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This Indian family sold their ancestral property for rescue dogs
Wonderful - Malvika January 13, 2020
This uplifting news comes to us from Mount Abu, India.
A dog lover, Jyoti Khandelwal from Mount Abu has turned out to be a saviour and hero for stray dogs. The hill station in district Sirohi has a rescue centre for dogs and Jyoti made it possible by selling a part of her ancestral house.
She along with her brother and sister have kept their careers aside and they are now spending time with rescue dogs. All the three are highly educated but for their unconditional love for dogs they didn't mind leaving their career.
The story goes back to the time when a dog gave birth to six little puppies in Jyoti's neighbourhood around four years back. The siblings used to hear the whining of the pups during midnight because of the extreme cold weather. They went ahead and brought the pups and the dog home. It was then that the idea of a rescue centre hit their mind.
They sold off a part of their ancestral house in order to fund the rescue centre. Their residence on Abu Road is where they operate the rescue centre on the top two floors and they use the ground floor as their residence. The use a rental income to support the finances of the rescue centre.
Today the rescue centre has around 60 dogs that have been rescued after an accident or illness. Jyoti also registered her centre as an NGO and named her sister the president. She did so with the motive to get some help from government vet hospitals when needed.
The sisters said that they do love other animals as well but they feel that street dogs are the most neglected ones. They also remember a time when they called up the "Nagar Palika" informing them about an injured dog and they were mocked very badly. The authority at the palika said that they had better jobs to perform than this.
It is truly a great initiative by the Khandelwal siblings and we hope they go a long way.
Image Credits: Times Of India
India dog lover, dog rescued, dog savior, doglovers, dogsavior, India news, inspiring news, inspiring news india, inspiringnews, Mount Abu news, uplifting news, uplifting news India
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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Great twinset looks great with the pop of colour and floral print!
What a nice outfit! I love the navy blue combined with white... and the touch that brings the floral print is just perfect.
I just discovered your blog and am your new follower!
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{
"redpajama_set_name": "RedPajamaC4"
}
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/**
* @fileoverview Enforce all defaultProps are declared and non-required propTypes
* @author Vitor Balocco
* @author Roy Sutton
*/
'use strict';
// ------------------------------------------------------------------------------
// Requirements
// ------------------------------------------------------------------------------
const RuleTester = require('eslint').RuleTester;
const rule = require('../../../lib/rules/default-props-match-prop-types');
const parsers = require('../../helpers/parsers');
const parserOptions = {
ecmaVersion: 2018,
sourceType: 'module',
ecmaFeatures: {
jsx: true,
},
};
const ruleTester = new RuleTester({ parserOptions });
// ------------------------------------------------------------------------------
// Tests
// ------------------------------------------------------------------------------
ruleTester.run('default-props-match-prop-types', rule, {
valid: parsers.all([
// stateless components
{
code: `
function MyStatelessComponent({ foo, bar }) {
return <div>{foo}{bar}</div>;
}
MyStatelessComponent.propTypes = {
foo: React.PropTypes.string.isRequired,
bar: React.PropTypes.string.isRequired
};
`,
},
{
code: `
function MyStatelessComponent({ foo, bar }) {
return <div>{foo}{bar}</div>;
}
MyStatelessComponent.propTypes = {
foo: React.PropTypes.string,
bar: React.PropTypes.string.isRequired
};
MyStatelessComponent.defaultProps = {
foo: "foo"
};
`,
},
{
code: `
function MyStatelessComponent({ foo, bar }) {
return <div>{foo}{bar}</div>;
}
`,
},
{
code: `
function MyStatelessComponent({ foo, bar }) {
return <div>{foo}{bar}</div>;
}
MyStatelessComponent.propTypes = {
bar: React.PropTypes.string.isRequired
};
MyStatelessComponent.propTypes.foo = React.PropTypes.string;
MyStatelessComponent.defaultProps = {
foo: "foo"
};
`,
},
{
code: `
function MyStatelessComponent({ foo, bar }) {
return <div>{foo}{bar}</div>;
}
MyStatelessComponent.propTypes = {
bar: React.PropTypes.string.isRequired
};
MyStatelessComponent.defaultProps = {
bar: "bar"
};
`,
options: [
{ allowRequiredDefaults: true },
],
},
{
code: `
function MyStatelessComponent({ foo, bar }) {
return <div>{foo}{bar}</div>;
}
MyStatelessComponent.propTypes = {
bar: React.PropTypes.string.isRequired
};
MyStatelessComponent.propTypes.foo = React.PropTypes.string;
MyStatelessComponent.defaultProps = {};
MyStatelessComponent.defaultProps.foo = "foo";
`,
},
{
code: `
function MyStatelessComponent({ foo }) {
return <div>{foo}</div>;
}
MyStatelessComponent.propTypes = {};
MyStatelessComponent.propTypes.foo = React.PropTypes.string;
MyStatelessComponent.defaultProps = {};
MyStatelessComponent.defaultProps.foo = "foo";
`,
},
{
code: `
const types = {
foo: React.PropTypes.string,
bar: React.PropTypes.string.isRequired
};
function MyStatelessComponent({ foo, bar }) {
return <div>{foo}{bar}</div>;
}
MyStatelessComponent.propTypes = types;
MyStatelessComponent.defaultProps = {
foo: "foo"
};
`,
},
{
code: `
const defaults = {
foo: "foo"
};
function MyStatelessComponent({ foo, bar }) {
return <div>{foo}{bar}</div>;
}
MyStatelessComponent.propTypes = {
foo: React.PropTypes.string,
bar: React.PropTypes.string.isRequired
};
MyStatelessComponent.defaultProps = defaults;
`,
},
{
code: `
const defaults = {
foo: "foo"
};
const types = {
foo: React.PropTypes.string,
bar: React.PropTypes.string.isRequired
};
function MyStatelessComponent({ foo, bar }) {
return <div>{foo}{bar}</div>;
}
MyStatelessComponent.propTypes = types;
MyStatelessComponent.defaultProps = defaults;
`,
},
// createReactClass components
{
code: `
var Greeting = createReactClass({
render: function() {
return <div>Hello {this.props.foo} {this.props.bar}</div>;
},
propTypes: {
foo: React.PropTypes.string.isRequired,
bar: React.PropTypes.string.isRequired
}
});
`,
},
{
code: `
var Greeting = createReactClass({
render: function() {
return <div>Hello {this.props.foo} {this.props.bar}</div>;
},
propTypes: {
foo: React.PropTypes.string,
bar: React.PropTypes.string.isRequired
},
getDefaultProps: function() {
return {
foo: "foo"
};
}
});
`,
},
{
code: `
var Greeting = createReactClass({
render: function() {
return <div>Hello {this.props.foo} {this.props.bar}</div>;
},
propTypes: {
foo: React.PropTypes.string,
bar: React.PropTypes.string
},
getDefaultProps: function() {
return {
foo: "foo",
bar: "bar"
};
}
});
`,
},
{
code: `
var Greeting = createReactClass({
render: function() {
return <div>Hello {this.props.foo} {this.props.bar}</div>;
}
});
`,
},
// ES6 class component
{
code: `
class Greeting extends React.Component {
render() {
return (
<h1>Hello, {this.props.foo} {this.props.bar}</h1>
);
}
}
Greeting.propTypes = {
foo: React.PropTypes.string,
bar: React.PropTypes.string.isRequired
};
Greeting.defaultProps = {
foo: "foo"
};
`,
},
{
code: `
class Greeting extends React.Component {
render() {
return (
<h1>Hello, {this.props.foo} {this.props.bar}</h1>
);
}
}
`,
},
{
code: `
class Greeting extends React.Component {
render() {
return (
<h1>Hello, {this.props.foo} {this.props.bar}</h1>
);
}
}
Greeting.propTypes = {
bar: React.PropTypes.string.isRequired
};
Greeting.propTypes.foo = React.PropTypes.string;
Greeting.defaultProps = {
foo: "foo"
};
`,
},
{
code: `
class Greeting extends React.Component {
render() {
return (
<h1>Hello, {this.props.foo} {this.props.bar}</h1>
);
}
}
Greeting.propTypes = {
bar: React.PropTypes.string.isRequired
};
Greeting.propTypes.foo = React.PropTypes.string;
Greeting.defaultProps = {};
Greeting.defaultProps.foo = "foo";
`,
},
{
code: `
class Greeting extends React.Component {
render() {
return (
<h1>Hello, {this.props.foo} {this.props.bar}</h1>
);
}
}
Greeting.propTypes = {};
Greeting.propTypes.foo = React.PropTypes.string;
Greeting.defaultProps = {};
Greeting.defaultProps.foo = "foo";
`,
},
// edge cases
// not a react component
{
code: `
function NotAComponent({ foo, bar }) {}
NotAComponent.defaultProps = {
bar: "bar"
};
`,
},
{
code: `
class Greeting {
render() {
return (
<h1>Hello, {this.props.foo} {this.props.bar}</h1>
);
}
}
Greeting.defaulProps = {
bar: "bar"
};
`,
},
// external references
{
code: `
const defaults = require("./defaults");
const types = {
foo: React.PropTypes.string,
bar: React.PropTypes.string
};
function MyStatelessComponent({ foo, bar }) {
return <div>{foo}{bar}</div>;
}
MyStatelessComponent.propTypes = types;
MyStatelessComponent.defaultProps = defaults;
`,
},
{
code: `
const defaults = {
foo: "foo"
};
const types = require("./propTypes");
function MyStatelessComponent({ foo, bar }) {
return <div>{foo}{bar}</div>;
}
MyStatelessComponent.propTypes = types;
MyStatelessComponent.defaultProps = defaults;
`,
},
{
code: `
MyStatelessComponent.propTypes = {
foo: React.PropTypes.string
};
MyStatelessComponent.defaultProps = require("./defaults").foo;
function MyStatelessComponent({ foo, bar }) {
return <div>{foo}{bar}</div>;
}
`,
},
{
code: `
MyStatelessComponent.propTypes = {
foo: React.PropTypes.string
};
MyStatelessComponent.defaultProps = require("./defaults").foo;
MyStatelessComponent.defaultProps.bar = "bar";
function MyStatelessComponent({ foo, bar }) {
return <div>{foo}{bar}</div>;
}
`,
},
{
code: `
import defaults from "./defaults";
MyStatelessComponent.propTypes = {
foo: React.PropTypes.string
};
MyStatelessComponent.defaultProps = defaults;
function MyStatelessComponent({ foo, bar }) {
return <div>{foo}{bar}</div>;
}
`,
parserOptions: Object.assign({ sourceType: 'module' }, parserOptions),
},
{
code: `
import { foo } from "./defaults";
MyStatelessComponent.propTypes = {
foo: React.PropTypes.string
};
MyStatelessComponent.defaultProps = foo;
function MyStatelessComponent({ foo, bar }) {
return <div>{foo}{bar}</div>;
}
`,
parserOptions: Object.assign({ sourceType: 'module' }, parserOptions),
},
// using spread operator
{
code: `
const component = rowsOfType(GuestlistEntry, (rowData, ownProps) => ({
...rowData,
onPress: () => ownProps.onPress(rowData.id),
}));
`,
},
{
code: `
MyStatelessComponent.propTypes = {
...stuff,
foo: React.PropTypes.string
};
MyStatelessComponent.defaultProps = {
foo: "foo"
};
function MyStatelessComponent({ foo, bar }) {
return <div>{foo}{bar}</div>;
}
`,
},
{
code: `
MyStatelessComponent.propTypes = {
foo: React.PropTypes.string
};
MyStatelessComponent.defaultProps = {
...defaults,
};
function MyStatelessComponent({ foo, bar }) {
return <div>{foo}{bar}</div>;
}
`,
},
{
code: `
class Greeting extends React.Component {
render() {
return (
<h1>Hello, {this.props.foo} {this.props.bar}</h1>
);
}
}
Greeting.propTypes = {
...someProps,
bar: React.PropTypes.string.isRequired
};
`,
},
{
code: `
class Greeting extends React.Component {
render() {
return (
<h1>Hello, {this.props.foo} {this.props.bar}</h1>
);
}
}
Greeting.propTypes = {
foo: React.PropTypes.string,
bar: React.PropTypes.string.isRequired
};
Greeting.defaultProps = {
...defaults,
bar: "bar"
};
`,
},
// with Flow annotations
{
code: `
type Props = {
foo: string
};
class Hello extends React.Component {
props: Props;
render() {
return <div>Hello {this.props.foo}</div>;
}
}
`,
features: ['types'],
},
{
code: `
type Props = {
foo: string,
bar?: string
};
class Hello extends React.Component {
props: Props;
render() {
return <div>Hello {this.props.foo}</div>;
}
}
Hello.defaultProps = {
bar: "bar"
};
`,
features: ['types'],
},
{
code: `
class Hello extends React.Component {
props: {
foo: string,
bar?: string
};
render() {
return <div>Hello {this.props.foo}</div>;
}
}
Hello.defaultProps = {
bar: "bar"
};
`,
features: ['types'],
},
{
code: `
class Hello extends React.Component {
props: {
foo: string
};
render() {
return <div>Hello {this.props.foo}</div>;
}
}
`,
features: ['types'],
},
{
code: `
function Hello(props: { foo?: string }) {
return <div>Hello {props.foo}</div>;
}
Hello.defaultProps = { foo: "foo" };
`,
features: ['types'],
},
{
code: `
function Hello(props: { foo: string }) {
return <div>Hello {foo}</div>;
}
`,
features: ['types'],
},
{
code: `
const Hello = (props: { foo?: string }) => {
return <div>Hello {props.foo}</div>;
};
Hello.defaultProps = { foo: "foo" };
`,
features: ['types'],
},
{
code: `
const Hello = (props: { foo: string }) => {
return <div>Hello {foo}</div>;
};
`,
features: ['types'],
},
{
code: `
