can skew the dQ/dx distribution resulting in non-exponential shapes. Second-order polynomial and exponential plus constant functions are also used to fit to the final distribution. Once the Q\textsubscript{A}/Q\textsubscript{C} charge ratio is calculated, assuming an exponential behavior, one can translate it into an electron lifetime using Equation~\ref{eq:charge_to_elife}. It was observed that for most of the runs, the final Q\textsubscript{A}/Q\textsubscript{C} value is greater than 1 due to space charge effects.
\begin{equation}
\frac{Q_A}{Q_C} = \frac{f(2200~\mu s)}{f(0~\mu s)}\label{eq:qa_qc}
\end{equation}
In Equation 3, the numerator and denominator stand for the amount of charge arriving at the anode after 2.2 ms and charge leaving the cathode at 0 ms, respectively.
\begin{equation}
\frac{Q_A}{Q_C} = \exp(\frac{-t_{\mathrm{drift}}}{\tau})\label{eq:charge_to_elife}
\end{equation}
Here $t_{\mathrm{drift}}$ should be replaced by 2200 $\mu$s to get $\tau$.
\begin{figure}[htb]
\begin{minipage}[h]{0.3\textwidth}
\includegraphics[width=7cm, height=5cm]{Fig_6.png}
\subcaption{\textsl{}}
\end{minipage} \hspace{0.2\textwidth}
\begin{minipage}[h]{0.3\textwidth}
\includegraphics[width=7cm, height=5cm]{Fig_7.png}
\subcaption{\textsl{}}
\end{minipage}
\caption{(a) dQ/dx vs drift time scatter plot. The full drift time window (2.2 ms) is split into 100 $\mu$s wide time bins. Regions between the dashed vertical lines represent smaller time bins. (b) Distribution of most probable dQ/dx values (from each smaller time bin) as a function of drift time. Distribution is fit with a second-order polynomial to get the final Q\textsubscript{A}/Q\textsubscript{A} charge ratio.}
\label{fig:splitting_final_dqdx}
\end{figure}
\begin{figure}[htb]
\begin{minipage}[h]{0.3\textwidth}
\includegraphics[width=7cm, height=5cm]{Fig_8.png}
\subcaption{\textsl{}}
\end{minipage} \hspace{0.2\textwidth}
\begin{minipage}[h]{0.3\textwidth}
\includegraphics[width=7cm, height=5cm]{Fig_8_1.png}
\subcaption{\textsl{}}
\end{minipage}
\caption{(a) dQ/dx distribution produced for the time bin 400 $\mu$s to 500 $\mu$s. (b) dQ/dx distribution produced for the time bin 2100 $\mu$s to 2200 $\mu$s. Distributions are fit with a Landau convoluted wih a Gaussian to extract the most probable dQ/dx value in that time range.}
\label{fig:small_dqdx_dis}
\end{figure}
\section{Space Charge Effects}
Since MicroBooNE is surface-based, a lot of cosmic activity is present in the detector. As a result of slow moving positive argon ions get accumulated leading to distortions in the applied electric field both in its magnitude (Figure 7(a)) and direction. Once the directionality of the electric field is impacted, it can affect reconstructed position of ionization electron clusters (Figure 7(b)) and calorimetric information (dQ/dx). \iffalse Equation~\ref{eq:dq_dx_change_direction} shows how the dQ/dx values are modified due to directionality change in the electric field.\fi The magnitude change of the electric field directly affects electron-ion recombination. The lower the applied electric field the stronger the recombination gets. From MicroBooNE simulations, it is estimated that the electric field close to the cathode increases by $\sim$12$\%$ whereas at the anode it decreases by $\sim$5$\%$. The modified box model is used to describe the electron-ion recombination\cite{recomb} in MicroBooNE where it shows dQ/dx values increasing by $\sim$3.55$\%$ close to the cathode and decreasing by $\sim$1.2$\%$ close the to anode. (See section 9.3 for more details on recombination). \\
In order to see the impact of space charge effects on the final measurement, a 3D space charge model is implemented in simulation and isotropic single muon monte carlo samples are created with a very high electron lifetime switching off all other effects such as diffusion which can affect the attenuation mesurement. The positive slope of Figure 8(b) shows final Q\textsubscript{A}/Q\textsubscript{A} charge ratio can be greater than 1 due to space charge effects.
\begin{figure}[htb]
\begin{minipage}[h]{0.3\textwidth}
\includegraphics[scale=0.6]{Fig_9.png}
\subcaption{\textsl{}}
\end{minipage} \hspace{0.2\textwidth}
\begin{minipage}[h]{0.3\textwidth}
\includegraphics[scale=0.6]{Fig_10.png}
\subcaption{\textsl{}}
\end{minipage}
\caption{(a) Illustration of simulated effect of space charge on the X component of the electric field as a function of the X and Y coordinates.. (b) Simulated effect of space charge on the reconstructed Y position of an electron cluster as a function of the X and Y coordinates. Both of these plots are created for the central Z region where space charge effects are thought to be maximal.}
\label{fig:mag_dir_simulate}
\end{figure}
\begin{figure}[htb]
\begin{minipage}[h]{0.3\textwidth}
\includegraphics[width=7cm, height=5cm]{Fig_11.png}
\subcaption{\textsl{}}
\end{minipage} \hspace{0.2\textwidth}
\begin{minipage}[h]{0.3\textwidth}
\includegraphics[width=7cm, height=5cm]{Fig_12.png}
\subcaption{\textsl{}}
\end{minipage}
\caption{(a) dQ/dx distribution of the monte carlo sample with no space charge effects. (b) dQ/dx distribution of the monte carlo sample with space charge effects including distortions in both the direction and magnitude of the electric field.}
\label{fig:sp_mc}
\end{figure}
\subsection{Space Charge Correction}
As space charge effects (SCE) significantly impact the final measurement, a correction is derived using the following procedure.
\begin{description}
\item[$\bullet$] The correction \enquote{C} is derived for dQ/dx values.
\end{description}
The correction for dQ/dx values is defined for all 22 drift bins using Equation 5 based on the two dQ/dx distributions in Figure 8.
\begin{equation}
C = \frac {(\frac {dQ}{dx})_{\mathrm{SCE=ON}} - (\frac {dQ}{dx})_{\mathrm{SCE=OFF}}} {(\frac {dQ}{dx})_{\mathrm{SCE=ON}}} \label{eq:cor_1}
\end{equation}
\begin{description}
\item[$\bullet$] A third order polynomial fit is used to extract corrections.
\end{description}
To make the derived corrections uniform, the correction \enquote{C} is plotted as a function of drift time and the resultant distribution is fit with a third order polynomial (Figure 9). The final corrections \enquote{$\tilde{\mathrm{C}}$} for each bin are extracted using the polynomial fit. This space charge correction can be applied to data using Equation~\ref{eq:cor_dqdx}.
\begin{equation}\label{eq:cor_dqdx}
(\frac{dQ}{dx})_{\mathrm{corrected}} = \frac{dQ}{dx}(1-\tilde{C})\label{eq:cor_dqdx}
\end{equation}
Here, dQ/dx stands for the uncorrected value from data including space charge effects.
\begin{figure}[htb]
\begin{center}
\includegraphics[scale=0.4]{Fig_13.png}
\end{center}
\caption{Plot of space charge corrections versus drift time. A third order polynomial is used to fit the points to extract corrections for all 22 bins.}
\end{figure}\label{fig:pol3_fit_11}
\section{Data Sample}
Cosmic muon data ranging from 02/16/2016 to 04/21/2016 is used in this analysis. Each dataset has approximately 5000 events processed to study the variation of electron attenuation on a daily basis. The time difference between two consecutive data sets is $\sim$24 hours and the detector spans a range of low purity conditions to high purity conditions during the selected time period. Some of the datasets are missing in this time period due to data processing problems.
\section{Variation of Q\textsubscript{A}/Q\textsubscript{C}}
The plot in the Figure 10 shows the variation of the Q\textsubscript{A}/Q\textsubscript{C} charge ratio over time by analyzing 56 datasets described in section 7 using the procedure explained in section 5 with and without the space charge corrections derived in section 6.1.
\begin{figure}[htb]
\begin{center}
\includegraphics[scale=0.6]{Fig_14.png}
\end{center}
\caption{Variation of Q\textsubscript{A}/Q\textsubscript{C} over time with and without space charge corrections. Colored bands show the regions of missing data due to data processing problems.}
\end{figure}\label{fig:inivar}
\section{Systematic Uncertainties}
A number of possible sources of systematic uncertainty are considered in this analysis. Among them, the dominant systematic uncertainty comes from the space charge correction
used in the analysis. The other systematics that effect this measurement in a minor way are recombination and diffusion. Both data-drive and monte-carlo-based techniques are used in addressing systematics.
\subsection{Space charge correction uncertainty}
Space charge corrections described in section 6.1 are derived from the 3D space charge model implemented in the simulations which is only valid for the central Z region. So to account for space charge model dependency, a large uncertainty is associated to the final Q\textsubscript{A}/Q\textsubscript{C} charge ratio values corresponding to 50$\%$ of the difference of the two Q\textsubscript{A}/Q\textsubscript{C} charge ratios before and after the space charge correction. This is done separately for all 56 data sets where the systematic varies between 1.4$\%$ to 7.5$\%$ with an average uncertainty of 4.6$\%$. The systematic is assigned to be 5.0$\%$ of the final Q\textsubscript{A}/Q\textsubscript{C} charge ratio for all data sets.
\subsection{Diffusion}
Diffusion can smear the ionization electron cloud as they drift from cathode to anode by affecting the charge collected by the anode plane wires. In particular, the transverse component of the diffusion which is perpendicular to the drift direction can leak the charge to neighboring wires from the target wire inducing modifications to the reconstructed dQ/dx values. Any modifications in dQ/dx values directly impacts the measured Q\textsubscript{A}/Q\textsubscript{C} charge ratio. To calculate the systematic uncertainty introduced by diffusion, two monte carlo single isotropic muon samples are used, where in one both the longitudinal and transverse diffusion components are turned on while in the other the diffusion effects are turned off. The systematic is assigned to be the difference between Q\textsubscript{A}/Q\textsubscript{C} charge ratios with and without diffusion, which is 2.0$\%$ of the final value.
\subsection{Recombination}
As explained before, in the current MicroBooNE simulation the electron-ion recombination is described by modified box model. Equation~\ref{eq:recombination_mbm} shows the relationship between the recombination factor \enquote{$R_{box}$} and the electric field.
\begin{equation}
R_{box} = \frac {ln(\alpha + \frac {\beta_p} {\rho\varepsilon}\cdot\frac{dE}{dx} )} { \frac {\beta_p} {\rho\varepsilon}\cdot\frac{dE}{dx}}\label{eq:recombination_mbm}
\end{equation}
Here $\varepsilon$ is the electric field of MicroBooNE (0.273 kV/cm), $\rho$ is liquid argon density (1.38 g/cm\textsuperscript{3} at a pressure 18.0 psia), $\beta$\textsubscript{p}=0.212 +/- 0.002 (kV/cm)(g/cm\textsuperscript{2})/MeV and $\alpha$=0.93+/-0.02. The values for $\beta$\textsubscript{p} and $\alpha$ were calculated by Argoneut experiment which had an operating electric field of 0.481 kV/cm\cite{recomb}.\\
To account for the model dependency of recombination in the electron attenuation measurement, two single isotropic muon monte carlo samples are generated, one having the default values for $\beta$\textsubscript{p} and $\alpha$ in the recombination model whereas in the other these two values are maximally changed ( by 0.01 (kV/cm)(g/cm\textsuperscript{2})/MeV and 0.1 respectively). In both of these samples all the other effects are turned off such as diffusion with the exception the change in the magnitude of the electric field due to space charge effects. The percentage difference of the Q\textsubscript{A}/Q\textsubscript{C} charge ratios between samples with modified parameter and default parameter settings is found to be 1.0$\%$ with respect to the Q\textsubscript{A}/Q\textsubscript{C} charge ratio extracted for default parameter setting. This is assigned to be the systematic value coming from recombination model dependency. Table 1 summarizes all the systematics considered in the analysis.
\begin{table}[b]\label{table:sys_summary}
\centering
\begin{tabular}{ l|ccc}
\hline
Systematic & Uncertainity ($\%$)\\
\hline
Space charge correction & 5.0\\
Recombination & 1.0\\
Diffusion & 2.0\\
\hline
Total & 5.5\\
\hline
\end{tabular}
\caption{Systematic uncertainties in the final Q\textsubscript{A}/Q\textsubscript{C} charge ratio. The total is calculated by adding individual systematics in quadrature.}
\end{table}
\section{Results}
Figure 11 shows the variation of the Q\textsubscript{A}/Q\textsubscript{C} charge ratio over time in MicroBooNE with both statistical and systematic uncertainties folded in and the space charge correction. It can be observed that the Q\textsubscript{A}/Q\textsubscript{C} charge ratio is very high even taking into account the systematic uncertainties. During stable purity conditions, (excluding the two dip regions in the figure) the Q\textsubscript{A}/Q\textsubscript{C} value changes between 0.88+/-0.04 and 1.01+/-0.05. Using Equation~\ref{eq:charge_to_elife} the lowest corresponding electron lifetime in this period can be found to be 18 ms which corresponds to a maximum charge loss of 12$\%$ and an O\textsubscript{2} equivalent contamination level of 17 ppt. The lowest Q\textsubscript{A}/Q\textsubscript{C} value of the entire time period studied is 0.72+/-0.03 which corresponds to a maximum charge loss of 28$\%$, an electron lifetime of 6.8 ms, and an O\textsubscript{2} equivalent contamination level of 44 ppt.
\begin{figure}[htb]\label{fig:final_var}
\begin{center}
\includegraphics[scale=0.7]{Fig_18.png}
\end{center}
\caption{Variation of Q\textsubscript{A}/Q\textsubscript{C} charge ratio over time with both statistical and systematic uncertainties and space charge corrections. Colored bands show the region where missing data lies.}
\end{figure}
\section{Summary and Conclusions}
This is of the first measurement of the liquid argon purity in MicroBooNE expressed in terms of Q\textsubscript{A}/Q\textsubscript{C} charge ratio (ratio of collected charge to deposited charge) using cosmic ray muons. During the time period of 02/16/2016 - 04/21/2016, Q\textsubscript{A}/Q\textsubscript{C} values range from 0.72+/-0.03 to 1.01+/-0.05 indicating a very high liquid argon purity inside the detector. The lowest Q\textsubscript{A}/Q\textsubscript{C} charge ratio recorded in the entire duration is 0.72+/-0.03 which corresponds to an electron lifetime of 6.8 ms and an O\textsubscript{2} equivalent contamination level of 44 ppt resulting in a charge loss of 28$\%$ for the full drift path. Systematic uncertainties dominate over statistical fluctuations in the entire time period studied. During stable purity conditions the lowest Q\textsubscript{A}/Q\textsubscript{C} charge ratio observed is 0.88+/-0.04 with a corresponding electron lifetime of 18 ms and an O\textsubscript{2} equivalent contamination level of 17 ppt. These results are indicative of MicroBooNE having a very high electron lifetime which clearly exceeds the early design goal of 3 ms.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 510 |
Q: mpd listening on port 6600 I find on netstat that mpd is listening on port 6600.
I play all my media files on VLC. I don't remember installing this, and its not in my startup applications.
Is mpd even needed?
A: mpd is the Music Player Daemon, and should not be needed by VLC. It should be safe to uninstall with sudo apt-get remove mpd.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,822 |
Eagle Air Iceland () est une compagnie aérienne islandaise, dont le siège se situe à l'aéroport de Reykjavik.
Histoire
Flotte
Au mois de , Eagle Air Iceland exploite les appareils suivants:
Destinations
Notes et références
Annexes
Articles connexes
Transport en Islande
Aéroport de Reykjavik
Liste des aéroports en Islande
Liens externes
Site officiel en islandais
Site officiel en anglais
Compagnie aérienne ayant son siège en Islande | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 131 |
Q: Deserialización Json atributo @nil: "true" newtonsoft Estoy deserializando un json, el cual algunos parametros cuando son nulos tiene una propiedad @nil, cuando realiza la deserialización genera error, el Json tiene la siguiente estructura:
{
"ArrayOfDatosDetallados": {
"DatosDetallados": {
"APELLIDO1": "ARTEA",
"APELLIDO2": {
"@nil": "true"
},
"DEPTO_DECLA": "NARIÑO (52)",
"DEPTO_OCU": "NARIÑO (52)",
"DISCAPACIDAD": "NINGUNA",
"DOCUMENTO": 67027563,
"ESTADO_BINARIO": 1,
"ESTADO_TRANSACCION": "EXITOSA",
"ETNIA": {
"@nil": "true"
}
}
}
}
El proceso que estoy realizando es el siguiente:
JsonConvert.DeserializeObject<VivantoMoviagro>(
response["ArrayOfDatosDetallados"]["DatosDetallados"].ToString(),
new JsonSerializerSettings
{
NullValueHandling = NullValueHandling.Ignore,
MissingMemberHandling = MissingMemberHandling.Ignore
});
El error generado es el siguiente:
Message:
Test method Banagrario.Agrobac.IntegrationTests.VivantoTest.VivantoClientTest.DocumentoVivantoOK threw exception:
Newtonsoft.Json.JsonReaderException: Unexpected character encountered while parsing value: {. Path 'ETNIA', line 13, position 12.
Stack Trace:
JsonTextReader.ReadStringValue(ReadType readType)
JsonTextReader.ReadAsString()
JsonReader.ReadForType(JsonContract contract, Boolean hasConverter)
JsonSerializerInternalReader.PopulateObject(Object newObject, JsonReader reader, JsonObjectContract contract, JsonProperty member, String id)
JsonSerializerInternalReader.CreateObject(JsonReader reader, Type objectType, JsonContract contract, JsonProperty member, JsonContainerContract containerContract, JsonProperty containerMember, Object existingValue)
JsonSerializerInternalReader.CreateValueInternal(JsonReader reader, Type objectType, JsonContract contract, JsonProperty member, JsonContainerContract containerContract, JsonProperty containerMember, Object existingValue)
JsonSerializerInternalReader.Deserialize(JsonReader reader, Type objectType, Boolean checkAdditionalContent)
JsonSerializer.DeserializeInternal(JsonReader reader, Type objectType)
JsonConvert.DeserializeObject(String value, Type type, JsonSerializerSettings settings)
JsonConvert.DeserializeObject[T](String value, JsonSerializerSettings settings)
UpdateDelegates.UpdateAndExecute3[T0,T1,T2,TRet](CallSite site, T0 arg0, T1 arg1, T2 arg2)
La propiedad Etnia es de tipo string.
El proceso de deserialización funciona cuando las propiedades Etnia o Apellido2 traen algún valor.
Saludos.
A: Cordial saludo,
la solución plantada fué realizar una método de extensión donde se verifica si el dato del JToken tiene como propiedad @nil para asignar el valor de null, y definir una clase que herede de la clase JsonConverter, al implementar la clase abstracta utilizamos el método ReadJson donde se valida mediante el método de extensión si tiene tal atributo y devolver null.
public static partial class JTokenExtensions
{
public static bool WasNilXmlElement(this JToken token)
{
if (token == null)
return true;
if (token.Type == JTokenType.Null)
return true;
var obj = token as JObject;
if (obj != null)
{
// Check if all properties were translated from XML attributes
// and one was translated from xsi:nil = true
// There might be namespaces present as well, e.g.
// "@xmlns:p3": "http://www.w3.org/2001/XMLSchema-instance"
if (obj.Properties().All(p => p.Name.StartsWith("@"))
&& obj.Properties().Any(p => p.Name == "@nil" || p.Name.EndsWith(":nil") && p.Value.ToString() == "true"))
return true;
}
return false;
}
}
y la clase Converter queda de la siguiente maneja.
public class NullableStructConverter<T> : JsonConverter where T : class
{
public override bool CanConvert(Type objectType)
{
return objectType == typeof(T);
}
public override object ReadJson(JsonReader reader, Type objectType, object existingValue, JsonSerializer serializer)
{
var token = JToken.Load(reader);
if (token.Type == JTokenType.Null)
return null;
if (token.WasNilXmlElement())
return null;
return token.ToObject<T>();
}
public override bool CanWrite { get { return false; } }
public override void WriteJson(JsonWriter writer, object value, JsonSerializer serializer)
{
throw new NotImplementedException();
}
}
y ya solo nos queda llamar este Converter en nuestro de Deserialización, de la siguiente manera
JsonConvert.DeserializeObject<Entidad>(
strJson,
new JsonSerializerSettings { Converters = { new NullableStructConverter<string>() } });
Es importarte especificar al Converter el tipo de dato, para controlar que no se aplique a todas las propiedades del Json. Tambien es una forma de convertir el Converter como atributo y especificarlo a nivel de las propiedades en la entidad.
Si tienen una solución más sencilla me gustaria conocerla.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,638 |
EXPRESS INFORMER > Magazine > Nicole Kidman back in Australia because of mother's health
Nicole Kidman back in Australia because of mother's health
The "Being Ricardos" star told NPR's Terry Gross during a conversation on the program "Fresh Air" that she had returned to her native country because of her mother, Janelle Kidman's, current health situation.
"We're down here primarily to take care of my mother and to have her surrounded by her grandchildren," the younger Kidman said.
Gross also asked how Kidman was doing regarding the Omicron surge.
"It's running wild in Australia," Kidman replied.
Despite that, she said, the family was able to have a safe outing with her 81-year-old mother.
"We were able to take her into the gallery after hours and show her the Matisse exhibit, which coming from a mother who's raised me in the arts, it was soothing balm," Kidman said. "Matisse was soothing balm last night."
Tyson Fury insists he would knock Anthony Joshua out in three rounds as he slams Derek Chisora
Novak Djokovic's visa saga divides opinion | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,434 |
Q: Maven docker plugin for spring boot build error I have a simple maven project with multi modules (only one for the moment) for Spring Boot microservices
In the root folder the pom.xml contains
<?xml version="1.0" encoding="UTF-8"?>
<project xmlns="http://maven.apache.org/POM/4.0.0" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://maven.apache.org/POM/4.0.0 http://maven.apache.org/xsd/maven-4.0.0.xsd">
<modelVersion>4.0.0</modelVersion>
<groupId>be.demo.microservices</groupId>
<artifactId>master</artifactId>
<version>1.0.0-SNAPSHOT</version>
<packaging>pom</packaging>
<modules>
<module>hello-service</module>
</modules>
</project>
In the hello-service folder, the pom.xml contains
<?xml version="1.0" encoding="UTF-8"?>
<project xmlns="http://maven.apache.org/POM/4.0.0" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
xsi:schemaLocation="http://maven.apache.org/POM/4.0.0 http://maven.apache.org/xsd/maven-4.0.0.xsd">
<modelVersion>4.0.0</modelVersion>
<groupId>be.demo.microservices</groupId>
<artifactId>hello-service</artifactId>
<version>1.0.0-SNAPSHOT</version>
<packaging>jar</packaging>
<description>Demo project for Spring Boot</description>
<parent>
<groupId>org.springframework.boot</groupId>
<artifactId>spring-boot-starter-parent</artifactId>
<version>2.0.1.RELEASE</version>
<relativePath/> <!-- lookup parent from repository -->
</parent>
<properties>
<project.build.sourceEncoding>UTF-8</project.build.sourceEncoding>
<project.reporting.outputEncoding>UTF-8</project.reporting.outputEncoding>
<java.version>1.8</java.version>
<docker.image.prefix>springio</docker.image.prefix>
</properties>
<dependencies>
<dependency>
<groupId>org.springframework.boot</groupId>
<artifactId>spring-boot-starter-actuator</artifactId>
</dependency>
<dependency>
<groupId>org.springframework.boot</groupId>
<artifactId>spring-boot-starter-web</artifactId>
</dependency>
<dependency>
<groupId>org.springframework.boot</groupId>
<artifactId>spring-boot-devtools</artifactId>
<scope>runtime</scope>
</dependency>
<dependency>
<groupId>org.springframework.boot</groupId>
<artifactId>spring-boot-starter-test</artifactId>
<scope>test</scope>
</dependency>
</dependencies>
<build>
<plugins>
<plugin>
<groupId>org.springframework.boot</groupId>
<artifactId>spring-boot-maven-plugin</artifactId>
</plugin>
<plugin>
<groupId>com.spotify</groupId>
<artifactId>docker-maven-plugin</artifactId>
<version>1.0.0</version>
<configuration>
<imageName>${docker.image.prefix}/${project.artifactId}</imageName>
<imageTags>
<imageTag>${project.version}</imageTag>
<imageTag>latest</imageTag>
</imageTags>
<baseImage>openjdk:8-jdk-alpine</baseImage>
<entryPoint>["sh","-c","java -Djava.security.egd=file:/dev/./urandom -jar /${project.build.finalName}.jar"]</entryPoint>
<resources>
<resource>
<targetPath>/</targetPath>
<directory>${project.build.directory}</directory>
<include>${project.build.finalName}.jar</include>
</resource>
</resources>
</configuration>
</plugin>
</plugins>
</build>
</project>
When building the project with mvn clean package docker:build I have
...
[INFO] --- spring-boot-maven-plugin:2.0.1.RELEASE:repackage (default) @ hello-service ---
[INFO]
[INFO] --- docker-maven-plugin:1.0.0:build (default-cli) @ hello-service ---
[INFO] Using authentication suppliers: [ConfigFileRegistryAuthSupplier]
[INFO] Copying D:\dev\microservices\demo\hello-service\target\hello-service-1.0.0-SNAPSHOT.jar -> D:\dev\microservices\demo\hello-service\target\docker\hello-service-1.0.0-SNAPSHOT.jar
[INFO] Building image springio/hello-service
Step 1/3 : FROM openjdk:8-jdk-alpine
---> 224765a6bdbe
Step 2/3 : ADD /hello-service-0.0.1-SNAPSHOT.jar //
---> 259b1622e107
Step 3/3 : ENTRYPOINT ["sh","-c","java -Djava.security.egd=file:/dev/./urandom -jar /hello-service-1.0.0-SNAPSHOT.jar"]
---> Running in e86870c47b6a
Removing intermediate container e86870c47b6a
---> 847840116367
ProgressMessage{id=null, status=null, stream=null, error=null, progress=null, progressDetail=null}
Successfully built 847840116367
Successfully tagged springio/hello-service:latest
[INFO] Built springio/hello-service
[INFO] Tagging springio/hello-service with 1.0.0-SNAPSHOT
[INFO] Tagging springio/hello-service with latest
[INFO]
[INFO] ------------------------------------------------------------------------
[INFO] Building master 1.0.0-SNAPSHOT
[INFO] ------------------------------------------------------------------------
[INFO]
[INFO] --- maven-clean-plugin:2.5:clean (default-clean) @ master ---
[INFO]
[INFO] --- docker-maven-plugin:1.0.0:build (default-cli) @ master ---
[INFO] Using authentication suppliers: [ConfigFileRegistryAuthSupplier]
[INFO] ------------------------------------------------------------------------
[INFO] Reactor Summary:
[INFO]
[INFO] hello-service ...................................... SUCCESS [ 9.881 s]
[INFO] master ............................................. FAILURE [ 0.102 s]
[INFO] ------------------------------------------------------------------------
[INFO] BUILD FAILURE
[INFO] ------------------------------------------------------------------------
[INFO] Total time: 10.650 s
[INFO] Finished at: 2018-04-18T13:33:49+02:00
[INFO] Final Memory: 54M/520M
[INFO] ------------------------------------------------------------------------
[ERROR] Failed to execute goal com.spotify:docker-maven-plugin:1.0.0:build (default-cli) on project master: Exception caught: Must specify baseImage if dockerDirectory is null -> [Help 1]
[ERROR] .....
So the service is packaged, the image is built but I don't understand why in the master pom maven is trying to build an image even if there is no docker plugin inside it ?
A: By default this docker-maven-plugin is bind with maven's package build phase.
Check following link bind-docker-commands-to-maven-phases
You can skip Docker goals bound to Maven phases with:
-DskipDockerBuild to skip image build
-DskipDockerTag to skip image tag
-DskipDockerPush to skip image push
-DskipDocker to skip any Docker goals
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 88 |
Many of you have probably heard of matcha tea – it is growing in popularity and has many benefits to us busy human beings.
Wondering what these great and glorious benefits are?
Matcha tea is popular among Zen monks due to it being great at reducing stress.
Zen Buddhists drank this tea as it would assist them in remaining alert during their times in meditation, while also helping them to keep calm during these long hours.
Matcha tea is great for boosting your immune system and contains many more antioxidants than you probably originally thought.
One cup of matcha tea contains the same amount of antioxidants as ten cups of normal tea.
Cancer is a world-wide known problem, so finding out that matcha tea can assist in preventing this, has no doubt helped boost its popularity.
This is due to the content of antioxidants that this drink contains (known as catechins), one of the most powerful being EGCG (epigallocatechin gallate) which is thought to be a strong anti-carcinogen. What these antioxidants to is search the body for dangerous free radicals.
Research has shown that a cup of matcha tea has 137 times the amount of EGCG antioxidants unlike a regular green tea.
People living in Okinawa, Japan have been recognised for living the longer. Matcha green tea has a great input in this, due to these people regularly consuming the drink.
Matcha tea is also useful in combating inflammation and oxidation.
Further studies have shown that matcha tea (or other extract of green teas) dramatically lowers the serum total cholesterol along with LDL cholesterol concentrations.
Weight loss and dieting is becoming popular with people wanting to be fit and healthy, especially with Christmas approaching and the new year not far behind.
Therefore, it is interesting to find out that matcha tea which is high in catechins contain thermogenic properties and is also able to assist in promoting fat oxidation.
It is proven that consuming green tea increases the bodies calorie burning rate from 10% up to 35-43% of the daily energy expenditure.
Also, by drinking matcha green tea, your body will burn 25% more fat during exercise.
Matcha tea is rich in chlorophyll due to the tea leaves growing in the shade. What this means is that chlorophyll apparently helps to detoxify the body. While this has yet to be scientifically proven, it is still useful to know that it can clear the harmful toxins from your body.
Unlike your average green tea, matcha contains 5 times more L-theanine which can create alpha wave activity in your brain which will assist in relieving any stress which was originally fogging up your brain, while helping you to relax and lower your blood pressure.
Being high in fiber matcha tea is great for stabilising blood sugar levels and assisting in easing any constipation.
If you are suffering from tiredness at work in the afternoon, then matcha green tea is the perfect pick-me-up!
This drink will increase your concentration levels and is the perfect coffee replacement, especially since you won't suffer from any headaches like usually would when the caffeine leaves your system.
Matcha green tea is available as a concentrated powder and can be found at health food stores. It is recommended to buy only organic matcha, because it is produced without any artificial fertilisers, herbicides or pesticides.
Do not add matcha green tea powder to boiling water because it will taste "grassy." Boil the water and let it sit for 5 minutes before adding the tea.
It might take some time to get used to matcha's flavor. Give in to the power of the matcha tea! | {
"redpajama_set_name": "RedPajamaC4"
} | 6,742 |
Lahcen Samsam Akka (born 14 June 1942) is a Moroccan athlete. He competed in the men's shot put at the 1964 Summer Olympics and the 1972 Summer Olympics.
References
External links
1942 births
Living people
Athletes (track and field) at the 1964 Summer Olympics
Athletes (track and field) at the 1972 Summer Olympics
Moroccan male shot putters
Olympic athletes of Morocco
Place of birth missing (living people)
Mediterranean Games silver medalists for Morocco
Mediterranean Games medalists in athletics
Athletes (track and field) at the 1971 Mediterranean Games | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,441 |
This guide will detail how to install Cachet on your server.
## Download the source code with Git
> **Check out the latest version!**
> The tags below are examples of what will be shown.
> You should always run git checkout on the latest tag.
```
$ cd /var/www # Or wherever you chose to install web applications to
$ git clone https://github.com/CachetHQ/Cachet.git
$ cd Cachet
$ git tag -l
v2.3.1
v2.3.10
v2.3.11
v2.3.12
v2.3.13
v2.3.14
git checkout v2.3.14
```
## Editing the configuration file
By default Cachet comes with a `.env.example` file. You'll need to copy this
file to `.env` regardless of what environment you're working on.
> On Windows you can use `copy .env.example .env` if you can't do it using the
> explorer.
It's now just a case of editing this new .env file and setting the values of your setup.
> **Environment Configuration Notice**
> Any values with spaces in them should be contained within double quotes.
The `.env` file set environment variables that will be used by the application.
> **SQLite hosts**
> If you're using SQLite then your .env file should not contain a
> `DB_HOST` key. You'll also need to touch ./database/database.sqlite
> and give it the required permissions.
## Installing Composer
Cachet uses dependencies, so it's required to have Composer installed.
Composer can be installed following the [official guide][1]
## Installing dependencies
```bash
composer install --no-dev -o
```
If you are installing Cachet as a contributor, you can forget the `--no-dev`
option.
> **Tip for Windows users**
> If you're stuck at the Composer stage, you can run
> `composer install --no-dev -o --no-scripts`
> which usually fixes any issues on Windows servers.
## Using the install command
Cachet comes with an installation command that will:
- Run migrations
- Run seeders (of which there are none)
```bash
php artisan cachet:install
```
> Never change the `APP_KEY` after installation on production environment.
> This will result in all of your encrypted/hashed data being lost.
> **Getting a 500 - Internal Server Error?**
> If you get a 500 error when visiting your status page, you may need to
> run `chmod -R 755 .env bootstrap/cache storage`.
> Also if you set value `file` for `CACHE_DRIVER` and `SESSION_DRIVER` parameters in `.env` file run `chmod -R 755 bootstrap/cachet`.
> Finally run `rm -rf bootstrap/cache/*` for delete old cache.
## Running Cachet on Apache
> **Required Apache Modules**
> You need to enable `mod_rewrite` for Apache. On Debian-based systems you can do this by
>
> `sudo a2enmod rewrite`
Once Cachet is setup, the Apache installation is as simple as creating a
new Virtual Host entry in the httpd-vhosts.conf file.
```
<VirtualHost *:80>
ServerName cachet.dev
# Or whatever you want to use
ServerAlias cachet.dev
# Make this the same as ServerName
DocumentRoot "/var/www/Cachet/public"
<Directory "/var/www/Cachet/public">
Require all granted
# Used by Apache 2.4
Options Indexes FollowSymLinks
AllowOverride All
Order allow,deny
Allow from all
</Directory>
</VirtualHost>
```
Restart Apache by running the following:
`sudo service apache2 restart`
If you also need HTTPS on apache you will need to get the ssl mod installed
and the default ssl conf file enabled. See DigitalOcean's [documentation][2].
## Running Cachet on nginx
- You'll need to install php5-fpm - [DigitalOcean][3] has a nice LEMP installation tutorial
- Generate your SSL key+certificate
- Create a new vhost such as `/etc/nginx/sites-enabled/cachet.conf:`
```
# Upstream to abstract backend connection(s) for php
upstream php {
server unix:/tmp/php-cgi.socket;
server 127.0.0.1:9000;
}
server {
server_name cachet.mycompany.com; # Or whatever you want to use
listen 80 default;
rewrite ^(.*) https://cachet.mycompany.com$1 permanent;
}
# HTTPS server
server {
listen 443;
server_name cachet.mycompany.com;
root /var/vhost/cachet.mycompany.com/public;
index index.php;
ssl on;
ssl_certificate /etc/ssl/crt/cachet.mycompany.com.crt; # Or wherever your crt is
ssl_certificate_key /etc/ssl/key/cachet.mycompany.com.key; # Or wherever your key is
ssl_session_timeout 5m;
# Best practice as at March 2014
ssl_protocols TLSv1 TLSv1.1 TLSv1.2;
ssl_prefer_server_ciphers on;
ssl_ciphers "ECDHE-RSA-AES128-GCM-SHA256:ECDHE-ECDSA-AES128-GCM-SHA256:ECDHE-RSA-AES256-GCM-SHA384:ECDHE-ECDSA-AES256-GCM-SHA384:DHE-RSA-AES128-GCM-SHA256:DHE-DSS-AES128-GCM-SHA256:kEDH+AESGCM:ECDHE-RSA-AES128-SHA256:ECDHE-ECDSA-AES128-SHA256:ECDHE-RSA-AES128-SHA:ECDHE-ECDSA-AES128-SHA:ECDHE-RSA-AES256-SHA384:ECDHE-ECDSA-AES256-SHA384:ECDHE-RSA-AES256-SHA:ECDHE-ECDSA-AES256-SHA:DHE-RSA-AES128-SHA256:DHE-RSA-AES128-SHA:DHE-DSS-AES128-SHA256:DHE-RSA-AES256-SHA256:DHE-DSS-AES256-SHA:DHE-RSA-AES256-SHA:AES128-GCM-SHA256:AES256-GCM-SHA384:AES128-SHA256:AES256-SHA256:AES128-SHA:AES256-SHA:AES:CAMELLIA:DES-CBC3-SHA:!aNULL:!eNULL:!EXPORT:!DES:!RC4:!MD5:!PSK:!aECDH:!EDH-DSS-DES-CBC3-SHA:!EDH-RSA-DES-CBC3-SHA:!KRB5-DES-CBC3-SHA";
ssl_buffer_size 1400; # 1400 bytes, within MTU - because we generally have small responses. Could increase to 4k, but default 16k is too big
location / {
add_header Strict-Transport-Security max-age=15768000;
try_files $uri /index.php$is_args$args;
}
location ~ \.php$ {
include fastcgi_params;
fastcgi_pass unix:/var/run/php5-fpm.sock;
fastcgi_param SCRIPT_FILENAME $document_root$fastcgi_script_name;
fastcgi_index index.php;
fastcgi_keep_conn on;
add_header Strict-Transport-Security max-age=15768000;
}
}
```
Start php5-fpm and nginx and you're done!
[1]: https://getcomposer.org/download/
[2]: https://www.digitalocean.com/community/tutorials/how-to-create-a-ssl-certificate-on-apache-for-ubuntu-14-04
[3]: https://www.digitalocean.com/community/tutorials/how-to-install-linux-nginx-mysql-php-lemp-stack-on-ubuntu-12-04
| {
"redpajama_set_name": "RedPajamaGithub"
} | 742 |
<!DOCTYPE html>
<html lang="en">
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<meta charset="UTF-8" />
<meta name="robots" content="index, follow, all" />
<title>Thelia\Handler\ExportHandler | Thelia 2 API</title>
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placeholder="Search">
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<li><a href="../../Thelia/Handler.html">Handler</a></li>
<li>ExportHandler</li>
</ol>
</div>
<div id="page-content">
<div class="page-header">
<h1>ExportHandler</h1>
</div>
<p> class
<strong>ExportHandler</strong>
</p>
<div class="description">
<p>Class ExportHandler</p> </div>
<h2>Methods</h2>
<div class="container-fluid underlined">
<div class="row">
<div class="col-md-2 type">
</div>
<div class="col-md-8 type">
<a href="#method___construct">__construct</a>(
<abbr title="Thelia\Handler\Symfony\Component\EventDispatcher\EventDispatcherInterface">EventDispatcherInterface</abbr> $eventDispatcher,
<abbr title="Thelia\Handler\Symfony\Component\DependencyInjection\ContainerInterface">ContainerInterface</abbr> $container)
<p>Class constructor</p> </div>
<div class="col-md-2"></div>
</div>
<div class="row">
<div class="col-md-2 type">
null|<a href="../../Thelia/Model/Export.html"><abbr title="Thelia\Model\Export">Export</abbr></a>
</div>
<div class="col-md-8 type">
<a href="#method_getExport">getExport</a>(
integer $exportId,
boolean $dispatchException = false)
<p>Get export model based on given identifier</p> </div>
<div class="col-md-2"></div>
</div>
<div class="row">
<div class="col-md-2 type">
null|<a href="../../Thelia/Model/Export.html"><abbr title="Thelia\Model\Export">Export</abbr></a>
</div>
<div class="col-md-8 type">
<a href="#method_getExportByRef">getExportByRef</a>(
string $exportRef,
boolean $dispatchException = false)
<p>Get export model based on given reference</p> </div>
<div class="col-md-2"></div>
</div>
<div class="row">
<div class="col-md-2 type">
null|<a href="../../Thelia/Model/ExportCategory.html"><abbr title="Thelia\Model\ExportCategory">ExportCategory</abbr></a>
</div>
<div class="col-md-8 type">
<a href="#method_getCategory">getCategory</a>(
integer $exportCategoryId,
boolean $dispatchException = false)
<p>Get export category model based on given identifier</p> </div>
<div class="col-md-2"></div>
</div>
<div class="row">
<div class="col-md-2 type">
<a href="../../Thelia/Core/Event/ExportEvent.html"><abbr title="Thelia\Core\Event\ExportEvent">ExportEvent</abbr></a>
</div>
<div class="col-md-8 type">
<a href="#method_export">export</a>(
<abbr title="Thelia\Handler\Thelia\Model\Export">Export</abbr> $export,
<abbr title="Thelia\Handler\Thelia\Core\Serializer\SerializerInterface">SerializerInterface</abbr> $serializer,
<abbr title="Thelia\Handler\Thelia\Core\Archiver\ArchiverInterface">ArchiverInterface</abbr> $archiver = null,
<abbr title="Thelia\Handler\Thelia\Model\Lang">Lang</abbr> $language = null,
boolean $includeImages = false,
boolean $includeDocuments = false,
null|array $rangeDate = null)
<p>Export</p> </div>
<div class="col-md-2"></div>
</div>
</div>
<h2>Details</h2>
<div id="method-details">
<div class="method-item">
<h3 id="method___construct">
<div class="location">at line 49</div>
<code>
<strong>__construct</strong>(
<abbr title="Thelia\Handler\Symfony\Component\EventDispatcher\EventDispatcherInterface">EventDispatcherInterface</abbr> $eventDispatcher,
<abbr title="Thelia\Handler\Symfony\Component\DependencyInjection\ContainerInterface">ContainerInterface</abbr> $container)</code>
</h3>
<div class="details">
<div class="method-description">
<p>Class constructor</p> </div>
<div class="tags">
<h4>Parameters</h4>
<table class="table table-condensed">
<tr>
<td>
<abbr title="Thelia\Handler\Symfony\Component\EventDispatcher\EventDispatcherInterface">EventDispatcherInterface</abbr></td>
<td>$eventDispatcher</td>
<td>An event dispatcher interface</td>
</tr>
<tr>
<td>
<abbr title="Thelia\Handler\Symfony\Component\DependencyInjection\ContainerInterface">ContainerInterface</abbr></td>
<td>$container</td>
<td></td>
</tr>
</table>
</div>
</div>
</div>
<div class="method-item">
<h3 id="method_getExport">
<div class="location">at line 65</div>
<code>
null|<a href="../../Thelia/Model/Export.html"><abbr title="Thelia\Model\Export">Export</abbr></a>
<strong>getExport</strong>(
integer $exportId,
boolean $dispatchException = false)</code>
</h3>
<div class="details">
<div class="method-description">
<p>Get export model based on given identifier</p> </div>
<div class="tags">
<h4>Parameters</h4>
<table class="table table-condensed">
<tr>
<td>
integer</td>
<td>$exportId</td>
<td>An export identifier</td>
</tr>
<tr>
<td>
boolean</td>
<td>$dispatchException</td>
<td>Dispatch exception if model doesn't exist</td>
</tr>
</table>
<h4>Return Value</h4>
<table class="table table-condensed">
<tr>
<td>
null|<a href="../../Thelia/Model/Export.html"><abbr title="Thelia\Model\Export">Export</abbr></a></td>
<td></td>
</tr>
</table>
<h4>Exceptions</h4>
<table class="table table-condensed">
<tr>
<td><a target="_blank" href="http://php.net/ErrorException"><abbr title="ErrorException">ErrorException</abbr></a></td>
<td></td>
</tr>
</table>
</div>
</div>
</div>
<div class="method-item">
<h3 id="method_getExportByRef">
<div class="location">at line 93</div>
<code>
null|<a href="../../Thelia/Model/Export.html"><abbr title="Thelia\Model\Export">Export</abbr></a>
<strong>getExportByRef</strong>(
string $exportRef,
boolean $dispatchException = false)</code>
</h3>
<div class="details">
<div class="method-description">
<p>Get export model based on given reference</p> </div>
<div class="tags">
<h4>Parameters</h4>
<table class="table table-condensed">
<tr>
<td>
string</td>
<td>$exportRef</td>
<td>An export reference</td>
</tr>
<tr>
<td>
boolean</td>
<td>$dispatchException</td>
<td>Dispatch exception if model doesn't exist</td>
</tr>
</table>
<h4>Return Value</h4>
<table class="table table-condensed">
<tr>
<td>
null|<a href="../../Thelia/Model/Export.html"><abbr title="Thelia\Model\Export">Export</abbr></a></td>
<td></td>
</tr>
</table>
<h4>Exceptions</h4>
<table class="table table-condensed">
<tr>
<td><a target="_blank" href="http://php.net/ErrorException"><abbr title="ErrorException">ErrorException</abbr></a></td>
<td></td>
</tr>
</table>
</div>
</div>
</div>
<div class="method-item">
<h3 id="method_getCategory">
<div class="location">at line 121</div>
<code>
null|<a href="../../Thelia/Model/ExportCategory.html"><abbr title="Thelia\Model\ExportCategory">ExportCategory</abbr></a>
<strong>getCategory</strong>(
integer $exportCategoryId,
boolean $dispatchException = false)</code>
</h3>
<div class="details">
<div class="method-description">
<p>Get export category model based on given identifier</p> </div>
<div class="tags">
<h4>Parameters</h4>
<table class="table table-condensed">
<tr>
<td>
integer</td>
<td>$exportCategoryId</td>
<td>An export category identifier</td>
</tr>
<tr>
<td>
boolean</td>
<td>$dispatchException</td>
<td>Dispatch exception if model doesn't exist</td>
</tr>
</table>
<h4>Return Value</h4>
<table class="table table-condensed">
<tr>
<td>
null|<a href="../../Thelia/Model/ExportCategory.html"><abbr title="Thelia\Model\ExportCategory">ExportCategory</abbr></a></td>
<td></td>
</tr>
</table>
<h4>Exceptions</h4>
<table class="table table-condensed">
<tr>
<td><a target="_blank" href="http://php.net/ErrorException"><abbr title="ErrorException">ErrorException</abbr></a></td>
<td></td>
</tr>
</table>
</div>
</div>
</div>
<div class="method-item">
<h3 id="method_export">
<div class="location">at line 152</div>
<code>
<a href="../../Thelia/Core/Event/ExportEvent.html"><abbr title="Thelia\Core\Event\ExportEvent">ExportEvent</abbr></a>
<strong>export</strong>(
<abbr title="Thelia\Handler\Thelia\Model\Export">Export</abbr> $export,
<abbr title="Thelia\Handler\Thelia\Core\Serializer\SerializerInterface">SerializerInterface</abbr> $serializer,
<abbr title="Thelia\Handler\Thelia\Core\Archiver\ArchiverInterface">ArchiverInterface</abbr> $archiver = null,
<abbr title="Thelia\Handler\Thelia\Model\Lang">Lang</abbr> $language = null,
boolean $includeImages = false,
boolean $includeDocuments = false,
null|array $rangeDate = null)</code>
</h3>
<div class="details">
<div class="method-description">
<p>Export</p> </div>
<div class="tags">
<h4>Parameters</h4>
<table class="table table-condensed">
<tr>
<td>
<abbr title="Thelia\Handler\Thelia\Model\Export">Export</abbr></td>
<td>$export</td>
<td></td>
</tr>
<tr>
<td>
<abbr title="Thelia\Handler\Thelia\Core\Serializer\SerializerInterface">SerializerInterface</abbr></td>
<td>$serializer</td>
<td></td>
</tr>
<tr>
<td>
<abbr title="Thelia\Handler\Thelia\Core\Archiver\ArchiverInterface">ArchiverInterface</abbr></td>
<td>$archiver</td>
<td></td>
</tr>
<tr>
<td>
<abbr title="Thelia\Handler\Thelia\Model\Lang">Lang</abbr></td>
<td>$language</td>
<td></td>
</tr>
<tr>
<td>
boolean</td>
<td>$includeImages</td>
<td></td>
</tr>
<tr>
<td>
boolean</td>
<td>$includeDocuments</td>
<td></td>
</tr>
<tr>
<td>
null|array</td>
<td>$rangeDate</td>
<td></td>
</tr>
</table>
<h4>Return Value</h4>
<table class="table table-condensed">
<tr>
<td>
<a href="../../Thelia/Core/Event/ExportEvent.html"><abbr title="Thelia\Core\Event\ExportEvent">ExportEvent</abbr></a></td>
<td></td>
</tr>
</table>
</div>
</div>
</div>
</div>
</div>
<div id="footer">
Generated by <a href="http://sami.sensiolabs.org/">Sami, the API Documentation Generator</a>.
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,392 |
{"url":"https:\/\/tex.stackexchange.com\/questions\/359939\/how-should-i-make-blur-shadows-appear-with-their-nodes-in-a-forest-tree-in-beame","text":"# How should I make blur shadows appear with their nodes in a Forest tree in Beamer? [closed]\n\nA little while ago, I asked How can I make drop shadows appear with their nodes in a Forest tree in Beamer? At the time, I just eliminated the shadows and went with a shodowless tree. Later, however, I found a very awkward method which worked. A while after that, I received Symbol 1 provided a real answer.\n\nNow I am trying to adapt that solution for my real uses. First, I wanted to make the code conditional on the loading of the appropriate library, so that I can include it in my standard Beamer configuration files.\n\n\\makeatletter\n% ateb Symbol 1: https:\/\/tex.stackexchange.com\/a\/357412\/\n\\def\\tikzopacityregister{1}%\n\\tikzset{\nopacity\/.append code={\n\\pgfmathsetmacro\\tikzopacityregister{#1*\\tikzopacityregister}%\n},\nopacity aux\/.code={% this is the original definition of opacity\n},\n}%\n}{}\n\\makeatother\n\n\nI couldn't find something like \\if@tikzlibrary@loaded, so I went with testing for the existence of a PGF key. If there's a way of testing whether a library is loaded, that would obviously be preferable.\n\nSecond, I want to make the code work with shadows.blur, so I can use blur shadow as well as drop shadow. The following seems to work, but I'm not sure if it is even vaguely reasonable as a method.\n\nopacity\/.append code={\n\\pgfmathsetmacro\\tikzopacityregister{#1*\\tikzopacityregister}%\n},\n\n\nI tried to append something to, say shadow opacity or to set this when opacity is set. However, I couldn't get this to work and ended up with the above. Again, not great as it relies on an internal implementation detail rather than just the public interface.\n\nI also still get thin outlines of the shadows in this case, but that may be a viewer artefact. (They are still there at 1600% magnification, however, which is the most Okular provides.)\n\nWhat is the correct way to do this?\n\n\\documentclass{beamer}\n\\usepackage{forest}\n\\tikzset{% set up for transitions using tikz with beamer overlays - developed by Daniel (https:\/\/tex.stackexchange.com\/a\/55849\/) and, in earlier form, by Matthew Leingang (https:\/\/tex.stackexchange.com\/a\/6155\/) and modified for this use, I think by Qrrbrbirlbel (https:\/\/tex.stackexchange.com\/a\/112471\/)\ninvisible\/.style={opacity=0,text opacity=0},\nvisible on\/.style={alt=#1{}{invisible}},\nalt\/.code args={<#1>#2#3}{%\n\\alt<#1>{\\pgfkeysalso{#2}}{\\pgfkeysalso{#3}} % \\pgfkeysalso doesn't change the path\n},\n}\n\\forestset{%\nvisible with edge from\/.style={% based on visible on, developed by Qrrbrbirlbel (https:\/\/tex.stackexchange.com\/a\/112471\/)\n\/tikz\/visible on=<#1->,\n\/tikz\/every label\/.append style={visible on=<#1->},\nedge={\/tikz\/visible on=<#1->},\n},\n}\n\\makeatletter\n% ateb Symbol 1: https:\/\/tex.stackexchange.com\/a\/357412\/\n\\def\\tikzopacityregister{1}%\n\\tikzset{\nopacity\/.append code={\n\\pgfmathsetmacro\\tikzopacityregister{#1*\\tikzopacityregister}%\n},\nopacity aux\/.code={% this is the original definition of opacity\n},\n}%\n}{}\n\\makeatother\n\\begin{document}\n\n\\begin{frame}\n\\begin{forest}\nfor tree={\ndraw,\nfill=white,\n},\nbefore typesetting nodes={\nfor tree={\ntempcounta\/.option=level,\ntempcounta'+=1,\nvisible with edge from\/.register=tempcounta,\n}\n}\n[first slide\n[second slide[third slide][third slide]]\n[second slide]\n]\n\\end{forest}\n\\end{frame}\n\\end{document}\n\n\n## closed as off-topic by user36296, Troy, Bobyandbob, user31729, Stefan PinnowMar 18 '18 at 17:04\n\n\u2022 This question does not fall within the scope of TeX, LaTeX or related typesetting systems as defined in the help center.\nIf this question can be reworded to fit the rules in the help center, please edit the question.\n\n\u2022 If I recall correctly, blur is eventually overlapping lines with various width. In case there is opacity\u22601, you have to make a transparency group for those lines. \u2013\u00a0Symbol 1 Mar 23 '17 at 2:56\n\u2022 @Symbol1 Hmmm. Thanks. I think I'll live with the lines then as I have no idea how to apply a transparency group here. \u2013\u00a0cfr Mar 23 '17 at 2:59\n\u2022 Or maybe I should stick to sharp shadows here. \u2013\u00a0cfr Mar 23 '17 at 3:01\n\u2022 As for checking library, you probably want to see tikz.code.tex line 5317 and 18. \u2013\u00a0Symbol 1 Mar 23 '17 at 3:13\n\u2022 I'm voting to close this question as issue of the pdf viewer \u2013\u00a0user36296 Mar 18 '18 at 16:13","date":"2019-09-21 15:48:45","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\": 0.7070478200912476, \"perplexity\": 1995.6250619760021}, \"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-2019-39\/segments\/1568514574532.44\/warc\/CC-MAIN-20190921145904-20190921171904-00446.warc.gz\"}"} | null | null |
{"url":"http:\/\/stm.sciencemag.org\/content\/5\/173\/173sr2?ijkey=c54786fd81ac939d8557951a4bc6c16621d39279&keytype2=tf_ipsecsha","text":"State of the Art ReviewRadiation Oncology\n\n# New Paradigms and Future Challenges in Radiation Oncology: An Update of Biological Targets and Technology\n\nSee allHide authors and affiliations\n\nScience Translational Medicine\u00a0 20 Feb 2013:\nVol. 5, Issue 173, pp. 173sr2\nDOI: 10.1126\/scitranslmed.3005148\n\n## Abstract\n\nRadiation oncology exploits the biological interaction of radiation within tissue to promote tumor death while minimizing damage to surrounding normal tissue. The clinical delivery of radiation relies on principles of radiation physics that define how radiation energy is deposited in the body, as well as technology that facilitates accurate tumor targeting. This review will summarize the current landscape of recent biological and technological advances in radiation oncology, describe the challenges that exist, and offer potential avenues for improvement.\n\n## Introduction\n\nThe field of radiation oncology\u2014where ionizing radiation is used to treat a variety of cancers as well as benign conditions\u2014was born shortly after the discovery of x-rays and their effects on tissue in 1895 (1). At first, treatments were typically delivered in single doses using low-energy cathode ray tubes or radium-filled glass tubes positioned close to tumors. Technological developments between 1920 and 1945 focused on improving beam output and energy. However, the low energy (200 to 500 kV) of x-rays used in that period was associated with skin toxicity because of poor penetration, thereby limiting the use of radiotherapy for deep tumors. The development of cobalt-60 units and linear accelerators in the 1950s to deliver \u201csupervoltage\u201d radiation energies (\u22651 MeV) was a critical advance because these high-energy x-rays could penetrate further to reach deeper seated tumors. Fractionated therapy, which dates back to the 1920s, divides treatment into multiple small doses rather than one large radiation dose and has allowed for further improvement in tolerance of normal tissues to treatment. Advances in technology, imaging, and cancer biology over subsequent decades have pushed the field of radiation oncology closer toward the idealized goal of noninvasively achieving maximal local cancer control with minimal normal tissue toxicity.\n\n## Basic Principles and Current Uses of Radiotherapy\n\nCancer may grow locally or spread systemically through lymphatic or hematogenous routes. Successful treatment requires therapy targeted toward all sites of involvement. The three major modalities of cancer therapy\u2014surgery, radiation therapy (RT), and chemotherapy\u2014can be used alone or in combination to address all sites at risk for harboring disease. In a curative-intent approach, surgery and\/or RT are generally used to address local-regional areas of risk. Surgery remains the most commonly used modality to treat local disease. By its nature, surgery can be both therapeutic and diagnostic because tumor excision expeditiously provides tissue for histologic examination and staging. RT as a sole modality can sometimes offer a noninvasive alternative to the therapeutic role of surgery, with the possibility for organ preservation, such as with bladder and laryngeal cancer. As an adjuvant therapy, RT can facilitate resection when given before surgery, or treat microscopic residual disease when given after surgery, such as treatment after breast-conserving lumpectomy. On the other hand, chemotherapy is given to treat known metastatic disease or as an adjuvant to reduce the risk of potential micrometastasis. Chemotherapy is often also combined with RT to act as a radiosensitizer for the purpose of increasing local control. The optimal use of each modality of cancer therapy is tailored according to the cancer cell histology, anatomic location, stage of cancer, and other patient factors. Some common diseases that serve as examples of the role of radiotherapy in integrated multimodality cancer treatment are summarized in Table 1. It is crucial for treatment approaches to consider quality of life in survivorship because each modality carries a different set of risks that need to be balanced against one another to provide the optimal risk-benefit ratio for each individual patient.\n\nTable 1\n\nMultidisciplinary management of several common cancers treated with curative intent.\n\nView this table:\n\nEmpiric clinical observations in the early 20th century demonstrated that daily radiation exposures can induce death in cancerous cells while allowing for normal tissue recovery, provided that the daily doses [expressed in units of gray (Gy)] are relatively small. This observation formed the basis for fractionated RT. Many mechanistic explanations for this effect have been proposed by radiobiologists. The ability of normal tissue to repopulate itself with healthy cells clearly represents one important component of normal tissue tolerance of fractionated radiotherapy. A common course of RT can average 6 to 8 weeks of treatment, with five to six daily treatments per week.\n\nAt the tissue level, the impact of RT on tissue function with increasing radiation dose can be graphically represented by a sigmoid-shaped curve (Fig. 1). The sigmoidal curve that describes tumor control probability is situated to the left of the sigmoidal curve that describes normal tissue complication probability. The degree of separation between these curves defines a therapeutic window, in which the dose of radiation is predicted to eradicate tumor while maintaining normal tissue tolerance. The sigmoidal relationship between dose and response implies that for any given tissue, there is a dose threshold above or below which incremental changes in dose yield little additional impact. However, within a critical range on the steep portion of the curve (such as cumulative doses of 40 to 100 Gy when given with \u201cconventional\u201d fractionation of 1.8 to 2.0 Gy per fraction), small increases in dose may result in large increases in clinical impact. Dose prescriptions in RT are determined with consideration to the unique relationship between the dose-response curves for the specific tumor and surrounding normal tissues, which vary widely for each clinical circumstance. The figure is an oversimplification because families of curves may exist for the tumor clones comprising a tumor and for the complex components of normal tissues intertwined with the tumor.\n\nStandard chemotherapeutic agents are the most common agents used for increasing the local efficacy of radiotherapy; however, this review will focus on more recent strategies of identifying targeted radiosensitizers, particularly those that have been tested in the clinic. It is important to remember that inhibition of specific proteins can frequently generate sensitization of cells in culture, but ultimately, a drug must improve the therapeutic index of radiation to be clinically useful (see illustration of this concept in Fig. 1).\n\nThe molecular pathophysiology of radiotherapy: Sensors, transducers, and effectors of DNA damage. One gray generates about 105 ionization events per cell, producing about 1000 to 2000 single-strand DNA breaks (SSBs) and 40 double-strand DNA breaks (DSBs) per nucleus. A large and growing list of non-DNA repair\/checkpoint\u2013related mechanisms contributes to cellular responses to radiation. Although ionizing radiation is known to generate DNA base damage, SSBs, and DSBs, the DSBs are generally thought to represent the principal lethal events and the most critical lesions to radiotherapy. DSBs initiate a complex set of cellular responses including DNA damage recognition and transduction of the signal, resulting in many downstream effects including cell cycle checkpoint activation, induction and coordination of stress response genes, DNA repair, and\/or activation of the apoptotic cascade (Fig. 2).\n\nTargeting of DSB response and repair. The MRN protein complex (Mre11, Rad50, Nbs1), a principal sensor of DNA damage, accumulates at DSBs very rapidly after radiation and participates in activating ataxia-telangiectasia mutated (ATM) protein. Although ATM and ATR (ATM- and RAD3-related) perform partially overlapping functions, ATM preferentially recognizes DSBs, whereas ATR preferentially senses replication-blocking DNA lesions. Both ATM and ATR phosphorylate downstream targets that regulate cell cycle checkpoints and apoptosis, as well as other forms of cellular responses like senescence, autophagy, and DNA repair. One centrally important phosphorylation target is the chromatin protein histone H2AX because chromatin structure subsequently becomes less condensed and allows for the recruitment of repair proteins. Because H2AX phosphorylation is easily detectable with fluorescent microscopic methods, it has become a common marker for DSB induction and resolution (repair) in both experimental and clinical settings. After DNA damage recognition, phosphorylated H2AX promotes the recruitment of other sensor\/effector proteins including 53BP1, MDC1, and BRCA1, which regulate the processing of damaged DNA ends in preparation for repair. This process is regulated by several ubiquitin and SUMO ligases (PIAS1, PIAS4, RNF4, and RNF8) and the PSMD4 proteasome.\n\nHomologous recombination (HR) and nonhomologous end-joining (NHEJ) repair pathways are the two pathways that contribute to the repair of DSBs. Both repair pathways occur after the recruitment of the sensor\/effector proteins discussed above. HR involves the identification of a stretch of homologous DNA and replication of the missing genetic information from this homologous DNA template (2). Alternatively, the NHEJ pathway processes the broken DNA ends and religates them, frequently by making use of a region of microhomology (3). These pathways appear to have somewhat overlapping and complementary roles.\n\nMany components of the DSB repair pathways have been investigated as therapeutic targets, and some chemical inhibitors might be considered as possible lead compounds in oncology drug development (Fig. 3). For example, mirin is a chemical inhibitor of the MRN complex (5). Consistent with the upstream role of MRN in sensing and signaling DNA damage, mirin generates a broad range of cellular effects, including inhibition of ATM activation, loss of G2-M cell cycle checkpoint, down-regulation of HR, and down-regulation of NHEJ repair efficiency (5, 6). This lack of cancer cell specificity may limit the utility of mirin in the clinic. Similar upstream signaling functions have been targeted with inhibitors of the family of phosphatidylinositol 3-kinase (PI3K)\u2013related kinases (PIKK), which include ATM, ATR, mammalian target of rapamycin (mTOR), human suppressor of morphogenesis in genitalia-1 (hSMG-1), DNA-PKcs, and transformation\/transcription domain\u2013associated protein (TRRAP) (7). Several broad-spectrum inhibitors of PIKKs have been developed, including wortmannin or LY294002; however, at least some have toxicities that limit their utility in the clinical setting. More specific inhibitors of ATM and\/or ATR include KU-55933, CGK733, NU6027, and CP466722. KU-55933 has been the focus of particular attention because it is a specific adenosine triphosphate\u2013competitive inhibitor of ATM that is capable of sensitizing cells to radiation and several chemotherapeutic drugs, including etoposide, doxorubicin, and camptothecin (8).\n\nSeveral inhibitory compounds have also been developed to modulate specific DSB repair pathways. HR has been inhibited by specifically targeting RAD51 protein or RAD51 paralogs [see Budke et. al. and references therein (9)]. Other drugs have also been shown to lower HR efficiency by nonspecifically reducing RAD51 protein levels. These strategies are promising because RAD51 protein is highly expressed in many human cancers (10, 11) and because HR inhibition has been shown to promote preferential sensitization of tumor cells relative to normal cells (12, 13). These observations suggest that human tumors may develop \u201caddictions\u201d to abnormally high RAD51 levels that can be exploited pharmacologically. Likewise, compounds have been developed to target components of the NHEJ repair. Again, this can be accomplished by blocking PIKKs. However, more targeted NHEJ inhibitors have also been developed, including several different DNA-PK inhibitors that include NU7441, Vanillin, SU11752, IC87102, IC87361, NU7026, CC-115, and Salvicine.\n\nRelatively few of these inhibitors of DSB repair have transitioned into clinical trials, perhaps because of a lack of commercial interest and the recognition of radiotherapy as fertile ground for pharmaceutical development. One interesting new strategy that is being developed clinically by DNA Therapeutics involves small DNA molecules that act as DNA bait. One such agent, termed Dbait or DT01, mimics DNA double-strand breaks and acts to disorganize damage signaling and DNA repair (14). DT01 is currently being studied in a phase 1 trial for patients with metastatic melanoma.\n\nPoly(adenosine diphosphate\u2013ribose) polymerase inhibitors. Poly[adenosine diphosphate (ADP)\u2013ribose] polymerase inhibitors (PARPi) are currently being studied in numerous clinical trials with chemotherapy and a few trials with radiotherapy. The PARP family of proteins is defined by their capacity to modify target proteins by the covalent addition of poly(ADP-ribose) polymers. PARP1 is the most abundant of this protein family, and it accounts for about 80% of PARP activity in cells. PARP1 and PARP2 have DNA binding domains, and their catalytic function is activated when they bind sites of DNA damage. When PARP1 becomes activated, it generates long and branching poly(ADP-ribose) chains on histones and other proteins located near DNA breaks. These polymer scaffolds are important in recruiting other DNA repair proteins (for example, the base excision repair protein XRCC1) to the break site. PARPi compounds have generated intense interest by the oncology community since 2005, following the demonstration of synthetic lethality in BRCA-defective cells (15, 16). Synthetic lethality is a concept whereby a tumor is defective in one survival pathway, and inhibition of an escape pathway is an effective cytotoxic strategy. A common feature of these drugs is the exquisite hypersensitivity observed with HR-defective tumor cells, including triple-negative breast cancers, which exhibit epigenetic deregulation of HR.\n\nSeveral studies have demonstrated that PARPi compounds can radiosensitize tumors in preclinical models (1719). Ionizing radiation, as discussed earlier, induces both SSBs and DSBs. Although the PARPi-mediated mechanism of radiosensitization remains somewhat unclear, PARP inhibitors are known to sensitize cells to agents that generate SSBs (20). Chalmers and colleagues have pointed out two key aspects of PARPi effects pertinent to this issue: (i) radiosensitization occurs primarily in replicating cells and (ii) PARPi compounds delay rather than abolish SSB repair (18). These factors suggest that components of replication machinery may collide with unrepaired PARP-bound SSB lesions, thereby generating more toxic lesions than the starting SSBs. Other possible mechanisms of PARPi-mediated radiosensitization may include reoxygenation of hypoxic tumors; PARPi compounds have demonstrated vasoactive properties and could potentially counteract the radioresistance associated with hypoxia. Finally, recent studies have shown that chronic hypoxia in tumors can generate a reduction in HR protein expression and function. Hence, hypoxia within tumors may generate specific anatomic compartments that behave like BRCA-defective tumors, wherein a contextual synthetic lethality occurs for PARPi (21).\n\nSeveral clinical trials are presently evaluating combinations of PARPi drugs plus radiotherapy, either with or without chemotherapeutic drugs for diseases including rectal, brain, and breast cancers (http:\/\/clinicaltrials.gov). Again, it is unknown whether these agents will improve the therapeutic index of radiotherapy, and specific disease site trials will be necessary to determine potential efficacy.\n\nHistone deacetylase inhibitors. Histone deacetylases (HDACs) represent a family of at least 18 enzymes that remove acetyl groups from lysine residues in core histone proteins. Deacetylation exposes the positive charges on histones, thereby increasing their interaction with the negatively charged phosphate backbone of DNA. This results in more compacted chromatin that is relatively inaccessible to modulation by transcription factors. This pathway has relevance to radiation biology, given that HDAC inhibitors stimulate radiation-induced cell cycle arrest, apoptosis, and DSB formation (22, 23). The mechanism is unclear, but studies suggest that HDAC inhibitors suppress DNA repair efficiency.\n\nTraditional HDAC-inhibitory drugs, like valproic acid, have a long history as mood stabilizers and antiepileptics. Several second-generation inhibitors have more recently been developed, including vorinostat, belinostat, and panobinostat. Ongoing clinical trials are evaluating these drugs plus radiotherapy in many tumor types and in a wide range of clinical settings, including fractionated radiotherapy, stereotactic radiosurgery (SRS), and together with an infused [131I]MIBG radiopharmaceutical. One recently completed phase 1 study showed that vorinostat (at 300 mg once daily) was tolerable in combination with a palliative radiotherapy course to the pelvis, consisting of 30 Gy over 2 weeks (24). Unfortunately, efficacy results from phase 2 trials are not available.\n\nCell cycle arrest after radiation. It has been known for decades that mammalian cells arrest in G1 and G2 after radiation exposure. The pathways that govern cell cycle checkpoints were first discovered in yeast by Hartwell, Nurse, and Hunt, for which they received the 2001 Nobel Prize. Briefly, radiation-induced activation of ATM and ATR leads to phosphorylation of the downstream effector kinases Chk1 and Chk2. These in turn phosphorylate the phosphatase CDC25A, which in turn becomes degraded and thus unable to dephosphorylate and activate CDK2; the net result is cell cycle arrest. Additionally, Chk2 phosphorylation of p53 activates expression of p21, which also induces cell cycle arrest.\n\nTargeted inhibition of the cell cycle checkpoint machinery has been explored as a method to sensitize cells to DNA damage. Some of these strategies have been based on the tendency of tumor cells to have abnormal G1 checkpoint functions but intact G2 checkpoint mechanisms. Targeted inhibition of G2 checkpoints can promote progression of damaged cells to mitotic catastrophe, and this effect may be tumor-specific because normal tissues are preferentially protected by intact G1 checkpoints. This may also potentially counteract the radiation resistance that has been observed in some cancer stem cells, such as CD133-positive glioma stem cells (25).\n\nA number of chemical inhibitors have been developed to target Chk1 and\/or Chk2, including UCN-01, AZD7762 (AstraZeneca), XL844 (EXEL-9844, Exelixis), LY2606368 (Eli Lilly), and PF-00477736 (Pfizer). All of these compounds have been tested in early-phase clinical trials against solid tumors. Many of these are being tested in combination with various chemotherapeutic drugs. However, it remains an open question as to whether these drugs can be safely combined with radiotherapy, and whether they can improve the therapeutic index.\n\nModifiers of cellular death after radiation. Cells subsequently face a critical period, particularly if radiation damage is not completely repaired (26). These critically injured cells may progress unrepaired to mitotic catastrophe or simply lose proliferative capacity by undergoing senescence. Radiation-induced senescence is commonly associated with activation of the p16\/RB and p53\/p21 tumor suppressor pathways. Alternatively, cells may progress to apoptotic death, which classically involves p53-dependent activation of the caspase cascade. Apoptotic death can also occur after activation of cell surface \u201cdeath receptors\u201d [receptors of tumor necrosis factor (TNF), Fas, or TRAIL (TNF-related apoptosis-inducing ligand)], which themselves can be up-regulated by radiation. Death receptors can also generate cytoprotective effects by activating transcription factor nuclear factor \u03baB (NF\u03baB), which increases expression of genes that promote proliferation or oppose apoptosis.\n\nPathways of intracellular signaling after radiation. Most of the radiation-induced reactive oxygen species (ROS) interact with cellular contents other than DNA. ROS such as superoxide and hydroxyl radicals are also known to deplete cellular stores of antioxidants like glutathione (27). The resulting cellular stresses stimulate a complex set of signaling cascades [reviewed in detail by Dent and colleagues (28)].\n\nRadiation can also activate signaling pathways that cells normally use to respond to mitogens, which in turn promote survival, antiapoptotic responses, and transcriptional changes. The net effect can be variable and cell-specific; however, a common theme includes activation of cell surface receptors, like the ErbB family that includes epidermal growth factor receptor (EGFR). Receptor activation subsequently signals downstream pathways, including the mitogen-activated protein kinase (MAPK) superfamily of cascades [extracellular signal\u2013regulated kinase (ERK), c-Jun N-terminal kinase (JNK), and p38] and the PI3K pathways. These pathways deliver antiapoptotic signals via Akt and ERK signaling. Radiation can also activate these pathways via autocrine mechanisms, like through the production of transforming growth factor\u2013\u03b1 (TGF-\u03b1), which binds and activates EGFR (28). Radiation also activates the proinflammatory cytokines including TNF-\u03b1 and interleukin-6 (IL-6) (29), which in part may account for bystander effect (discussed below).\n\nFinally, radiation-induced damage of the plasma membrane induces the breakdown of sphingomyelin to ceramide, which is proapoptotic independent of DNA damage (30). Radiation also activates cytosolic phospholipase A2 (cPLA2), an enzyme that recognizes phospholipids on the cell membrane and degrades them into inflammatory products like arachidonic acid and eventually eicosanoids. Lysophosphatidylcholine is one such product formed by cPLA2, and its production leads to activation of Akt and enhanced cell death (31).\n\nOne interesting strategy for radiosensitization has been the blockade of progrowth signaling from receptor tyrosine kinases, such as insulin-like growth factor 1 receptor (IGF-1R) and EGFR. Numerous antibodies and chemicals have been developed to target different levels of this signaling cascade. One successful pharmacologic effort was the inhibition of EGFR during radiotherapy for head and neck cancers, using the chimeric (mouse\/human) monoclonal antibody cetuximab. A randomized trial demonstrated benefit of weekly cetuximab in addition to 6 to 7 weeks of radiotherapy. The updated data show a 5-year overall survival of 45% with cetuximab\/radiotherapy and 36% with radiotherapy alone (32). Many related strategies are being evaluated in this and other anatomic tumor sites, using inhibitory antibodies or small molecule\u2013based tyrosine kinase inhibitors.\n\nBystander effects and importance of tumor microenvironment. Nonirradiated cells often exhibit stress responses, after even low-dose (<0.1 Gy) radiation exposures to neighboring cells. This \u201cbystander effect\u201d has been well described in detail previously (33, 34). Bystander responses appear to be cell type\u2013specific; in general, they consist of a broad range of effects including gene induction, genomic instability, differentiation, and changes in apoptotic potential. These processes are mediated, at least in part, by diffusible substances, given that the effects occur when bystander cells are physically separated from the irradiated cells. Some authors have additionally implicated cell-cell contacts (gap junctions) as contributing to this effect.\n\nThis illustrates an important concept\u2014that one must consider radiation effects in the context of the entire tumor microenvironment, rather than simply the sensitivities of individual cancer cells. This notion is further demonstrated by the contribution of host stromal components within tumors to radiation responsiveness. Studies in mice, for example, show that host-derived blood vessels are a key determinant of tumor control with radiotherapy (35).\n\nImpact of cancer stem cells (tumor-initiating cells) within irradiated tumors and normal tissues. Although many researchers disagree over various aspects of cancer stem cell biology and nomenclature, there clearly exists a population of tumor cells that exhibit an exclusive ability for self-renewal and differentiation into the heterogeneous lineages that promote tumor maintenance. These concepts have been supported by three recent important papers demonstrating the existence of cancer stem cells in mouse models of brain, skin, and intestinal tumors (3638). Results of these studies indicate that targeting cancer stem cells may improve therapeutic outcomes. In response to fractionated radiation, cell populations become enriched for cells expressing putative markers of \u201cstemness.\u201d Specific phenotypes vary based on tumor type and methodology, but common features of these cells after radiation often include increased survival, reduced apoptosis, and rapid resolution of H2AX foci (that is, fast repair of DSBs). For example, one study demonstrated an enrichment of glial cancer stem cells after radiation, and that these cells exhibited preferential activation of the DNA damage checkpoint responses and increases in DNA repair capacity (25).\n\nThe basic mechanisms of radiation resistance remain unclear in stem cells; however, aldehyde dehydrogenase 1 (ALDH1) may represent a partial explanation. High expression of ALDH1 protein in tumors is an adverse prognostic factor, perhaps because its aldehyde-catabolizing activity confers a survival advantage. Also, aldehyde-catabolizing enzymes are known to cooperate with Fanconi anemia genes, suggesting that ALDH activity might play a role in repairing adducts on DNA (39). Hence, because ALDH1 appears to be preferentially expressed in cancer stem cells, this may represent a therapeutic target to reverse radioresistance in stem cell clones. Another interesting feature of these cells is their propensity to repair DSBs using HR repair. Breast cancer\u2013derived stem cells (CD24, ESA+) were \u201ceffectively sterilized\u201d by inhibition of the HR pathway, whereas nonsorted cells from the same cell line (MDA-MB231) were unaffected (40). This finding suggests that HR inhibitors represent a promising therapeutic strategy for depleting cancer stem cell reservoirs in tumors. Also, screening of small-molecule libraries led to the identification of salinomycin and thioridazine, both of which appear to target cancer stem cells (41, 42) and may be used to improve radiotherapy.\n\nNormal tissues also contain stem cell niches that likely influence their ability to tolerate radiotherapy. Thomas Helleday\u2019s group recently reported a study, in which punch biopsies were taken from normal skin during the first and last weeks of a 5-week clinical radiotherapy course (43). The week 5 skin biopsies showed a significant enrichment for proliferating cells that were undergoing HR (Ki67+ and RAD51 focus+), as well as epidermal stem cells (\u03b21 integrin+ and Ki67). These results suggest that tumor tissue and normal tissues share some features in terms of stem cell responses, which may pose challenges to targeting cancer stem cells. In some situations, the anatomic locations of normal stem cell niches are known and can perhaps be spared from radiation exposure. One example is the hippocampus, in which neural stem cells are known to reside. An ongoing trial by the Radiation Therapy Oncology Group (RTOG) is prospectively testing whether reduced exposure to the hippocampus can decrease neurocognitive toxicity associated with whole-brain radiotherapy. Another example comes from stem cell niches located within the walls of larger ducts in salivary glands (44), which could potentially be excluded from radiotherapy treatment target volumes.\n\nTargeting tumor hypoxia and redox conditions. The investigation of tumor hypoxia as an effector and biomarker of tumor resistance has exploded in the past decade, and modalities that exploit tumor hypoxia have been widely tested. Tumors are known to outgrow their blood supply, thereby generating regions of necrosis that are surrounded by areas of hypoxia. Hypoxia promotes activation of the hypoxia-inducible transcription factor (HIF) family of proteins that regulate a variety of downstream genes that promote angiogenesis, cell survival, anaerobic energy metabolism, and treatment resistance (45). Hypoxia can also select for highly aggressive tumor cell clones. This topic is particularly relevant to radiotherapy because the presence of molecular oxygen during delivery of ionizing radiation enhances radiation-induced cell kill by 2.5- to 3.5-fold. The commonly accepted mechanism for this observation is that oxygen \u201cfixes\u201d free radical\u2013induced DNA damage into a permanent state. Hence, hypoxic regions of tumor are generally considered radiation-resistant, and reversal of hypoxia has long been a goal for radiosensitization.\n\nA newer class of radiation sensitizers has been developed to target hypoxia on the basis of reduction\/oxidation conditions that predominate in anaerobic environments. Tirapazamine and porfiromycin are hypoxic cell cytotoxins\/sensitizers, which act as bioreductive alkylating agents (53). Tirapazamine differs from oxygen mimetics in that it does not \u201cfix\u201d radiation damage, but instead, it is metabolized into a highly reactive radical species in anaerobic conditions. Preclinical data regarding tirapazamine were extremely promising; however, several phase 3 trials failed to demonstrate significant clinical benefit in combination with radiotherapy (54, 55). This disappointing outcome may reflect poor penetration of tirapazamine into hypoxic tumor regions. In an interesting related development, RAD51 and other HR-related DNA repair proteins have been shown to be transcriptionally down-regulated in response to chronic hypoxia (56). Contrary to traditional concepts of hypoxia, this down-regulation of DNA repair may actually render cells more sensitive to radiation.\n\nRelated efforts have used porphyrin-like macrocycles that form complexes with large metal cations and participate with the cellular redox cycle. Motexafin gadolinium (also known as MGd) is one such drug that inhibits antioxidant proteins such as thioredoxin reductase, and it exhibits preferential localization to tumor over normal tissue (45). In combination with whole-brain radiotherapy, MGd provided modest improvements for patients with brain metastases from lung cancer (57); however, these effects have not been impressive enough for Xcytrin to gain U.S. Food and Drug Administration (FDA) approval. MGd is also being evaluated in ongoing or completed clinical trials involving radiotherapy for other central nervous system tumors (glioblastoma multiforme and pediatric brainstem gliomas), as well as various carcinomas including pancreaticobiliary, non\u2013small cell lung, and head and neck cancers.\n\nPharmacologic approaches are being developed to target cellular signaling that occurs in response to hypoxia and redox status in tumors. Many of these have focused on the transcription factor HIF-1, using agents that modulate its transcription, stability, association with binding partners, or signal transduction (45). EZN-2968 (Enzon Pharmaceuticals), an antisense oligonucleotide to HIF-1\u03b1, demonstrated safety and potential activity in phase 1 testing; however, data from subsequent testing are still awaited.\n\nA somewhat related strategy arose from a small interfering RNA screen, looking for tumor-selective radiosensitizing targets in head and neck cancers. Surprisingly, knockdown of uroporphyrinogen decarboxylase, a regulator of heme synthesis, sensitized head and neck cancer cells to radiation and some cytotoxic agents such as cisplatin, paclitaxel, and 5-fluorouracil (5-FU). This may occur due to alterations in iron homeostasis, which promotes an increased production of ROS (58). The role of hypoxia in tumor killing by radiotherapy remains to be resolved. Perhaps, biomarkers or noninvasive physical measures of hypoxia will determine which patients might benefit from modification of hypoxia.\n\nAntiangiogenic drugs and radiotherapy. New blood vessel development is required for tumors to grow beyond 1 to 2 mm. Vascular endothelial growth factor (VEGF) is a key proangiogenic growth factor that is secreted by solid tumors and acts through one of three VEGF receptors (VEGFRs). The most widely studied VEGF inhibitor is a humanized monoclonal antibody (bevacizumab) that binds to and inhibits the activity of human VEGF. Preclinical data have demonstrated that blockade of VEGF signaling increases the antitumor effects of radiation. Additional studies have suggested that VEGF protein levels can become up-regulated in tumors in response to ionizing radiation, suggesting that VEGF might mediate the development of tumor endothelial cell radioresistance (59, 60). The therapeutic combination of radiotherapy plus VEGF inhibition could be considered counterintuitive because antagonism of tumor vasculature might be expected to increase tumor hypoxia. However, tumor angiogenesis has proven to be a dysregulated process that generates networks of tortuous and hyperpermeable vessels, resulting in spatial heterogeneity in tumor oxygenation and elevated interstitial fluid pressure. Current research in preclinical models has demonstrated that VEGF blockade can \u201cnormalize\u201d the tumor vasculature, thereby reducing tumor hypoxia and interstitial pressure and improving the metabolic profile of the tumor microenvironment (61).\n\nGene therapy and radiotherapy. There has been extensive preclinical work over the past 15 years on combining gene therapy with radiotherapy. Replication-defective viruses have been used to deliver radiosensitizing prodrugs, cytokines, tumor suppressor genes, and immune-activating compounds. Other related strategies have used replication-competent viruses to generate oncolytic antitumor effects. The following section will focus on gene therapy modalities that have been tested with radiotherapy in clinical trials.\n\nOther trials have used enzyme\/prodrug strategies. For example, virus-directed expression of herpes simplex virus thymidine kinase (HSV-tk) can phosphorylate the prodrug ganciclovir into a toxic metabolite that interferes with DNA replication, leading to chain terminations and SSBs. A phase 3 randomized trial was unable to show a survival benefit with this strategy, when it was combined with surgery and radiation for glioblastoma multiforme (69), although the agent was delivered after therapy. Related strategies involve the use of an additional \u201csuicide gene\u201d to viral vectors. Freytag and colleagues have tested replication-competent adenovirus that carries both HSV-tk and cytosine deaminase, which converts the prodrug 5-fluorocytosine into 5-FU (70). This gene therapy strategy with prostate radiotherapy appears to be tolerable, but efficacy remains to be demonstrated in phase 2 to 3 trials.\n\nOncolytic viruses have also been combined with radiotherapy, based on interesting preclinical evidence that some viruses can cooperate with radiotherapy by preferentially infecting and lysing cancer cells. Replication-competent HSV-1 has been genetically modified to allow safe use with radiotherapy. For example, a modified HSV-1, called G207, has been safely administered via intratumoral injection immediately before radiotherapy in recurrent or progressive malignant gliomas. More recently, a genetically modified HSV-1 encoding granulocyte macrophage colony-stimulating factor was combined with chemoradiotherapy in a phase 1\/2 trial for locally advanced head and neck cancer (71). Also, intratumoral injections of a reovirus were recently combined with palliative radiotherapy in a phase 1 trial (72).\n\nImmune modulation and radiotherapy. Preclinical evidence shows that radiation can stimulate antitumor responses by the immune system, which has important potential implications clinically. Radiation generates an inflammatory microenvironment in tumors, whereby damaged cells increase antigen presentation and immune recognition. More specifically, radiation-induced immune effects include an elevation of major histocompatibility complex class I (MHC-I) expression, changes in antigenic peptide repertoire, and decreases in regulatory T cells. Radiation also up-regulates cytokines and adhesion molecules that recruit and activate CD8+ cytotoxic T lymphocytes and dendritic cells. The evolving understanding of these processes may allow for them to be exploited therapeutically (7375).\n\nNational Cancer Institute researchers reported a randomized phase 2 trial that treated prostate cancer with standard radiotherapy, with or without a gene therapy\u2013based vaccine that encodes prostate-specific antigen (PSA) (76). The therapy consisted of a \u201cpriming\u201d vaccination with recombinant vaccinia (rV) PSA plus rV containing a T cell costimulatory molecule, followed by monthly booster vaccines with recombinant fowlpox PSA. Most of the vaccinated patients did successfully develop increases in PSA-targeted T cells. Other clinical trials have explored related strategies of radiotherapy in combination with different vaccines, intratumoral injection of immature dendritic cells, or adoptive immunotherapy via infusion with expanded tumor-infiltrating lymphocytes [reviewed by Kamrava and colleagues (73)]. Many of these strategies have demonstrated an impressive induction of immune responses; however, it remains unclear whether these effects will translate to improved clinical outcomes. A recent phase 1 trial reported that patients with metastatic melanoma or kidney cancer treated with high-dose IL-2 demonstrated impressive tumor responses when one to three tumors were treated with stereotactic body radiotherapy compared with historical controls treated with high-dose IL-2 alone (77).\n\n### Molecular predictors of outcome in radiotherapy\n\nIn the past decade, we have witnessed an explosion in new diagnostic technologies capable of subclassifying malignancies and predicting clinical outcomes, based on genetic and epigenetic tumor characteristics. Such studies have fueled a vision in the oncology community of \u201cpersonalized medicine,\u201d a system by which individualized treatment courses can be tailored to specific patient and tumor characteristics. These studies represent an expansive work that cannot be adequately encompassed by this review, and most of these studies have focused on predictors of overall prognosis.\n\nMany studies have reported gene expression signatures that predict overall tumor behavior for different cancer types, and some of these signatures have been validated with clinical data. However, few have attempted to use expression array technology to predict radiocurability per se. One such gene expression signature, called the radiosensitivity index (RSI), consists of 10 genes (AR, cJun, STAT1, PKC, RelA, cABL, SUMO1, CDK1, HDAC1, and IRF1) that associate with radiosensitivity within a collection of human cancer cell lines (97). This signature has been clinically validated in five independent clinical data sets, which include rectal, esophageal, head and neck, and breast cancers treated with chemoradiation or radiotherapy alone (98). A related approach identified a chemotherapy\/radiation resistance signature, using tumors generated in mice. This interferon-related gene signature analysis was evaluated retrospectively in clinical breast cancer data sets, and it successfully predicted the efficacy of adjuvant chemotherapy and local-regional control after radiation (99).\n\nAn alternative approach is to look for associations between radiosensitivity and genetic alterations, using methods like comparative genomic hybridization (CGH) arrays or next-generation sequencing. CGH and SNP (single-nucleotide polymorphism) arrays measure copy number variations and can quantify global levels of genomic instability. Ishkanian and co-workers have used CGH on prostate cancer biopsy specimens to identify deletions of chromosome regions that normally harbor DNA repair genes (including PARP1, ATM, DNA-PKcs, p53, Rb, and RAD17). Detection of these gene losses might help identify tumors susceptible to radiation or other specific treatments (100). Similar work showed NKX3.1 haploinsufficiency to predict risk for prostate cancer recurrence after radiotherapy, and it also predicted for recurrence after prostatectomy (101). These and other emerging methods will need to be validated more thoroughly in prospective clinical trials, but the findings are encouraging. Furthermore, next-generation sequencing of tumor genomes will presumably revolutionize these types of approaches as we move forward.\n\nAlthough the excitement of these new methods has echoed throughout the entire oncology community, it remains unclear whether personalized medicine can live up to our collective expectations. These newer methods generally rely on \u201crepresentative\u201d tumor biopsies, which can underestimate tumor cell heterogeneity. A recent cautionary study demonstrated alarmingly high levels of intratumor heterogeneity in renal cell cancers, in terms of both gene expression levels and DNA sequence (102). More than 60% of all somatic mutations were not detectable across all regions of a given tumor. Hopefully, many of the \u201cdriver mutations\u201d (which form the basis of many new targeted therapies) will prove to be better represented throughout tumors. These data suggest that our existing therapies, conventional chemotherapy drugs and radiotherapy, will continue to serve important roles.\n\nTable 2\n\nModalities in RT. IMRT, intensity-modulated RT; SBRT, stereotactic body RT; SRS, stereotactic radiosurgery. Proton therapy illustration is courtesy of University of Florida Proton Therapy Institute.\n\nView this table:\n\n### Intensity-modulated RT\n\nStandard radiation planning is performed with axial computed tomography (CT) imaging and three-dimensional (3D) planning software, in which multiple beams are arranged to pass through the target volume. With intensity-modulated RT (IMRT), the intensity of each beam is modulated to provide a tighter approximation of high-dose radiation to the intended target, a characteristic known as conformality. IMRT planning uses an iterative computer algorithm approach, which has input variables that specify dose coverage and normal tissue\u2013sparing goals, and output that defines the shape of the multiple beam \u201csegments\u201d that are delivered. This high degree of conformality is especially useful for irregularly shaped targets or disease in close proximity to critical structures that are sensitive to high doses of radiation (103).\n\nClinical use of IMRT has become widespread and has enabled the escalation of dose and reduction of toxicity. IMRT has become fairly standard in the organ-sparing treatment of head and neck cancers with RT. The protection of parotid and submandibular salivary glands from radiation dose (sparing goal <26 Gy) has led to improvements in quality of life through a reduction in treatment-related xerostomia (104). IMRT improves the ability to treat irregularly shaped targets such as advanced nasopharynx tumors that can wrap around the brainstem or approximate the temporal lobes of the brain. Furthermore, because beam intensity can be modulated, IMRT can enable differential dosing within a singular volume to treat variable levels of disease risk. This technique of \u201cdose painting\u201d or \u201csimultaneous integrated boost\u201d planning offers the oncologist more latitude to individualize plans by tailoring dose to match the appropriate burden of disease (105). In tandem with concurrent chemotherapy (often including 5-FU or cisplatin) and altered fractionation (such as twice daily treatment with smaller fraction sizes), the therapeutic window can potentially be further widened (106).\n\nDespite the many predicted advantages of IMRT, there are only a few instances in which IMRT has been proven to reduce morbidity compared to conventional RT in a randomized setting (107), and potential disadvantages exist. Related to the conformality of IMRT planning, accurate treatment setup including immobilization becomes more critical to avoid underdosing of the target volume. Target volumes that are subject to motion or deformation over the course of treatment require more advanced technology for daily target localization and, in some cases, may be more appropriately treated with a 3D plan. Additionally, IMRT requires more resources to plan and deliver treatment, and it results in significantly more low-dose exposure to tissues outside the target area because of the increased number of beam angles and cumulative \u201cbeam on\u201d time required to deliver IMRT. There exists some concern that the exposure of more normal tissue to low doses of radiation might increase the risk of normal tissue complications, specifically radiation-induced malignancy years after RT (108). These concerns, along with additional costs, explain why IMRT has not entirely supplanted more traditional forms of RT for all disease sites.\n\n### Image-guided RT, hypofractionated therapy, and ablative therapy (SRS, stereotactic body RT)\n\nAlthough CT-based RT planning provides a more precise ability to target radiation to soft tissue areas, inaccuracies in the reproducibility of daily setup and organ motion (such as respiratory motion) force the radiation oncologist to treat a volume larger than the actual tumor. Advancements in technology have addressed both positional error and physiologic motion to allow this setup margin to be reduced. Linear accelerators are now commonly fitted with various on-board imaging devices, which can provide diagnostic quality kilovolt x-rays or cone beam CT. Soft tissue targets that are not easily visible on an x-ray can potentially be implanted with radio-opaque \u201cfiducial\u201d markers to better guide treatment delivery with pretreatment imaging (109). With the regular implementation of any of these imaging modalities, known as image-guided RT (IGRT), positioning setup error can be reduced, thereby enabling the use of small treatment volumes. Finally, commercially available software\/hardware packages have been developed to account for physiologic motion such as respiratory motion.\n\nWhereas the development of IMRT after 3D conformal RT could be considered evolutionary, in that IMRT was an incremental advancement in planning which arose from hardware (multileaf collimator) and software (inverse-planning) innovations, perhaps the implementation of SRS and SBRT could be considered more revolutionary. IMRT is typically used to achieve maximal conformality of the high-dose region, especially in cases where the target volume is irregularly shaped. On the other hand, SBRT typically prioritizes the steepness of the dose gradient between the target area and adjacent critical normal structures. The use of multiple beam angles, sometimes arranged with the treatment table rotated off axis, can help position this dose gradient in a favorable location. SBRT techniques allow for \u201chotspots\u201d of increased dose within the target volume, which can significantly exceed the intended prescription dose; in certain cases, the hotspot within the target can be >50% the prescription dose. This dose inhomogeneity further contributes to the steepness of the dose gradient. Notably, IMRT and SBRT techniques are not mutually exclusive\u2014IMRT techniques can be used to deliver more favorable and homogeneous dose distributions. However, a combined approach comes at the expense of added complexity of planning and delivery, and an increase in treatment time could compromise the reproducibility of setup because of intrafraction patient movement.\n\nThe success of hypofractionated, image-guided RT to date has challenged conventional paradigms in radiation oncology. SBRT and SRS shift priority toward tumor ablation rather than normal tissue preservation, along the same lines of a surgical approach. Hence, the emphasis of hypofractionated RT is increasingly focused on local rather than local-regional therapy. Clinical data to date have provided reason for cautious optimism, given the high rates of local control and reasonably low rates of severe toxicity that have been reported (118). However, the best use of SBRT remains to be defined. Perhaps most importantly, appropriate dose limits to normal tissues need to be redefined with clinical outcome data, given that the bulk of the existing dose-volume analyses of normal tissue complications is derived from the conventional fractionation era (119). It is important to note that only a small subset of tumors is amenable to SBRT. Factors such as tumor volume (115), method of daily localization, and proximity and nature of adjacent normal tissue (120) may play critical roles in determining whether SBRT or traditional fractionated radiotherapy techniques are more appropriate.\n\n### Brachytherapy\n\nBrachytherapy represents a modality of treatment that could potentially outperform external beam RT for particular clinical situations. By varying the strength of the radioactive sources and manipulating the location and exposure time, it is possible to achieve highly conformal and dose-intense radiation distributions within tumors while generating relatively low exposures to adjacent normal organs. Source placement can be interstitial, in which sources are placed directly into the target tissue. Alternatively, in intracavitary brachytherapy, sources are placed in a space next to the target tissue, such as a body cavity or a body lumen. Because of the direct visualization of sources within the treatment site, concerns with setup uncertainty and organ motion are minimal.\n\nDespite the dosimetric advantages of brachytherapy, this modality of radiotherapy is not used as commonly as external beam radiation. This is because only a small subset of cases present with well-defined areas of risk, which can be accessed in a relatively noninvasive fashion. Given this practical restriction, cancers of the cervix, prostate, breast, and skin are the most common sites to be treated with brachytherapy. In some situations, brachytherapy is combined with a course of external beam RT, which precedes, follows, or interdigitates with brachytherapy. Cervical cancer, for example, may include a course of conventional fractionation radiation and concurrent chemotherapy to treat the pelvic lymph nodes, cervical parametria, and cervix over 6 weeks of daily treatment. Additional intracavitary brachytherapy procedures are done to deliver dose to the primary cervical tumor. In prostate cancer, brachytherapy presents an alternative approach to the narrow therapeutic window. Some studies suggest that disease outcomes after brachytherapy are superior to those after dose-escalated external beam RT (121), although this hypothesis has not been formally tested. Note that many anatomic locations are not suitable for brachytherapy, such that the therapeutic ratio for external beam radiation exceeds that of brachytherapy. Additionally, the high doses delivered by brachytherapy carry the potential risk of increased morbidity if sources are misplaced or migrate (122), so a certain level of technical proficiency is required.\n\n### Particle therapy\n\nParticle therapy (123) is a form of external beam radiotherapy in which particles are delivered rather than x-rays. Beams of electrons, protons, carbon ions, or neutrons are generated by a particle accelerator and delivered to the target volume. The use of proton therapy in particular has grown in recent years because of its unique interaction in tissue. The underlying physics is characterized by a dose distribution known as the Bragg peak, whereby radiation is deposited immediately before the proton comes to rest in tissue, at a depth that can be manipulated by varying its energy. Photons, by comparison, deposit a trail of in-transit dose as the x-ray beams enter and exit the body, typically requiring multiple beams to be arranged so that their paths intersect over the target area. This exposure of tissue within each beam path can have negative clinical consequences for normal tissues. In clinical practice, to treat an entire tumor at a given depth, protons of different energies must be used. This \u201cspread out Bragg peak\u201d offers an improved ability to minimize radiation exposure to normal tissues situated adjacent to the target volume, especially for tissues behind or deeper to the tumor. These reductions in total body exposure (termed integral dose) can translate to reduced complication risks, including the risk of radiation-induced malignancies. Proton beam therapy has had a rapidly growing role in pediatric malignancies and particularly pediatric brain tumors. Tumors located near the spine, like chordomas or paraspinal chondrosarcomas, are a well-defined indication for proton beam therapy, because it allows these relatively radioresistant tumors to be aggressively treated while respecting the radiation tolerance of the spinal cord (124).\n\nAnother attribute that distinguishes particle therapies from x-ray beams is that some types of particle beams generate higher biologic effectiveness per unit of energy deposited. For example, the relative biologic effectiveness of protons is 1 to 1.2 times that of photons, whereas for neutrons or carbon ions it could be 4 to 10 times that of photons. The difference in biologic effect is related to the rate at which energy is deposited in tissue, which is a function of both mass and charge of the radiation. This radiobiological difference could decrease the dependency of treatment success on the oxygen levels and cell cycle phase of the tumor target. It is also possible that heavy ion particle therapy could have a more desirable effect on blood vessels that supply tumors. However, further research is necessary to determine whether these radiobiological differences will translate into a better therapeutic index relative to photon-based therapy. Neutron radiotherapy, for example, provided these theoretical advantages, but significant normal tissue damage has limited its clinical applicability.\n\nA clear challenge to particle therapy is the high cost to build treatment centers, which require significantly more physical space and highly trained support staff than do photon-based centers. As of 2013, the International Particle Therapy Cooperative Group (125) reported that only 36 centers are using proton therapy and only 6 centers are using carbon ion therapy worldwide. Between 1969 and 2011, particle therapy has compromised <1% of external beam radiation treatment. This may change in the future, however, because there are currently plans to build 25 more particle facilities worldwide within the next 2 years.\n\n## Conceptual and Therapeutic Bottlenecks and Future Directions to Improve Radiotherapy\n\n### Radioresistance prevents local control of some tumor types\n\nIn tumors like glioblastoma, the probability of local tumor control remains miserably poor, despite the use of advanced radiotherapy technologies and very high doses of radiotherapy. This has been well illustrated by unsuccessful trials of dose escalation, including one study increasing doses to >200 Gy using brachytherapy (126). Many possible mechanisms have been proposed for this intrinsic treatment resistance, including prevalence of radioresistant tumor stem cell clones, hypoxia, and uncontrolled signaling by growth factor receptors and\/or interferons. Although some of these hypotheses may prove to be correct, the current supporting data are not compelling and a complete explanation is likely to be much more complex. For example, unknown mechanisms may stimulate repopulation of malignant clones to brain tumors from sanctuary sites, which are located outside the irradiated volume. Another example of unyielding radioresistance is observed in very large tumors, like the typical presentation of locally advanced lung cancer. A recent trial for non\u2013small cell lung cancer showed that radiation dose escalation by almost 25% above the standard dose offered no measurable improvement to patients (117). One probable explanation is that the tumor cell burden is simply too large to be controlled in these tumors, such that high enough radiation doses cannot be achieved without incurring major complications to adjacent normal tissues. Another likely explanation is that larger tumors have greater genomic heterogeneity and, hence, a greater likelihood for harboring treatment-resistant clones.\n\nAvenues to overcome this resistance will require a better understanding of the underlying biological problems. They will likely also require a better arsenal of drugs that can specifically sensitize tumor cells or protect normal tissues. It is critical to widen the therapeutic window for radiotherapy at the biological level, particularly in situations where the physical and technical advances could be nearing a plateau. Some of the newer drug approaches discussed earlier may hold promise in this respect, particularly if issues related to drug pharmacokinetics and timing of administration relative to radiation are solved. However, we are left with the impression that there are many uncoordinated and competing research efforts. This probably occurs in part because radiation oncology is a relatively small field, relative to the much larger field of cancer biology research. Relatively few radiotherapy-modifying agents have progressed to testing in clinical trials, and the interest by large drug companies to invest in this class of agents appears somewhat underwhelming. Most of the current trials testing radiosensitizers involve HDAC and PARP inhibitors, neither of which was developed with this specific use in mind.\n\n### Uncontrolled systemic disease can overshadow the importance of local control\n\nFor a malignancy to be cured, it must be controlled at both the local and systemic levels. However, the ability to prevent distant metastases remains an unmet challenge in many disease types, such as lung and pancreatic cancers. As systemic chemotherapies continue to improve outcomes in these diseases, the previously unnoticed importance of local control can become more obvious. Recent studies in breast cancer demonstrate this interdependent relationship well. Older studies showed that chest wall radiotherapy improved the regional control of locally advanced breast cancers after mastectomy; these improvements were often clinically dismissed because they did not translate to improvements in overall survival. However, more recent studies done in the context of more effective systemic chemotherapy have shown that postmastectomy chest wall RT does improve survival (127, 128). The magnitude of survival benefit from radiotherapy is similar or greater than that of chemotherapy. This highlights the fact that uncontrolled systemic disease can overshadow the importance of local control, such that oncologists may view local management as secondary. This, in turn, can reduce enthusiasm for improving local\/regional modalities like radiotherapy.\n\nAn exciting new development is the use of SBRT to ablate sites of metastatic disease, which represents a 180\u00b0 shift in paradigm. It is based on the observation that some cancers present with only a few sites of metastatic disease, a condition termed oligometastases (129, 130). It remains unclear which patients will benefit the most from SBRT to metastases. However, there is evidence for the concept, based on surgical series that achieved cures after resection of limited liver or lung metastases in patients with colon cancer and sarcomas, respectively. It is probably unrealistic to believe that local therapies will ever be able to address micrometastases, so SBRT for metastases and systemic chemotherapies should continue to evolve together. Nonetheless, impressive outcomes have been reported for SBRT in this metastatic setting. One unmet challenge is the ability to determine which tumors will behave in an oligometastatic manner and which are likely to develop widespread metastases. Advances in imaging would be helpful in this regard, so that oligometastases can be detected and treated when the disease burden is lowest. Recent microRNA signatures have begun to identify patients with oligometastasis who remain oligometastatic, and this might be useful to select optimal candidates for curative surgery or radiotherapy (131). Another exciting idea is the use of SBRT to augment antitumor immunotherapy. Although clinical data are limited (79) and this approach might be limited to particular cancers, this relatively unexplored strategy has a huge potential payoff if it is successful.\n\n### A more individualized approach is needed in radiotherapy\n\nTreatment guidelines are available for most types of cancer, such as those put forth by the National Comprehensive Cancer Network (132). In most cases, the decision to deliver radiation and the key parameters of radiation (for example, dose and volume) are primarily based on the stage of disease at presentation. Management decisions tend not to be personalized toward individual biology or sensitivity to treatment. As an example, locally advanced rectal cancer is typically treated with radiosensitizing chemotherapy and 50.4 Gy, followed by total mesorectal excision, and then adjuvant chemotherapy. Although 10 to 20% of tumors are essentially sterilized by the initial preoperative chemoradiotherapy, almost all patients complete the preplanned course of therapy including surgical excision and full-dose adjuvant chemotherapy. If, however, the treatment-sensitive tumors could be identified early, this information could be used to guide subsequent management. For example, systemic therapy could possibly be deintensified, and perhaps, surgical resection could be eliminated for some patients. By contrast, patients found to have treatment-refractory tumors could be offered even more intensive therapies.\n\nThe definition of such predictive markers remains yet another unmet challenge for many cancer types, but some promising leads are beginning to emerge. One example is carcinoma of the head and neck treated with chemoradiotherapy, where the presence of human papillomavirus carries a favorable prognosis (133). Several ongoing trials are testing whether the intensity of chemoradiotherapy can be safely reduced in this subpopulation. To move forward as a field, it will be important to start focusing on biomarkers that predict responsiveness to specific therapies, as opposed to simple prognostic markers. This concept has already been realized for some chemotherapies, like KRAS mutation status for predicting EGFR inhibitor sensitivity (134) or MGMT gene silencing for predicting temozolomide sensitivity (135). Similar markers that predict sensitivity to radiation are being developed, but these have not yet found widespread clinical use. Such innovations will help transition radiotherapy away from a \u201cone-size fits all\u201d mentality and allow us to start the inevitable transition to personalized medicine.\n\n### The allocation of radiotherapy resources needs to be evidence-based\n\n1. Acknowledgments: We apologize to colleagues whose contributions could not be included in this review due to space constraints. Funding: This work was supported by funding from the NIH [CA142642-02 2010-2015 (P.P.C.) and P50 CA090386 2001-2014 (R.R.W. and S.L.L.)], P01 CA071933 (R.R.W.), the Ludwig Center for Metastasis Research, the Center for Radiation Therapy, the Chicago Tumor Institute, a Young Investigator Award from the Prostate Cancer Foundation (S.L.L.), and The Foglia Foundation. Competing interests: S.L.L. and P.P.C. declare no competing interests. R.R.W. has a commercial organizational interest in Catherex, Magi, RefleXion, and OncoSenescence. He has consulted for MedImmune ($1500 one-time consultant fee) and Bristol-Myers ($4900 one-time consultant fee) and has a grant for \\$100,000 to study the effect of immune modulators with radiotherapy. R.R.W. has patents in gene therapy and the use of tyrosine kinase inhibitors with DNA damaging agents.","date":"2019-03-23 12:44:08","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\": 0.38727372884750366, \"perplexity\": 6717.0936870267105}, \"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-2019-13\/segments\/1552912202804.80\/warc\/CC-MAIN-20190323121241-20190323143241-00126.warc.gz\"}"} | null | null |
Q: PeopleSoft Query Manager - isolating a row Okay so I'm struggling here. We have a table that keeps track of a certain user ID. One row has the ID, a second row has the inactive ID. It looks kind of like this:
B.MISC_INFO
Date
B.MISC_VALUE
Active
1/1/20
BXXXX
Inactive
1/1/21
BXXXX
Active
1/1/22
B2XXX
I create a report using query manager and it pulls in both active statuses. I need it to only pull in the active status without a corresponding inactive status (in the example above, the 'B2XXX' value).
Right now the SQL on the view SQL tab looks like this:
SELECT DISTINCT A.EMPLID, A.NAME, B.MISC_INFO, B.MISC_VALUE
FROM ((PS_EMPLOYEES A INNER JOIN PS_EMPLMT_SRCH_QRY A1 ON (A.EMPLID = A1.EMPLID AND A.EMPL_RCD = A1.EMPL_RCD AND A1.OPRID = 'XXXXXXXX' )) LEFT OUTER JOIN PS_FTI_EMP_MISC2 B ON A.EMPLID = B.EMPLID AND B.MISC_INFO = 'Active' )
WHERE ( ( A.EFFDT =
(SELECT MAX(A_ED.EFFDT) FROM PS_EMPLOYEES A_ED
WHERE A.EMPLID = A_ED.EMPLID
AND A.EMPL_RCD = A_ED.EMPL_RCD
AND A_ED.EFFDT <= SUBSTRING(CONVERT(CHAR,GETDATE(),121), 1, 10))
AND A.EFFSEQ =
(SELECT MAX(A_ES.EFFSEQ) FROM PS_EMPLOYEES A_ES
WHERE A.EMPLID = A_ES.EMPLID
AND A.EMPL_RCD = A_ES.EMPL_RCD
AND A.EFFDT = A_ES.EFFDT)
AND B.MISC_VALUE LIKE 'B%' ))
I've thought about writing a case statement but I can't figure it out.
A: There are a couple of ways to do this. If you are on PeopleTools 8.56 or later, you can first write a query that pulls all active statuses. Drop the effective dated logic for this one. You would then use composite query to query your query. This will treat the inner query like an inline view. You would then apply effective dated logic in composite query.
Another option is to put the active status criteria in the effective dated logic. For this to work, you won't be able to use standard effective dated logic. You will need to create your own subquery criteria on the EFFDT field.
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"redpajama_set_name": "RedPajamaStackExchange"
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<Setting name="launchType" value="trigger" /> <!-- startup|trigger|periodic -->
<Setting name="enabled" value="true" /> <!-- true|false -->
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<Task id="1" name="FilesLoader" description="Loading SQL Server scripts" enabled="false">
<Setting name="file" value="/opt/wexflow/WexflowTesting/SqlServer/script1.sql" />
<Setting name="file" value="/opt/wexflow/WexflowTesting/SqlServer/script2.sql" />
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<Task id="2" name="Sql" description="SQL Server" enabled="false">
<Setting name="type" value="sqlserver" />
<Setting name="connectionString" value="Data Source=localhost;Initial Catalog=HELLOWORLD;Integrated Security=True" />
<!--<Setting name="sql" value="UPDATE [dbo].[Data] SET [Description] = 'Hello World Description 1! updated' WHERE Id = 11;" />-->
<Setting name="selectFiles" value="1" />
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<Task id="3" name="FilesLoader" description="Loading SQLite scripts" enabled="true">
<Setting name="file" value="/opt/wexflow/WexflowTesting/sqlite/script1.sql" />
<Setting name="file" value="/opt/wexflow/WexflowTesting/sqlite/script2.sql" />
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<Task id="4" name="Sql" description="SQlite" enabled="true">
<Setting name="type" value="sqlite" />
<Setting name="connectionString" value="Data Source=/opt/wexflow/WexflowTesting/sqlite/HelloWorld.db;Version=3" /><!-- https://www.connectionstrings.com/sqlite/ -->
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<Setting name="selectFiles" value="3" />
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<Setting name="file" value="/opt/wexflow/WexflowTesting/mysql/script1.sql" />
<Setting name="file" value="/opt/wexflow/WexflowTesting/mysql/script2.sql" />
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<Setting name="type" value="mysql" />
<Setting name="connectionString" value="Server=localhost;Database=helloworld;Uid=root;Pwd=password;" />
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<Setting name="selectFiles" value="5" />
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<Setting name="type" value="postgresql" />
<Setting name="connectionString" value="User ID=postgres;Password=password;Host=localhost;Port=5432;Database=helloworld;" />
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<Setting name="sql" value="UPDATE DATA SET Description = 'Hello World Description 1! updated' WHERE Id = 1;" />
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<Setting name="connectionString" value="Data Source=HelloWorld;User ID=root;Password=password;" />
<Setting name="sql" value="UPDATE DATA SET Description = 'Hello World Description 1! updated' WHERE Id = 1;" />
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<Setting name="sql" value="UPDATE DATA SET Description = 'Hello World Description 1! updated' WHERE Id = 1;" />
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 764 |
Фи́линское — село в Вачском районе Нижегородской области, административный центр Филинского сельсовета.
История
В конце XIX — начале XX века на территории нынешнего села существовал погост Кубовский и 4 деревни: Филинская, Кожинка, Нехайка, Турлово.
Церковь пророка святого Илии в Кубовом погосте упоминается в писцовых книгах 1629-30 годов. Деревни Филинская, Кожинка, Нехайка, Турлово упоминаются в составе Кубовского прихода в окладных книгах 1676 года.
В конце XIX — начале XX века деревни Филинское и Кожинка входили в состав Новосельской волости, деревни Нехайка, Турлово и Кубов погост — в состав Монаковской волости Муромского уезда Владимирской губернии. В 1926 году в деревне Филинское числилось 62 двора, в Кожинке — 29 дворов, в Нехайке — 76 дворов, в Турлово — 62 двора.
С 1929 года деревня Филинское являлась центром Филинского сельсовета Вачского района Горьковского края, с 1936 года — в составе Горьковской области, с 2009 года — в составе Филинского сельсовета.
Население
Филинское в наши дни
Построен вновь филинский храм в честь святого пророка Божиего Илии с приделом в честь преподобного Сергия Радонежского.
Село газифицировано и телефонизировано. В нём установлен «красный» таксофон с номером (83173) 72-401. В селе имеются средняя школа, участковая больница, дом культуры, библиотека, ветеринарный пункт.
В Филинском действует предприятие по переработке слюдяного сырья и выпуску продукции на основе слюды, производству элементов обогрева и ТЭНов ОАО «Слюда».
Филинское стоит на автодороге Муром — Нижний Новгород. Доехать до Филинского можно по вышеупомянутой автодороге на автомобиле или автобусами Нижний Новгород-Филинское, Павлово—Филинское, Павлово—Клин и др. Или из Нижнего Новгорода автобусом, следующим до Выксы, Кулебак или Навашино.
Примечания
Населённые пункты Вачского района | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,891 |
// Generated by the protocol buffer compiler. DO NOT EDIT!
// source: google/pubsub/v1/pubsub.proto
package com.google.pubsub.v1;
public interface ListTopicsResponseOrBuilder
extends
// @@protoc_insertion_point(interface_extends:google.pubsub.v1.ListTopicsResponse)
com.google.protobuf.MessageOrBuilder {
/**
*
*
* <pre>
* The resulting topics.
* </pre>
*
* <code>repeated .google.pubsub.v1.Topic topics = 1;</code>
*/
java.util.List<com.google.pubsub.v1.Topic> getTopicsList();
/**
*
*
* <pre>
* The resulting topics.
* </pre>
*
* <code>repeated .google.pubsub.v1.Topic topics = 1;</code>
*/
com.google.pubsub.v1.Topic getTopics(int index);
/**
*
*
* <pre>
* The resulting topics.
* </pre>
*
* <code>repeated .google.pubsub.v1.Topic topics = 1;</code>
*/
int getTopicsCount();
/**
*
*
* <pre>
* The resulting topics.
* </pre>
*
* <code>repeated .google.pubsub.v1.Topic topics = 1;</code>
*/
java.util.List<? extends com.google.pubsub.v1.TopicOrBuilder> getTopicsOrBuilderList();
/**
*
*
* <pre>
* The resulting topics.
* </pre>
*
* <code>repeated .google.pubsub.v1.Topic topics = 1;</code>
*/
com.google.pubsub.v1.TopicOrBuilder getTopicsOrBuilder(int index);
/**
*
*
* <pre>
* If not empty, indicates that there may be more topics that match the
* request; this value should be passed in a new `ListTopicsRequest`.
* </pre>
*
* <code>string next_page_token = 2;</code>
*/
java.lang.String getNextPageToken();
/**
*
*
* <pre>
* If not empty, indicates that there may be more topics that match the
* request; this value should be passed in a new `ListTopicsRequest`.
* </pre>
*
* <code>string next_page_token = 2;</code>
*/
com.google.protobuf.ByteString getNextPageTokenBytes();
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,119 |
Q: Сделать 2 ресайза подпяд. Битрикс Загружается оригинальная картинка в свойство "оригинальная картинка". Есть 3 свойства "большой квадрат", "маленький квадрат" и "прямоугольник"
так же есть 3 временных свойства "большой квадрат (временное)", "маленький квадрат (временное)" и "прямоугольник (временное)"
проблема в том, что картинки после второго ресайза добавляются те же самые, что и после первого. размеры одинаковые, хотя пути разные. в чем может быть проблема?
код
class AddImage{
function AddImageFunc($arFields){
$el = new CIBlockElement;
$element_id = $arFields['ID'];
// первый ресайз
$res = CIBlockElement::GetList(
array('sort'),
array('ID' => $element_id),
false,
false,
array("ID", "NAME", "PROPERTY_ORIGINAL")
);
while($ob = $res->GetNextElement()){
$arFieldsElement = $ob->GetFields();
}
$original_image_id = $arFieldsElement['PROPERTY_ORIGINAL_VALUE'];
$big_r1 = CFile::ResizeImageGet(
$original_image_id,
array('width'=>820, 'height'=>820),
BX_RESIZE_IMAGE_PROPORTIONAL_ALT,
true
);
$small_r1 = CFile::ResizeImageGet(
$original_image_id,
array('width'=>410, 'height'=>410),
BX_RESIZE_IMAGE_PROPORTIONAL_ALT,
true
);
$rectangle_r1 = CFile::ResizeImageGet(
$original_image_id,
array('width'=>820, 'height'=>410),
BX_RESIZE_IMAGE_PROPORTIONAL_ALT,
true
);
$big_file_array = CFile::MakeFileArray($big_r1['src']);
$small_file_array = CFile::MakeFileArray($small_r1['src']);
$rectangle_file_array = CFile::MakeFileArray($rectangle_r1['src']);
$PROP = array(
'BIG_TIME' => array("n0" => $big_file_array),
'SMALL_TIME' => array("n0" => $small_file_array),
'RECTANGLE_TIME' => array("n0" => $rectangle_file_array)
);
$el->Update($element_id, array('PROPERTY_VALUES' => $PROP));
// второй ресайз
$res = CIBlockElement::GetList(
array('sort'),
array('ID' => $element_id),
false,
false,
array("ID", "NAME", "PROPERTY_BIG_TIME", "PROPERTY_SMALL_TIME", "PROPERTY_RECTANGLE_TIME")
);
while($ob = $res->GetNextElement()){
$arFieldsElement = $ob->GetFields();
}
$big_image_id = $arFieldsElement['PROPERTY_BIG_TIME_VALUE'];
$small_image_id = $arFieldsElement['PROPERTY_SMALL_TIME_VALUE'];
$recangle_image_id = $arFieldsElement['PROPERTY_RECTANGLE_TIME_VALUE'];
$big_r2 = CFile::ResizeImageGet(
$big_image_id,
array('width'=>820, 'height'=>820),
BX_RESIZE_IMAGE_EXACT,
true
);
$small_r2 = CFile::ResizeImageGet(
$small_image_id,
array('width'=>410, 'height'=>410),
BX_RESIZE_IMAGE_EXACT,
true
);
$rectangle_r2 = CFile::ResizeImageGet(
$recangle_image_id,
array('width'=>820, 'height'=>410),
BX_RESIZE_IMAGE_EXACT,
true
);
$big_file_array2 = CFile::MakeFileArray($big_r2['src']);
$small_file_array2 = CFile::MakeFileArray($small_r2['src']);
$rectangle_file_array2 = CFile::MakeFileArray($rectangle_r2['src']);
$PROP2 = array(
'BIG' => array("n0" => $big_file_array2),
'SMALL' => array("n0" => $small_file_array2),
'RECTANGLE' => array("n0" => $rectangle_file_array2)
);
$el->Update($element_id, array('PROPERTY_VALUES' => $PROP2));
}
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,152 |
define({
"_widgetLabel": "Eğik Görüntüleyici",
"locateButtonLabel": "Eğik resimleri görüntülemek için ana harita üzerinde bir nokta seçin.",
"clearButtonLabel": "Tüm grafikleri temizleyin",
"zoomButtonLabel": "Eğik görüntü yayılımına yakınlaştırın",
"syncButtonLabel": "Nadir haritasını senkronize et",
"rasterListLabel": "Kullanılabilir eğik resimleri görüntüleyin",
"measureWidgetLabel": "Yüksekliği ölçün",
"intialScaleLabel": "Eğik resimler için başlangıç ölçeğini seçin",
"smallBuildingLabel": "Kentsel Alan: Küçük Yapılar",
"buildingLabel": "Kentsel Alan: Binalar",
"azimuthChangeNotification": "Seçilen yönde veri yok."
}); | {
"redpajama_set_name": "RedPajamaGithub"
} | 4,043 |
\section{Training Algorithm for ISDA}
\section*{Appendex}
\section{Implementation Details of ISDA. }
\label{grad}
\textbf{Dynamic estimation of covariance matrices.}
During the training process using $\overline{\mathcal{L}}_{\infty}$, covariance matrices are estimated by:
\begin{equation}
\label{ave}
\bm{\mu}_j^{(t)} = \frac{n_j^{(t-1)}\bm{\mu}_j^{(t-1)} + m_j^{(t)} {\bm{\mu}'}_j^{(t)}}
{n_j^{(t-1)} +m_j^{(t)}},
\end{equation}
\begin{equation}
\label{cv}
\begin{split}
\Sigma_j^{(t)}
= \frac{n_j^{(t-1)}\Sigma_j^{(t-1)} + m_j^{(t)} {\Sigma'}_j^{(t)}}
{n_j^{(t-1)} +m_j^{(t)}}
+ \frac{n_j^{(t-1)}m_j^{(t)} (\bm{\mu}_j^{(t-1)} - {\bm{\mu}'}_j^{(t)})
(\bm{\mu}_j^{(t-1)} - {\bm{\mu}'}_j^{(t)})^T}
{(n_j^{(t-1)} +m_j^{(t)})^2},
\end{split}
\end{equation}
\begin{equation}
\label{sum}
n_j^{(t)} = n_j^{(t-1)} + m_j^{(t)}
\end{equation}
where $\bm{\mu}_j^{(t)}$ and $\Sigma_j^{(t)}$ are the estimates of average values and covariance matrices of the features of $j^{th}$ class at $t^{th}$ step. ${\bm{\mu}'}_j^{(t)}$ and ${\Sigma'}_j^{(t)}$ are the average values and covariance matrices of the features of $j^{th}$ class in $t^{th}$ mini-batch. $n_j^{(t)}$ denotes the total number of training samples belonging to $j^{th}$ class in all $t$ mini-batches,
and $m_j^{(t)}$ denotes the number of training samples belonging to $j^{th}$ class only in $t^{th}$ mini-batch.
\textbf{Gradient computation.} In backward propagation, gradients of $\overline{\mathcal{L}}_{\infty}$ are given by:
\begin{equation}
\label{g_1}
\frac{\partial{\overline{\mathcal{L}}_{\infty}}}{\partial b_j} =
\frac{\partial{\overline{\mathcal{L}}_{\infty}}}{\partial z_j} =
\begin{cases}
\frac{e^{z_{y_i}}}{\sum_{j=1}^{C}e^{z_{j}}}-1, &j = y_i \\
\frac{e^{z_{j}}}{\sum_{j=1}^{C}e^{z_{j}}}, &j \neq y_i
\end{cases},
\end{equation}
\begin{equation}
\label{g_2}
\frac{\partial{\overline{\mathcal{L}}_{\infty}}}{\partial \bm{w}^{T}_{j}} =
\begin{cases}
(\bm{a}_{i} + \sum_{n=1}^{C}[(\bm{w}^{T}_{n} - \bm{w}^{T}_{y_{i}})\Sigma_{i}])
\frac{\partial{\overline{\mathcal{L}}_{\infty}}}{\partial z_j}, &j = y_i \\
(\bm{a}_{i} + (\bm{w}^{T}_{j} - \bm{w}^{T}_{y_{i}})\Sigma_{i})
\frac{\partial{\overline{\mathcal{L}}_{\infty}}}{\partial z_j}, &j \neq y_i
\end{cases},
\end{equation}
\begin{equation}
\label{g_3}
\frac{\partial{\overline{\mathcal{L}}_{\infty}}}{\partial a_k} = \sum_{j=1}^{C}
w_{jk} \frac{\partial{\overline{\mathcal{L}}_{\infty}}}{\partial z_j}, 1 \leq k \leq A,
\end{equation}
where $w_{jk}$ denotes $k^{th}$ element of $\bm{w}_{j}$. ${\partial{\overline{\mathcal{L}}_{\infty}}}/{\partial \bm{\Theta}}$ can be obtained through the backward propagation algorithm using ${\partial{\overline{\mathcal{L}}_{\infty}}}/{\partial \bm{a}}$.
\section{Training Details}
On CIFAR, we implement the ResNet, SE-ResNet, Wide-ResNet, ResNeXt, DenseNet and PyramidNet.
The SGD optimization algorithm with a Nesterov momentum is applied to train all models. Specific hyper-parameters for training are presented in Table \ref{Training_hp}.
\begin{table*}[h]
\scriptsize
\centering
\vskip -0.2in
\caption{Training configurations on CIFAR. `$l_r$' donates the learning rate.}
\label{Training_hp}
\setlength{\tabcolsep}{0.5mm}{
\vspace{5pt}
\renewcommand\arraystretch{1.15}
\begin{tabular}{c|c|c|c|c|c|c}
\hline
Network & Total Epochs & Batch Size & Weight Decay & Momentum & Initial $l_r$ & $l_r$ Schedule \\
\hline
ResNet & 160 & 128 & 1e-4 & 0.9 & 0.1 & Multiplied by 0.1 in $80^{th}$ and $120^{th}$ epoch. \\
\hline
SE-ResNet & 200 & 128 & 1e-4 & 0.9 & 0.1 & Multiplied by 0.1 in $80^{th}$, $120^{th}$ and $160^{th}$ epoch. \\
\hline
Wide-ResNet & 240 & 128 & 5e-4 & 0.9 & 0.1 & Multiplied by 0.2 in $60^{th}$, $120^{th}$, $160^{th}$ and $200^{th}$ epoch. \\
\hline
DenseNet-BC & 300 & 64 & 1e-4 & 0.9 & 0.1 & Multiplied by 0.1 in $150^{th}$, $200^{th}$ and $250^{th}$ epoch. \\
\hline
ResNeXt & 350 & 128 & 5e-4 & 0.9 & 0.05 & Multiplied by 0.1 in $150^{th}$, $225^{th}$ and $300^{th}$ epoch. \\
\hline
Shake Shake &\multirow{1}{*}{1800}&\multirow{1}{*}{64}&\multirow{1}{*}{1e-4}&\multirow{1}{*}{0.9}&\multirow{1}{*}{0.1}&\multirow{1}{*}{Cosine learning rate.} \\
\hline
PyramidNet &\multirow{1}{*}{1800}&\multirow{1}{*}{128}&\multirow{1}{*}{1e-4}&\multirow{1}{*}{0.9}&\multirow{1}{*}{0.1}&\multirow{1}{*}{Cosine learning rate.} \\
\hline
\end{tabular}}
\end{table*}
On ImageNet, we train ResNet for 120 epochs using the same l2 weight decay and momentum as CIFAR, following \cite{huang2016deep}. The initial learning rate is set as 0.1 and divided by 10 every 30 epochs. The size of mini-batch is set as 256.
All baselines are implemented with the same training configurations mentioned above.
Dropout rate is set as 0.3 for comparison if it is not applied in the basic model, following the instruction in \cite{Srivastava2014DropoutAS}. For noise rate in disturb label, 0.05 is adopted in Wide-ResNet-28-10 on both CIFAR-10 and CIFAR-100 datasets and ResNet-110 on CIFAR 10, while 0.1 is used for ResNet-110 on CIFAR 100. Focal Loss contains two hyper-parameters $\alpha$ and $\gamma$. Numerous combinations have been tested on the validation set and we ultimately choose $\alpha=0.5$ and $\gamma=1$ for all four experiments.
For L$_q$ loss, although \cite{Zhang2018GeneralizedCE} states that $q=0.7$ achieves the best performance in most conditions, we suggest that $q=0.4$ is more suitable in our experiments, and therefore adopted.
For center loss, we find its performance is largely affected by the learning rate of the center loss module, therefore its initial learning rate is set as 0.5 for the best generalization performance.
For generator-based augmentation methods, we apply the GANs structures introduced in \cite{arjovsky2017wasserstein, mirza2014conditional, odena2017conditional, chen2016infogan} to train the generators.
For WGAN, a generator is trained for each class in CIFAR-10 dataset. For CGAN, ACGAN and infoGAN, a single model is simply required to generate images of all classes. A 100 dimension noise drawn from a standard normal distribution is adopted as input, generating images corresponding to their label. Specially, infoGAN takes additional input with two dimensions, which represent specific attributes of the whole training set. Synthetic images are involved with a fixed ratio in every mini-batch. Based on the experiments on the validation set, the proportion of generalized images is set as $1/6$.
\section{Reversing Convolutional Networks}
To explicitly demonstrate the semantic changes generated by ISDA, we propose an algorithm to map deep features back to the pixel space. Some extra visualization results are shown in Figure \ref{Extra}.
An overview of the algorithm is presented in Figure \ref{Reversing}.
As there is no closed-form inverse function for convolutional networks like ResNet or DenseNet, the mapping algorithm acts in a similar way to \cite{mahendran2015understanding} and \cite{Upchurch2017DeepFI}, by fixing the model and adjusting inputs to find images corresponding to the given features. However, given that ISDA augments semantics of images in essence, we find it insignificant to directly optimize the inputs in the pixel space. Therefore, we add a fixed pre-trained generator $\mathcal{G}$, which is obtained through training a wasserstein GAN \cite{arjovsky2017wasserstein}, to produce images for the classification model, and optimize the inputs of the generator instead. This approach makes it possible to effectively reconstruct images with augmented semantics.
\begin{figure*}
\begin{center}
\centerline{\includegraphics[width=\columnwidth]{Reverse_Algorithm.pdf}}
\caption{Overview of the algorithm. We adopt a fixed generator $\mathcal{G}$ obtained by training a wasserstein gan to generate fake images for convolutional networks, and optimize the inputs of $\mathcal{G}$ in terms of the consistency in both the pixel space and the deep feature space.}
\label{Reversing}
\end{center}
\vskip -0.2in
\end{figure*}
\begin{figure*}
\begin{center}
\centerline{\includegraphics[width=\columnwidth]{extra_result.pdf}}
\caption{Extra visualization results.}
\label{Extra}
\end{center}
\vskip -0.3in
\end{figure*}
The mapping algorithm can be divided into two steps:
\textbf{Step I. }Assume a random variable $\bm{z}$ is normalized to $\hat{\bm{z}}$ and input to $\mathcal{G}$, generating fake image $\mathcal{G}(\hat{\bm{z}})$. $\bm{x}_{i}$ is a real image sampled from the dataset (such as CIFAR). $\mathcal{G}(\hat{\bm{z}})$ and $\bm{x}_{i}$ are forwarded through a pre-trained convolutional network to obtain deep feature vectors $f(\mathcal{G}(\hat{\bm{z}}))$ and $\bm{a}_{i}$. The first step of the algorithm is to find the input noise variable $\bm{z}_{i}$ corresponding to $\bm{x}_{i}$, namely
\begin{equation}
\label{ra1}
\bm{z}_{i} = \arg\min_{\bm{z}} \|f(\mathcal{G}(\hat{\bm{z}})) - \bm{a}_{i}\|_{2}^{2} +
\eta\|\mathcal{G}(\hat{\bm{z}}) - \bm{x}_{i}\|_{2}^{2},\
s.t.\ \hat{\bm{z}} = \frac{\bm{z} - \overline{\bm{z}}}{std(\bm{z})},
\end{equation}
where $ \overline{\bm{z}}$ and $std(\bm{z})$ are the average value and the standard deviation of $\bm{z}$, respectively.
The consistency of both the pixel space and the deep feature space are considered in the loss function, and we introduce a hyper-parameter $\eta$ to adjust the relative importance of two objectives.
\textbf{Step II. }We augment $\bm{a}_{i}$ with ISDA, forming $\tilde{\bm{a}}_{i}$ and reconstructe it in the pixel space. Specifically, we search for $\bm{z}_{i}'$ corresponding to $\tilde{\bm{a}}_{i}$ in the deep feature space, with the start point $\bm{z}_{i}$ found in Step I:
\begin{equation}
\label{ra2}
\bm{z}_{i}' = \arg\min_{\bm{z'}} \|f(\mathcal{G}(\hat{\bm{z}}')) - \tilde{\bm{a}}_{i}\|_{2}^{2},\
s.t.\ \hat{\bm{z}}' = \frac{\bm{z'} - \overline{\bm{z'}}}{std(\bm{z'})}.
\end{equation}
As the mean square error in the deep feature space is optimized to 0, $\mathcal{G}(\hat{\bm{z}_{i}}')$
is supposed to represent the image corresponding to $\tilde{\bm{a}}_{i}$.
The proposed algorithm is performed on a single batch. In practice, a ResNet-32 network is used as the convolutional network. We solve Eq. (\ref{ra1}), (\ref{ra2}) with a standard gradient descent (GD) algorithm of 10000 iterations. The initial learning rate is set as 10 and 1 for Step I and Step II respectively, and is divided by 10 every 2500 iterations. We apply a momentum of 0.9 and a l2 weight decay of 1e-4.
\section{Extra Experimental Results}
\begin{figure}[htp]
\begin{center}
\subfigure[ResNet-110 on CIFAR-10]{
\label{fig:evaluationC10}
\includegraphics[width=0.45\columnwidth]{Re110C10.pdf}
}
\subfigure[ResNet-110 on CIFAR-100]{
\label{fig:evluationC100}
\includegraphics[width=0.45\columnwidth]{Re110C100.pdf}}
\caption{Comparison with state-of-the-art image classification methods.}
\label{compare}
\end{center}
\vskip -0.2in
\end{figure}
Curves of test errors of state-of-the-art methods and ISDA are presented in Figure \ref{compare}. ISDA outperforms other methods consistently, and shows the best generalization performance in all situations. Notably, ISDA decreases test errors more evidently in CIFAR-100, which demonstrates that our method is more suitable for datasets with fewer samples. This observation is consistent with the results in the paper. In addition, among other methods, center loss shows competitive performance with ISDA on CIFAR-10, but it fails to significantly enhance the generalization in CIFAR-100.
\section{Introduction}\label{introduction}
\IEEEPARstart{A}{n} audio-visual event (AVE) often refers as an event that is both audible and visible in a video segment, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, a sound source appears in an image ({\em visible}) while the sound it makes also exists in the audio portion ({\em audible}). The AVE localization task is to find these video segments which contain an audio-visual event and classify it into a certain category.
We illustrate this task in Fig.~\ref{fig:AVEL_task}.
It belongs to the research topic of audio-visual scene understanding. It uses both audio and vision inputs to
answer {\em if an event happens in both modalities at different video segments}.
The task must explore unconstrained videos (events in real life) that are not limited to the temporal consistency of lip reading or other human-making sounds.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.48\textwidth]{./figures/fig1-gd.pdf}
\end{center}
\vspace{-0.4cm}
\caption{An illustration of the AVE localization task.
Each video segment is composed of an audio and a visual component.
In this example,
the ``hum'' of the bus exists in all the segments (audio modality), but
the visual images of the ``bus'' appear only in the third and fourth segments (visual modality).
So only these two segments (red boxes) are localized as \emph{bus event}, the remaining are recognized as \emph{background}.
A localization system is expected to analyze and utilize the audio-visual pairs (audio-visual correspondence-aware), determine whether a video segment contains an audio-visual \emph{event} (event-aware), and further identify its event category (category-aware).
}
\label{fig:AVEL_task}
\vspace{-3mm}
\end{figure}
Recent literature has shown that by fusing multi-modality information can lead to better deep feature representation, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, audio-visual fusion~\cite{arandjelovic2017look} and text-visual fusion~\cite{radford2021learning}. However, building a large scale multi-modality pre-training datasets would require heavy manual labors to clean and annotate the raw video sets. To relief the manual labor, recent work either focuses on learning from noise supervision~\cite{cheng2019noise, jia2021scaling} or tries to automatically filter out irrelevant samples~\cite{tian2018audio}.
In this work, we devote to explore better deep feature representation for AVEs, which is served for the latter purpose.
We study how to effectively leverage audio and visual information for event localization. In the current AVE localization work, two relations are considered: intra-modal and cross-modal relations. The former often addresses temporal relations in one single modality while the later also takes audio and visual relations into account. Pioneer works~\cite{lin2019dual, tian2018audio} often try to regress the class by directly concatenating features from synchronized audio-visual pairs;
their accuracy is often unsatisfying. The following works~\cite{wu2019dual, xu2020MM, xuan2020cross} utilize a self-attention mechanism to explicitly encode the temporal relations within intra-modality and some of them~\cite{ramaswamy2020makes, ramaswamy2020see, xu2020MM, xuan2020cross} also aggregate better audio-visual feature representations by encoding cross-modal relations. However, these methods aggregate all the audio and visual components,
often ignore the interference caused by irrelevant audio-visual segment pairs during the fusion process. In this paper, we solve this problem in the cross-modal interaction by considering the relations from different perspectives: intra- and inter-videos. More importantly, we devote to feature aggregation and enhancement from high-relevant (positive) samples and can obtain better AVE localization accuracy.
About our observations, we argue and detail that the AVE localization task has three main challenges below.
(1) {\em Unconstrained audio-visual relevance matching}. On one hand, the sound-maker is often occluded by some event-irrelevant objects or even be out of the screen, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, the humming sound but accompanied by an announcer as shown in Fig.~\ref{fig:AVEL_task}. On the other hand, there are usually multiple objects contained in the visual scene which could be sound-makers or not, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, the humming accompanied by the bus, man, and road, also as shown in Fig.~\ref{fig:AVEL_task}; the audio signal is also inevitably mixed with other noises.
Such scenarios in unconstrained videos make it hard to match the audio-visual segment pairs in a flexible and accurate manner.
2) {\em Temporal inconsistency in AVE videos}. In real-life videos, the audio and visual signals are processed by an independent workflow.
This spawns the research on the audio-visual synchronization problem~\cite{chung2016out, korbar2018cooperative, Khosravan2019OnAM} and brings the temporal inconsistency issue. Such issue lets the segment event judgment (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, AVE or background) difficult. Especially for the weakly-supervised setting (refer Sec.~\ref{sec:problem_define} for the setting details), AVE localization task still asks to parse from segment-level but only given video-level label.
(3) {\em Distinction of similar but different representations.} For the AVE localization task, it not only requires locating the event along the timeline but also must identify its category. In order to obtain discriminative feature representations, we must constraint the representations learning to be category-aware, such as distinguishing the videos displaying musical instruments guitar and violin although these two events are with a negligible difference.
\textbf{The proposed Contrastive Positive Sample Propagation model (CPSP).} To deal with aforementioned challenges, as shown in Fig.~\ref{fig:HLPSA}, we propose a new {\bf \em CPSP method}
that enables the localization system to encode discriminative representations by activating the \emph{positive instances in the audio-visual data} from three levels, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, the most relevant audio-visual pairs (\emph{pair-level}), the segments containing an AVE consistently in audio and vision modalities (rather than the background, \emph{segment-level}), the videos belonging to the same event category (rather than other categories, \emph{video-level}). By exploiting these positive samples, the learned audio-visual representation encourages our model to be AVC (audio-visual correspondence)-aware, event-aware, and category-aware, which exactly matches the goal of AVE localization task. We introduce the details next.
Specifically, as shown in Fig.~\ref{fig:system_flow}, we propose a new {\bf \em Positive Sample Propagation (PSP) module}. In a nutshell, PSP constructs an all-pair similarity map between each audio and visual segment and cuts off the entries that are below a pre-set similarity threshold, and then aggregates the audio and visual features without considering the negative and weak entries in an online fashion. Through various visualizations, we show that the PSP allows the most relevant features that are not necessarily synchronized to be aggregated in an online fashion.
It is noteworthy that our PSP is \emph{AVC-aware}, which is different from existing cross-modal feature fusion methods~\cite{ramaswamy2020makes, ramaswamy2020see, xu2020MM, xuan2020cross}.
A concise illustration of the PSP has been shown in Fig.~\ref{fig:HLPSA} and details can be seen from Fig.~\ref{fig:PSP_process}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.48\textwidth]{./figures/fig2-gd-v5.pdf}
\end{center}
\vspace{-0.6cm}
\caption{An illustration of the proposed contrastive positive sample propagation (CPSP) method. The rounded rectangle in gray color represents a background segment, otherwise it means that the segment describes an event.
The CPSP considers the AVE localization task from three aspects and works in a contrastive manner: 1) The \emph{pair-level} positive sample propagation (PSP) aims to select the most relevant audio-visual segment pairs for cross-modal feature aggregation (\emph{AVC-aware}). 2) The \emph{segment-level} positive sample activation (PSA$_S$) aims to distinguish the positive segments that contain an audio-visual event from the background segments (\emph{event-aware}). 3) The \emph{video-level} positive sample activation (PSA$_V$) activates the videos sharing the same event category as the positive samples (\emph{category-aware}).
}
\label{fig:HLPSA}
\vspace{-0.3cm}
\end{figure}
Apart from the PSP, it is also significant to effectively address the difficulties: 1) complex visual or audio backgrounds in an unconstrained video make it difficult to localize an AVE along the time line, and 2) localizing and recognizing an AVE category requires the model to deeply exploit the representative features among different AVE videos. Thus, as illustrated in Fig.~\ref{fig:system_flow}, we propose a new {\bf \em Positive Sample Activation (PSA) module} to constraint the representation of target segments containing an AVE to be possibly distinguishable from both the backgrounds in the same video (intra-video) and the segments of other videos with different event categories (inter-video). Specifically, the PSA is conducted from the segment-level (PSA$_S$) and video-level (PSA$_V$) whereas the PSA$_S$ is designed to address the former purpose, while the PSA$_V$ is proposed to solve the latter. As illustrated in Fig.~\ref{fig:HLPSA}, we show that the PSA$_S$ is \emph{event-aware} and the PSA$_V$ is \emph{category-aware}. Details are introduced in Sec.~\ref{sec:contrastive_learning}.
It is nontrivial to build positive and negative connections between complex visual scenes and intricate sounds. We utilize the principle of the {\em contrastive learning} without additional annotations to optimize the audio-visual representation learning. Three new contrastive losses, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, the $\mathcal{L}_{\text{avpsp}}$ for the PSP, $\mathcal{L}_{\text{spsa}}$ for the PSA$_S$, and $\mathcal{L}_{\text{vpsa}}$ for the PSA$_V$, are proposed for gathering positive samples while pushing the negatives away from them in the feature space.
They are different from these audio-visual related works~\cite{afouras20ssl, ma2021contrastive, wu2021exploring} that take the synchronized audio-visual segments as positive samples, which is not compatible with AVE localization. In our work, we explore the positive (high relevant) AVC pairs and consider performing contrastive techniques from more levels, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, adding the segment-level and video-level contrasting.
To the best of our knowledge, we are the first to introduce contrastive learning to solve the AVE localization problem and provide a comprehensive discussion about the three levels of positive instances covering both intra- and inter- AVE video correlations as shown in fig.~\ref{fig:HLPSA}.
\textbf{The improvement of backbone for fully and weakly supervised AVE localization.} Besides the proposed CPSP,
we analyze the fully and weakly supervised AVE localization, and further propose two improvements that work under each setting, respectively.
On the one hand, an audio-visual pair similarity loss based on the PSP $\mathcal{L}_{\text{avpsp}}$ is introduced under the fully supervised setting that encourages the network to learn high correlated features for the audio-visual pair if they contain the same event.
On the other hand, we propose a weighting branch in the weakly supervised setting, which gives temporal weights to the segment features. We evaluate these two improvements on the standard AVE dataset~\cite{tian2018audio} and the experimental results demonstrate their effectiveness. It is noteworthy that these two improvements are not only helpful in the proposed CPSP method but also can be applied to other localization networks (relevant results have been shown in Sec.~\ref{sec:evaluation_improvements}).
To summarize, the proposed CPSP including the PSP and PSA modules devotes to better deep representation by imposing contrastive constraints on the localization system from three semantic levels (positive audio-visual instances of AVC pairs, segments, and videos), which covers both intra- and inter- AVE video correlations. The main contributions are summarized as follows:
\begin{itemize}
\item The proposed PSP allows to explore the most relevant (positive) \emph{pair-level} features that are not necessarily synchronized but semantic-related to be aggregated and enables to encode more distinguishable audio-visual representations.
\item The proposed PSA explicitly activates the positive samples from additional \emph{segment-level} and \emph{video-level} rather than directly sending the fused audio-visual features into the final classifier such as in previous works\cite{xu2020MM, xuan2020cross, ramaswamy2020makes, ramaswamy2020see}.
\item The improvements of backbone proposed for the fully and weakly supervised settings consistently benefit our localization system and are also advantageous in other networks.
\item Extensive experiments demonstrate the effectiveness of following designs and our method achieves new state-of-the-art performances under both settings.
\end{itemize}
At last, we remind that the PSP module was first introduced in our previous work~\cite{zhou2021positive}.
Compared to the preliminary version, in this paper, we have made improvements in six aspects:
(1) in addition to the PSP, we expand it to the CPSP by adding the Positive Sample Activation (PSA, including PSA$_S$ and PSA$_V$) scheme that systematically exploiting segment-level and video-level positive audio-visual instances; (2) we perform a comprehensive survey of relevant works about the contrastive learning in the field of audio-visual representation learning in Sec.~\ref{sec:related_work}; (3) two new contrastive objective losses are designed and introduced into the AVE localization in Sec.~\ref{sec:contrastive_learning}; we add the discussion about the objectives in Sec.~\ref{sec:discussion};
(4) we implement a self-supervised contrastive method SSPSP and give more analysis by comparing it with the CPSP in Sec.~\ref{sec:quanti2}.
(5) we release a large-scale VGGSound-AVEL100k dataset for AVEL task.
The videos are sampled from VGGSound~\cite{chen2020vggsound} where the video-level categories are given and we provide segment-level annotations.
Considerable experiments are conducted on this large dataset to evaluate our model and more detailed analyses are provided in Sec.~\ref{sec:quanti2}.
(6) we extend the proposed method on the LLP~\cite{tian2020avvp} dataset collected for a similar but more challenging audio-visual video parsing (AVVP) task in Sec.~\ref{sec:avvp} and the results also demonstrate the effectiveness and generalization ability of our method.
In brief, the CPSP proposed in this paper makes the localization framework much more comprehensive and extensive experiments make the CPSP more convincing.
The rest of this paper is organized as follows. Sec.~\ref{sec:related_work} provides an overview of the related works. Sec.~\ref{sec:problem_define} introduces two settings of the AVE localization problem, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, the fully and weakly supervised tasks. Sec.~\ref{sec:method} elaborates on the proposed CPSP. Discussions on the CPSP methodology are shown in Sec.~\ref{sec:discussion}. The experimental results and analyses are presented in Sec.~\ref{sec:experiment}, and conclusion are given in Sec.~\ref{sec:conclusion}.
\vspace{-3mm}
\section{Related Work}\label{sec:related_work}
\textbf{Audio-visual correspondence (AVC)} aims to predict whether a given visual image corresponds to a short audio recording.
The task is asked to judge whether the audio and visual signals describe the same object,
\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, dog {\em v.s.} bark, cat {\em v.s.} meow.
It is a self-supervised problem since the visual image is usually accompanied by the corresponding sound in video data.
Existing methods try to evaluate the correspondences by measuring the audio-visual similarity~\cite{arandjelovic2017look, arandjelovic2018objects, aytar2016soundnet, cheng2020look, fayek2020large}.
It will get a large similarity score if the audio-visual pair is corresponding, otherwise, a low score.
This is in line with our focused AVE localization problem since the synchronized audio-visual pair of a target {\em event} segment must be corresponding.
The difference is that AVC tackles the correspondence of an audio and an image, rather than an audio and a video in our AVE localization task.
So, we are motivated to tackle the abundant audio-visual segment pairs in AVE localization problem by further exploring and exploiting the audio-visual similarity.
\textbf{Sound source localization (SSL)} aims to localize those visual regions which are relevant to the provided audio signal.
It is related to {\em sound source separation} problem, which mainly focuses on the event of people speech~\cite{darrell2000audio, xu2018single, afouras2018deep, jenrungrot2020cone, afouras20ssl, gao2021visualvoice} or musical instrument playing~\cite{parekh2017motion, pu2017audio, zhao2018sound, gao2018learning, zhao2019sound, gao2019co}.
For SSL, there is usually a condition that the sound source must appear in the visual image.