const Hello = function(props: { foo?: string }) {
return <div>Hello {props.foo}</div>;
};
Hello.defaultProps = { foo: "foo" };
`,
features: ['types'],
},
{
code: `
const Hello = function(props: { foo: string }) {
return <div>Hello {foo}</div>;
};
`,
features: ['types'],
},
{
code: `
type Props = {
foo: string,
bar?: string
};
type Props2 = {
foo: string,
baz?: string
}
function Hello(props: Props | Props2) {
return <div>Hello {props.foo}</div>;
}
Hello.defaultProps = {
bar: "bar",
baz: "baz"
};
`,
features: ['types'],
},
{
code: `
type PropsA = { foo?: string };
type PropsB = { bar?: string, fooBar: string };
type Props = PropsA & PropsB;
class Bar extends React.Component {
props: Props;
static defaultProps = {
foo: "foo",
}
render() {
return <div>{this.props.foo} - {this.props.bar}</div>
}
}
`,
features: ['types'],
},
{
code: `
import type Props from "fake";
class Hello extends React.Component {
props: Props;
render () {
return <div>Hello {this.props.name.firstname}</div>;
}
}
`,
features: ['types'],
},
{
code: `
type Props = any;
const Hello = function({ foo }: Props) {
return <div>Hello {foo}</div>;
};
`,
features: ['types'],
},
{
code: `
import type ImportedProps from "fake";
type Props = ImportedProps;
function Hello(props: Props) {
return <div>Hello {props.name.firstname}</div>;
}
`,
features: ['types'],
},
{
code: `
type DefaultProps1 = {|
bar1?: string
|};
type DefaultProps2 = {|
...DefaultProps1,
bar2?: string
|};
type Props = {
foo: string,
...DefaultProps2
};
function Hello(props: Props) {
return <div>Hello {props.foo}</div>;
}
Hello.defaultProps = {
bar1: "bar1",
bar2: "bar2",
};
`,
features: ['flow'],
},
{
code: `
type DefaultProps1 = {|
bar1?: string
|};
type DefaultProps2 = {|
...DefaultProps1,
bar2?: string
|};
type Props = {
foo: string,
...DefaultProps2
};
class Hello extends React.Component<Props> {
render() {
return <div>Hello {props.foo}</div>;
}
}
Hello.defaultProps = {
bar1: "bar1",
bar2: "bar2",
};
`,
features: ['flow'],
},
// don't error when variable is not in scope
{
code: `
import type { ImportedType } from "fake";
type Props = ImportedType;
function Hello(props: Props) {
return <div>Hello {props.name.firstname}</div>;
}
`,
features: ['types'],
},
// make sure error is not thrown with multiple assignments
{
code: `
import type ImportedProps from "fake";
type NestedProps = ImportedProps;
type Props = NestedProps;
function Hello(props: Props) {
return <div>Hello {props.name.firstname}</div>;
}
`,
features: ['types'],
},
// don't error when variable is not in scope with intersection
{
code: `
import type ImportedProps from "fake";
type Props = ImportedProps & {
foo: string
};
function Hello(props: Props) {
return <div>Hello {props.name.firstname}</div>;
}
`,
features: ['types'],
},
{
code: `
import type { FieldProps } from 'redux-form';
type Props = FieldProps & {
name: string,
type: string,
label?: string,
placeholder?: string,
disabled?: boolean,
};
TextField.defaultProps = {
label: '',
placeholder: '',
disabled: false,
};
`,
features: ['types'],
},
]),
invalid: parsers.all([
// stateless components
{
code: `
function MyStatelessComponent({ foo, bar }) {
return <div>{foo}{bar}</div>;
}
MyStatelessComponent.propTypes = {
foo: React.PropTypes.string,
bar: React.PropTypes.string.isRequired
};
MyStatelessComponent.defaultProps = {
baz: "baz"
};
`,
errors: [
{
messageId: 'defaultHasNoType',
data: { name: 'baz' },
line: 10,
column: 11,
},
],
},
{
code: `
function MyStatelessComponent({ foo, bar }) {
return <div>{foo}{bar}</div>;
}
MyStatelessComponent.propTypes = forbidExtraProps({
foo: React.PropTypes.string,
bar: React.PropTypes.string.isRequired
})
MyStatelessComponent.defaultProps = {
baz: "baz"
};
`,
settings: {
propWrapperFunctions: ['forbidExtraProps'],
},
errors: [
{
messageId: 'defaultHasNoType',
data: { name: 'baz' },
line: 10,
column: 11,
},
],
},
{
code: `
function MyStatelessComponent({ foo, bar }) {
return <div>{foo}{bar}</div>;
}
const propTypes = {
foo: React.PropTypes.string,
bar: React.PropTypes.string.isRequired
};
MyStatelessComponent.propTypes = forbidExtraProps(propTypes);
MyStatelessComponent.defaultProps = {
baz: "baz"
};
`,
settings: {
propWrapperFunctions: ['forbidExtraProps'],
},
errors: [
{
messageId: 'defaultHasNoType',
data: { name: 'baz' },
line: 11,
column: 11,
},
],
},
{
code: `
function MyStatelessComponent({ foo, bar }) {
return <div>{foo}{bar}</div>;
}
MyStatelessComponent.propTypes = {
foo: React.PropTypes.string,
bar: React.PropTypes.string.isRequired
};
MyStatelessComponent.defaultProps = {
baz: "baz"
};
`,
errors: [
{
messageId: 'defaultHasNoType',
data: { name: 'baz' },
line: 10,
column: 11,
},
],
options: [{ allowRequiredDefaults: true }],
},
{
code: `
function MyStatelessComponent({ foo, bar }) {
return <div>{foo}{bar}</div>;
}
MyStatelessComponent.propTypes = {
foo: React.PropTypes.string,
bar: React.PropTypes.string.isRequired
};
MyStatelessComponent.defaultProps = {
bar: "bar"
};
MyStatelessComponent.defaultProps.baz = "baz";
`,
errors: [
{
messageId: 'requiredHasDefault',
data: { name: 'bar' },
line: 10,
column: 11,
},
{
messageId: 'defaultHasNoType',
data: { name: 'baz' },
line: 12,
column: 9,
},
],
},
{
code: `
const types = {
foo: React.PropTypes.string,
bar: React.PropTypes.string.isRequired
};
function MyStatelessComponent({ foo, bar }) {
return <div>{foo}{bar}</div>;
}
MyStatelessComponent.propTypes = types;
MyStatelessComponent.defaultProps = {
bar: "bar"
};
`,
errors: [
{
messageId: 'requiredHasDefault',
data: { name: 'bar' },
line: 12,
column: 11,
},
],
},
{
code: `
const defaults = {
foo: "foo"
};
function MyStatelessComponent({ foo, bar }) {
return <div>{foo}{bar}</div>;
}
MyStatelessComponent.propTypes = {
foo: React.PropTypes.string.isRequired,
bar: React.PropTypes.string
};
MyStatelessComponent.defaultProps = defaults;
`,
errors: [
{
messageId: 'requiredHasDefault',
data: { name: 'foo' },
line: 3,
column: 11,
},
],
},
{
code: `
const defaults = {
foo: "foo"
};
const types = {
foo: React.PropTypes.string.isRequired,
bar: React.PropTypes.string
};
function MyStatelessComponent({ foo, bar }) {
return <div>{foo}{bar}</div>;
}
MyStatelessComponent.propTypes = types;
MyStatelessComponent.defaultProps = defaults;
`,
errors: [
{
messageId: 'requiredHasDefault',
data: { name: 'foo' },
line: 3,
column: 11,
},
],
},
// createReactClass components
{
code: `
var Greeting = createReactClass({
render: function() {
return <div>Hello {this.props.foo} {this.props.bar}</div>;
},
propTypes: {
foo: React.PropTypes.string,
bar: React.PropTypes.string.isRequired
},
getDefaultProps: function() {
return {
baz: "baz"
};
}
});
`,
errors: [
{
messageId: 'defaultHasNoType',
data: { name: 'baz' },
line: 12,
column: 15,
},
],
},
{
code: `
var Greeting = createReactClass({
render: function() {
return <div>Hello {this.props.foo} {this.props.bar}</div>;
},
propTypes: {
foo: React.PropTypes.string.isRequired,
bar: React.PropTypes.string
},
getDefaultProps: function() {
return {
foo: "foo"
};
}
});
`,
errors: [
{
messageId: 'requiredHasDefault',
data: { name: 'foo' },
line: 12,
column: 15,
},
],
},
// ES6 class component
{
code: `
class Greeting extends React.Component {
render() {
return (
<h1>Hello, {this.props.foo} {this.props.bar}</h1>
);
}
}
Greeting.propTypes = {
foo: React.PropTypes.string,
bar: React.PropTypes.string.isRequired
};
Greeting.defaultProps = {
baz: "baz"
};
`,
errors: [
{
messageId: 'defaultHasNoType',
data: { name: 'baz' },
line: 14,
column: 11,
},
],
},
{
code: `
class Greeting extends React.Component {
render() {
return (
<h1>Hello, {this.props.foo} {this.props.bar}</h1>
);
}
}
Greeting.propTypes = {
foo: React.PropTypes.string.isRequired,
bar: React.PropTypes.string
};
Greeting.defaultProps = {
foo: "foo"
};
`,
errors: [
{
messageId: 'requiredHasDefault',
data: { name: 'foo' },
line: 14,
column: 11,
},
],
},
{
code: `
class Greeting extends React.Component {
render() {
return (
<h1>Hello, {this.props.foo} {this.props.bar}</h1>
);
}
}
Greeting.propTypes = {
bar: React.PropTypes.string.isRequired
};
Greeting.propTypes.foo = React.PropTypes.string.isRequired;
Greeting.defaultProps = {};
Greeting.defaultProps.foo = "foo";
`,
errors: [
{
messageId: 'requiredHasDefault',
data: { name: 'foo' },
line: 14,
column: 9,
},
],
},
{
code: `
class Greeting extends React.Component {
render() {
return (
<h1>Hello, {this.props.foo} {this.props.bar}</h1>
);
}
}
Greeting.propTypes = {
bar: React.PropTypes.string
};
Greeting.propTypes.foo = React.PropTypes.string;
Greeting.defaultProps = {};
Greeting.defaultProps.baz = "baz";
`,
errors: [
{
messageId: 'defaultHasNoType',
data: { name: 'baz' },
line: 14,
column: 9,
},
],
},
{
code: `
class Greeting extends React.Component {
render() {
return (
<h1>Hello, {this.props.foo} {this.props.bar}</h1>
);
}
}
Greeting.propTypes = {};
Greeting.propTypes.foo = React.PropTypes.string.isRequired;
Greeting.defaultProps = {};
Greeting.defaultProps.foo = "foo";
`,
errors: [
{
messageId: 'requiredHasDefault',
data: { name: 'foo' },
line: 12,
column: 9,
},
],
},
{
code: `
class Greeting extends React.Component {
render() {
return (
<h1>Hello, {this.props.foo} {this.props.bar}</h1>
);
}
}
const props = {
foo: React.PropTypes.string,
bar: React.PropTypes.string.isRequired
};
Greeting.propTypes = props;
const defaults = {
bar: "bar"
};
Greeting.defaultProps = defaults;
`,
errors: [
{
messageId: 'requiredHasDefault',
data: { name: 'bar' },
line: 15,
column: 11,
},
],
},
{
code: `
class Greeting extends React.Component {
render() {
return (
<h1>Hello, {this.props.foo} {this.props.bar}</h1>
);
}
}
const props = {
foo: React.PropTypes.string,
bar: React.PropTypes.string
};
const defaults = {
baz: "baz"
};
Greeting.propTypes = props;
Greeting.defaultProps = defaults;
`,
errors: [
{
messageId: 'defaultHasNoType',
data: { name: 'baz' },
line: 14,
column: 11,
},
],
},
// ES6 classes with static getter methods
{
code: `
class Hello extends React.Component {
static get propTypes() {
return {
name: React.PropTypes.string.isRequired
};
}
static get defaultProps() {
return {
name: "name"
};
}
render() {
return <div>Hello {this.props.name}</div>;
}
}
`,
errors: [
{
messageId: 'requiredHasDefault',
data: { name: 'name' },
line: 10,
column: 15,
},
],
},
{
code: `
class Hello extends React.Component {
static get propTypes() {
return {
foo: React.PropTypes.string,
bar: React.PropTypes.string
};
}
static get defaultProps() {
return {
baz: "world"
};
}
render() {
return <div>Hello {this.props.bar}</div>;
}
}
`,
errors: [
{
messageId: 'defaultHasNoType',
data: { name: 'baz' },
line: 11,
column: 15,
},
],
},
{
code: `
const props = {
foo: React.PropTypes.string
};
const defaults = {
baz: "baz"
};
class Hello extends React.Component {
static get propTypes() {
return props;
}
static get defaultProps() {
return defaults;
}
render() {
return <div>Hello {this.props.foo}</div>;
}
}
`,
errors: [
{
messageId: 'defaultHasNoType',
data: { name: 'baz' },
line: 6,
column: 11,
},
],
},
{
code: `
const defaults = {
bar: "world"
};
class Hello extends React.Component {
static get propTypes() {
return {
foo: React.PropTypes.string,
bar: React.PropTypes.string.isRequired
};
}
static get defaultProps() {
return defaults;
}
render() {
return <div>Hello {this.props.bar}</div>;
}
}
`,
errors: [
{
messageId: 'requiredHasDefault',
data: { name: 'bar' },
line: 3,
column: 11,
},
],
},
// ES6 classes with property initializers
{
code: `
class Greeting extends React.Component {
render() {
return (
<h1>Hello, {this.props.foo} {this.props.bar}</h1>
);
}
static propTypes = {
foo: React.PropTypes.string,
bar: React.PropTypes.string.isRequired
};
static defaultProps = {
bar: "bar"
};
}
`,
features: ['class fields'],
errors: [
{
messageId: 'requiredHasDefault',
data: { name: 'bar' },
line: 13,
column: 13,
},
],
},
{
code: `
class Greeting extends React.Component {
render() {
return (
<h1>Hello, {this.props.foo} {this.props.bar}</h1>
);
}
static propTypes = {
foo: React.PropTypes.string,
bar: React.PropTypes.string
};
static defaultProps = {
baz: "baz"
};
}
`,
features: ['class fields'],
errors: [
{
messageId: 'defaultHasNoType',
data: { name: 'baz' },
line: 13,
column: 13,
},
],
},
{
code: `
const props = {
foo: React.PropTypes.string,
bar: React.PropTypes.string.isRequired
};
const defaults = {
bar: "bar"
};
class Greeting extends React.Component {
render() {
return (
<h1>Hello, {this.props.foo} {this.props.bar}</h1>
);
}
static propTypes = props;
static defaultProps = defaults;
}
`,
features: ['class fields'],
errors: [
{
messageId: 'requiredHasDefault',
data: { name: 'bar' },
line: 7,
column: 11,
},
],
},
{
code: `
const props = {
foo: React.PropTypes.string,
bar: React.PropTypes.string
};
const defaults = {
baz: "baz"
};
class Greeting extends React.Component {
render() {
return (
<h1>Hello, {this.props.foo} {this.props.bar}</h1>
);
}
static propTypes = props;
static defaultProps = defaults;
}
`,
features: ['class fields'],
errors: [
{
messageId: 'defaultHasNoType',
data: { name: 'baz' },
line: 7,
column: 11,
},
],
},
// edge cases
{
code: `
let Greetings = {};
Greetings.Hello = class extends React.Component {
render () {
return <div>Hello {this.props.foo}</div>;
}
}
Greetings.Hello.propTypes = {
foo: React.PropTypes.string.isRequired
};
Greetings.Hello.defaultProps = {
foo: "foo"
};
`,
errors: [
{
messageId: 'requiredHasDefault',
data: { name: 'foo' },
line: 12,
column: 11,
},
],
},
{
code: `
var Greetings = ({ foo = "foo" }) => {
return <div>Hello {this.props.foo}</div>;
}
Greetings.propTypes = {
foo: React.PropTypes.string.isRequired
};
Greetings.defaultProps = {
foo: "foo"
};
`,
errors: [
{
messageId: 'requiredHasDefault',
data: { name: 'foo' },
line: 9,
column: 11,
},
],
},
// with Flow annotations
{
code: `
class Hello extends React.Component {
props: {
foo: string,
bar?: string
};
render() {
return <div>Hello {this.props.foo}</div>;
}
}
Hello.defaultProps = {
foo: "foo"
};
`,
features: ['types'],
errors: [
{
messageId: 'requiredHasDefault',
data: { name: 'foo' },
line: 14,
column: 11,
},
],
},
{
code: `
function Hello(props: { foo: string }) {
return <div>Hello {props.foo}</div>;
}
Hello.defaultProps = {
foo: "foo"
}
`,
features: ['types'],
errors: [
{
messageId: 'requiredHasDefault',
data: { name: 'foo' },
line: 6,
column: 11,
},
],
},
{
code: `
type Props = {
foo: string
};
function Hello(props: Props) {
return <div>Hello {props.foo}</div>;
}
Hello.defaultProps = {
foo: "foo"
}
`,
features: ['types'],
errors: [
{
messageId: 'requiredHasDefault',
data: { name: 'foo' },
line: 10,
column: 11,
},
],
},
{
code: `
const Hello = (props: { foo: string, bar?: string }) => {
return <div>Hello {props.foo}</div>;
};
Hello.defaultProps = { foo: "foo", bar: "bar" };
`,
features: ['types'],
errors: [
{
messageId: 'requiredHasDefault',
data: { name: 'foo' },
line: 5,
column: 32,
},
],
},
{
code: `
type Props = {
foo: string,
bar?: string
};
type Props2 = {
foo: string,
baz?: string
}
function Hello(props: Props | Props2) {
return <div>Hello {props.foo}</div>;
}
Hello.defaultProps = { foo: "foo", frob: "frob" };
`,
features: ['flow'], // TODO: FIXME: change to "types" and fix failures
errors: [
{
messageId: 'requiredHasDefault',
data: { name: 'foo' },
line: 15,
column: 32,
},
{
messageId: 'defaultHasNoType',
data: { name: 'frob' },
line: 15,
column: 44,
},
],
},
{
code: `
type PropsA = { foo: string };
type PropsB = { bar: string };
type Props = PropsA & PropsB;
class Bar extends React.Component {
props: Props;
static defaultProps = {
fooBar: "fooBar",
foo: "foo",
}
render() {
return <div>{this.props.foo} - {this.props.bar}</div>
}
}
`,
features: ['types'],
errors: [
{
messageId: 'defaultHasNoType',
data: { name: 'fooBar' },
},
{
messageId: 'requiredHasDefault',
data: { name: 'foo' },
},
],
},
{
code: `
class SomeComponent extends React.Component {
render() {
return <div />;
}
}
SomeComponent.propTypes = {
"firstProperty": PropTypes.string.isRequired,
};
SomeComponent.defaultProps = {
"firstProperty": () => undefined
};
`,
errors: [
{
messageId: 'requiredHasDefault',
data: { name: 'firstProperty' },
},
],
},
{
code: `
type DefaultProps = {
baz?: string,
bar?: string
};
type Props = {
foo: string,
...DefaultProps
}
function Hello(props: Props) {
return <div>Hello {props.foo}</div>;
}
Hello.defaultProps = { foo: "foo", frob: "frob", baz: "bar" };
`,
features: ['flow'],
errors: [
{
messageId: 'requiredHasDefault',
data: { name: 'foo' },
line: 15,
column: 32,
},
{
messageId: 'defaultHasNoType',
data: { name: 'frob' },
line: 15,
column: 44,
},
],
},
{
code: `
type DefaultProps = {
baz?: string,
bar?: string
};
type Props = {
foo: string,
...DefaultProps
}
class Hello extends React.Component<Props> {
render() {
return <div>Hello {props.foo}</div>;
}
}
Hello.defaultProps = { foo: "foo", frob: "frob", baz: "bar" };
`,
features: ['flow'],
errors: [
{
messageId: 'requiredHasDefault',
data: { name: 'foo' },
line: 17,
column: 32,
},
{
messageId: 'defaultHasNoType',
data: { name: 'frob' },
line: 17,
column: 44,
},
],
},
{
code: `
export type SharedProps = {|
disabled: boolean,
|};
type Props = {|
...SharedProps,
focused?: boolean,
|};
class Foo extends React.Component<Props> {
static defaultProps = {
disabled: false
};
};
`,
features: ['flow'],
errors: [
{
messageId: 'requiredHasDefault',
data: {
name: 'disabled',
},
},
],
},
]),
});
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,154
|
package main
import (
"encoding/json"
"fmt"
rethink "github.com/christopherhesse/rethinkgo"
"github.com/gorilla/mux"
"io/ioutil"
"net/http"
)
const RDB_HOST = "localhost"
const RDB_PORT = "28015"
const TODO_DB = "todoapp"
const TODO_TABLE = "todos"
var session *rethink.Session
type Todo struct {
Id string `json:"id,omitempty"`
Title string `json:"title"`
Order int `json:"order"`
Done bool `json:"done"`
}
func getTodos() (todos []Todo, err error) {
err = rethink.Table(TODO_TABLE).Run(session).All(&todos)
return
}
func getTodo(id string) (todo Todo, err error) {
err = rethink.Table(TODO_TABLE).Get(id).Run(session).One(&todo)
return
}
func (todo *Todo) Update() (response rethink.WriteResponse, err error) {
err = rethink.Table(TODO_TABLE).Get(todo.Id).Update(todo).Run(session).One(&response)
return
}
func (todo *Todo) Delete() (rethink.WriteResponse, error) {
var response rethink.WriteResponse
err := rethink.Table(TODO_TABLE).Get(todo.Id).Delete().Run(session).One(&response)
return response, err
}
func setupDatabase() (err error) {
err = rethink.DbCreate(TODO_DB).Run(session).Exec()
if err != nil {
// TODO: Check if the failure is that the database already exists
return
}
err = rethink.TableCreate(TODO_TABLE).Run(session).Exec()
if err != nil {
// TODO: Check if the failure is that the table already exists
return
}
return nil
}
func todoListHandler(w http.ResponseWriter, r *http.Request) {
todos, err := getTodos()
if err != nil {
fmt.Println("Unable to fetch todos:", err)
}
header := w.Header()
header["Content-Type"] = []string{"application/json"}
responseBody, err := json.Marshal(todos)
if err != nil {
fmt.Println("Error marshalling todos:", err)
}
fmt.Fprintf(w, "%s", responseBody)
}
func todoCreateHandler(w http.ResponseWriter, r *http.Request) {
var todo Todo
var response rethink.WriteResponse
todoBody, err := ioutil.ReadAll(r.Body)
if err != nil {
fmt.Println("Error reading request body:", err)
}
err = json.Unmarshal(todoBody, &todo)
if err != nil {
fmt.Println("Error unmarshalling request body:", err)
}
err = rethink.Table(TODO_TABLE).Insert(todo).Run(session).One(&response)
if err != nil {
fmt.Println("Error inserting record:", err)
}
responseBody := map[string]string{}
responseBody["id"] = response.GeneratedKeys[0]
responseBodyText, err := json.Marshal(responseBody)
if err != nil {
fmt.Println("Unable to marshal response:", err)
}
fmt.Fprintf(w, "%s", responseBodyText)
}
func todoDetailHandler(w http.ResponseWriter, r *http.Request) {
vars := mux.Vars(r)
id := vars["id"]
todo, err := getTodo(id)
if err != nil {
fmt.Println("Unable to fetch todo %s: %s", id, err)
}
header := w.Header()
header["Content-Type"] = []string{"application/json"}
responseBody, err := json.Marshal(todo)
if err != nil {
fmt.Println("Error marshalling todo:", err)
}
fmt.Fprintf(w, "%s", responseBody)
}
func todoUpdateHandler(w http.ResponseWriter, r *http.Request) {
vars := mux.Vars(r)
id := vars["id"]
var todo Todo
todoBody, err := ioutil.ReadAll(r.Body)
if err != nil {
fmt.Println("Error reading request body:", err)
}
err = json.Unmarshal(todoBody, &todo)
if err != nil {
fmt.Println("Error unmarshalling request body:", err)
}
// Make sure that the Id of the todo matches
if todo.Id == "" {
todo.Id = id
} else if todo.Id != id {
fmt.Println("Attempted to update a todo without a matching id!")
}
response, err := todo.Update()
if err != nil {
fmt.Println("Unable to update todo %s: %s (%s)", id, err, response)
}
header := w.Header()
header["Content-Type"] = []string{"application/json"}
responseBody, err := json.Marshal(response)
if err != nil {
fmt.Println("Error marshalling response:", err)
}
fmt.Fprintf(w, "%s", responseBody)
}
func todoDeleteHandler(w http.ResponseWriter, r *http.Request) {
vars := mux.Vars(r)
id := vars["id"]
todo, err := getTodo(id)
if err != nil {
fmt.Println("Unable to fetch todo %s: %s", id, err)
}
response, err := todo.Delete()
if err != nil {
fmt.Println("Unable to delete todo %s: %s (%s)", id, err, response)
}
header := w.Header()
header["Content-Type"] = []string{"application/json"}
responseBody, err := json.Marshal(response)
if err != nil {
fmt.Println("Error marshalling response:", err)
}
fmt.Fprintf(w, "%s", responseBody)
}
func staticHandler(w http.ResponseWriter, r *http.Request) {
http.ServeFile(w, r, r.URL.Path[1:])
}
func indexHandler(w http.ResponseWriter, r *http.Request) {
body, err := ioutil.ReadFile("templates/index.html")
if err != nil {
fmt.Println("Error loading index template:", err)
}
fmt.Fprintf(w, "%s", body)
}
func main() {
var err error
connect_string := fmt.Sprintf("%s:%s", []byte(RDB_HOST), []byte(RDB_PORT))
session, err = rethink.Connect(connect_string, TODO_DB)
if err != nil {
fmt.Println("Error connecting:", err)
return
}
err = setupDatabase()
if err != nil {
fmt.Println("Unable to set up database:", err)
}
r := mux.NewRouter()
// API handlers
r.HandleFunc("/todos/{id}", todoDetailHandler).Methods("GET")
r.HandleFunc("/todos/{id}", todoUpdateHandler).Methods("PUT")
r.HandleFunc("/todos/{id}", todoDeleteHandler).Methods("DELETE")
r.HandleFunc("/todos", todoListHandler).Methods("GET")
r.HandleFunc("/todos", todoCreateHandler).Methods("POST")
// Front-end handlers
r.PathPrefix("/static/").Handler(http.StripPrefix("/static/", http.FileServer(http.Dir("static/"))))
r.HandleFunc("/", indexHandler)
http.ListenAndServe("0.0.0.0:8000", r)
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,263
|
When Elvis Presley had his first recording, though not that professional, for his mothers birthday, he had no idea of how his songs would travel far and wide. But just as good as they were through, his songs definitely soared higher than expected and everything about him was phenomenal, from his songs to his costumes.
And what could be more descriptive of Elvis Presley than his costumes. Elvis Presley's costumes were his unique trademark, making him more popular than the rest of his co-entertainers.
There are lots of people who still go crazy over Elvis Presley. Just look at how many people still line up whenever there is a sale on Elvis memorabilia.
So if you have ever found yourself in the midst of preparing for a particular costume party and not just a simple jail-house-rock kind of thing, whether it is a Halloween or birthday party, you probably want to use Elvis Presley costumes not just because they are an all time favorite for any kind of party but you are a die-hard Elvis fan.
If you really want to reflect the typical Elvis Presley by using his costumes, you'd better start with the hair and make-up. You need to find those Elvis look-alike wigs that are being sold on the market or in novelty stores. You may opt for that standard Elvis look with sleekly combed hair with matching wide sideburns or the Rock Star wig with soft, classic rock star waves.
For the perfect finish, you may add some foundation if your skin is darker than Elvis's.
Elvis was popular and widely recognized for his trademark sunglasses. It is one of those items that when not included, you can't simply say you are wearing an Elvis Presley costume.
Of course, to complete the Elvis Presley costume, you need to wear clothes that reflect the archetypal Elvis look. You can wear long sleeves and sleek, bell bottom or flared pants. Just be sure to emphasize the wide collar and vinyl-looking fabric. Top it off with complete accessories such as rhinestones, sequins, belt, glitters, and gold studs.
All of these things represent a typical Elvis Presley costume. Costumes can make or break your look at the party. With an Elvis Presley costume, you'll definitely stand out.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,871
|
Q: Can't launch Kodi - No display found + few simple questions I installed kodi on my bananian, a variation of debian, on banana pro via apt-get. I can't launch it tough. As the title states, in the error log file it says: ERROR: GLX Error: No Display found. Whole log file below. I tried googling it, but no luck there.
############## Kodi CRASH LOG ###############
################ SYSTEM INFO ################
Date: Tue Apr 5 21:05:08 UTC 2016
Kodi Options:
Arch: armv7l
Kernel: Linux 3.4.108-bananian #2 SMP PREEMPT Thu Aug 13 06:08:25 UTC 2015
Release: Debian GNU/Linux 8 (jessie)
############## END SYSTEM INFO ##############
############### STACK TRACE #################
gdb not installed, can't get stack trace.