In other words, it mainly focuses on the \emph{visual} localization while the AVE localization devotes to the \emph{temporal} localization.
SSL has two settings: the single and multiple sound source(s) localization.
For the single SSL, the localization map can be easier obtained in an unsupervised manner by directly computing the similarity between the audio feature and visual feature map.
The multiple SSL is more challenging that requires to accurately locate the sound-maker when there are multiple sound sources~\cite{qian2020multiple, hu2019deep, hu2020discriminative, zhou2022avs}.
The class activation mapping (CAM)~\cite{zhou2016CAM} is helped to realize class-aware object localization.
Qian \emph{et al}\onedot~\cite{qian2020multiple}
adapt the Grad-CAM~\cite{selvaraju2017grad} to disentangle class-specific features for multiple SSL.
Hu \emph{et al}\onedot~\cite{hu2019deep} maps audio and visual features into respective K cluster centers and take the center distance as a supervision to rank the paired audio-visual objects.
Hu \emph{et al}\onedot~\cite{hu2020discriminative} first learn the object semantics in single SSL then use that to help with multiple SSL.
Recent methods use contrastive techniques to utilize the discriminative sound characteristics and diverse object appearances.
Senocak \emph{et al}\onedot~\cite{senocak2021TAPMI} propose a triplet loss working in an unsupervised manner.
Afouras \emph{et al}\onedot~\cite{afouras20ssl} utilize
a contrastive loss to train the model in a self-supervised learning way.
Both of these methods~\cite{afouras20ssl, senocak2021TAPMI}
need to construct positive and negative audio-visual pair samples.
Considering the positive and negative samples can also be obtained in AVE localization task,
we propose to exploit the abundant audio-visual instances for contrastive learning.
\textbf{Audio-visual event localization (AVEL)} aims to distinguish those segments including an
audio-visual event from a long video.
Different from the acoustic event classification~\cite{hershey2017cnn, kong2018audio, kumar2018knowledge, mcfee2018adaptive} or video classification~\cite{karpathy2014large, long2018attention, long2018multimodal, wang2018appearance, tran2019video} making a prediction based on the whole audio or video embedding, AVE localization requires to judge the audio-visual correspondence and event category for each segment.
Currently, the AVEL task is a supervised problem with weak labels or full labels. The former merely contains video-level labels, and the latter refers to both segment-level and video-level annotations.
Existing works mainly focus on the audio-visual fusion process.
A dual multimodal residual network is proposed in~\cite{tian2018audio}.
Lin \emph{et al}\onedot~\cite{lin2019dual} adapt a bi-directional LSTM~\cite{schuster1997bidirectional} to fuse audio and visual features in a seq2seq manner.
These methods simply concatenate the synchronized audio-visual features during fusion.
Subsequent work utilizes a bilinear method~\cite{ramaswamy2020makes, ramaswamy2020see} or a joint co-attention strategy~\cite{xuan2020cross, Duan_2021_WACV} to capture cross-modal relations
between both synchronized and unsynchronized audio-visual pairs.
Self-attention mechanism is also widely used to encode temporal relation in both the AVEL~\cite{wu2019dual, xu2020MM, xuan2020cross} and \emph{audio-visual video parsing (a new weakly supervised audio-visual related task)}~\cite{tian2020avvp, wu2021explore, lin2021exploring}.
Lin \emph{et al}\onedot~\cite{Lin_2020_ACCV} design an audio-visual transformer to describe local spatial and temporal information.
The visual frame is divided into patches and adjacent frames are utilized, making the model complicated and computationally intensive.
Xu \emph{et al}\onedot~\cite{xu2020MM} attempt to use the concatenating audio-visual features as the supervision
then the feature of each modality is updated by separate modules.
Unlike these, the proposed CPSP method has a further in-depth study on the abundant audio-visual pairs, event and background segments, and similar videos but with different categories, activating the most relevant ones.
Relying on these positive samples, more distinguished audio-visual features can be obtained after feature aggregation.
\textbf{Contrastive learning in audio-visual field.}
The technique of contrastive learning turns out to be an effective solution that is widely used in self-supervised learning~\cite{oord2018CPC, he2020momentum, chen2020simple, morgado2021audio, harwath2018jointly} and various weakly supervised tasks~\cite{zhang2020counterfactual, zhang2021cola, gupta2020contrastive, ki2020sample}.
Seeing data in a large batch size, the model learns discriminative representations by identifying the positive or negative samples. The key of contrastive learning is how to construct the positive and negative samples.
Recently, some researchers start to explore injecting the label information to accurately select more positive/negative samples for better representation learning.
Such supervised contrastive learning has shown its superiority in both computer vision~\cite{SCL2020, zeng2021modeling} and natural language processing tasks~\cite{sedghamiz2021supcl, gunel2020supervised}.
In the audio-visual related works~\cite{afouras20ssl, ma2021contrastive, wu2021exploring}, they usually adopt the self-supervised contrastive manner, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, directly selecting the positive sample from the synchronized audio-visual segment and contrast them with the negatives come from different timestamps.
However, this is not compatible with AVE localization: an audio and visual segment can be regarded as a positive sample as long as they describe the \emph{same event}, and vice versa.
To effectively distinguish an AVE, it is vital to construct exact positive and negative samples from the video segments for contrastive learning.
Inspired by these, we propose the PSA in a supervised manner that explicitly performs contrastive learning between segments and videos with the segment/video-level label prior.
To the best of our knowledge, we are the first to utilize the contrastive learning to solve AVE localization problem and we explore comprehensive contrastive strategies from different levels.
\section{Problem statement}\label{sec:problem_define}
AVE localization aims to find out those segments
containing an audio-visual event \cite{tian2018audio}.
In other words, AVE localization is expected to decide whether each synchronized audio-visual pair depicts an event.
Besides, AVE localization needs to identify the event category for each segment.
Specifically, a video sequence $S$ is divided into $T$ non-overlapping yet continuous segments $\{S_t^v, S_t^a\}^{T}_{t=1}$, and each segment is one-second in length. $S^v$ and $S^a$ are the visual and audio components, respectively. We consider two settings of this task, to be described below.
\textbf{Fully-supervised AVE localization.}
Under the fully supervised setting, the event label of every video segment is given, indicating
whether the segment denotes an event and which category the event belongs to.
We denote the event label of the $t^{\text{th}}$ segment as $\bm{y}_t = { \{ y_t^c | y_t^c \in \{0, 1\}, \sum_{c=1}^{C} y_t^c = 1\} \in \mathbb{R}^C}$,
where $C$ is the number of categories (including the {\em background}).
Then, the label for the entire video can be written as $\bm{Y}^{\text{fully}} = [\bm{y}_1; \bm{y}_2; ...; \bm{y}_T] \in \mathbb{R}^{T \times C}$.
Through $\bm{Y}^{\text{fully}}$, we know whether an arbitrary synchronized audio-visual pair at time $t$ is an event: if the $1$ of its event label $\bm{y}_t$ is at the entry of a certain event instead of the \emph{background},
the pair describes an event and otherwise does not.
\begin{figure*}[t]
\begin{center}
\setlength{\abovecaptionskip}{0.cm}
\includegraphics[width=\textwidth]{./figures/fig_main_pipeline_for_ieee_adding_losses.pdf}
\end{center}
\vspace{-0.5cm}
\caption{System Flow. We first extract and encode video and audio features through existing modules such as AVGA~\cite{tian2018audio} and Bi-LSTM.
The proposed {\em positive sample propagation} (PSP) takes the LSTM encoded features as input;
an affinity matrix is computed before selecting the connections of audio-visual segment pairs using thresholding.
In this module,
audio and visual features are aggregated by feature propagation through the {\em pair-level positive} connections.
Then, the fused features are further processed under the constraints of the proposed {\em positive sample activation} (PSA), where the {\em segment-level} (PSA$_S$) enforces features of these video segments containing the same event ({\em positive}) to be possibly closer and be away from the background segments while the {\em video-level} (PSA$_V$) encourages representations of the videos sharing the same event category ({\em positive}) to be similar. Details are described in Sec.~\ref{sec:contrastive_learning}.
In the last stage, we classify the event into predefined categories.
For the supervised setting, apart from the commonly used cross-entropy (CE) loss, we further propose an audio-visual pair similarity loss based on the PSP which enforces similar features between them when they contain an event. For the weakly supervised setting, we introduce another FC layer that gives weights to different video segments: higher weights are given event-containing segments.
The whole system is optimized by the basic classification loss with additional proposed contrastive objectives, detailed in Sec.~\ref{sec:objective_function}.
}
\vspace{-0.3cm}
\label{fig:system_flow}
\end{figure*}
\textbf{Weakly-supervised AVE localization.}
We adapt the weakly-supervised setting following \cite{lin2019dual,xuan2020cross}, where the label $\bm{Y}^{\text{weak}} \in \mathbb{R}^{1 \times C}$ is the average pooling value of $\bm{Y}^{\text{fully}}$ along the column.
It implies the proportion of video segments that contain an event.
This setting is different from the fully supervised one because the event label of each segment $y_t$ is unknown, making the problem more challenging.
\section{Our method}\label{sec:method}
The overall pipeline of our system is illustrated in Fig.~\ref{fig:system_flow},
which includes four modules:
a feature extraction and encoding module (Sec.~\ref{sec:feature_extractor}),
a positive sample propagation module (Sec.~\ref{sec:PSP}), a
positive sample activation module (Sec.~\ref{sec:contrastive_learning}), and a classification module (Sec.~\ref{sec:classification}).
In the {\em feature extraction and encoding} module,
the audio-guided visual attention (AVGA~\cite{tian2018audio}) is adapted for early fusion to make the model focus on those visual regions closely related to the audio component.
Then Bi-LSTM is utilized to encode temporal relations in video segments for each modality.
The LSTM encoded audio and visual features are sent to the proposed {\em positive sample propagation (PSP)} module.
PSP is able to select those positive connections of audio-visual segment pairs by measuring the cross-modal similarity with thresholding, $i.e.$, the pair-level contrastive constraint.
Audio and visual features are aggregated by feature propagation through the positive connections.
The fused audio-visual features after PSP are further processed by two contrastive constraints, {\em i.e.}, the \emph{
positive sample activation} from both segment-level (PSA$_S$) and video-level (PSA$_V$), which refine the audio-visual features for segments containing an event in a video and different videos but sharing the same event category, respectively.
The updated features are then sent to the final {\em classification} module, predicting which video segments contain an event and the event category.
\subsection{Feature extraction and encoding}\label{sec:feature_extractor}
The visual and synchronized audio segments are processed by pretrained convolutional neural networks (CNNs).
We denote the resulting visual feature as $ \pmb V \in \mathbb{R}^{T \times N \times d_v } $,
where $d_v$ is the feature dimension, $N = H \times W$, $H$ and $W$ are the height and width of the feature map, respectively.
The extracted audio feature is denoted as $ \pmb A \in \mathbb{R}^{T \times d_a} $, where $d_a$ denotes feature dimension.
We then directly adapt AGVA~\cite{tian2018audio} for multi-modal early fusion. AVGA allows the model to focus on visual regions that are relevant to the audio component.
To encode the temporal relationship in video sequences, the visual and
audio features after AVGA are further sent to two independent Bi-LSTMs.
The updated visual and audio features are
represented as $\bm{v}^{\text{lstm}}\in \mathbb{R}^{T \times d_l}$ and $\bm{a}^{\text{lstm}} \in \mathbb{R}^{T \times d_l}$, respectively.
\subsection{Positive sample propagation (PSP)}\label{sec:PSP}
PSP allows the network to learn more representative features
by exploiting the similarities of synchronized and unsynchronized audio-visual segment pairs.
It involves three steps.
In \emph{all-pair connection construction},
all the audio-visual pairs are connected.
As shown in Fig.~\ref{fig:PSP_process},
here we only display the connections of one visual segment for simplicity,
{\em i.e.}, $\langle v_1 \leftrightarrow a_1/a_2/a_3/a_4 \rangle$.
The strength of these connections are measured
by the similarity between the audio-visual components $\langle{\bm{a}^{\text{lstm}}, \bm{v}^{\text{lstm}}} \rangle$,
computed by,
\begin{equation}
\bm{\beta}^{\text{va}} = \frac{(\bm{v}^{\text{lstm}}{\bm{W}_1^v})(\bm{a}^{\text{lstm}}{\bm{W}_1^a})^{\top}} {\sqrt{d_l}}, \quad
\bm{\beta}^{\text{av}} = (\bm{\beta}^{\text{va}})^{\top},
\end{equation}
where $\bm{W}_1^v \mbox{ and } \bm{W}_1^a \in \mathbb{R}^{d_l \times d_h}$
are learnable parameters of linear transformations,
implemented by a linear layer, and
$d_l$ is the dimension of the audio or visual feature.
$\bm{\beta}^{\text{va}} \mbox{ and } \bm{\beta}^{\text{av}} \in \mathbb{R}^{T \times T}$ are the similarity matrices.
Second, we \emph{prune the negative and weak connections}.
Specifically, the connections constructed in the first step
are divided into three groups according to the similarity values: negative, weak, and positive.
As a classification task, the success of AVE localization highly depends on the richness and correctness of training samples for each class. That is, we aim to collect possibly many and relevant \emph{positive} connections.
We achieve this goal by filtering out the weak and negative ones, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, $v_1 \leftrightarrow a_3$ and $v_1 \leftrightarrow a_4$ as shown in Fig.~\ref{fig:PSP_process}.
We begin with processing all the audio-visual pairs
with the ReLU activation function,
cutting off connections with negative similarity values.
Row-wise $\ell_1$ normalization is then performed,
yielding the normalized similarity matrices
$\bm{\beta}^{\text{va}}$ and $\bm{\beta}^{\text{va}}$.
\begin{figure}[t]
\begin{center}
\setlength{\abovecaptionskip}{0.cm}
\includegraphics[width=0.45\textwidth]{./figures/figure_PSP_illustration_add_weak_annotation.pdf}
\end{center}
\vspace{-0.5cm}
\caption{An illustration of the proposed PSP.
In this example, only the first two video segments contain an audio-visual event, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, \emph{motorcycle}.
``$\surd$'' denotes the audio or visual segment that describes the event, while ``$\times$'' means not.
The red lines denote connections of audio-visual pairs,
solid lines represent connections formed by relevant pairs,
while dotted lines denote irrelevant pairs.
The thickness of line reflects the similarity of the audio-visual pair.
$v_1 \leftrightarrow a_4$ is a {\em negative} connection, formed by irrelevant audio-visual pair with negative similarity value.
$v_1 \leftrightarrow a_3$ and $v_1 \leftrightarrow a_1/a_2$ are {\em weak} and {\em positive connections} respectively,
determined via similarity.
``Step-1'' corresponds to the all-pair connection construction,
while ``Step-2'' denotes the pruning of the negative and weak connections,
and the green arrow indicates the {\em positive} direction of feature propagation in ``Step-3''.
}
\label{fig:PSP_process}
\vspace{-0.3cm}
\end{figure}
\begin{figure*}[t]
\subfigure[Illustration of the {\em segment-level} positive sample activation (PSA$_S$). On the \textbf{left},
only the first five video segments contain the event \emph{aircraft};
for these segments, take anyone as an \emph{anchor}, the remaining four segments can be regarded as its \emph{positive} samples, while the last five segments constitute the \emph{negative} samples.
Next, as illustrated on the \textbf{right}, in PSA$_S$, the positive samples are gathered around the anchor while pushed away from the negative ones.]{
\centering
\includegraphics[width={\textwidth}]{./figures/fig_subset_bg_anno.pdf}
\label{fig:subset_bg}
}
\vspace{-2mm}
\newline
\subfigure[Illustration of the {\em video-level} positive sample activation (PSA$_V$). On the \textbf{left},
all the segments in the video contain the event \emph{guitar}.
Videos sharing the same event category \emph{guitar} are treated as the \emph{positive} samples, while the \emph{negative} samples come from those holding other event categories.
As illustrated on the \textbf{right}, for an anchor video, we merely select one positive sample belonging to the same event category with the largest Euclidean distance, and negative samples classified to other categories but with top-$K$ smallest distances.
The positive and top-$K$ hard negative samples are online selected.
]{
\centering
\includegraphics[width={\textwidth}]{./figures/fig_subset_ae_anno.pdf}
\label{fig:subset_ae}
}
\vspace{-5mm}
\caption{We show two types of video data and illustrate the
contrastive constraints (PSA$_S$ and PSA$_V$) that are proposed to further
exploit more positive samples from both segment-level and video-level, respectively. The green boxes represent an \emph{event} happens in the visual segment, while gray boxes mean not.}
\vspace{-2mm}
\label{fig:contrastive_strategy}
\end{figure*}
The negative and weak connections are presumably featured by smaller similarity values, so we simply adapt a thresholding method, written as,
\begin{equation}
\begin{split}
\bm{\gamma}^{\text{va}} = \bm{\beta}^{\text{va}} \mathbb{I}(\bm{\beta}^{\text{va}}-\tau), \\
\bm{\gamma}^{\text{av}} = \bm{\beta}^{\text{av}} \mathbb{I}(\bm{\beta}^{\text{av}}-\tau),
\label{threshold_operation_1}
\end{split}
\end{equation}
where $\tau$ is the hyper-parameter, controlling how many connections will be pruned. $\mathbb{I(\cdot)}$ is an indicator function, which outputs $1$ when the input
is greater than or equal to $0$, and otherwise outputs $0$. After thresholding, row-wise $\ell_1$ normalization is again performed to obtain the final similarity matrices $\bm{\gamma}^{\text{va}},$ $\bm{\gamma}^{\text{av}}\in \mathbb{R}^{T \times T}$.
\emph{Online feature aggregation}. The above step identifies audio (visual) components with high similarities with a given visual (audio) component, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, $v_1 \leftrightarrow a_1$ and $v_1 \leftrightarrow a_2$ shown in Fig.~\ref{fig:PSP_process}. This is essentially a positive sample propagation process that can be utilized to update the features of audio or visual components.
Particularly, given the connection weights $\bm{\gamma}^{\text{av}}$ and $\bm{\gamma}^{\text{va}}$, the audio and visual features $\bm{a}^{\text{psp}}$ and $\bm{v}^{\text{psp}}$ are respectively updated as,
\begin{equation}
\begin{split}
\bm{a}^{\text{psp}} = \overbrace {\bm{\gamma}^{\text{av}}(\bm{v}^{\text{lstm}}{\bm{W}_2^v})}^{\bm{v}^{\text{pos}}} + \bm{a}^{\text{lstm}},\\
\bm{v}^{\text{psp}} = \overbrace {\bm{\gamma}^{\text{va}}(\bm{a}^{\text{lstm}}{\bm{W}_2^a})}^{\bm{a}^{\text{pos}}} + \bm{v}^{\text{lstm}},
\end{split}
\label{eq:3}
\end{equation}
where $\bm{W}_2^{a}, \bm{W}_2^{v} \in \mathbb{R}^{d_l \times d_l}$ are parameters defining linear transformations, and
$\bm{a}^{\text{psp}}, \bm{v}^{\text{psp}} \in \mathbb{R}^{T \times d_l}$.
Generally, the audio (visual) feature $\bm{a}^{\text{psp}}$ ($\bm{v}^{\text{psp}}$) is enhanced by the propagated positive support from the other modality.
This practice allows us to learn more discriminative audio-visual representations, displayed in Fig.~\ref{fig:feature_distribution}. More discussions are provided in Sec. \ref{sec:discussion}.
\subsection{Positive sample activation (PSA)}\label{sec:contrastive_learning}
PSA is designed to make the model more event-aware and category-aware. It involves two steps.
We activate more positive samples from both segment and video levels. We introduce the PSA$_S$ and PSA$_V$ with contrastive strategies below. Before that, we first fuse the audio and visual features $\bm{a}^{\text{psp}}$ and $\bm{v}^{\text{psp}}$ into an integrated audio-visual feature $\bm{f}$ as follows:
\begin{equation}\label{eq:av_fusion}
\bm{f} = \frac {1}{2}[{\mathcal{N}(\bm{v}^{\text{psp}}{\bm{W}_3^v}) + \mathcal{N}({\bm{a}^{\text{psp}}}{\bm{W}_3^a})}],
\end{equation}
where $\bm{f}\in \mathbb{R}^{T \times d_l}$ is the feature of video segments, $ \mathcal{N}(\cdot) $ represents layer normalization,
$ \bm{W}_3^v, \bm{W}_3^a \in \mathbb{R}^{d_l \times d_l} $ represent learnable parameters in the linear layers. $\bm{f}$ can be used to represent the segment feature and be summarized to the video feature.
\textbf{\subsubsection{Segment-level positive sample activation (PSA$_S$).}}
To make the model be event-aware, we design a contrastive strategy from the segment-level.
As shown in Fig.~\ref{fig:subset_bg}, there are two sets of segments: segments depicting an audio-visual event constitute the \emph{event set}, while remaining segments form the \emph{background set}.
We present a contrastive strategy to perceive the difference between these two video segment sets.
Take arbitrary segment from the event set as an anchor, the remaining ones in the event set are regarded as its \emph{positive} samples, while the segments in the background set are treated as \emph{negative} samples.
As shown in the right of Fig.~\ref{fig:subset_bg}, positive samples should be pulled together to the anchor, while the negative ones are pushed away.
The segment-level contrastive objective takes the following form,
\begin{equation}\label{eq:loss_scon}
\begin{gathered}
\begin{split}
&\mathcal{L}_{\text{spsa}} \!= \\
& \!-\!
\frac{1}{N_i^e}\! \sum_{i=1}^{N_i^e}\! \log(\frac{\text{exp}(\frac{\text{sim}(\bm{f}_i, \bm{f}_j)}{\eta})}{\text{exp}(\frac{\text{sim}(\bm{f}_i, \bm{f}_j)}{\eta})\! + \!\frac{1}{N_i^{\text{bg}}}\!\sum_{k=1}^{N_i^{\text{bg}}}\!\text{exp}(\frac{\text{sim}(\bm{f}_i, \bm{f}_k)}{\eta})}\!), \\
\end{split}
\end{gathered}
\end{equation}
where features $\bm{f}_i$ and $\bm{f}_j$ belongs to the event set ($i \ne j$), $\bm{f}_i$ is the segment anchor, $\bm{f}_j$ is one of $\bm{f}_i$'s \emph{positive} samples, $\bm{f}_k$ denotes a \emph{negative} sample comes from the background set.
$N_i^e$ and $N_i^{\text{bg}}$ are the total numbers of the event and background segments, respectively.
$\text{sim}(\cdot, \cdot)$ computes the dot product of the $\ell_2$ normalized vectors (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, cosine similarity);
$\eta$ is a temperature parameter controlling the concentration level of feature distribution.
\textbf{\subsubsection{Video-level positive sample activation (PSA$_V$).}}
To make the model be category-aware, we design another online contrastive strategy from video-level. As shown in Fig.~\ref{fig:subset_ae}, there are some instrument-related events in dataset ({\em e.g.} guitar, violin, mandolin, banjo, ukulele), which are hard to distinguish since they are similar in vision and sound.
A main challenge for AVE localization is to correctly distinguish the event category.
Specifically, for each data batch during training process, we compute the Euclidean distance between the video samples.
Taking a video as an anchor, we select the \emph{positive} sample identified with the same category and the largest distance.
Similarly, we select \emph{negative} videos that have different event categories from the anchor and the top-$K$ closest distances,
where $K$ is a hyper-parameter controlling the number of \emph{negative} samples.
The model is expected to correctly recognize those similar but hard to learn samples: the positive sample should gather around the anchor video while the negative ones should be pushed farther.
To this end, the contrastive objective for video-level positive sample activation can be formulated as,
\begin{equation}\label{eq:loss_vcon}
\mathcal{L}_{\text{vpsa}} = \text{max}(0, d(\overline{\bm{f}}^a, \overline{\bm{f}}^p) - \frac{1}{K}\sum_{k=1}^{K}d(\overline{\bm{f}}^{a}, \overline{\bm{f}}_k^n) + \theta),
\end{equation}
where $d(\cdot, \cdot)$ computes the Euclidean distance between the $\ell_2$ normalized vectors, $\overline{\bm{f}}^a$ is the feature vector of an anchor video, obtained by averaging feature of video segments $\bm{f}$ (Eq.~\ref{eq:av_fusion}) along the temporal dimension, $\overline{\bm{f}}^p$ and $\overline{\bm{f}}^n$ are the features of positive and negative samples.
$K$ controls the number of the negative samples, $\theta$ denotes the minimum margin that the positive and negative samples should maintain.
\vspace{7mm}
\subsection{Classification}\label{sec:classification}
The fused audio-visual feature $\bm{f} \in \mathbb{R}^{T \times d_l}$
is send to the classifier for prediction. We detail the classifier and the objective function for the fully and weakly supervised settings below.
\subsubsection{ Classifier}\label{sec:classification}
For the fully supervised setting, as shown in Fig.~\ref{fig:system_flow}, the fused feature is further processed by two FC layers.
The classifier prediction $\bm{o}^{\text{fully}}\in \mathbb{R}^{T \times C}$ can be obtained through a softmax function.
For the weakly supervised setting, different from existing methods~\cite{lin2019dual, tian2018audio, xuan2020cross},
we add a weighting branch on the fully supervised classification module (Fig.~\ref{fig:system_flow}). It is essentially another FC layer that enables the model to further capture the differences between synchronized audio-visual pairs.
This process is summarized below,
\begin{equation
\left\{\begin{split}
& \bm{f}^h = {\bm{f}\bm{W}_4^{\text{weak}} \bm{W}_5^{\text{weak}}}, \\
& \bm{\phi} = \sigma(\bm{f}^h \bm{W}_6^{\text{weak}} ), \\
& \bm{o}^{\text{weak}} = s(f_{\text{avg}}(\bm{f}^h \odot \bm{\Phi})),
\end{split}\right.
\label{eq:o^weak}
\end{equation}
where $\bm{W}_4^{\text{weak}} \in \mathbb{R}^{d_l \times d_h}$, $\bm{W}_5^{\text{weak}} \in \mathbb{R}^{d_h \times C}$,
$\bm{W}_6^{\text{weak}} \in \mathbb{R}^{C\times 1} $ are learnable parameters in the FC layers, and $\bm{f}^h \in \mathbb{R}^{T \times C}$.
$\sigma$ and $s$ denote the sigmoid and softmax operators, respectively.
$\bm{\phi} \in \mathbb{R}^{T \times 1}$ weighs the importance of the temporal video segments, and $\bm{\Phi} \in \mathbb{R}^{T\times C}$ is obtained by duplicating $\bm{\phi}$ for
$C$ times. $\odot$ is the element-wise multiplication,
$f_{\text{avg}}$ is the average operation along the temporal dimension.
The final prediction $\bm{o}^{\text{weak}} \in \mathbb{R}^{1 \times C}$.
\subsubsection{Objective function}\label{sec:objective_function}
\textbf{Fully supervised setting.} Given the network output $\mathbf{o}^{\text{fully}}$ and ground truth $\mathbf{Y}^{\text{fully}}$, we adapt the cross entropy (CE) loss
as the basic objective function, written as,
\begin{equation}
\mathcal{L}_{\text{ce}} = -\frac {1}{TC} \sum_{t=1}^{T} \sum_{c=1}^{C} \bm{Y}_{tc}^{\text{fully}} {\log(\bm{O}_{tc}^{\text{fully}})}.
\label{eq:softmax}
\end{equation}
Recall that each row of $\bm{Y}^{\text{fully}}$ contains a one-hot event label vector, describing the category of each video segment (synchronized audio-visual pair).
As such, this classification loss allows the network to predict which \emph{event category} a video segment contains.
Apart from the CE loss, we propose a new loss item, named audio-visual pair similarity loss based on the PSP $\mathcal{L}_{\text{avpsp}}$. In principle, it asks the network to produce similar features for a pair of audio and visual components if the pair \emph{contains an event} (contrasting from background) during PSP.
Specifically, for a video composed of $T$ segments, we define label vector $\bm{G} = { \{g_t | g_t \in \{ 0, 1\}, t=1,2,...,T\} \in \mathbb{R}^{1 \times T}} $,
where $g_t$ represents whether the $t^{\text{th}}$ segment is an event or background. Next, $\ell_1$ normalization is performed on $\bm{G}$.
We then compute the $\ell_1$ normalized similarity vector $\bm{S} \in \mathbb{R}^{1 \times T}$ between the visual and audio features
\begin{equation}
\begin{split}
\bm{S} &= \frac {\bm{v}^{\text{psp}} \odot \bm{a}^{\text{psp}}} {\left \| \bm{v}^{\text{psp}} \odot \bm{a}^{\text{psp}} \right\|_1},
\end{split}
\label{eq:similarity}
\end{equation}
where $\|\cdot\|_1$ calculates the $\ell_1$ norm of a vector. The proposed loss $\mathcal{L}_{\text{avpsp}}$ is then written as,
\begin{equation}\label{eq:avps}
\begin{split}
\mathcal{L}_{\text{avpsp}} &= \mathcal{L}_{\text{MSE}}(\bm{S}, \bm{G}),
\end{split}
\end{equation}
where $\mathcal{L}_{\text{MSE}}(\cdot, \cdot)$ computes the mean squared error between two vectors.
Combining Eq. \ref{eq:avps} and Eq. \ref{eq:softmax}, the objective function for fully-supervised setting $\mathcal{L}_{\text{fully}}$ can be computed by:
\begin{equation}
\mathcal{L}_{\text{fully}} = \mathcal{L}_{\text{ce}}
+ \lambda_1{\mathcal{L}_{\text{avpsp}}}.
\label{eq:fully_loss}
\end{equation}
When refining the fused features with PSA$_S$ and PSA$_V$ jointly, the overall objective function $\mathcal{L}_{\text{fully}}^{r}$ can be computed by,
\begin{equation}
\mathcal{L}_{\text{fully}}^{r} =
\mathcal{L}_{\text{fully}} +
\lambda_2{\mathcal{L}_{\text{spsa}}} +
\lambda_3{\mathcal{L}_{\text{vpsa}}},
\label{eq:rf_loss}
\end{equation}
where $\lambda_1$, $\lambda_2$, $\lambda_3$ are hyper-parameters to balance the losses.
The PSA$_S$ and PSA$_V$ can also be added to the vanilla PSP separately and such manner is slightly superior than joint training, we will discuss these two strategies in Sec. 6.4.1.
\textbf{Weakly supervised setting.} For this setting,
following the practice in ~\cite{lin2019dual, xu2020MM}, we adapt the binary cross entropy (BCE) loss as the basic classification loss, formulated as,
\begin{equation}
\mathcal{L}_{\text{weak}} = \mathcal{L}_{\text{BCE}}(\bm{o}^{\text{weak}}, \bm{Y}^{\text{weak}}).
\label{eq:weak_bce_loss}
\end{equation}
It is worth mentioning that the segment-level event label should be known to distinguish the positive and negative segments in the video,
thus, PSA$_S$ is not applicable for weakly supervised AVE localization.
The PSA$_V$ can be used in both fully and weakly supervised settings since both of them provide the video-level event label. When combined with the loss item of PSA$_V$, the overall objective can be written as,
\begin{equation}
\mathcal{L}_{\text{weak}}^{r} = \mathcal{L}_{\text{weak}} + \lambda_4{\mathcal{L}_{\text{vpsa}}},
\label{eq:rw_loss}
\end{equation}
where $\lambda_4$ is a balance weight.
\section{Discussion}\label{sec:discussion}
\textbf{Detailed examination and meanings of $\bm{v}^{\text{pos}}$ and $\bm{a}^{\text{pos}}$.}
The computation of $\bm{v}^{\text{pos}}$ ($\bm{a}^{\text{pos}}$) is shown in Eq. \ref{eq:3}.
Take $\bm{v}^{\text{pos}}$ for example, the $i^{\text{th}}$ row $\bm{v}_{i}^{\text{pos}}$ is the weighted sum of the visual feature $\bm{v}_j^{\text{lstm}} (j=1,2,...,T)$ after linear transformation.
Here the weight, denoted as $\bm{\gamma}_{i}^{\text{av}}$, is exactly the similarity between the audio feature $\bm{a}_i$ and features of all the visual components.
Note that some elements of $\bm{\gamma}_{i}^{\text{av}}$ are zeros since the negative and weak connections are pruned during PSP, so $\bm{v}_i^{\text{pos}}$ is the aggregation result of those {\em positive} visual features which
are most relevant to $\bm{a}_i$.
\textbf{Physical meanings of $\bm{v}^{\text{psp}}$ and $\bm{a}^{\text{psp}}$.}
Take $\bm{a}^{\text{psp}}$ for example. From Eq. \ref{eq:3}, we find that $\bm{a}^{\text{psp}}$ is composed of two features: the original audio feature $\bm{a}^{\text{lstm}}$ and the aggregation of positive visual features $\bm{v}^{\text{pos}}$.
As discussed above, those positive visual features have large audio-visual similarity values, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, small vector angles and similar vector directions.
Therefore, after being added to $\bm{v}^{\text{pos}}$, the magnitude and direction of vectors representing original audio feature $\bm{a}^{\text{lstm}}$ will be changed to reflect that during training. Such an adjustment in the distribution of audio representation can be verified by the visualization results in Fig. \ref{fig:feature_distribution}.
\textbf{Why an additional FC layer in the weakly supervised setting?} When fully supervised, clear supervision is known for each segment.
For the weakly supervised setting, both the ground truth label $\bm{Y}^{\text{weak}} \in \mathbb{R}^{1 \times C}$ and the prediction $\bm{o}^{\text{weak}}\in \mathbb{R}^{1 \times C}$ are obtained through an average pooling operation along the temporal dimension.
Without knowing the supervision of each segment, the baseline approach considers all temporal video segments to have similar weights when calculating the loss.
It makes it harder for the model to focus on video segments that contain an event.
In our design, through the sigmoid activation function,
we obtain the weights of temporal video segments.
As such, our model can better distinguish these temporal sequences and thus help locate which segments contain an event.
\textbf{Implications of $\mathcal{L}_{\text{avpsp}}$.
}
As shown in Eq. \ref{eq:softmax}, the classification loss $\mathcal{L}_{\text{ce}}$ prompts the model to calculate the loss between the output probabilities and the ground truth label.
In comparison, $\mathcal{L}_{\text{avpsp}}$ allows the network to be aware of \emph{whether an event exists in an audio-visual pair} (pair-level contrasting).
Specifically, if $g_t$ is equal to $1$, the synchronized audio-visual feature should have a higher similarity, and otherwise lower.
Therefore, for an audio (visual) component, $\mathcal{L}_{\text{avpsp}}$ provides another auxiliary constraint so that the model can better select the most relevant visual (audio) components for feature aggregation during PSP.
Note that $\mathcal{L}_{\text{avpsp}}$ cannot be adapted to the weakly supervised setting, where the label $g_t$ of each segment is unknown. To summarize, $\mathcal{L}_{\text{ce}}$ and $\mathcal{L}_{\text{avpsp}}$ serve as strong supervisions, especially in the fully supervised setting.
\textbf{Implications of $\mathcal{L}_{\text{spsa}}$ and $\mathcal{L}_{\text{vpsa}}$}.
The design of $\mathcal{L}_{\text{spsa}}$ in Eq.~\ref{eq:loss_scon} and $\mathcal{L}_{\text{vpsa}}$ in Eq.~\ref{eq:loss_vcon} are intended to further advance the audio-visual representation learning. $\mathcal{L}_{\text{spsa}}$ allows the network to be aware of \emph{whether an event exists in a segment unit}, and $\mathcal{L}_{\text{vpsa}}$ allows the network to be aware of \emph{whether an exact event category exists in a video}.
Here, the network is enforced to learn the discriminative representation capability with the most relevant segments and the same category samples.
In fact, these two losses can be regarded as the soft supervisions since they are controlled by the hyper-parameters (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, $\eta$ for $\mathcal{L}_{\text{spsa}}$, $K$ and $\theta$ for $\mathcal{L}_{\text{vpsa}}$). The positive segment and video samples respectively activated by the PSA$_S$ ($\mathcal{L}_{\text{spsa}}$) and PSA$_V$ ($\mathcal{L}_{\text{vpsa}}$) are beneficial for the classifier training and this can be confirmed by the visualization examples shown in Figs.~\ref{fig:PSP_PSPCL_a},~\ref{fig:PSP_PSPCL_b}.
To summarize, $\mathcal{L}_{\text{avpsp}}$ (PSP) and $\mathcal{L}_{\text{spsa}}$ (PSA$_S$) exploit the positive clues of intra-video correlation (using pair and segment event labels), $\mathcal{L}_{\text{vpsa}}$ (PSA$_V$) focuses on the positive clues of inter-video correlation (using video category label). As the same reason for $\mathcal{L}_{\text{avpsp}}$, $\mathcal{L}_{\text{spsa}}$ is used for fully supervised setting while $\mathcal{L}_{\text{vpsa}}$ is not limited to this constraint.
\section{Experiment}
\label{sec:experiment}
\subsection{Experimental setup}\label{sec:expriment_setup}
\textbf{Dataset.}
\textbf{(1) AVE dataset~\cite{tian2018audio}}. Following the existing works \cite{lin2019dual, tian2018audio, xu2020MM, xuan2020cross}, we use the public AVE dataset for localization.
This dataset contains 4,143 videos,
which cover various real-life scenes and can be divided into 28 event categories, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, church bell, male speech, acoustic guitar, and dog barking.
Each video sample is evenly partitioned into 10 segments, and the duration of each segment is one-second.
The audio-visual event boundary on the segment level and the event category on the video level are provided. Keeping consistent with prior work, 3,339 videos are used for training, while both the validation and test set contains 402 videos.
\textbf{(2) VGGSound-AVEL100k dataset}. We construct a new large-scale VGGSound-AVEL100k datatset for AVEL task, in which the videos are sampled from VGGSound~\cite{chen2020vggsound}.
VGGSound-AVEL100k contains 101,072 videos that spans 141 audio-visual event categories covering more scenes in real-life that do not appear in the AVE dataset, such as motorboat, electric shaver, sharpen knife, \emph{etc}\onedot} \def\vs{\emph{vs}\onedot.
The ratio of train/validation/test split percentages are set as 60/20/20\footnote{The large-scale VGGSound-AVEL100k dataset for AVEL task is available at \href{https://drive.google.com/drive/folders/1en1dks1GYiGaDS9Ar-QtJmmyoOdzEsQj?usp=sharing}{\emph{https://drive.google.com/drive/folders/1en1dks1GYiGaDS9Ar-QtJmmyoOdzEsQj?usp=sharing}}. We give more details in Appendix.~\ref{sec:100k_details}
}.
\textbf{(3) LLP dataset~\cite{tian2020avvp}.} It is collected for a more challenging audio-visual video parsing (AVVP) task where only video-level labels are given. It contains 11,849 videos and each video in this dataset contains multiple audio and visual events. The AVVP task requires to predict what events happen in both audio and visual tracks, separately.
We extend the proposed CPSP in the weakly supervised setting to this task to evaluate the generalization ability of our method.
\textbf{Evaluation metric.} The category label of each segment is predicted in both fully and weakly
supervised settings. Following~\cite{lin2019dual, tian2018audio,xu2020MM,xuan2020cross}, we adopt
the classification accuracy of each segment as the evaluation metric.
\textbf{Training procedure and configuration.}
We have to deal with all the video data of different types (as shown in Fig.~\ref{fig:contrastive_strategy}). For convenience, we use $\mathcal{D}_{\text{bg}}$ to denote this type of video that contains both AVE and background segments (Fig.~\ref{fig:subset_bg}) and use $\mathcal{D}_{\text{ae}}$ to represent another type of video contains pure AVE segments belonging to a certain event category (Fig.~\ref{fig:subset_ae}). We initialize the proposed localization system (Fig.~\ref{fig:system_flow}) with the objective function $\mathcal{L}_{\text{fully}}$ (Eq.~\ref{eq:fully_loss}) and $\mathcal{L}_{\text{weak}}$ (Eq.~\ref{eq:weak_bce_loss}) on the dataset benchmark ($\mathcal{D}_{\text{bg}}$ \& $\mathcal{D}_{\text{ae}}$). Then we further refine the audio-visual features with PSA$_S$ on subset $\mathcal{D}_{\text{bg}}$
and PSA$_V$ on subset $\mathcal{D}_{\text{ae}}$ in Eqs.~\ref{eq:loss_scon} and \ref{eq:loss_vcon}, respectively; their corresponding usages (objective functions) for fully and weakly supervised settings are introduced in Eqs.~\ref{eq:rf_loss} and \ref{eq:rw_loss}.
More related details are discussed in Sec.~\ref{sec:evaluation_PSA}. We abbreviate the initialized network as {\textbf {PSP}}, and the refined network as {\textbf {CPSP}} in the following experiment evaluation.
The parameters are tuned on the validation set with the final model is tested on a held-out test set. The results are reported in Sec.~\ref{sec:quanly} and ~\ref{sec:quanti2}.
\textbf{Implementation details.}
(1) \emph{Visual feature extractor.}
For fair comparison, we use the VGG-19~\cite{simonyan2014very} pretrained on ImageNet~\cite{krizhevsky2017imagenet} to extract the visual features.
Specifically, 16 frames are sampled from each one-second video segment.
We extract the visual feature map for each frame from the \emph{pool-5} layer in VGG-19 with the size of $7\times7\times512 $ and then use the average map as
the visual feature for this segment.
(2) \emph{Audio feature extractor.}
For audio features, we first process the raw audio into log-mel spectrograms and then use the VGGish, a VGG-like network~\cite{hershey2017cnn}
pretrained on AudioSet~\cite{gemmeke2017audio}, to extract the acoustic feature with the dimension of 128.
(3) \emph{Hyper-parameter settings.}
The temperature parameter $\eta$ in Eq.~\ref{eq:loss_scon} is set to 0.1.
Impacts of the number of negative samples $K$ and the margin $\theta$ in Eq.~\ref{eq:loss_vcon} are discussed in Sec.~\ref{sec:evaluation_PSA}.
The weights of $\lambda_1$ in Eq. \ref{eq:fully_loss} and $\lambda_2$, $\lambda_3$ in Eq.~\ref{eq:rf_loss} is empirically set to 100, 0.01, 1, respectively.
$\lambda_4$ in Eq.~\ref{eq:rw_loss} is set to 0.005.
These hyper-parameters remain the same on the AVE and the VGGSound-AVEL100k datasets in our experiments. In addition, the batch size in our experiments is set to 128. We use Adam~\cite{kingma2014Adam} as the optimizer, and dropout technique is used in all the linear layers (Fig.~\ref{fig:system_flow}) with the drop rate set to 0.1.
As for the experiments on LLP dataset for video parsing, the batch size is set to 16 same as baseline method HAN~\cite{tian2020avvp}, more implementation details are introduced in Sec.~\ref{sec:avvp}.
\begin{table}[t]
\caption{Comparison with the state-of-the-art methods under both the fully and weakly supervised settings. We report the accuracy(\%) measured on the AVE and the VGGSound-AVEL100k datasets. * indicates the number is reproduced by us.}
\begin{center}
\setlength{\tabcolsep}{3.5mm}
\begin{tabular}{l|c|c|c|c}
\toprule[0.8pt]
\multirow{2}{*}{Method} & \multicolumn{2}{c|}{AVE} & \multicolumn{2}{c}{VGGSound-AVEL100k} \\ \cline{2-5}
& fully & weakly & fully & weakly \\ \hline
AVEL~\cite{tian2018audio} & 68.6 & 66.7 & 55.7* & 46.2* \\
AVSDN~\cite{lin2019dual} & 72.6* & 67.3* & - & - \\
CMAN~\cite{xuan2020cross} & 73.3* & 70.4* & - & - \\
DAM~\cite{wu2019dual} & 74.5 & - & - & - \\
AVRB~\cite{ramaswamy2020see} & 74.8 & 68.9 & - & -\\
AVIN~\cite{ramaswamy2020makes} & 75.2 & 69.4 & - & - \\
RFJCA~\cite{Duan_2021_WACV} & 76.2 & - & - & - \\
AVT~\cite{Lin_2020_ACCV} & 76.8 & 70.2 & - & - \\
CMRA~\cite{xu2020MM} & 77.4 & 72.9 & 57.1* & 46.8*\\
MPN~\cite{yu2021mpn} & 77.6 & 72.0 & - & -\\ \hline
PSP~\cite{zhou2021positive}(Ours) & 77.8 & 73.5 & 58.3 & 47.4 \\
CPSP(Ours) & {\bf 78.6} & {\bf 74.2} & \textbf{59.9} & \textbf{48.4} \\ \bottomrule[0.8pt]
\end{tabular}
\end{center}
\label{table_4}
\end{table}
\subsection{Comparison with state of the arts}
We compare our method with the state of the arts in Table~\ref{table_4} by evaluating on the AVE and the VGGSound-AVEL100k datasets. Taking the results on the AVE dataset for example, compared with the baseline method AVEL~\cite{tian2018audio}, the CPSP exceeds it by 10.0\% and 7.5\% under the fully and weakly supervised settings, respectively. Such superiority is also proved by the results shown in Table \ref{table_1}. Also, our method exceeds those SOTAs~\cite{xu2020MM, xuan2020cross, ramaswamy2020makes, ramaswamy2020see} that focus on the cross-modal feature fusion using all of the audio-visual pairs. This indicates the necessity of the positive pair selection in PSP.
CMRA~\cite{xu2020MM} has comparable performance with the PSP method, but the CPSP is superior than CMRA on both datasets in both settings. Also, the CPSP exceeds the PSP method by a large margin.
This again demonstrates the effectiveness of the PSA performing additional contrastive learning from both segment-level and video-level.
Such advantages of the proposed CPSP method can also be observed on the large-scale VGGSound-AVEL100k dataset.
For example, the CPSP exceeds the competitive CMRA and vanilla PSP by a large margin.
We also notice that all the methods have a performance drop on the large-scale VGGSound-AVEL100k compared to the AVE dataset, we speculate VGGSound-AVEL100k contains much more videos with more event categories that makes the problem more challenging.
Nevertheless, our method is more superior and robust under all the settings, which can be attributed to our system design.
\begin{figure*}[t]
\begin{center}
\includegraphics[width=\textwidth]{./figures/fig_distance_between_event_and_bg_segments.pdf}
\end{center}
\vspace{-0.6cm}
\caption{Euclidean distances between the centroids of event and the background segments for each event category in the fully supervised setting. We respectively evaluate the segment features learned by the PSP and CPSP (merely w. PSA$_S$)
in fully supervised setting.
Larger distance of the CPSP demonstrates the benefit of PSA$_S$ helping to encode features of event and background that are easier to distinguish.
This experiment is conducted on the AVE dataset.
}
\label{fig:distance_between_event_and_bgs}
\vspace{-3mm}
\end{figure*}
\subsection{Quantitative analysis - main modules}
\label{sec:quanly}
Here we test the effects of the PSP and PSA modules. Ablation experiments are mainly conducted on the AVE dataset.
\subsubsection{Evaluation of the proposed PSP module}\label{sec:evaluation_psp}
\textbf{The effectiveness of the PSP encoding} can be verified through an ablation study in Table~\ref{table_2}. In Table \ref{table_2}, we denote the method without PSP, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, removing it from the localization network (Fig.~\ref{fig:system_flow}), as ``w/o PSP''.
We observe from the table that
the performance on the AVE dataset drops in both the fully supervised and weakly supervised settings significantly. Specifically, the accuracy decrease is 4.1\% (from 77.8\% to 73.7\%) and 3.3\% (from 73.5\% to 70.2\%) for the two settings, respectively. This experiment clearly validates the PSP.
\textbf{Comparison with alternative pair-level positive sample selection methods.} In our method, we emphasize that weak and negative samples are filtered out. Here, we compare this strategy with two variants: (1) all connections are used (denoted as ``ASP''); (2) only negative ones are removed, while weak connections are remained (denoted as ``WPSP''). Results are shown in Table \ref{table_2}. We have two main observations.
First, when all samples are propagated, the accuracy of ``ASP'' drops by 1.9\% and 2.3\% on the fully and weakly supervised settings, respectively. This shows that it is essential to have a selection process before feature aggregation instead of utilizing all the connections.
Second, although we merely remove the negative connections (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, with a similarity value below $\tau = 0$), the system of ``WPSP'' is inferior to the full method. Specifically, the classification accuracy decreases by 1.8\% and 2.3\% under the fully and weakly supervised settings, which validates the effectiveness of filtering out the negative connections.
\begin{table}[t]
\caption{Ablation studies of the proposed PSP, measured by accuracy(\%) on the AVE dataset. ``w/o'' denotes ``without''. ``ASP'' means retaining all connections ($\tau = -\infty$), while ``WPSP'' uses the weak and positive ones ($\tau = 0$). ``SAPSP'' represents adding self-attention to the feature extractor.}
\setlength{\tabcolsep}{4mm}
\begin{center}
\begin{tabular}{l|c|c}
\toprule[0.8pt]
Method & Fully-supervised & Weakly-supervised \\
\hline
w/o PSP & 73.7 & 70.2 \\
ASP & 75.9 & 71.2\\
WPSP & 76.0 & 71.2 \\
SAPSP & 75.4 & 70.8 \\\hline
PSP (ours) & {\bf 77.8} & {\bf 73.5} \\
\bottomrule[0.8pt]
\end{tabular}
\end{center}
\label{table_2}
\vspace{-3mm}
\end{table}
\begin{table}[t]
\caption{Impact of various values of $\tau$ on the system accuracy evaluated on the AVE dataset. Results on the two settings are shown.}
\setlength{\tabcolsep}{1.95mm}
\small
\begin{center}
\begin{tabular}{l|c|c|c|c|c}
\toprule[0.8pt]
$\tau$ & 0 & 0.025 & 0.075 & 0.095 & 0.115 \\
\hline
Fully-supervised & 76.0 & 76.1 & 75.3 & {\bf 77.8} & 76.6 \\
Weakly-supervised & 71.2 & 71.7 & 70.4 & {\bf 73.5} & 72.8 \\
\bottomrule[0.8pt]
\end{tabular}
\end{center}
\label{table_3}
\vspace{-5mm}
\end{table}
\begin{figure*}[htbp]
\begin{center}
\includegraphics[width=\textwidth]{./figures/fig_acc_for_ae_video.pdf}
\end{center}
\vspace{-0.6cm}
\caption{Classification accuracy for videos containing no background segments in the fully supervised setting.
Compared with the PSP, the CPSP (merely w. PSA$_V$)
has overall superior performance in predicting the correct event categories.
This experiment is conducted on the AVE dataset.
}
\label{fig:acc_for_ae_videos}
\vspace{-0.5cm}
\end{figure*}
\textbf{Sensitivity to hyper-parameter $\tau$.} The selection process is controlled by $\tau$, determining how many connections will be cut off.
Its influence on the system accuracy is shown in Table~\ref{table_3}. We observe that the accuracy generally remains stable when $\tau$ varies between 0 and 0.115 and that the highest accuracy is achieved when $\tau=0.095$.
For different videos, the proportion of segments that are cut off highly depends on the video itself. If the whole video contains the same event of interest, it is likely that most will be retained in training; if a video contains lots of background, the same threshold will cut off more of its content.
Such a connection pruning (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, positive pair selection) process
in PSP can be clearly observed from the visualization example in Fig.~\ref{fig:localization_example}.
\textbf{Comparison with adding self-attention \cite{vaswani2017attention} to the feature extractor.}
Self-attention~\cite{vaswani2017attention} is widely used in existing methods
~\cite{tian2020avvp, wu2019dual, xu2020MM, xuan2020cross} to capture relationships within single modality.
To explore whether it is useful in our system, we add a self-attention module before the Bi-LSTMs in the feature extractor module and denote it as the ``SAPSP'' method.
As shown in Table \ref{table_2}, the performance surprisingly decreases by
2.4\% and 2.7\%
under fully and weakly supervised settings, respectively.
This indicates that in our system, it is not required to add additional intra-modal verification through self-attention before the PSP module.
We speculate that the PSP is sufficient to describe the cross-modality while implicitly reveals the intra-modality correlations.
\vspace{3mm}
\subsubsection{Evaluation of the proposed PSA module}\label{sec:evaluation_PSA}
\textbf{Effectiveness of the PSA$_S$.} PSA$_S$ is expected to constraint the model to learn consistent features for the video segments containing the same event, while possibly be distinguishable from the background segments. We reflect its effect by the distance of centroids of the event and background segments in feature space.
For videos in the dataset, we encode the segment features by the PSP and CPSP (merely equipped with PSA$_S$ for fair comparison), respectively.
Specifically, for each same event category, we filter out and average the features of event and background segments respectively; we take the two obtained vectors as event and background centroids. We calculate their Euclidean distance. Results on the AVE dataset are presented in Fig.~\ref{fig:distance_between_event_and_bgs}.
We can see that the distances between event and background segments are increased in most of the categories (24 out of 28) using the CPSP method. For example, for the event of \emph{female}, \emph{guitar}, and \emph{bus}, the distances increase by around 33\%.
This verifies the benefit of the PSA$_S$ that activates the event segments such that they can be better recognized from the backgrounds.
\textbf{Effectiveness of the PSA$_V$.}
As introduced in Sec.~\ref{sec:contrastive_learning},
PSA$_V$ aims to distinguish the positive video from the top-$K$ closest but negative samples, and the Euclidean distance between the video-level representations is expected to be no less than the margin $\theta$ (Eq.~\ref{eq:loss_vcon}). Here, we test CPSP merely with
PSA$_V$ for fair comparison.
We conduct a study on the AVE dataset to explore the impacts of parameters $K$ and $\theta$ in PSA$_V$.
First, we empirically fix the number of negatives $K$ to 4 and sample $\theta$ from \{0.2, 0.4, 0.6\}.
As shown in Table~\ref{table:parameter_VLPSR}, the performance is gradually improved as $\theta$ increases. And the best accuracy is achieved when $\theta=0.6$ for both settings (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, $78.31\%$ for fully supervised, $74.20\%$ for weakly supervised).
We speculate that this is a relatively large margin to better distinguish the positive and negative samples.
Next, we fix $\theta$ to 0.6 and test $K$ with values \{1, 2, 4, 6\}.
As observed from the table, $K=4$ is the optimal setup. This means four negative video samples are selected from a batch of data during training to compare with the positive one.
In this way, the model enables to simultaneously compare videos of multiple event categories at once in an online fashion. We set $K=4$ and $\theta=0.6$ for all of our other experiments on both the AVE and the VGGSound-AVEL100k dataset when conducting PSA$_V$.
It is worthy to note that the performances of almost all the setups in the CPSP exceed the results of the PSP (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, obtained from the case without PSA$_V$, $77.8\%$ and $73.5\%$ accuracy for fully and weakly supervised settings, respectively).
\begin{table}[t]
\begin{center}
\caption{Parameter study of the $K$ and $\theta$ in PSA$_V$. We report the performance of the CPSP under different PSA$_V$ setups in both fully and weakly supervised settings.
Experiments are conducted on the AVE dataset. The \textbf{bold-faced} results represent the optimal performance is achieved under that setup.}
\begin{tabular}{l|l|c|c}
\toprule[0.8pt]
\multicolumn{2}{l|}{Parameter setup} & Fully-supervised & Weakly-supervised \\ \hline
\multirow{3}{*}{$K=4$} & $\theta=0.2$ & 77.61 & 74.10 \\
& $\theta=0.4$ & 78.10 & 74.18 \\
& $\theta=0.6$ & \textbf{78.31} & \textbf{74.20} \\ \hline
\multirow{4}{*}{$\theta=0.6$} & $K=1$ & 78.20 & 74.15 \\
& $K=2$ & 78.13 & 74.15 \\
& $K=4$ & \textbf{78.31} & \textbf{74.20} \\
& $K=6$ & 78.23 & 74.10 \\
\bottomrule[0.9pt]
\end{tabular}
\label{table:parameter_VLPSR}
\end{center}
\vspace{-5mm}
\end{table}
To clarify the effect of PSA$_V$ more clearly, here we report the classification accuracy for each event category under the fully supervised setting on subset $\mathcal{D}_{\text{ae}}$ of AVE dataset, where videos in $\mathcal{D}_{\text{ae}}$ contain no background segments (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot. having definite video-level category).
The results are shown in Fig.~\ref{fig:acc_for_ae_videos}, the CPSP (merely equipped with PSA$_V$ here) has better performance in most event categories (26 out of 28).
This verifies that PSA$_V$ can further help to predict the accurate event category, which is contributed to the video-level contrastive learning that makes the learned features more distinguishable for videos owing to different categories.
\begin{table*}[t]
\caption{
Results on the AVE and VGGSound-AVEL100k datasets are reported.
We list details of the experimental configurations, \emph{i.e.}, the objective function (Objective), the type of video data used in the objective optimization (Data), the initialized model
(Init.), the trained model (Return), and the learning rate (Lr).}
\vspace{-5mm}
\begin{center}
\resizebox{\textwidth}{!}{
\begin{tabular}{llllllllllc}
\toprule[0.8pt]
\multicolumn{11}{c}{Fully-supervised setting} \\ \cline{1-11}
\multirow{2}{*}{Method} & \multicolumn{4}{c}{\multirow{1}{*}{Objective}} &\multicolumn{1}{l}{\multirow{2}{*}{Data}} & \multicolumn{1}{l}{\multirow{2}{*}{Init.}} & \multicolumn{1}{l}{\multirow{2}{*}{Return}}& \multicolumn{1}{l}{\multirow{2}{*}{Lr}} & \multicolumn{2}{c}{Accuracy} \\ \cline{2-5}
\cline{10-11}
& \multicolumn{2}{c}{$\mathcal{L}_{\text{ce}}$\&$\mathcal{L}_{\text{avpsp}}$} &$\mathcal{L}_{\text{spsa}}$ & $\mathcal{L}_{\text{vpsa}}$ & \multicolumn{1}{l}{} & \multicolumn{1}{l}{} & \multicolumn{1}{l}{} & \multicolumn{1}{l}{} & AVE & VGGSound-AVEL100k \\ \hline
PSP & \multicolumn{2}{c}{\checkmark} & & & $\mathcal{D}_{\text{bg}}$\& $\mathcal{D}_{\text{ae}}$ & Xavier~\cite{glorot2010xavier} & $\mathcal{M}_{\text{fully}}^{\emph{\text{p}}}$ & $10^{-3}$ & 77.8 & 58.3 \\ \hline
CPSP$_S$ & \multicolumn{2}{c}{\checkmark} &\checkmark & & $\mathcal{D}_{\text{bg}}$ & $\mathcal{M}_{\text{fully}}^\emph{\text{p}}$ & $\mathcal{M}_{\text{fully}}^\emph{\text{sp}}$ & $10^{-4}$ & 78.2 & 59.6 \\
CPSP$_V$ & \multicolumn{2}{c}{\checkmark} &\multirow{1}{*}{} &\multirow{1}{*}{\checkmark} & \multirow{1}{*}{$\mathcal{D}_{\text{ae}}$} & $\mathcal{M}_{\text{fully}}^\emph{\text{p}}$ & $\mathcal{M}_{\text{fully}}^\emph{\text{vp}}$ & \multirow{1}{*}{$10^{-5}$} & 78.3 & 59.8\\ \hline
CPSP(join) & \multicolumn{2}{c}{\checkmark} &\multirow{1}{*}{\checkmark} &\multirow{1}{*}{\checkmark} &$\mathcal{D}_{\text{bg}}$\& $\mathcal{D}_{\text{ae}}$ & $\mathcal{M}_{\text{fully}}^\emph{\text{p}}$ & $\mathcal{M}_{\text{fully}}^\emph{\text{vsp}}$ & \multirow{1}{*}{$10^{-5}$} & 78.4 & 59.8 \\
CPSP(sepa) & \multicolumn{2}{c}{\checkmark} &\multirow{1}{*}{\checkmark} &\multirow{1}{*}{\checkmark} & \multirow{1}{*}{$\mathcal{D}_{\text{bg}} \rightarrow\mathcal{D}_{\text{ae}}$} & $\mathcal{M}_{\text{fully}}^\emph{\text{sp}}$ & $\mathcal{M}_{\text{fully}}^\emph{\text{vsp}}$ & \multirow{1}{*}{$10^{-5}$} & \textbf{78.6} & \textbf{59.9} \\ \hline
\hline
\multicolumn{11}{c}{Weakly-supervised setting} \\
\cline{1-11}
\multirow{2}{*}{Method} & \multicolumn{4}{c}{\multirow{1}{*}{Objective}} &\multicolumn{1}{l}{\multirow{2}{*}{Data}} & \multicolumn{1}{l}{\multirow{2}{*}{Init.}} & \multicolumn{1}{l}{\multirow{2}{*}{Return}}& \multicolumn{1}{l}{\multirow{2}{*}{Lr}} & \multicolumn{2}{c}{Accuracy} \\ \cline{2-5}
\cline{10-11}
& \multicolumn{2}{c}{$\mathcal{L}_{\text{bce}}$} & \multicolumn{2}{c}{$\mathcal{L}_{\text{vpsa}}$} & \multicolumn{1}{l}{} & \multicolumn{1}{l}{} & \multicolumn{1}{l}{} & \multicolumn{1}{l}{} & AVE & VGGSound-AVEL100k \\ \hline
PSP & \multicolumn{2}{c}{\checkmark} & \multicolumn{2}{c}{} & $\mathcal{D}_{\text{bg}}$ \& $\mathcal{D}_{\text{ae}}$ & Xavier~\cite{glorot2010xavier} & $\mathcal{M}_{\text{weak}}^\emph{\text{p}}$ & $10^{-3}$ & 73.5 & 47.4 \\
CPSP & \multicolumn{2}{c}{\checkmark} & \multicolumn{2}{c}{\checkmark} & $\mathcal{D}_{\text{ae}}$ & $\mathcal{M}_{\text{weak}}^\emph{\text{p}}$ & $\mathcal{M}_{\text{weak}}^\emph{\text{vp}}$ & $10^{-5}$ &\textbf{74.2} & \textbf{48.4}\\ \bottomrule[0.8pt]
\end{tabular}}
\end{center}\label{table:training_strategy}
\vspace{-5mm}
\end{table*}
\begin{table}[t]
\caption{Method comparison on the AVE dataset under two settings. We evaluate 1) loss $\mathcal{L}_{\text{avpsp}}$ under the fully supervised setting, and 2) the weighting branch under the weakly supervised setting. The two improvements are implemented on top of our system and AVEL \cite{tian2018audio}. Under AVEL, * denotes that the number is produced by us. We use \textbf{bold} font to show the higher performance brought by our technique.}
\setlength{\tabcolsep}{3mm}
\begin{center}
\begin{tabular}{l|l|c|c}
\toprule[0.8pt]
\multicolumn{1}{l|}{Setting} & Method & PSP~\cite{zhou2021positive}(ours) & AVEL~\cite{tian2018audio} \\ \hline
\multirow{2}{*}{fully} & $\mathcal{L}_{\text{ce}}$ & 76.6 & 69.8* \\
& $\mathcal{L}_{\text{ce}} + \lambda_1\mathcal{L}_{\text{avpsp}}$ & {\bf 77.8} & \bf 71.3* \\ \hline
\multirow{2}{*}{weakly} & w/o weight. branch & 71.6 & 66.9* \\
& w/ weight. branch & {\bf 73.5} & \bf 69.2* \\ \bottomrule[0.8pt]
\end{tabular}
\end{center}
\label{table_1}
\vspace{-3mm}
\end{table}
\subsubsection{Evaluation of the improvement in fully/weakly setting}\label{sec:evaluation_improvements}
\textbf{Effectiveness of the pair similarity loss $\mathcal{L}_{\text{avpsp}}$ in fully supervised setting.}
We respectively adapt $\mathcal{L}_{\text{ce}}$
and $\mathcal{L}_{\text{ce}}+\lambda_1\mathcal{L}_{\text{avpsp}}$ as the objective function and test them for model training. Two baselines are used: our PSP system and the AVEL system \cite{tian2018audio}. Results are presented in Table \ref{table_1}.
We can clearly see that $\mathcal{L}_{\text{avpsp}}$ improves the accuracy when the system is fully supervised. The improvement is 1.2\% and 1.5\% for PSP and AVEL, respectively. These results confirm the role of $\mathcal{L}_{\text{avpsp}}$ as an auxiliary restriction to help to select the positive audio-visual pairs for feature aggregation.
\textbf{Improvement from the additional FC in the weakly supervised setting.}
In the weakly supervised setting, the major difference between our classification module and traditional methods~\cite{lin2019dual, tian2018audio, xuan2020cross} consists in the weighting branch (Fig. \ref{fig:system_flow}).
To evaluate its effectiveness, we also implement this branch on top of the PSP and AVEL baselines.
The results are shown in the last two rows of Table \ref{table_1}. We find that the performance
of PSP and AVEL is improved by
1.9\% and 2.3\%,
respectively.
We argue that the additional weighting branch within the designed classification module allows the model to give different weights to the temporal sequences, thus benefiting the localization of the target video segments.
These results confirm the effectiveness of the proposed improvements and show their robustness in other localization network. We refer readers to Sec. \ref{sec:discussion} for methodological discussions on the this technique.
\subsection{Quantitative analysis - contrastive manner in CPSP}\label{sec:quanti2}
In this subsection, we first compare the PSP (without contrastive learning) and the CPSP (with contrastive learning), and then discuss the supervised CPSP and self-supervised PSP (named SSPSP).
\subsubsection{Comparison of the PSP and CPSP}
To reveal the impact of each contrastive loss, we list five training modes in Table~\ref{table:training_strategy}:
\textcircled{1} the vanilla PSP~\cite{zhou2021positive},
\textcircled{2} the CPSP with sole PSA$_S$ (denoted as {\bf CPSP$_S$}),
\textcircled{3} the CPSP with sole PSA$_V$ (denoted as {\bf CPSP$_V$}),
\textcircled{4} the CPSP with both PSA$_S$ and PSA$_V$ by jointly training (denoted as {\bf CPSP(join)}), and
\textcircled{5} the CPSP with both PSA$_S$ and PSA$_V$ by separately training (first PSA$_S$ then PSA$_V$, denoted as {\bf CPSP(sepa)}).
As shown in Table~\ref{table:training_strategy}: (1) Compared with the vanilla PSP, the performances are improved by the CPSP with PSA$_S$ and PSA$_V$ in both fully and weakly supervised settings for both datasets. Take the VGGSound-AVEL100k dataset for example, after utilizing the PSA$_S$ or PSA$_V$ separately for the fully supervised setting, the accuracy of \textbf{CPSP$_S$} and \textbf{CPSP$_V$} increase by $1.3\%$ and $1.5\%$ (from $58.3\%$ to $59.6\%$ and $59.8\%$), respectively;
(2) When jointing the PSA$_S$ and PSA$_V$ together, the performances keep stable under both combination strategies, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, \textbf{CPSP(join)} and \textbf{CPSP(sepa)}.
The results of \textbf{CPSP(join)} and \textbf{CPSP(sepa)} are comparable and \textbf{CPSP(join)} performs slightly worse. We argue that it may be a little confusing for \textbf{CPSP(join)} to train with the PSA$_S$ and PSA$_V$ simultaneously (totally different contrastive goals).
With \textbf{CPSP(sepa)}, the best accuracy can achieve $78.6\%$ and $59.9\%$ for the AVE and the VGGSound-AVEL100k datasets, respectively.
The CPSP is flexible to be utilized: with single independent PSA$_S$ or PSA$_V$ module, or with both modules under either training setup.
Anyway, these benefit from the proposed PSA by activating the model
to distinguish (1) positive and negative segments in PSA$_S$,
and (2) event categories in PSA$_V$, the model gains a more robust capability to correctly identify the event location and its categories.
This is consistent with the goal of AVE localization thus can promise better results, which can also be verified by the qualitative examples as shown in Figs.~\ref{fig:PSP_PSPCL_a},~\ref{fig:PSP_PSPCL_b}.
\begin{table}[t]
\caption{Quality analysis of the encoded features of videos in the AVE dataset. We measure three clustering metrics (SC, CH, DBI) in two ways: ``Mean" denotes evaluating two clusters (event and background) in each event category, while ``All" refers to $C-1$ clusters, where $C-1$ is the number of all the event categories except background.
}
\begin{center}
\setlength{\tabcolsep}{2.5mm}
\begin{tabular}{l|c|c|c|c|c|c}
\toprule[0.8pt]
\multicolumn{1}{c|}{\multirow{2}{*}{Method}} & \multicolumn{2}{c|}{SC $\uparrow$} & \multicolumn{2}{c|}{CH $\uparrow$} & \multicolumn{2}{c}{DBI $\downarrow$} \\ \cline{2-7}
\multicolumn{1}{c|}{} & Mean & All & Mean & All & Mean & All \\ \hline
PSP & 0.17 & 0.19 & 16.64 & 194.12 & 1.62 & 2.07\\
CPSP & {\bf 0.21} & {\bf 0.22} & {\bf 18.29} & {\bf 195.51} & {\bf 1.54} & {\bf 1.95} \\ \bottomrule[0.8pt]
\end{tabular}
\end{center}
\label{table_sc_ch_dbi}
\vspace{-3mm}
\end{table}
\begin{table*}[t]
\caption{Comparison with the baseline methods on the test set of LLP dataset.
$\dag$ denotes the results reported in the paper HAN~\cite{tian2020avvp}.}
\vspace{-3mm}
\begin{center}
\begin{threeparttable}
\begin{tabular}{lp{0.6cm}<{\centering}p{0.6cm}<{\centering}p{0.6cm}<{\centering}p{1.4cm}<{\centering}p{1.5cm}<{\centering}p{0.6cm}<{\centering}p{0.6cm}<{\centering}p{0.6cm}<{\centering}p{1.4cm}<{\centering}p{1.5cm}<{\centering}}
\toprule[0.8pt]
\multirow{3}{*}{Method} & \multicolumn{5}{c}{Segment-level} & \multicolumn{5}{c}{Event-level}\\ \cmidrule(r){2-6} \cmidrule(r){7-11}
& A & V & AV & Type@AV & Event@AV & A & V & AV & Type@AV & Event@AV\\ \midrule
AVEL~\cite{tian2018audio} $\dag$ & 47.2 & 37.1 & 35.4 & 39.9 & 41.6 & 40.4 & 34.7 & 31.6 & 35.5 & 36.5 \\
AVSDN~\cite{lin2019dual} $\dag$ & 47.8 & 52.0 & 37.1 & 45.7 & 50.8 & 34.1 & 46.3 & 26.5 & 35.6 & 37.7 \\
HAN~\cite{tian2020avvp} & \textbf{60.1} & 52.9 & 48.9 & 54.0 & 55.4 & 51.3 & 48.9 & 43.0 & 47.7 & 48.0 \\ \midrule
PSP~\cite{zhou2021positive} & 54.2 & 54.7 & 48.3 & 52.4 & 52.5 & 46.8 & 50.2 & 42.8 & 46.6 & 45.6 \\
CPSP (Ours) & 58.5 & \textbf{57.8} & \textbf{52.6} & \textbf{56.3} & \textbf{55.8} & \textbf{51.6} & \textbf{54.0} & \textbf{46.5} & \textbf{50.7} & \textbf{49.9} \\
\bottomrule[0.8pt]
\end{tabular
\end{threeparttable}
\end{center}
\label{table:CPSP_on_LLP}
\vspace{-4mm}
\end{table*}
Moreover, we introduce three widely-used clustering metrics, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, Silhouette Coefficient (SC)~\cite{ROUSSEEUW198753}, Calinski-Harabasz Index (CH)~\cite{calinski1974}, and Davies-Bouldin Index (DBI)~\cite{DaviesB79}. These metrics validate the data clustering quality from the intra-class aggregation and inter-class separation.
Here, we use them to evaluate the event aggregation and background separation of features learned by the PSP and CPSP.
At first, we have a brief introduction:
(1) SC~\cite{ROUSSEEUW198753} calculates the difference of intra-class and inter-class dissimilarities and divides it by the maximum value of these two dissimilarities;
larger score denotes that clusters are dense while well separated;
(2) CH~\cite{calinski1974} is the ratio of the covariance of the intra-class data to the covariance of the inter-class data; higher score means better performance;
(3) DBI~\cite{DaviesB79} represents the average similarity between clusters; a lower value indicates better separation between clusters.
Next, we measure these metrics (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, SC, CH, and DBI) in two ways: as shown in Table~\ref{table_sc_ch_dbi}, the ``Mean'' denotes that we first split the data into event and background segments (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, 2 clusters, binary separation) in each event category, and then average the metrics over all the categories; the ``All'' refers to $C-1$ clusters, including all the different event categories except background (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, multi-class event separation).
In other words, we adopt ``Mean'' to measure the clustering effect with the partition of event and background, while ``All'' with the partition of all the event categories.
At last, experiment is performed on the AVE dataset.
As observed from Table~\ref{table_sc_ch_dbi}, for all of the metrics, CPSP is better scored than PSP in any measurement method. This demonstrates that the positive features learned through CPSP are better clustered thus making it easier to distinguish the event segments from backgrounds, and also performing better for classifying videos with different categories.
\subsubsection{Comparison of the CPSP and SSPSP}
The contrastive learning is always conducted in a self-supervised manner in the audio-visual field \cite{afouras20ssl, ma2021contrastive, wu2021exploring}. Specifically, the synchronized audio-visual segment pair is regarded as a positive sample and otherwise is negative. There is a drawback that this manner will inevitably bring false negatives of the audio-visual pair depicting the same event (existing AVC) but at different timestamp.
We are curious about the effect of using such self-supervised manner in AVE localization task.
So we introduce the self-supervised learning into our positive sample propagation, and denote it as ``SSPSP'' method.
In ``SSPSP'', all unsynchronized features (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, audio feature $\bm{a}^{\text{psp}}$ and visual feature $\bm{v}^{\text{psp}}$) are treated as negative instances sampling from a batch of data during training.
The corresponding contrastive objective can be written as $\mathcal{L}_{\text{ss}}$ below.
We first train a vanilla PSP with $\mathcal{L}_{\text{fully}}$ (Eq.~\ref{eq:fully_loss}), and then inject the self-supervised learning to the PSP. The total objective function can be computed by
\begin{equation
\left\{
\begin{gathered}
\begin{split}
& \mathcal{L}_{\text{sspsp}} =
\mathcal{L}_{\text{ce}} +
\lambda_\text{2}^{'}{\mathcal{L}_{\text{ss}}},\\
& \mathcal{L}_{\text{ss}} =
- \frac{{1}}{N^b T}\! \sum_{{\emph{i}}=\text{1}}^{{N^b T}}\! \log(\frac{\text{exp}(\frac{\text{sim}(\bm{a}_\emph{i}^{\text{psp}}, \bm{v}_\emph{i}^{\text{psp}})}{\eta})}{ \!\sum_{\emph{j}=\text{1}}^{{N^b T}}\!\text{exp}(\frac{\text{sim}(\bm{a}_\emph{i}^{\text{psp}}, \bm{v}_\emph{j}^{\text{psp}})}{\eta^{'}})}\!),
\end{split}
\end{gathered}
\right.
\label{eq:loss_SS}
\end{equation}
where $N^b$ denotes the number of videos in a batch and $T$ is the number of segment in each video; thus, $N^b \cdot T$ denotes the total segment number in a batch. $\text{sim}(\cdot, \cdot)$ computes the dot product of the $\ell_2$ normalized vectors (\emph{i.e.}, cosine similarity). The $i$ and $j$ are the indexes of the video segment, note that $j$ can also index the $i$-th segment. $\eta^{'}$ is a temperature parameter controlling the concentration level of feature distribution; it is set to 0.3 in our experiments. $\lambda_2^{'}$ is the weight to balance these two losses and is empirically set to 0.01.
\begin{table}[t]
\caption{Comparison of different contrastive manners (CPSP v.s. self-supervised SSPSP) under two settings. We report the accuracy(\%) measured on the AVE and the VGGSound-AVEL100k datasets.
}
\vspace{-7mm}
\begin{center}
\setlength{\tabcolsep}{4.5mm}
\begin{tabular}{l|cc|cc}
\toprule[0.8pt]
\multirow{2}{*}{Method} & \multicolumn{2}{c|}{AVE} & \multicolumn{2}{c}{VGGSound-AVEL100k} \\ \cline{2-5}
& fully & weakly & fully & weakly \\ \hline
PSP & 77.8 & 73.5 & 58.3 & 47.4 \\
SSPSP & 78.2 & 73.8 & 58.8 & 48.0 \\
CPSP & \textbf{78.6} & \textbf{74.2} & \textbf{59.9} & \textbf{48.4} \\ \bottomrule[0.8pt]
\end{tabular}
\end{center}
\label{table:SSPSP}
\vspace{-0.5cm}
\end{table}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=\textwidth]{./figures/fig9_PSP_AVEL_example_reverse.pdf}
\end{center}
\vspace{-0.5cm}
\caption{A qualitative example of pair-level propagation in PSP.
For the video on the \textbf{left}, only the first three segments simultaneously contain the visual and audio signals of the event \emph{frying food}.
The green boxes represent ground truth labels.
The blue and orange boxes indicate predictions
of AVEL~\cite{tian2018audio} and the PSP method, respectively. Besides, we visualize the attention effect on the images. It is clear that our method produces more accurate localization.
On the \textbf{right}, we visualize the audio-visual similarity matrices $\bm{\gamma}^{\text{va}}$ and $\bm{\gamma}^{\text{av}}$ (Eq.~\ref{eq:3}) after PSP. For $\bm{\gamma}^{\text{va}}$, the x-axis and y-axis correspond to audio and visual features, respectively, and for $\bm{\gamma}^{\text{av}}$ the order is reversed. The red bounding box in $\bm{\gamma}^{\text{va}}$ shows that
all the visual components are highly correlated with the first three audio components containing the sound of the event.
Besides, negative and weak connections are cut off to 0 in $\bm{\gamma}^{\text{va}}$ and $\bm{\gamma}^{\text{av}}$. The color bar corresponds to the similarity strength, with red denoting high similarities and blue for low similarities.
}
\label{fig:localization_example}
\vspace{-2mm}
\end{figure*}
\begin{figure*}[htbp]
\begin{center}
\setlength{\abovecaptionskip}{0.cm}
\includegraphics[width=\textwidth]{./figures/feature_distribution_PSP.pdf}
\vspace{-3mm}
\caption{TSNE \cite{maaten2008visualizing} visualization of audio and visual feature distributions.
The data all come from the validation set of AVE dataset under the fully supervised setting.
(\textbf{Row 1:}) audio features.
(\textbf{Row 2:}) visual features.
(\textbf{Column 1:}) the CNN features.
(\textbf{Column 2:}) features after Bi-LSTM encoding. (\textbf{Column 3:}) features after PSP encoding.
We observe that features after PSP are much better clustered into individual classes than the Bi-LSTM and CNN features.
Different colors represent different classes.
Best view in color and zoom in.}
\vspace{-3mm}
\label{fig:feature_distribution}
\end{center}
\end{figure*}
\begin{figure*}[htbp]
\begin{center}
\includegraphics[width=\textwidth]{./figures/fig10_a_v5_PSP_CPSP_example.pdf}
\caption{Localization results of a qualitative example (sampling from $\mathcal{D}_{\text{bg}}$ of AVE dataset) under both fully and weakly supervised settings. In this example, the first five video segments contain the audio-visual event \emph{Rodents}.
In either setting, PSP wrongly predicts some segments as \emph{Background} (gray boxes), the CPSP method has the correct results (green boxes). The classification probability maps confirm this.
We also compare the features learned by the PSP and CPSP using the metrics aforementioned in Table~\ref{table_sc_ch_dbi}. These metrics reflect the clustering quality of features.
The results show that the features learned by the CPSP are more distinguishable.
It also verifies the segment-level event-awareness capability of the CPSP.}
\label{fig:PSP_PSPCL_a}
\end{center}
\vspace{-3mm}
\end{figure*}
The experimental result is shown in Table~\ref{table:SSPSP}, where SSPSP is conducted with the optimal experiment setup.
We can find that the performance of the SSPSP is comparable with the CPSP on the AVE dataset but is much lower on the large-scale VGGSound-AVEL100k dataset.
On VGGSound-AVEL100k, our CPSP method surpasses SSPSP by 1.1\% and 0.4\% under fully and weakly supervised settings, respectively.
This reflects that such self-supervised contrastive method is not robust for audio-visual event localization.\footnote{We provide more experimental results and analyses in the appendix~\ref{supp_sspsp} that show the SSPSP is much more sensitive to the data distribution and training batch size, \emph{etc}.} In fact, the self-supervised SSPSP indeed ignores the semantic alignment of audio-visual pairs when constructing positive-negative samples which is vital for AVEL.
Unlike SSPSP, the proposed CPSP uses reliable audio-visual pairs to construct positive and negative samples.
This makes the CPSP more superior.
\subsection{Quantitative analysis - generalization to AVVP task}\label{sec:avvp}
In this subsection, we extend the proposed CPSP method to a related and more challenging audio-visual video parsing (AVVP) task.
We adopt the baseline method HAN~\cite{tian2020avvp} specifically designed for this task as the backbone and replace its core hybrid attention network for aggregating audio-visual features by the proposed PSP module.
As for the objective optimization, we keep the loss items proposed in ~\cite{tian2020avvp} and introduce the proposed video-level contrastive objective ${\mathcal{L}_{\text{vpsa}}}$
under the weakly-supervised labels (given only video-level labels) to adapt our CPSP model for AVVP.
Notably, since there are multiple categories of events in each video in AVVP task, there are some differences from AVEL to AVVP when constructing the positive and negative sets for contrastive learning.
Specifically, for a certain video in a batch during training, videos in the negative set can be selected from those remaining videos that have completely irrelevant event categories from it. But it is hard to select the positive samples requiring completely the same labels.
Therefore, we consider to define a co-occurrence ratio $r$ to measure the coincidence degree of event categories between two videos.
For example, given a video $Vid_a$ with event labels \{\emph{barking}, \emph{speech}\} and its pair video $Vid_b$ with labels \{\emph{speech}, \emph{music}, \emph{clapping}\}, $r$ is calculated by the proportion of co-occurrence event labels (\{\emph{speech}\}) to the total event labels of $Vid_a$ (\{\emph{barking}, \emph{speech}\}), \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, $r$ is 1/2.
In this way, the ratio $r$ indicates that the positive samples are expected to contain as many events as possible that are appeared in the reference video (large $r$).
We consider to set a threshold $\mu$ to construct the positive samples.
For any pair of videos, we first compute the ratio $r$ between them.
If the $r$ is greater or equal than pre-set $\mu$ ($r \ge \mu)$, the pairwise video is selected as positive sample.
As for the negative samples, the ratio $r$ is strictly
equal to zero which means there are no events overlapping between the two videos.
The hyper-parameters $\mu$ and $\theta$, $K$ in Eq.~\ref{eq:loss_vcon} are empirically set to 0.6, 0.4 and 4 for AVVP, respectively.
And we provide an ablation study on the threshold $\mu$ in the appendix~\ref{supp_avvp}.
With the above setup, we train our CPSP model on LLP dataset~\cite{tian2020avvp} from scratch for AVVP.
For fair comparison, we use the same evaluation metrics as in HAN ~\cite{tian2020avvp}, referring to ``A'' and ``V'' (the F-score of audio events and visual events, respectively), ``AV'' (the F-score of audio-visual co-occurrence events, namely AVEs in AVEL task), ``Type@AV'' (the averaged result of ``A'', ``V'', and ``AV''), and ``Event@AV'' (the F-score of audio-visual events where mIoU is set to 0.5).
From Table~\ref{table:CPSP_on_LLP}, we have three observations:
\textbf{first}, our PSP and CPSP methods surpass the baseline methods from audio-visual event localization task (AVEL~\cite{tian2018audio}, AVSDN~\cite{lin2019dual}) by a large margin for audio-visual video parsing (AVVP).
\textbf{Second}, compared to the vanilla PSP, the proposed CPSP with the video-level contrastive objective $\mathcal{L}_{\text{vpsa}}$ improves the performances significantly. For example, the metric ``Type@AV'' and ``Event@AV'' are improved by 3.9\% and 3.3\% for the segment-level, and they are 4.1\% and 4.3\% for the event-level, respectively. This again demonstrates the benefits of the proposed contrastive strategy.
\textbf{Third}, compared to the HAN~\cite{tian2020avvp} that is specially designed for AVVP, CPSP is even more superior that has better performances, especially having obvious performance superiority on ``V'' and ``AV''.
This reflects that the CPSP not only keeps a strong ability to recognize the audio-visual events but can also competently identify separate audio events and visual events.
In short, the proposed CPSP is still effective and advanced in both the AVEL and more challenging AVVP tasks providing more evidence of the model generalization.
\subsection{Qualitative analysis}\label{visualization}
\subsubsection{Visualization of the effectiveness of PSP}
\textbf{Propagated audio and visual components in PSP.}
We start by presenting an example of audio-visual event localization in Fig.~\ref{fig:localization_example}.
The event in this sample is difficult to predict because the visual images are changeable
and the audio signals are mixed with background noise.
From the figure, we have three observations.
(1). While both our method and AVEL \cite{tian2018audio} use the AVGA attention, we show that the PSP enables better attention to visual regions closely related to sound sources.
As displayed in Fig.~\ref{fig:localization_example}, for the event of {\em frying food},
our attended regions include both the frying chicken thighs and the pot, especially in the first four segments.
In comparison, AVEL only finds the thighs and very small receptive fields.
(2). Our method has a better prediction result.
AVEL seems to make decisions merely according to synchronized audio-visual segments while
our method can pay attention to visual and audio components that are at different time stamps. For example,
AVEL incorrectly regards the fifth and sixth segments as the {\em frying food} event, ignoring the third and fourth segments which are more relevant to the event.
(3). We visualize the similarity matrices $\bm{\gamma}^{\text{va}}$ and $\bm{\gamma}^{\text{av}}$ in Fig.~\ref{fig:localization_example}.
We find that only a small percentage of all the audio-visual connections are retained after PSP selection
and are closely related to the event. For example,
they tend to build strong connections (large similarity values)
between the first three audio components and the first four visual components containing the scene of "flying food",
\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, feature propagation merely takes place in these event-related audio-visual pairs.
Such a propagation mechanism is critical for AVE localization because
more discriminative audio-visual features can be identified with these {\em positive} connections and subsequently used in classifier training.
Through back-propagation, it allows the model to be able to attend on more sound-relevant visual regions and visual-relevant audio segments.
\begin{figure*}[t]
\begin{center}
\includegraphics[width=\textwidth]{./figures/fig11_b_v5_PSP_CPSP_example.pdf}
\vspace{-5mm}
\caption{Weakly supervised setting is a more challenging setting.
We display the AVE localization results of two examples that all of the video segments contain the event (sampling from $\mathcal{D}_{\text{ae}}$ of AVE dataset).
We have two observations:
1) as shown in example (a), the PSP classifies incorrect event category in the orange box, while the CPSP provides accurate predictions in green boxes;
2) for example (b), even both the PSP and CPSP have exact predictions, the CPSP gives larger probabilities to the ground truth category.
The classification probability maps confirm this.
This indicates that the features encoded by the CPSP contain more category-aware semantics related to the ground truth thus facilitates the event category classification.}
\label{fig:PSP_PSPCL_b}
\end{center}
\vspace{-5mm}
\end{figure*}
\textbf{Feature distribution in the PSP.}
We then visualize the data distribution of features processed by different stages in our framework using TSNE~\cite{maaten2008visualizing}.
As shown in Fig.~\ref{fig:feature_distribution},
we first find that the CNN-based audio and visual features are not very well clustered.
This is because they are at a relatively low level in the network hierarchy encoding limited semantics.
Then, after Bi-LSTM, features of some categories (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, \emph{Rodents} and \emph{Frying food}) can be better clustered compared with the CNN features, but most are still disordered and highly mixed.
Further, after PSP, the features are much better clustered: cohesive within the same class and divergent between different classes.
This reflects that the audio-visual representations gain stronger discriminative abilities along the pipeline of our method.
\subsubsection{Visualization of the effectiveness of CPSP}
Here, we display some examples to explore the classification capability of the CPSP, where compared with the PSP, the CPSP introduces the contrastive constraints PSA$_S$ and PSA$_V$.
\textbf{Segment-level event-aware.}
First, in Fig.~\ref{fig:PSP_PSPCL_a}, we show a video example. We perform the PSP and CPSP in both fully and weakly supervised settings and report the AVE localization results.
The CPSP has more accurate predictions in both settings. Using the PSP, some segments containing the event are incorrectly classified to the background (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, the second and fourth segments in the fully supervised setting, the first two segments in the weakly supervised setting) while the CPSP outputs all the correct results.
We display the classification probability map to see what happens. In the fully supervised setting, the second and fourth segments are predicted to the \emph{background} with high probabilities by the PSP but this result is overturned by the CPSP. Similar phenomenon is observed in the weakly supervised setting, which performs slightly lower probabilities than full supervision
due to its poor knowledge (segment-level event labels).
We also use the evaluation metrics mentioned in Table~\ref{table_sc_ch_dbi} to test the discriminability of the segment features. As shown in the histograms, the CPSP has superior performances under all of the indicators in both settings. This reflects that positive event-aware semantics of segment features are aggregated and discriminative from the backgrounds. We speculate this is attributed to the PSA$_S$ that enforces the CPSP to learn more \emph{event-aware} semantics.
\textbf{Video-level category-aware.} We further display two examples containing no backgrounds and conduct the localization under the more challenging weakly supervised setting.
As shown in Fig.~\ref{fig:PSP_PSPCL_b}, for example (a), the PSP incorrectly classify the seventh segment to the \emph{Helicopter} (orange box; the ground truth is \emph{Aircraft}); it's even easily confused by human.CPSP takes efforts to modify the wrongly predicted result generated by the PSP (orange bounding box).
This is reflected in the classification probability map with a high probability of \emph{Aircraft} at the seventh segment.
In example (b), both the PSP and CPSP provide accurate predictions, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, all the segments are classified to the \emph{Car} event.
But the classification probability map tells that the CPSP gives higher probabilities to the \emph{Car} category (red bounding box), making the video segments more recognizable from the similar \emph{Truck} or \emph{background}.
These two examples demonstrate the CPSP is more \emph{category-aware} thanks to the PSA$_V$ that enables to encode features including more semantics related to the ground truth category thus distinguishing from other categories.
\section{Conclusion}\label{sec:conclusion}
For the AVE localization problem, we propose a contrastive positive sample propagation (CPSP) method that comprehensively explores three levels of positive samples for distinguishable audio-visual representation learning.
Specifically, the pair-level PSP identifies and exploits the most relevant audio and visual pairs when fusing the cross-modal features. We find that negative and weak connections, even though with small weights, have a detrimental effect on the system, and thus need to be completely removed.
The segment-level PSA$_S$ and video-level PSA$_V$ provide additional contrastive constraints to refine the features encoded by the PSP. The PSA$_S$ enforces the model to be event-aware by gathering the positive segments containing an AVE and being far away from the backgrounds. The PSA$_V$ is actually an online hard sample learning that contrasts the positive video from negatives according to the event category thus makes the model to be category-aware.
We show that such pair-level, segment-level and video-level positive sample propagation and activation method are beneficial to the classifier training.
To evaluate the model generalization ability, we collect a large-scale VGGSound-AVEL100k dataset and extend our method to a more challenging audio-visual video parsing task.
Extensive experimental results validate the effectiveness of the proposed CPSP method.
In addition, this paper covers a comprehensive study on the contrastive learning manners with different supervisions, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, fully-, weakly-, and self-supervised.
\ifCLASSOPTIONcompsoc
\section*{Acknowledgments}
\else
\fi
We would like to thank the reviewers for their constructive suggestions.
This work was supported by the National Natural Science Foundation of China (72188101, 61725203, 62020106007, and 62272144), and the Major Project of Anhui Province (202203a05020011).
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,688 |
The iPhone Measure app level function as a measuring device for the weight bearing lunge test in adults: a reliability study
Helen A. Banwell ORCID: orcid.org/0000-0001-5730-16111,2,
Hayley Uden2,
Nicole Marshall2,
Carlie Altmann2 &
Cylie M. Williams2,3,4
Journal of Foot and Ankle Research volume 12, Article number: 37 (2019) Cite this article
Ankle joint range of motion is a frequently assessed measure used by health care clinicians who manage lower limb pathologies to identify ankle equinus and/or other joint motion concerns that may negatively impact on function. The purpose of this study was to assess a new iPhone application (the level function of the 'Measure application'), for measuring the weightbearing ankle lunge test in a healthy adult population (reliability) and measuring known angles (validity) when compared to a digital inclinometer.
To determine intra-rater reliability, inter-rater reliability and concurrent validity, 168 measures were conducted on 21 participants. Participants were preconditioned prior to assessment, and two experienced raters measured ankle dorsiflexion range of motion in the knee extended and knee flexed positions of the weight bearing lunge test, using an iPhone level function (of the Measure application) and a digital inclinometer in a randomised order, over two timepoints. Concurrent validity was also determined by comparison of measures of the two devices at known surface angles (0 and 15 degrees) in multiple planes. Reliability and validity were determined with intraclass correlation coefficients, concurrent validity was explored with the Bland Altman plot and an intraclass correlation coefficient. The Standard Error of the Mean and the minimal detectable change were also explored.
The intra-rater reliability using the iPhone and inter-rater reliability using the digital inclinometer, in the knee extended position, were ICC 0.85 respectively, indicating good reliability. All other intra-rater reliability and inter-rater reliability for both devices and both leg positions were over ICC 0.90, indicating excellent reliability. Concurrent validity between the two devices on a flat and known angle surface were ICC 1.0 (Limits of Agreement − 1.0 to 0.61), indicating excellent validity, with good validity demonstrated by a Bland Altman plot of all measures in all positions (ICC of 0.84 (Limits of agreement = − 4.51 to 6.49)).
The use of the iPhone level measure, within the Measurement App has demonstrated to be an easy and reliable measurement tool to determine ankle joint dorsiflexion during the weightbearing lunge test in healthy adults.
A reduced range of ankle joint motion (i.e. ankle equinus) has been shown to have a negative impact on lower limb function and economy of gait in healthy and pathological populations [1,2,3,4,5,6]. Clinicians involved in the assessment, diagnosis and management of foot and leg conditions often identify restrictions of ankle joint motion and prescribe interventions, such as stretch and strengthening programs, with re-assessment of measures used to determine success [1, 5]. This requires the measure used to be repeatable and consistent. The identification of reduced ankle joint motion can be measured clinically via weightbearing and/or non-weightbearing methods, with the weight bearing lunge test deemed the preferred method due to improved capture of full joint excursion [7,8,9,10].
In clinical practice digital inclinometers are a frequently used measuring tool, which have proven to be reliable and valid for the weight bearing lunge test [8, 11] and are comparable to two-dimensional motion capture systems [12]. However, digital inclinometers may be considered costly for the average clinician and are not often accessible by clients/carers who may wish to assess range of motion changes at home. With the advances in technology, some applications (Apps) have been reported as suitable substitutes. Specifically, the Tiltmeter App and the iHandy App (available on smart phones/tablets) have been shown as reliable measures of ankle joint dorsiflexion [13, 14]. These have the additional benefits of being cheap, easily accessible and quick to administer [13]. Unfortunately, with rapidly changing technology, these Apps become outdated and unsupported, as demonstrated with the recent discontinuation of the Tiltmeter App for iPhone users. With Apple's recent software upgrade (operating systems IOS 7 and above) a new Measure App which includes a 'level' function has been introduced. This level function, if reliable, would potentially be a suitable alternative to the discontinued Tiltmeter App with the additional bonus of being included in the Apple App suite (that is, it is standardly installed/upgraded with each software upgrade). Furthermore, in Australia, iPhone users account for 45% of the smart phone market share (8.6 million users) [15], meaning the Measure App is freely accessible to a large population of smartphone users. To be confident in its use in the clinical setting, however, determination of the psychometric properties is required.
The primary aim of this study was to determine the intra and inter-rater reliability of the level function of the Measure App compared to a digital inclinometer. The secondary aim was to determine the concurrent validity of the two tools (i.e. how well does the level function measure when compared to the digital inclinometer).
The study design was to determine intra-rater and inter-rater reliability of the weight bearing lunge test with both the knee extended and knee flexed, using the digital inclinometer and the iPhone Measure App. The study was also designed to determine the concurrent validity between the two tools.
Two podiatrists (CA and NM) conducted all measurements. Both raters (CA and NM) had 8 years clinical experience, have post-graduate research training and use the WBL measurement technique routinely during clinical practice. Raters were involved in the development of the protocol, reviewed the final protocol and practiced the measure on two participants (not included in the final study) 1 week prior to conducting the study to allow open discussion regarding procedure.
A convenience sample of 21 participants were recruited from the University of South Australia podiatry student cohort. Students were alerted to the study by email correspondence outlining the study aims and disseminating participant information sheets and consent forms for participants to consider in their own time. To minimise the risk of coercion, all correspondence informed students that involvement in the study was voluntary and could be withdrawn at any time, and participants indicated their willingness to be involved by returning a signed consent form to an administrator external to the podiatry course. Participants were excluded from the study if they had: foot pain or injury within the past 6 months; any past foot or ankle surgery; or a neurological or inflammatory condition affecting gait. Ethics approval was obtained from the University of South Australia's Human Research Ethics Committee (Approval number 201357).
Two tools were compared within this study. The Geo Fennel S-Digit Mini Inclinometer (digital inclinometer), (GSR Laser Tools, Perth, Australia); and the level function available via the Measure App, a free App available on the iPhone smart phone (operating systems IOS 7 and above). For this study the iPhone 6S was used (Apple Inc., Cupertino, CA, USA). Prior to testing, the digital inclinometer and iPhone Measure measures were compared for consistency on identical hard static flat and angled surfaces in multiple planes across three trials per angle. Prior to testing, the digital inclinometer was calibrated in accordance to industry requirements (Laser-Liner, UK), the iPhone was calibrated to zero degrees by placing it with its long axis on the floor.
Participants were introduced to the study as a group and the WBL technique was explained and demonstrated. Prior to testing, each participant was required to hold a static WBL test stance in the knee flexed and knee extended position for 30 s each, three times. This preconditioning technique was chosen to allow participants to adopt the position easily. The WBL test protocol used during testing was consistent with Bennell et al. [16] as follows:
Participants stood with their hands shoulder width apart against the wall in front of them.
The participants right leg was placed as far back as comfortably possible behind them whilst keeping their right heel to the ground, parallel to the left leg and perpendicular to the wall
The rater assisted the participant to move their right foot back until the lunge position could be held whilst the heel remained on the floor and the knee aligned over the second toe [16]
WBL measures were then taken with the knee extended (Fig. 1) and the knee flexed (Fig. 2).
A single measure was taken at each time point, in each position by each of the raters.
Weightbearing lunge test – knee extended position
Weightbearing lunge test – knee flexed position
To measure the WBL, the short arm of the device was placed flat against the posterior heel, approximately one-centimetre superior to the posterior calcaneal tuberosity and held perpendicular to the shank of the tibia until the measure (in degrees) remained fixed (Figs. 1 and 2). The degree was determined by the long axis of the device relative to the horizontal (zero degrees). This is consistent with the method of measurement and position of measuring devices in similar studies [13, 17].
Testing occurred over one four-hour session. The order of participants and the measuring device used were randomised by computer table [18] and administered independently to the raters (HB). Measures were collected for the right foot only to satisfy the assumption of data independence [19]. To minimise recall, participants were measured behind a partition that allowed the practitioner to visualise the person from their knees down only. The author group considered the sample size large enough to ensure raters were unable to remember the result; and the time space between retesting participants (minimum of 30 min) was appropriate to not cause fatigue to the target muscle group.
Participant data were described in means (SD) and frequencies (%). The raw data from each rater, at the two timepoints, and each measured position, were normally distributed. Systematic error between timepoints were explored with t-tests. Significant differences between timepoints were considered where p < 0.05. The intra-rater reliability between timepoints for equipment was determined using the raw data with the intraclass correlation coefficient (ICC) (Model 3,1), 95% confidence interval (95% CI), Standard Error of the Mean (SEM) and the minimal detectable change (MDC). The SEM provided a measure of the variability and its calculation assisted in determining the MDC. The SEM was calculated with the raw data with the following formula: SEM = SD√(1-r) where r was the ICC for intra-rater reliability and SD was of the SD of measurement [20]. The MDC was calculated as the magnitude of change necessary in order to provide confidence that the change is not a result of random measurement error. The MDC was calculated as MDC = 1.96 x SEM x √2 [20]. The interrater reliability determined with all raw data collected from two raters, for each position and each measurement tool using ICCs (Model 2,2) 95% CI's and Standard Error of the Mean (SEM). Concurrent validity was explored with the ICC and Bland Altman plot between the devices in both leg positions. The Bland Altman plot was used as a graphical display of agreement between measurement. It was used to assess the degree of agreement between the tools in all positions and by both raters, across the two timepoints. It also helps to identify the presence of bias. The Bland Altman was also used to calculate the mean difference between measures, the limits of agreement and the 95% confidence interval for the limits of agreement [21].
A minimum sample size of 18 was calculated to provide 80% power of detecting a ICC of 0.6 with a two-tailed alpha = 0.05 for the intra-rater reliability analysis [22]. The following ranges were used to report ICC data: < 0.5 = poor reliability, 0.5 to 0.75 = moderate reliability, 0.76 to 0.9 = good reliability, and > 0.90 = excellent reliability [22]. All data were analysed with Stata 15 [23].
Twenty-one participants met the eligibility criteria, gave informed consent to be part of the study and recorded their age, weight (kg) and height (cm), (Table 1). One hundred and sixty-eight measures were recorded.
Table 1 Participant characteristics
There were no differences between measures at each time point (p > 0.05). The intra-rater reliability for the tools were calculated for both raters (Table 2). The intra-rater reliability of the digital inclinometer was excellent for both leg positions (ICC = 0.91 to 0.97). The level function had good to excellent intra-rater reliability (ICC = 0.85 to 0.95), the lesser was with the knee extended (Table 2). There was also good to excellent inter-rater reliability between the raters with each tool. The digital inclinometer inter-rater reliability was good to excellent (ICC = 0.85 to 0.96). The level function had excellent inter-rater reliability between raters (ICC = 0.94 to 0.98). The lesser of both ICC scores was in relationship to the knee being in the extended position (Table 3).
Table 2 Outcomes of intra-rater reliability of the iPhone level function and the digital inclinometer for the weightbearing lunge test
Table 3 Outcomes of inter-rater reliability of the iPhone level function and the digital inclinometer for the weightbearing lunge test
Initial concurrent validity, determined between the digital inclinometer and level function on static hard flat and angled (15 degrees) surfaces, was ICC of 1.0 (limits of agreement − 1.0 to 0.61), indicating excellent reliability. There was acceptable concurrent validity between the two devices, and in all leg positions as demonstrated in the Bland Altman plot (Fig. 3). The ICC between all measures, in all positions was good (ICC = 0.84, Mean Difference = 0.99, Limits of agreement = − 4.51 to 6.49) and at least 90% of the plots were within ±1.96 SD (Fig. 3).
Bland Altman plot of concurrent validity outcomes
To the best of our knowledge this is the first use of the new iPhone level function within the Measure App to review reliability in ankle joint range of motion measures. The outcomes of the study suggest the tool is comparable to digital inclinometers and can be used to measure the weight bearing lunge test in healthy adult populations with confidence.
The weight bearing lunge test with the knee extended and knee flexed has high levels of reliability [7] and is regularly used in research to assess joint range of motion. This includes studies in Charcot Marie Tooth Disease [24], children's heel pain [6], idiopathic toe walking [25], dancers [17] and changes in plantar pressures in a diabetic population at risk of ulceration [26]. The results from this present study determined intra and inter-rater reliability of all measures were deemed good or excellent. Validity of the level function was also determined as an acceptable comparison to the digital inclinometer, with a low bias and a mean difference close to zero. Within a healthy adult population, the weight bearing lunge test, along with the use of the level function within the iPhone Measure App, can be confidently introduced into clinical practice for quantifying ankle dorsiflexion range of motion.
Similar to previous studies, the knee flexed position demonstrated higher reliability than the straight leg position [13]. The authors proposed that the lower scores with a straight leg may be due to either mechanical placement issues, participant force differences (where potentially more force is placed on the posterior soft tissue structures resulting in increased participant discomfort) or an unknown order effect not dispersed via randomising of participants. With the knee flexed, the measure is presumed to be more of capsular stiffness and less soft tissue impact therefore higher reliability scores were obtained. However, for these reasons the knee extended weight bearing lunge is considered more clinically applicable and is the encouraged measure for research and clinical practice purposes [9, 27, 28].
Whilst these findings encourage clinicians to use the readily accessible technology within their clinic to confidently aid assessment, consideration needs to be given for infection control concerns and phone design. Specifically, mobile phones have previously been shown as an infection hazard [29], the iPhone used within this study did not have a cover and had a flat base. These factors aided positioning but required the phone to be cleaned between and after testing. A cover would not eliminate that cleaning schedule but may alter the flatness of the surface and skin contact. However, these concerns are minimal and can be rectified by following standard cleaning schedules that apply to all other multiple use assessment items used on intact skin.
There are a number of limitations to this study. Experienced raters conducted all measures. Alternative studies on reliability have included a novice rater to compare, therefore care should be taken in considering how these results may apply to the learner user. Additionally, we have suggested that this App is unlikely to change due to its inclusion in the Apple App suite, however, there is still the risk that changes to its function may occur, including but not limited to: the App being removed from the iPhone software; phone case shape variation, or; updates to the Measure App format with changes to the level functionality. Android phone users will need to consider alternative measure Apps as the Measure App is not available on the Android platform. The study was powered with an ICC of 0.6, which indicates moderate reliability [22]. Whilst the research team determined this as an acceptable level, other researchers or clinicians may consider this as low. This should be considered when applying these results in practice or research in the future. Lastly, the mean values of weight bearing lunge were lower than other reported ranges [4, 5], however, comparable to other published values in normative populations [6, 7]. This also highlights that researchers and clinicians should consider the placement of measurement equipment for the weightbearing lunge. Specifically, placement of measurement equipment at tibia's anterior surface [30] may elicit different results to the equipment's position as used within this study, and outcomes may not be comparable. It is unknown what impact this may also have on measured reliability.
Future research in the use of this technology for measurement should include understanding the reliability in children and in pathological populations, where there is (potential for) a smaller surface area for device placement. There is also the potential to consider including family/carers in future assessment of this and alternative measuring Apps to determine appropriateness of non-health professional's ability to determine success where interventions have been prescribed to improve ankle flexibility.
Using the iPhone level measure, within the Measurement App has demonstrated to be an easy to use and reliable measurement tool for healthy adults. Clinicians should consider how the use of this technology may assist in their clinical practice to assess and measure treatment outcomes.
Raw data is available from authors and request.
ICC:
intraclass coefficient
standard error of measurement
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James AM, Williams CM, Luscombe M, Hunter R, Haines TP. Factors associated with pain severity in children with calcaneal Apophysitis (sever disease). J Pediatrs. 2015;167(2):455–9.
Powden CJ, Hoch JM, Hoch MC. Reliability and minimal detectable change of the weight-bearing lunge test: a systematic review. Man Ther. 2015;20:524–32.
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Menz H. Two feet, or one person? Problems associated with statistical analysis of paired data in foot and ankle medicine. Foot. 2004;13(1):2–5.
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Kang MH, Oh JS. Relationship between Weightbearing ankle dorsiflexion passive range of motion and ankle kinematics during gait. J Am Podiatr Med Assoc. 2017;107:39–45.
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International Centre for Allied Health Evidence (iCAHE), University of South Australia, Adelaide, South Australia, 5001, Australia
Helen A. Banwell
School of Health Sciences, University of South Australia, Adelaide, South Australia, 5001, Australia
, Hayley Uden
, Nicole Marshall
, Carlie Altmann
& Cylie M. Williams
Allied Health, Peninsula Health, Frankston, Victoria, 3199, Australia
Cylie M. Williams
School of Primary and Allied Health, Monash University, Frankston, Victoria, 3199, Australia
Search for Helen A. Banwell in:
Search for Hayley Uden in:
Search for Nicole Marshall in:
Search for Carlie Altmann in:
Search for Cylie M. Williams in:
HB and CMW conceived the study. Data collection and analysis was conducted by HB, HU, NM, CA. All authors contributed to the manuscript and reviewed and approved the final submission.
Correspondence to Helen A. Banwell.
Approval was gained by the University of South Australia's Human Research Ethics Committee (Approval number 201357).
Authors obtained written consent from participants within both photographs.
CMW is an Associate Editor of the Journal of Foot and Ankle Research. It is journal policy that editors are removed from the peer review and editorial decision-making process for the papers that they have co-authored. All other authors declare that they have no competing interests.
Banwell, H.A., Uden, H., Marshall, N. et al. The iPhone Measure app level function as a measuring device for the weight bearing lunge test in adults: a reliability study. J Foot Ankle Res 12, 37 (2019) doi:10.1186/s13047-019-0347-9 | {
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Q: Simple Matplotlib animate not working I am trying to run this code below but it is not working properly. I've followed the documentation from matplotlib and wonder what is wrong with this simple code below. I am tryting to animate this into jupyter notebook with anaconda distro. My python version is 2.7.10.
import numpy as np
from matplotlib import pyplot as plt
from matplotlib import animation
fig = plt.figure()
def init():
m = np.zeros(4800)
m[0] = 1.6
return m
def animate(i):
for a in range(1,4800):
m[a] = 1.6
m[a-1] = 0
return m
anim = animation.FuncAnimation(fig, animate, init_func=init,
frames=200, interval=20, blit=True)
plt.show()
A: You need to create an actual plot. Just updating a NumPy array is not enough.
Here is an example that likely does what you intend. Since it is necessary to access the same objects at multiple places, a class seems better suited as it allows to access instance attributes via self:
import numpy as np
from matplotlib import pyplot as plt
from matplotlib import animation
class MyAni(object):
def __init__(self, size=4800, peak=1.6):
self.size = size
self.peak = peak
self.fig = plt.figure()
self.x = np.arange(self.size)
self.y = np.zeros(self.size)
self.y[0] = self.peak
self.line, = self.fig.add_subplot(111).plot(self.x, self.y)
def animate(self, i):
self.y[i - 1] = 0
self.y[i] = self.peak
self.line.set_data(self.x, self.y)
return self.line,
def start(self):
self.anim = animation.FuncAnimation(self.fig, self.animate,
frames=self.size, interval=20, blit=False)
if __name__ == '__main__':
ani = MyAni()
ani.start()
plt.show()
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,071 |
Q: Android Studio: No build variant found for ':expo'. error I Encountered a error saying No variants found for ':expo'. Check build files to ensure at least one variant exists.
I Already download all the SDK's Clear the Project multi times but nothing work
Here is the build.gradle code
buildscript {
ext {
buildToolsVersion = "29.0.2"
minSdkVersion = 23
compileSdkVersion = 31
targetSdkVersion = 30
ndkVersion = "21.4.7075529"
supportLibVersion = "29.0.0"
}
repositories {
google()
mavenCentral()
maven { url 'https://developer.huawei.com/repo/' }
}
dependencies {
classpath('com.android.tools.build:gradle:4.2.2')
classpath 'com.google.gms:google-services:4.3.2'
classpath 'com.huawei.agconnect:agcp:1.4.2.301'
// NOTE: Do not place your application dependencies here; they belong
// in the individual module build.gradle files
}
}
allprojects {
repositories {
maven {
// All of React Native (JS, Obj-C sources, Android binaries) is installed from npm
url("$rootDir/../node_modules/react-native/android")
}
maven {
// Android JSC is installed from npm
url("$rootDir/../node_modules/jsc-android/dist")
}
maven {
// expo-camera bundles a custom com.google.android:cameraview
url "$rootDir/../node_modules/expo-camera/android/maven"
}
maven { url 'https://developer.huawei.com/repo/' }
mavenCentral {
// We don't want to fetch react-native from Maven Central as there are
// older versions over there.
content {
excludeGroup "com.facebook.react"
}
}
google()
maven { url 'https://www.jitpack.io' }
}
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,658 |
Content marketing analytics are critical for success. We know that.
Yet, most (75% according to Znet) can't calculate the ROI of their digital marketing as a whole, let alone their content marketing.
Measuring what matters improves decision-making and optimizes market performance. But, what should you measure? Measure what matters.
Sounds simple, doesn't it? But, what matters?
In the era of digital marketing, consumers increasingly interact with your online marketing before they convert — stats from Brandpoint show 63% of consumers check out a brand online before buying. How does this interaction contribute to market performance?
The answer is different for different businesses because they have different goals. For some clients interested in generating leads, we measure leads and conversion of those leads into business. For other clients, we measure sales or email subscriptions.
We call these bottom of the funnel actions because there are usually a number of activities leading to the bottom of the funnel. Check out the typical marketing (sales) funnel on the right to see just a few possible actions preceding actual goals. Often we talk about bottom of the funnel metrics as KPIs or Key Performance Indicators. If you're looking for a list of digital KPIs, check out my list and feel free to add to it.
Because we don't get to the bottom of the funnel where we reach our goals without doing a good job at the top of the funnel, we need to start by measuring top of the funnel activities leading our target market down the funnel.
Let's take a look at content marketing metrics that matter.
And, content marketing replaces many earlier forms of marketing communication like TV (folks just skip through commercials), newspapers (folks don't read newspapers any more), and even recent search strategies (content marketing is the new SEO).
I included the graphic below to show where you might get metrics feeding into your content marketing analytics report. I use a Cognos dashboard from IBM to bring these metrics into a single place to enhance usability. Google Analytics (for blog metrics), Facebook Insights, and Twitter and Pinterest Analytics all have the ability to export data into CSV files that you import into Cognos — or other dashboard software.
It's important to assess both the performance of individual pieces of content as well as the performance of your content marketing strategy as a whole. And, for my money, trends are much more important than point metrics.
In addition, you'll need to add cost and production data — and your content marketing calendar helps with this task if you set it up to capture metrics such as producing content in a timely fashion, time spend on crafting content (which should translate into costs), content backlog (Curate includes a great formula for tracking backlog), and any distribution costs to share the content, such as using Facebook or Twitter sponsored posts or creating PR related to the post.
1. Don't get fooled with vanity metrics, content marketing analytics go far beyond simply tracking likes and followers. For instance, reach assesses the marketing impact of having a community more effectively than vanity metrics.
2. Attribution is hard. While digital media gives unprecedented access to what consumers think and how they behave, you'll never really know why they did something. Assuming consumer rationality is a wrong, so you'll never know what drives a consumer to buy. That means you can't attribute the sale to the last touch — the last click leading up to a sale — those of you using Google Analytics to create funnels know how different results are between using last touch and attributing sales to the entire pipeline.
3. No one tool gives you all the metrics you need to construct a content marketing analytics report — although MOZ has something they call 1Metric that creates a composite score for individual pieces of content based on traffic, social shares, and link data.
4. You should craft content for each stage in the consumer journey, so some pieces of content aren't designed to create sales. These pieces of content simply move consumers along their journey. That means you need to evaluate each piece of content against the individual goals for the content, not against the goals for the entire campaign. In crafting your content marketing analytics report, be sure to reflect goal achievement rather than holding each piece of content up to the same standard.
5. Not all consumers are created equal — commerce isn't a democracy. Some consumers buy lots and some consumers buy a little, so they differ in revenue. Some consumers require a lot of hand-holding and others don't, so they differ in the costs to serve them. That's the concept of CLV or customer lifetime value. When constructing your content marketing analytics report, you should include a measure of how effective content is at generating sales from consumers with a high CLV (see here to calculate CLV).
Be sure to take a look at the graphic from Curata — it shows various metrics to help construct your content marketing analytics report. I have a few issues with the graphic, such as including vanity metrics rather than reach, but it's very good overall.
Also, check out a recent post where I share an alternative — the 4 factor model for measuring social media metrics. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,795 |
German policymakers are striking back at the United States, after a US Treasury report blasted the country's massive trade surplus. Sensitivities in Berlin are still high over US spying on Chancellor Angela Merkel.
Volkswagen cars are covered with protective covers before they are loaded for export. Germany's trade surplus has skyrocketed beyond that of China.
A day after a US Treasury Department report bluntly denounced Germany's economic model, accusing it of hampering the euro zone recovery and hurting global growth, Germany called the conclusions "incomprehensible" and challenged the US to "analyze its own economic situation."
The Treasury's semiannual currency report criticized Germany's overreliance on exports, a high current-account surplus and weak domestic demand. These factors "have hampered rebalancing at a time when many other euro-area countries have been under severe pressure," the report concluded, citing budget tightening in the euro-zone periphery. "The net result has been a deflationary bias for the euro area, as well as for the world economy."
The German Economics Ministry responded in a statement with equally harsh words, saying that Germany's surplus is "a sign of the competitiveness of the German economy and global demand for quality products from Germany."
The report -- traditionally a forum for ridiculing alleged Chinese currency manipulation -- noted that Germany's nominal current account surplus for 2012 was greater than that of China. Germany's surplus rose to $238.5 billion in 2012, compared to China's $193.1 billion, according to the World Bank.
German experts largely agree with the Treasury's assessment of Germany's export-fueled economy, and suggest that increased investment could spur domestic demand. But some question the emphasis on Germany's role in creating the euro zone's economic imbalances.
"The point about the huge current account surplus is accurate. It's nothing new, but they are being a bit more aggressive," Simon Juncker, an economist at the German Institute for Economic Research, said of the Treasury report.
However Stormy-Annika Mildner, an expert on trans-Atlantic economic relations at the German Institute for International and Security Affairs, cautioned that the Treasury may be saddling Germany with too much of the blame. "The report could have been more balanced in pointing the finger," she said.
Germany's dependence on exports is only one piece of the puzzle, Mildner said. Structural problems in other European countries, including limited labor market flexibility in Mediterranean countries and the lack of fiscal union in the euro zone, also add to the continent's economic imbalances.
Still, Mildner acknowledged that limited German domestic demand in Germany continues to hamper overall euro-zone growth. She suggested that increased government investment in infrastructure could facilitate new production and services, thereby boosting consumer spending.
"Increased public investment could be financed by deficits where interest rates are close to zero," said Guntram Wolff, director of Brussels-based think tank Bruegel. On the corporate side, he says, the government could provide tax credits to businesses to incentivize investment.
Despite rising wages over the past few years, German household consumption remains relatively weak. Demand is also strangled by a heavy tax burden. "High taxes, fees and social contributions are weighing on households' disposable income," said Junker.
But a shift in German economic policy may be on the horizon, as the country's main political parties negotiate a new "grand coalition" government between the center-left Social Democrats and the center-right Christian Democrats. And this could be just the right moment in German politics for "tough language" from the US, Juncker said.
On a trans-Atlantic level, however, the Treasury report may only complicate relations between Germany and the US, which were strained earlier this month following allegations that the US spied on Chancellor Angela Merkel's cell phone.
The Treasury's strong words could also undermine EU-US free trade negotiations set to resume next month, said Mildner. She added: "The skin is getting thinner on this side of the Atlantic."
time some nation stood up to the usa,congratulations Germany for the courage.
Germans are hard working and clever people and should be admired for their well deserved success. This sour grapes from the Americans and they should keep their nose out of German affairs.
3. Rather like a firm blaming customers for not buying!
The US Dollar is in reality not worth the Paper it is printed on! The only value it has, is the psychological necessity for an internationally used currency to be stable. Were the USA to admit the impossibility of ever paying back its debts, to those countries which hold huge Dollar reserves, the world market would collapse. But instead of taking the obviously needed action, The US government seeks every opportunity to put the blame on other countries.
US corporate short term profit driven outsourcing is now exposed as a strategic failure by having destroyed the production infrastructure. Ultimate stupidity to state that Germany should have been as stupid as the US.
Are we Americans that stupid to criticize the successful economic policies of DE. We need to learn from Germany. If America needs a role model, and it does, Germany is that model. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,291 |
Georgia High School Senior Awarded $2.2M in Scholarships from 80 Colleges
by Andrea Blackstone April 2, 2021 April 2, 2021 1764
Aylah Birks is a high school senior from Georgia who received approximately $2.2 million in scholarship offers. She was also accepted into 84 colleges. Image- LinkedIn
Student loan debt remains a big hurdle for U.S. college graduates. Yahoo! news reported that during an interview with Politco, White House Chief of Staff Ron Klain confirmed that President Joe Biden is exploring options on canceling a large chunk of debt held by federal student loan borrowers.
Fortunately, one student named Aylah Birks will not face having to repay any student loans. Birks proves staying committed to achieving academic excellence can pay off in big ways. 11Alive reported the Twiggs County High School senior from Georgia received approximately $2.2 million in scholarship offers. She was also accepted into 84 colleges.
Birks holds a 4.0 GPA, is a finalist for the Gates Scholarship and she already achieved her goal of becoming a published author. According to Birk's interview with 11Alive, her book Perspectives Through the Looking Glass is a hybrid book between poetry and personal stories is interactive. It gives the wordsmith the ability for the audience to start a conversation with her. Birks could also start a conversation about being determined at a young age.
"My story is not ending in Twiggs County, but it's only just beginning, and I think the most rewarding part is that I've made my family proud and I've made myself proud," Birks said in the 11Alive interview.
She has not always walked an easy path to achieving noteworthy accomplishments. When her mother faced the task of caring for an ill grandparent a few years ago, she was left to face several hurdles. However, Birks remained optimistic. Birks remained driven and took initiative to press forward.
"I especially could not participate in as many extracurricular activities as I would like to, but I still maintained that grasp in school and my mother couldn't be as present as she is now in my extracurricular activities because she was so busy supporting him and I completely understand," Birks said to 11Alive, also recalling her college application experience. "It was not easy and you will have no idea how many long nights I stayed up trying to find out the information for myself on college deadlines, college admission expectations, what to do, what not to do."
In the interview, Birks also mentioned that some people at school tried to shake her confidence. They attempted to pull her off a successful path, but she did not allow them to succeed in dragging her down. In the interview, Birks said that she is now an advocate for anti-bullying.
"I was often bullied and I didn't understand why, but when you're bullied…people see something in you that they don't see in themselves," Birks said to 11Alive.
Additionally, Clemson and Mercer are currently two of her top colleges choices. Burkes also remarked that she is determined to finish her high school career on a high note. The focused teenager informed 11Alive that she plans to major in behavioral neuroscience and legal studies with minors in communications and public health.
Andrea Blackstone | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,347 |
Q: Create IDs for each
* (including nestings) in a jQuery sortable list I have a sortable list with nested LI items. I'm looking for create an ID for each LI in the group.
Example. If I have this:
<li class="main"> Cat 0
<ul class="subCat">
<li>Subcat 0
<ul class="subCat">
<li>Sub-Subcat 0
<ul class="subCat"></ul>
</li>
</ul>
</li>
<li>Subcat 1</li>
</ul>
</li>
Ok, those ul.subCat are there to nest other li items. I want to make a function to add IDs to li elements and its li child elements. This function will be called by every order change.
The result should be something like this:
<li class="main" id="cat_0"> Cat 0
<ul class="subCat">
<li id="cat_0_0">Subcat 0
<ul class="subCat">
<li id="cat_0_0_0">Sub-Subcat 0
<ul class="subCat"></ul>
</li>
</ul>
</li>
<li id="cat_0_1">Subcat 1</li>
</ul>
</li>
And, for each li.main element repeat the story to reach 4 levels (0 to 3).
My actual code is this one:
// level 0
target = $('#ulMenu >li');
for( i=0; i<=target.length-1; i++ ){
target.eq(i).attr('id', 'cat_' + i).addClass('main');
}
// level 1
//------------------------------------------------------------------
$('#ulMenu >li.main').each(function(){
target = $(this).children('ul').children('li');
for( i=0; i<=target.length-1; i++ ){
father = target.eq(i).parent('ul').parent('li').attr('id').split('_')[1];
target.eq(i).attr('id', 'cat_' + padre + '_' + i);
}
I have to know how to add IDs to the rest of elements. I was trying but I can't found with the solution.
A: Something like this should do it...
$('#ulMenu').children('li').each(function(cat) {
$(this).attr('id', 'cat_' + cat).children('ul').children('li').each(function(sCat) {
$(this).attr('id', 'cat_' + cat + '_' + sCat).children('ul').children('li').each(function(ssCat) {
$(this).attr('id', 'cat_' + cat + '_' + sCat + '_' + ssCat);
});
});
});
Example: http://jsfiddle.net/6YR5p/2/
A: Here's a recursive solution that'll work to any depth you want:
function menuID(el, base) {
base = base || 'cat';
$(el).filter('li').each(function(i) {
this.id = base + '_' + i;
menuID($(this).children('ul').children('li'), this.id);
});
};
menuID('.main');
See http://jsfiddle.net/alnitak/XhnYa/
Alternatively, here's a version as a jQuery plugin:
(function($) {
$.fn.menuID = function(base) {
base = base || 'cat';
this.filter('li').each(function(i) {
this.id = base + '_' + i;
$(this).children('ul').children('li').menuID(this.id);
});
};
})(jQuery);
$('.main').menuID();
See http://jsfiddle.net/alnitak/5hkQU/
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,441 |
========
Glossary
========
.. glossary::
adapters
A pluggable class that allows a ChatBot instance to execute some kind of functionality.
logic adapter
An adapter class that allows a ChatBot instance to select a response to
storage adapter
A class that allows a chat bot to store information somewhere, such as a database.
input adapter
An adapter class that gets input from somewhere and provides it to the chat bot.
output adapter
An adapter class that returns a chat bot's response.
corpus
In linguistics, a corpus (plural corpora) or text corpus is a large
and structured set of texts. They are used to do statistical analysis
and hypothesis testing, checking occurrences or validating linguistic
rules within a specific language territory [1]_.
preprocessors
A member of a list of functions that can be used to modify text
input that the chat bot receives before the text is passed to
the logic adapter for processing.
statement
A single string of text representing something that can be said.
response
A single string of text that is uttered as an answer, a reply or
an acknowledgement to a statement.
untrained instance
An untrained instance of the chat bot has an empty database.
.. [1] https://en.wikipedia.org/wiki/Text_corpus | {
"redpajama_set_name": "RedPajamaGithub"
} | 9,332 |
Q: How to remove "[" and "]" signs from array in Javascript I'm using react.js for building my dashboard
I want to convert an array like this (old version) [ {...}, {...}, {...} ] into this (new version) {...}, {...}, {...} in javascript
So I can put the new version of the array inside a JSON array like this [ {...}, newArray ]
I know a map function returns an array and I know it's a silly question but I wonder how
here is my code:
const siteProfilesList = ['ABC', 'DEF', 'GHI']
const pagesList = ['Dashboard', 'Routes', 'Payload']
const siteProfileNavigationsList = siteProfilesList.map((item, index) => {
let menu = {}
menu['_tag'] = 'CSidebarNavDropdown'
menu['name'] = item
menu['_children'] = pagesList.map((pageItem, pageIndex) => {
let pageMenu = {}
pageMenu['_tag'] = 'CSidebarNavItem'
pageMenu['name'] = pageItem
pageMenu['to'] = `/${pageItem.toLowerCase()}/location=${item.toLowerCase()}`
return pageMenu
})
return menu
})
const navigations = [
{
_tag: 'CSidebarNavTitle',
_children: ['Site Profile']
},
siteProfileNavigationsList
]
export default navigations
I know it's a silly question but I just wonder about the solution.
A: Is that what you want? I use flat().
const siteProfilesList = ["ABC", "DEF", "GHI"];
const pagesList = ["Dashboard", "Routes", "Payload"];
const siteProfileNavigationsList = siteProfilesList.map((item, index) => {
let menu = {};
menu["_tag"] = "CSidebarNavDropdown";
menu["name"] = item;
menu["_children"] = pagesList.map((pageItem, pageIndex) => {
let pageMenu = {};
pageMenu["_tag"] = "CSidebarNavItem";
pageMenu["name"] = pageItem;
pageMenu[
"to"
] = `/${pageItem.toLowerCase()}/location=${item.toLowerCase()}`;
return pageMenu;
});
return menu;
});
const navigations = [
{
_tag: "CSidebarNavTitle",
_children: ["Site Profile"],
},
siteProfileNavigationsList,
];
console.log(navigations.flat());
A: I think what you're looking for is the destructuring spread syntax.
const arr = [x, y, z]
const anotherArr = [a, b]
const combined = [...anotherArr, ...arr] // [a, b, x, y, z]
The ... "removes" the brackets arround the array.
A: you don't need to remove the brackets, you just need to concatenate your two arrays https://www.w3schools.com/jsref/jsref_concat_array.asp
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,591 |
# ESSENTIAL
CALCULUS
# _with Applications_
RICHARD A. SILVERMAN
DOVER PUBLICATIONS, INC.
New York
_In Memory of_
A.G.S.
Copyright © 1977, 1989 by Richard A. Silverman.
All rights reserved.
This Dover edition, first published in 1989, is a corrected, slightly enlarged republication of the work first published by the W. B. Saunders Company, Philadelphia, 1977. The section "Supplementary Hints and Answers," originally issued in a separate instructor's manual, has been added to the Dover edition by the author, who has also corrected a number of errors in the original text.
**Library of Congress Cataloging-in-Publication Data**
Silverman, Richard A.
Essential calculus with applications / by Richard A. Silverman.
p. cm.
"Corrected, slightly enlarged republication of the work first published by the W. B. Saunders Company, Philadelphia, 1977. The section 'Supplementary Hints and Answers,' originally issued in a separate instructor's manual, has been added to the Dover edition by the author, who has also corrected a number of errors in the original text."
Includes index.
ISBN 0-486-66097-4
1. Calculus. I. Title.
QA303.S55432 1989
515—dc20 | 89-33584
---|---
|
CIP
Manufactured in the United States by Courier Corporation
66097412
www.doverpublications.com
TO THE INSTRUCTOR
The attributes and philosophy of this book are best described by giving a running synopsis of each of the six chapters. This summary is accompanied by open expressions of my pedagogical preferences. Like most authors, I tend to regard these not as idiosyncrasies, but as the only reasonable way to do things! If you disagree in spots, I hope you will attribute this lack of modesty to an excess of enthusiasm, an occupational hazard of those with the effrontery to write books.
The contents of Chapter 1 are often called "precalculus," and are in fact just what that term implies, namely, material that ought to be at one's mathematical fingertips before attempting the study of calculus proper. Opinions differ as to what such a background chapter should contain. Some authors cannot wait to get on with the main show, even at the risk of talking about derivatives to students who are still struggling with straight lines, while others seem unwilling to venture into the heartland of calculus without a year's supply of mathematical rations. I have tried to strike a happy medium by travelling light, but well-equipped. Thus there is a brief section on sets, a larger one on numbers, a little bit on mathematical induction, and quite a lot on inequalities and absolute values, two topics that always seem to give students trouble despite their precalculus character. There is a whole section on intervals, both finite and infinite. The last three sections of the chapter administer a modest dose of analytic geometry, with the emphasis on straight lines and their equations. It should not take long to bring all the students up to the mathematical level of Chapter 1, regardless of their starting points, and those few who are there already can spend their spare time solving extra problems while the others catch up!
The class is now ready to attack Chapter 2, and with it the study of differential calculus. The chapter begins with a rather leisurely and entirely concrete discussion of the function concept. It is my belief that many books adopt too abstract an approach to this important subject. Thus I do not hesitate to use terms like "variable" and "argument," which some may regard as old-fashioned, relegating the mapping and ordered pair definitions of function to the problems. At the same time, I find this a natural juncture to say a few words about functions of several variables. After all, why should one have to wait until the very end of the book to write a simple equation like _F_ ( _x_ , _y_ ) = 0? And what's wrong with a few examples of nonnumerical functions, which crop up all the time in the social sciences? While still in the first three sections of Chapter 2, the student encounters one-to-one functions and inverse functions, and then composite functions and sequences after specializing to numerical functions of a single variable. Graphs of equations and functions are treated in terms of solution sets, with due regard for parity of functions and its consequences for the symmetry of their graphs.
Having mastered the concept of function, in all its various manifestations, the student now arrives at Sec. 2.4, where derivatives and limits are introduced _simultaneously._ I am of the opinion that the novice can hardly develop any respect for the machinery of limits, without first being told that limits are needed to define _derivatives._ Here the development of the individual's understanding must recapitulate the actual historical evolution of the subject. For the same reason, I feel that no time should be wasted in getting down to such brass tacks as difference quotients, rates of change, and increments. Moreover, after defining the tangent to a curve, I find it desirable to immediately say something about differentials. This is a small price to pay for the ability to motivate the ubiquitous " _d_ notation," and differentials have many other uses too (for example, in Secs. 4.6 and 6.2).
It is now time for the student to learn more about limits. This is done in Sec. 2.6, where a number of topics are presented in quick order, namely, algebraic operations on limits, one-sided limits, the key concept of continuity, algebraic operations on continuous functions, and the fact that differentiability implies continuity. Armed with this information, one can now become a minor expert on differentiation, by mastering the material in Secs. 2.7 and 2.8. After establishing the basic differentiation formula ( _x_ _r_ )′ = _rx_ _r_ −1 for _r_ a positive or negative integer, I authorize the student to make free use of the same formula for _r_ an arbitrary real number. Why waste time justifying special cases when the "master formula" itself will be proved once and for all in Sec. 4.4? (However, in a concession to tradition, the validity of the formula for _r_ a rational number is established in the problems, in the usual two ways.) Following a brief discussion of higher derivatives, the student arrives next at the rule for differentiating an inverse function and the all-important chain rule. Unlike most authors, I use a proof of the chain rule which completely avoids the spurious difficulty stemming from the possibility of a vanishing denominator, and which has the additional merit of generalizing at once to the case of functions of several variables (see Sec. 6.3). The method of implicit differentiation is treated as a corollary of the chain rule, and I do not neglect to discuss what can go wrong with the method if it is applied blindly. Chapter 2, admittedly a long one, closes with a comprehensive but concise treatment of limits of other kinds, namely, limits involving infinity, asymptotes, the limit of an infinite sequence, and the sum of an infinite series. Once having grasped the concept of the limit of a function at a point, the student should have little further difficulty in assimilating these variants of the limit concept, and this seems to me the logical place to introduce them.
In Chapter 3 differentiation is used as a tool, and the book takes a more practical turn. I feel that the concept of velocity merits a section of its own, as do related rates and the concept of marginality in economic theory. It is then time to say more about the properties of continuous functions and of differentiable functions, and I do so in that order since the student is by now well aware that continuity is a weaker requirement than differentiability. The highly plausible fact that a continuous image of a closed interval is itself a closed interval leads to a quick proof of the existence of global extrema for a continuous function defined in a closed interval, with the intermediate value theorem as an immediate consequence. The connection between the sign of the derivative of a function at a point and its behavior in a neighborhood of the point is then used to prove Rolle's theorem and the mean value theorem, in turn. With the mean value theorem now available, I immediately exploit the opportunity to introduce the antiderivative and the indefinite integral, which will soon be needed to do integral calculus.
The chapter goes on to treat local extrema, including the case where the function under investigation may fail to be differentiable at certain points. Both the first and second derivative tests for a strict local extremum are proved in a straightforward way, with the help of the mean value theorem. The next section, on concavity and inflection points, is somewhat of an innovation, in that it develops a complete parallelism between the theory of monotonic functions and critical points, on the one hand, and the theory of concave functions and inflection points, on the other. The chapter ends with a discussion of concrete optimization problems, and the three solved examples in Sec. 3.8 are deliberately chosen to be nontrivial, so that the student can have a taste of the "real thing."
It is now Chapter 4, and high time for integral calculus. Here I prefer to use the standard definition of the Riemann integral, allowing the points _ξ i_ figuring in the approximating sum σ to be _arbitrary_ points of their respective subintervals. Students seem to find this definition perfectly plausible, in view of the interpretation of σ as an approximation to the area under the graph of the given function. Once the definite integral is defined, it is immediately emphasized that all continuous functions are integrable, and this fact is henceforth used freely. After establishing a few elementary properties of definite integrals, I prove the mean value theorem for integrals and interpret it geometrically. It is then a simple matter to prove the fundamental theorem of calculus. Next the function ln _x_ is defined as an integral, in the usual way, and its properties and those of its inverse function _e x_ are systematically explored. The related functions log _a_ _x_ , _a x_ and _x r_ are treated on the spot, and the validity of the formula ( _x r_)′ = _rx_ _r_ −1 for arbitrary real _r_ is finally proved, as promised back in Chapter 2. The two main techniques of integration, namely, integration by substitution and integration by parts, are discussed in detail. The chapter ends with a treatment of improper integrals, both those in which the interval of integration is infinite and those in which the integrand becomes infinite.
There are various ways in which integration can be used as a tool, but foremost among these is certainly the use of integration to solve differential equations. It is for this reason that I have made Chapter 5 into a brief introduction to differential equations and their applications. All the theory needed for our purposes is developed in Sec. 5.1, both for first-order and second-order equations. The next section is then devoted to problems of growth and decay, a subject governed by simple first-order differential equations. The standard examples of population growth, both unrestricted and restricted, are gone into in some detail, as is the topic of radioactive decay. The last section of this short chapter is devoted to problems of motion, where second-order differential equations now hold sway. Inclusion of this material may be regarded as controversial in a book like this, but I for one do not see anything unreasonable in asking even a business or economics student to devote a few hours to the contemplation of Newton's mechanics, a thought system which gave birth first to modern industrial society and then to the space age. In any event, those who for one reason or another still wish to skip Sec. 5.3 hardly need my permission to do so.
The last of the six chapters of this book is devoted to the differential calculus of functions of several variables. Here my intent is to highlight the similarities with the one-dimensional case, while not neglecting significant differences. For example, this is why I feel compelled to say a few words about the distinction between differentiable functions of several variables and those that merely have partial derivatives. However, I do not dwell on such matters. It turns out that much of the theory of Chapters 2 and can be generalized almost effortlessly to the _n_ -dimensional case, without doing violence to the elementary character of the book. In particular, as already noted, the proof of the chain rule in Sec. 6.3 is virtually the same as the one in Sec. 2.8. Chapter 6 closes with a concise treatment of extrema in _n_ dimensions, including the test for strict local extrema and the use of Lagrange multipliers to solve optimization problems subject to constraints. I stop here, because unlike some authors I see no point in reproducing the standard examples involving indifference curves, budget lines, marginal rates of substitution, and the like, to be found in every book on microeconomic theory. I conceive of this book as one dealing primarily with the common mathematical ground on which many subjects rest, and the applications chosen here are ones which shed most light on the kind of mathematics we are trying to do, not those which are most intriguing from other points of view.
The idea of writing this book in the first place was proposed to me by John S. Snyder, Jr. of the W. B. Saunders Co. Without his abiding concern, I find it hard to imagine that the book would ever have arrived at its present form. In accomplishing a total overhaul of an earlier draft, I was guided by helpful suggestions from a whole battery of reviewers, notably, Craig Comstock of the Naval Postgraduate School, John A. Pfaltzgraff of the University of North Carolina, J. H. Curtiss of the University of Miami, Carl M. Bruns of Florissant Valley Community College, David Brown of the University of Pittsburgh, and Maurice Beren of the Lowell Technological Institute. The last of these reviewers played a particularly significant role in my revision of Chapter 1. I would also like to thank my friend Neal Zierler for checking all the answers to the problems in the first draft of the book, and my copy-editor Lloyd Black for his patience in dealing with the kind of author who keeps reading proof, looking for trouble, until it is finally taken away from him once and for all. It has been a pleasure to work with all these fine people.
TO THE STUDENT
Calculus cannot be learned without solving lots of problems. Your instructor will undoubtedly assign you many problems as homework, probably from among those that do not appear in the Selected Hints and Answers section at the end of the book. But, at the same time, every hint or answer in that section challenges you to solve the corresponding problem, whether it has been assigned or not. This is the only way that you can be sure of your command of the subject. Problems marked with stars are either a bit harder than the others, or else they deal with side issues. However, there is no reason to shun these problems. They're neither that hard nor that far off the main track.
The system of cross references used in this book is almost self-explanatory. For example, Theorem 1.48 refers to the one and only theorem in Sec. 1.48, Example 2.43b refers to the one and only example in Sec. 2.43b, and so on. Any problem cited without a further address will be found at the end of the section where it is mentioned. The book has a particularly complete index to help you find your way around. Use it freely.
Mathematics books are not novels, and you will often have to read the same passage over and over again before you grasp its meaning. Don't let this discourage you. With a little patience and fortitude, you too will be doing calculus before long. Good luck!
CONTENTS
To the Instructor
To the Student
_Chapter 1_
MATHEMATICAL BACKGROUND
1.1. Introductory Remarks
1.2. Sets
1.3. Numbers
1.4. Inequalities
1.5. The Absolute Value
1.6. Intervals and Neighborhoods
1.7. Rectangular Coordinates
1.8. Straight Lines
1.9. More about Straight Lines
_Chapter 2_
DIFFERENTIAL CALCULUS
2.1. Functions
2.2. More about Functions
2.3. Graphs
2.4. Derivatives and Limits
2.5. More about Derivatives
2.6. More about Limits
2.7. Differentiation Technique
2.8. Further Differentiation Technique
2.9. Other Kinds of Limits
_Chapter 3_
DIFFERENTIATION AS A TOOL
3.1. Velocity and Acceleration
3.2. Related Rates and Business Applications
3.3. Properties of Continuous Functions
3.4. Properties of Differentiable Functions
3.5. Applications of the Mean Value Theorem
3.6. Local Extrema
3.7. Concavity and Inflection Points
3.8. Optimization Problems
_Chapter 4_
INTEGRAL CALCULUS
4.1. The Definite Integral
4.2. Properties of Definite Integrals
4.3. The Logarithm
4.4. The Exponential
4.5. More about the Logarithm and Exponential
4.6. Integration Technique
4.7. Improper Integrals
_Chapter 5_
INTEGRATION AS A TOOL
5.1. Elementary Differential Equations
5.2. Problems of Growth and Decay
5.3. Problems of Motion
_Chapter 6_
FUNCTIONS OF SEVERAL VARIABLES
6.1. From Two to _n_ Dimensions
6.2. Limits and Differentiation
6.3. The Chain Rule
6.4. Extrema in _n_ Dimensions
Tables
Selected Hints and Answers
Supplementary Hints and Answers
Index
_Chapter 1_
MATHEMATICAL BACKGROUND
1.1 INTRODUCTORY REMARKS
**1.11.** You are about to begin the study of calculus, a branch of mathematics which dates back to the seventeenth century, when it was invented by Newton and Leibniz independently and more or less simultaneously. At first, you will be exposed to ideas that you may find strange and abstract, and that may not seem to have very much to do with the "real world." After a while, though, more and more applications of these ideas will put in an appearance, until you finally come to appreciate just how powerful a tool calculus is for solving a host of practical problems in fields as diverse as physics, biology and economics, just to mention a few.
Why this delay? Why can't we just jump in feet first, and start solving practical problems right away? Why must the initial steps be so methodical and careful?
The reason is not hard to find, and it is a good one. You are in effect learning a new language, and you must know the meaning of key words and terms before trying to write your first story in this language, that is, before solving your first nonroutine problem. Many of the concepts of calculus are unfamiliar, and were introduced, somewhat reluctantly, only after it gradually dawned on mathematicians that they were in fact indispensable. This is certainly true of the central concept of calculus, namely, the notion of a "limit," which has been fully understood only for a hundred years or so, after having eluded mathematicians for millennia. Living as we do in the modern computer age, we can hardly expect to learn calculus in archaic languages, like that of "infinitesimals," once so popular. We must also build up a certain amount of computational facility, especially as involves _inequalities_ , before we are equipped to tackle the more exciting problems of calculus. And we must become accustomed to think both algebraically and geometrically at the same time, with the help of rectangular coordinate systems. All this "tooling up" takes time, but nowhere near as much as in other fields, like music, with its endless scales and exercises. After all, in calculus we need only train our minds, not our hands!
It is also necessary to maintain a certain generality in the beginning, especially in connection with the notion of a "function." The power of calculus is intimately related to its great generality. This is why so many different kinds of problems can be solved by the methods of calculus. For example, calculus deals with "rates of change" in general, and not just special kinds of rates of change, like "Velocity," "marginal cost" and "rate of cooling," to mention only three. From the calculus point of view, there are often deep similarities between things that appear superficially unrelated.
In working through this book, you must always have your pen and scratch pad at your side, prepared to make a little calculation or draw a rough figure at a moment's notice. Never go on to a new idea without understanding the old ideas on which it is based. For example, don't try to do problems involving "continuity" without having mastered the idea of a "limit." This is really a workshop course, and your only objective is to learn how to solve calculus problems. Think of an art class, where there is no premium on anything except making good drawings. That will put you in the right frame of mind from the start.
**1.12. Two key problems.** Broadly speaking, calculus is the mathematics of change. Among the many problems it deals with, two play a particularly prominent role, in ways that will become clearer to you the more calculus you learn. One problem is
(1) Given a relationship between two changing quantities, what is the rate of change of one quantity with respect to the other?
and the other, so-called "converse" problem is
(2) Given the rate of change of one quantity with respect to another, what is the relationship between the two quantities?
Thus, from the very outset, we must develop a language in which "relationships," whatever they are, can be expressed precisely, and in which "rates of change" can be defined and calculated. This leads us straight to the basic notions of "function" and "derivative." In the same way, the second problem leads us to the equally basic notions of "integral" and "differential equation." It is the last concept, of an equation involving "rates of change," that unleashes the full power of calculus. You might think of it as "Newton's breakthrough," which enabled him to derive the laws of planetary motion from a simple differential equation involving the force of gravitation. Why _does_ an apple fall?
We will get to most of these matters with all deliberate speed. But we must first spend a few sections reviewing that part of elementary mathematics which is an indispensable background to calculus. Admittedly, this is not the glamorous part of our subject, but first things first! We must all stand on some common ground. Let us begin, then, from a starting point where nothing is assumed other than some elementary algebra and geometry, and a little patience.
1.2 SETS
A little set language goes a long way in simplifying the study of calculus. However, like many good things, sets should be used sparingly and only when the occasion really calls for them.
**1.21.** A collection of objects of any kind is called a _set_ , and the objects themselves are called _elements_ of the set. In mathematics the elements are usually numbers or symbols. Sets are often denoted by capital letters and their elements by small letters. If _x_ is an element of a set _A_ , we may write _x_ ∈ _A_ , where the symbol ∈ is read "is an element of." Other ways of reading _x_ ∈ _A_ are " _x_ is a member of _A_ ," " _x_ belongs to _A_ ," and " _A_ contains _x_." For example, the set of all Portuguese-speaking countries in Latin America contains a single element, namely Brazil.
**1.22.** If every element of a set _A_ is also an element of a set _B_ , we write _A_ ⊂ _B_ , which reads " _A_ is a _subset_ of _B_." If _A_ is a subset of _B_ , but _B_ is not a subset of _A_ , we say that _A_ is a _proper subset_ of _B_. In simple language, this means that _B_ not only contains all the elements of _A_ , but also one or more extra elements. For example, the set of all U.S. Senators is a proper subset of the set of all members of the U.S. Congress.
**1.23. a.** One way of describing a set is to write its elements between curly brackets. Thus the set { _a_ , _b_ , _c_ } is made up of the elements _a_ , _b_ and _c_. Changing the order of the elements does _not_ change the set. For example, the set { _b_ , _c_ , _a_ } is the same as { _a_ , _b_ , _c_ }. Repeating an element does not change a set. For example, the set { _a_ , _a_ , _b_ , _c_ , _c_ } is the same as { _a_ , _b_ , _c_ }.
**b.** We can also describe a set by giving properties that uniquely determine its elements, often using the colon: as an abbreviation for the words "such that." For example, the set { _x_ : _x_ = _x_ 2} is the set of all numbers _x_ which equal their own squares. You can easily convince yourself that this set contains only two elements, namely 0 and 1.
**1.24. Union of two sets.** The set of all elements belonging to at least one of two given sets _A_ and _B_ is called the _union_ of _A_ and _B_. In other words, the union of _A_ and _B_ is made up of all the elements which are in the set _A_ or in the set _B_ , or possibly in both. We write the union of _A_ and _B_ as _A_ _B_ , which is often read " _A_ cup _B_ ," because of the shape of the symbol . For example, if _A_ is the set { _a_ , _b_ , _c_ } and _B_ is the set { _c_ , _d_ , _e_ }, then _A_ _B_ is the set { _a_ , _b_ , _c_ , _d_ , _e_ ).
**1.25. Intersection of two sets.** The set of all elements belonging to both of two given sets _A_ and _B_ is called the _intersection_ of _A_ and _B_. In other words, the intersection of _A_ and _B_ is made up of only those elements of the sets _A_ and _B_ which are in both sets; elements which belong to only one of the sets _A_ and _B_ do not belong to the intersection of _A_ and _B_. We write the intersection of _A_ and _B_ as _A_ _B_ , which is often read " _A_ cap _B_ ," because of the shape of the symbol . For example, if _A_ is the set { _a_ , _b_ , _c_ , _d_ } and _B_ is the set { _b_ , _d_ , _e_ , _f_ , _g_ }, then _A_ _B_ is the set { _b_ , _d_ }.
**1.26. Empty sets.** A set which has no elements at all is said to be an _empty set_ and is denoted by the symbol ∅. For example, the set of unicorns in the Bronx Zoo is empty.
By definition, an empty set is considered to be a subset of every set. This is just a mathematical convenience.
**1.27. Equality of sets.** We say that two sets _A_ and _B_ are _equal_ and we write _A_ = _B_ if _A_ and _B_ have the same elements. If _A_ is empty, we write _A_ = ∅. For example, { _x_ : _x_ = _x_ 2} = {0, 1}, as already noted, while { _x_ : _x_ ≠ _x_ } = ∅ since no number _x_ fails to equal itself!
PROBLEMS
**.** Find all the proper subsets of the set { _a_ , _b_ , _c_ }.
**.** Write each of the following sets in another way, by listing elements:
(a) { _x_ : _x_ = − _x_ }; (b) { _x_ : _x_ \+ 3 = 8}; (c) { _x_ : _x_ 2 = 9};
(d) { _x_ : _x_ 2 − 5 _x_ \+ 6 = 0}; (e) { _x_ : _x_ is a letter in the word "calculus"}.
**.** Let _A_ = {1, 2, {3}, {4, 5}}. Which of the following are true?
(a) 1 ∈ _A_ ; (b) _3_ ∈ _A_ ; (c) {2} ∈ _A_.
How many elements does _A_ have?
**.** Which of the following are true?
(a) If _A_ = _B_ , then _A_ ⊂ _B_ and _B_ ⊂ _A_ ; (b) If _A_ ⊂ _B_ and _B_ ⊂ _A_ , then _A_ = _B_ ; (c) { _x_ : _x_ ∈ _A_ } = _A_ ; (d) {all men over 80 years old} = ∅.
**.** Find the union of the sets _A_ and _B_ if
(a) _A_ = { _a_ , _b_ , _c_ }, _B_ = { _a_ , _b_ , _c_ , _d_ }; (b) _A_ = {1, 2, 3, 4}, _B_ = {−1, 0, 2, 3}.
**.** Find the intersection of the sets _A_ and _B_ if
(a) _A_ = {1, 2, 3, 4}, _B_ = {3, 4, 5, 6}; (b) _A_ = { _a_ , _b_ , _c_ , _d_ }, _B_ = { _f_ , _g_ , _h_ }.
**.** Given any set _A_ , verify that _A_ _A_ = _A_ _A_ = _A_.
**8.** Given any two sets _A_ and _B_ , verify that both _A_ and _B_ are subsets of _A_ _B_ , while _A_ _B_ is a subset of both _A_ and _B_.
**9.** Given any two sets _A_ and _B_ , verify that _A_ ∅ _B_ is always a subset of _A_ _B_. Can _A_ ∅ _B_ ever equal _A_ _B_?
**.** Given any two sets _A_ and _B_ , by the _difference A_ − _B_ we mean the set of all elements which belong to _A_ but not to _B_. Let _A_ = {1, 2, 3}. Find _A_ − _B_ if
(a) _B_ = {1, 2}; (b) _B_ = {4, 5}; (c) _B_ = ∅; (d) _B_ = {1, 2, 3}.
**11.** Which of the following sets are empty?
(a) { _x_ : _x_ is a letter before _c_ in the alphabet};
(b) { _x_ : _x_ is a letter after _z_ in the alphabet};
(c) { _x_ : _x_ \+ 7 = 7};
(d) { _x_ : _x_ 2 = 9 and 2 _x_ = 4}.
***12.** Which of the following sets are empty?
(a) The set of all right triangles whose side lengths are whole numbers;
(b) The set of all right triangles with side lengths in the ratio 5:12:13;
(c) The set of all regular polygons with an interior angle of 45 degrees;
(d) The set of all regular polygons with an interior angle of 90 degrees;
(e) The set of all regular polygons with an interior angle of 100 degrees.
Explain your reasoning.
_Comment_. A polygon is said to be _regular_ if all its sides have the same length and all its interior angles are equal.
***13.** Let _A_ = { _a_ , _b_ , _c_ , _d_ }, and let _B_ be the set of all subsets of _A_. How many elements does _B_ have?
1.3 NUMBERS
In this section we discuss numbers of various kinds, beginning with integers and rational numbers and moving on to irrational numbers and real numbers. The set of all real numbers is called the _real number system_. It is the number system needed to carry out the calculations called for in calculus.
**1.31. The number line.** Suppose we construct a straight line _L_ through a point _O_ and extend it indefinitely in both directions. Selecting an arbitrary unit of measurement, we mark off on the line to the right of _O_ first 1 unit, then 2 units, 3 units, and so on. Next we do the same thing to the left of _O_. The marks to the right of _O_ correspond to the _positive integers_ 1, 2, 3, and so on, and the marks to the left of _O_ correspond to the _negative integers_ −1, −2, −3, and so on. The line _L_ , "calibrated" by these marks, is called the _number line_ , and the point _O_ is called the _origin_ (of _L_ ). The direction from negative to positive numbers along _L_ is called the _positive direction_ , and is indicated by the arrowhead in Figure 1.
Figure 1.
**1.32. Integers**
**a.** The set of positive integers is said to be _closed_ under the operations of addition and multiplication. In simple language, this means that if we add or multiply two positive integers, we always get another positive integer. For example, 2 + 3 = 5 and 2 · 3 = 6, where 5 and 6 are positive integers. On the other hand, the set of positive integers is _not_ closed under subtraction. For example, 2 − 3 = −1, where −1 is a negative integer, rather than a positive integer.
The number 0 corresponding to the point _O_ in Figure 1 is called _zero_. It can be regarded as an integer which is neither positive nor negative. Following mathematical tradition, we use the letter _Z_ to denote the set of all integers, positive, negative and zero. The set _Z_ , unlike the set of positive integers, is closed under subtraction. For example, 4 − 2 = 2, 3 − 3 = 0 and 2 − 5 = −3, where the numbers 2, 0 and −3 are all integers, whether positive, negative or zero.
**b.** An integer _n_ is said to be an _even number_ if _n_ = 2 _k_ , where _k_ is another integer, that is, if _n_ is divisible by 2. On the other hand, an integer _n_ is said to be an _odd number_ if _n_ = 2 _k_ \+ 1, where _k_ is another integer, that is, if _n_ is not divisible by 2, or equivalently leaves the remainder 1 when divided by 2. It is clear that every integer is either an even number or an odd number.
**1.33. Rational numbers.** The set _Z_ is still too small from the standpoint of someone who wants to be able to _divide_ any number in _Z_ by any other number in _Z_ and still be sure of getting a number in _Z_. In other words, the set _Z_ is not closed under division. For example, 2 ÷ 3 = and −4 ÷ 3 = , where and − are fractions, not integers. Of course, the quotient of two integers is _sometimes_ an integer, and this fact is a major preoccupation of the branch of mathematics known as _number theory_. For example, 8 ÷ 4 = 2 and 10 ÷ −5 = −2. However, to make division possible in general, we need a bigger set of numbers than _Z_. Thus we introduce _rational numbers_ , namely fractions of the form _m/n_ , where _m_ and _n_ are both integers and _n is not zero_. Note that every integer _m_ , including zero, is a rational number, since _m_ /1 = _m_.
Let _Q_ (for "quotient") denote the set of all rational numbers. Then the set _Q_ is closed under the four basic arithmetical operations of addition, subtraction, multiplication and division, _provided that we never divide by zero_. It cannot be emphasized too strongly that _division by zero is a forbidden operation_ in this course. These matters are considered further in Problems 3 and 13.
Figure 2.
**1.34. Irrational numbers**
**a.** With respect to the number line, the rational numbers fill up the points corresponding to the integers and many but _not all_ of the points in between. In other words, there are points of the number line which do _not_ correspond to rational numbers. To see this, suppose we construct a right triangle _PP′O_ with sides _PP′_ and _P′O_ of length 1, as in Figure 2A. Then, by elementary geometry, the side _OP_ is of length (use the familiar Pythagorean theorem). Suppose we place the side _OP_ on the number line, as in Figure 2B, with the point _O_ coinciding with the origin of the line. Then the point _P_ corresponds to the number . But, as mathematicians concluded long ago, the number cannot be rational, and therefore _P_ is a point of the number line which does not correspond to a rational number.
**b.** By an _irrational number_ we simply mean a number, like , which is not rational. To demonstrate that is irrational, we argue as follows. First we digress for a moment to show that the result of squaring an odd number (Sec. 1.32b) is always an odd number. In fact, every odd number is of the form 2 _k_ \+ 1, where _k_ is an integer, and, conversely, every number of this form is odd. But, squaring the expression 2 _k_ \+ 1, we get
(2 _k_ \+ 1)2 = 4 _k_ 2 \+ 4 _k_ \+ 1 = 2(2 _k_ 2 \+ 2 _k_ ) + 1,
which is odd, since 2 _k_ 2 \+ 2 _k_ is itself an integer (why?).
Now, returning to the main argument, suppose is a rational number. Then _ _must be of the form _m/n_ , where _m_ and _n_ are positive integers and we can assume that the fraction _m/n_ has been reduced to lowest terms, so that _m_ and _n_ are no longer divisible by a common factor other than 1. (For example, the fraction is not in lowest terms, but the equivalent fraction is.) We can then write
Squaring both sides of (1), we have
or equivalently
Thus _m 2_ is an even number, being divisible by 2, and therefore the number _m_ itself must be even, since if _m_ were odd, _m_ 2 would also be odd, as shown in the preceding paragraph. Since _m_ is even, we can write _m_ in the form
where _k_ is a positive integer. Squaring both sides of (3), we have
Substituting (4) into (2), we get
4 _k_ 2 = 2 _n_ 2,
or equivalently
But then _n 2_ is an even number, and hence so is _n_ , for the reason just given in connection with _m_ 2 and _m_.
Thus we have managed to show that _m_ and _n_ are both even numbers. Therefore _m_ and _n_ are both divisible by 2. But this contradicts the original assumption that the fraction _m/n_ has been reduced to lowest terms. Since we run into a contradiction if we assume that is a rational number, we must conclude that is an irrational number. This fact was known to the ancient Greeks, who proved it in just the same way.
**c.** There are many other irrational numbers. For example, the square roots and are all irrational, and so is π, the ratio of the circumference of a circle to its diameter. For convenience, we use the letter _I_ to designate the set of all irrational numbers.
**1.35. The real number system.** Let _R_ be the set made up of _Q_ , the set of all rational numbers, and _I_ , the set of all irrational numbers. In other words, let _R_ be the union of _Q_ and _I_ , in the language of sets. Thus
_R_ = _Q_ _I_
in symbolic notation (Sec. 1.24). The set _R_ is called the _real number system_ , and its elements are called the _real numbers_. From now on, when we use the word "number" without further qualification, we will always mean a _real_ number.
**1.36. Properties of the real numbers**
Next we list several useful facts about real numbers. The student who finds these things interesting is encouraged to pursue them further by visiting the library and looking up a more detailed treatment of the subject.
**a.** There is one and only one point on the number line corresponding to any given real number, and, conversely, there is one and only one real number corresponding to any given point on the number line. For this reason, the number line is often called the _real line_. In mathematical language, we say that there is a _one-to-one correspondence_ between the real numbers and the points of the real line, or between the real number system and the real line itself.
**b.** Let _N_ be any positive integer, _no matter how large_. Then between any two distinct real numbers there are _N_ other real numbers. Since _N_ is as large as we please, this can be expressed mathematically by saying that between any two distinct real numbers there are _arbitrarily many_ real numbers, or better still, _infinitely many_ real numbers. In view of the one-to-one correspondence between the real numbers and the points of the real line, this fact is geometrically obvious, since between any two distinct points of the real line we can clearly pick as many other points as we please.
**c.** If a rational number is expressed in decimal form, the decimal either terminates in some digit from 1 to 9 or else the decimal does not terminate, but continues indefinitely with groups of repeated digits after a certain decimal place. For example, the rational numbers are represented by the terminating decimals 0.5, 0.2, 0.125, 0.1 and 0.0625, respectively, while the rational numbers are represented by the repeating decimals and . Here the dots ... mean "and so on forever," and the digits written beneath the horizontal line repeat over and over again. Actually, a terminating decimal can be regarded as a special kind of repeating decimal, namely, one with an endless run of zeros after a certain decimal place. Thus and so on.
**d.** Conversely, if a number in decimal form is a repeating decimal (which includes the case of a terminating decimal), then the number is a rational number, and it can be put in the form of a fraction _m/n_.
**e.** If an irrational number is expressed in decimal form, the decimal does not terminate, but continues indefinitely with _no_ groups of repeated digits. For example, ..., where the dots ... again mean "and so on forever," but this time with no groups of repeated digits. Conversely, if a number in decimal form is this kind of nonrepeating decimal, then the number is an irrational number.
**f.** It follows from the foregoing that there is a one-to-one correspondence between the real number system and the set of all decimals, repeating and nonrepeating.
**1.37. Mathematical induction**
**a.** In mathematics we often encounter assertions or formulas involving an arbitrary positive integer _n_. As an example, consider the formula
which asserts that the sum of the first _n_ odd integers equals the square of _n_. Here the dots ... indicate the missing terms, if any, and it is understood that the left side of (5) reduces to simply 1 if _n_ = 1, 1 + 3 if _n_ = 2, and 1 + 3 + 5 if _n_ = 3. To prove a formula like (5), we can use the following important technique, known as the principle of _mathematical induction_. Suppose that the formula (or assertion) is known to be true for _n_ = 1, and suppose that as a result of assuming that it is true for _n_ = _k_ , where _k_ is an arbitrary positive integer, we can prove that it is also true for _n_ = _k_ \+ 1. _Then the formula is true for all k_.
The reason why mathematical induction works is perfectly clear: First we choose _k_ = 1 and use the truth of the formula _n_ = _k_ = 1 to deduce its truth for _n_ = _k_ \+ 1 = 2. This shows that the formula is true for _n_ = 2. Playing the same game again, we now choose _k_ = 2 and use the truth of the formula for _n_ = _k_ = 2 to deduce its truth for _n_ = _k_ \+ 1 = 3. Doing this over and over again, we can prove the truth of the formula for every positive integer _n_ , _no matter how large_.
**b.** Thus, to prove formula (5), we first note that (5) is certainly true for _n_ = 1, since it then reduces to the trivial assertion that
1 = 12.
Suppose (5) holds for _n_ = _k_ , so that we have
Then, adding 2 _k_ \+ 1 to both sides of (6), where 2 _k_ \+ 1 is the next odd number after 2 _k_ − 1, we get
The expression on the right clearly equals ( _k_ \+ 1)2, so that (7) takes the form
1 + 3 + 5 + ... + (2 _k_ − 1) + (2 _k_ \+ 1) = ( _k_ \+ 1)2.
But this is just the form taken by (5) if _n_ = _k_ \+ 1, since then
2 _n_ − 1 = 2( _k_ \+ 1) − 1 = 2 _k_ \+ 1.
In this way, we have shown that if (5) is true for _n_ = _k_ , it is also necessarily true for _n_ = _k_ \+ 1. Therefore, by the principle of mathematical induction, (5) is true _for all n_ starting from _n_ = 1.
**c.** The truth of the assertion for _n_ = 1 is only needed to "get the induction started." This condition can be relaxed. For example, to give a rather wild example, suppose the assertion is known to be true for _n_ = 8, and suppose its truth for _n_ = _k_ implies its truth for _n_ = _k_ \+ 1. Then the assertion is true for all _n_ = 8, 9,..., that is, for all _n_ starting from 8. This is actually the situation in Problem 20.
PROBLEMS
**.** Is the set of negative integers closed under the operation of addition? Give numerical examples to show that the set of negative integers is not closed under the operations of subtraction, multiplication and division.
**.** Give numerical examples to show that
(a) The sum of two rational numbers is a rational number;
(b) The difference of two rational numbers is a rational number;
(c) The product of two rational numbers is a rational number;
(d) The quotient of two rational numbers is a rational number.
**.** Show algebraically that the set of rational numbers is closed under the operation of addition. Do the same for the operations of subtraction, multiplication and division.
**.** Which of the following exist?
(a) A largest positive integer;
(b) A smallest positive integer;
(c) A largest positive integer less than 100;
(d) A smallest positive integer greater than 100.
**.** Is the number 1 − rational or irrational? Explain your answer.
**.** Give an example to show that the sum of two irrational numbers can be a rational number. How about the difference of two irrational numbers?
**.** Give an example to show that the product of two irrational numbers can be a rational number. How about the quotient of two irrational numbers?
**.** What conclusions can you draw from Problems 6 and 7 about whether or not the set of irrational numbers is closed under the operations of addition, subtraction, multiplication and division?
**.** Prove that 0 · _c_ = 0 for every real number _c_.
**10.** Give examples other than those in the text of rational numbers which terminate when expressed in decimal form.
**11.** Give examples other than those in the text of rational numbers which continue indefinitely with groups of repeated digits when expressed in decimal form.
**.** Use mathematical induction to prove that
for all _n_ = 1, 2,...
**.** Let _a_ be any real number, possibly zero. Why is the expression _a_ /0 meaningless? In other words, why is division by zero impossible?
_Comment_. On the other hand, if _a_ ≠ 0, then 0/ _a_ is a perfectly respectable number, equal to 0.
**.** It can be shown (Sec. 1.4, Prob. 12) that between any two rational numbers there is another rational number. Illustrate this statement by inserting another rational number between .
***15.** Verify that by adding the corresponding decimals. What conclusions can you draw from this about any decimal with an endless run of nines after a certain decimal place?
***16.** Which rational number (in lowest terms) is expressed by the following repeating decimal?
***17.** Explain why a rational number, when expressed in decimal form, either terminates or continues indefinitely with groups of repeated digits.
***18.** It can be shown (Sec. 1.5, Prob. 13) that between any two irrational numbers there is another irrational number. Illustrate this statement by inserting another irrational number between = 1.414213562 ... and 1.414215784...
***19.** Use mathematical induction to prove that
for all _n_ = 1, 2,...
***20.** Verify that every integer greater than seven can be written as a sum made up of threes and fives exclusively. For example, 8 = 3 + 5, 9 = 3 + 3 + 3, 10 = 5 + 5, 11 = 3 + 3 + 5, and so on.
1.4 INEQUALITIES
**1.41.** Let _a_ and _b_ be any two numbers. Then there are only three, mutually exclusive possibilities:
(1) Either _a equals b_ , written _a_ = _b_ ;
(2) Or _a is greater than b_ , written _a_ > _b_ ;
(3) Or _a is less than b_ , written _a_ < _b_.
On the _real line_ , _a_ > _b_ simply means that the point corresponding to the number _a_ lies to the right of the point corresponding to the number _b_ , or, in simpler language, that "the point _a_ " lies to the right of "the point _b_ " (Sec. 1.56). Similarly, _a_ < _b_ means that the point _a_ lies to the left of the point _b_. Note that _a_ > _b_ and _b_ < _a_ mean exactly the same thing, and so do _a_ < _b_ and _b_ > _a_.
Another way of saying that _a_ is greater than _b_ is to say that if _b_ is subtracted from _a_ , we get a _positive_ number, that is, a number greater than zero, while if _a_ is subtracted from _b_ , we get a _negative_ number, that is, a number less than zero. In other words, _a_ > _b_ , _a_ − _b_ > 0 and _b_ − _a_ < 0 all mean exactly the same thing, and similarly, so do _a_ < _b_ , _b_ − _a_ > 0 and _a_ − _b_ < 0. We regard it as a known fact that if _a_ and _b_ are both positive numbers, then so are the sum _a_ \+ _b_ and the product _ab_.
Formulas involving the symbols > and < (or the symbols ≥ and ≤ to be introduced in Sec. 1.47) are called _inequalities_. There are several theorems about inequalities which are both intuitively reasonable and very easy to prove. We now prove some of these which are particularly useful.
**1.42.** THEOREM. _Adding the same number to each side of an inequality does not change the sense of the inequality. That is_ ,
_where c is any number at all, while_
_Proof_. To prove (1), we need only show that ( _a_ \+ _c_ ) − ( _b_ \+ _c_ ) > 0, which means exactly the same thing as _a_ \+ _c_ > _b_ \+ _c_. But
( _a_ \+ _c_ ) − ( _b_ \+ _c_ ) = ( _a_ − _b_ ) + ( _c_ − _c_ ) = ( _a_ − _b_ ) + 0 = _a_ − _b_ > 0,
since _a_ > _b_. The proof of (1′) is just as easy, and is left as an exercise.
The symbol is a modern way of indicating the end of a proof. The old-fashioned way is "Q.E.D.," which you may recall from elementary geometry.
**1.43.** THEOREM. _Multiplying both sides of an inequality by the same positive number does not change the sense of the inequality. That is_ ,
_While_
_Proof_. To prove (2), we need only show that _ac_ − _bc_ > 0, which means exactly the same thing as _ac_ > _bc_. But _a_ − _b_ is positive, since _a_ > _b_ , and _c_ is positive, by hypothesis. Therefore the product ( _a_ − _b_ ) _c_ = _ac_ − _bc_ is also positive, since the product of two positive numbers is a positive number. Thus _ac_ − _bc_ > 0, as required. The proof of (2′) is just as easy, and is left as an exercise.
**1.44.** THEOREM. _Multiplying both sides of an inequality by the same **negative** number **changes** the sense of the inequality. That is_,
_While_
_Proof_. To prove (3), we need only show that _ac_ − _bc_ < 0, which means exactly the same thing as _ac_ < _bc_. But _a_ − _b_ is positive, since _a_ > _b_ , and _c_ is negative, by hypothesis. Therefore the product ( _a_ − _b_ ) _c_ = _ac_ − _bc_ is negative, since the product of a positive number and a negative number is a negative number. Thus _ac_ − _bc_ < 0, as required. The proof of (3′) is just as easy, and is left as an exercise.
**1.45.** THEOREM. _If a_ > _b and b_ > _c_ , then _a_ > _c_.
_Proof_. By hypothesis, _a_ − _b_ > 0 and _b_ − _c_ > 0. But then
( _a_ − _b_ ) + ( _b_ − _c_ ) = _a_ − _c_ > 0,
since the sum of two positive numbers is positive. Alternatively, the theorem follows at once by examining the relative positions of _a_ , _b_ and _c_ , regarded as points on the real line (give the details).
**1.46.** THEOREM. _Let a and b be positive numbers such that a > b. Then_
_Proof_. To prove (4), we need only show that
Writing
we note that _a_ − _b_ > 0, since _a_ > _b_ , while _ab_ > 0, since _a_ > 0 and _b_ > 0. It follows that the expression on the right in (5) is positive, being the quotient of two positive numbers.
**1.47.** In dealing with inequalities, it is a great convenience to introduce the symbol ≥, which means "is either greater than or equal to," and the symbol ≤, which means "is either less than or equal to." Thus _a_ ≥ _b_ means " _a_ is either greater than or equal to _b_ ," while _a_ ≤ _b_ means " _a_ is either less than or equal to _b_." It is possible for both inequalities _a_ ≥ _b_ and _a_ ≤ _b_ to be valid simultaneously, but only if _a_ is actually equal to _b_ , since the three possibilities listed in Sec. 1.41 are mutually exclusive. In other words, _a_ ≥ _b_ and _a_ ≤ _b_ together imply _A_ = _B_.
Clearly _a_ ≥ _b_ means exactly the same thing as _a_ − _b_ ≥ 0, while _a_ ≤ _b_ means exactly the same thing as _a_ − _b_ ≤ 0.
**1.48.** Here is another theorem on inequalities, this time involving the symbol ≤ :
THEOREM. _If a_ ≤ _b and c_ ≤ _d_ , _then_
_Proof_. As just noted, _a_ ≤ _b_ means exactly the same thing as _a_ − _b_ ≤ 0, while _c_ ≤ _d_ means exactly the same thing as _c_ − _d_ ≤ 0. But the sum of two numbers which are negative or zero is itself a number which is negative or zero. In other words, the sum of two nonpositive numbers is a nonpositive number. It follows that
( _a_ − _b_ ) + ( _c_ − _d_ ) = ( _a_ \+ _c_ ) − ( _b_ \+ _d_ ) ≤ 0,
which means exactly the same thing as (6).
**1.49.** Inequalities are often combined. For example, _a_ ≥ _b_ > _c_ means that both inequalities _a_ ≥ _b_ and _b_ > _c_ hold simultaneously. Similarly, _d_ < _e_ ≤ _f_ means that both _d_ < _e_ and _e_ ≤ _f_ hold simultaneously. Thus we have
Give other examples involving the same numbers. Bear in mind that by , where _x_ is a positive number, we always mean the _positive_ square root of _x_. Thus, for example, equals 2, never − 2.
PROBLEMS
**.** Show that
(a) If _a_ > _b_ , then − _a_ < − _b_ ;
(b) If _a_ > _b_ and _c_ > _d_ , then _a_ \+ _c_ > _b_ \+ _d_.
**.** Given two unequal rational numbers _p_ = _m/n_ and _p_ ′ = _m_ ′/ _n_ ′, written with positive denominators (as is always possible), show that _p_ > _p_ ′ is equivalent to _mn_ ′ > _m_ ′ _n_ , while _p_ < _p′_ is equivalent to _mn′_ < _m′n_.
**.** Which is larger?
**.** Verify that if _a_ > _b_ > 0 and _c_ > _d_ > 0, then _ac_ > _bd_ > 0.
**.** Verify that if _a_ > 0, _b_ > 0 and _b_ 2 > _a_ 2, then _b_ > _a_. Use this to confirm that .
**.** Show that _a_ 2 > _a_ if _a_ > 1, while _a_ 2 < _a_ if 0 < _a_ < 1. When does _a_ 2 = _a_?
**7.** Verify that
Write this in another way, using the symbol ≤ instead.
**8.** Show that
(a) If _a_ ≥ _b_ , then − _a_ ≤ − _b_ ;
(b) If _a_ ≥ _b_ and _b_ ≥ _c_ , then _a_ ≥ _c_ ;
(c) If _a_ ≥ _b_ and _b_ > _c_ or if _a_ > _b_ and _b_ ≥ _c_ , then _a_ > _c_.
**9.** Verify that if _a_ ≥ _b_ and _c_ > 0, then _ac_ ≥ _bc_ , while if _a_ ≥ _b_ and _c_ < 0, then _ac_ ≤ _bc_.
**.** Given a number _x_ , the largest integer less than or equal to _x_ is called the _integral part_ of _x_ and is denoted by [ _x_ ], not to be confused with { _x_ }, the set whose only element is _x_. Find
**.** Let _n_ be an integer. Find
(a) [ _n_ ]; (b) [ _n_ \+ ] (c) [ _n_ − ]
***12.** Let _p_ and _q_ be two rational numbers such that _p_ < _q_. Show that the number ( _p_ \+ _q_ ) is also rational and lies between _p_ and _q_. Use this to show that there is no largest rational number less than 1, and no smallest rational number greater than 0. Can we change the word "rational" to "real" here?
***13.** Verify that
(a) _a_ 2 \+ _b_ 2 ≥ 2 _ab_ ; (b) ( _a_ \+ _b_ )2 ≤ 4 _ab_ ;
***14.** The _arithmetic mean_ of two positive numbers _x_ and _y_ is defined as
and the _geometric mean_ as
Verify that _g_ < _a_ unless _x_ = _y_ , in which case _g_ = _a_.
***15.** Use the preceding problem to show that of all rectangles with a given perimeter (combined side length), the square has the greatest area.
1.5 THE ABSOLUTE VALUE
**1.51.** By the _absolute value_ of a number _x_ we mean the number which equals _x_ if _x_ is nonnegative and − _x_ if _x_ is negative. If _x_ is expressed in decimal form, then the absolute value of _x_ is just the decimal without its minus sign if it has one. The absolute value of _x_ is denoted by | _x_ |, with two vertical lines (never with brackets). In other words,
Thus, for example, |2.2| = 2.2, |−3.1| = −(−3.1) = 3.1, |0| = 0. Note that |− _x_ | = | _x_ | for any number _x_. For example, |−3.1| = |3.l| = 3.1.
**1.52.** THEOREM. _The inequalities_
_and_
_hold for arbitrary numbers x and y._
_Proof_. To prove (2), we merely note that, by (1), _x_ = | _x_ | if _x_ ≥ 0, while _x_ = −| _x_ | if _x_ < 0. Therefore (2) holds in either case.
To prove (3), we write
as well as (2). It then follows from Theorem 1.48, with _a_ = _x_ , _b_ = | _x_ |, _c_ = _y_ , _d_ = | _y_ |, that
and from the same theorem, this time with _a_ = −| _x_ |, _b_ = _x_ , _c_ = −| _y_ |, _d_ = _y_ , that
But (5) and (5′) together imply (3). In fact, if _x_ \+ _y_ ≥ 0, then _x_ \+ _y_ = | _x_ \+ _y_ |, so that (5) is equivalent to (3), while if _x_ \+ _y_ < 0, then _x_ \+ _y_ = −| _x_ \+ _y_ |, so that (5′) becomes −| _x_ | − | _y_ | ≤ −| _x_ \+ _y_ |, which is again equivalent to (3).
**1.53.** According to (3), the absolute value of the sum of any two numbers is either less than or equal to the sum of the absolute values of the numbers. More concisely, "the absolute value of a sum cannot exceed the sum of the absolute values." You should convince yourself by testing the various possibilities that (3) reduces to the equality | _x_ \+ _y_ | = | _x_ | + | _y_ | when _x_ and _y_ have the same sign or at least one of the numbers _x_ and _y_ is zero, and that (3) can be replaced by the "strict" inequality | _x_ \+ _y_ | < | _x_ | + | _y_ | when _x_ and _y_ have opposite signs. Formula (3) is often called the "triangle inequality," for a reason we do not go into here.
**1.54. The coordinate of a point on the real line.** As we have seen in Sec. 1.36a, there is one and only one point on the real line corresponding to any given real number, and, conversely, there is one and only one real number corresponding to any given point on the real line. Thus, to specify a point _P_ on the line, we need only give the real number corresponding to _P_. This number is called the _coordinate_ of _P_.
Let _d_ be the distance between the origin _O_ , namely the point with coordinate zero, and the point _P_. Then _P_ has the coordinate _d_ if _P_ lies to the right of _O_ (see Figure 3A) and the coordinate − _d_ if _P_ lies to the left of _O_ (see Figure 3B). If the point _P_ coincides with the origin _O_ , its distance from _O_ is zero, and therefore so is its coordinate.
Figure 3.
Conversely, suppose _P_ has the coordinate _x_ , where _x_ is any real number. Then _P_ is just the point at distance | _x_ | from _O_ , lying to the right of _O_ if _x_ is positive (see Figure 4A) and to the left of _O_ if _x_ is negative (see Figure 4B). If _x_ = 0, then _P_ clearly coincides with _0_.
Figure 4.
**1.55. The distance between two points on the real line**
THEOREM. _Let P_ 1 _and P_ 2 _be two points on the real line with coordinates x_ 1 _and x_ 2, _respectively, and let d be the distance between P_ 1 _and P_ 2. _Then_
Figure 5.
Figure 6.
_Proof_. Let _P_ 1 and _P_ 2 both lie to the right of the origin _O_. Then formula (6) follows from Figure 5A if _P_ 1 lies to the right of _P_ 2, and from Figure 5B if _P_ 1 lies to the left of _P_ 2. On the other hand, if _P_ 1 or _P_ 2 (or both) lies to the left of _O_ , as in Figure 6, then we can replace _O_ by a new origin _O_ ′, a distance _c_ to the left of _O_ , such that _P_ 1 and _P_ 2 both lie to the right of _O_ ′. With respect to the new origin _O_ ′, the points _P_ 1 and _P_ 2 have coordinates _c_ \+ _x_ 1 and _c_ \+ _x_ 2, as is apparent from the figure, where the point _P_ 1 lies to the left of _O_ and the distance from _O_ ′ to _P_ 1 equals _c_ − | _x_ 1| = _c_ \+ _x_ 1. Therefore, by the first part of the proof, we now have
_d_ = |( _c_ \+ _x_ 1) − ( _c_ \+ _x_ 2)| = |( _c_ − _c_ ) + ( _x_ 1 − _x_ 2)| = | _x_ 1 − _x_ 2|,
so that formula (6) is still valid.
**1.56.** In talking about real numbers, we will make free use of geometrical language whenever it seems appropriate. In particular, we will usually say "the point _x_ " instead of "the point with coordinate _x_." The distance between the points _P_ 1 and _P_ 2 will often be denoted by | _P_ 1 _P_ 2|, as suggested by the absolute value in (6). The same "double vertical line notation" will also be used for the distance between points in the plane and in space.
PROBLEMS
**.** Verify that | _xy_ | = | _x_ | | _y_ |.
_Comment_. Thus "the absolute value of a product equals the product of the absolute values."
**2.** Show that
**3.** Verify that | _x_ |2 = _x_ 2 and | _x_ | = for every number _x_ , regardless of the sign of _x_.
**.** Show that
| _x_ \+ _y_ \+ _z_ | ≤ | _x_ | + | _y_ | + | _z_ |
for arbitrary numbers _x_ , _y_ , _z_.
**5.** More generally, show that
| _x_ 1 \+ _x_ 2 \+ ... + _x_ _n_ | ≤ | _x_ 1| + | _x_ 2| + ... + | _x_ _n_ |
for arbitrary numbers _x_ 1, _x_ 2, . . ., _x_ _n_.
**6.** Which points are at distance 2 from the point −1?
**.** Find the two points which are four times closer to the point −1 than to the point 4.
**.** When does the point _x_ 2 lie to the right of the point _x_? When does it lie to the left? When do the two points coincide?
**.** If _P_ 1 and _P_ 2 are the points with coordinates _x_ 1 and _x_ 2, verify that the point with coordinate ( _x_ 1 \+ _x_ 2) is the midpoint of the segment _P_ 1 _P_ 2.
***10.** Show that
| _x_ − _y_ | ≥ || _x_ | − | _y_ ||
for arbitrary _x_ and _y_. When does equality occur?
***11.** Solve the equation
(a) | _x_ − 1| = 2; (b) | _x_ − 1| = |3 − _x_ |; (c) | _x_ \+ 1| = |3 + _x_ |; (d) |2 _x_ | = | _x_ − 2|.
In other words, find the set of all numbers _x_ satisfying the equation.
***12.** What happens to the point (1 − _x_ ) _a_ \+ _xb_ as _x_ varies from 0 to 1?
***13.** Verify that between any two real numbers _x_ 1 and _x_ 2
(a) There is a rational number and an irrational number;
(b) There are infinitely many rational numbers and infinitely many irrational numbers.
1.6 INTERVALS AND NEIGHBORHOODS
**1.61. Intervals**
**a.** Let _a_ and _b_ be any two real numbers such that _a_ < _b_ , and consider the set _I_ of all real numbers _x_ such that _x_ is greater than _a_ but less than _b_. Then _I_ is called an _open interval_ and is denoted by the symbol ( _a_ , _b_ ). Note that _I_ does _not_ include the points _x_ = _a_ and _x_ = _b_ , called the _end points_ of the interval. We can also denote ( _a_ , _b_ ) by writing _a_ < _x_ < _b_ , it being understood that _a_ < _x_ < _b_ is shorthand for the _set I_ = { _x_ : _a_ < _x_ < _b_ }.
**b.** Suppose we enlarge the open interval ( _a_ , _b_ ) by including the end points _x_ = _a_ and _x_ = _b_. Then the resulting set is called a _closed interval_ and is denoted by the symbol [ _a_ , _b_ ], with square brackets instead of round brackets (parentheses). Since [ _a_ , _b_ ] is clearly the set of all _x_ such that _x_ is greater than or equal to _a_ but less than or equal to _b_ , we can also denote [ _a_ , _b_ ] by writing _a_ ≤ _x_ ≤ _b_ , this being shorthand for the set { _x_ : _a_ ≤ _x_ ≤ _b_ }.
**c.** Sometimes it is convenient to speak of intervals which include one end point but not the other. Thus we have the interval [ _a, b_ ), which includes the left end point _a_ but excludes the right end point _b_ , or the interval ( _a, b_ ], which includes the right end point _b_ but excludes the left end point _a_. Note the crucial difference between the meaning of a round bracket ( or ) and a square bracket [ or ]. We can also write [ _a, b_ ) as _a_ ≤ _x_ ≤ _b_ and ( _a_ , _b_ ] as _a_ < _x_ ≤ _b_. These intervals, which are neither open nor closed, might be regarded as "half open," but they might just as well be regarded as "half-closed." The intervals ( _a_ , _b_ ), [ _a_ , _b_ ], [ _a_ , _b_ ) and ( _a_ , _b_ ] are all assigned the same _length_ , namely _b_ − _a_.
**d.** The geometrical meaning of the various kinds of intervals is shown in Figure 7, where included end points are indicated by solid dots and excluded end points by hollow dots.
**1.62. Examples**
**a.** Find the open interval _a_ < _x_ < _b_ or ( _a_ , _b_ ) equivalent to
Figure 7.
SOLUTION . Each of the two inequalities in (1), namely
and
remains valid if we add the same number to both sides. Adding −2 to both sides of (2), or equivalently subtracting 2 from both sides, we get
Similarly, subtracting 2 from both sides of (3), we get
Combining (2′) and (3′), we find that the open interval equivalent to (1) is
−7 < _x_ < 1.
This is just the interval (−7, 1) in bracket form.
**b.** Find the closed interval _a_ ≤ _x_ ≤ _b_ or [ _a_ , _b_ ] equivalent to
SOLUTION . Again, each of the inequalities in (4), namely
remains valid if we add the same number to both sides. The only sensible choice of the number to be added is 5, of course, and this converts the inequalities (5) to
Combining these inequalities, we find that the closed interval equivalent to (4) is
6 ≤ _x_ ≤ 9,
or [6, 9] in bracket form.
**1.63. Neighborhoods**
**a.** By a _neighborhood_ of a point _c_ we mean any open interval with _c_ as its midpoint. Thus, for example, the intervals (−1, 1), ( −2, 2) and (−3, 3) are all neighborhoods of the origin of the real line, that is, of the point _x_ = 0. If we exclude the midpoint _c_ from any neighborhood of _c_ , the resulting set is called a _deleted neighborhood_ of _c_. Note that a deleted neighborhood is the union of _two_ open intervals, rather than a single open interval.
**b.** Let _δ_ (the Greek letter delta) denote any positive number. Then by the _δ-neighborhood_ of a point _c_ we mean the neighborhood of _c_ of length 2 _δ_. In other words, the _δ_ -neighborhood of _c_ is the open interval _c_ − _δ_ < _x_ < _c_ \+ _δ_ , or equivalently ( _c_ − _δ_ , _c_ \+ _δ_ ), shown in Figure 8A. The corresponding _deleted δ-neighborhood_ of _c_ is the union of intervals ( _c_ − _δ_ , _c_ ) ( _c_ , _c_ \+ _δ_ ), shown in Figure 8B, where the hollow dot indicates the missing point _c_. Note that ( _c_ − _δ_ , _c_ \+ _δ_ ) can also be described as the set of all _x_ whose distance from _c_ is less than _δ_ , and hence, by Theorem 1.55, as the set of all _x_ such that | _x_ − _c_ | < _δ_. Similarly, ( _c_ − _δ_ , _c_ ) ( _c_ , _c_ \+ _δ_ ) can be described as the set of all _x_ such that 0 < | _x_ − _c_ | < _δ_.
Figure 8.
There is nothing sacred about the use of the letter _δ_ in this context, apart from mathematical tradition, and we could use any other letter as well. A common choice is _ε_ (the Greek letter epsilon).
**c. Example.** Find the 1-neighborhood of the point 2.
SOLUTION . Here _c_ = 2, _δ_ = 1, and the neighborhood is just the open interval 2 − 1 < _x_ < 2 + 1, namely 1 < _x_ < 3 or (1, 3). The corresponding deleted 1-neighborhood is the set (1, 2) (2, 3).
**1.64. Infinite intervals**
**a.** In discussing intervals, it is convenient to introduce two new symbols. These are ∞, called ( _plus_ ) _infinity_ , and −∞, called _minus infinity_. The symbols ∞ and −∞ must not be thought of as numbers, even though they appear in inequalities. Using ∞ and −∞, we now define the following kinds, of intervals, where _c_ is an arbitrary number:
(1) The set of all numbers _x_ such that _x_ < _c_ , denoted by −∞ < _x_ < _c_ ;
(2) The set of all numbers _x_ such that _x_ ≤ _c_ , denoted by −∞ < _x_ ≤ _c_ ;
(3) The set of all numbers _x_ such that _x_ > _c_ , denoted by _c_ < _x_ < ∞;
(4) The set of all numbers _x_ such that _x_ ≥ _c_ , denoted by _c_ ≤ _x_ < ∞;
(5) The set of _all_ numbers _x_ , namely the whole real number system, denoted by −∞ < _x_ < ∞
In bracket notation, we denote these five kinds of intervals by (−∞, _c_ ), (−∞, _c_ ], ( _c_ , ∞), [ _c_ , ∞) and (−∞, ∞), respectively, with a round bracket for an excluded end point and a square bracket for an included end point, just as before. These intervals, involving ∞ and − ∞, are said to be _infinite_ , as opposed to the _finite_ intervals ( _a_ , _b_ ), [ _a_ , _b_ ], [ _a_ , _b_ ) and ( _a_ , _b_ ].
**b.** Since ∞ and −∞ are not numbers, we cannot allow either _x_ = ∞ or _x_ = −∞. Therefore it is meaningless to write intervals like −∞ ≤ _x_ < _c_ , _c_ ≤ _x_ ≤ ∞, −∞ ≤ _x_ ≤ ∞, etc., and no infinite interval written in bracket form can have a square bracket next to the symbol ∞ or −∞.
**c. Example.** Find the set of all _x_ such that
SOLUTION . According to Theorem 1.55, equation (6) means that the distance between the point _x_ and the point 1 minus the distance between the point _x_ and the point 2 equals 1. This happens when _x_ ≥ 2 and only then (what goes wrong if _x_ < 2?). Therefore the set of all _x_ satisfying (6) is the infinite interval 2 ≤ _x_ < ∞, or [2, ∞) in bracket form.
PROBLEMS
**.** What is the open interval, in bracket form, equivalent to −3 < _x_ −3 < −1? The closed interval equivalent to −2 ≤ _x_ \+ 1 ≤ 4?
**2.** What is the half-open interval, in bracket form, equivalent to 1 ≤ _x_ − 1 < 7? Equivalent to
**.** Find the set of all _x_ such that | _x_ − 1| + | _x_ − 2| = 1.
**4.** Find the -neighborhood of the point 3. Write the corresponding deleted neighborhood as a union of open intervals.
**.** What is the open interval, in bracket form, equivalent to −6 < 3 _x_ < 3? The closed interval equivalent to −6 ≤ −3 _x_ ≤ 3?
**.** Find a simpler way of writing
(a) (1, 3) {3}; (b) [1, 3) [3, ∞); (c) (−∞, 1) (0, ∞).
**.** Find a simpler way of writing
(a) [−2, 3] [−1, 1]; (b) [−1, 1] [1, 2]; (c) (−∞, 1] (−1, ∞).
***8.** In what interval is the expression
defined? How about
1.7 RECTANGULAR COORDINATES
As shown in Sec 1.54, any given point on a line can be uniquely specified by giving a single real number, called the _coordinate_ of the point. We now show how to uniquely specify any given point in a _plane_. This can be done by giving _two_ real numbers, again called the _coordinates_ of the point.
**1.71.** Suppose that at a convenient point in a plane (this page, say) we construct a pair of perpendicular lines, known as _coordinate lines_ or _coordinate axes_ , intersecting in a point _O_ , called the _origin_ (of coordinates). For convenience, we choose one of the lines parallel to the short dimension of the page, calling it the _x-axis_ and labelling it _Ox_ , and the other line parallel to the long dimension of the page, calling it the _y-axis_ and labelling it _Oy_. Each line is regarded as extending indefinitely in both directions, and each is equipped with a _positive direction_ , pointing to the right in the case of the _x_ -axis and upward in the case of the _y_ -axis, as indicated by the arrowheads in Figure 9A.
Figure 9.
The plane containing the pair of perpendicular lines _Ox_ and _Oy_ is called the _xy-plane_. The two coordinate lines divide the _xy_ -plane into four regions, called _quadrants_. These are indicated by the Roman numerals in Figure 9B, where I refers to the _first quadrant_ , II to the _second quadrant_ , and so on. Note that the quadrants are arranged in the _counterclockwise direction_ , that is, in the direction opposite to that in which the hands of a clock move.
**1.72.** We are now able to associate a pair of numbers with any given point _P_ in the _xy_ -plane, by making the following construction which you have probably already encountered before: Through the point _P_ we draw two straight lines, one perpendicular to the _x_ -axis, the other perpendicular to the _y_ -axis. Suppose that, as in Figure 10, the first line intersects the _x_ -axis in the point with coordinate _a_ , where the _x_ -axis is regarded as a number line with the indicated positive direction. Suppose also that the second line intersects the _y_ -axis in the point with coordinate _b_ , where the _y_ -axis is regarded as another number line with the indicated positive direction. Then the numbers _a_ and _b_ are called the _rectangular coordinates_ , or simply the _coordinates_ , of the point _P_. More exactly, _a_ is called the _abscissa_ or _x-coordinate_ of _P_ , while _b_ is called the _ordinate_ or _y-coordinate_ of _P_.
Conversely, given any pair of numbers _a_ and _b_ , to "plot" (that is, to find) the point _P_ in the _xy_ -plane with abscissa _a_ and ordinate _b_ , we simply reverse the above construction: We draw two straight lines, one perpendicular to the _x_ -axis through the point of the _x_ -axis with coordinate _a_ , the other perpendicular to the _y_ -axis through the point of the _y_ -axis with coordinate _b_. Then, as is immediately apparent, the point of intersection of these two lines is just the point _P_ with abscissa _a_ and ordinate _b_.
Figure 10.
**1.73.** The point _P_ with abscissa _a_ and ordinate _b_ may also be denoted by ( _a_ , _b_ ). The symbol ( _a_ , _b_ ) is called an _ordered pair_ , and is a special kind of two-element set of real numbers, namely one in which _the order of the elements matters_. Thus, although the ordinary sets { _a, b_ ) and { _b, a_ } are identical (Sec. 1.23a), the ordered pairs ( _a_ , _b_ ) and ( _b, a_ ) are different unless _a_ = _b_.
Do not confuse the ordered pair ( _a_ , _b_ ) with the same symbol ( _a_ , _b_ ) used to designate an open interval with end points _a_ and _b_. The context will always show which meaning is to be attached to the symbol ( _a_ , _b_ ). Although it would be nice to have different symbols for different things, this is a case where mathematical tradition must be respected.
Note that the origin _O_ has abscissa zero and ordinate zero. In other words, _O_ is the point (0, 0).
Figure 11.
**1.74. The distance between two points in the plane**
THEOREM. _Let P_ 1 = ( _x_ 1, _y_ 1) _and P_ 2 = ( _x_ 2, _y_ 2) _be two points in the xy-plane, and let |P_ 1 _P_ 2| _be the distance between them. Then_
_Proof_. Dropping perpendiculars from _P_ 1 and _P_ 2 to the _x_ \- and _y_ -axes, we find that _P_ 1 _P_ 2 is the hypotenuse of the right triangle _P_ 1 _QP_ 2 shown in Figure 11. Moreover, it is obvious that | _P_ 1 _Q_ | = | _AB_ | and | _QP_ 2| = | _CD_ |, where _A_ and _B_ have coordinates _x_ 1 and _x_ 2 regarded as points of the _x_ -axis, while _C_ and _D_ have coordinates _y_ 1 and _y_ 2 regarded as points of the _y_ -axis. Therefore, by the Pythagorean theorem,
| _P_ 1 _P_ 2|2 = | _P_ 1 _Q_ |2 \+ | _QP_ 2|2 = | _AB_ |2 \+ | _CD_ |2,
and hence
But, according to Theorem 1.55,
Combining (2) and (3), we get (1) at once.
Since | _x_ |2 = _x_ 2 for every number _x_ , regardless of the sign of _x_ , we can just as well write (1) in the equivalent form
**1.75. Examples**
**a.** Find the distance between the points _P_ 1 = (−1, 2) and _P_ 2 = (1, −2).
SOLUTION . According to (4),
**b.** Find the distance between two points _P_ 1 and _P_ 2 lying on the same line parallel to the _x_ -axis. On the same line parallel to the _y_ -axis.
SOLUTION . If _P_ 1 and _P_ 2 lie on the same line parallel to the _x_ -axis, then _P_ 1 and _P_ 2 have the same ordinate, say _b_ , so that _P_ 1 = ( _x_ 1, _b_ ) and _P_ 2 = ( _x_ 2, _b_ ). Therefore, in this case, formula (4) reduces to
This is just what we would expect from Theorem 1.55, which deals with the case where the common ordinate _b_ equals zero.
In the same way, it is easy to show that if _P_ 1 and _P_ 2 lie on the same line parallel to the _y-axis_ , so that _P_ 1 and _P_ 2 have the same _abscissa_ , then formula (4) reduces to
Give the details.
**c.** Is the triangle _ABC_ with vertices _A_ = (0, 0), _B_ = (3, 3), _C_ = (−1, 7) a right triangle?
SOLUTION . Yes. To see this, we first calculate the squares of the side lengths of _ABC_ , with the help of formula (4). As a result, we obtain
| _AB_ |2 = 32 \+ 32 = 18,
| _BC_ |2 = (−4)2 \+ 42 = 32,
| _AC_ |2 = (−1)2 \+ 72 = 50,
so that
| _AC_ |2 = | _AB_ |2 \+ | _BC_ |2
for the given triangle _ABC_. It follows from the converse of the Pythagorean theorem (explain) that _ABC_ is a right triangle with the side _AC_ as its hypotenuse.
PROBLEMS
**.** Plot the points _A_ = (2, 0), _B_ = (2, 2), _C_ = (0, 3), _D_ = (−2, 2), _E_ = (−2, 0), _F_ = (0, −1) on ordinary graph paper. Then join _A_ to _C_ , _B_ to _D_ , _C_ to _E_ , _D_ to _F_ , _E_ to _A_ , and finally _F_ to _B_. What is the resulting figure?
**.** Suppose the figure in the preceding problem is shifted one unit to the right and two units upward. Then _A_ , _B_ , _C_ , _D_ , _E_ , _F_ go into new points _A_ ′, _B_ ′, _C_ ′, _D_ ′, _E_ ′, _F_ ′. What are these new points?
**.** If the point ( _x_ , _y_ ) lies in the first quadrant, then _x_ > 0, _y_ > 0. Write similar conditions characterizing the other three quadrants.
**.** Find the distance between the pair of points
(a) (1, 3), (5, 7); (b) (−2, −3), (1, 1); (c) (1, 3), (1, 4); (d) (2, 4), (5, 4).
**5.** Give an example of four points, each in a different quadrant, whose distances from the origin are all equal.
**6.** Given two points _P_ 1 = ( _x_ 1, _y_ 1) and _P_ 2 = ( _x_ 2, _y_ 2), verify that the point with abscissa ( _x_ 1 \+ _x_ 2) and ordinate ( _y_ 1 \+ _y_ 2) is the midpoint of the segment _P_ 1 _P_ 2.
**.** Two vertices _A_ and _B_ of an isosceles triangle _ABC_ lie at the points (0, 1) and (10, 1). Find the abscissa of the point _C_ if | _AC_ | = | _BC_ |.
**.** Locate the points _A_ = (4, 1), _B_ = (3, 5), _C_ = (−1, 4) and _D_ = (0, 0). Show that _ABCD_ is a square. What is the side length of the square?
**9.** Find the midpoints of the sides of the square _ABCD_ in the preceding problem.
***10.** Find all points which are equidistant from the _x_ -axis, the _y_ -axis and the point (3, 6).
***11.** How many points of the form ( _m_ , _n_ ), where _m_ and _n_ are integers, lie inside the circle of radius with its center at the origin?
***12.** Given three noncollinear points _A_ = (0, 0), _B_ = ( _x_ , _y_ ) and _D_ = ( _x_ ′, _y_ ′), what choice of the point _C_ makes the quadrilateral _ABCD_ a parallelogram?
1.8 STRAIGHT LINES
**1.81. The slope of a line**
**a.** Let _L_ be any nonvertical straight line in the _xy_ -plane, and let _P_ 1 = ( _x_ 1, _y_ 1)and _P_ 2 = ( _x_ 2, _y_ 2) be any two distinct points of _L_. Then by the _slope_ of _L_ we mean the ratio
To interpret the slope geometrically, we draw the line through _P_ 1 parallel to the _x_ -axis and the line through _P_ 2 parallel to the _y_ -axis, intersecting in the point _A_ = ( _x_ 2, _y_ 1), as shown in Figure 12. Then the slope _m_ is just the ratio
of the length of the side _P_ 2 _A_ to the length of the side _P_ 1 _A_ in the right triangle _P_ 1 _AP_ 2.
**b.** It is important to note that the slope of a line _L_ does not depend on the particular choice of the points used to define the slope. To see this, let _P_ 3 and _P_ 4 be any two points on _L_ other than _P_ 1 and _P_ 2, and suppose the line through _P_ 3 parallel to the _x_ -axis intersects the line through _P_ 4 parallel to the _y_ -axis in the point _B_ , as shown in Figure 13, where _L_ , _P_ 1, _P_ 2 and _A_ are exactly the same as in Figure 12. Then the right triangles _P_ 1 _AP_ 2 and _P_ 3 _BP_ 4 are similar (why?), and therefore
Figure 12.
Figure 13.
Figure 14.
so that the formula
leads to exactly the same value of _m_ as formula (2).
**c.** The slope of the line _L_ in Figure 12 is clearly positive, since in this case both _x_ 2 − _x_ 1 and _y_ 2 − _y_ 1 are positive, and hence so is the ratio (1). However, a line may well have a _negative_ slope. For example, the ratio (1) is negative for the line _L_ shown in Figure 14. To see this, we need only note that _y_ 2 − _y_ 1 is now negative, while _x_ 2 − _x_ 1 is still positive.
Thus, in brief, if a line slopes _up_ to the right, its slope is _positive_ , while if a line slopes _down_ to the right, its slope is _negative_. *
**1.82. The inclination of a aline**
**a.** By the _inclination_ of a straight line _L_ in the _xy_ -plane we mean the smallest angle between the _x_ -axis and _L_ , as measured from the _x_ -axis to _L_ in the counterclockwise direction. The inclination, which we denote by _θ_ (the Greek letter theta), will be measured in _degrees_ , denoted by the symbol °. Moreover, any line parallel to the _x_ -axis, including the _x_ -axis itself, will be regarded as having the inclination 0° (zero degrees). Since vertical angles are equal (see Figure 15), it doesn't matter whether the measurement of _θ_ is started to the right or to the left of the point in which _L_ intersects the _x_ -axis. It is also clear from the figure that if _L_ makes the angle 180° + _θ_ with the _x_ -axis, then _L_ also makes the smaller angle _θ_ with the _x_ -axis. Therefore the inclination _θ_ of any line whatsoever lies in the half-open interval 0° ≤ _θ_ ≤ 180°.
Figure 15.
**b.** Now, according to formula (2), the slope _m_ of the line _L_ is just the ratio of the length of the side _P_ 2 _A_ to the length of the side _P_ 1 _A_ in the right triangle _P_ 1 _AP_ 2. Students who have had some elementary trigonometry will recall that this ratio ("the opposite side over the adjacent side") is also called the _tangent_ of the interior angle at _P_ 1 in the triangle _P_ 1 _AP_ 2. But this angle is precisely the inclination _θ_ of the line _L_ , since the side _P_ 1 _A_ is parallel to the _x_ -axis. Thus, in the notation of trigonometry, we have the formula
which is read as " _m_ equals the tangent of _θ_ ," giving the relation between the inclination of a line and its slope. The same argument as in Sec. 1.81b, based on the use of similar triangles, shows that the tangent of an angle depends only on the size of the given angle and not on the size of the right triangle containing the angle.
Figure 16 shows various lines, together with their inclinations and slopes, as related by formula (3). Note that although a vertical line has inclination 90°, its slope is undefined, since setting _x_ 1 = _x_ 2 in formula (1) would make the denominator zero. It is for this reason, of course, that _L_ was assumed to be nonvertical in Sec. 1.81a.
**c.** In connection with formula (3), it should be noted that if the angle _θ_ lies between 90° and 180°, then tan _θ_ is negative, as we would expect since the line _L_ then slopes down to the right and has negative slope. To calculate tangents between 90° and 180°, we use the formula
established in every course on trigonometry. For example,
tan 135° = tan (180° − 45°) = −tan 45° = −1,
tan 150° = tan (180° − 30°) = −tan 30° = ,
and so on.
Figure 16.
Figure 17.
**1.83. Examples**
**a.** Find the slope _m_ of the line going through the points (1, 3) and (4, 6).
SOLUTION . According to (1),
**b.** Find the inclination _θ_ of the line going through the same points.
SOLUTION . According to (3) and the definition of inclination, _θ_ is the smallest angle whose tangent is 1, namely 45°.
**c.** Find the slope _m_ of the line whose inclination is 15°.
SOLUTION . Here (3) gives
_m_ = tan 15° = 0.26795,
where we consult a table of tangents or use a pocket scientific calculator.
**d.** How are the slopes of a pair of perpendicular lines related?
SOLUTION . Let the lines be _L_ and L', as in Figure 17, where _L_ has slope _m_ and inclination θ, while _L'_ has slope _m'_ and inclination θ _' =_ 180° − α. Here a (the Greek letter alpha) is the other acute interior angle of the triangle P1P2P3. Then
On the other hand, by (4),
_m_ ′ = tan _θ_ ′ = tan (180° − _α_ ) = −tan _α_ ,
so that
Comparing (5) and (5′), we find that
or, equivalently,
In other words, _the slope of either of two perpendicular lines is the negative of the reciprocal of the slope of the other line_.
**e.** Find the slope of the line _L_ ′ perpendicular to the line _L_ going through the points (2, 3) and (4, 6).
SOLUTION . Let _m_ be the slope of _L_ and _m_ ′ the slope of _L_ ′. Then
so that
PROBLEMS
**.** Find the slope of the line going through the pair of points
(a) (−2, 4), (−3, −7); (b) (−2, 6), (1, 5); (c) (2, 3), (2, 5); (d) .
**.** Let _L_ be a line with slope _m_ and _L_ ′ a line with slope _m_ ′. When are _L_ and _L_ ′ parallel?
**.** Find the inclination of the line going through the pair of points
(a) (2, 4), (4, 6); (b) (2, 3), (2, 5); (c) (2, −4), (4, −6); (d) .
**.** Find the slope of the line with inclination
(a) 20°; (b) 100°; (c) 165°.
**5.** Find the slope of every line parallel to the line going through the points (1, −4) and (−2, 5).
**.** Find the slope of every line perpendicular to the line going through the points (1, 3) and (−3, −1).
**7.** Show that the line _L_ going through the points (1, 3) and (2, 5) is perpendicular to the line _L_ ′ going through the points (4, 6) and (2, 7).
***8.** How is the line going through the points related to the line going through the points ?
1.9 MORE ABOUT STRAIGHT LINES
**1.91. The equation of a straight line**
**a.** By the term "equation of a straight line" we mean a mathematical statement, in equation form, which expresses the relationship between the _x_ -coordinate and the _y_ -coordinate of every point on the line. To find such a statement, we reason as follows: Let _L_ be a nonvertical straight line with slope _m_ , going through a fixed point _P_ 1 = ( _x_ 1, _y_ 1), and let _P_ = ( _x_ , _y_ ) be an arbitrary point on _L_ , as in Figure 18. Then, expressing the slope of _L_ in terms of the coordinates of _P_ 1 and _P_ , we get
It follows that
_y_ − _y_ 1 = _m_ ( _x_ − _x_ 1),
Figure 18.
or equivalently
**b.** Despite its appearance, the right side of (1) does not actually depend on the particular choice of the fixed point _P_ 1 = ( _x_ 1, _y_ 1) on the line _L_. To see this, suppose we replace _P_ 1 by another fixed point _P_ 2 = ( _x_ 2, _y_ 2) on _L_ , as in the figure. Then we get
instead of (1). But we can easily show that _the two expressions y_ 1 − _mx_ 1 and _y_ 2 − _mx_ 2 _are equal_. In fact, suppose we calculate the slope of _L_ with the help of the points _P_ 1 and _P_ 2, relying on the fact, proved in Sec. 1.81b, that the slope of a line _L_ does not depend on the particular choice of the points used to define the slope. The result is
so that
_y_ 2 − _y_ 1 = _m_ ( _x_ 2 − _x_ 1),
or
_y_ 2 − _mx_ 2 = _y_ 1 − _mx_ 1,
as claimed.
Since the expression _y_ 1 − _mx_ 1 in (1) does not depend on the particular choice of the point _P_ 1 = ( _x_ 1, _y_ 1)on _L_ , we might just as well denote _y_ 1 − _mx_ 1 by a single symbol, say _b_. Equation (1) then becomes simply
which is the desired _equation of a straight line with slope m_. The geometric meaning of the term _b_ will be given in a moment (Sec. 1.92b).
**1.92. The intercepts of a straight line**
**a.** Let _L_ be a straight line _other than the coordinate axes themselves_ , and suppose _L_ intersects the _x_ -axis in a point ( _a_ , 0). Then _a_ is called an _x-intercept_ of _L_. Similarly, if _L_ intersects the _y_ -axis in a point (0, _b_ ), we call _b_ a _y-intercept_ of _L_. By an _intercept_ we mean either an _x_ -intercept or a _y_ -intercept. Consulting Figure 19, we see that
(1) If _L_ is neither vertical nor horizontal, then _L_ has exactly two intercepts, namely an _x_ -intercept and a _y_ -intercept (as in Figure 19A);
(2) If _L_ is vertical or horizontal, then _L_ has just one intercept, an _x_ -intercept but no _y_ -intercept if _L_ is vertical (as in Figure 19B) and a _y_ -intercept but no _x_ -intercept if _L_ is horizontal (as in Figure 19C).
Figure 19.
**b.** Now let _L_ be the straight line with equation (2), where we assume that _L_ is neither horizontal nor vertical, so that _L_ has both an _x_ -intercept and a _y_ -intercept. To find these intercepts, we need only note that the substitution _x_ = 0 in (2) gives
_y_ = _m_ · 0 + _b_ = _b_ ,
while the substitution _y_ = 0 gives
0 = _mx_ \+ _b_
or
_mx_ = − _b_ ,
so that
(why is _m_ nonzero in this case?). Therefore _L_ intersects the _x_ -axis in the point (− _b_ / _m_ , 0) and the _y_ -axis in the point (0, _b_ ). But this just means that the line _L_ has − _b_ / _m_ as its _x_ -intercept and _b_ as its _y_ -intercept.
**c.** If the line _L_ is vertical and hence parallel to the _y_ -axis, every point of _L_ has the same abscissa regardless of its ordinate. Suppose this abscissa is _a_. Then every point _P_ = ( _x_ , _y_ ) on _L_ satisfies the simple equation
in which the ordinate _y_ does not appear at all. Note that (3) is _not_ a special case of (2). This is hardly surprising, since in deriving (2) the case where _L_ is vertical, and hence has no slope (recall Sec. 1.82b), was excluded from the outset.
**d.** If the line _L_ is horizontal and hence parallel to the _x_ -axis, every point of _L_ has the same ordinate regardless of its abscissa. Suppose this ordinate is _b_. Then every point _P_ = ( _x_ , _y_ ) on _L_ satisfies the simple equation
in which the abscissa _x_ does not appear at all. This time (4) _is_ a special case of (2). In fact, to get (4) from (2) we need only make the substitution _m_ = 0. This is hardly surprising, since every horizontal line has zero slope.
**e.** All three equations (2), (3) and (4) can be combined into the single "master equation"
where _A_ , _B_ and _C_ are _constants_ , that is, fixed numbers, and at least one of the numbers _A_ and _B_ is nonzero. In fact, if _A_ ≠ 0 and _B_ = 0, equation (5) becomes
_Ax_ \+ _C_ = 0,
or
which is of the form (3) with _a_ = − _C_ / _A_ , while if _A_ = 0 and _B_ ≠ 0, (5) becomes
_By_ \+ _C_ = 0,
or
which is of the form (4) with _b_ = − _C/B_. In any case, if _B_ ≠ 0 we can divide both sides of (5) by _B_ , obtaining
or
which is of the form (2) with slope _m_ = − _A_ / _B_ and _y_ -intercept _b_ = − _C_ / _B_.
**f.** Note that none of the equations for a straight line involves powers of _x_ and _y_ higher than the first power. In mathematics such an equation is said to be _linear_ (in _x_ and _y_ ). This term stems from the fact that every linear equation in _x_ and _y_ is the equation of some straight line in the _xy_ -plane. Another way of writing the equation of a straight line, involving the intercepts, is given in Problem 12.
By "the line _y_ = _mx_ \+ _b_ " we mean, of course, the line _with equation y_ = _mx_ \+ _b_. Similarly, "the line _Ax_ \+ _By_ \+ _C_ = 0" means the line _with equation Ax_ \+ _By_ \+ _C_ = 0, and so on.
**1.93. Examples**
**a.** Find the line with slope 2 going through the point (3, 1).
SOLUTION Using (1) with _m =_ 2, _x_ 1 = 3 and _y_ 1 = 1, we get
_y_ = 2 _x_ \+ (1 − 2 · 3) = 2 _x_ − 5.
**b.** Find the line going through the points (1, 3) and (4, 6).
SOLUTION . The line has slope
Therefore, choosing (1, 3) as the point ( _x_ 1, _y_ 1) in (1), we get
_y_ = _x_ \+ (3 − 1 · 1) = _x_ \+ 2.
Naturally, the same result is obtained if we choose (4, 6) as the fixed point on the line (check this).
**c.** Find the slope and intercepts of the line
SOLUTION . Since (6) is of the form (2), with _m_ = 3 and _b_ = 2, the line has slope 3, _y_ -intercept 2 and _x_ -intercept − _b_ / _m_ =
**d.** Find the line _L_ which goes through the point (1, 3) and is parallel to the line (6).
SOLUTION . Being parallel to (6), _L_ has the same slope as (6), namely 3. Using (1), with _m_ = 3, _x_ 1 = 1 and _y_ 1 = 3, we get
Note that _L_ goes through the origin of the _xy_ -plane (why?).
**e.** Find the line _L_ ′ which goes through the point (1, 3) and is _perpendicular_ to the line (6).
SOLUTION . Being perpendicular to (6), _L_ ′ must have slope (recall Sec. 1.83d). Thus, instead of (7), we now have
We can also write (8) as
_x_ \+ 3 _y_ − 10 = 0,
which is of the form (5), with _A_ = 1, _B_ = 3, _C_ = −10.
**f.** Find the point of intersection of the lines (6) and (8).
SOLUTION . The abscissa _x_ 1 of the point of intersection of the lines (6) and (8) is characterized by the fact that both lines have the same ordinate _y_ 1 when _x_ = _x_ 1. It follows that _x_ 1 is the solution of the equation
obtained by setting the right side of (6) equal to the right side of (8). Solving (9) for _x_ , we get
or
To get _y_ 1, the ordinate of the point of intersection, we substitute (10) into (6), or into (8), obtaining
Therefore the lines (6) and (8) intersect in the point
PROBLEMS
**.** Find the line with slope 2 going through the point
(a) (1, 0); (b) (0, 1); (c) (1, 1); (d) (0, 0); (e) (1, −1).
**.** Find the line going through the pair of points
(a) (2, −5), (3, 2); (b) (c) (−3, 1), (7, 11); (d) (5, 3), (−1, 6).
**.** Find the line with slope _m_ and _y_ -intercept _b_ if
(a) _m_ = −1, _b_ = 1; (b) _m_ = 3, _b_ = 0; (c) _m_ = 0, _b_ = −2; (d)
**.** Find the slope _m_ , _x_ -intercept _a_ and _y_ -intercept _b_ of the line
(a) _y_ = 3 _x_ − 6; (b) _y_ = 2 _x_ \+ 4; (c) _y_ = − _x_ \+ 3; (d) _y_ = 2.
**.** Do the same for the line
(a) 5 _x_ − _y_ \+ 4 = 0; (b) 3 _x_ \+ 2 _y_ = 0; (c) 2 _y_ − 6 = 0; (d) _x_ \+ _y_ \+ 1 = 0.
**.** Find the line which goes through the point (2, −4) and is parallel to the line _y_ = 2 _x_ \+ 3.
**.** Find the line which goes through the point (1, 2) and is perpendicular to the line going through the points (2, 4) and (3, 5). What is the point of intersection of these two lines?
**.** What is the area of the triangle lying between the coordinate axes and the line 2 _x_ \+ 5 _y_ − 20 = 0?
**9.** Does the point (2, 3) lie above the line _y_ = 2 _x_ \+ 1 or below it?
**10.** What is the relationship between the lines 2 _x_ \+ 3 _y_ − 1 = 0 and 4 _x_ \+ 6 _y_ \+ 3 = 0? Between the lines 2 _x_ \+ 5 _y_ − 4 = 0 and 15 _x_ − 6 _y_ \+ 5 = 0?
**.** Find the line joining the origin to the point of intersection of the lines _x_ \+ 2 _y_ − 3 = 0 and _x_ − 3 _y_ \+ 7 = 0.
***12.** Show that the equation of the line with _x_ -intercept _a_ and _y_ -intercept _b_ is
assuming that _a_ and _b_ are both nonzero.
***13.** Find the line with _x_ -intercept _a_ and _y_ -intercept _b_ if
(a) _a_ = 1, _b_ = 2; (b) _a_ = −3, _b_ = −1; (c)
***14.** There are two lines, each with equal intercepts, going through the point (2, 3). Find them.
***15.** What is the (perpendicular) distance between the parallel lines 3 _x_ − 4 _y_ − 10 = 0 and 6 _x_ − 8 _y_ \+ 5 = 0?
***16.** Let _d_ be the (perpendicular) distance between the point _P_ 1 = ( _x_ 1, _y_ 1) and the line _Ax_ \+ _By_ \+ _C_ = 0. Show that
***17.** Find the distance between
(a) The point (3, 1) and the line 3 _x_ \+ 4 _y_ − 3 = 0;
(b) The point (1, 1) and the line 5 _x_ − 12 _y_ \+ 72 = 0;
(c) The point (1, −2) and the line _x_ − 2 _y_ − 5 = 0.
*Note that in interpreting the slope geometrically (and in proving Theorem 1.74), we have tacitly assumed that the line _L_ slopes _up_ to the right and that the point _P_ 1 lies to the _left_ of the point _P_ 2. As an exercise, consider the modifications required in the other cases.
_Chapter 2_
DIFFERENTIAL CALCULUS
### 2.1 FUNCTIONS
**2.11. Constants and variables.** The quantities encountered in mathematics fall into two broad categories, namely "fixed quantities," called _constants_ , and "changing quantities," called _variables_. A constant "takes only one value" in the course of a given problem, while a variable "takes two or more values" in the course of one and the same problem.
For example, let _L_ be the straight line with slope _m_ and _y_ -intercept _b_. Then, according to Sec. 1.9, _L_ has the equation
Here _m_ and _b_ are constants characterizing the given line _L_ , while _x_ and _y_ are variables, namely the abscissa and ordinate of a point which is free to change its position along _L_.
As another example, suppose a stone is dropped from a high tower. Let _s_ be the distance fallen by the stone and _t_ the elapsed time after dropping the stone. Then, according to elementary physics, _s_ and _t_ are variables which are related, at least for a while, by the formula
where _g_ is a constant known as the "acceleration due to gravity." To a good approximation, this formula becomes
if _s_ is measured in feet and _t_ in seconds.
**2.12. Related variables and the function concept**
**a.** A great many problems arising in mathematics and its applications involve _related variables_. This means that there are at least two relevant variables, and the value of one of them depends on the value of the other, or on the values of the others if there are more than two. For example, the position of a spy satellite depends on the elapsed time since launching, the cost of producing a commodity depends on the quantity produced, the area of a rectangle depends on both its length and its width, and so on.
Actually, the situation we have in mind is where knowledge of the values of all but one of the variables _uniquely_ determines the value of the remaining variable, in a crucial sense to be spelled out in a moment (Sec. 2.12b). The variables whose values are chosen in advance are called _independent variables_ , and the remaining variable, whose value is determined by the values of the other variables, is called the _dependent variable_. The "rule" or "procedure" leading from the values of the independent variables to the value of the dependent variable, regardless of how this is accomplished, is called a _function_. We then say that the dependent variable "is a function of" the independent variables. The independent variables are often called the _arguments_ of the function.
Thus, for example, if _x_ and _y_ are the abscissa and ordinate of a variable point on the line with slope _m_ and _y_ -intercept _b_ , then _y_ is a function of _x_ , as described by formula (1). Similarly, the distance traversed by a falling stone is a function of time, as described by formula (2).
**b.** Consider a function of one independent variable, say _x_ , and let _y_ be the dependent variable. Then the function assigns a value of _y_ to each value of _x_. Expressed somewhat differently, the function establishes a correspondence between the values of _x_ and those of _y_. This correspondence must be such that each value of _x uniquely_ determines the corresponding value of _y_. This simply means that _to each value of x there corresponds one and only one value of y_. On the other hand, the same value of _y_ may well correspond to more than one value of _x_. The situation is the same for several independent variables _x_ 1, _x_ 2, ..., _x n_ if for "value of _x_ " we read "set of values of _x_ 1, _x_ 2, ..., _x n_."
Thus if
_y_ = x2,
then _y_ is a function of _x_ , since to each value of _x_ there corresponds one and only one value of _y_. On the other hand, each positive value of _y_ corresponds to _two_ values of _x_. For example, the value _y_ = 4 corresponds to the two values _x_ = 2 and _x_ = −2. This clearly prevents _x_ from being a function of _y_ , since every positive value of _y_ fails to uniquely determine the corresponding value of _x_.
As another example, let _A_ be the area of the rectangle of length _l_ and width _w_. Then _A_ is a function of _l_ and _w_ , since the relation between these variables is described by the simple formula
leading to one and only one value of _A_ for any given pair of values of _l_ and _w_. Solving (3) for _l_ , we get
which shows that _l_ is a function of _A_ and _w_. Similarly,
which shows that _w_ is a function of _A_ and _l_.
**c.** Do not jump to the conclusion that all related variables are numbers, and that all functions involve the use of formulas or numerical calculations. For example, the name of a car's owner is a function of the inscription on the car's license plate, and the rule or procedure describing this function is just this: Look up the plate in the motor vehicle records of the state in question, and find the owner's name.
On the other hand, it is true that examples like this are a bit "offbeat." Almost all the functions considered in this book involve variables which take only numerical values.
**2.13. Function notation**
**a.** The fact that one variable, say _y_ , is a function of another variable, say _x_ , can be indicated by writing
_y_ = _f_ ( _x_ ),
pronounced " _y_ equals _f_ of _x_." Here _f_ is a letter denoting the function, that is, the rule or procedure (usually, but not always, involving some formula) leading from the values of _x_ to the values of _y_. Suppose we give the independent variable _x_ the value _c_. Then the corresponding, value of _y_ is denoted by _f_ ( _c_ ), and is called the _value of the function _f_ at c_. This is a bit fussy, and it is simpler to use the same letter to denote both the independent variable and its values. We can then call _f_ ( _x_ ) the _value off at x_.
Although, strictly speaking, _f_ ( _x_ ) is the value of the function _f_ at _x_ , we will often talk about "the function _y_ = _f_ ( _x_ )" or simply "the function _f_ ( _x_ )." Suppose _y_ = _f_ ( _x_ ) = _x_ 2, for example. Then we might talk about "the function _f_ ( _x_ ) = _x_ 2," "the function _y_ = _x_ 2," or simply "the function _x_ 2."
**b.** There is nothing sacred about the use of the letter _f_ to denote a function, apart from its being the first letter of the word "function," and other letters will do just as well. Common choices are Latin letters like _g_ , _h_ , _F_ , etc., or Greek letters like _φ_ (phi), _ψ_ (psi), Φ (capital phi), etc. Sometimes the letter is chosen to suggest a geometrical or physical quantity under discussion. Thus _A_ is often used for area, _V_ for volume, _t_ for time, and so on.
**c.** Functions of several variables are indicated in the same way. Thus _f_ ( _s_ , _t_ ) means a function of two independent variables _s_ and _t_ , Φ( _u_ , _v_ , _w_ ) means a function of three independent variables _u_ , _v_ and _w_ , and so on.
**2.14. The domain and range of a function**
**a.** There is still one thing missing in our definition of a function, for we have yet to specify the set of values taken by the independent variable (or variables). This set is called the _domain_ of the function. For example, returning to the problem of the falling stone, we observe that formula (2) does not describe the motion of the stone for all values of _t_ , but only until the stone hits the ground. If the stone is dropped from a height of 64 feet, say, it hits the ground after falling for 2 seconds (64 = 16 · 22) and is subsequently motionless. In other words, formula (2) is valid only during the time interval 0 ≤ _t_ ≤ 2, a fact we can make explicit by writing
instead of (2). The subsequent behavior of the stone is described by the formula
_s_ = 64,
or, more exactly, by
We can also write (5) as
in terms of the infinite interval 2 < _t_ < ∞. Incidentally, this shows the desirability of considering _constant functions_ , that is, functions which take only one value,
**b.** Formulas (4) and (5) can be combined into the single formula
Moreover, noting that the stone is motionless before it is dropped, as well as after it hits the ground, we have the even more comprehensive formula
Formulas (4), (6) and (7) all describe different functions, in the sense that in each case the domain of the function, that is, the set of allowed values of the independent variable _t_ , is different.
**c.** Another way of saying that a function _f_ has the domain _D_ is to say that _f is defined in D_. Thus the function (6) is defined in the interval 0 ≤ _t_ ≤ ∞, that is, for all nonnegative _t_ , while the function (7) is defined in the interval −∞ < _t_ < ∞, that is, for all _t_ , positive, negative and zero.
**d.** By the _range_ of a function we mean the set of all values taken by the function, or, equivalently, the set of all values taken by the _dependent_ variable. For example, all three functions (4), (6) and (7) have the same range, namely the interval 0 ≤ _s_ ≤ 64. On the other hand, the range of the constant function (5) is the set whose only element is the number 64.
**2.15. Examples**
**a.** Let
Find _f_ (0), _f_ (1) and _f_ (2).
SOLUTION. To find _f_ (0), we merely substitute _x_ = 0 into (8), obtaining
and similarly
On the other hand, the quantity
"does not exist," since there is no real number whose square is negative. There is a sense in which meaning can be ascribed to "imaginary numbers" like , but such an extension of the concept of number lies beyond the scope of this book, in which all numbers are assumed to be _real_ (Sec. 1.35).
Whenever a function _f_ ( _x_ ) is specified by an explicit formula like (8), we will understand the domain of _f_ ( _x_ ) to be the _largest_ set of numbers _x_ for which the formula makes sense. In the present case, this set is just the interval −1 ≤ _x_ ≤ 1, since 1 − _x 2_ is negative for any other value of _x_ and we do not take square roots of negative numbers. Note that any smaller set can serve as the domain of a function whose values are given by the same formula (8), but in such cases we will always explicitly indicate the domain, as in the formula
where the domain is now the _smaller_ interval 0 < _x_ < 1.
**b.** Is the area of a rectangle a function of its perimeter?
SOLUTION. No, since knowledge of the perimeter of a rectangle does not uniquely determine its area. For example, the rectangle of length 15 and width 3 has perimeter 15 + 3 + 15 + 3 = 36 and area 15 · 3 = 45, while the square of side 9 has the same perimeter 9 + 9 + 9 + 9 = 36 and a different area 92 = 81.
**c.** Turning to a function of two variables, let
Find _g_ (1, 2), _g_ (2, 1) and _g_ (1, 1). What is the domain of _g_ ( _x_ , _y_ )?
SOLUTION. Easy substitutions give
and
On the other hand, _g_ (1, 1) fails to exist, since
and division by zero is impossible. The domain of _g_ ( _x_ , _y_ ) is the set of all pairs of numbers _x_ and _y_ such that _x_ ≠ _y_ , since _x_ = _y_ leads to division by zero. Regarding each such pair of numbers as the rectangular coordinates of a point ( _x_ , _y_ ) in the _xy_ -plane, we see that the domain of _g_ ( _x_ , _y_ ) is the set of all points in the _xy_ -plane except those on the line _y_ = _x_.
**2.16. One-to-one functions and inverse functions**
**a.** Let _y_ be a function of _x_ , or, in symbols, _y_ = _f_ ( _x_ ). Then, as in Sec. 2.12b, _x uniquely determines y_ , that is, _to each value of _x_ there corresponds one and only one value of y_. On the other hand, there is nothing so far to prevent more than one value of _x_ from corresponding to one and the same value of _y_. However, suppose we now impose an _extra requirement_ on the function _y_ = _f_ ( _x_ ), namely that not only should _x_ uniquely determine _y_ but also that _y should uniquely determine x_. Then not only does there correspond one and only one value of _y_ to each value of _x_ , but also _to each value of _y_ there corresponds one and only one value of x_. A function _y_ = _f_ ( _x_ ) of this special type is called a _one-to-one function._
**b.** Let _y_ = _f_ ( _x_ ) be a one-to-one function. Then there is a simple rule leading from the _dependent_ variable _y_ back to the _independent variable x_. In fact, let _y_ 1 be any given value of _y_. Then _x_ 1, the corresponding value of _x_ , is just the unique value of _x_ to which the function _y_ = _f_ ( _x_ ) assigns the value _y_ 1. Thus the rule leading from _y_ to _x_ is just as much a function as the original rule leading from _x_ to _y_. This new function, leading from _y_ to _x_ , is denoted by _x_ = _f_ −1( _y_ ) and is called the _inverse function_ , or simply the _inverse_ , of the original function _y_ = _f_ ( _x_ ). Never make the mistake of confusing the inverse function _f_ −1( _y_ ) with the reciprocal 1/ _f_ ( _y_ ).
Let _f_ ( _x_ ) be a one-to-one function defined in some interval _I_. Then we simply say that _f_ ( _x_ ) is _one-to-one in I_.
**c. Example.** The function
with domain −∞ < _x_ < ∞ and range 0 ≤ _y_ < ∞, is not one-to-one, since to each positive value of _y_ there correspond two values of _x_ , namely and − (as always, means the _positive_ square root of _y_ ). But suppose we restrict the domain of (9) to nonnegative values of _x_. Then to each value of _y_ there corresponds precisely one value of _x_ , namely . In other words, the function
_y_ = _f_ ( _x_ ) = _x_ 2 (0 ≤ _x_ < ∞)
_is_ one-to-one, with inverse
_x_ = _f_ −1( _y_ ) = (0 ≤ _y_ < ∞)
More generally, it is easy to see that the function _y_ = _x_ 2 is one-to-one in any interval in which _x_ is of fixed sign, but not in any interval in which _x_ changes sign.
##### PROBLEMS
**.** If _f_ ( _x_ ) = _x_ 2 \+ 3 _x_ \+ 6, find _f_ (0), _f_ (1), _f_ (2) and .
**2.** If _φ_ ( _t_ ) = | _t_ | + 3 _t_ 2, find _φ_ (−2), _φ_ (−1), _φ_ (0) and _φ_ ( ).
**.** Let
Find _g_ (−1), _g_ (0), _g_ (1), and _g_ (1/ ).
**.** Find the domain and range of the function
**.** Let
Find _f_ (3, 1), _f_ (0, 1), _f_ (1, 0), _f_ ( _a_ , _a_ ) and _f_ ( _a_ , − _a_ ).
**.** Find the domain and range of the function
**.** Is the number of hairs on your head a function of time?
**.** Is a man's birthday a function of his first name? Of his Social Security number, assuming that he has one?
**9.** Let _P_ be the closing price of a given security traded on the stock exchange, and let _d_ be the date. Then _P_ is a function of _d_. Given _d_ , how do you find _P_? Is the function ever undefined?
**.** Is the weight of a first-class letter a function of its postage?
**.** Is the area of a square a function of its perimeter?
**12.** Is the number of a page of this book a function of the number of commas on the page?
**.** "The area of a right triangle is a function of two variables." True or false?
**.** "The area of a parallelogram is a function of two variables." True or false?
**15.** Express the volume _V_ of a brick as a function of its length _l_ , width _w_ and height _h_.
**.** Let
Find _f_ (l, 1, 1), _f_ (4, 1, 9), _f_ (1, 9, 1) and _f_ (4, 9, 16).
**17.** The function given by the table
is familiar from everyday life. What is it? Fill in the missing entries in the table. Find a formula relating _y_ to _x_ and one relating _x_ to _y_.
**.** Is the inverse of a one-to-one function always a one-to-one function?
**.** Let _X_ be the domain and _Y_ the range of a one-to-one function _y_ = _f_ ( _x_ ). What are the domain and range of the inverse function _x_ = _f_ −1( _y_ )?
**.** Which of the following functions are one-to-one?
Find the inverse of each one-to-one function.
**.** "The position of a clock's hands is a one-to-one function of the time of day." True or false?
***22.** Find the domain and range of the function _y_ = _x_ ], where [ _x_ ] is the integral part of _x_ ([Sec. 1.4, Prob. 10).
***23.** Verify that the following formal definitions of function, domain, value and range agree in all essentials with those given in the text:
Given any two nonempty sets _X_ and Y, let _f_ be a set of ordered pairs ( _x_ , _y_ ) with _x_ ∈ _X_ and _y_ ∈ _Y_ such that for every _x_ ∈ _X_ there is one and only one ordered pair ( _x_ , _y_ ) ∈ _f_ with _x_ as its first element. Then _f_ is said to be a _function_ defined in _X_ , and _X_ is called the _domain_ of _f_. If ( _x_ , _y_ ) is an ordered pair in _f_ , then _y_ , the second element of the pair, is called the _value_ of _f_ at _x_ , written _f_ ( _x_ ). The set of all values of a function _f_ , that is, the set { _f_ ( _x_ ): _x_ ∈ _X_ }, is called the _range_ of _f_.
***24.** Is the set _Y_ in the preceding problem always the range of _f_?
***25.** How many different functions are there with domain _X_ = {1, 2,..., _n_ } and range _Y_ = { _a_ , _b_ }?
***26.** Let the function _f_ be defined as a set of ordered pairs ( _x_ , _y_ ), as in Problem 23. When is _f_ one-to-one? If _f_ is one-to-one, how is the inverse _f_ −1 obtained?
***27.** We can exhibit the behavior of a function _f_ with domain _X_ and range _Y_ by drawing a diagram like Figure 1, where _X_ and _Y_ are represented by disks, the values of the independent and dependent variables _x_ and _y_ are represented by points inside the disks, and each value of _x_ is connected by an arrow to the corresponding value of _y_. The arrows show quite explicitly how _f_ "maps" or "carries" the values of _x_ into those of _y._
The function represented in Figure 1 is not one-to-one, since two arrows terminate on the same point _y_ 1. Modify Figure 1 in such a way as to make it represent a one-to-one function.
Figure 1.
***28.** What is the simplest way of converting the "mapping diagram" of a one-to-one function _f_ into the analogous diagram for the inverse function _f_ −1?
***29.** We say that there is a _one-to-one correspondence_ between two sets _A_ and _B_ if there is a one-to-one function with domain _A_ and range _B_. Two sets are said to _have the same number of elements_ if there is a one-to-one correspondence between them. Show that the set of all even numbers has the same number of elements as the set of all odd numbers.
***30.** A set _A_ is said to _have n elements_ if there is a one-to-one correspondence between _A_ and the set {1, 2, ..., _n_ } made up of the first _n_ positive integers. If a set _A_ contains _n_ elements, where _n_ is some positive integer, we say that _A_ is _finite_ ; otherwise _A_ is said to be _infinite_. (An empty set is regarded as finite.) Which of the following sets are finite and which infinite?
(a) The set of all cells in a human body;
(b) The set of all integers less than 1,000;
(c) The set of all integers greater than 1,000,000;
(d) The set of all right triangles whose side lengths are integers.
### 2.2 MORE ABOUT FUNCTIONS
A variable whose values are all real numbers is called a _real variable_ , and a function whose values are all real numbers is called a _numerical function_. Calculus is primarily concerned with numerical functions of one or more real variables, and these are the only functions to be considered in the rest of this book. Thus, from now on, when we use the words "function" and "variable" without further qualification, we will always mean a _numerical_ function and a _real_ variable, just as the word "number" always means a _real_ number.
**2.21.** Two functions _f_ ( _x_ ) and _g_ ( _x_ ) are said to be _identically equal_ , and we write _f_ ( _x_ ) ≡ _g_ ( _x_ ), if the functions have the same domain _X_ and if _f_ ( _x_ ) = _g_ ( _x_ ) for all _x_ in _X_. Note the distinction between the ordinary equals sign = and the sign ≡ with three bars.
For example, the two functions
and
are both identically equal to the constant function 1. An equation like
involving the sign ≡, is called an _identity._
**2.22. Composite functions**
**a.** Functions are often combined by letting the arguments of one function equal the values of another. In this way, we get _composite functions_ like _f_ ( _g_ ( _x_ )) and _g_ ( _f_ ( _x_ )). For example, suppose
Then straightforward substitution shows that
and
_g_ ( _f_ ( _x_ )) = [ _f_ ( _x_ )]2 = (1 + _x_ )2.
In the same way,
_f_ ( _f_ ( _x_ )) = 1 + _f_ ( _x_ ) = 2 + _x_
and
_g_ ( _g_ ( _x_ )) = [ _g_ ( _x_ )]2 = _x_ 4.
Things are not always this simple. For example, suppose
_f_ ( _x_ ) = , _g_ ( _x_ ) = −| _x_ |.
Then
fails to exist for every value of _x_ except _x_ = 0. This shows that a composite function is defined only for values of the independent variable such that the values of the "inner function" belong to the domain of the "outer function."
Never make the mistake of confusing the composite function _f_ ( _g_ ( _x_ )) with the product function _f_ ( _x_ ) _g_ ( _x_ ). For example, the product of the functions (1) is
_f_ ( _x_ ) _g_ ( _x_ ) = (1 + _x_ ) _x_ 2 = _x_ 2 \+ _x_ 3,
which is not the same as the composite function (2).
**b.** Let _y_ = _f_ ( _x_ ) be a one-to-one function, with inverse _x_ = _f_ −1( _y_ ). Substituting _y_ = _f_ ( _x_ ) into _x_ = _f_ −1( _y_ ), and then substituting _x_ = _f_ −1( _y_ ) into _y_ = _f_ ( _x_ ), we get the important pair of identities
involving the composite functions _f_ −1( _f_ ( _x_ )) and _f_ ( _f_ −1( _y_ )). These formulas tell us that each of the functions _f_ and _f_ −1 "nullifies" the action of the other.
**2.23. Sequences**
**a.** A function _f_ whose domain is the set {1, 2, ...} of all positive integers is called an _infinite sequence_ , or simply a _sequence_ , and the values of _f_ at _n_ = 1, 2, ... are called the _terms_ of the sequence. More informally, a sequence is a rule or procedure assigning a number to every positive integer. We can write a sequence by listing some of its terms
where _f_ ( _n_ ) is called the _general term_ of the sequence. The first set of dots in (4) means "and so on up to," while the second set means "and so on forever." It is customary to save a lot of parentheses by simply writing
instead of (4). A more concise way of specifying the sequence (5) is to write its general term inside curly brackets:
{ _f n_}.
Do not confuse { _f n_} in this context with the set whose only element is _f n_. Note that the terms of a sequence are always listed in such a way that the integers 1, 2, ..., _n_ , ... appear in their natural order, increasing from left to right, as in (4) and (5).
**b.** Again there is nothing sacred about the letter _f_ , and other letters will do just as well. Common choices are small Latin letters like _a_ , _b_ , _c_ , _s_ and _x_. Although, strictly speaking, _x n_ is the general term of the sequence { _x_ _n_ }, we will often talk about "the sequence _x_ _n_." For example, suppose { _x_ _n_ } is the sequence such that _x n_ = _n_ 2. Then we might talk about "the sequence _x n_ = _n_ 2," or simply "the sequence _n_ 2."
**c.** The "law of formation" of a sequence is often given explicitly as a formula for its general term, as in the above example of the sequence _x n_ = _n_ 2. A sequence may also be given _recursively_ , that is, by showing how each term can be obtained from terms with lower subscripts. For example, suppose that
Then
_x_ 1 = 1, _x_ 2 = _x_ 1 \+ 2 = 3, _x_ 3 = _x_ 2 \+ 3 = 6, _x_ 4 = _x_ 3 \+ 4 = 10, ...
This sequence can be written as 1, 3, 6, 10, ..., but this does little more than suggest how the sequence was actually arrived at. A rule like (6) is called a _recursion formula._
##### PROBLEMS
**.** What is the largest set of numbers _x_ for which the following identities are true?
**2.** Find values of _a_ and _b_ in the formula _f_ ( _x_ ) = _ax_ 2 \+ _bx_ \+ 5 such that
_f_ ( _x_ \+ 1) − _f_ ( _x_ ) ≡ 8 _x_ \+ 3.
**.** Let _f_ and _g_ be two numerical functions with the same domain _X_. Then by the _sum F_ \+ _g_ we mean the function with domain _X_ whose value at every point _x_ ∈ _X_ is just the sum of the value of _f_ at _x_ and the value of _g_ at _x_. More concisely,
( _f_ \+ _g_ )( _x_ ) ≡ _f_ ( _x_ ) + _g_ ( _x_ ).
Other algebraic operations are defined similarly. Thus
and so on.
Suppose that
Find the values of _f_ \+ _g_ , _f_ − _g_ , _fg_ , _f_ 3 and _f_ / _g_ all at _x_ = 5.
**4.** Let
Find _f_ ( _f_ ( _x_ )), _f_ ( _g_ ( _x_ )), _g_ ( _f_ ( _x_ )) and _g_ ( _g_ ( _x_ )).
**.** Let
Find _h_ ( _h_ ( _h_ (2))).
**.** "In general, _f_ ( _g_ ( _x_ )) ≡ _g_ ( _f_ ( _x_ ))." True or false?
**.** Write the first five terms of the sequence { _a n_} with general term
**.** Let { _x_ _n_ } be the sequence 1, 3, 5, . . ., 2 _n_ − 1, ... of all odd numbers written in increasing order, and let _s_ 1 = _x_ 1, _s_ 2 = _x_ 1 \+ _x_ 2, ..., _s n_ = _x_ 1 \+ _x_ 2 \+ . . . + _x n_, ... Write the first few terms of the sequence { _s n_}, and find a simple expression for its general term.
***9.** A sequence { _x_ _n_ } is specified by the following rule: Its first two terms equal 1, and the remaining terms are given by the recursion formula
_x n_ = _x_ _n_ −1 \+ _x_ _n_ −2 ( _n_ = 3, 4, . . .).
Write the first eight terms of this sequence, known as the _Fibonacci sequence._
***10.** Find the terms _a_ 1, _a_ 3, _a_ 4 and _a_ 7 of the sequence { _a n_} determined by the formula = 1. _a_ 1 _a_ 2 ... _a n_ ...
***11.** Let
Prove that _f_ ( _x_ ) ≡ _g_ ( _x_ ).
***12.** Does _f_ ( _x_ ) _g_ ( _x_ ) ≡ 0 always imply _f_ ( _x_ ) ≡ 0 or _g_ ( _x_ ) ≡ 0?
***13.** Let
Find _f_ ( _g_ (2), _h_ (2)).
### 2.3 GRAPHS
**2.31. a.** Let _F_ ( _x_ , _y_ ) be a numerical function of two real variables _x_ and _y_. Then by the _solution set_ of the equation
we mean the set of all ordered pairs ( _x_ , _y_ ) for which (1) holds. For example, if
_F_ ( _x_ , _y_ ) = _xy_ ,
then (1) becomes the equation
which implies that either _x_ = 0 or _y_ = 0 (or both). Therefore the solution set of (2) is the set of all ordered pairs of the special form ( _x_ , 0) or (0, _y_ ), where _x_ and _y_ are arbitrary numbers.
Similarly, if
_F_ ( _x_ , _y_ ) = _x_ 2 \+ _y_ 2,
we get the equation
But (3) implies that both _x_ = 0 and _y_ = 0. Therefore the solution set of (3) is the set whose only element is the pair (0, 0).
**b.** Let _S_ be the solution set of equation (1). Then there is a simple way of "drawing a picture" of _S_. First we introduce a "system of rectangular coordinates," that is, we set up perpendicular axes _Ox_ and _Oy_ in the plane, as in Sec. 1.71. Next we plot all the elements of _S_ as points in the _xy_ -plane; since all the elements of _S_ are ordered pairs of numbers, this can be done in the way described in Sec. 1.72. These points make up a "picture," called the _graph of S_ , or, equivalently, the _graph of equation_ (1). For example, the graph of equation (2) consists of the coordinate axes themselves, while the graph of equation (3) consists of a single point, namely the origin of coordinates.
**c.** We can apply the same technique to a function
of a single variable _x_. Let _S_ be the set of all ordered pairs ( _x_ , _y_ ) for which (4) holds. Then, plotting all the elements of _S_ as points in the _xy_ -plane, we get a "picture," called the _graph of S_ , or, equivalently, the _graph of the function_ (4). For example, according to Sec. 1.9, the graph of the function
_y_ = _mx_ \+ _b_
is just the straight line with slope _m_ and _y_ -intercept _b._
**d.** The word "graph" will also be used as a verb, meaning "find the graph of." Note that (4) is a special case of (1), corresponding to the choice _F_ ( _x_ , _y_ ) = _y_ − _f_ ( _x_ ). Thus the graph of a function is the graph of a special kind of equation.
**e.** The graph of an equation or function typically looks like a "curve," possibly made up of several "pieces." With the help of calculus methods, we will eventually become proficient at "curve sketching," learning how to draw the graph of a function without explicitly plotting more than a few points. The graph of an equation _F_ ( _x_ , _y_ ) = 0 or of a function _y_ = _f_ ( _x_ ) is often simply called "the curve _F_ ( _x_ , _y_ ) = 0" or "the curve _y_ = _f_ ( _x_ )."
**2.32. Examples**
**a.** Graph the equation
SOLUTION. Since _x_ 2 \+ _y_ 2 is the square of the distance between the point ( _x_ , _y_ ) and the origin _0_ (Sec. 1.74), the point ( _x_ , _y_ ) belongs to the graph of (5) when the distance between ( _x_ , _y_ ) and _O_ equals 1, and only then. Therefore the graph of (5) is the circle of radius 1 with its center at _O_ , as shown in Figure 2.
**b.** Graph the equation
SOLUTION. First we "complete the squares" in (6), by noting that
_x_ 2 − 6 _x_ \+ _y_ 2 − 4 _y_ \+ 9 = ( _x_ 2 − 6 _x_ \+ 9) + ( _y_ 2 − 4 _y_ \+ 4) − 4
= ( _x_ − 3)2 \+ ( _y_ − 2)2 − 4,
so that (6) is equivalent to
( _x_ − 3)2 \+ ( _y_ − 2)2 = 4.
But the expression on the left is just the square of the distance between the variable point ( _x_ , _y_ ) and the fixed point (3, 2). Therefore the graph of (6) is the circle of radius = 2 with its center at the point (3, 2), as shown in Figure 3. Note that the x-axis is tangent to the circle at the point (3, 0).
Figure 2.
Figure 3.
**c.** Graph the function
SOLUTION. If _x_ ≥ 0, then | _x_ | = _x_ and (7) reduces to the straight line
_y_ = _x_
with slope 1 going through the origin, while if _x_ < 0, then | _x_ | = − _x_ and (7) reduces to the straight line
_y_ = − _x_
with slope −1 going through the origin. Therefore the graph of the function (7) is the curve shown in Figure 4, made up of "pieces" of the lines _y_ = _x_ and _y_ = − _x_. Note that the curve has a sharp "corner" at the origin.
A function like _y_ = | _x_ |, whose graph is made up of pieces of two or more straight lines, is said to be _piecewise linear._
Figure 4.
Figure 5.
**d.** Graph the function
SOLUTION. The graph of (8) is the curve shown in Figure 5, known as a _parabola_. The _x_ -axis seems to be tangent to the curve at the origin, and moreover the curve "opens upward" along its whole extent. These ideas will be made precise later, when we introduce the concepts of the "tangent to a curve" and "concavity."
The curve _y_ = _x_ 2 has another interesting property, namely it is _symmetric in the y-axis_. This simply means that for every point _P_ of the curve on one side of the _y_ -axis, there is another point _Q_ of the curve on the other side such that the _y_ -axis is the perpendicular bisector of the line segment _PQ_. To see that this is true, we merely note that changing _x_ to − _x_ has no effect on the value of _y_ = _x_ 2, since (− _x_ )2 = _x_ 2. But the points _P_ = ( _x_ , _y_ ) and _Q_ = (− _x_ , _y_ ) clearly lie on opposite sides of the _y_ -axis, and the _y_ -axis is the perpendicular bisector of the horizontal segment _PQ_ , as the figure makes apparent.
A function _f_ ( _x_ ) is said to be _even_ if _f_ (− _x_ ) ≡ _f_ ( _x_ ), where it is tacitly assumed that the domain of _f_ ( _x_ ) contains − _x_ whenever it contains _x_. We have just shown that the function _f_ ( _x_ ) = _x_ 2 is even and that its graph is symmetric in the _y_ -axis. Clearly, the graph of every other even function has the same symmetry property. For example, the function _f_ ( _x_ ) = | _x_ | is even, since |− _x_ | ≡ | _x_ |, and hence the graph of _y_ = | _x_ | is symmetric in the _y_ -axis, as is apparent from Figure 4.
**e.** Graph the function
SOLUTION. The graph of (9) is the curve shown in Figure 6, known as a _cubical parabola_. This curve seems to "open downward" to the left of the origin and "upward" to the right of the origin, while changing from "downward" to "upward" at the origin itself, which is accordingly called an "inflection point" of the curve. All these ideas will be made precise later, in connection with our discussion of "concavity."
The curve _y_ = _x_ 3 has another interesting property, namely it is _symmetric in the origin_. This simply means that for every point _P_ of the curve, there is another point _Q_ of the curve such that the origin is the midpoint of the line segment _PQ_ (thus _P_ and _Q_ are, so to speak, on "opposite sides" of _O_ ). To see that this is true, we note that changing the sign of _x_ also changes the sign of _y_ = _x_ 3, since (− _x_ )3 = − _x_ 3. But the points _P_ = ( _x_ , _y_ ), _Q_ = (− _x_ , − _y_ ) and _O_ = (0, 0) are collinear, as follows at once from the observation that the slope of the line through _O_ and _P_ has the same value _y/x_ as the slope of the line through _Q_ and _O_. Moreover, _O_ is clearly the midpoint of the segment _PQ_ , since
A function _f_ ( _x_ ) is said to be _odd_ if _f_ (− _x_ ) ≡ − _f_ ( _x_ ), where it is again assumed that the domain of _f_ ( _x_ ) contains − _x_ whenever it contains _x_. We have just shown that the function _f_ ( _x_ ) = _x_ 3 is odd and that its graph is symmetric in the origin. Clearly, the graph of every other odd function has the same symmetry property. For example, every line _y_ = _mx_ through the origin has this property.
The problem of evenness versus oddness plays an important role in applied mathematics and physics. The question "What is the parity of _f_ ( _x_ )?" simply means "Is _f_ ( _x_ ) even or odd?"
Figure 6.
**2.33. Increasing and decreasing functions**
**a.** Suppose the graph of a function _f_ ( _x_ ) _rises_ steadily as a variable point _P_ on the graph moves from left to right, with its abscissa in some interval _I_. Then _f_ ( _x_ ) is said to be _increasing in I_. For example, the functions | _x_ | and _x_ 2 are both increasing in the interval 0 ≤ _x_ < ∞, as we see at once from Figures 4 and 5, while the function _x_ 3 is increasing on the whole real line, that is, in the whole interval −∞ < _x_ < ∞, as we see from Figure 6. Similarly, if the graph of _f_ ( _x_ ) _falls_ steadily as _P_ moves from left to right with its abscissa in an interval _I_ , we say that _f_ ( _x_ ) is _decreasing in I_. For example, the functions | _x_ | and _x 2_ are both decreasing in the interval −∞ < _x_ ≤ 0, as is again apparent from Figures 4 and 5.
**b.** The graph of the constant function _f_ ( _x_ ) ≡ 1 is simply the horizontal line _y_ = 1, which neither rises nor falls. Therefore this function is neither increasing nor decreasing, in _every_ interval. The same is true of any other constant function.
**c.** It is easy to give a purely algebraic definition of increasing and decreasing functions. Thus a function _f_ defined in an interval _I_ is said to be _increasing in I_ if _f_ ( _x_ ) < _f_ ( _x_ ′) whenever _x_ and _x_ ′ are two points of _I_ such that _x_ < _x_ ′. Similarly, _f_ is said to be _decreasing in I_ if _f_ ( _x_ ) > _f_ ( _x_ ′) whenever _x_ and _x_ ′ are two points of _I_ such that _x_ < _x_ ′.
##### PROBLEMS
**.** What is the graph of the equation _x_ 2 − _y_ 2 = 0?
**2.** What is the graph of the equation _x_ 2 \+ _y_ 2 \+ 2 _x_ − 2 _y_ \+ 1 = 0?
**.** What is the equation of the circle of radius 2 with its center at the point (− 2, 3)?
**.** "No line parallel to the _y_ -axis can intersect the graph of a _function y_ = _f_ ( _x_ ) in more than one point." True or false?
**.** "No line parallel to the _y_ -axis can intersect the graph of an _equation F_ ( _x_ , _y_ ) = 0 in more than one point." True or false?
**.** Neither of the graphs in Figures 2 and 3 is the graph of a function. Why not?
**.** What is special about the graph of a _one-to-one_ function _y_ = _f_ ( _x_ )?
**8.** Let _G_ be the graph of the function _f_ ( _x_ ). Describe the graphs of the functions _f_ ( _x_ ) + _c_ and _f_ ( _x_ \+ _c_ ).
**.** Which of the following functions are even and which are odd?
(a) _y_ ≡ 2; (b) _y_ = _x_ \+ 1;
**.** Show that the function
is even if _n_ is even and odd if _n_ is odd.
_Comment_. By _x n_ we mean, of course, the _nth power_ of _x_ , that is,
**11.** Show that the product of two functions of the same parity is even, while the product of two functions of different parity is odd.
**.** "If the function _f_ ( _x_ ) is increasing in an interval _I_ , then the function − _f_ ( _x_ ) is decreasing in _I_ , and conversely." True or false?
**.** What is the equation of the circle circumscribed about the square with vertices (0, 0), (0, 1), (1, 0) and (1, 1)?
**.** Given the graph of a one-to-one function _y_ = _f_ ( _x_ ), how does one find the graph of the inverse function _x_ = _f_ −1( _y_ )?
**.** Show that if a function _f_ is increasing, then _f_ is one-to-one, with an increasing inverse _f_ −1.
**.** Show that if a function _f_ is decreasing, then _f_ is one-to-one, with a decreasing inverse _f_ −1
***17.** Graph the function _y_ = | _x_ \+ 1| + | _x_ − 1|. Is the function piecewise linear? Where is the function increasing and where decreasing? What happens in the interval −1 ≤ _x_ ≤ 1?
***18.** Graph the function _y_ = | _x_ | + | _x_ \+ 1| + | _x_ \+ 2|. Is the function piecewise linear? Where does the graph have corners? Where is the function increasing and where decreasing?
***19.** Show that if 0 < _x_ < _x_ ′, then 0 < _x n_ < _x_ ′ _n_ for all _n_ = 1, 2, ...
***20.** Show that if _n_ is even, then the function (10) is increasing in the interval 0 ≤ _x_ < ∞ and decreasing in the interval −∞< _x_ ≤ 0, while if _n_ is odd, then the function is increasing in the whole interval (−∞, ∞).
### 2.4 DERIVATIVES AND LIMITS
**2.41. An instructive calculation**
**a.** We now make a little calculation, leading us straight to the heart of our subject. Consider the function
defined for all nonzero values of _h_. Do not be disconcerted by our use of the "off beat" letter _h_ for the independent variable; this is a deliberate choice, made to avoid "tying up" the letter _x_ , which will be needed a little later. We cannot allow _h_ = 0 in (1), because _Q_ ( _h_ ) would then reduce to the expression
which is meaningless. In fact, if 0/0 is to make sense, it must mean the one and only number _c_ such that 0 · _c_ = 0. But _every_ number _c_ has this property! For this reason, the expression 0/0 is often called an _indeterminate form._
Despite the fact that _Q_ (0) itself is meaningless, the function _Q_ ( _h_ ) is perfectly meaningful for values of _h_ which are as close as we please to 0, _whether these values of h be positive or negative_. Thus it is only natural to ask: What happens to _Q_ ( _h_ ) as _h_ gets "closer and closer" to the forbidden value _h_ = 0?
**b.** To answer this question, we first carry out the algebraic operations in the numerator of _Q_ ( _h_ ), obtaining
We then divide the numerator by the denominator _h_ , which is permissible since _h_ ≠ 0. This gives the simple formula
_Q_ ( _h_ ) = 2 + _h_ ( _h_ ≠ 0).
We now observe that as _h_ gets "closer and closer" to 0, _Q_ ( _h_ ) in turn gets "closer and closer" to 2. In fact, the distance between _Q_ ( _h_ ) and 2, regarded as points of the real line, is just
| _Q_ ( _h_ ) − 2| = |(2 + _h_ ) − 2| = | _h_ |
(Theorem 1.55), and | _h_ | is certainly very small whenever _h_ is very near 0, for the simple reason that | _h_ | is just the distance between _h_ and 0.
The fact that _Q_ ( _h_ ) gets "closer and closer" to 2 as _h_ gets "closer and closer" to 0 is summarized by writing
In words, (2) says that "the limit of _Q_ ( _h_ ) as _h_ approaches zero equals 2." This is our first encounter with the concept of a _limit_ , about which we will say much more later. An equivalent way of writing (2) is
_Q_ ( _h_ ) → 2 as _h_ → 0,
which says that " _Q_ ( _h_ ) approaches 2 as _h_ approaches zero."
**c.** We have just killed two birds with one stone. Not only have we calculated the limit of _Q_ ( _h_ ) as _h_ approaches zero, but as we will see in a moment, we have also calculated something called "the derivative of the function _f_ ( _x_ ) = _x_ 2 at the point _x_ = 1." In fact, both the limit and the derivative equal 2.
**2.42. The derivative concept**
**a.** Let _f_ ( _x_ ) be a function defined in some neighborhood of a point _x_ 0. Then by the _difference quotient_ of _f_ ( _x_ ) at _x_ 0, we mean the new function
of the variable _h_. The letter _Q_ stands for "quotient," and _x 0_ has a subscript zero to show that it is a _fixed_ value of the argument _x_. For example, if _f_ ( _x_ ) = _x_ 2 and _x_ 0 = 1, then (3) reduces to
which is nothing other than the expression (1).
**b.** Let _Q_ ( _h_ ) be the difference quotient of the function _f_ ( _x_ ) at the point _x_ 0. Then by the _derivative of f_ ( _x_ ) _at the point x_ 0, denoted by _f_ ′( _x_ 0) and pronounced " _f_ prime of _x_ zero," we mean the limit
provided that the limit "exists" (that is, makes sense). Here, as in formula (2), the expression (4) means the number, if any, which " _Q_ ( _h_ ) approaches as _h_ approaches zero." Combining (3) and (4), we find that
It is important to note that _f_ ′( _x_ 0) is a number, not a function. Suppose once again that _f_ ( _x_ ) = _x_ 2 and _x_ 0 = 1. Then
Thus the derivative of the function _f_ ( _x_ ) = _x_ 2 at the point _x_ = 1 exists and equals 2. This justifies the claim made in Sec. 2.41c.
**c.** The derivative _f_ ′( _x_ 0) is also called the _rate of change of y_ = _f_ ( _x_ ) _with respect to x at the point x_ 0. The reason for this designation is not hard to find. The numerator of the difference quotient (3) is just the change
_f_ ( _x_ 0 \+ _h_ ) − _f_ ( _x_ 0)
in the dependent variable _y_ = _f_ ( _x_ ) when the independent variable _x_ is changed from _x_ 0 to _x_ 0 \+ _h_ , while the denominator of (3) is just the change
( _x_ 0 \+ _h_ ) − _x_ 0 = _h_
in _x_ itself. Therefore the difference quotient itself becomes
and the derivative is the "limiting value" of this "change ratio" as the change in the independent variable _x_ gets "smaller and smaller," that is, "approaches zero." As for the word "rate," which suggests something changing with respect to time, it is a metaphor borrowed from problems involving motion, where the independent variable is indeed time (usually denoted by _t_ ), and the dependent variable changes with respect to time at a certain "rate."
**d.** In Sec. 1.12 we described calculus as the "mathematics of change" and formulated the two basic types of problems with which calculus deals. The first of these problems was stated in the following unsophisticated language:
(1) Given a relationship between two changing quantities, what is the rate of change of one quantity with respect to the other?
We are now in a position to restate this problem in more precise language:
(1′) Given a function _y_ = _f_ ( _x_ ), what is the rate of change of _y_ with respect to _x_?
The study of this problem is the province of a branch of calculus known as _differential calculus_ and always involves the calculation of a derivative.
**2.43. Examples**
**a.** Find the derivative of the function _f_ ( _x_ ) = _x_ 2 at an arbitrary point _x_ 0.
SOLUTION. For this function we have
which becomes
after doing a little algebra. But as _h_ gets "closer and closer" to 0, the quantity 2 _x_ 0 \+ _h_ gets "closer and closer" to 2 _x_ 0, for the simple reason that the distance between the points 2 _x_ 0 \+ _h_ and 2 _x_ 0 is just
|(2 _x_ o \+ _h_ ) − 2 _x_ 0| = | _h_ |.
Therefore
so that finally
_f_ ′( _x_ 0) = 2 _x_ 0.
Note that _f′_ ( _x_ 0) = 2 when _x_ 0 = 1, as we already know from the preliminary calculation made in Sec. 2.42b.
**b.** Let
where _m_ and _b_ are constants. Find the derivative of _f_ ( _x_ ) at an arbitrary point _x_ 0.
SOLUTION. In this case,
The last step calls for finding the number to which _m_ gets "closer and closer" as _h_ → 0, but this can only be the number _m_ itself, since _m_ is a constant! Choosing _m_ = 0 in (7), we find that _the derivative of any constant function f_ ( _x_ ) ≡ _b equals_ 0 _at every point x_ 0. Choosing _m_ = 1, _b =_ 0 in (7), we find that _the derivative of the function f_ ( _x_ ) = _x equals_ 1 _at every point x_ 0.
You will recognize (7) as the equation of the straight line with slope _m_ and _y_ -intercept _b_. Since the derivative of (7) equals _m_ at every point _x_ 0, you may begin to suspect that the derivative has something to do with slope. Indeed it has. In Sec. 2.52d we will see that the derivative _f_ ′( _x_ 0) is just the slope of the _tangent to the curve y_ = _f_ ( _x_ ) at the point with abscissa _x_ 0. Of course, this will require that we first decide what is meant by the "tangent to a curve." Note that the word "tangent" as used here has nothing to do with the same word as used in trigonometry (Sec. 1.82b). In fact, the tangent to a curve is a line, while the tangent of an angle is a number.
**2.44. Limits**
**a.** So far we have only considered limits of the form
where the function _Q_ ( _h_ ) is the difference quotient associated with another function _f_ ( _x_ ). This, of course, is the special kind of limit leading to the notion of a derivative. More generally, we can consider the limit of an arbitrary function _f_ ( _x_ ) as its argument _x_ approaches an arbitrary point _x_ 0, provided that _f_ ( _x_ ) is defined in some _deleted_ neighborhood of _x_ 0 (Sec. 1.63a). Thus we say that a function _f_ ( _x_ ) _approaches the limit A as x approaches x_ 0, or that _f_ ( _x_ ) _has the limit A at x_ 0, if _f_ ( _x_ ) gets "closer and closer" to _A_ as _x_ gets "closer and closer" to _x_ 0 without ever actually coinciding with _x_ 0. This fact is expressed by writing
or
_f_ ( _x_ ) → _A_ as _x_ → _x_ 0.
Put somewhat differently, (8) means that _f_ ( _x_ ) is "arbitrarily near" _A_ for all _x_ which are "sufficiently near" _x_ 0, or, equivalently, that | _f_ ( _x_ ) − _A_ | is "arbitrarily small" for all "sufficiently small" (but nonzero) values of | _x_ − _x_ 0|.
**b.** Can this rather intuitive definition of a limit be made mathematically exact? Yes, it can, by resorting to the following procedure, invented by Cauchy in the early nineteenth century, which involves two positive numbers, traditionally called _ε_ (the Greek letter epsilon) and _δ_ (the Greek letter delta). What does it really mean to say that | _f_ ( _x_ ) − _A_ | is "arbitrarily small" for all "sufficiently small" | _x_ − _x_ 0|? Just this: Suppose somebody we call the "challenger" presents us with any positive number _ε_. he pleases. Then we must be able to find another positive number _δ_ such that | _f_ ( _x_ ) − _A_ | < _ε_ for all _x_ (≠ _x_ 0) satisfying the inequality | _x_ − _x_ 0| < _δ_. At this point, you may well ask: What has all this to do with the numbers | _f_ ( _x_ ) − _A_ | and | _x_ − _x_ 0| being small? The answer is simply that we allow our challenger to present us with any positive number whatsoever, in particular, with a number which is as small as he pleases (that is, "arbitrarily small"). We must then find a corresponding number _δ_ , which in general cannot be "too large" (and hence is "sufficiently small") such that | _f_ ( _x_ ) − _A_ | < _ε_ whenever | _x_ − _x_ 0| < _δ_.
Thus, once again, in more concise language, to say that _f_ ( _x_ ) has the limit _A_ at _x_ 0 means that, _given any ε_ > 0, _we can find a number δ_ > 0 _such that_ | _f_ ( _x_ ) − _A_ | < _ε whenever_ 0 < | _x_ − _x_ 0| < _δ_. Here the formula 0 < | _x_ − _x_ 0| < _δ_ is just a neat way of writing | _x_ − _x_ 0| < _δ_ and _x_ ≠ _x_ 0 at the same time, since | _x_ − _x_ 0| > 0 is equivalent to _x_ ≠ _x_ 0.
You may find this definition a bit strange, but we urge you to master it anyway. It is a most valuable tool, which will help you keep many calculations brief and to the point (see Probs. 12-15, for example).
**c.** The fact that _x_ is not allowed to take the value _x_ 0 in the definition of the limit of _f_ ( _x_ ) at _x_ 0 is crucial. It shows that the limit (if any) of _f_ ( _x_ ) at _x_ 0 has nothing to do with the value of _f_ ( _x_ ) at _x_ = _x_ 0, since this value does not even enter into the definition of the limit. In fact, a function can have a limit even at a point _x_ 0 where it fails to be _defined_! For example, the limit of the function
as _h_ → 0 is the derivative _f_ ′( _x_ 0), a fundamental concept of calculus, and yet _Q_ ( _h_ ) is undefined at _h_ = 0, where it reduces to the indeterminate form 0/0.
**d.** It is often convenient to talk about having a limit without specifying what the limit is. Thus we say that a function _f_ ( _x_ ) _has a limit at x_ 0 if there is some number _A_ such that _f_ ( _x_ ) → _A_ as _x_ → _x_ 0.
**2.45. Examples**
**a.** Let _x_ 0 be an arbitrary point. Then
as we would certainly hope! This can be seen at once by using " _ε_ , _δ_ language." In fact, given _ε_ > 0, we need only choose _δ_ = _ε_. It is then self-evident that | _x_ − _x_ 0| < _ε_ whenever 0 < | _x_ − _x_ 0| < _δ_.
**b.** The constant function _f_ ( _x_ ) ≡ _A_ approaches the limit _A_ as _x_ approaches an arbitrary point _x_ 0. In fact, in this case | _f_ ( _x_ ) − _A_ | ≡ | _A_ − _A_ | ≡ 0, so that, given any _ε_ > 0, we have | _f_ ( _x_ ) − _A_ | < _ε_ for _all x_ , and in particular for all _x_ such that 0 < | _x_ − _x_ 0| < _δ_ , where _δ_ > 0 is arbitrary.
**c.** The function _f_ ( _x_ ) = 3 _x_ approaches the limit 6 as _x_ → 2. In fact, given any _ε_ > 0, choose _δ_ = _ε_ /3. We then have
| _f_ ( _x_ ) − 6| = |3 _x_ − 6| = 3| _x_ − 2| < 3 _δ_ = _ε_
whenever 0 < | _x_ − 2| < _δ_.
**d.** Show that
SOLUTION. It seems plausible enough that as a number gets "closer and closer" to 2, its square must get "closer and closer" to 4, and this is an acceptable, if somewhat crude, solution. A better solution is based on the use of " _ε_ , _δ_ language" and goes as follows: Given any _ε_ > 0, let _δ_ be the smaller of the two numbers 1 and _ε_ /5. Then
whenever 0 < | _x_ − 2| < _δ_ , since our choice of _δ_ automatically forces _x_ to satisfy the extra condition | _x_ \+ 2| < 5, as well as | _x_ \- 2| < _ε_ /5. To see this, note that | _x_ − 2| < _δ_ certainly implies | _x_ − 2| < 1, or equivalently 1 < _x_ < 3, so that 3 < _x_ \+ 2 < 5, which in turn implies | _x_ \+ 2| < 5.
Actually, even this solution is only a "stopgap measure." In Sec. 2.61 we will show that
and then (10) will be an immediate consequence of (9), with _x_ 0 = 2.
**e.** Does the function
have a limit at _x_ = 0?
SOLUTION. No. If _x_ >0, then | _x_ | = _x_ and _f_ ( _x_ ) = 1, while if _x_ <0, then | _x_ | = − _x_ and _f_ ( _x_ ) = −1. Therefore _f_ ( _x_ ) takes both values 1 and −1 in every deleted neighborhood of _x_ = 0. But then _f_ ( _x_ ) can hardly be "arbitrarily near" some number _A_ for all _x_ "sufficiently near" 0, even if we pick _A_ = 1 or _A_ = −1. This intuitive solution seems plausible, but its crudity is rather distressing. Again " _ε_ , _δ_ language" comes to the rescue, providing us with a solution which is both simple and perfectly sound. Suppose _f_ ( _x_ ) has a limit _A_ at _x_ = 0. Then, choosing _ε_ = , we can find a number _δ_ > 0 such that | _f_ ( _x_ ) − _A_ | < _ε_ = whenever 0 < | _x_ | < _δ_. Let _x_ 1 = _δ_ , _x_ 2 = − so that, in particular 0 < | _x_ 1| < _δ_ , 0 < | _x_ 2| < _δ_. Then, on the one hand,
_f_ ( _x_ 1)= 1, _f_ ( _x_ 2)= −1,
while, on the other hand,
But these requirements are incompatible. In fact, using the triangle inequality (3), p. 14, we find that
while, at the same time,
| _f_ ( _x_ 1) − _f_ ( _x_ 2)| = |1 − (−1)| = 2.
Thus the assumption that _f_ ( _x_ ) has a limit at _x_ = 0 has led to the absurd conclusion that 2 < 1. Therefore _f_ ( _x_ ) does not have a limit at _x_ = 0.
**f.** Does the function _f_ ( _x_ ) = | _x_ | have a derivative at _x_ = 0?
SOLUTION. No, since
or
if we denote the independent variable by _x_ instead of by _h_ , which is our privilege. But we have just shown that this limit fails to exist.
##### PROBLEMS
**1.** Let _f_ ( _x_ ) = _ax_ 2 \+ _bx_ \+ _c_ , where _a_ , _b_ and _c_ are constants. Verify that _f_ ′( _x_ 0) = 2 _ax_ 0 \+ _b_ at every point _x_ 0.
**2.** Let _f_ ( _x_ ) = x3. Verify that _f_ ′( _x_ 0) = at every point _x_ 0.
_Comment_. Problems 1 and 2 are just "warming up exercises." The technique of evaluating derivatives will be developed more systematically in Sec. 2.7.
**.** Is the formula _f_ ′( _α_ \+ _β_ ) = _f_ ′( _α_ ) + _f_ ′( _β_ ) true for _f_ ( _x_ ) = _mx_ \+ _b_? For _f_ ( _x_ ) = _x_ 2?
**.** Can _f_ ′( _x_ 0) ever equal _f_ ( _x_ 0)?
**.** Show that _f_ ( _x_ ) → _A_ as _x_ → _x_ 0 and _f_ ( _x_ ) − _A_ → 0 as _x_ → _x_ 0 mean exactly the same thing.
**.** Find the limit
**.** Does the function
have a limit at the point _x_ = 1? At the point _x_ = 2?
**.** Find the limit of the function
at the point
(a) _x_ = −1; (b) _x_ = 0; (c) _x_ = .
**.** To find the number _δ_ in the " _ε_ , _δ_ language" must we know the number ε? Is _δ_ a function of _ε_?
**.** Show that if _f_ ( _x_ ) → 0 as _x_ → _x_ 0, then | _f_ ( _x_ )| → 0 as _x_ → _x_ 0, and conversely.
***11.** Show that if _f_ ( _x_ ) → _A_ as _x_ → _x_ 0, then | _f_ ( _x_ )| → | _A_ | as _x_ → _x_ 0. Is the converse true?
The following problems are all easily solved with the help of " _ε_ , _δ_ language." Make sure that you understand what these problems mean, even if you don't work them out.
***12.** Show that if _f_ ( _x_ ) has a limit at _x_ 0, then the limit is unique. In other words, show that _f_ ( _x_ ) cannot have more than one limit at _x_ 0.
***13.** Show that changing the value of a function _f_ ( _x_ ) at any point _x_ 1 ≠ _x_ 0 has no effect on the limit (if any) of _f_ ( _x_ ) at _x_ 0.
_Comment_. Thus only the values of _f_ ( _x_ ) in the "immediate vicinity" of _x_ 0 have any effect on the "limiting behavior" of _f_ ( _x_ ) at _x_ 0.
***14.** Show that if _f_ ( _x_ ) → _A_ as _x_ → _x_ 0, then there is a deleted neighborhood of _x_ 0 in which | _f_ ( _x_ )| < | _A_ | + 1.
_Comment_. Thus a function cannot become "too large" in absolute value near a point where it has a limit.
***15.** Show that if _f_ ( _x_ ) → _A_ ≠ 0 as _x_ → _x_ 0, then there is a deleted neighborhood of _x_ 0 in which _f_ ( _x_ ) has the same sign as _A_ and | _f_ ( _x_ )| > | _A_ |.
_Comment_. Thus a function cannot change sign or even become "too small" in absolute value near a point where it has a _nonzero_ limit.
### 2.5 MORE ABOUT DERIVATIVES
**2.51. Increment notation.** The meaning of the derivative can be made even clearer by using a somewhat different notation. Given a function _y_ = _f_ ( _x_ ), let the change in the independent variable, namely the difference between the final and initial values of _x_ , be denoted by Δ _x_ , instead of by _h_ as in Sec. 2.42c. Here Δ _x_ , where Δ is the Greek capital letter delta, must be thought of as a _single entity_ , pronounced "delta _x_ ," and _not_ as the product of the separate symbols Δ and _x_. Let the corresponding change in the dependent variable, namely the difference between the final and initial values of _y_ , be denoted by Δ _y_ ("delta _y_ "), that is, let
Δ _y_ = _f_ ( _x_ 0 \+ Δ _x_ ) − _f_ ( _x_ 0).
Then formula (6), p. 54, takes the particularly suggestive form
where the reason for the term "difference quotient" is now staring you in the face. We also call Δ _x_ the _increment of x_ and Δ _y_ the _increment of y_. We will favor this "increment notation" from now on, because of the way it identifies the quantities which are actually being changed.
In terms of increment notation, formula (5), p. 53, defining the derivative of _f_ ( _x_ ) at the point _x_ 0 takes the form
Bear in mind that in writing Δ _x_ → 0, we impose no restriction on the sign of Δ _x_ , which is free to take both positive and negative values.
**2.52. The tangent to a curve**
**a.** In keeping with the remarks in Example 2.43b, we now decide what is meant by the "tangent to a curve." You already know how the tangent is defined in the case where the curve is a circle _C_. In fact, according to elementary geometry, the tangent to a circle _C_ at a point _P_ 0 is the line which intersects _C_ in the point _P_ 0 and _in no other point_. A moment's thought shows that this property of the tangent to a circle is useless for defining the tangent to a general curve. For example, in the case of the parabola _y_ = _x_ 2 graphed in Figure 7, there are _two_ lines, namely the _x_ -axis and the _y_ -axis, intersecting the curve in the origin _O_ and in no other point. But common sense rejects the idea of the _y_ -axis being the tangent to the curve at _O_ , although the _x_ -axis seems a perfectly plausible candidate for the tangent to the curve at _O_.
As this example suggests, the key property of the tangent is that it "hugs the curve very closely at the point of tangency." For example, this seems to be true of the line _T_ in Figure 7, which represents the tangent to the parabola at the point _P_ 0. We now give a precise mathematical meaning to this qualitative idea. As you may suspect, the notions of limit and derivative will play a prominent role here.
Figure 7.
**b.** Thus let _P_ 0 = ( _x_ 0, _y_ 0) be a fixed point and _P_ = ( _x_ , _y_ ) a variable point of a given curve _y_ = _f_ ( _x_ ), and let _S_ be the straight line going through the points _P_ 0 and _P_. Such a line is called a _secant_ ( _line_ ) of the curve _y_ = _f_ ( _x_ ). The slope of _S_ is just
or equivalently
in terms of the increments
Δ _x_ = _x_ − _x_ 0, Δ _y_ = _y_ − _y_ 0 = _f_ ( _x_ 0 \+ Δ _x_ ) − _f_ ( _x_ 0).
The geometrical meaning of these increments is shown in Figure 8, which is drawn for the case where Δ _x_ and Δ _y_ are both positive. As an exercise, sketch similar figures for the other three choices of the signs of Δ _x_ and Δ _y_. All of these figures are equiv alent from the standpoint of illustrating the construction of the tangent to the curve _y_ = _f_ ( _x_ ) at _P_ 0.
Figure 8.
Figure 9.
**c.** We now vary the point _P_ along the curve, making _P_ move "closer and closer" to the fixed point _P_ 0, and at the same time allowing _P_ to move freely from one side of _P_ 0 to the other. Then Δ _x_ approaches zero, and at the same time the secant through _P_ 0 and _P_ varies, taking first one position and then another. Suppose the limit
exists. Then the straight line through _P_ 0 with slope _m_ is called the _tangent_ ( _line_ ) _to the curve y_ = _f_ ( _x_ ) _at the point P_ 0. In other words, the tangent at _P_ 0 is the "limiting position" of the secant through _P_ 0 and _P_ as the variable point _P_ approaches the fixed point _P_ 0, taking positions on both sides of _P_ 0. This behavior is illustrated in Figure 9, where the secants go through a sequence of positions _S_ l, _S_ 2, _S_ 3, _S_ 4, ..., getting "closer and closer" to the limiting tangent line _T_. This figure also shows that, unlike the case of a circle, the tangent to a general curve _C_ may well intersect _C_ in points other than the point of tangency _P_ 0.
**d.** Finally, substituting (1) into (2), we find that
where the limit on the right is, of course, just the derivative _f_ ′( _x_ 0) of the function _f_ ( _x_ ) at the point _x_ 0. Thus we have proved the following key result of differential calculus: _The curve y_ = _f_ ( _x_ ) _has a tangent T at the point P_ 0 = ( _x_ 0, _f_ ( _x_ 0)) _if the derivative f_ ′( _x_ 0) _exists, and only in this case. The slope of T is then equal to f′_ ( _x_ 0). This slope is often simply called the _slope of the curve at P_ 0.
According to Sec. 1.91a, the equation of the straight line with slope _m_ going through the point ( _x_ 0, _y_ 0) is just
_y_ = _m_ ( _x_ − _x_ 0) + _y_ 0.
Therefore the tangent to the curve _y_ = _f_ ( _x_ ) at the point ( _x_ 0, _y_ 0) = ( _x_ 0, _f_ ( _x_ 0)) has the equation
**2.53. Examples**
**a.** Find the tangent _T_ to the parabola _y_ = _x 2_ at the point _P_ 0 = ( _x_ 0, _y_ 0).
SOLUTION. Here _f_ ( _x_ ) = _x_ 2, and hence
as calculated in Example 2.43a. Substituting (4) into (3), we find that _T_ has the equation
since _y_ 0 = . Setting _y_ = 0 and solving for _x_ , we see at once that the line _T_ has the _x_ -intercept _x_ 0/2. Thus, to construct the tangent to the parabola _y_ = _x_ 2 at a point _P_ 0 other than the origin, we need only drop the perpendicular _P_ 0 _B_ to the _x_ -axis, bisect the segment _OB_ , and then draw the line _T_ joining the midpoint _A_ of _OB_ to the point _P_ 0, as shown in Figure 7. If _P_ 0 is the origin, then _x_ 0 = 0 and (5) reduces to the equation _y_ = 0. Therefore in this case _T_ is the line _y_ = 0, namely the _x_ -axis, as conjectured in Sec. 2.52a.
**b.** Find the tangent _T_ to the curve _y_ = | _x_ | at the point ( _x_ 0, _y_ 0).
SOLUTION. Here we have _f_ ( _x_ ) = | _x_ |. If _x_ 0 > 0, then
since _f_ ( _x_ ) = _x_ for all _x_ in a suitable neighborhood of _x_ 0. Substituting (6) into (3), we find that _T_ has the equation
_y_ = 1 · ( _x_ − _x_ 0) + _x_ 0 = _x_ ,
since _y_ 0 = | _x_ 0| = _x_ 0. Therefore in this case _T_ is just the line _y_ = _x_ , as is geometrically apparent from Figure 10A. On the other hand, if _x_ 0 < 0, we have
Figure 10.
instead of (6), since now _f_ ( _x_ ) = − _x_ for all _x_ in a suitable neighborhood of _x_ 0, and substitution of (7) into (3) gives
_y_ = −1 · ( _x_ − _x_ 0) − _x_ 0 = − _x_ ,
since in this case _y_ 0 = | _x_ 0| = − _x_ 0. Therefore _T_ is now just the line _y_ = − _x_ , as is geometrically apparent from Figure 10B.
If _x_ 0 = 0, then _f_ ′( _x_ 0) fails to exist, as shown in Example 2.45f. Therefore the curve _y_ = | _x_ | has no tangent at the origin _O_. The reason for this is geometrically evident from Figure 10C and is associated with the presence of the sharp "corner" of the graph of _y_ = | _x_ | at the origin. The secant drawn through the origin _O_ and a variable point _P_ of the curve _y_ = | _x_ | can hardly approach a "limiting position" as _P_ approaches _O_ , since the secant coincides with the line _y_ = _x_ whenever _P_ lies to the right of _O_ (in the first quadrant) and with the perpendicular line _y_ = − _x_ whenever _P_ lies to the left of _O_ (in the second quadrant). In fact, suppose _P_ approaches _O_ through a sequence of positions _on both sides of O_ , in keeping with the definition of the tangent at _O_. Then the secant changes its inclination by a full 90° (in either the clockwise or the counterclockwise direction) every time _P_ goes from one side of _O_ to the other, and this "wild" behavior is clearly inconsistent with the secant approaching any tangent line at all as _P_ approaches _O_.
**2.54. Differentiation**
**a.** The process leading from a function to its derivative is called _differentiation_ , with respect to the independent variable. Another way of saying that a function _y_ = _f_ ( _x_ ) has a derivative at a point _x_ 0 is to say that _f_ ( _x_ ) is _differentiable at x_ 0. If _f_ ( _x_ ) is differentiable at every point of an interval _I_ , we say that _f_ ( _x_ ) is _differentiable in I_. For example, the function _f_ ( _x_ ) = | _x_ | is differentiable in both intervals −∞ < _x_ < 0 and 0 < _x_ < ∞, although it fails to be differentiable at the point _x_ = 0. Whenever we call a function _differentiable_ , without further qualification, we always mean differentiable at some point or in some interval, where the context makes it clear just what is meant.
**b.** Suppose _f_ ( _x_ ) is differentiable in an interval _I_. Then the derivative
exists for every _x_ 0 in _I_. Hence there is a _new function_ defined in _I_ , whose value at every point _x_ 0 is just _f_ ′( _x_ 0). This new function, which we denote by ( _f_ ( _x_ ))′, or simply by _f_ ′( _x_ ), is called the _derivative of f_ ( _x_ ), with no mention of a point _x_ 0. It will always be clear from the context whether the term "derivative" refers to a _function_ or to a _number_ , namely the value of the derivative function at some point. The results of Examples 2.43a and 2.43b can now be written more concisely as
( _x_ 2)′ = 2 _x_ , ( _mx_ \+ _b_ )′ = _m_.
The derivative of a function _y_ = _f_ ( _x_ ) is sometimes denoted simply by _y_ ′.
**2.55. The differential**
**a.** We will usually write (8) in the form
dropping the subscript zero in three places. This is done with the understanding that _x_ is held _fixed_ during the evaluation of the limit. The numerator of the difference quotient in (9) is often denoted by Δ _f_ ( _x_ ) instead of by Δ _y_. This has the advantage of allowing us to make explicit the point _x_ at which the increment Δ _f_ ( _x_ ) = _f_ ( _x_ \+ Δ _x_ ) − _f_ ( _x_ ) is taken. We call Δ _f_ ( _x_ ) the _increment of the function f at the point x_ , where, as usual, Δ _f_ is to be thought of as a single entity. In terms of this notation, (9) takes the form
Equation (10) can also be written as
(Sec. 2.4, Prob. 5), or equivalently as
Where
The numerator of _α_ (Δ _x_ ) is just the error made in replacing Δ _f_ ( _x_ ) by _f_ ′( _x_ ) Δ _x_ , and, according to (11), this error is small in absolute value compared to |Δ _x_ | if |Δ _x_ | is small. Thus it is often a good approximation to replace Δ _f_ ( _x_ ) by the quantity _f_ ′( _x_ ) Δ _x_ , called the _differential of the function f at the point x_. For the differential we introduce the special notation
which stresses the connection between _df_ ( _x_ ) and the increment Δf(x). As in the case of Δ _f_ , the symbol d _f_ must be thought of as a _single entity_ , pronounced "dee f," and _not_ as the product of the separate symbols _d_ and f. In the case of a function written as _y_ = _f_ ( _x_ ), we can write _dy_ for _df_ ( _x_ ), just as we write Δ _y_ for Δ _f_ ( _x_ ). Then (12) takes the form
We will often drop arguments for brevity, writing _f_ ′ for _f_ ′( _x_ ), _df_ for _df_ ( _x_ ), and so on. Thus formulas like _df_ = _f_ ′( _x_ ) Δ _x_ or _dy_ = _f_ ′ Δ _x_ shouldn't bother you a bit.
**b. Example.** Find the increment Δ _y_ and the differential _dy_ of the function _y_ = _x_ 2 for _x_ = 20, Δ _x_ = 0.1. What is the error of the approximation Δ _y_ ≈ _dy_? (The symbol ≈ means "is approximately equal to.")
SOLUTION.
Δ _y_ = ( _x_ \+ Δ _x_ )2 − _x_ 2 = 2 _x_ Δ _x_ \+ (Δ _x_ )2,
_dy_ = (x2)′ Δ _x_ = 2 _x_ Δ _x_ ,
and hence
Δ _y_ = 2(20)(0.1) + (0.1)2 = 4.01,
_dy_ = 2(20)(0.1) = 4.00
for _x_ = 20, Δ _x_ = 0.1. Thus the error of the approximation Δ _y_ ≈ _dy_ is only 0.01, about 0.25% of the actual value of Δ _y_.
**2.56. The _d_ notation**
**a.** For the function _y_ = _f_ ( _x_ ) = _x_ , we have _dy_ = _dx_ , _f_ ′( _x_ ) = 1, so that (13) reduces to
_dx_ = 1 · Δ _x_ = Δ _x_.
In other words, the increment and the differential of the _independent_ variable are equal. We can now write
instead of (12), and
instead of (13). These formulas lead at once to an important new way of writing derivatives, called the " _d_ notation," which we will use freely from now on. Solving (14) for _f_ ′( _x_ ), we get
or, without the arguments,
This formula is read as " _f_ prime equals dee _f_ by dee _x_." Similarly, if _y_ = _f_ ( _x_ ), it follows from (15) and the last sentence of Sec. 2.54b that
The advantage of the new notation, in which _f_ ′ becomes _df/dx_ and _y_ ′ becomes _dy/dx_ is that the independent variable is now indicated explicitly. Thus there is a distinction between
which is not so easily made in the old notation; here _x_ , _t_ , _u_ , ... indicate different symbols for the independent variable. To distinguish between the different derivatives (16) verbally, we call _df/dx_ the derivative of _f with respect to x_ , _df/dt_ the derivative of _f with respect to t_ , and so on. Such distinctions are often crucial.
**b.** Having learned enough about differentials to appreciate how the " _d_ notation" arises, there is no further need to think of derivatives as ratios of differentials. Rather you should regard _d/dx_ as a single entity, pronounced "dee by dee _x_ " and called the _differentiation operator_ , which has the effect of differentiating with respect to _x_ any function written after it; for simplicity, the function is often written after the first letter _d_ in _d/dx_ rather than after the whole expression. Thus, for example,
where the prime denotes differentiation with respect to _x_. Similar remarks apply to differentiation operators like _d/dt_ , _d/du_ , and so on, where the independent variable is indicated by another letter.
##### PROBLEMS
**.** Find the increment Δ _x_ of the independent variable and the corresponding increment Δ _y_ of the dependent variable for the function _y_ = 1/ _x_ 2 if _x_ is changed from 0.01 to 0.001.
**2.** Let _u_ = _u_ ( _x_ ) and _v_ = _v_ ( _x_ ) be two functions of _x_ , where, for simplicity, we use the same letters to denote both the functions and the dependent variables. Show that Δ( _u_ \+ _v_ ) = Δ _u_ \+ Δ _v_ ;.
**3.** Verify that the tangent to the line _y_ = _mx_ \+ _b_ at every point of the line is just the line itself, as is to be expected.
**.** Does the curve _y_ = _x_ 2 have two different tangents which are parallel? Does the curve _y_ = _x_ 3?
**5.** Does the curve _y_ = _x_ 2 have two different tangents which are perpendicular? Does the curve _y_ = _x_ 3?
**.** Find the tangent to the curve _y_ = _x_ 2 going through the point (2, 0). Note that (2, 0) is not a point of the curve.
**.** At what point of the curve _y_ = _x_ 2 is the tangent parallel to the secant drawn through the points with abscissas 1 and 3?
**.** Find the value of _x_ such that _df_ ( _x_ ) = 0.8 if _f_ ( _x_ ) = _x_ 2 and Δ _x_ = −0.2.
**.** Find the increment Δ _y_ and the differential _dy_ of the function _y_ = _x_ 3 if _x_ = 1 and
(a) Δ _x_ = l; (b) Δ _x_ = 0.1; (c) Δ _x_ = 0.01; (d) Δ _x_ = 0.001.
In each case, find the error _E_ = Δ _y_ − _dy_ made in replacing Δ _y_ by _dy_ , both as a number and as a percentage of Δ _y_. What happens as Δ _x_ gets smaller?
**.** When is Δ _y_ ≈ _dy_ a bad approximation?
***11.** Let _u_ = _u_ ( _x_ ) and _v_ = _v_ ( _x_ ) be the same as in Problem 2. Write two different expressions for Δ( _uv_ ) in terms of Δ _u_ and Δ _v._
***12.** For what values of _b_ and _c_ does the curve _y_ = _x_ 2 \+ _bx_ \+ _c_ have the line _y_ = _x_ as a tangent at the point with abscissa 2?
***13.** Where is the function _y_ = | _x_ \+ 1| + | _x_ − 1| differentiable? Where does the graph of the function fail to have a tangent?
***14.** How much would the earth's surface area increase if its radius were increased by 1 foot?
***15.** What is the geometrical meaning of the differential _dy_?
### 2.6 MORE ABOUT LIMITS
**2.61. Algebraic operations on limits.** The following basic theorem on limits is used time and again in calculus. It merely says that "the limit of a sum equals the sum of the limits," and similarly with the word "sum" replaced by "difference," "product" and "quotient."
THEOREM. _If_
_then_
_provided that B_ ≠ 0 _in the last formula_.
_Proof_. As you might expect, " _ε_ , _δ_ language" is the thing to use here. Since _f_ ( _x_ ) → _A_ as _x_ → _x_ 0 and _g_ ( _x_ ) → _B_ as _x_ → _x_ 0, then, given any _ε_ > 0, we can find numbers _δ_ 1 > 0 and _δ_ 2 > 0 such that | _f_ ( _x_ ) − _A_ | < _ε_ /2 whenever 0 < | _x_ − _x_ 0| < _δ_ 1 and | _g_ ( _x_ ) − _B_ | < ε _/2_ whenever 0 < | _x_ − _x_ 0| < _δ_ 2. (Yes, we really mean _ε_ /2 here, not _ε_.) Let _δ_ be the smaller of the two numbers _δ_ 1 and _δ_ 2. Then, using our old standby, the triangle inequality (3), p. 14, we have
whenever 0 < | _x_ − _x_ 0| < _δ_. But this is just " _ε_ , _δ_ language" for the statement that _f_ ( _x_ ) + _g_ ( _x_ ) → _A_ \+ _B_ as _x_ → _x_ 0. Thus we have proved (2). To prove (3), we need only note that
whenever 0 < | _x_ − _x_ 0| < _δ_.
To prove (4) and (5) we argue in the same way. For example to prove (4) we show that, given any _ε_ > 0, there is a _δ_ > 0 such that | _f_ ( _x_ ) _g_ ( _x_ ) − _AB_ | < _ε_ whenever 0 < | _x_ − _x_ 0| < _δ_ , and similarly for (5). However, the details are not very instructive, and for that reason are left to Problem 15. You can work through this rather difficult problem if you have a special interest in mathematical technique. Otherwise, persuade yourself of the validity of formulas (4) and (5) by thinking of their intuitive meaning.
**2.62. Examples**
**a.** Show that
SOLUTION. We already know from Example 2.45a that
a result which is almost obvious. Therefore, using (4) twice, we have
and
(by Example 2.45b, "the limit of a constant equals the constant itself"). It then follows from (2) that
**b.** Evaluate
SOLUTION. We have
where in the last step we cancel the common factor _x_ − 3 of the numerator and denominator, relying on the fact that _x_ approaches 3 without ever being _equal_ to 3, so that _x_ − 3 is never zero. Using (3) and (6), we have the easy calculations
It then follows from (5) that
**2.63. Continuous functions**
**a.** Once again we stress that the _limit_ of a function _f_ ( _x_ ) at a point _x_ 0 is something quite different from the _value_ of _f_ ( _x_ ) at _x_ 0, and in fact _f_ ( _x_ ) may not even be _defined_ at _x_ 0. There is a special name for a function "nice enough" to have a limit at _x_ 0 which equals its value at _x_ 0: Such a function is said to be _continuous at x_ 0. In other words, we say that _f_ ( _x_ ) is continuous at _x_ 0 if
Of course, this presupposes that _f_ ( _x_ ) is defined in some _nondeleted_ neighborhood of _x_ 0, so that both sides of (7) make sense. If formula (7) does not hold, we say that _f_ ( _x_ ) is _discontinuous at x_ 0.
**b.** If a function _f_ ( _x_ ) is continuous at every point of an interval _I_ , we say that _f_ ( _x_ ) is _continuous in I_. When we call a function _continuous_ , without further qualification, we always mean continuous at some point or in some interval, where the context makes it clear just what is meant. A function is said to be _discontinuous_ , without further qualification, if it is discontinuous at one or more points. The property of being continuous is called _continuity_ , and plays an important role in calculus.
**c.** Algebraic operations on continuous functions are governed by the following rule:
THEOREM. _If the functions f_ ( _x_ ) _and g_ ( _x_ ) _are both continuous at x_ 0, _then so are the sum f_ ( _x_ ) + _g_ ( _x_ ), _the difference f_ ( _x_ ) − _g_ ( _x_ ), _the product f_ ( _x_ ) _g_ ( _x_ ) _and the quotient f_ ( _x_ )/ _g_ ( _x_ ), _provided that g_ ( _x_ 0) ≠ 0 _in the case of the quotient._
_Proof_. This is an immediate consequence of the corresponding theorem on limits. Instead of (1), we now have
But then formulas (2) through (5) become
provided that _g_ ( _x_ 0) ≠ 0 in the last formula, and this is exactly what is meant by saying that the functions _f_ ( _x_ ) + _g_ ( _x_ ), _f_ ( _x_ ) − _g_ ( _x_ ), _f_ ( _x_ ) _g_ ( _x_ ) and _f_ ( _x_ )/ _g_ ( _x_ ) are continuous at _x_ 0.