############# END STACK TRACE ###############
################# LOG FILE ##################
21:05:03 T:2973908992 NOTICE: special://profile/ is mapped to: special://masterprofile/
21:05:03 T:2973908992 NOTICE: -----------------------------------------------------------------------
21:05:03 T:2973908992 NOTICE: Starting Kodi from Debian (15.2 Git: (unknown)). Platform: Linux ARM (Thumb) 32-bit
21:05:03 T:2973908992 NOTICE: Using Debug Kodi from Debian x32 build
21:05:03 T:2973908992 NOTICE: Kodi from Debian compiled from 15.2+dfsg1-1~bpo8+1 by GCC 4.9.2 for Linux ARM (Thumb) 32-bit version 3.16.7 (200711)
21:05:03 T:2973908992 NOTICE: Running on Debian GNU/Linux 8 (jessie), kernel: Linux ARM 32-bit version 3.4.108-bananian
21:05:03 T:2973908992 NOTICE: FFmpeg version: 2.8.1-1~bpo8+1
21:05:03 T:2973908992 NOTICE: Host CPU: ARMv7 Processor rev 4 (v7l), 2 cores available
21:05:03 T:2973908992 NOTICE: ARM Features: Neon disabled
21:05:03 T:2973908992 NOTICE: special://xbmc/ is mapped to: /usr/share/kodi
21:05:03 T:2973908992 NOTICE: special://xbmcbin/ is mapped to: /usr/lib/arm-linux-gnueabihf/kodi
21:05:03 T:2973908992 NOTICE: special://masterprofile/ is mapped to: /root/.kodi/userdata
21:05:03 T:2973908992 NOTICE: special://home/ is mapped to: /root/.kodi
21:05:03 T:2973908992 NOTICE: special://temp/ is mapped to: /root/.kodi/temp
21:05:03 T:2973908992 NOTICE: The executable running is: /usr/lib/arm-linux-gnueabihf/kodi/kodi.bin
21:05:03 T:2973908992 NOTICE: Local hostname: bananapi
21:05:03 T:2973908992 NOTICE: Log File is located: /root/.kodi/temp/kodi.log
21:05:03 T:2973908992 NOTICE: -----------------------------------------------------------------------
21:05:03 T:2973908992 DEBUG: ConsoleKit.Manager: org.freedesktop.DBus.GLib.UnmappedError.GDbusErrorQuark.Code2 - GDBus.Error:org.freedesktop.DBus.Error.ServiceUnknown: The name org.freedesktop.PolicyKit1 was not provided by any .service files
21:05:03 T:2973908992 DEBUG: UPower: org.freedesktop.DBus.Error.ServiceUnknown - The name org.freedesktop.UPower was not provided by any .service files
21:05:03 T:2973908992 DEBUG: ConsoleKit.Manager: org.freedesktop.DBus.GLib.UnmappedError.GDbusErrorQuark.Code2 - GDBus.Error:org.freedesktop.DBus.Error.ServiceUnknown: The name org.freedesktop.PolicyKit1 was not provided by any .service files
21:05:03 T:2973908992 DEBUG: DeviceKit.Power: org.freedesktop.DBus.Error.ServiceUnknown - The name org.freedesktop.DeviceKit.Disks was not provided by any .service files
21:05:03 T:2973908992 DEBUG: UPower: org.freedesktop.DBus.Error.ServiceUnknown - The name org.freedesktop.UPower was not provided by any .service files
21:05:03 T:2973908992 NOTICE: load settings...
21:05:04 T:2973908992 DEBUG: CSettings: loaded settings definition from special://xbmc/system/settings/settings.xml
21:05:04 T:2973908992 DEBUG: CSettings: loaded settings definition from special://xbmc/system/settings/linux.xml
21:05:04 T:2973908992 ERROR: PulseAudio: Failed to connect context
21:05:04 T:2973908992 NOTICE: PulseAudio might not be running. Context was not created.
21:05:04 T:2906121040 NOTICE: Thread FDEventMonitor start, auto delete: false
21:05:04 T:2973908992 INFO: CAESinkALSA - Unable to open device "surround71" for playback
21:05:04 T:2973908992 INFO: CAESinkALSA - Unable to open device "surround51" for playback
21:05:04 T:2973908992 INFO: CAESinkALSA - Unable to open device "surround71" for playback
21:05:04 T:2973908992 INFO: CAESinkALSA - Unable to open device "surround40" for playback
21:05:04 T:2973908992 INFO: CAESinkALSA - Unable to open device "surround51" for playback
21:05:04 T:2973908992 INFO: CAESinkALSA - Unable to open device "surround71" for playback
21:05:04 T:2973908992 INFO: CAESinkALSA - Unable to open device "pulse" for playback
21:05:04 T:2973908992 NOTICE: Found 1 Lists of Devices
21:05:04 T:2973908992 NOTICE: Enumerated ALSA devices:
21:05:04 T:2973908992 NOTICE: Device 1
21:05:04 T:2973908992 NOTICE: m_deviceName : @
21:05:04 T:2973908992 NOTICE: m_displayName : Default (sunxi-CODEC sunxi PCM)
21:05:04 T:2973908992 NOTICE: m_displayNameExtra:
21:05:04 T:2973908992 NOTICE: m_deviceType : AE_DEVTYPE_PCM
21:05:04 T:2973908992 NOTICE: m_channels : FL,FR
21:05:04 T:2973908992 NOTICE: m_sampleRates : 8000,11025,16000,22050,32000,44100,48000,96000,192000
21:05:04 T:2973908992 NOTICE: m_dataFormats : AE_FMT_S16NE,AE_FMT_S16LE
21:05:04 T:2973908992 NOTICE: No settings file to load (special://xbmc/system/advancedsettings.xml)
21:05:04 T:2973908992 NOTICE: No settings file to load (special://masterprofile/advancedsettings.xml)
21:05:04 T:2973908992 NOTICE: Default DVD Player: dvdplayer
21:05:04 T:2973908992 NOTICE: Default Video Player: dvdplayer
21:05:04 T:2973908992 NOTICE: Default Audio Player: paplayer
21:05:04 T:2973908992 NOTICE: Disabled debug logging due to GUI setting. Level 0.
21:05:04 T:2973908992 NOTICE: Log level changed to "LOG_LEVEL_NORMAL"
21:05:04 T:2973908992 NOTICE: Loading player core factory settings from special://xbmc/system/playercorefactory.xml.
21:05:04 T:2973908992 DEBUG: CPlayerCoreConfig::<ctor>: created player DVDPlayer for core 1
21:05:04 T:2973908992 DEBUG: CPlayerCoreConfig::<ctor>: created player oldmplayercore for core 1
21:05:04 T:2973908992 DEBUG: CPlayerCoreConfig::<ctor>: created player PAPlayer for core 3
21:05:04 T:2973908992 DEBUG: CPlayerSelectionRule::Initialize: creating rule: system rules
21:05:04 T:2973908992 DEBUG: CPlayerSelectionRule::Initialize: creating rule: hdhomerun/mms/udp
21:05:04 T:2973908992 DEBUG: CPlayerSelectionRule::Initialize: creating rule: lastfm/shout
21:05:04 T:2973908992 DEBUG: CPlayerSelectionRule::Initialize: creating rule: rtmp
21:05:04 T:2973908992 DEBUG: CPlayerSelectionRule::Initialize: creating rule: rtsp
21:05:04 T:2973908992 DEBUG: CPlayerSelectionRule::Initialize: creating rule: streams
21:05:04 T:2973908992 DEBUG: CPlayerSelectionRule::Initialize: creating rule: aacp/sdp
21:05:04 T:2973908992 DEBUG: CPlayerSelectionRule::Initialize: creating rule: mp2
21:05:04 T:2973908992 DEBUG: CPlayerSelectionRule::Initialize: creating rule: dvd
21:05:04 T:2973908992 DEBUG: CPlayerSelectionRule::Initialize: creating rule: dvdimage
21:05:04 T:2973908992 DEBUG: CPlayerSelectionRule::Initialize: creating rule: sdp/asf
21:05:04 T:2973908992 DEBUG: CPlayerSelectionRule::Initialize: creating rule: nsv
21:05:04 T:2973908992 DEBUG: CPlayerSelectionRule::Initialize: creating rule: radio
21:05:04 T:2973908992 NOTICE: Loaded playercorefactory configuration
21:05:04 T:2973908992 NOTICE: Loading player core factory settings from special://masterprofile/playercorefactory.xml.
21:05:04 T:2973908992 NOTICE: special://masterprofile/playercorefactory.xml does not exist. Skipping.
21:05:04 T:2973908992 INFO: creating subdirectories
21:05:04 T:2973908992 INFO: userdata folder: special://masterprofile/
21:05:04 T:2973908992 INFO: recording folder:
21:05:04 T:2973908992 INFO: screenshots folder:
21:05:04 T:2829050704 NOTICE: Thread ActiveAE start, auto delete: false
21:05:04 T:2896163664 NOTICE: Thread AESink start, auto delete: false
21:05:04 T:2896163664 INFO: CActiveAESink::OpenSink - initialize sink
21:05:04 T:2896163664 DEBUG: CActiveAESink::OpenSink - trying to open device ALSA:@
21:05:04 T:2896163664 INFO: CAESinkALSA::Initialize - Attempting to open device "@"
21:05:04 T:2896163664 INFO: CAESinkALSA::Initialize - Opened device "sysdefault"
21:05:04 T:2896163664 INFO: CAESinkALSA::InitializeHW - Your hardware does not support AE_FMT_FLOAT, trying other formats
21:05:04 T:2896163664 INFO: CAESinkALSA::InitializeHW - Using data format AE_FMT_S16NE
21:05:04 T:2896163664 DEBUG: CAESinkALSA::InitializeHW - Request: periodSize 2048, bufferSize 8192
21:05:04 T:2896163664 DEBUG: CAESinkALSA::InitializeHW - Got: periodSize 2048, bufferSize 8192
21:05:04 T:2896163664 DEBUG: CAESinkALSA::InitializeHW - Setting timeout to 186 ms
21:05:04 T:2896163664 DEBUG: CAESinkALSA::GetChannelLayout - Input Channel Count: 2 Output Channel Count: 2
21:05:04 T:2896163664 DEBUG: CAESinkALSA::GetChannelLayout - Requested Layout: FL,FR
21:05:04 T:2896163664 DEBUG: CAESinkALSA::GetChannelLayout - Got Layout: FL,FR (ALSA: none)
21:05:04 T:2896163664 DEBUG: CActiveAESink::OpenSink - ALSA Initialized:
21:05:04 T:2896163664 DEBUG: Output Device : Default (sunxi-CODEC sunxi PCM)
21:05:04 T:2896163664 DEBUG: Sample Rate : 44100
21:05:04 T:2896163664 DEBUG: Sample Format : AE_FMT_S16NE
21:05:04 T:2896163664 DEBUG: Channel Count : 2
21:05:04 T:2896163664 DEBUG: Channel Layout: FL,FR
21:05:04 T:2896163664 DEBUG: Frames : 2048
21:05:04 T:2896163664 DEBUG: Frame Samples : 4096
21:05:04 T:2896163664 DEBUG: Frame Size : 4
21:05:04 T:2973908992 NOTICE: Running database version Addons19
21:05:04 T:2973908992 DEBUG: SECTION:LoadDLL(special://xbmcbin/system/libcpluff-arm.so)
21:05:04 T:2973908992 DEBUG: Loading: /usr/lib/arm-linux-gnueabihf/kodi/system/libcpluff-arm.so
21:05:04 T:2973908992 NOTICE: ADDONS: Using repository repository.xbmc.org
21:05:04 T:2885677904 NOTICE: Thread RemoteControl start, auto delete: false
21:05:04 T:2885677904 INFO: LIRC Process: using: /dev/lircd
21:05:04 T:2885677904 INFO: LIRC Connect: connect failed: No such file or directory
21:05:04 T:2885677904 INFO: CRemoteControl::Process - failed to connect to LIRC, will keep retrying every 5 seconds
21:05:04 T:2973908992 INFO: CKeyboardLayoutManager: loading keyboard layouts from special://xbmc/system/keyboardlayouts...
21:05:04 T:2973908992 DEBUG: CKeyboardLayoutManager: keyboard layout "Arabic QWERTY" successfully loaded
21:05:04 T:2973908992 DEBUG: CKeyboardLayoutManager: keyboard layout "Norwegian QWERTY" successfully loaded
21:05:04 T:2973908992 DEBUG: CKeyboardLayoutManager: keyboard layout "Portuguese (Portugal) QWERTY" successfully loaded
21:05:04 T:2973908992 DEBUG: CKeyboardLayoutManager: keyboard layout "Hebrew QWERTY" successfully loaded
21:05:04 T:2973908992 DEBUG: CKeyboardLayoutManager: keyboard layout "Hebrew ABC" successfully loaded
21:05:04 T:2973908992 DEBUG: CKeyboardLayoutManager: keyboard layout "Russian ЙЦУКЕН" successfully loaded
21:05:04 T:2973908992 DEBUG: CKeyboardLayoutManager: keyboard layout "Russian АБВ" successfully loaded
21:05:04 T:2973908992 DEBUG: CKeyboardLayoutManager: keyboard layout "Lithuanian AZERTY" successfully loaded
21:05:04 T:2973908992 DEBUG: CKeyboardLayoutManager: keyboard layout "Lithuanian QWERTY" successfully loaded
21:05:04 T:2973908992 DEBUG: CKeyboardLayoutManager: keyboard layout "Ukrainian ЙЦУКЕН" successfully loaded
21:05:04 T:2973908992 DEBUG: CKeyboardLayoutManager: keyboard layout "Ukrainian АБВ" successfully loaded
21:05:04 T:2973908992 DEBUG: CKeyboardLayoutManager: keyboard layout "Spanish QWERTY" successfully loaded
21:05:04 T:2973908992 DEBUG: CKeyboardLayoutManager: keyboard layout "Romanian QWERTY" successfully loaded
21:05:04 T:2973908992 DEBUG: CKeyboardLayoutManager: keyboard layout "Turkish QWERTY" successfully loaded
21:05:04 T:2973908992 DEBUG: CKeyboardLayoutManager: keyboard layout "Danish QWERTY" successfully loaded
21:05:04 T:2973908992 DEBUG: CKeyboardLayoutManager: keyboard layout "Bulgarian ЯВЕРТЪ" successfully loaded
21:05:04 T:2973908992 DEBUG: CKeyboardLayoutManager: keyboard layout "Bulgarian АБВ" successfully loaded
21:05:04 T:2973908992 DEBUG: CKeyboardLayoutManager: keyboard layout "Swedish QWERTY" successfully loaded
21:05:04 T:2973908992 DEBUG: CKeyboardLayoutManager: keyboard layout "German QWERTZ" successfully loaded
21:05:04 T:2973908992 DEBUG: CKeyboardLayoutManager: keyboard layout "Hungarian QWERTZ" successfully loaded
21:05:04 T:2973908992 DEBUG: CKeyboardLayoutManager: keyboard layout "Polish QWERTY" successfully loaded
21:05:04 T:2973908992 DEBUG: CKeyboardLayoutManager: keyboard layout "English QWERTY" successfully loaded
21:05:04 T:2973908992 DEBUG: CKeyboardLayoutManager: keyboard layout "English AZERTY" successfully loaded
21:05:04 T:2973908992 DEBUG: CKeyboardLayoutManager: keyboard layout "English ABC" successfully loaded
21:05:04 T:2973908992 DEBUG: CKeyboardLayoutManager: keyboard layout "Greek QWERTY" successfully loaded
21:05:04 T:2973908992 DEBUG: UDisks: org.freedesktop.DBus.Error.ServiceUnknown - The name org.freedesktop.UDisks was not provided by any .service files
21:05:04 T:2973908992 DEBUG: DeviceKit.Disks: org.freedesktop.DBus.Error.ServiceUnknown - The name org.freedesktop.DeviceKit.Disks was not provided by any .service files
21:05:04 T:2973908992 DEBUG: Selected UDev as storage provider
21:05:04 T:2973908992 ERROR: GLX Error: No Display found
21:05:04 T:2973908992 FATAL: CApplication::Create: Unable to init windowing system
21:05:04 T:2973908992 DEBUG: PVRManager - destroyed
21:05:04 T:2885677904 DEBUG: Failed to connect to LIRC. Giving up.