**d.** COROLLARY. _If the functions f_ 1( _x_ ), _f_ 2( _x_ ),..., _f_ _n_ ( _x_ ) _are all continuous at x_ 0, _then so are the sum f_ 1( _x_ ) + _f_ 2( _x_ ) + . . . + _f n_( _x_ ) _and the product f_ 1( _x_ ) _f_ 2( _x_ ) . . . _f_ _n_ ( _x_ )
_Proof_. For example, if
_h_ ( _x_ ) = _f_ 1( _x_ ) + _f_ 2( _x_ ) + _f_ 3( _x_ ),
then _h_ ( _x_ ) = _g_ ( _x_ ) + _f_ 3( _x_ ), where _g_ ( _x_ ) = _f_ 1( _x_ ) + _f_ 2( _x_ ). One application of the theorem just proved shows that _g_ ( _x_ ) is continuous at _x_ 0, being the sum of two functions _f_ 1( _x_ ) and _f_ 2( _x_ ) which are continuous at _x_ 0, and then another application of the theorem shows that _h_ ( _x_ ) is continuous at _x_ 0, being the sum of two functions _g_ ( _x_ ) and _f_ 3( _x_ ) which are continuous at _x_ 0. The proof for products is virtually the same.
**2.64. Examples**
**a.** Any constant function is continuous _everywhere_ , that is, at every point _x_ 0. In fact, if _f_ ( _x_ ) ≡ _c_ , then
Moreover, the function _x_ is continuous everywhere, since, by Example 2.45a,
**b.** By a _polynomial_ we mean a function of the form
_P_ ( _x_ ) = _a_ 0 \+ _a_ 1 _x_ \+ _a_ 2 _x_ 2 \+ . . . + _a nxn_,
where _a_ 0, _a_ 1 _a_ 2, ..., _a n_ are arbitrary constants and _n_ is a positive integer, called the _degree_ of the polynomial (if _a n_ ≠ 0). Each term in the sum is continuous, by the corollary and the preceding example, and hence so is the sum itself, by the corollary again. Thus _P_ ( _x_ ) is defined and continuous everywhere, that is, in the whole interval (−∞, ∞).
**c.** By a _rational function_ we mean a quotient of two polynomials
where _n_ and _N_ are in general different. It follows from the preceding example and the theorem that _R_ ( _x_ ) exists and is continuous except at the points (if any) where the denominator of (8) equals zero. For example, the rational function
is continuous in the whole interval (−∞, ∞), while
is continuous everywhere except at the two points _x_ = ± 1.
**d.** The function | _x_ | is continuous everywhere. In fact,
since _x_ → 0 and | _x_ | → 0 mean exactly the same thing, while
if _x_ 0 > 0, and
if _x_ 0 < 0. In the last two calculations, we use the fact that _x_ has the same sign as _x_ 0 if _x_ is "sufficiently close" to _x_ 0.
**e.** The function
which can also be written
is discontinuous at _x_ = 0, since, as shown in Example 2.45e, it has no limit at _x_ = 0. Moreover, there is no way of defining _f_ ( _x_ ) at _x_ = 0 which will make _f_ ( _x_ ) continuous at _x_ = 0, since the failure of _f_ ( _x_ ) to have a limit at _x_ = 0 has nothing to do with its value at _x_ = 0 (Sec. 2.44c).
**f.** The situation is different for the function
Here
and the function fails to be continuous at _x_ = 0 only because the point _x_ = 0 has been excluded, rather artificially, from the domain of _g_ ( _x_ ). However, we can make _g_ ( _x_ ) continuous at _x_ = 0 by the simple expedient of setting _g_ (0) equal to 0, _by definition_. In this way, we can "remove the discontinuity" at _x_ = 0, replacing the discontinuous function (10) by the function _x_ which is continuous everywhere. On the other hand, if we set _g_ (0) equal to 1, say, instead of 0, we get a new function
Figure 11.
which is still discontinuous at _x_ = 0, since
**g.** To get a better idea of the behavior of discontinuous functions, we graph the three functions (9), (10) and (11) in Figure 11, using hollow dots to indicate "missing points." The solid dot in Figure 11C indicates the "isolated point" (0, 1) belonging to the graph of the function (11). There is something about all these graphs that shows us at a glance that each is the graph of a discontinuous function, namely each graph behaves "pathologically" at _x_ = 0. Either the graph has no point at all with abscissa 0, as in Figures 11A and 11B, or there is such a point, as in Figure 11C, but it does not fall where it "ought to," namely at the origin. Moreover, the graph in Figure 11A has a "jump discontinuity" at _x_ = 0, since a point moving along the graph of _f_ ( _x_ ) from left to right has to make a "sudden jump" at _x_ = 0 in order to get from the line _y_ = −1 to the line _y_ = 1. The graph of a function which is continuous in some interval cannot have "gaps" and "jumps" like these. Thus you can think of the graph of a continuous function as one which can be drawn without lifting pen from paper.
**2.65. One-sided limits**
**a.** Taking another look at Figure 11A, we are led at once to the idea of a _one-sided limit_ , for we can hardly help noticing that the function _y_ = _f_ ( _x_ ) would have a limit at _x_ = 0, equal to either −1 or 1, if we were to insist that _x_ approach the origin _O from one side or the other_ , either from the left of _O_ , taking only negative values, or from the right of _O_ , taking only positive values. In fact, under these circumstances, we could forget about the behavior of the function _y_ = _f_ ( _x_ ) on the other side of _O_ , regarding _y_ = _f_ ( _x_ ) as either the constant function _y_ ≡ −1 on the left of _O_ or the constant function _y_ ≡ 1 on the right of _O_. If _x_ approaches a point _x_ 0 from the left, we write _x_ → _x_ 0−, while if _x_ approaches _x_ 0 from the right, we write _x_ → _x_ 0+. Thus we have just observed that in the case of the function (9), _f_ ( _x_ ) → −1 as _x_ → 0− and _f_ ( _x_ ) → 1 as _x_ → 0 +, or equivalently
where the first limit is called the _left-hand limit_ of _f_ ( _x_ ) at _x_ = 0 and the second is called the _right-hand limit_ of _f_ ( _x_ ) at _x_ = 0.
**b.** As you may have already guessed, there is also a kind of continuity involving one-sided limits. In fact, if
we say that _f_ ( _x_ ) is _continuous from the left at x_ 0, while if
we say that _f_ ( _x_ ) is _continuous from the right at x_ 0. For example, the function _f_ ( _x_ ) defined by (9) has the one-sided limits (12) at the point _x_ = 0, so that _f_ ( _x_ ) can be made continuous from the left at _x_ = 0 by setting _f_ (0) = − 1 or continuous from the right by setting _f_ (0) = 1 (remember that _f_ (0) was not defined originally). However, there is clearly no way to make this function continuous both from the left and from the right at _x_ = 0, since the one-sided limits (12) are _different._
**c.** We now make a small improvement in our definition of a function which is continuous in an interval _I_. In Sec. 2.63b we insisted that such a function be continuous at every point of _I_. We now relax this requirement a bit, but _only_ at the end points of _I_. Suppose _I_ contains its _left_ end point, call it _a_. Then we require only that _f_ ( _x_ ) be continuous _from the right_ at _a_. This makes sense, since a point which stays inside _I_ can only approach _a_ from the right. Similarly, if _I_ contains its _right_ end point, call it _b_ , we now require only that _f_ ( _x_ ) be continuous _from the left_ at _b_. Again this makes sense, since a point inside _I_ can only approach _b_ from the left.
For example, the function
differing from (9) only by having _x_ ≥ 0 instead of _x_ > 0, is regarded as continuous in the closed interval 0 ≤ _x_ ≤ 1, even though it is not continuous at _x_ = 0, because it is continuous at every point _x_ > 0 and continuous _from the right_ at _x_ = 0.
**2.66.** We have already encountered a function | _x_ | which fails to have a derivative at a point (namely _x_ = 0) where it is continuous. On the other hand, _a function is automatically continuous at any point where it has a derivative_. In fact, suppose _f_ ( _x_ ) has a derivative _f_ ′( _x_ 0) at a point _x_ 0. Then
with the help of Theorem 2.61 and the definition of the derivative _f_ ′( _x_ 0). But
and therefore
or equivalently
since _f_ ( _x_ 0) is a constant. But this is just another way of writing the formula
expressing the continuity of _f_ ( _x_ ) at _x_ 0 (let _x_ = _x_ 0 \+ Δ _x_ , Δ _x_ = _x_ − _x_ 0).
##### PROBLEMS
**1.** Deduce from (4) that if _f_ ( _x_ ) → _A_ as _x_ → _x_ 0, then _cf_ ( _x_ ) → _cA_ as _x_ → _x_ 0, where _c_ is an arbitrary constant.
**.** Show that if _f_ 1( _x_ ) → _A_ 1, _f_ 2( _x_ ) → _A_ 2, ..., _f_ _n_ ( _x_ ) → _A_ _n_ as _x_ → _x_ 0, then _f_ 1( _x_ ) + _f_ 2( _x_ ) + . . . + _f n_( _x_ ) → _A_ 1 \+ _A_ 2 \+ . . . + _A n_ and _f_ 1( _x_ ) _f_ 2( _x_ )... _f n_( _x_ ) → _A_ 1 _A_ 2... _A n_ as _x_ → _x_ 0.
**3.** What goes wrong in Example 2.62b if we try to evaluate the limit _A_ directly by using Theorem 2.61, without making the preliminary factorization?
**.** Evaluate
**.** At what points does the function
fail to be continuous?
**6.** What choice of _f_ (0) makes the function
continuous at _x_ = 0?
**.** Find the one-sided limits at _x_ = 2 of the function
**.** Graph the function _f_ ( _x_ ) = _x_ ], where [ _x_ ] is the integral part of _x_ ([Sec. 1.4, Prob. 10). At what points is _f_ ( _x_ ) discontinuous?
**.** Verify that the function _f_ ( _x_ ) = [ _x_ ] is continuous from the right at every point of the real line.
**.** Is the function
continuous in the interval 0 ≤ _x_ ≤ 2? In the interval 0 ≤ _x_ ≤ 1? In 0 < _x_ < 2?
**11.** Show that the ordinary limit
exists if both one-sided limits
exist and are equal, and conversely. Show that if the ordinary limit exists, then
**12.** Show that _f_ ( _x_ ) is continuous at _x_ 0 if _f_ ( _x_ ) is continuous both from the left and from the right at _x_ 0, and conversely.
**.** Do the considerations of Sec. 2.65c apply to _open_ intervals?
***14.** Show that if _f_ ( _x_ ) is continuous at _x_ 0, then so is | _f_ ( _x_ )|.
***15.** Prove formulas (4) and (5) with the help of Sec. 2.4, Problems 14 and 15.
### 2.7 DIFFERENTIATION TECHNIQUE
So far we have only calculated the derivatives of a few functions, resorting each time to the definition of the derivative as a limit. This is, of course, very inefficient, and what we really want are ways to evaluate derivatives simply and methodically, without the need to always go back to first principles. To this end, we now prove a number of easy theorems, each establishing an important differentiation rule. As in Sec. 2.55a, we will avoid the use of subscripts to keep the notation as simple as possible. For the same reason, we will often leave out arguments of functions, writing _f_ instead of _f_ ( _x_ ), _F_ instead of _f_ ( _x_ ), and so on.
**2.71. a. T HEOREM (Derivative of a sum or difference).** _If the functions f and g are both differentiable at x, then so is the sum F_ = _f_ \+ _g and the difference G_ = _f_ − _g. The derivatives of F and G at x are given by_
_or equivalently by_
_in the "d notation."_
_Proof_. By the definition of a derivative, we have
But "the limit of a sum is the sum of the limits" (Theorem 2.61), and therefore
which proves (1). The proof of (2) is virtually the same.
**b.** By an _algebraic sum_ we mean a sum whose terms can have either sign.
COROLLARY. _If the functions f_ 1, _f_ 2, ..., _f n are all differentiable at x_, _then so is the algebraic sum F_ = _f_ 1 ± _f_ 2 ± . . . ± _f n_. _The derivative of F at x is given by_
_or equivalently by_
_Proof_. Here we can choose any combination of pluses and minuses in _F_ = _f_ 1 ± _f_ 2 \+ . . . ± _f_ _n_ , just as long as we pick the same combination in (3). To prove (3), we merely apply Theorem 2.71a repeatedly. For example, if _F_ = _f_ 1 \+ _f_ 2 − _f_ 3, then _F_ = _g_ − _f_ 3, where _g_ = _f_ 1 \+ _f_ 2, and therefore
_F_ ′( _x_ ) = _g_ ′( _x_ ) − _f_ ′3( _x_ ), _g_ ′( _x_ ) = _f_ ′1( _x_ ) + _f_ ′2( _x_ ),
which together imply
_F_ ′( _x_ ) = _f_ ′1( _x_ ) + _f_ ′2( _x_ ) − _f_ ′3( _x_ ),
and similarly for other combinations.
In other words, to calculate the derivative of the algebraic sum of two or more differentiable functions, we differentiate the sum "term by term." This fact can be expressed even more simply by writing
( _f_ 1 ± _f_ 2 ± . . . ± _f n_)′ = _f_ ′1 ± _f_ ′2 ± . . . ± _f_ ′ _n_.
**2.72. a. T HEOREM (Derivative of a product).** _If the functions f and g are both differentiable at x, then so is the product F = fg. The derivative of F at x is given by_
_or equivalently by_
_Proof_. Again, by definition,
where we repeatedly use Theorem 2.61. It follows that
where we use the definitions of the derivatives _f_ ′( _x_ ) and _g_ ′( _x_ ) as well as the fact that _f_ ( _x_ ) is a constant in this calculation. But _g_ is continuous at _x_ , by Sec. 2.66, and therefore
Substituting (6) into (5), we get the desired formula (4).
**b.** COROLLARY. _If the functions f_ 1, _f_ 2, . . ., _f n are all differentiable at x_, _then so is the product F_ = _f_ 1 _f_ 2... _f n. The derivative of F is given by_
_or equivalently by_
_Proof_. This time we apply Theorem 2.72a repeatedly. For example, if _F_ = _f_ 1 _f_ 2 _f_ 3, THEN _f_ = _gf_ 3 where _g_ = _f_ 1 _f_ 2. Therefore, by two applications of the theorem,
In other words, to calculate the derivative of the product of two or more differentiable functions, we add the result of differentiating the first factor and leaving the other factors alone to the result of differentiating the second factor and leaving the others alone, and then if necessary we add this sum to the result of differentiating the third factor and leaving the others alone, and so on until all the factors have been differentiated.
**2.73.** THEOREM **(Derivative of a quotient)**. _If the functions f and g are both differentiable at x_ , _then so is the quotient F_ = _f_ / _g_ , _provided that g_ ( _x_ ) ≠ 0. _The derivative of F at x is given by_
_or equivalently by_
_Proof_. This time we have
where Theorem 2.61 has been used twice. Taking the limits called for in (8) and using formula (6) again, we get the desired formula (7), where, of course, _g_ 2( _x_ ) is shorthand for [ _g_ ( _x_ )]2.
**2.74. Examples**
**a.** We have already shown in Examples 2.43a and 2.43b that
where _c_ is an arbitrary constant. Of course, each of these formulas is an almost effortless application of the definition of the derivative as a limit.
**b.** If _c_ is an arbitrary constant, then
SOLUTION. By Theorem 2.72a,
But this immediately implies (10), since _dc_ / _dx_ = 0, by the first of the formulas (9).
**c.** Show that
where _x_ 0 = 1, by definition.
SOLUTION. Formula (11) holds for _n_ = 1 and _n_ = 2. In fact, for these values it reduces to the last two of the formulas (9). Suppose (11) holds for _n_ = _k_ , so that
Then, by Theorem 2.72a,
Thus if formula (11) holds for _n_ = _k_ , it also holds for _n_ = _k_ \+ 1. But the formula holds for _n_ = 1 (or, for that matter, for _n_ = 2), and therefore it holds for all _n_ = 1, 2, ..., by mathematical induction (Sec. 1.37).
**d.** Differentiate the polynomial
_P_ ( _x_ ) = _a_ 0 \+ _a_ 1 _x_ \+ _a_ 2 _x_ 2 \+ . . . + _a nxn_.
SOLUTION. Using the corollary to Theorem 2.71a, together with formulas (10) and (11), we get
This is, of course, another polynomial, whose degree is one less than that of the original polynomial _P_ ( _x_ ).
**e.** Show that
SOLUTION. Using Theorem 2.73, we have
with the help of (11). Suppose we set
in accordance with the usual definition of negative powers. Then (12) can be written in the form
But, apart from the necessary stipulation that _x_ ≠ 0, this is just formula (11) with − _n_ instead of _n_. Thus we see that (11) remains valid for _negative_ integers. Note that (11) also holds for _n_ = 0 (and _x_ ≠ 0), since it then reduces to the formula
which merely expresses the fact that the derivative of the constant 1 equals 0.
We have just shown that formula (11) is valid for any integer _n_ , positive, negative or zero. Remarkably enough, it can be shown that (11) remains valid even when _n_ is an arbitrary real number. This is worth writing down as a separate formula:
Here, of course, we assume that _x r_ and _x_ _r_ −1 are both defined. If _r_ is irrational, we must require that _x_ > 0, but if _r_ is rational, _x r_ and _x_ _r_ −1 may well be defined for all _x_ or for all _x_ ≠? 0 (see Probs. 5-7). We will use formula (14) freely from now on, and it should be committed to memory. The proof of (14), and of formulas (15) and (16) below, as well as the reason for the requirement _x_ > 0 if _r_ is irrational, will be given in Sec. 4.45, where we will decide just what is meant by _x r_ in the first place! Formula (14) is used in conjunction with the natural extension of (13) to the case of an arbitrary real number _r_ :
**f.** Differentiate .
SOLUTION. First we note that . To see this, we use the formula
valid for arbitrary real numbers _r_ and _s_. Choosing _r_ = , _s_ = 2 in (16), we find that
( _x_ 1/2)2 = _x_ ,
which implies .
Applying formula (14), with _r_ = , we get
where in the last step we use (15). Thus, finally,
**2.75. Higher derivatives**
**a.** Let _f_ ( _x_ ) be differentiable in an interval _I_ , with derivative _f_ ′( _x_ ), and suppose _f_ ′( _x_ ) is itself differentiable in _I_. Then the function
is called the _second derivative_ of _f_ ( _x_ ), written _f_(2)( _x_ ) or _f_ ″( _x_ ). Similarly, if _f_ ″( _x_ ) is differentiable in _I_ , the function
is called the _third derivative_ of _f_ ( _x_ ), written _f_ (3)( _x_ ) or _f_ ″′( _x_ ). More generally, by the _derivative of order n of f_ ( _x_ ), or briefly the _nth derivative of f_ ( _x_ ) denoted by _f_ ( _n_ )( _x_ ), we mean the function
assuming that the derivative _f_ ( _n_ −1)( _x_ ) of order _n_ − 1 exists and is itself differentiable in _I_. We also write
_f_ ( _x_ ) = _f_ (0)( _x_ ),
that is, _f_ ( _x_ ) is the result of not differentiating _f_ ( _x_ ) at all!
**b.** In terms of the " _d_ notation," _f_ ( _n_ )( _x_ ) is written as
Note that in the numerator the exponent _n_ is attached to the symbol _d_ , while in the denominator it is attached to the independent variable _x_. The expression _d n_/ _dx n_ should be thought of as a single entity calling for _n_ -fold differentiation of any function written after it. Similarly, _d nf_( _x_ )/ _dx n_ should be regarded as just another way of writing _f_ ( _n_ )( _x_ ), without attempting to ascribe separate meaning to the different symbols making up the expression. Higher derivatives of the dependent variable are defined in the natural way. Thus, if _y_ = _f_ ( _x_ ), we have
**c. Example.** If _y_ = _x_ 4, then
It is clear that in this case all derivatives of order _n_ > 5 also equal zero. Note that the fourth derivative of _x_ 4 equals 4 · 3 · 2 · 1 = 24. More generally,
where we use the symbol _n_!, pronounced " _n_ factorial," as shorthand for the product of the first _n_ positive integers.
##### PROBLEMS
**.** Differentiate
(a) _x_ 4 \+ 3 _x_ 2 − 6; (b) 2 _ax_ 3 − _bx_ 2 \+ _c_ ;
**.** Differentiate
(a) ( _x_ − _a_ )( _x_ − _b_ ); (b) _x_ ( _x_ − _a_ )( _x_ − _b_ ); (c) (1 + 4 _x_ 2)(1 + 2 _x_ 2); (d) (2 _x_ − 1)( _x_ 2 − 6 _x_ \+ 3).
**.** Differentiate
**.** "The derivative of a rational function is also a rational function." True or false?
**5.** Given any positive integer _n_ , by the _nth root_ of _x_ , denoted by or _x_ 1/ _n_ (with the conventions ), we mean either the unique _nonnegative_ number whose _n_ th power equals _x_ if _n_ is even, or the unique number (possibly negative) whose _n_ th power equals _x_ if _n_ is odd. Show that this definition is in keeping with formula (16). Show that if _n_ is odd, then is defined for all _x_ and is an odd function, while if _n_ is even, then is defined only for _x_ ≥ 0.
**6.** Given any positive integers _m_ and _n_ , where the fraction _m/n_ is in lowest terms, let
_by definition_. Show that this is in keeping with formula (16). Show that if _n_ is odd, then _x m/n_ is defined for all _x_ and is an even function if _m_ is even and an odd function if _m_ is odd, while if _n_ is even, then _x m/n_ is defined only for _x_ ≥ 0.
**7.** Let _x m/n_ be the same as in the preceding problem, and let
_by definition_. Show that this is in keeping with formula (15). Show that if _n_ is odd, then _x_ − _m_ / _n_ is defined for all _x_ ≠ 0 and is an even function if _m_ is even and an odd function if _m_ is odd, while if _n_ is even, then _x_ − _m_ / _n_ is defined only for _x_ > 0.
**.** Differentiate
**.** "The _nth_ derivative of a polynomial of degree _n_ is a nonzero constant." True or false?
**.** Find the first _n_ derivatives of the function _y_ = 1/ _x_.
**11.** Given two functions _f_ and _g_ with third derivatives, evaluate ( _fg_ )′″.
**.** Let _y_ = _x_ (2 _x_ − 1)2( _x_ \+ 3)3. Find _y_ (6) and _y_ (7) with as little work as possible.
***13.** Show that the segment of any tangent to the curve _y_ = 1/ _x_ cut off by the coordinate axes is bisected by the point of tangency.
***14.** Why are the denominators in (8) all nonzero, as required?
### 2.8 FURTHER DIFFERENTIATION TECHNIQUE
**2.81. a.** The concept of an inverse function was introduced in Sec. 2.16. The next rule shows how to express the derivative of an inverse function in terms of the derivative of the original function.
THEOREM. _Let f be a one-to-one function with inverse g_ = _f_ -1. _Suppose f is differentiable at x_ , _with derivative f_ ′( _x_ ) ≠ 0, _and suppose g is continuous at y_ = _f_ ( _x_ ). _Then g is differentiable at y, with derivative_
_Proof_. If
_y_ = _f_ ( _x_ ), _y_ \+ Δ _y_ = _f_ ( _x_ \+ Δ _x_ ),
then
_x_ = _g_ ( _y_ ), _x_ \+ Δ _x_ = _g_ ( _y_ \+ Δ _y_ ),
so that, in particular,
Since _g_ is continuous at _y_ , Δ _y_ → 0 implies Δ _x_ 0. But then
where the denominator _f_ ( _x_ \+ Δ _x_ ) − _f_ ( _x_ ) cannot vanish since _f_ is one-to-one. Therefore
**b.** In the " _d_ notation," (1) becomes
More concisely, we have
or equivalently
in terms of the variables _y_ = _f_ ( _x_ ) and _x_ = _g_ ( _y_ ). All three formulas resemble algebraic identities, but they do not, of course, constitute a proof of our theorem. They do show, however, that the " _d_ notation" is so apt that it tends to suggest true theorems!
**c.** In order to use Theorem 2.81a, we must somehow know that the inverse function _g_ = _f_ −1 is continuous at _x_ , so that Δ _y_ → 0 will imply Δ _x_ → 0. In every case of interest, this will follow from the following fact, a complete proof of which is beyond the scope of this book (for the easy part of the proof, see Sec. 2.3, Probs. 15 and 16): _If _f_ is continuous and one-to-one in a closed interval_ [ _a_ , _b_ ], _then there are only two possibilities:_
(1) _f is increasing in_ [ _a_ , _b_ ] _and its inverse function f_ −1 _is increasing and continuous in the closed interval_ [ _f_ ( _a_ ), _f_ ( _b_ )];
(2) _f is decreasing in_ [ _a_ , _b_ ] _and its inverse function f_ −1 _is decreasing and continuous in the closed interval_ [ _f_ ( _b_ ), _f_ ( _a_ )].
The meaning of this assertion is illustrated by Figure 12A for the case of increasing _f_ (and _f_ −1) and by Figure 12B for the case of decreasing _f_ (and _f_ -−1).
**d. Example.** The function
_y_ = _f_ ( _x_ ) = _x_ 2
is one-to-one and continuous in every closed interval _a_ ≤ _x_ ≤ _b_ , where _a_ ≥ 0, by Examples 2.16c and 2.64b, and we already know that _f_ ( _x_ ) is increasing in _a_ ≤ _x_ ≤ _b_ , since 0 ≤ _x_ 1 < _x_ 2 implies (why?). It follows from the italicized assertion that the inverse function
is increasing and continuous in the interval _a_ 2 ≤ _y_ ≤ _b_ 2. In particular, _f_ −1 is continuous at every point _y_ ≥ 0, since every such point belongs to an interval of the type _a_ 2 ≤ _y_ ≤ _b_ 2, where _a_ ≥ 0.
Figure 12.
Having proved the continuity of _x_ = , we can now use Theorem 2.81a to differentiate , without recourse to formula (14), p. 78. In fact,
which is the same as the result of Example 2.74f, except that the roles of _x_ and _y_ have been reversed (why is this?).
**2.82. a.** The concept of a composite function was introduced in Sec. 2.22. The next rule, one of the most important in calculus, shows how to express the derivative of a composite function in terms of the derivatives of its "constituent functions."
THEOREM. _Let f and g be two functions such that f is differentiable at x and g is differentiable at f_ ( _x_ ). _Then the composite function F, defined by F_ ( _x_ ) ≡ _g_ ( _f_ ( _x_ )), _is differentiable at x_ , _with derivative_
_Proof_. Let _y_ = _f_ ( _x_ ) and _z_ = _g_ ( _y_ ) = _F_ ( _x_ ). Since _f_ is differentiable at _x_ and _g_ is differentiable at _y_ = _f_ ( _x_ ), both limits
exist. But then
or equivalently
where
(recall Sec. 2.55a). Using (3) to write the increments of _y_ and _z_ in terms of the increments of _x_ and _y_ , we get
The trick now is to substitute the expression for Δ _y_ into the formula for Δ _z_. This gives
which is beginning to look a little like (2). It follows from the expression for Δ _y_ (or from the continuity of _f_ at _x_ ) that Δ _x_ → 0 implies Δ _y_ → 0, so that Δ _x_ → 0 implies both _α_ (Δ _x_ ) → 0 and _β_ (Δ _y_ ) → 0. Therefore, dividing (4) by Δ _x_ and taking the limit as Δ _x_ → 0, we find that
At the same time,
which implies
Comparing (5) and (6), we immediately get (2).
**b.** Thus, to differentiate the composite function _g_ ( _f_ ( _x_ )), we multiply the result of differentiating _g_ with respect to _its_ argument _f_ ( _x_ ) by the result of differentiating _f_ with respect to _its_ argument _x_. Roughly speaking, we "peel off" the layers of parentheses one by one, differentiating each function encountered on the way. This procedure applies equally well to more than two functions. For example, if _F_ ( _x_ ) ≡ _h_ ( _g_ ( _f_ ( _x_ ))), then
if _f_ , _g_ and _h_ are differentiable at _x_ , _f_ ( _x_ ) and _g_ ( _f_ ( _x_ )), respectively.
Theorem 2.82a is called the _chain rule_ , a term suggesting the process of differentiation just described. The term is even more suggestive in the case of functions of several variables (see Sec. 6.3).
**c.** In the " _d_ notation," (2) becomes
in terms of the variables _y_ = _f_ ( _x_ ), _z_ = _g_ ( _y_ ). Similarly, introducing variables _y_ = _f_ ( _x_ ), _z_ = _g_ ( _y_ ), _u_ = _h_ ( _z_ ), we can write (7) in the form
Do not make the mistake of regarding these formulas as trivial algebraic calculations, involving nothing more than cancelling _dy_ and _dz_ from the numerators and denominators. We have not done away with the need for proving the chain rule, but have merely written it in a very suggestive way, which, in particular, makes it very easy to remember.
**d.** In connection with composite functions, it should be noted that "a continuous function of a continuous function is continuous." More exactly, _if f and g are two functions such that f is continuous at x_ 0 _and g is continuous at f_ ( _x_ 0), _then the composite function F, defined by F_ ( _x_ ) ≡ _g_ ( _f_ ( _x_ )), _is continuous at x_ 0. This is easily shown with the help of " _ε_ , _δ_ language." Since _g_ is continuous at _f_ ( _x_ 0), given any _ε_ > 0, we can find a number _δ_ 1 > 0 such that
whenever | _f_ ( _x_ ) − _f_ ( _x_ 0)| < _δ_ 1. (Note that there is now no need to require that _f_ ( _x_ ) ≠ _f_ ( _x_ 0) or _x_ ≠ _x_ 0, since _g_ ( _f_ ( _x_ 0)) and _f_ ( _x_ 0) are defined.) But since _f_ is continuous at _x_ 0, we can also find a number _δ_ > 0 such that | _f_ ( _x_ ) − _f_ ( _x_ 0)| < _δ_ 1 whenever | _x_ − _x_ 0| < _δ_. Therefore (8) holds whenever | _x_ − _x_ 0| < _δ_ , that is, _F_ ( _x_ ) → _F_ ( _x_ 0) as _x_ → _x_ 0. In other words, _F_ is continuous at _x_ 0, as asserted.
Thus, to prove the continuity of the function
we use the continuity of _g_ ( _x_ ) = , established in Example 2.81d, and the continuity of _f_ ( _x_ ) = 1 + _x_ 2, established in Example 2.64b, together with the observation that _F_ ( _x_ ) ≡ _g_ ( _f_ ( _x_ )). In fact, _F_ ( _x_ ) is continuous in the whole interval (−∞, ∞), since 1 + _x_ 2 ≥ 1 for all _x_ , while is continuous for all _x_ ≥ 0.
**2.83. Examples**
**a.** Differentiate
SOLUTION. Here _F_ ( _x_ ) ≡ _g_ ( _f_ ( _x_ )), where _g_ ( _x_ ) = _x_ 100, _f_ ( _x_ ) = 1 + _x_ −2. Therefore
_g_ ′( _x_ ) = 100 _x_ 99, _f_ ′( _x_ ) = −2 _x_ −3.
by Examples 2.74c and 2.74e, or, if you prefer, by formula (14), p. 78. The chain rule then gives
It would be the height of folly to actually calculate the right side of (9) explicitly and then differentiate!
**b.** Differentiate
SOLUTION. By the chain rule,
where we use Example 2.74f twice.
**c.** Given a function _y_ = _f_ ( _x_ ), find the derivative of _y n_.
SOLUTION. By the chain rule,
or, more concisely,
**d.** If
find _y_ ′.
SOLUTION. Rather than solve (11) for _y_ as a function of _x_ and then differentiate _y_ , we differentiate (11) with respect to _x_ and then solve for _y_ ′. Thus
or
2 _x_ – _y_ – _xy_ ′ + 3 _y_ 2 _y_ ′ = 0,
with the help of (10). Solving for _y_ ′, we find that
Since we make no attempt to express _y_ explicitly as a function of _x_ , this process is called _implicit differentiation_. In the present case, direct calculation of _y_ from the cubic equation (11), followed by differentiation of the resulting expression for _y_ , would lead at once to a mass of tedious and completely unnecessary calculations.
**2.84.** Two remarks must be made in connection with implicit differentiation:
**a.** The method cannot be used blindly, since it gives a formal answer for _y_ ′ even in cases where _y_ (and hence _y_ ′) fails to exist! For example, the solution set (Sec. 2.31a) of the equation
_x_ 2 \+ _y_ 2 = _a_
is empty if _a_ < 0, and yet implicit differentiation of this equation gives
2 _x_ \+ 2 _yy_ ′ = 0,
and hence
regardless of the sign of _a_.
**b.** Extra work is required to evaluate the derivative _y_ ′ at a particular point _x_ = _x_ 0. For example, to evaluate (12) at _x_ = 1, we need the value of _y_ at _x_ = 1. Substituting _x_ = 1 into (11), we get
1 − _y_ \+ _y_ 3 = 1
or
_y_ 3 = _y_ ,
which has three solutions _y_ = 0 and _y_ = ± 1. The corresponding values of _y_ ′ are
Here, of course, stands for the value of _y_ ′ corresponding to _x_ = _x_ 0, _y_ = _y_ 0. We will often find this kind of "single vertical bar notation" useful.
##### PROBLEMS
**.** Use Theorem 2.81a to differentiate . Why is this function continuous everywhere? Check the result by using formula (14), p. 78.
**.** Which is larger, Justify your answer.
**.** Differentiate
**.** Let _y_ = (2 _x_ \+ 3)100. Find _y_ ′| _x_ = 0 with as little work as possible.
**5.** Show that "differentiation changes parity," which means that the derivative of an even function is odd, while the derivative of an odd function is even.
**6.** Use the chain rule and the rule for differentiating a product to deduce the rule for differentiating a quotient.
**.** Differentiate
**.** Differentiate
where _r_ and _s_ are arbitrary real numbers.
**.** Verify that
**10.** Verify that
**.** Where is the function continuous? How about the function ?
**12.** If
find _y_ ′. Evaluate _y_ ′| _x_ = 1.
**.** If
find _y_ ′. Evaluate _y_ ′| _x_ = 1.
**.** Use implicit differentiation to find _y_ ″ if _x myn_ = 1, where _m_ and _n_ are nonzero integers.
**.** We have already assumed the validity of the formula
where _r_ is an arbitrary real number (this will be proved in Sec. 4.45). Use the chain rule to verify (15) for the case where _r_ is an arbitrary _rational_ number _m_ / _n_ , starting from the fact that _y_ = _x m/n_ is equivalent to _y_ = _t m_, where _t_ = _x_ 1/ _n_.
**16.** Use implicit differentiation to verify (15) for rational _r_ , this time starting from the fact that _y_ = _x_ _m_ / _n_ is equivalent to _y n_ = _x m_.
***17.** If _x_ 2 \+ _y_ 2 = 25, find the values of _y_ ′, _y_ ″ and _y_ ′″ at the point (3, 4).
***18.** Solve Problem 13 by first finding an explicit formula for _y_ as a function of _x._
***19.** "If
find _y_ ′." Why is this an impossible assignment?
***20.** Heeding the warning in Sec. 2.84a, verify the existence of the derivatives in Sec. 2.84b and Problem 12.
### 2.9 OTHER KINDS OF LIMITS
**2.91. Limits involving infinity**
**a.** The graph of the function
is shown in Figure 13A. Examining this graph, we see that _f_ ( _x_ ) has a number of interesting "limiting properties" of a kind not yet encountered:
(a) As _x_ takes "smaller and smaller" positive values, _y_ takes "larger and larger" positive values;
(b) As _x_ takes "smaller and smaller" negative values, _y_ takes "larger and larger" negative values;
(c) As _x_ takes "larger and larger" positive values, _y_ takes "smaller and smaller" positive values;
(d) As _x_ takes "larger and larger" negative values, _y_ takes "smaller and smaller" negative values.
Figure 13.
By a "small" or "large" negative number, we mean, of course, a negative number of "small" or "large" absolute value.
These properties of _f_ ( _x_ ) all express a kind of limiting behavior in which "largeness" plays a role, as well as "smallness." How do we modify the language of limits to cover situations of this type? Very simply. If a variable, say _x_ , takes "larger and larger" positive values, we say that " _x_ approaches (plus) infinity" and write _x_ → ∞, while if _x_ takes "larger and larger" negative values, we say that " _x_ approaches minus infinity" and write _x_ → −∞. This is in keeping with the use of the symbols ∞ and −∞ in writing infinite intervals (Sec. 1.64). Once again, we emphasize that ∞ and −∞ are not numbers, so that we can never have _x_ = ∞ or _x_ = −∞.
We can now express the four listed properties of the function (1) much more concisely:
(a) As _x_ → 0 +, _y_ → ∞, or
(b) As _x_ → 0−, _y_ → −∞, or
(c) As _x_ → ∞, _y_ → 0 (more exactly, _y_ → 0+), or
(d) As _x_ → −∞, _y_ → 0 (more exactly, _y_ → 0−), or
In (a) and (b) we have "infinite limits," and in (c) and (d) we have "limits at infinity," as opposed to the "finite limits"
considered previously, where _A_ and _x_ 0 are both _numbers_ , rather than one of the symbols ∞, −∞, a fact often emphasized by calling _A_ and _x_ 0 "finite." There are also ordinary, "two-sided" infinite limits. For example, it is clear from Figure 13B that
We can also have infinite limits at infinity. For example, _x_ 2 takes "arbitrarily large" positive values when _x_ takes "arbitrarily large" values of either sign (see Figure 5, p. 49), and therefore
Similarly,
(see Figure 6, p. 50).
We will sometimes say that a variable "becomes infinite." This simply means that it approaches either (plus) infinity or minus infinity. A function _f_ ( _x_ ) is said to become infinite at a point _x_ 0 if _y_ = _f_ ( _x_ ) becomes infinite as _x_ approaches _x_ 0.
**b.** All this can be made mathematically exact by using a version of the " _ε_ , _δ_ language" in which letters other than _ε_ and _δ_ are used for numbers that are typically _large_ , since _ε_ and _δ_ have a built-in connotation of _smallness_. For example, _f_ ( _x_ ) → ∞ as _x_ → _x_ 0 means that, given any _M_ > 0, _no matter how large_ , we can find a number _δ_ > 0 such that _f_ ( _x_ ) > _M_ whenever 0 < | _x_ − _x_ 0| < _δ_ , _f_ ( _x_ ) → ∞ as _x_ → ∞ means that, given any _M_ > 0, we can find a _suitably large_ number _L_ > 0 such that _f_ ( _x_ ) > _M_ whenever _x_ > _L_ , _f_ ( _x_ ) → _A_ as _x_ → −∞ means that, given any number _ε_ > 0, we can find a number _L_ > 0 such that | _f_ ( _x_ ) − _A_ | < _ε_ whenever _x_ < − _L_ , and so on.
**c.** Every problem involving infinite limits or limits at infinity can be reduced to an analogous problem involving a finite limit at a finite point. To see this, we observe that if
or equivalently
then _x_ → ∞ is equivalent to _t_ → 0+, while _x_ → −∞ is equivalent to _t_ → 0−. In fact, if _x_ takes "larger and larger" positive values, then its reciprocal _t_ takes "smaller and smaller" positive values, and conversely, while if _x_ takes "larger and larger" negative values, _t_ takes "smaller and smaller" negative values, and conversely. It follows that _f_ ( _x_ ) → _A_ as _x_ → ∞ is equivalent to _f_ (1/ _t_ ) → _A_ as _t_ → 0+, while _f_ ( _x_ ) → _A_ as _x_ → −∞ is equivalent to _f_ (1/ _t_ ) → _A_ as _t_ → 0−. By virtually the same argument, _f_ ( _x_ ) → ∞ as _x_ → _x_ 0 is equivalent to 1/ _f_ ( _x_ ) → 0 + as _x_ → _x_ 0, while _f_ ( _x_ ) → −∞ as _x_ → _x_ 0 is equivalent to 1/ _f_ ( _x_ ) → 0− as _x_ → _x_ 0.
**2.92. Examples**
**a.** If
Find
SOLUTION. We make the substitution (2) and investigate the behavior of the resulting function of _t_ as _t_ → 0±. Thus
and similarly
This behavior as _x_ → _±_ ∞ is apparent from the graph of _f_ ( _x_ ), shown in Figure 14.
**b.** If
Find
Figure 14.
SOLUTION. Noting that _f_ ( _x_ ) is undefined at _x_ = 1, we go over to the function
which is perfectly well-behaved at _x_ = 1. Clearly _x_ 2 − 1 → 0 as _x_ → 1+, and moreover _x_ 2 − 1 > 0 if _x_ > 1 (why?). Therefore _g_ ( _x_ ) = _x_ 2 − 1 → 0+ as _x_ → 1+. It follows from the last sentence of Sec. 2.91c that
In virtually the same way, we see that _g_ ( _x_ ) → 0− as _x_ → 1 −, and hence
This behavior as _x_ → 1± is apparent from the graph of _f_ ( _x_ ), shown in Figure 15. From the graph we also deduce at a glance that
As this example illustrates, and as is quite generally true, a rational function approaches infinity at precisely those points where its denominator equals zero, provided, of course, that all common factors of the numerator and denominator have been cancelled out.
**2.93. Asymptotes**
**a.** Suppose a function _f_ ( _x_ ) becomes infinite at certain points, or is defined in an infinite interval, so that the argument _x_ becomes infinite. The graph of _f_ ( _x_ ) then consists of one or more parts, called "infinite branches," which "extend out to infinity" in one direction or another. For example, the function graphed in Figure 15 has three such branches, namely the part of the graph to the left of the line _x_ = − 1, the part of the graph between the lines _x_ = − 1 and _x_ = 1, and the part of the graph to the right of the line _x_ = 1.
Figure 15.
Figure 16.
Now suppose an infinite branch of _f_ ( _x_ ) approaches a straight line _L_ (without touching it) as _x_ approaches infinity in one or both directions, or as _x_ approaches certain "exceptional points" from one or both sides. Then _L_ is called an _asymptote_ of _f_ ( _x_ ), and the function _f_ ( _x_ ), or its graph, is said to approach _L asymptotically._
**b. Horizontal asymptotes.** If the horizontal line _y_ = _y_ 0 is an asymptote of _f_ ( _x_ ), then the distance between the point ( _x_ , _f_ ( _x_ )) and the line _y_ = _y_ 0 approaches 0 as _x_ → ∞ or as _x_ → −∞. But this distance is just | _f_ ( _x_ ) − _y_ 0|, and therefore at least one and possibly both of the formulas
must hold. Thus, to find the horizontal asymptotes (if any) of _f_ ( _x_ ), we need only examine the limiting behavior of _f_ ( _x_ ) as _x_ → ±∞. For example, the line _y_ = 1 is a horizontal asymptote of the function _f_ ( _x_ ) graphed in Figure 14, while the function
whose graph is the "S-shaped" curve shown in Figure 16, has two horizontal asymptotes, namely the lines _y_ = ±1. (How can this also be seen without drawing a graph?) It is clear that a function can have no more than two horizontal asymptotes.
**c. Vertical asymptotes.** If the vertical line _x_ = _x_ 0 is an asymptote of _f_ ( _x_ ), then the distance between the points ( _x_ , _f_ ( _x_ )) and the line _x_ = _x_ 0 approaches 0 as _x_ → _x_ 0 \+ or as _x_ → _x_ 0−. This is automatically true for any function _f_ ( _x_ ), but in the case of an asymptote, _f_ ( _x_ ) must at the same time become infinite, since an asymptote is defined only for an infinite branch. Therefore, excluding the case (of no practical interest) where _f_ ( _x_ ) does not stay of fixed sign as _f_ ( _x_ ) approaches its asymptote, we see that at least one and possibly two of the formulas
must hold. Thus, in looking for the vertical asymptotes of _f_ ( _x_ ), we can confine our attention to the points (if any) at which _f_ ( _x_ ) becomes infinite. Note that _f_ ( _x_ ) is necessarily undefined at any such point. For example, the lines _x_ = 1 and _x_ = −1 are vertical asymptotes of the function _f_ ( _x_ ) graphed in Figure 15.
An example of a function with an asymptote which is neither horizontal nor vertical is given in Problem 20.
**2.94. The limit of a sequence**
**a.** We say that a sequence { _x n_} _approaches_ ( _or has_ ) _a limit A as n approaches infinity_ if the general term _x n_ gets "closer and closer" to _A_ as _n_ gets "larger and larger." This fact is expressed by writing
or _x n_ → _A_ as _n_ → ∞. Put somewhat differently, (3) means that | _x n_ − _A_ | is "arbitrarily small" for all "sufficiently large" _n_. Better still, in the natural analogue of the " _ε_ , _δ_ language," (3) means that, given any _ε_ > 0, _no matter how small_ , we can find an integer _n_ 0 such that | _x_ _n_ − _A_ | < _ε_ whenever _n_ ≥ _n_ 0, that is, for all _n_ starting from _n_ 0. Clearly, this also means that every _ε_ -neighborhood of _A_ , namely every open interval of the form ( _A_ − _ε_ , _A_ \+ _ε_ ), contains all the terms of the sequence _x n_ starting from some value of _n_ , where this value, of course, depends on the choice of _ε_. Choosing _ε_ = 1, we find that all the terms of the sequence _x n_ starting from some value of _n_ fall in the interval ( _A_ − 1, _A_ \+ 1). This fact will be used in a moment.
**b.** A sequence is said to be _convergent_ if it has a finite limit as _n_ → ∞ and _divergent_ otherwise. If a sequence is convergent, with limit _A_ , we also say that the sequence _converges to A_. A sequence _x n_ is said to be _bounded_ if there is some number _M_ > 0 such that | _x_ _n_ | < _M_ for all _n_ = 1, 2, ... and _unbounded_ if no such number exists. (For emphasis, we sometimes write "for all _n_ = 1, 2, ..." instead of the equivalent phrase "for all _n_ ") For example, the sequence _x_ _n_ = 1/ _n_ is bounded, since 0 < _x_ _n_ ≤ 1 for all _n_ , while the sequence _x_ _n_ = _n_ is unbounded, since there is clearly no number _M_ > 0 such that | _x_ _n_ | = _n_ < _M_ for all _n._
**c.** _A convergent sequence is necessarily bounded_. In fact, if { _x_ _n_ } is a convergent sequence, with limit _A_ , then there is an integer _n_ 0 such that all the terms _x_ _n_ 0, _x_ _n_ 0 + 1, _x_ _n_ 0 + 2, ..., that is, all the terms _x n_ starting from _n_ 0, lie in the interval ( _A_ − 1, _A_ \+ 1). By choosing _M_ > 0 large enough we can see to it that the interval (− _M_ , _M_ ), with its midpoint at the origin, contains the interval ( _A_ − 1, _A_ \+ 1), together with the remaining terms _x_ 1, _x_ 2, ..., _x_ _n_ 0 _ 1, some or all of which may not lie in ( _A_ − 1, _A_ \+ 1). But then | _x_ _n_ | < _M_ for all _n_ = 1, 2,..., so that the sequence is indeed bounded, as claimed.
Since a convergent sequence is necessarily bounded, _an unbounded sequence is necessarily divergent._
**d.** A sequence _x n_ is said to be _increasing_ if _x_ _n_ < _x_ _n_ \+ 1 for all _n_ and _decreasing_ if _x_ _n_ > _x_ _n_ \+ 1 for all _n_. By a _monotonic sequence_ we mean either an increasing sequence or a decreasing sequence. An important tool in the study of sequences is the following key proposition, whose proof lies beyond the scope of this book: _A bounded monotonic sequence is necessarily convergent._
**e.** Algebraic operations on convergent sequences obey the same rules as algebraic operations on limits of functions (why?). For example, if _x_ _n_ → _A_ and _y n_ → _B_ as _n_ → ∞, then _x_ _n_ \+ _y n_ → _A_ \+ _B_ and _x nyn_ → _AB_ as _n_ → ∞.
**2.95. Examples**
**a.** The sequence
is convergent, with limit 0. In fact, given any _ε_ > 0, let _n_ 0 be any integer greater than 1/ _ε_. Then | _x_ _n_ − 0| = | _x_ _n_ | = 1/ _n_ < _ε_ for all _n_ ≥ _n_ 0, since 1/ _n_ ≤ 1/ _n_ 0 < _ε_ for such _n_ (use Theorem 1.46 twice). Note that this sequence is bounded and decreasing, so that its convergence follows from the proposition in Sec. 2.94d. However, the proposition does not tell us how to _find_ the limit.
**b.** The sequence
_x n_ = _n_! = _n_ ( _n_ − 1) . . . 2 · 1
is unbounded and hence divergent. In fact, to make | _x_ _n_ | larger than any given positive number _M_ , we need only choose _n_ > _M_ , since then | _x_ _n_ | = _n_! > _n_ > _M_.
**c.** A bounded sequence need not be convergent. For example, the sequence
which looks like
−1, 1, −1, 1, −1, 1,...,
is obviously bounded, since | _x n_| = 1 for all _n_. On the other hand, the sequence is divergent. To see this, take any proposed limit _A_ , and make _ε_ so small that the interval _I_ = ( _A_ − _ε_ , _A_ \+ _ε_ ) fails to contain at least one of the points 1 and −1. Clearly this can always be done, even if _A_ = 1 or _A_ = −1. Then all the terms of (4) with even _n_ lie outside _I_ if _I_ fails to contain the point _x_ = 1, while all the terms of (4) with odd _n_ lie outside _I_ if _I_ fails to contain the point _x_ = −1. Thus, in any event, the sequence (4) cannot be convergent.
**d.** The sequence
is convergent for −1 < _a_ < 1. To see this, suppose first that 0 < _a_ < 1. Then the sequence is decreasing, since
_x_ _n_ \+ 1 = _a_ _n_ \+ 1 = _ax_ _n_ < _x_ _n_
for all _n_. Moreover, the sequence is bounded, since
0 < _x n_ < _x_ 1 = _a_
for all _n_. It follows from the proposition in Sec. 2.94d that the sequence is convergent. Let the limit of the sequence as _n_ → ∞ be _A_. To find _A_ , we note that
But since _a_ ≠ 1, this is possible only if _A_ = 0. Therefore
Next, if −1 < _a_ < 0, we have 0 < | _a_ | < 1, and therefore | _a_ | _n_ approaches 0 as _n_ → ∞, by formula (6) with | _a_ | instead of _a_. But then, since | _a n_| = | _a_ | _n_ , it follows that _a n_ also approaches 0 as _n_ → ∞ (explain further), that is
Combining this with (6) and the obvious fact that the sequence _a n_ converges to 0 if _a_ = 0, since all its terms then equal 0, we get
Finally, we note that the sequence _a n_ converges to 1 if _a_ = 1, since all its terms then equal 1.
**e.** The sequence (5) is divergent for _a_ = −1 and | _a_ | > 1. For _a_ = −1 the sequence reduces to the sequence (4), which has already been shown to be divergent. If | _a_ | > 1, we first write
| _a_ | = 1 + (| _a_ | − 1).
Then
| _a_ | _n_ = [1 + (| _a_ | − 1)] _n_ ≥ 1 + _n_ (| _a_ | − 1) > _n_ (| _a_ | − 1),
with the help of the inequality
(1 + _x_ ) _n_ ≥ 1 + _nx_ ( _x_ > −1),
proved in Problem 16. Since | _a_ | − 1 is positive, the product _n_ (| _a_ | − 1) is greater than any given number _M_ > 0 for all _n_ greater than
It follows that the sequence (5) is unbounded and hence divergent if | _a_ | > 1, as claimed.
**2.96. The sum of an infinite series**
**a. Summation notation.** First we introduce a concise way of writing sums, involving the symbol (capital Greek sigma). Let _p_ and _q_ be nonnegative integers such that _p_ ≤ _q_ , and let _f_ ( _n_ ) be a function defined for all integers _n_ from _p_ to _q_. Then
is shorthand for the sum
_f_ ( _p_ ) + _f_ ( _p_ \+ 1) + ... + _f_ ( _q_ ).
The symbol _n_ is a "dummy index" (of summation), in the sense that it can be replaced by any other symbol without changing the meaning of (8). For example,
If _p_ = _q_ , the sum (8) reduces to the single term _f_ ( _p_ ).
**b.** Given a sequence { _x n_}, the expression
involving the terms of the sequence, is called an _infinite series_ , or simply a _series_ , with _terms x_ 1, _x_ 2, ..., _x n_, ... The symbol ∞ on top of the summation sign means that the sum on the right "goes on forever." This is also expressed by the second set of dots . . . on the right. Suppose the sequence
of _partial sums_ of the terms of the series (9) is convergent, with limit _s_. Then we say that the series (9) is _convergent_ , with _sum s_. By the same token, if the sequence of partial sums (10) is divergent, we call the series (9) _divergent_ and assign it no sum at all.
**c. Example.** Investigate the convergence of the _geometric series_
SOLUTION. In other words, we are asked to find the values of _a_ for which the series (11) is convergent and the values for which it is divergent. Here the sum of the first _n_ terms of the series is just
Multiplying _s n_ by _a_ , we get
_as n_ = _a_ \+ _a_ 2 \+ . . . + _a n_.
Subtracting _as n_ from (12), we find that all but two terms cancel out, leaving
_s n_ − _as n_ = (1 − _a_ ) _s n_ = 1 − _a n_.
Therefore
provided that _a_ ≠ 1. It follows from Examples 2.95d and 2.95e that _s n_ is convergent with limit
if −1 < _a_ < 1 and divergent if _a_ = −1 or if | _a_ | > 1. If _a_ = 1, formula (13) breaks down (why?), but in this case the series (11) reduces to simply
so that _s n_ = _n_. Being unbounded, the sequence _s n_ is divergent, and hence so is the series (14).
Thus, to summarize, the geometric series (11) is convergent, with sum
if −1 < _a_ < 1 and divergent otherwise. For example,
While
##### PROBLEMS
**.** Evaluate
**.** Evaluate
**.** If
find all the one-sided limits of _f_ ( _x_ ) at 0, 1 and 2.
**4.** Verify that
**.** Evaluate
**.** If
verify that _f_ ( _x_ ) → 1 as _x_ → ∞. Find all positive _x_ such that | _f_ ( _x_ ) − 1| < 0.001.
**7.** If
verify that _f_ ( _x_ ) → as _x_ → ∞. Find all negative _x_ such that | _f_ ( _x_ ) − | < 0.001.
**.** If
verify that _f_ ( _x_ ) → ∞ as _x_ → 3+ and _f_ ( _x_ ) → −∞ as x → 3−. Find all _x_ such that _f_ ( _x_ ) > 1000 and all _x_ such that _f_ ( _x_ ) < −1000.
**.** Find all asymptotes of the function
**10.** Given any positive integer _n_ , find a function _f_ ( _x_ ) with _n_ vertical asymptotes.
**.** Give an example of a function approaching a horizontal asymptote from one direction only.
**12.** Give an example of a function approaching a vertical asymptote from one side only.
**.** Find the limit of the sequence
**.** Which of the following sequences are convergent?
**15.** Starting from what value of _n_ are the terms of the sequence within 10−6 of its limit?
**.** Use mathematical induction to verify that
(1 + _x_ ) _n_ ≥ 1 + _nx_
for all _n_ = 1, 2, ... if _x_ > −1.
**.** Evaluate
**.** Write the following expressions out in full, and then calculate their numerical values:
**.** Find the sum of the series
**.** Verify that the function
has neither horizontal nor vertical asymptotes. Convince yourself that _f_ ( _x_ ) has the line _y_ = _x_ /2 as an asymptote.
***21.** Give an example of a convergent sequence of rational numbers with an irrational limit.
***22.** Verify that the _harmonic series_
is divergent.
***23.** Show that if the series (9) is convergent, then _x n_ → 0 as _n_ → ∞. Is the converse true?
_Chapter 3_
DIFFERENTIATION AS A TOOL
3.1 VELOCITY AND ACCELERATION
**3.11.** By a _particle_ we mean an object whose actual size can be ignored in a given problem, and which can therefore be idealized as a point. There are problems in which the earth itself can be regarded as a particle, just as there are problems in which a pinhead is a complicated structure made up of vast numbers of tiny particles.
Consider the motion of a particle along a straight line. Let _s_ be the particle's distance at time _t_ from some fixed reference point, where _s_ is positive if measured in a given direction along the line and negative if measured in the opposite direction. Then the particle's motion is described by some _distance function_
Here, for simplicity, we denote the dependent variable and the function by the same letter, a common practice. In the language of physics, (1) is the _equation of motion_ of the particle.
We now ask a key question: How fast is the particle going? There are two answers, depending on whether we ask about a given _interval_ of time or about a given _instant_ of time. In the first case, we get the _average velocity_ , which is a difference quotient. In the second case, we get the _instantaneous velocity_ , which is a derivative.
**3.12. Average velocity**
**a.** By the _average velocity_ of the particle with equation of motion (1), _over the interval from t to t_ \+ Δ _t_ , we mean the function of _two_ variables
It is meaningless to ask for the average velocity at a given instant _t_ without specifying the _averaging time_.
**b.** To be useful, an average velocity should not be too "crude," that is, Δ _t_ should not be too large. Consider, for example, a particle whose equation of motion is described (in part) by the table
The particle is actually moving back and forth rather dramatically, but you would never know it calculating the "two-second averages"
which seems to suggest that the particle is moving slowly and steadily in the positive direction with a velocity of 1 foot per second! Choosing a shorter averaging time Δ _t_ = 1, we get
_v_ av(0, 1) = −10, _v_ av(1, 1) = 12, _v_ av(2, 1) = −10, _v_ av(3, 1) = 12.
This gives a better picture of the particle's motion. At least, it shows that the direction of the particle's motion changes. But how do we know that it's an accurate picture? After all, everything depends on what the particle is doing between the times of measurement.
**3.13. Instantaneous velocity**
**a.** By the _instantaneous velocity_ (or "true velocity") of the particle with equation of motion (1), _at the time t_ , we mean the function of _one_ variable
obtained by taking the limit of the average velocity as the averaging time Δ _t_ "goes to zero." This is, of course, just the derivative
of the distance function _s_ ( _t_ ) with respect to the time _t_. Since our averaging time is now "infinitesimal," we can rest assured that no details of the particle's motion have been overlooked in calculating _v_ ( _t_ ).
From now on, when we talk about "velocity" without further qualification, we mean _instantaneous_ velocity. The quantity called "speed" in common parlance is just the absolute value of the velocity.
**b.** Suppose a stone is dropped from a high tower, and let distance be measured vertically downward from the initial position of the stone. Then, as in Sec. 2.11, the stone, regarded as a particle, has the equation of motion
where _s_ is measured in feet and _t_ in seconds, provided that the stone has not yet hit the ground. The stone's velocity at time _t_ is just
Note that _v_ is an increasing function of time (Sec. 2.33), and is in fact directly proportional to _t_.
**3.14. Acceleration**
**a.** Suppose we differentiate the velocity function _v_ ( _t_ ) itself. This gives a new function
called the _acceleration_ of the particle at the time _t_. Since _v_ ( _t_ ) = _s_ ′( _t_ ) is the first derivative (that is, the ordinary derivative) of the distance function _s_ ( _t_ ), the acceleration is just the second derivative of _s_ ( _t_ ):
Both velocity and acceleration are, of course, _rates of change_ (Sec. 2.42c), the first the rate of change of the distance with respect to time, the second the rate of change of the velocity with respect to time. Negative acceleration is often called _deceleration_.
**b.** The acceleration corresponding to the velocity (3), and hence in turn to the distance function (2), is just
Thus the acceleration of the falling stone has the constant value of 32 feet per second per second (more concisely, 32 ft/sec2), the so-called "acceleration due to gravity."
**3.15. Example.** As will be shown in Sec. 5.32c, a stone thrown vertically upward from ground level with an initial velocity of _v_ 0 ft/sec (at the time _t_ = 0) has the equation of motion
where, as usual, _t_ is the time in seconds, and _s_ is now the height of the stone above the ground, in feet. Suppose _v_ 0 = 96 ft/sec. At what time does the stone stop rising and begin to fall? What is the maximum height reached by the stone?
SOLUTION. Substituting _v_ 0 = 96 in (4), we get
Differentiating with respect to _t_ , we then find that
The stone stops rising and begins to fall when its velocity changes from positive to negative values. This change occurs at the precise instant when the velocity equals zero. Setting _v_ = 0 in (6) and solving for _t_ , we find that this happens at the time
Thus the stone rises for 3 seconds, comes to rest instantaneously, and then falls down for 3 more seconds, finally hitting the ground 6 seconds after being thrown upward (note that _s_ (6) = 0).
To find the maximum height achieved by the stone, we make the substitution _t_ = 3 in equation (5). This gives
_s_ = 96 · 3 − 16 · 9 = 144.
Thus the stone rises to a height of 144 feet before beginning to fall back to the ground. To get the stone's acceleration, we differentiate (6) once again, obtaining
Thus the acceleration has the constant value of −32 ft/sec2.
Figure 1.
We graph the distance function (5) in Figure 1A and the velocity function (6) in Figure 1B. There is no reason to make the horizontal and vertical units the same in these graphs, and we have not done so. The curve in Figure 1A is an upside-down version of the curve in Figure 5, p. 49, with a shift and a scale change, and is again called a _parabola_. Do not make the mistake of confusing this curve with the stone's trajectory! In fact, the stone's trajectory is just the vertical line segment 0 ≤ _s_ ≤ 144, traversed once in the upward direction and once in the downward direction. It is clear from the figure that the function _s_ ( _t_ ) is increasing for the first 3 seconds of the stone's motion and decreasing for the next 3 seconds, while _v_ ( _t_ ) is decreasing for the whole 6 seconds. Note that the stone hits the ground at the same _speed_ as its initial speed.
As we will see in Sec. 3.3, the fact that the derivative _v_ ( _t_ ) = _s_ ′( _t_ ) equals zero for the value of _t_ at which the function _s_ ( _t_ ) achieves its maximum is not just a special feature of this problem, but rather reflects a general property of differentiable functions.
PROBLEMS
**.** Suppose a particle moving along a straight line has the equation of motion
_s_ = 10 _t_ \+ 5 _t_ 2,
where _s_ is measured in feet and _t_ in seconds. Find the average velocity of the particle over the interval from 20 to 20 + Δ _t_ for Δ _t_ = 1, 0.1 and 0.01. What is the particle's instantaneous velocity at the time _t_ = 20?
**2.** Suppose a particle moving along a straight line has the equation of motion
(units unspecified). Find the particle's velocity _v_ and acceleration _a_ at the time _t_. When does the direction of motion of the particle change? When does the particle return to its initial position (at _t_ = 0)?
**.** What can be said about the motion of a particle whose equation of motion contains powers of _t_ greater than 2?
**.** A stone is thrown vertically upward with an initial velocity of 32 ft/sec by a man standing at the edge of a roof 48 feet above the ground. Find the time when the stone hits the ground, assuming that it misses the roof on the way down. How fast is the stone going when it hits the ground?
**5.** How high should the roof be in the preceding problem if the stone is to hit the ground 4 seconds later? 5 seconds later?
**.** The equation of motion of a car starting from rest is
where _s_ is measured in feet and _t_ in seconds. Interpret _k_. Find _k_ if the car reaches a speed of 60 mi/hr in 10 seconds flat. How long do you expect equation (7) to be valid?
**.** A car is going _v_ 0 mi/hr when its brakes are suddenly applied. Suppose its subsequent motion is described by the equation
Interpret _k_. Find _k_ if _v_ 0 = 60 mi/hr and the car brakes to a complete stop in 22 seconds.
***8.** Show that the distance travelled by a car after its brakes are suddenly applied is proportional to the square of its speed _v_ 0.
***9.** The graph of _s_ ( _t_ ) in Figure 1A is symmetric in the line _t_ = 3, that is, reflection in this line does not change the graph. What does this mean physically?
***10.** After _t_ seconds a braked flywheel rotates through an angle of
_θ_ = _θ_ ( _t_ ) = _a_ \+ _bt_ − _ct_ 2
degrees, where _a_ , _b_ and _c_ are positive constants. Suitably define and then determine the flywheel's angular velocity and angular acceleration. When does the flywheel stop rotating?
3.2 RELATED RATES AND BUSINESS APPLICATIONS
**3.21.** First we consider a class of problems involving _related rates_. In such problems we are given the rate of change of one quantity (usually with respect to time), and we are asked to find the rate of change of another related quantity.
**a. Example.** A large spherical balloon is losing air at the rate of one tenth of a cubic foot per second (more concisely, 0.1 ft3/sec). How fast is the radius of the balloon decreasing when its diameter is 6 feet?
SOLUTION. Let _R_ be the radius and _V_ the volume of the balloon. Then
by elementary geometry. Since the size of the balloon is changing, both _V_ and _R_ are functions of time. We could express this fact by writing _V_ = _V_ ( _t_ ), _R_ = _R_ ( _t_ ), but it is better to just bear in mind that _V_ and _R_ depend on time. Differentiating (1) with respect to time (identical functions have identical derivatives), we get
with the help of Example 2.83c. We then solve this equation for _dR/dt_ , obtaining
or
since _dV/dt_ = −0.1 ft3/sec, according to the statement of the problem. Note that _dV/dt_ is negative because air is being _lost_. When the balloon's diameter is 6 feet, its radius is 3 feet. Substituting _R_ = 3 into (2), we find that at that moment
or equivalently
(inches per minute). Thus _R_ is decreasing at the rate of about 0.64 in/min, a rather slow leak for a large balloon. Note that _dR/dt_ is itself a function of the radius. In fact, the smaller the balloon, the larger _dR/dt_ , as shown by (2).
**b. Example.** A ladder 20 feet long is leaning against a wall. Suppose the bottom of the ladder is pulled away from the wall at a constant rate of 6 ft/min. How fast is the top of the ladder moving down the wall when
(a) The bottom of the ladder is 12 feet from the wall;
(b) The top of the ladder is 12 feet from the ground?
SOLUTION. Idealizing the ladder as a straight line segment, we introduce rectangular coordinates as shown in Figure 2, where _x_ is the distance between the wall and the bottom of the ladder, and _y_ is the height above ground of the top of the ladder. By the Pythagorean theorem,
Figure 2.
Since the position of the ladder is changing, both _x_ and _y_ are functions of the time _t_ , a fact that could be emphasized by writing _x_ = _x_ ( _t_ ), _y_ = _y_ ( _t_ ). To find _dy/dt_ , we use the technique of implicit differentiation. Thus we differentiate (3) with respect to _t_ , obtaining the equation
which we then solve for _dy/dt_. The result is
or
since _dx/dt_ = 6 ft/min, according to the statement of the problem. Note that _dx/dt_ is positive because the bottom of the ladder is moving _away from_ the wall, and _dy/dt_ is negative because the top of the ladder is moving _down_ the wall. Therefore the top of the ladder moves down the wall at the rate
when the bottom of the ladder is 12 feet from the wall, and at the rate of
when the top of the ladder is 12 feet from the ground.
**3.22. a.** The word "marginal" is encountered repeatedly in business and economics, in expressions like "marginal cost," "marginal revenue," "marginal profit," etc The second word in each expression is always some function, and the word "marginal" merely calls for taking the derivative of this function, with respect to the independent variable. For example, the _total cost_ to a firm of producing a quantity _Q_ of some commodity is some function of _Q_ , called the _cost function_ and denoted by _C_ ( _Q_ ). The derivative of the cost function, namely
is called the _marginal cost_ , denoted by _MC_ ( _Q_ ). Here we follow the convention, standard in economic theory, of denoting certain functions by pairs of capital letters, like _MC_ for "marginal cost," _AR_ for "average revenue," and so on. (Do not think of these pairs as products!) In this notation, (4) takes the form
**b.** In writing _C_ ( _Q_ ) and its derivative _MC_ ( _Q_ ), we tacitly assume that the units of _Q_ are such that _C_ ( _Q_ ) is defined for arbitrary _Q_ ≥ 0, and not just for the integers _Q_ = 0, 1, 2, ... This assumption is certainly appropriate for oil, measured in pints or gallons, or for salt, measured in pounds or tons, but it is absurd for aircraft carriers. For TV sets the assumption makes sense if the output is large and if we are not too literal-minded. Thus if the answer to a production problem is "Make 31.5 TV sets a day," we can either make 63 sets in 2 days or else settle for making 31 or 32 sets a day. The same remark applies to the application of calculus methods to a host of other problems involving objects that come one at a time, like members of an animal population.
**c. Example.** The cost function _C_ ( _Q_ ) is typically the sum of a constant term, representing certain fixed costs, called the _overhead_ , which are independent of the output _Q_ , and a variable term which depends on the actual value of _Q_. Prove that the marginal cost is independent of the overhead. Find the marginal cost _MC_ ( _Q_ ) corresponding to the commonly used model of a _cubic_ cost function
where _a_ , _b_ , _c_ and _d_ are constants. What can be said about the constant _d_?
SOLUTION. If _C_ ( _Q_ ) = _f_ ( _Q_ ) + _k_ , where _k_ is the overhead and hence a constant, then clearly
so that _MC_ ( _Q_ ) is independent of the overhead. For the cost function (6), the constant _d_ is the overhead and therefore must be positive. Differentiating (6), we get the corresponding marginal cost
_MC_ ( _Q_ ) = 3 _aQ_ 2 \+ 2 _bQ_ \+ _c_.
**d. Example.** The function
is called the _average cost_. Express the marginal cost in terms of the average cost. Show that the derivative of the average cost equals zero when the marginal cost equals the average cost, and only then.
SOLUTION. Combining (5) and (7), we get the formula
expressing the marginal cost in terms of the average cost. The derivative of the average cost equals
by the rule for differentiating a quotient, and this equals zero when
and only then. But _C_ ′( _Q_ ) = _MC_ ( _Q_ ), so that (8) can be written as
_MC_ ( _Q_ ) _Q_ − _C_ ( _Q_ ) = 0,
or equivalently
This proves the assertion made in the statement of the example.
PROBLEMS
**.** Air is being pumped into a large spherical balloon at the rate of 10 ft3/min. How fast is the radius of the balloon increasing when its diameter is 4 feet?
**2.** Two ships _A_ and _B_ sail away from a point _P_ along perpendicular routes. Ship _A_ is going 15 mi/hr, while ship _B_ is going 20 mi/hr. Suppose that at a certain time _A_ is 5 miles from _P_ and _B_ is 10 miles from _P_. How fast are the ships moving apart 1 hour later?
**.** The radius of a circle is increasing at a constant rate. Is the same true of its area? Of its circumference?
**4.** A point moves away from the origin in the first quadrant along the curve . Which coordinate, _x_ or _y_ , is increasing faster?
**.** The length of one side of a rectangle increases at 2 in/sec, while the length of the other side decreases at 3 in/sec. At a certain moment the first side is 20 inches long and the second side is 50 inches long. Is the area of the rectangle increasing or decreasing at this moment? How fast?
**6.** A man 6 feet tall walks at a speed of 4 ft/sec toward a street light 18 feet above the ground. How fast is the length of the man's shadow decreasing? Does the answer depend on his distance from the light?
**.** Let _R_ ( _Q_ ) be the _total revenue_ received by a firm from the sale of a quantity _Q_ of some commodity. Then the derivative _R_ ′( _Q_ ) is called the _marginal revenue_ , denoted by _MR_ ( _Q_ ), and the function
is called the _average revenue_. Express the marginal revenue in terms of the average revenue.
**8.** Suppose the curve of average cost is a straight line
_AC_ ( _Q_ ) = _a_ − _mQ_ ( _a_ > 0, _m_ > 0),
with negative slope (average cost typically decreases as output increases). Find the curve of marginal cost.
**.** Suppose the demand for a commodity produced by a monopolistic firm is described by the function _Q_ = _Q_ ( _P_ ), where _Q_ is the quantity demanded at the price _P_. What does the truism "The greater the price, the less the demand" tell us about the function _Q_ ( _P_ )?
**.** Let _Q_ ( _P_ ) be the same as in the preceding problem. Then the firm's total revenue is clearly _R_ ( _Q_ ) = _PQ_ ( _P_ ), so that its _profit_ is just
Π( _Q_ ) = _R_ ( _Q_ ) − _C_ ( _Q_ ) = _PQ_ ( _P_ ) − _C_ ( _Q_ ),
where _C_ ( _Q_ ) is the firm's total cost function. Show that _Q_ = _Q_ ( _P_ ) has an inverse function _P_ = _P_ ( _Q_ ), allowing us to also write the profit as
Π( _Q_ ) = _QP_ ( _Q_ ) − _C_ ( _Q_ ).
***11.** How fast is the surface area of the balloon in Problem 1 increasing when its diameter is 8 feet?
***12.** In the ladder problem of Example 3.21b, find the acceleration of the top of the ladder when the bottom of the ladder is 16 feet from the wall.
3.3 PROPERTIES OF CONTINUOUS FUNCTIONS
So far, one of our chief concerns has been to acquire the technique of differentiation, so that we can find the derivative _f_ ′ of a given function _f_. But what do we do with _f_ ′ once we have found it? As we will see later in this chapter, knowledge of _f_ ′ can tell us a great deal about the behavior of the original function _f_. Knowledge of the second derivative _f_ ″ also turns out to be a great asset in many cases, because of the further light it sheds on the behavior of _f_.
Remarkably enough, there is also much that can be deduced from the mere fact that a function is _continuous_ in an interval, especially in a _closed_ interval, as we now show.
**3.31. The continuous image of a closed interval**
**a.** The properties of a function continuous in a _closed_ interval (Sec. 2.65c) all stem from the following key proposition, whose proof is a bit too hard for a first course in calculus:
THEOREM. _If f is continuous in a closed interval I_ = [ _a_ , _b_ ] _and if f is not a constant function, then the range of f, namely the set of all values taken by f at the points of I, is itself a closed interval_.
This fact is expressed by saying that "a continuous function maps a closed interval into a closed interval," or that "the continuous image of a closed interval is a closed interval."
**b.** We must insist that _f_ be nonconstant, since the function _f_ ( _x_ ) ≡ _C_ maps _I_ into the single point _C_ , hardly a closed interval! A function which is continuous in an _open_ interval can map the interval into any other kind of interval, open, closed or half-open (see Prob. 10). Thus it is crucial to the validity of the theorem that the interval _I_ be _closed_. The set of values taken by _f_ at the points of _I_ will be denoted by _f_ ( _I_ ). Do not think of _f_ ( _I_ ) as the value of _f_ at _I_ , which is meaningless. In fact, _f_ ( _I_ ) is not a number, but rather the set .
**c.** What this means geometrically is shown in Figure 3, where the solid curve is the graph of a function _f_ continuous in a closed interval _I_ = [ _a_ , _b_ ] Suppose we drop perpendiculars from all the points of the curve _y_ = _f_ ( _x_ ) onto the _y_ -axis. Then, according to our theorem, the resulting points completely "fill up" some closed interval [ _m_ , _M_ ] on the _y_ -axis, as shown in the figure. Note that in general several points of [ _a_ , _b_ ] correspond to the same point of [ _m_ , _M_ ]. For example, the points _r_ and _s_ shown in the figure are both "mapped" by _f_ into the same point _l_ [ _m_ , _M_ ].
Figure 3.
**3.32. Maxima and minima**
**a.** Let _f_ be a function defined in an interval _I_ , and suppose there is a point _p_ _I_ such that _f_ ( _x_ ) ≤ _f_ ( _p_ ) for all _x_ _I_. Then _f_ ( _p_ ) is called the _maximum_ of _f_ in _I_ , and we say that _f_ has this maximum at the point _p_. Similarly, suppose there is a point _q_ _I_ such that _f_ ( _x_ ) ≥ _f_ ( _q_ ) for all _x_ _I_. Then _f_ ( _q_ ) is called the _minimum_ of _f_ in _I_ , and we say that _f_ has this minimum at the point _q_. We also say that " _f_ takes its maximum value at _p_ and its minimum value at _q_." Note that _f_ may well take its maximum (or minimum) value, provided there is one, at more than one point of _I_.
The word _extremum_ refers to either a maximum or a minimum, and the phrase _extreme value_ refers to either a maximum value or a minimum value. The words "maximum," "minimum" and "extremum" have Latin plurals, namely "maxima," "minima" and "extrema." Extrema of a different kind will be introduced in Sec. 3.51, and will be known as _local_ extrema, as opposed to the kind of extrema considered here, which are often called _global_ extrema.
**b.** For example, if _I_ is the closed interval −1, 1], the function _x_ 2 has its maximum in _I_ , equal to 1, at both points _x_ = ±1, and its minimum in _I_ , equal to 0, at the point _x_ = 0. In the same interval, the function _x_ 3 has its maximum, equal to 1, at the point _x_ = 1, and its minimum, equal to −1, at the point _x_ = −1. A constant function, say _f_ ( _x_ ) ≡ _k_ , has a maximum and a minimum, both equal to _k_ , at _every_ point of any interval. On the other hand, the function _x_ has neither a maximum nor a minimum in the _open_ interval (0, 1), since there is no largest number less than 1 and no smallest number greater than 0 ([Sec. 1.4, Prob. 12), and the same is true of the functions _x 2_ and _x_ 3 in this interval. All three functions _x_ , _x 2_ and _x_ 3 have a minimum, equal to 0, in the half-open interval [0, 1), at the point _x_ = 0, but again none of them has a maximum in [0, 1). As we now see, this failure to have one or both extrema cannot occur if the function is continuous in a _closed_ interval.
**c.** THEOREM. _If f is continuous in a closed interval I_ = [ _a_ , _b_ ], _then I contains points p and q such that_
_for all _x_ I. In other words, f has both a maximum and a minimum in I, at the points p and q, respectively._
_Proof_. We can immediately exclude the case of a constant function, for which the theorem is trivially true. Thus, suppose _f_ is not a constant function. Then, by Theorem 3.31a, the range of _f_ , namely the set _f_ ( _I_ ), is a closed interval. Let this interval be [ _m_ , _M_ ]. Then, for all _x_ _I_ , we have _f_ ( _x_ ) [ _m_ , _M_ ], or equivalently
by the very meaning of _f_ ( _I_ ). Let _p_ be any point of _I_ such that _f_ ( _p_ ) = _M_ and _q_ any point of _I_ such that _f_ ( _q_ ) = _m_. Again, such points _p_ and _q_ exist by the very meaning of _f_ ( _I_ ). We can then write (2) in the form (1).
**d.** Naturally, _M_ is the maximum of _f_ in _I_ , and _m_ is the minimum of _f_ in _I_. Interpreted geometrically, the theorem means that the graph of a function _f_ continuous in a closed interval _I_ must have both a "highest point" _P_ = ( _p_ , _M_ ) and a "lowest point" _Q_ = ( _q_ , _m_ ), as illustrated by Figure 3. It is important to note that _f_ may take one or both of its extreme values at end points of _I_ (give examples), although this is not the case for the function shown in Figure 3.
**3.33. The intermediate value theorem**
**a.** Let _k_ be any number between _f_ ( _a_ ) and _f_ ( _b_ ). Then, since both _f_ ( _a_ ) and _f_ ( _b_ ) belong to the interval _f_ ( _I_ ) = [ _m_ , _M_ ], so does _k_. Therefore _k_ is a value of _f_ taken at some point _c_ in the interval [ _a_ , _b_ ]. But _c_ cannot be one of the end points _a_ and _b_ , since _k_ lies between _f_ ( _a_ ) and _f_ ( _b_ ), and therefore cannot coincide with either _f_ ( _a_ ) and _f_ ( _b_ ). The situation is illustrated in Figure 3.
**b.** This key property of continuous functions deserves a name of its own:
THEOREM **(Intermediate value theorem).** _If f is continuous in an interval I, which need not be closed, and if f takes different values f_ ( _α_ ) _and f_ ( _β_ ) _at two points α and β of I, then f takes every value between f_ ( _α_ ) _and f_ ( _β_ ) _at some point between α and β._
_Proof_. In view of the preceding remarks, we need only show that _f_ is continuous in [ _α_ , _β_ ] if _α_ < _β_ , or in [ _β_ , _α_ ] if _β_ < _α_. But, by hypothesis, _f_ is continuous in an interval _I_ containing _α_ and _β_. Therefore _f_ is certainly continuous in the closed interval with end points _α_ and _ß_ , since this is the smallest interval containing both _α_ and _β_.
PROBLEMS
**.** Find the extrema, if any, of the function _f_ ( _x_ ) = 1/ _x_ in the interval
(a) (0, 1); (b) (0, 1]; (c) [1, 2]; (d) (0, ∞); (e) (−∞, −1].
**.** Find the extrema, if any, of the function _f_ ( _x_ ) = _x_ ], where [ _x_ ] is the integral part of _x_ ([Sec. 1.4, Prob. 10), in the interval
(a) (0, 1); (b) (0, 1]; (c) (−1, 0]; (d) (0, ∞); (e) (−∞, ∞).
**3.** Verify that if a function _f_ is increasing in a closed interval _I_ = [ _a_ , _b_ ], then _f_ has global extrema in _I_ , even if _f_ is discontinuous. Where does _f_ take its extreme values? How about the case of decreasing _f_?
**.** "If _f_ is continuous in an interval _I_ and if _f_ takes values _f_ ( _α_ ) and _f_ ( _β_ ) with opposite signs at two points _α_ and _β_ of _I_ , then _f_ equals zero at some point between _α_ and _β_ " True or false? Why?
**.** Is the function
continuous? Does it map the closed interval −1 ≤ _x_ ≤ 1 into a closed interval?
**6.** Does the function (3) map the closed interval into a closed interval?
**.** Investigate the global extrema of the function (3) in the interval −1 ≤ _x_ ≤ 1.
***8.** Give an example of a continuous function mapping a finite interval into an infinite interval.
***9.** Give an example of a continuous function mapping an infinite interval into a finite interval.
***10.** Give an example of a continuous function mapping an open interval into
(a) An open interval; (b) A half-open interval; (c) A closed interval.
3.4 PROPERTIES OF DIFFERENTIABLE FUNCTIONS
**3.41. a.** Having investigated the properties of continuous functions, we now return to the study of differentiable functions. We begin by establishing the following interesting fact: _If f is differentiable at a point p, with derivative f′(p), and if f′(p) is_ **positive,** _then _f_ is_ **increasing** _in some neighborhood of p_. To show this, we observe that just as in Sec. 2.55a
by the definition of the derivative _f_ ′( _p_ ), or equivalently
in terms of the difference quotient
In " _ε_ , _δ_ language" this means that, given any _ε_ > 0, there is a _δ_ > 0 such that
| _Q_ (Δ _x_ ) − _f_ ′( _p_ )| < _ε_
whenever 0 < |Δ _x_ | < _δ_. Let _ε_ = _f_ ′( _p_ ) > 0. Then there is a _δ_ > 0 such that
| _Q_ (Δ _x_ ) − _f_ ′( _p_ )| < _f_ ′( _p_ ),
or equivalently
− _f_ ′( _p_ ) < _Q_ (Δ _x_ ) − _f_ ′( _p_ ) < _f_ ′( _p_ ),
whenever 0 < |Δ _x_ | < _δ_. But then
whenever 0 < |Δ _x_ | < _δ_. Therefore the difference quotient _Q_ (Δ _x_ ) = Δ _f_ ( _p_ )/Δ _x_ is _positive_ whenever 0 < Δ _x_ < _δ_ or − _δ_ < Δ _x_ < 0, which means that _the increments_ Δ _f_ ( _p_ ) _and_ Δ _x have the same sign_ under these conditions. In other words,
if 0 < Δ _x_ < _δ_ , while
if − _δ_ < Δ _x_ < 0. Changing the sign of Δ _x_ in (3), we find that
if 0 < Δ _x_ < _δ_. Combining (2) and (4), we finally get
_f_ ( _p_ − Δ _x_ ) < _f_ ( _p_ ) < _f_ ( _p_ \+ Δ _x_ )
if 0 < Δ _x_ < _δ_ , which shows at once that _f_ is _increasing_ in some neighborhood of the point _p_ , namely in the neighborhood ( _p_ − _δ_ , _p_ \+ _δ_ ) or in any smaller neighborhood.
**b.** In virtually the same way, we can show that _if f is differentiable at a point p, with derivative f_ ′( _p_ ), _and if f_ ′( _p_ ) _is_ **negative,** _then f is_ **decreasing** _in some neighborhood of p_. This time we choose _ε_ = − _f_ ′( _p_ ) > 0, obtaining first
_f_ ′( _p_ ) < _Q_ (Δ _x_ ) − _f_ ′( _p_ ) < − _f_ ′( _p_ )
and then
2 _f_ ′( _p_ ) < _Q_ (Δ _x_ ) < 0,
instead of (1), whenever 0 < |Δ _x_ | < _δ_. Therefore _Q_ (Δ _x_ ) is _negative_ whenever 0 < Δ _x_ < _δ_ or − _δ_ < Δ _x_ < 0, which means that _the increments_ Δ _f_ ( _p_ ) _and_ Δ _x have opposite signs_ under these conditions. In other words,
if 0 < Δ _x_ < _δ_ , while
if − _δ_ < Δ _x_ < 0. Changing the sign of Δ _x_ in (6), we find that
if 0 < Δ _x_ < _δ_. Combining (5) and (7), we finally get
_f_ ( _p_ − Δ _x_ ) > _f_ ( _p_ ) > _f_ ( _p_ \+ Δ _x_ )
if 0 < Δ _x_ < _δ_ , which shows at once that _f_ is _decreasing_ in some neighborhood of the point _p_ , namely in the neighborhood ( _p_ − _δ_ , _p_ \+ _δ_ ) or in any smaller neighborhood.
**c.** By an _interior point_ of an interval _I_ (open, closed or half-open), we mean any point of _I_ other than its end points. A function _f_ is said to _vanish_ at a point _c_ if _f_ ( _c_ ) = 0. In other words, "to vanish" means the same thing as "to equal zero." If a function _f_ vanishes at every point of an interval _I_ , we say that _f vanishes identically_ in _I_. This extra vocabulary will come in handy time and again.
**3.42. Rolle's theorem**
**a.** Our next result is a stepping stone on the way to another, more important result, called the "mean value theorem," but it is of considerable interest in its own right.
THEOREM **(Rolle's theorem).** _Let f be continuous in the closed interval_ [ _a_ , _b_ ] _and differentiable, with derivative f_ ′, _in the open interval_ ( _a_ , _b_ ). _Suppose that f_ ( _a_ ) = _f_ ( _b_ ) = _k_. _Then there is a point c_ ( _a_ , _b_ ) _such that f_ ′( _c_ ) = 0.
_Proof_. Do not be put off by the formal language. A more informal way of stating the theorem is the following: Let _f_ be continuous in a closed interval _I_ = [ _a_ , _b_ ] and differentiable at every interior point of _I_. Suppose _f_ takes the same value at both end points of _I_. Then the derivative _f_ ′ vanishes at some interior point of _I_.
To prove the theorem, we first observe that _f_ has both a maximum _M_ and a minimum _m_ in _I_ = [ _a_ , _b_ ], by Theorem 3.32c, where _m_ ≤ _k_ ≤ _M_ , since _f_ equals _k_ at the points _a_ and _b_. If _m_ = _k_ = _M_ , then _f_ reduces to the constant function _f_ ( _x_ ) ≡ _k_ , whose derivative vanishes at _every_ interior point of _I_ , and the theorem is proved. Otherwise, we have either _m_ < _k_ or _M_ > _k_. Suppose _M_ > _k_ , and let _c_ be a point of _I_ such that _f_ ( _c_ ) = _M_. Then _c_ ( _a_ , _b_ ), that is, _c_ is an interior point of _I_ , so that the derivative _f_ ′( _c_ ) exists. If _f_ ′( _c_ ) ≠ 0, then _f_ ′( _c_ ) is either positive or negative. In the first case, _f_ is increasing in some neighborhood of _c_ , by Sec. 3.41a, while in the second case, _f_ is decreasing in some neighborhood of _c_ , by Sec. 3.41b. In either case, the neighborhood contains values of _f_ larger than _M_ , so that _M_ cannot be the maximum of _f_. It follows that _f_ ′( _c_ ) = 0. The proof for _m_ < _k_ is almost identical (give the details).
**b.** Rolle's theorem has a simple geometrical interpretation. It merely says that if the end points of the curve
have the same ordinate, so that _f_ ( _a_ ) = _f_ ( _b_ ), then the slope of the tangent to the curve vanishes and hence is _horizontal_ at some "intermediate point," that is, at some point of the curve other than its end points. This situation is illustrated by Figure 4, which shows that the curve can actually have horizontal tangents at more than one intermediate point, in particular, at points other than those with the maximum and minimum ordinates _M_ and _m_.
**3.43. The mean value theorem**
**a.** If _f_ ( _a_ ) ≠ _f_ ( _b_ ), we can no longer assert that the curve (8) has a horizontal tangent at some intermediate point. However, we can now assert that at some intermediate point the curve has a tangent with the same slope as the chord joining its end points _A_ = ( _a_ , _f_ ( _a_ )) and _B_ = ( _b_ , _f_ ( _b_ )), that is, a tangent with slope
as illustrated by Figure 5.
Figure 4.
Figure 5.
THEOREM **(Mean value theorem).** _Let f be continuous in the closed interval_ [ _a_ , _b_ ] _and differentiable, with derivative f_ ′, _in the open interval_ ( _a_ , _b_ ). _Then there is a point c_ ( _a_ , _b_ ) _such that_
_or equivalently_
_Proof_. We introduce a new function
_g_ ( _x_ ) = _f_ ( _x_ ) − _kx_ ,
choosing the constant _k_ in such a way that _g_ ( _x_ ) has the same value at both end points _a_ and _b_. Then _k_ must satisfy the equation
_g_ ( _a_ ) = _f_ ( _a_ ) − _ka_ = _f_ ( _b_ ) − _kb_ = _g_ ( _b_ ),
with solution
With this choice of _k_ , _g_ ( _x_ ) satisfies all the conditions of Rolle's theorem. But then there is a point _c_ ( _a_ , _b_ ) such that
_g_ ′( _c_ ) = _f_ ′( _c_ ) − _k_ = 0,
that is, such that
**b.** COROLLARY **(Mean value theorem in increment form)**. _If f is differentiable in an interval I containing the points x and x_ \+ Δ _x_ , _then the increment of f at x can be written in the form_
_where_ 0 < _α_ < 1.
_Proof_. Here it is not necessary to state explicitly that _f_ is continuous in _I_ , since this follows automatically from the assumption that _f_ is differentiable in _I_ (Sec. 2.66). Choosing _a_ = _x_ , _b_ = _x_ \+ Δ _x_ in (10), we obtain
where _c_ lies between _x_ and _x_ \+ Δ _x_ , regardless of the sign of Δ _x_. Therefore the number
is always positive and lies in the interval (0, 1), or equivalently
Comparing (12) and (13), we get (11).?
PROBLEMS
**.** Check the validity of Rolle's theorem for the function
_f_ ( _x_ ) = ( _x_ − 1)( _x_ − 2)( _x_ − 3).
In other words, verify that the derivative _f_ ′ vanishes at a point in the interval (1, 2) and at a point in the interval (2, 3).
**2.** The function
_f_ ( _x_ ) = | _x_ | (− _a_ ≤ _x_ ≤ _a_ )
takes the same value | _a_ | at both points _x_ = ± _a_ , but the derivative _f_ ′ does not vanish at any point _c_ (− _a_ , _a_ ). Why doesn't this contradict Rolle's theorem?
**.** Show that the mean value theorem (10) remains valid even if _a_ > _b_.
**4.** At what points of the curve _y_ = _x_ 3 is the tangent parallel to the chord joining the points _A_ = (−3, −9) and _B_ = (3, 9)?
**.** According to formula (10),
_f_ (2) − _f_ (1) = _f_ ′( _c_ ) (1 < _c_ < 2).
Find _c_ if _f_ ( _x_ ) = 1/ _x_.
**6.** According to formula (11),
_f_ (1 + Δ _x_ ) − _f_ (1) = _f_ ′(1 + _α_ Δ _x_ ) Δ _x_ (0 < _α_ < 1).
Find _α_ if _f_ ( _x_ ) = _x_ 3, Δ _x_ = −1.
**.** Justify the following "kinematic interpretation" of the mean value theorem: If a train traverses the distance between two stations at an average velocity _v_ av, then there is a moment when the train's instantaneous velocity equals _v_ av.
***8.** Let
Find two points _c_ satisfying formula (10) for _a_ = 0, _b_ = 2.
***9.** Show that the square roots of any two consecutive integers exceeding 24 differ by less than 0.1.
3.5 APPLICATIONS OF THE MEAN VALUE THEOREM
**3.51.** We already know that the derivative of a constant function vanishes everywhere. More concisely, if _f_ ( _x_ ) ≡ constant, then _f_ ′( _x_ ) ≡ 0. But, conversely, does _f_ ′( _x_ ) ≡ 0 imply _f_ ( _x_ ) ≡ constant? Yes, if the domain of _f_ is an interval, as we now show.
THEOREM. _If f is differentiable in an interval I, and if the derivative f_ ′ _vanishes identically in I, then f_ ( _x_ ) ≡ _constant in I, that is, f has the same value at every point of I._
_Proof_. Let _x_ 0 be a fixed point of _I_ , and let _x_ be any other point of _I_. Then, by the mean value theorem,
_f_ ( _x_ ) − _f_ ( _x_ 0) = _f_ ′( _c_ )( _x_ − _x_ 0),
where _c_ lies between _x_ 0 and _x_. But _f_ ′( _c_ ) = 0, since _c_ belongs to _I_ , and therefore _f_ ( _x_ ) − _f_ ( _x_ 0) = 0 or _f_ ( _x_ ) = _f_ ( _x_ 0). Since _x_ is an arbitrary point of _I_ , it follows that _f_ ( _x_ ) = _f_ ( _x_ 0) for every _x_ _I_.
**3.52. Antiderivatives**
**a.** Given a function _f_ ( _x_ ) defined in an interval _I_ , suppose _f_ ( _x_ ) is another function defined in _I_ such that
for every _x_ _I_. Then _F_ ( _x_ ) is said to be an _antiderivative_ of _f_ ( _x_ ), in the interval _I_. For example, is an antiderivative of _x_ 2 in (−∞, ∞), since
for every _x_ (−∞, ∞). If _F_ ( _x_ ) is an antiderivative of _f_ ( _x_ ) in _I_ , then so is
where _C_ is an arbitrary constant, since
The next proposition shows that there are no other antiderivatives of _f_ ( _x_ ).
**b.** THEOREM. _Let F_ ( _x_ ) _be any antiderivative of f_ ( _x_ ) _in an interval I. Then every other antiderivative of f_ ( _x_ ) _in I is of the form_ (1).
_Proof_. Let _G_ ( _x_ ) be any other antiderivative of _f_ ( _x_ ) in _I_ , and let _H_ ( _x_ ) = _G_ ( _x_ ) − _f_ ( _x_ ). Then
_H_ ′( _x_ ) = _G_ ′( _x_ ) − _F_ ( _x_ ) = _f_ ( _x_ ) − _f_ ( _x_ ) = 0
for every _x_ _I_ , that is, the derivative _H_ ′ vanishes identically in _I_. It follows from the preceding theorem that _H_ has the same value, call it _C_ , at every point of _I_. In other words,
_H_ ( _x_ ) = _G_ ( _x_ ) − _F_ ( _x_ ) ≡ _C_ ,
which is equivalent to (1).
**3.53. The indefinite integral**
**a.** Let _F_ ( _x_ ) be an antiderivative of _f_ ( _x_ ) in _I_ , so that _F_ ′( _x_ ) = _f_ ( _x_ ) for every _x_ _I_. Then, as just shown, the "general antiderivative" of _f_ ( _x_ ) in _I_ is of the form
_F_ ( _x_ ) + _C_ ,
where _C_ is an arbitrary constant. This expression is also called the _indefinite integral_ of _f_ ( _x_ ), and is denoted by
Thus
_by definition_ , so that the indefinite integral is defined only to within an arbitrary "additive constant." The symbol ∫ is called the _integral sign_. The operation leading from the function _f_ ( _x_ ), called the _integrand_ , to the expression (2) is called ( _indefinite_ ) _integration_ , with respect to _x_ , the argument _x_ is called the _variable of integration_ , and the constant _C_ is called the _constant of integration_. Note that the expression behind the integral sign in (2) is the product of the integrand _f_ ( _x_ ) and the differential _dx_ of the variable of integration. Recalling Sec. 2.55a, we recognize this product as the differential
_dF_ ( _x_ ) = _F_ ′( _x_ ) _dx_ = _f_ ( _x_ ) _dx_
of the antiderivative _F_ ( _x_ ), so that (3) can also be written as
∫ _dF_ ( _x_ ) = _F_ ( _x_ ) + _C_.
In writing (3), it is tacitly assumed that the formula is an identity for all _x_ in some underlying interval _I_ in which _f_ ( _x_ ) and _F_ ( _x_ ) are both defined; however, _I_ is usually left unspecified. The convention is to give _f_ and _F_ the same argument, in keeping with the formula _F_ ′( _x_ ) = _f_ ( _x_ ), but we could just as well write
∫ _f_ ( _t_ ) _dt_ = _F_ ( _t_ ) + _C_ ,
say, replacing _x_ by some other symbol (here _t_ ) in both sides of (3). Differentiating (3), we get
so that
**b.** Since a function _f_ ( _x_ ) is obviously an antiderivative of its own derivative _f_ ′( _x_ ), we have
∫ _f_ ′( _x_ ) _dx_ = _f_ ( _x_ ) + _C_.
This formula can be used to derive an integration formula from any differentiation formula. For example, the formula
(Sec. 2.74e), valid for arbitrary real _r_ ≠ −1, leads at once to the formula
if _r_ ≠ −1. Choosing in (5), we get the formula
valid in the interval (0, ∞), but in no larger interval, while the choice gives the formula
valid in the whole interval (−∞, ∞).
**c.** THEOREM. _Suppose f_ ( _x_ ) _and g_ ( _x_ ) _have indefinite integrals in the same interval I. Then_
_where a and b are arbitrary constants._
_Proof_. It follows from (4), applied to the function _af_ ( _x_ ) + _bg_ ( _x_ ), that
On the other hand, by the usual rules of differentiation,
where we use (4) twice more. Thus the two sides of (6) have the same derivative in _I_ , and hence can differ only by an arbitrary constant, by the same argument as in the proof of Theorem 3.52b. But then (6) holds, since the indefinite integral on the left is defined only to within an arbitrary "additive constant."
By virtually the same argument, you can easily convince yourself of the validity of the more general formula
where _f_ 1( _x_ ), _f_ 2( _x_ ), . . ., _f_ _n_ ( _x_ ) are functions which have indefinite integrals in the same interval, and _c_ 1, _c_ 2, . . ., _c_ _n_ are arbitrary constants.
**d. Example.** Evaluate
SOLUTION. By (7), we have
with the help of (5). Note that the constants of integration contributed by each of the three integrals separately can be combined into a single constant of integration _C_.
Figure 6.
**3.54.** In Sec. 3.41c we showed that if _f_ is differentiable at a point _x_ 0, with derivative _f_ ′( _x_ 0), and if _f_ ′( _x_ 0) is positive, then _f_ is increasing in some neighborhood of _x_ 0. We now use the mean value theorem to establish the following closely related result: _If f is differentiable in an interval I, with derivative f_ ′, _and if f_ ′ _is_ **positive** _at every interior point of I, then f is_ **increasing** _in I_. To see this, let _x_ 1 and _x_ 2 be any two points of _I_ such that _x_ 1 < _x_ 2. Then, by the mean value theorem,
for some point _c_ between _x_ 1 and _x_ 2. Therefore _c_ is an interior point of _I_ , so that _f_ ′( _c_ ) > 0. The right side of (8) is positive, being the product of two positive numbers, and hence _f_ ( _x_ 2) − _f_ ( _x_ 1) is also positive. In other words, _x_ 1 < _x_ 2 implies _f_ ( _x_ 1) < _f_ ( _x_ 2), which means that _f_ is increasing in _I_ , as claimed.
Virtually the same argument shows that _if f is differentiable in an interval I, with derivative f_ ′, _and if f_ ′ _is_ **negative** _at every interior point of I, then f is_ **decreasing** _in I_. Give the details.
For example, the function _f_ ( _x_ ) = ( _x_ − 1)3 shown in Figure 6A is increasing in both intervals (−∞, 1] and [1, ∞), since _f_ ′( _x_ ) = 3( _x_ − 1)2 > 0 if _x_ < 1 or _x_ > 1. Therefore _f_ ( _x_ ) is increasing in the whole interval (−∞, ∞). Similarly, the function _g_ ( _x_ ) = 1 − _x_ 4 shown in Figure 6B is increasing in (−∞, 0], since _g_ ′( _x_ ) = − 4 _x_ 3 > 0 if _x_ < 0, and decreasing in [0, ∞), since _g_ ′( _x_ ) = −4 _x_ 3 < 0 if _x_ > 0.
PROBLEMS
**.** The function
is not a constant, but the derivative _f_ ′ vanishes at every point of the domain of _f_. Why doesn't this contradict Theorem 3.51?
**.** Find all antiderivatives of the function
**.** Verify that
∫ _dx_ = ∫ 1 · _dx_ = _x_ \+ _C_.
**.** Show that if
∫ _f_ ( _x_ ) _dx_ = _F_ ( _x_ ) + _C_ ,
then
for arbitrary constants _a_ ≠ 0 and _b_.
**.** Evaluate
**.** "The indefinite integral of a polynomial of degree _n_ is a polynomial of degree _n_ \+ 1." True or false?
**7.** Is the derivative of an increasing function necessarily increasing?
**.** In which intervals are the following functions increasing? Decreasing?
(a) 2 + _x_ − _x_ 2; (b) 3 _x_ − _x_ 3;
***9.** What can be said about the function _f_ if _f_ ( _n_ )( _x_ ) ≡ 0?
3.6 LOCAL EXTREMA
**3.61. a.** Figure 7 shows the graph of a function _f_ continuous in a closed interval _a_ , _b_ ]. Just as in Figure 3, [p. 110, _f_ takes its maximum in [ _a_ , _b_ ], equal to _M_ , at the "highest point" of the graph, namely _P_ = ( _p_ , _M_ ), and its minimum in [ _a_ , _b_ ], equal to _m_ , at the "lowest point" of the graph, namely _A_ = ( _a_ , _m_ ), which this time happens to be an end point of the graph. But now there also seems to be something special about the behavior of the graph at certain other points, namely _Q_ , _R_ and _S_. In fact, _Q_ is "higher" than all "nearby points" of the graph, although not as high as _P_ , while each of the points _R_ and _S_ is "lower" than all "nearby points" of the graph, although not as low as _A_.
**b.** We now make these qualitative notions precise. Let _f_ be a function defined in an interval _I_ , and suppose there is a point _p_ _I_ such that _f_ ( _x_ ) ≤ _f_ ( _p_ ) for all _x_ "sufficiently near" _p_ , that is, for all _x_ in some neighborhood of _p_. Then _f_ is said to have a _local maximum_ , equal to _f_ ( _p_ ), at the point _p_. Similarly, suppose there is a point _q_ _I_ such that _f_ ( _x_ ) ≥ _f_ ( _q_ ) for all _x_ in some neighborhood of _q_. Then _f_ is said to have a _local minimum_ , equal to _f_ ( _q_ ), at the point _q_. The term _local extremum_ refers to a _local maximum_ or a _local minimum_.
Figure 7.
**c.** Make sure you understand the crucial distinction between _local extrema_ , as just defined, and the kind of extrema defined in Sec. 3.32a. The latter, which will henceforth be called _global extrema_ whenever there is any possibility of confusion, involve comparison of a proposed extremum _f_ ( _p_ ) with the value of _f_ at _every point_ of the interval _I_ in which _f_ is defined, while the former only require comparison of _f_ ( _p_ ) with the values of _f_ at points "sufficiently near" _p_ , or, for that matter, "arbitrarily near" _p_. The adjective "global," which suggests the "overall" behavior of _f_ in the whole interval _I_ , and the adjective "local," which suggests the behavior of _f_ in the "immediate vicinity" of the point _p_ , are well-suited to emphasize this distinction. In other books you will often encounter the terms _absolute extremum_ and _relative extremum_ , as synonyms for global extremum and local extremum.
**3.62. a.** Unlike the case of a global maximum or minimum, a function can have several distinct (that is, different) local maxima or minima. On the other hand, as noted in Sec. 3.32a, a function can take its global extremum at more than one point. For example, the function _f_ shown in Figure 7 has distinct local maxima at the points _p_ and _q_ , and distinct local minima at the points _r_ and _s_. The global minimum of _f_ at the point _a_ is not a local minimum, since _f_ is not defined in a neighborhood of _a_ (it is not enough to be defined on only one side of _a_ ). For the same reason, a function defined in an interval _I_ can have local extrema only at _interior_ points of _I_ , that is, at points of _I_ other than the end points of _I_. The function _f_ in Figure 7 has neither a global extremum nor a local extremum at the point _b_ , but _b_ is still special in the sense that it is an end point of the interval [ _a_ , _b_ ]. (In this regard, see Prob. 1.) Note that _f_ has both a global maximum and a local maximum at the point _p_. In fact, if a function _f_ is defined in an interval _I_ , then any global extremum of _f_ at an interior point of _I_ is automatically a local extremum of _f_. (Why is this so?)
**b.** A local maximum _f_ ( _p_ ) is said to be _strict_ if _f_ ( _x_ ) < _f_ ( _p_ ), with < instead of ≤, for all _x_ "sufficiently near" _p_ but not equal to _p_ , that is, for all _x_ in some _deleted_ neighborhood of _p_ (Sec. 1.63a). Similarly, a local minimum _f_ ( _q_ ) is said to be _strict_ if _f_ ( _x_ ) > _f_ ( _q_ ), with > instead of ≥, for all _x_ in some deleted neighborhood of _q_. For example, all four local extrema of the function in Figure 7 are strict. On the other hand, a constant function has both a local maximum and a local minimum at every point, but none of these extrema is strict.
**3.63. a.** In practical problems involving local extrema, we will always be concerned with a function _f_ defined in some interval _I_ , where _f_ is differentiable at every point of _I_ with the possible exception of certain special points. We say that _f_ fails to have a derivative at a point _p_ , or that the derivative _f_ ′( _p_ ) fails to exist, if the limit defining _f_ ′( _p_ ) either does not exist or is infinite.
THEOREM. _If f has a local extremum at a point p, then either f fails to have a derivative at p, or f_ ′( _p_ ) _exists and equals zero._
_Proof_. Either _f_ ′( _p_ ) exists or it does not. Suppose _f_ ′( _p_ ) exists and is nonzero. Then _f_ ′( _p_ ) is either positive or negative. In the first case, _f_ is increasing in some neighborhood of _p_ , by Sec. 3.41a, while in the second case, _f_ is decreasing in some neighborhood of _p_ , by Sec. 3.41b. In either case, this neighborhood, _or any smaller neighborhood_ , contains values of _f_ larger than _f_ ( _p_ ) and values of _f_ smaller than _f_ ( _p_ ), so that _f_ ( _p_ ) cannot be either a local maximum or a local minimum of _f_. It follows that _f_ ′( _p_ ) = 0.
This argument, of course, closely resembles that used in the proof of Rolle's theorem.
**b.** Interpreted geometrically, the theorem says that if a function _f_ has a local extremum at a point _p_ , then either the graph of _f_ has no tangent at the point _P_ = ( _p_ , _f_ ( _p_ )), if _f_ ′( _p_ ) fails to exist, or the graph of _f_ has a _horizontal_ tangent at _P_ , if _f_ ′( _p_ ) = 0. These two possibilities are illustrated in Figure 8A, where each of the functions has a (strict) local maximum at _p_.
**c.** By a _critical point_ of a function _f_ we mean either a point where _f_ has no derivative or a point where the derivative of _f_ vanishes, and by a _stationary point_ of _f_ we mean a point where the derivative of _f_ vanishes. Thus a critical point of _f_ is either a point where _f_ has no derivative or a stationary point of _f_. According to the theorem, if _f_ has a local extremum at _p_ , then _p_ is a critical point of _f_. On the other hand, if _p_ is a critical point of _f_ , there is no necessity for _f_ to have a local extremum at _p_. This is illustrated by Figure 8B, which shows two functions, each with a critical point at _p_ , but neither with a local extremum at _p_.
Thus what we really want are conditions on a function _f_ which _compel f_ to have a local extremum at a given critical point _p_. (We will always assume that _f_ is continuous at _p_.) Such conditions will now be presented in the form of two tests for a local extremum, one called the _first derivative test_ , the other called the _second derivative test_.
Figure 8.
**3.64. The first derivative test**
**a.** Suppose _f_ is differentiable in a deleted neighborhood of a point _p_ , and suppose the derivative _f_ ′ has one sign to the left of _p_ and the opposite sign to the right of _p_. Then _f_ ′ is said to _change sign in going through p_ , from the sign on the left of _p_ to the sign on the right of _p_. Note that we do not require _f_ to be differentiable at the point _p_ itself and in fact _f_ ′( _p_ ) may fail to exist.
THEOREM **(First derivative test for a local extremum).** _Let p be a critical point of f, and suppose f_ ′ _changes sign in going through p. Then f has a strict local extremum at p. The extremum is a maximum if f_ ′ _changes sign from plus to minus, and a minimum if f_ ′ _changes sign from minus to plus_.
_Proof_. Suppose _f_ ′ changes sign from plus to minus in going through _p_ , in a deleted _δ_ -neighborhood of _p_ , that is, in the union of intervals ( _p_ − _δ_ , _p_ ) ( _p_ , _p_ \+ _δ_ ) (Sec. 1.63b). By the mean value theorem in increment form,
_f_ ( _p_ \+ Δ _x_ ) − _f_ ( _p_ ) = _f_ ′( _p_ \+ _α_ Δ _x_ )Δ _x_ (0 < _α_ < 1).
Therefore
if − _δ_ < Δ _x_ < 0, since then _f_ ′( _p_ \+ _α_ Δ _x_ ) > 0, Δ _x_ < 0, and similarly
if 0 < Δ _x_ < _δ_ , since then _f_ ′( _p_ \+ _α_ Δ _x_ ) < 0, Δ _x_ > 0. In other words,
if 0 < |Δ _x_ | < _δ_ , so that _f_ has a strict local maximum at _p_. On the other hand, if _f_ ′ changes sign from minus to plus in going through _p_ , then we get > instead of < in (1), (2) and (3), so that _f_ has a strict local minimum at _p_ (check the details).
**b.** Interpreted geometrically, the first derivative test says that if the slope of the tangent to the graph of _f_ at a variable point _P x_ = ( _x_ , _f_ ( _x_ )) changes from plus to minus as _P x_ goes through the point _P_ = ( _p_ , _f_ ( _p_ )), then _f_ has a strict local maximum at _p_ even if there is no tangent to the graph of _f_ at _P_. (What is the analogous statement for the case where the slope of the tangent changes sign from minus to plus?) That this is actually so is apparent from Figure 8A. On the other hand, it is easy to see that the slope of the tangent to the graph of both functions in Figure 8B does not change sign in going through _P_ , and is in fact positive on both sides of _P_. This is because both functions are increasing in a neighborhood of _p_ , and hence cannot have a local extremum at _p_.
**c. Example.** Find the local extrema of the function
SOLUTION. Differentiating (4), we get
with the help of formulas (14) and (15), p. 78, and the formula _x rxs_ = _x_ _r_ \+ _s_ , to be proved in Sec. 4.45a. Therefore _f_ has two critical points, the point _x_ = 0 at which the derivative _f_ ′ fails to exist (check this directly), and the point at which _f_ ′ vanishes. It follows from (5) that
_f_ ′( _x_ ) > 0 if _x_ < 0,
_f_ ′( _x_ ) < 0 if 0 < _x_ <
_f_ ′( _x_ ) > 0 if _x_ >
(the cube root of a negative number is negative). Thus _f_ ′ changes sign from plus to minus in going through _x_ = 0 and from minus to plus in going through . Therefore, by the first derivative test, _f_ has a strict relative maximum, equal to 0, at _x_ = 0, and a strict relative minimum, equal to
at , as confirmed by Figure 9.
**3.65. a.** The next test is applicable only when the first derivative _f_ ′( _p_ ) and second derivative _f_ ″( _p_ ) both exist, but this is the most common situation.
THEOREM **(Second derivative test for a local extremum).** _Let p be a stationary point of f, and suppose f_ ″( _p_ ) _exists and is nonzero. Then f has a strict local extremum at p. The extremum is a maximum if f_ ″( _p_ ) < 0 _and a minimum if f_ ″( _p_ ) > 0.
_Proof_. Since the second derivative _f_ ″( _p_ ) exists, _f_ is differentiable in a neighborhood of _p_ , that is, _f_ ′ exists in a neighborhood of _p_. Moreover, _f_ ′( _p_ ) = 0, since _p_ is a stationary point of _f_. Applying the argument in Sec. 3.41 to the derivative _f_ ′ instead of to the function _f_ itself, we see that _f_ ′ is decreasing in a neighborhood of _p_ if _f_ ″( _p_ ) < 0 and increasing in a neighborhood of _p_ if _f_ ″( _p_ ) > 0. Since _f_ ′( _p_ ) = 0, it follows that _f_ ′ changes sign from plus to minus in going through _p_ if _f_ ″( _p_ ) _<_ 0 and from minus to plus if _f_ ″( _p_ ) > 0. The second derivative test is now an immediate consequence of the first derivative test.
**b.** The function _f_ may or may not have a local extremum at _p_ if _f_ ″( _p_ ) = 0. For example, _f_ ″(0) = 0 if _f_ ( _x_ ) = _x_ 3 or if _f_ ( _x_ ) = ± _x_ 4, but in the first case _f_ has no local extremum at _x_ = 0, being increasing in (−∞, ∞), by Sec. 3.54, while in the second case _f_ clearly has a strict local extremum at _x_ = 0, in fact a minimum if _f_ ( _x_ ) = _x_ 4 and a maximum if _f_ ( _x_ ) = − _x_ 4.
Figure 9.
**c. Example.** Find the local extrema of the function
_f_ ( _x_ ) = 3 _x_ 5 − 5 _x_ 3.
SOLUTION. Here _f_ is differentiable for all _x_ , and the only critical points of _f_ are stationary points. These are the roots of the equation
_f_ ′( _x_ ) = 15 _x_ 4 − 15 _x_ 2 = 15 _x_ 2( _x_ − 1)( _x_ \+ 1) = 0,
namely the points _x_ = −1, 0, 1. Calculating the second derivative, we get
_f_ ″( _x_ ) = 60 _x_ 3 − 30 _x_ = 30 _x_ (2 _x_ 2 − 1),
so that
_f_ ″(−1) = −30 < 0, _f_ ″(0) = 0, _f_ ″(1) = 30 > 0.
Therefore, by the second derivative test, _f_ has a strict local maximum, equal to 2, at _x_ = −1, and a strict local minimum, equal to −2, at _x_ = 1. Although the second derivative test does not work at the point _x_ = 0, it is easy to see that _f_ has no extremum at _x_ = 0. In fact,
_f_ ′( _x_ ) = 15 _x_ 2( _x_ 2 − 1) < 0
if −1 < _x_ < 1. Therefore _f_ is decreasing in [−1, 1], by Sec. 3.54, and hence can have no extremum at _x_ = 0. You should confirm all this by sketching a graph of _f_ ( _x_ ).
PROBLEMS
**.** Verify the following rule for finding the global extrema of a function _f_ continuous in a closed interval [ _a_ , _b_ ]: Let _x_ 1, _x_ 2, . . ., _x_ _n_ be all the points of the open interval ( _a_ , _b_ ) at which _f_ has local extrema. Then the largest of the numbers
_f_ ( _a_ ), _f_ ( _x_ 1), _f_ ( _x_ 2), . . ., _f_ ( _x_ _n_ ), _f_ ( _b_ )
is the global maximum of _f_ in [ _a_ , _b_ ], while the smallest of these numbers is the global minimum of _f_ in [ _a_ , _b_ ].
**.** By investigating all critical points, find the local extrema, if any, of
(a) _y_ = | _x_ |; (b) _y_ = 2 + _x_ − _x_ 2; (c) _y_ = _x_ 3 − 2 _x_ 2 \+ 3 _x_ − 1.
**.** Do the same for
(a) _y_ = 2 _x_ 2 − _x_ 4; (b) _y_ = _x_ \+ ; (c) _y_ = _x_ 1/3(1 − _x_ )2/3.
**.** Find the global extrema of
**5.** Show that |3 _x_ − _x_ 3| ≤ 2 if | _x_ | ≤ 2.
**.** Show that the function
has no strict local extrema, regardless of the values of _a_ , _b_ , _c_ , _d_.
***7.** Find the local extrema of the function
_y_ = _x_ _m_ (1 − _x_ ) _n_ ,
where _m_ and _n_ are positive integers.
***8.** What value of _c_ minimizes the maximum of the function _f_ ( _x_ ) = | _x_ 2 \+ _c_ | in the interval [−1, 1]?
***9.** Suppose the function
has a local extremum, equal to −1, at the point _x_ − 2. Find _a_ and _b_ , and show that the extremum is a maximum.
***10.** Which term of the sequence
is the largest?
3.7 CONCAVITY AND INFLECTION POINTS
**3.71. a.** Let _f_ be continuous in an interval _I_ and differentiable at a point _p_ _I_ , and let _y_ = _T_ ( _x_ ) be the equation of the tangent to the curve _y_ = _f_ ( _x_ ) at the point with abscissa _p_. (For brevity, we will henceforth say "at the point _p_ " or simply "at _p_ ," instead of "at the point with abscissa _p_.") Then, according to Sec. 2.52d,
_y_ = _T_ ( _x_ ) = _f_ ′( _p_ )( _x_ − _p_ ) + _f_ ( _p_ ).
Suppose that _f_ ( _x_ ) > _T_ ( _x_ ) in some deleted neighborhood of _p_ , so that the curve _y_ = _f_ ( _x_ ) lies _above_ its tangent at _p_ in this neighborhood, as shown in Figure 10A. Then _f_ is said to be _concave upward at p_. Similarly, suppose that _f_ ( _x_ ) < _T_ ( _x_ ) in some deleted neighborhood of _p_ , so that the curve _y_ = _f_ ( _x_ ) lies _below_ its tangent at _p_ in this neighborhood, as shown in Figure 10B. Then _f_ is said to be _concave downward at p_. If _f_ is concave upward (or downward) at every point of the interval _I_ , we say that _f_ is _concave upward_ (or _downward_ ) _in I_.
**b.** A point _p_ is said to be an _inflection point_ of the function _f_ if the curve _y_ = _f_ ( _x_ ) lies on one side of its tangent (at _p_ ) if _x_ < _p_ and on the other side of its tangent if _x_ > _p_. The two ways in which this can happen are illustrated in Figures 11A and 11B. If _p_ is an inflection point of _f_ , we also say that _f_ has an inflection point at _p_.
Figure 10.
Figure 11.
**3.72.** With the help of the mean value theorem, it is not hard to show (see Prob. 11) that if _f_ ′ exists and is _increasing_ in some neighborhood of _p_ , then _f_ is concave _upward_ at _p_ , while if _f_ ′ exists and is _decreasing_ in some neighborhood of _p_ , then _f_ is concave _downward_ at _p_. It can also be shown (see Prob. 12) that if _f_ ′ exists in some neighborhood of _p_ and has a strict local extremum at _p_ , then _p_ is an inflection point of _f_. Using these facts, we can develop a complete parallelism between the theory of increasing (or decreasing) functions and critical points, on the one hand, and the theory of upward (or downward) concavity and inflection points, on the other hand, with the first derivative _f_ ′ now playing the role of the function _f_ , and the second derivative _f_ ″ now playing the role of the first derivative _f_ ′. Thus you can easily convince yourself of the validity of the following propositions. In every case, the proof is the exact analogue of a proof that has already been given.
(1) _If f_ ″( _p_ ) _exists and is positive, then f_ ′ _is increasing in some neighborhood of p, so that f is concave upward at p_. This is the analogue of the result in Sec. 3.41a.
(2) If _f_ ″( _p_ ) _exists and is negative, then f_ ′ _is decreasing in some neighborhood of p, so that f is concave downward at p_. This is the analogue of the result in Sec. 3.41b.
(3) If f has an inflection point at p, then either _f_ ″( _p_ ) _fails to exist or f_ ″( _p_ ) _exists and equals zero_. This is the analogue of Theorem 3.63a.
(4) Given that _f_ ″ _exists in a deleted neighborhood of p, suppose f_ ″( _p_ ) _either fails to exist or equals zero, and suppose f_ ″ _changes sign in going through p. Then f has an inflection point at p_. This second derivative test for an inflection point is the analogue of the first derivative test for a local extremum (Theorem 3.64a).
(5) If _f_ ″( _p_ ) = 0 _and if the third derivative f_ ″′( _p_ ) _exists and is nonzero, then f has an inflection point at p_. This third derivative test for an inflection point is the analogue of the second derivative test for a local extremum (Theorem 3.65a).
**3.73. Examples**
**a.** Find the inflection points and investigate the concavity of the function
_f_ ( _x_ ) = _x_ 4 − 2 _x_ 3 \+ 3 _x_ − 4.
SOLUTION. Here _f_ ″ exists for all _x_. Therefore, by Proposition (3), the only candidates for inflection points of _f_ are the roots of the equation
_f_ ″( _x_ ) = 12 _x_ 2 − 12 _x_ = 12 _x_ ( _x_ − 1) = 0,
namely the points _x_ = 0 and _x_ = 1. By Propositions (1) and (2), _f_ is concave upward in the interval (−∞, 0), since _f_ ″( _x_ ) > 0 if _x_ < 0, concave downward in the interval (0, 1), since _f_ ″( _x_ ) < 0 if 0 < _x_ < 1, and concave upward in the interval (1, ∞), since _f_ ″( _x_ ) > 0 if _x_ > 1. Therefore _x_ = 0 and _x_ = 1 are both inflection points of _f_ , by Proposition (4). This also follows from Proposition (5), since _f_ ′″( _x_ ) = 24 _x_ − 12, and hence _f_ ′″(0) = −12 ≠ 0, _f_ ′″(1) = 12 ≠ 0.
**b.** Graph the function
SOLUTION. Since _f_ is even, the graph of _f_ is symmetric in the _y_ -axis (Example 2.32d), and we need only study the behavior of _f_ in the interval [0, ∞). To find the extrema of _f_ , we solve the question
obtaining two nonnegative stationary points _x_ = 0 and _x_ = . Since
we have
Figure 12.
It follows from Theorem 3.65a that _f_ has a strict local maximum, equal to _f_ (0) = 6, at the point _x_ = 0, and a strict local minimum, equal to , at the point _x_ = . The only point in [0, ∞) which can be an inflection point of _f_ is the nonnegative solution of the equation
namely the point _x_ = 2. This point is actually an inflection point, by Proposition (4), since _f_ ″( _x_ ) < 0 in [0, 2) and _f_ ″( _x_ ) > 0 in (2, ∞). At the same time, we note that _f_ is concave downward in the interval [0, 2) and concave upward in the interval (2, ∞). Moreover _f_ ″′( _x_ ) = _x_ , and hence _f_ ″′(2) = 2 ≠ 0. Therefore the fact that _x_ = 2 is an inflection point of _f_ also follows from Proposition (5).
The function _f_ has no asymptotes (Sec. 2.93), since it does not become infinite at any finite points and does not approach a finite limit as _x_ → ±∞. In fact, _f_ ( _x_ ) → ∞ as _x_ → ±∞. To draw an accurate graph of _f_ , we need a few more values of _f_ besides _f_ (0) = 6 and . The following three will suffice:
Plotting the corresponding points and connecting them by a "smooth curve," we get the graph shown in Figure 12, after using the symmetry of the graph in the _y_ -axis.
**c.** It is apparent from the previous example that we can form no clear idea of the behavior of a function without first locating all its extrema and inflection points, as well as examining it for possible asymptotes and testing it for parity (evenness or oddness). Figure 13 shows what can go wrong if we try to graph a function without doing this first. The solid curve is the "true graph" and the dashed curve is the quite misleading result of connecting five points of the graph by a "smooth curve."
Figure 13.
PROBLEMS
**1.** "The function _f_ is concave upward at _p_ if it has a strict local minimum at _p_ and concave downward at _p_ if it has a strict local maximum at _p_." True or false?
**.** What does the condition _f_ ″( _p_ ) = 0 by itself tell us about concavity at _p_ or the presence of an inflection point at _p?_
**3.** Must a function have a local extremum between two consecutive inflection points?
**.** Find the inflection points and investigate the concavity of the function _y =_ 2 _x_ 4 − 3 _x_ 2 \+ 2 _x_ \+ 2.
**5.** Do the same for the function
**.** For what value of _c_ does the function _y_ = _x_ 3 \+ _cx_ 2 \+ 1 have an inflection point at _x_ = 1?
**7.** A point _P_ = ( _p_ , _f_ ( _p_ )) is said to be an inflection point of the _curve y_ = _f_ ( _x_ ) if _p_ is an inflection point of the _function f_. For what values of _a_ and _b_ is the point (1, 3) an inflection point of the curve _y_ = _ax_ 3 \+ _bx_ 2?
**.** Graph the function _y_ = ( _x_ \+ 1)( _x_ − 1)2, after first investigating extrema, concavity, inflection points, asymptotes, etc
***9.** Do the same for the function
***10.** Show that the three inflection points of the function (1) are collinear, that is, lie on the same straight line.
***11.** Show that if _f_ ′ exists and is increasing in some neighborhood of _p_ , then _f_ is concave upward at _p_ , while if _f_ ′ exists and is decreasing in some neighborhood of _p_ , then _f_ is concave downward at _p_.
***12.** Show that if _f_ ′ exists in some neighborhood of _p_ and has a strict local extremum at _p_ , then _p_ is an inflection point of _f_.
***13.** Suppose _f_ and its first and second derivatives _f_ ′ and _f_ ″ are continuous in an interval _I_ , Justify the following statement: _f_ vanishes at a point _p_ if the sign of _f_ changes in passing through _p_ , _f_ has a local extremum at _p_ if the sign of _f_ ′ changes in passing through _p_ , _f_ has an inflection point at _p_ if the sign of _f_ ″, and hence the concavity of _f_ , changes in passing through _p_.
3.8 OPTIMIZATION PROBLEMS
A host of practical problems involve the determination of _largest_ size, _least_ cost, _shortest_ time, _greatest_ revenue, and so on. Problems of this type ask for the "best value" of some variable, and hence are called _optimization problems_. Many of them can be solved with the help of the powerful tools developed in the last few sections. There is no universal rule that works in all cases, and as in all "word problems," there is no substitute for using a little common sense early in the game before trying to turn some computational crank. The following examples will give you a good idea of how to go about solving optimization problems.
**3.81. Example**. A square box with no top is made by cutting little squares out of the four corners of a square sheet of metal _c_ inches on a side, and then folding up the resulting flaps, as shown in Figure 14. What size squares should be cut out to make the box of largest volume?
SOLUTION. Let _x_ be the side length of each little square. Then the volume of the box in cubic inches is just
Moreover,
Figure 14.
since it is impossible to cut away either overlapping squares or squares of negative side length. Our problem is thus to determine the value of _x_ at which the function (1) takes its global maximum in the interval (2). Since _V_ is differentiable for all _x_ , the only critical points of _V_ , and hence, by Sec. 3.63c, the only points at which _V_ can have a local extremum in the whole interval (−∞, ∞) are the solutions of the equation
namely _x_ = _c_ /6 and _x_ = _c_ /2. Moreover, since _V_ is positive in the open interval
and vanishes at the end points _x_ = 0 and _x_ = _c_ /2, the global maximum of _V_ in the closed interval (2), guaranteed by Theorem 3.32c and by the "physical meaning" of the problem, must be at an interior point of (2), and hence must be a local maximum of _V_. But _x_ = _c_ /6 is the only interior point of (2) at which _V_ can have a local extremum, and therefore it is apparent without any further tests that _V_ takes its maximum in (2) at the point _x_ = _c_ /6. This can be confirmed by noting that
and then applying the second derivative test (Theorem 3.65a).
Thus, finally, the largest box is obtained by cutting squares of side length _c_ /6 out of the corners of the original sheet of metal. The volume of the resulting box equals
**3.82. Example.** An island lies _l_ miles offshore from a straight beach. Down the beach _h_ miles from the point nearest the island, there is a group of vacationers who plan to get to the island by using a beach buggy going α mi/hr, trailing a motorboat which can do _β_ mi/hr. At what point of the beach should the vacationers transfer from the buggy to the boat in order to get to the island in the shortest time?
SOLUTION. The geometry of the problem is shown in Figure 15, where the vacationers start at _A_ , the island is at _C_ , and _x_ is the distance between the point _P_ at which they launch the boat and the point _B_ of the beach nearest the island. The time it takes to get to the island is given by the formula
(the time taken equals the distance travelled divided by the speed), where the boat leaves from the starting point _A_ if _x_ = _h_ and from the point _B_ nearest the island if _x_ = 0. Differentiating (3) with respect to _x_ , we get
where _k_ = _β_ / _α_. If _k_ ≥ 1, that is, if _β_ ≥ α, then _dT_ / _dx_ is negative for all _x_ (0, _h_ ), and hence _T_ is decreasing in [0, _h_ ], by Sec. 3.54. In this case, _T_ takes its global minimum in [0, _h_ ] at _x_ = _h_ , so that the vacationers should forget about the buggy and go straight to the island by boat.
The same is true if _k_ < 1, provided that
In fact, if _k_ < 1, the equation
has the unique solution _x_ = _x_ 0, where _x_ 0 lies outside the interval (0, _h_ ) if (4) holds. Therefore _dT_ / _dx_ is again negative at every point of (0, _h_ ), since _dT_ / _dx_ cannot change sign in (0, _h_ ) and _dT_ / _dx_ is clearly negative for small enough _x_. But then _T_ is again decreasing in [0, _h_ ]. Thus, in this case too, the vacationers should go straight to the island by boat.
Figure 15.
However, if
then _x_ 0 lies in the interval (0, _h_ ). Moreover, in this case, _dT/dx_ is negative at every point of (0, _x_ 0) and positive at every point of ( _x_ 0, _h_ ). Therefore _T_ must have a local minimum at _x_ 0, by the first derivative test (Theorem 3.64a). But then _T_ takes its global minimum in [0, _h_ ] at _x_ 0 (why?). This means that the vacationers should now stop the buggy and launch the boat at the point with coordinate _x_ 0, as measured from _B_.
**3.83. Example.** A monopolistic firm has a total revenue function
_R_ ( _Q_ ) = − _AQ_ 2 \+ _BQ_
(Sec. 3.2, Prob. 7) and a total cost function
_C_ ( _Q_ ) = _aQ_ 2 \+ _bQ_ \+ _c_
(Sec. 3.22a), where the coefficients _A_ , _B_ , _a_ , _b_ , _c_ are all positive constants and _B_ > _b_. The government wishes to levy an excise tax on the commodity produced by the firm. What tax rate should the government impose on the firm's output to maximize the tax revenue _T_ = _rQ_ , knowing that the firm will add the tax to its costs and adjust its output to maximize the profit after taxes?
SOLUTION. The cost and profit after taxes are
_C_ _T_ ( _Q_ ) = _C_ ( _Q_ ) + _rQ_ = _aQ_ 2 \+ ( _b_ \+ _r_ ) _Q_ \+ _c_
and
(Sec. 3.2, Prob. 10). Differentiating (5) with respect to _Q_ , with _r_ regarded as a constant, and setting the result equal to zero, we get the equation
whose only solution is
Since
it follows from the second derivative test that the output level (6) actually maximizes the firm's profit after taxes, at the tax rate _r_.
Knowing that the firm will maximize its profit after taxes, the government chooses its tax rate _r_ to maximize the revenue
calculated at the output level (6). To maximize _T_ as a function of the tax rate _r_ , which is now regarded as variable, we differentiate (7) with respect to _r_ , obtaining
The optimum tax rate _r_ 0 is the solution of the equation _dT_ / _dr_ = 0, namely
By the second derivative test, _r_ 0 actually maximizes the government's revenue, since
PROBLEMS
**.** Among all rectangles of a given area _A_ , find the one with the smallest perimeter.
**.** Find the right triangle of largest area, given that the sum of one leg of the triangle and the hypotenuse is a constant _c_.
**.** What is the largest volume of a right circular cone of slant height _l_?
**4.** What is the largest volume of a right circular cylinder inscribed in a sphere of radius _R?_
**.** Given two points _A_ = (0, 3) and _B_ = (4, 5), find the point _P_ on the _x_ -axis for which the distance | _AP_ | + | _PB_ | is the smallest.
**.** Find the least amount of sheet metal needed to make a cylindrical cup of a given volume _V_.
**7.** In Example 3.82, should the vacationers ever take the buggy all the way to the point _B_ nearest the island?
**8.** The results of _n_ measurements of an unknown quantity _x_ are _x_ 1, _x_ 2, . . ., _x n_. What value of _x_ minimizes the expression ( _x_ − _x_ 1)2 \+ ( _x_ − _x_ 2)2\+ ... +( _x_ − _x n_)2?
**.** Two ships, originally at distances _a_ and _b_ from a point _P_ , sail toward _P_ with speeds α and _β_ along straight line routes making an angle of 90° with each other. At what time _t_ is the distance between the two ships the smallest? What is the distance _d_ of closest approach?
**.** Given a point _P_ = ( _a_ , _b_ ) in the first quadrant, find the line through _P_ which cuts off the triangle of least area from the quadrant.
**11.** Let _R_ ( _Q_ ) be the total revenue received by a monopolistic firm from the sale of a quantity _Q_ of some commodity, and let _C_ ( _Q_ ) be the firm's total cost function. The firm wants to adjust its output to maximize its profit
Show that the profit function (8) is maximized at any output level such that
(a) Marginal revenue ( _MR_ ) equals marginal cost ( _MC_ );
(b) Marginal revenue is increasing more slowly than marginal cost.
**.** Suppose a monopolistic firm has a total revenue function _R_ ( _Q_ ) = 1200 _Q_ − 10 _Q_ 2 and a total cost function _C_ ( _Q_ ) = _Q_ 3 − 60 _Q_ 2 \+ 1500 _Q_ \+ 1000. What output level maximizes the firm's profit?
***13.** "Normal cost conditions" are characterized by three properties:
(a) There are certain fixed costs (the overhead);
(b) Total cost increases with output;
(c) Marginal cost is always positive; as the output increases, the marginal cost first decreases and then increases.
Suppose a firm has a cubic total cost function
_C_ ( _Q_ ) = _aQ_ 3 \+ _bQ_ 2 \+ _cQ_ \+ _d_.
Show that
_a_ > 0, _b_ < 0, _c_ > 0, _d_ > 0, _b_ 2 < 3 _ac_
under normal cost conditions.
***14.** For which chord _BC_ parallel to the tangent to a circle at a point _A_ is the area of the triangle _ABC_ largest?
***15.** What is the largest surface area (including the top and bottom) of a right circular cylinder inscribed in a sphere of radius _R_?
***16.** Given a point _P_ inside an acute angle, let _L_ be the line segment through _P_ cutting off the triangle of least area from the angle. Show that _P_ bisects the part of _L_ inside the angle. Show that this property also characterizes the point _P_ in Problem 10.
***17.** According to _Fermat's principle_ , the path taken by a ray of light which leaves a point _A_ and passes through a point _B_ after being reflected by a plane mirror is such as to minimize the time taken to traverse the whole path from _A_ to the mirror to _B_. According to the _law of reflection_ , the _angle of incidence_ (between the incident ray and the perpendicular to the mirror) equals the _angle of reflection_ (between the reflected ray and the perpendicular to the mirror). Deduce the law of reflection from Fermat's principle.
_Chapter 4_
INTEGRAL CALCULUS
4.1 THE DEFINITE INTEGRAL
The study of calculus is closely associated with the study of limits of various kinds. So far we have encountered the limit of a function at a point, the limit of a sequence, and the sum of an infinite series, as well as one-sided limits, infinite limits, limits at infinity, and asymptotes. We now consider still another kind of limit, leading to the concept of the _definite integral_. This kind of limit comes up time and again in problems involving the "summation of a very large number of individually small terms." The prototype of all such problems is the problem of finding the "area under a curve."
**4.11. The area under a curve**
**a.** Let
_y_ = _f_ ( _x_ )( _a_ ≤ _x_ ≤ _b_ )
be a function which is continuous and nonnegative in a closed interval [ _a_ , _b_ ]. Then, by the _area under the curve y_ = _f_ ( _x_ ), from _x_ = _a_ to _x_ = _b_ , we mean the area _A_ of the plane region bounded by the curve _y_ = _f_ ( _x_ ), the _x_ -axis, and the lines _x_ = _a_ and _x_ = _b_. This can also be described as the _area between the curve y_ = _f_ ( _x_ ) and the _x-axis_ , from _x_ = _a_ to _x_ = _b_.
We can think of the region as a kind of trapezoid with three straight sides and one curved side, unless _f_ ( _a_ ) = 0 or _f_ ( _b_ ) = 0, in which case one or both of the vertical sides may shrink to a point. Such regions are not considered in elementary geometry. Thus, in the process of calculating _A_ , we must decide what is meant by _A_ in the first place!
**b.** With this in mind, we divide the interval [ _a_ , _b_ ] into a large number _n_ of small subintervals [ _x_ _i_ − 1, _x_ _i_ ], by introducing points of subdivision _x_ 1, _x_ 2, . . ., _x_ _n_ − 1 such that
_a_ = _x_ 0 < _x_ 1 < _x_ 2 < . . . < _x_ _n_ − 1 < _x_ _n_ = _b_ ,
where, in the interest of a uniform notation, the end points _a_ and _b_ of the original interval [ _a_ , _b_ ] are assigned alternative symbols _x_ 0 and _x_ _n_ , as if they were points of subdivision too. Let
Δ _x_ _i_ = _x i_ − _x_ _i_ − 1 ( _i_ = 1, 2, ..., _n_ )
be the length of the _i_ th subinterval, and let _λ_ be the maximum length of all the sub-intervals, that is, the largest of the numbers Δ _x_ 1, Δ _x_ 2, ..., Δ _x_ _n_. We denote this by
Figure 1.
writing
_λ_ = max { _x_ 1 − _x_ 0, _x_ 2 − _x_ 1, ..., _x_ _n_ − _x_ _n_ − 1} = max {Δ _x_ 1, Δ _x_ 2, . . ., Δ _x_ n}
(see Prob. 1). The lines _x_ = _x_ 0, _x_ = _x_ 1, _x_ = _x_ 2, . . ., _x_ = _x_ _n_ − 1, _x_ = _x n_ divide the region into _n_ narrow strips, as shown in Figure 1. Being continuous, _f_ ( _x_ ) does not change "much" in the interval [ _x_ _i_ − 1, _x_ _i_ ], and hence it seems like a good approximation to regard _f_ ( _x_ ) as having the constant value _f_ ( _ξ_ _i_ ) in [ _x_ _i_ − 1, _x_ _i_ ], where _ξ_ _i_ is an _arbitrary_ point of [ _x_ _i_ − 1, _x_ _i_ ]. This is equivalent to replacing the strips, with curved tops, by the shaded rectangles shown in the figure. The sum of the areas of these rectangles is given by
where we use the summation notation introduced in Sec. 2.96a. It seems reasonable to regard (1) as a good approximation to the area _A_ of the region, where the approximation gets "better and better" as the bases of the rectangles all get "smaller and smaller," that is, as the number _λ_ , the largest of these bases, gets "smaller and smaller." This suggests that we _define A_ as the limit
and this is exactly what we will do.
**4.12. a.** These considerations lead naturally to the following definition: Given a function _f_ ( _x_ ) defined in a closed interval [ _a_ , _b_ ], let _x_ 1, _x_ 2, ..., _x_ _n_ − 1 be points of subdivision of [ _a_ , _b_ ] such that
_a_ = _x_ 0 < _x_ 1 < _x_ 2 < . . . < _x_ _n_ − 1 < _x_ _n_ = _b_ ,
and let _ξ_ _i_ be an arbitrary point of the subinterval [ _x_ _i_ − 1, _x_ _i_ ], of length Δ _x _i__ = _x_ _i_ − _x_ _i_ − 1. Suppose the sum
approaches a finite limit as
_λ_ = max { _x_ 1 − _x_ 0, _x_ 2 − _x_ 1, . . ., _x_ _n_ − _x_ _n_ − 1} = max {Δ _x_ 1, Δ _x_ 2, ..., Δ _x_ _n_ }
approaches zero. Then the limit is called the _definite integral_ of _f_ ( _x_ ) from _a_ to _b_ , denoted by
and the function _f_ ( _x_ ) is said to be _integrable_ in [ _a_ , _b_ ], or over [ _a_ , _b_ ].
It should be noted that here we do not require the function _f_ ( _x_ ) to be continuous in [ _a_ , _b_ ], and in fact, the integral (4) exists even in cases where _f_ ( _x_ ) is discontinuous. This matter will be discussed further in Sec. 4.14.
**b.** The quantities _σ_ and _λ_ depend, of course, on the choice of the points of subdivision
To emphasize this, we can write _σ_ = _σ_ ( _X_ ) and _λ_ = _λ_ ( _X_ ), where _X_ is the set of points (5). Loosely speaking, _X_ is a "partition" of the interval [ _a_ , _b_ ], and the quantity _λ_ = _λ_ ( _X_ ) is a measure of the "fineness" of the partition. What does it mean to say that _σ_ approaches the limit (4) as _λ_ → 0? Just this: The quantity
is "arbitrarily near" zero for all _X_ such that _λ_ ( _X_ ) is "sufficiently small," regardless of the choice of the points
_ξ_ _i_ ∈ _x_ _i_ − 1, _x_ _i_ .
or in " _ε_ , _δ_ language," which is particularly appropriate here, given any _ε_ > 0, we can find a number _δ_ > 0 such that
whenever 0 < _λ_ ( _X_ ) < _δ_.
**c.** This is a different kind of limit than those considered so far, but it is handled in the same way. For example, let _σ_ be the sum (3) and let _c_ be an arbitrary constant. Then
by elementary algebra, and
by the usual rule for the limit of a product, provided that the limit on the right exists. Therefore
provided that _f_ ( _x_ ) is integrable in [ _a_ , _b_ ].
Similarly, let
where _g_ ( _x_ )is another function defined in [ _a_ , _b_ ]. Then
by elementary algebra, and
by the usual rule for the limit of a sum, provided that both limits on the right exist. Therefore
provided that both functions _f_ ( _x_ ) and _g_ ( _x_ )are integrable in [ _a_ , _b_ ].
By repeated application of (6) and (7), we arrive at the more general formula
where _f_ 1( _x_ ), _f_ 2( _x_ ), ..., _f_ n( _x_ ) are functions which are all integrable in [ _a_ , _b_ ], and _c_ 1, _c_ 2, . . ., _c_ _n_ are arbitrary constants. For example,
and similarly for more than three terms. Formula (8) is, of course, the exact analogue for definite integrals of formula (7), p. 119, for indefinite integrals.
**d.** Note the distinction between the _definite_ integral (4), which is a _number_ , and the _indefinite_ integral
∫ _f_ ( _x_ ) _dx_ ,
which is a _function_. A definite integral always has two numbers attached to the integral sign, like the numbers _a_ and _b_ in (4), called the _lower limit of integration_ and the _upper limit of integration_ , respectively, where in these expressions, the word "limit" is used in the loose, colloquial sense, meaning "boundary" or "extent," and not in the precise technical sense in which it is used elsewhere in this book. The numbers _a_ and _b_ are, of course, the end points of the interval _a_ , _b_ ], called the _interval of integration_. Otherwise, the terminology is the same for both definite and indefinite integrals. Thus the function _f_ ( _x_ ) in (4) is again called the _integrand_ , as in [Sec. 3.53a, the operation leading from _f_ ( _x_ ) to the number (4) is called ( _definite_ ) _integration_ , with respect to _x_ , and the argument _x_ is called the _variable of integration_. Since definite integration is an operation producing a number from a given function _f_ ( _x_ ), the symbol _x_ is a "dummy variable," in the sense that it can be replaced by any other symbol without changing the meaning of (4). For example,
The situation is exactly the same as for a dummy index of summation (Sec. 2.96a). Things are different for indefinite integration, since our convention is to give the indefinite integral, which is an antiderivative, the same argument as the integrand. Thus
and in this sense
∫ _x dx_ ≠ ∫ _t dt_.
**4.13. Examples**
**a.** Comparing Secs. 4.11b and 4.12a, we find that the area under the curve _y_ = _f_ ( _x_ ) from _a_ to _b_ is given by the formula
More generally, let _f_ ( _x_ ) and _g_ ( _x_ )be two functions defined and continuous in the same interval [ _a_ , _b_ ], and suppose _f_ ( _x_ ) ≥ _g_ ( _x_ )for every _x_ ∈ [ _a_ , _b_ ], as shown in Figure 2. Then how do we define the area _A_ of the plane region _DCEF_ bounded by the lines _x_ = _a_ , _x_ = _b_ and the curves _y_ = _f_ ( _x_ ), _y_ = _g_ ( _x_ )? Clearly, if the precise definition of area is to be compatible with the everyday meaning of area, we must insist that area be "additive" in the following sense: The area of a figure Φ made up of two other figures Φ1 and Φ2, which have no points in common except possibly parts of their boundaries, must equal the sum of the separate areas of Φ1 and Φ2. As applied to Figure 2, this means that
(Area of _abCD_ ) + (Area of _DCEF_ ) = Area of _abEF_ ,
and therefore
_A_ = Area of _DCEF_ = (Area of _abEF_ ) − (Area of _abCD_ ).
Since
Figure 2.
it follows that
Note that if _g_ ( _x_ ) ≡ 0, then the "lower curve" is just the _x_ -axis, and (10) reduces to formula (9) for the area under the curve _y_ = _f_ ( _x_ ).
**b.** Evaluate
SOLUTION. Here we are integrating the constant function _f_ ( _x_ ) ≡ 1. Thus
since _f_ ( _ξ_ _i_ ) = 1 for each _ξ_ _i_. But
where all the terms in the sum on the right vanish except the first and the last, leaving
Therefore
so that, finally,
**c.** Evaluate
SOLUTION. Here the integrand is _f_ ( _x_ ) = x. Therefore
Suppose we choose _ξ_ _i_ to be the midpoint of the interval [ _x_ _i_ − 1, _x_ _i_ ], so that
(Sec. 1.5, Prob. 9). Then
since the sum again "telescopes," reducing to simply . Thus, finally,
In Sec. 4.24 we will establish a general technique for evaluating definite integrals, which will allow us to completely bypass "brute force calculations" like those just made in deriving formulas (11) and (12).
**4.14.** We now come to a crucial question: Which functions defined in a closed interval [ _a_ , _b_ ] are integrable in [ _a_ , _b_ ]? In other words, if we take a function _f_ ( _x_ ) defined in [ _a_ , _b_ ] and form the sum (3), when does the sum approach a finite limit as _λ_ → 0? The answer is well beyond the scope of this book, and cannot even be expressed in the language of elementary calculus. However, the fact that we can't describe the _largest_ set of integrable functions shouldn't bother us very much, since the following key proposition presents us with a _huge_ set of integrable functions: _If f_ ( _x_ ) _is continuous in_ [ _a_ , _b_ ], _then f_ ( _x_ ) _is integrable in_ [ _a_ , _b_ ]. The proof of this proposition is not particularly difficult, but it does require a deeper study of continuous functions than it would be profitable to pursue here.
The importance of continuity in calculus again stands revealed. Recall some of the other nice properties of continuous functions, presented in Sec. 3.3. It should be noted that there are integrable functions which are not continuous. An example of such a function is given in Problem 10. With a little ingenuity, we can also construct a function which fails to be integrable (see Prob. 11).
PROBLEMS
**.** Given a set _A_ , all of whose elements are numbers, suppose _A_ contains a largest element, that is, an element _M_ such that _x_ ≤ _M_ for all _x_ ∈ _A_. Then _M_ is called the _maximum_ of _A_ , denoted by max _A_. Find max _A_ if
(a) (b) _A_ = { _x_ : _x_ 3 − 2 _x_ 2 \+ _x_ = 0};(c) _A_ = { _x_ : 0 < _x_ < 1}.
**2.** Let _f_ be continuous in [ _a_ , _b_ ]. What is the number max { _f_ ( _x_ ): _a_ ≤ _x_ ≤ _b_ }?
**3.** As in Sec. 4.12a, let _λ_ = max {Δ _x_ 1, Δ _x_ 2, ..., Δ _x_ _n_ }. Does _n_ → ∞ imply _λ_ → 0? Does _λ_ → 0 imply _n_ → ∞?
**.** What is the smallest value of _λ_ = max {Δ _x_ 1, Δ _x_ 2, . . ., Δ _x_ _n_ } for all choices of the points of subdivision _x_ 1, _x_ 2, . . ., _x_ _n_ − 1? Does _λ_ have a largest value?
**5.** Show that formula (9) leads to the correct expressions for the area of a rectangle and for the area of a right triangle.
**6.** Test formula (10) by using it to calculate the area of the trapezoid bounded by the lines _x_ = 2, _x_ = 4, _y_ = 1 and _y_ = _x_.
**.** Can negative area be defined in a meaningful way?
**8.** How can we tell at once that the function _f_ ( _x_ ) = 1/ _x_ is integrable in every closed interval that does not contain the point _x_ = 0?
***9.** Find max _A_ if _A_ = { _a_ , _a_ 2, _a_ 3, ...} and 0 ≤ _a_ ≤ 1. What happens if _a_ > 1? If _a_ is negative?
***10.** The function
is discontinuous at _x_ = 0. Verify that _f_ ( _x_ ) is integrable in [−1, 1].
***11.** Let
Show that
(a) _f_ ( _x_ ) is discontinuous at every point _c_ ;
(b) _f_ ( _x_ ) fails to be integrable in every interval [ _a_ , _b_ ].
4.2 PROPERTIES OF DEFINITE INTEGRALS
**4.21.** First we consider what happens when the interval of integration is "split up."
**a.** THEOREM. _If f is continuous in_ [ _a_ , _b_ ] _and if c is an interior point of_ [ _a_ , _b_ ], _then_
_Proof_. As before, we divide the interval [ _a_ , _b_ ] into a large number of small subintervals, by introducing points of subdivision, but this time we insist that one of the points of subdivision be the fixed point _c_. In other words, we now choose points of subdivision _x_ _i_ ( _i_ = 1, . . ., _n_ − 1) such that
_a_ = _x_ 0 < _x_ 1 < . . . < _x_ _m_ − 1 < _x m_ = _c_ < _x_ _m_ \+ 1 < . . . < _x_ _n_ − 1 < _x_ _n_ = _b_ ,
where the subscript _m_ depends, of course, on the number of points _x i_ which are less than _c_. Every such "partition" of [ _a_ , _b_ ] automatically gives rise to a partition of the interval [ _a_ , _c_ ], made up of the points _x_ 1, ..., _x_ _m_ − 1, and a partition of the interval [ _c_ , _b_ ], made up of the points _x_ _m_ \+ 1, ..., _x_ _n_ − 1. Correspondingly, the sum
used to define the integral of _f_ from _a_ to _b_ , can be written as
_σ_ = _σ_ ′ + _σ_ ″,
in terms of the sums
needed to define the integral of _f_ from _a_ to _c_ and the integral of _f_ from _c_ to _b_. (Here Δ _x _i__ and _ξ_ _i_ have the same meaning as in Sec. 4.12a.)
Now let
_λ_ = max {Δ _x_ 1, ..., Δ _x_ _n_ },
_λ_ ′ = max {Δ _x_ 1, ..., Δ _x_ m},
_λ_ ″ = max {Δ _x_ _m_ \+ 1, Δ _x_ _n_ }.
Then clearly _λ_ → 0 implies _λ_ ′ → 0 and _λ_ ″ → 0, so that
where the existence of all three integrals follows from the assumption that _f_ is continuous in [ _a_ , _b_ ], and hence in [ _a_ , _c_ ] and [ _c_ , _b_ ] as well.
**b.** So far, in writing the integral
it has been assumed that _a_ < _b_. We now allow the case _a_ ≥ _b_ , setting
_by definition_. Suppose _b_ = _a_ in (2). Then
which implies
The merit of the definition (2) is shown by the following extension of the preceding result:
**c.** THEOREM. _If f is continuous in an interval containing the points a, b and c, then_
_Proof_. Formula (4) is an immediate consequence of (2) and (3) if two or three of the points _a_ , _b_ and _c_ coincide. Moreover, (4) reduces to (1) if _a_ < _c_ < _b_. The other cases can be dealt with by using (2) together with (1). For example, if _c_ < _b_ < _a_ , then, by (1),
and hence, by (2),
which implies
The remaining cases _a_ < _b_ < _c_ , _b_ < _a_ < _c_ , _b_ < _c_ < _a_ and _c_ < _a_ < _b_ are treated similarly (give the details).
**4.22. The mean value theorem for integrals**
The mean value theorem of Sec. 3.43a expresses the difference between the values of a differentiable function at two points _a_ and _b_ in terms of the derivative of the function at some point of [ _a_ , _b_ ], in fact, at an interior point of [ _a_ , _b_ ]. There is a similar proposition expressing the definite integral from _a_ to _b_ of a continuous function in terms of the value of the function at some point of [ _a_ , _b_ ]:
**a.** THEOREM **(Mean value theorem for integrals)**. _If f is continuous in_ [ _a_ , _b_ ], _then there is a point c_ ∈ [ _a_ , _b_ ] _such that_
_Proof_. As in Sec. 3.32, let _M_ be the maximum and _m_ the minimum of _f_ in [ _a_ , _b_ ], taken at points _p_ and _q_ , respectively, and let
be the sum involved in the definition of the integral in (5). Clearly
_m_ Δ _x_ _i_ ≤ _f_ ( _ξ_ _i_ ) Δ _x_ _i_ ≤ _M_ Δ _x_ _i_ ,
and therefore
or
since
Taking the limit of (6) as _λ_ → 0, we get
with the help of Problem 13, or equivalently
Thus the quantity
is a number belonging to the interval _m_ , _M_ ]. It follows from the intermediate value theorem ([Sec. 3.33b) that there is a point _c_ , either equal to _p_ or _q_ if _h = M_ or _h_ = _m_ , or lying between _p_ and _q_ if _m_ < _h_ < _M_ , but in any event certainly in the interval [ _a_ , _b_ ], such that
Comparing (8) and (9), we immediately obtain (5).
Note that formula (5) remains true for _b_ < _a_ , provided that _f_ is continuous in [ _b_ , _a_ ]. In fact, we then have
**b.** The mean value theorem for integrals has a simple geometrical interpretation. Suppose _f_ ( _x_ ) ≥ 0, as in Figure 3. Then, by Example 4.13a, the area of the region _abCD_ bounded by the curve _y_ = _f_ ( _x_ ), the _x_ -axis and the lines _x_ = _a_ and _x_ = _b_ is given by the integral
Figure 3.
According to (5), there is a point _c_ ∈ [ _a_ , _b_ ] such that the rectangle with base _b_ − _a_ and altitude _h_ = _f_ ( _c_ ) has the same area as _abCD_. How this comes about is shown in the figure, where the dark parts of _abCD_ are "compensated" by the shaded parts of the rectangle _abEF_. The number _h_ , equal to
is called the _mean value_ or _average_ of the function _f_ ( _x_ ) over the interval [ _a_ , _b_ ].
**4.23.** According to Sec. 4.14, every continuous function has a definite integral. We now use the mean value theorem for integrals to prove that every continuous function has an antiderivative and hence an indefinite integral.
**a.** THEOREM. _Let f be continuous in an interval I, and let_
_where x_ 0 _is a fixed point of I and x is a variable point of I. Then_ Φ _is an antiderivative of f in I._
_Proof_. There is a slight technicality here, namely, if _I_ has end points _a_ and _b_ , then Φ′( _a_ ) is defined by the right-hand limit
if _a_ ∈ _I_ , while Φ′( _b_ ) is defined by the left-hand limit
if _b_ ∈ _I_. This is simply because _x_ must belong to _I_ , and hence can approach _a_ only from the right and _b_ only from the left.
To get on with the proof, we first note that the existence of the integral (10) follows from the continuity of _f_ , by Sec. 4.14. Suppose _x_ and _x_ \+ Δ _x_ both belong to _I_. Then
by Theorem 4.21c, and hence
Applying the mean value theorem for integrals to the right side of (11), which is independent of the fixed point _x_ 0, we get
where _x_ ≤ _c_ ≤ _x_ \+ Δ _x_ or _x_ \+ Δ _x_ ≤ _c_ ≤ _x_ , depending on whether Δ _x_ is positive or negative. But then _c_ → _x_ as Δ _x_ → 0, and therefore _f_ ( _c_ ) → _f_ ( _x_ ) as Δ _x_ → 0, by the continuity of _f_. It follows that
that is, Φ is an antiderivative of _f_ in _I_.
**b.** The content of this key theorem can be written concisely as
Notice how the presence of the letter _x_ in the upper limit of integration forces us to use another letter for the variable of integration. Here we use _t_ , but any letter other than _x_ would do just as well. The theorem has a simple geometrical interpretation: Suppose _I_ = [ _a_ , _b_ ] and _f_ ( _t_ ) ≥ 0, as in Figure 4, and let _x_ 0 = _a_. Then Φ( _x_ ) is the shaded area under the curve _y_ = _f_ ( _t_ ) from _t_ = _a_ to _t_ = _x_ , which varies from
to
as _x_ varies from _a_ to _b_ , and the rate of change of this area with respect to _x_ at a given point of [ _a_ , _b_ ] equals the "height" of the curve at the given point.
**c.** COROLLARY. _If f is continuous in an interval I, then F has an indefinite integral in 1_.
_Proof_. Since the function (10) is an antiderivative of _f_ in _I_ , we have
where _C_ is an arbitrary constant.
Figure 4.
**4.24.** The fundamental theorem of calculus
The following key theorem reveals the connection between differential and integral calculus. At the same time, it gives us a powerful tool for evaluating definite integrals.
**a.** THEOREM **(Fundamental theorem of calculus)**. _If f is continuous in_ [ _a_ , _b_ ] _and if F is any antiderivative of f in_ [ _a_ , _b_ ], _then_
_Proof_. By the preceding theorem, the function
is an antiderivative of _f_ in _a_ , _b_ ]. Let _F_ be any other antiderivative of _f_ in [ _a_ , _b_ ]. Then, by [Theorem 3.52b,
Φ( _x_ ) = _F_ ( _x_ ) + _C_ ,
where _C_ is a constant. To determine _C_ , we note that
which implies
_C_ = − _F_ ( _a_ ).
Therefore
Φ( _x_ ) = _F_ ( _x_ ) + _C_ = _F_ ( _x_ ) − _f_ ( _a_ ).
Changing the dummy variable of integration from _t_ to _x_ , we then get
Note that formula (13) remains true for _b_ < _a_ , provided that _f_ is continuous in [ _b_ , _a_ ]. In fact, we then have
**b.** A little extra notation comes in handy here. Given any function _φ_ ( _x_ ) defined for _x_ = _a_ and _x_ = _b_ , let
denote the _difference φ_ ( _b_ ) − _φ_ ( _a_ ). With this notation, we can write (13) compactly as
Moreover, since
we can also write (14) as
This formula shows the connection between the definite and indefinite integrals of _f_ ( _x_ ) very explicitly.
**4.25. Examples**
**a.** Evaluate
SOLUTION. It follows from (14) and formula (5), p. 119, that
if _r_ ≠ −1. Choosing _r_ = 0 and _r_ = 1 in (15), we immediately get the results of Examples 4.13b and 4.13c. The interval [ _a_ , _b_ ] (or [ _b_ , _a_ ] if _b_ < _a_ ) must not contain the point _x_ = 0 if _r_ is negative, since otherwise _x r_ will fail to be continuous in [ _a_ , _b_ ],
**b.** Suppose
_MC_ ( _Q_ ) = 3 _Q_ 2 − 100 _Q_ \+ 1200
is the marginal cost of producing a commodity at output level _Q_. Find the total cost function _C_ ( _Q_ ) if the overhead is 1000 money units. Express the cost (exclusive of overhead) of producing the second 10 units of the commodity as an integral, and evaluate it. Find the average cost (inclusive of overhead) of producing 20 units of the commodity.
SOLUTION. According to Sec. 3.22a,
Therefore
where _k_ is a constant of integration. But _k_ = _C_ (0) = 1000, and hence
_C_ ( _Q_ ) = _Q_ 3 − 50 _Q_ 2 \+ 1200 _Q_ \+ 1000.
The cost of producing the second 10 units is
while the average cost of producing 20 units is
PROBLEMS
**1.** Verify that
by direct calculation.
**.** Evaluate
**3.** Verify that
**.** Find the definite integral from 0 to 2 of the function
**.** Find the area _A_ between the curves _y_ = and _y_ = _x_ 2 (see Figure 5). Why is the part of the curve _y_ = _x_ 2 lying in the first quadrant the reflection of the curve _y_ = in the line _y_ = _x_?
**6.** Find the area between the line _x_ \+ _y_ = 2 and the curve _y_ = _x_ 2.
**.** According to formula (5),
where _c_ ∈ [1, 7]. Find _c_ if _f_ ( _x_ ) = _x_.
**8.** Verify that the average of the function _f_ ( _x_ ) = _x_ over the interval [ _a_ , _b_ ] is just the midpoint of the interval.
**.** Consider a particle with equation of motion _s_ = _s_ ( _t_ ). We now have two definitions of the average velocity of the particle over the interval [ _a_ , _b_ ], namely
Figure 5.
(Sec. 3.12a) and
(Sec. 4.22b), where _v_ = _v_ ( _t_ ) is the particle's instantaneous velocity. Show that the two definitions are equivalent.
**10.** Is it true that
**.** Why is the function Φ( _x_ ) defined by (10) continuous in the interval _I_?
**.** Show that if _σ_ ( _λ_ ) ≥ 0 and _σ_ ( _λ_ ) → _σ_ 0 as _λ_ → 0, then _σ_ 0 ≥ 0.
**.** Show that if _A_ ≤ _σ_ ( _λ_ ) ≤ _B_ and _σ_ ( _λ_ ) → σ0 as _λ_ → 0, then _A_ ≤ _σ_ 0 ≤ _B_.
**.** Evaluate
**.** Show that if _f_ is continuous in [ _a_ , _b_ ] and if _A_ ≤ _f_ ( _x_ ) ≤ _B_ for every _x_ ∈ [ _a_ , _b_ ], then
**16.** Show that if _f_ is continuous and nonnegative in [ _a_ , _b_ ], that is, nonnegative at every point _x_ ∈ [ _a_ , _b_ ], then
**.** Show that if _f_ 1 and _f_ 2 are continuous in [ _a_ , _b_ ] and if _f_ 1( _x_ ) ≤ _f_ 2( _x_ ) for every _x_ ∈ [ _a_ , _b_ ], then
***18.** Show that if _f_ is continuous and nonnegative in [ _a_ , _b_ ], and if _f_ is nonzero at some point _c_ ∈ [ _a_ , _b_ ], then
***19.** Let _f_ be continuous and nonnegative in [ _a_ , _b_ ], and suppose that
Show that _f_ has the constant value 0 in [ _a_ , _b_ ].
***20.** Let _f_ 1 and _f_ 2 be the same as in Problem 17, and suppose that in addition _f_ 1 _f_ 2 in [ _a_ , _b_ ], that is, suppose that _f_ 1( _c_ ) ≠ _f_ 2( _c_ ) for at least one point _c_ ∈ [ _a_ , _b_ ]. Show that
***21.** Verify that
***22.** Show that if _f_ is continuous in [ _a_ , _b_ ], then
***23.** Show that we can always choose the point _c_ in formula (5) to be an _interior_ point of [ _a_ , _b_ ], that is, a point of ( _a_ , _b_ ).
4.3 THE LOGARITHM
**4.31.** One of the most important functions in mathematics is the _natural logarithm_ , or simply the _logarithm_ , denoted by ln _x_ and defined for all positive _x_ by the formula
The expression on the right is just the definite integral from the fixed point _t_ = 1 to the variable point _t_ = _x_ of the function 1/ _t_. In geometrical terms, if _x_ > 1, then ln _x_ is the area under the curve _y_ = 1/ _t_ from _t_ = 1 to _t_ = _x_ , as in Figure 6A. If 0 < _x_ < 1, then, since
ln _x_ is the _negative_ of the area under the curve _y_ = 1/ _t_ from _t_ = _x_ to _t_ = 1, as in Figure 6B. Thus ln _x_ > 0 if _x_ > 1, while ln _x_ < 0 if 0 < _x_ < 1. Moreover,
Since
Figure 6.
**4.32.** It follows from Theorem 4.23a that the function ln _x_ is _differentiable_ in the interval (0, ∞), with derivative
In particular, this shows that ln _x_ is _continuous_ in (0, ∞), for the reason given in Sec. 2.66. Moreover, the function ln _x_ is _increasing_ in (0, ∞). In fact, suppose that 0 < _x_ 1 < _x_ 2 < ∞. Then
Applying the mean value theorem for integrals to the last integral on the right, we get
where _x_ 1 ≤ _c_ ≤ _x_ 2, so that
In other words, _x_ 1 < _x_ 2 implies ln _x_ 1 < ln _x_ 2, so that ln _x_ is increasing, as claimed.
**4.33.** The next property of ln _x_ is so important that it deserves a theorem of its own:
**a.** THEOREM. _Let a and b be arbitrary positive numbers. Then_
_Proof_. Using the chain rule to differentiate the composite function ln ( _ax_ ), we find that
Therefore both functions ln ( _ax_ ) and ln _x_ have the same derivative 1/ _x_. In other words, ln ( _ax_ ) and ln _x_ are both antiderivatives of 1/ _x_. It follows from Theorem 3.52b that
where _C_ is a constant. To determine _C_ , we set _x_ = 1 in (5), obtaining
ln _a_ = ln 1 + _C_ = _C_ ,
with the help of (2). Therefore (5) becomes
ln ( _ax_ ) = ln _x_ \+ ln _a_.
Setting _x_ = _b_ in this formula, we immediately get (4).
**b.** In particular, (4) implies
ln ( _a_ 2) = ln _a_ \+ ln _a_ = 2 ln _a_ ,
ln ( _a_ 3) = ln ( _a_ 2) + ln _a_ = 2 ln _a_ \+ ln _a_ = 3 ln _a_ ,
and, more generally,
where, as always,
It also follows from (2) and (4) that
so that
Therefore
Note that (6) holds for every integer, positive, negative or zero, if we make the usual definitions
In fact,
by (6) and (7), while
ln _a_ 0 = ln 1 = 0 = 0 . ln _a_ ,
by (2).
**c.** Choosing _a_ = 2, say, in formula (6), we get
ln (2 _n_ ) = _n_ ln 2,
where ln 2 > 0. Given any positive number _M_ , no matter how large, let _n_ be any integer greater than _M_ /ln 2. Then
ln _x_ > ln (2 _n_ ) = _n_ ln 2 > _M_
whenever _x_ > 2 _n_ , since ln _x_ is increasing. This means that
(Sec. 2.91b). Moreover,
since, by Sec. 2.91c,
where we use (7) and (8).
According to (8) and (9), ln _x_ takes "arbitrarily large" values of both signs. Since ln _x_ is continuous, it follows from the intermediate value theorem (Sec. 3.33b) that ln _x_ takes _every_ value. In other words, the range of the function ln _x_ is the whole interval (−∞, ∞).
**4.34.** Let _e_ be the number such that
or equivalently
so that the area under the curve _y_ = 1 _/t_ from _t_ = 1 to _t_ = _e_ is precisely 1. The number _e_ , called the _base of the natural logarithms_ , is a constant of great importance in calculus and its applications. It turns out that _e_ is irrational and equals
_e_ = 2.7182818284...
As we will see in Sec. 4.51a,
that is, _e_ is the limit of the sequence
**4.35.** Figure 7 shows the graph of the function _y_ = ln _x_. It is apparent from the figure that ln _x_ is increasing in (0, ∞), has the range (−∞, ∞), and satisfies formulas (2) and (10). Note also that ln _x_ is a one-to-one function, like every increasing function (Sec. 2.3, Prob. 15), and has the _y_ -axis as its only asymptote. Moreover, ln _x_ is concave downward in the whole interval (0, ∞), by Sec. 3.72, Proposition (2), since
for every _x_ ∈ (0, ∞).
Figure 7.
**4.36. The function log _a_ _x_**
**a.** We now introduce the function log _a_ _x_ , where _x_ is any positive number and _a_ is positive but different from 1. This function, called the _logarithm to the base a_ , is defined by the formula
and has properties very similar to those of the function ln _x_ , to which it reduces for _a_ = _e_. For example, it follows at once from (12) that
with the help of (2), (4) and (7). For _a_ = 10, we get the _common logarithm_ log10 _x_ of elementary mathematics, usually denoted by log _x_ without the subscript 10.
**b.** If _a_ and _b_ are two positive numbers different from 1, then
so that
Setting _x_ = _a_ in (13), we find that
1 = log _a_ _b_ · log _b_ _a_ ,
or equivalently
In particular, choosing _b_ = _e_ in (14), we get
**c.** The derivative of the function log _a_ _x_ is easily calculated. In fact,
with the help of (15).
PROBLEMS
**1.** Why can't the integral defining ln _x_ be evaluated by using formula (15), p. 150?
**.** Are the functions ln ( _x_ 2) and 2 ln _x_ identical?
**.** Find the domain of
(a) ;(b)ln (ln _x_ );(c) ln (ln (ln (ln _x_ ))).
**.** Differentiate
(a)ln ( _x_ 3 − 2 _x_ \+ 5);(b) _x_ ln _x_ ;(c)(ln _x_ )3;(d)ln (ln _x_ ).
**.** Differentiate
**.** According to the mean value theorem (Sec. 3.43a),
_f_ (2) − _f_ (1) = _f_ ′( _c_ ),
where 1 < _c_ < 2. Find _c_ if _f_ ( _x_ ) = ln _x_.
**7.** Show that the tangent to the curve _y_ = ln _x_ at the point _e_ goes through the origin.
**.** What is the fourth derivative of the function _y_ = _x_ 2 ln _x_?
**9.** Verify that
∫ ln _x dx_ = _x_ ln _x_ − _x_ \+ _C_.
**.** Find the area of the region bounded by the _x_ -axis, the line _x_ = _e_ and the curve _y_ = ln _x_.
**11.** In which intervals is the function _x_ 2 − ln ( _x_ 2) increasing? Decreasing?
**.** Where does the function _x_ − ln _x_ have its global minimum in (0, ∞)? Does it have a maximum in (0, ∞)?
**13.** Find the inflection points and investigate the concavity of the function ln (1 + _x_ 2).
***14.** Find the domain of
(a) ;(b)log10 (1 − log10 ( _x_ 2 − 5 _x_ \+ 16)).
***15.** Show that
if _a_ > 1, while
if 0 < _a_ < 1.
***16.** Verify that the function is odd.
***17.** Use the mean value theorem to verify that
if 0 < _a_ < _b_.
***18.** Show that
4.4 THE EXPONENTIAL
**4.41. a.** The logarithm function
defined in the preceding section, has domain (0, ∞) and range (−∞, ∞). Moreover, it is increasing, one-to-one and continuous in the whole interval (0, ∞), and hence in every closed subinterval _a_ , _b_ ] (0, ∞). Therefore, by the proposition cited in [Sec. 2.81c, the inverse function
is increasing and continuous in the interval [ln _a_ , ln _b_ ]. But α = ln _a_ → −∞ as _a_ → 0+, while _β_ = ln _b_ → ∞ as _b_ → ∞, by Sec. 4.33c. Therefore the function (1) is increasing and continuous in every closed subinterval [α, _β_ ] (−∞, ∞), and hence in the whole interval (−∞, ∞).
**b.** In studying the function (1), it is natural to preserve the custom of denoting the independent variable by _x_ and the dependent variable by _y_. Thus we now write
_y_ = ln−1 _x_ ,
instead of (1). This function, which is one of the most important in mathematics, deserves a name and notation of its own. It is called the _exponential to the base e_ , or simply the _exponential_ , and is denoted by
_y_ = exp _x_.
Another, even more common notation for the exponential will be introduced in a moment. The function exp _x_ is defined and _positive_ for all _x_. This follows from the fact that the range of exp _x_ is just the domain of ln _x_ , namely the interval (0, ∞).
**c.** Being the inverse of the function ln _x_ , the exponential satisfies the formulas
To see this, we use the formulas (3), p. 43, changing _y_ to _x_ in the second formula, and writing ln for _f_ and exp for _f_ −1. It follows from the formulas
ln 1 = 0, ln _e_ = 1
that
and
where _e_ is the number introduced in Sec. 4.34.
**4.42.** The following theorem expresses one of the key properties of the exponential:
**a.** THEOREM. _Let x and y be arbitrary real numbers. Then_
_Proof_. Let _a_ = exp _x_ , _b_ = exp _y_ , so that _x_ = ln _a_ , _y_ = ln _b_. Then
_x_ \+ _y_ = ln _a_ \+ ln _b_ = ln ( _ab_ ),
by Theorem 4.33a, and hence
exp ( _x_ \+ _y_ ) = _ab_ = (exp _x_ )(exp _y_ ).
**b.** In particular, (4) and (5) together imply
exp (2) = (exp l)(exp 1) = _e_ . _e_ = _e_ 2,
exp (3) = (exp 2)(exp 1) = _e_ . _e_ = _e_ = _e_ 3,
and, more generally,
Since exp _x_ coincides for _x_ = _n_ with _e n_, the nth power of the number _e_ , it is natural to write
even when _x_ is a real number rather than a positive integer. Thus (7) is to be regarded as the _definition_ of the function _e x_, but one which is particularly appropriate, because of (6). In terms of this notation, formulas (3) and (4) become
while (5) takes the form
Choosing _y_ = − _x_ in (9), we get
_e_ _x_ _e_ − _x_ = _e_ _x_ − _x_ = _e_ ° = 1,
so that
This is in keeping with the usual definition of negative powers for the case where _x_ is a positive integer.
**4.43. a.** Given any positive number _M_ , no matter how large, we have
_e_ _x_ > _e_ ln _M_ = _M_
whenever _x_ > ln _M_ , since _e_ _x_ is increasing. It follows that
Moreover,
with the help of (10). Therefore, by (11),
for the reason given in Sec. 2.91c.
**b.** To differentiate the function _e x_, we use Theorem 2.81a, noting that all the conditions of the theorem are satisfied (check this). Thus, writing _y_ = _e_ _x_ , _x_ = ln _y_ , we have
so that
As this formula shows, the function _e x_ has the remarkable property of being equal to its own derivative, and therefore of being unaffected by any number of differentiations. Thus, for example,
**c.** Figure 8 shows the graph of the function _e x_. It is apparent from the figure that _e x_ is increasing in (−∞, ∞), has the range (0, ∞), and satisfies the formulas (8). Note also that _e x_ has the _x_ -axis as its only asymptote, and is concave upward in the whole interval (−∞, ∞), by Sec. 3.72, Proposition (1), since
for every _x_ ∈ (−∞, ∞).
**4.44. The function _a_ _x_**
**a.** We now introduce the function _a x_, where _a_ is positive and _x_ is arbitrary. This function, called the _exponential to the base a_ , is defined by the formula
Figure 8.
and has properties very similar to those of the function _e x_, to which it reduces for _a_ = _e_. For example, it follows at once from (14) that
with the help of (8), (9) and (10). In particular, if _n_ is a positive integer,
so that _a x_ coincides for _x_ = _n_ with the _n_ th power of the number _a_ , as we would expect from the notation.
Taking the logarithm of both sides of (14), we find that
ln ( _a x_) = ln ( _e_ _x_ ln _a_ ).
Therefore
which generalizes formula (6), p. 154, from the case _x_ = _n_ , where _n_ is an integer, to the case of arbitrary real _x_.
**b.** The following proposition gives another key property of _a x_:
THEOREM. _Let a be positive, and let x and y be arbitrary real numbers. Then_
_Proof_. By (14),
( _e_ _x_ ) _y_ = _e_ _y_ ln _e_ _x_ = _e_ _yx_ = _e_ _xy_ ,
so that (16) holds for _a_ = _e_. Therefore
( _a_ _x_ ) _y_ = ( _e_ _x_ ln _a_ ) _y_ = _e_ _xy_ ln _a_ = _a_ _xy_.
**c.** The derivative of the function _a x_ is easily calculated. In fact,
with the help of (13) and the chain rule.
**4.45. The function _x r_**
**a.** Let _x_ be positive, and let _r_ be an arbitrary real number. Then, changing _a_ to _x_ and _x_ to _r_ in (14), we get the formula
_defining_ the rth power of _x_. The function _x r_ has the domain (0, ∞) and is continuous, being a continuous function of a continuous function (Sec. 2.82d). It follows from (17) that
with the help of (10). Similarly, if _r_ and _s_ are arbitrary real numbers, then
( _x_ _r_ ) _s_ = ( _e_ _r_ ln _x_ ) _s_ = _e_ _rs_ ln _x_ ,
because of (16), with _a_ = _e_ , so that
while
_x_ _r_ \+ _s_ = _e_ ( _r_ \+ _s_ ) ln _x_ = _e_ _r_ ln _x_ \+ _s_ ln _x_ = _e_ _r_ ln _x_ _e_ _s_ ln _x_ ,
because of (9), so that
**b.** Suppose _r_ is a rational number _m_ / _n_ , possibly an integer. Then the function (17), which is defined only for _x_ > 0, coincides in (0, ∞) with the function _x m/n_ introduced in Sec. 2.7, Problems 5–7, which may well be defined for all _x_ ≥ 0 or even for all _x_. This follows at once from formulas (18) and (19), and the way the function _x m/n_ was previously defined. The merit of (17) is, of course, that we are now able to define _x r_ for _irrational r_. Another virtue of (17) is that we can now _construct_ the _n_ th root , a number whose existence has so far been tacitly _assumed_. In fact, if _x_ > 0, then is simply the perfectly well-defined number.
_x_ 1/ _n_ = _e_ (1/ _n_ ) ln _x_.
If _n_ is odd, we then set , and the definition of is completed by noting that 0 _n_ = 0 for every positive integer _n_ , whether odd or even, so that .
There is nothing to prevent us from _defining_ 0 _r_ = 0 for a positive irrational number _r_. In fact, this is the only sensible definition, since if _r_ > 0, then _r_ ln _x_ → −∞ as _x_ → 0+, by formula (9), p. 155, so that _x_ _r_ = _e_ _r_ ln _x_ → 0 as _x_ → 0+, by formula (12).
**c.** We are now ready to prove the key formula (14), p. 78.
THEOREM. _Let r be an arbitrary real number. Then_
_Proof_. By (13) and the chain rule,
with the help of (18) and (20).
Thus we have at last proved all the formulas given in Sec. 2.74e.
PROBLEMS
**1.** Verify that the graph of the function _ke x_ ( _k_ > 0) can be obtained by shifting the graph of _e x_ along the _x_ -axis.
**.** Differentiate
(a) _e_ 4 _x_ \+ 5;(b) _e_ −3 _x_ ;(c) _xe_ _x_ ;(d) _e x_(1 − _x_ 2).
**.** Differentiate
**.** According to the mean value theorem (Sec. 3.43b),
_f_ (1 + Δ _x_ ) − _f_ (1) = _f_ ′(1 + αΔ _x_ ) Δ _x_ ,
where 0 < α < 1. Find α if _f_ ( _x_ ) = _e x_, Δ _x_ = 1.
**5.** Verify that
**.** Find the area between the curves _y_ = _e x_ and _y_ = _e_ − _x_ from _x_ = ln 2 to _x_ = ln 3.
**7.** Find the local extrema of the function _y_ = ( _x_ \+ 1)10 _e_ − _x_.
**.** Find the inflection points and investigate the concavity of the function _y_ = _e_ − _x_ 2/2. Graph this function.
**.** How is the function _a x_ defined in Sec. 4.44a related to the function log _a_ _x_ defined in Sec. 4.36a?
**10.** Solve the equation 2 _x_ − 2 _x_ = 0.
**.** An advisor to a certain king was asked what he would like as a reward for interpreting one of the king's dreams. He asked for a chessboard with one grain of rice on the first square, twice as much rice on the second square as on the first, twice as much on the third square as on the second, and so on. Why did the king have his advisor executed for insolence?
**.** Differentiate
(a) _x_ 10 _x_ ;(b) _a_ _x_ 2;(c) _a_ _x_ _x_ _a_ ;
***13.** Show that
if _a_ > 1, while
if 0 < _a_ < 1.
***14.** Show that
if _r_ > 0, while
if _r_ < 0.
***15.** Find the inverse of the function
***16.** For what values of _c_ does the function _e x_ \+ _cx_ 3 have an inflection point?
**17.** Use differentiation to confirm that the fourteenth term of the sequence
is the largest.
4.5 MORE ABOUT THE LOGARITHM AND EXPONENTIAL
**4.51. The number _e_**
**a.** To derive formula (11), p. 156, we start from the fact that
Expressing this derivative as a limit, with _h_ instead of Δ _x_ for brevity, we have
so that
Therefore
with the help of formula (15), p. 162. Thus the function
is continuous at _h_ = 0. But then, by Sec. 2.82d, the composite function _e_ _f_ ( _h_ ) is also continuous at _h_ = 0, since _e x_ is continuous at _x_ = 1. It follows that
or equivalently
Suppose _h_ = 1/ _n_ , where _n_ is a positive integer. Then _h_ → 0 implies _n_ → ∞, and (2) takes the form
in agreement with formula (11), p. 156.
**b.** There is a more general formula
valid for an arbitrary real number _r_. (Note that (4) reduces to (3) for _r_ = 1.) The formula holds for _r_ = 0, since it then reduces to the trivial equality
Let _h_ = _r_ / _n_ , where _r_ ≠ 0. Then _h_ → 0+ as _n_ → ∞ if _r_ > 0, while _h_ → 0− as _n_ → ∞ if _r_ < 0. Therefore
where _f_ ( _h_ ) is the function (1). As already noted, the function _e_ _f_ ( _h_ ) is continuous at _h_ = 0, where it has the value _e_. Therefore the composite function [ _e_ _f_ ( _h_ )] _r_ is also continuous at _h_ = 0, since _x_ _r_ is continuous at _x_ = _e_. It follows that
which is equivalent to (4).
**4.52. Compound interest**
The following three examples explore this very practical topic, and show how the exponential function gets into the act:
**a. Example.** Suppose money invested at an annual interest rate _r_ , or equivalently at 100 _r_ percent, is compounded _N_ times a year. Show that a principal of _P_ dollars will grow to
dollars at the end of _t_ years. Show that
dollars must be invested now to become worth _A_ dollars in _t_ years.
SOLUTION. Let _A m_ be the amount in the bank at the end of the _m_ th interest period, assuming that no money is withdrawn after the initial deposit. Then
since the interest is computed on the accrued amount at a rate equal to _r_ , the nominal annual interest rate, divided by _N_ , the number of compoundings per annum. The initial amount _A_ 0 is, of course, just the principal _P_. Therefore the amount _A_ in the bank after _t_ years, that is, after _Nt_ interest periods, is equal to
which proves (5), since _A_ 0 = _P_. To get (6), we solve (5) for _P_ , obtaining
**b. Example.** Suppose interest is "compounded continuously," that is, suppose the number of compoundings _N_ per annum becomes "arbitrarily large." Show that formula (5) for the "compound amount" _A_ becomes
while formula (6) for the "present value" _P_ becomes
SOLUTION. Here we have
Let _n_ = _Nt_ , so that _N_ → ∞ implies _n_ → ∞. Then
with the help of (4). This proves (7). To get (8), we solve (7) for _P_ , obtaining
**c. Example.** Let _P_ = $1,000, _r_ = 6%, _t_ = 1 year. Test the accuracy of formula (7), as compared with the exact formula (5).
SOLUTION. The results are given in the following table for annual, semiannual, quarterly, monthly, daily, and continuous compounding (the last indicated by ∞):
**4.53. Logarithmic differentiation**
In calculating derivatives, it is often helpful to first take logarithms and then differentiate the result. The following two examples show the power of this technique, called _logarithmic differentiation_.
**a. Example.** Differentiate
SOLUTION. Taking logarithms, we get
Differentiation of (9) then gives
where the prime denotes differentiation with respect to _x_ and we repeatedly use trie chain rule. Multiplying by _y_ , we get the desired derivative
The quantity
is called the _logarithmic derivative_ of _y_.
**b. Example.** Differentiate
_y_ = _u_ _v_ ,
where _u_ and _v_ are both differentiable functions of _x_.
SOLUTION. By logarithmic differentiation, we have
Therefore
For example, if _y_ = _x_ _x_ , so that _u_ = _v_ = _x_ , then (10) reduces to
_y_ ′ = _x_ _x_ _x_ ′ ln _x_ \+ _xx_ _x_ −1 _x_ ′ = _x_ _x_ (ln _x_ \+ 1),
since _x_ ′ = ( _d_ / _dx_ ) _x_ = 1.
**4.54. Elasticity**
**a.** Next we introduce a concept of considerable interest in business and economics. Given a function _y_ = _f_ ( _x_ ), by the derivative of _y_ with respect to _x_ we mean, of course, the limit as Δ _x_ → 0 of the difference quotient
Let the _proportional change in y_ be defined by
Then the logarithmic derivative of _y_ with respect to _x_ is the limit as Δ _x_ → 0 of the ratio
Since
Similarly, defining the _proportional change in x_ by
we can go a step further and introduce the limit as Δ _x_ → 0 of the ratio
We then get a quantity
called the _elasticity_ of the function _y_ = _f_ ( _x_ ), at the point _x_. As a measure of the "change in _y_ due to a change in _x_ ," the elasticity has the merit of being independent of the units of _x_ and _y_ , which are divided out in forming the ratios (11) and (12). This is a great convenience in certain business problems, where, for example, _y_ might be the quantity of a commodity demanded at the price _x_. Changing the units of _x_ from dollars to pesos, say, or the units of _y_ from bushels to carloads, would then have no effect on the elasticity of the demand curve.
Note that we can also regard the elasticity _ε_ _yx_ as the limit
This follows at once from the fact that
**b.** There is another way of writing (13) as a kind of "double logarithmic derivative" of the form
where the expression on the right is the ratio of the differential of ln _y_ to the differential of ln _x_. To verify (14), we recall from Sec. 2.55a that if _y_ = _f_ ( _x_ ), then _dy_ , the differential of _f_ ( _x_ ), is given by the formula
_dy_ = _f_ ′( _x_ ) _dx_.
Therefore
While
by the chain rule, so that
as claimed.
**c. Example.** Suppose the demand for a commodity produced by a monopolistic firm is described by the function _Q_ = _Q_ ( _P_ ), where _Q_ is the quantity demanded at the price _P_. Then
is called the _elasticity of demand_ , at the price _P_. Do not be disconcerted by the extra minus sign in (15). Its sole purpose is to make the elasticity come out positive, in keeping with economic convention and in anticipation of the fact that a demand curve typically has negative slope (Sec. 3.2, Prob. 9). The demand is said to be _elastic_ if _ε_ _D_ > 1 and _inelastic_ if _ε_ _D_ < 1.
Suppose the demand function is
_Q_ = _Q_ ( _P_ ) = 60 − 3 _P_.
Find _ε_ _D_. When is the demand elastic? Inelastic? Express the firm's marginal revenue (Sec. 3.2, Prob. 7) in terms of the price and the elasticity of demand. Why should the firm adjust its price to keep the demand elastic?
SOLUTION. Here
so that the demand is elastic ( _ε_ _D_ > 1) if 10 < _P_ < 20 and inelastic ( _ε_ _D_ < 1) if 0 < _P_ < 10. The firm's total revenue is
_R_ ( _Q_ ) = _PQ_ ( _P_ ) = _QP_ ( _Q_ ),
as in Sec. 3.2, Problem 10, where _P_ ( _Q_ ) is the inverse function of _Q_ ( _P_ ). Correspondingly, the firm's marginal revenue is
where we use the fact that
by the rule for differentiating an inverse function (Sec. 2.81b). The firm certainly wants to operate at an output level where marginal revenue is positive, so that more revenue would be received if the output were increased slightly. It should therefore make _ε_ _D_ > 1 in (16). In other words, other things being equal, it should keep the demand elastic, by choosing a price in the range 10 < _P_ < 20.
PROBLEMS
**.** Evaluate
**.** Evaluate
**3.** It was shown in Sec. 4.51a that
Use this to show that
**.** Use (17) to show that
**5.** Use (18) to show that
**.** Use (17)−(19) to evaluate
**.** How much is a principal of $1,000 worth in 5 years if it is compounded quarterly at an annual interest rate of 8%?
**8.** A principal which is being compounded continuously doubles in 10 years. What is the annual interest rate?
**.** How long does it take $10,000 compounded continuously at an annual interest rate of 7% to grow to $25,000?
**10.** What amount of money must be invested now to be worth $10,000 in 5 years if compounded continuously at an annual interest rate of 6%?
**.** Justify calling
the _effective_ annual interest rate, as opposed to the _nominal_ annual rate _r_.
**.** Interpret the number _e_ in the language of finance.
**.** Use logarithmic differentiation to find the derivative of
**.** Differentiate
(a) _x_ _x_ 2;(b) _x_ 1/ _x_ ;(c)(ln _x_ ) _x_ ;(d) _e_ _x_ _x_.
**15.** Show that the function _y_ = _x_ _x_ has no inflection points.
**.** Verify the following "chain rule for elasticities":
_ε_ _zx_ = _ε_ _zy_ _ε_ _yx_.
**17.** What is the elasticity of the function _e_ _ax_?
**.** Given an example of a function with constant nonzero elasticity.
**19.** Show that if _f_ ( _x_ ) has elasticity _ε_ _yx_ , then _xf_ ( _x_ ) has elasticity 1 + _ε_ _yx_.
***20.** If _C_ = _C_ ( _Q_ ) is a firm's total cost function (Sec. 3.22), then
(without a minus sign) is called the _elasticity of cost_ , at the output _Q_. Verify that
(a) If _ε_ _C_ < 1 at a given output, then average cost ( _AC_ ) is greater than marginal cost ( _MC_ ), and average cost decreases as output increases;
(b) If _ε_ _C_ > 1 at a given output, then average cost is less than marginal cost, and average cost increases as output increases.
***21.** Consider the functions
and
called the _hyperbolic cosine_ of _x_ and the _hyperbolic sine_ of _x_ , respectively. Graph the functions , cosh _x_ and sinh _x_ in the same system of rectangular coordinates. Show that
(a) cosh _x_ ≥ 1 for all _x_ , cosh 0 = 1;
(b) sinh _x_ > 0 if _x_ > 0, sinh _x_ < 0 if _x_ < 0, sinh 0 = 0;
(c)
(d)
***22.** It can be shown that the number _e_ is the sum of the "rapidly convergent" series
By referring to the value of _e_ given in Sec. 4.34, show that the sum of just 8 terms of this series gives a value of _e_ which is accurate to 4 decimal places.
4.6 INTEGRATION TECHNIQUE
The technique of integration is inherently more difficult than that of differentiation. Thus, while it is no trick at all to become a minor expert on differentiation, it is the easiest thing in the world to write down integrals that would stump even a professional mathematician. However, two powerful methods of integration are available at the level of this course. We now discuss these methods and use them to evaluate a number of integrals which may appear quite intractable at first glance.
**4.61. Integration by substitution**
**a.** We begin by observing that if
∫ _g_ ( _t_ ) _dt_ = _G_ ( _t_ ) + _C_ ,
then
for every differentiable function _t_ = _t_ ( _x_ ). Here the common practice of denoting the dependent variable and the function by the same letter, _t_ in this case, is particularly appropriate. To verify (1), we merely note (as in Sec. 3.53c) that both sides of (1) have the same derivative. In fact,
by the very definition of the indefinite integral as an antiderivative of its integrand, while
by the chain rule and the fact that _G_ ( _t_ ) is an antiderivative of _g_ ( _t_ ).
Now suppose we want to evaluate an integral
which does not look like anything familiar, but which can be recognized as being of the form
in terms of some function _g_ ( _t_ ) of a new variable _t_ = _t_ ( _x_ ), where _g_ ( _t_ ) is a function which is more easily integrated than _f_ ( _x_ ) itself. Then it follows from (1) that
Integration by substitution is also known as integration by _change of variables_ , for a self-evident reason.
**b.** Recalling from Sec. 2.55a that the differential of _t_ is given by
we can write (3) simply as
∫ _g_ ( _t_ ) _dt_ ,
where it is understood that the substitution _t_ = _t_ ( _x_ ) will eventually be made. The fact that (2) and (3) are equivalent then takes the concise form
∫ _f_ ( _x_ ) _dx_ = ∫ _g_ ( _t_ ) _dt_.
The advantage of writing the expression behind an integral sign as a product of a function (the integrand) and a _differential_ is now apparent for the first time. In fact, if we change variables, then formula (5), or its analogue
for the case of a substitution _x_ = _x_ ( _t_ ), _automatically_ multiplies the old integrand by the appropriate "correction factor," without any need for a separate calculation involving the chain rule.
**c.** For example, to evaluate
let _t_ = 1 + _x_ 2, so that _dt_ = 2 _x dx_ , or equivalently _x dx_ = _dt_. Then
where the integral on the right can be recognized at once as being equal to ln _t_. Therefore, by (4),
after going back to the original variable _x_ and introducing a constant of integration _C_.
Once you have got the idea of how the technique of integration by substitution works, you can omit some of the intermediate steps, even leaving out explicit introduction of the auxiliary variable _t_. Thus a more concise way of evaluating (7) is
where the whole expression 1 + _x_ 2 is treated as a variable of integration.
**d.** To evaluate a _definite_ integral by substitution, we first evaluate the corresponding indefinite integral, and then use the fundamental theorem of calculus (Sec. 4.24a). Thus, for example,
There is also a more direct method of evaluating a definite integral by substitution (see Examples 4.62c and 4.62d).
**e.** Instead of recognizing (2) as being of the form (3), involving a differentiable substitution _t_ = _t_ ( _x_ ), we can also try making a differentiable substitution _x_ = _x_ ( _t_ ) directly in the integral (2), thereby "transforming" it into
with the aid of (6). Again, this will help only if the new integral (8) is easier to evaluate than the original integral (2), which means that the substitution _x_ = _x_ ( _t_ ) must be chosen intelligently.
**4.62. Examples**
**a.** Evaluate
SOLUTION. The substitution _x_ = _t_ 2 seems a good choice, since it gets rid of the radical. With this substitution, = _t_ ,
so that _dx_ = 2 _t dt_. Therefore
or
after going back to the original variable _x_ and introducing a constant of integration _C_. The expression can be replaced by , if you prefer.
Alternatively, you might recognize that
is of the form
if _t_ = , but this requires a good eye. The fact that each of the functions _x_ = _t_ 2 and _t_ = is the inverse of the other is, of course, no accident (see Prob. 18).
**b.** Evaluate
SOLUTION. If _t_ = ln _x_ , then _dt_ = _dx_ / _x_ and
On the other hand, if we choose the "inverse substitution" _x_ = _e t_, then _dx_ = _e t dt_ and
and we get the same answer again.
**c.** Evaluate
SOLUTION. Using (9), we have
On the other hand, suppose that instead of first evaluating the indefinite integral (9), we try to calculate (10) from scratch. Then, just as before, we observe that the substitution _t_ = ln _x_ reduces the expression behind the integral sign in (10) to _t dt_. This suggests writing
But what are the appropriate limits of integration α and _β_? The answer is simple enough: As the variable _x_ varies from 1 to _e_ in the left side of (11), the variable _t_ = ln _x_ varies from ln 1 = 0 to ln _e_ = 1 in the right side, and hence α = 0, _β_ = 1. This is not only plausible, but perfectly correct, as shown in Problem 19. Therefore we can write
without bothering to calculate the indefinite integral (9). Note that there is now no need to return from _t_ to the original variable _x_. In fact, once the second integral in (12) has been evaluated, the first integral is automatically known, since both are _definite_ integrals and therefore _numbers_.
**d.** Starting from the definition
give another proof of the formula
ln ( _ab_ ) = ln _a_ \+ ln _b_ ( _a_ , _b_ positive),
already established in Theorem 4.33a.
SOLUTION. We have
where in the last step we go over to a new variable _u_ = _at_ and make the corresponding change in the limits of integration. Therefore
after returning to the original dummy variable _t_. It follows that
with the help of Theorem 4.21c.
**4.63. Integration by parts**
**a.** We now consider another important integration technique. Let _u_ ( _x_ ) and _v_ ( _x_ ) be two differentiable functions such that _u_ ′( _x_ ) _v_ ( _x_ ) and _u_ ( _x_ ) _v_ ′( _x_ ) both have antiderivatives. Differentiating the product _u_ ( _x_ ) _v_ ( _x_ ) with respect to _x_ , and omitting arguments for simplicity, we have
( _uv_ )′ = _u_ ′ _v_ \+ _uv_ ′,
so that
Multiplying (13) by _dx_ and integrating with respect to _x_ , we get
∫ _uv_ ′ _dx_ = ∫ ( _uv_ )′ _dx_ − ∫ _vu_ ′ _dx_.
But
∫ ( _uv_ )′ _dx_ = _uv_ \+ _C_ ,
and hence
∫ _uv_ ′ _dx_ = _uv_ − ∫ _uv_ ′ _dx_ ,
where _C_ is absorbed into the other constants of integration. In terms of the differentials
_du_ = _u_ ′ _dx_ , _dv_ = _v_ ′ _dx_ ,
(13) takes the even simpler form
Equation (14), called the formula for _integration by parts_ , is well worth memorizing. It is one of the most valuable tricks of the trade, often allowing us to express difficult integrals in terms of easy ones.
**b.** To find a corresponding formula for definite integrals, we merely observe that
(justify the last step). Therefore
**4.64. Examples**
**a.** Evaluate
∫ ln _x dx_.
SOLUTION. This integral is of the form ∫ _u dv_ if we choose _u_ = ln _x_ , _dv_ = _dx_. We then have _du_ = _dx_ / _x_ , _v_ = _x_ , and hence, by (14),
∫ ln _x dx_ = _x_ ln _x_ − ∫ _dx_ = _x_ ln _x_ − _x_ \+ _C_ ,
where a constant of integration is supplied in the last step. It would be pointless to include a constant of integration _C_ in going from _dv_ to _v_ , since _C_ would be cancelled out automatically in the expression _uv_ − ∫ _v du_.
**b.** Evaluate
∫ _x_ ln _x dx_.
SOLUTION. There are various possibilities here. We can choose _u_ = _x_ , _dv_ = ln _x dx_ , or _u_ = ln _x_ , _dv_ = _x dx_ , or even _u_ = _x_ ln _x_ , _dv_ = _dx_. The only good choice is _u_ = ln _x_ , _dv_ = _x dx_ , since only this choice makes ∫ _v du_ simpler than ∫ _u dv_ , which is the whole point of integration by parts. We then have _du_ = _dx_ / _x_ , , and hence, by (14),
**c.** Evaluate
SOLUTION. We can start from (16), writing
Alternatively, we can start from (15), writing
**d.** Evaluate
∫ _x_ 2 _e_ _x_ _dx_.
SOLUTION. Let _u_ = _x_ 2, _dv_ = _e_ _x_ _dx_. Then _du_ = 2 _x dx_ , _v_ = _e x_, and therefore
To evaluate the integral on the right, we integrate by parts _again_ , this time choosing _u_ = _x_ , _dv_ = _e x dx_, _du_ = _dx_ , _v_ = _e x_:
Substituting (18) into (17) and supplying a constant of integration, we find that
You should get into the habit of checking formulas like this by differentiating the expression on the right. In this case,
which confirms (19).
PROBLEMS
**.** Let _f_ ( _x_ ) be continuous and even in [− _a_ , _a_ ]. Show that
**2.** Let _f_ ( _x_ ) be continuous and odd in [− _a_ , _a_ ]. Show that
**3.** Verify that
where _m_ and _n_ are positive integers.
**.** Use integration by substitution to evaluate
**.** Show that
**.** Use (21) to evaluate
**7.** Find the average of the function _y_ = ln _x_ over the interval [1, _e_ ].
**.** Use integration by parts to evaluate
**.** Use integration by parts to verify that
**.** Evaluate
**.** Evaluate
∫ (2 _x_ \+ 3 _x_ )2 _dx_ ,
after first showing that _a xbx_ = ( _ab_ ) _x_.
**.** Verify that
**.** Use (22) to evaluate
**.** Find the area between the curves _y_ = ln _x_ and _y_ = ln2 _x_.
***15.** Show that
***16.** Use the substitution to show that
***17.** Evaluate
***18.** Given two functions _f_ ( _x_ ) and _g_ ( _t_ ), with antiderivatives _F_ ( _x_ ) and _G_ ( _t_ ), let _t_ = _t_ ( _x_ ) be a differentiable one-to-one function, with a differentiable inverse _x_ = _x_ ( _t_ ), such that one of the formulas
holds. Show that the other formula also holds.
***19.** Let
∫ _f_ ( _x_ ) _dx_ = ∫ _g_ ( _t_ ) _dt_
be shorthand for both of the formulas (23), depending on whether we replace _dt_ by _t_ ′( _x_ ) _dx_ or _dx_ by _x_ ′( _t_ ) _dt_. Show that
***20.** Evaluate the integral (20).
***21.** Let _P_ ( _x_ ) be a polynomial of degree _n_. Verify that
***22.** Let the quantity of a commodity demanded by the market at price _P_ be _Q_ = _Q_ ( _P_ ), where _Q_ ( _P_ ) is a decreasing function, and let _P_ 1 be the actual market price. Then the total revenue received from the sale of the commodity is
where _Q_ 1 = _Q_ ( _P_ 1) and _P_ = _P_ ( _Q_ ) is the inverse of the function _Q_ = _Q_ ( _P_ ). Since some consumers are willing to pay more than _P_ 1 for the commodity, the total revenue from the sale of a quantity _Q_ 1 of the commodity would be greater than (24) by some amount _S_ , known as the _consumer's surplus_ , if the price of the commodity were gradually lowered from _P_ 0, the price at which demand just begins, to the actual market price _P_ 1. Show that
or equivalently
***23.** Use both (25) and (26) to calculate the consumer's surplus _S_ for the demand function _Q_ = 100 ln ( _P_ 0/ _P_ ) and market price _P_ 1 = _P_ 0/ _e_.
4.7 IMPROPER INTEGRALS
**4.71. a.** In introducing the concept of the definite integral
it was assumed from the outset that _f_ ( _x_ ) is defined at every point of the closed interval _a_ , _b_ ], where, of course, _a_ and _b_ are finite numbers ([Sec. 1.64a). Thus, at this stage of the game, neither of the integrals
and
makes sense, the first because the upper limit of integration is infinite, the second because the integrand is not defined at _x_ = 0 and in fact approaches infinity as _x_ → 0+ . However, there is a simple way of ascribing meaning to both of these integrals, which are called "improper" to distinguish them from the ordinary or "proper" integrals considered up to now. As we will see in a moment, the device is essentially the same in both cases: First we calculate the integral over a _finite_ interval in which the integrand is well-defined, and then we take the limit of the resulting _proper_ integral as the interval of integration is suitably enlarged. For simplicity, we will consider only continuous integrands.
**b.** Suppose _f_ ( _x_ ) is continuous, and hence integrable (Sec. 4.14) in every interval [ _a_ , _X_ ], where _a_ is fixed and _X_ > _a_ is variable, and suppose the limit
exists and is finite. Then the improper integral
is said to be _convergent_ and is assigned the value (3). On the other hand, if the limit (3) is infinite or fails to exist, we call the integral (4) _divergent_ and assign it no value at all.
Similarly, suppose _f_ ( _x_ ) is continuous in every interval [ _X_ , _a_ ], where _a_ is fixed and _X_ < _a_ is variable, and suppose the limit
exists and is finite. Then the improper integral
is said to be _convergent_ and is assigned the value (5). Again we call the integral (6) _divergent_ if the limit (5) is infinite or fails to exist. We can also consider the case where _both_ limits of integration are infinite. Thus, suppose _f_ ( _x_ ) is continuous in every finite interval, and suppose both improper integrals
are convergent for an arbitrary finite point _c_. Then the improper integral
is said to be _convergent_ , and is assigned the value
Of course, this definition depends on the fact that both the sum (9), and the convergence or divergence of the integrals (7), are independent of the choice of the point _c_ (see Prob. 5). On the other hand, the integral (8) is said to be _divergent_ if either of the integrals (7) is divergent.
**c.** We now turn to improper integrals of the type (2), where the integrand becomes infinite at one or more points of the interval of integration. Suppose _f_ ( _x_ ) approaches infinity as _x_ → _a_ +, at the same time that _f_ ( _x_ ) is continuous, and hence integrable, in every interval [ _a_ \+ _ε_ , _b_ ], where _a_ and _b_ > _a_ are fixed and _ε_ > 0 is variable (but less than _b_ − _a_ ). Suppose further that the limit
exists and is finite. Then the improper integral
is said to be _convergent_ and is assigned the value (10). As before, we call the integral (11) _divergent_ if the limit (10) is infinite or fails to exist. Similarly, if _f_ ( _x_ ) becomes infinite at the other end point _b_ , we set
by definition, while if _f_ ( _x_ ) becomes infinite at an _interior_ point _c_ ∈ ( _a_ , _b_ ), we write
provided that both integrals on the right are convergent.
**4.72. Examples**
**a.** The integral (1) is convergent. In fact
and hence
This number can also be regarded as the area of the "infinite region" under the curve _y_ = 1/ _x_ 2 from 1 to ∞, by a natural extension of the definition of the area under a curve for the finite case (Sec. 4.11).
**b.** The integral
is divergent, since
**c.** The integral
is convergent. In fact,
since _e_ − _X_ → 0 as _X_ → ∞, and similarly
Therefore, as in (9),
**d.** The integral (2) is convergent. In fact,
and hence
**e.** The integral
is divergent, since
**f.** The integral
is convergent. In fact, using (12) with _c_ = 1, we get
(let _t_ = 1 − _x_ , _u_ = _x_ − 1). But both integrals on the right equal 2, by formula (13), and hence
**g.** Since the integral (14) is divergent, so is the integral
(why?). Suppose we make the mistake of calculating this integral formally, ignoring the fact that the integrand becomes infinite at the origin. Then we get the absurd result
seemingly an example of a positive function with a negative integral!
PROBLEMS
**.** Investigate the improper integral
for arbitrary _r_.
**.** Evaluate
**.** Investigate the improper integral
for arbitrary _r_.
**.** Evaluate
**.** Let _f_ ( _x_ ) be continuous in every finite interval. Show that the two integrals
are either both convergent or both divergent, and similarly for
Show that
Show that the sum (12) is also independent of _c_.
**.** How does the theory of improper integrals resemble the theory of infinite series?
**.** First define and then find the area _A_ between the curves _y_ = _x_ −1/2 and _y_ = _x_ −1/3 from _x_ = 0 to _x_ = 1.
***8.** First define and then find the area _A_ between the curves _y_ = cosh _x_ and _y_ = sinh _x_ in the first quadrant.
_Chapter 5_
INTEGRATION AS A TOOL
5.1 ELEMENTARY DIFFERENTIAL EQUATIONS
**5.11.** The equations
are all called _differential equations_ , because each contains at least one derivative
of an "unknown" function _y_ = _f_ ( _x_ ). Note that the second equation does not contain the function _y_ itself, while the third equation does not contain the independent variable _x_ , although _x_ is, of course, present implicitly as the argument of the function _y_ = _f_ ( _x_ ) and its fourth derivative _y_ (4) = _f_ (4)( _x_ ). A differential equation is said to be _of order n_ if it contains the _n_ th derivative _y_ ( _n_ ) = _f_ ( _n_ )( _x_ ), but no derivatives of higher order. Thus the equations (1) are of orders 1, 2 and 4, respectively.
In this book we have no intention of doing more than scratch the surface of the vast subject of differential equations and their applications. In fact, we will consider only the most elementary differential equations, of either the first or the second order.
**5.12. a.** Consider the _first-order differential equation_
where _F_ ( _x_ , _y_ ) is a function of two variables, which may reduce to a function of _x_ alone, to a function of _y_ alone, or even to a constant. By a _solution_ of (2) we mean any function _y_ = _φ_ ( _x_ ) such that
holds for all values of _x_ in some interval. We write _φ_ ( _x_ ) instead of _f_ ( _x_ ) here, because the solution is regarded as a "known" function.
For example, _y_ = _e_ _x_ 2 is a solution of the differential equation
since
in every interval. Moreover, if _C_ is an arbitrary constant, then _y_ = _Ce_ _x_ 2 is also a solution of (3), since
We call
the _general solution_ of (3), because every solution of (4) can be obtained from (4) by making a suitable choice of _C_ (see Example 5.13d).
More generally, let
be a solution of the differential equation (2), involving an "arbitrary constant" _C_ (which is temporarily variable!), and suppose every solution of (2) can be obtained from (5) by making a suitable choice of _C_. Then (5) is called the _general solution_ of (2), and each solution of (2), corresponding to a particular choice of _C_ in (5), is called a _particular solution_ of (2). For example, giving _C_ the values 0 and in (5), we get two particular solutions
_y_ ≡ 0,
of equation (3).
**b.** In Sec. 1.12 we posed two key problems of calculus. The meaning of the first problem was clarified in Sec. 2.42d. The second problem was originally stated in the following unsophisticated language:
(2) Given the rate of change of one quantity with respect to another, what is the relationship between the two quantities?
We are now in a position to restate this problem elegantly in more precise language:
(2′) Solve the differential equation _dy_ / _dx_ = _F_ ( _x_ , _y_ ).
Here, of course, _dy_ / _dx_ is the rate of change, and to give this rate of change we will in general have to know the values of _both_ variables _x_ and _y_.
**5.13. Examples**
**a.** The simplest first-order differential equation is
and its general solution is just
This follows at once from the meaning of the indefinite integral as the "general antiderivative" of _f_ ( _x_ ). Here it is better to write the general solution of (6) in the form
which makes the arbitrary constant of integration explicit. We then think of ∫ _f_ ( _x_ ) _dx_ as any _fixed_ antiderivative of _f_ ( _x_ ). We will follow this convention in any problem involving differential equations.
**b.** Find the particular solution of the differential equation
satisfying the condition
SOLUTION. First we use (7) to find the general solution of (8), obtaining
Then we use the condition (9) to determine the constant _C_ in (10). Thus
so that . Substituting this value of _C_ into (10), we get the desired particular solution
The fact that (11) satisfies both (8) and (9) is easily verified by direct calculation.
More generally, by an _initial condition_ for the differential equation (2), we mean a condition of the form
where _x_ 0 and _y_ 0 are given numbers (the word "initial" stems, by analogy, from the common situation where the independent variable is the _time_ ). If (2) has the general solution _φ_ ( _x_ , _C_ ), the particular solution of (2) satisfying (12) has the value of _C_ obtained by solving
_φ_ ( _x_ 0, _C_ ) = _y_ 0.
**c.** Solve the differential equation
subject to the initial condition (12).
SOLUTION. This equation, or the equivalent equation
with
is said to have _separated variables_. Multiplying (13) by _g_ ( _y_ ) _dx_ , we get
where the left side involves only the variable _y_ and the right side involves only the variable _x_ ; it is in this sense that the variables are "separated." To solve (14), we merely integrate both sides, obtaining
∫ _g_ ( _y_ ) _dy_ = ∫ _f_ ( _x_ ) _dx_ \+ _C_
(one constant of integration is enough), or
where _G_ ( _y_ ) is any antiderivative of _g_ ( _y_ ) and _F_ ( _x_ ) is any antiderivative of _f_ ( _x_ ). To determine _C_ , we impose the initial condition (12), which says that _y_ = _y_ 0 when _x_ = _x_ 0. Thus
_G_ ( _y_ 0) = _F_ ( _x_ 0) + _C_ ,
so that
_C_ = _G_ ( _y_ 0) − _F_ ( _x_ 0).
Substituting this expression for _C_ into (15), we get
The unique solution of the differential equation (13) satisfying the initial condition (12) is then obtained by solving (16) for _y_ as a function of _x_ , call it _y_ = _φ_ ( _x_ ). In the cases to be considered here, we will always be able to do this without difficulty.
The function _y_ = _φ_ ( _x_ ) determined by (16) clearly satisfies the initial condition (12), which is "built into" equation (16). To verify that _y_ = _φ_ ( _x_ ) actually satisfies the differential equation (13), we need only use the chain rule to differentiate (16) with respect to _x_. This gives
so that
which is equivalent to (13).
**d.** Find the general solution of the differential equation (3).
SOLUTION. To separate variables, we divide (3) by _y_ , obtaining
Multiplying (17) by _dx_ and integrating, we then get
or
where we denote the arbitrary constant of integration by _k_ , saving the symbol _C_ for later. In writing ln | _y_ |, we use the fact that if _y_ is negative, then ln | _y_ | = ln (− _y_ ) has the same derivative 1/ _y_ as ln _y_ (check this). Taking the exponential of both sides of (18), we find that
| _y_ | = _e_ _x_ 2 \+ _k_ = _e_ _k_ _e_ _x_ 2 = _Ce_ _x_ 2.
where _C_ = _e k_ is now an arbitrary _positive_ constant (why?). But | _y_ | can never vanish, since _Ce_ _x_ 2 > 0. Therefore _y_ is either positive or negative for all _x_. Thus we can take the vertical bars off | _y_ |, obtaining formula (4), by simply allowing _C_ to take arbitrary negative values, as well as arbitrary positive values. In dividing (3) by _y_ , we have tacitly assumed that _y_ is nonvanishing. Thus the solution _y_ ≡ 0 may have been lost in solving (17) instead of (3), and indeed it has, as we see at once by substituting _y_ ≡ 0 into (3). Therefore the general solution of (3) is obtained by allowing _C_ to take any value in (4), _including zero_.
**e.** The simplest _second-order_ differential equation is
Integrating (19), we get
where _C_ 1 is an arbitrary constant of integration and _F_ ( _x_ ) = ∫ _f_ ( _x_ ) _dx_ is any fixed antiderivative of _f_ ( _x_ ). Observe that (20) is now a _first-order_ differential equation, and is in fact of the form (6). Integrating (20) in turn, we get
where _C_ 2 is another arbitrary constant of integration. Thus the _general_ solution of the differential equation (19) involves _two_ arbitrary constants, and this is a characteristic feature of the general solution of a _second-order_ differential equation. Therefore, to single out a _particular_ solution of (19), we must impose _two_ initial conditions, since this will give us _two_ algebraic equations which we can solve for the _two_ constants _C_ 1 and _C_ 2..
**f.** Find the particular solution of the differential equation
satisfying the initial conditions
SOLUTION. Note that one initial condition involves the function _y_ , while the other involves its derivative _y_ ′. Integrating (21) twice with respect to _x_ , we get first
and then
Substituting the second of the conditions (22) into (23) and the first into (24), we find that
Solving for _C_ 1 and _C_ 2, we then obtain
Thus the particular solution of the differential equation (21) satisfying the conditions (22) is just
Instead of imposing one condition on _y_ and the other on _y_ ′ at the same point, we can impose two conditions on _y_ at two different points. In this case, the conditions are called _boundary conditions_ rather than _initial conditions_. For example, to find the particular solution of (21) satisfying the boundary conditions
we solve the equations
for _C_ 1 and _C_ 2. This gives
and leads to the particular solution
PROBLEMS
**.** Show that equation (2) is a special case of the even more general first-order differential equation Φ( _x_ , _y_ , _y_ ′) = 0, where Φ( _x_ , _y_ , _z_ ) is a function of three variables.
**2.** Find the particular solution of the differential equation _y_ ′ = − _y_ / _x_ satisfying the initial condition _y_ | _x_ = 2 = 1.
**.** Find the particular solution of the differential equation _y_ ′ + 2 _xy_ = 0 satisfying the initial condition _y_ | _x_ = 0 = 1.
**4.** Find the particular solution of the differential equations satisfying the initial condition _y_ | _x_ = _e_ = 1.
**.** Show that all but one of the solutions of the differential equation _y_ ′2 = 4 _y_ are given by the formula _y_ = ( _x_ \+ _C_ )2, where _C_ is an arbitrary constant. What is the extra solution?
**6.** Find the general solution of the differential equation _y_ ″ = ln _x_.
***7.** By making the preliminary substitutions _y_ ′ = _p_ , _y_ ″ = _p_ ( _dp_ / _dy_ ), find the particular solution of the differential equation _y_ ″ = 2 _y_ 3 satisfying the initial conditions _y_ | _x_ = 0 = 1, _y_ ′| _x_ = 0 = 1.
***8.** A differential equation of the form
is said to be _homogeneous_. Solve this equation by separation of variables, after making the substitution _y_ = _ux_.
***9.** Find the particular solution of the homogeneous differential equation _x_ \+ _y_ \+ _xy_ ′ = 0 satisfying the initial condition _y_ | _x_ = 1 = 0.
***10.** A curve goes through the point (−1, −1) and has the property that the _x_ -intercept of the tangent to the curve at every point _P_ is the square of the abscissa of _P_. What is the curve?
5.2 PROBLEMS OF GROWTH AND DECAY
**5.21. a.** Suppose the dependence of one variable, say _y_ , on another variable, say _t_ , is described by an "exponential law"
where _y_ 0 > 0 and _r_ are constants. Then the rate of change of _y_ (with respect to _t_ ) is given by the derivative
Thus _y_ satisfies the simple differential equation
that is, _the rate of change of the variable y is proportional to the value of y_. The function _e rt_ is an _increasing_ function of _t_ if _r_ is _positive_ , since then _t_ 1 < _t_ 2 implies _rt_ 1 < _rt_ 2 and hence _e_ _rt_ 1 < _e_ _rt_ 2; in this case, we say that _y grows exponentially_ (with _t_ ), or that _y_ is an _exponentially increasing_ function of _t_. On the other hand, _e rt_ is a _decreasing_ function of _t_ if _r_ is _negative_ , since then _t_ 1 < _t_ 2 implies _rt_ 1 > _rt_ 2 and hence _e_ _rt_ 1 > _e_ _rt_ 2; in this case, we say that _y decays_ (or _falls off_ ) _exponentially_ (with _t_ ) or that _y_ is an _exponentially decreasing_ function of _t_.
**b.** Setting _t_ = 0 in (1), we find that
Thus the constant _y_ 0 is just the _initial value_ of _y_ , that is, the value of _y_ at the time _t_ = 0. In other words, (1) is the particular solution of the differential equation (2) satisfying the initial condition (3). This can, of course, be seen directly. In fact, separating variables in (2), we get
so that
or
where _k_ is a constant of integration. Taking the exponential of both sides of (4), we find that
where _C_ = _e_ _k_ > 0. But | _y_ | can never vanish, since _Ce_ _rt_ > 0. Therefore _y_ is either positive or negative for all _t_. Since _y_ 0 > 0, by assumption, the initial condition (3) can be satisfied only if _y_ is positive, and then | _y_ | = _y_ , _C_ = _y_ 0, so that (5) reduces to(1).
**c.** It follows from (2) that
or equialently
that is, _r_ is the _logarithmic derivative_ of _y_ (Sec. 4.53a). Thus _r_ is not the rate of change of _y_ , but rather the rate of change of _y_ divided by the "current" value of _y_. The quantity _dy_ / _dt_ is called the _growth rate_ , whether it be positive or negative, while _r_ is called the _proportional growth rate_. The word "proportional" can be dropped if _r_ is given in _percent_ per unit time, since it is then clear that we can only be talking about a proportional growth rate. Sometimes _r_ is simply called the _rate_ , when there is no possibility of confusion.
**5.22. Population growth**
**a. Example.** A population grows exponentially at the rate _r_. How long does it take the population to double?
SOLUTION. Here _r_ is positive and we have
where _N_ (for "number") is the population at time _t_ and _N_ 0 is the population at time _t_ = 0. The function _N_ is, of course, just the particular solution of the differential equation
satisfying the initial condition
Let _T_ be the _doubling time_ , that is, the time at which the population is twice its initial value _N_ 0. Then
_N_ 0 _e_ _rT_ = 2 _N_ 0,
or
Note that _T_ is independent of the population size. Taking logarithms, we get
_rT_ = ln 2,
or
Thus the doubling time is inversely proportional to the proportional growth rate _r_ , which makes sense ("the faster the growth, the shorter the doubling time"). Suppose _r_ is measured in percent per year and _t_ in years, as in studies of human population. Then we get the rule of thumb
For example, the average annual growth rate of the population of Brazil during the period 1961–1968 was 3%. At this rate the population of Brazil will double in about years.
**b.** If _T_ is the doubling time of a population, then the population will double _every T_ years, as long as it is growing at the rate _r_. To see this, we merely note that
_N_ ( _T_ ) = _e_ _rT_ _N_ 0 = 2 _N_ 0,
_N_ (2 _T_ ) = _e_ _r_ 2 _T_ _N_ 0 = ( _e_ _rT_ )2 _N_ 0 = 22 _N_ 0 = 4 _N_ 0,
with the help of (6) and (9), and, more generally,
_N_ ( _nT_ ) = _e_ _nrT_ _N_ 0 = ( _e_ _rT_ ) _n_ _N_ 0 = 2 _n_ _N_ 0.
**c.** The differential equation (7) merely says that the rate of change of the population is proportional to the present size of the population. This is perfectly plausible. In fact, on the one hand, we must have
where _B_ is the _birth rate_ and _D_ the _death rate_. On the other hand, both _B_ and _D_ are proportional to the population size _N_ (large cities have more maternity wards and more cemeteries than small towns). Therefore _B_ − _D_ is also proportional to _N_. Comparing (7) and (7′), we find that
In other words, the proportional annual growth rate of population is just the per capita excess of the birth rate over the death rate.
**d.** Eventually, of course, population growth must stop, due to lack of food, spread of infectious disease, loss of fertility due to overcrowding, wars fought for dwindling resources, or whatever. It turns out that these effects of "overpopulation" are described remarkably well in many cases by introducing an extra term − _sN_ 2 in the right side of (7), where _s_ (like _r_ ) is a positive constant. The resulting "growth equation" then becomes
instead of (7), subject to the same initial condition (8).
To solve (11), we separate variables and integrate, obtaining
where _c_ is a constant of integration. The integral on the left is not hard to evaluate. In fact, setting α = _s_ / _r_ , we have
Therefore (12) becomes
where _k_ = _rc_ , or
where _C_ = _e k_. Applying the initial condition (8), we get
so that
where the vertical bars can now be dropped, since 1 − α _N_ and 1 − α _N_ 0 have the same sign (in evaluating the integral on the left in (12), it was tacitly assumed that 1 − α _N_ ≠ 0 for all _t_ ≥ 0). Doing this and solving for _N_ , we finally obtain
or, even more simply,
where
Note that _N_ 1 ≠ _N_ 0, since 1 − α _N_ 0 ≠ 0.
Graphing _N_ as a function of time, we get the "S-shaped" curve shown in Figure 1 for the case _N_ 1 = 50 _N_ 0. Note that the population growth is now restricted. In fact, if _t_ is large enough, then _e_ − _rt_ ≈ 0 and (13) is close to the "stable" population level _N_ 1 given by (14), where _N_ 1 is independent of the size of the initial population. The validity of formula (13) has been confirmed by many observations, both of human populations and of experimental populations of bacteria, fruit flies, etc.
Figure 1.
**5.23. Radioactivity**
**a.** Suppose it takes 2 days for 50% of the radioactivity emitted by a radioactive substance to disappear. How long does it take for 99% of the radioactivity to disappear?
SOLUTION. For simplicity, we assume that the radioactivity is entirely due to a single radioactive substance. It is known from physics that the rate of change of the mass of a substance undergoing radioactive decay is proportional at each instant of time _t_ to the mass _m_ = _m_ ( _t_ ) of the substance actually present; the situation resembles that of a sterile population which is "dying off," except that an atom of radioactive substance, unlike a person, can have an arbitrarily large "longevity." Thus _m_ = _m_ ( _t_ ) is the particular solution of the differential equation
satisfying the initial condition
where _m_ 0 is the mass of the radioactive substance present at time _t_ = 0. Since the proportional growth rate _r_ is now negative, we replace it by − _k_ , where _k_ is positive. The differential equation (15) then becomes
The solution of (17), subject to the initial condition (16), is, of course, just the function
describing the exponential decay of the amount of radioactive substance.
We now use the data of the problem to determine the number _k_ , called the _proportional decay rate_ , or simply the _decay constant_. Since 50% of the radioactivity disappears in 2 days, we have, measuring _t_ in days,
This implies
or
With this value of _k_ , (18) becomes
_m_ = _m_ 0 _e_ −( _t_ /2) ln 2.
Disappearance of 99% of the radioactivity means that . Thus 99% of the radioactivity will disappear after _τ_ days if _τ_ satisfies the equation
or
Solving (19) for _τ_ , we get
Actually, we could have estimated the value of _τ_ at once by the following argument: Half the radioactivity disappears in 2 days, half of what's left disappears in 2 more days, leaving one fourth the original amount after 4 days, half of what's now left disappears after another 2 days, leaving one eighth the original amount after 6 days, and so on. Hence the original amount is left after 12 days, and is left after 14 days, so that is left after about 13 days. You will recognize this reasoning, based on repeated _halving_ , as the exact analogue of the treatment of repeated _doubling_ , given in Sec. 5.22b.
**b.** The time it takes a radioactive substance to decay to one half its original amount is called the _half-life_ of the substance, and is independent of the amount originally present. The connection between the half-life _T_ and the decay constant _k_ is just
as we see at once by solving the equation
for _T_. Note that (20) is the same as formula (10) for the doubling time, with _r_ replaced by _k_.
PROBLEMS
**.** The number of bacteria in a culture doubles every hour. How long does it take a thousand bacteria to produce a million? What is the number _N_ of bacteria in the culture at time _t_?
**2.** The world's population, equal to 3.6 billion in 1970, is growing exponentially at the rate of about 2.1% per year. Estimate the world's population in the year 1984. In the year 2001.
**.** An exponentially growing population increases by 20% in 5 years. What is its doubling time?
**4.** One fourth of a radioactive substance disintegrates in 20 years. What is its half-life?
**.** The average amount of radium in the earth's crust is about 1 atom in 1012. Does it make sense to assume that this is the radium left over from a larger amount present at the time the earth was formed? The half-life of radium is 1620 years, and the age of the earth is estimated at 4.6 billion years.
**6.** Let _N_ = _N_ ( _t_ ), _N_ 0 and _N_ 1 be the same as in formula (13). Show that
(a)If _N_ 0 < _N_ 1, _N_ ( _t_ ) is increasing in [0, ∞);
(b)If _N_ 0 > _N_ 1, _N_ ( _t_ ) is decreasing in [0, ∞);
(c)In both cases, _N_ ( _t_ ) → _N_ 1 as _t_ → ∞;
(d)If _N_ 0 < _N_ 1, _N_ ( _t_ ) has an inflection point at _t_ = _t_ 0, where _N_ ( _t_ 0) = .
**7.** Suppose consumption grows exponentially at the rate of _r_ % per year, while population grows exponentially at the rate of _s_ % per year How does the per capita consumption behave?
**.** According to _Newton's law of cooling_ , a body at temperature _T_ cools at a rate proportional to the difference between _T_ and the temperature of the surrounding air. Suppose the air temperature is 20° (Centigrade) and the body cools from 100° to 60° in 20 minutes. How long does it take the body to cool to 30°?
**9.** The absorption of daylight by sea water is described by the exponential law
_I_ = _I_ 0 _e_ − _μx_ ,
where _I_ 0 is the intensity of light at the surface of the sea and _I_ is its intensity at the depth _x_. Find the constant _μ_ , called the _absorption coefficient_ , if the intensity of light at a depth of 5 meters is one thousandth of its intensity at the surface.
***10.** Solve the growth equation
differing from (11) in the sign of the right side. Show that in this case, corresponding to a birth rate proportional to the square of the population size, the population is destined for extinction if its initial size _N_ 0 is less than _N_ 1 = _r_ / _s_. Show that if _N_ 0 > _N_ 1, then _N_ → ∞ as
***11.** Solve the growth equation
corresponding to a birth rate proportional to the population size, together with an "immigration" rate _s_.
***12.** Radioactive carbon 14 ("radiocarbon"), with a half-life of 5570 years, is continually being produced in the upper atmosphere by the action of cosmic rays on nitrogen. Incorporated in carbon dioxide, the radiocarbon is mixed into the lower atmosphere, and is absorbed first by plants, during photosynthesis, and then by animals eating the plants. As long as they are alive, the plants and animals take in fresh radiocarbon, but when they die, the process ceases and the radiocarbon in their tissues slowly disintegrates, dropping to half its original amount in 5570 years. This fact leads to a method, called _radiocarbon dating_ , for estimating the ages of such things as fossil organisms and old bits of wood and charcoal. For example, the age of a sliver of a mummy case can be estimated by comparing the amount of radioactivity in the sliver with the amount of radioactivity in a piece of fresh wood of the same kind and size.
Suppose a Geiger counter records _m_ disintegrations from an old specimen of unknown age _τ_ during the same period in which it records _n_ (> _m_ ) disintegrations from a similar contemporary sample. Show that
***13.** Heartwood from a giant sequoia tree has only 75% of the radioactivity of the younger outer wood. Estimate the age of the tree.
5.3 PROBLEMS OF MOTION
**5.31.** Consider the motion of a particle moving along a straight line _L_. As in Sec. 3.11, let _s_ be the particle's distance at time _t_ from some fixed reference point, where _s_ is positive if measured in a given direction along the line and negative if measured in the opposite direction. Suppose the particle is subject to a force _F_ , acting along the line _L_. Then _Newton's second law of motion_ states that
where _m_ is the particle's mass and
is the particle's acceleration (Sec. 3.14a). The deceptively simple formula (1) is actually a second-order differential equation, with far-reaching physical consequences. As we now illustrate by a series of examples, once _F_ is known, we can determine the particle's position as a function of time by solving (1), subject to appropriate initial conditions.
**5.32. Examples**
**a.** Find the motion of a particle in the absence of any external forces.
SOLUTION. In this case there are no forces, so that _F_ = 0 in (1). It follows that
after cancelling out the mass, which plays no role here. We now solve (2) by two consecutive integrations. Recalling that
where
is the particle's instantaneous velocity (Sec. 3.13a), we first write (2) in the form
Integrating (3), we find that
where _C_ 1 is a constant of integration, and hence
Integrating (5) in turn, we get
where _C_ 2 is another constant of integration.
We must now determine the constants _C_ 1 and _C_ 2. This is done by taking account of the _initial conditions_ of the problem, namely
_v_ | _t_ = 0 = _v_ 0, _s_ | _t_ = 0 = _s_ 0,
where _v_ 0 and _s_ 0 are the velocity and position of the particle at the initial time _t_ = 0. Setting _t_ = 0, _v_ = _v_ 0 in (4) and _t_ = 0, _s_ = _s_ 0 in (6), we find at once that
_C_ 1 = _v_ 0, _C_ 2 = _s_ 0.
Therefore (4) and (6) become
_v_ = _v_ 0
and
_s_ = _v_ 0 _t_ \+ _s_ 0,
where it will be noted that _s_ has the constant value _s_ 0 if _v_ 0 = 0. Thus we have proved _Newton's first law of motion:_ Unless acted upon by an external force, a body at rest ( _v_ 0 = 0) remains at rest and a body in motion ( _v_ 0 ≠ 0) continues to move with constant velocity along a straight line. It is shown in a course on mechanics that this conclusion remains true for a particle free to move in three-dimensional space, rather than just along some line _L_.
**b.** Find the motion of a stone of mass _m_ dropped from a point above the earth's surface.
SOLUTION. We regard the stone as a particle, neglecting its size (Sec. 3.11). Let _s_ = _s_ ( _t_ ) be the stone's position, as measured along a vertical axis with the positive direction pointing downward and the origin at the initial position of the stone. By elementary physics, the force acting on the stone is
_F_ = _mg_ ,
where _g_ is the acceleration due to gravity (approximately 32 ft/sec2) and we neglect the effect of air resistance. Thus, in this case, Newton's second law reduces to
which says that the acceleration has the constant value _g_. Integrating (7) twice, we get first
and then
This time the initial conditions are
_v_ | _t_ = 0 = 0, _s_ | _t_ = 0 = 0,
since the stone is "dropped" (that is, released with no initial velocity) from the point chosen as origin. Setting _t_ = 0, _v_ = 0 in (8) and _t_ = 0, _s_ = 0 in (9), we immediately get _C_ 1 = _C_ 2 = 0. Thus, finally,
and
at least until the stone hits the ground.
**c.** Find the motion of a stone thrown vertically upward with initial velocity _v_ 0.
SOLUTION. We now find it more convenient to measure the stone's position along a vertical axis with the positive direction pointing _upward_. This has the effect of changing _g_ to − _g_ in (8) and (9), since the acceleration due to gravity points _downward_. The initial conditions are now
_v_ | _t_ = 0 = _v_ 0, _s_ | _t_ = 0 = 0.
Setting _t_ = 0, _v_ = _v_ 0 in (8) and _t_ = 0, _s_ = 0 in (9), we get _C_ 1 = _v_ 0, _C_ 2 = 0. Thus, in this case, (8) and (9) reduce to
and
(recall Example 3.15, where _v_ 0 = 96 ft/sec).
**d.** Find the motion of a falling stone of mass _m_ subject to air resistance.
SOLUTION. It is shown in physics that the effect of air resistance can be approximated by a force
_F_ = − _kv_ ( _k_ > 0),
proportional to the stone's velocity and acting in the direction opposite to its motion. The force acting on the stone is now the sum of two forces, its weight _mg_ and the air resistance − _kv_. Thus, in this case, Newton's second law ( _F_ = _ma_ ) gives
where _s_ is measured _downward_ again, as in (7). Introducing the constants
we can write (10) in the form
Separating variables in (11) and integrating, we find that
where _c_ is a constant of integration. Therefore
ln | _v_ − _v_ 1| = −α _t_ \+ _c_ ,
or
since _v_ − _v_ 1 < 0 (why?), where _C_ = _e c_. Applying the initial condition _v_ | _t_ = 0 = 0 (the stone is dropped from rest), we get _C_ = _v_ 1 so that (12) becomes
Figure 2.
The behavior of _v_ as a function of time is shown in Figure 2. After falling for _T_ seconds, where _T_ is three or four times larger than 1/α, the stone effectively attains its _terminal velocity v_ 1, which is never exceeded. Note that _v_ 1 = _mg_ / _k_ is proportional to the weight of the falling object, and hence is much smaller for falling feathers than for falling bricks! Nevertheless, feathers and bricks fall in exactly the same way _in a vacuum_.
To find the stone's equation of motion, we integrate (13), obtaining
Applying the initial condition _s_ | _t_ = 0 = 0, we find that _C_ = − _v_ 1/α. Therefore
at least until the stone hits the ground.
**5.33. Work and energy**
**a.** Suppose a particle of mass _m_ , moving along a straight line _L_ , is acted upon by a force _F_ = _F_ ( _s_ ) which is a continuous function of its position _s_. Then, according to Newton's second law,
or, by the chain rule,
if we think of the velocity _v_ as a function of _s_ rather than _t_. Let
_v_ 0 = _v_ ( _s_ 0), _v_ 1 = _v_ ( _s_ 1)
be the particle's velocity at two different positions _s_ 0 and _s_ 1. Then, integrating (14) with respect to _s_ from _s_ 0 to _s_ 1, we get
or
In other words, as a result of the action of the force, the quantity , called the _kinetic energy_ of the particle, increases by an amount
called the _work_ done by the force on the particle in moving it from _s_ 0 to _s_ 1.
**b.** In the absence of any force, _F_ ≡ 0 and the work (16) vanishes. Then (15) reduces to
so that the kinetic energy remains unchanged, or, in the language of physics, is _conserved_. If _F_ ≡ constant, (16) becomes
Thus, in this case, the work equals the product of the force _F_ and the "displacement" _s_ 1 − _s_ 0, as taught in elementary physics.
**c.** If _F_ = _mg_ , _s_ 0 = 0, _v_ 0 = 0, we have the problem of the falling stone, as in Example 5.32b. Then (16) gives _W_ = _mgs_ 1 and (15) reduces to
after dropping the subscript 1 twice. Solving this equation for _v_ , we get
The same result can be obtained by eliminating _t_ from formulas (8′) and (9′), but here we have used the concepts of work and kinetic energy to find the connection between the stone's velocity and its position without bothering to express either as a function of time.
**d.** Now let _V_ ( _s_ ) be any antiderivative of the function − _F_ ( _s_ ), where the existence of _V_ ( _s_ ) is guaranteed by the assumed continuity of _F_ ( _s_ ) and Theorem 4.23a. Then it follows from (15) and the fundamental theorem of calculus that
or equivalently
The function _V_ ( _s_ ) is called the _potential energy_ (of the particle), and the sum of the kinetic energy and the potential energy _V_ = _V_ ( _s_ ) is called the _total energy E_ = _T_ \+ _V_. Thus equation (18) says that the total energy remains unchanged, or synonymously, is _conserved_ , in the presence of any force _F_ = _F_ ( _s_ ). In the absence of any force, _F_ ≡ 0, _V_ ≡ constant, and then formula (18) for the conservation of the total energy reduces to formula (17) for the conservation of the kinetic energy. Note that the potential energy _V_ , being an antiderivative, is defined only to within an arbitrary "additive constant," and hence the same is true of the total energy _E_ = _T_ \+ _V_. This leads to no difficulties, since formula (18) remains valid if we replace _V_ ( _s_ ) by _V_ ( _s_ ) + _C_ , where _C_ is an arbitrary constant.
**e.** If _F_ = − _mg_ and _s_ 0 = 0, _v_ 0 ≠ 0, we have the problem of the stone thrown upward with initial velocity _v_ 0, as in Example 5.32c. In this case, _V_ = _mgs_ is an antiderivative of − _F_ = _mg_ , and (18) takes the form
after dropping the subscript 1 in two places. To find the maximum height reached by the stone, we set _v_ = 0 in (19) and solve for _s_ , obtaining
The same result can be obtained by solving (8″) for the time _t_ at which _v_ vanishes, and then substituting this value of _t_ into (9″).
**f. Example.** With what velocity _v_ 0 must a rocket be fired vertically upward in order to completely escape the earth's gravitational attraction?
SOLUTION. According to _Newton's law of gravitation_ , the force attracting the rocket back to earth is given by the "inverse square law"
where _k_ is a positive constant, _M_ is the mass of the earth, _m_ is the mass of the rocket, and _s_ is the distance between the rocket (regarded as a particle) and the center of the earth. Here, of course, we choose the _s_ -axis vertically upward along a line going through the center of the earth, and the minus sign in (20) expresses the fact that the force of gravitation is _attractive_ , pulling the rocket back to earth.
The work done on the rocket by the earth's gravitational pull as the rocket leaves the surface of the earth and goes off to a remote point in outer space is given by the integral
where _s_ 0 equals _R_ , the radius of the earth, and _s_ 1 is a very large number. Therefore
after dropping the negligibly small number _kMm_ / _s_ 1. The work _W_ equals the change
in the rocket's kinetic energy in going from the earth's surface to outer space. Since we are looking for the smallest value of the rocket's initial velocity that will allow it to escape the earth's gravitational pull, we choose _v_ 1 = 0 as the rocket's final velocity, so that the rocket arrives in outer space with its initial velocity _v_ 0 completely lost. Equating (21) and (22), with _v_ 1 = 0, we get
Therefore _v_ 0 is given by the formula
and is independent of the rocket's mass _m_.
To evaluate (23), we observe that the force acting on the rocket at the earth's surface is − _kMm_ / _R_ 2 by (20) and − _mg_ in terms of the constant _g_ , the "acceleration due to gravity," which figures in terrestrial problems involving gravitation. Equating these two expressions, we find that the "universal gravitational constant" _k_ equals
Substituting this into (23), we get
Since, to a good approximation, _R_ = 4000 miles and _g_ = 32 ft/sec2, we finally have
(there are 5280 feet in a mile). In the language of rocketry, the quantity _v_ 0 is called the earth's _escape velocity_. A rocket fired upward with a velocity less than _v_ 0 must eventually "fall" back to earth, unless it is "captured" by the gravitational attraction of some other celestial body.
PROBLEMS
**.** Find the equation of motion _s_ = _s_ ( _t_ ) of a particle of mass _m_ acted upon by a constant force _F_ , given that the particle is initially at rest at the point _s_ = 0.
**2.** A particle of mass _m_ moves under the action of a constant force _F_. Suppose the particle's position at time _t_ = _t_ 0 is _s_ = _s_ 0. What velocity _v_ 0 must the particle have at time _t_ = _t_ 0 in order to arrive at the point _s_ = _s_ 1 at time _t_ = _t_ 1?
**.** Suppose a particle of mass _m_ is subject to a force _F_ = _kt_ , proportional to the time that has elapsed since the onset of the motion. Find the resulting equation of motion, assuming that the particle starts from the point _s_ = 0 with initial velocity _v_ 0.
**4.** With what velocity must a stone be thrown vertically upward from ground level to reach a maximum height of 64 feet? How many seconds after it is thrown will the stone hit the ground?
**.** The acceleration due to gravity on the moon is about 5 ft/sec2, as compared with 32 ft/sec2 on the earth. Suppose a man can jump 5 feet high on the earth. How high can he jump on the moon?
**6.** Which has more kinetic energy, a one-ounce bullet going 500 mi/hr or a ten-ton truck going 1 mi/hr? What happens if the bullet goes 600 mi/hr?
**.** According to _Hooke's law_ , the tension in a stretched spring equals _ks_ , where _k_ is a positive constant and _s_ is the length of the spring minus its unstretched length. Show that the potential energy _V_ of the stretched spring equals _ks 2_.
**8.** A particle is attracted to each of two fixed points with a force proportional to the distance between the particle and the point. How much work is done in moving the particle from one point to another along the line connecting them? Assume that the constant of proportionality _k_ is the same for both points.
**.** Show that to reach an altitude of _h_ miles, a rocket must be fired vertically upward with velocity
where _g_ is the acceleration due to gravity and _R_ is the earth's radius. Verify that this expression approaches the value (25) as _h_ → ∞.
**10.** If a rocket is fired vertically upward with a velocity of 1 mi/sec, how high will it rise?
**.** Estimate the moon's escape velocity, given that the moon has approximately the radius and the mass of the earth.
***12.** Two men stand on the edge of a roof _h_ feet above the ground. The first man throws a stone downward with velocity _v_ 0 ft/sec, while the second simultaneously throws another stone upward with the same velocity. Show that both stones hit the ground with the same velocity _v_ l, but naturally at different times _t_ 1 and _t_ 2. Find _v_ 1, _t_ 1, _t_ 2 and Δ _t_ = _t_ 2 − _t_ 1.
***13.** A spider hangs from the ceiling by a single strand of web. Suppose the spider's weight doubles the unstretched length of the strand, stretching it from _s_ to 2 _s_. Show that to climb back to the ceiling, the spider need only do 75% of the work required to climb an inelastic strand of length 2 _s_.
_Chapter 6_
FUNCTIONS OF SEVERAL VARIABLES
6.1 FROM TWO TO _n_ DIMENSIONS
**6.11. Rectangular coordinates in space**
**a.** Rectangular coordinates in space are the natural extension of rectangular coordinates in the plane (Sec. 1.7). Suppose we construct three mutually perpendicular lines _Ox_ , _Oy_ and _Oz_ , known as the _coordinate axes_ , intersecting in a point _0_ , called the _origin_ (see Figure 1). Just as in the plane, each line is regarded as extending indefinitely in both directions, and each is equipped with a _positive direction_ , as indicated by the arrowheads in the figure. The coordinate axes _Ox_ , _Oy_ and _Oz_ are called the _x-axis_ , the _y-axis_ and the _z-axis_ , respectively. These axes determine three mutually perpendicular _coordinate planes_ , the _xy-plane_ containing the _x_ and _y_ -axes, the _yz-plane_ containing the _y_ and _z_ -axes, and the _xz-plane_ containing the _x_ and _z_ -axes. In Figure 1 the _yz_ -plane is the plane of the paper, and the positive _x_ -axis points out from the paper at right angles to the _yz_ -plane.
Figure 1.
Figure 2.
**b.** We now associate three numbers with any given point _P_ in space, by making the following construction, analogous to the construction in Sec. 1.72: Through the point _P_ we draw three planes, one perpendicular to the _x_ -axis, another perpendicular to the _y_ -axis, and a third perpendicular to the _z_ -axis. Suppose that, as in Figure 2, the first plane intersects the _x_ -axis in the point with coordinate _a_ , the second plane intersects the _y_ -axis in the point with coordinate _b_ , and the third plane intersects the _z_ -axis in the point with coordinate _c_. Then the numbers _a_ , _b_ and _c_ are called the _rectangular coordinates_ , or simply the _coordinates_ , of the point _P_. More exactly, _a_ is called the _x-coordinate_ of _P_ , _b_ is called the _y-coordinate_ of _P_ , and _c_ is called the _z-coordinate_ of _P_. Figure 2 is, of course, just the three-dimensional analogue of Figure 10, p. 21.
**c.** The point _P_ with _a_ , _b_ and _c_ as its _x_ , _y_ and _z_ -coordinates may also be denoted by ( _a_ , _b_ , _c_ ). The symbol ( _a_ , _b_ , _c_ ) is called an _ordered triple_ , and is a special kind of three-element set of real numbers, namely one in which _the order of the elements matters_. More generally, an _n_ -element set of real numbers in which the order of the elements matters is called an _ordered n-tuple_ , and is denoted by ( _a_ 1, _a_ 2, ..., _a_ _n_ ).
**d. Example.** Find the distance | _P_ 1 _P_ 2| between two points _P_ 1 = ( _x_ 1, _y_ 1, _z_ 1) and _P_ 2 = ( _x_ 2, _y_ 2, _z_ 2) in space.
SOLUTION. Consulting Figure **3** which generalizes Figure 11, p. 22, we find that
| _P_ 1 _P_ 2|2 = | _P_ 1 _Q_ |2 \+ | _QP_ 2|2,
by the Pythagorean theorem. Therefore
where we use the fact that | _P_ 1 _Q_ | = | _AB_ |. But _A_ = ( _x_ 1, _y_ 1), _B_ = ( _x_ 2, _y_ 2), regarded as points in the _xy_ -plane, and hence
Figure 3.
by the formula for the distance between two points in the plane (Sec. 1.74). Substituting (2) into (1), and noting that | _QP_ 2| = | _z_ 1 − _z_ 2|, we finally get
For example, the distance between the points _P_ 1 = (3, 1, 9) and _P_ 2 = (−1, 4, −3) is
**6.12.** By _n-dimensional space_ , or simply _n-space_ , we mean the set, denoted by _R_ _n_ , of all ordered _n_ -tuples ( _x_ 1, _x_ 2, . . ., _x_ _n_ ) of real numbers. If _n_ = 1, we have the real number system _R_ 1 = _R_. Thus one-space _R_ 1 is the line, two-space _R_ 2 is the plane, and three-space _R_ 3 is ordinary three-dimensional space. Here, of course, we rely on the one-to-one correspondence between _R_ and the points of the line (Sec. 1.36a), and the analogous one-to-one correspondence between _R_ 2 and the plane (Sec. 1.72), and between _R_ 3 and space (see Prob. 1). The elements of _R_ _n_ are called "points," just as in the case of one, two and three dimensions.
The _distance_ between two points _P_ 1 = ( _a_ 1, _a_ 2, ..., _a_ _n_ ) and _P_ 2 = ( _b_ 1, _b_ 2, ..., _b_ _n_ ) in _n_ -space is defined by
or, more concisely, by
When we set _n_ equal to 1, 2 and 3 in formula (3), it reduces in turn to the formula for the distance between two points on the line, in the plane, and in space (check this).
**6.13.** By a (numerical) _function of n variables_ we simply mean a function _f_ ( _x_ 1, _x_ 2, ..., _x_ _n_ ) whose domain is some subset of _n_ -space. For simplicity, we will usually restrict the number of independent variables _x_ l, _x_ 2, ..., _x_ _n_ to two or three. Other things being equal, we write _x_ , _y_ for _x_ l, _x_ 2 if _n_ = 2 and _x_ , _y, z_ for _x_ l, _x_ 2, _x_ 3 if _n_ = 3, to preserve the standard labelling of the coordinate axes in the plane and in space.
When dealing with functions of several variables _x_ l, _x_ 2, ..., _x_ _n_ , we often want to be vague about the actual number of variables. We then write _f_ ( _P_ ) instead of _f_ ( _x_ 1, _x_ 2, ..., _x_ _n_ ), where _P_ = ( _x_ l, _x_ 2, ..., _x_ _n_ ) is a variable point of _n_ -space.
**6.14.** Next we generalize the considerations of Sec. 2.3 to the case of three dimensions. By the _solution set_ of the equation
where _F_ ( _x_ , _y_ , _z_ ) is a function of three variables, we mean the set _S_ of all ordered triples ( _x_ , _y_ , _z_ ) for which (4) holds. Suppose we introduce a three-dimensional system of rectangular coordinates, by setting up perpendicular axes _Ox_ , _Oy_ and _Ox_ , as in Sec. 6.11a, and then plot all the elements of _S_ as points in space. These points make up a "three-dimensional picture," called the _graph of S_ , or, equivalently, the _graph of equation_ (4). The same technique can be applied to a function
of two variables. Let _S_ be the set of all ordered triples ( _x_ , _y_ , _z_ ) for which (5) holds. Then, plotting all the elements of _S_ as points in space, we get a "picture," called the _graph of S_ , or, equivalently, the _graph of the function_ (5). Note that (5) is a special case of (4), corresponding to the choice _F_ ( _x_ , _y_ , _z_ ) = _z_ − _f_ ( _x_ , _y_ ).
The graph of an equation (4) or function (5) typically looks like a "surface," possibly made up of several "pieces." We will often refer to these graphs as "the surface _F_ ( _x_ , _y_ , _z_ ) = 0," or "the surface _z_ = _f_ ( _x_ , _y_ )"
**6.15. Examples**
**a.** Graph the equation
Figure 4.
SOLUTION. Since _x 2 \+ y2_ \+ _z_ 2 is the square of the distance between the point ( _x_ , _y_ , _z_ ) and the origin _O_ = (0, 0, 0), the point ( _x_ , _y_ , _z_ ) belongs to the graph of (6) when the distance between ( _x_ , _y_ , _z_ ) and _0_ equals 1, and only then. Therefore the graph of (6) is the sphere of unit radius with its center at _O_ , shown in Figure 4.
**b.** Graph the function
SOLUTION. Here every point of the circle
_x_ 2 \+ _y_ 2 = _C_ ( _C_ ≥ 0)
corresponds to the same value of _z_ , namely _C_. Equivalently, every plane _z_ = _C_ ( _C_ ≥ 0) parallel to the _xy_ -plane intersects the graph of (7) in a circle, namely the circle of radius (why the square root?) with its center on the _z_ -axis. The value _C_ = 0 gives rise to the "degenerate circle" _x_ 2 \+ _y_ 2 = 0, consisting of the single point _O_ = (0, 0, 0). Thus the graph of (7) is the surface shown in Figure 5. This surface intersects the _xz_ -plane ( _y_ = 0) in the parabola _z_ = _x_ 2 and the _yz_ -plane ( _x_ = 0) in the parabola _z_ = _y_ 2, as we find by substituting first _y_ = 0 and then _x_ = 0 into (7). It is easy to see (how?) that the surface (7) is "generated" (that is, "swept out") by rotating either of these parabolas about the _z_ -axis. For this reason, the surface (7) is called a _paraboloid of revolution_.
Figure 5.
**c.** Graph the function
SOLUTION. The graph of (8) is the right circular cone shown in Figure 6, with its vertex at the point (0, 0, 1). How is this deduced from (8)?
Figure 6.
**6.16. Regions and neighborhoods**
**a.** In general, the domain _D_ of a function _z_ = _f_ ( _x_ , _y_ ) can be an arbitrary set of points in the _xy_ -plane, but in the simple cases considered here _D_ will always be either the whole plane, or a subset of the plane bounded by one or more curves, parts or all of which make up the _boundary_ of _D_. Suppose _D_ is _connected_ , in the sense that any point of _D_ can be joined to any other point of _D_ by a curve which never leaves _D_. (For example, the map of Ohio is connected, but not the map of Hawaii.) Then _D_ is said to be a _region_. A region is said to be _closed_ if it contains its boundary and _open_ if it does not. More generally, a region may contain some but not all of its boundary points. A region is said to be _finite_ if it lies entirely inside some circle _x_ 2 \+ _y_ 2 = _r_ 2 of sufficiently large radius _r_ ; otherwise the region is said to be _infinite_. The symbol _R_ will henceforth be used to denote a _region_ , rather than the real number system.
**b.** The domain of a function _z_ = _f_ ( _x_ , _y_ ) is understood to be the _largest_ set of points ( _x_ , _y_ ) for which the function is defined, just as in the case of one variable (Sec 2.15a). Thus the domain of the function
is the set of all points ( _x_ , _y_ ) such that the square root makes sense, that is, such that _x_ 2 \+ _y_ 2 ≤ 1 (what is the graph of the function?). This is the finite region consisting of the "unit circle"
and its interior. This region is closed, since it contains its boundary, namely the circle (9). We can talk about "the region _x_ 2 \+ _y_ 2 ≤ 1," just as we talk about "the interval −1 ≤ _x_ ≤ 1."
On the other hand, the domain of the function
is the set of all points ( _x_ , _y_ ) such that _x_ \+ _y_ > 0, or equivalently _y_ > − _x_. Thus the domain of (10) is the infinite region _R_ consisting of all points lying to the right of the line _y_ = − _x_ (sketch a figure). The line _y_ = − _x_ is the boundary of _R_ , but is not contained in _R_ , since ln 0 is meaningless. Therefore _R_ is an open region.
**c.** Given any fixed point _P_ 0 = ( _a_ , _b_ ) in the plane, let _N_ be the set of all points _P_ = ( _x_ , _y_ ) such that
where _δ_ is a positive number and | _PP_ 0| is the distance between _P_ and _P_ 0. In terms of coordinates, _N_ is the set of all points ( _x_ , _y_ ) such that
that is, the interior of the _circle_
of radius _δ_ with its center at ( _a_ , _b_ ). Since _N_ does not contain the circle (13) itself, _N_ is an open region. A region of this kind is called a _neighborhood_ of the point _P_ 0 = ( _a_ , _b_ ). Clearly _N_ is the two-dimensional generalization of the one-dimensional neighborhood | _x_ − _a_ | < _δ_ (Sec. 1.63b).
The merit of the inequality (11), as opposed to (12), is that it leaves the number of independent variables unspecified. If _n_ = 3, we write
_P_ 0 = ( _a_ , _b_ , _c_ ), _P_ = ( _x_ , _y_ , _z_ ).
Then the set _N_ of all points _P_ satisfying (11) is the interior of the _sphere_
( _x_ − _a_ )2 \+ ( _y_ − _b_ )2 \+ ( _z_ − _c_ )2 = _δ_ 2
of radius _δ_ with its center at ( _a_ , _b_ , _c_ ). We again call _N_ a neighborhood, this time of the point _P_ 0 = ( _a_ , _b_ , _c_ ) in three-space. Similarly, the set _N_ of all points _P_ = ( _x_ 1, _x_ 2, ..., _x_ _n_ ) in _n_ -space satisfying the inequality (11) is the "interior" of the " _n_ -dimensional sphere"
of radius _δ_ with its center at ( _a_ 1, _a_ 2, ..., _a_ _n_ ). Naturally we again call _N_ a neighborhood, this time of the point _P_ 0 = ( _a_ 1, _a_ 2, ..., _a_ _n_ ) in _n_ -space.
By a _deleted neighborhood_ of a point _P_ 0 we mean any neighborhood of _P_ 0 with the point _P_ 0 itself excluded.
PROBLEMS
**.** We have shown how to find the coordinates of a given point in _R_ 3. How does one find the point in _R_ 3 with given coordinates?
**2.** If a cube has the points (1, 1, 1), (1, −1, − 1), (−1, 1, −1) and (−1, −1, −1) as four of its vertices, what are the other four?
**.** Find the distance from the origin to the point
(a)(4, −2, −4);(b)(−4, 12, 6);(c)(12, 16, −15).
**4.** Verify that the points (3, −1, 6), (− 1, 7, −2) and (1, −3, 2) are the vertices of a right triangle.
**.** Which of the points lie on the surface of the sphere of radius 2 with its center at the origin?
**6.** Show that the sphere _x_ 2 \+ _y_ 2 \+ _z_ 2 − 4 _x_ − 6 _y_ − 2 _z_ \+ 13 = 0 is tangent to the _xy_ -plane.
**.** _By a surface of revolution_ we mean any geometrical figure in _R_ 3 generated by rotating a plane curve about a straight line lying in its plane. Can the same surface of revolution be generated by rotating a given plane curve about two different axes?
**.** Describe the surface of revolution obtained by rotating the line _x_ = 0, _y_ = _a_ about
(a)The _y_ -axis;(b)The _z_ -axis.
**.** Find the piecewise linear curves in which the cone (8) intersects
(a)The _xz_ -plane;(b)The _yz_ -plane.
**.** Justify thinking of intervals as "one-dimensional regions."
**11.** Write inequalities describing the finite open region bounded by the lines _x_ = ±1, _y_ = ±1.
***12.** Find the two points of the _x_ -axis at distance 12 from the point (−3, 4, 8).
***13.** Find the distance between the two points _P_ 1 = (1, 1, 1, 1) and _P_ 2 = (0, 2, 0, 2) of _R_ 4. Which is closer to the origin of four-space?
***14.** What is the surface _x_ 2 \+ _y_ 2 = _z_ 2?
***15.** Why is the domain of the function _f_ ( _x_ , _y_ ) = ln ( _x_ 2 − _y_ 2) not a region?
6.2 LIMITS AND DIFFERENTIATION
**6.21. Limits and continuity**
**a.** To define the limit of a function of several variables, we need only make slight changes in the definition of the limit of a function of a single variable, given in Sec. 2.44a. Thus we say that a (numerical) function _f_ ( _P_ ) of several variables, defined in a deleted neighborhood of a point _P_ 0, _approaches the limit A as P approaches P_ 0, or that _f_ ( _P_ ) _has the limit A at P_ 0, if _f_ ( _P_ ) gets "closer and closer" to _A_ as _P_ gets "closer and closer" to _P_ 0 without ever coinciding with _P_ 0. This fact is expressed by writing
or
_f_ ( _P_ ) → _A_ as _P_ → _P_ 0.
In the " _ε_ , _δ_ language" introduced in Sec. 2.44b, (1) means that, given any _ε_ > 0, we can find a number _δ_ > 0 such that | _f_ ( _P_ ) − _A_ | < _ε_ whenever 0 < | _PP_ 0| < _δ_. As in Sec. 2.44d, it is often convenient to talk about having a limit without specifying what the limit is. Thus we say that a function _f_ ( _P_ ) _has a limit at P_ 0 if there is some number _A_ such that _f_ ( _P_ ) → _A_ as _P_ → _P_ 0.
**b.** In two dimensions, we have _f_ ( _P_ ) = _f_ ( _x_ , _y_ ) and _P_ 0 = ( _a_ , _b_ ), say. We can then write the limit (1) more explicitly as a "double limit"
Do not make the mistake of confusing (2) with the "iterated limit"
which means something quite different.
**c.** Just as in the case of a function of one variable (Sec. 2.63a), a function _f_ ( _P_ ) of several variables, defined in a neighborhood of a point _P_ 0, is said to be _continuous at P_ 0 if
If a function _f_ ( _P_ ) is continuous at every point of a region _R_ , we say that _f_ ( _P_ ) is _continuous in R_. When we call a function continuous, without further qualification, we always mean continuous at some point or in some region, where the context makes it clear just what is meant.
**6.22. Partial derivatives**
**a.** Let _f_ be a function of _n_ variables defined in a neighborhood of a point ( _x_ 1, _x_ 2, ..., _x_ _n_ ). Then by the _partial derivative of f with respect to x_ _i_ _at_ ( _x_ 1, _x_ 2, ..., _x_ _n_ ), denoted by the expression
we mean the limit
where _x_ _i_ is given an increment Δ _x_ _i_ , _but all the other variables are held fixed_ , provided this limit exists and is finite. Clearly there are _n_ such partial derivatives, corresponding to the _n_ subscripts 1, 2, ..., _n_. Thinking of the symbol ∂/∂ _x_ _i_ as a single entity, whose effect is to form the partial derivative with respect to _x_ _i_ of any function written after it, we can also write (3) as
The symbol ∂ is still pronounced "dee," even though we are now dealing with a "curved dee."
**b.** To evaluate (3), we need only treat all the independent variables except _x_ _i_ as if they are constants. Thus no extra technique is required to calculate partial derivatives. For example, the function
_f_ ( _x_ , _y_ ) = _xe_ _xy_
of two variables has two partial derivatives, namely
calculated by regarding _y_ as a constant and then treating ∂/∂ _x_ like _d/dx_ , and
calculated by regarding _x_ as a constant and treating ∂/∂ _y_ like _d/dy_.
**c.** There are other ways of writing (3). If _u_ = _f_ ( _x_ 1, _x_ 2, ..., _x_ _n_ ), we can abbreviate (3) to ∂ _f/_ ∂ _x_ _i_ or ∂ _u/_ ∂ _x_ _i_. We can also write (3) as
or simply _f_ _xi_ or _u_ _xi_ , where the subscript _x_ _i_ calls for (partial) differentiation with respect to _x_ _i_. We can go a step further, and write just
_f_ _i_ ( _x_ 1, _x_ 2, ..., _x_ _n_ ),
instead of (4), or simply _f_ _i_ or _u_ _i_ , where the subscript _i_ calls for differentiation with respect to the _i_ th argument, whatever it be called. For example, if
_u_ = _f_ ( _r_ , _s_ , _t_ ) = _rs_ 2 ln _t_ ,
then _f_ _t_ ( _r_ , _s_ , _t_ ), _f_ _t_ , _u_ _t_ , _f_ 3, and _u_ 3 are all ways of writing the same partial derivative
**d.** Partial derivatives of higher order are defined in the natural way. For example, if _z_ = _f_ ( _x_ , _y_ ), we have four _second partial derivatives_ , two of the form
and two "mixed" derivatives of the form
one obtained by differentiating _z_ first with respect to _y_ and then with respect to _x_ , the other obtained by carrying out the differentiations in the opposite order. Similarly, if _u_ = _f_ ( _x_ , _y_ , _z_ ), then
and so on.
**e. Example.** If
_f_ ( _x_ , _y_ , _z_ ) = _xy_ ln _z_ ,
then
Note that the value of each mixed partial derivative is independent of the order of differentiation. It is not hard to show that this is always true for a function with continuous first and second partial derivatives, but the proof will not be given here.
**6.23. Differentiable functions and differentials**
**a.** It will be recalled from Sec. 2.55a that if _f_ is a function of a single variable, and if _f_ has a derivative _f_ ′( _x_ ) at a point _x_ , then the expression
where Δ _f_ ( _x_ ) = _f_ ( _x_ \+ Δ _x_ ) − _f_ ( _x_ ) is the increment of _f_ at _x_ , approaches zero as Δ _x_ → 0. Therefore, if _f_ has a derivative _f_ ′( _x_ ) at the point _x_ , we can write the increment Δ _f_ ( _x_ ) in the form
Δ _f_ ( _x_ ) = _f_ ′( _x_ ) Δ _x_ \+ α(Δ _x_ ) Δ _x_ ,
where
We can also write the increment as
Δ _f_ ( _x_ ) = _df_ ( _x_ ) + α(Δ _x_ ) Δ _x_ ,
in terms of _df_ ( _x_ ) = _f_ ′( _x_ )Δ _x_ , the differential of _f_ at the point _x_.
Conversely, if Δ _f_ ( _x_ ) can be represented in the form
where _A_ is a constant and (5) holds, then the derivative _f_ ′( _x_ ) exists and equals _A_. To see this, we need only divide (6) by Δ _x_ and take the limit as Δ _x_ → 0, obtaining
Thus, in the case of a function _f_ of one variable, having a differential _df_ ( _x_ ) and having a derivative _f_ ′( _x_ ) are equivalent properties, both described by saying that _f_ is _differentiable_ at _x_.
**b.** In the case of a function of several variables, the situation is rather different. For such a function, having a differential (in a sense to be defined in a moment for a function of two variables) and having partial derivatives are _not_ equivalent properties. Indeed, the mere fact of having partial derivatives does not guarantee having a differential (it does, though, if the partial derivatives are _continuous_ ). On the other hand, having a differential does guarantee having partial derivatives. Thus, having a differential is a "stronger" requirement than having partial derivatives, and we will use the word "differentiable" exclusively in the sense of having a differential. These things are worth knowing, but, with the exception of Theorem 6.23d, we omit the proofs, which are tedious and rather technical. They can be found in a more advanced course, together with the proof alluded to at the end of Sec. 6.22e.
**c. Definition.** Given a function _z_ = _f_ ( _x_ , _y_ ) of two variables, by the _increment of f at the point_ ( _x_ , _y_ ) we mean the expression
Suppose that for every point ( _x_ \+ Δ _x_ , _y +_ Δ _y_ ) in some neighborhood of ( _x_ , _y_ ), we can write Δ _f_ ( _x_ , _y_ ) in the form
analogous to (6), where _A_ and _B_ are constants, and
Then we say that _f has a differential_ or is _differentiable at_ ( _x_ , _y_ ), and the expression _A_ Δ _x_ \+ _B_ Δ _y_ in (8) is called the ( _total_ ) _differential of f at_ ( _x_ , _y_ ), denoted by _df_ ( _x_ , _y_ ), so that
For brevity, we will often write Δ _f_ and _df_ for Δ _f_ ( _x_ , _y_ ) and _df_ ( _x_ , _y_ ).
**d.** THEOREM. _If f is differentiable at_ ( _x_ , _y_ ), _then_
(1) _f is continuous at_ ( _x_ , _y_ );
(2) _f has partial derivatives f x and fy at_ ( _x_ , _y_ );
(3) _The increment_ Δ _f can be written in the form_
(4) _The differential df can be written in the form_
_Proof_. It follows at once from (8) and (9) that
and therefore
which expresses the continuity of _f_ at ( _x_ , _y_ ) in increment notation. Setting Δ _y_ = 0 in (8), we get
_f_ ( _x_ \+ Δ _x_ , _y_ ) − _f_ ( _x_ , _y_ ) = _A_ Δ _x_ \+ α(Δ _x_ , 0)Δ _x_ ,
with the help of (7), and hence
(see Prob. 1), that is, the partial derivative _f_ _x_ ( _x_ , _y_ ) exists and equals _A_. Similarly, setting Δ _x_ = 0 in (8), we get
_f_ ( _x_ , _y_ \+ Δ _y_ ) − _f_ ( _x_ , _y_ ) = _B_ Δ _y_ \+ _β_ (0, Δ _y_ )Δ _y_ ,
and hence
so that _f_ _y_ ( _x_ , _y_ ) exists and equals _B_. Replacing _A_ and _B_ in (8) and (10) by _f_ _x_ ( _x_ , _y_ ) and _f_ _y_ ( _x_ , _y_ ), we get (11) and (12).
**e.** It follows from (12) that
In other words, the increments and the differentials of the _independent_ variables are equal (recall Sec. 2.56a). Thus we can write (12) in the form
_df_ ( _x_ , _y_ ) = _f_ _x_ ( _x_ , _y_ ) _dx_ \+ _f_ _y_ ( _x_ , _y_ ) _dy_ ,
or, even more concisely, as
The generalization of (13) to the case of a function _f_ ( _x_ 1, _x_ 2, ..., _x_ _n_ ) of _n_ variables is just
and is proved in much the same way (we omit the details).
**f. Example.** Estimate the quantity
SOLUTION. If
then
_Q_ = _f_ ( _x_ \+ Δ _x_ , _y_ \+ Δ _y_ , _z_ \+ Δ _z_ ),
where
_x_ = 2, _y_ = 4, _z_ = 4,
Δ _x_ = −0.02, Δ _y_ = 0.02, Δ _z_ = −0.04.
Therefore
_Q_ = _f_ ( _x_ , _y_ , _z_ ) + Δ _f_ ( _x_ , _y_ , _z_ ) ≈ _f_ ( _x_ , _y_ , _z_ ) + _df_ ( _x_ , _y_ , _z_ ),
where we have approximated the increment Δ _f_ by the differential _df_. Using (13′) with _n_ = 3, we have
and therefore
An exact calculation shows that
to four decimal places.
PROBLEMS
**.** Show that formula (2) implies
**.** Show that
does not exist.
**.** Does ( _x_ , _y_ ) → ( _a_ , _b_ ) imply _x_ → _a_ , _y_ → _b_ , and conversely? Justify your answer.
**.** Suppose _f_ ( _x_ , _y_ ) is independent of _y_ , so that _f_ ( _x_ , _y_ ) ≡ _g_ ( _x_ ), and suppose _g_ ( _x_ ) is continuous at the point _a_. Show that _f_ ( _x_ , _y_ ) is continuous at the point ( _a_ , _b_ ) for arbitrary _b_.
**.** Suppose _f_ ( _x_ , _y_ ) is independent of _x_ , so that _f_ ( _x_ , _y_ ) ≡ _h_ ( _y_ ), and suppose _h_ ( _y_ ) is continuous at the point _b_. Show that _f_ ( _x_ , _y_ ) is continuous at the point ( _a_ , _b_ ) for arbitrary _a_.
**.** Let _g_ ( _x_ ) be continuous at the point _a_ , while _h_ ( _y_ ) is continuous at the point _b_. Show that the functions _f_ ( _x_ ) ± _g_ ( _y_ ), _f_ ( _x_ ) _g_ ( _y_ ) and _f_ ( _x_ )/ _g_ ( _y_ ) are all continuous at ( _a_ , _b_ ), provided that _g_ ( _b_ ) ≠ 0 in the last case.
**.** Use continuity to evaluate
**.** Define and evaluate the limit
**9.** Where is the function continuous?
**.** Where does the function
fail to be continuous?
**.** Find ∂ _z_ /∂ _x_ and ∂ _z_ /∂ _y_ if
**.** Find ∂ _u_ /∂ _x_ , ∂ _u_ /∂ _y_ and ∂ _u_ /∂ _z_ if
**.** Find
**.** Verify by direct calculation that ∂2 _z_ /∂ _x_ ∂ _y_ = ∂2 _z_ /∂ _y_ ∂ _x_ if
(a) _z_ = ln ( _x_ \+ _y_ );(b) _z_ = ln ( _xy_ ).
**.** Verify directly from the definition of the differential that each of the following functions is differentiable in the whole _xy_ -plane:
(a) _f_ ( _x_ , _y_ ) = _x_ 2 \+ _y_ 2;(b) _f_ ( _x_ , _y_ ) = _xy_.
**.** Find the (total) differential of
(a) _z_ = _xy_ − _x_ 2 _y_ 3 \+ _x_ 3 _y_ ;(b) _z_ = _y_ _x_.
**.** Use differentials to estimate
***18.** Verify that the function
satisfies the equation
known as _Laplace's equation_.
_Comment_. An equation like this, involving one or more partial derivatives of a function, is called a _partial_ differential equation, as opposed to the _ordinary_ differential equations considered in Chapter 5.
***19.** Is the function differentiable at the origin?
***20.** According to the _perfect-gas law_ , the pressure _p_ , the volume _V_ and the temperature _T_ (in degrees Kelvin) of a confined gas are related by the formula _pV_ = _kT_ , where _k_ is a constant of proportionality. Show that
_Comment_. This formula should cure you of any temptation to treat partial derivatives like fractions.
6.3 THE CHAIN RULE
**6.31. a.** We now generalize Theorem 2.82a on the derivative of a composite function to the case of a function of _n_ variables. For simplicity, we choose _n_ = 2, just as in Theorem 6.23d, but the result is readily extended to the case _n_ > 2.
THEOREM **(Chain rule)**. _Suppose x and y are functions of a single variable, both differentiable at t_ , _and suppose f is a function of two variables, differentiable at_ ( _x_ ( _t_ ), _y_ ( _t_ )). _Then the composite function F, defined by F_ ( _t_ ) ≡ _f_ ( _x_ ( _t_ ), _y_ ( _t_ )), _is differentiable at t, with derivative_
_Proof_. The proof is the natural generalization of the proof of Theorem 2.82a. Let _z_ = _f_ ( _x_ , _y_ ). Since _x_ and _y_ are differentiable at _t_ , then, as in Sec. 6.23a,
where _λ_ (Δ _t_ ) → 0, _μ_ (Δ _t_ ) → 0 as Δ _t_ → 0. Moreover, since _f_ is differentiable at ( _x_ ( _t_ ), _y_ ( _t_ )), then, by Theorem 6.23d,
Δ _z_ = _f_ ( _x_ \+ Δ _x_ , _y_ \+ Δ _y_ ) − _f_ ( _x_ , _y_ )
= [ _f_ _x_ ( _x_ , _y_ ) + α(Δ _x_ , Δ _y_ )]Δ _x_ \+ [ _f_ _y_ ( _x_ , _y_ ) + _β_ (Δ _x_ , Δ _y_ )]Δ _y_ ,
where we temporarily drop the argument _t_ in many places, and α(Δ _x_ , Δ _y_ ), _β_ (Δ _x_ , Δ _y_ ) satisfy the conditions (9), p. 219. Substituting the expressions for Δ _x_ and Δ _y_ into the formula for Δ _z_ , we get
It follows from the expressions for Δ _x_ and Δ _y_ (or from the continuity of _x_ and _y_ at _t_ ) that Δ _t_ → 0 implies Δ _x_ → 0, Δ _y_ → 0, so that Δ _t_ → 0 implies not only _λ_ (Δ _t_ ) → 0, _μ_ (Δ _t_ ) → 0, but also α(Δ _x_ , Δ _y_ ) → 0, _β_ (Δ _x_ , Δ _y_ ) → 0. Therefore, dividing (3) by Δt and taking the limit as Δ _t_ → 0, we find that
where we reinstate the argument _t_ in four places. On the other hand,
Δ _z_ = _f_ ( _x_ \+ Δ _x_ , _y_ \+ Δ _y_ ) − _f_ ( _x_ , _y_ )
= _f_ ( _x_ ( _t_ ) + _x_ ( _t_ \+ Δ _t_ ) − _x_ ( _t_ ), _y_ ( _t_ ) + _y_ ( _t_ \+ Δ _t_ ) − _y_ ( _t_ )) − _f_ ( _x_ ( _t_ ), _y_ ( _t_ ))
= _f_ ( _x_ ( _t_ \+ Δ _t_ ), _y_ ( _t_ \+ Δ _t_ )) − _f_ ( _x_ ( _t_ ), _y_ ( _t_ )) = _F_ ( _t_ \+ Δ _t_ ) − _F_ ( _t_ ),
which implies
Comparing (4) and (5), we get (1).
**b.** Formula (1) can be written more concisely as
We can simplify (6) even further by changing _F_ to _f_ :
After all, since _f_ is a function of two variables, the fact that we write an _ordinary_ derivative _df/dt_ on the left in (7) means that each argument of _f_ is being thought of as a function of a single variable, namely _t_. With this understanding, we can do without the extra symbol _F_ , which was introduced only to make the distinction between _f_ ( _x_ , _y_ ) and _f_ ( _x_ ( _t_ ), _y_ ( _t_ )) more explicit.
**c.** The case where _x_ and _y_ are functions of _several_ variables presents no difficulties. For example, suppose _x_ = _x_ ( _t_ , _u_ ), _y_ = _y_ ( _t_ , _u_ ), where _x_ and _y_ are differentiable functions of two variables. If _u_ is held fixed, _x_ and _y_ reduce to functions of a single variable _t_ , and we can apply the theorem without further ado, obtaining
where all three ordinary derivatives in (7) now become partial derivatives. Similarly, holding _t_ fixed, we get
Here again we might have introduced a composite function _F_ , defined by
_F_ ( _t_ , _u_ ) ≡ _f_ ( _x_ ( _t_ , _u_ ), _y_ ( _t_ , _u_ )),
but it is simpler to regard _f_ ( _x_ , _y_ ) and _f_ ( _x_ ( _t_ , _u_ ), _y_ ( _t_ , _u_ )) as being the same function, written in terms of different independent variables.
**d.** The generalization of formula (7) to the case of a function _f_ ( _x_ 1, _x_ 2, ..., _x_ _n_ ) whose _n_ arguments depend on a single new independent variable _t_ is given by
and is proved in much the same way (we omit the lengthy details). Similarly, the generalization of (8) and (8′) to the case of a function _f_ ( _x_ 1, _x_ 2, ..., _x_ _n_ ) whose _n_ arguments depend on _m_ new independent variables _t_ 1, _t_ 2, ..., _t_ _m_ is given by
The last two formulas are the "master" chain rules, of which all previous versions (including the first formula in Sec. 2.82c) are merely special cases. Note the following common features of (9) and (10):
(1) The right side contains _n_ terms, one for each "intermediate" variable _x_ 1, _x_ 2, . . ., _x_ _n_ ,
(2) Each of these terms is a product of two derivatives, with the intermediate variable appearing in the denominator of one derivative and in the numerator of the other.
**6.32. Examples**
**a.** Let _f_ be any differentiable function of two variables. Prove that the function _u_ = _f_ ( _x_ − _y_ , _y_ − _z_ ) satisfies the partial differential equation
SOLUTION. Let _s_ = _x_ − _y_ , _t_ = _y_ − _z_. Then, by the chain rule,
and adding these three equations, we get (11). Here, of course, we have used the formulas
**b.** Given a function _F_ ( _x_ , _y_ ) of two variables, suppose the equation
defines _y_ as an "implicit" function of _x_ , in the sense that there exists a function _y_ = _y_ ( _x_ ) such that
_F_ ( _x_ , _y_ ( _x_ )) = 0
for all _x_ in some interval _I_. Then, assuming that the functions _F_ ( _x_ , _y_ ) and _y_ ( _x_ ) are both differentiable, we can use the chain rule to differentiate (12) with respect to _x_ :
Solving (13) for _dy/dx_ , we get
provided that _F_ _y_ ( _x_ , _y_ ) ≠ 0. This is, of course, just a more official version of the technique of implicit differentiation introduced in Sec. 2.83d. For example, the equation
_x_ 2 − _xy_ \+ _y_ 3 = 1
considered there is of the form (12) if
_F_ ( _x_ , _y_ ) = _x_ 2 − _xy_ \+ _y_ 3 − 1.
We then have
_F_ _x_ ( _x_ , _y_ ) = 2 _x_ − _y_ , _F_ _y_ ( _x_ , _y_ ) = − _x_ \+ 3 _y_ 2,
so that (14) becomes
which is precisely formula (12), p. 86.
**c.** The technique of implicit differentiation can also be used to calculate partial derivatives. Thus, given a function _F_ ( _x_ , _y_ , _z_ ) of three variables, suppose the equation
defines _z_ as an "implicit" function of _x_ and _y_ , in the sense that there exists a function _z_ = _z_ ( _x_ , _y_ ) such that
_F_ ( _x_ , _y_ , _z_ ( _x_ , _y_ )) = 0
for all ( _x_ , _y_ ) in some region _R_. Then, assuming that the functions _F_ ( _x_ , _y_ , _z_ ) and _z_ ( _x_ , _y_ ) are both differentiable, we can use the chain rule to differentiate (15) with respect to _x_ and _y_ (dropping arguments for simplicity):
since
Solving (16) and (16′) for ∂ _z_ /∂ _x_ and ∂ _z_ /∂ _y_ , we get
provided, of course, that _F_ _z_ ( _x_ , _y_ , _z_ ) ≠ 0.
**d.** Given that
find ∂ _z_ /∂ _x_ and ∂ _z_ /∂ _y_.
SOLUTION. Here _F_ ( _x_ , _y_ , _z_ ) is just the left side of (18). Therefore, by (17),
provided that _z_ ≠ ln 2.
PROBLEMS
**.** Find _dz/dt_ by both the chain rule and by direct substitution if
(a) _z_ = _x_ 2 \+ _xy_ 2, _x_ = _e_ _t_ , _y_ = 1/ _t_ ;
(b) _z_ = _e_ _u_ \+ _v_ ln ( _u_ \+ _v_ ), _u_ = 2 _t_ 2, _v_ = 1 − 2 _t_ 2
**.** Find ∂ _z_ /∂ _x_ and ∂ _z_ /∂ _y_ in two different ways if
(a) _z_ = _u_ \+ _v_ 2, _u_ = _x_ 2, _v_ = ln ( _x_ \+ _y_ );
**.** Given that _z_ = _f_ ( _x_ , _y_ ), express ∂ _z_ /∂ _x_ and ∂ _z_ /∂ _y_ in terms of ∂ _z_ /∂ _u_ and ∂ _z_ /∂ _u_ if
(a) _u_ = _px_ \+ _qy_ , _v_ = _rx_ \+ _sy_ ;(b) _u_ = _xy_ , _v_ = _y_ / _x_.
**.** Given that
_z_ 2 − 2 _xyz_ \+ 3 = 0,
find _z_ _x_ = ∂ _z_ /∂ _x_ and _z_ _y_ = ∂ _z_ /∂ _y_. Evaluate these derivatives at the point _x_ = 1, _y_ = 2.
**.** A function _f_ ( _x_ , _y_ ) with domain _D_ is said to be _homogeneous of degree k_ if ( _x_ , _y_ ) ∈ _D_ implies ( _tx_ , _ty_ ) ∈ _D_ for all _t_ > 0 and if
for all ( _x_ , _y_ ) ∈ _D_ and _t_ > 0. Show that each of the following functions is homogeneous, and find its degree:
**.** Show that if _f_ ( _x_ , _y_ ) is homogeneous of degree _k_ and differentiable at every point of its domain, then
a result known as _Euler's theorem on homogeneous functions_. Verify by direct calculation that each of the functions in the preceding problem satisfies (20).
***7.** Let
_x_ 2 \+ 2 _y_ 2 \+ 3 _z_ 2 \+ _xy_ − _z_ − 9 = 0.
Use repeated implicit differentiation to find ∂2 _z_ /∂ _x_ 2, ∂2 _z_ /∂ _x_ ∂ _y_ , ∂2 _z_ /∂ _y_ ∂ _x_ and ∂2 _z_ /∂ _y_ 2 at the point _x_ = 1, _y_ = −2, _z_ = 1.
***8.** Show that if _f_ is continuous in [ _a_ , _b_ ] and if _a_ ≤ _u_ ( _x_ ) ≤ _v_ ( _x_ ) ≤ _b_ , then
provided that _u_ and _v_ are differentiable.
6.4 EXTREMA IN _n_ DIMENSIONS
**6.41. a.** In this section we favor the notation _X_ = ( _x_ 1, _x_ 2, ..., _x_ _n_ ) for a variable point in _n_ -space, reserving the symbol _P_ for _a fixed_ point of _n_ -space. Global and local extrema of a function _f_ ( _X_ ) = _f_ ( _x_ 1, _x_ 2, ..., _x_ _n_ ) of _n_ variables are defined exactly as in Secs. 3.32a and 3.61b, with _x_ , _p_ and _q_ replaced by _n_ -dimensional points _X_ , _P_ and _Q_ , respectively. As in Sec. 3.62b, if _f_ ( _X_ ) has a local extremum at _P_ , the extremum is said to be _strict_ if _f_ ( _X_ ) ≠ _f_ ( _P_ ) for all _X_ ≠ _P_ in some neighborhood of _P_ , that is, for all _X_ in some _deleted_ neighborhood of _P_ (Sec. 6.16c).
In _R_ _n_ we have the following analogue of Theorem 3.32c, which we state without proof:
THEOREM. _If f is continuous in a finite closed region R, then R contains points P and Q such that_
_f_ ( _Q_ ) ≤ _f_ ( _X_ ) ≤ _f_ ( _P_ )
_for all X_ ∈ _R. In other words, f has both a maximum and a minimum in R, at the points P and Q, respectively_.
For example, the function
whose graph is the upper half of the sphere shown in Figure 4, p. 212, has a global minimum, equal to 0, at every point of the circle _x_ 2 \+ _y_ 2 = 1, but no local minima (why not?), and both a global maximum and a strict local maximum, equal to 1, at the origin _O_ = (0, 0). As another example, the function
whose graph is the paraboloid of revolution shown in Figure 5, p. 212, has both a global minimum and a strict local minimum, equal to 0, at the origin _O_ , but no global or local maxima. This can be seen by inspection of Figures 4 and 5, and will be verified in Sec. 6.42 by using partial differentiation.
**b.** There is a natural generalization of Theorem 3.63a for functions of several variables:
THEOREM. _If f_ ( _X_ ) = _f_ ( _x_ 1, _x_ 2, ..., _x_ _n_ ) _has a local extremum at a point P_ = ( _a_ 1, _a_ 2, ..., _a_ _n_ ), _then either f_ ( _X_ ) _is nondifferentiable at P, or the partial derivatives of f_ ( _X_ ) _all vanish at P_ :
_Proof_. Obviously, _f_ ( _X_ ) is either nondifferentiable at _P_ or differentiable at _P_. In the latter case, the partial derivatives of _f_ ( _X_ ) at _P_ all exist, by the natural generalization of Theorem 6.23d to the case of _n_ variables. But then all _n_ functions
of a single variable are differentiable, the first at _a_ 1, the second at _a_ 2, and so on, with derivatives
The first of the functions (4) has a local extremum at _a_ 1, the second has a local extremum at _a_ 2, and so on (why?). Therefore, applying Theorem 3.63a _n_ times, we find that the left sides of all _n_ equations (5) vanish. But then (3) holds.
Note that the theorem reduces to Theorem 3.63a if _n_ = 1.
**c.** By analogy with Sec. 3.63c, by a _critical point_ of a function _f_ of _n_ variables we mean either a point where _f_ is nondifferentiable or a point where the condition (3) holds, and by a _stationary point_ of _f_ we mean a point where (3) holds. Thus a critical point of _f_ is either a point where _f_ is nondifferentiable or a stationary point of _f_. According to the theorem, if _f_ has a local extremum at _P_ , then _P_ is a critical point of _f_. On the other hand, just as in the case of one variable (Sec. 3.63c), if _P_ is a critical point of _f_ , there is no necessity for _f_ to have a local externum at _P_. For example, consider the function
The partial derivatives
vanish at the origin _O_ = (0, 0), which is therefore a critical point of _f_ , in fact a stationary point. But _f_ does not have a local extremum at _O_. In fact,
so that, by the second derivative test for functions of a single variable (Theorem 3.65a), the function _f_ ( _x_ , 0) has a local minimum at _O_ , while the function _f_ (0, _y_ ) has a local maximum at _O_ , and this obviously prevents _f_ ( _x_ , _y_ ) from having either a local minimum or a local maximum at _O_.
Thus what we really want are conditions on a function _f_ which _compel f_ to have a local extremum at a given point _P_. For the case of a function of two variables such conditions are given by the following two-dimensional generalization of Theorem 3.65a, which we state without proof:
THEOREM. _Suppose f_ ( _x_ , _y_ ) _has continuous second partial derivatives in a neighborhood of a critical point P_ = ( _a_ , _b_ ), _and let_
_Then f_ ( _x_ , _y_ ) _has a strict local maximum at P if D_ > 0, _A_ < 0, _and a strict local minimum at P if D_ > 0, _A_ > 0, _but no extremum at P if D_ < 0.
If _D_ = 0, there may or may not be an extremum at _P_. For example, _D_ = 0 at the origin _O_ = (0, 0) if _f_ ( _x_ , _y_ ) = _x_ 2 \+ _y_ 4 or if _f_ ( _x_ , _y_ ) = _x_ 2 \+ _y_ 3, but in the first case the function _f_ ( _x_ , _y_ ) clearly has a local minimum at _O_ , since it vanishes at _O_ and is positive everywhere else, while in the second case _f_ ( _x_ , _y_ ) has no extremum at _O_ , since _f_ (0, _y_ ) = _y_ 3 is increasing in the interval −∞ < _y_ < ∞.
**6.42. Examples**
**a.** The function (1) is differentiable for _x_ 2 \+ _y_ 2 < 1, with partial derivatives
Therefore, by Theorem 6.41b, the only critical point of _f_ in the open disk _x_ 2 \+ _y_ 2 < 1 is at the point _O_ = (0, 0), where these partial derivatives vanish. Moreover, as you can easily verify,
It follows from Theorem 6.41c that _f_ has a strict local maximum, equal to 1, at the point _O_ , as already observed.
**b.** The function (2) is differentiable in the whole _xy_ -plane, with partial derivatives
Again the only critical point of _f_ , this time in the whole plane, is at the origin _O_ = (0, 0). We now have
Therefore, by Theorem 6.41c, _f_ has a strict local minimum, equal to 0, at the origin. Actually, this fact is obvious without making any calculations at all, since the function _f_ is positive everywhere except at the origin, where it vanishes.
**c.** Inspection of Figure 6, p. 213, shows that the function
has both a global maximum and a strict local maximum, equal to 1, at the origin _O_ = (0, 0). Therefore, by Theorem 6.41b, _O_ is a critical point of _f_ , but this time not because _0_ is a stationary point of _f_ , but because _f_ is nondifferentiable at _0_. In fact, the derivative
fails to exist, for the reason given in Sec. 2.45e, and the same is true of the derivative ∂ _f_ (0, 0)/∂ _y_.
**d.** Find the local extrema of the function
_f_ ( _x_ , _y_ ) = _x_ 3 \+ _y_ 3 − 3 _xy_.
SOLUTION. Solving the system
we find that _f_ has two critical points, namely (0, 0) and (1, 1). Since
we have
_A_ = 0, _B_ = −3, _C_ = 0, _D_ = _AC_ − _B_ 2 = −9
at (0, 0), and
_A_ = 6, _B_ = −3, _C_ = 6, _D_ = _AC_ − _B_ 2 = 27
at (1, 1). It follows from Theorem 6.41c that _f_ has no extremum at (0, 0) and a strict local minimum, equal to −1, at (1, 1).
**e.** Find the brick of largest volume with a given surface area 2 _c_.
SOLUTION. Let _x_ be the length, _y_ the width and _z_ the height of the brick. Then the brick has volume
and surface area
2( _xy_ \+ _xz_ \+ _yz_ ) = 2 _c_
(there are two faces of area _xy_ , two of area _xz_ and two of area _yz_ ). Thus our problem is to find the largest value of (7), subject to the "side condition" or "constraint"
Suppose we solve (8) for _z_ , obtaining
Substituting (9) into (7), we then get
and we can solve the problem by maximizing (10) as a function of the _two_ variables _x_ and _y_. This approach leads to no particular difficulties (see Probs. 9 and 10), but we now solve the problem by another method, which is both more elegant and of interest in its own right.
Suppose we multiply the constraint (8) by a new variable _λ_ , called a _Lagrange multiplier_ , and add the result to (7), obtaining a new function
of _four_ variables _x_ , _y_ , _z_ and _λ_. We then look for local extrema of _V*_ , by the usual technique of setting the partial derivatives of _V*_ equal to zero:
The fact that we get back the constraint (8) as the last of these equations is, of course, essential to the success of the method. Suppose the four equations (12) can be solved for _λ_ as a function of _x_ , _y_ and _z_. Then, substituting _λ_ back into the first three equations, we get three equations in _x_ , _y_ and _z_ , whose solutions are just the values of _x_ , _y_ and _z_ for which the original function _V_ achieves its local extrema, subject to the constraint (8). To see this, we merely observe that if the equations (12) hold, then the constraint (8) is automatically satisfied, so that the term in parentheses in (11) vanishes, and the "unconstrained extrema" of _V*_ reduce to the "constrained extrema" of _V_.
We now solve the equations (12) for _λ_. To this end, we multiply the first equation by _x_ , the second by _y_ , and the third by _z_. We then add the results and invoke the fourth equation, namely the constraint, obtaining
3 _xyz_ \+ 2 _λ_ ( _xy_ \+ _xz_ \+ _yz_ ) = 3 _xyz_ \+ 2 _λc_ = 0,
which determines _λ_ as a function of _x_ , _y_ and _z_ :
Substituting (13) back into the first three equations (12), we get
or equivalently,
since _x_ , _y_ and _z_ are nonzero, because of their physical meaning. The first and second of the equations (14) imply _x_ ( _y_ \+ _z_ ) = _y_ ( _x_ \+ _z_ ) and hence _x_ = _y_ , while the second and third imply _y_ ( _x_ \+ _z_ ) = _z_ ( _x_ \+ _y_ ) and hence _y_ = _z_ , so that
Substituting (15) into the constraint (8), we get 3 _x_ 2 = _c_. It follows that
It is "physically obvious" that the function _V_ must have a maximum, rather than a minimum, at the point (16), since there are long, skinny bricks with surface area 2 _c_ which have "arbitrarily small" volume (see Prob. 5). Thus, finally, we find that the brick of largest volume with surface area 2 _c_ is the cube of side ( _c_ /3)1/2 and volume ( _c_ /3)3/2.
PROBLEMS
**.** By investigating all critical points, find the local extrema, if any, of
(a) _z_ = 3 _x_ \+ 6 _y_ − _x_ 2 − _xy_ \+ _y_ 2;(b) _z_ = _x_ 2 − _xy_ \+ _y_ 2 − 2 _x_ \+ _y_ ;
(c) _z_ = 2 _x_ 3 − _xy_ 2 \+ 5 _x_ 2 \+ _y_ 2.
**.** Do the same for
(a) _z_ = 2 _x_ 3 \+ _xy_ 2 − 216 _x_ ;
(d) _z_ = ( _x_ − _y_ \+ 1)2.
**3.** Find the global extrema of the function _z_ = _x_ 2 − _y_ 2 in the closed disk _x_ 2 \+ _y_ 2 ≤ 4.
**.** Find the global extrema of the function _z_ = (2 _x_ 2 \+ 3 _y_ 2) _e_ − _x_ 2− _y_ 2 in the closed disk _x_ 2 \+ _y_ 2 ≤ 1.
**.** Let _V_ be the same as in Example 6.42e. Show that _V_ → 0 as _z_ → ∞.
**.** Suppose that in Example 6.42e we _subtract λ_ times the constraint (8) from the function (7), obtaining the new function
_V*_ = _xyz_ − _λ_ ( _xy_ \+ _xz_ \+ _yz_ − _c_ )
instead of (11). Does this have any effect on the final answer?
**7.** Use a Lagrange multiplier to show that _c n_ is the largest value of the product of _n_ positive numbers _x_ 1, _x_ 2, . . ., _x_ _n_ with a given sum _nc_.
***8.** Use a Lagrange multiplier to show that
for arbitrary positive numbers _x_ 1, _x_ 2, . .., _x_ _n_ , thereby generalizing Sec. 1.4, Problem 14.
***9.** Solve Example 6.42e by maximizing (10).
***10.** Use Theorem 6.41c to confirm that the function _V_ = _xyz_ has a strict local maximum at the point (16), subject to the constraint (8).
***11.** Use a Lagrange multiplier to derive formula (11), p. 34, for the distance _d_ between a point _P_ 1 = ( _x_ 1, _y_ 1) and the line _Ax_ \+ _By_ \+ _C_ = 0.
***12.** Suppose a firm produces two commodities. Then the _total cost_ to the firm of producing a quantity _Q_ 1 of the first commodity and a quantity _Q_ 2 of the second commodity is some function of _Q_ 1 and _Q_ 2, called the _cost function_ and denoted by _C_ ( _Q_ 1, _Q_ 2). There are now two _marginal costs_ , _MC_ 1( _Q_ 1, _Q_ 2) = ∂ _C_ ( _Q_ 1, _Q_ 2)/∂ _Q_ 1, the marginal cost of the first commodity, and _MC_ 2( _Q_ 1, _Q_ 2) = ∂ _C_ ( _Q_ 1, _Q_ 2)/∂ _Q_ 2, the marginal cost of the second commodity. This is, of course, just the natural extension of the considerations of Sec. 3.22a to the case of a two-commodity firm.
Suppose the firm's cost function is
Find the marginal costs _MC_ 1( _Q_ 1, _Q_ 2) and _MC_ 2( _Q_ 1, _Q_ 2). Suppose further that the two commodities are sold at predetermined prices _P_ 1 and _P_ 2, chosen to make them sell in a competitive market. Write an expression for the firm's profit Π( _Q_ 1, _Q_ 2), which is now a function of two variables. What output levels of the two commodities maximize this profit? Verify that your answer actually leads to a maximum. What condition must the prices satisfy?
***13.** In the preceding problem, suppose profit is to be maximized subject to a constraint of the form _Q_ 1 \+ _Q_ 2 = _q_ > 0. For example, an automobile manufacturer may want to make a given total number of sedans and station wagons. What output levels maximize the profit in this case? What condition must now be satisfied by the prices and the number _q_?
TABLES
Table 1. GREEK ALPHABET
Table 2. EXPONENTIAL FUNCTIONS
Table 3. NATURAL LOGARITHMS
* Take tabular value – 10.
Table 4. ELEMENTARY DIFFERENTIATION RULES*
* Here _c_ , _r_ and _a_ > 0 are arbitrary constants, _f_ and _g_ are functions of the independent variable _x_ , _f_ −1 is the inverse of _f_ , and _F_ is a function of the dependent variable _y_. The prime denotes differentiation with respect to _x_.
SELECTED HINTS AND ANSWERS
Chapter 1
**Sec. 1.2**
**.** { _a_ }, { _b_ }, { _b_ }, { _a_ , _b_ }, { _a_ , _c_ }, { _b_ , _c_ }, ∅.
**.** (a){0};(c){3, −3};(e){a, c, 1, s, u}.
**.** Only (a) is true. _A_ has 4 elements.
**.** All but (d) are true.
**.** (a){ _a_ , _b_ , _c_ , _d_ }.
**.** (a){3, 4}.
**.** Trivial, but give details anyway.
**.** (a) {3};(c) {1, 2, 3}.
**.** Only (c) and (e) are empty.
**.** 16.
**Sec. 1.3**
**.** Yes. −1 − (−2) = 1, (−1)(−1) = 1, − 1 ÷ −1 = 1.
**.**
**.** Let _m_ / _n_ and _m_ ′/ _n_ ′ be two rational numbers, where _m_ , _n_ (≠0), _m_ ′, _n_ ′ (≠0) are integers. Then , where _mn_ ′ + _m_ ′ _n_ and _nn_ ′ (≠0) are again integers, and similarly for subtraction and multiplication. If _m_ ′ ≠ 0 as well, so that _m_ ′/ _n_ ′ ≠ 0, then , where _mn_ ′ and _m_ ′ _n_ (≠0) are again integers.