############### END LOG FILE ################
############ END Kodi CRASH LOG #############
Also, I have a couple of questions that bother me and I couldn't really find the answer on the internet. Perhaps I didn't know how to word it, because this seems like some really simple stuff.
This bananian distro doesn't ship with any gui. When it boots, there is just plain shell. Should I be able to run a fullscreen app with no gui? (I installed xfce before trying to launch kodi, asking just because I don't know how exactly it works) How to force system to launch xfce, if I want it to? I also installed xorg, but same with this, not quite sure why I need it and what is the exact role of this tool.
A: You do need a GUI so that you can see the Kodi interface. Using Xfce will work. Sounds like you are having issues with figuring out how to run Xfce. I found this article on Xfce with Debian. Below is the section pertaining to running Xfce, but if you view the article, it will go through the installation process as well as information on what Xfce is. In basic terms, it is a desktop environment.
With a display manager
For GDM, KDM and LightDM choose xfce4-session.
For slim boot session in /etc/slim.conf:
login_cmd exec ck-launch-session /bin/bash -login /etc/X11/Xsession %session
You also can install xdm to login in start Xfce.
From the console
If you don't use a login manager but start Xfce from console, you need
to take care of few stuff in order to get a complete Xfce session with
full permission (mount, suspend/shutdown/hibernate etc.) This is
because Debian now uses ?PolicyKit/ConsoleKit to manage policies for
things like device and power management.
*
*only use startx, without any argument
*don't use a .xinitrc, add in file ~/.xsession:
exec ck-launch-session startxfce4
This is because ?ConsoleKit ships an init script
(/etc/X11/Xsession.d/90consolekit), but the /etc/X11/Xsession.d/
scripts are only executed if you don't use any .xinitrc. See startx
(1) for more information.
Then you need to fine-tune your pam installation so ?ConsoleKit can be
sure that your user is correctly authenticated. For that, you need to
install libpam-ck-connector and put before pam_ck_connector.so in
/etc/pam.d/common-session.
session optional pam_loginuid.so
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,523
|
require "simplecov"
SimpleCov.start do
add_filter "/spec/"
end
require "rspec"
require "timecop"
require "fakefs/safe"
require "fakefs/spec_helpers"
$:.unshift File.expand_path("../../lib", __FILE__)
def mock_export_error(message)
lambda { yield }.should raise_error(Foreman::Export::Exception, message)
end
def mock_error(subject, message)
mock_exit do
mock(subject).puts("ERROR: #{message}")
yield
end
end
def foreman(args)
capture_stdout do
begin
Foreman::CLI.start(args.split(" "))
rescue SystemExit
end
end
end
def forked_foreman(args)
rd, wr = IO.pipe("BINARY")
Process.spawn("bundle exec bin/foreman #{args}", :out => wr, :err => wr)
wr.close
rd.read
end
def fork_and_capture(&blk)
rd, wr = IO.pipe("BINARY")
pid = fork do
rd.close
wr.sync = true
$stdout.reopen wr
$stderr.reopen wr
blk.call
$stdout.flush
$stdout.close
end
wr.close
Process.wait pid
buffer = ""
until rd.eof?
buffer += rd.gets
end
end
def fork_and_get_exitstatus(args)
pid = Process.spawn("bundle exec bin/foreman #{args}", :out => "/dev/null", :err => "/dev/null")
Process.wait(pid)
$?.exitstatus
end
def mock_exit(&block)
block.should raise_error(SystemExit)
end
def write_foreman_config(app)
File.open("/etc/foreman/#{app}.conf", "w") do |file|
file.puts %{#{app}_processes="alpha bravo"}
file.puts %{#{app}_alpha="1"}
file.puts %{#{app}_bravo="2"}
end
end
def write_procfile(procfile="Procfile", alpha_env="")
File.open(procfile, "w") do |file|
file.puts "alpha: ./alpha" + " #{alpha_env}".rstrip
file.puts "\n"
file.puts "bravo:\t./bravo"
file.puts "foo_bar:\t./foo_bar"
file.puts "foo-bar:\t./foo-bar"
end
File.expand_path(procfile)
end
def write_env(env=".env", options={"FOO"=>"bar"})
File.open(env, "w") do |file|
options.each do |key, val|
file.puts "#{key}=#{val}"
end
end
end
def without_fakefs
FakeFS.deactivate!
ret = yield
FakeFS.activate!
ret
end
def load_export_templates_into_fakefs(type)
without_fakefs do
Dir[File.expand_path("../../data/export/#{type}/**/*", __FILE__)].inject({}) do |hash, file|
next(hash) if File.directory?(file)
hash.update(file => File.read(file))
end
end.each do |filename, contents|
FileUtils.mkdir_p File.dirname(filename)
File.open(filename, "w") do |f|
f.puts contents
end
end
end
def resource_path(filename)
File.expand_path("../resources/#{filename}", __FILE__)
end
def example_export_file(filename)
FakeFS.deactivate!
data = File.read(File.expand_path(resource_path("export/#{filename}"), __FILE__))
FakeFS.activate!
data
end
def preserving_env
old_env = ENV.to_hash
begin
yield
ensure
ENV.clear
ENV.update(old_env)
end
end
def normalize_space(s)
s.gsub(/\n[\n\s]*/, "\n")
end
def capture_stdout
old_stdout = $stdout.dup
rd, wr = IO.method(:pipe).arity.zero? ? IO.pipe : IO.pipe("BINARY")
$stdout = wr
yield
wr.close
rd.read
ensure
$stdout = old_stdout
end
RSpec.configure do |config|
config.treat_symbols_as_metadata_keys_with_true_values = true
config.color_enabled = true
config.order = 'rand'
config.include FakeFS::SpecHelpers, :fakefs
config.mock_with :rr
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,155
|
\section{Discrepancy Theory and Signed Domination}
Originated from number theory, discrepancy theory is, generally
speaking, the study of irregularities of distributions in various
settings. Classical combinatorial discrepancy theory is devoted to
the problem of partitioning the vertex set of a hypergraph into
two classes in such a way that all hyperedges are split into
approximately equal parts by the classes, i.e. we are interested
in measuring the deviation of an optimal partition from perfect,
when all hyperedges are split into equal parts. It may be pointed
out that many classical results in various areas of mathematics,
e.g. geometry and number theory, can be formulated in these terms.
Combinatorial discrepancy theory was introduced and studied by
Beck in \cite{beck1}. Also, studies on discrepancy theory have
been conducted in \cite{beck3,beck2, beck4} and \cite{sos}.
Let ${\cal H}=(V,{\cal E})$ be a hypergraph with the vertex set $V$
and the hyperedge set ${\cal E}=\{E_1,...,E_m\}.$ One of the main
problems in classical combinatorial discrepancy theory is to colour
the elements of $V$ by two colours in such a way that all of the
hyperedges have almost the same number of elements of each colour.
Such a partition of $V$ into two classes can be represented by a
function
$$f:V\rightarrow \{+1, -1\}.$$
For a set $E\subseteq V$, let us define the {\em imbalance} of $E$ as follows:
$$
f(E)=\sum_{v \in E} f(v).
$$
First defined by Beck \cite{beck1}, the
{\em discrepancy of ${\cal H}$ with respect to} $f$ is
$$
{\cal D}({\cal H},f)=\max_{E_i \in {\cal E}}|f(E_i)|
$$
and the {\em discrepancy} of ${\cal H}$ is
$$
{\cal D}({\cal H})=\min_{\tiny f:\; V\rightarrow\{+1,-1\}}{\cal
D}({\cal H},f).
$$
Thus, the discrepancy of a hypergraph tells us how well all its
hyperedges can be partitioned. Spencer \cite{spe1} proved a
fundamental ``six-standard-deviation" result that for any hypergraph
${\cal H}$ with $n$ vertices and $n$ hyperedges,
$$
{\cal D}({\cal H}) \le 6 \sqrt n.
$$
As shown in \cite{alo}, this bound is best possible up to a constant
factor. More precisely, if a Hadamard matrix of order $n>1$ exists,
then there is a hypergraph ${\cal H}$ with $n$ vertices and $n$
hyperedges such that
$$
{\cal D}({\cal H}) \ge 0.5 \sqrt n.
$$
It is well known that a Hadamard matrix of order between $n$ and
$(1-\epsilon)n$ does exist for any $\epsilon$ and sufficiently large
$n$. The following important result, due to Beck and Fiala
\cite{beck2}, is valid for a hypergraph with any number of
hyperedges:
$$
{\cal D}({\cal H}) \le 2\Delta -1,
$$
where $\Delta$ is the maximum degree of vertices of ${\cal H}$. They
also posed the discrepancy conjecture that for some constant $K$
$$
{\cal D}({\cal H}) < K \sqrt\Delta.
$$
Another interesting aspect of discrepancy was discussed by
F\"{u}redi and Mubayi in their fundamental paper \cite{fur}. A
function $g: V\rightarrow \{+1, -1\}$ is called a {\em signed
domination function} (SDF) of the hypergraph ${\cal H}$ if
$$
g(E_i)=\sum_{v \in E_i} g(v) \ge 1
$$
for every hyperedge $E_i\in \cal E$, i.e. each hyperedge has a
positive imbalance. The {\em signed discrepancy} of ${\cal H}$,
denoted by ${\cal SD}({\cal H})$, is defined in the following way:
$$
{\cal SD}({\cal H})=\min_{{\mathrm SDF} g} g(V),
$$
where the minimum is taken over all signed domination functions of
${\cal H}$. Thus, in this version of discrepancy, the success is
measured by minimizing the imbalance of the vertex set $V$, while
keeping the imbalance of every hyperedge $E_i\in \cal E$ positive.
One of the main results in this context, formulated in terms of
hypergraphs, is due to F\"{u}redi and Mubayi \cite{fur}:
\begin{thm}[\cite{fur}] \label{f_m}
Let ${\cal H}$ be an $n$-vertex hypergraph with hyperedge set
${\cal E}= \{E_1,...,E_m\}$, and suppose that every hyperedge has
at least $k$ vertices, where $k \ge 100.$ Then
$$
{\cal SD}({\cal H}) \le 4{\sqrt{\ln k \over k}}n + {1 \over k}m.
$$
\end{thm}
This theorem can be easily re-formulated in terms of graphs by
considering the neighbourhood hypergraph of a given graph. Note that
$\gamma_s(G)$ is the signed domination number of a graph $G$, which is
formally defined in the next section.
\begin{thm}[\cite{fur}] \label{fur_m_thm}
If $G$ has $n$ vertices and minimum degree $\delta \ge 99$, then
$$
\gamma_s(G) \le \left(4\sqrt{{\ln(\delta+1) \over \delta+1}} + {1 \over \delta+1} \right)n.
$$
\end{thm}
Moreover, F\"{u}redi and Mubayi \cite{fur} found quite good upper
bounds for very small values of $\delta$ and, using Hadamard
matrices, constructed a $\delta$-regular graph $G$ of order
$4\delta$ with
$$
\gamma_s(G) \ge 0.5 \sqrt \delta - O(1).
$$
This construction shows that the upper bound in Theorem
\ref{fur_m_thm} is off from optimal by at most the factor of
$\sqrt{\ln\delta}$. They posed an interesting conjecture that, for
some constant $C$,
$$
\gamma_s(G) \le {C \over \sqrt\delta} n,
$$
and proved that the above discrepancy conjecture, if true, would
imply this upper bound for $\delta$-regular graphs. A strong result
of Matou\v{s}ek \cite{mat} shows that the bound is true, but the
constant $C$ in his proof is big making the result of rather
theoretical interest.
The lower bound for the signed domination number given in the
theorem below is formulated in terms of the degree sequence of a
graph. Other lower bounds are also known, see Corollaries
\ref{haas-zhang}, \ref{lb-Cor1} and \ref{lb-Cor2}.
\begin{thm}[\cite{dun1}] \label{dunb_low}
Let $G$ be a graph with degrees $d_1 \le d_2 \le...\le d_n.$ If $k$ is the smallest integer for which
$$
\sum_{i=0}^{k-1}d_{n-i} \ge 2(n-k) + \sum_{i=1}^{n-k}d_{i},
$$
then
$$\gamma_s(G) \ge 2k-n.$$
\end{thm}
In this paper, we present new upper and lower bounds for the
signed domination number, which improve the above theorems and
also generalise three known results formulated in Corollaries
\ref{haas-zhang}, \ref{lb-Cor1} and \ref{lb-Cor2}. Note that our
results can be easily re-formulated in terms of hypergraphs.
Moreover, we rectify F\"{u}redi--Mubayi's conjecture formulated
above as follows: for some $C\le 10$ and $\alpha$, $0.18\le \alpha
< 0.21$,
$$
\gamma_s(G) \le \min\Big\{{n\over \delta^\alpha}, {Cn\over
\sqrt\delta}\Big\}.
$$
\section{Notation and Technical Results
All graphs will be finite and undirected without loops and multiple
edges. If $G$ is a graph of order $n$, then
$V(G)=\{v_1,v_2,...,v_n\}$ is the set of vertices in $G$ and $d_i$
denotes the degree of $v_i$. Let $N(x)$ denote the neighbourhood of
a vertex $x$. Also, let $ N(X)=\cup_{x\in X} {N(x)} $ and $
N[X]=N(X)\cup X$. Denote by $\delta(G)$ and $\Delta(G)$ the minimum
and maximum degrees of vertices of $G$, respectively. Put
$\delta=\delta(G)$ and $\Delta=\Delta(G)$.
A set $X$ is called a {\it dominating set} if every vertex not in
$X$ is adjacent to a vertex in $X$. The minimum cardinality of a
dominating set of $G$ is called the \emph{domination number}
$\gamma(G)$. The domination number can be defined equivalently by
means of a {\it domination function}, which can be considered as a
characteristic function of a dominating set in $G$. A function $f :
V(G) \rightarrow \{0, 1\}$ is a domination function on a graph $G$
if for each vertex ${v \in V(G)}$,
\begin{equation}\label{exfirst}
\sum_{x \in N[v]}f(x) \ge 1.
\end{equation}
The value $\sum_{v \in V(G)} f(v)$
is called the weight $f(V(G))$ of the function $f$. It is obvious
that the minimum of weights, taken over all domination functions on
$G$, is the domination number $\gamma(G)$ of $G$.
It is easy to obtain different variations of the domination number
by replacing the set \{0,1\} by another set of numbers. If \{0,1\}
is exchanged by \{-1,1\}, then we obtain {\it the signed
domination number}. A signed domination function of a graph $G$
was defined in \cite{dun1} as a function $f : V(G) \rightarrow
\{-1, 1 \}$ such that for each $v \in V(G)$, the expression
(\ref{exfirst}) is true. The signed domination number of a graph
$G$, denoted $\gamma_s(G)$, is the minimum of weights $f(V(G))$,
taken over all signed domination functions $f$ on $G$. A research
on signed domination has been carried out in
\cite{dun1}--\cite{hen2} and \cite{mat}.
Let $d \ge 2$ be an integer and $0\le p \le 1.$ Let us denote
$$
f(d,p)=\sum_{m=0}^{\lceil 0.5d \rceil} (\lceil 0.5d \rceil-m+1)\pmatrix{d+1 \cr m}p^m(1-p)^{d+1-m}.
$$
We will need the following technical results:
\begin{lem}[\cite{fur}] \label{fur_m}
If $d$ is odd, then $$f(d+1,p) < 2(1-p)f(d,p).$$ If $d$ is even, then
$$f(d+1,p)<\left(2p+(1-p){d+4\over d+2}\right)f(d,p).$$
In particular, if $$2(1-p)\left(2p+(1-p){d+4 \over
d+2}\right)<1,$$ then \begin{displaymath}\max_{d \ge
\delta}{f(d,p)\in \{f(\delta,p),f(\delta+1,p)\}}.
\end{displaymath}
\end{lem}
\begin{lem}[\cite{cher}] \label{chernoff}
Let $p \in [0,1]$ and $X_1,...,X_k$ be mutually independent random
variables with
\begin{eqnarray*}
&{\mathbf P}[X_i=1-p]=p, \\
&{\mathbf P}[X_i=-p]= 1-p.
\end{eqnarray*}
If $X= X_1+...+X_k$ and $c>0$, then
$$
{\mathbf P}[X < -c] < e^{-{c^{2} \over 2pk}}.
$$
\end{lem}
Let us also denote
$$
{\widetilde d}_{0.5} = \pmatrix{\delta'+1 \cr {\lceil 0.5 \delta'
\rceil}},
$$
where
\begin{displaymath}
\delta'= \cases{\\
\delta& if $\delta$ is odd;\cr\\
\delta+1& if $\delta$ is even.\\
}\\
\end{displaymath}
\section{Upper Bounds for the Signed Domination Number
The following theorem provides an upper bound for the signed
domination number, which is better than the bound of Theorem
\ref{fur_m_thm} for `relatively small' values of $\delta$. For
example, if $\delta(G)=99$, then, by Theorem \ref{fur_m_thm},
$\gamma_{s}(G) \le 0.869n,$ while Theorem \ref{main_signed} yields
$\gamma_{s}(G) \le 0.537n$. For larger values of $\delta$, the
latter result is improved in Corollaries \ref{c1}--\ref{c3}.
\begin{thm}\label{main_signed}
For any graph $G$ with $\delta > 1,$
\begin{equation}\label{boundsigned}
\gamma_{s}(G) \le \left(1-{{2\widehat{\delta}} \over
{(1+{\widehat{\delta}})}^{1+1/{\widehat{\delta}}}\,{\widetilde
d}_{0.5}^{\;1/{\widehat{\delta}}}} \right) n,
\end{equation}
where
$ \widehat{\delta} = \lfloor 0.5 \delta \rfloor$.
\end{thm}
\noindent{\bf Proof: } Let $A$ be a set formed by an independent choice of vertices
of $G$, where each vertex is selected with the probability
$$p = 1- {1 \over (1+\widehat{\delta})^{1/\widehat{\delta}}\;{\widetilde
d}_{0.5}^{\;1/\widehat{\delta}}}.$$ For $m\ge0$, let us denote by $B_m$ the set of
vertices $v\in V(G)$ dominated by exactly $m$ vertices of $A$ and
such that $|N[v]\cap A|< \lceil 0.5 d_v \rceil +1$, i.e.
$$
|N[v]\cap A| = m \le \lceil 0.5 d_v \rceil.
$$
Note that each vertex $v\in V(G)$ is in at most one of the sets
$B_m$ and $0\le m\le \lceil 0.5 d_v \rceil$. Then we form a set $B$
by selecting $\lceil 0.5 d_v \rceil - m + 1 $ vertices from $N[v]$
that are not in $A$ for each vertex $v \in B_m$ and adding them to
$B$. We construct the set $D$ as follows: $D=A\cup B$. Let us assume
that $f$ is a function $f : V(G) \rightarrow \{-1, 1\}$ such that
all vertices in $D$ are labelled by $1$ and all other vertices by
$-1$. It is obvious that $f(V(G))=|D| - (n - |D|)$ and $f$ is a
signed domination function.
The expectation of $f(V(G))$ is
\begin{eqnarray*}
{\mathbf E}[f(V(G))] &=& 2{\mathbf E}[|D|]- n\\
&=& 2({\mathbf E}[|A|] + {\mathbf E}[|B|]) - n\\
&\le& 2 \, \sum_{i=1}^{n}{\mathbf P}(v_i\in A)+ 2
\sum_{i=1}^{n}\sum_{m=0}^{\lceil
0.5 d_i \rceil}(\lceil 0.5 d_i \rceil - m + 1) {\mathbf P}(v_i\in B_m) - n \\
&=& 2pn + 2\sum_{i=1}^{n}\sum_{m=0}^{\lceil 0.5 d_i \rceil}(\lceil
0.5 d_i \rceil - m + 1) \pmatrix{d_i+1 \cr m} p^m (1-p)^{d_i+1-m}
- n\\
&\le& 2pn + 2\sum_{i=1}^{n}{\max_{d_i \ge \delta}f(d_i,p)-n}.
\end{eqnarray*}
It is not difficult to check that
$2(1-p)(2p+(1-p)(d+4)/(d+2)) < 1$ for any $d \ge \delta \ge 2$. By Lemma \ref{fur_m},
$$
\max_{d \ge \delta}{f(d,p) \in \{f(\delta,p),f(\delta+1,p)}\}.
$$
The last inequality implies $2(1-p)<1$.
Therefore, by Lemma \ref{fur_m},
$$
\max_{d \ge \delta}{f(d,p)}=f(\delta,p)
$$
if $\delta$ is odd. If $\delta$ is even, then we can prove that
$$
\max_{d \ge \delta}{f(d,p)}=f(\delta+1,p).
$$
Thus,
$$
\max_{d \ge \delta}{f(d,p)}=f(\delta',p).
$$
Therefore,
$$
{\mathbf E}[f(V(G))] \le 2pn + 2n\sum_{m=0}^{\lceil 0.5 \delta'
\rceil}(\lceil 0.5 \delta' \rceil - m + 1) \pmatrix{\delta'+1 \cr
m} p^m (1-p)^{\delta'+1-m} - n.
$$
Since
$$(\lceil 0.5 \delta' \rceil - m + 1) \pmatrix{\delta'+1 \cr m} \le \pmatrix{\delta'+1 \cr \lceil 0.5 \delta' \rceil} \pmatrix{\lceil 0.5 \delta' \rceil \cr m},$$
we obtain
\begin{eqnarray*}
{\mathbf E}[f(V(G))]&\le& 2pn + 2n\sum_{m=0}^{\lceil 0.5 \delta' \rceil} \pmatrix{\delta'+1 \cr \lceil 0.5 \delta' \rceil} \pmatrix{\lceil 0.5 \delta' \rceil \cr m} p^m (1-p)^{\delta'+1-m} - n\\
&=& 2pn + 2n\pmatrix{\delta'+1 \cr \lceil 0.5 \delta' \rceil} (1-p)^{\delta'-\lceil 0.5 \delta' \rceil+1} \sum_{m=0}^{\lceil 0.5 \delta' \rceil} \pmatrix{\lceil 0.5 \delta' \rceil \cr m} p^m (1-p)^{\lceil 0.5 \delta' \rceil-m} - n\\
&=& 2pn + 2n{\widetilde d}_{0.5}(1-p)^{\delta'-\lceil 0.5 \delta' \rceil+1} - n.
\end{eqnarray*}
Taking into account that $\delta'-\lceil0.5 \delta'\rceil=\lfloor0.5\delta'\rfloor=\lfloor0.5\delta\rfloor=\widehat{\delta},$ we have
\begin{eqnarray*}
{\mathbf E}[f(V(G))] &\le& 2pn + 2n{\widetilde
d}_{0.5}(1-p)^{\widehat{\delta}+1} - n\\
&\le& \left(1-{2\;\widehat{\delta} \over
{(1+\widehat{\delta})}^{1+1/\widehat{\delta}} \,{\widetilde
d}_{0.5}^{\;1/\widehat{\delta}}} \right) n,
\end{eqnarray*}
as required. The proof of Theorem \ref{main_signed} is complete.
\hfill$\rule{.05in}{.1in}$\vspace{.3cm}
Our next result and its corollaries give a modest improvement of
Theorem \ref{fur_m_thm}. More precisely, the upper bound of Theorem
\ref{main_2} is asymptotically 1.63 times better than the bound of
Theorem \ref{fur_m_thm}, and for $\delta=10^6$ the improvement is
1.44 times.
\begin{thm}\label{main_2}
If $\delta(G) \ge 10^6$, then
$$
\gamma_{s}(G) \le {\sqrt{6\ln(\delta+1)}+ 1.21 \over \sqrt{\delta +1}}n.
$$
\end{thm}
\noindent{\bf Proof: } Denote $\delta^{+}=\delta+1$, $N_v=N[v]$ and $n_v=|N_v|.$ Let $A$ be a set formed by an independent choice of vertices
of $G$, where each vertex is selected with the probability
$$p= 0.5 + \sqrt{1.5\ln{\delta^{+}/\delta^{+}}}. $$
Let us construct two sets $Q$ and $U$ in the following way: for each
vertex $ v \in V(G)$, if $|N_v\cap A| \le 0.5 n_v,$ then we put $v
\in U$ and add $\lfloor 0.5 n_v +1 \rfloor $ vertices of $N_v$ to
$Q$. Furthermore, we assign ``+" to $A\cup Q$, and ``--" to all
other vertices. The resulting function $g:V(G)\rightarrow \{-1, 1\}$
is a signed domination function, and
$$
g(V(G))=2|A \cup Q| - n \le 2|A| + 2|Q|-n.
$$
The expectation of $g(V(G))$ is
\begin{eqnarray}
\nonumber
{\mathbf E}[g(V(G))] &\le& 2{\mathbf E}[|A|]+2{\mathbf E}[|Q|]-n\\
\label{1}&=& 2pn - n + 2{\mathbf E}[|Q|].
\end{eqnarray}
It is easy to see that $|Q| \le \sum_{v \in \; U}\lfloor 0.5n_v+1\rfloor,$ hence
\begin{equation} \label{4}
{\mathbf E}[|Q|]\le\sum_{v \in V(G)}\lfloor
0.5n_v+1\rfloor\;{\mathbf P}[v \in U],
\end{equation}
where
$$
{\mathbf P}[v\in U]={\mathbf P}[|N_v \cap A| \le 0.5 n_v].
$$
Let us define the following random variables
\begin{displaymath}
X_w= \cases{\\
1-p& if $w \in A$\cr\\
-p& if $w \notin A$\\
}\\
\end{displaymath}
and let $X_v^{*}=\sum_{w \in N_v}X_w.$
We have
$$
|N_v \cap A| \le 0.5 n_v \mbox{\quad if and only if
\quad}X_v^{*}\le(1-p)0.5 n_v+(-p)0.5 n_v.
$$
Thus,
$$
{\mathbf P}[|N_v \cap A| \le 0.5 n_v]= {\mathbf P}[
X_v^{*}\le(0.5-p)n_v].
$$
By Lemma \ref{chernoff},
$$
{\mathbf P}[X_v^{*} \le \left(0.5 - p\right)n_v] <
e^{-{1.5n_v\ln{\delta^{+}/\delta^{+}} \over 1 +
\sqrt{6\ln{\delta^{+}/\delta^{+}}}}}.
$$
For $n_v \ge \delta^{+}>10^{6}$, let us define
$$
y(n_v, \delta^{+})={1.5n_v\ln{\delta^{+}/\delta^{+}} \over 1 + \sqrt{6\ln{\delta^{+}/\delta^{+}}}}-\ln(2.25n_v^{1.5})+1.
$$
The function $y(n_v, \delta^{+})$ is an increasing function of $n_v$ and
$y(\delta^{+},\delta^{+})>0$ for $\delta^+>10^6.$
Hence $y(n_v, \delta^{+})\ge y(\delta^{+}, \delta^{+})>0$ and
$$
{1.5n_v\ln{\delta^{+}/\delta^{+}} \over 1 +
\sqrt{6\ln{\delta^{+}/\delta^{+}}}} > \ln(2.25n_v^{1.5}) -1.
$$
We obtain
$${\mathbf P} [|N_v\cap A| \le 0.5 n_v] < e^{1-\ln(2.25n_v^{1.5})} = {e \over 2.25n_v^{1.5}},$$
and, using inequality (\ref{4}),
$$
2{\mathbf E}[|Q|]\le 2\sum_{v\in V(G)}{e(0.5n_v+1) \over
2.25n_v^{1.5}} \le {e(\delta +3)n \over 2.25(\delta+1)^{1.5}}\le
{1.21 \over \sqrt{\delta+1}}n.