**.** All but (a) exist.
**.** Irrational.
**.** , where both terms are irrational.
**.** is rational, and so is .
**.** The set of irrational numbers is not closed under any of the operations.
**.** On the one hand, 0 · _c_ \+ 1 · _c_ = 0 · _c_ \+ _c_ , while on the other hand, 0 · _c_ \+ 1 · _c =_ (0 + 1) · _c_ = 1 · _c_ = _c_ , so that 0 · _c_ \+ _c_ = _c_. Now subtract _c_ from both sides.
**.** The formula holds for _n_ = 1, since . Suppose the formula is true for _n_ = _k_ , so that 1 + 2 + . . . + _k_ = . Then 1 + 2 + . . . + _k_ \+ ( _k_ \+ 1) = , so that the formula also holds for _n_ = _k_ \+ 1.
**.** By definition, _a_ /0 is the number _c_ such that 0 · _c_ = _a_. But 0 · _c_ = 0 for all _c_ (Prob. 9), and hence there is no such _c_ unless _a_ = 0. If _a_ = 0, we get 0/0 which is meaningless, not because there is _no_ number _c_ such that 0 · _c_ = 0, but rather because _every_ number _c_ has this property!
**.**
**.** = 0.3333... + 0.1666... = 0.4999... Let _x_ = 0.4999... Then 10 _x_ = 4.9999..., and hence 9 _x_ = 10 _x_ − _x_ = 4.9999... − 0.4999... = 4.5000... = 4.5, so that _x_ = 4.5/9 = 0.5. Thus a decimal with an endless run of nines after a certain place represents the same rational number as the "next highest decimal," which always terminates.
**.** Use the same reasoning as in Prob. 15.
**.** 1.414214673...
**.** Suppose _k_ is a sum of threes and fives exclusively. Then this sum either contains a five or it does not. In the first case, replace a five by 2 threes. In the second case, there are at least 3 threes, since _k_ exceeds 7, by hypothesis, and we can replace 3 threes by 2 fives. To start the induction (Sec. 1.37c), note that 8 = 5 + 3.
**Sec. 1.4**
**.** (a) _a_ − _b_ > 0, and hence − _a_ − (− _b_ ) < 0.
**.** _p_ > _p_ ′ means _p_ − _p_ ′ > 0, or equivalently . But _nn_ ′ > 0, and hence _mn_ ′ − _m_ ′ _n_ > 0, or equivalently _mn_ ′ > _m_ ′ _n_.
**.**
**.** By Theorem 1.43 twice, _ac_ > _bc_ and _bc_ > _bd_. Hence _ac_ > _bd_ , by Theorem 1.45.
**.** Clearly _a_ ≠ _b._ If _a_ > _b_ , then, by Prob. 4, with _c_ = _a_ , _d_ = _b_ , we have _a_ 2 > _b_ 2, contrary to _b_ 2 > _a_ 2.
**.** An immediate consequence of Theorem 1.43. If _a_ 2 = _a_ , then _a_ = 0 or _a_ = 1.
**.** (a) 0;(c) 1;(e) −1.
**.** (b) _n_.
**.** If _p_ < _q_ , then . Adding first _p_ and then _q_ to both sides of the last inequality, we get and . Let _r_ and _s_ be rational numbers such that _r_ < 1, _s_ > 0. Then and _ _are rational numbers such that _r_ < _r_ ′ < 1, 0 < _s_ ′ < _s_. The argument works equally well for real _p_ , _q_ , _r_ , _s_ , _r_ ′, _s_ ′.
**.** (b) Show that ( _a_ \+ _b_ )2 ≥ 4 _ab_ is equivalent to ( _a_ − _b_ )2 ≥ 0.
**Sec. 1.5**
**.** Examine cases. For example, if _x_ > 0, _y_ < 0, then | _x_ | = _x_ , | _y_ | = − _y_ , | _xy_ | = − _xy_.
**.** By two applications of the inequality (3), | _x_ \+ _y_ \+ _z_ | = |( _x_ \+ _y_ ) + _z_ | ≤ | _x_ \+ _y_ | + | _z_ | ≤ | _x_ | + | _y_ | + | _z_ |.
**.** 0, .
**.** _x_ 2 lies to the right of _x_ if _x_ > 1 or if _x_ < 0, and to the left of _x_ if 0 < _x_ < 1. _x_ 2 and _x_ coincide if _x_ = 0 or _x_ = 1.
**.**
**.** First replace _x_ by _x_ − _y_ in (3), and then replace _y_ by _y_ − _x_. Equality occurs under the same conditions as for (3) itself, namely if _x_ and _y_ have the same sign or if one (or both) of the numbers _x_ and _y_ is zero.
**.** The point moves from _a_ to _b_.
**.** (a) If _n_ is any integer greater than 1/| _x_ 1 − _x_ 2|, and if _c_ is the irrational number , then at least one of the rational numbers..., −3/2 _n_ , −1/2 _n_ , l/2 _n_ , 3/2 _n_ , ... and at least one of the irrational numbers..., − 3 _c_ /2 _n_ , − _c_ /2 _n_ , _c_ /2 _n_ , 3 _c_ /2 _n_ , ... falls between _x_ 1 and _x_ 2;(b) Apply (a) repeatedly.
**Sec. 1.6**
**.** (0, 2). [−3, 3].
**.** The interval 1 ≤ _x_ ≤ 2.
**.** (−2, 1). [−1, 2].
**.** (a) (1, 3]; (c) (−∞, ∞).
**.** (b) {1}.
**.** (0, ∞), (−∞, 1].
**Sec. 1.7**
**.** A six-pointed star.
**.** _A_ ′ = (3, 2), _B_ ′ = (3, 4), _C_ ′ = (1, 5), _D_ ′ = (−1, 4), _E_ ′ = (−1, 2), _F_ ′ = (1, 1). Each abscissa is increased by 1 and each ordinate by 2.
**.** _x_ < 0, _y_ > 0 in the second quadrant, _x_ < 0, _y_ < 0 in the third, _x_ > 0, _y_ < 0 in the fourth.
**.** (a) ; (c) 1.
**.** 5.
**.** | _AB_ | = | _BC_ | = | _CD_ | = | _DA_ | = and | _AC_ | = | _BD_ | .
**.** (3, 3), (15, 15).
**.** 21.
**.** _C_ = ( _x_ \+ _x_ ′, _y_ \+ _y_ ′).
**Sec. 1.8**
**.** (a) 11; (c) Slope undefined.
**.** When _m_ = _m_ ′.
**.** (a) 45°; (c) 135°.
**.** (a) tan 20° = 0.36397; (c) tan 165° = −tan 15° = −0.26795.
**.** −1.
**.** The lines are perpendicular.
**Sec 1.9**
**.** (a) _y_ = 2 _x_ − 2;(c) _y_ = 2 _x_ − 1;(e) _y_ = 2 _x_ − 3.
**.** (a) _y_ = 7 _x_ − 19;(c) _y_ = _x_ \+ 4.
**.** (a) _y_ = − _x_ \+ 1;(c) _y_ = −2.
**.** (a) _m_ = 3, _a_ = 2, _b_ = −6;(c) _m_ = −1, _a_ = 3, _b_ = 3.
**.** (a) _m_ = 5, _a_ = , _b_ = 4;(c) _m_ = 0, _a_ undefined, _b_ = 3.
**.** _y_ = 2 _x_ − 8.
**.** _y_ = − _x_ \+ 3. .
**.** 20.
**.** _y_ = −2 _x_.
**.** The line has slope _m_ = − _b_ / _a_. Substitute this value of _m_ in formula (2).
**.** (b) _x_ \+ 3 _y_ \+ 3 = 0.
**.** _y_ = − _x_ \+ 5, _y_ = .
**.** .
**.** (a) 2;(c) 0.
Chapter 2
**Sec. 2.1**
**.** _f_ (0) = 6, _f_ (1) = 10, _f_ (2) = 16, .
**.** does not exist.
**.** (a) Domain all _x_ such that | _x_ | ≥ 3, range all _y_ ≥ 0;(c) Domain all _x_ ≠ 3, range all _y_ ≠ 0.
**.** _f_ (3, 1) = , _f_ (0, 1) = 2, _f_ (1, 0) = , _f_ ( _a_ , _a_ ) = −1, _f_ ( _a_ , − _a_ ) = 1.
**.** Domain all points in the _xy_ -plane except the origin, range −∞ < _z_ < ∞.
**.** Yes.
**.** No. Yes.
**.** No.
**.** Yes.
**.** True.
**.** False.
**.** _f_ (1, 1, 1) = 3, _f_ (4, 1, 9) = , _f_ (1, 9, 1) = , _f_ (4, 9, 16) = .
**.** Yes.
**.** The inverse function has domain _Y_ and range _X_.
**.** All but (e). The inverses are: (a) _x_ = _y_ (b) _x_ = 1/ _y_ ;(c) _x_ = 1 + (1/ _y_ );(d) _x_ = _y_ 2 ( _y_ ≥ 0).
**.** False.
**.** Domain −∞ < _x_ < ∞, range the set of all integers.
**.** No, only when every _y_ _Y_ is the second element of a pair ( _x_ , _y_ ) _f_.
**.** 2 _n_.
**.** When no two ordered pairs in _f_ have the same second element. To get _f_ −1, write all the pairs in _f_ in reverse order.
**.** Delete either the point _x_ 1 and the arrow joining _x_ 1 to _y_ 1 or the point _x_ ′1 and the arrow joining _x_ ′1 to _y_ 1.
**.** Reverse the directions of all the arrows.
**.** (a) Finite;(c) Infinite.
**Sec. 2.2**
**.** (a) _x_ ≠ 0;(b) _x_ ≥ 0;(c) _x_ ≥ 0.
**.**
**.** 2.
**.** False.
**.**
**.** 1, 4, 9,..., _n_ 2, ... (recall Sec. 1.37a).
**.** 1, 1, 2, 3, 5, 8, 13, 21.
**.** Examine the three cases −∞ < _x_ < −1, −1 ≤ _x_ ≤ 1 and 1 < _x_ < ∞ separately.
**.**
**Sec. 2.3**
**.** The pair of intersecting lines _y_ = _x_ and _y_ = − _x_.
**.** _x_ 2 \+ _y 2_ \+ 4 _x_ − 6 _y_ \+ 9 = 0.
**.** True. If the line _x_ = _c_ intersects the graph in more than one point, then the function takes more than one value at _x_ = _c_ , contrary to the definition of a function.
**.** False. Consider circles, for example.
**.** See Prob. 4.
**.** No line parallel to either the _x_ -axis or the _y_ -axis can intersect the graph in more than one point.
**.** (a), (d) and (f) are even, (c) and (e) are odd, (b) is neither even nor odd.
**.** Note that (−1) _n_ = 1 if _n_ is even, while (−1) _n_ = −1 if _n_ is odd.
**.** True.
**.** _x_ 2 \+ _y_ 2 − _x_ − _y_ = 0.
**.** Reflect _G_ in the line _y_ = _x_. Then interchange the labelling of the coordinate axes.
**.** If _x_ ≠ _x_ ′, then either _x_ < _x_ ′ or _x_ ′ < _x_. Since _f_ is increasing, _f_ ( _x_ ) < _f_ ( _x_ ′) in the first case and _f_ ( _x_ ′) < _f_ ( _x_ ) in the second case, so that in any event _f_ ( _x_ ) ≠ _f_ ( _x_ ′). Therefore _f_ is one-to-one, with an inverse _f_ −1. Let _y_ = _f_ ( _x_ ), _y_ ′ = _f_ ( _x_ ′), so that _x_ = _f_ −1( _y_ ), _x_ ′ = _f_ −1( _y_ ′), and suppose _y_ < _y_ ′. Then _x_ ≠ _x_ ′, since _f_ −1 is itself one-to-one, but _x_ ′ < _x_ is impossible, since then _y_ ′ < _y_. Therefore _x_ < _x_ ′, so that _f_ −1 is also increasing.
**.** Use Probs. 12 and 15.
**.** See Figure 1A.
**.** See Figure 1B.
**.** Suppose 0 < _x_ < _x_ ′ and 0 < _x k_ < _x_ ′ _k_. Then 0 < _x_ _k_ \+ 1 < _x_ ′ _k_ \+ 1, by Sec. 1.4, Prob. 4. The result now follows by mathematical induction (Sec. 1.37).
**.** Use Prob. 19 and the symmetry of the curve _y_ = _x_ _n_ (recall Prob. 10).
Figure 1.
**Sec. 2.4**
**.** No, unless _m_ = 0. Yes.
**.** Yes.
**.** If | _f_ ( _x_ ) − _A_ | is "arbitrarily small," then so is |[ _f_ ( _x_ ) − _A_ ] − 0|, and conversely.
**.** (a) 0;(c) 1;(e) No limit.
**.** No. Yes.
**.** (b) 0.
**.** In general, yes. No.
**.** If | _f_ ( _x_ ) − 0| is "arbitrarily small," then so is || _f_ ( _x_ )| − 0| = || _f_ ( _x_ )|| = | _f_ ( _x_ )|, and conversely.
**.** No.
**.** Suppose both _f_ ( _x_ ) → _A_ 1 as _x_ → _x_ 0 and _f_ ( _x_ ) → _A_ 2 as _x_ → _x_ 0, where _A_ 1 ≠ _A_ 2. Then, choosing , we can find numbers _δ_ 1 > 0 and _δ_ 2 > 0 such that | _f_ ( _x_ 1) − _A_ 1| < _ε_ whenever 0 < | _x_ − _x_ 0| < _δ_ 1 and | _f_ ( _x_ ) − _A_ 2| < _ε_ whenever 0 < | _x_ − _x_ 0| < _δ_ 2. Let _δ_ be the smaller of the two numbers _δ_ 1 and _δ_ 2. Then | _A_ 1 − _A_ 2| = | _A_ 1 − _f_ ( _x_ ) + _f_ ( _x_ ) − _A_ 2| ≤ | _A_ 1 − _f_ ( _x_ )| + | _f_ ( _x_ ) − _A_ 2| < 2 _ε_ = | _A_ 1 − _A_ 2| whenever 0 < | _x_ − _x_ 0| < _δ_. But this is impossible!
**.** Let _f_ ( _x_ ) → _A_ as _x_ → _x_ 0, and let _f_ 1( _x_ ) = _f_ ( _x_ ) everywhere except at _x_ = _x_ 1. Then, given any _ε_ > 0, there is a _δ_ > 0 such that | _f_ ( _x_ ) − _A_ | < _ε_ whenever 0 < | _x_ − _x_ 0| < _δ_. Let _δ_ 1 be the smaller of the numbers _δ_ and | _x_ 0 − _x_ 1| ≠ 0. Then, given any _ε_ > 0, we have | _f_ 1( _x_ ) − _A_ | = | _f_ ( _x_ ) − _A_ | < _ε_ whenever 0 < | _x_ − _x_ 0| < _δ_ 1, so that _f_ 1( _x_ ) → _A_ as _x_ → _x_ 0.
**.** Choosing _ε_ = 1, we can find a _δ_ > 0 such that | _f_ ( _x_ ) − _A_ | < 1 whenever 0 < | _x_ − _x_ 0| < _δ_. But | _f_ ( _x_ ) − _A_ | ≥ | _f_ ( _x_ )| − | _A_ |, with the help of Sec. 1.5, Prob. 10. Therefore 0 < | _x_ − _x_ 0| < _δ_ implies | _f_ ( _x_ )| − | _A_ | < 1, or equivalently | _f_ ( _x_ )| < | _A_ | + 1.
**.** If _A_ > 0, choose . Then there is a _δ_ > 0 such that 0 < | _x_ − _x_ 0| < _δ_ implies | _f_ ( _x_ ) − _A_ | < _A_ , or equivalently , so that, in particular, _f_ ( _x_ ) > 0 and . If _A_ < 0, choose . Then there is a _δ_ > 0 such that 0 < | _x_ − _x_ 0| < _δ_ implies , or equivalently , so that, in particular, _f_ ( _x_ ) < 0 and , or equivalently .
**Sec. 2.5**
**.** Δ _x_ = −0.009, Δ _y_ = 990,000.
**.** No. Yes, the tangents at any pair of points are parallel.
**.** _y_ = 0 or _y_ = 8 _x_ − 16.
**.** (2, 4).
**.** −2.
**.** (a) Δ _y_ = 7, _dy_ = 3, _E_ = 4, about 57% of Δ _y_ ;(c) Δ _y_ = 0.030301, _dy_ = 0.03, _E_ = 0.000301, about 1% of Δ _y_. The approximation of Δ _y_ by _dy_ improves as Δ _x_ gets smaller.
**.** When |Δ _x_ | is too large or when _f_ ′( _x_ ) = 0.
**.** Δ( _uv_ ) = Δ _u_ ( _x_ ) _v_ ( _x_ \+ Δ _x_ ) + _u_ ( _x_ ) Δ _v_ ( _x_ ) = _u_ ( _x_ \+ Δ _x_ ) Δ _v_ ( _x_ ) + Δ _u_ ( _x_ ) _v_ ( _x_ ).
**.** _b_ = −3, _c_ = 4.
**.** Recall Sec. 2.3, Prob. 17.
**.** About 19 square miles, almost as large as Manhattan Island.
**.** In Figure 2, drawn for the case Δ _x_ > 0, 0 < _dy_ < Δ _y_ , we have _dy_ = | _AB_ |, Δ _y_ = | _AQ_ |. Thus _dy_ is the increment of the ordinate of the tangent _T_ to the curve _y_ = _f_ ( _x_ ), while Δ _y_ is the increment of the ordinate of the curve itself.
Figure 2.
**Sec. 2.6**
**.** Apply formulas (2) and (4) repeatedly.
**.** (a) 0;(c) .
**.** At _x_ = 1, 2.
**.**
**.** See Figure 3. At _x_ = 0, ± 1, ±2,...
Figure 3.
**.** Examine the figure, noting that the solid dots belong to the graph.
**.** No. Yes. No.
**.** No, since such intervals do not contain their end points.
**.** We have
Given any _ε_ > 0, there are positive numbers _δ_ 1, _δ_ 2, _δ_ 3 such that | _f_ ( _x_ )| < | _A_ | + 1 whenever 0 < | _x_ − _x_ 0| < _δ_ 1, | _f_ ( _x_ ) − _A_ | < whenever 0 < | _x_ − _x_ 0| < _δ_ 2, and | _g_ ( _x_ ) − _B_ | < whenever 0 < | _x_ − _x_ 0| < _δ_ 3. Let _δ_ be the smallest of the numbers _δ_ 1, _δ_ 2, _δ_ 3. Then
whenever 0 < | _x_ − _x_ 0| < _δ_.
Similarly,
Given any _ε_ > 0, there are positive numbers _δ_ 1, _δ_ 2, _δ_ 3 such that | _g_ ( _x_ )| > whenever 0 < | _x_ − _x_ 0| < _δ_ 1, | _f_ ( _x_ ) − _A_ | < whenever 0 < | _x_ − _x_ 0| < _δ_ 2, and | _g_ ( _x_ ) − _B_ | < whenever 0 < | _x_ − _x_ 0| < _δ_ 3. Let _δ_ 3 be the smallest of the numbers _δ_ 1, _δ_ 2, _δ_ 3. Then
whenever 0 < | _x_ − _x_ 0| < _δ_.
**Sec 2.7**
**.** (a) 4 _x_ 3 \+ 6 _x_ ;
**.** (a) 2 _x_ − ( _a_ \+ _b_ );(c) 32 _x_ 3 \+ 12 _x_.
**.**
**.** True.
**.**
**.** True.
**.**
**.** _y_ (6) = 4 · 6!, _y_ (7) = 0.
**Sec. 2.8**
**.**
**.**
**.**
**.** 200 · 399.
**.**
**.**
**.** Start from
**.** In the interval 0 ≤ _x_ < ∞. In the whole interval −∞ < _x_ < ∞.
**.**
**.**
**.** where we make free use of formulas (15) and (16), p. 78.
**.**
**.** Solving (16) for _y_ 2, we get . Hence the only real solution of (15) is _x_ = _y_ = 0, and it is meaningless to talk about _y_ ′.
**Sec. 2.9**
**.** (a) 1;
**.** (a) −∞;(c) ∞.
**.** _f_ ( _x_ ) → ∞ as _x_ → 0+, 1−, 2+, while _f_ ( _x_ ) → −∞ as _x_ → 0−, 1+, 2−.
**.**
**.** _x_ > 2998.
**.**
**.** (a) _x_ = −2, _y_ = ;(c) _x_ = ±2, _y_ = 1.
**.** See Figure 16.
**.** (a) 1;(c) .
**.** None.
**.** The formula obviously holds for _n_ = 1. Suppose it holds for _n_ = _k_ , so that (1 + _x_ ) _k_ ≥ 1 + _kx_. Then it holds for _n_ = _k_ \+ 1, since (1 + _x_ ) _k_ \+ 1 = (1 + _x_ ) _k_ (1 + _x_ ) ≥ (1 + _kx_ )(1 + _x_ ) = 1 + ( _k_ \+ 1) _x_ \+ _kx_ 2 ≥ 1 + ( _k_ \+ 1) _x_.
**.** 1 if | _a_ | > 1, 0 if | _a_ | < 1, if _a_ = 1.
**.** (a) ;(c) 21 \+ 32 \+ 43 \+ 54 = 700.
**.** (a) 2;(c) 1. Note that .
**.** There are no points at which _f_ ( _x_ ) → ±∞, and _f_ ( _x_ ) does not approach a finite limit as _x_ → ±∞. However,
**.** Let _x n_ equal to _n_ decimal places.
**.** Note that _x n_ = _s_ _n_ \+ 1 − _s n_ → _s_ − _s_ = 0 as _n_ → ∞.
Chapter 3
**Sec. 3.1**
**.** The average velocities are 215, 210.5 and 210.05 ft/sec. The instantaneous velocity is 210 ft/sec.
**.** The acceleration is variable.
**.** The stone hits the ground 3 seconds later, travelling at a speed of 64 ft/sec.
**.** The car is accelerating with a constant acceleration of _k_ = 8.8 ft/sec2. Equation (7) can only be valid when the car's speed is well below its top speed.
**.** The car is decelerating with a constant deceleration of _k_ = 4 ft/sec2.
**.** The stone's motion during the last 3 seconds is the "reverse" of its motion during the first 3 seconds (make this precise).
**.** The flywheel has angular velocity _θ_ ′( _t_ ) = _b_ − 2 _ct_ and angular acceleration _θ_ ″( _t_ ) = −2 _c_.
**Sec 3.2**
**.** 5/8π ≈ 0.2 ft/min = 2.4 in/min.
**.** No. Yes.
**.** Increasing, at 40 in2/sec.
**.**
**.** _Q_ ( _P_ ) is a decreasing function of _P_.
**.** Since _Q_ = _Q_ ( _P_ ) is decreasing, it has a decreasing inverse _P_ = _P_ ( _Q_ ), by Sec. 2.3, Prob. 16, so that _PQ_ = _PQ_ ( _P_ ) = _QP_ ( _Q_ ).
**.** ft/min2.
**Sec. 3.3**
**.** (a) No extrema;(c) A maximum equal to 1 at _x_ = 1, a minimum equal to at _x_ = 2;(e) No maximum, a minimum equal to −1 at _x_ = −1.
**.** (a) A maximum and a minimum, both equal to 0, at every point of (0, 1);(c) A maximum equal to 0 at _x_ = 0, a minimum equal to −1 at every point of (−1, 0);(e) No extrema.
**.** True, by the intermediate value theorem.
**.** No. Yes.
**.** No maximum, a minimum equal to −1 at _x_ = 0.
**.** The function _f_ ( _x_ ) = 1/ _x_ maps (0, 1) Into (1, ∞).
**.** The function graphed in Figure 14, p. 91, maps (−∞, ∞) into (1, 2).
**.** Let
These functions are all continuous in (−2, 2), say; _f_ maps (−2, 2) into (−2, 2), _g_ maps (−2, 2) into [0, 2), and _h_ maps (−2, 2) into [−1, 1].
**Sec. 3.4**
**.** _f_ ′( _x_ ) = 3 _x_ 2 − 12 _x_ \+ 11 = 0 if .
**.** The formula _f_ ( _a_ ) − _f_ ( _b_ ) = _f_ ′( _c_ )( _a_ − _b_ ) is equivalent to (10), but now _c_ ( _b_ , _a_ ).
**.** _c_ = .
**.** Apply formula (9) to the train's distance function _s_ = _s_ ( _t_ ).
**.** Apply the mean value theorem to the function .
**Sec. 3.5**
**.** The domain of _f_ is not an interval.
**.** (b) −(1/ _x_ ) + _C_.
**.** Trivial, but worthy of note.
**.** Use the chain rule.
**.**
**.** True.
**.** (b) Increasing in [−1, 1], decreasing in (−∞, − 1] and [1, ∞).
**.** _f_ is a polynomial of degree less than _n_.
**Sec. 3.6**
**.** Reread Sec. 3.62a.
**.** (a) Minimum _y_ = 0 at _x_ = 0;(c) No extrema.
**.** (b) Maximum _y_ = − 2 at _x_ = −1, minimum _y_ = 2 at _x_ = 1.
**.** (a) Maximum _y_ = 66 at _x_ = 10, minimum _y_ = 2 at _x_ = 2;
(c) Maximum _y_ = 3 at _x_ = −1, minimum _y_ = 1 at _x_ = 1.
**.** _y_ ′ ≠ 0 if _ad_ − _bc_ ≠ 0, _y_ ≡ constant if _ad_ − _bc_ = 0.
**.** _c_ = − .
**.**
**Sec. 3.7**
**.** Nothing. Explain.
**.** Inflection points at _x_ = ± , concave upward in , concave downward in , concave upward in .
**.** _c_ = −3.
**.** See Figure 4.
**.** See Figure 5.
**.** Let _y_ = _T_ ( _x_ ) be the tangent to the curve _y_ = _f_ ( _x_ ) at _x_ = _p_. Then, by the mean value theorem in increment form (Sec. 3.43b),
where Δ _x_ = _x_ − _p_ and 0 < α < 1. If _f_ ′ is increasing in a _δ_ -neighborhood of _p_ , then _f_ ′( _p_ \+ αΔ _x_ ) − _f_ ′( _p_ ) < 0 if − _δ_ < Δ _x_ < 0, while _f_ ′( _p_ \+ αΔ _x_ ) − _f_ ′( _p_ ) > 0 if 0 < Δ _x_ < _δ_ , so that, in either case, the right side of (i) is positive. Therefore _f_ ( _x_ ) > _T_ ( _x_ ) in the _δ_ -neighborhood, so that _f_ is concave upward at _p_. The proof for decreasing _f_ ′ is virtually the same.
**.** Again we start from (i). Suppose the extremum is a maximum. Then _f_ ′( _p_ \+ αΔ _x_ ) − _f_ ′( _p_ ) < 0 if − _δ_ < Δ _x_ < 0, while _f_ ′( _p_ \+ αΔ _x_ ) − _f_ ′( _p_ ) < 0 if 0 < Δ _x_ < _δ_ , so that the right side of (i) is positive to the left of p and negative to the right of _p_. Therefore _f_ ( _x_ ) > _T_ ( _x_ ) to the left of _p_ , while _f_ ( _x_ ) < _T_ ( _x_ ) to the right of _p_ , so that _p_ is an inflection point of _f_. The proof for the case of a minimum is virtually the same.
Figure 4.
Figure 5.
**Sec. 3.8**
**.** The square of side .
**.** The triangle with legs _c_ /3 and .
**.** .
**.**
**.**
**.**
**.** The line ( _x_ /2 _a_ ) + ( _y_ /2 _b_ ) = 1, with _x_ -intercept 2 _a_ and _y_ -intercept 2 _b_.
**.** _Q_ = 30.
**.** The chord whose distance from the point _A_ equals of the diameter of the circle.
**.**
Chapter 4
**Sec. 4.1**
**.** (a) 2;(c) max _A_ does not exist.
**.** ( _b_ − _a_ )/ _n_. No, although _λ_ < _b_ − _a_.
**.** Yes. Define the area _A_ between the curve _y_ = _f_ ( _x_ ) and the _x_ -axis from _x_ = _a_ to _x_ = _b_ by the integral , whether or not _f_ ( _x_ ) is nonnegative. Then _A_ < 0 if more area lies "above" the curve than "below" it.
**.** max _A_ = _a_ if 0 ≤ _a_ ≤ 1, max _A_ does not exist if _a_ > 1, max _A_ = _a_ 2 if −1 ≤ _a_ < 0, max _A_ does not exist if _a_ < − 1.
**.** Show that 0 ≤ _σ_ ≤ _λ_ , where _σ_ is the sum (3), regardless of the choice of the points _ξ_ 1, _ξ_ 2, ..., _ξ_ _n_. Therefore _σ_ → 0 as _λ_ → 0.
**.** (b) Choosing all the points _ξ_ 1, _ξ_ 2, ..., _ξ_ n in the sum (3) to be rational, we have _σ_ = 1, and choosing them all to be irrational, we have _σ_ = −1, regardless of the size of _λ_. Therefore _σ_ cannot approach a limit as _λ_ → 0.
**Sec. 4.2**
**.**
**.**
**.** . See Sec. 2.3, Prob. 14.
**.** _c_ = 4.
**.** Use the fundamental theorem of calculus, noting that _v_ = _ds_ / _dt_.
**.** An immediate consequence of formula (11).
**.** Given any _ε_ > 0, there is a _δ_ > 0 such that | _σ_ ( _λ_ ) − _σ_ 0| < _ε_ , or equivalently _σ_ 0 − _ε_ < _σ_ ( _λ_ ) < _σ_ 0 \+ _ε_ , whenever 0 < | _λ_ | < _δ_. If _σ_ 0 < 0, choose _ε_ = − _σ_ 0 > 0. Then there is a _δ_ > 0 such that 2 _σ_ 0 < _σ_ ( _λ_ ) < 0 whenever 0 < | _λ_ | < _δ_ , which contradicts _σ_ ( _λ_ ) ≥ 0. Therefore _σ_ 0 ≥ 0.
**.** Apply Prob. 12 to the functions _σ_ ( _λ_ ) − _A_ and _B_ − _σ_ ( _λ_ ).
**.** (b) − _f_ ( _a_ ).
**.** Follow the argument used to prove formula (7).
**.** Apply Prob. 16 to the function _f_ = _f_ 2 − _f_ 1.
**.** Note that , where equality occurs only if _x_ = 0 or _x_ = 2.
**Sec. 4.3**
**.** No, since In ( _x_ 2) is defined for all _x_ ≠ 0, while 2 In _x_ is defined only for _x_ > 0.
**.** (a) 4 ≤ _x_ ≤ 6;(c) _x_ > _e_.
**.**
**.**
**.**
**.** _y_ (4) = −2/ _x_ 2.
**.** 1.
**.** At _x_ = 1. No.
**.** (a) _x_ ≥ 1 if _a_ > 1, 0 < _x_ ≤ 1 if 0 < _a_ < 1.
**.** Use (7).
**.** Note that .
**Sec. 4.4**
**.** (a) 4 _e_ 4 _x_ \+ 5;(c) _e x_(1 + _x_ ).
**.** (a) 2 _xe_ _x_ 2;
**.** α = ln ( _e_ − 1) ≈ 0.54.
**.**
Figure 6.
**.** Inflection points at _x_ = ± 1, concave upward in (-∞, − 1), concave downward in (−1, 1), concave upward in (1, ∞). The graph is the 'bell-shaped" curve shown in Figure 6.
**.** They are inverses of each other.
**.** The advisor had the effrontery to ask for more than 18 billion billion billion grains of rice! Show this by using formula (13), _p_. 97, to evaluate the sum 1 + 2 + 22 \+ . . . + 263.
**.** (a) 10 _x_ (1 + _x_ ln 10);
**.** Use (14), (11) and (12), noting that ln _a_ > 0 if _a_ > 1, ln _a_ < 0 if 0 < _a_ < 1.
**.** Use (17), (11) and (12), together with formulas (8) and (9), p. 155.
**.** _c_ ≤ − _e_ /6, _c_ > 0.
**Sec. 4.5**
**.** (a) 1/ _e_ ;(c) 1 _/e_.
**.** (a) _e_ ;(c) _e_ 2.
**.**
**.**
**.** $1,485.95.
**.** About 13 years and a month.
**.**
**.** One dollar grows to _e_ dollars in one year if compounded continuously at an annual interest rate of 100%.
**.**
**.**
**.** By the ordinary chain rule,
**.** _x r_.
**.** Note that
**.** See Figure 7. The various properties of cosh _x_ and sinh _x_ are easy consequences of those of _e x_ and _e_ s− _x_.
**Sec. 4.6**
**.**
**.**
**.** Let _t_ = _f_ ( _x_ ), noting that if _t_ < 0, then
**.** (a) ln (1 + _x_ 2) + _C_ ;(c) ln |ln _x_ | + _C_
**.**
**.**
**.**
Figure 7.
**.**
**.**
**.**
**.** 3 − _e_.
**.** See Sec. 4.5, Prob. 21d.
**.**
**.** The formulas (23) are equivalent to
The substitution _x_ = _x_ ( _t_ ) transforms the first of these formulas into the second, while the substitution _t_ = _t_ ( _x_ ) transforms the second into the first, since _t_ ( _x_ ( _t_ )) ≡ _t_ , _x_ ( _t_ ( _x_ )) ≡ _x_ (Sec. 2.22b).
**.** By the fundamental theorem of calculus,
with the help of (i).
**.** Repeated integration by parts gives
**.** Integrate by parts repeatedly, noting that _P_ ( _n_ \+ 1)( _x_ ) ≡ 0, since _P_ ( _x_ ) is of degree _n_.
**Sec. 4.7**
**.** The integral is divergent if _r_ ≤ 1 and equals _a_ 1 − _r_ /( _r_ − 1) if _r_ > 1.
**.** (b) Divergent.
**.** The integral is divergent if _r_ ≥ 1 and equals _a_ 1 − _r_ /(1 − _r_ ) if _r_ < 1.
**.**
**.** Note that
so that one limit is finite when the other is finite, and only then. Similarly
Moreover,
where, for brevity, we omit the expression _f_ ( _x_ ) _dx_ behind the integral signs.
**.** For example, stands in the same relation to the improper integral as the partial sum to the infinite series . Develop the analogy further.
**.** .
Chapter 5
**Sec. 5.1**
**.** Let Φ( _x_ , _y_ , _z_ ) = _z_ − _F_ ( _x_ , _y_ ).
**.** _y_ = _e_ − _x_ 2.
**.** Take the square root and then separate variables. The extra solution is _y_ ≡ 0.
**.** We have _p_ ( _dp_ / _dy_ ) = 2 _y_ 3, and hence ∫ _p dp_ = ∫2 _y_ 3 _dy_ \+ _C_ 1 or _p_ 2 = _y_ 4 \+ _C_ 1. Application of the initial conditions gives _C_ = 0. Therefore _p_ 2 = _y_ 4, or _y_ ′ = _y_ 2, so that _y_ = 1/(1 − _x_ ), after solving this first-order equation and applying the initial conditions again.
**.** The general solution is ln | _x_ | = Φ( _y_ / _x_ ) + _C_ , where Φ( _u_ ) is any antiderivative of the function 1/[ _f_ ( _u_ ) − _u_ ].
**.** Solving the differential equation _x_ − ( _y_ / _y_ ′) = _x_ 2, subject to the initial condition _y_ | _x_ = −1 = −1, we get _y_ = 2 _x_ /(l − _x_ ).
**Sec. 5.2**
**.**
**.**
**.** No.
**.** 1 hr.
**.** To get the solution of (21), change _e_ − _rt_ to _e rt_ in formula (13).
**.**
**.** If the fresh specimen has N0 radioactive atoms, the old specimen has _N_ 0 _e_ − _kτ_ atoms, where _k_ = (ln 2)/5570 is the decay constant of radiocarbon. Therefore _n_ = α _N_ 0, _m_ = α _N_ 0 _e_ − _kτ_ , where α is some constant of proportionality. But then _n_ / _m_ = _e kτ_.
**Sec. 5.3**
**.** _s_ = _Ft_ 2/2 _m_.
**.** _s_ = ( _kt_ 3/6 _m_ ) + _v_ 0 _t_.
**.** 32 ft.
**.**
**.** As in Example 5.33f, the work done on the rocket by the earth's gravitational pull is
with the help of (24). Therefore .
**.** About 1.5 mi/sec.
Chapter 6
**Sec. 6.1**
**.** The point _P_ in _R_ 3 with _x_ , _y_ and _z_ -coordinates _a_ , _b_ and _c_ is the unique point of intersection of the planes _x_ = _a_ , _y_ = _b_ and _z_ = _c_.
**.** (a) 6;(c) 25.
**.**
**.** Yes. Consider a sphere.
**.** (a) The plane _y_ = _a_.
**.** (a) _z_ = 1 − | _x_ |(−1 ≤ _x_ ≤ 1).
**.** Intervals are connected.
**.** (5, 0, 0), (−11, 0, 0).
**.** A pair of right circular cones with their common vertex at the origin (make a sketch).
**.** The domain of _f_ is the set of all points ( _x_ , _y_ ) such that | _x_ | > | _y_ |. This set is not connected (why not?).
**Sec. 6.2**
**.** Formula (2) means that, given any _ε_ > 0, there is a _δ_ > 0 such that | _f_ ( _x_ , _y_ ) − _A_ | < _ε_ whenever . Therefore | _f_ ( _x_ , _b_ ) − _A_ | < _ε_ whenever whenever | _y_ − _b_ | < _δ_.
**.** Use Prob. 1, first setting _y_ = 0 and then _x_ = 0.
**.** Yes. Use the inequalities , .
**.**
**.** See Prob. 4.
**.** Use the analogue of Theorem 2.63c for functions of two variables.
**.** (b) 1.
**.** The triple limit is another way of writing ( _x_ , _y_ , _z_ ) → (0, 1, _e_ ). By the three-dimensional analogue of Prob. 6, the function is continuous at (0, 1, _e_ ). Therefore _A_ = ln _e_ = 1.
**.** On the cylinder _x_ 2 \+ _y_ 2 = 1.
**.**
**.**
**.** (a) −1/ _x_ 2.
**.**
**.** (a) Δ _f_ ( _x_ , _y_ ) = ( _x_ \+ Δ _x_ )2 \+ ( _y_ \+ Δ _y_ )2 − _x_ 2 − _y_ 2 = 2 _x_ Δ _x_ \+ 2 _y_ Δ _y_ \+ (Δ _x_ )2 \+ (Δ _y_ )2 is of the form (8) with _A_ = 2 _x_ , _B_ = 2 _y_ , α(Δ _x_ , Δ _y_ ) = Δ _x_ , _β_ (Δ _x_ , Δ _y_ ) = Δ _y_.
**.** (a) _dz_ = ( _y_ − 2 _xy_ 3 \+ 3 _x_ 2 _y_ ) _dx_ \+ ( _x_ − 3 _x_ 2 _y_ 2 \+ _x_ 3) _dy_.
**.** (a) 5.022.
**.** First solve for each of the three variables as a function of the other two.
**Sec. 6.3**
**.**
**.**
**.**
**.**
**.** (a) 2;(c) ;(e) −1.
**.** Differentiate (19) with respect to _t_ , and then set _t_ = 1.
**.**
**Sec. 6.4**
**.** (b) Minimum _z_ = −1 at ( _x_ , _y_ ) = (1, 0).
**.** (b) Minimum _z_ = 0 at ( _x_ , _y_ ) = (2, 4).
**.** Maximum _z_ = 3/ _e_ at ( _x_ , _y_ ) = (0, ±1), minimum _z_ = 0 at ( _x_ , _y_ ) = (0, 0),
**.** If _z_ → ∞, then _x_ → 0, _y_ → 0, since otherwise _xz_ \+ _yz_ → ∞. Therefore
with the help of (10) and the inequality , valid for positive _x_ and _y_.
**.** No.
**.** Solve the equations
for positive _x_ and _y_.
**.** It follows from (i) that
Let
Then
so that _D_ = _AC_ − _B_ 2 is positive at the point , while _A_ is negative.
**.** The (perpendicular) distance _d_ between _P_ 1 and the line _L_ with equation _Ax_ \+ _By_ \+ _C_ = 0 is, of course, also the minimum distance between _P_ 1 and a variable point _P_ = ( _x_ , _y_ ) of _L_. Minimizing subject to the condition _Ax_ \+ _By_ \+ _C_ = 0 is equivalent to minimizing _δ_ 2 subject to the same condition. Let _u_ = _δ_ 2 − _λ_ ( _Ax_ \+ _By_ \+ _C_ ), where _λ_ is a Lagrange multiplier. Setting the partial derivatives of _u_ with respect to _x_ , _y_ and _λ_ equal to zero, we get
The last equation is just the equation of _L_. It follows from the first two equations that ( _x_ 2 − _x_ 1)/ _A_ = ( _y_ 2 − _y_ 1/ _B_ , where _P_ 2 = ( _x_ 2, _y_ 2) is the point of _L_ minimizing _u_ and hence _δ_. Let this last ratio be denoted by _q_. Then _x_ 2 − _x_ 1 = _Aq_ , _y_ 2 − _y_ 1 = _Bq_ , and hence _d_ , the minimum value of _δ_ , equals . But _P_ 2 lies on _L_ , and hence _Ax_ 2 \+ _By_ 2 \+ _C_ = _A_ ( _Aq_ \+ _x_ 1) + _B_ ( _Bq_ \+ _y_ 1) + _C_ = 0, so that _q_ = −( _Ax_ 1 \+ _By_ 1 \+ _C_ )/( _A_ 2 \+ _B_ 2). Substituting this value of _q_ into the formula for _d_ , we get the required answer.
**.** _MC_ 1( _Q_ 1, _Q_ 2) = 6 _Q_ 1 \+ 2 _Q_ 2, _MC_ 2( _Q_ 1, _Q_ 2) = 2 _Q_ 1 \+ 6 _Q_ 2. The profit Π( _Q_ 1, _Q_ 2) = _P_ 1 _Q_ 1 \+ _P_ 2 _Q_ 2 − _C_ ( _Q_ 1, _Q_ 2) is maximized when . Note that , while . The larger price must not exceed three times the smaller price.
SUPPLEMENTARY HINTS AND ANSWERS
Chapter 1
Sec. 1.2
. (b){5};(d){2, 3}.
. (b){−1, 0, 1, 2, 3, 4}.
. (b)
. If x belongs to A, then x certainly belongs to A or B. If x belongs to both A and B, then x certainly belongs to A.
. If x belongs to A and B, then x certainly belongs to A or B, in fact to both. Yes, if A = B.
. (b){1, 2, 3};(d) .
. (b)and (d).
. (a)The triangle with sides 3, 4, 5 is a right triangle, and so is the triangle with sides 3n, 4n, 5n, where n is any positive integer;(b)Note that 52 \+ 122 = 132;(c)The interior angles of a regular polygon all equal , and hence cannot be smaller than 60°;(d)The square is a regular polygon;(e)There is no positive integer n such that .
Sec. 1.3
.
. If were rational, then , where m and n are integers. But then , where n - m and n are integers. This is impossible, since is irrational.
.
.
.
. Let the rational number be , and carry out the long division. Each step of the division gives a remainder less than n. If 0 is obtained at any step, the decimal representing terminates. Otherwise, since there are at most n - 1 nonzero remainders, one of the remainders must eventually repeat. But then the same group of digits must repeat in the quotient, provided we are in the part of the quotient past the decimal point.
. The formula holds for n = 1, since . Suppose the formula is true for n = k, so that . Then . so that the formula also holds for n = k + 1.
Sec. 1.4
. (b)a - b > 0, c - d > 0, and hence (a - b) + (c - d) = (a + c) - (b + d) > 0.
.
. Note that . We can also write .
. (a)Use Prob. la and the fact that a = b implies -a = -b;(b)Examine cases, using Theorem 1.45;(c)Same hint.
. Use Theorem 1.4 3 and the fact that a = b implies ac = bc.
. (b)1;(d)1;(f)-2.
. (a)n;(c)n - 1.
. (a)Start from (a - b)2 ≥ 0;(c)Start from (a - 1)2 ≥ 0, and use Theorem 1.43 to divide by a.
. Use Prob. 13b, noting that equality occurs when a = b and only then. Also use Prob. 5.
. A rectangle of length x and width y has perimeter p = 2(x + y) and area A = xy. In terms of the arithmetic mean a and geometric mean g, we have . Holding a (or p) fixed, we get the greatest value of g (or ), and hence of , when x = y.
Sec. 1.5
. An immediate consequence of formula (1).
. If x ≥ 0, then |x| = x and |x|2 = x2, while if x < 0, then |x| = -x and |x|2 = (-x)2 = x2.
. Use mathematical induction (Sec. 1.37) and an argument like that in Prob. 4.
. -3, 1.
. (b)x = 2; .
Sec. 1.6
.
.
. (b)[1, ∞).
. (a)[−1, 1];(c)(−1, 1].
Sec. 1.7
. (b)5;(d)3.
. (1, 1), (−1, 1), (−1, −1), (1, −1).
.
. is the midpoint of AB, is the midpoint of BC, and so on.
Sec. 1.8
.
. (b)90°;(d)0°.
. (b)tan 100° = -tan 80° = -5.67128.
. -3.
. Let L have slope m and L' slope m'. Then m = 2, so that .
Sec. 1.9
. (b)y = 2x + 1;(d)y = 2x.
.
. (b)y = 3x; .
. (b)m = 2, a = -2, b = 4;(d)m = 0, a undefined, b = 2.
. (b)m = , a = 0, b = 0;(d)m = −1, a = −1, b = −1.
. Below it.
. The first two lines are parallel, the second two are perpendicular.
. (a)2x + y - 2 = 0;(c)4x + 8y - 1 = 0.
. Let L be the line Ax + by + C = 0, and let P2 = (x2, y2) be the foot of the perpendicular dropped from P1 to L. Then . Since the slope of L equals , the slope of the line L' through P1 and P2 equals . Hence the equation of L' is . Since P2 lies on L', we have or . Let this last ratio be denoted by q. Then x2 \- x1 = Aq, y2 \- y1 = Bq, and . But P2 also lies on L, and hence Ax2 \+ By2 \+ C = A(Aq + x1) + B(Bq + y1) + C = 0, so that q = -(Ax1 \+ By1 \+ C)/(A2 \+ B2). Substituting this value of q into the formula for d, we get the required answer.
. (b)5.
Chapter 2
Sec. 2.1
. φ(-2) = 14, φ(−1) = 4, φ(0) = 0, .
. (b)Domain -3 ≤ x ≤ 3, range 0 ≤ y ≤ 3;(d)Domain all x ≠ -5, range all y ≠ 0.
. Take an evening paper dated d, and look up P in the financial section. The function is undefined on days when the exchange is closed.
. No.
. V = wh.
. Let x be the temperature in degrees Centigrade and y the temperature in degrees Fahrenheit. Then , . The missing entries are x = 40 and y = 176.
. Convince yourself that every "rule" or "procedure" associating a unique value of y with each given value of x is in effect a set of ordered pairs of the type described.
. The one-to-one function f(n) = n + 1 maps the even numbers into the odd numbers.
. (b)Finite;(d)Infinite.
Sec 2.2
. a = 4, b = −1.
. f(f(x)) = x, f(g(x)) = 1/x2, g(f(x)) = 1/x2, g(g(x)) = x4.
.
. a1 = 4, a3 = 4, a4 = 2, a7 = 5.
12.No. For example, let
Sec. 2.3
. The circle of radius 1 with its center at the point (−1, 1).
. The graph of f(x) + c is obtained by shifting G a distance c upward if c > 0 and a distance |c| downward if c < 0. The graph of f (x + c) is obtained by shifting G a distance c to the left if c > 0 and a distance |c| to the right if c < 0.
. For example, if f(x) and g(x) are odd, then f(-x) ≡ -f(x), g(-x) ≡ -g(x), and hence f(-x)g(-x) ≡ f(x)g(x), so that f(x)g(x) is even.
. Yes, The function is increasing in the interval 1 ≤ x < ∞, decreasing in the interval - ∞ < x ≤ −1, and constant in the interval −1 ≤ x ≤ 1.
. Yes. The graph has corners at the points (-2, 3), (−1, 2) and (0, 3). The function is increasing in the interval −1 ≤ x < ∞ and decreasing in the interval -∞ < x ≤ 1.
Sec. 2.4
.
.
. For example, if f(x) = x2, then f'(x0) = f(x0) for x0 = 0 or 2.
. (b)1;(d)0.
. (a)1;(c)2.
. If |f(x) - A| is "arbitrarily small," then so ||f(x)| - |A||, since ||f(x)| - |A|| ≤ |f(x) - A|, by Sec. 1.5, Prob. 10. The converse is false, for example, does not exist (Example 2.45e), but .
Sec. 2.5
. Δ(u + v) = [u(x + Δx) + v(x + Δx)] - [u(x) + v(x)] = [u(x + Δx) - u(x)] + [v(x + Δx) - v(x)] = Δu + Δv.
. Here f(x) = mx + b, f(x0) = mx0 \+ b, f'(x0) = m, so that (3) becomes y = m(x - x0) + mx0 \+ b = mx + b.
. Yes, the tangents at any pair of points such that are perpendicular. No.
. (b)Δy = 0.331, dy = 0.3, E = 0.031, about 9% of Δy;
(d)Δy = 0.003003001, dy = 0.003, E = 0.000003001, about 0.1% of Δy.
. In the intervals - ∞ < x < −1, −1 < x < 1, 1 < x < ∞. At the points (−1, 2), (1, 2).
. If S is the surface area of the earth and R its radius (≈ 4000 miles), then S = 4πR2. Therefore square miles.
Sec. 2.6
. Let g(x) ≡ c.
. We get the indeterminate form .
. (b)-3;(d)23.
.
. If |f(x) - A| is "arbitrarily small" both for all "sufficiently small" x0\- x > 0 and all "sufficiently small" x - x0 > 0, then |f(x) - A| is "arbitrarily small" for all "sufficiently small" |x - x0| > 0, and conversely.
. Use Prob. 11.
. Use Sec. 2.4, Prob. 11.
Sec. 2.7
. (b)6ax2 \- 2bx;
. (b)3x2 \- 2(a + b)x + ab;(d)6x2 \- 26x + 12.
.
. If n is odd and tn = x, then (-t)n = -x, so that .
. If n is odd, then .
. Use Prob. 6 and the fact that division by zero is impossible.
.
. f'''g + 3f''g' + 3f'g" + g'''.
. The tangent T to the curve y = 1/x at the point P0 = (x0, 1/x0) has equation . Therefore T has x-intercept 2x0 and y-intercept 2/x0. Now use Sec. 1.7, Prob. 6.
. By Sec. 2.66, g is continuous at x. But g(x) ≠ 0, by hypothesis, and hence g(x + Δx) ≠ 0 for all "sufficiently small" | Δx|, by Sec. 2.4, Prob. 15.
Sec. 2.8
. Use the fact that is an increasing function.
.
. For example, if f is even, then f(-x) ≡ f(x), so that , and hence -f'(-x) ≡ f'(x) or f'(-x) ≡ -f'(x).
.
.
. Note that .
.
.
. Solving the quadratic equation (14) for y, we get , and hence , so that
. In the theory of equations, it is shown that the cubic equation y3 \+ ay + b = 0 has three distinct real roots if . There is only one real root if . Use this to investigate equations (11) and (13).
Sec. 2.9
. (b)−1.
. (b)∞.
. The limit is a product of five limits, all equal to .
. x < −1000.
. (b)x = -d/c, y = a/c.
. , where the constants al,a2, . . ., an are all different.
. Consider the separate branches of the function graphed in Figure 15.
. (b)0;(d)No limit.
. 20.
. (b)1! + 2! + 3! + 4! + 5! + 6! = 1 + 2 + 6 + 24 + 120 + 720 = 873.
. (b)1.
. Let . Then , and so on. Therefore , and so on. Thus the sequence sn is unbounded and hence divergent.
. No, as shown by the example of the harmonic series.
Chapter 3
Sec. 3.1
. v = t2 \- 4t + 3, a = 2t - 4. The direction of motion changes at t = 1 and t = 3. The particle returns to its initial position at t = 3.
. 128 ft. 240 ft.
. Differentiating (8), we find that the velocity after braking is v0 \- kt. Hence it takes a time equal to v0/k to bring the car to a stop, during which it travels a distance equal to .
. The flywheel stops rotating when t = b/2c.
Sec. 3.2
.
. x increases faster if x < 4, y increases faster if x > 4, x and y increase at the same rate if x = 4.
. 2 ft/sec. No.
. The curve of marginal cost is the straight line
MC(Q) = a - 2mQ.
. 5 ft2/min.
Sec. 3.3
. (b)No maximum, a minimum equal to 1 at x = 1;(d)No extrema.
. (b)A maximum equal to 1 at x = 1, a minimum equal to 0 at every point of (0, 1);(d)No maximum, a minimum equal to 0 at every point of (0, 1).
. If f is increasing in [a, b], f has its minimum at a and its maximum at b, while if f is decreasing in [a, b], f has its maximum at a and its minimum at b.
. No.
Sec. 3.4
. f is not differentiable at x = 0.
. (1, 1), (−1, −1).
.
.
Sec. 3.5
.
.
. No.
. (a)Increasing in , decreasing in ;(c)Increasing in [−1, 1], decreasing in (−∞, −1] and [1, ∞) .
Sec. 3.6
. (b)Maximum .
. (a)Maximum y = 1 at x = ±1, minimum y = 0 at x = 0;(c)No extremum at x = 0, maximum , minimum y = 0 at x = 1.
. (b)Maximum y = 100.01 at x = 0.01, 100, minimum y = 2 at x = 1;
(d)Maximum y = 132 at x = −10, minimum y = 0 at x = 1, 2.
. |3x - x3| has its maximum in [-2, 2] at the points x = ±1, ±2.
. Minimum y = 0 at x = 0 if m is even and no extremum at x = 0 if m is odd, maximum y = mnnn/(m + n)m+n at x = m/(m + n), minimum y = 0 at x = 1 if n is even and no extremum at x = 1 if n is odd.
. Solve the equations y|x=2 = −1, Y'| x=2 = 0 to get a = 1, b = 0. Then show that y"|x=2 < 0.
Sec. 3.7
. True.
. For example, if f(x) = x3, g(x) = x4, h(x) = -x4, then f"(0) = g"(0) = h" (0) = 0, but f has an inflection point at x = 0, g is concave upward at x = 0, h is concave downward at x = 0.
. No.
. Inflection points at x = 0, ±3a, concave upward in (-∞, -3a), concave downward in (-3a, 0), concave upward in (0, 3a), concave downward in (3a, ∞) .
.
. The points have abscissas 1, , the solutions of the equation x3 \+ 3x2 \- 3x - 1 = 0.
. Reread Secs. 3.33b, 3.64a and 3.72.
Sec. 3.8
.
. No, even if the buggy is much faster than the boat.
.
. If Π(Q) has a local extremum at Q = Q0, then Π'(Q0) = R'(Q0) - C'(Q0) = MR(Q0) - MC(Q0) = 0, so that MR(Q0) = MC(Q0). This extremum will be a maximum if Π"(Q0) = R"(Q0) - C"(Q0) < 0, that is, if MR'(Q0) < MC'(Q0).
. Overhead is positive, and hence d > 0. The marginal cost is MC(Q) = 3aQ2 \+ 2bQ + c, with first derivative MC'(Q) = 6aQ + 2b and second derivative MC"(Q) = 6a. Therefore MC(Q) has a local minimum at Q0 = -b/3a if a > 0. But Q0 > 0, and hence b < 0. Moreover, , and hence b2 < 3ac, which, in particular, implies c > 0.
. Let the sides of the angle be the x-axis and the line y = mx. The line through P = (a, b) with slope X intersects the x-axis in the point and the line y = mx in the point , forming a triangle of area m(a λ - b)2/2λ(λ - m).
. Choosing A = (a, b), P = (x, 0) and B = (c, d), minimize (the speed of light can be cancelled out). The minimum is achieved when x satisfies the condition . Now use similar triangles.
Chapter 4
Sec. 4.1
. (b)1.
. The global maximum of f in [a, b], whose existence is guaranteed by Theorem 3.32c.
. No. Yes, since the number of subintervals cannot be less than the integral part of (b - a)/λ.
. If f(x) = c, the region bounded by the curve y = f(x), the x-axis, and the lines x = a and x = b is a rectangle of length b - a and width c; then , by (6) and (11). If a = 0, f(x) = cx, the region bounded by the curve y = f (x), the x-axis, and the lines x = 0 and x = b is a right triangle with legs b and cb; then , by (6) and (12). Both results are in keeping with elementary geometry.
. . But , by elementary geometry.
. f(x) is continuous in every such interval.
. (a)Use the same argument as in Example 2.45e, noting that f(x) takes both values 1 and −1 in every deleted neigborhood of c, since every such neighborhood contains both rational and irrational points (this is a consequence of Sec. 1.5, Prob. 13).
Sec. 4.2
.
. (b)6;(d)1.
.
.
.
. Yes, provided that x > 0.
. (a)0;(c)f(b).
. Choose A = 0 in Prob. 15.
. Clearly f(c) > 0. Suppose a < c < b. Then there is an interval [c - δ, c + δ] such that f(x) > 0 for every _x_ ∈ [c - δ, c + δ] (why?). By Theorem 4.21a,
The first and third integrals on the right are nonnegative, by Prob. 16, while the second integral is positive, by the mean value theorem for integrals. Therefore the integral on the left is also positive. The proof is even simpler if c = a or c = b.
. Use Prob. 18.
. Apply Prob. 18 to the function f = f1 \- f2.
. Use Prob. 17, noting that -|f(x)| ≤ f(x) ≤ | f(x)|, where, by Sec. 2.6, Prob. 14, |f(x)| is continuous and hence integrable in [a, b].
. The assertion is obviously true if f is a constant function. Otherwise f takes values between its maximum M and its minimum m in [a, b]. But then M - f(x) > 0 at some point in [a, b], and hence , by Prob. 18, or equivalently . In the same way, we find that . Continuing as in the proof of Theorem 4.22a, we observe that the point c is now known to lie between the points p and q at which f takes its maximum and minimum, so that c ∈ (a, b).
Sec. 4.3
. Because r = −1.
. (b)x > 1.
.
.
. The tangent has equation y = x/e.
. x ln x - x is an antiderivative of ln x.
. Increasing in [−1, 0) and [1, ∞), decreasing in (0, 1] and (−∞, −1].
. Inflection points at x = ±1, concave downward in (-∞, -1), concave upward in (−1, 1), concave downward in (1, ∞).
. (b)2 < x < 3.
. Use (12), (8) and (9), noting that ln a > 0 if a > 1, while ln a < 0 if 0 < a < 1.
. Let f(x) = ln x. Then ln b - ln a = (b - a)f'(c) , where a < c < b.
Sec. 4.4
. If c = ln k, then kex = ecex = ec+x.
. (b)-3e-3x;(d)ex(1 - 2x - x2).
.
. eax/a is an antiderivative of eax.
. Maximum y = 1010e-9 at x = 9, minimum y = 0 at x = 1.
. x = 1, 2.
.
.
. The function has its maximum at . Now compare y14 with y15.
Sec. 4.5
. (b)e.
. (b)e2.
.
.
. (b)ln 4.
.
. $7,408.18
.
.
. y" is nonvanishing.
. ax.
. If y = f(x), then . The fuction xf(x) = xy has elasticity .
. The sum of 8 terms of the series is 2.71825...
Sec. 4.6
. Let x = -t.
. Let x = 1 - t.
.
.
.
.
. (b)1;(d)e - 2.
.
. Note that .
.
. Divide the price range P1, P0] into n equal small units ΔP = (p0 \- p1), where ΔP is just large enough so that each successive price drop causes more of the commodity to be sold. Let ΔQi be the extra quantity sold when the price is lowered from P0 \- (i - 1)ΔP to P0 \- iΔP. Then the total revenue received in the course of the staged price drop is just , which approximates the integral in [(25). To convert (25) into (26), integrate by parts.
.
Sec. 4.7
.
. (a)Divergent;(c)6.
. By Prob. 3, .
. By Sec. 4.6, Prob. 15,
Chapter 5
Sec. 5.1
. y = 2/x.
. y = (x ln x - x + 1)2.
.
.
Sec. 5.2
. 4.8 billion. 6.9 billion.
.
. (a),(b)and(c)follow at once from formula (13). To prove (d), differentiate (11), obtaining N" = rN' - 2sNN' = (r - 2sN)N' = (r - 2sN)(r - sN)N, where the expression on the right is positive for N < r/2s, zero for N = r/2s, and negative for r/2s < N < r/s. Now use Sec. 3.72, Proposition (4).
. Per capita consumption is constant if r = s, grows exponentially at the rate of r - s percent per year if r > s, and decays exponentially at the rate of s - r percent per year if r < s.
.
. Let . Then , and hence N* = Cert, or . Applying the initial condition N|t=0 = N0, we get .
. About 2310 yrs.
Sec. 5.3
.
. 64 ft/sec. 4 sec.
. The truck has more kinetic energy in the first case, the bullet in the second.
. Let the fixed points be s = ±a. Then the force is F(s) = -k(s + a) - k(s - a) = -2ks. Hence the work done in going from -a to a is .
. About 84 mi.
.
. Since the spider's weight mg stretches the strand by an amount s, the tension ks in the strand satisfies the condition ks = mg, so that k = mg/s. Therefore the potential energy of the stretched strand is mgs (Prob. 7). As a result of the spider's climb, the potential energy of the system consisting of the spider and the strand changes by , since the strand is no longer stretched after the climb. This is the work W1 done by the spider in climbing up the strand, to be compared with the work W2 = 2mgs done by the spider in climbing up an inelastic strand of length 2s.
Chapter 6
Sec. 6.1
. (1, 1, −1), (1, −1, 1), (−1, 1, 1), (−1, −1, 1).
. (b)14.
. Show that the side lengths satisfy the Pythagorean theorem.
. Complete the squares, as in Sec. 2.32b.
. (b)The cylinder x2 \+ y2 = a2.
. (b)z = 1 - |y| (−1 ≤ y ≤ 1).
. −1 < x < 1, −1 < y < 1. The region is an "open square."
. |P1P2| = 2. P1 is closer to the origin.
Sec. 6.2
. If the limit in question exists, then , which is impossible.
.
. In the first and third quadrants of the xy-plane.
.
.
. (b)(1 + 3xyz + x2y2z2)exyz.
.
. (b)Δf(x, y) = (x + Δx)(y + Δy) - xy = yΔx + xΔy is of the form (8) with A = y, B = x, α(Δx, Δy) = β(Δx, Δy) = 0.
. (b)dz = yx ln y dx + xyx−1dy.
. (b)108.432.
. Let . Then . Interchanging the roles of x and y, we get .
. No, since the partial derivatives fx(0, 0) and fy(0, 0) do not exist (why not?).
Sec. 6.3
. (b)0.
.
.
. (b)1;(d)0.
.
Sec. 6.4
. (a)No extrema;(c)Minimum z = 0 at (x, y) = (0, 0), no extrema at (x, y) = (1, ±4), .
. (a)Maximum z = 864 at (x, y) = (-6, 0), minimum z = -864 at (x, y) = (6, 0), no extrema at (x, y) = ;(c)Minimum z = 0 at every point of the line y = x + 1.
. Maximum z = 4 at (x, y) = (±2, 0), minimum z = -4 at (x, y) = (0, ±2).
. Let u = x1x2 ... xn, u* = u - λ(x1 \+ x2 \+ ... + xn \- nc). Then, at any critical point.
so that
which implies x1 = ... = xn = c.
. Maximize subject to the condition x1 \+ x2 \+ ... + xn = nc.
. The profit is maximized when , . The absolute value of the price difference must not exceed 4q.
INDEX
Abscissa,
Absolute value,
Absorption coefficient,
Acceleration,
due to gravity, , ,
Air resistance,
Algebraic sum,
Angle of incidence,
Angle of reflection,
Annual interest rate,
effective,
nominal,
Antiderivative,
existence of,
general,
Area
between a curve and the _x_ -axis,
between two curves, 141–142
negative,
under a curve,
Arguments (of a function),
Arithmetic mean,
Asymptote,
horizontal,
vertical,
Average, of a function,
Average velocity,
Averaging time,
Base of natural logarithms ( _e_ ),
Bell-shaped curve,
Birth rate,
Boundary,
Boundary conditions,
Bounded sequence,
Calculus
differential, ff.
first key problem of, ,
fundamental theorem of,
integral, ff.
second key problem of, ,
Cauchy, A. L.,
Chain rule, ,
Change of variables,
Closed interval,
continuous image of,
Closed region,
Completing the square,
Composite function,
continuity of,
derivative of,
Compound amount,
Compound interest,
Concavity
downward,
in an interval,
at a point,
tests for,
upward,
Conservation of energy,
Constant(s), ff.
of integration,
vs. variables,
Constraint,
Consumer's surplus,
Continuity, ff.
in a closed interval, consequences of, 109–111
of a composite function,
as a consequence of differentiability,
of functions of several variables,
geometrical meaning of,
in increment notation,
in an interval, ,
of an inverse function,
from the left,
at a point, ,
of a polynomial,
of a rational function,
in a region,
from the right,
Continuous function(s), ff.
algebraic operations on, 68–71
integrability of,
inverse of,
product of, 68–69
properties of, 109–111
quotient of,
sum or difference of, 68–69
Convergent improper integral, ,
Convergent sequence,
Coordinate axes, ,
positive directions of, ,
Coordinate planes,
Coordinates, ff.
on a line,
origin of, , ,
in a plane,
rectangular, system of,
in space,
Cost
average,
elasticity of,
marginal, , 106–107,
total, ,
Cost function, 106–107,
cubic, ,
Counterclockwise direction, of quadrants,
Critical point, ,
Cubical parabola,
Curve(s), ff.
area between two, 141–142
area under, 137–138
inflection point of, ,
sketching of, ,
symmetric,
tangent to, ,
Death rate,
Decay constant,
Decimals,
nonrepeating,
repeating,
terminating,
Decreasing function
exponentially,
in an interval, 50–51,
in a neighborhood,
Decreasing sequence,
Definite integral(s), ff.
evaluation of, 142–143, 149–150, 173–178
existence of, ,
vs. indefinite integral,
properties of, 144–145
Definite integration,
Deleted neighborhood. ,
Delta-neighborhood,
deleted,
Demand,
elastic,
elasticity of,
inelastic,
Dependent variable, ff.
differential of, ,
increment of,
value of,
Derivative(s), , ff.
of a composite function,
of a constant function,
higher,
of an inverse function,
logarithmic,
_n_ th,
of the _n_ th power of _x_ , 77–78
of order _n_ ,
partial ( _see_ Partial derivatives)
of a polynomial,
of a product,
of a quotient,
of the _r_ th power of _x_ , ,
second,
of a sum or difference,
third,
Difference quotient, ,
Differentiability
in an interval,
at a point,
Differentiable functions, ff.
properties of, 112–116
of several variables,
Differential(s), ,
of the dependent variable,
of a function, ,
of the independent variable(s), ,
total,
Differential equation, , ff.
first-order,
homogeneous,
initial conditions for, , , ,
of order _n_ ,
order of,
partial,
second-order,
with separated variables,
solution of,
general, ,
particular, ,
Differentiation, ff.
implicit, 86–87
logarithmic, 167–168
operator,
partial,
rules, table of,
Discontinuity, removal of,
Discontinuous function,
Distance
between a point and a line,
between two points
on a line,
in _n_ -Space,
in a plane,
in space,
Distance function,
Divergent improper integral, ,
Divergent sequence,
Division by zero, impossibility of, ,
_d_ notation,
Domain, ,
of a function of two variables,
Double limit,
Doubling time,
Dummy index, ,
Dummy variable,
_e_ (base of natural logarithms), , 165–166
Elasticity,
of cost,
of demand,
Element (of a set),
Empty set,
End of proof symbol,
End points,
Energy
conservation of,
kinetic,
potential,
total,
Epsilon-delta language, ff.
Equation of motion,
Error of approximation,
Escape velocity,
Even function,
Even number,
Excise tax,
Exponential growth and decay,
Exponential law,
Exponential(s), ff.
to the base _a_ , 161–162
to the base _e_ ,
graph of,
properties of, 159–161
table of,
Extreme value,
Extremum (extrema), ff.
absolute,
constrained,
global, , ,
local, , 121–126,
first derivative test for, ,
necessary condition for, ,
second derivative test for, ,
relative,
unconstrained,
Fermat's principle,
Fibonacci sequence,
Finite interval,
Finite region,
Finite set,
First derivative test,
Force,
Fractions,
Function(s), ff.
algebraic operations on, 44–46
arguments of,
asymptote of,
average of,
composite,
concave downward,
concave upward,
continuous ( _see_ Continuous functions)
critical point of,
decreasing, 50–51
derivative of, ,
differentiable,
differential of,
differentiation of,
discontinuous,
domain of, , ,
elasticity of,
even,
formal definition of,
graph of, ,
identically equal,
increasing, 50–51
increment of,
inflection point of,
integrable,
inverse,
limit of,
as a mapping, 41–42
mean value of,
notation for,
numerical,
of _n_ variables, ,
odd,
one-to-one,
of one variable,
parity of,
piecewise linear,
range of,
rate of change of,
rational,
of several variables ( _see_ Functions of several variables)
stationary point of,
sum of,
value of, ,
vanishing,
Function(s) of several variables, ,
ff. continuous,
critical point of,
differentiable,
differential of,
domain of, , ,
extrema of, 228–233
graph of,
homogeneous,
increment of,
limit of,
partial derivatives of,
stationary point of,
value of, ,
Fundamental theorem of calculus,
General solution,
arbitrary constants in, ,
General term,
Geometric mean,
Geometric series,
sum of,
Global extrema, , ,
Graph
asymptotes of,
of a continuous function,
of a discontinuous function,
of an equation, ,
of a function, ,
of a one-to-one function,
symmetric
in the origin,
in the y-axis,
Gravitation, ,
Greek alphabet,
Growth rate,
proportional,
Half-life,
Harmonic series,
Higher derivatives, 79–80
Homogeneous functions,
degree of,
Euler's theorem on,
Hooke's law,
Hyperbolic cosine,
graph of,
Hyperbolic sine,
graph of,
Identical equality,
Identity,
Implicit differentiation, 86–87
Improper integral(s), 181–184
convergent, ,
divergent, ,
evaluation of, 183–184
vs. proper integral,
Inclination,
Increasing function
exponentially,
in an interval, 50–51,
in a neighborhood,
Increasing sequence,
Increment(s)
of the dependent variable,
approximated by differential, 64–65, 220–221
of a function, ,
of the independent variable(s), , ,
notation for, , ,
Indefinite integral(s), ff.
vs. definite integral, 140–141
evaluation of, 118–119, 173–178
existence of,
properties of,
Indefinite integration,
Independent variable(s), ff.
differential(s) of, ,
increment(s) of, , ,
value(s) of,
Indeterminate form, ,
Index of summation,
dummy, ,
Induction, mathematical, 8–9
Inequalities, , 10–13
greater than,
greater than or equal to,
less than,
less than or equal to,
Infinite branches,
Infinite interval,
Infinite limits,
Infinite region,
Infinite sequences ( _see_ Sequences)
Infinite series ( _see_ Series)
Infinite set,
Infinity,
limits at,
minus, ,
plus, ,
Inflection point,
necessary condition for,
tests for,
Initial conditions, , ,
vs. boundary conditions,
Initial value,
Instantaneous velocity, 100–101
Integers,
negative,
positive,
Integrability,
of continuous functions,
Integrable function,
Integral part,
graph of,
Integral sign,
Integral(s), , 118–119, ff.
definite ( _see_ Definite integrals)
improper ( _see_ Improper integrals)
indefinite ( _see_ Indefinite integrals)
mean value theorem for,
proper,
Integrand, ,
Integration
constant of,
definite,
indefinite,
interval of,
lower limit of,
by parts, 176–178
by substitution, 173–176
upper limit of,
variable of, ,
Interest
compounded _N_ times per year,
continuously compounded, 166–167
rate of,
Interior point,
Intermediate value theorem,
Intersection, (of two sets),
Interval, ff.
closed,
continuity in, ,
differentiability in,
end points of,
finite,
half-closed,
half-open,
infinite,
of integration,
length of,
open,
partition of,
Inverse (function),
continuity of,
derivative of,
Inverse square law,
Irrational numbers,
decimal expression of,
Irrationality of , 6–7
Iterated limit,
Jump discontinuity,
Kinetic energy,
conservation of,
Lagrange multiplier, ,
Laplace's equation,
Law of reflection,
Left-hand limit,
Leibniz, G. W.,
Limit(s), , 51ff.
algebraic operations on, 66–68
definition of
in epsilon-delta language,
informal,
double,
finite,
of a function of several variables,
infinite,
at infinity,
iterated,
in _n_ dimensions,
left-hand,
one-sided,
of a product, 66–67
of a quotient, 66–67
right-hand,
of a sequence,
of a sum or difference, 66–67
uniqueness of,
Lines ( _see_ Straight lines)
Logarithmic derivative, ,
double,
Logarithmic differentiation, 167–168
Logarithm(s), ff.
to the base _a_ ,
common,
natural,
base of,
graph of,
properties of, 153–157
table of,
Mapping diagram,
Marginal cost, , 106–107,
Marginal profit,
Marginal revenue, ,
Mathematical induction, 8–9
Maximum
in an interval,
local, ,
strict, ,
of a numerical set,
Mean value of a function,
Mean value theorem,
applications of, 117–120
in increment form,
for integrals,
Minimum
in an interval,
local, ,
strict, ,
Minus infinity, ,
Monotonic sequence,
bounded, convergence of,
Neighborhood, ,
deleted, ,
Newton, I., ,
Newton's first law of motion,
Newton's law of cooling,
Newton's law of gravitation,
Newton's second law of motion,
_n_ -dimensional sphere,
_n_ factorial,
Nonrepeating decimal,
Normal cost conditions,
_n_ -space,
points in,
_n_ th derivative,
_n_ th power,
_n_ th root, ,
Number line ( _see_ Real line)
Number theory,
Number(s), 4–8
decimal expression of,
even,
irrational, 6–7
negative,
odd,
positive,
rational,
real, 7–8
Numerical function,
Odd function,
Odd number,
One-sided limits,
One-to-one correspondence, ,
between real numbers and decimals,
between real numbers and points of line,
between sets,
One-to-one function,
continuous,
Open interval,
Open region,
Optimization problems,
Ordered _n_ -tuple,
Ordered pair,
Ordered triple,
Ordinate,
Origin (of coordinates), , , ,
Overhead,
Parabola, , , ,
cubical,
tangent to,
Paraboloid of revolution,
Parity,
Partial derivative(s),
of higher order,
mixed,
equality of,
second,
Partial differential equation,
Partial sum,
Particle,
Particular solution,
Partition,
fineness of,
Perfect-gas law,
Perpendicular lines, slopes of,
Piecewise linear function,
Plus infinity, ,
Points
on a line, ,
in _n_ -space,
in a plane,
in space,
of subdivision, ,
Polynomial,
continuity of,
degree of,
derivative of,
Population growth, 193–195
Positive direction, , ,
Potential energy,
Present value,
Profit,
marginal,
Proportional changes,
Pythagorean theorem, , , ,
converse of,
Quadrant(s),
first,
second,
Q.E.D., symbol for,
Radioactivity, 196–197
Radiocarbon dating,
Range (of a function), ,
Rate(s)
birth,
of change, , , ff.
related, 104–106
of cooling, ,
death,
growth,
interest,
Rational function,
continuity of,
Rational numbers,
decimal expression of,
Real line, ff.
coordinates of points on,
distance between two points of,
origin of,
positive direction of,
Real number system, ,
Real numbers, ff.
decimal expression of,
Real variable,
Rectangular coordinates ( _see_ Coordinates)
Recursion formula,
Region,
closed,
finite,
infinite,
open,
Regular polygon,
Repeating decimal,
Revenue
average, ,
marginal,
total, ,
Right-hand limit,
Rocket, motion of, 205–206
Rolle's theorem, 113–114
_r_ th power of _x_ , 162–163
derivative of, ,
Secant (line),
Second derivative test,
Second-order differential equation,
Separation of variables,
Sequence(s), ff.
bounded,
convergent,
decreasing,
divergent,
Fibonacci,
general term of,
increasing,
limit of,
monotonic,
terms of,
unbounded,
Series, 96–97
convergent,
divergent,
geometric,
harmonic,
partial sums of,
sum of,
terms of,
Set(s), ff.
closed, under algebraic operations, ,
connected,
difference of,
elements of,
empty,
equality of,
finite,
infinite,
intersection of,
with _n_ elements,
one-to-one correspondence between,
with the same number of elements,
subset of,
proper,
union of,
Side condition,
Slope,
in terms of inclination,
negative,
positive,
Solution set, ,
Speed,
Square root,
S-shaped curve,
Stationary point, ,
Straight line(s), 24–33
equations of, 29–31
inclination of,
intercepts of,
perpendicular, slopes of,
slope of,
Subset,
proper,
Summation notation,
Summation sign,
Surface,
of revolution,
Symmetry of a curve (or graph)
in the origin,
in the _y_ -axis,
Tangent (of an angle), ,
Tangent (line)
to a circle,
to a curve, ,
horizontal,
to a parabola,
Terminal velocity,
Terminating decimal,
Total differential,
Total energy,
Triangle inequality,
Unbounded sequence,
Union,
Universal gravitational constant,
Vanishing function,
identically,
Variable(s), ff.
change of,
vs. constants,
dependent,
independent,
of integration, ,
proportional changes in,
real,
related,
separated,
separation of,
values of,
Velocity, , ff.
angular,
average,
escape,
initial, ,
instantaneous,
terminal,
true,
Work,
_x_ -axis, ,
_x_ -coordinate, ,
_x_ -intercept,
_xy_ -plane, ,
_xz_ -plane,
_y_ -axis, ,
_y_ -coordinate, ,
_y_ -intercept,
_yz_ -plane,
_z_ -axis,
_z_ -coordinate,
Zero,
| {
"redpajama_set_name": "RedPajamaBook"
} | 3,146 |
using System;
namespace Xango.Mvc.Extensions
{
public static class TypeExtensions
{
public static void SetId(this Type type, object obj, object value)
{
var property = type.GetType().GetProperty("Id");
if (null == property)
throw new InvalidOperationException(string.Format("{0} não possui uma propriedade Id.", type));
property.SetValue(obj, value, null);
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,662 |
Customs Officer dragged to court for Drug Trafficking
Nigeria Customs Service (NCS) officer, Ugwoke Emmanuel, was on Thursday arraigned by the National Drugs Law Enforcement Agency (NDLEA) on drug trafficking charges before a Federal High Court sitting in Lagos.
The court read a two-count charge of unlawful selling and transporting of 384 kilograms of cannabis, or marijuana.
Kenyan Pastor With Mysterious Skin Disease Rejected By Parents
Silvester Karanja suffers from a mysterious skin disease and narrated his challenges recently in a vernacular Kenyan TV station. The father of three grew up with his grandparents after facing rejection from his stepfather, mother and siblings.
Vendor Who Uses Urine To Wash Cucumbers Caught Red-handed
A roadside vendor was caught using his Urine to wash cucumbers he sold to unsuspecting members of the public. This shocking incident occurred in Owerri, Imo state, recently.
EFCC Arraigns former chairman, Lamorde's Impostor
The Economic and Financial Crimes Commission, EFCC, has arraigned one Promise Tochi Nwalozie a. k. a Ibrahim Lamorde and Ohaji Ujunwa Raymond, before Justice H. I. O Oshomah of the Federal High Court, sitting in Port Harcourt, Rivers State on a 9-count charge bordering on conspiracy and obtaining money by false presence to the tune of $1,330.00 (USD).
INEC officially writes Melaye, begins recall process July 3
The Independent National Electoral Commission says it has formally notified Senator Dino Melaye of the demand by the people of his constituency to recall him from the Senate.
INEC's National Commissioner and member Information and Voter Education Committee, Madam Mohammed Haruna, disclosed the commission's decision in a statement in Abuja.
Prince Philip has left London hospital, Buckingham Palace says
Prince Philip, the 96-year-old husband of Britain's Queen Elizabeth, was discharged from hospital on Thursday after treatment for an infection.
Philip, who is also known as the Duke of Edinburgh, was admitted to hospital on Tuesday as a precautionary measure for the treatment of an infection arising from a pre-existing condition.
Heartbroken girlfriend hangs herself 2 months after boyfriend committed suicide
A heartbroken girlfriend hanged herself just two months after after her boyfriend took his own life.
Hotel receptionist Laura Bedernyak, 30, killed herself after the death of her football player boyfriend Kyle Jones,29.
Photos: Former Thai Prime Minister pictured crying as she prays on her 50th birthday at a temple
Yingluck Shinawatra, the ousted Prime Minister of Thailand is seen crying as she was praying on the occasion of her 50th birthday at the Golden Mount Temple on Wednesday, June 21. More photos after the cut...
Father of 6 arrested for having sex with neighbor's goat
A 42-years-oldfather of six children, Babangida Garba has been arrested in Funtua Local government area of Katsina State for turning a goat to sex object. Garba was paraded along side other suspects for various criminal offences by the state Commissioner of Police, CP Usman Abdullahi in the police headquarters on Thursday.
(Photo) American man remanded in Kirikiri prison for defrauding Nigerians
An Ikeja High Court on Thursday remanded an American citizen, Marco Ramirez, at Kirikiri Maximum Prison for allegedly defrauding three Nigerians in a $565,000 green card and investment scam. .
Mr. Ramirez was remanded following his not-guilty plea to a 16-count charge of obtaining money under false presences. He was arraigned by the EFCC. .
Six European countries deport 34 Nigerians
No fewer than 34 Nigerians arrived the Murtala Muhammed International Airport Lagos on Thursday after their deportation from six European countries for violating immigration laws.
The Nigerians were deported from Switzerland, Germany, Iceland, Austria, Belgium and Hungary.
The deportees, comprising of 32 males and two females, arrived at the Murtala Muhammed International Airport (MMIA) Lagos at about 6.30am on board a chartered Airblue Panorama aircraft.
Real Madrid is bigger than Ronaldo – Figo
Former Portugal international, Luis Figo, has said "nobody is indispensable", amid heavy speculation that his compatriot Cristiano Ronaldo could leave Real Madrid this summer.
Ronaldo has refused to comment on his future , after reports emerged that he wants a move away from the Santiago Bernabeu.
Tragedy as Gunpowder Explodes, Burns Teenager's Hands in Bizarre Ritual to Recover Lost Laptop
The palms of a 16-year-old boy identified as Yaw Prince, got burnt after a failed 'spiritual exercise' by a fetish priest to identify a laptop thief. The incident happened at Akyem Sekyere in the Eastern Region of Ghana.
According to a report by Punch, the teenage victim was accused of stealing a laptop and was sent to a 22-year-old fetish priest known as Kwasi Afari, to use spiritual means to establish if it was Yaw who stole it.
Driver Miraculously Escapes Death as Container Crushes Car in Rivers
An unnamed motorists based in the South-south part of Nigeria has escaped death yesterday - 21st of June, 2017 by the whiskers after a container being conveyed to an undisclosed location by a truck accidentally fell on the vehicle.
South African Actress, Celia Kriel Shot By Unknown Gunmen
Popular South African actress, Celia Kriel, was shot by unknown gunmen in the early hours of Wednesday 21 June, 2017.
The actress who appeared in Afrikaans soapie 7de Laan, was shot at home in Randburg. She is said to be in a critical but stable condition after she was rushed to the hospital, Buzz SA reported.
Nigeria to become world's 3rd most populous country in 2050
Nigeria has been projected to be the world's third most populous country by the year 2050, UN Department of Economic and Social Affairs reports.
The report, titled 'World Population Prospects: The 2017 Revision', said with such development, Nigeria would overtake the United States in terms of population just as world population would reach 9.8 billion people.
Married teacher banned from classrooms for life over her affair with 18-year-old student
A married teacher has been banned from the classroom for life for having a nine-month affair with an 18-year-old student, whom she nicknamed 'The A Team'.
Amena Nazam-Khan, 36, who taught at Tong High School in Bradford, used Facebook and a mobile phone app to sext the schoolboy and send naked pictures of herself.
Chelsea council chief resigns over Grenfell Tower blaze
The chief executive of Kensington and Chelsea council Nicholas Holgate has quit following a barrage of criticism over its response to the Grenfell Tower fire tragedy.
Holgate stepped down saying it would have been a "distraction" if he had stayed in his post after the "heart-breaking tragedy", which left at least 79 feared dead.
Actress Iretiola Doyle Opens Up on Getting Pregnant As A Teenager
The ace actress Iretiola Doyle recently opened up on some aspects of her teenage years.
While speaking recently at Harvesters International Christian Centre, Gbagada, Lagos, Doyle revealed that her family members, including her mother, were hard on her when she got pregnant at the age of 17.
North Korea calls Trump a psychopath
North Korea on Thursday called US President Donald Trump a "psychopath" as tensions soar following the death of American student Otto Warmbier, who was evacuated in a coma from North Korean detention last week.
Pyongyang's official Rodong Sinmun newspaper said the US president was in a "tough situation" at home and claimed he was toying with the idea of a preemptive strike on North Korea to divert attention from a domestic political crisis.
Governors tell agitators: Nigeria must remain one
Governors told Acting President Yemi Osinbajo yesterday that they are united with the Federal Government to keep Nigeria as one indivisible entity.
"It has been unanimously decided that the unity of this country is sacrosanct, is non-negotiable and we have all agreed to work together to educate people," Oyo State Governor Abiola Ajimobi said. He was briefing reporters on the outcome of the governors' meeting with Prof. Osinbajo at the Presidential Villa in Abuja.
Nomoreloss' wife Phoenix battles chronic illness one year after husband's death
One year after she buried her husband, wife of late Nigerian singer Nomoreloss, Adeola Phoenix Osinuga, is battling a chronic disease called fibromyalgia.
A chronic disease that has kept her in constant daily pain and kept her away from her radio job with Rhythm FM.
Suspected herdsmen rape, behead woman in Edo
A woman identified as Margaret Udiamehi has been raped and beheaded by suspected herdsmen at Ekpoma, Esan West local government area.
The body of the victim was found on Tuesday night by a search party after she failed to return home from the farm while the police discovered her severed head on Wednesday morning at a different location.
Single mother of 4 dupes Facebook lover N5million (photo)
A single mother of four, Aishat Akintunde, 33, has been arrested in Lagos for allegedly defrauding her Facebook lover of money and properties valued at about N5 million.
Police alleged that Akintunde, a resident of Egbado road, Dalemo, Alakuko, Alagbada, Lagos and hailed from Abeokuta, Ogun State, Nigeria allegedly succeeded in defrauding her France-based victim, Ayobami Adeniyi after she promised to marry him and also helped him purchase a filling station in Nigeria.
Police to guard prayer grounds, parks during Eid el-Fitr
The FCT Police Commissioner, Mr Musa Kimo, on Wednesday ordered water-tight security at all prayer grounds and recreational parks across the Federal Capital Territory before and during the Eid el-Fitri celebration.
A statement by the spokesman of the Command, ASP Usen Omorodion, said that Kimo also directed Area Commanders and Divisional Police Officers to deploy adequate personnel to the points mentioned to forestall any security breach.
Melanin Poppin- Beverly Naya and Beverly Osu in stunning photoshoot
Actress Beverly Naya and Big Brother Africa star,Beverly Osu were paired together for a stunning shoot by photographer ,Sunmisola Olorunnisola..
Their skin is goals!
More below
Man forced to wed live-in lover who died during child birth
A man in Mozambique's southern province of Inhambane had to pay the traditional bride price, locally known as "lobolo", for a dead woman at the weekend, Mozambique's state radio reports.
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Customs Officer dragged to court for Drug Traffick...
Kenyan Pastor With Mysterious Skin Disease Rejecte...
Vendor Who Uses Urine To Wash Cucumbers Caught Red...
INEC officially writes Melaye, begins recall proce...
Prince Philip has left London hospital, Buckingham...
Heartbroken girlfriend hangs herself 2 months afte...
Photos: Former Thai Prime Minister pictured crying...
Father of 6 arrested for having sex with neighbor'...
(Photo) American man remanded in Kirikiri prison f...
Tragedy as Gunpowder Explodes, Burns Teenager's Ha...
Driver Miraculously Escapes Death as Container Cru...
South African Actress, Celia Kriel Shot By Unknown...
Nigeria to become world's 3rd most populous countr...
Married teacher banned from classrooms for life ov...
Chelsea council chief resigns over Grenfell Tower ...
Actress Iretiola Doyle Opens Up on Getting Pregnan...
Nomoreloss' wife Phoenix battles chronic illness o...
Single mother of 4 dupes Facebook lover N5million ...
Police to guard prayer grounds, parks during Eid e...
Melanin Poppin- Beverly Naya and Beverly Osu in st...
Man forced to wed live-in lover who died during ch... | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 590 |
Сільське господарство Білорусі — скорочуваний сектор білоруської економіки, важлива галузь, що забезпечує 7,5 % ВВП країни та 17,1 % інвестицій в основний капітал (2010). У сільському господарстві зайнято 9,7 % населення.
Сучасний стан
Основні показники
У 2010 р. сільське господарство забезпечило 7,5 % ВВП країни (12225 млрд рублів), у 2000 р. — 11,6 %. При цьому на сільське господарство в 2010 р. довелося 17,1 % всіх інвестицій в основний капітал (у 2022 р. — 6,8 %).
У 2010 р. в сільському господарстві зайнято 9,7 % населення країни (у 2000 р. — 14,1 %). Середня зарплата в сільському господарстві є найнижчою серед усіх галузей і становить 815200 рублів (2010; 67 % середньореспубліканського рівня).
Основу сільського господарства становлять колгоспи і радгоспи, в основному перейменовані і діючі на ринковій основі з активною державною підтримкою. На їх частку припадає 99,6 % виробництва льоноволокна, 98,6 % цукрових буряків, 93,6 % зерна, 86,8 % м'яса, 86,5 % молока, 67,7 % яєць, 12,9 % овочів, 11, 1 % картоплі, 7 % вовни. Господарства населення, які практично не користуються державною підтримкою (не рахуючи низьких перехресно-субсидованих тарифів на комунальні послуги), виробляють 88,7 % вовни, 86,9 % картоплі, 81 % овочів, 32,2 % яєць, 13,3 % молока, 12,7 % м'яса. Фермерські господарства (особисті господарства, оформлені, як юридичні особи) не грають великої ролі і виробляють 6,1 % овочів, 4,3 % вовни, 2 % картоплі, 1,4 % зерна та цукрових буряків, 0,5 % м'яса.
Сукупна площа сільськогосподарських земель на початок 2011 р. — 8897,5 тис. га (5510,5 тис. га — орні землі, 3240,6 тис. га — лугові землі). 16,4 % земель меліоровані. З 5510,5 тис. га орних земель 4698,2 тис. га перебувають у користуванні сільгоспорганізацій, 682,1 тис. га — у користуванні громадян (у 2001 р. — 1022 тис. га), з яких 640 тис. га відведено під особисті підсобні господарства, 31,6 тис. га — під дачі, 85,4 тис. га — у користуванні фермерських господарств.
У 2010 р. всіма господарствами країни вироблено сільськогосподарської продукції на 35,6 трлн рублів (у поточних цінах). 55,2 % становила продукція рослинництва, 44,8 % — тваринництва. Близько 2/3 продукції вироблено в сільгоспорганізаціях (менше половини сукупної продукції рослинництва, майже 90 % продукції тваринництва), близько 1/3 — в особистих господарствах населення, близько 1 % — у фермерських господарствах. Частка сільгоспорганізацій у виробництві зерна становить 93,6 %, картоплі — 11,1 %, овочів — 12,9 %.
За 2000-10 рр. кількість техніки в господарствах помітно скоротилася. Кількість тракторів знизилося з 72,9 тис. у 2000 р. до 47,3 тис. на початок 2011-го, кількість вантажних автомобілів — з 46,3 тис. до 25,1 тис., зернозбиральних комбайнів — з 17,1 тис. до 11,4 тис., силосу-і кормозбиральних комбайнів — з 7,2 тис. до 2,6 тис. Збільшилася кількість бурякозбиральних комбайнів і тракторних обприскувачів і запилювачів. Більшість сільськогосподарської техніки — власного виробництва (Мінський тракторний завод, Лідсельмаш, Гомсельмаш та інші). Щорічне виробництво тракторів за 2000-10 рр. зросла з 22 470 до 44 370, тракторів потужністю більше 100 к.с. — з 2617 до 9454, зернозбиральних комбайнів — з 445 до 2035. Значна частина нової техніки поставляється на експорт. У результаті зниження загальної кількості сільгосптехніки, забезпеченість угідь тракторами знизилася з 15 тракторів на 1000 га ріллі у 2000 р. до 10 в 2010 році.
Велика частина використовуваних добрив — місцевого виробництва (найбільший виробник калійних добрив — «Білоруськалій», азотних — «Гродно Азот»). За 2000-10 рр. виробництво азотних добрив зросла з 597 до 761 тис. тонн, фосфорних — з 87 до 192 тис. тонн, калійних — з 3,4 до 5,2 млн тонн, вапнякового і доломітового борошна — з 1,5 до 1,9 млн тонн. Внесення мінеральних добрив під сільськогосподарські культури збільшилася за цей же період з 850 до 1323 тис. тонн, включаючи збільшення кількості азотних добрив з 270 до 463 тис. тонн, фосфорних з 119 до 230 тис. тонн, калійних з 462 до 630 тис. тонн. У розрахунку на один гектар орних земель приріст склав з 169 до 284 кг. Сукупна внесення органічних добрив збільшилася з 35,9 до 43,2 млн тонн.
Підготовкою фахівців у сфері сільського господарства займаються Білоруський державний аграрний технічний університет (Мінськ), Білоруська державна сільськогосподарська академія (місто Горки, Могилевська область) і Гродненський державний аграрний університет, підрозділи інших вузів, ряд спеціалізованих середніх спеціальних навчальних закладів.
Статистичні показники
Виробництво деяких видів сільськогосподарської продукції на душу населення:
Середня врожайність зернових, ц/га (дані Белстата и БСЭ):
Загальна посівна площа, тис. га:
Посівна площа за областями, тис. га:
Посівна площа під різні культури (червоний — зернові, золотий — картопля, зелений — кормові, сірий — інші):
Структура посівних площ (2010):
Примітки
Посилання
CIA World Factbook: Belarus
Library of Congress Country Studies: Belarus
Міністерство сільського господарства і продовольства Республіки Білорусь
Новини сільського господарства в Білорусі
Перелік державних організацій, підпорядкованих Міністерству сільського господарства і продовольства Республіки Білорусь
Сайти управлінь та об'єднань Аграрно -промислового комплексу
Провідні підприємства експортери Аграрно -промислового комплексу
Інформаційно- консультаційна служба Аграрно -промислового комплексу | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 31 |
Onside er et dansk fodboldmagasin, der bliver sendt hver søndag aften på TV3+ efter aftenens livekamp fra Superligaen. Programmet sender reportager og interviews fra alle rundens kampe i Superligaen, samt fra andre store sportsbegivenheder, især boksning og Formel 1.
Den nuværende vært er Signe Vadgaard, der også har fungeret på stationens Premier League-dækning. Tidligere har Camilla Martin også været frontfigur på udsendelserne gennem mange år. Rækken af tidligere værter på programmet tæller Peter Palshøj, Mette Cornelius, Carsten Werge, Per Frimann, Dan Engelsted, Jakob Kjeldbjerg, Lotte Thor Høgsberg Helle Smidstrup og Dan Hirsch Sørensen.
Kilder
Eksterne henvisninger
Onside.dk
TV3-programmer
Sportsprogrammer fra Danmark
Danske tv-serier fra 2010'erne | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,833 |
\section{Introduction}
Let us begin by recalling the Sharkovskii Theorem \cite{ShU,Stefan,Block}:
\begin{theorem}[Sharkovskii]\label{th:shar}
Define an ordering `$\triangleleft$' of natural numbers:
\begin{equation}\label{eq:order}
\begin{array}{r@{\ \triangleleft\ }c@{\ \triangleleft\ }c@{\ \triangleleft\ }c@{\ \triangleleft\ }l}
3\triangleleft 5 \triangleleft 7 \triangleleft 9 \triangleleft\ \dots & 2\cdot 3 & 2 \cdot 5 & 2 \cdot 7 & \dots
\\
\dots & 2^2\cdot 3 & 2^2 \cdot 5 & 2^2 \cdot 7 & \dots
\\
\dots & 2^3\cdot 3 & 2^3 \cdot 5 & 2^3 \cdot 7 & \dots
\\
\multicolumn{5}{c}{\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots }
\\
\dots & 2^{k+1} & 2^k & 2^{k-1} & \ldots\ \triangleleft 2^2 \triangleleft 2 \triangleleft 1.
\\
\end{array}
\end{equation}
Let $f: I \to \mathbb{R}$ be a continuous map of an interval. If $f$ has an $n$-periodic point and $n\triangleleft m$, then $f$ also has an $m$-periodic point.
\end{theorem}
The paper addresses the following question: Can the Sharkovskii theorem be carried over to multidimensional dynamics? Without additional assumptions the answer is negative.
However, if the map is in some sense close to 1-dimensional then one can hope for a positive answer.
The typical situation in which such a result can be of interest is the following: consider a Poincar\'e map $F$ for some ODE with the following
properties: there exists an attracting set which is close to being one-dimensional. Then we have an approximate 1D map $f$ which may posses some periodic
orbits to which the Sharkovskii theorem applies, implying the existence of some periodic points for $f$. We would like to infer that $F$ has
also orbits with the same periods. The result of this type has been given in \cite{PZszarI, PZmulti}, but it can hardly
be directly applied to an explicit example (the R\"ossler system considered in our paper) as it is difficult to obtain any reasonable quantitative condition
on the difference between $F$ and $f$ from the proof in \cite{PZszarI}.
However, in our recent paper \cite{AGPZrossler}, where we consider two sets of parameters in the R\"ossler system with attracting periodic orbits: 3-periodic for the first set and 5-periodic for the other, we proved the existence of the periods implied by the Sharkovskii theorem quite easily.
Just as in \cite{PZszarI}, the proof relies on the construction of a set of covering relations, but due to the fact that orbit we started with was attracting the construction was much simpler in this case.
The goal of this work the generalize this observation, which is the main abstract result in this paper.
\begin{theorem}
\label{thm:sh-manyD}
Consider a continuous map $F:I\times \overline{B}(0,R)\to \inte \left(I\times \overline{B}(0,R)\right)$, where $I\subset \mathbb{R}$ is a closed interval and $\overline{B}(0,R) \subset \mathbb{R}^{n-1}$ a closed ball of radius $R$. Let us denote by $(x,y)$
points in $I\times \overline{B}(0,R)$.
Suppose that $F$ has an $n$-periodic point $(x_0,y_0) \in \mathbb{R} \times \mathbb{R}^{n-1}$ with least period $n$ and denote its orbit by $\{(x_0,y_0), (x_1,y_1)=F(x_0,y_0), \dots, (x_{n-1},y_{n-1}) = F^{n-1}(x_0,y_0), (x_n,y_n)=(x_0,y_0)\}\subset \inte I\times \overline{B}(0,R)$.
Suppose that there exist $\delta_0$, $\delta_1$, \dots, $\delta_{n-1}>0$ such that
\[
\forall\: i\in\{0,\dots,n-1\} \qquad F\left([x_i\pm\delta_i]\times\overline{B}(0,R)\right) \subset (x_{i+1}\pm\delta_{i+1})\times B(0,R).
\]
Then for every natural number $m$ succeeding $n$ in the Sharkovskii order \eqref{eq:order} $F$ has a point with the least period $m$.
\end{theorem}
The geometric situation in which the above theorem is applicable is as follows:
consider a map $F: \mathbb{R}^n \to \mathbb{R}^n$, assume that $F$ has an nearly one-dimensional attractor $\mathcal{A}$, by which we mean that in suitable coordinates $\mathcal{A}$ is contained in $I \times \overline{B}(0,R)$ for some small $R>0$ and $F\left(I\times \overline{B}(0,R)\right)\subset \inte \left(I\times \overline{B}(0,R)\right)$, and $F$ has an attracting orbit of period $m$ in $\inte \left(I\times \overline{B}(0,R)\right)$, which basin of attraction contains $C_i$ products of interval and a ball $[x_i -\delta_i,x_i+\delta] \times \overline{B}(0,R)$ such that $F(C_i) \subset C_{i+1}$ for $i=0,\dots,n-1$.
The essential difference between the construction used here and in \cite{PZszarI} is the version of the proof of the Sharkovskii theorem
which is used to construct some covering relations. In the present work we use the ideas from the proof of Burns and Hasselblatt \cite{Burns}
while in \cite{PZszarI} is was based the so-called \u{S}tefan cycle \cite{Stefan, Block}.
As an application of Theorem~\ref{thm:sh-manyD} we study the R\"ossler system \cite{AGPZrossler} for different values of parameters with attracting periodic orbits of $3$, $5$ and $6$. We verify the assumptions of Theorem~\ref{thm:sh-manyD}, hence we obtain an infinite number of periodic orbits. For precise statements see Sec~\ref{sec:App}). The proofs for the R\"ossler system are computer-assisted, written in C++ with the use of CAPD (Computer-Assisted Proofs in Dynamics) library \cite{capd,capd-article} for interval arithmetic, differentiation and ODE integration.
The content of the paper can be described as follows. In Section \ref{sec:BHproof} we recall some ideas and facts from the proof of the Sharkovskii theorem by Burns and Hasselblatt \cite{Burns}. In Section~\ref{sec:coveringRn} we discuss the notion of covering relations, which is the main tool used in this work to obtain periodic orbits. In Section~\ref{sec:grids} we define a notion of a \emph{contracting grid} around a periodic orbit. In Section~\ref{sec:mainThm} we prove the main theorem. Finally in Section~\ref{sec:App} we apply our result to the R\"ossler system from \cite{AGPZrossler}.
\subsection*{Notation:}
\begin{itemize}
\item We use the common notation for the closure, interior, and boundary of a topological set $A\subset\mathbb{R}^k$, which are $\overline{A}$, $\inte A$, and $\partial A$, respectively.
\item $\pi_i$ denotes the projection onto the $i$th coordinate in $\mathbb{R}^N$.
\item By an `$n$-periodic orbit' or `point', we understand an orbit or a point with basic period $n$.
\end{itemize}
\section{Proof of Sharkovskii's theorem by Burns \& Hasselblatt}\label{sec:BHproof}
First we introduce some necessary definitions taken directly from \cite{Burns}.
\subsection{Interval covering relation `$\longrightarrow$'}
Let $\mathcal{O} \subset \mathbb{R}$ be a set of $n$ points.
By an \emph{$\mathcal{O}$-interval} we understand any interval $J\subset [\min\mathcal{O}, \max\mathcal{O}]$ of positive length with the endpoints in $\mathcal{O}$, that is, $\partial J \subset \mathcal{O}$.
Let $f:\mathcal{O} \to \mathcal{O}$ be an $n$-periodic permutation with every $x\in \mathcal{O}$ being an $n$-periodic point for $f$.
In the paper we will use a stronger notion of a one-dimensional covering relation between intervals than the classical one used by Block \textit{et al.} \cite{Block}. Our definition appears in the paper of Burns and Hasselblatt \cite{Burns} as the \emph{$\mathcal{O}$-forced covering} between $\mathcal{O}$-intervals, which we simply call the covering between $\mathcal{O}$-intervals:
\begin{definition}[($\mathcal{O}$-forced) covering relation]\label{def:Oforced_covering}
An $\mathcal{O}$-interval $I$ \emph{covers} an $\mathcal{O}$-interval $J$ (denoted by $I \overset{f}{\rightarrow} J $ or simply $I \rightarrow J $), if
there exists a $\mathcal{O}$-subinterval $K\subset I$ such that
\begin{equation}
\min f(\partial K) \leq \min J \text{ \qquad and \qquad } \max J \leq \max f(\partial K).
\end{equation}
\end{definition}
The above relation fulfills the Itinerary Lemma \cite{Block,Burns}, which is crucial in proving the existence of periodic orbits in the proofs of Sharkovskii's theorem:
\begin{theorem}[Itinerary Lemma]
\label{th:1d-covering}
Let $f: I \to \mathbb{R}$ be a continuous map on an interval $I \subset\mathbb{R}$ and $\mathcal{O}\subset I$ be an $n$-periodic point for $f$. Assume that we have a sequence of $\mathcal{O}$-intervals $J_j \subset I$ for $j=0,\dots,m-1$ such that
\begin{equation}\label{eq:1d_loop}
J_0 \overset{f}{\longrightarrow} J_1 \overset{f}{\longrightarrow} J_2 \overset{f}{\longrightarrow} \dots \overset{f}{\longrightarrow} J_{m-1} \overset{f}{\longrightarrow} J_0.
\end{equation}
Then there exists a point $x \in J_0$, such that $f^j(x) \in J_j$ for $j=1,\dots,m-1$ and $f^m(x)=x$.
\end{theorem}
A point which fulfills the thesis of Theorem \ref{th:1d-covering} is said to \emph{follow the loop} \eqref{eq:1d_loop}. A loop of length $m$ (such as \eqref{eq:1d_loop}), we will call shortly an $m$-loop.
In general, from Theorem \ref{th:1d-covering} we do not know whether the point's period is fundamental. To make sure that the point $x\in J_0$ following the loop \eqref{eq:1d_loop} is indeed $m$-periodic, we must add some assumptions on the $\mathcal{O}$-intervals forming the loop of covering relations. Particular criteria may put additional assumptions on the intervals $J_i$ or some restrictions on their order. We use the following criterion, which includes all the cases studied in \cite{Burns}:
\begin{definition}[\cite{Burns}, Lemma 2.6\footnote{It is a more specific definition than the one of an \emph{elementary loop} from \cite[Def. 2.4]{Burns}.}]\label{def:basic_loop}
We call a loop \eqref{eq:1d_loop} a \emph{non-repeating loop}, if it fulfills the following conditions:
\begin{enumerate}
\item it is not followed by any endpoint $x\in\bigcup_{i=0}^{m-1} \partial J_i$;
\item $\displaystyle \inte J_0 \cap \bigcup_{i=1}^{m-1} J_{i} = \varnothing$.
\end{enumerate}
\end{definition}
\begin{lemma}[\cite{Burns}, Lemma 2.6]\label{lem:non-repeating}
If a loop \eqref{eq:1d_loop} is non-repeating, then every point following it has period $m$.\qed
\end{lemma}
\subsection{Proposition 6.1 of \cite{Burns}}
Let now $f: I \to \mathbb{R}$ be a continuous map on an interval $I \subset\mathbb{R}$ and $\O\subset I$ be an $n$-periodic orbit for $f$.
The following theorem is Proposition 6.1 of \cite{Burns} for the map $f$.
\begin{theorem}[\cite{Burns}]\label{th:Burns}
For every number $m$ succeeding $n$ in Sharkovskii's order \eqref{eq:order} (\textit{i.e.} $n\triangleleft m$) there exists a non-repeating $\mathcal{O}$-forced $m$-loop of $\mathcal{O}$-intervals $J_i$, $i=0,\dots, m-1$:
\begin{equation}
J_0 \overset{f}{\longrightarrow} J_1 \overset{f}{\longrightarrow} J_2 \overset{f}{\longrightarrow} \dots \overset{f}{\longrightarrow} J_{m-1} \overset{f}{\longrightarrow} J_0\text{,}
\end{equation}
which proves the existence of an $m$-periodic point for $f$ in the interval $ [\min \mathcal{O},\max\mathcal{O}]$.
\end{theorem}
Since the above theorem is crucial for our construction, we present here the outline of the proof.
\begin{remark}[Sketch of the proof]
The proof of above theorem, which is the main result of \cite{Burns}, relies by induction on either:
\begin{itemize}
\item constructing the so-called \u{S}tefan sequence of some length $2\leq l\leq n$ of $\mathcal{O}$-intervals $J_i$, $i=0,\dots,l-1$, which form the diagram of $\mathcal{O}$-forced covering relations ($\inte J_0$ is disjoint with all other $J_i$'s):
\vspace{3mm}
\begin{equation}\label{eq:diagOdd}
\begin{tikzcd}
&
J_1
\arrow[r, dashed]
\arrow[loop, distance=2.5em, in=100, out=170]
&
\text{\rotatebox[origin=c]{-20}{$\ldots$}}
\arrow[rd, dashed]
&
\\
J_0
\arrow[d, shift left]
\arrow[ru]
\arrow[rru, "\text{\tiny to $J_{l-2k-1}$}"', dashed]
\arrow[rrr] \arrow[rrdd]
& & &
J_{l-5}
\arrow[d]
\\
J_{l-1}
\arrow[u, shift left]
& & &
J_{l-4}
\arrow[ld]
\\
&
J_{l-2}
\arrow[lu]
&
J_{l-3}
\arrow[l]
&
\end{tikzcd}
\end{equation}
From the above diagram \eqref{eq:diagOdd} one deduces the existence of $m$-periodic points, from non-repeating loops:
\begin{itemize}
\item $m=1$ from \qquad $
J_1 \longrightarrow J_1$,
\item $m \geq l$ from \qquad $
J_0 \rightarrow J_1 \underbrace{\dots \rightarrow J_1}_{\text{$(m-l) \times$ `$\rightarrow J_1$'}} \rightarrow J_2 \rightarrow \dots \rightarrow J_{l-1} \rightarrow J_0$,
\item even $m<l$ from \qquad $
J_0 \rightarrow J_{l-m+1} \rightarrow J_{l-m+2} \rightarrow \dots \rightarrow J_{l-1} \rightarrow J_0$.
\end{itemize}
\item or reducing to the case of the orbit of length $n/2$ for the map $f^2$, if the construction of a \u{S}tefan sequence is impossible.
\end{itemize}
\end{remark}
Although the authors formulate their Proposition 6.1 in a more general way, the loops which appear in the proof of Theorem \ref{th:Burns} are in fact non-repeating and the proof relies on that fact. Let us review shortly the induction on the length of $\mathcal{O}$, $n\in\mathbb{N}$ presented in the proof of \cite[Proposition 6.1]{Burns}:
\begin{enumerate}
\item For $n=1$ there are no loops of length $m\triangleright n$.
\item For $n=2$ the constructed loop is of the form
\[
J_1 \rightarrow J_1\text{,}
\]
which is non-repeating.
\item Suppose now that for all lengths of $\mathcal{O}$ smaller that $n$ the loops constructed in the proof are non-repeating. For the length $n>2$ and a number $m\triangleright n$ there are two possible cases.
\begin{enumerate}
\item We are able to construct a \u{S}tefan sequence and the non-repeating $m$-loop is of the form:
\[
J_0 \rightarrow \underbrace{\dots \rightarrow J_1}_{\geq 0 \text{ times}} \rightarrow J_1 \rightarrow J_2 \rightarrow \dots \rightarrow J_{k} \rightarrow J_0\text{,}
\]
with $\inte J_0 \cap \bigcup_{i=1}^{k} J_i = \varnothing$ and $k\geq 1$.
\item We cannot construct a \u{S}tefan sequence, but then either $m=1$ and the case is trivial, or $n$, $m$ are even. Hence, by induction, we have a non-repeating loop of length $\frac{m}{2}$ for the map $f^2$:
\begin{equation}\label{eq:even_loop}
J_0 \overset{f^2}\longrightarrow J_{1} \overset{f^2}\longrightarrow J_{2} \overset{f^2}\longrightarrow \dots \overset{f^2}\longrightarrow J_{\frac{m}2-1} \overset{f^2}\longrightarrow J_0.
\end{equation}
Next, the construction from the proof of \cite[Proposition 6.1]{Burns} extends the above $\frac{m}2$-loop \eqref{eq:even_loop} to the following $m$-loop for $f$:
\begin{equation*
J_0
\overset{f}\longrightarrow J'_{0}
\overset{f}\longrightarrow J_{1}
\overset{f}\longrightarrow J'_{1}
\overset{f}\longrightarrow
\dots
\overset{f}\longrightarrow J_{\frac{m}2-1}
\overset{f}\longrightarrow J'_{\frac{m}2-1}
\overset{f}\longrightarrow J_0\text{,}
\end{equation*}
where $\bigcup_{i=0}^{m/2} J_i \cap \bigcup_{i=0}^{m/2} J'_i = \varnothing$. In particular, $J_0 \cap J'_{i} = \varnothing$ for $i=0,\dots,\frac{m}2-1$.
\end{enumerate}
\end{enumerate}
\subsection{Proper covering}
Let us introduce now a more specific notion of covering between $\mathcal{O}$-intervals, which we can easily compare later to the notion of horizontal covering between segments (Section \ref{sec:coveringRn}).
\begin{definition}[Proper covering of intervals]
Assume that $I, J \subset \mathbb{R}$ are $\mathcal{O}$-intervals. We say that
$I$ \emph{$f$-covers $J$ properly} (denoted\footnote{The same symbol is used in \cite{Burns}, but for a different notion.} by $I \overset{f}{\rightarrowtail} J$ or simply $I \rightarrowtail J $), if
\begin{equation}\label{eq:proper_cover}
\min f(\partial I) \leq \min J \text{ \qquad and \qquad } \max J \leq \max f(\partial I).
\end{equation}
\end{definition}
It is easy to see from the definition of $\mathcal{O}$-forced covering (Definition \ref{def:Oforced_covering}) that the following statement is true.
\begin{lemma}\label{lem:proper_covering}
If $I\rightarrow J$ is an $\mathcal{O}$-forced covering relation, then there exists an $\mathcal{O}$-subinterval $K\subset I$ such that $K\rightarrowtail J$.\qed
\end{lemma}
We can finally replace $\mathcal{O}$-forced loops by loops of proper covering relations.
\begin{lemma}\label{lem:proper_loop}
For every number $m$ succeeding $n$ in Sharkovskii's order \eqref{eq:order} (\textit{i.e.} $n\triangleleft m$) there exists a non-repeating $m$-loop of proper covering relations between $\mathcal{O}$-intervals $K_i$, $i=0,\dots, m-1$:
\begin{equation}\label{eq:proper_loop}
K_0 \overset{f}{\rightarrowtail} K_1 \overset{f}{\rightarrowtail} K_2 \overset{f}{\rightarrowtail} \dots \overset{f}{\rightarrowtail} K_{m-1} \overset{f}{\rightarrowtail} K_0\text{.}
\end{equation}
\end{lemma}
\begin{proof}
Fix $m\triangleright n$ and consider the non-repeating $\mathcal{O}$-forced loop obtained from Theorem \ref{th:Burns}:
\begin{equation*}
J_0 \overset{f}{\longrightarrow} J_1 \overset{f}{\longrightarrow} J_2 \overset{f}{\longrightarrow} \dots \overset{f}{\longrightarrow} J_{m-1} \overset{f}{\longrightarrow} J_m=J_0\text{.}
\end{equation*}
Now apply Lemma \ref{lem:proper_covering} to every relation $J_i \rightarrow J_{i+1}$, $i=0,\dots, m-1$. We obtain $m$ $\mathcal{O}$-intervals $K_i\subset J_i$, for $i=0,\dots, m-1$ which fulfill the proper covering relations:
\[
K_i \rightarrowtail J_{i+1}\text{, } \quad i=0,\dots, m-1.
\]
Observe now that also each of the proper covering relations $K_i \rightarrowtail K_{i+1}$ are true, because each $K_{i+1}$ is a subinterval of $J_{i+1}$.
Finally, note that the loop
\begin{equation*}
K_0 \overset{f}{\rightarrowtail} K_1 \overset{f}{\rightarrowtail} K_2 \overset{f}{\rightarrowtail} \dots \overset{f}{\rightarrowtail} K_{m-1} \overset{f}{\rightarrowtail} K_0\text{}
\end{equation*}
is non-repeating, because both conditions (1) and (2) from Definition \ref{def:basic_loop} are fulfilled if we replace the $\mathcal{O}$-intervals by their $\mathcal{O}$-subintervals.
\end{proof}
\section{Horizontal covering relation `$\Longrightarrow$'}\label{sec:coveringRn}
In order to have the Itinerary Lemma in many dimensions we need a good notion of covering where we have a direction of possible expansion
and an apparent contraction in other directions.
For this end we recall the notion of covering for h-sets in $\mathbb{R}^N$ from \cite{PZszarI,PZmulti,AGPZrossler}.
It is, in fact, a particular case of the similar notion from \cite{GZ}, but with exactly one exit (or 'unstable') direction. For such a covering relation a special version of Itinerary Lemma is true (Th.\ \ref{th:periodic} below) and we use it to prove the existence of periodic points for multidimensional maps.
\begin{definition}
An \emph{h-set} is a hyper-cuboid $S=[a_1,b_1]\times \dots \times [a_N,b_N]\subset \mathbb{R}^N$ for some $a_i<b_i$, $i=1,\dots,N$, with the following elements distinguished:
\begin{itemize}
\item its left face \quad $L(S) = \{x\in\partial S \,:\,x_1=a_1\}$,
\item its right face \quad $R(S) = \{x\in\partial S \,:\,x_1=b_1\}$,
\item its horizontal boundary \quad $H(S) = \overline{\partial S \setminus (L(S)\cup R(S))}$,
\item its left side \quad $\mathcal{L}(S) = \{x\in\mathbb{R}^N \,:\,x_1<a_1\}$,
\item its right side \quad $\mathcal{R}(S) = \{x\in\mathbb{R}^N \,:\,x_1>b_1\}$.
\end{itemize}
\end{definition}
Now we define a horizontal covering relation between h-sets.
\begin{definition}
\label{def:cov}
Let $S$, $S'$ be two h-sets and $f: V \to \mathbb{R}^N$ be a continuous map on an open neighborhood of $S\subset V\subset\mathbb{R}^N$.
We say that $S$ \emph{$f$-covers $S'$ horizontally} and denote by $S \overset{f}{\Longrightarrow} S'$ if
\begin{equation}\label{eq:cond9}
f(S) \subset \left(\mathcal{L}(S')\cup S'\cup \mathcal{R}(S')\right) \setminus H(S')\text{,}
\end{equation}
and one of the two conditions hold:
\begin{equation}\label{eq:cond10}
\begin{aligned}
&\text{either } & f(L(S))\subset \mathcal{L}(S') &\text{\quad and \quad} f(R(S))\subset \mathcal{R}(S')\text{,}
\\
&\text{or } & f(L(S))\subset \mathcal{R}(S') &\text{\quad and \quad} f(R(S))\subset \mathcal{L}(S').
\end{aligned}
\end{equation}
\end{definition}
See Fig.\ \ref{fig:cover} for the illustration of horizontal covering in $\mathbb{R}^2$ and Fig.\ \ref{fig:cover3d} for covering in $\mathbb{R}^3$.
\begin{figure}[h]
\includegraphics[height=7cm]{covering2d}
\caption{\label{fig:cover}Horizontal covering between 2D h-sets $S \overset{f}{\Longrightarrow} S'$}
\end{figure}
\begin{figure}[h]
\includegraphics[width=0.7\textwidth]{cover3d}
\caption{\label{fig:cover3d}Horizontal covering between 3D h-sets $S \overset{f}{\Longrightarrow} S'$}
\end{figure}
Let us emphasize that the above conditions can be easily checked with the use of computer via interval arithmetic and `$<$', `$>$' relations.
The following theorem might be understood as a version of Itinerary Lemma (Theorem~\ref{th:1d-covering}).
\begin{theorem}[\cite{PZszarI}]\label{th:periodic}
Suppose that there occurs a loop of $m$ horizontal $f$-coverings between h-sets $S_i\subset\mathbb{R}^N$, $i=0,\dots,m-1$:
\[
S_0 \overset{f}{\Longrightarrow} S_1\overset{f}{\Longrightarrow} \dots \overset{f}{\Longrightarrow} S_{m-1} \overset{f}{\Longrightarrow} S_m = S_0\text{,}
\]
then there exists $x\in \inte S_0$ such that $f^m(x)=x$ and
\[
\text{for } i=1,\dots ,m-1: \qquad f^i(x)\in \inte S_i.
\]
\end{theorem}
\section{Grids}
\label{sec:grids}
The goal of this section is to introduce the notion of \emph{contracting grid}. In the context of Theorem~\ref{thm:sh-manyD} a contracting grid is a set of cubes of the form $C_i=[x_i-\delta_i,x_i+\delta_i] \times \overline{B}(0,R)$ for $i=0,\dots,n-1$ and the h-sets $S_{ij}=[x_i+\delta_i,x_{j}-\delta_j] \times \overline{B}(0,R)$ for $i,j$, such that $x_i < x_j$, lying between $C_i$'s. Due to the fact that for each $i$ $F(C_i) \subset \inte C_{i+1}$, we obtain, using the ideas from the proof of the Sharkovskii theorem recalled in Section~\ref{sec:BHproof}, a rich set
of horizontal covering relations, which gives us all periods.
\subsection{Model grid}
Let now $\mathcal{O} = \{1, 5,\dots, 4n-3\}$ be the set of $n$ points on the real line $Ox_1$ which, together with $Ox_k$ lines, $k=2,\dots, N$, span the Euclidean space $\mathbb{R}^N = \{x=(x_1, \dots, x_N) \;:\; x_i \in \mathbb{R},\; i=1,\dots, N\}$.
\begin{definition}
By a \emph{model grid} $\mathcal{G}$ enclosing $\mathcal{O}$ we understand an $N$-dimensional full closed hyper-cuboid
restricted by the following hyper-planes in $\mathbb{R}^N$:
\begin{itemize}
\item $\{x_1 = 0\}$ and $\{x_1 = 4n-2\}$;
\item $\{x_k = 1\}$ and $\{x_k = -1\}$, for $k= 2,\dots,N$,
\end{itemize}
with the following elements distinguished:
\begin{itemize}
\item \emph{inner cubes}: $C_k$, $k\in\mathcal{O}$: \hfil $C_k = \{x\in\mathcal{G} : k-1\leq x_1\leq k+1\}$.
\\Note that each inner cube $C_k$ contains a single point from $\mathcal{O}$, that is, the point $k$.
\item \emph{outer segments}: $S_{ij}$, $i$, $j\in\mathcal{O}$, $i< j$: \hfil $S_{ij} = \{x\in\mathcal{G} : i+1\leq x_1\leq j-1\}$.
Each outer segment has on its boundary two vertical faces common with two inner cubes $C_i$ and $C_j$. We will also say that the segment $S_{ij}$ \emph{lies between the cubes} $C_i$ and $C_j$ or \emph{lies between points} $i$, $j\in\mathcal{O}$.
\\
Note also that there is a natural one-to-one correspondence between outer segments and $\mathcal{O}$-intervals on the real line $Ox_1$. The segment $S_{ij}$ corresponding to an $\mathcal{O}$-interval $I_{ij}=[i,j]$ we will denote by $S(I_{ij}) = S_{ij}$.
\end{itemize}
\end{definition}
For the illustration of the notion of the model grid see Fig. \ref{fig:model_grid}.
\begin{figure}[h]
\includegraphics[height=5cm]{model_grid}
\caption{\label{fig:model_grid}A model grid enclosing four points (red) in $\mathbb{R}^3$.
Two segments are marked in gray: $S_{1,5}$ lying between $1$ and $5$, and $S_{5,13}$ between $5$, $13$. White cubes are the sets $C_1$, $C_5$ and $C_{13}$.}
\end{figure}
\begin{remark}
Note that we can naturally provide a structure of an h-set for a model segment $S=S_{kl}$ lying between the points $k < l \in \{1,5,\dots,4n-3\}$:
\begin{itemize}
\item $L(S) = \{x\in\partial S \,:\,x_1=k+1\}$,\quad $R(S) = \{x\in\partial S \,:\,x_1=l-1\}$,
\item $H(S) = \{x\in\partial S : \exists_{i=2,\dots N} |x_i|=1\}$,
\item $\mathcal{L}(S) = \{x\in\mathbb{R}^N \,:\,x_1<k+1\}$, \quad $\mathcal{R}(S) = \{x\in\mathbb{R}^N \,:\,x_1>l-1\}$.
\end{itemize}
\end{remark}
\subsection{Contracting grid}
Let $U$ be an open subset of $\mathbb{R}^N$.
Consider a set of $n$ points $P = \{p_1, p_2, \dots, p_n\} \subset U$.
\begin{definition}
A set $G\subset U$ containing $P$ is called a \emph{grid enclosing $P$} if there exists a homeomorphism $h : V\to U$, where $V$ is some open neighborhood of the model grid $\mathcal{G}$ in $\mathbb{R}^N$, such that $P=h(\mathcal{O})$ and $G =h(\mathcal{G})$.
We also define all elements mapped from the model grid:
inner cubes and outer segments, as the images of the model elements through $h$.
\end{definition}
Let now $F: U \to \mathbb{R}^N$ be continuous map and $P$ be an $n$-periodic orbit for $F$:
\[
P = \{p_1, p_2, \dots ,p_n, p_{n+1}=p_1\}
\text{, \quad where \quad}
p_1\mapsto p_2\mapsto \dots \mapsto p_n\mapsto p_1.
\]
Suppose that there exists a grid $G = h(\mathcal{G})\subset U$ enclosing the periodic orbit $P$ and denote $F_h = h^{-1}\circ F \circ h : V \to V$.
\begin{definition} We call a grid $G$ a \emph{contracting grid} if
\begin{equation}\label{eq:contracting}
F(G) \subset \inte G
\quad \text{ and } \quad
\mathop{\forall}_{i=1,\dots,n} F(h(C_i)) \subset \inte h(C_{F_h(i)}) .
\end{equation}
\end{definition}
\begin{theorem}\label{th:1to2}
Assume that an $n$-periodic orbit $P$ for $F$ is enclosed by a contracting grid $G= h(\mathcal{G})$ and $\mathcal{O} = h^{-1}(P)$.
Define $f:\mathcal{O}\to\mathcal{O}$ by $f=F_h|_{\mathcal{O}}$.
If two $\mathcal{O}$-intervals $I$, $I'$ fulfill the proper covering relation $I \overset{f}{\rightarrowtail} I'$, then also the corresponding segments $S(I)$, $S(I')$ fulfill the horizontal covering relation
\begin{equation}
S(I) \overset{F_h}{\Longrightarrow} S(I').
\end{equation}
\end{theorem}
\begin{proof}
Denote by $i<j\in \mathcal{O}$ the endpoints of $I$ and by $k<l\in \mathcal{O}$ the endpoints of $I'$, so that $I=I_{ij}$, $I'=I_{kl}$, $S(I) = S_{ij}$ and $S(I') = S_{kl}$. From the proper covering $I_{ij} \overset{f}{\rightarrowtail} I_{kl}$ we know that either $f(i)\leq k$ and $f(j)\geq l$, or $f(i)\geq l$ and $f(j)\leq k$.
\begin{itemize}
\item \underline{Condition \eqref{eq:cond9}:} The grid is contracting, so $F_h(\mathcal{G})\subset \inte \mathcal{G}$. In particular, $F_h(S(I)) \subset \inte \mathcal{G}\subset\left(\mathcal{L}(S(I'))\cup S(I')\cup \mathcal{R}(S(I'))\right) \setminus H(S(I'))$.
\item \underline{Condition \eqref{eq:cond10}:}
Consider just the case $f(i)\leq k$ and $f(j)\geq l$, for the other one is analogous.
From the condition \eqref{eq:contracting}, $F_h(C_i)\subset \inte C_{f(i)}$ and $F_h(C_j)\subset \inte C_{f(j)}$. In particular, $L(S_{ij})\subset C_i$ and $R(S_{ij})\subset C_j$, so $\max \pi_1(F_h(L(S_{ij})))< f(i) +1 \leq k+1$ and $\min \pi_1(F_h(R(S_{ij})))> f(j) -1 \geq l-1$, which means that $F_H(L(S_{ij})) \subset \mathcal{L}(S_{kl})$ and $F_H(R(S_{ij})) \subset \mathcal{R}(S_{kl})$.
\end{itemize}
\end{proof}
\section{The main theorem}
\label{sec:mainThm}
We have the following result.
\begin{theorem}\label{th:Final}
Let $F:U\to \mathbb{R}^N$ be a continuous map on an open set $U\subset\mathbb{R}^N$ and $p_0\in U$ be an $n$-periodic point of $F$ for $n>1$. Suppose that the orbit $O = \{p_0, p_1=F(p_0),\dots p_{n-1} = F^{n-1}(p_0)\}$ is enclosed in a contracting grid $G= h(\mathcal{G})$.
Then for every $m\in\mathbb{N}$ such that $n\triangleleft m$ in Sharkovskii's order \eqref{eq:order}, $F$ has also an $m$-periodic point.
\end{theorem}
\begin{proof}~
Define $f:\mathcal{O}\to\mathcal{O}$ by $f=F_h|_{\mathcal{O}}$, where $\mathcal{O} = h^{-1}(O)$.
From Lemma \ref{lem:proper_loop} we deduce the existence of a non-repeating $m$-loop of proper covering relations between $\mathcal{O}$-intervals $K_i$:
\begin{equation}\label{eq:final_loop}
K_0 \overset{f}{\rightarrowtail} K_1 \overset{f}{\rightarrowtail} K_2 \overset{f}{\rightarrowtail} \dots \overset{f}{\rightarrowtail} K_{m-1} \overset{f}{\rightarrowtail} K_0\text{.}
\end{equation}
Next, from Theorem \ref{th:1to2} we deduce that the following loop of horizontal covering relations occurs:
\begin{equation}\label{eq:Final_Loop}
S(K_0)
\overset{F_h}{\Longrightarrow}
S(K_1)
\overset{F_h}{\Longrightarrow}
S(K_2)
\overset{F_h}{\Longrightarrow}
\dots
\overset{F_h}{\Longrightarrow}
S(K_{m-1})
\overset{F_h}{\Longrightarrow}
S(K_0)\text{,}
\end{equation}
from which follows the existence of a point $x\in S(K_0)$, such that $F_h^m(x)=x$.
The loop \eqref{eq:final_loop} is non-repeating, so one easily observes that the corresponding loop \eqref{eq:Final_Loop} fulfills the condition
\begin{equation}
S(K_0) \cap \bigcup_{i=1}^{m-1} S(K_i) = \varnothing\text{,}
\end{equation}
which implies that $m$ is the least period for $x$.
Therefore, the point $h(x)$ is $m$-periodic for $F$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:sh-manyD}] Observe that under assumptions we have an $n$-periodic orbits which is enclosed
by a contracting grid. The assertion then follows from Theorem~\ref{th:Final}.
\end{proof}
\section{Examples of application: the R\"ossler system}\label{sec:App}
In \cite{AGPZrossler} we proved the existence of $m$-periodic orbits of all $m$ succeeding $n$ in Sharkovskii order \eqref{eq:order} in the R\"ossler system \cite{Rossler76} with an attracting $n$-periodic orbit
\begin{equation}\label{eq:rossler}
\begin{cases}
x'=-y-z,
\\
y'=0.2 y+x,
\\
z'=z (x-a)+0.2
\end{cases}\text{,}
\end{equation}
for two sets of parameters:
\begin{itemize}
\item $a=5.25$, for which $n=3$ and
\item $a=4.7$, for which $n=5$.
\end{itemize}
The proof requires finding some explicit special sets (using some ad-hoc trial and error approach) and proving some covering relations between them by computer-assisted methods. However, with Theorem \ref{th:Final}, a part of the proof is easier: the only thing to find and prove its existence is a contracting grid.
We are convinced that it should be possible to find a contracting grid for a wide variety of the parameter $a$'s values for which the system \eqref{eq:rossler} has an attracting periodic orbit.
In this section, we study four values of the parameter $a$: apart from the two cases treated in \cite{AGPZrossler}, we also compare the $6$-periodic orbits appearing in cases $a=4.381$ and $a=5.42$ (see the bifurcation diagram on Fig.\ \ref{fig:bif}). What is interesting, in the latter case one can prove the existence of even more periodic orbits than follow from Theorem \ref{th:Final}.
This is a reflection of the following fact about forcing relation between periods of interval maps. The set of forced periods depends on the pattern (the permutation
induced on the orbit) and the periodic orbits of the same period may force different sets of periods. The Sharkovskii Theorem gives only a lower bound
on the set of forced periods.
\begin{figure}[h]
\includegraphics[width=0.9\textwidth]{bif_new}
\caption{\label{fig:bif}The bifurcation diagram for the Poincar\'e map of the system \eqref{eq:rossler} with $b=0.2$.
The cases of attracting periodic orbits for $a=4.7$ and $a=5.25$ (red) are treated in \cite{AGPZrossler} and in Subsections \ref{ss:rossler3}, \ref{ss:rossler5}.
The two other cases of $6$-periodic orbits (blue) are studied in Subsections \ref{ss:rossler6}, \ref{ss:rossler6_2}. }
\end{figure}
In all the four cases, we denote by $\Pi$ the half-plane $\{x=0, y<0\}$ with induced coordinates $(y,z)$, and $P$ is a Poincar\'e map of the system \eqref{eq:rossler} on section $\Pi$, that is the map
\[
P(y,z) = \pi_{(y,z)}\left( \Phi_{T(y,z)}\left(x=0,y,z\right)\right)\text{,}
\]
where $\pi_{(y,z)}$ is the projection on the $(y,z)$ plane, $\Phi_t$ is the dynamical system induced by considered system and $T=T(y,z)$ is a return time, if well-defined.
\subsection{Case $a=5.25$}\label{ss:rossler3}
Let $a=5.25$. Then the system \eqref{eq:rossler} has an attracting $3$-periodic point for $P$ \cite[Lemma 4]{AGPZrossler}.
\begin{lemma}\label{lem:Roessler525}
Let $M=\left[\smallmatrix -1.& 0.000656767 \\ -0.000656767 & -1. \endsmallmatrix\right]$. The parallelogram $G_3$ in the $(y, z)$ coordinates on the section $\Pi$ (see
Fig. \ref{fig:grid3}):
\[
G_3= \bmatrix -6.38401 \\ 0.0327544 \endbmatrix
+ M \cdot
\bmatrix \pm 3.63687 \\ \pm 0.0004 \endbmatrix
\]
is a contracting grid for the attracting $3$-periodic orbit from \cite[Lemma 4]{AGPZrossler}, with the inner cubes $C_i$, $i=1,2,3$, defined by $C_i = C'_i \cap G_3$, where
\begin{align*}
&C'_1= \bmatrix -3.46642 \\ 0.0346316 \endbmatrix
+ M \cdot
\bmatrix \pm 0.072 \\ \pm 0.00048 \endbmatrix
\text{,}
\\
&C'_2= \bmatrix -6.26401 \\ 0.0326544 \endbmatrix
+ M \cdot
\bmatrix \pm 0.162 \\ \pm 0.00066 \endbmatrix
\text{,}
\\
&C'_3= \bmatrix -9.74889\\ 0.0307529 \endbmatrix
+ M \cdot
\bmatrix \pm 0.036 \\ \pm 0.00072 \endbmatrix.
\end{align*}
\end{lemma}
\begin{figure}[h]
\includegraphics[width=0.8\textwidth]{grid3}
\caption{\label{fig:grid3}The contracting grid $G_3$ from Lemma \ref{lem:Roessler525} and its image through $P$.
\newline The supersets of $C_i$, $i=1,2,3$ and their images are marked in red, green and blue.}
\end{figure}
\begin{proof}
Computer-assisted, \cite{proof}.
\end{proof}
From the above Lemma and Theorem~\ref{th:Final} we obtain
\begin{theorem}(Compare to \cite[Theorem 5]{AGPZrossler})\label{th:r3}
The R\"ossler system \eqref{eq:rossler} with $a=5.25$ has $m$-periodic orbits for any $m\in \mathbb{N}$ in the set $G_3$ defined in Lemma \ref{lem:Roessler525}.
\end{theorem}
\subsection{Case $a=4.7$}\label{ss:rossler5}
Let $a=4.7$. Then the system \eqref{eq:rossler} has an attracting $5$-periodic point for $P$ \cite[Lemma 7]{AGPZrossler}.
\begin{lemma}\label{lem:Roessler47}
The parallelogram $G_5$ in the $(y, z)$ coordinates on the section $\Pi$ (see
Fig.\ \ref{fig:grid5}):
\[
G_5= \bmatrix -6.1885 \\ 0.0356707 \endbmatrix
+ \bmatrix -1. & 0.000778356 \\ -0.000778356 & -1. \endbmatrix \cdot
\bmatrix \pm 2.68797 \\ \pm 0.0004 \endbmatrix
\]
is a contracting grid for the attracting $5$-periodic orbit from \cite[Lemma 7]{AGPZrossler}, with the inner cubes $C_i$, $i=1,\dots,5$, defined by $C_i = C'_i \cap G_5$, where
\begin{align*}
&C'_1= \bmatrix -3.86108 \\ 0.0375827 \endbmatrix
+ \bmatrix 0.0693366 & 1. \\ -0.997593 & 0.000984231 \endbmatrix \cdot
\bmatrix \pm 0.0006 \\ \pm 0.00138 \endbmatrix
\text{,}
\\
&C'_2= \bmatrix -6.82009 \\ 0.0350822 \endbmatrix
+ \bmatrix 0.7879108 & 1. \\ 0.615789 & 0.0007307 \endbmatrix \cdot
\bmatrix \pm 0.0012 \\ \pm 0.0024 \endbmatrix
\text{,}
\\
&C'_3= \bmatrix -7.83056 \\ 0.0343732 \endbmatrix
+ \bmatrix 0.8138516 & 1. \\ 0.581073 & 0.000671 \endbmatrix \cdot
\bmatrix \pm 0.0012 \\ \pm 0.0042 \endbmatrix
\text{,}
\\
&C'_4= \bmatrix -5.75153 \\ 0.0359038 \endbmatrix
+ \bmatrix 0.9997319 & 1. \\ -0.023153 & 0.0008062 \endbmatrix \cdot
\bmatrix \pm 0.0228 \\ \pm 0.01116 \endbmatrix
\text{,}
\\
&C'_5= \bmatrix -8.73615 \\ 0.0337875 \endbmatrix
+ \bmatrix 0.8997843 & 1. \\ 0.436335 & 0.00062508 \endbmatrix \cdot
\bmatrix \pm 0.00144 \\ \pm 0.000744 \endbmatrix.
\end{align*}
\end{lemma}
\begin{figure}[h]
\includegraphics[width=0.8\textwidth]{grid5}
\caption{\label{fig:grid5}The contracting grid $G_5$ from Lemma \ref{lem:Roessler47}
and its image through $P$. The supersets of $C_i$, $i=1,\dots,5$ and their images are marked in red, orange, green, blue and purple.}
\end{figure}
\begin{proof}
Computer-assisted, \cite{proof}.
\end{proof}
From the above Lemma and Theorem~\ref{th:Final} we obtain
\begin{theorem}(Compare to \cite[Theorem 6]{AGPZrossler})\label{th:r5}
The R\"ossler system \eqref{eq:rossler} with $a=4.7$ has $m$-periodic orbits for any $m\in \mathbb{N}\setminus\{3\}$ in the set $G_5$ defined in Lemma \ref{lem:Roessler47}.
\end{theorem}
\begin{remark}
In fact, Theorem~\ref{th:r5} is only a part of \cite[Theorem 6]{AGPZrossler}, because it does not exclude the possibility that a $3$-periodic orbit exists in $G_5$.
\end{remark}
\subsection{Case $a=4.381$}\label{ss:rossler6}
Consider $a=4.381$. From Fig. \ref{fig:bif} it is clear that the R\"ossler system has an attracting $6$-periodic orbit.
\begin{lemma}\label{lem:6per}
The Poincar\'e map $P$ of the system \eqref{eq:rossler} with $a=4.381$ on the section $\Pi$ has a 6-periodic orbit $\mathcal{O}^6=\{p_1^6,p_2^6,p_3^6,p_4^6,p_5^6,p_6^6\}$, contained in the following rectangles in the $(y,z)$ coordinates on the section $\Pi$:
\begin{equation}
\begin{aligned}\label{eq:6-per}
p_1^6 \in & -7.448265140_{33532}^{244187} \times 0.03638524011_{881493}^{973746}, \\
p_2^6 \in & -5.432682771_{276081}^{080253} \times 0.0381210024_{7833106}^{8150609}, \\
p_3^6 \in & -8.146150765_{219835}^{118602} \times 0.03585533157_{361669}^{606319}, \\
p_4^6 \in & -4.05248247_{1003891}^{0816507} \times 0.03953831884_{313481}^{723778},\\
p_5^6 \in & -6.98865159_{7169091}^{6889441} \times 0.03675237717_{289087}^{31595},\\
p_6^6 \in & -6.38538092_{5198882}^{4637889} \times 0.0372584624_{5305077}^{617641}.
\end{aligned}
\end{equation}
\end{lemma}
\begin{proof}
Computer-assisted \cite{proof}, via Interval Newton Method \cite{N}.
\end{proof}
\begin{lemma}\label{lem:Roessler4381}
The parallelogram $G_6$ in the $(y, z)$ coordinates on the section $\Pi$ (see
Fig.\ \ref{fig:grid6}):
\[
G_6= \bmatrix -5.99932 \\ 0.0376868 \endbmatrix
+ \bmatrix -1. & -0.000899679 \\ 0.000899679 & -1. \endbmatrix \cdot
\bmatrix \pm 2.24683 \\ \pm 0.00022 \endbmatrix
\]
is a contracting grid for the attracting $6$-periodic orbit \eqref{eq:6-per} from Lemma \ref{lem:6per}, with the inner cubes $C_i$, $i=1,\dots,6$, defined by $C_i = C'_i \cap G_6$, where
\begin{align*}
&C'_1= \bmatrix -7.44827 \\ 0.0363852 \endbmatrix
+ \bmatrix1. & 0.8498 \\ 0.0007825 & 0.527106 \endbmatrix \cdot
\bmatrix \pm 0.00225 \\ \pm 0.0005 \endbmatrix
\text{,}
\\
&C'_2= \bmatrix -5.43268 \\ 0.038121 \endbmatrix
+ \bmatrix 1.23042 & 0.696746 \\ -0.00567154 & -0.0240555 \endbmatrix \cdot
\bmatrix \pm 0.00509 \\ \pm 0.015 \endbmatrix
\text{,}
\\
&C'_3= \bmatrix -8.14614 \\ 0.0358553 \endbmatrix
+ \bmatrix 1. & 0.907289 \\ 0.000736978 & 0.420507 \endbmatrix \cdot
\bmatrix \pm 0.000265 \\ \pm 0.00085 \endbmatrix
\text{,}
\\
&C'_4= \bmatrix -4.05249 \\ 0.0395383 \endbmatrix
+ \bmatrix 0.999999 & 0.155044 \\ 0.00111181 & -0.987908 \endbmatrix \cdot
\bmatrix \pm 0.000485 \\ \pm 0.00035 \endbmatrix
\text{,}
\\
&C'_5= \bmatrix -6.98865 \\ 0.0367524 \endbmatrix
+ \bmatrix 1. & 0.827066 \\ 0.000815525 & 0.562105 \endbmatrix \cdot
\bmatrix \pm 0.000712 \\ \pm 0.0005 \endbmatrix
\text{,}
\\
&C'_6= \bmatrix -6.38538 \\ 0.0372585 \endbmatrix
+ \bmatrix 1. & 0.834783 \\ 0.000863145 & 0.550579 \endbmatrix \cdot
\bmatrix \pm 0.00149 \\ \pm 0.0006 \endbmatrix.
\end{align*}
\end{lemma}
\begin{figure}[h]
\includegraphics[width=0.8\textwidth]{grid6}
\caption{\label{fig:grid6}The contracting grid from Lemma \ref{lem:Roessler4381}
and its image through $P$. The supersets of $C_i$, $i=1,\dots,6$ and their images are marked in red, orange, green, cyan, blue and purple.}
\end{figure}
\begin{proof}
Computer-assisted, \cite{proof}.
\end{proof}
From the above Lemma and Theorem~\ref{th:Final} we obtain
\begin{theorem}\label{th:r6}
The R\"ossler system \eqref{eq:rossler} with $a=4.381$ has $m$-periodic orbits for any even $m\in 2\mathbb{N}$ and $m=1$, in the set $G_6$ defined in Lemma \ref{lem:Roessler4381}.
\end{theorem}
\begin{remark}
The above result cannot be strengthened to include odd periods because the permutation of the orbit with respect to the location on the model 1-dimensional manifold is as follows:
\[
\begin{tikzpicture}
\foreach \x/\l in {1/4, 2/2, 3/6, 4/5, 5/1, 6/3}
{
\coordinate (p\l) at (\x,0);
\coordinate (pg\l) at (\x,0.2);
\coordinate (pd\l) at (\x,-0.13);
\draw (\x,0.1) -- (\x,-0.1) node[below]{$p_\l$};
}
\draw[->] (0,0) -- (7,0);
\begin{scope}[>={stealth}]
\foreach \b/\e in {4/5, 2/3, 6/1}
\draw (pg\b) edge[->,bend left=40] (pg\e);
\foreach \b/\e in {5/6, 1/2, 3/4}
\draw (pd\b) edge[->,bend left=40] (pd\e);
\end{scope}
\end{tikzpicture}
\]
\end{remark}
Note that this is the case of an even-periodic orbit for which a \u{S}tefan sequence cannot be constructed (`all points switch sides', see details in \cite{Burns}) and using the method from the proof of Burns and Hasselblatt we obtain non-repeating loops of covering relations only of even length or a self-covering.
However, the next case shows the situation, in which one can prove the existence of odd-periodic points for an even-periodic attracting orbit. As we have mentioned, the difference lies in the permutation of the orbit.
\subsection{Case $a=5.42$}\label{ss:rossler6_2}
Let now see another case with an attracting $6$-periodic orbit for the R\"ossler system. Consider $a=5.42$ (see Fig. \ref{fig:bif}).
\begin{lemma}\label{lem:6per2}
The Poincar\'e map $P$ of the system \eqref{eq:rossler} with $a=4.381$ on the section $\Pi$ has a 6-periodic orbit $\mathcal{O}^6=\{p_1^6,p_2^6,p_3^6,p_4^6,p_5^6,p_6^6\}$, contained in the following rectangles in the $(y,z)$ coordinates on the section $\Pi$:
\begin{equation}
\begin{aligned}\label{eq:6-per2}
p_1^6 \in & -3.3303887279_{60296}^{4934} \times 0.033810081022_{70888}^{86536}, \\
p_2^6 \in & -6.0438781482_{33535}^{13811} \times 0.0319883054102_{6752}^{8062}, \\
p_3^6 \in & -9.930004688_{71182}^{693574} \times 0.029985122265_{72283}^{83182}, \\
p_4^6 \in & -3.561109751_{505439}^{469876} \times 0.033636111425_{11445}^{66204},\\
p_5^6 \in & -6.450138010_{324274}^{261234} \times 0.0317509776490_{3493}^{7362},\\
p_6^6 \in & -10.061811798_{91221}^{88145} \times 0.02992604451_{798922}^{853264}.
\end{aligned}
\end{equation}
\end{lemma}
\begin{proof}
Computer-assisted \cite{proof}, via Interval Newton Method \cite{N}.
\end{proof}
\begin{lemma}\label{lem:Roessler542}
The parallelogram $G_6$ in the $(y, z)$ coordinates on the section $\Pi$ (see
Fig.\ \ref{fig:grid6_2}):
\[
G_6= \bmatrix -6.60556 \\ 0.0317909 \endbmatrix
+ \bmatrix -1. & -0.000573253 \\0.000573253 & -1. \endbmatrix \cdot
\bmatrix \pm 3.57445 \\ \pm 0.00035 \endbmatrix
\]
is a contracting grid for the attracting $6$-periodic orbit \eqref{eq:6-per2} from Lemma \ref{lem:6per2}, with the inner cubes $C_i$, $i=1,\dots,6$, defined by $C_i = C'_i \cap G_6$, where
\begin{align*}
&C'_1= \bmatrix -3.33039 \\ 0.0338101 \endbmatrix
+ \bmatrix 1. & 0.0114844 \\ 0.000763188 & -0.999934 \endbmatrix \cdot
\bmatrix \pm 0.0015225 \\ \pm 0.000525 \endbmatrix
\text{,}
\\
&C'_2= \bmatrix -6.04388 \\ 0.0319883 \endbmatrix
+ \bmatrix 1. & 0.566012 \\ 0.000593828 & -0.824397 \endbmatrix \cdot
\bmatrix \pm 0.0029925 \\ \pm 0.0005775 \endbmatrix
\text{,}
\\
&C'_3= \bmatrix -9.93 \\ 0.0299851 \endbmatrix
+ \bmatrix 1. & 0.866643 \\ 0.000450065 & 0.498928 \endbmatrix \cdot
\bmatrix \pm 0.0021 \\ \pm 0.00105 \endbmatrix
\text{,}
\\
&C'_4= \bmatrix -3.56111 \\ 0.0336361 \endbmatrix
+ \bmatrix 1. & 0.0148011 \\ 0.000745026 & -0.99989 \endbmatrix \cdot
\bmatrix \pm 0.0043575 \\ \pm 0.000525 \endbmatrix
\text{,}
\\
&C'_5= \bmatrix -6.45014 \\ 0.031751 \endbmatrix
+ \bmatrix 1. & 0.999296 \\ 0.000574732 & -0.0375247 \endbmatrix \cdot
\bmatrix \pm 0.00945 \\ \pm 0.013125 \endbmatrix
\text{,}
\\
&C'_6= \bmatrix -10.0618 \\ 0.029926 \endbmatrix
+ \bmatrix 1. & 0.887687 \\ 0.000446372 & 0.460448 \endbmatrix \cdot
\bmatrix \pm 0.00084 \\ \pm 0.0011025 \endbmatrix.