$$
Thus, (\ref{1}) yields
\begin{eqnarray*}
{\mathbf E}[g(V(G))]&\le& 2pn - n + {1.21n \over \sqrt{\delta +1}}\\
&=& {\sqrt{6\ln(\delta+1)}+ 1.21 \over \sqrt{\delta +1}}n,
\end{eqnarray*}
as required. The proof of Theorem \ref{main_2} is complete. \hfill$\rule{.05in}{.1in}$\vspace{.3cm} \\
\begin{cor} \label{c1}
If $24,000 \le \delta,$ then
$$
\gamma_s(G) \le {\sqrt{6.8\ln(\delta+1)}+0.32 \over \sqrt{\delta+1}}n.
$$
\end{cor}
\noindent{\bf Proof: } Putting $p=0.5 + \sqrt{1.7\ln{\delta^+}/\delta^+}$ in the
proof of Theorem \ref{main_2}, we obtain by Lemma \ref{chernoff},
$$
{\mathbf P}[X_v^{*} \le \left(0.5 - p\right)n_v] <
e^{-{1.7n_v\ln{\delta^{+}/\delta^{+}} \over 1 +
\sqrt{6.8\ln{\delta^{+}/\delta^{+}}}}}.
$$
Let us define the following function:
$$
y(n_v, \delta^+)= {1.7n_v\ln\delta^+/\delta^+ \over 1+
\sqrt{6.8\ln\delta^+/\delta^+}} - \ln(3.14n_v^{1.5})
$$
for $n_v\ge \delta^+ > 24,000$. The function $y(n_v, \delta^{+})$ is
an increasing function of $n_v$ and $y(\delta^{+},\delta^{+})>0$ for
$\delta^+>24,000.$ Hence $y(n_v, \delta^{+})\ge y(\delta^{+},
\delta^{+})>0$ and
$$
{1.7n_v\ln\delta^+/\delta^+ \over 1+ \sqrt{6.8\ln\delta^+/\delta^+}}
> \ln(3.14n_v^{1.5}).
$$
We obtain
$$
2{\mathbf E}[|Q|]\le 2\sum_{v\in V(G)}{0.5n_v+1 \over
3.14n_v^{1.5}} \le {(\delta +3)n \over 3.14(\delta+1)^{1.5}}\le
{0.32 \over \sqrt{\delta+1}}n.
$$
Thus, (\ref{1}) yields
\begin{eqnarray*}
{\mathbf E}[g(V(G))]&\le& 2pn - n + {0.32n \over \sqrt{\delta +1}}\\
&=& {\sqrt{6.8\ln(\delta+1)}+ 0.32 \over \sqrt{\delta +1}}n,
\end{eqnarray*}
as required. The proof is complete. \hfill$\rule{.05in}{.1in}$\vspace{.3cm}
\begin{cor} \label{c2}
If $1,000 \le \delta \le 24,000,$ then
$$
\gamma_s(G) \le {\sqrt{\ln(\delta+1)(11.8-0.48\ln\delta)}+0.25 \over \sqrt{\delta+1}}n.
$$
\end{cor}
\noindent{\bf Proof: } It is similar to the proof of Corollary \ref{c1} if we put
$p=0.5 + \sqrt{(2.95-0.12\ln\delta)\ln{\delta^+}/\delta^+}$ and
consider the the following function for $1,000 \le \delta \le
24,000$:
$$
y(n_v, \delta^+)= {(2.95-0.12\ln\delta)n_v\ln\delta^+/\delta^+
\over 1+ \sqrt{(11.8-0.48\ln\delta)\ln\delta^+/\delta^+}} -
\ln(4.01n_v^{1.5}).
$$
\hfill$\rule{.05in}{.1in}$\vspace{.3cm}
\begin{cor} \label{c3}
If $230 \le \delta \le 1,000,$ then
$$
\gamma_s(G) \le {\sqrt{\ln(\delta+1)(18.16-1.4\ln\delta)}+0.25 \over \sqrt{\delta+1}}n.
$$
\end{cor}
\noindent{\bf Proof: } It is similar to the proof of Corollary \ref{c1} if we put
$p=0.5 + \sqrt{(4.54-0.35\ln\delta)\ln{\delta^+}/\delta^+}$ and
consider the following function for $230 \le \delta \le 1,000$:
$$
y(n_v, \delta^+)= {(4.54-0.35\ln\delta)n_v\ln\delta^+/\delta^+ \over
1+ \sqrt{(18.16-1.4\ln\delta)\ln\delta^+/\delta^+}} -
\ln(4.04n_v^{1.5}).
$$
\hfill$\rule{.05in}{.1in}$\vspace{.3cm}
We believe that F\"{u}redi--Mubayi's conjecture, saying that
$\gamma_s(G) \le {Cn \over \sqrt\delta}$, is true for some small
constant $C$. However, as the Peterson graph shows, $C>1$, i.e.
the behaviour of the conjecture is not good for relatively small
values of $\delta$. Therefore, we propose the following rectified
conjecture, which, roughly speaking, consists of two functions for
`small' and `large' values of $\delta$.
\begin{conj}
For some $C\le 10$ and $\alpha$, $0.18\le \alpha < 0.21$,
$$
\gamma_s(G) \le \min\Big\{{n\over \delta^\alpha}, {Cn\over
\sqrt\delta}\Big\}.
$$
\end{conj}
The above results imply that if $C=10$ and $\alpha=0.13$, then
this upper bound is true for all graphs with $\delta\le 96\times
10^4$.
\section{A Lower Bound for the Signed Domination Number
The following theorem provides a lower bound for the signed
domination number of a graph $G$ depending on its order and a
parameter $\lambda$, which is determined on the basis of the degree
sequence of $G$ (note that $\lambda$ may be equal to $0$, in this
case we put $\sum_{i=1}^{\lambda}=0$). This result improves the
bound of Theorem \ref{dunb_low} and, in some cases, it provides a
much better lower bound. For example, let us consider a graph $G$
consisting of two vertices of degree 5 and $n-2$ vertices of degree
3. Then, by Theorem \ref{dunb_low},
$$\gamma_s(G) \ge 0.25n-1,$$
while Theorem \ref{LowerBound} yields
$$\gamma_s(G) \ge 0.5n-1.$$
\begin{thm} \label{LowerBound}
Let $G$ be a graph with $n$ vertices and degrees $d_1\le d_2\le
... \le d_n$. Then
$$\gamma_s(G) \ge n-2\lambda,$$
where $\lambda\ge 0$ is the largest integer such that
$$
\sum_{i=1}^\lambda \left\lceil{d_i\over2}+1\right\rceil \le
\sum_{i=\lambda+1}^n \left\lfloor {d_i\over2}\right\rfloor.
$$
\end{thm}
\noindent{\bf Proof: } Let $f$ be a signed domination function of minimum weight of
the graph $G$. Let us denote
$$
X = \{v\in V(G): f(v)=-1\},
$$
and
$$
Y = \{v\in V(G): f(v)=1\}.
$$
We have
$$
\gamma_s(G) = f(V(G)) = |Y|-|X| = n - 2|X|.
$$
By definition, for any vertex $v$,
$$
f[v]=\sum_{u\in N[v]}f(u) \ge 1.
$$
Therefore, for all $v\in V(G)$,
$$
|N[v]\cap Y|-|N[v]\cap X| \ge 1.
$$
Using this inequality, we obtain
$$
|N[v]|=\deg (v)+1 = |N[v]\cap Y|+|N[v]\cap X| \le 2|N[v]\cap Y|-1.
$$
Hence
$$
|N[v]\cap Y| \ge {\deg(v)\over2} +1.
$$
Since $|N[v]\cap Y|$ is an integer, we conclude
$$
|N[v]\cap Y| \ge \left\lceil{\deg(v)\over2}\right\rceil +1
$$
and
$$
|N[v]\cap X| = \deg(v)+1 - |N[v]\cap Y| \le
\left\lfloor{\deg(v)\over2}\right\rfloor.
$$
Denote by $e_{X,Y}$ the number of edges between the parts $X$ and
$Y$. We have
$$
e_{X,Y} = \sum_{v\in X} |N[v]\cap Y| \ge \sum_{v\in X}
\left(\left\lceil{\deg(v)\over2}\right\rceil +1\right) \ge
\sum_{i=1}^{|X|} \left(\left\lceil{d_i\over2}\right\rceil
+1\right).
$$
Note that if $X=\emptyset$, then we put $\sum_{i=1}^0g(i)=0$. On
the other hand,
$$
e_{X,Y} = \sum_{v\in Y} |N[v]\cap X| \le \sum_{v\in Y}
\left\lfloor{\deg(v)\over2}\right\rfloor \le \sum_{i=n-|Y|+1}^n
\left\lfloor d_i/2\right\rfloor = \sum_{i=|X|+1}^n \lfloor
d_i/2\rfloor.
$$
Therefore, the following inequality holds:
$$
\sum_{i=1}^{|X|} \left(\left\lceil{d_i\over2}\right\rceil
+1\right) \le \sum_{i=|X|+1}^n
\left\lfloor{d_i\over2}\right\rfloor.
$$
Since $\lambda\ge 0$ is the largest integer such that
$$
\sum_{i=1}^{\lambda} \left(\left\lceil{d_i\over2}\right\rceil
+1\right) \le \sum_{i=\lambda+1}^n
\left\lfloor{d_i\over2}\right\rfloor,
$$
we conclude that
$$
|X| \le \lambda.
$$
Thus, $$\gamma_s(G) = n-2|X| \ge n-2\lambda.$$ The proof is
complete. \hfill$\rule{.05in}{.1in}$\vspace{.3cm}
Theorem \ref{LowerBound} immediately implies the following known
results:
\begin{cor} [\cite{haa1} and \cite{zha1}] \label{haas-zhang}
For any graph $G$,
$$
\gamma_s(G) \ge \left({\lceil0.5\delta\rceil -
\lfloor0.5\Delta\rfloor +1 \over \lceil0.5\delta\rceil +
\lfloor0.5\Delta\rfloor +1} \right)n.
$$
\end{cor}
Note that Haas and Wexler \cite{haa1} formulated the above bound
only for graphs with $\delta\ge 2$, while Zhang et al. \cite{zha1}
proved a weaker version without the ceiling and floor functions.
\begin{cor} [\cite{hen2}] \label{lb-Cor1}
If $\delta$ is odd and $G$ is $\delta$-regular, then
$$
\gamma_s(G) \ge {2n \over {\delta+1}}.
$$
\end{cor}
\begin{cor} [\cite{dun1}] \label{lb-Cor2}
If $\delta$ is even and $G$ is $\delta$-regular, then
$$
\gamma_s(G) \ge {n \over {\delta+1}}.
$$
\end{cor}
Disjoint unions of complete graphs show that these lower bounds are
sharp whenever $n/(\delta+1)$ is an integer, and therefore the bound
of Theorem \ref{LowerBound} is sharp for regular graphs.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,578
|
PILATES ~ students work on a progressive and flowing classical mat work programme across a range of increasingly challenging levels developing core strength and stability while challenging the body to work through a full range of movement.
PILATES FOR 65+ ~ general mobility exercises are followed by a programme to improve core strength and stability. To participate in this mat based class it is important you can safely get up and down on a mat and be able to move easily from sitting to lying on the mat.
YOGA ~ traditional Hatha yoga classes offering a wide range of stretching and balancing exercises combined with a focus on breathing, mindfulness and relaxation.
SMALL GROUP CLASSES ~ Why not get together with a group of friends and we can create a Bespoke class for your group. It might be pilates or yoga, or our unique Bespoke Blend of the two!
PILATES & STRETCH ~ A traditional Pilates class which has been extended to include a varied programme of yoga based stretches for mobility. This "Bespoke blend" of strength and stretch will focus for approximately 50 minutes on building core strength and stability and for 25 minutes on improving flexibility.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,811
|
Gastrotheca prasina es una especie de anfibio anuro de la familia Hemiphractidae.
Distribución geográfica
Esta especie es endémica de Minas Gerais en Brasil. Habita en Jequitinhonha a 989 m sobre el nivel del mar en el bosque atlántico.
Publicación original
Teixeira, Dal Vechio, Recoder, Carnaval, Strangas, Damasceno, de Sena & Rodrigues, 2012: Two new species of marsupial tree-frogs genus Gastrotheca Fitzinger, 1843 (Anura, Hemiphractidae) from the Brazilian Atlantic Forest. Zootaxa, n.º3437, p. 1–23.
Referencias
Enlaces externos
prasina
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,077
|
'use strict';
module.exports = require('./lib/tesseract');
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 311
|
«Сыны свободы» () — революционная американская организация, боровшаяся за самоопределение североамериканских колоний. Основана в 1765 году Сэмюэлем Адамсом. Одной из акций «Сынов свободы» стало Бостонское чаепитие.
Название
Наиболее распространенным в революционные годы и утвердившимся в историографии является название «Сыны свободы». Но подобные организации имели и иные самоназвания. Например, наименование «товарищи» было распространено в Пенсильвании, а в Коннектикуте организация называлась «тесной компанией»; также в некоторых случаях встречалось название «распорядители».
Происхождение
В соответствии с давней традицией в американских городах существуют неформальные группы людей, объединяющиеся для того, чтобы совместно оказывать политическое влияние на местные органы самоуправления. В Бостоне второй половины XVIII известны по меньшей мере две такие группы: «Девять лояльных» и «Клуб бостонского комитета», «политическая организация купцов, ремесленников, нескольких адвокатов и врачей». К 1765 г. они объединились на почве оппозиции акту о гербовом сборе и другим решениям британского правительства, непопулярным в колониях.
Состав
Массовой базой революционных организаций были ремесленники, мастеровые, плотники, столяры, печатники, кораблестроители, чеканщики, конопатчики, канатчики, каменщики, матросы и др. Руководителями были по большей части торговцы и ремесленные мастера. Эти люди не имели избирательных прав, поэтому прибегали к радикальным методам для влияния на власть.
Лидером новой организации, получившей название «Сыны свободы», стал Сэмюэл Адамс. Его девизом стала знаменитая фраза «Нет налогам без представительства». Позже в организацию вошли также представители из других американских городов.
Цели
«Сыны свободы» боролись против колониальных властей, осуществляли бойкот английских товаров, противодействовали размещению и переброске британских войск. Боролись за предоставление избирательных прав рабочим и за тайное голосование. Активно содействовали проведению 1-го Континентального конгресса (1774), сыгравшего важную роль в объединении колоний для борьбы за независимость. Добивались от консервативных богатых купцов соглашения о совместных действиях, что позволяло проводить более последовательную революционную политику.