\end{align*}
\end{lemma}
\begin{figure}[h]
\includegraphics[width=0.8\textwidth]{grid6_2}
\caption{\label{fig:grid6_2}The contracting grid from Lemma \ref{lem:Roessler542}
and its image through $P$. The supersets of $C_i$, $i=1,\dots,6$ and their images are marked in red, orange, green, cyan, blue and purple.}
\end{figure}
\begin{proof}
Computer-assisted, \cite{proof}.
\end{proof}
From the above Lemma and Theorem~\ref{th:Final} we obtain
\begin{theorem}
The R\"ossler system \eqref{eq:rossler} with $a=5.42$ has $m$-periodic orbits for any even $m\in 2\mathbb{N}$ and $m=1$, in the set $G_6$ defined in Lemma \ref{lem:Roessler542}.
\end{theorem}
\begin{remark}
In this case, we are able to strengthen the above result. Note that the permutation of the orbit with respect to the location on the model 1-dimensional manifold is the following:
\begin{equation}\label{eq:6-permutation}
\begin{tikzpicture}
\foreach \x/\l in {1/6, 2/3, 4/5, 5/2, 7/4, 8/1}
{
\coordinate (p\l) at (\x,0);
\coordinate (pg\l) at (\x,0.2);
\coordinate (pd\l) at (\x,-0.13);
\draw (\x,0.1) -- (\x,-0.1) node[below]{$p_\l$};
}
\foreach \b/\e/\l in {4/2/0, 5/3/1}
\draw[ultra thick] (p\b) -- (p\e) node[midway,above]{$I_\l$};
\draw[->] (0,0) -- (9,0);
\begin{scope}[>={stealth}]
\foreach \b/\e in {1/2, 2/3, 4/5, 5/6}
\draw (pg\b) edge[->,bend right=40] (pg\e);
\foreach \b/\e in {3/4, 6/1}
\draw (pd\b) edge[->,bend right] (pd\e);
\end{scope}
\end{tikzpicture}
\end{equation}
and, although the orbit is even-periodic, we are able to construct a \u{S}tefan sequence. One of the shortest possible diagrams of 1-dimensional covering relations that we obtain involves only two $\mathcal{O}$-intervals: $I_0 = [p_2,p_4]$ and $I_1 = [p_3,p_5]$, as denoted on \eqref{eq:6-permutation}. Then the diagram is
\begin{equation}\label{eq:diag2}
\begin{tikzcd}
I_0
\arrow[r, rightarrow, bend left=15]
&
I_1
\arrow[l, rightarrow, bend left=15]
\arrow[rightarrow, loop, distance=2em, in=-20, out=20]
\end{tikzcd}
\text{,}
\end{equation}
and all covering relations are proper. Therefore, using the grid from Lemma \ref{lem:Roessler542}, we can construct a similar diagram of horizontal 2-dimensional covering relations:
\begin{equation}
\begin{tikzcd}
S(I_0)
\arrow[r, Rightarrow, bend left=10]
&
S(I_1)
\arrow[l, Rightarrow, bend left=10]
\arrow[Rightarrow, loop, distance=2em, in=-10, out=15]
\end{tikzcd}
\text{,}
\end{equation}
from which follows the existence of $m$-periodic points for $P$, for all natural $m$.
\end{remark}
Therefore we obtain
\begin{theorem}
The R\"ossler system \eqref{eq:rossler} with $a=5.42$ has $m$-periodic orbits for any $m\in \mathbb{N}$ in the set $G_6$ defined in Lemma \ref{lem:Roessler542}.
\end{theorem}
\section*{Acknowledgment}
The second author is supported by the Funding Grant NCN UMO-2016/22/A/ST1/00077.
\bibliographystyle{plain}
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} | 567 |
Q: How to sort Json file based on value of object? I have below json file named all, and contain below data
[{"name":{"common":"Dominican Republic","official":"Dominican Republic","nativeName":{"spa":{"official":"República Dominicana","common":"República Dominicana"}}},"region":"Americas","area":48671.0,"flags":{"png":"https://flagcdn.com/w320/do.png","svg":"https://flagcdn.com/do.svg"}},{"name":{"common":"Heard Island and McDonald Islands","official":"Heard Island and McDonald Islands","nativeName":{"eng":{"official":"Heard Island and McDonald Islands","common":"Heard Island and McDonald Islands"}}},"region":"Antarctic","area":412.0,"flags":{"png":"https://flagcdn.com/w320/hm.png","svg":"https://flagcdn.com/hm.svg"}},{"name":{"common":"Ghana","official":"Republic of Ghana","nativeName":{"eng":{"official":"Republic of Ghana","common":"Ghana"}}},"region":"Americas","area":238533.0,"flags":{"png":"https://flagcdn.com/w320/gh.png","svg":"https://flagcdn.com/gh.svg"}}]
I want to sort this file based on name.official value, I wrote below codes but it does not work.
how can I achieve that?
const path = "../all.json;
getCuntries(path);
async function getCuntries(path) {
const response = await fetch(path);
const scrampled = await response.json();
const parsed = await JSON.parse(JSON.stringify(scrampled));
const data = parsed.sort(GetSortOrder("name"));
console.log(data)
}
function GetSortOrder(prop) {
return function(a, b) {
if (a[prop][1] > b[prop][1]) {
return 1;
} else if (a[prop][1] < b[prop][1]) {
return -1;
}
return 0;
}
}
A: Your comparator method is a bit messed up. First, comparing strings with > and < operators may lead to unexpected behaviour sometimes and this is not the recommended way to compare strings. And second (which is probably the root cause), you should access object values with their keys and not with indexes (like you're doing with [1]). This should work - make it the body of function (a, b):
const aPassedProp = a[prop];
const bPassedProp = b[prop];
return aPassedProp.official.localeCompare(bPassedProp.official);
The localeCompare method can compare your strings in an appropriate way: you can learn more here.
A: You can't access an object key by index. As it's is not specified if the object key should be dynamic, here a simple solution where it's not:
const data = [
{ "name": { "official": "Republic of Ghana"} },
{ "name": { "official": "Heard Island and McDonald Islands" } },
{ "name": { "official": "Dominican Republic" } },
]; // shortened for readability
const sortArray = (a, b) => {
const aCompare = a.name.official;
const bCompare = b.name.official;
if (aCompare === bCompare) {
return 0;
}
return aCompare < bCompare ? -1 : 1;
};
data.sort(sortArray);
console.log(data);
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,373 |
Q: При запуске jar файла выдает ошибку : Error: JavaFx runtime components are missing, and are required to run this application Продолжил бороться с данной проблемой. Скачал последний javaFX-sdk версии 17.0.0.1 (September 2021).
Через батник пытаюсь открыть таким образом:
java --module-path C:\Program_Files\JavaFX\javafx-sdk\lib --add-modules javafx.controls,javafx.fxml,javafx.graphics -jar C:\Users\USER\IdeaProjects\keys\out\artifacts\keys_jar\keys.jar
Через intellij приложение запускается без проблем.
Если нужна дополнительная информация, только скажите.
A: Я запускаю так:
java --module-path c:\Java\javafx-sdk-11.0.2\lib --add-modules ALL-MODULE-PATH -jar [fullpath]\myjar.jar
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,691 |
\section{Introduction}
The HI mass function (HIMF) is the density distribution of HI masses of galaxies in the Universe and represents a key component in understanding how collapsed structures form. HI surveys are complementary to optical surveys, and the galaxy luminosity functions they deliver, because they have fundamentally different selection effects and thus detect a different component of the underlying galaxy population. Together the luminosity functions and mass functions that these surveys calculate offer important constraints on the population of galaxies that simulations of structure formation generate.
Detailed studies of the HIMF have only become possible in the last decade or so, as previously sample sizes were too small and selection effects too poorly understood. With the advent of wide area, blind surveys like HIPASS \citep[HI Parkes All Sky Survey;][]{Barnes+2001} and ALFALFA \citep[Arecibo Legacy Fast ALFA survey;][]{Giovanelli+2005} precise determination of the HIMF in the local Universe has become possible, with both HIPASS and ALFALFA \citep{Zwaan+2005,Martin+2010} indicating that the HIMF is well fit by a Schechter function \citep[an analytic expression for the mass distribution of collapsed objects in an expanding universe,][]{Press+Schechter1974,Schechter1976}, with a low-mass slope of approximately -1.3 and a `knee' mass of almost $10^{10}$ \Msol. The large area and source counts of these surveys have also allowed studies of environmental dependence that are not restricted to 10s or 100s of objects and a handful of nearby groups.
Although many studies looking for environmental dependence have been carried out \citep[for example][]{Rosenberg+2002,Springob+2005,Zwaan+2005,Stierwalt+2009,Moorman+2014}, it is still important to ask why any environmental dependence is expected at all? There are many processes and properties that are known to depend on a galaxy's environment, here we will briefly discuss a few that we expect to be the most influential on a galaxy's HI content. First, due to their mass and tendency to cluster, more massive dark matter (DM) halos are generally found in more overdense regions. Thus, the `knee' mass ($M_{*}$) of a Schechter function fit to the HIMF, would be expected to increase towards more dense regions of the Universe. Secondly, voids can be considered as more slowly evolving sections of our Universe \citep{Peebles2001,Tinker+Conroy2009}. This means that by isolating the void galaxies in a sample, you are effectively probing the HIMF at a previous time, where systems are likely to be lower mass and more numerous, assuming a hierarchical model of galaxy formation. Therefore, it would be expected that the low-mass slope would steepen within lower density regions. In addition to these two effects, in the most dense regions (galaxy clusters) galaxies will be unable to retain their neutral gas due to the harassment and ram pressure stripping they experience, and so might be expected to be HI-deficient with respect to galaxies in the field; while galaxies in voids are likely more prone to background UV heating than those in the field \citep{Hoeft+2006}. Given all of the above, some environmental dependence in the shape of the HIMF is expected, however there are numerous competing affects, making the exact nature of the dependence difficult to predict. To complicate matters further, most studies have thus far produced marginal and/or conflicting results.
Using the Arecibo Dual Beam Survey \citep[ADBS;][]{Rosenberg+2000} \citet{Rosenberg+2002} found that the HIMF low-mass slope ($\alpha$) was flatter in Virgo than the $\sim$-1.5 value found in the rest of the survey. However, the paper points out that small number statistics and distance errors make their results somewhat uncertain. \citet{Springob+2005} also found (at low significance) that both $\alpha$ and $M_{*}$ decrease in high density environments, from their analysis of an optically selected sample from the Arecibo General Catalog \citep{Springob+2005b}. However, more recently, \citet{Stierwalt+2009} used an early ALFALFA release to show essentially the opposite result, that the low-mass slope in the dense Leo region was steeper than other measurements of the HIMF at the time \citep[though, given the quoted error, is now consistent with that of the global ALFALFA HIMF;][]{Martin+2010}. There are also a number or other results from surveys of individual groups \citep{Verheijen+2001,Kovac+2005,Freeland+2009,Pisano+2011} which generally imply that the low-mass slope is flatter in galaxy groups.
\citet{Zwaan+2005} concluded that $\alpha$ steepened in high density environments, based on data from HIPASS. However, unlike all other studies, the proximity to other HI galaxies was used to define environment (rather than an optically selected reference catalogue). HI surveys are known to be incomplete for galaxies in the densest environments, which combined with the fact that HIPASS is not a volume limited catalogue, makes a comparison with this result difficult; but we note that attempting to perform a similar experiment with ALFALFA did not result in any apparent environmental dependence in the HIMF. Most recently \citet{Moorman+2014} used the 40\% ALFALFA catalogue ($\alpha$.40) to search for environmental dependence based on void and wall regions defined using the method devised by \citet{Hoyle+Vogeley2002}. They found no evidence of any change in $\alpha$, but contrary to \citet{Springob+2005} $M_{*}$ was found to increase in denser regions. This represents the most statistically significant result of large scale environmental dependence in the HIMF to date, which is in part due to the greatly larger sample size that ALFALFA provides. Since that study, data from 30\% more of ALFALFA's nominal area ($\sim$7,000 deg$^{2}$) have been reduced, and $\sim$7,000 additional high signal-to-noise HI sources have been extracted.
In this paper we choose to focus on a local definition of galaxy environment, rather than defining voids, walls and clusters, for two reasons. First, because the majority of the additional 30\% added to the ALFALFA catalogue since the \citet{Moorman+2014} study is not within the SDSS (Sloan Digital Sky Survey) spectroscopic footprint, making defining voids problematic; and secondly because related optical and theoretical works \citep{Berlind+2005,Blanton+2006,Tinker+Conroy2009} find that galaxy properties are most closely related to a galaxy's host halo, and may even be almost independent of its large scale environment. Obviously the two are not independent, but if a galaxy's properties depend mostly on its host halo mass rather than its ``assembly bias" \citep[the idea that haloes of a given mass, but which assemble at different times, will cluster differently, e.g. see][]{Wechsler+2006}, then the strongest signal of any change in the mass function would presumably arise from a measure of local environment, rather than large scale structure (LSS).
We use a combination of SDSS data release 8 \citep{Aihara+2011} and the 2MASS Redshift Survey \citep[2MRS;][]{Huchra+2012} as reference catalogues to define the local density of ALFALFA galaxies based on the separation of their projected nearest neighbours in these catalogues. This allows us to split the HI sources into quartiles of differing environment and calculate the HIMF for each environment separately. 2MRS allows us to make use of the full ALFALFA 70\% sample, while the superior depth of SDSS permits smaller scale environments to be probed.
In the following section we give a brief overview of the ALFALFA survey, in \S\ref{sec:env} we describe our definitions of environment, \S\ref{sec:HIMFcalc} outlines how the HIMF is calculated, and our results are presented in \S\ref{sec:results}. The implications of these results are discussed in \S\ref{sec:discuss}, and finally we draw our conclusions in \S\ref{sec:conclude}.
\section{The ALFALFA sample}
\label{sec:alfalfa_sample}
Observations for the main ALFALFA survey were completed in October 2012 after over 7 years of observing with the 305 m Arecibo radio telescope in Puerto Rico. The final ALFALFA footprint covers approximately 6,900 deg$^{2}$ on the sky, and is broken up into two contiguous regions: one ranges from $\sim$7.5 hr RA to $\sim$16.5 hr RA in the Arecibo Spring sky, and the other from $\sim$22 hr RA to $\sim$3 hr RA in the Arecibo Fall sky. While the Spring ALFALFA region has almost complete overlap with SDSS spectroscopy, in the Fall sky there are only a few stripes where spectra are available. The drift scan observing strategy of ALFALFA proved extremely successful with over 95\% of observing time spent with the ``shutter open", including all start-up, shutdown and calibration procedures. A matched filtering algorithm \citep{Saintonge+2007} is used to help identify sources, but all ALFALFA spectra are ultimately extracted by a person, and the current progress is over 70\% complete, yielding over 20,000 high signal-to-noise (S/N) sources and counting. Over 99\% of these HI sources have identified optical counterparts (with matching redshifts where optical spectra exist).\footnotemark{}
\footnotetext{The ALFALFA 70\% catalogue is publicly available at \url{http://egg.astro.cornell.edu/alfalfa/data/index.php}}
In order to calculate the HIMF it is essential to have HI masses for the ALFALFA sources, which in turn necessitates distance measurements for every source. ALFALFA uses a flow model developed by \cite{Masters2005} to convert recessional velocities below 6,000 \kms \ to distances. Distances to galaxies beyond 6,000 \kms \ are calculated instead assuming Hubble flow, with $H_{0}$ = 70 \kms$\,\mathrm{Mpc^{-1}}$. In addition, 303 sources are assigned to regions of the Virgo cluster by matching to the VCC \citep[Virgo Cluster Catalog;][]{Binggeli+1985}, 1,130 sources are assigned to groups from 2MRS \citep{Crook+2007} and given the mean velocity of the group members, and 63 (1,646) sources are given their primary (secondary) distances from the literature. Note that in this article we only consider galaxies in the 70\% ALFALFA catalogue within the range of distances 1,000-15,000~\kms$/H_0$.
Once distances to ALFALFA galaxies have been calculated, their HI masses can be computed through the usual equation:
\begin{equation}
\frac{M_{\mathrm{HI}}}{\mathrm{M}_{\odot}} = 2.356 \times 10^{5} D_{\mathrm{Mpc}}^{2} S_{21} \;\; .
\end{equation}
In the equation above, $D_{\mathrm{Mpc}}$ is the distance to the galaxy in Mpc and $S_{21}$ is its integrated flux in Jy \kms.
\section{Quantifying Environment}
\label{sec:env}
The term `environment' has no objective definition, and different studies have used drastically different methods to describe it quantitatively. On one extreme we can find techniques that characterise the environment based on the morphology of the cosmic web, classifying galaxies as void, wall, and filament objects \citep[e.g.][]{Hoyle+Vogeley2004,Rojas+2004,Hoyle+2005}. On the other extreme, it is possible to characterise the most immediate surroundings of a galaxy based on its status as a central or satellite galaxy \citep[e.g.][]{Carollo+2013}. In this article we choose to study the dependence of the HIMF on the \textit{local} environment of ALFALFA sources, as traced by the proximity of neighbouring galaxies. More specifically, we employ the widely-used nearest neighbour (NN) and fixed aperture (FA) methods to quantify the environment \citep[e.g.][]{Muldrew+2012}. The former method calculates a local density based on the distance between the target galaxy and its $N^\mathrm{th}$ nearest neighbour. The latter is instead based on the number of objects found within a region of fixed size surrounding the target galaxy.
Each method of environment characterisation has its own set of advantages and drawbacks, and there are often trade-offs between a method's physical motivation and its simplicity. Our choice to use the NN and FA methods is based on the fact that these two methods are purely observational, and have a clear and intuitive definition. Sections \ref{sec:reference_catalogue}--\ref{sec:fixed_aperture} below contain a detailed description of the methods' implementation in the context of the ALFALFA sample.
\subsection{An external reference catalogue for environment characterisation}
\label{sec:reference_catalogue}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{a70+SDSS_spring_15000_png_convert.eps}
\caption{
\textit{Left panel}: Coneplot of ALFALFA galaxies in the Spring region of the sky. \textit{Right panel}: Coneplot of the SDSS galaxies in the reference volume-limited catalogue, within the same volume as the ALFALFA sample. The environment of each ALFALFA galaxy in the left panel is calculated based on the position of neighbours in the reference catalogue shown in the right panel (refer to \S\ref{sec:reference_catalogue} for details).
}
\label{fig:a70+SDSS_cone}
\end{figure*}
The simplest way to find neighbouring galaxies for the ALFALFA sources would be to search within the ALFALFA catalogue itself. This approach has been previously used by \citet{Zwaan+2005} to measure the environment of galaxies detected by the HIPASS blind HI survey. Even though straightforward, this methodology comes with two important observational disadvantages. First, any blind HI survey produces a nearly flux-limited\footnotemark{} sample. As the left panel of Figure \ref{fig:a70+SDSS_cone} shows, the number of detections in such a sample drops in the outer parts of the survey, since only the most HI massive galaxies remain visible at these large distances. Consequently, a bias is introduced in the measurement of environment, whereby galaxies appear systematically more isolated with increasing distance. Second, galaxies located in the central regions of clusters and rich groups are known to be HI-deficient with respect to their peers in the field \citep[][for a review]{Haynes+1984}. This means that HI-selected samples are biased against the highest density regions of the cosmic web. This effect can be clearly seen either directly in the spatial distribution of ALFALFA galaxies near clusters (see figure 6 in \citealp{Haynes+2011}), or indirectly in the clustering properties and the colour-magnitude diagram of ALFALFA galaxies (see figure 20 in \citealp{Papastergis+2013} and figure 10 in \citealp{Huang+2012b}, respectively).
\footnotetext{In reality, the detection limit of a blind HI survey depends both on the integrated flux and the width of a galaxy's HI profile (see section 6 in \citealp{Haynes+2011}). However, the width dependence of the detection limit is mild enough such that the detectability of a galaxy by ALFALFA depends primarily on its HI mass.}
In this article we remedy these shortcomings by defining the environment of ALFALFA galaxies based on an external reference catalogue. The catalogue we use has two important properties:
\begin{enumerate}
\item \textit{It is optically selected.} In particular, we use galaxies from the spectroscopic database of the eighth data release of the Sloan Digital Sky Survey (SDSS DR8; \citealp{Aihara+2011}). This property ensures that we trace the environment well even in high density regions where gas-deficiency becomes an issue.
\item \textit{It is volume-limited.} We include in the reference catalogue only galaxies that are brighter than $M_r = -18.9$. Given the apparent magnitude limit for the SDSS spectroscopic sample ($m_r = 17.75$) and the maximum distance cut for the ALFALFA sample ($\approx$214 Mpc), these galaxies are bright enough to constitute a volume-complete sample within the ALFALFA volume. In turn, this ensures that environment is measured consistently regardless of the distance at which the ALFALFA galaxy is located.
\end{enumerate}
The right panel of figure \ref{fig:a70+SDSS_cone} shows the spatial distribution of the SDSS reference catalogue. As expected from its volume-limited nature, the number of objects in the reference catalogue grows steadily with increasing distance. Note that in order to avoid edge effects, the reference catalogue is slightly more extended in the radial direction than the ALFALFA sample, covering the distance range 500 -- 15,500~\kms$/H_0$. We remind the reader that distance cuts are quoted in terms of recessional velocity, but they actually refer to distances that are estimated as described in \S\ref{sec:alfalfa_sample}. In order to avoid edge effects in the plane of the sky as well, the reference catalogue must have more than complete sky overlap with the ALFALFA sample. Figure \ref{fig:sky_cov} shows the footprints of the ALFALFA sample and the SDSS reference catalogue in the Spring region of the sky, and details the complicated sky mask that is necessary to maximise the number of ALFALFA galaxies while maintaining high levels of overlap with the reference catalogue. Keep in mind that, given the poor spectral coverage of SDSS in the Fall region of the sky, a different reference catalogue is necessary to study the 70\% ALFALFA sample over its full sky extent (see \S\ref{sec:2MRS}).
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{sky_coverage.eps}
\caption{The sky positions of the sources in the ALFALFA 1,000-15,000 \kms \ sample (small blue points), and the 500-15,500 \kms \ SDSS reference catalogue (large, overlapping grey points). The thick red line is the cut that is applied to the ALFALFA sample when comparing with SDSS, in order to ensure there is more than complete overlap.}
\label{fig:sky_cov}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{SDSSNN3_env_hist.eps}
\caption{
Histogram of the 3rd nearest neighbour density, $\Sigma_3$, for ALFALFA galaxies. The density of each ALFALFA galaxy is calculated based on the proximity of neighbouring objects in an SDSS volume-limited reference catalogue (refer to \S\ref{sec:reference_catalogue} \& \S\ref{sec:nearest_neighbour}). Different colours and hatching styles mark the four quartiles of the distribution, which from light blue to dark red (light to dark colours, and left to right) contain galaxies situated in progressively denser environments.
}
\label{fig:NN3_hist}
\end{figure}
Defining environment in this way, based on a volume-limited reference catalogue avoids the need to place harsh flux cuts on the ALFALFA sample (to make it volume-limited), as its sensitivity and completeness are well understood \citep{Haynes+2011} and can be corrected for independently of our external definition of environment, as will be described in \S\ref{sec:HIMFcalc}.
\subsection{Nearest Neighbour Environment}
\label{sec:nearest_neighbour}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{a70_spring_15000_NNquart1and4.eps}
\caption{
Coneplots of ALFALFA galaxies belonging to the lowest density quartile (\textit{left panel}) and highest density quartile (\textit{right panel}) of the nearest neighbour density distribution (see figure \ref{fig:NN3_hist}). Note the marked difference in clustering between these two environmental subsamples.
}
\label{fig:SDSS_NN_cone}
\end{figure*}
We calculate a nearest neighbour density for each ALFALFA galaxy based on the projected distance to the third closest galaxy in the reference SDSS catalogue. First, we record the sky position of all objects in the reference catalogue that have a recessional velocity within $\pm$500 \kms \ from the recessional velocity of the target ALFALFA galaxy. We then identify the third nearest object in the plane of the sky, and calculate its projected separation at the distance of the ALFALFA galaxy, $R_3$. The projected nearest neighbour density can then be calculated as
\begin{equation}
\Sigma_{3} = \frac{3}{\mathrm{\pi} R_{3}^{2}} \;\;\; .
\end{equation}
When identifying neighbours, we exclude any object in the reference catalogue that is located within 5 arcsec and $\pm$70 \kms \ from the ALFALFA galaxy; such an object corresponds (almost always) to the counterpart of the ALFALFA galaxy in SDSS. Throughout this article, $\Sigma_{3}$ will be used to characterise the local environment via the NN method, and will often be referred to as simply `the environment' or `local density'.
Figure \ref{fig:NN3_hist} shows the distribution of $\Sigma_3$ for the ALFALFA galaxies. Based on the distribution's approximately lognormal shape, we divide the ALFALFA sample into four quartiles which contain objects residing in increasingly denser environments. Figure \ref{fig:SDSS_NN_cone} shows coneplots of the ALFALFA galaxies belonging to the lowest and highest density quartile (left and right panel, respectively). Reassuringly, the difference in clustering between the two environmental subsamples is clearly visible by eye. Sources in the densest environment are grouped together in clumps and filaments, whereas the sources in the least dense environment are distributed almost uniformly in space. This is an excellent indication that the NN method is splitting the ALFALFA galaxies into environmental subsamples in a sensible way.
\subsection{Fixed Aperture Environment}
\label{sec:fixed_aperture}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{SDSSFA_env_hist.eps}
\caption{
Histogram of the number of SDSS neighbours within the fixed aperture, \nfa, for galaxies in the ALFALFA sample (see \S\ref{sec:fixed_aperture}). The green (leftmost) bar denotes the lowest density subsample, \nfa$=0$. The crimson, purple and black bars (left to right) represent instead the ALFALFA galaxies located in the densest 25\%, 10\% and 5\% environments, according to the fixed aperture method (\nfa$\geq3$, \nfa$\geq 6$ and \nfa$\geq 9$, respectively). Note that these three all overlap as the densest 25\% includes both the densest 10\% and 5\%. The final bin contains counts for all ALFALFA sources with 20 or more SDSS neighbours within the fixed aperture. The white bars correspond to galaxies with $0<$\nfa$<3$.
}
\label{fig:FA_hist}
\end{figure}
In addition to the NN method described above, we also adopt a fixed aperture approach as a complementary way to measure the environment of ALFALFA galaxies. In particular, we count the number of galaxies in the reference catalogue that lie within a radius of 1 Mpc and a velocity range of $\pm$500 \kms \ from the position and velocity of our target ALFALFA galaxy. The fixed aperture environment is thus characterised simply by a natural number, $N_\mathrm{FA}$. As with the nearest neighbour method, we exclude possible optical counterparts from the count (any object that is within 5\arcsec \ and $\pm$70 \kms \ from the ALFALFA galaxy).
Figure \ref{fig:FA_hist} shows the distribution of fixed aperture environment, \nfa, for the ALFALFA sample. Unlike in the case of nearest neighbour densities, the distribution of \nfa \ has a power law form. This means that the fixed aperture method provides a rather coarse description of environment at low densities; for example, the lowest FA density subsample (\nfa $=0$) contains already 38\% of the total sample. On the other hand, the FA method is better for isolating the ALFALFA galaxies that reside in the highest density environments. The preceding points are visually demonstrated by the two coneplots in Figure \ref{fig:FA_cone}: The left panel shows the lowest FA density subsample of ALFALFA. This sample is successful at tracing low density environments in general, but cannot discriminate between galaxies located in voids and galaxies located in parts of filaments with low local density. On the other hand, the right panel shows the top 5\% of ALFALFA galaxies in terms of FA density (\nfa $\geq 9$). This latter sample does an excellent job at tracing the locations of the largest clusters and groups in the survey volume.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{a70_spring_15000_FA_low+5pdense.eps}
\caption{
Coneplots of ALFALFA galaxies belonging to the lowest density subsample (\textit{left panel}) and highest density subsample (\textit{right panel}) of fixed aperture environment (refer to figure \ref{fig:FA_hist}). The latter sample demonstrates that the fixed aperture method can be used to probe the largest groups and clusters in the survey volume.
}
\label{fig:FA_cone}
\end{figure*}
\subsection{2MRS Nearest Neighbour Environment}
\label{sec:2MRS}
In order to study the environment of the 70\% ALFALFA sample in both the Spring and Fall regions of the sky, we need a reference catalogue that covers the entire celestial sphere. To this end, we follow the same approach described in \S\ref{sec:reference_catalogue}, but now using the all-sky 2MASS Redshift Survey (2MRS) as reference. We select galaxies in the 2MRS that are brighter in the $K$-band than $M_K = -24.9$. Given the 2MRS apparent magnitude limit of $m_K = 11.75$, this cut makes the 2MRS catalogue volume-limited over the entire volume probed by ALFALFA out to 15,000 \kms/$H_{0}$.
Compared to the SDSS spectroscopic survey, the 2MRS survey is much shallower. This means that the 2MRS-based reference catalogue is limited to much brighter objects than the SDSS-based one, and consequently it is much sparser in space. This fact affects the way in which environment is measured with the nearest neighbour method. In particular, the third nearest neighbour in the 2MRS catalogue is usually so far apart from the target ALFALFA galaxy that it does not provide a good measure of local environment. As a result, when using the 2MRS catalogue as reference we calculate local densities based on the distance to the nearest neighbour, $R_1$; the corresponding density is then $\Sigma_1 = 1/\pi R_1^2$.
Figure \ref{fig:NN1_hist} shows the distribution of $\Sigma_1$ for the full ALFALFA sample. We follow the same process described in \S\ref{sec:nearest_neighbour} and split the distribution into four quartiles, containing galaxies located in progressively denser environments. Note that despite the change in nearest neighbour rank, the scale over which environment is probed by 2MRS is still larger than in the SDSS case. This is evident by the shift in the location of the distribution peak between figures \ref{fig:NN3_hist} and \ref{fig:NN1_hist}; the former peaks at $\sim$0.16 Mpc$^{-2}$, and the latter at $\sim$0.01 Mpc$^{-2}$. The difference in scale over which environment is probed is also reflected in the spatial distribution of the 2MRS environmental subsamples. This is clearly visible in Figure \ref{fig:2MRS_NN_cone}, which plots the spatial distribution of the lowest and highest density 2MRS quartiles (left and right panel, respectively). By comparing with the corresponding panels in figure \ref{fig:SDSS_NN_cone}, one can immediately recognise that the 2MRS environmental subsamples follow more closely the cosmic LSS than their SDSS counterparts. For example, large filaments are more starkly defined in the highest density 2MRS sample than in the highest density SDSS sample. At the same time, galaxies in the lowest density 2MRS sample actively avoid the locations of large filaments, an effect that is not present in the corresponding SDSS sample (see figure \ref{fig:SDSS_NN_cone}).
\section{Calculating HIMFs}
\label{sec:HIMFcalc}
The HI mass function (HIMF) is defined as the number density of galaxies as a function of their HI mass, $\phi(M_{HI})$. Galaxies span several orders of magnitude in terms of their HI mass, so the HIMF is customarily measured in logarithmic mass intervals as
\begin{eqnarray}
\phi(M_\mathrm{HI}) = \frac{dN_{\mathrm{gal}}}{dV \: d\log_{10}(M_\mathrm{HI})} \;\;\; .
\end{eqnarray}
\noindent
In the equation above, $dN_{\mathrm{gal}}$ is the average number of galaxies in a cosmic box of volume $dV$, whose HI mass lies within a small logarithmic bin centred around $M_{\mathrm{HI}}$.
Since the ALFALFA sample is (roughly) flux-limited, the measurement of the HIMF is not a simple counting exercise. For example, there are many more detections in ALFALFA with \mhi $= 10^{10}$ \Msol \ than with \mhi $= 10^8$ \Msol, but the former sources can be detected out to much larger distances than the latter. Once the sensitivity limits of the survey are known \citep[section 6]{Haynes+2011}, this effect can be compensated for by weighting each source according to the maximum volume over which it is detectable by the survey (`$1/V_\mathrm{max}$' method).
The $1/V_\mathrm{max}$ method has the advantage of being intuitive and simple to implement, but has one major limitation: it is unbiased only if the galactic population is distributed in an approximately uniform way within the survey volume. This is definitely not the case for the ALFALFA survey, where large-scale structure is clearly present in the spatial distribution of galaxies (see figure \ref{fig:a70+SDSS_cone}). For this reason, we use in this article a more sophisticated method to calculate the HIMF, referred to as the `$1/V_\mathrm{eff}$' method \citep{Zwaan+2005}. More specifically, the HIMF can be calculated within logarithmic mass bins as
\begin{eqnarray}
\phi_i = \frac{1}{\Delta m_\mathrm{HI}} \cdot \sum_j \frac{1}{V_{\mathrm{eff},j}} \;\;\; ,
\end{eqnarray}
\noindent
where the summation runs over all galaxies $j$ that belong to mass bin $i$. Accordingly, the counting error on the HIMF can be calculated as
\begin{eqnarray}
\sigma_{\phi_i}^2 = \frac{1}{\Delta m_\mathrm{HI}^2} \cdot \sum_j \frac{1}{V_{\mathrm{eff},j}^2} \;\;\; .
\label{eq:veff_err}
\end{eqnarray}
\noindent
In the equations above, $\Delta m_\mathrm{HI}$ is the logarithmic width of the mass bin (i.e. $\Delta \log_{10}(M_\mathrm{HI}/M_\odot)$), while $V_{\mathrm{eff},j}$ is the `effective volume' available to galaxy $j$. The effective volume is determined through a maximum-likelihood statistical technique, and takes into account both the survey sensitivity limits and the fluctuations of galaxy counts with distance induced by the large-scale structure in the survey volume. As a result, the $1/V_\mathrm{eff}$ method is fairly robust against bias caused by inhomogeneities in the spatial distribution of galaxies. Full details of the implementation of the $1/V_\mathrm{eff}$ method in the context of the ALFALFA survey can be found in \citet[Appendix B]{Martin+2010} and \citet[\S 3.1]{Papastergis+2011}, and references therein.
There are two important technical differences between the measurement of the HIMF of various environmental subsamples in this work, and the measurement of the overall HIMF of ALFALFA \citep{Martin+2010}. First, it is very difficult to determine the actual survey volume occupied by each environmental subsample (see figure \ref{fig:SDSS_NN_cone}). As a result, we do not attempt to compute absolute normalisations for the environmental HIMFs, but rather we compare the HIMF shape among the various subsamples. Second, the spatial distribution of different environmental subsamples can be drastically dissimilar (see e.g. figure \ref{fig:FA_cone}). As a result, the effective volumes for galaxies that belong to a specific subsample are computed based on the spatial distribution of the other subsample members only (rather than the whole ALFALFA sample).
The method described above for the measurement of the HIMF is fully non-parametric. However, previous studies \citep[e.g.][]{Zwaan+2003,Zwaan+2005,Martin+2010} have shown that the HIMF can be described very well by a specific functional form, referred to as the `Schechter function' \citep{Schechter1976}:
\begin{eqnarray}
\phi(M_\mathrm{HI}) & = & \frac{dN_\mathrm{gal}}{dV \: d\log_{10}(M_\mathrm{HI})} = \nonumber \\
& = & \ln(10) \: \phi_\ast \: \left( \frac{M_\mathrm{HI}}{M_\ast} \right)^{\alpha+1} \: e^{-\left( \frac{M_\mathrm{HI}}{M_\ast}\right)} \;\;\; .
\end{eqnarray}
\noindent
The Schechter function describes a power law of logarithmic slope $\alpha+1$ at the low-mass end ($M_\mathrm{HI} \ll M_\ast$), which transitions to an exponential drop off at the high-mass end ($M_\mathrm{HI} \gg M_\ast$). The parameter $M_\ast$ is therefore the value of mass corresponding to the transition `knee' of the HIMF, while $\phi_\ast$ controls the normalisation of the HIMF. In this work, we determine the best fit Schechter parameters for the measured HIMFs by ordinary least squares minimisation\footnotemark{}. As explained in the previous paragraph, the value of $\phi_\ast$ in the environmental HIMFs is arbitrary, and only the two shape parameters ($M_\ast$ and $\alpha$) are physically relevant in this case. Note that the two shape parameters are covariant, such that the fit error is best depicted as an ellipse in the $\{M_\ast,\alpha\}$ plane. Lastly, keep in mind that the errors on the fit parameters depend on the errorbars of individual HIMF datapoints. These errorbars are computed through Eqn. \ref{eq:veff_err}, and represent the statistical counting error only. As a result, systematic uncertainties are not included in the fit error values quoted in this article. The robustness of the $1/V_\mathrm{eff}$ method is discussed further in appendix \ref{sec:HIMF_check}.
\footnotetext{The best fit parameters are determined by the \texttt{scipy.optimize.curve\_fit} routine written in the \texttt{Python} programming language. The minimisation is performed in linear space, assuming Gaussian errors with a magnitude determined by Eqn. \ref{eq:veff_err}.}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{2MRSNN1_env_hist.eps}
\caption{
Similar to figure \ref{fig:NN3_hist}, but referring to the environment as defined by the 2MRS reference catalogue. Keep in mind that in the case of 2MRS the nearest neighbour density is calculated based on the distance to the closest neighbour (\S\ref{sec:2MRS}). Once again, different colours (shades) mark the four quartiles of the distribution, increasing in density left to right.
}
\label{fig:NN1_hist}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{a70_2MRS_15000_NNquart1and4.eps}
\caption{
Same as figure \ref{fig:SDSS_NN_cone}, but showing coneplots of the lowest (\textit{left panel}) and highest (\textit{right panel}) density quartiles of the 2MRS nearest neighbour environment (see figure \ref{fig:NN1_hist}). Note that the 2MRS is an all-sky survey, and therefore it can be used to measure the environment in both the Spring and Fall portions of the ALFALFA footprint.
}
\label{fig:2MRS_NN_cone}
\end{figure*}
\section{Results}
\label{sec:results}
\subsection{SDSS Reference Catalogue}
The following subsection is concerned with the results obtained when defining an ALFALFA galaxy's environment based on the SDSS reference catalogue that extends from 500-15,500 \kms/$H_{0}$, this includes both the NN and FA methods for defining environment (see \S\ref{sec:env}).
\subsubsection{Nearest Neighbour Density}
\label{sec:NN3_results}
The nearest neighbour density calculated by the 3rd SDSS neighbour above the volume limiting absolute magnitude cut was used to define quartiles of environment for the ALFALFA galaxies. The galaxies from each quartile were used to calculate the HIMF for that environment (as described in \S\ref{sec:HIMFcalc}) and were compared to the HIMF calculated from all four quartiles combined.
\begin{figure*}
\centering
\includegraphics[width=\columnwidth]{SDSS_HIMF_1000-15000kms_Veff_plot.eps}
\includegraphics[width=\columnwidth]{SDSS_HIMF_1000-15000kms_err_ell_plot.eps}
\caption{\textit{Left panel}: The HIMFs of each environment density quartile in the ALFALFA sample. The solid coloured lines represent Schechter function fits of each quartile in nearest neighbour density, calculated using the 3rd nearest neighbour in the associated SDSS catalogue. In order of most to least dense they are dark red, gold, green, light blue, or equivalently, top to bottom (or dark to light shades). The dashed grey lines show the HIMF of the full sample, and are offset to aid readability. The error bars represent the counting errors only, and neglect errors in the input masses and velocity widths. \textit{Right panel}: The 2-$\sigma$ error ellipses of the Schechter function fit parameters of the HIMFs in the left plot. The colour scheme is identical to the left plot and the hatching styles are as follows: positively sloped, vertical cross, diagonal cross, negatively sloped, in order of increasing density quartiles. The grey filled ellipse represents the fit to the full sample.}
\label{fig:NN3_results}
\end{figure*}
Figure \ref{fig:NN3_results} shows the HIMF for each of the four ALFALFA quartiles (left) and the 2-$\sigma$ errors ellipses of the fit to the Schechter function parameters of each quartile (right). There is a clear trend of the lowest environmental density quartile (light blue) HIMF function falling below that of the full sample at the high mass end, and this switches to lying above it for the highest density quartile (dark red), with the middle two quartiles falling between the two extremes. There is also a much weaker dependence on the low-mass slope, with the quartiles appearing to produce a marginally flatter slope as the local density decreases.
Theses results seem to indicate that the `knee' mass of the HIMF is indeed a function of nearest neighbour environment (as defined by the SDSS reference catalogue in \S\ref{sec:nearest_neighbour}) with the value of $\log M_{*}/\mathrm{M_{\odot}}$ changing from $9.81 \pm 0.02$ to $10.00 \pm 0.03$ between the lowest and highest density quartiles (of the ALFALFA sample). There is also a suggestion of a trend in the low-mass slope, although this is much less pronounced. The error ellipses in figure \ref{fig:NN3_results} appear to move progressively further right (flatter low-mass slope) with decreasing density. However, this trend is not statistically significant as all the ellipses overlap in $\alpha$, indicating that they are consistent within 2-$\sigma$. Fitting a vertical line (fixed $\alpha$ value) to the ellipses results in a reduced $\chi^{2}$ value of 1.2, indicating that assuming no change in $\alpha$ is a reasonable model for the data (the equivalent $\chi^{2}$ value, assuming no change in $M_{*}$, is 13). It should also be noted that the Schechter fit is based only on the counting errors when calculating the HIMF, thus the error ellipses are likely underestimates of the true errors, as they do not include distance uncertainties (probably the largest single source of error). Furthermore, this apparent shift is in the direction that you would expect $\alpha$ to be driven by the change in $M_{*}$, due to the covariance between the two parameters. This is also opposite to the trend between environment and $\alpha$ that is expected (steeper in low density environments).
\subsubsection{Fixed Aperture Environment}
\label{sec:results_FA}
\begin{figure*}
\centering
\includegraphics[width=\columnwidth]{Fix_ap_HIMF_1000-15000kms_Veff_plot.eps}
\includegraphics[width=\columnwidth]{Fix_ap_HIMF_1000-15000kms_err_ell_plot.eps}
\caption{\textit{Left panel}: The HIMFs of each environment defined by the fixed aperture method in the ALFALFA sample. The solid coloured lines represent the Schechter function fits of the four different environments, those with 0 neighbours within the fixed aperture, with 3 or more, 6 or more, or 9 or more. The respective colours are green, crimson, purple, and black or equivalently, bottom to top (or light shades to dark shades). The last three of these samples approximately corresponds to the 25, 10 and 5 percent most dense environments. The dashed grey lines show the HIMF of the full sample, and are offset to aid readability. The error bars represent the counting errors only. \textit{Right panel}: The 2-$\sigma$ error ellipses of the Schechter function fit parameters of the HIMFs in the left plot. The colour scheme is identical to the left plot and the hatching styles are as follows: positively sloped, vertical cross, diagonal cross, negatively sloped, in order of increasing density quartiles. The grey filled ellipse represents the fit to the full sample.}
\label{fig:HIMF_FA}
\end{figure*}
In Figure \ref{fig:HIMF_FA} we show the measured HIMFs and error ellipses for four environmental subsamples defined via the fixed aperture method (refer to \S\ref{sec:fixed_aperture}). In particular, the four sub-samples correspond to galaxies that belong to the lowest density FA environment (zero neighbours within the fixed aperture), and galaxies that belong to the 25\%, 10\% and 5\% densest environments in terms of FA neighbours. Figure \ref{fig:HIMF_FA} shows that there is no clear dependence of the low-mass slope on environment, in agreement with the findings of \S\ref{sec:NN3_results}. However, the environmental dependence of the `knee' mass is more complicated than before. In particular, we do observe a shift in the value of $M_\ast$ between the lowest density and 25\% densest FA sub-samples, that is compatible with the trend seen in figure \ref{fig:NN3_results}. However, the trend does not extend consistently to the two highest density FA sub-samples; instead the value of $M_\ast$ for the 10\% and 5\% densest FA sub-samples is actually slightly lower than for the 25\% sub-sample.
At first glance, the results of figures \ref{fig:NN3_results} and \ref{fig:HIMF_FA} regarding the environmental dependence of $M_\ast$ may seem inconsistent with each other. However, this is most probably not the case, because the two densest FA subsamples probe a higher density regime than the fourth quartile of NN environmental density (refer to \S\ref{sec:fixed_aperture}). We therefore interpret the observed $M_\ast$ trend with FA environment as the result of HI-deficiency affecting galaxies in the highest density regions of the ALFALFA volume. According to this interpretation, the extrapolation of the environmental $M_\ast$ trend observed for the NN subsamples into the highest density environments fails, because the processes responsible for HI-deficiency inhibit the formation of galaxies with high HI masses in these crowded environments.
\subsection{2MRS Reference Catalogue}
\label{sec:2MRS_NN}
\begin{figure*}
\centering
\includegraphics[width=\columnwidth]{2MRS_HIMF_1000-15000kms_Veff_plot.eps}
\includegraphics[width=\columnwidth]{2MRS_HIMF_1000-15000kms_err_ell_plot.eps}
\caption{Identical to figure \ref{fig:NN3_results} except that here nearest neighbour environment quartiles are defined using the first neighbour in the 2MRS reference catalogue.}
\label{fig:HIMF_2MRS_NN}
\end{figure*}
The 1st nearest neighbour in the volume-limited 2MRS catalogue was used to define quartiles of environmental density for the ALFALFA galaxies (refer to \S\ref{sec:2MRS}).
The ALFALFA sample that can be used in the 2MRS analysis contains about 50\% more galaxies than the sample used in the SDSS analysis, as 2MRS is all sky survey. Figure \ref{fig:HIMF_2MRS_NN} shows the HIMF Schechter parameters calculated for each quartile of neighbour density (in 2MRS). Despite having a greater number of sources to compute the HIMFs, and therefore smaller error ellipses, no consistent trend in either $M_{*}$ or $\alpha$ is evident; all four quartiles are consistent with the global sample at 2-$\sigma$ confidence. This result has been checked to be robust against cosmic variance and the colour of the reference sample (see appendix for details).
The fundamental difference between the SDSS-based and 2MRS-based environmental measures is the scales that they probe. As argued in \S\ref{sec:2MRS}, the environment defined using 2MRS is probing a larger scale than that defined using SDSS. This is because 2MRS is a shallower survey, which leads to larger separations between sources. In addition, using the 2MRS catalogue to define the environment results in a better separation of our ALFALFA sample based on the position of galaxies in the LSS. For example, filaments and clusters are starkly defined in figure \ref{fig:2MRS_NN_cone} (right panel), while in the left panel there are clear gaps in the corresponding positions. Given these differences between environment defined using SDSS and 2MRS, and the fact that a trend between environment and $M_{*}$ is only measured when using SDSS, the most straightforward interpretation of our results is that an HI-selected galaxy's characteristic HI mass ($M_{*}$) increases with the density of its local environment, but is independent of its position relative to large scales structures. In addition, we find that the faint end slope of HI-selected galaxies is universal, having no significant dependence on any measure of environment we explored.
\section{Discussion}
\label{sec:discuss}
The notion that local environment is the primary factor for determining a galaxy's properties is not a new idea, in fact it is the fundamental assumption underlying the very successful HOD formalism. There are also optical based experiments which have found similar results: \citet{Berlind+2005} compared simulations and the SDSS to demonstrate that galaxy properties are strongly correlated with the host halo mass, and that this is the parameter that most environment measures based on local galaxy density, are tracing; \citet{Blanton+2006} studied the environment of SDSS galaxies on different scales and found that only environment within $\sim$1 Mpc is important for determining a galaxy's star formation rate and colour. Our results fit well with these theoretical and optical results, however there are still a number of tensions with theory and other HI observations. Below we review some literature results regarding the environmental dependence of the HIMF and discuss cases where there exists tension between these studies and the results of this work.
\subsection{Comparison with previous HI survey results}
\label{sec:discussion_surveys}
The first study on the environmental dependence of the HIMF based on a large-area blind HI survey was performed by \citet{Zwaan+2005}, using the HIPASS dataset. Contrary to our results, they found that the low-mass slope, $\alpha$, becomes steeper with increasing environmental density, while the `knee' mass, $M_\ast$, is roughly independent of the environment (see their figure 3). The comparison between the HIPASS result of \citet{Zwaan+2005} and the ALFALFA result obtained in \S\ref{sec:results} is not straightforward, because the two studies define the NN environment in different ways. In particular, \citet{Zwaan+2005} find neighbours for the HIPASS galaxies in the HIPASS catalogue itself. This decision was dictated by the fact that there is no large-area spectroscopic survey at optical wavelengths that covers the HIPASS footprint. As explained in \S\ref{sec:reference_catalogue}, this neighbour definition makes the consistent computation of environmental density throughout the survey volume very difficult to achieve in practice. In Appendix \ref{sec:HIPASS_mock} we show that if an environmental trend in $\alpha$, equivalent to that found in HIPASS by \citet{Zwaan+2005}, was present in the ALFALFA dataset, it would have been easily detected by the current analysis.
Another important difference between the HIPASS and ALFALFA nearest neighbour definitions is the scale over which they probe the environment. More specifically, the HIPASS catalogue is much sparser than the SDSS reference catalogue used for environment definition in this work. If not due to the observational limitations therefore, the HIPASS trend should be driven by the large-scale environment of galaxies, rather than the local environment probed in this article. However, this interpretation of the HIPASS result is also open to question. For example, \citet{Moorman+2014} have recently measured the HIMF separately for the ALFALFA galaxies that reside in voids and for those that reside in walls/filaments. They find a difference in the HIMF measured for the two environmental samples that is similar to the environmental trend found in \S\ref{sec:NN3_results}. In particular, the wall/filament HIMF has a higher `knee' mass than the void HIMF, but only a marginally steeper low-mass slope (refer to their figure 8). Given that the \citet{Moorman+2014} environment definition also refers to large scales ($\sim$10 Mpc), their result seems to contradict the HIPASS finding.
Our results make an intriguing addition to those of \citet{Moorman+2014} because we detect a very similar trend in $M_{*}$, but associated with local, rather than large scale, environment. The reason for this apparent contradiction is not clear, however we note that it could be resolved if the separation of galaxies between void and wall objects in \citet{Moorman+2014} is correlated with the local environment of the galaxies more than naively expected based on the size of these cosmic structures; in that case, it would be natural for the \citet{Moorman+2014} result to be closely related to the result obtained by considering SDSS-based local densities.
An additional complication is added by the fact that the \citet{Moorman+2014} trend is not detected in the present work when environment is defined on relatively large scales with 2MRS-based densities (see \S\ref{sec:2MRS_NN}). Again, the reason for this tension is not entirely clear, although (as above) if the void and wall samples of \citet{Moorman+2014} were sufficiently correlated with local density, then a trend associated with local environment could be masquerading as one with large scale environment -- a false trend that we would not necessarily expect to see with 2MRS neighbour densities. Alternatively, it is possible that 2MRS could be missing the large scale component of a real trend associated with both local and large scale. 2MRS clearly separates out the densest LSS into the 4th quartile of neighbour density, but if the separation between the remaining 3 quartiles was extremely noisy, then trends could be suppressed.
\subsection{Comparison to the HIMF in groups}
\label{sec:group_HIMF}
\citet{Verheijen+2001, Kovac+2005, Freeland+2009, Pisano+2011} studied the HIMF in galaxy groups and all came to essentially the same conclusion; that the low-mass slop is flat in groups. Given these consistent findings, it is perplexing that we see no evidence for variation of the low-mass slope, as in the field it has been shown by both HIPASS \citep{Zwaan+2005} and ALFALFA \citep{Martin+2010} that it is not flat (both surveys measure $\alpha \approx -1.3$).
Assuming that a non-negligible fraction of ALFALFA's detections are galaxies in groups \citep[][find that approximately 25\% of ALFALFA galaxies are in groups]{Hess+2013}, such that any trend would not be drowned out, then the findings above suggest that the nearest neighbour definition of environment is not consistently separating groups from the rest of the sample. If this were not the case, then there would need to be an inconsistency in how the wide field and targeted surveys are calculating the HIMF, in order to explain these seemingly contradictory findings.
This apparent shortcoming in the nearest neighbour method could be explained if the surface number density of galaxies in groups is approximately independent of group size. As our method cannot distinguish regions of the same surface density, under these assumptions, it would be incapable of separating groups of different sizes and we would be blind to any trend associated with group size. Therefore, if the low mass slope varies with group size, our analysis might not reveal this. Alternatively, as the surveys which have measured a flat low-mass slope in groups are mostly interferometric surveys (that resolve many of their sources), an uncertain detection threshold associated with HI surface density could result in an erroneous slope. A more detailed study of the HIMF in groups is required to test these hypotheses and compare the two existing methodologies.
\section{Conclusions}
\label{sec:conclude}
We have used the 70\% ALFALFA sample to search for dependence of the HIMF on galactic environment. In particular, we defined the environment of ALFALFA galaxies based on the neighbours found in both SDSS and 2MRS volume limited reference catalogues. We find that the Schechter function `knee' mass ($\log{M_{*}/\mathrm{M_{\odot}}}$) is dependent on environment, with its value shifting from $9.81 \pm 0.02$ to $10.00 \pm 0.03$ between the lowest and highest density quartiles. However, this dependence was only observed when defining environment based on the SDSS reference catalogue, not 2MRS. Using a fixed aperture measure of environment with SDSS, we also found tentative evidence for a decrease in $M_{*}$ in the highest density environments, in agreement with the notion that galaxies in clusters should become HI-deficient.
In \S\ref{sec:env} we demonstrated that using our approach, 2MRS both measures environment on a larger scale than SDSS, and is more effective at separating large scale structures into different environment density quartiles. This strongly suggests that the dependence we are seeing is on local environment, rather than large scale, supporting the fundamental assumption of the HOD formalism, that a galaxy's properties are only dependent on the mass of its host halo. However, this is in tension with a previous ALFALFA-based study \citep{Moorman+2014} which found a similar trend in $M_{*}$, but based on separating galaxies which reside in walls and voids.
Although the true resolution remains unclear we offered two potential explanations for this discrepancy between our results and those of \citet{Moorman+2014}. If void and wall environments are sufficiently correlated with local densities such that trends are expected with either definition of environment, then the results would be in agreement. Alternatively, if the 2MRS densities used in this paper were to be incapable of distinguishing low density environments then trends associated with large scales might be hidden from our analysis.
In all of the tests we performed we detected no significant dependence of the the low-mass slope ($\alpha$) on environment. Again, this appears in conflict with existing results, both from HIPASS \citep{Zwaan+2005} and from several studies of galaxy groups (which measure $\alpha \sim -1$). The steepening of $\alpha$ with denser environments that was observed in HIPASS is not directly comparable to this article due to different methodology (see \S\ref{sec:reference_catalogue}), and in appendix \ref{sec:HIPASS_mock} we demonstrate that we would be capable of detecting an equivalent trend if it existed in our data. As an explanation to resolve the tension with the findings of group HI studies, we suggest that the inability of the nearest neighbour environment to separate different sized groups of the same projected surface density, might be responsible for our null result. If the low-mass slope was a function of group size and most groups had similar surface densities, then this would explain the observations. An alternative explanation could be inconsistent methodologies resulting from uncertain surface brightness limits in narrow field surveys. A more complete understanding of the HIMF in groups is needed to test these hypotheses.
\section*{Acknowledgements}
The authors acknowledge the work of the entire ALFALFA collaboration in observing, flagging, and extracting the catalogue of galaxies that this paper makes use of. The ALFALFA team at Cornell is supported by NSF grants AST-0607007 and AST-1107390 to RG and MPH and by grants from the Brinson Foundation. EP is supported by a NOVA postdoctoral fellowship at the Kapteyn Institute. MGJ would like to thank Kelley Hess for useful discussions about HI in groups.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,360 |
Q: Trying to multiply a float in bash not working I have this script that is to rescale images to a percentage value
#!/bin/bash
percent=$1
echo $percent
for img in `find *.png`;
do
echo Processing file $img
width=$( mdls $img | grep kMDItemPixelWidth | tail -n1 | cut -d= -f2 )
height=$( mdls $img | grep kMDItemPixelHeight | tail -n1 | cut -d= -f2 )
newWidth=$((width*percent))
newHeight=$((height*percent))
echo $newWidth $newHeight
sips -z $newWidth $newHeight $img
done
My bash is configured to accept commas as decimal separators.
So, whey I type
rescale 0,3019
I am trying to rescale the images to 30.19% of their values
the problem is that the line
echo $newWidth $newHeight
shows me the values as they were multiplied by 3019. Strangely the first echo
echo $percent
shows me 0,3019 (the correct value)
what am I missing?
A: To your headline: bash can only multiply integers.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,616 |
\section{Introduction}
\label{sec:intro}
The first and most radical departure of string theory from a theory
of elementary point particles is in the nature of its elementary degrees of
freedom. The rest of the structure of string theory rather conservatively
follows established principles: relativistic invariance
(generalized to include supersymmetry), quantum
mechanics, locality of interaction, and internal mathematical consistency.
Together, these result in new symmetries and properties that
open up conceptual possibilities inconceivable in elementary point
particle theories.
In this note we focus on a few such properties that make it possible
to imagine a resolution of the initial
singularity problem in cosmology. As will be evident, this possibility
is directly traceable
to the basic degrees of freedom of a string.
In the next section we begin with a detailed discussion of the
degrees of freedom in string theory, contrasting them with
elementary point particle theory, and describing the nature of
`particles' in string theory. The `string uncertainty principle'
(which modifies the Heisenberg uncertainty principle to state that
there is a minimum observable size in a world whose fundamental
constituents are strings) is introduced
in terms of string wavefunctions. The `t-duality' symmetry
(indistinguishability of very large and very small universes)
is described. Section III discusses some features of the thermodynamics
of an ideal gas of strings at very high energy densities. This suggests
the possibility that temperature and pressure will have finite
limiting values in a string universe. In section IV the preceding
material is applied to string cosmology. We discuss a thought
experiment for measuring the size of
a very small universe in the context of string theory. This
prompts a shift of perspective regarding what one means by a small universe,
and from the new vantage point
the initial singularity problem disappears (Brandenberger and Vafa 1989).
We conclude with some
cautionary remarks and some questions of possible relevance to
observations. While much of the material presented here is a review
of existing literature, there are some points which do not seem
to have appeared elsewhere. One is
the expression and physical interpretation of the pressure
of an ideal string gas. Another is the derivation of the string uncertainty
principle from string wavefunctions.
\section{The basic degrees of freedom of a string}
\label{sec:dof}
\noindent {\bf Classical string modes in flat euclidean space}:
The classical configuration of a scalar particle is described by
specifying its position, a point in space, say ${\bf R}^d$.
The classical configuration of a
closed bosonic string is similarly described by specifying its position,
in this case a closed curve embedded in space.
The latter can be described by specifying
a continuous map ${\bf x}$ from the unit circle into space,
${\bf x}:S^1 \rightarrow {\bf R}^d$.
If $\sigma \in [0,\pi]$ is a coordinate on $S^1$ and
$x^i \in (-\infty,\infty), \ i=1,\ldots,d$ are coordinates of ${\bf R}^d$,
the map ${\bf x}$ is specified by $d$ continuous functions
$x^i(\sigma)$ that are periodic, $x^i(0)=x^i(\pi)$. The point
${\bf x}(\sigma)=(x^1(\sigma),\ldots,x^d(\sigma))$
traces out a closed curve in space as $\sigma$ traverses
from $0$ to $\pi$.
It is convenient to Fourier decompose the functions $x^i(\sigma)$:
\begin{equation}
\label{fourier}
x^i(\sigma) = x^i_0 + \sum_{n=1}^\infty(x^i_n \cos 2n\sigma +
\tilde{x}^i_n \sin 2n\sigma).
\end{equation}
The infinite-tuple
$({\bf x}_0,\{{\bf x}_n,\tilde{\bf x}_n\})$
equivalently describes the map ${\bf x}$ or the classical string
configuration. ${\bf x}_0 = (1/{\pi})\int_0^\pi d\sigma {\bf x}(\sigma)$
is a `centre of mass' coordinate describing the average position
of the string. ${\bf x}_n,\tilde{\bf x}_n$ describe the extension
of the string in space. It is evident that a string has an infinitely
richer repertoire of classical configurations than a point particle.
E.g., in the configuration where all ${\bf x}_n,\tilde{\bf x}_n$ are zero
the string has no extension and
becomes a point particle at the point ${\bf x}_0$ in space.
If $x^1_1=\tilde{x}^2_1=r$ and all other $x^i_n,\tilde{x}^i_n$ are
zero, the string configuration is a circle of radius $r$ in a plane
parallel to the $x^1x^2$ plane with centre ${\bf x}_0$. If some of
the $x^i_n,\tilde{x}^i_n$ with higher $n$ are also non-zero, the
configuration will become a `wiggly' circle, and wiggles become
finer with increasing $n$ (typical radius of curvature of a wiggle
caused by the $n^{th}$ mode is $\sim r/n$).
The length of the string in a classical configuration specified by
a map ${\bf x}$ is given by $l = \int_0^\pi d\sigma |d{\bf x}/d\sigma|$,
and it is evident that this can be expressed in terms of the
${\bf x}_n,\tilde{\bf x}_n$.
\vskip 0.3cm
\noindent{\bf String wavefunctions; particle like states}:
In the quantum theory a state can be described by a wavefunction, a
complex valued function over the configuration space. For a point
particle the configuration space is just ${\bf R}^d$ (real space),
hence a wavefunction is a map $\psi :{\bf R}^d \rightarrow {\bf C}$.
For a string let us take the configuration space to be the space
$Q$ of infinite-tuples $({\bf x}_0,\{{\bf x}_n,\tilde{\bf x}_n\})$
or, equivalently, the set of maps ${\bf x}:s^1 \rightarrow {\bf R}^d$.
Then a string wavefunction is a map $\psi :Q \rightarrow {\bf C}$,
assigning, to every string configuration ${\bf x}$ or infinite-tuple
$({\bf x}_0,\{{\bf x}_n,\tilde{\bf x}_n\})$, the complex number
$\psi[{\bf x}]\equiv \psi({\bf x}_0,\{{\bf x}_n,\tilde{\bf x}_n\})$.
(We ignore for the moment the subtlety that strictly $\psi$ ought to
be a map from $Q$ modulo diffeomorphisms of the circle.)
Just as for a point particle $\psi^*({\bf x})\psi({\bf x})$ stands
for the probability density of finding the particle at the point
${\bf x}$ in space when it is in the state $\psi$, so similarly
$\psi^*({\bf x}_0,\{{\bf x}_n,\tilde{\bf x}_n\})
\psi({\bf x}_0,\{{\bf x}_n,\tilde{\bf x}_n\})$ represents the
probability density (in the infinite dimensional space $Q$) of finding
the string in the configuration $({\bf x}_0,\{{\bf x}_n,\tilde{\bf x}_n\})$
when it is in the state $\psi$.
It is instructive to consider a few sample string wavefunctions to see
what the string would look like in the corresponding states.
\vskip 0.1cm \noindent
{\bf 1}. $\psi[{\bf x}] = \delta^d({\bf x}_0-{\bf X}_0) \prod_{n=1}^{\infty}
\delta^d({\bf x}_n-{\bf X}_n)\delta^d(\tilde{\bf x}_n-\tilde{\bf X}_n)$.
\hfill\break
This wavefunction has support only over a single classical configuration
described by the infinite-tuple $({\bf X}_0,\{{\bf X}_n,\tilde{\bf X}_n\})$.
Thus an `observation' of the components of the infinite-tuple
$({\bf x}_0,\{{\bf x}_n,\tilde{\bf x}_n\})$ in this state
(assuming such an experiment can be devised, a point we return
to later) would yield a definite answer for each component, with
the conclusion that in this quantum state the string looks like
a classical string at the configuration
$({\bf X}_0,\{{\bf X}_n,\tilde{\bf X}_n\})$.
In particular if the ${\bf X}_n,\tilde{\bf X}_n$ are all zero, the
string would appear to be a point particle (with no extension) localized
at ${\bf X}_0$.
\vskip 0.1cm \noindent
{\bf 2}. $\psi[{\bf x}] = \delta^d({\bf x_0}-{\bf X}_0)
[\prod_{i=1}^d \phi_i(x^i_1)\tilde{\phi}_i(\tilde{x}^i_1)]
[\prod_{n=2}^{\infty}\delta^d({\bf x}_n)\delta^d(\tilde{\bf x}_n)]$.
\hfill\break
Case i) $\phi_i = \tilde{\phi}_i = \phi$ for all $i$ and $\phi$ is a function
that has support only in a small region around the origin (e.g.,
$\phi(x)=e^{-x^2/a^2}$). In such a state, the fourier components
for $n=1$ have a spread of order $a$, and consequently the string
will appear to have an average position still ${\bf X}_0$ but will
now have an indefinite extension in space of order $a$. If $a$
is small compared to the probes, it will still appear particle-like
(a fuzz of size $a$ around ${\bf X}_0$). If $\prod_{i=1}^d \phi(x_1^i)
= \phi(|{\bf x}_1|)$ (as in the example), the fuzz will appear
spherically symmetric.\hfill\break
Case ii) The $\phi_i$ still have support in a small region of order
$a$ but are now different for different $i$. Then the
`fuzz of size $a$ around ${\bf X}_0$' will no
longer appear isotropic. (The spin of the photon, graviton, etc.,
in string theory is due to such internal spatial structure of the
corresponding states. There $a$ is of order Planck length.)
It should also be evident that the visual picture
is not qualitatively very different if some of the higher modes
are also given a fuzz around the origin. Since these states
are not eigenstates of the ${\bf x}_n,\tilde{\bf x}_n$, they are also
not eigenstates of the length operator. When higher modes are allowed
to be nonzero, the expectation value of $l$ increases (more
wiggles in the classical configurations over which the wavefunction
has support).
\vskip 0.1cm \noindent
{\bf 3}. $\psi[{\bf x}] = e^{-i{\bf p.x_0}}
[\prod_{i=1}^d \phi_i(x^i_1)\tilde{\phi}_i(\tilde{x}^i_1)]
[\prod_{n=2}^{\infty}\delta^d({\bf x}_n)\delta^d(\tilde{\bf x}_n)]$.
\hfill\break
If $\phi_i$ are as in 2 this wavefunction represents the same particle-like
string state, but now with a definite centre of mass momentum ${\bf p}$ rather
than a definite centre of mass position ${\bf X}_0$.
The above discussion made no reference to string dynamics. We only
discussed the degrees of freedom in string theory and how particle
like states can be imagined out of strings. It turns out that the
most natural dynamics for strings in fact makes such states appear
as eigenstates of the energy. The latter turn out to be plane waves
for the centre of mass
coordinate and wavefunctions localized around the origin (in fact
harmonic oscillator wavefunctions) for the wiggle modes.
\vskip 0.3cm
\noindent {\bf Dynamics:}
As a single string moves, its trajectory traces out a two dimensional
surface embedded in space. This can be described by introducing a
fictitious parameter $\tau$ taking values in the interval
${\cal{T}} = [\tau_1,\tau_2]$ of the real line
and letting ${\bf x}$ be a map from ${\cal{T}} \times S^1$
into space. The image of ${\cal{T}} \times S^1$ under the map is a string
trajectory. To specify the dynamics for a single string we need
to introduce an action for every such trajectory. It is rather
cumbersome to introduce a relativistic invariant action in terms
of the space variables ${\bf x(\tau,\sigma})$ alone. It is more convenient to
let the time coordinate $x^0$ also be a function of $\tau$ and
$\sigma$ and to write an action for $x(\tau,\sigma) \equiv
(x^0(\tau,\sigma),x^1(\tau,\sigma),\ldots,x^d(\tau,\sigma))
\equiv (x^{\mu}(\tau,\sigma)), \ \mu =0,1,\ldots,d$, which
describes a string trajectory, or `worldsheet', in $d+1$ dimensional
Minkowski spacetime. The action is taken to be proportional to
the area of the worldsheet (in analogy with the
action of a relativistic point particle's trajectory, which is
proportional to the length of the corresponding worldline).
\begin{equation}
\label{action1}
S[x]=-{1 \over {2\pi\alpha'}}\times {\rm area\ of\ worldsheet}
=- {1 \over {2\pi\alpha'}}\int_{\tau_1}^{\tau_2} d\tau\int_0^\pi d\sigma
\sqrt{-\det \gamma},
\end{equation}
where $\gamma_{\alpha \beta}\equiv \partial_{\alpha}x^\mu
\partial_{\beta}x^\nu \eta_{\mu \nu}$ is the metric on the
worldsheet induced from the spacetime metric $\eta_{\mu\nu}
= {\rm diag}(1,-1,\ldots,-1)$, and $\alpha,\beta$ refer to
worldsheet coordinates $\xi \equiv (\xi^\alpha) \equiv (\xi^0,\xi^1)
\equiv (\tau,\sigma)$.
This dynamics has relativistic invariance in that one can define
the generators of spacetime translations $P^\mu$, and of rotations
and boosts $M^{\mu\nu}$ (see Scherk (1975) for a review)
whose Poisson brackets satisfy the Poincare
algebra. The $P^\mu$ are canonically conjugate to the centre of
mass coordinates $x^\mu_0$. The proportionality constant
${1 \over {2\pi\alpha'}} \equiv T$ is called the string tension
or mass per unit length
since the energy $P^0$ of any static classical configuration
$(x^0(\tau,\sigma) = \tau, x^i(\tau,\sigma) = x^i(\sigma))$
turns out to be $T l$, where
$l$ is the length.
The action (\ref{action1})\ is nonlinear in the derivatives
$\partial_\alpha x^\mu$. But it has an infinite local symmetry
corresponding to the reparametrizations $\xi \rightarrow \xi' =
\xi'(\xi)$ which can be used to bring $\gamma_{\alpha\beta}$
into the form $\gamma_{\alpha\beta}(\xi)=\eta_{\alpha\beta}\rho(\xi)$,
with $\eta_{\alpha\beta}={\rm diag}(1,-1)$. In this `conformal gauge'
$\sqrt{-\det \gamma}=\rho=(1/2)\gamma_{\alpha\beta}\eta^{\alpha\beta}$
and the action reduces to the free scalar field form $S = -(1/4\pi\alpha')
\int \ d\tau d\sigma[(\partial_\tau x)^2-(\partial_\sigma x)^2]$.
Substituting the mode expansion
$x^\mu(\tau,\sigma)=x^\mu_0(\tau)+\sum_{n=1}^\infty
[x^\mu_n(\tau)\cos 2n\sigma + \tilde{x}^\mu_n (\tau)\sin 2n\sigma]$
in this action yields $S = \int \ d\tau L$ with
\begin{equation}
\label{lagrangian}
L = -{1 \over {4\alpha'}}\dot{x}_0^2 - {1 \over {8\alpha'}}\sum_{n=1}^\infty
[({\dot{x}}_n^2 - 4n^2 x_n^2) + ({\dot{\tilde{x}}}_n^2
-4n^2{\tilde{x}}_n^2)],
\end{equation}
where the dot denotes derivative w.r.t. $\tau$. Thus the area law
dynamics automatically prescribes a free particle like role for
the centre of mass mode and a simple harmonic oscillator like
role for the wiggle modes $x_n,\tilde{x}_n$ with frequency $2n$.
It is then evident that the quantum states $\psi$ of the system in question
(a single free string in Minkowski spacetime) will be specified
by the set of quantum numbers $(p,\{N_n,\tilde{N}_n\})$ where $p$ is
a momentum conjugate to the centre of mass
mode (and is the eigenvalue of the spacetime translation generator $P$)
and $N_n,\tilde{N}_n$ are harmonic oscillator excitation level
quantum numbers for the modes $x_n,\tilde{x}_n$.
The conformal gauge does not fix the freedom of reparametrizations
completely. All the $x^\mu$ are not independent variables. One
can show that the independent variables can be taken to be
$({\bf x}_0,\{x^I_n,\tilde{x}^I_n\})$ where $I$ goes only over
the `transverse' spatial indices $I=1,\ldots,d-1$. Classically,
once these are known as functions of $\tau$, all others,
$(\{x^d_n,\tilde{x}^d_n\})$ and $(x^0,\{x^0_n,\tilde{x}^0_n\})$,
are determined as functions of $\tau$ by the constraints and hence the string
worldsheet is determined. (Roughly speaking, diffeos of $(\tau,\sigma)$
eat up two spacetime coordinates $x^0,x^d$ excepting the zero mode of $x^d$.)
\vskip 0.3cm
\noindent {\bf The spectrum}:
Quantum mechanically, this means
that string states are characterized by the set of
quantum numbers $({\bf p},\{N_n^I,\tilde{N}_n^I\})$, the
other quantum numbers being determined in terms of them. In particular
the quantum number $p^0 \equiv \epsilon$, eigenvalue of the energy
$P^0$, is given by
\begin{equation}
\label{spectrum1}
\epsilon^2={\bf p}^2 + (2/\alpha')[-({{d-1} \over {12}}) +
\sum_{I=1}^{d-1}\sum_{n=1}^\infty
n(N_n^I + \tilde{N}_n^I)].
\end{equation}
The closure of the quantum Lorentz algebra fixes $d=25$.
This defines the spectrum of the free closed string in ${\bf R}^d$.
The wavefunction of this state in the basis of independent coordinate variables
then follows from inspection of (\ref{lagrangian}):
\begin{equation}
\label{wave1}
\psi_{({\bf p},\{N_n^I,\tilde{N}_n^I\})}({\bf x}_0,\{x_n^I,\tilde{x}_n^I\}) =
e^{-i{\bf p \cdot}{\bf x}_0} \prod_{I=1}^{d-1}\prod_{n=1}^\infty \ \
H_{N_n^I}(\sqrt{n \over {2\alpha'}} x_n^I)
H_{\tilde{N}_n^I}(\sqrt{n \over {2\alpha'}} \tilde{x}_n^I)
e^{-{n \over {4\alpha'}}[(x_n^I)^2 + (\tilde{x}_n^I)^2]},
\end{equation}
where $H_m(x)$ is the $m^{th}$ Hermite polynomial.
\footnote{There is an additional constraint $f(\{N_n^I,\tilde{N}_n^I\})=0$
on the oscillators coming from the fact that there is no preferred
point in the $\sigma$ direction along the string. The form of $f$ is
more complicated than the usual $f=\sum n(N_n^I - \tilde{N}_n^I)$
because the $x_n,\tilde{x}_n$ defined here do not represent the left
and right moving modes respectively. We henceforth assume that
$({\bf p},\{N_n^I,\tilde{N}_n^I\})$ in (\ref{spectrum1})\ and
(\ref{wave1}) are such that this constraint is satisfied.}
It is interesting to compare this with the third wavefunction discussed
earlier. The $\phi_i(x_1)$ there are replaced by harmonic oscillator
wavefunctions whose width is order $\sqrt{\alpha'}$. The $\delta$-functions
of the higher $n$ modes are also replaced by harmonic oscillator wavefunctions
of width $\sim \sqrt{\alpha'/n}$. Thus a string in a state with quantum numbers
$({\bf p},\{N_n^I,\tilde{N}_n^I\})$ in which the
$N_n^I,\tilde{N}_n^I$ are not too large, when observed via probes of
energy ${\buildrel < \over \sim} \ \ {\alpha'}^{-1/2}$, will effectively
appear to be a particle with some internal structure of size $\sim
\sqrt{\alpha'}$
and momentum ${\bf p}$. (A large value of $N_n^I$ would elongate the size
in the $I^{th}$ direction to $\sim \sqrt{N_n^I\alpha'/n}$.
These are sometimes referred to as the `really stringy' states.)
It is natural to define the mass $M$ of a
state as $M^2 = \epsilon^2 - {\bf p}^2$. Eq. (\ref{spectrum1})\ then
gives the mass formula in terms the oscillator excitations of the state. Thus
different states in the spectrum of a single string can be identified with
various particle species having different masses (which come in units of
$\alpha'^{-1/2}$) and momenta.
In addition to the length scale $\sqrt{\alpha'}$, string theory has a dimensionless
coupling constant $g$, which represents the amplitude that two strings
touching each other will fuse into a single string, or the reverse
process. Between such joinings or splittings, strings travel freely
according to (\ref{action1}). These rules essentially specify string
perturbation theory completely. The effective interactions of
massless particles in the string spectrum (gravitons, dilatons and
antisymmetric tensor particles in the bosonic string and also photons
or gauge particles in the heterotic string) can be determined from
these considerations (see, e.g., Green, Schwarz and Witten 1987).
In particular the gravitational constant in $d$ spatial dimensions
is given by $G = g^2 {\alpha'}^{(d-1)/2}$. Thus Newton's constant
($G$ in $d=3$) is given by $G_N = g^2 \alpha'$ (assuming higher
dimensions compactify to radii $\sim \sqrt{\alpha'}$). In other words, string
theory reproduces classical Einstein gravity at low energies if we
choose its two parameters $\alpha'$ and $g$ to satisfy
$g\sqrt{\alpha'} = l_p$ (Planck length). Note that the `string length scale'
$\sqrt{\alpha'}$ is $\sim l_p$ if $g \sim O(1)$ (strong coupling) and is much larger than
$l_p$ if $g \ll 1$ (weak coupling).
\vskip 0.3cm \noindent
{\bf The size of strings; string uncertainty principle}:
What is the size of the string in the state (\ref{wave1})? Consider the
transverse `mean-square spread' operator
(Mitchell and Turok 1987; Karliner, Klebanov and Susskind 1988)
$q \equiv \int_0^\pi d\sigma
({x^I(\sigma) - x^I_0})^2$. (\ref{wave1})\ is not an eigenstate of this,
but we can ask for its expectation value. Consider the ground state of
all the oscillator modes, $N_n^I=\tilde{N}_n^I=0$ (the scalar tachyon).
The expectation value of $q$ in this state
is $\langle q \rangle \sim \alpha'\sum_{n=1}^\infty 1/n$ which
diverges logarithmically because each of the
$x_n^I,\tilde{x}_n^I$ modes makes a finite contribution.
This divergence is empirically unobservable because an experiment does
not observe $q$ or the $x_n$ directly. A typical experiment involves
scattering a probe off the string. In order for the probe to `see' the
$x_n$ mode it must interact with it and excite it from its ground state.
This would cost energy $\sim \sqrt{n/\alpha'}$ from (\ref{spectrum1}).
A probe with a finite energy $E$
would only excite a finite number of oscillator modes; therefore the infinite
sum in $q$ should be cutoff at a finite value of $n$ depending upon the
energy of the probe. For $E \ll {\alpha'}^{-1/2}$, none of the oscillator
modes will be excited and the string will effectively look like a point
particle. Probes with $E \sim {\alpha'}^{-1/2}$ will see the state as a
fuzz of size $\sqrt{\alpha'}$. For probes with energy
$E \gg {\alpha'}^{-1/2}$ the fuzz size will increase. The maximum fuzz
size is obtained if all the energy of the probe goes into exciting
only the $n=1$ mode. Its excitation level
is then $N_1^I \sim \alpha' E^2$ from (\ref{spectrum1}), and
the consequent root mean square spread in space of the target string
wavefunction (through its ${\bf x}_1$ and $\tilde{\bf x}_1$ modes)
is $\sqrt{N_1^I \alpha'}\sim \alpha' E$. The
size {\em grows} with the energy of the probe. This is a new term
that must be added to the usual uncertainty in position
$\Delta x \sim 1/E$ coming from Heisenberg's uncertainty principle.
Setting $\alpha' = G_N /g^2$ (in $3+1$ dimensions)
and putting back units we get the `string uncertainty principle'
\begin{equation}
\label{stringup}
\Delta x \sim {{\hbar c} \over E} + {{G_N E} \over {g^2 c^4}}.
\end{equation}
Minimizing this w.r.t. $E$, one finds the smallest observable
length scale in string theory ${\Delta x}_{min} \sim l_p/g \sim \sqrt{\alpha'}$.
Here we assumed that all the energy goes into exciting only the $n=1$
modes. If the energy is shared with the higher modes whose
wavefunctions are more strongly localized, the spread will
be smaller. For example if one assumes that each mode $n$ upto
some maximum $n_{max}$ is excited to its first excited level
($N_n = 1$ for $n \leq n_{max}$ and zero thereafter), then one finds
$n_{max} \sim \sqrt{\alpha'} E$ and the sum in $q$ should be cutoff at this
value. Then, instead of $\Delta x \sim \alpha' E$, one gets
$\Delta x \sim [\alpha' \ln (\sqrt{\alpha'} E)]^{1/2}$, modifying the second
term in (\ref{stringup}).
Different choices putting in less energy into the higher modes
than the second case would
yield $\Delta x$ between these two values.
For all these choices it remains true that
${\Delta x}_{min} \sim \sqrt{\alpha'}$.
These two forms of the string uncertainty principle were conjectured,
respectively, by
Gross (1989) and Amati, Ciafaloni and Veneziano (1989) from studies of string
scattering amplitudes at high energies at all loops
(Gross and Mende 1988; Amati, Ciafaloni and Veneziano 1988).
It is interesting that both forms as well as intermediate ones
can be derived by elementary considerations
of string wavefunctions using different assumptions of how the
energy is distributed among the oscillator modes of the target. One can
ask, what determines the actual distribution of the probe kinetic energy
among the various oscillator modes of the target?
This needs further investigation.
The analysis of
Amati, Ciafaloni and Veneziano (1989) suggests that the scattering angle
plays a role in determining it.
{}From the above it is evident
that the smallest observable length of any object in a string universe
(where everything, objects and probes, is ultimately made of strings)
is of the order of $\sqrt{\alpha'}$. This is a direct consequence
of the new wiggle degrees of freedom of strings.
\vskip 0.3cm
\noindent {\bf Gravitational collapse, black holes, random walks}:
In the above scattering experiment if too much
energy is deposited in the higher $n$ modes of the target string, its
size can become smaller than its Schwarzchild radius and it
can suffer gravitational collapse. For example,
an interesting choice of energy distribution among the modes of the
target is to assume that it is thermal. That is,
assume that all oscillator states of the target having a total
energy $E$ are equally likely. (To avoid a violation of unitarity,
the probe, also stringy, carries away all the correlations.)
What is $\langle q \rangle$ in such
an ensemble? This question has been investigated by Mitchell and Turok (1987)
and Aharonov, Englert and Orloff (1987)
in a different context. It turns out that $\langle q \rangle \sim
\alpha'^{3/2} E$. Taking $r = \langle q \rangle^{1/2}$ to be
the size of the string, such a string would suffer gravitational
collapse if its Schwarzchild radius exceeded $r$, or its energy
exceeded $\alpha'^{-1/2} g^{-4}$ (in $3+1$ dimensions). The entropy
$S$ of this `black hole', given by $\sim \sqrt{\alpha'} E$ (see next section)
would be $S \sim \alpha' r^2$, proportional to its `area'.
The expectation value of the length of the string in the thermal
state is $\langle l \rangle \sim \alpha' E$, as for a classical
string configuration (or a cosmic string). Thus in this
state the string resembles a random walk in space with
step length $\sqrt{\alpha'}$, since $l \sim \alpha'^{-1/2} r^2$ (Mitchell and
Turok 1987; Aharonov, Englert and Orloff 1987).
\vskip .3cm
\noindent {\bf String spectrum in a compact space}:
Spacetime itself is characterized by a metric (at least on scales
familiar to us), to determine which
we must measure lengths. If the smallest measureable length is in
principle $\sim \sqrt{\alpha'}$, this must ultimately reflect on the
smallest conceivable size of the universe in string theory. To study
this more precisely, we now consider strings in a finite sized space.
Consider a toroidal compactification of space with a radius $R$,
i.e., the coordinate $x^i$ of space ($i=1,\ldots, d$) is identified
with $x^i + 2\pi w R$ with $w$ an integer. While we
describe this special case for simplicity and clarity, many of the
consequences for cosmology discussed later are valid for a much
larger class of compactifications. Then classical string configurations
have another mode, modifying (\ref{fourier}) to
\begin{equation}
\label{fourier1}
x^i(\sigma) = x^i_0 + 2L^i\sigma + \sum_{n=1}^\infty(x^i_n \cos 2n\sigma +
\tilde{x}^i_n \sin 2n\sigma),
\end{equation}
where $L^i = w^i R$ with $w^i$ an integer. As $\sigma$ runs from
$0$ to $\pi$, $x^i$ runs from $x^i(0)$ to $x^i(0) + 2\pi R w^i$, the
string therefore winds around the universe in the $i^{th}$ direction
$w^i$ times.
This adds a term $L^2/\alpha'$ to (\ref{lagrangian}), and $L^2/\alpha'^2$
to (\ref{spectrum1}). In compact space $p^i=m^i/R$ is also quantized
($m^i$ integer), and the spectrum is now given by (Green, Schwarz and Brink
1982)
\begin{equation}
\label{spectrum2}
\epsilon^2={{\bf m}^2 \over R^2} + {{{\bf w}^2 R^2} \over {\alpha'^2}}
+ (2/\alpha')[-2 + \sum_{I=1}^{d-1}\sum_{n=1}^\infty
n(N_n^I + \tilde{N}_n^I)].
\end{equation}
This spectrum maps into itself under the transformation
\begin{equation}
\label{duality}
R \rightarrow \tilde{R} \equiv \alpha'/R.
\end{equation}
This is evident from the fact that the state with quantum numbers
$({\bf m},{\bf w},\{N_n^I,\tilde{N}_n^I\})$
in a universe of radius $R$
has the same energy as the state
$({\bf w},{\bf m},\{N_n^I,\tilde{N}_n^I\})$
in a universe of radius $\tilde{R}$. The interchange ${\bf m} \leftrightarrow
{\bf w}$ together with the transformation (\ref{duality})\ does
not alter the r.h.s. of (\ref{spectrum2}). Thus at the level of the
free spectrum, string theory does not distinguish between a universe
of size $R$ and a universe of size $\tilde{R}$. This symmetry is also
respected by string interactions: the amplitude of a process
in a universe of size $R$
with a given set of external states
is the same as the amplitude in a universe of size $\tilde{R}$
of the `dual' set of states (obtained by
interchanging ${\bf m}$ and ${\bf w}$ quantum numbers for
each state in the first set). This symmetry, known as target-space
duality or `t-duality' was found by Kikkawa and Yamasaki (1984),
Sakai and Senda (1986), Nair, Shapere, Strominger and Wilczek (1987),
Sathiapalan (1987), and Ginsparg and Vafa (1987).
\vskip 0.3cm \noindent
{\bf A new periodic spatial coordinate and wavefunctions of winding states}:
In addition to the coordinate $x^i_0$ which is compact with
period $2\pi R$, there exists another spatial coordinate $\tilde{x}^i_0$
in string theory with period $2\pi \tilde{R}$ (Sathiapalan 1987).
This is just the conjugate variable to the operator
$\hat{L}^i \equiv (1/2\pi \alpha')\int_0^\pi d\sigma \ \partial_\sigma x^i$
whose eigenvalue is $L^i/\alpha'= w^i/\tilde{R}$ (just as $x^i_0$ is conjugate to
$\hat{P}^i$ whose eigenvalue is $p^i=m^i/R$). Formally, define
$|\tilde{\bf x}_0\rangle \equiv \sum_{\bf w}e^{i\hat{\bf L} \cdot \tilde{\bf x}_0}|{\bf w}\rangle$
and $\hat{\tilde{\bf x}}_0|\tilde{\bf x}_0\rangle \equiv \tilde{\bf x}_0|\tilde{\bf x}_0\rangle$,
where the sum goes over ${\bf w} \in {\bf Z}^d$ and
$|{\bf w}\rangle$ denotes $|{\bf m},{\bf w},\{N_n^I,\tilde{N}_n^I\}
\rangle$ for brevity. It follows that
$[\hat{\tilde{x}}^i_0,\hat{L}^j] = i\delta^{ij}$.
Since ${\bf w}$ is quantized on an integer lattice, it is easy to see that
$|\tilde{\bf x}_0 + 2\pi {\bf n}\tilde{R}\rangle = |\tilde{\bf x}_0\rangle$ for any
${\bf n} \in {\bf Z}^d$. I.e., the points $\tilde{x}^i_0$ and
$\tilde{x}^i_0 + 2\pi \tilde{R}$ in this `dual space' are physically
indistinguishable.
The wavefunctions are now given by
$\psi_{({\bf m},{\bf w},\{N_n^I,\tilde{N}_n^I\})}
({\bf x}_0,\tilde{\bf x}_0,\{x_n^I,\tilde{x}_n^I\}) =
\langle {\bf x}_0,\tilde{\bf x}_0,\{x_n^I,\tilde{x}_n^I\}|
{\bf m},{\bf w},\{N_n^I,\tilde{N}_n^I\}\rangle$, where the r.h.s differs from
that of (\ref{wave1}) by the factor $e^{-i{\bf p \cdot x_0}}$ being replaced
by $e^{-i({\bf m \cdot x_0}/R + {\bf w \cdot \tilde{\bf x}_0}/\tilde{R})}$. The physical
significance of this `dual position coordinate' will be discussed in
the last section.
\section{Statistical mechanics of strings at high energy densities}
\label{sec:statmech}
\noindent {\bf The partition function and the density of states}:
In order to study the very early universe in the context of string
theory, it is important to know how a very hot gas of superstrings
behaves. Consider the thermal partition function of a string gas:
\begin{equation}
\label{Z}
Z(\beta,R)=\sum_{\alpha}\ e^{-\beta E_{\alpha}(R)}.
\end{equation}
Here $\alpha = (N, a_1,\ldots,a_N)$ labels a state with $N$ strings,
the quantum numbers of the $k^{th}$ string being given by $a_k$.
Each $a_k$ in turn stands for the full set of quantum numbers
$({\bf m, w},\{N_n^I,\tilde{N}_n^I\})$ for the $k^{th}$ string. $E_{\alpha}(R)$
is the energy of the multi-string state $\alpha$ in a universe of
radius $R$, and in the ideal gas approximation is given by the
sum of the individual single string energies:
$E_{\alpha}(R) = \sum_{k=1}^N \epsilon_{a_k}(R)$
where $\epsilon_{a_k}(R)$ is given by (\ref{spectrum2}) for
closed bosonic strings. (For superstrings the formula for $\epsilon$
is modified by additional degrees of freedom but retains the
essential character needed for subsequent discussion.) The sum
over $\alpha$ includes a sum over all individual string states for
a fixed $N$ and a sum over $N$ from zero to infinity.
This partition function has a number of interesting properties.
First, it has singularities in the complex $\beta$ plane (other
than the usual $\beta=0$ singularity) even at finite volume. In
point particle field theories, singularities, which are usually signatures of
phase transitions, arise only in the thermodynamic limit. In the
string case they arise at finite volume because even a single
string has infinite degrees of freedom. The location of the right
most singularity,
$\beta_0$ ($\equiv 1/T_H$, where $T_H$ is known as the Hagedorn
temperature (Hagedorn 1965; Huang and Weinberg 1970)),
is proportional to the only length scale in the theory,
$\beta_0 =c_0\sqrt{\alpha'}$. The proportionality constant is
independent of the size of the box
(or universe) and other details of compactification
(Antoniadis, Ellis and Nanopoulos 1987; Axenides, Ellis and Kounnas 1988)
but dependent only
on the type of string theory, bosonic ($c_0=4\pi$), type II superstring
($c_0=2\sqrt{2}\pi$), or heterotic ($c_0=(2+\sqrt{2})\pi$).
As long as space is compact, the singularity is universally a simple
pole (Brandenberger and Vafa 1989; Deo, Jain and Tan (DJT) 1989a). There
is a representation of $Z(\beta)$ due to O'Brien and Tan (1987)
(see also Maclain and Roth 1987; McGuigan 1988)
which is useful in determining its analytic
structure in the complex $\beta$ plane. It turns out that there is
an infinite number of singularities to the left of $\beta_0$ (DJT 1989a)
whose locations in general depend upon the radius of universe.
For universes much larger than $\sqrt{\alpha'}$
(and also, by duality, for universes much smaller than $\sqrt{\alpha'}$),
a number of these singularities approach $\beta_0$.
Second, since the spectrum exhibits duality, so do the partition
function and density of states
$\Omega(E,R)= \sum_{\alpha}\delta (E-E_{\alpha}(R))$:
\begin{equation}
\label{Zduality}
Z(\beta,R)=Z(\beta,\tilde{R})\quad \quad
{\rm and} \quad \quad \Omega(E,R)= \Omega(E,\tilde{R}).
\end{equation}
This follows from the
fact that for every $\alpha$ there exists an $\tilde{\alpha}$
(obtained from $\alpha$ by interchanging the momentum and winding numbers
of every string in the state $\alpha$) such that
$E_{\alpha}(R)=E_{\tilde{\alpha}}(\tilde{R})$.
Third,
the behaviour of $Z(\beta)$ near $\beta_0$ is such that at temperatures
close to the Hagedorn temperature, fluctuations
are large and invalidate the use of the canonical ensemble for
deducing the thermodynamic properties of the string gas. One is forced
to use the more fundamental microcanonical ensemble, defined by $\Omega(E,R)$
(Frautschi 1971; Carlitz 1972; Mitchell and Turok 1987; Turok 1989).
Finally, since
$Z$ and $\Omega$ are related by a Laplace transform, the
leading large energy behaviour of $\Omega(E)$
is controlled by the behaviour of $Z(\beta)$ near its singularities,
and can be determined by a contour deformation technique
(DJT 1989a, 1991).
At large radius ($R \gg \sqrt{\alpha'}$) and at energy densities
above the `Hagedorn energy density' $\rho_0 \sim \alpha'^{-(\bar{d}+2)/2}$,
the density of states is given by (Deo, Jain, Narayan, Tan 1992 (DJNT))
\begin{equation}
\label{density}
\Omega(E,R) \simeq \beta_0 \ e^{\beta_0 E + a_0 V}
[ 1-\delta(E,R)], \quad \quad
\delta(E,R)= {{(\beta_0 E)^{2\bar{d} - 1}} \over {(2\bar{d}-1)!}}
e^{-(\beta_0-\beta_1)(E-\rho_0 V)}.
\end{equation}
Here we use the notation that $d$ represents the total number of
spatial dimensions, all of them compact ($d=25$ for bosonic strings
and $9$ for superstrings and heterotic strings). $\bar{d}$ is the
number of spatial dimensions that have large radius $R \gg \sqrt{\alpha'}$;
the remaining $d-\bar{d}$ dimensions are assumed to have radii $\sim
\sqrt{\alpha'}$. $V=(2\pi R)^{\bar{d}}$ is the volume of the large
dimensions. $a_0$ is a constant of order $\sim \alpha'^{-\bar{d}/2}$.
$\beta_1$ is the singularity of $Z(\beta,R)$ closest
to $\beta_0$; $\beta_0-\beta_1 \sim \alpha'^{3/2}/R^2$. The formula
(\ref{density})\ is valid for $\bar{d} > 2$ and for energy density
$\rho \equiv E/V$ greater than $\rho_0$. $\rho-\rho_0$ should be
large enough (greater than $O(\sqrt{\alpha'} R^{2-\bar{d}})$) so that
$\delta \ll 1$.
\vskip 0.3cm \noindent
{\bf Thermodynamic properties; physical interpretation in terms of
degrees of freedom}:
The thermodynamic properties of the gas are determined by (\ref{density}).
The entropy $S \equiv \ln \Omega$ is given by
\begin{equation}
\label{entropy}
S(E,R) \simeq \beta_0 E + a_0 V + \ln (1-\delta),
\end{equation}
from which one finds the temperature
$T \equiv [(\partial S/\partial E)_V]^{-1}$ to be
\begin{equation}
\label{temp}
T(E,R) \simeq T_H(1-{{\beta_0-\beta_1} \over {\beta_0}}\delta),
\end{equation}
and the pressure $p \equiv T(\partial S/\partial V)_E$
\begin{equation}
\label{pressure}
p(E,R) \simeq T_H a_0 \big( 1-\delta{{\beta_0-\beta_1}\over \beta_0}
[1+{{\beta_0\rho_0}\over{a_0 \bar{d}}}
(2{\rho \over \rho_0} + \bar{d} - 2)]\big) .
\end{equation}
Thus both the temperature and pressure of the string universe
reach asymptotic values determined by the string length scale
$\alpha'$ at energy densities above Hagedorn; corrections to
these asymptotic values are exponentially suppressed above these
energy densities. The physical reasons for this are as follows.
The leading contribution to the density of states of a string gas grows as
the exponential of a linear function of $E$,
unlike for a gas of point particles where it grows exponentially
with a sublinear function. This is because
the number of oscillator states at a fixed large value of
$\bar{N}\equiv \sum_{I=1}^{d-1}\sum_{n=1}^\infty
n(N_n^I + \tilde{N}_n^I)$, grows as
$\sim e^{c_1\sqrt{\bar{N}}}$.
This is just the Hardy-Ramanujam asymptotic formula for
the number of partitions of a large positive integer $\bar{N}$
into non-negative integers, a result in number theory. Thus, even
for a single string the density of states grows exponentially
$\sim e^{\beta_0 \epsilon}$ with energy (since $\sqrt{\bar{N}} \sim
\sqrt{\alpha'} \epsilon$ from (\ref{spectrum2})). By contrast the contribution to
the density of states from the momentum and winding modes is
very small. E.g., for a single particle, for which only momentum modes
contribute, the density of states grows only as a power $\epsilon^{\bar{d}-1}$.
Thus at large energies it is entropically
favourable for the energy to go into oscillator modes rather than
momentum or winding modes. A term in the entropy of the gas that is
linear in energy
gives rise to a constant, i.e., energy independent temperature.
The form of the subleading corrections
(which is due to an interplay between oscillator, momentum and
winding modes) tells us that $T_H$ is
an upper limiting temperature.
Figure 1 displays the behaviour of temperature as
a function of energy for an ideal gas of strings and
contrasts it with an ideal gas of point particles.
The reason for the asymptotic pressure is as follows:
The leading term $\beta_0 E$ in the entropy
does not contribute to the pressure because it is independent
of the volume at constant energy; this is because the
oscillator mode contribution
to the energy (\ref{spectrum2})\ is volume independent.
The second term, $a_0 V$, should be
interpreted as the contribution of momentum modes.
It is proportional to volume
just like for a gas of point particles, due to the translational
degrees of freedom. (A pure winding mode gas
by contrast will give a contribution proportional to $1/V$,
which is subleading for large radii.)
For an ordinary point particle gas the entropy also depends upon
the energy density $S \simeq c_2 [\rho^{d/(d+1)}]V$, from which the
usual expression $p = \gamma \rho$ with
$\gamma = 1/d$ follows. In the string
case above, the coefficient of $V$ is just a constant, $a_0$.
The physical reason is that above Hagedorn energy density,
the energy density in momentum modes is a constant independent
of the total energy density. If more energy is pumped into
the box, it goes primarily into oscillator modes, which
are entropically favoured, than into momentum modes. Conversely, if
some energy is taken out of the box (keeping the total density still
above Hagedorn), it is primarily extracted from the oscillator modes
keeping the energy in the momentum modes essentially the same. Equivalently,
if one expands the volume slightly keeping energy the same
(this is what is implied by the derivative $(\partial/\partial V)_E$),
energy flows from the oscillators to the momentum modes to keep the
energy density in the latter constant. Thus the energy density in
momentum modes (which are the contributors to pressure, consisting
of small strings bouncing around like particles) is independent of
the volume or the total energy density (as long as the latter is
above Hagedorn) and hence is the pressure.
The above argument seems to be consistent with our present picture
of how the total energy of the gas is distributed among various
strings.
In the energy and radius domain under discussion, the
string gas can be considered to be consisting of broadly
speaking two `components', of energies $E_1$ and $E_2$
with $E=E_1+E_2$.
The first component consists of a few ($\sim \ln [R/\sqrt{\alpha'}]$)
large strings which capture most of the energy of the gas
($E_1 \gg E_2$ provided $E \gg \rho_0 V$).
`Large' strings are those whose energies are $O(R^2 \alpha'^{-3/2})$
or greater. Most of their energy is due to oscillator modes and
the wavefunctions of these strings spread across the
whole universe (recall from the previous section that the size
of a thermal string of energy $\epsilon$ is $\sqrt{\alpha'^{3/2} \epsilon}$,
hence spread is $\sim R$ for $\epsilon \sim R^2 \alpha'^{-3/2}$).
If one adopts a classical picture, the universe is stuffed with
space filling brownian walks (see Salomonson and Skagerstam 1986;
Mitchell and Turok 1987). The second
component has fixed total energy $E_2 \sim \rho_0 V$ and
consists of many ($\sim V\alpha'^{-\bar{d}/2}$) small strings
`Small' ranges in size
from $O(\sqrt{\alpha'})$ to $< O(R)$, and in energy from
zero to $< O(R^2\alpha'^{-3/2}$). A crucial property of the
gas is that as more energy is pumped into
the box, it goes into the first component, leaving $E_2$ fixed.
This was qualitatively anticipated by Frautschi (1971), Carlitz (1972),
Mitchell and Turok (1987), Aharanov, Englert and Orloff (1987), and
Bowick and Giddings (1989), and made
quantitatively explicit in DJT (1989b, 1991) and DJNT. This picture is
unaltered
by the introduction of conservation laws for the total winding number
and momentum, even though additional subleading terms arise in the
density of states.
\vskip 0.3cm \noindent
{\bf Duality; thermodynamics in small spaces}:
We have so far discussed the case of large $E$ and large radius $R
\gg \sqrt{\alpha'}$. What happens at very small radii?
This is immediately answered by duality. At $R \ll \sqrt{\alpha'}$,
the r.h.s. of (\ref{density})\ has the same form but with $V$ replaced by
$\tilde{V} \equiv (2\pi\tilde{R})^{\bar{d}}$ (now $\beta_0-\beta_1
\sim \alpha'^{3/2}/{\tilde{R}^2}$). The same is therefore true of
temperature and pressure (we now define
$p \equiv T(\partial S/\partial \tilde{V})_E$).
At $R \sim \sqrt{\alpha'}$ we find
that the leading behaviour of the density of states is still given
by (\ref{density}), but now $V$ is replaced by a slowly varying
function of $R$ of order $\alpha'^{\bar{d}/2}$, and $\beta_0-\beta_1
\sim \sqrt{\alpha'}$ (DJT 1989a, DJNT). The
temperature as a function of $E$ is still given by (\ref{temp})\
with these replacements. However the pressure needs to be
appropriately defined and interpreted in this domain
(since $S(E,R)$ has to have an extremum at the duality radius,
both definitions $p \equiv T(\partial S/\partial V)_E$
and $p \equiv T(\partial S/\partial \tilde{V})_E$ imply that
$p$ passes through a zero at $R = \sqrt{\alpha'}$).
\vskip 0.3cm \noindent
{\bf Inconsistency of string thermodynamics in non-compact spaces}:
Finally we remark that
string thermodynamics seems to be internally consistent
only in a compact space. The reason is that in a noncompact
space to define the density of states we have to consider an
artificial box of large volume $V$ to confine the gas and
later take the thermodynamic limit. This is problematic
in string theory because strings are extended objects,
they can in principle extend from one wall to another, and
render the entropy inextensive. One can see the problem
explicitly at high energy densities $E > \rho_0 V$ when
there exist a few strings in the gas whose individual
energy is a significant fraction of the total energy $E$.
The spread of their wavefunction is therefore $\sim \sqrt{\alpha'^{3/2} E}$.
Let the number of large dimensions (of radius $R \gg \sqrt{\alpha'}$) be $\bar{d}$.
Then since $E > \alpha'^{-(\bar{d}+1)/2}R^{\bar{d}}$, these
strings have a size $\alpha'^{(2-\bar{d})/4} R^{\bar{d}/2}$.
Thus for $\bar{d} > 2$ these strings have a spread much greater
than $R$, the size of the universe itself. In a compact universe
this is not a problem; the string can wrap around the universe
many times. But if the universe were to be noncompact in these
$\bar{d}$ directions, then we find that these strings hit
the walls of the artificial box with nowhere to expand, leading
to an inconsistency of interpretation (see also DJNT).
\section{Implications for superstring cosmology and initial singularities}
\label{sec:cosmology}
\noindent {\bf Absence of a temperature singularity}:
We now discuss how the above considerations might impinge on cosmology.
Let us follow our present universe (assumed compact in all dimensions
but with three large dimensions)
backwards in time according to the standard model of cosmology.
At the epoch where the energy density in the large dimensions
is above $\rho_0 \sim \alpha'^{-2}$ but the radius is still much greater
than $\sqrt{\alpha'}$
(this is quite natural in the standard model at early epochs),
let us assume that
the standard model physics is replaced by string theory,
and use the ideal gas approximation (\ref{density}). Then as
we proceed to smaller radii and hence higher energy densities, the temperature
and pressure being governed by equations (\ref{temp})\ and (\ref{pressure})
no longer increase indefinitely (as they would in any point particle
theory) but flatten out. The temperature remains flat as $R$ approaches
$\sqrt{\alpha'}$ and well into the domain $R \ll \sqrt{\alpha'}$
(as long as $E > \rho_0 \tilde{V}$ or $\tilde{\rho} \equiv E/\tilde{V} >
\rho_0$).
As $R$ declines further (i.e., $\tilde{R}$ increases) the temperature
{\em falls}. This is shown in figure 2. The behaviour of temperature
as a function of radius (at fixed energy or fixed entropy) is symmetric
about $R = \sqrt{\alpha'}$. At very small radius it does not diverge
as it does for a universe made of elementary point particles, but
behaves just as for a very large universe. The string universe has
no temperature singularity.
\vskip 0.3cm \noindent
{\bf Physical interpretation of a small universe}:
What is the physics of this bizarre behaviour? This was discussed
by Brandenberger and Vafa (1989), even before the precise expression
(\ref{density}) for the density of states was known. They asked the
question: how would one measure the size of the universe if it were
very small? For a large periodic box one can imagine sending a light signal
(a localized photon wave-packet) and measuring the time it takes to come back.
But this experiment would fail in very small box. The energy of a
momentum mode goes as $m/R$, and a superposition of many such modes
is needed to create a localized wave-packet, thereby making it more
and more energetically difficult to send a wave-packet in smaller universe.
In string theory the photon is a massless state with some momentum quantum
number $\bf m$, winding number zero, and a single oscillator excitation
(the term $[-2 + \bar{N}]$ in (\ref{spectrum2}), or its analogue for
heterotic strings, is zero). In today's universe (assumed large) these
are easily excited, but it would be energetically very difficult
to create photons and send them around in a universe of size
$R \ll \sqrt{\alpha'}$ (see (\ref{spectrum2})). On the other hand, in
a very small universe, particles `dual' to the photon, with quantum
numbers ${\bf m}=0$, some winding number {\bf w}, and the same oscillator
quantum numbers as the photon would be easily excited. Indeed these
would constitute the `light' particles of the very small universe. An
observer in this very small universe would hardly think of sending
photons to measure the size of his universe (just as we would not
contemplate sending winding modes around); he would use a superposition
of the `dual photon' modes. By sending such modes he would be measuring
the extent of the `dual position coordinate' $\tilde{x}^i$ (recall that
$\tilde{x}^i$ is to winding
modes what position $x^i$ is to momentum modes). But, as discussed
earlier, that extent is just ${2\pi \tilde{R}}$; hence observers in a universe
of size $R \ll \sqrt{\alpha'}$ would find its radius to be not $R$ but
$\tilde{R} = \alpha'/R \gg \sqrt{\alpha'}$.
Indeed in a universe with $R \ll \sqrt{\alpha'}$ all momentum modes
would be energetically difficult to excite. Everything - signals,
apparatus, observer - would be made from particles that have zero
{\bf m} quantum number (in our present large universe everything is made
of zero {\bf w} quantum number). Since string theory has
duality as a symmetry of the spectrum as well as the interactions,
the dual particles would interact with each other exactly the way
normal particles do in our present universe. The observers in a very
small universe would not therefore know that they are in a universe
much smaller than $\sqrt{\alpha'}$, their physics would be identical
to ours (and for that matter nor do we know whether our universe
is very large or very small compared to $\sqrt{\alpha'}$).
It is therefore no surprise that temperature has the behaviour shown
in figure 2. As radius goes much smaller than $\sqrt{\alpha'}$,
the universe actually {\em expands, as seen by the modes
that are excited
in it}. This also makes it evident that there are no physical singularities
in the energy density, pressure or curvature as $R \rightarrow 0$. In a very
small universe, the physical energy density is not $E/V$ but
$E/\tilde{V}$ (which goes to zero and not infinity as $R \rightarrow 0$),
since the physical volume of the universe is $\tilde{V}$.
In string theory the smallest {\em physical} size of the universe
is $\sqrt{\alpha'}$.
Note that the arguments leading to the string uncertainty
principle - that the smallest observable size of an elementary string is
$\sqrt{\alpha'}$ - and the arguments leading to the same minimum physical size of the
universe both make essential use of probes in thought experiments. Also
note the difference: while the former argument uses
the oscillator modes, the latter rests on the duality between
momentum and winding modes (although the limiting temperature and pressure
depend again on the oscillators). All these
modes are simultaneously forced upon us as soon as we accept strings as the
elementary constituents of nature, and all are governed by a single scale
parameter that appears in (\ref{action1}).
\vskip 0.3cm \noindent
{\bf A cosmological scenario without initial singularities}:
Brandenberger and Vafa sketch the following scenario.
Let us assume that
at some point in the future our universe stops expanding and starts
contracting and heating up. As the energy density increases to the Hagedorn
energy density, stringy effects will take over and the temperature
will flatten out. If it continues to contract through the duality
radius and
comes out the `other side', then dual (analogues of winding) modes
will take over. The universe will cool
and `expand' and give rise to dual nucleons, galaxies, stars, planets, life,
etc.
What appears to us to be the `big crunch' will
be a `big bang' for the dual observers. The process could repeat
giving rise to an oscillatory universe. `Our own' big bang was just one
such periodic occurrence.
Of course much more work is needed to justify any such {\em dynamical}
scenario. We have been concerned with just those aspects which hinge
only on the {\em degrees of freedom}. A body of literature
now exists which also deals with the time evolution of the metric
and other low energy modes in string theory
in the cosmological context (see Tseytlin and Vafa (1992), Gasperini (1997),
the contribution by Bose (1997) to these proceedings, and references therein).
Perhaps it would be worthwhile to revisit some of this
in the light of the expression for the pressure of a string gas
presented here, since pressure as part of the energy momentum tensor
is a source in the field equations.\footnote{This suggestion arose in
discussions with S. Kalyana Rama.}
Nevertheless the above scenario is important in that it at least
allows us to {\em imagine} how initial singularities might be avoided in
string theory.
It is important to emphasize that singularities are avoided not by
recourse to quantum gravity (spacetime has all along been treated
classically) but simply by a reinterpretation of what it means to
talk of a small universe in the light of string theory.
In point particle theories, classical imagination fails at $R=0$.
This is an example of how the new
degrees of freedom in string theory allow
(or rather, necessitate) a new perspective on our ideas of spacetime,
in this case specifically on our notion of the size of the universe.
It should be mentioned that while we have explicitly discussed
the case of a toroidal compactification for simplicity, the t-duality
symmetry which makes this reinterpretation possible holds for
a much larger class of string models (and is expected to be a
symmetry in a nonperturbative formulation of string theory).
A limiting temperature and pressure in the ideal gas approximation
also seem to be a universal feature of strings in compact
spaces.
\vskip 0.3cm \noindent {\bf Cautionary remarks}:
At this point some caveats are in order. Thermodynamics
in the presence of gravity must take into account the
Jeans instability. At constant energy density a sufficiently
large volume will be susceptible to gravitational collapse. This
places an upper limit on the value of $R$ for which our thermodynamic
considerations are valid. Second, the results are
based on an ideal gas approximation, used in a regime of
high energy densities, greater than the string energy scale itself.
This is justified only if the coupling is weak ($g \ll 1$).
Even for a fixed weak coupling the approximation can be expected to break
down at sufficiently high energy densities, at which point
non-perturbative effects will need to be taken into account.
This places a lower limit on $R$ for the validity of the
approximation. Thus there is possibly a window $R \in (R_1,R_2)$,
$\sqrt{\alpha'} < R_1 \leq R_2$ (and the `dual window' $\tilde{R} \in (R_1,R_2)$)
in which one can expect this to be valid. The
window expands in both directions as coupling becomes weaker
(see Atick and Witten (1988) for related arguments).
In addition to string interaction and nonperturbative effects in the
region of $R$ close to $\sqrt{\alpha'}$, we face the uncertainty of interpretation
of spacetime itself at such small scales. For sufficiently large
(or sufficiently small) $R$, spacetime may be treated classically,
as we have done. But this is questionable
near the duality radius. This is the regime where the universe as
well as its elementary constituents have the same `size'.
This problem awaits a better understanding
of spacetime in string theory.
\vskip 0.3cm \noindent
{\bf Possible observational consequences, further questions}:
Assuming that there was an era in the past where
the ideal string gas approximation was valid, could there be
some observable relic? From the picture of how energy is distributed
in the string gas it seems likely that density fluctuations
would have a different character in the stringy era, and as seeds
for later structure formation could have observable consequences.
Second, it would be interesting to look for signatures of compactness at
very large scales in the universe.\footnote{I thank T. Souradeep
for informing me that such analyses of the data are possible.}
Apart from a resolution of the singularity problem,
string thermodynamics also seems to be internally consistent only
in compact spaces. A compact universe is even otherwise natural in string
theory, since the extra dimensions in any case have to be compact.
At a more theoretical level, it may be worthwhile to investigate
dynamical mechanisms based on string modes (see Brandenberger and
Vafa (1989) for a proposal) for why only three spatial dimensions
are large. Also it is of interest to study how the recent progress
in our understanding of some non-perturbative aspects of string theory
affects the above considerations.
Note added: After this writeup was submitted for the proceedings, I
became aware of other papers (Yoneya 1989; Konishi, Paffuti and
Provero 1990; Kato 1990; Susskind 1994) which attempt to derive the
string uncertainty principle. The argument presented in the
present article is different from those given in these papers.
Other recent references of related interest are Li and Yoneya 1997,
which argues that the string uncertainty principle is consistent
with D-brane dynamics, as well as Barbon and Vazquez-Mozo 1997 and
Lee and Thorlacius 1997, which attempt to include D-branes within
superstring statistical mechanics. For literature on a minimal length
in the context of quantum gravity without invoking string theory see
references in the review by Garay 1995, as well as Padmanabhan 1997.
I thank N. Deo, C-I Tan and C. Vafa for discussions in which most
of my understanding of string thermodynamics and cosmology was
developed, S. Kalyana Rama for getting me interested in the pressure
of an ideal string gas and pointing out some recent references,
and the participants and organizers of
the Conference on Big Bang and Alternative Cosmologies for a
stimulating meeting.
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Carea rhodophila är en fjärilsart som beskrevs av Walker 1865. Carea rhodophila ingår i släktet Carea och familjen trågspinnare. Inga underarter finns listade i Catalogue of Life.
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rhodophila | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,225 |
{"url":"https:\/\/math.stackexchange.com\/questions\/2737447\/show-operator-is-well-defined","text":"# Show operator is well defined\n\nI'm supposed to show that the following operator\n\n$$T:L^{1}(0,2)\\rightarrow L^{1}(0,2),\\ (Tf)(x):=\\int_{0}^{x}tf(t)\\ \\text{d}t$$\n\nis well defined. The first thing I'm supposed to do is to make sure that $(Tf)(x)$ makes sense, i.e.,\n\n$$\\int_{0}^{x}|tf(t)|\\ \\text{d}t<\\infty\\ \\ \\text{for all }f\\in L^{1}(0,2),\\ x\\in [0,2]$$\n\nHere's my idea: I can manipulate a little with the above expression\n\n$$\\int_{0}^{x}|tf(t)|\\ \\text{d}t\\leq \\int_{0}^{2}|t||f(t)|\\ \\text{d}t=\\int_{0}^{2}t|f(t)|\\ \\text{d}t$$\n\nBecause we're integrating from 0 to $x$ and over $|tf(t)|$, then the first inequality in the above expression makes sense. Furthermore, if we integrate from 0 to $2$, then $|t|=t$, since we have positive limits of integration.\n\nBut after here, I get stuck. Any hints on how to proceed would be highly appreciated!\n\n\u2022 As for the convergence of the integral, notice that $2|f(t)|$ is integrable and dominates $tf(t)$ on $[0, 2]$. To show that $Tf$ is $L^1$, Fubini's theorem may help you. \u2013\u00a0Sangchul Lee Apr 14 '18 at 23:12\n\u2022 I think we can safely say that $\\int_0^2 t|f(t)|dt \\leq \\int_0^2 2|f(t)|dt .$ \u2013\u00a0Dalamar Apr 14 '18 at 23:14\n\u2022 That makes sense. Thanks for the help! \u2013\u00a0James Apr 15 '18 at 16:13\n\nHint: $x\\in[0,2]$ and so $t\\in[0,x]\\subset[0,2$. Thus $$\\int_{0}^{x}|tf(t)|\\ \\text{d}t\\leq \\int_{0}^{2}|t||f(t)|\\ \\text{d}t\\le2\\int_{0}^{2}|f(t)|\\ \\text{d}t<\\infty.$$","date":"2021-01-16 22:08:20","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\": 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\": 0.954676628112793, \"perplexity\": 159.36015313609187}, \"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-2021-04\/segments\/1610703507045.10\/warc\/CC-MAIN-20210116195918-20210116225918-00458.warc.gz\"}"} | null | null |
Q: Converting Incoming Byte Array File Type with FFMPEG - C# ASP.NET Using FFMPEG (FFlib .NET Library), is it possible to convert a stored byte array into another byte array, using the API rather than CLI, resulting in a different file type? I.e. .mov to .mp4.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,375 |
Q: How to check for existing meta tag before generating a new one with javascript (or jquery)? I'm developing a widget that creates basic SEO tags that could already exist on a page so I'm wanting to make sure the widget checks for existing tags before the new tag is generated.
For example, one of the tags that the widget produces is
<meta name="description" content="Description of the webpage"/>
Since this tag is one that most pages already have, is there a way to use javascript to check for the tag before writing it?
A: Here is the plain Javascript solution (no libraries):
Check for the meta description element using:
document.querySelector("meta[name=description]");
If found, access its content attribute using .getAttribute("content").
Demo:
if(document.querySelector("meta[name=description]")){
if(document.querySelector("meta[name=description]").getAttribute("content") === "Description of the webpage")
console.log("meta description with content 'Description of the webpage' found");
// do something
}
else{
console.log("meta description not found");
// do something
}
<meta name="description" content="Description of the webpage"/>
Read up:
*
*How to check if element exists in the visible DOM?
*How do I get the information from a meta tag with javascript?
Source
A: Sure its possible. Just use something like:
var description = $('meta[name=description]').attr("content");
Check the fiddle: https://jsfiddle.net/034ghy1y/
A: This will return the first meta tag found on a page.
var tag = document.getElementsByTagName("meta")
if(tag[0]) {
console.log('found ' + tag[0]);
} else {
console.log("Not found")
}
http://plnkr.co/edit/Gj9x5l8yS9xh2BoWkAm8?p=preview
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,574 |
\section{Introduction}
\label{sec_intro}
\input{intro}
\section{Preliminaries}
\label{sec_prelim}
\input{defs}
\subsection{Parity Games}
\label{subsec_paritygames}
\input{paritygames}
\subsection{Delay Games}
\label{subsec_delaygames}
\input{delaygames}
\subsection{Prompt LTL}
\label{subsec_prompt}
\input{prompt}
\subsection{The Alternating-color Technique}
\label{subsec_altcolor}
\input{altcolor}
\section{Delay Games with Prompt-LTL Winning Conditions}
\label{sec_promptdelaygames}
\input{promptdelaygames}
\section{Lower Bounds for LTL and Prompt-LTL Delay Games}
\label{sec_lowerbounds}
\input{lowerbounds}
\subsection{Lower Bounds on Lookahead}
\label{subsec_lowerbounds_la}
\input{lowerbounds_la}
\subsection{Lower Bounds on the Bound \boldmath$k$}
\label{subsec_lowerbounds_k}
\input{lowerbounds_k}
\subsection{Lower Bounds on Complexity}
\label{subsec_lowerbounds_cc}
\input{lowerbounds_cc}
\section{Delay Games on Non-deterministic, Universal, and Alternating Automata}
\label{sec_nonaltuniv}
\input{omegaregular}
\vspace{-2.8em}
\section{Conclusion}
\label{sec_conc}
\input{conclusion}
\bibliographystyle{splncs03}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,752 |
{"url":"https:\/\/diracseashore.wordpress.com\/2009\/08\/24\/","text":"Feeds:\nPosts\n\n## The correlation length of\u00a0notation\n\nWell, I\u2019m buried up to my head in work trying to finish up a paper. I have been doing this for a while. A project that should have ended with a 20 page paper ballooned on me, and now I am writing a paper on the edge of 50 pages.\n\nWhich brings me to the title of the post. Fifty pages is a long paper. There is an additional problem in writing such a long paper. This is that by the time I\u2019m writing the end, I\u2019ve somewhat forgotten what I\u2019ve written before. Mre specifically, not so much the content, but the way in which it was written. And even more specifically, what precise notation was used.\n\nFor example, just to give you a feel,\u00a0 was it ${\\mathbb C}$ or ${\\bf C}$, or was it ${\\cal C}$? Or was it $A,B$, rather than $W,Z$.\n\nI\u2019m writing stream of consciousness, so it can\u00a0 be fixed later when I\u2019m combing through the paper. Moreover, the longer the paper, the more symbols and fonts one needs, and they can\u00a0 start overlapping. This means that notation degenerates as I\u2019m writing a paper. It\u2019s not consistent for more than I can work on in one day. About five pages that is. So that is what I will call the correlation length of notation: the amount of pages that one writes before the notation mutates and starts getting disordered.\n\nSo, on a 50 page paper, there are ten correlation lengths of notation, so the end looks nothing like the beginning in terms of notation. This gets worse with more authors. And don\u2019t even get me started on writing books. To fix this, one has to \u2018cool down the system\u2019 so that it becomes ordered (I\u2019m making an analogy with ferromagnetism here). This requires time and many passes. So a paper of length $m$ seems to take of order of $m^2$ time to write it down. Maybe that critical exponent is different.\n\nThere is always a plan B: give a guide to notation changes so that the work is piled on the reader. What do you think of this strategy? This seems to be the way for books, because there are many conventions that overlap: they come from different developments by different authors. Fortunately our brains seem to be able to read contextually, and $E$ can be energy, and electric field and $e$ can be the electric charge and the Euler constant all of them in the same equation, when it becomes obvious how to interpret it. Isn\u2019t it?","date":"2020-02-23 20:30:36","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\": 0, \"img_math\": 9, \"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\": 0.8440650701522827, \"perplexity\": 377.0670972448275}, \"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-2020-10\/segments\/1581875145839.51\/warc\/CC-MAIN-20200223185153-20200223215153-00010.warc.gz\"}"} | null | null |
When using our Contact form we collect the following information which is directly provided by the user, their name, email address, contact number and message itself. The information provided is only used to reply or for direct contact with the user. This information is never used by third party services or sold. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,966 |
{"url":"https:\/\/economics.stackexchange.com\/questions\/32639\/regarding-the-expenditure-function-underlying-a-bliss-point","text":"# Regarding the Expenditure Function Underlying a Bliss Point\n\nI've been looking at expenditure systems and have been really interested in the behaviour of the demand system that underlies bliss points:\n\nConsider the bliss point utility function of the following form:\n\n$$U(x_1,x_2)=-(x_1-\\delta_1)^2-(x_2-\\delta_2)^2$$\n\nfor two dimensions the corresponding hicksian demands are:\n\n$$x_1^c=\\delta_1-\\left[\\frac{\\bar{U}}{1+\\frac{p_2}{p_1}}\\right]^\\frac{1}{2}$$ $$x_2^c=\\delta_2-\\left[\\frac{\\bar{U}}{1+\\frac{p_1}{p_2}}\\right]^\\frac{1}{2}$$\n\nIt follows that the expenditure function is: $$e(p_1,p_2,\\bar{U})=p_1\\delta_1-p_1\\left[\\frac{\\bar{U}}{1+\\frac{p_2}{p_1}}\\right]^\\frac{1}{2}+p_2\\delta_2-p_2\\left[\\frac{\\bar{U}}{1+\\frac{p_1}{p_2}}\\right]^\\frac{1}{2}$$\n\nObviously expenditure functions can be much larger. however I'm having a hard time for generating a expenditure function for a number of $$n$$ goods.\n\ntldr What would the hicksian demands look like for the utility function: $$U(\\mathbf{x})=-\\sum_{i=1}^n(x_i-\\delta_i)^2$$","date":"2019-11-12 01:43:30","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\": 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\": 6, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9500274062156677, \"perplexity\": 1481.834439269936}, \"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-2019-47\/segments\/1573496664469.42\/warc\/CC-MAIN-20191112001515-20191112025515-00322.warc.gz\"}"} | null | null |
\section{Introduction}
\label{sec:intro}
The Large Area Telescope \citep[LAT;][]{atwood09} aboard the \textit{Fermi Gamma-ray Space Telescope}
has discovered a large number of $\gamma$-ray point sources, of which many are unidentified or even
unassociated with any known potential counterpart \citep{Ackermann2012}. The LAT can localize most of these sources well enough that they can be covered in a single pointing with the large primary beams of radio telescopes at low-frequencies, allowing them to be searched efficiently. Targeted radio searches of unassociated LAT point sources by the \textit{Fermi} Pulsar Search Consortium (PSC) have resulted in the discovery
of 43 radio millisecond pulsars \citep[MSPs;][]{Ray12}.
MSPs are thought to evolve from normal pulsars in binary systems via transfer of
angular momentum from companions. Thus, the majority of MSPs are naturally expected to be
in binaries ($\sim$ 83\% being the binary fraction for MSPs in the Galactic field\footnote{\url{http://astro.phys.wvu.edu/GalacticMSPs/}}). Binary systems where the pulsar wind
evaporates the companion are one way to form isolated MSPs.
Such systems where the interaction is ongoing are called black-widow (BW) pulsars. Many exhibit long eclipses ($\sim$ 10\%
of the orbital period, apparently larger than the companion's Roche lobe) that are believed to be caused by the material blown from the
very low mass companion ($M_c \ll 0.1 M_{\odot}$) by the pulsar wind.
There were two such eclipsing BW systems in the Galactic field known before
the launch of \textit{Fermi} $-$ PSR B1957$+$20 \citep{Fruchter88} and PSR J2051$-$0827
\citep{Stappers96}. The BW pulsars are found to have higher values of spin-down energy-loss rate
($\dot{E} \sim$ 10$^{34}$ erg s$^{-1}$) compared to other MSPs, making these systems good candidates for pulsed
$\gamma$-ray emission \citep{Mallory2011}.
Among 43 new MSPs found in \textit{Fermi}-directed searches there are at least 10 BWs \citep{Ray12}.
This Letter describes the discovery and follow-up study of an eclipsing BW MSP,
J1544+4937, with the GMRT.
\section{Observations and search analysis}
\label{sec:obs_analysis}
As a part of the PSC search effort, we observed mid- and high-Galactic-latitude unassociated
\textit{Fermi} point sources
with the GMRT at 607 MHz. The GMRT Software Back-end \citep[GSB;][]{Roy10} produces
simultaneous incoherent and coherent filter-bank outputs of 512$\times$0.0651 MHz sampled every 61.44 $\mu$s.
The wider incoherent beam of the GMRT (40\arcmin~at 607 MHz) can easily cover error-circles associated
with the \textit{Fermi} sources. In addition a coherent beam that is $3\times$ more sensitive and narrower (1.5\arcmin~at 607 MHz
using the central core of the GMRT) can be useful if the pulsar happens to be near the pointing center.
One of the targets was \textsc{Fermi J1544.2+4941}, a $\gamma$-ray source from an unpublished internal source list created
by the LAT Collaboration using 18 months of data in preparation for the 2FGL catalog \citep{Nolan2012}. The source
location (J2000) from that analysis (and used for our telescope pointing) was R.A. = 236\degr.074, Decl. = 49\degr.695, with
a 95\% confidence error-circle of radius 9.5\arcmin. This source is very weak, with a likelihood test statistic \citep[TS;][]{Mattox1996} of 26.2
in the 18 month analysis, and did not make the significance cut to be included in the 2FGL catalog itself.
We processed the data on an IUCAA HPC cluster with Fourier-based acceleration search methods using PRESTO \citep{Ransom02}.
We investigated trial dispersion measures
(DMs) ranging from 0 pc cm$^{-3}$ up to 350 pc cm$^{-3}$. A linear drift of up to 200 Fourier-frequency bins for the
highest summed harmonic was allowed. The powerline, 50 Hz, and its subsequent harmonics were
excised. Using parameters of 32 MHz bandwidth, 10\% duty-cycle, incoherent array gain of 2.3 K/Jy,
for 30 minutes of observing, we estimate the search sensitivity as (92K $+T_\mathrm{sky}$)/(335K) mJy for a 5$\sigma$
detection at 607 MHz. Considering $|b|>$ 5\degr, where $T_\mathrm{sky}$ $\sim$ 10--45 K, our search sensitivity is 0.3--0.4 mJy.
In a 30-minute pointing on 2011 February 1, towards \textsc{Fermi J1544.2+4941}\, we discovered a binary MSP of period 2.16 ms with
significant acceleration of 2.25 m s$^{-2}$ at a DM of 23.2 pc cm$^{-3}$.
\section{Follow-up timing}
\label{sec:timing}
We localized J1544$+$4937 with an accuracy of 5\arcsec~(positions listed in Table~\ref{tab:params}) using continuum
imaging for the full GMRT array followed by multi-pixel beamforming \citep{Roy12}, which allowed us to have sensitive
follow-up studies using the coherent array.
We estimate a flux of 5.4 mJy at 322 MHz, and a spectral index of $-$2.3.
We started the regular timing campaign for J1544$+$4937 in April 2011 at 322 MHz with the same coherent filter-bank.
With the derived position from the multi-pixel search and an \textit{a priori} binary model predicted by \cite{Bhattacharyya08},
we obtained phase-connected time-of-arrivals (TOAs) from TEMPO\footnote{\url{http://tempo.sourceforge.net}}, using the
JPL DE405 solar system ephemeris \citep{Standish04}.
The binary timing model used is ELL1 \citep{Lange01}, since J1544$+$4937 is in a very low eccentricity system.
This MSP is in a very compact binary with an orbital period of 2.9 hours. We derive a minimum companion mass (for 90\degr\ orbital
inclination) of 0.017 M$_{\odot}$ using the Keplerian mass function, assuming a pulsar mass
of 1.4 M$_{\odot}$.
J1544$+$4937 is eclipsed for about 13\% of the orbit at 322 MHz (Sec.\ref{sec:eclipse_ch}). The best-fit timing model
(MJD 55680.927--56332.90) is obtained excluding the TOAs around the eclipse phase (0.05$-$0.35). We achieved a
post-fit rms timing residual of 6.9 $\mu$s from 652 days of timing (Fig.~\ref{fig:residual}). There are still un-modeled residuals,
which can be partially absorbed by proper motion fit. However the inclusion of proper motion reduces the LAT detection significance,
indicating that more timing data are required to improve the model.
We estimate a precise DM equal to 23.2258(11) pc cm$^{-3}$ by a timing fit using 322 and 607 MHz TOAs from non-eclipsing binary phases. Ephemeris,
position and derived parameters are listed in Table~\ref{tab:params}.
\begin{figure}[htb]
\includegraphics[width=4.0in]{Fig1.pdf}
\caption{Post-fit timing residuals of J1544$+$4937 considering non-eclipsing binary phases. \label{fig:residual}}
\end{figure}
\begin{deluxetable}{ll}
\tabletypesize{\scriptsize}
\tablewidth{0pt}
\tablecaption{Parameters of J1544$+$4937
\label{tab:params}}
\tablehead{
\colhead{Parameter} & \colhead{Value\tablenotemark{a}}
}
\startdata
\hline
\multicolumn{2}{c}{Interferometric position\tablenotemark{b}} \\
\hline
Right ascension (J2000)\dotfill & 15$^\mathrm{h}$44$^\mathrm{m}$04\fs166$\pm$0\fs3\\
Declination (J2000)\dotfill & +49\degr37\arcmin57\farcs45$\pm$4\farcs7\\
Offset from survey beam center\dotfill & 4.3\arcmin \\
\hline
\multicolumn{2}{c}{Parameters from radio timing} \\
\hline
Right ascension (J2000)\dotfill & 15$^\mathrm{h}$44$^\mathrm{m}$04\fs48722 (2)\\
Declination (J2000)\dotfill & +49\degr37\arcmin55\farcs2545 (2) \\
Position Epoch (MJD)\dotfill & 51544.0 \\
Pulsar period $P$ (ms)\dotfill & 2.15928839043289 (5) \\
Pulsar frequency $f$ (Hz)\dotfill & 463.11553585462 (1)\\
Frequency derivative $\dot{f}$ (Hz s$^{-1}$)\dotfill & $-$6.29 (1)$\times$10$^{-16}$ \\
Period Epoch (MJD)\dotfill & 56007.0\\
Dispersion measure $DM$ (cm$^{-3}$ pc)\dotfill & 23.2258 (11)\\
Binary model\dotfill & ELL1\\
Orbital period $P_{b}$ (days)\dotfill & 0.1207729895 (1) \\
Projected semi-major axis $x$ (lt-s)\dotfill & 0.0328680 (4) \\
Epoch of ascending node passage $T_{ASC}$ (MJD)\dotfill & 56124.7701121 (2)\\
Span of timing data (MJD)\dotfill & 652 \\
Number of TOAs\dotfill & 280 \\
Post-fit residual rms ($\mu$s)\dotfill & 6.9\\
Reduced chi-square\dotfill & 2.7 \\
\hline
\multicolumn{2}{c}{Derived parameters} \\
\hline
Mass function $f$ (M$_{\odot}$)\dotfill & 0.0000026132\\
Min companion Mass $m_{c}$ (M$_{\odot}$)\dotfill & 0.017\\
DM distance\tablenotemark{c} (kpc)\dotfill & 1.2\\
Flux density at 322 MHz (mJy)\dotfill & 5.4\\
Flux density at 607 MHz (mJy)\dotfill & 1.2\\
Spectral index\dotfill & $-$2.3\\
Surface magnetic field $B_{s}$ ($10^{8}$ G)\dotfill & 0.805 (1) \\
Spin down luminosity \.{E} (10$^{34}$ erg s$^{-1}$)\dotfill & 1.150 (8) \\
Characteristic age $\tau$ (Gyr)\dotfill & 11.65 (3) \\
\hline
\multicolumn{2}{c}{$\gamma$-ray parameters\tablenotemark{d}} \\
\hline
Photon flux ($>0.1$ GeV, cm$^{-2}$ s$^{-1}$)\dotfill & 1.6(8) $\times 10^{-9}$\\
Energy flux ($>0.1$ GeV, erg cm$^{-2}$ s$^{-1}$)\dotfill & 2.1(6) $\times 10^{-12}$\\
Luminosity, $L_\gamma/f_\Omega$ ($>0.1$ GeV, erg cm$^{-2}$ s$^{-1}$)\dotfill & 3.6 $\times 10^{32}$\\
Efficiency, $\eta_\gamma/f_\Omega$ ($>0.1$ GeV)\dotfill & 0.03 \\
\enddata
\tablenotetext{a}{Errors in the last digit are in parentheses.}
\tablenotetext{b}{\cite{Roy12}}
\tablenotetext{c}{\cite{Cordes02}}
\tablenotetext{d}{phase-averaged}
\end{deluxetable}
\begin{figure}[htb]
\includegraphics[width=3.5in]{Fig2.pdf}
\caption{Phase-aligned radio and LAT $\gamma$-ray light-curves of J1544+4937. The horizontal dashed lines are an estimate of the background level from sources other than the pulsar. The 322 MHz profile is broadened by incoherent dedispersion across the channel bandwidth of 0.0651 MHz, introducing $\sim 373 \mu$s smearing. \label{fig:lc}}
\end{figure}
\section{$\gamma$-ray pulsations}
\label{sec:gammaray}
\begin{figure}[htb]
\includegraphics[width=6.0in]{Fig3.pdf}
\caption{Variation of timing residuals and electron column density (TOAs of 90 s time resolution) around eclipse phase
at 322 MHz (top) and 607 MHz (bottom). \label{fig:pulsars_all_group}}
\end{figure}
\begin{figure}[htb]
\includegraphics[width=6.0in]{Fig4.pdf}
\caption{Example of additional short-duration absorption beyond the eclipse phase at 322 MHz on 2012 June 3.
Left: variation of timing residuals and electron column density at 42 s time resolution. Right: phaseogram
showing short-duration absorption. \label{fig:short_eclipse}}
\end{figure}
The radio timing position of J1544+4937 is 4.3\arcmin\ from the LAT localization of \textsc{Fermi J1544.2+4941}.
This is well within the radius of the 95\% confidence error circle, suggesting an association
between the pulsar and the LAT source, which we sought to confirm with detection of pulsations.
In addition, the $\gamma$-ray detectability metric \.{E}$^{1/2}$/$d^2$ for J1544$+$4937 is
7.4 $\times$ 10$^{16}$ erg$^{1/2}$ kpc$^{-2}$ s$^{-1/2}$, which is comparable to other $\gamma$-ray detected MSPs \citep[Fig.~12 of][]{abdo10}.
Because of the low significance of the LAT source, a phase-averaged spectral analysis of the region
is required to optimize the sensitivity of the H-test for pulsed significance using photon probability weighting \citep{Kerr2011}.
We performed a binned likelihood spectral analysis of the region using the \textit{Fermi} Science
Tools\footnote{\url{http://fermi.gsfc.nasa.gov/ssc/data/analysis/software/}}. We selected LAT data with
reconstructed energies in the range 100 MeV to 20 GeV collected between 2008 August 4 and 2013 February 6.
We used the P7SOURCE\_V6 instrument response functions and excluded events at a zenith angle $>100$\degr\
and times when the LAT was in the SAA or the rocking angle was greater than 52\degr. We modeled\footnote{\url{http://fermi.gsfc.nasa.gov/ssc/data/access/lat/BackgroundModels.html}} a region of radius 15\degr\ using a source list from the 2FGL catalog with
an additional source added at the location of the pulsar. We modeled the pulsar with an exponentially
cutoff power-law spectrum. Because it is so faint, we fitted only for the power-law index and normalization,
keeping the cutoff energy fixed at 2.5 GeV, a value
typical of other MSPs \citep{abdo10}. The spectral model included both the isotropic and Galactic diffuse
contributions with normalizations free in the fit. We detected the source with a TS of 38.2, corresponding to $\sim 6\sigma$. The spectral index and cutoff energy are very poorly constrained for this weak source, so
to estimate the uncertainty on the integrated flux we repeated the spectral fit with the cutoff frozen at 0.7,
2.5, and 5.0 GeV, and with the index frozen at $1.3$ (the average value for LAT-detected MSPs) and the cutoff free.
We find that the uncertainty due to the spectral shape is about $3\times 10^{-10}$ cm$^{-2}$ s$^{-1}$ for the
photon flux and $2\times 10^{-13}$ erg cm$^{-2}$ s$^{-1}$ for the energy flux. These are smaller than the
statistical errors for this faint source and are added in quadrature to compute the errors on the fluxes
reported in Table \ref{tab:params}. Using the fitted energy flux ($G$), and the DM distance ($d$), we compute
the $\gamma$-ray luminosity $L_\gamma = 4 \pi f_\Omega G d^2$ and efficiency $\eta = L_\gamma/\dot{E}$, where
the beaming factor $f_\Omega$ is assumed to be 1 \citep{Watters09}.
Using this model for the continuum emission from the region, we search for pulsed $\gamma$-ray emission by selecting events within
2\degr\ of the source and computed the probability that they originated
from the source with the \textit{Fermi} Science Tool \texttt{gtsrcprob}. Using these probabilities as weights and
pulse phases computed using the radio timing model, we computed a weighted H-test of 37.1, corresponding to a detection
significance of 5.1 $\sigma$ and confirming that J1544+4937 is indeed a $\gamma$-ray pulsar and identifying the
LAT source as this MSP. The LAT weighted light-curves in two energy bands, phase aligned to the radio profiles, are
shown in Fig. \ref{fig:lc}. For this faint source, it is difficult to determine the peak multiplicity, but
the $\gamma$-ray emission seems to be mainly between phases 0.4 and 0.6, relative to the single radio peak. Fitting
a single gaussian peak to the light-curve gives a peak at phase $0.54 \pm 0.02$ with a width of $0.25 \pm 0.05$.
\section{X-ray and optical observations}
\label{sec:xray_optical}
We analyzed 5 {\it Swift}/XRT observations (totaling 9.3 ks) of this field from 2013 March 19--24. No X-ray counterpart is
detected, providing a flux upper limit of $\sim$ 1.7$\times$10$^{-14}$ ergs cm$^{-2}$ s$^{-1}$ for 0.3$-$10 keV.
We have checked for an optical counterpart for J1544$+$4937 using archival
SDSS data. This field was imaged \citep{Adelman08} on MJD 52046.294, at binary
phase 0.57. No optical counterpart was detected. The strongest upper
limits were $g^\prime > 22.5$, $r^\prime >22.0$ and $i^\prime >21.5$ (90\% CL).
We also obtained a 300s H$\alpha$ image with the MiniMo camera on the 3.5m WIYN telescope
on MJD 55975.521 \citep{Brownsberger13}. No stellar counterpart or extended emission was
seen with an H$\alpha$ flux upper limit of $2.5 \times 10^{-16}$ erg\,cm$^{-2}\,s^{-1}$ (90\% CL)
for a compact $< 5^{\prime\prime}$ nebula.
\cite{Breton13} have obtained optical photometry of the heated companions of a
number of {\it Fermi} MSPs. They find a typical heating efficiency
$\eta \sim 0.15$. For systems with a very strong wind $\eta$ approaches unity
(e.g. J1810$+$1744, \cite{Breton13}; J1311$-$3430, \cite{Romani12}).
\cite{Breton13} give an estimate for the irradiation temperature
$T_{Irr}=(\eta {\dot E}_{SD}/4\pi\sigma a^2)^{1/4}$, which is 4370$\,K$ considering $\eta =0.15$
and 6360$\,K$ considering $\eta = 0.6$ for J1544$+$4937.
If we assume that the deep, variable radio eclipses imply a near Roche lobe filling companion,
and consider a typical inclination angle $=60^\circ$, an unheated (backside)
companion temperature of 2500\,K and $\eta= 0.15$, we can use the binary light-curve
synthesis program `Icarus' \citep{Breton11} to predict magnitudes at
phase $\phi_B=0.57$ of $g^\prime \approx 25.2$, $r^\prime \approx 23.9$ and
$i^\prime \approx 23.1$ (magnitudes at maximum are 23.0, 21.9 and 21.5, respectively).
Thus the observed SDSS limits are not constraining. However if
the irradiation efficiency is higher, the fluxes can be detectable; for example
with $\eta= 0.6$ we expect $g^\prime \approx 22.4$, $r^\prime \approx 21.8$ and
$i^\prime \approx 21.6$ at $\phi_B=0.57$. These are comparable to our observed
magnitude limits, so higher efficiencies are ruled out unless the Roche lobe filling
factor is small or the source distance is larger. These estimates also depend weakly
on the observer inclination and the secondary unheated temperature and composition
Our H$\alpha$ limit for J1544$+$4937 corresponds to $<0.15$ of the highest
surface brightness $\sim 5^{\prime\prime}$ patch of the B1957$+$20 bow-shock.
The average flux ratio is even more constraining, with an
upper limit of $\sim 0.03$ of the total bow-shock flux. However, given the
small fraction of MSPs that show optical bow-shocks (likely due to the small
filling factor of the neutral interstellar medium), the non-detection at such a
large distance from the Galactic plane (b$\sim$ 50\degr) is not surprising.
\section{Eclipse characteristics}
\label{sec:eclipse_ch}
Fig.~\ref{fig:pulsars_all_group} presents the timing residuals (and electron column densities) around the eclipse phase at
322 and 607 MHz, with 90s time resolution. The effect of the eclipses is generally
seen from 0.18 to 0.31 orbital phase (eclipse-zone hereafter) at 322 MHz. The eclipses are
centered at binary phase 0.24 with a duration of around 22 minutes.
We estimate the radius of the companion's Roche lobe, R$_L$ \citep{Eggleton83},\\
\begin{equation}
{R_{L}}=\frac{0.49 a q^{2/3}}{0.6q^{2/3}+ \ln(1+q^{1/3})} \sim 0.13 R_{\odot}
\label{eqn1}
\end{equation}
where $q=m_c/m_p$ is the mass ratio of the companion and the pulsar, and $a$ is the separation of the companion from the pulsar
($a \sim 1.2 R_{\odot}$ for J1544$+$4937, indicating an extremely compact binary).
The opaque portion of the companion's orbit is 0.98 R$_{\odot}$, much larger than R$_L$ of 0.13~R$_{\odot}$, so the
volume occupied by the eclipsing material is well outside the companion's Roche lobe, and thus is not gravitationally bound to the companion.
This confirms that this binary is a BW, where the pulsar is ablating its companion, creating a significant amount of intrabinary
material that obscures the pulsar's emission.
Our sample consisted of six eclipses at 322 MHz where the pulsar emission was fully obscured by companion and its wind
and two 607 MHz observing sessions covering the full orbit (Fig.~\ref{fig:pulsars_all_group}). At 607 MHz we detect the
MSP throughout the 322 MHz eclipse-zone. However, we observe a flux fading at 607 MHz near the superior conjunction
(orbital phase 0.24).
Significant delays in pulse arrival times are observed at 607 MHz during the eclipse-zone at 322 MHz. Maximum delay
in pulse arrival time at 607 MHz near the eclipse superior conjunction is around 300 $\mu$s, which corresponds to an increase
in DM of 0.027 pc cm$^{-3}$ and an added electron density, N$_e$ of 8$\times$10$^{16}$ cm$^{-2}$.
\citet{Thompson94} (T94 hereafter) elucidate a collection of eclipse mechanisms. According to them, eclipsing due to refraction of the radio beam
demands an order of magnitude higher group delay ($\sim$ few tens of milliseconds) than we observe for J1544$+$4937
(250 $\mu$s at 322 MHz egress). Using N$_e \sim$8$\times$10$^{16}$ cm$^{-2}$ observed during superior conjunction at 607 MHz and
absorption length about twice the size of the eclipse-zone, according to equation 11 of T94 we find that free-free absorption is possible
(absorption optical depth $\tau_{ff} > 1$) if the plasma temperature T$\leq 4 \times {f_{cl}}^{2/3}$ K, where ${f_{cl}}^{2/3}$
is the clumping factor. This demands either a very low temperature or a very high value of the clumping factor, both of which are not physically
achievable. Eclipsing by pulse smearing (due to the increase of N$_e$ along the line-of-sight) can be ruled out, as the excess electron column
density inferred from 607 MHz predicts 373 $\mu$s smearing of pulses at 322 MHz near superior conjunction (considering
incoherent dedispersion), which is less than one fifth of pulse period. Since J1544$+$4937 has a narrow main-pulse, and no significant
profile evolution is apparent at the eclipse boundary, pulse broadening due to scattering can reduce the detectability but cannot explain
the eclipse. In addition, since J1544$+$4937 is relatively weak, nearby and has a shallower spectrum than B1957$+$20, the
expected induced Compton scattering optical depth is much less than one (equation 26 of T94). Another eclipse mechanism considered by
T94 is cyclotron-synchrotron absorption of the radio waves by non-relativistic/relativistic electrons, which requires a magnetic field
in the vicinity of the companion. We calculate a magnetic field B $\sim$ 11 G and corresponding cyclotron absorption
frequency $\sim$ 31 MHz (equations 35, 37 of T94, assuming a moment-of-inertia of 10$^{45}$ g cm$^2$). Thus 322 and 607 MHz will
correspond to 10th and 20th harmonics of the cyclotron resonance. For a fixed temperature, the optical depth for cyclotron absorption drops with
harmonics, which may explain the lack of absorption seen at 607 MHz.
A modified model is proposed by \cite{Khechinashvili00} based on kinematic treatment of cyclotron damping, assuming white-dwarf companions
with reasonably strong surface magnetic fields. Further observations over a wider radio spectrum may help to
investigate the frequency-dependent degree of damping predicted by this model.
We observe a temporal variation of the ingress phase at 322 MHz. For two eclipses, the ingress phase is shifted to 0.16 (from 0.18
for the other four eclipses), whereas there are no apparent shifts in the egress phase. This corresponds to an increase of the opaque portion by
0.15 R$_{\odot}$ and N$_e >$1.2$\times$10$^{16}$ cm$^{-2}$ at the ingress boundary. Such asymmetric increase of eclipse duration
may indicate that our line-of-sight is probing a wind zone where there is systematic outflow of eclipse material.
We also observe strong phase modulations and additional short-duration absorptions at ingress and egress, in time-series
data at higher resolution. The durations of these features are in general around 10--20 s, and hence they are not seen in
Fig.~\ref{fig:pulsars_all_group}. However, in one of the observing epochs these modulations lasted longer $-$ phase modulation of
duration 100 s, followed by a short-duration absorption of $\sim$ 180 s, then regular emission resumes for 500 s, after which the
eclipse starts (Fig.~\ref{fig:short_eclipse}). Fragmented blobs of plasma randomly oriented around eclipsing zone, obscuring
radiation from the pulsar, can explain these short-duration absorptions.
\section{Discussion}
\label{sec:discussion}
We report the GMRT discovery of an eclipsing BW pulsar, at the position of an unassociated LAT source,
\textsc{Fermi J1544.2+4941}. This is the first Galactic field MSP discovered at the GMRT. The detection of pulsed $\gamma$-rays from this pulsar
demonstrates it as the source powering \textsc{Fermi J1544.2+4941}. Due to the limited significance of the source in $\gamma$-rays additional data are needed before conclusions on peak multiplicity and system geometry can be drawn.
The implied efficiency ($\sim$ 3\%) of converting spin-down energy into $\gamma$-rays is typical of LAT-detected MSPs.
This is the first discovery of a radio MSP in a LAT source fainter than the 2FGL catalog limit \citep{Ray12}. Since the
radio pulsar is relatively bright, this provides strong justification to continue these searches as new LAT sources
are revealed in analyses of longer datasets. The radio flux is uncorrelated with the $\gamma$-ray flux \citep{Ackermann2012},
so even faint new LAT sources can harbor bright radio MSPs.
Eclipsing BW pulsars have the potential of providing information on the evolutionary connection between the low-mass X-ray
binaries and isolated MSPs.
With long term monitoring of this pulsar we aim to estimate $\dot P_{b}$ and its higher derivatives, which can provide an estimate
of the life span of the system.
\citet{Bates11} noted that for BW systems the measured value of $\dot {E}$/a$^2$
is an order of magnitude higher than for other MSP binaries, indicative of greater energy flux needed to ablate the companion. For
J1544$+$4937 we calculate $\dot {E}$/a$^2$ $\sim$ 1.5 $\times$10$^{33}$ erg lt-s$^{-2}$s$^{-1}$,
which is similar to other BW systems in the Galactic field.
Dual frequency observations presented in this paper suggest that cyclotron
absorption by the plasma formed via interaction of the pulsar wind with ablated material can obscure the pulsed emission.
However, exploring the radio spectrum on either side to probe the reduced/increased opaqueness of the stellar wind during the
eclipse phase may provide better insight into the plausible eclipse mechanism.
\acknowledgments
The \textit{Fermi} LAT Collaboration acknowledges support from a number of agencies and institutes for both development and the operation of the LAT as well as scientific data analysis. These include NASA and DOE in the United States, CEA/Irfu and IN2P3/CNRS in France, ASI and INFN in Italy, MEXT, KEK, and JAXA in Japan, and the K.~A.~Wallenberg Foundation, the Swedish Research Council and the National Space Board in Sweden. Additional support from INAF in Italy and CNES in France for science analysis during the operations phase is also gratefully acknowledged. We acknowledge support of telescope operators of the GMRT, which is run by the National Centre for Radio Astrophysics of the Tata Institute of Fundamental Research.
We thank the {\it Swift} team at Pennsylvania State University, especially Abe Falcone. We acknowledge help of C. Cheung in interpreting the XRT data.We thank D. Thompson and T. Johnson for their comments and R. Breton for a discussion of heating fluxes.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,208 |
#Russia #North Korea #trade
N. Korea-Russia border trade nearly triples in Q1 on improving ties
North Korea 08:57 July 09, 2019
SEOUL, July 9 (Yonhap) -- North Korea's trade with Russia's Far Eastern region nearly tripled in the first quarter from a year earlier, data showed Tuesday, amid intensifying efforts by the two neighbors to strengthen bilateral relations.
According to the data by South Korea's Consulate General in Vladivostok, the trade between the North and Russia's Far Eastern region totaled US$10.69 million during the January-March period, up from $3.72 million tallied a year earlier.
Russia's exports to the North came to $10.67 million, while the North's exports to the Russian border region amounted to $20,000, the data showed.
Oil-related products, including refinery fuel, accounted for the largest portion of the Far Eastern region's exports to the North, with $9.08 million worth of such goods shipped across the border.
The spike in border trade came as Pyongyang is apparently pushing to consolidate its ties with its neighboring countries, including Russia, to ease the impact of crushing global sanctions.
North Korean leader Kim Jong-un traveled to the Russian border town of Vladivostok in late April for his first-ever summit with Russian President Vladimir Putin. They agreed to boost bilateral cooperation in various areas.
kokobj@yna.co.kr
Latest News North Korea
News Focus North Korea
(News Focus) Seoul's push for individual tours to North meaningful but many hurdles ahead: experts
Surging U.S.-Iran tensions feared to dilute U.S. focus on N.K. nuke talks
Biegun expected to seek breakthrough in N.K. talks ahead of year-end deadline
Tensions rise to perilous point in U.S.-NK nuke diplomacy
Shorter firing interval indicates N.K.'s super-large rocket launcher almost ready for operation: experts
HOME North Korea | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,822 |
While issues of Cape Town's readiness, safety and security and transport infrastructure have dominated global headlines in the run-up to the 2010 FIFA World Cup, fears of being fleeced are now top-of-mind for scores of potential World Cup visitors. Cape Town Tourism's head of marketing, Lianne Burton, explains why.
Speaking to a cross-section of members of the UK travel trade at the World Travel Market in London in November, I was struck by the fact that concerns about greedy accommodation providers and international and domestic airlines ripping off World Cup visitors were as widespread as those about safety and security. When England qualified, for instance, a spate of press reports in the UK encouraged excited fans to stay home and watch the World Cup on TV, arguing that unrealistic pricing would put the World Cup out of most mere mortals' reach. The astronomical prices of luxury private villas on the Atlantic Seaboard were cited, out of context, to add fuel to the fire. While these reports were undoubtedly exaggerated, they indicate that the issue of overpricing during the 2010 World Cup is a hot media topic.
And it's not surprising. Battered by the worst recession since the 1930s, travellers worldwide are particularly price-sensitive. Global tourism figures for January to October 2009 are down by 8% compared with the corresponding period last year, and the European Travel Commission has warned that there is unlikely to be a strong travel rebound, with recovery likely to be subdued.
Add to this the fact that Cape Town is a long-haul destination, and the subject of accessible pricing becomes even more critical to our global attractiveness, World Cup or not.
According to Cape Town Tourism's representative in London, Mary Tebje of MTA Tourism Leisure consultants, 'the downturn in travel and tourism has been especially pronounced for long-haul travel, with a move towards increased short-haul trips and short leisure breaks.' The UK Office of National Statistics reported that the number of overseas visits by UK residents (Cape Town's key source market), fell by 12% in the last 12 months to July 2009. Interestingly enough, affordable long-haul destinations like Mexico, Thailand, the Dominican Republic and Jamaica have bucked the trend, showing an increase in British visitors, year-on-year. Even traditionally well-heeled travellers are trading down from luxury to mid-priced travel options.
So where does Cape Town fit into this picture?
Cape Town's reputation as a leading destination is underpinned by the fact that we offer pristine natural beauty, cultural diversity, a fascinating political history, a cosmopolitan vibe and a sophisticated tourism infrastructure – all at an affordable price by global standards. This value positioning has cemented our place as one of the top long-haul destinations for travellers from the UK, USA, Germany and the Netherlands. But it is not guaranteed.
The opportunity presented by the 2010 FIFA World Cup has seen major infrastructural investment in Cape Town as a Host City. Cape Town is undoubtedly ready to welcome the world – some 350 000 visitors from traditional and non-traditional markets are expected to visit Cape Town – in June 2010.
This 'memory' will become the new definition of our destination brand, and will drive tourism growth for years to come. In tourism terms, as with infrastructural improvements, the World Cup opportunity is all about legacy. We have a once-in-a-lifetime chance to reinforce our positioning as a unique, value-for-money destination. If we adopt a short-term, 'get rich quick' attitude and hike prices unreasonably, visitors will become negative brand ambassadors, spreading the word that Cape Town is officially overpriced. This will seal our fate alongside many cities, including Sydney, for instance, that have seen a drop in tourism after hosting mega events. In contrast, responsible pricing practices will ensure that visitors will return time and time again to Cape Town.
Sydney experienced a staggering decline in visitor numbers in the three years after hosting the Olympics in 2000, with 'greed' being singled out as a key factor.In September 2009 Cape Town Tourism conducted a pricing strategy workshop for its members, in partnership with US consultancy Myriad Marketing, highlighting the strategic importance of responsible pricing practices during the World Cup and citing global best and worst practice examples to drive home the message. Sydney, for instance, experienced a staggering decline in visitor numbers in the three years after hosting the Olympics in 2000, with 'greed' being singled out as a key factor and a painful lesson learnt.
As a helpful guideline, Cape Town Tourism is encouraging local tourism establishments and operators to peg their June/July 2010 World Cup rates somewhere in the region of their high-season 2010 rates, and certainly not more than 15% above next year's high season rates. We would like to think that local airlines will share our responsible approach.
It's hard not to fall in love with Cape Town. But if we rip off World Cup visitors, we will almost certainly have to settle for being a one-night stand. Like many lost loves, it will be a wasted opportunity that might haunt us for years to come. | {
"redpajama_set_name": "RedPajamaC4"
} | 316 |
I once heard a transgender woman give a talk about the process of socially transitioning to being recognized as a woman. She discussed various decisions she made in taking some final critical steps toward the social identity of woman. She talked at length about her hair. She asked, "What kind of woman am I and how is my haircut going to indicate that?" She talked about being preoccupied with her hair for a long time as she attempted to figure out a cut and style that "felt right." But what struck me the most was her discussion of carrying a purse.
She said that getting used to carrying a purse everywhere was one of the more challenging elements of the transition. If asked what I thought would be a significant everyday challenge if I were a woman, I don't think purse would have been high on my list. But, it was high on hers. She discussed remembering to bring it, how to carry it, norms surrounding purse protection in public, but also more intimate details like: what belongs in a purse?
Purses and wallets are gendered spaces. There's nothing inherent in men's and women's constitutions that naturally recommends carrying money and belongings in different containers. Like the use of urinals in men's restrooms, wallets and purses are a way of producing understandings of gender difference rather than as a natural consequence of differences.
I got the idea for this post after reading Christena Nippert-Eng's book, Islands of Privacy — a sociological study of privacy in everyday life. One chapter deals specifically with wallets and purses. In it, Nippert-Eng discusses one way she interviewed her participants about privacy. She used participants' wallets and purses as a means of getting them to think more critically about privacy. Participants were asked to empty the contents of their wallets and purses and to form two piles with the contents: "more private" and "more public." As they sifted through the contents of their wallets and purses, they talked about why they carried what they carried as well as how and why they thought about it as public or private.
After collecting responses, she documented all of the contents and created categories and distinctions between objects based on how people thought about them as public or private. One question that was clearly related to privacy was whether the objects were personally meaningful to the participant. Invariably, objects defined as more personally meaningful were also considered more private.
Another question that routinely arose as participants made sense of the objects they carry around everyday was how damaging it might be for participants if a specific object was taken. Based on this findings, she creates a useful table delineating participants concerns surrounding and understandings of the objects they carry with them (see left).
Just for clarification, there's sort of a sliding scale of privacy going from most to least private as one proceeds from the bottom left cell to the top right cell. Thus, items classified by participants in the lower left cell (1) are the most private objects. Here, participants identified things like prescription medications, letters from friends, and a variety of personally meaningful objects that were thought of as completely private and carried only for the self.
Other items were still considered private, but "less private" than objects in cell 1 because they were shared selectively. Consider cell 2. While credit cards, bank cards, memberships, credit cards and money were all classified as "private," individual's also thought of them as "more public" than object in cell 1 because they were required to share these objects with institutions throughout their lives.
Similarly, some objects were thought of as "private," but were also carried to share with certain others, such as photographs of children (cell 4). Finally, items classified in the top right cell (3) are the most public objects in wallets and purses—carried for the self and, potentially, "anyone" else. Items here include things like tissues, lip balm, money classified as "extra," gum, breath mints, etc.
Objects from most of the cells exist in both wallets and purses, but not all of them. The contents of cell 3 (containing the "most public" objects in wallets and purses) are inequitably distributed between wallets and purses. As Nippert-Eng writes, "This is the one category of objects that is overwhelmingly absent for participants who carry only wallets, yet universally present for those who carry purses" (here: 130). She also found that some of her participants only carried objects all fitting the same cell in the above table. These participants — universally "wallet carriers" in her sample — carry only objects necessary for institutional transactions (cell 2).
This is, I believe, a wonderful analysis of one of the more subtle ways in which gender is accomplished in daily life. Certain objects are simply more likely to be carried in purses. Interestingly, this class of "feminine" objects are also objects that play a critical role in social interactions. Indeed, many of us are able to travel without these objects because we can "count on" purse-carriers as having them. Things like packs of gum, tissues, breath mints and more might seem like inconsequential objects. But, they play a crucial role in social interactions, and many of us count on purse-carriers to provide us with these objects when we are "in need." It's an aspect of care work by which some (those carrying purses) care for others (those without purses). And if they're any good at it, the caring goes virtually unacknowledged, though potentially highly acknowledged when these objects are absent in purses. Children routinely ask their mothers for objects they presume they'll be carrying in their purses. Indeed, these objects may be carried in anticipation of such requests. It's a small aspect of doing gender, but a significant element of social interactions and life.
When I was learning about interviewing and ethnography, I was told to always carry a pack of gum, a pack of cigarettes (something "lite"), and a lighter. My professor told me, "It opens people up. It's a small gesture that comforts people–puts them at ease." These are the ways you might want people to feel if you're asking them to "open up" for you. I still remember my first foray into "the field." I bought my gum and cigarettes (objects I don't typically carry) and the first thought I had was, "Where the heck am I going to keep these things?" What I didn't realize at the time was that I was asking an intensely gendered question.
Tristan Bridges is a sociologist of gender and sexuality at the College at Brockport (SUNY). Dr. Bridges blogs about some of this research and more at Inequality by (Interior) Design. You can follow him on twitter @tristanbphd.
One glaring problem with the wallet/purse dichotomy is that non-purse users frequently distribute those Column 3 items, among others, into other places outside of their wallets.
Non-purse-users carry things like phones, keys, tissues, cigarettes, lighters, mints, gum, and loose change in their pockets, backpacks, and briefcases. It is beyond obvious that most of these items won't be found in wallets, as they don't fit there.
One thing that does happen when changing from typical men's clothing to form-fitting women's clothing is that pockets get smaller and less accommodating to bulky objects, making the purse seem like a better idea. Of course, I know plenty of women who prefer not to use purses at all, as they make it far too easy to have many valuable items lost or stolen at once. As an accessory, I'd say they are more closely related to the gendering of clothes than to gendered discrepancies of what one needs to carry around.
As for urinals, they aren't merely a way of justifying toilet segregation; they do happen to save a great deal of water and space, to an extent that actually makes a difference when large numbers of people are being accommodated.
Minor nitpick, but in the second paragraph after the image you list "credit cards" twice.
Also, I think it is kind of silly to say that people with wallets don't keep the contents of cell 3 in there. You know who else doesn't store tissues and gum in their wallets? People with purses!
Lots of wallet users also have gum and cigarettes on them, that's what pockets are for. It seems like a rather glaring omission to not have asked participants to empty their pockets, as if keeping your lighter there were somehow significantly different than keeping your lighter in your purse.
There crops up from time to time a discussion of why more men don't get custody of children, why we see women as more "maternal" and therefore better deserving of the children. And I am usually on the mother's side, even though I think the male is every bit as capable and caring. I think this article is the first time I've really figured out why. Because of a social process of nurture (not nature) that causes a woman's default setting to be "caretaker," while the male default setting is "he who takes," even on the smallest level.
Given a choice between someone who looks after others and facilitates communication as a matter of social instinct and those who do not, I know who I, if I were a judge, would choose. And we see this sort of behavior so often it becomes invisible, we really only pick it up on a subconscious level. If mens' default setting switched so that they were also doing these tiny invisible social cues, would our attitude on their paternal instincts change as well?
I know it's not a direct comment to the article, but I found the idea intriguing.
I am a cis woman, and I carry a big purse. I have a lot of 'category three' objects that are not covered here and that wouldn't fit in pockets (as suggested by fellow commenters), like a first aid kit, a sewing kit, a flashlight, pads and tampons, and a variety of tools. I often am asked for objects from my 'purse of holding' by my adult friends (and I am most often asked by men, as they cannot carry any of this for themselves), and they pretty much expect me to have whatever they need. However, these same people will often mock me for carrying such a large purse.
Seeing the gendered nature of carrying potentially-communal items explained in this article really oriented this somewhat uncomfortable experience for me as a gesture of unexamined male privilege and even possibly unconscious sexism from my friends. I will continue to carry my purse, because I like being prepared, but now I feel more equipped to broach with people with the larger context of their mockery. Thank you.
Andrew and Japaniard point out pockets should have been included in the study, and while I think they are correct, my suspicion is that wouldn't have changed the results all that much. The men I know don't carry gum, tissues, band aids, breath mints - and certainly not all of those (as I do!). There are likely some men who do carry these social interaction objects in their pockets, but I don't find this happens "frequently" or that "lots" of wallet carriers do this. A more complete study including pockets would be nice - but I think Dr. Bridges' analysis is bang-on.
I've never carried a purse in my 30 odd years. What's interesting is how distressing this decision is to some people. My mother used to buy me a new purse for every holiday. If I ever misplaced my debit card, my father would tell me it's because I don't have a purse (I also don't carry a wallet, but I was never pressured to do so.) Other relatives, mostly female, would ask why I didn't carry one or would buy me one. I don't 'code' female in many other ways (v. short hair in a 'male' style, absolutely TERRIBLE fashion sense), but this was the only one I ever got a lot of flack for.
In my social circle it is usual for everyone (men and women) to carry bags of some type (often bicycle pannier bags, for ease of attaching to bicycles but also rucksacks, handbags, and other options) at most times. I'm not really sure how one would distinguish a "purse" from any other sort of "bag" into which one might put one's phone/wallet/keys/tissues/novel/knitting/etc.
I don't often spend time with men who have gone out with only what they can carry in their pockets; and even then most men seem to fit their phones, keys, etc. into pockets (but not of course into their wallets - wallets clearly do not have space to contain large phones) and could presumably manage tissues, gum, etc. if they felt like it.
It often irritates me that men's formal wear has useful pockets, whilst women's formal wear comes with the expectation that one will carry a decorative bag (often too small to contain my usual wallet, keys, phone; let alone novel, knitting, bike repair kit...).
It occurs to me that, especially for men, at least one object that could be considered helpful for others has almost totally disappeared; the handkerchief. This object was ubiquitous before WWI, and now who carries one? That's a shame, really, because it was the only object that a man would carry routinely whose purpose could be classified as at least box 4, or maybe box 3 with respect to women and children (a man would be expected to have their own).
A couple months ago, I got some fast food at the mall, and the teenager behind the counter said, "You're the second lady with a legit wallet I've seen this week. Most of 'em carry these big bags." I'm not sure what to think of the fact that he felt moved to comment; certainly nobody but my mother has ever commented on the fact that I don't carry a purse before.
I note you don't mention tampons a whole area of interest for women and girls and a rich field for study of interactions etc. Did you buy a 'man-bag'?
The third gender: parents of small children. Everything goes in the diaper bag.
style and make you amazing.
I've always been interested in how other women learn to carry purses, since I actually got taught to by my mom after I lost my asthma inhaler at a roller rink. I got taken out for it first bra style and told to keep an eye on it in public and such.
Where do keys go in this graph? They kind of change, since sometimes they are carried just by one person in my family if we're going out, and sometimes they're just for me. They seem to change with gender too, since a lot of self defense items for women are key chains, and I would imagine you have a lot less keychains if you don't have a purse.
I used to travel about one week a month, to meet a group of 10-20 colleagues at different workshops. Everyone had all sorts of bags, but we all tried to leave them at the hotel, etc. One of my colleagues and I quickly fell into a routine of relying on each other for items. He always had bandaids and aspirin, I always had antacids and tissues. He carried cash, I carried change. He checked luggage containing booze, I checked luggage containing snacks. We never coordinated any of this, but it worked out beautifully.
FWIW - I carry my wallet in my purse. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,955 |
local parent, ns = ...
local global = GetAddOnMetadata(parent, 'X-oUF')
local _VERSION = '@project-version@'
if(_VERSION:find('project%-version')) then
_VERSION = 'devel'
end
local oUF = ns.oUF
local Private = oUF.Private
local argcheck = Private.argcheck
local error = Private.error
local print = Private.print --luacheck: no unused
local unitExists = Private.unitExists
local styles, style = {}
local callback, objects, headers = {}, {}, {}
local elements = {}
local activeElements = {}
local PetBattleFrameHider = CreateFrame('Frame', (global or parent) .. '_PetBattleFrameHider', UIParent, 'SecureHandlerStateTemplate')
PetBattleFrameHider:SetAllPoints()
PetBattleFrameHider:SetFrameStrata('LOW')
RegisterStateDriver(PetBattleFrameHider, 'visibility', '[petbattle] hide; show')
local function updateActiveUnit(self, event)
-- Calculate units to work with
local realUnit, modUnit = SecureButton_GetUnit(self), SecureButton_GetModifiedUnit(self)
-- _GetUnit() doesn't rewrite playerpet -> pet like _GetModifiedUnit does.
if(realUnit == 'playerpet') then
realUnit = 'pet'
elseif(realUnit == 'playertarget') then
realUnit = 'target'
end
if(modUnit == 'pet' and realUnit ~= 'pet') then
modUnit = 'vehicle'
end
if(not unitExists(modUnit)) then return end
-- Change the active unit and run a full update.
if(Private.UpdateUnits(self, modUnit, realUnit)) then
self:UpdateAllElements(event or 'RefreshUnit')
return true
end
end
local function evalUnitAndUpdate(self, event)
if(not updateActiveUnit(self, event)) then
return self:UpdateAllElements(event)
end
end
local function iterateChildren(...)
for i = 1, select('#', ...) do
local obj = select(i, ...)
if(type(obj) == 'table' and obj.isChild) then
updateActiveUnit(obj, 'iterateChildren')
end
end
end
local function onAttributeChanged(self, name, value)
if(name == 'unit' and value) then
if(self.hasChildren) then
iterateChildren(self:GetChildren())
end
if(not self:GetAttribute('oUF-onlyProcessChildren')) then
updateActiveUnit(self, 'OnAttributeChanged')
end
end
end
local frame_metatable = {
__index = CreateFrame('Button')
}
Private.frame_metatable = frame_metatable
for k, v in next, {
--[[ frame:EnableElement(name, unit)
Used to activate an element for the given unit frame.
* self - unit frame for which the element should be enabled
* name - name of the element to be enabled (string)
* unit - unit to be passed to the element's Enable function. Defaults to the frame's unit (string?)
--]]
EnableElement = function(self, name, unit)
argcheck(name, 2, 'string')
argcheck(unit, 3, 'string', 'nil')
local element = elements[name]
if(not element or self:IsElementEnabled(name)) then return end
if(element.enable(self, unit or self.unit)) then
activeElements[self][name] = true
if(element.update) then
table.insert(self.__elements, element.update)
end
end
end,
--[[ frame:DisableElement(name)
Used to deactivate an element for the given unit frame.
* self - unit frame for which the element should be disabled
* name - name of the element to be disabled (string)
--]]
DisableElement = function(self, name)
argcheck(name, 2, 'string')
local enabled = self:IsElementEnabled(name)
if(not enabled) then return end
local update = elements[name].update
if(update) then
for k, func in next, self.__elements do
if(func == update) then
table.remove(self.__elements, k)
break
end
end
end
activeElements[self][name] = nil
return elements[name].disable(self)
end,
--[[ frame:IsElementEnabled(name)
Used to check if an element is enabled on the given frame.
* self - unit frame
* name - name of the element (string)
--]]
IsElementEnabled = function(self, name)
argcheck(name, 2, 'string')
local element = elements[name]
if(not element) then return end
local active = activeElements[self]
return active and active[name]
end,
--[[ frame:Enable(asState)
Used to toggle the visibility of a unit frame based on the existence of its unit. This is a reference to
`RegisterUnitWatch`.
* self - unit frame
* asState - if true, the frame's "state-unitexists" attribute will be set to a boolean value denoting whether the
unit exists; if false, the frame will be shown if its unit exists, and hidden if it does not (boolean)
--]]
Enable = RegisterUnitWatch,
--[[ frame:Disable()
Used to UnregisterUnitWatch for the given frame and hide it.
* self - unit frame
--]]
Disable = function(self)
UnregisterUnitWatch(self)
self:Hide()
end,
--[[ frame:IsEnabled()
Used to check if a unit frame is registered with the unit existence monitor. This is a reference to
`UnitWatchRegistered`.
* self - unit frame
--]]
IsEnabled = UnitWatchRegistered,
--[[ frame:UpdateAllElements(event)
Used to update all enabled elements on the given frame.
* self - unit frame
* event - event name to pass to the elements' update functions (string)
--]]
UpdateAllElements = function(self, event)
local unit = self.unit
if(not unitExists(unit)) then return end
assert(type(event) == 'string', "Invalid argument 'event' in UpdateAllElements.")
if(self.PreUpdate) then
--[[ Callback: frame:PreUpdate(event)
Fired before the frame is updated.
* self - the unit frame
* event - the event triggering the update (string)
--]]
self:PreUpdate(event)
end
for _, func in next, self.__elements do
func(self, event, unit)
end
if(self.PostUpdate) then
--[[ Callback: frame:PostUpdate(event)
Fired after the frame is updated.
* self - the unit frame
* event - the event triggering the update (string)
--]]
self:PostUpdate(event)
end
end,
} do
frame_metatable.__index[k] = v
end
local function onShow(self)
evalUnitAndUpdate(self, 'OnShow')
end
local function updatePet(self, event, unit)
local petUnit
if(unit == 'target') then
return
elseif(unit == 'player') then
petUnit = 'pet'
else
-- Convert raid26 -> raidpet26
petUnit = unit:gsub('^(%a+)(%d+)', '%1pet%2')
end
if(self.unit ~= petUnit) then return end
evalUnitAndUpdate(self, event)
end
local function updateRaid(self, event)
local unitGUID = UnitGUID(self.unit)
if(unitGUID and unitGUID ~= self.unitGUID) then
self.unitGUID = unitGUID
self:UpdateAllElements(event)
end
end
-- boss6-8 exsist in some encounters, but unit event registration seems to be
-- completely broken for them, so instead we use OnUpdate to update them.
local eventlessUnits = {
boss6 = true,
boss7 = true,
boss8 = true,
}
local function isEventlessUnit(unit)
return unit:match('%w+target') or eventlessUnits[unit]
end
local function initObject(unit, style, styleFunc, header, ...)
local num = select('#', ...)
for i = 1, num do
local object = select(i, ...)
local objectUnit = object:GetAttribute('oUF-guessUnit') or unit
local suffix = object:GetAttribute('unitsuffix')
-- Handle the case where someone has modified the unitsuffix attribute in
-- oUF-initialConfigFunction.
if(suffix and not objectUnit:match(suffix)) then
objectUnit = objectUnit .. suffix
end
object.__elements = {}
object.style = style
object = setmetatable(object, frame_metatable)
-- Expose the frame through oUF.objects.
table.insert(objects, object)
-- We have to force update the frames when PEW fires.
-- It's also important to evaluate units before running an update
-- because sometimes events that are required for unit updates end up
-- not firing because of loading screens. For instance, there's a slight
-- delay between UNIT_EXITING_VEHICLE and UNIT_EXITED_VEHICLE during
-- which a user can go through a loading screen after which the player
-- frame will be stuck with the 'vehicle' unit.
object:RegisterEvent('PLAYER_ENTERING_WORLD', evalUnitAndUpdate, true)
if(not isEventlessUnit(objectUnit)) then
object:RegisterEvent('UNIT_ENTERED_VEHICLE', evalUnitAndUpdate)
object:RegisterEvent('UNIT_EXITED_VEHICLE', evalUnitAndUpdate)
-- We don't need to register UNIT_PET for the player unit. We register it
-- mainly because UNIT_EXITED_VEHICLE and UNIT_ENTERED_VEHICLE don't always
-- have pet information when they fire for party and raid units.
if(objectUnit ~= 'player') then
object:RegisterEvent('UNIT_PET', updatePet)
end
end
if(not header) then
-- No header means it's a frame created through :Spawn().
object:SetAttribute('*type1', 'target')
object:SetAttribute('*type2', 'togglemenu')
object:SetAttribute('toggleForVehicle', true)
if(isEventlessUnit(objectUnit)) then
oUF:HandleEventlessUnit(object)
else
oUF:HandleUnit(object)
end
else
-- update the frame when its prev unit is replaced with a new one
-- updateRaid relies on UnitGUID to detect the unit change
object:RegisterEvent('GROUP_ROSTER_UPDATE', updateRaid, true)
if(num > 1) then
if(object:GetParent() == header) then
object.hasChildren = true
else
object.isChild = true
end
end
if(suffix == 'target') then
oUF:HandleEventlessUnit(object)
end
end
Private.UpdateUnits(object, objectUnit)
styleFunc(object, objectUnit, not header)
object:HookScript('OnAttributeChanged', onAttributeChanged)
-- NAME_PLATE_UNIT_ADDED fires after the frame is shown, so there's no
-- need to call UAE multiple times
if(not object.isNamePlate) then
object:SetScript('OnShow', onShow)
end
activeElements[object] = {}
for element in next, elements do
object:EnableElement(element, objectUnit)
end
for _, func in next, callback do
func(object)
end
-- Make Clique kinda happy
if(not object.isNamePlate) then
_G.ClickCastFrames = _G.ClickCastFrames or {}
_G.ClickCastFrames[object] = true
end
end
end
local function walkObject(object, unit)
local parent = object:GetParent()
local style = parent.style or style
local styleFunc = styles[style]
local header = parent:GetAttribute('oUF-headerType') and parent
-- Check if we should leave the main frame blank.
if(object:GetAttribute('oUF-onlyProcessChildren')) then
object.hasChildren = true
object:HookScript('OnAttributeChanged', onAttributeChanged)
return initObject(unit, style, styleFunc, header, object:GetChildren())
end
return initObject(unit, style, styleFunc, header, object, object:GetChildren())
end
--[[ oUF:RegisterInitCallback(func)
Used to add a function to a table to be executed upon unit frame/header initialization.
* self - the global oUF object
* func - function to be added
--]]
function oUF:RegisterInitCallback(func)
table.insert(callback, func)
end
--[[ oUF:RegisterMetaFunction(name, func)
Used to make a (table of) function(s) available to all unit frames.
* self - the global oUF object
* name - unique name of the function (string)
* func - function or a table of functions (function or table)
--]]
function oUF:RegisterMetaFunction(name, func)
argcheck(name, 2, 'string')
argcheck(func, 3, 'function', 'table')
if(frame_metatable.__index[name]) then
return
end
frame_metatable.__index[name] = func
end
--[[ oUF:RegisterStyle(name, func)
Used to register a style with oUF. This will also set the active style if it hasn't been set yet.
* self - the global oUF object
* name - name of the style
* func - function(s) defining the style (function or table)
--]]
function oUF:RegisterStyle(name, func)
argcheck(name, 2, 'string')
argcheck(func, 3, 'function', 'table')
if(styles[name]) then return error('Style [%s] already registered.', name) end
if(not style) then style = name end
styles[name] = func
end
--[[ oUF:SetActiveStyle(name)
Used to set the active style.
* self - the global oUF object
* name - name of the style (string)
--]]
function oUF:SetActiveStyle(name)
argcheck(name, 2, 'string')
if(not styles[name]) then return error('Style [%s] does not exist.', name) end
style = name
end
--[[ oUF:GetActiveStyle()
Used to get the active style.
* self - the global oUF object
--]]
function oUF:GetActiveStyle()
return style
end
do
local function iter(_, n)
-- don't expose the style functions.
return (next(styles, n))
end
--[[ oUF:IterateStyles()
Returns an iterator over all registered styles.
* self - the global oUF object
--]]
function oUF.IterateStyles()
return iter, nil, nil
end
end
local getCondition
do
local conditions = {
raid40 = '[@raid26,exists] show;',
raid25 = '[@raid11,exists] show;',
raid10 = '[@raid6,exists] show;',
raid = '[group:raid] show;',
party = '[group:party,nogroup:raid] show;',
solo = '[@player,exists,nogroup:party] show;',
}
function getCondition(...)
local cond = ''
for i = 1, select('#', ...) do
local short = select(i, ...)
local condition = conditions[short]
if(condition) then
cond = cond .. condition
end
end
return cond .. 'hide'
end
end
local function generateName(unit, ...)
local name = 'oUF_' .. style:gsub('^oUF_?', ''):gsub('[^%a%d_]+', '')
local raid, party, groupFilter, unitsuffix
for i = 1, select('#', ...), 2 do
local att, val = select(i, ...)
if(att == 'oUF-initialConfigFunction') then
unitsuffix = val:match('unitsuffix[%p%s]+(%a+)')
elseif(att == 'showRaid') then
raid = val ~= false and val ~= nil
elseif(att == 'showParty') then
party = val ~= false and val ~= nil
elseif(att == 'groupFilter') then
groupFilter = val
end
end
local append
if(raid) then
if(groupFilter) then
if(type(groupFilter) == 'number' and groupFilter > 0) then
append = 'Raid' .. groupFilter
elseif(groupFilter:match('MAINTANK')) then
append = 'MainTank'
elseif(groupFilter:match('MAINASSIST')) then
append = 'MainAssist'
else
local _, count = groupFilter:gsub(',', '')
if(count == 0) then
append = 'Raid' .. groupFilter
else
append = 'Raid'
end
end
else
append = 'Raid'
end
elseif(party) then
append = 'Party'
elseif(unit) then
append = unit:gsub('^%l', string.upper)
end
if(append) then
name = name .. append .. (unitsuffix or '')
end
-- Change oUF_LilyRaidRaid into oUF_LilyRaid
name = name:gsub('(%u%l+)([%u%l]*)%1', '%1')
-- Change oUF_LilyTargettarget into oUF_LilyTargetTarget
name = name:gsub('t(arget)', 'T%1')
name = name:gsub('p(et)', 'P%1')
name = name:gsub('f(ocus)', 'F%1')
local base = name
local i = 2
while(_G[name]) do
name = base .. i
i = i + 1
end
return name
end
do
local function styleProxy(self, frame)
return walkObject(_G[frame])
end
-- There has to be an easier way to do this.
local initialConfigFunction = [[
local header = self:GetParent()
local frames = table.new()
table.insert(frames, self)
self:GetChildList(frames)
for i = 1, #frames do
local frame = frames[i]
local unit
-- There's no need to do anything on frames with onlyProcessChildren
if(not frame:GetAttribute('oUF-onlyProcessChildren')) then
RegisterUnitWatch(frame)
-- Attempt to guess what the header is set to spawn.
local groupFilter = header:GetAttribute('groupFilter')
if(type(groupFilter) == 'string' and groupFilter:match('MAIN[AT]')) then
local role = groupFilter:match('MAIN([AT])')
if(role == 'T') then
unit = 'maintank'
else
unit = 'mainassist'
end
elseif(header:GetAttribute('showRaid')) then
unit = 'raid'
elseif(header:GetAttribute('showParty')) then
unit = 'party'
end
local headerType = header:GetAttribute('oUF-headerType')
local suffix = frame:GetAttribute('unitsuffix')
if(unit and suffix) then
if(headerType == 'pet' and suffix == 'target') then
unit = unit .. headerType .. suffix
else
unit = unit .. suffix
end
elseif(unit and headerType == 'pet') then
unit = unit .. headerType
end
frame:SetAttribute('*type1', 'target')
frame:SetAttribute('*type2', 'togglemenu')
frame:SetAttribute('oUF-guessUnit', unit)
end
local body = header:GetAttribute('oUF-initialConfigFunction')
if(body) then
frame:Run(body, unit)
end
end
header:CallMethod('styleFunction', self:GetName())
local clique = header:GetFrameRef('clickcast_header')
if(clique) then
clique:SetAttribute('clickcast_button', self)
clique:RunAttribute('clickcast_register')
end
]]
--[[ oUF:SpawnHeader(overrideName, template, visibility, ...)
Used to create a group header and apply the currently active style to it.
* self - the global oUF object
* overrideName - unique global name to be used for the header. Defaults to an auto-generated name based on the name
of the active style and other arguments passed to `:SpawnHeader` (string?)
* template - name of a template to be used for creating the header. Defaults to `'SecureGroupHeaderTemplate'`
(string?)
* visibility - macro conditional(s) which define when to display the header (string).
* ... - further argument pairs. Consult [Group Headers](http://wowprogramming.com/docs/secure_template/Group_Headers.html)
for possible values.
In addition to the standard group headers, oUF implements some of its own attributes. These can be supplied by the
layout, but are optional.
* oUF-initialConfigFunction - can contain code that will be securely run at the end of the initial secure
configuration (string?)
* oUF-onlyProcessChildren - can be used to force headers to only process children (boolean?)
--]]
function oUF:SpawnHeader(overrideName, template, visibility, ...)
if(not style) then return error('Unable to create frame. No styles have been registered.') end
template = (template or 'SecureGroupHeaderTemplate')
local isPetHeader = template:match('PetHeader')
local name = overrideName or generateName(nil, ...)
local header = CreateFrame('Frame', name, PetBattleFrameHider, template)
header:SetAttribute('template', 'SecureUnitButtonTemplate, SecureHandlerStateTemplate, SecureHandlerEnterLeaveTemplate')
for i = 1, select('#', ...), 2 do
local att, val = select(i, ...)
if(not att) then break end
header:SetAttribute(att, val)
end
header.style = style
header.styleFunction = styleProxy
header.visibility = visibility
-- Expose the header through oUF.headers.
table.insert(headers, header)
-- We set it here so layouts can't directly override it.
header:SetAttribute('initialConfigFunction', initialConfigFunction)
header:SetAttribute('_initialAttributeNames', '_onenter,_onleave,refreshUnitChange,_onstate-vehicleui')
header:SetAttribute('_initialAttribute-_onenter', [[
local snippet = self:GetAttribute('clickcast_onenter')
if(snippet) then
self:Run(snippet)
end
]])
header:SetAttribute('_initialAttribute-_onleave', [[
local snippet = self:GetAttribute('clickcast_onleave')
if(snippet) then
self:Run(snippet)
end
]])
header:SetAttribute('_initialAttribute-refreshUnitChange', [[
local unit = self:GetAttribute('unit')
if(unit) then
RegisterStateDriver(self, 'vehicleui', '[@' .. unit .. ',unithasvehicleui]vehicle; novehicle')
else
UnregisterStateDriver(self, 'vehicleui')
end
]])
header:SetAttribute('_initialAttribute-_onstate-vehicleui', [[
local unit = self:GetAttribute('unit')
if(newstate == 'vehicle' and unit and UnitPlayerOrPetInRaid(unit) and not UnitTargetsVehicleInRaidUI(unit)) then
self:SetAttribute('toggleForVehicle', false)
else
self:SetAttribute('toggleForVehicle', true)
end
]])
header:SetAttribute('oUF-headerType', isPetHeader and 'pet' or 'group')
if(_G.Clique) then
SecureHandlerSetFrameRef(header, 'clickcast_header', _G.Clique.header)
end
if(header:GetAttribute('showParty')) then
self:DisableBlizzard('party')
end
if(visibility) then
local type, list = string.split(' ', visibility, 2)
if(list and type == 'custom') then
RegisterAttributeDriver(header, 'state-visibility', list)
header.visibility = list
else
local condition = getCondition(string.split(',', visibility))
RegisterAttributeDriver(header, 'state-visibility', condition)
header.visibility = condition
end
end
return header
end
end
--[[ oUF:Spawn(unit, overrideName)
Used to create a single unit frame and apply the currently active style to it.
* self - the global oUF object
* unit - the frame's unit (string)
* overrideName - unique global name to use for the unit frame. Defaults to an auto-generated name based on the unit
(string?)
oUF implements some of its own attributes. These can be supplied by the layout, but are optional.
* oUF-enableArenaPrep - can be used to toggle arena prep support. Defaults to true (boolean)
--]]
function oUF:Spawn(unit, overrideName)
argcheck(unit, 2, 'string')
if(not style) then return error('Unable to create frame. No styles have been registered.') end
unit = unit:lower()
local name = overrideName or generateName(unit)
local object = CreateFrame('Button', name, PetBattleFrameHider, 'SecureUnitButtonTemplate')
Private.UpdateUnits(object, unit)
self:DisableBlizzard(unit)
walkObject(object, unit)
object:SetAttribute('unit', unit)
RegisterUnitWatch(object)
return object
end
--[[ oUF:SpawnNamePlates(prefix, callback, variables)
Used to create nameplates and apply the currently active style to them.
* self - the global oUF object
* prefix - prefix for the global name of the nameplate. Defaults to an auto-generated prefix (string?)
* callback - function to be called after a nameplate unit or the player's target has changed. The arguments passed to
the callback are the updated nameplate, if any, the event that triggered the update, and the new unit
(function?)
* variables - list of console variable-value pairs to be set when the player logs in (table?)
--]]
function oUF:SpawnNamePlates(namePrefix, nameplateCallback, nameplateCVars)
argcheck(nameplateCallback, 3, 'function', 'nil')
argcheck(nameplateCVars, 4, 'table', 'nil')
if(not style) then return error('Unable to create frame. No styles have been registered.') end
if(_G.oUF_NamePlateDriver) then return error('oUF nameplate driver has already been initialized.') end
local style = style
local prefix = namePrefix or generateName()
-- Because there's no way to prevent nameplate settings updates without tainting UI,
-- and because forbidden nameplates exist, we have to allow default nameplate
-- driver to create, update, and remove Blizz nameplates.
-- Disable only not forbidden nameplates.
NamePlateDriverFrame:HookScript('OnEvent', function(_, event, unit)
if(event == 'NAME_PLATE_UNIT_ADDED' and unit) then
self:DisableBlizzard(unit)
end
end)
local eventHandler = CreateFrame('Frame', 'oUF_NamePlateDriver')
eventHandler:RegisterEvent('NAME_PLATE_UNIT_ADDED')
eventHandler:RegisterEvent('NAME_PLATE_UNIT_REMOVED')
eventHandler:RegisterEvent('PLAYER_TARGET_CHANGED')
if(IsLoggedIn()) then
if(nameplateCVars) then
for cvar, value in next, nameplateCVars do
SetCVar(cvar, value)
end
end
else
eventHandler:RegisterEvent('PLAYER_LOGIN')
end
eventHandler:SetScript('OnEvent', function(_, event, unit)
if(event == 'PLAYER_LOGIN') then
if(nameplateCVars) then
for cvar, value in next, nameplateCVars do
SetCVar(cvar, value)
end
end
elseif(event == 'PLAYER_TARGET_CHANGED') then
local nameplate = C_NamePlate.GetNamePlateForUnit('target')
if(nameplateCallback) then
nameplateCallback(nameplate and nameplate.unitFrame, event, 'target')
end
-- UAE is called after the callback to reduce the number of
-- ForceUpdate calls layout devs have to do themselves
if(nameplate) then
nameplate.unitFrame:UpdateAllElements(event)
end
elseif(event == 'NAME_PLATE_UNIT_ADDED' and unit) then
local nameplate = C_NamePlate.GetNamePlateForUnit(unit)
if(not nameplate) then return end
if(not nameplate.unitFrame) then
nameplate.style = style
nameplate.unitFrame = CreateFrame('Button', prefix..nameplate:GetName(), nameplate)
nameplate.unitFrame:EnableMouse(false)
nameplate.unitFrame.isNamePlate = true
Private.UpdateUnits(nameplate.unitFrame, unit)
walkObject(nameplate.unitFrame, unit)
else
Private.UpdateUnits(nameplate.unitFrame, unit)
end
nameplate.unitFrame:SetAttribute('unit', unit)
if(nameplateCallback) then
nameplateCallback(nameplate.unitFrame, event, unit)
end
-- UAE is called after the callback to reduce the number of
-- ForceUpdate calls layout devs have to do themselves
nameplate.unitFrame:UpdateAllElements(event)
elseif(event == 'NAME_PLATE_UNIT_REMOVED' and unit) then
local nameplate = C_NamePlate.GetNamePlateForUnit(unit)
if(not nameplate) then return end
nameplate.unitFrame:SetAttribute('unit', nil)
if(nameplateCallback) then
nameplateCallback(nameplate.unitFrame, event, unit)
end
end
end)
end
--[[ oUF:AddElement(name, update, enable, disable)
Used to register an element with oUF.
* self - the global oUF object
* name - unique name of the element (string)
* update - used to update the element (function)
* enable - used to enable the element for a given unit frame and unit (function)
* disable - used to disable the element for a given unit frame (function)
--]]
function oUF:AddElement(name, update, enable, disable)
argcheck(name, 2, 'string')
argcheck(update, 3, 'function', 'nil')
argcheck(enable, 4, 'function')
argcheck(disable, 5, 'function')
if(elements[name]) then return error('Element [%s] is already registered.', name) end
elements[name] = {
update = update;
enable = enable;
disable = disable;
}
end
oUF.version = _VERSION
--[[ oUF.objects
Array containing all unit frames created by `oUF:Spawn`.
--]]
oUF.objects = objects
--[[ oUF.headers
Array containing all group headers created by `oUF:SpawnHeader`.
--]]
oUF.headers = headers
if(global) then
if(parent ~= 'oUF' and global == 'oUF') then
error('%s is doing it wrong and setting its global to "oUF".', parent)
elseif(_G[global]) then
error('%s is setting its global to an existing name "%s".', parent, global)
else
_G[global] = oUF
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,235 |
package org.apache.pinot.core.util;
import java.util.HashSet;
import java.util.List;
import java.util.Set;
import javax.annotation.Nullable;
import org.apache.commons.lang3.StringUtils;
import org.apache.pinot.core.data.function.FunctionEvaluator;
import org.apache.pinot.core.data.function.FunctionEvaluatorFactory;
import org.apache.pinot.spi.config.table.ingestion.FilterConfig;
import org.apache.pinot.spi.config.table.ingestion.IngestionConfig;
import org.apache.pinot.spi.config.table.ingestion.TransformConfig;
import org.apache.pinot.spi.data.FieldSpec;
import org.apache.pinot.spi.data.Schema;
import org.apache.pinot.spi.data.readers.GenericRow;
/**
* Utility methods for extracting source and destination fields from ingestion configs
*/
public class IngestionUtils {
/**
* Extracts all fields required by the {@link org.apache.pinot.spi.data.readers.RecordExtractor} from the given TableConfig and Schema
* Fields for ingestion come from 2 places:
* 1. The schema
* 2. The ingestion config in the table config. The ingestion config (e.g. filter) can have fields which are not in the schema.
*/
public static Set<String> getFieldsForRecordExtractor(@Nullable IngestionConfig ingestionConfig, Schema schema) {
Set<String> fieldsForRecordExtractor = new HashSet<>();
extractFieldsFromIngestionConfig(ingestionConfig, fieldsForRecordExtractor);
extractFieldsFromSchema(schema, fieldsForRecordExtractor);
return fieldsForRecordExtractor;
}
/**
* Extracts all the fields needed by the {@link org.apache.pinot.spi.data.readers.RecordExtractor} from the given Schema
* TODO: for now, we assume that arguments to transform function are in the source i.e. no columns are derived from transformed columns
*/
private static void extractFieldsFromSchema(Schema schema, Set<String> fields) {
for (FieldSpec fieldSpec : schema.getAllFieldSpecs()) {
if (!fieldSpec.isVirtualColumn()) {
FunctionEvaluator functionEvaluator = FunctionEvaluatorFactory.getExpressionEvaluator(fieldSpec);
if (functionEvaluator != null) {
fields.addAll(functionEvaluator.getArguments());
}
fields.add(fieldSpec.getName());
}
}
}
/**
* Extracts the fields needed by a RecordExtractor from given {@link IngestionConfig}
*/
private static void extractFieldsFromIngestionConfig(@Nullable IngestionConfig ingestionConfig, Set<String> fields) {
if (ingestionConfig != null) {
FilterConfig filterConfig = ingestionConfig.getFilterConfig();
if (filterConfig != null) {
String filterFunction = filterConfig.getFilterFunction();
if (filterFunction != null) {
FunctionEvaluator functionEvaluator = FunctionEvaluatorFactory.getExpressionEvaluator(filterFunction);
if (functionEvaluator != null) {
fields.addAll(functionEvaluator.getArguments());
}
}
}
List<TransformConfig> transformConfigs = ingestionConfig.getTransformConfigs();
if (transformConfigs != null) {
for (TransformConfig transformConfig : transformConfigs) {
FunctionEvaluator expressionEvaluator =
FunctionEvaluatorFactory.getExpressionEvaluator(transformConfig.getTransformFunction());
fields.addAll(expressionEvaluator.getArguments());
fields.add(transformConfig
.getColumnName()); // add the column itself too, so that if it is already transformed, we won't transform again
}
}
}
}
/**
* Returns false if the record contains key {@link GenericRow#SKIP_RECORD_KEY} with value true
*/
public static boolean shouldIngestRow(GenericRow genericRow) {
return !Boolean.TRUE.equals(genericRow.getValue(GenericRow.SKIP_RECORD_KEY));
}
public static Long extractTimeValue(Comparable time) {
if (time != null) {
if (time instanceof Number) {
return ((Number) time).longValue();
} else {
String stringValue = time.toString();
if (StringUtils.isNumeric(stringValue)) {
return Long.parseLong(stringValue);
}
}
}
return null;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,635 |
Roofing Company Dumfries and Galloway: Do I need a new roof?
When your roof is coming to the end of its life, it's important that you reinstate it before the Scottish weather breaks through.
At Carnegie Contracts, our roofing services team specialise in helping customers pick the right replacement for their existing roof. We also guarantee that your new roof will keep your home protected for a long time.
Roofing Company Dumfries and Galloway: What type of roof do I need?
No matter if you have a slate roof or a tile roof, we can help you to pick a brand new one, which not only delivers its principal function but also looks they way you envisioned it would seem.
We fit and can supply a wide selection of tiles or slates from various manufacturers.
Formed in 1997, we've got real experience in both slate roof replacement and tile roof replacement. At Carnegie Contracts, we can offer you all of the help and guidance you demand for your new roof.
Roofing Services in Dumfries and Galloway: What sort of roofing services do I want?
Properties in Dumfries and Galloway and House's and or other Scottish homes.
Do your tiles or slates have to do the function of making a watertight barrier, which keeps your property dry, however they need to seem amazing from instalment, also.
Don't forget, that picking even stone roofing, slate or clay tile might be decided by the local planning authority. Your house may have to be in keeping with other properties. They may even need to approve your samples, before giving the go ahead.
At Carnegie Contracts, we think the tile finish is crucial to the overall appearance of your home. In Dumfries and Galloway, sand-faced tiles are more popular. Smooth-faced characteristic in Dumfries and Galloway.
Sand-faced concludes create a rustic look but can weather faster. Smooth-faced finishes provide a slippery surface which helps prevent moss from growing and surface run-off.
Roof profiles that are different are typical throughout the country. With basic tiles common. S shaped pantiles are the most visible in the Scotland. Whatever type of tile you choose for your house, it's significant to stick with basic tiles or a look common to your area.
If you're looking for Roofing Company Dumfries and Galloway, phone us on 0141 280 2928 or 0131 510 2622 at Carnegie Contracts now. We can offer an accurate on-site inspection plus a quotation that is realistic. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,541 |
Lori talks to Michael Gurian, about his most recent book LESSONS OF LIFELONG INTIMACY: Building A Stronger Marriage Without Losing Yourself -- The Nine Principles Of A Balanced And Happy Relationship.
This episode is also brought to you by Next Issue, Next issue is like Netflix for magazines…only better - with access to all the latest issues. NextIssue.com has all the best, most up to date magazines on the newsstand but delivered to your phone or tablet. Go to NextIssue.com/LOVE for your free trial. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,852 |
{"url":"https:\/\/paligo.net\/docs\/en\/index-title.html","text":"Index Title\n\nAbstract\n\nLearn how to add a title element to an Index in Paligo. Typically, index titles are used when you what to give the index topic a different name than \"Index\".\n\nBy default, the index element does not include a title (PDF). This is because we recommend that for an index, you create a topic called \"Index\" and add the index element to that. In that scenario, there is no need for the index to have a separate title to the topic.\n\nBut it is also possible to add an index to a topic that is not called \"Index\". For example, you could add an index to a topic called \"Reference\".\n\nIn this scenario, it can be a good idea to include an index title as well. The title will make it easier for your readers to find your index, as there is no \"Index\" topic. In the following image, an index has been added to a \"References\" topic, and the index has a title called \"Index\".\n\nIf you are going to use an index title, you should consider:\n\nWhen you add an index element to a topic, it does not include an index title. If you want to add an index title, you can either:\n\n\u2022 Add the title element inside the index element\n\n\u2022 Set Paligo to automatically generate an index title for you, when you publish.\n\nIf you want an automatic index title, see Automatic Index Title for PDF Outputs . For HTML5 outputs, an index title is included by default - if you have added a title manually, Paligo will use that. Otherwise, it will create an automatic title for you.\n\nNote\n\nThe index title is shown in HTML5 outputs by default. But you can hide it by using CSS (see Index Title for HTML5 Outputs).\n\n1. Open the index topic in the Paligo editor.\n\n2. Select the index element.\n\n3. Use the element context menu to add the the title element.\n\n4. Enter the text for your index title inside the title element.\n\n5. Select Save.\n\nAutomatic Index Title for PDF Outputs\n\nPaligo can automatically add a title to your index. The automatic title has the text \"Index\" and if you publish to other languages, a translation of \"Index\" is provided as well.\n\nTo use an automatic index title for PDF outputs:\n\n1. Create a PDF layout. Alternatively, you can edit an existing PDF layout.\n\n2. In the PDF layout, select General > Glossary and Index.\n\n3. Set Index auto title to Enable.\n\nWhen you publish, Paligo will check to see if your index has a title.\n\n\u2022 If the index already has a title, Paligo will use that title. It will not generate an automatic title.\n\n\u2022 If the index does not have a title, Paligo will add a title element with \"Index\" as the title text.\n\nIf you set Index auto title to Disable, Paligo will not create an automatic index title.\n\n4. Select Save.\n\nWhen you publish to PDF with this layout, the output will include\/exclude an automatic index title.\n\nIndex Title included in PDF Bookmarks\n\nPDFs can have bookmarks that act like a table of contents in a side panel, where your topics are shown in order. If you have an index, it's likely that you will want a link to the index to appear here.\n\nIf your index is inside a topic called \"Index\", you will not need to take any action. The \"Index\" topic will appear in the bookmarks by default.\n\nBut if your index is inside a topic with a different title, there will be no obvious way for the reader to access the index. For example, let's say you have added your index to a topic called \"references\". In the published PDF, the bookmarks will only show \"References\", which makes it harder for your readers to find the index.\n\nTo fix this, you can set Paligo to include the index's title in the bookmarks as well. If your index does not have a title, you can add one or you can set Paligo to generate one automatically.\n\n1. Create a PDF layout. Alternatively, you can edit an existing PDF layout.\n\n2. In the PDF layout, select General > Glossary and Index.\n\n3. Enable the Index title or auto title in bookmarks setting to get Paligo to include the title of the index element in the bookmarks.\n\nNote\n\nThis setting will only work if your index has a title, or you have set Paligo to generate a title automatically.\n\n4. Select Save.\n\nWhen you publish to PDF with this layout, Paligo includes the index title in the bookmarks. It is a subsection of its parent topic.\n\nIndex Title for HTML5 Outputs\n\n\u2022 Your index contains a title element\n\n\u2022 Your index has no title element, but your HTML5 layout is set to generate an index title automatically. The automatic title is called \"Index\" (or a translation of Index for other languages).\n\nThis can mean that your HTML index has two \"Index\" titles, one for the topic and one for the index element. To hide the title for the index element, use CSS:\n\n1. Create or edit an existing custom CSS file and add the following code:\n\n.index .titlepage{\ndisplay: none;\n}\n\n2. Create an HTML5 Help Center layout. Alternatively, you can edit an existing HTML5 Help Center layout.\n\n.index .titlepage{\n}","date":"2022-09-24 23:32:36","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\": 0.47980034351348877, \"perplexity\": 1968.4206827133912}, \"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-2022-40\/segments\/1664030333541.98\/warc\/CC-MAIN-20220924213650-20220925003650-00648.warc.gz\"}"} | null | null |
Die Geschichte Sienas beginnt mit den Etruskern, von denen die Stadt wahrscheinlich gegründet und Saena genannt wurde. Einige Gräber jenes Zeitalters sind außerhalb der Porta Camollia gefunden worden. Siena wurde, nachdem es unter römische Herrschaft gefallen war, in der Regierungszeit des Augustus oder etwas früher eine römische Colonia. Die Stadt trug zu dieser Zeit den Namen Saena Iulia und dasselbe Wappen wie Rom: die Wölfin und die Zwillinge. Aber ihre eigentliche Bedeutung erlangte die Stadt im Mittelalter. Wenige Zeugnisse aus der römischen Epoche oder aus den ersten Jahrhunderten des Christentums sind erhalten (außer der Legende des St. Ansanus), und überhaupt keine aus dem Zeitraum, der der langobardischen Periode vorausging.
Legende der Stadtgründung
Der Sage nach mussten die Kinder des Remus, Senius und Aschius, nach dem Konflikt ihres Vaters mit Romulus aus Rom flüchten. Sie gelangten an die Ufer des Flusses Tressa und errichteten hier an einer Anhöhe das Castel Senio (heute Castelvecchio). Sie entfachten ein Feuer, das schwarzen und weißen Rauch entwickelte und so zu den Stadtfarben Sienas (Balzana) wurde.
Mittelalterliche Frühzeit
Es gibt dokumentarische Indizien, dass es im 7. Jahrhundert während der Regierung von Rotaris (oder Rotari) einen Bischof Mauro in Siena gab. Versuche, frühere Bischöfe bis ins 5. Jahrhundert zurückzuverfolgen, haben nur vage und widersprüchliche Ergebnisse geliefert. Dokumentiert sind die Konflikte mit dem Bischof von Arezzo, als sich in den ersten Jahren des 8. Jahrhunderts um die Zugehörigkeit einiger Pieven im Raum der Crete Senesi gestritten wurde. Unter den Langobarden war die Stadtregierung in den Händen eines Gastaldo, unter den Karolingern eines Grafen, dessen Autorität allmählich und im Laufe von Ereignissen, die denen in anderen italienischen Städten ähnelten, auf den Bischof überging. 1055 setzte Kaiser Heinrich III. einen Provisor im Sieneser Territorium ein, einen Salamone Piccolomini, womit diese Familie erstmals in die Geschichte Sienas eingriff.
Die bischöfliche Macht ging wiederum allmählich auf städtische Konsuln über. Schriftliche Überlieferungen zeigen eine Konsulregierung in Siena von 1125 bis 1212; die Anzahl der Konsuln variierte zwischen drei und zwölf. Diese Regierung aus gentiluomini oder Adligen änderte sich allmählich zur Beteiligung der popolani oder unteren Schichten, deren Bemühungen zum Aufstieg an die Macht stetig wuchsen. So erhielten sie 1137 einen dritten Teil der Regierung durch die Wiederherstellung eines allgemeinen Rats aus 100 Adligen und 50 popolani. Im selben Jahr fiel die Festung von Staggia Senese in den seneser Machtbereich. Durch die Lage an der Via Francigena und dem damit verbundenen Bevölkerungszuwachs wurden ab der Mitte des 12. Jahrhunderts die Stadtmauern von Siena von der damals nur auf dem Hügel Terzo di Città gelegenen Stadt bis zur heutigen Porta Camollia erweitert. 1199 versetzte die Einsetzung eines fremden Podestà (einer Form der Regierung, die 1212 permanent wurde) dem Konsulnamt einen schweren Schlag und löschte es bald aus. 1233 erhob sich das Volk wiederum gegen die Adligen in der Hoffnung, sie völlig aus den Ämtern zu verdrängen.
Der Streit war im Wesentlichen wirtschaftlicher Natur, da das Volk die Steuerbefreiung des Adels abschaffen wollte. Der Versuch war nicht ganz erfolgreich, aber die Regierung war nun durch die Bildung eines höchsten Magistrats aus 24 Bürgen, 12 Adligen und 12 popolani zu gleichen Teilen zwischen den Ständen aufgeteilt. Während der Herrschaft der Adligen und der gemischten Herrschaft von Adligen und popolani wurde die Gemeinde Sienas durch glückliche Aneignungen von Land in der Nachbarschaft und durch die Unterwerfung von Feudalherren, wie die Scialenghi, Aldobrandeschi, Pannocchieschi, Visconti di Campiglia d'Orcia etc. vergrößert.
Konflikt mit Florenz
Vermehrte militärische Aktivitäten begannen am Beginn des 13. Jahrhunderts. Zunächst errichtete der Podestà Guelfo da Porcari zwischen 1213 und 1219 Monteriggioni als defensiven Stützpunkt an der Grenze zu Florenz, 1221 wurden Orte der Aldobrandeschi in der Maremma erobert, Grosseto 1224 eingenommen. Die Stadtmauern von Montefollonico als strategischer Punkt in der Grenzverteidigung zu Montepulciano wurden 1234 verstärkt.
Da Florenz und Siena neues Territorium suchten sowie Grenzstreitigkeiten, insbesondere mit Poggibonsi und Montepulciano, zu ständigen Feindseligkeiten führten, war der große Konflikt vorgezeichnet. Die Sienesen stellten sich auf die ghibellinische Seite, um die rivalisierende Republik zu provozieren, und die deutschen Kaiser, angefangen mit Friedrich Barbarossa, belohnten ihre Treue mit verschiedenen Privilegien (Münzrecht, Gerichtsbarkeit, Konsulwahl). Die Unruhen hielten an, kleinere Kriege und schnelle Wiederversöhnungen zwischen Florenz und Siena folgten, bis 1254/55 ein verbindlicherer Frieden und ein Bündnis geschlossen wurden. Aber dieser Vertrag verschärfte trotz seiner scheinbaren Stabilität innerhalb weniger Jahre den Streit. Denn 1258 beschwerten sich die Florentiner über die Verletzung der Bedingungen durch Siena, indem es aus Florenz verbannten Ghibellinen Zuflucht gewährt habe. Als die Sienesen sich weigerten, diesen Protesten nachzugeben, machten sich beide Staaten an umfangreiche Kriegsvorbereitungen.
Siena wandte sich an Manfred, erhielt von ihm eine starke Gruppe deutscher Reitern unter dem Kommando von Conte Giordano, und ersuchte ebenso seine ghibellinischen Verbündeten um Hilfe. Florenz rüstete eine mächtige Bürgerarmee aus, deren originale Register noch in dem Band Il libro di Montaperti in den Florentiner Archiven erhalten sind. Die vom Podestà von Florenz und zwölf Capitani geführte Armee machte sich im April 1260 optimistisch auf den Marsch in die feindlichen Territorien und errang am 18. Mai bei Santa Petronilla außerhalb der nördlichen Stadtmauern Sienas einen unbedeutenden Sieg. Aber in einem zweiten und wichtigeren Feldzug, an dem die Milizen anderer guelfischen Städte der Toskana teilnahmen, wurden die Florentiner am 4. September 1260 bei Montaperti entscheidend geschlagen.
Diese Niederlage brach die Macht von Florenz für viele Jahre und löschte scheinbar die Florentiner Guelfen aus. Aber die Schlacht bei Benevent (1266) und die Begründung der Dynastie des Karl von Anjou auf dem neapolitanischen Thron setzte der ghibellinischen Vorherrschaft in der Toskana ein Ende. Das ghibellinische Siena spürte bald die Auswirkungen bei der Niederlage seiner Armee bei Colle di Val d'Elsa (1269) gegen die vereinigten Kräfte der guelfischen Exilanten, Florentiner und Franzosen, und beim Tod seines mächtigen Bürgers Provenzano Salvani (von Dante erwähnt) in dieser Schlacht, der der führende Geist der Regierung zur Zeit des Siegs von Montaperti gewesen war.
Innenpolitische Veränderungen (1269–1355)
Für einige Zeit blieb Siena der ghibellinischen Sache noch treu. Dennoch begannen guelfische und demokratische Ansichten sich Bahn zu brechen. Die Ghibellinen wurden bei mehreren Gelegenheiten aus der Stadt vertrieben, und selbst, als eine vorübergehende Versöhnung der beiden Parteien ihnen die Rückkehr ermöglichte, erlangten sie nicht ihren früheren Einfluss.
Inzwischen hatte sich die populäre Partei zunehmend Macht im Staat errungen. Empört über die Tyrannei der Salimbeni und anderer mit den Ghibellinen verbündeter Patrizierfamilien beschloss sie 1277 den Ausschluss aller Adligen aus dem obersten Magistrat (der 1270 aus 64 statt 24 Mitgliedern bestand) und bestand darauf, dass dieser Rat allein aus guelfischen Händlern und Männern der Mittelklasse gebildet werden solle. Diese Verfassung wurde 1280 durch die Verkleinerung des obersten Magistrats auf 15 Mitglieder bekräftigt, die allesamt aus einfacheren Schichten stammten, und wurde endgültig 1285 (und 1287) durch die Einsetzung des Magistrats der Neun sanktioniert, der nur aus Bürgern bestand.
Dieser Neunerrat führte die Regierung für rund siebzig Jahre, und seine Herrschaft war klug und friedlich. Die Territorien des Staats wurden vergrößert, ein freundschaftliches Bündnis mit Florenz gepflegt, der Handel blühte. Das Gebiet für den seneser Hafen in Talamone wurde 1303 erworben und der Hafen ausgebaut. 1321 wurde die Universität Siena gegründet bzw. durch das Heranziehen von Gelehrten aus Bologna wiederbelebt, als offizielles Gründungsjahr der Universität wird heute 1240 angegeben; die Hauptgebäude, die heute die Stadt schmücken, wurden begonnen; und die wohltätigen Institutionen gediehen.
Seit 1327 stand der Condottiere Guidoriccio da Fogliano im Dienste der Stadt. Unter seiner Führung wurde Konflikte mit der Republik Pisa und den Aldobrandeschi in der Maremma ausgetragen. 1328 eroberte er gegen Castruccio Castracani nach siebenmonatiger Belagerung den Ort Montemassi. Dies war der Seneser Regierung Anlass, bei Simone Martini das Fresko Guidoriccio da Fogliano all'assedio di Montemassi (Guidoriccio da Fogliano bei der Belagerung von Montemassi) in Auftrag zu geben, welches sich im Saal der Landkarten (Sala del Mappamondo) im Rathaus von Siena befindet.
Die Pest (1348)
Im Jahr 1348 brach die Pest in Siena aus, zwei Drittel der Bevölkerung starben. Davon sollte sich die Stadt nicht mehr erholen und konnte sich danach auch nicht mehr gegen den alten Rivalen Florenz behaupten.
Der Zwölferrat (1355–1368)
Aber inzwischen hatte die Exklusivität einer einzigen Bürgergruppe, aus deren Reihen sich der Hauptmagistrat rekrutierte, die Regierung in eine geschlossene Oligarchie umgewandelt und den Hass aller anderen Gruppen geschürt. Adlige, Richter, Notare und das einfache Volk erhoben sich häufig in Revolten, während die Neun ihren Staat durch eine starke Bürgermiliz verteidigten (1295–1309), die in terzieri (Sektionen) und contrade (Stadtteile innerhalb der Stadtmauern) gegliedert war und diese Versuche gewaltsam unterdrückte. Aber 1355 gab die Ankunft Karls IV. in Siena den Aufrührern neuen Mut. Von der kaiserlichen Autorität unterstützt, stürzten sie die Regierung der Neun und richteten einen Magistrat der Zwölf aus der untersten Schicht ein. Diese neuen Herrscher standen in einem gewissen Ausmaß unter dem Einfluss der Adligen, die die Rebellion geschürt hatten, aber letztere wurden wieder bald von jeder Beteiligung an der Regierung ausgeschlossen.
Damit begann der Kampf um die Vorherrschaft, der für viele Jahre zwischen den verschiedenen Bürgergruppen ausgetragen wurde, die dort ordini oder monti genannt wurden. Die unteren strebten nach der Macht, während die höheren Klassen im Amt danach strebten, alle Macht in ihren eigenen Händen zu behalten oder sie in Proportion zu der relativen Stärke jedes monte aufzuteilen.
Die Zwölf, die den Rat der Neun ersetzt hatten (so wie diese vorher den Rat der Adligen ersetzt hatten), bestanden sowohl als Individuen als auch als Partei aus unwissenden, unfähigen, stürmischen Männern, die weder den Staat mit fester Hand führen noch der Republik zu Wohlstand verhelfen konnten. Sie brachen rasch mit dem Adel, für dessen Manöver sie zunächst nützliche Werkzeuge gewesen waren, und spalteten sich dann in zwei Parteien nach Familienverbünden, die eine auf Seiten der Tolomei, die heftigere andere auf Seiten der Salimbeni und der Noveschi (Anhänger der Neun), die immer noch Einfluss in der Stadt hatten und wahrscheinlich diese Uneinigkeit schürten und jede sich bietende Chance ausnutzten, die ihnen vielleicht wieder zur Macht verhelfen konnte.
1359 konnte der Krieg mit Perugia beendet werden. In den Friedensverhandlungen wurde Montepulciano dem Gebiet von Perugia zugeschlagen, Siena erhielt die Herrschaft über Cortona.
Erneute Umwälzungen (1368–1369)
1368 gelang es den Gegnern der Zwölf, sie gewaltsam aus dem Palazzo Pubblico zu verjagen und durch eine Regierung aus 13 Adligen und 3 Noveschi zu ersetzen. Diese Regierung dauerte nur 22 Tage vom 2. bis zum 24. September an und wurde mühelos durch die dominante Partei der Dodicini (Anhänger der Zwölf) gestürzt, die von den Salimbeni und vom Pöbel unterstützt und von Kaiser Karl IV. begünstigt wurden. Die Adligen wurden überwältigt und sowohl aus der Stadt als auch von der Macht verjagt; aber die Herrschaft der Zwölf wurde zu einem Ende gebracht und das Recht der Beteiligung an der Regierung auf eine weitere Bürgerklasse ausgedehnt. Denn nach der Ausweisung der Dreizehn aus dem Palast bildete ein Rat aus 124 Plebejern einen neuen Magistrat aus zwölf difensori (Verteidiger), der aus 5 Mitgliedern des popolo minuto oder dem niedrigsten Volk (das nun erstmals in die Regierung aufgenommen wurde), 4 Mitgliedern aus den Dodicini und 3 Mitgliedern aus den Noveschi bestand. Er war jedoch von nur kurzer Dauer, denn die Dodicini waren mit ihrem Anteil nicht zufrieden, und im Dezember desselben Jahrs (1368) taten sie sich mit dem popolo minuto zusammen, um die drei Noveschi aus dem Palast zu werfen.
Aber der neue Volksstand, der seine Vorherrschaft im Rat der riformatori bereits behauptet hatte, verdrängte nun die Dodicini und behielt für fünf Tage (11.–16. Dezember) die Regierung in seinen eigenen Händen. Aus Angst vor dem Kaiser, der auf seinem Weg nach Rom zwei Monate zuvor durch Siena gekommen war und auf seiner Rückreise dort wieder haltmachen sollte, versuchte er seine Feinde durch die Gründung eines neuen Rats aus 150 riformatori zu beschwichtigen. Dieser ersetzte die 12 difensori durch einen neuen 15-köpfigen obersten Rat, der aus 8 Popolani, 4 Dodicini sowie 3 Noveschi bestand. Aus dieser Erneuerung datiert die Bildung eines neuen Standes, des monte dei riformatori; dieser Titel wurde von da an allen Bürgern verliehen, die die Regierung reformiert hatten und seit 1368 an ihr beteiligt waren.
Der turbulente Kampf der Zwölf und der Salimbeni, die mit diesen Änderungen unzufrieden waren, wendete sich rasch gegen die neue Regierung. Diesmal half ihnen Karl IV. aktiv; nachdem er aus Rom zurückgekehrt war, schickte er seine Miliz, kommandiert vom kaiserlichen Vikar Malatesta da Rima, um den Palazzo Pubblico anzugreifen. Aber das Sieneser Volk, vom Rat der Fünfzehn zu den Waffen gerufen, widersetzte sich entschlossen, schlug die kaiserlichen Truppen in die Flucht, eroberte das Banner und setzte den Kaiser im Salimbeni-Palast gefangen. Daraufhin einigte sich Karl mit der Regierung, gewährte ihr ein kaiserliches Patent und verließ die Stadt, für seine Demütigung durch die Schenkung einer großen Geldsumme getröstet.
Trotz seiner breiten Basis und großen Energie gelang es dem monte dei riformatori, dem Herz der neuen Regierung, nicht, zufriedenstellend mit den Angriffen von gegnerischen Parteien und verräterischen Verbündeten umzugehen. Um sie besser in den Griff zu bekommen, schaffte er 1369 einen Polizeichef mit dem Titel esecutore sowie einen zahlreichen Verband aus Popolani – die casata grande des Volks als Bollwerk gegen die Adligen, die aus der Verbannung zurückgerufen worden waren, und die nun, wenn auch durch strenge Regulierungen gefesselt, in Staatsämter wählbar waren.
Aber der Appetit des niederen Volks nach Macht wurde durch die Einsetzung der riformatori in den Hauptposten der Macht eher angeregt als gesättigt. Unter den Wollkämmerern der untersten Klasse, die in den abschüssigen Gassen um die Porta Ovile wohnten, gab es eine Organisation, die sich als Gesellschaft des Wurms bezeichnete. Während der Hungersnot von 1371 erhob sich diese Gesellschaft zur Revolte, plünderte die Häuser der Reichen, stürmte den Palazzo Pubblico, vertrieb aus dem Rat der Fünfzehn die 4 Mitglieder der Dodicini und die 3 Mitglieder der Noveschi. Als sie sich in ihr Quartier zurückgezogen hatten, wurden sie plötzlich von erzürnten Bürgern (Noveschi und Dodicini) angegriffen, die in Häuser und Werkstätten einbrachen und viele Einwohner ohne Rücksicht auf Alter oder Geschlecht umbrachten. Daraufhin rächten die populären Dachdecker diese Missetaten durch viele Exekutionen auf der Piazza. Dieser Aufruhr wurde erst durch erneute Änderungen im Rat der Fünfzehn gebremst. Er wurde nun aus 12 aus dem größeren Volk und 3 Noveschi gebildet. Die Dodicini wurden aus der Stadt verbannt.
Zeit der Riformatori (1369–1385)
Inzwischen hatte die Regierung auch mit Schwierigkeiten außerhalb der Stadtmauern zu kämpfen. Die benachbarten Herren überfielen und plünderten die Territorien der Stadt; ernste Verletzungen wurden durch die Händlergruppen zugefügt, insbesondere die Betrons und Gascons. Die rivalisierenden Ansprüche Carlo di Durazzos und Ludwigs von Anjou auf das neapolitanische Königreich verursachten erneute Unruhen in der Toskana. Die Sieneser Regierung erhoffte sich den Besitz der Stadt Arezzo, die zuerst von Durazzos Leuten und dann von Enguerrand VII. de Coucy für Ludwig von Anjou besetzt wurde. Aber während die Sieneser noch von der Eroberung träumten, verkaufte der französische General die Stadt überraschend an die Florentiner, deren Verhandlungen mit bewundernswertem Geschick geführt worden waren (1384).
Der aufkommende Ärger der Sieneser, insbesondere der Mittelklasse, gegen ihre Herrscher wurde durch diese herbe Enttäuschung zum Höhepunkt gebracht. Ihre Unzufriedenheit war allmählich durch verschiedene Entscheidungen in der Innen- und Außenpolitik während der sechzehnjährigen Herrschaft der riformatori angeschwollen, und die Konzessionen an die Anhänger der Zwölf und deren erneute Wählbarkeit ins Amt hatten nicht dabei geholfen, sie zu versöhnen. Schließlich brach die Revolte aus und gewann im März 1385 die Oberhand. Die riformatori wurden von der Macht ausgeschlossen und aus der Stadt verbannt.
Der Handel Sienas erlitt durch das Exil so vieler Handwerkerfamilien erheblichen Schaden. Die Fünfzehn wurden durch einen neuen obersten Magistrat aus 10 Prioren ersetzt, die in den folgenden Proportionen gewählt wurden: 4 aus den Dodicini, 4 aus den Noveschi und 2 aus dem gewöhnlichen Volk. All diejenigen, die an der Regierung Anteil gehabt hatten oder unter den riformatori im Rat gesessen hatten, wurden ausgeschlossen. So begann ein neuer Stand oder monte del popolo, bestehend aus Familien derselben Klasse wie die riformatori, die aber vorher nicht an der Regierung beteiligt gewesen waren. Sie genossen aber nur sehr begrenzte Privilegien.
Visconti und Florenz (1385–1409)
1387 führte erneuter Streit mit Florenz über Montepulciano zu einem offenen Krieg, der durch die Einmischung des ehrgeizigen Herzogs von Mailand Gian Galeazzo Visconti in die toskanischen Angelegenheiten noch verschärft wurde. Die Sieneser schlossen 1389 mit ihm ein Bündnis, akzeptierten zehn Jahre später seine Vorherrschaft und gaben die Freiheiten ihres Staates auf. Aber 1402 erleichterte der Tod Gian Galeazzos ihr Joch. In diesem Jahr wurde das erste Komplott gegen die Herrschaft der Visconti, das von den Dodicini und den Salimbeni ausgebrütet und von den Florentinern geschürt worden war, gewaltsam unterdrückt und führte zur Verdrängung der Dodicini aus dem Amt; aber im folgenden Jahr hob eine als Konsequenz aus den Aufständen geschaffene balia die Vorherrschaft des Herzogs auf und stellte die Freiheiten der Stadt wieder her.
Während dieses Zeitraums hatte der oberste Magistrat eine populärere Form angenommen. Durch die teilweise Wiederzulassung der riformatori und den Ausschluss der Dodicini, war die permanente balia nun zusammengesetzt aus 9 Prioren (3 aus den Noveschi, 3 aus dem Volk und 3 aus den riformatori) und einem Kapitän des Volks, der abwechselnd aus jedem der drei monti gewählt wurde. Am 11. April 1403 wurde mit den Florentinern Frieden geschlossen, und Siena erfreute sich mehrerer Jahre friedlichen Wohlstands.
Auseinandersetzung mit Neapel und dem Papsttum (1409–1452)
Aber das große westliche Schisma, das dann die christliche Welt aufwühlte, brachte wieder Unruhen nach Siena. Als Konsequenz aus den Entscheidungen des Konzils von Pisa hatten sich Florenz und Siena gegen Gregor XII. erklärt (1409). Als Unterstützer des Papstes nahm Ladislaus von Neapel daher die Gelegenheit wahr, um Überfälle auf das Territorium Sienas durchzuführen, es zu verwüsten und die Stadt zu bedrohen. Die Sieneser leisteten erbittert Widerstand, bis der Tod dieses Monarchen 1414 sie von den Angriffen befreite.
Ein erneuter Krieg mit Florenz brach 1431 aus, verursacht durch die florentinischen Bemühungen um Lucca und fortgesetzt infolge der Florentiner Allianz mit Venedig und Papst Eugen IV. und der Sieneser Allianz mit dem Herzog von Mailand und dem römischen König Sigismund. Dieser Monarch machte auf dem Weg nach Rom zu seiner Krönung in Siena halt und erhielt einen äußerst fürstlichen Empfang.
1433 unterzeichneten die gegnerischen Bündnisse einen Friedensvertrag, und obwohl er nachteilig für die Sieneser war und sie oft versucht waren, ihn zu brechen, hielten sie sich treu an seine Bedingungen. Während dieser vergleichsweise ruhigen Periode wurde Siena durch den Besuch von Papst Eugen IV. (1433) und den des Kaisers Friedrich III. geehrt. Der Kaiser empfing dort seine Braut Eleanor von Portugal aus den Händen von Bischof Aeneas Sylvius Piccolomini, seinem Sekretär und Historiker (1452). Dieses Zusammentreffen ist auf der Denkmalssäule festgehalten, die man außerhalb des Camollia-Tors sehen kann.
1452–1480
1452 wurden die Feindseligkeiten gegen Florenz wieder aufgenommen, als Antwort auf Invasionen und Zerstörungen auf Sieneser Territorium, die von Florentiner Truppen in ihren Konflikten mit Alfons von Neapel begangen wurden. Dieser hatte 1447 die Toskana zu seinem Schlachtfeld gemacht. 1454 wurde ein weiteres Mal mit Florenz Frieden geschlossen. Als Nächstes lag Siena mehrere Jahre lang mit Aldobrandino Orsini, Graf von Pitigliano und mit Jacopo Piccinini im Krieg und erlitt viele Katastrophen wegen Verrat durch seine Generäle.
Ungefähr um die gleiche Zeit war die Republik noch ernsterer Gefahr ausgesetzt, denn einige ihrer führenden Bürger verschworen sich, die Macht zu ergreifen und die Stadt unter die Vorherrschaft Alfonsos zu bringen, so wie sie zuvor unter der des Herzogs von Mailand gewesen war. Aber das Komplott kam ans Licht; ihre Rädelsführer wurden enthauptet und viele andere ins Exil geschickt (1456). Der Tod Alfonsos beendete schließlich alle Gefahren von dieser Seite.
Während dieser kritischen Zeiten wurde die Regierung des Staats durch einen neuen exekutiven Magistrat gestärkt, der balia benannt wurde und der von 1455 an unabhängig von den Prioren oder dem Konsistorium handelte. Bis dahin war er lediglich ein provisorisches Komitee gewesen, das letzterem anhing. Aber von nun an hatte die balia die oberste Rechtsprechung in allen Angelegenheiten des Staates, obwohl sie immer, bis zum Fall der Republik, nominell den Charakter eines außerordentlichen Magistrats bewahrte.
Die Wahl von Aeneas Sylvius Piccolomini, der den Namen Pius II. annahm, auf den Papststuhl im Jahr 1458 erfreute die Sieneser außerordentlich. Als Kompliment für ihren gefeierten Mitbürger gaben sie dem Ersuchen der Adligen nach und ließen sie wieder an der Regierung teilhaben. Diese widerwillig eingeräumte Konzession blieb nur einige Jahre in Kraft, und nach dem Tod des Papstes (1464) wurde sie aufgehoben, außer für Mitglieder des Hauses Piccolomini, die man als Popolani zählte und die all ihre Privilegien behalten durften. Inzwischen gärte neue Uneinigkeit unter den Plebejern an der Spitze der Angelegenheiten. Die Bank Monte dei Paschi di Siena wurde 1472 gegründet und ist heute die älteste noch existierende Bank der Welt.
Die Revolution von 1480 und ihre Folgen
Die Pazzi-Verschwörung 1478 führte zu einem Krieg, in dem Florenz und Mailand dem Papst und dem König von Neapel gegenüberstanden, und der mit dem Frieden vom 13. März 1480 zu Ende ging. Daraufhin kam Alfonso, Herzog von Kalabrien, der in der Toskana an der Seite seines Vaters Ferdinand gekämpft hatte, zu einer Übereinkunft mit Siena. In der gleichen Weise wie sein Großvater Alfonso versuchte er die Herrschaft über die Stadt und die Zurückrufung der exilierten Rebellen von 1456 zu erreichen. Die Noveschi (zu deren Stand die meisten Rebellen gehörten) begünstigten seine Ansprüche, aber die Riformatori waren gegen ihn.
Viele aus dem Volk standen auf der Seite der Noveschi, erhoben sich am 22. Juni 1480 zur Revolte und organisierten, unterstützt von Soldaten des Herzogs, die Regierung zu ihrem eigenen Vorteil neu. Sie teilten die Macht zwischen den beiden Ständen der Noveschi und der Popolani und schlossen die Riformatori aus. Sie wurden ersetzt durch einen neuen und heterogenen Stand, genannt aggregati, der aus Adligen, Exilanten von 1456 und Bürgern aus anderen Ständen bestand, die nie zuvor im Amt gewesen waren.
Aber dieser gewaltsame und gefährliche Umsturz der inneren Freiheiten der Republik dauerte nicht lange. Ein Dekret des neapolitanischen Königs (1482) beraubte Siena bestimmter Territorien zugunsten von Florenz, was sie diesem Monarchen völlig entfremdete. Inzwischen war der monte der Noveschi, der Hauptunterstützer der Revolution von 1480, wachsendem Hass und Neid seines früheren Verbündeten ausgesetzt, des monte del popolo, der sich seiner überlegenen Stärke und Zahl bewusst war und nun versuchte, die Noveschi zu vernichten und an ihrer Stelle an die Macht zu treten.
Dies wurde durch eine Reihe von Aufständen zwischen dem 7. Juni 1482 und dem 20. Februar 1483 vollbracht. Der monte del popolo ergriff den Löwenanteil an der Regierung; die Riformatori wurden zurückgerufen, die aggregati abgeschafft und die Noveschi zu ewiger Verbannung aus der Regierung und aus der Stadt verurteilt. Aber in perpetuo war eine Worthülse in jenen turbulenten italienischen Republiken. Die Noveschi mit ihren mächtigen Verbindungen, Fähigkeiten und Traditionen gewannen zunehmend Einfluss im Exil; und fünf Jahre später, am 22. Juli 1487, kehrten sie triumphierend nach Siena zurück, zerstreuten die wenigen Anhänger des popolo, die Widerstand leisteten, ermordeten den Kapitän des Volks, reorganisierten den Staat und setzten ihn unter den Schutz der Jungfrau Maria. Da ihre eigene Vorherrschaft durch ihre numerische Stärke und ihren Einfluss sichergestellt war, erteilten sie den anderen monti gleichen Anteil an der Macht.
Pandolfo Petrucci (1487–1512)
Unter den zurückgekehrten Exilanten war Pandolfo Petrucci, Anführer der Noveschi, der bald an der Spitze der Regierung stehen sollte. Während der Vorherrschaft dieses Mannes (der wie Lorenzo de Medici den Beinamen il Magnifico erhielt) erfreute sich Siena vieler Jahre des Aufschwungs. Streng genommen war Petrucci nie Herr des Staates und ließ die etablierte Regierungsform intakt; aber er übte despotische Autorität aufgrund der Stärke seines Charakters und dem ständigen Anwachsen seiner persönlichen Macht aus. Er stützte seine Außenpolitik auf das Bündnis mit Florenz und Frankreich und leitete die inneren Angelegenheiten des Staates mittels eines Rats (collegio) der Balia, der zwar gelegentlich neuorganisiert wurde, um die rivalisierenden Parteien zu beschwichtigen, der aber seinem Willen immer untergeordnet war.
Auf ähnliche Weise gewann er durch die Annahme des Kapitänsamts (1495) und später durch den Kauf mehrerer abgelegener Burgen von der ausgelaugten Kommune (1507) an Macht. Er schreckte auch nicht vor Mord und Rachetaten zurück. Die Ermordung seines Schwiegervaters Niccolò Borghesi (1500) ist ein unauslöschlicher Fleck auf seinem Namen. Er widerstand aller Opposition innerhalb des Staats, bis er schließlich in seinem Kampf mit Cesare Borgia abgesetzt und 1502 aus Siena verstoßen wurde. Durch die freundliche Vermittlung der Florentiner und des französischen Königs wurde am 29. März 1503 die Verbannung aufgehoben. Er behielt die Macht bis zu seinem Tod im Alter von sechzig Jahren am 21. Mai 1512 und wurde auf öffentliche Kosten mit fürstlichen Zeremonien in der Basilica dell'Osservanza beerdigt. In seine Regierungszeit fiel der Bruch der Bruna-Staumauer im Jahr 1492.
Petruccis Nachfolger (1512–1524)
Nach seinem Ableben hielt die Vorherrschaft seiner Familie in Siena nicht lange an. Pandolfo hatte nicht die nötigen Qualitäten, um eine Dynastie wie die der Medici zu begründen. Ihm fehlte der hohe Intellekt eines Cosimo oder eines Lorenzo, und die Atmosphäre des freiheitsliebenden Siena mit seinen ständig wechselnden Parteien war in keiner Weise für seine Absicht geeignet. Sein ältester Sohn Borghese Petrucci war unfähig, hochmütig und äußerst korrupt. Er blieb nur vier Jahre an der Spitze und floh 1516 schmachvoll. Durch die Gunst Leos X. folgte ihm sein Cousin Raffaele Petrucci nach, vorher Bischof von Grosseto, Gouverneur von Sant'Angelo und später Kardinal.
Dieser Petrucci war ein erbitterter Gegner von Pandolfos Kindern. Er veranlasste, dass Borghese und ein jüngerer Sohn namens Fabio zu Rebellen erklärt wurden, während ein dritter Sohn, Kardinal Alfonso, auf Anordnung Leos X. am 16. Juli 1517 erdrosselt wurde. Er war ein tyrannischer Herrscher und starb plötzlich 1522. Im folgenden Jahr bestand Clemens VII. auf der Wiederberufung Fabio Petruccis, aber zwei Jahre später vertrieb ihn ein erneuter Volksaufstand für immer aus Siena. Die Stadt stellte sich dann unter den Schutz Kaiser Karls V., schuf einen Magistrat aus zehn Konservatoren der Freiheiten des Staates (Dezember 1524) und vereinigte die verschiedenen monti in einem einzigen.
Unter der Herrschaft des Kaisers (1524–1552)
Die sogenannte freie Regierung, die Untertan des Reichs war, dauerte 27 Jahre. Dabei musste sie sich schon 1526 gegen eine Belagerung durch die Fiorentiner und Clemens VII. wehren. Durch den militärischen Sieg der Seneser in der sogenannten Battaglia di Camollia am 25. Juli 1526 konnte die Belagerung beendet werden. Danach wurde Baldassare Peruzzi von 1527 bis 1532 als Architetto della Repubblica eingesetzt, um die Verteidigungsanlagen von Siena und der dazugehörigen Provinz auszubauen. So wurden die Stadtmauern verstärkt und es entstanden fünf neue Bastionen (Bastione di Porta Laterina, Bastione di San Marco, Bastione di San Prospero, Bastione San Viene und das Fortino delle Donne), wovon heute noch drei bestehen. Zudem wurden die Festungen in Sarteano und Torrita di Siena ausgebaut. Die gewünschte Protektion Spaniens wog danach immer schwerer, bis sie eine Tyrannei wurde. Die kaiserlichen Legaten und die Kapitäne der spanischen Garde in Siena quetschten sowohl die Regierung als auch das Volk durch andauernde Erpressung und unangebrachte Einmischung in die Funktionen der Balia aus. Karl V. reiste 1535 durch Siena und wurde wie in allen anderen Städten des versklavten Italiens mit größtem Pomp empfangen; aber er brachte weder Frieden noch Freiheit mit sich.
Von 1527 bis 1545 wurde die Stadt durch Parteikämpfe und gewaltsame Revolten gegen die Noveschi zerrissen und war ein Ort häufigen Blutvergießens, während die Streitsucht und schlechte Regierung der Sieneser für große Unzufriedenheit in der Toskana sorgten. Die Balia wurde mehrere Male durch kaiserliche Vertreter wiedereingesetzt, 1530 von Don Lopez di Soria und Alfonso Piccolomini, Herzog von Amalfi, 1540 von Granvella (oder Granvelie) und 1548 von Don Diego Hurtado de Mendoza (1503–1575). Aber die Regierung war so schlecht wie zuvor, und der Hass gegen die spanische Herrschaft vergrößerte sich. Als 1549 Don Diego die Absicht des Kaisers ankündigte, eine Festung (Zitadelle Cittadella imperiale) in Siena zu errichten, um die Bürger zu kontrollieren, entlud sich der allgemeine Hass in entrüsteten Protesten.
Der Historiker Orlando Malavolti und andere spezielle Gesandte wurden 1550 mit einer Petition zum Kaiser geschickt, die von mehr als tausend Bürgern unterzeichnet worden war, und die ihn ersuchte, ihnen eine solch schreckliche Gefahr zu ersparen. Aber ihre Mission scheiterte: sie kehrten ungehört zurück. Inzwischen hatte Don Diego die Fundamente der Zitadelle legen lassen und trieb das Projekt voran. Daraufhin traten bestimmte Sieneser Bürger in Rom – angeführt von Aeneas Piccolomini (ein Verwandter Pius' II.) – in Verhandlungen mit Vertretern des französischen Königs, und nachdem sie mit deren Hilfe Männer und Geld gesammelt hatten, marschierten sie auf Siena und erzwangen sich am 26. Juli 1552 den Weg durch das neue Tor (die Porta Romana). Die Stadtmenschen erhoben sich, ermutigt und verstärkt durch die Hilfe von außen, sofort zur Revolte, griffen die spanischen Truppen an, entwaffneten sie und trieben sie dazu, Zuflucht in der Zitadelle zu suchen (28. Juli). Und schließlich wurden die Spanier durch ein Abkommen mit Cosimo I. de' Medici, Herzog von Florenz, am 5. August 1552 weggeschickt, und die Sienesen nahmen ihre Festung in Besitz.
Unter französischem Einfluss (1552–1555)
Die Regierung wurde nun unter dem Schutz französischer Vertreter wiedereingesetzt. Die Balia wurde abgeschafft, da ihr Name durch die spanische Tyrannei anrüchig geworden war, und durch einen ähnlichen Magistrat ersetzt, der capitani del popolo e reggimento genannt wurde. Siena frohlockte über seine wiedergewonnene Freiheit, aber bald zogen sich wieder dunkle Wolken zusammen. Zuerst wurde der Kaiser durch den Einfluss Frankreichs auf die Politik der Republik erzürnt. Dann überlegte sich Cosimo, der nicht weniger eifersüchtig auf die Franzosen war, den Plan, Siena in sein eigenes Herrschaftsgebiet einzuverleiben.
Die ersten feindlichen Handlungen der kaiserlichen Truppen im Val di Chiana (1552–1553) richteten wenig Schaden an. Aber als Cosimo mit einer vom Marquis von Marignano befehligten Armee auf das Feld trat, war der Untergang Sienas besiegelt. Am 26. Januar 1554 eroberte Marignano die Festung von Porta Camollia (die die gesamte Bevölkerung Sienas, einschließlich der Frauen unter Laudomia Forteguerri, zu bauen geholfen hatte) und belagerte die Stadt. Am 2. August desselben Jahres errang er in der Schlacht von Scannagallo, einem Wassergraben südlich von Marciano della Chiana zwischen Pozzo della Chiana und Santa Luce (heute Ortsteile von Foiano della Chiana) im Val di Chiana einen vollständigen Sieg über die Sieneser und französischen Truppen unter Piero Strozzi, dem Florentiner Verbannten und späterem Marschall von Frankreich.
Inzwischen wurde Siena nachdrücklich belagert. Ein ruhmreicher Bericht der Leiden der Bevölkerung findet sich im Tagebuch des Sieneser Historikers Sozzini und in den Kommentaren von Blaise de Monluc, dem französischen Vertreter in Siena. Am 17. April 1555 war die Stadt durch Nahrungsmittelknappheit infolge mehrmonatiger Belagerung zur Kapitulation gegenüber dem florentinisch-kaiserlichen Heer gezwungen. Am 21. April traten die spanischen Truppen des Kaisers durch die Tore. Daraufhin gaben viele Patrioten die Stadt auf, nahmen Zuflucht in Montalcino und pflegten dort eine schattenhafte Form einer Republik bis zum Frieden von Cateau-Cambrésis 1559.
Siena als Teil des Großherzogtums Toskana
Cosimo I. de' Medici wurde durch ein auf den 3. Juli 1557 datiertes Patent von Philipp II. von Spanien mit dem Staat Siena belehnt und nahm am 19. desselben Monats formal Besitz von der Stadt. Ein Generalleutnant wurde als Vertreter seiner Autorität ernannt; der Rat der Balia wurde mit zwanzig vom Herzog gewählten Mitgliedern wiedergegründet. Das Konsistorium und der allgemeine Rat blieben bestehen, wurden aber ihrer politischen Autonomie beraubt. Auf diese Weise wurde Siena vom Florentiner Staat annektiert und wurde ein integraler Teil des Großherzogtums Toskana. Dennoch behielt es für mehr als zwei Jahrhunderte eine separate Verwaltung, bis die allgemeinen Reformen des Großherzogs Pietro Leopoldo, die französische Herrschaft, und schließlich die Restauration alle Unterschiedene zwischen den Sieneser und Florentiner Regierungssystemen hinwegfegten. Cosimo beauftragte 1560 Baldassare Lanci mit der Errichtung einer neuen Festung in der Nähe der zerstörten Zitadelle, nun Fortezza di Santa Barbara und Fortezza Medicea genannt.
Am 26. Mai 1798 wurde Siena von einem Erdbeben erschüttert. Die größten Schäden entstanden an der Basilica di San Domenico, deren Campanile und das Dach einstürzten. Die Basilica di San Clemente in Santa Maria dei Servi wurde ebenfalls schwer beschädigt. Das Beben der Stärke 8,5 auf der Mercalliskala beschädigte zudem die Chiesa di San Cristoforo und das Theater Teatro dei Rinnovati im Palazzo Pubblico. Die Truppen von Napoleon Bonaparte besetzten die Stadt am 29. März 1799 und blieben bis 1814.
Ab dem 19. Jahrhundert
Die Bahnstrecke nach Empoli wurde 1849 eingeweiht, 1862 kam die Verbindung nach Chiusi hinzu. Die Strecke nach Grosseto entstand am Anfang der 1870er Jahre. Der Bahnhof befand sich damals als Kopfbahnhof kurz außerhalb der Barriera di San Lorenzo. Der heutige Durchgangsbahnhof wurde am 25. November 1935 eingeweiht. 1859 war Siena die erste toskanische Stadt, die für eine Einverleibung ins Piemont und die Monarchie Viktor Emanuel II. stimmte. Diese Entscheidung vom 17. Juni war der erste Schritt zur Vereinigung Italiens. Von 1858 bis 1868 wurde der Piazza del Campo umgestaltet. Der Bildhauer Tito Sarrocchi ersetzt die von 1409 bis 1419 entstandenen Figuren des Jacopo della Quercia am Brunnen Fonte Gaia. Der Brunnen wurde 9,60 Meter nach Westen und 1,60 Meter nach Süden versetzt und befindet sich nun zentral auf der Piazza vor dem Palazzo Casino dei Nobili. Die alten Figuren des Jacopo della Quercia befinden sich heute im Fienile genannten Saal im Museum von Santa Maria della Scala, wo neben den Originalfiguren auch die Gipsabdrücke zur Gestaltung der Kopien des Sarrocchi ausgestellt sind. Ab 1895 entstand für die moderne Wasserversorgung das Aquädukt Acquedotto del Vivo (auch Acquedotto di Siena), welches 1914 eingeweiht wurde.
Während des Zweiten Weltkrieges erlitt Siena verhältnismäßig geringe Schäden. Die Basilika dell'Osservanza, außerhalb liegend, wurde am 23. Januar 1944 bei einem Bombenangriff der Streitkräfte der Vereinigten Staaten bis auf Ausnahme der Fassade und einigen Seitenmauern fast vollständig zerstört, später (1945 bis 1949) unter Verwendung der nicht zerstörten Steine aber fast originalgetreu wieder aufgebaut. In den 1960er Jahren entstand in Siena innerhalb der Stadtmauern der erste Verkehrsberuhigte Bereich in Italien. Das neue Krankenhaus außerhalb der Stadtmauern, Policlinico Santa Maria le Scotte genannt, welches das von Santa Maria della Scala ersetzte, wurde in den 1980er Jahren in Betrieb genommen und Santa Maria della Scala wurde zu einem Museumskomplex umgebaut.
Literatur
Mario Aschieri: Storia di Siena. Dalle origini ai giorni nostri. Edizioni Biblioteca dell'Immagine, Pordenone 2013, ISBN 978-88-6391-138-1
William M. Bowsky: A Medieval Italian Commune: Siena under the Nine, 1287–1355. University of California Press, Berkeley/London 1981, ISBN 978-0-52004256-8.
Langton Douglas: A History of Siena. Betti Editrice, Siena 2000 (Org. London 1902), ISBN 88-86417-61-6
Emanuele Repetti: SIENA (SENAE, anticamente SAENA) nella Val-d-Arbia. In Dizionario Geografico Fisico Storico della Toscana (1833–1846), Onlineausgabe der Universität Siena (pdf, ital.)
Weblinks
Siena in der Enciclopedia Italiana (1936)
Siena in der Enciclopedie on line.
Einzelnachweise
Siena
Siena
Siena | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,100 |
Hovik Demircsjan (örményül: Հովիկ Դեմիրճյան, görögül: Χοβίκ Ντεμιρτζιάν) vagy művésznevén Hovig (Nicosia, 1989. január 3. –) örmény származású ciprusi énekes. Ő képviselte Ciprust a 2017-es Eurovíziós Dalfesztiválon Kijevben, Gravity című dalával. A döntőben 68 pontot sikerült összegyűjtenie, így a 21. helyezést érte el.
Élete
1989. január 3-án született Nicosiában, Cipruson, örmény származású. Marketingtevékenységet folytatott, de hamar lemondott, hogy énekes lehessen. A jazz-en gitározni és zongorázni tanult. Első nagy sikere a második helyezés a larnacai zenei versenyen. Cipruson a "The Music Messenger" becenéven vált ismertté. 2009. június 1-jén kiadta a "Δεν μού μιλάς - ἱστορία έχει τελειώσει" zenei videót a görög X-faktor előkészítéséhez. A dalt Argiro Christodoulidou írta és a Kostas Voniatis rendezte a videót.
Karrier
2009-ben részt vett a görög X-Faktor második évadában. Az élő műsorokba bejutva Níkosz Múratídisz mentorlásában a 16 és 24 év közötti fiúk csapatjában versenyzett. A tehetségkutatóban végül a 7. helyen végzett. Az X-Faktor után visszatért Ciprusra, hogy szólókarrierjét felépítse hazájában is. Számos görög nyelvű dalt jelentetett meg ez idő alatt. Demircsjan kétszer vett részt az Eurovíziós Dalfesztivál ciprusi előválogatójában: először 2010-ben, ekkor harmadik lett a Goodbye című dalával, majd öt évvel később a negyedik helyen végzett a Stone in a Riverrel. 2016. október 21-én a CyBC ciprusi közszolgálati televízió bejelentette, hogy Hovig képviseli a szigetországot a 2017-es Eurovíziós Dalfesztiválon, Kijevben. A dalát a svéd producer–zeneszerző Thomas G:son írta.
Diszkográfia
Single
Fordítás
Források
Az Eurovíziós Dalfesztivál résztvevői
1989-ben született személyek
Élő személyek
Nicosiaiak | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,610 |
Hipón (/ hʊpoʊ /; Griego: Ἵππων, Hippon, siglo V a C) fue un filósofo griego presocrático. Se le describe varias veces como procedente de Rhegium, Metaponto, Samos, y Crotona, y es posible que haya más de un filósofo con este nombre.
Filosofía
A pesar de ser un filósofo natural, Aristóteles se negó a colocarlo entre los otros grandes filósofos presocráticos "debido a la tontería de su pensamiento". En algún momento Hipón fue acusado de ateísmo, pero como sus obras han desaparecido, no podemos estar seguros de por qué fue así acusado. También fue acusado de impiedad por el poeta cómico Cratino en su Panopta, y según Clemente de Alejandría, Hipón supuestamente ordenó que los siguientes versos se inscribieran en su tumba:
Según Hipólito, Hipón sostuvo que el agua y el fuego eran los elementos primarios, el fuego emanaba del agua, y después se desarrollaba generando el universo. Simplicio también decía que Hipón pensó que el agua era el principio de todas las cosas. La mayoría de los relatos de su filosofía sugieren que estaba interesado en asuntos biológicos. Pensó que hay un nivel adecuado de humedad en todos los seres vivos, y la enfermedad se produce cuando la humedad está fuera de equilibrio. También vio el alma como surgiendo de la mente y el agua. Un escolio medieval sobre las nubes de Aristófanes atribuye a Hipón la visión de que los cielos eran como la cúpula (πνιγεύς) de un horno que cubre la Tierra.
Véase también
Anexo:Catálogo de pitagóricos de Jámblico
Notas
Filósofos de la Antigua Grecia del siglo V a. C. | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 765 |
\section{Introduction}
Transmission policies that guarantee a timely access to wireless sensor data represent an important component for Internet of Things (IoT) sensor networks, which can be applied to environmental monitoring, industrial automation, and several other critical systems. This demand for timely information has motivated the definition of Age of Information (AoI)~\cite{kaul12}, which has been analyzed by a wide range of research studies over the past years, manifesting the relevance of timeliness in sensor networks~\cite{kosta17,yates21}. In addition to the AoI, a number of related metrics have been proposed, including the Value of Information (VoI)~\cite{ayan19}, which captures the error of the estimated sensor values at the sink, thereby allowing the policy to take into account the dynamics of the observed processes. The Age of Incorrect Information (AoII)~\cite{10.1109/TNET.2020.3005549} is a proposal that attempts to combine the two metrics, being defined as the product of the AoI and VoI.
Beyond the analysis of AoII in a variety of systems, there is a significant challenge of implementing policies and protocols that reduce AoII for low-power IoT sensor networks. A simple solution for minimizing AoI is to use a pull-based scheme, in which the receiver can poll sensors and schedule the one with the highest age. However, this solution has two issues: firstly, the receiver knows the AoI for each sensor, but not the VoI, so it can only optimize for the statistical value, and might schedule sensors that do not actually have valuable information even if their age is high. Secondly, polling can be an energy-intensive process, and many IoT communication technologies do not even support it. Indeed, in order for the gateway to be able to initiate the communication, sensors need to listen for requests at any time, which can quickly deplete their batteries.
The optimization of AoI and AoII is often performed using Markov models, which can optimize expected performance. In~\cite{KSA21}, the authors analyze Whittle index-based heuristics for minimizing AoI, achieving asymptotic optimality when the number of users is large. Whittle index policies can also be used for AoII, as in~\cite{9518209}, which considers a multi-sensor scenario with an infinite time horizon. Many recent works also involve the use of threshold based policies. \cite{Bjoshi} proposes threshold-type policies to minimize the AoII with estimation at the receiver in absence of updates for a single source observing an autoregressive Markov process. A policy which is a randomized combination of two deterministic threshold policies is shown to be optimal in \cite{YA21}, where the authors study problem of minimizing AoII with power constraints in the presence of an unreliable channel. \cite{delay} shows the optimality of a threshold based pre-emptive policy and achieves the minimum AoII under a uniform and bounded delay distribution.
However, most of these works either only deal with the first issue by considering statistical optimization, so that the system is still based on a centralized scheduling, or consider single-sensor systems, in which collisions are nonexistent. In real push-based communications, sensors need to be able to independently access the channel, while coordinating with each other in a distributed manner. This problem is significantly more challenging, as the information each sensor has is severely limited. An aggressive strategy can cause the network to become unstable, leading to extremely frequent collisions, while a too conservative one can starve the receiver, leading to an extremely high error rate. This decentralized setting has been studied in the context of the basic AoI metric. Threshold based policies for minimizing AoI in slotted ALOHA have been proposed in \cite{9162973,vavascan21}, where the sensors stay silent until they have a certain AoI, after which they transmit with a constant probability.
The objective of this work is to minimize AoII in a decentralized setting in which a group of IoT devices communicate to the sink over a shared slotted ALOHA random access channel. To this end, we formulate the AoII minimization task as a non-convex optimization problem, proposing a gradient-based algorithm to obtain an approximate solution. Our results show that the proposed algorithm manages to find policies that significantly outperform state-independent policies, and suggest that the optimal policy is a threshold-like function that depends both on the current AoI and the VoI.
The remainder of the paper is organized as follows. First, we present the system model in Sec.~\ref{sec:sysmodel}. We then analyze the AoII evolution in Sec.~\ref{sec:aoii_evolution} under an arbitrary policy, and derive an upper bound on the average AoII which we use in Sec.~\ref{sec:aoii_optim} to optimize the policy. Finally, we present numerical results in Sec.~\ref{sec:results} and conclude the paper in Sec.~\ref{sec:conclusion}.
\section{System Model}\label{sec:sysmodel}
The system we consider involves a remote BS that aims to collect observations from $N$ different sensors which observe independent, identically distributed discrete Markov processes. Specifically, at time $t=1,2,\ldots$ the $i$-th sensor observes $X_i(t) \in \mathbb{Z}$, a value governed by the random walk with the transition probability diagram depicted in Fig.~\ref{fig:2}. Note that $p_r+2p_t =1$, to ensure that they represent a valid probability space. Each sensor is responsible for communicating its observed value to the BS over a shared medium, whose access is regulated by slotted ALOHA: in every slot, any sensor may access the channel, and interference is destructive, so that if multiple sensors transmit in the same time slot, no packet is received. We assume that the BS acknowledges the successfully received packets, and will denote by $\hat{X}_i(t)$ the state of the Markov process from sensor $i$ that was most recently communicated to the BS. We do not assume that the nodes follow a specific retransmission strategy, but instead model retransmissions implicitly as part of the policy.
To jointly characterize the freshness and accuracy of the information the BS has, we define the AoII for sensor $i$ as
\begin{equation}
A_i(t) = f_i(t)g_i(t),
\end{equation}
where $f_i(t)$ corresponds to the penalty for information freshness (age), while $g_i(t)$ accounts for a penalty that occurs due to the difference in the actual information.
We will work with a linear time penalty given as $f_i(t) = t - U_i(t)$, where $U_i(t)$ corresponds to the last time instance when $X_i$ and $\Hat{X}_i$ were equal, i.e.,
\begin{equation}
U_i(t) = \max \{t_i | t_i \leq t , X_i(t_i) = \Hat{X}_i(t_i) \}.
\end{equation}
For simplicity we define $\Hat{X}_i(0)=X_i(0)=0$.
Note that, according to this definition, $f_i(t)$ is \emph{not} the same as the time elapsed since last successful transmission by sensor $i$, but the time elapsed since the observed state was the same as the most recently transmitted state. Clearly, $f_i(t)$ is computable by sensor $i$, which observes the state transitions. The penalty term $g_i(t)$ is simply given by the difference between the two:
\begin{equation}
g_i(t) = |X_i(t)-\Hat{X}_i(t)|.
\end{equation}
The objective of this work is to devise a common transmission policy for all sensors, i.e, a mapping $\pi:\mathbb{Z}^2\rightarrow[0,1]$ that assigns a transmission probability for every pair $(f,g)$ of age and correctness penalties. Specifically, we seek a policy that minimizes the expected AoII averaged across the $N$ sensors:
\begin{equation}\label{eq:expected_aoii}
\bar{A}=\mathbb{E}\left[\limsup_{T\to\infty}\frac{1}{TN}\sum_{t=1}^T\sum_{i=1}^N A_i(t)\middle\vert\pi\right],
\end{equation}
where the expectation is over the state evolution and the decisions of the transmission policy.
\begin{figure}
\centering
\begin{tikzpicture}[font=\sffamily]
\node[state] at (0, 0) (S_m) {0};
\node[state] at (2, 0) (S_1) {1};
\node[state] at (-2, 0) (S_m2) {-1};
\node[draw=none, right=of S_1] (m) {$ \cdots$};
\node[draw=none, left=of S_m2] (k) {$ \cdots$};
\draw[every loop, line width=0.3mm,
>=latex,draw=black,
fill=black]
(m) edge[bend left=20, auto=left] node {$p_t$} (S_1)
(S_1) edge[bend left=20, auto=left] node {$p_t$} (m)
(k) edge[bend left=20, auto=left] node {$p_t$} (S_m2)
(S_m2) edge[bend left=20, auto=left] node {$p_t$} (k)
(S_m) edge[bend left=20, auto=left] node {$p_t$} (S_1)
(S_1) edge[bend left=20, auto=left] node {$p_t$} (S_m)
(S_m2) edge[bend left=20, auto=left] node {$p_t$} (S_m)
(S_m) edge[bend left=20, auto=left] node {$p_t$} (S_m2)
(S_1) edge[loop below, auto=left] node {$p_r$} (S_1)
(S_m) edge[loop below, auto=left] node {$p_r$} (S_m)
(S_m2) edge[loop below, auto=left] node {$p_r$} (S_m);
\end{tikzpicture}
\caption{Process observed at each sensor.}
\label{fig:2}
\end{figure}
\section{AoII Evolution Analysis}\label{sec:aoii_evolution}
In this section, we derive the transition probabilities of the Markov chain describing the evolution of the AoII under a given policy $\pi$, which we will use in the next section to optimize the transmission policy. However, as the optimization problem is significantly complicated by the dependence/collisions among the sensors over the shared channel, at first we relax this constraint and assume that each user sees a fixed probability of success, computed as a steady state approximation. Consequently, we will focus on the evolution of $(f_i(t),g_i(t))$ for an arbitrary user $i$. Similar approximations have been used previously~\cite{LD12,BJK05} to analyse the performance of various random access protocols.
\subsection{Markov Model Formulation}\label{sec:markov_chain}
Since $f_i(t)$ and $g_i(t)$ are countably infinite, we truncate the Markov chain to $f_i(t)\leq F$, $g_i(t) \leq G$, setting fixed values $F$ and $G$, so that the mapping function becomes $\pi:\{0,\ldots,F\}\times\{0,\ldots,G\}\rightarrow[0,1]$. Assuming that $F$ and $G$ are sufficiently large, this assumption comes with a negligible impact on the optimal policy.
We define the truncated state as $f^{\tau}_i(t) = \min\{f_i(t),F\}$ and $g^{\tau}_i(t) = \min\{g_i(t),G\}$ so that the state space for each node is given by
\begin{equation}
\mathcal{S} = \left\{(f,g)\;|\;f \leq F, \;g \leq \min(G,f) \right\}.
\end{equation}
Let $\pi(f^{\tau}_i(t),g^{\tau}_i(t))$ denote the probability that sensor $i$ transmits in state $(f^{\tau}_i(t),g^{\tau}_i(t))$. Denote further by $s_i(t)\in\mathcal{S}$ the tuple $(f^{\tau}_i(t),g^{\tau}_i(t))$, and by $q(s_i(t))$ denote the probability of successful transmission. Note that $\pi(s_i)\geq q(s_i)\,\forall s_i$, as not all transmissions are successful.
Any state for which $g^{\tau}_i(t)=0$ is collapsed into state $(0,0)$, due to the definition of the age penalty. We can then define the probability function $P(s,s')$, mapping the transition probabilities.
We also define the symbol $[x+y]_T=\min(T,x+y)$ to simplify the notation below.
The simplest case is state $(0,0)$, in which we have:
\begin{equation}
P((0,0),s')=\begin{cases}
2p_t(1-q((0,0))), &f'=g'=1;\\
p_r+q((0,0)), & f'=g'=0,
\end{cases}
\end{equation}
where $s'=(f',g')$. If we consider state $(f,1)$, we get:
\begin{equation}
P((f,1),s')\!=\!\begin{cases}
p_t(1-q((f,1))), &f'\!=[f\!+\!1]_F,g'\!=2;\\
p_r(1-q((f,1))), &f'\!=[f\!+\!1]_F,g'\!=1;\\
p_t\!+\!(1\!-p_t)q((f,1)), &f'\!=g'\!=0,
\end{cases}
\end{equation}
In all other cases, we have:
\begin{equation}
P(s,s')=\begin{cases}
p_t(1-q(s)), &\begin{aligned} &f'=[f+1]_F,\\
&g'\in\{[g+1]_G,g-1\};\end{aligned}\\
p_r(1-q(s)), &f'=[f+1]_F,g'=g;\\
q((s)), & f'=g'=0.
\end{cases}
\end{equation}
The chain on $\mathcal{S}$ defined by the transition probabilities above is a truncated version of the true AoII evolution.
Additionally, it is finite, aperiodic, and irreducible, i.e., every state can be reached from $(0,0)$ with positive probability and $(0,0)$ can be reached from every state with positive probability. Given these conditions, the chain has a stationary distribution $\phi(f,g)$.
\subsection{Probability of Successful Transmission}
The probability of successful transmission $q(s)$ used in the previous section depends on the policy $\pi$. Recall that we consider the case in which all sensors have the same policy, but act in an entirely independent fashion, and that we for simplicity assume that each user sees a fixed probability of success in any given state. Before deriving an expression for $q(s)$, we first present the following intermediate result.
\begin{lemma}\label{lemma:gminf}
For any reachable state $(f,g)$ with $f \neq F$, we have $g \leq f$.
\end{lemma}
\begin{proof}
In order for $g$ to have a certain value, there must have been at least $g$ transitions starting from value 0. These transitions would require at least $f$ steps, as the chain can only increase or decrease by 1 in each step.
\end{proof}
Assuming that all sensors are in steady state, except the sensor of interest, we have $q(f,g) = \pi(f,g)\ell^{N-1}$,
where $\ell$ corresponds to the probability that no other sensor is transmitting. Using Lemma~\ref{lemma:gminf}, the value of $\ell$ can be computed as:
\begin{equation}
\ell = \sum_{f=0}^{F}\sum_{g=1}^{\min(f,G)}\phi(f,g)(1-\pi(f,g)).
\end{equation}
\subsection{Upper Bound on the AoII}
We conclude the section by presenting an upper bound on the AoII.
We can give the expected truncated AoII (which is a lower bound to the real AoII) by applying the ergodic theorem of Markov chains:
\begin{equation}
\mathbb{E}[fg] \geq \sum_{f=0}^{F}\sum_{g=0}^{\min(f,G)}fg\phi(f,g).
\end{equation}
Note that this value depends on $\pi$, as the steady state distribution is a function of the transmission policy. As the truncated AoII is a lower bound to the real AoII, it is not suitable for reliability-oriented optimization, as using it leads to optimistic policies and does not return a reliable estimate of the AoII.
We can then derive the following result.
\begin{theorem}\label{theo:upper_bound}
For any policy $\pi:\mathcal{S}\to[0,1]$ and a stationary distribution $\phi$ for the Markov chain derived in Section~\ref{sec:markov_chain}, the average truncated AoII $\mathbb{E}[fg]$ is upper bounded by
\begin{equation}
\begin{aligned}
J(\pi,\phi)= FG\phi(F,G) +\sum_{f=1}^{F-1}\left[\phi(f,G)f^2+\sum_{g=1}^{\mathclap{\min(f,G-1)}}fg\phi(f,g)\right]\\
+\sum_{g=1}^{G-1}G\phi(F,g)\left(F+\frac{(1-\min_{g\in\{1,\ldots,G-1\}}q(F,g))}{\min_{g\in\{1,\ldots,G-1\}}q(F,g)}\right)\\
+ \left(F+G+\frac{2(1-q((F,G))}{q((F,G))}\right)\frac{1-q((F,G))}{q((F,G))}\phi(F,G).
\end{aligned}
\end{equation}
\end{theorem}
\begin{proof}
Let us consider the two subsets of $\mathcal{S}$ given by
\begin{align}
\mathcal{S}_F=\left\{(F,g):0<g<G\right\};
\mathcal{S}_G=\left\{(f,G):G\leq f<F\right\}; \nonumber
\end{align}
The only possible transitions out of these classes are to states $(0,0)$, for successful transmission, or $(F,G)$, if the other limit is reached. Transitions within each class are possible, as defined by $P$. The non-truncated AoII, $A_i(t)=f_i(t)g_i(t)$, is different from the truncated AoII only if the truncated chain is in one of the states belonging to $\mathcal{S}_f$, $\mathcal{S}_g$ or $(F,G)$.
The probability of exiting $\mathcal{S}_f$ through a successful transmission at any given time, denoted as $\omega_F$, is given by
\begin{equation}
\omega_F=\frac{\sum_{s\in\mathcal{S}_f}\phi(s)q(s)}{\sum_{s\in\mathcal{S}_f}\phi(s)}.
\end{equation}
This is lower-bounded by $\min_{s\in\mathcal{S}_F}q(s)$, so that the geometric distribution with parameter $\min_{s\in\mathcal{S}_F}q(s)$ represents an upper bound to the time spent in $\mathcal{S}_F$.
The AoII after being in $\mathcal{S}_F$ for $i$ slots is upper-bounded by $(F+i)G$, as all states in $\mathcal{S}_F$ have an error smaller than $G$. We can adopt the same approach for state $(F,G)$, as the AoII after remaining in the state for $i$ steps is upper-bounded by $(F+i)(G+i)$. Finally, we consider class $\mathcal{S}_G$: thanks to Lemma~\ref{lemma:gminf}, the AoII in state $(f,G)$ is upper-bounded by $f^2$. We can then join the pieces to obtain the bound
\begin{multline}
\mathbb{E}[A_i(t)]
\leq \sum_{f=1}^{F-1}\quad\quad\sum_{g=1}^{\mathclap{\min(G-1,f)}}\phi(f,g)fg\; + \; \underbrace{\sum_{f=1}^{F-1}\phi(f,G)f^2}_{\text{U.B. for states in } \mathcal{S}_G}\\ +\underbrace{\sum_{i=0}^{\infty}\left[ (\min_{s\in\mathcal{S}_F}q(s))(1-\min_{s\in\mathcal{S}_F}q(s))^i(F+i)G \sum_{s\in\mathcal{S}_F}\phi(s)\right] }_{\text{U.B. for states in } \mathcal{S}_F}\\
\underbrace{\sum_{i=0}^{\infty} \phi((F,G))(q((F,G)))(1-q((F,G)))^i(F+i)(G+i)}_{\text{U.B. for state }(F,G)}
\end{multline}
We can solve the second term in the sum as follows:
\begin{multline}
\sum_{i=0}^{\infty}\left[ (\min_{s\in\mathcal{S}_F}q(s))^i(1-\min_{s\in\mathcal{S}_F}q(s))(F+i)G \sum_{s\in\mathcal{S}_F}\phi(s)\right]=\\
G\sum_{s\in\mathcal{S}_F}\phi(s)\left(F+\frac{1-\min_{s\in\mathcal{S}_F}q(s)}{\min_{s\in\mathcal{S}_F}q(s)}\right)=\\
\sum_{g=1}^{G-1}G\phi(F,g)\left(F+\frac{(1-\min_{s\in\mathcal{S}_F}q(s)}{\min_{s\in\mathcal{S}_F}q(s)}\right).
\end{multline}
Finally, the series giving the upper bound for state $(F,G)$ can be solved as below, omitting the term $\phi((F,G))$:
\begin{multline}
\sum_{i=0}^{\infty} q((F,G))(1-q((F,G)))^i(F+i)(G+i)=\\
\left[FG+\frac{(1-q((F,G))}{q((F,G))}\left(F+G+\frac{2(1-q((F,G))}{q((F,G))}\right)\right].
\end{multline}
If we sum the components, we obtain the value of $J(\pi,\phi)$.
\end{proof}
Theorem~\ref{theo:upper_bound} provides a closed-form upper bound on the system under the steady state assumption, which we can use to search for a policy $\pi^*$ that minimizes approximated AoII. Such a solution is expected to also have a small actual AoII.
\section{AoII optimization}\label{sec:aoii_optim}
\subsection{Problem Definition}
We can now define an optimization problem over the policy space. Our objective is to find the policy $\pi^*$ that minimizes the average expected AoII, and can be defined as
\begin{equation}
\pi^*=\argmin_{\pi:\mathcal{S}\rightarrow[0,1]}
\mathbb{E}\left[\limsup_{T\to\infty}\frac{1}{TN}\sum_{t=1}^T\sum_{i=1}^N f_i(t)g_i(t)\middle\vert\pi\right].
\end{equation}
As mentioned, we approximate the minimization using the bound $J(\pi,\phi)$ derived in Theorem~\ref{theo:upper_bound}. To this end, we define the following problem:
\begin{mini}
{(\pi,\phi)}{J(\pi,\phi)}
{}{}
\addConstraint{\phi P = P,\ \ \|\phi\|_1=1}
\addConstraint{0 \leq \phi(s) \leq 1\,\forall s\in\mathcal{S}}
\addConstraint{0 \leq \pi(s) \leq 1\,\forall s\in\mathcal{S}.}\label{eq:mini1}
\end{mini}
Here, the constraint $\phi P = P$ represents the equality conditions for the steady state distribution, where $P$ here is the transition probability matrix for the Markov chain derived in Section~\ref{sec:markov_chain}, which is a function of both $\pi$ and $\phi$. The other constraints ensure that $\phi$ and $\pi(s)$ are probability distributions.
Because of the constraints, \eqref{eq:mini1} is difficult to solve directly. Instead, we consider the Lagrangian relaxation to the above problem given as
\begin{mini}[2]
{(\pi,\phi)}{J(\pi,\phi)+k_1c_1 +k_2c_2+k_3c_3+k_4c_4,}
{}{}
\label{opt1}
\end{mini}
where the four penalty terms are defined as:
\begin{equation}
\begin{aligned}
c_1 &= (\|\phi P - \phi\|_2)^2;\\
c_2 &= \|\pi-1\|^2\mathbb{I}\{\pi>1\}+\|\pi\|^2\mathbb{I}\{\pi<0\};\\
c_3 &=(\|\phi\|_1- 1)^2;\\
c_4 &= (\|\phi-\Hat{1}\|)^2\mathbb{I}\{\phi>1\}+\|\phi\|^2\mathbb{I}\{\phi<0\},
\end{aligned}
\end{equation}
where $k_1,k_2,k_3,k_4$ are nonnegative Lagrange multipliers that penalize the violation of the constraints. As the Lagrange multiplier $k_i$ is increased, the constraint becomes tighter, i.e. the resulting policy chosen by the algorithm has lower $c_i$.
\subsection{Optimization Algorithm}
Solving the relaxed minimization problem on a computer involves rewriting the objective function as
\begin{mini}[2]
{(\pi,\phi)}{J(\pi,\phi)+\sum_{i=1}^4k_i\rho(c_i - \epsilon_i),}
{}{}
\label{opt2}
\end{mini}
where $\rho(x)=\max(0,x)$ is the rectified linear unit (ReLU) function, $\epsilon_i$ is a tolerance level for constraint $i$ quantifying satisfactory performance on constraint $i$. Ideally, in problem (\ref{opt1}) one should increase $k_i$ in steps to $\infty$ which ensures that the solution obtained have $c_i \rightarrow 0$ asymptotically. However trying to make $k_i c_i$ arbitrarily small can result in very high values of $J(\pi,\phi)$, and is a procedure of high complexity.
Using tolerances can help us avoid having to increase $k_i$ to $\infty$ trying to get arbitrarily small $c_i$ as anything below the tolerance level $\epsilon_i$ will fetch a zero penalty. Thus increasing $k_i$ will not change the solution. In our experiments we set $k_i$ to a constant large value in the initialization itself. However our experiments suggest that leaky ReLU, $\rho(x)=\max(\rho_a x,x)$ ($\rho_a\ll 1$), leads to better convergence than the regular ReLU.
\begin{algorithm}[tb]
\SetAlgoLined
\footnotesize
\caption{Gradient descent based minimization}\label{alg:agd}
Input: initial $\phi_i$, $\pi_i$, consts. $K_1, K_2, K_3, K_4$, $\epsilon_1$, $\epsilon_2$, $\epsilon_3$ ,$\epsilon_4$ \\
\For{$i\leq \text{max steps}$}{
$U(\pi, \phi)=J(\pi,\phi)+K_1 \rho(c_1-\epsilon_1) +K_2\rho(c_2-\epsilon_2) + K_3\rho(c_3-\epsilon_3) +K_4\rho(c_4-\epsilon_4)$\\
$\pi \leftarrow \pi - \alpha_{\pi}\frac{\nabla_\pi U(\pi,\phi)}{\|\nabla_\pi U(\pi,\phi)\|_2}\\
\phi \leftarrow \phi -
\alpha_{\phi}\frac{\nabla_\phi
U(\pi,\phi)}{\|\nabla_\phi U(\pi,\phi)\|_2}$\\}
\Return $\; \pi, \phi$
\label{Algo}
\end{algorithm}
The method used for minimization is based on gradient descent with a normalized gradient since the raw gradient can be very large at certain points, and using this is likely to result in probabilities that are either negative or greater than one. $\alpha_{\phi}$ and $\alpha_{\pi}$ are the learning rates for $\phi$ and $\pi$, respectively.
\subsection{Threshold Initialization}
Since the problem is complicated non-convex, the initialization of $\pi$ and $\phi$ play a central role in obtaining a good policy. We initialize $\pi$ as a threshold policy, which has demonstrated to work well in practice. Specifically, we set $\pi(f,g) = 0$ for $fg < \tau$ and $\pi(f,g) = p$ for $fg \geq \tau$, where $\tau$ is a suitably chosen threshold. In our experiments we set $\tau$ equal to the mean AoII obtained using Algorithm~\ref{alg:agd} with initialization $\phi(f,g)\; \propto\; \frac{1}{fg}$, (normalized such that the sum is 1) and $\pi(f,g) \; \propto \;fg$, scaled such that $\pi \phi = 0.9$ transmission attempts, and $p=\frac{5}{N}$.
After empirically finding a good $\tau$ for a given value of $N$ and $p_t$, it can be extrapolated to other values of $p_t$ using a simple relation threshold $\tau\; \propto \; \sqrt{p_t}$, which can be shown to approximate the real AoII.
\section{Numerical Results}\label{sec:results}
In order to verify the quality of our optimization, we tested it in a Monte Carlo simulation over $10^5$ steps for each scenario. We compare our optimization with two state-independent benchmarks, which transmit regardless of the value of $f$ and $g$, as long as $g>0$, i.e., there is new information to send. The first strategy is to always transmit with probability $1/N$, and is dubbed PT1, while the second limits the load to $E$ by having sensors transmit with probability $E/N$.
\begin{figure}
\input{plots/aoii}
\caption{Average AoII as a function of $p_t$.}
\label{fig:aoii}
\end{figure}
\begin{figure}
\input{plots/energy}
\caption{Average load as a function of $p_t$ for the dual policy.}
\label{fig:energy}\vspace{-0.4cm}
\end{figure}
A leaky ReLU with a slope $\rho_a=10^{-6}$ was used in all computations. We used tolerance levels
$\epsilon_1=10^{-3}$, %
$\epsilon_2=10^{-6}$, $\epsilon_3=10^{-5}$, and $\epsilon_4=10^{-6}$, and penalties $K_1=10^{8}$, %
$K_2=10^{11}$, $K_3=10^{10}$, and $K_4=10^{11}$. In most of our experiments the value of $J(\pi,\phi)$ is in the order of $10^3$; thus, these values ensure that the product $K_iP_i$ is about two orders of magnitude greater than $J(\pi,\phi)$ whenever $c_i>\epsilon_i$.
We simulated the resulting policy for different values of $N$ and $p_t$, after initializing all sensors in state $(0,0)$. The time average AoII (the true AoII, not the truncated version) is shown in Fig.~\ref{fig:aoii}. As expected, the AoII increases with $p_t$ and $N$, and the growth seems to be approximately proportional to $\sqrt{p_t}$: as we will discuss later, this behavior is reflected by the chosen threshold. However, the effect of the number of sensors $N$ depends on $p_t$, as the total load on the network has a non-linear effect: as we need to maintain the slotted ALOHA system in its stability range to avoid a complete collapse, the time $f$ between subsequent transmissions can significantly increase, and so will the error $g$ as transitions in the chain accumulate. The AoII will be a product of these, so the effect compounds, making systems with faster transitions much more sensitive to an increased number of nodes. However, the steady state approximation becomes more accurate with $N$, so the dual policy gets closer to the optimum.
Fig.~\ref{fig:energy} shows the average total number of transmissions across all sensors per time slot obtained from these simulations. Interestingly, when transitions are rare, the total number of transmissions is almost independent from $N$: as the channel is less loaded, the priority is to avoid collisions, as sensors in larger networks are able to transmit less often and still obtain a good AoII performance. When $p_t>0.2$, transitions become frequent, and holding back transmissions enough to avoid congestion becomes suboptimal: in larger networks, sensors can achieve a better AoII by attempting to transmit and failing with a relatively high probability, rather than waiting and risking wasting some slots. This matches our earlier intuition of a non-linearity in the optimal behavior, that becomes more pronounced as the network approaches full load and becomes severely congested. Note that the number of transmissions is less than $1$ in all cases, never reaching higher than 0.85: as such, the optimal policy is also more energy-efficient than state-independent slotted ALOHA approaches. Note that we can come up with policies with lower energy consumption ($\approx 0.5-0.6$) for higher values of $p_t$. This can be done by adding an energy constraint with a penalty($p_e = \|\pi \phi-0.5\|_2^2\mathbb{I}(\pi \phi > 0.5))$ to the objective function. However, these policies result in a higher AoII (about double the AoII we obtain without the constraint).
\begin{figure}
\centering
\input{plots/pt1}
\caption{Performance improvement over the PT1 policy as a function of $p_t$.}
\label{fig:pt1}\vspace{-0.4cm}
\end{figure}
Next, we compare the dual policy with the PT1 and PTE benchmarks: Figs.~\ref{fig:pt1} and \ref{fig:pte} show that the performance improvement in terms of the average AoII is between 75\% and 85\%, i.e., the AoII achieved by the dual policy is about 5 times lower than for state-independent strategies. PT1 performs better than PTE, but while PTE has the same energy efficiency as our policy by design, the load generated by PT1 is always higher, causing a higher energy cost for sensors. The overall improvement is slightly higher for larger networks, but even for smaller networks the performance gain is significant.
\begin{figure}
\centering
\input{plots/pte}
\caption{Performance improvement over the PTE policy as a function of $p_t$.}
\label{fig:pte}
\end{figure}
Finally, we can visualize the strategy itself: we take the case with $N=100$ and $p_t=0.3$ and plot the transmission probability as a colormap in Fig.~\ref{fig:p_visual}. We can see that the policy almost never transmits if the AoI is lower than 30, and that probability increases as $f$ and $g$ increases, but is very high if $f$ is saturated: in that case, the real AoII is unbounded, and transmission is relatively urgent. The maximum transmission rate in that case is close to 0.08, which is much higher than the normal rate. The case in which $g>f$, which is shown as having transmission probability 0 in the colormap, is never reached in practice, as we discussed above. We can also see the steady state probability for each state, mapped in Fig.~\ref{fig:ssd_visual}: in general, states with a relatively high age $f$ are reached often, but only if the error $g$ is very low. The unlucky case in which a sequence of transitions quickly leads the AoII to increase also has a relatively high probability, but as soon as the overall AoII passes the threshold, the transmission probability is correspondingly high.
\section{Conclusions and Future Work}\label{sec:conclusion}
In this work, we analyzed the optimization of the AoII in a slotted ALOHA network, in which sensors need to distributedly transmit updates about independent Markov processes. We consider a dual optimization based on a steady state approximation of other sensors to find a high-performance strategy to minimize AoII, starting from a threshold-based policy and gradually improving it.
We found that the policy outperforms naive approaches that do not take the sensor state into account and benchmark threshold policies, and that sensors can successfully coordinate to reduce AoII, even though the distributed scenario is much harder than a centralized one.
In future work, we plan to consider more advanced scenarios in which sensors' observations may be correlated, as well as dual approaches that combine polling and unprompted updates, taking the best from each approach to deal with complex environments which cannot be perfectly modeled.
\begin{figure}[t]
\centering
\includegraphics[width = 0.9\columnwidth]{plots/p1.png}
\caption{Visualization of the policy used ($p_t\!=\!0.3,N\!=\!100$).}
\label{fig:p_visual}\vspace{-0.4cm}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width = 0.9\columnwidth]{plots/v1.png}
\caption{Steady-state distribution (optimized policy, $p_t\!=\!0.3,N\!=\!100$).}
\label{fig:ssd_visual}\vspace{-0.4cm}
\end{figure}
\section*{Acknowledgment}
This work was supported by the Villum Investigator grant ``WATER'' from the Velux Foundation, Denmark. The work of A. E. Kal{\o}r was supported by the Independent Research Fund Denmark under Grant 1056-00006B. The work of F. Chiariotti was supported by the European Union under the Italian National Recovery and Resilience Plan (NRRP) of NextGenerationEU, under the REDIAL Young Researchers grant and the partnership PE0000001 - program
``RESTART''.
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,785 |
Update: Maxis says that a fix has been found and those unable to install the game yesterday will be able to do so now.
It's been nearly six months since SimCity launched on the PC where it was met with a whole host of issues. Maxis and EA eventually got things under control and even offered early adopters a free game for their troubles but now it looks as if the just-released Mac version is having some launch issues of its own.
One might suspect there wouldn't be any problems with the Mac version as it isn't a direct port of the PC game but rather a fully native version designed using OpenGL. That apparently isn't the case as a number of Mac users have reportedly been unable to install the game while others aren't having much luck getting the game to launch after installation.
A Mac install FAQ has been posted in the EA forums to help walk through some of the more common issues including a "mutexAlert" error which can be fixed by switching the OS to English. Gamers that aren't able to install the digital version after downloading it are encouraged to re-install Origin and opt-in for the beta version.
And if you're running Mac OS X 10.7.4 or the upcoming 10.9 beta, you're currently out of luck. The post indicates a solution is in the pipeline, however.
EA has yet to release an official statement on the matter as of writing. It is worth pointing out that anyone that purchased the PC version of the game can also download the Mac version free of charge and vice-versa. Perhaps Mac gamers can get a little solace in the fact that they could play the game on a PC until the issues are hammered out. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,373 |
>RR-A308
Telecommuting and Work in the COVID-19 Pandemic
Are Workers Returning to the Workplace or Staying in Their Home Offices?
by Philip Armour, Katherine Grace Carman, Kathleen J. Mullen, Shanthi Nataraj
Coronavirus Disease 2019 (COVID-19),
Employment and Unemployment,
Labor Markets,
Photo by len4ik/Adobe Stock
We know from previous RAND Corporation research that the ability to work from home protected jobs at the onset of stay-at-home orders in early spring 2020; by May, only 6 percent of workers who had the option to telecommute prior to the onset of the coronavirus disease 2019 (COVID-19) pandemic had lost their jobs compared with about 25 percent of workers who did not have that option. But is telecommuting still protective of employment months later? And is the nature of telecommuting changing? Answers to these questions depend on what kinds of jobs workers had before the pandemic.
In the first week of May, the middle of June, and the middle of September 2020, we fielded surveys in the nationally representative RAND American Life Panel (ALP) to explore how workers' lives changed as a result of the pandemic. We focus on the 1,015 individuals who were working for pay or profit in February 2020 and who responded to all three survey waves.
Being Able to Telecommute Still Protects Employment, but the Gap Is Narrowing
The ability to make an immediate shift to telecommuting varied widely by occupation: Those in legal, scientific, arts, or media professions mostly telecommuted in May, while those in protective services, maintenance, construction, food preparation, and transportation had to leave home to keep working or lost their jobs. However, this difference is narrowing (see Figure 1).
Figure 1. Ability to Telecommute Pre-Pandemic Affected Employment
Could telecommute in February job (n = 545)
100% 95% 95% 94%
Could not telecommute in February job (n = 470)
SOURCE: Authors' analysis of ALP data. The sample was limited to the 1,015 respondents who were working in February 2020 and who responded to the May, June, and September survey waves. Responses are weighted using sampling weights as described in Carman and Nataraj, 2020.
Although 77 percent of those who could not telecommute in February were working in May, this number rose to 84 percent in September. Meanwhile, there has been little change in employment among those who could telecommute before the pandemic: Nearly 95 percent of workers who could telecommute before the pandemic were employed in both May and September.
Telecommuting Patterns Are Shifting, but the Nature of the Shift Depends on the Type of Job a Worker Has
How has telecommuting changed over the course of the pandemic? To understand how telecommuting has changed within existing jobs, we focused on workers who still hold the same jobs that they had in February. In Figures 2 and 3, we break down trends in telecommuting across two measures: (1) whether workers are telecommuting at all (i.e., at least one day per week) and (2) how many days workers are telecommuting if they are telecommuting at all.
Figure 2. Percentage of Employees Telecommuting at Least One Day in the Past Week
Could telecommute in February (n = 462)
Could not telecommute in February (n = 300)
SOURCE: Authors' analysis of ALP data. The sample was limited to the 762 respondents who were working in February 2020; who responded to the May, June, and September waves; and who were working in their February jobs during all three waves. Responses are weighted using sampling weights as described in Carman and Nataraj, 2020.
Figure 3. Average Number of Days Spent Telecommuting in the Past Week for Those Who Telecommuted at Least Once During the Week
Could telecommute in February
Could not telecommute in February
SOURCE: Authors' analysis of ALP data. The sample is further limited from that in Figure 2 to those who were telecommuting in any given month (among those who could not telecommute in February, n = 48 in May, n = 63 in June, and n = 62 in September; among those who could telecommute in February, n = 414 in May, n = 382 in June, and n = 363 in September). Responses are weighted using sampling weights as described in Carman and Nataraj, 2020.
Figures 2 and 3 tell contrasting stories about trends in telecommuting for those with jobs that previously allowed telecommuting versus those with jobs that did not have this option. In Figure 2, we see that for those who could not telecommute in February, but who kept their jobs during the pandemic, only one in five telecommuted at all in May—and that number stayed the same in September. In contrast, those who already had the option to telecommute had a different experience: More than 90 percent of these respondents were telecommuting in May, but more and more are returning to the workplace; only 78 percent telecommuted at least one day per week in September.
Figure 3 shows that the workers who were able to telecommute in February and who continue to work from home work remotely essentially full time, or five days per week, on average. Those who could not telecommute in February saw a slight decline in the number of days they spent telecommuting as the pandemic progressed, although it is not clear whether this trend indicates that the increase in telecommuting for this group was temporary and the number of days spent telecommuting will decline over time or that they will continue to work at home several days per week.
In Figures 4 and 5, we delve into how telecommuting has changed by occupation. We limit this analysis to those who remained in their February jobs in both May and September. Figure 4 shows rates of telecommuting at all in May and September for the occupations with decreasing rates of telecommuting, while Figure 5 shows occupations with increasing rates of telecommuting or experiencing only small changes in rates of telecommuting.
Figure 4. Percentage of Employees Telecommuting at Least One Day in the Past Week, by Occupation: Decreasing Rates
SOURCE: Authors' analysis of ALP data. The sample was limited to the 768 respondents who were working in February 2020; who responded to the May, June, and September waves; and who were working in their February jobs in both May and September. Only occupations with at least 25 respondents are shown. This figure shows occupations for which the decrease in the percentage of respondents reporting any telecommuting between May and September was statistically significant at the 5-percent level. Responses are weighted using sampling weights as described in Carman and Nataraj, 2020. These occupational classifications are from the U.S. Bureau of Labor Statistics, 2020.
Figure 5. Percentage of Employees Telecommuting at Least One Day in the Past Week, by Occupation: Steady or Increasing Rates
Computer or mathematics
Health care support
Health care practitioner
SOURCE: Authors' analysis of ALP data. The sample was limited to the 768 respondents who were working in February 2020; who responded to the May, June, and September waves; and who were working in their February jobs in both May and September. Only occupations with at least 25 respondents are shown. This figure shows occupations that did not exhibit a statistically significant decrease in telecommuting. Responses are weighted using sampling weights as described in Carman and Nataraj, 2020. These occupational classifications are from U.S. Bureau of Labor Statistics, 2020.
NOTE: "Computer or mathematics" refers to computer and information analysts and research scientists, database and network administrators, and software and web developers.
Unsurprisingly, there has been a dramatic reduction in telecommuting among teachers from May to September 2020, once the new school year began and many districts restarted full or partial in-person instruction. However, managers, administrative support staff, and those in business and financial operations also have exhibited substantial declines in telecommuting since May, indicating that rates of telecommuting were temporary adjustments.
However, other occupations continue to have unchanged or potentially increasing rates of telecommuting. Those working in computer or mathematical occupations continue to telecommute at high and persistent rates. The percentage of health care support workers and salespeople reporting any telecommuting increased, although the change was not statistically significant.
The ability to switch rapidly to telecommuting at the start of the pandemic saved many workers' jobs. As the pandemic continues, we are seeing evidence that some workers who were previously working from home are starting to go back to their workplaces, although most who are able to telecommute continue to do so.
To what extent are these changes likely to persist even after the need for social distancing is over? The differing changes in telecommuting across occupations suggest that there are likely to be some long-term impacts. There is some evidence that telecommuting might be spreading to such occupations as health care and sales; it was less prevalent in these occupations at the start of the pandemic. This might indicate ongoing adjustments that facilitate telecommuting, which could allow for more telecommuting over time. Our research team will continue to track the evolving nature of work in the coming months.
Katherine Grace Carman and Shanthi Nataraj, 2020 American Life Panel Survey on Impacts of COVID-19: Technical Documentation, Santa Monica, Calif.: RAND Corporation, RR-A308-1, 2020. As of November 24, 2020: https://www.rand.org/pubs/research_reports/RRA308-1.html
Armour, Philip, Katherine Grace Carman, Kathleen J. Mullen, and Shanthi Nataraj, The COVID-19 Pandemic and the Changing Nature of Work: Lose Your Job, Show Up to Work, or Telecommute? Santa Monica, Calif.: RAND Corporation, RR-A308-4, 2020. As of November 24, 2020: https://www.rand.org/pubs/research_reports/RRA308-4.html
U.S. Bureau of Labor Statistics, "2018 Standard Operational Classification System," webpage, April 17, 2020. As of November 30, 2020: https://www.bls.gov/soc/2018/major_groups.htm
This report describes a subset of results from three survey waves fielded in May, June, and September 2020 through the ALP to explore how workers' lives changed as a result of the COVID-19 pandemic. A technical description of the survey that includes a description of the ALP, the objectives of the survey, and information about the fielding of the survey are presented in Katherine Grace Carman and Shanthi Nataraj, 2020 American Life Panel Survey on Impacts of COVID-19: Technical Documentation, Santa Monica, Calif.: RAND Corporation, RR-A308-1, 2020.
This study was undertaken by RAND Education and Labor, a division of the RAND Corporation that conducts research on early childhood through postsecondary education programs, workforce development, and programs and policies affecting workers, entrepreneurship, and financial literacy and decisionmaking. Questions about this report should be directed to the lead author, Philip Armour, at parmour@rand.org, and questions about RAND Education and Labor should be directed to educationandlabor@rand.org.
Funding for this research was provided by unrestricted gifts from RAND supporters and income from operations.
DOI: https://doi.org/10.7249/RRA308-11
Document Number: RR-A308-11
Armour, Philip, Katherine Grace Carman, Kathleen J. Mullen, and Shanthi Nataraj, Telecommuting and Work in the COVID-19 Pandemic: Are Workers Returning to the Workplace or Staying in Their Home Offices?. Santa Monica, CA: RAND Corporation, 2020. https://www.rand.org/pubs/research_reports/RRA308-11.html.
Armour, Philip, Katherine Grace Carman, Kathleen J. Mullen, and Shanthi Nataraj, Telecommuting and Work in the COVID-19 Pandemic: Are Workers Returning to the Workplace or Staying in Their Home Offices?, Santa Monica, Calif.: RAND Corporation, RR-A308-11, 2020. As of January 12, 2021: https://www.rand.org/pubs/research_reports/RRA308-11.html | {
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} | 2,973 |
Q: How to get response in Spring webclient I have the following method which calls a rest api using webclient in a library.
public Flux<Offer[]> getOffersByIds(List<String> ids) {
return Flux.fromStream(batch(ids, 100)).flatMap(batch -> getOffersForBatch(batch));
}
private Mono<Offer[]> getOffersForBatch(List<String> ids) {
return webClient.get().uri(URL + PROMOTIONS_ENDPOINT, uriBuilder -> uriBuilder
.queryParam("ids", String.join(",", ids))
.build())
.header("Accept-Language", "en-GB")
.retrieve()
.onStatus(HttpStatus::is5xxServerError, response -> Mono.error(new RetryableException("api error")))
.bodyToMono(Offer[].class)
.retryWhen(Retry.backoff(3, Duration.ofSeconds(5))
.jitter(0.75)
.filter(throwable -> throwable instanceof RunException)
.onRetryExhaustedThrow(((retryBackoffSpec, retrySignal) -> {
throw new ServerException("service failed after max retries");
})));
}
public static <T> Stream<List<T>> batch(List<T> source, int length) {
if (length <= 0)
throw new IllegalArgumentException("length = " + length);
int size = source.size();
if (size <= 0)
return Stream.empty();
int fullChunks = (size - 1) / length;
return IntStream.range(0, fullChunks + 1).mapToObj(
n -> source.subList(n * length, n == fullChunks ? size : (n + 1) * length));
}
In the service, I am trying to invoke the api:
offService.getOffersByIds(ids)
.subscribe(
success -> log.info("Success:" + success.toString()),
error -> log.error("Failure:" + error.getMessage()),
() -> log.error("No value")
);
But I don't see any logs related to response
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,059 |
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