Формы борьбы
«Сыны свободы» организовывали акции протеста, подавали петиции властям, а также прибегали к открытым насильственным действиям против британских властей (поджоги, нападения на чиновников).
Организовывали бойкот английских товаров. Нарушителей бойкота они вымазывали смолой и валяли в перьях. Саботаж против англичан зачастую лишал бойкотистов заработка, но это не останавливало их: например, когда в 1768 году в Бостоне были расквартированы британские войска, плотники и каменщики отказались строить для них казармы даже за повышенную плату, хотя не имели в то время иной работы. Бостонцев поддержали и нью-йоркские рабочие, которых власти пытались привлечь к этому строительству.
Газета «Меркурии» (Нью-Порт, Род-Айленд) 26 сентября 1774 года, выступая против консервативных священников, заявляла: «мастеровые и деревенские мужланы (позор называть их такими именами!) являются настоящими и полными хозяевами короля, лордов, палаты общин и духовенства». Другим методом борьбы с консервативными проповедниками было их разжалование своими конгрегациями.
После Бостонской бойни «Сыны свободы» стали готовить запасы оружия и снаряжения и проводить сборы для военной подготовки, организовали систему разведки за войсками британцев. Благодаря этому им удавалось захватывать оружейные склады до прихода войск и укрывать революционеров. Именно разведывательный комитет в ночь на 18 апреля 1775 года установил, что 800 солдат выступили из Бостона, чтобы захватить военные склады патриотов Конкорды, которые заранее уже были надежно перепрятаны. Оповещенные ополченцы дали отпор британцам в сражениях при Лексингтоне и Конкорде. После этих события патриоты стали еще активнее захватывать британские военные припасы, при этом им иногда удавалось уговорить солдат дезертировать.
Фактически им также удалось сорвать введение гербового сбора в колониях. В марте 1776 года, после отмены закона о гербовом сборе, организация «Сыны свободы» самораспустилась. Однако члены организации и в дальнейшем боролись против произвола колониальных властей, используя легальные и нелегальные методы.
Символика
Члены организации носили на груди медаль с изображением дерева свободы. У них был собственный флaг, полосы нa флaге символизировaли количество присоединившихся колоний к оргaнизaции, оригинaл флaгa не сохрaнился.
Известные члены организации
Сэмюэл Адамс — сборщик налогов, Бостон. Лидер организации.
Джон Адамс — юрист, в будущем, второй президент США, Массачусетс
Чарльз Томсон — секретарь, Филадельфия
Хаим Соломон — финансовый брокер, Нью-Йорк
Томас Янг — врач, Бостон
Пол Ревир — чеканщик по серебру, Бостон
Джозеф Варрен — врач и солдат, Бостон
Бенджамин Эдес — журналист и издатель «Бостонской газеты», Бостон
Александр МакДоугалл — капитан каперского судна, Нью-Йорк
Патрик Генри — юрист, Бостон
Джон Хэнкок — купец, Бостон
Айзек Сирс — капитан каперского судна, Нью-Йорк
Джон Лэмб — торговец, Нью-Йорк
Джеймс Отис младший — юрист, Массачусетс
Маринуй Виллетт — столяр и солдат, Нью-Йорк
Сайлэс Даунер — занятие неизвестно
Уильям Маккей — купец, Бостон
Джонатан Тремиан — подмастерье, Вирджиния
Бенедикт Арнольд — бизнесмен и солдат, Англия
Кристофер Гадсден — торговец, Южная Каролина
Джеймс Свaн — финaнсист
Оливер Уоллкот — государственный деятель
«Сыны свободы» в кино
«Сыны свободы» (2015) — американский телевизионный мини-сериал производства History Channel, глaвным героем является Сэмюэль Адамс, рaсскрывaет историю зaрождения и создaния оргaнизaции.
«Джон Адамс» (2008) — американский телевизионный мини-сериал производства HBO о Джоне Адамсе.
Примечания
Библиография
Dawson, Henry Barton. The Sons of Liberty in New York (1859) 118 pages; online edition
Foner, Philip Sheldon. Labor and the American Revolution (1976) Westport, CN: Greenwood. 258 pages.
Labaree, Benjamin Woods. The Boston Tea Party (1964).
Maier, Pauline. "Reason and Revolution: The Radicalism of Dr. Thomas Young, " American Quarterly Vol. 28, No. 2, (Summer, 1976), pp. 229—249 in JSTOR
, a Marxist interpretation
Walsh, Richard. Charleston's Sons of Liberty: A Study of the Artisans, 1763—1789 (1968)
Американская революция
История тринадцати колоний
Войны за независимость
Национально-освободительные движения
Тайные общества США
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Jobless and without pay, migrant workers are returning home. But they have no recourse for compensation
Labour migrant experts say the concerned Nepali embassies and the government should start recording the details of the returning migrant workers to take the matter to the court.
Chandan Kumar Mandal
In January, when Kalim Miya migrated to Sharjah, the third-largest city in the United Arab of Emirates, he paid nearly 35 times more than the price ceiling set by the government to go overseas for employment.
For a taxi driver's job, Miya from Laxmi Bazar, Gorkha, had paid Rs345,000 to a Kathmandu-based recruitment agency.
When Miya returned home on June 22 in an evacuation flight, he had no money on him.
"I did not get even one-month's salary during my time in Sharjah," Miya told the Post.
After returning home, Miya has been staying in a quarantine facility at Shree Manakamana Deaf Residential Secondary School in Gorkha Municipality-9.
"I had taken some money with myself when I left for Sharjah. Once I reached there, I had to keep asking for money from my family to survive," Miya said.
Thirty-year-old Miya was evacuated along with 14 of his friends from Sharjah. All of them had worked for Emirates Cab, a taxi service provider, and they were not paid during their stint in Sharjah.
As the coronavirus pandemic reached the shores of Sharjah, they were put out of their jobs. Unpaid and without jobs, they scrambled for food and shelter for months. There were times when they slept in public parks at night and frequented the Nepal embassy during the day to ask for help but without avail.
After languishing in a foreign country for months, they were finally evacuated. Their air fares were paid by the recruiting agency in Kathmandu.
"I am home all right but I don't know how I am going to pay off my loans. I had thought I would clear my loans with my earnings in the UAE, but I ended racking up loans instead," said Miya. "I have no hope that I will be compensated for my loss. I had invested a lot of money to go abroad. Now here I am, without a single rupee."
When Miya and his friends boarded their return flight, no one asked them if they had been paid by their company. Once they landed in Nepal, they were asked to fill forms detailing their health and their personal information and taken to quarantine.
"No one inquired about our salaries and facilities," Miya said.
Miya and his friends are among the cohort of thousands of migrant workers who are returning home after losing their jobs and without being paid from the Covid-19 hit labour destination countries.
While the government's evacuation plan has prioritised bringing back those workers who have been languishing in labour hosting countries without jobs and pay, there is no guarantee of them ever being compensated for their losses.
Labour migration experts have been expressing concern about the migrant workers returning home without pay and without prospect of ever recovering their pay.
Jeevan Baniya, a labour migration expert, said following the Covid-19 pandemic, cases of workers getting their contracts terminated, employers not paying their workers companies getting shut have become widespread in labour destination countries.
"Such a situation has not only left workers without their salaries but also their provident funds and other gratuities," said Baniya, also the assistant director of Centre for the Study of Labour and Mobility (CESLAM) at Social Science Baha, a non-governmental think tank. "As thousands of affected workers have already returned facing hardship and many more in the process of returning, compensating them is going to be a lasting issue."
According to Covid-19 Crisis Management Centre (CCMC), the authority currently overseeing repatriation, a total of 15,052 Nepalis, which mostly include migrant workers from the Persian Gulf countries and Malaysia, have returned home as of Wednesday.
According to the Pravasi Nepali Coordination Committee (PNCC), a non-governmental organisation working for the welfare of migrant workers, 22 percent of the workers, who had contacted the organisation to seek help to return home, complained about arbitrary contract breach, unpaid salaries, bonus and other company facilities.
"Even before the pandemic, Nepali migrant workers had been facing the problem of unpaid wages and unexplained termination from their job," said Kul Prasad Karki, PNCC chairperson. "The pandemic has only made their situation worse. First, their companies robbed them of their pays, and after that, they were forced to pay for their repatriation, hotel quarantine and Covid-19 screening."
Despite all of these, there has been no assurance whatsoever from the government regarding compensating these workers who have the legal right to claim their unpaid salaries, Kari said.
"The government has not come up with any roadmap to ensure the workers that they will get their money back, nor offered them any legal recourse to claim their salaries," said Karki.
Baniya, the assistant director of Centre for the Study of Labour and Mobility, said the prospect of providing compensation to the migrant workers will begin only with a robust documentation of the returning workers by Nepali missions in the concerned labour destination countries and the government agencies in Nepal.
"Nepali missions, on behalf of Nepali workers or even workers, can file a complaint against employers demanding unpaid remuneration at the local court. The Nepal government itself can also fight a case," said Baniya. "But for that to happen, we need proper documentation which can serve as evidence to seek legal remedies for workers."
Labour migration experts say it is the responsibility of the state and the recruiting agencies to ensure compensation for the returning workers. By not keeping the record of the workers and their grievances, they say the government is only weakening their case.
So far, the Nepali embassies in labour destination countries have only been collecting the details of workers who wish to return home through online applications.
"Nepali missions abroad are not resourceful enough to handle repatriation and the grievances handling at this time. But someone has to do this, whether in labour destination countries or here in Nepal," said Baniya. "The pandemic should not be an excuse for employers and recruitment companies to exploit workers and their human rights."
According to the Foreign Employment Act, if any employer does not provide employment as per the terms prescribed in the agreement, the worker or his or her agent may file a complaint at the Department of Foreign Employment, along with evidence for compensation. Upon inquiry, the department can ask the recruiting agencies to provide compensation for all expenses incurred by the worker while applying for foreign employment.
There are cases of employers paying their workers following formal complaints. As many as 34 Nepali workers who had returned empty-handed from Qatar three years ago. They were provided with their unpaid salaries in February. The workers were employed by Mercury MENA, a company building stadium for Qatar World Cup in 2022.
Labour migration experts say the Nepal government first needs to acknowledge the problem of migrant workers in order to address their grievances.
"The Labour Ministry can advise the CCMC also to record the grievances of returnee workers. The records can be used for claiming compensation. International labour laws guarantee that workers should be paid even during crises," said Baniya. "Later, we can use diplomatic negotiations to get our workers' rights ensured. We can also internationalise the issue of migrant workers' rights and gain access to justice by building pressure through the Colombo Process, a forum of labour sending countries, as well as other migrant rights conventions. It can set a precedent for the future."
As for now, Miya, who is observing staying quarantine in his village, plans to file a complaint against the recruiting agency to seek compensation.
"Once I can travel to Kathmandu, I will file a complaint at the Department of Foreign Employment," said Miya. "I have little hope of recovering my salaries. But Nepali embassies should not issue labour demands for such companies that cheat poor workers like me."
Chandan Kumar Mandal is the environment and migration reporter for The Kathmandu Post, covering labour migration and governance, as well as climate change, natural disasters, and wildlife.
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Q: unable to view previous version items in sharepoint I am working on SharePoint file versions and i created a SharePoint site on which i have uploaded a 5Mb size of excel file. I tried to create some version of by editing the file. After creating some versions in version history. I tried to view version of file with the version ID of 2.0 but it is not showing me hyperlink on name to view the file.
Here,
Versions of file:
Version of ID = 2.0
Can someone please tell me how can i view previous versions of the file in view mode?
A: Modify the libraryName in code below, then add the code into script editor web part in document library DispForm.aspx page.
<script src="https://code.jquery.com/jquery-1.12.4.min.js" type="text/javascript"></script>
<script type="text/javascript">
$(function(){
var libraryName="Shared Documents";
setNameField(libraryName);
})
function setNameField(libraryName){
var itemID=getUrlParameter("ID");
var versionID=getUrlParameter("VersionNo");
if(itemID!=""&&versionID!=""){
$.ajax({
url: _spPageContextInfo.webAbsoluteUrl + "/_api/lists/getbytitle('"+libraryName+"')/Items("+itemID+")/versions("+versionID+")?$select=FileLeafRef,FileRef",
type: "GET",
headers: {
"Accept": "application/json;odata=verbose",
},
success: function (data) {
var versionFileUrl=data.d.FileRef.replace(libraryName,"_vti_history/"+versionID+"/"+libraryName);
var linkFileHtml="<a href='"+versionFileUrl+"'>"+data.d.FileLeafRef+"</a>";
$(".ms-standardheader:contains('Name')").closest("td").next().html(linkFileHtml);
},
error: function (data) {
//alert("Error");
}
});
}
}
function getUrlParameter(name) {
name = name.replace(/[\[]/, '\\[').replace(/[\]]/, '\\]');
var regex = new RegExp('[\\?&]' + name + '=([^&#]*)');
var results = regex.exec(decodeURIComponent(location.search));
return results === null ? '' : decodeURIComponent(results[1].replace(/\+/g, ' '));
}
</script>
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Léglise Saint-Siméon est une église catholique située dans la commune de Bouliac, dans le département de la Gironde, en France.
Localisation
L'église se trouve au centre du bourg de Bouliac.
Historique
L'église du , dédiée à Siméon le Stylite, remplace un oratoire construit sur un cimetière paléochrétien du . Elle se compose d'une nef, un chevet et une sacristie. Le chevet et les chapiteaux du portail sont les seuls restes de l'église originelle.
Au , pendant la guerre de Cent Ans, l'église a été restaurée sur ordre de Pey Berland, archevêque de Bordeaux et ancien curé de Saint-Siméon, entre 1413 et 1422. Les murs gouttereaux nord et sud de la nef sont percés chacun par trois baies. Les colonnettes ont toutes des chapiteaux à décor végétal.
Puis, pour protéger le village contre la menace des troupes françaises, l'église a été aussi fortifiée par Pey Berland : un clocher fortifié, une barbacane donnait accès à un chemin de ronde sur la nef bordée par un parapet crénelé qui a disparu, au-dessus de l'abside, une chambre forte percée d'archères en croix pattées (voir l'aquarelle de Léo Drouyn). Cette fortification au-dessus du chevet est encore visible aujourd'hui. Cependant, le village de Bouliac ne fut jamais attaqué.
En 1649, elle résiste aux assauts du duc d'Epernon, en lutte avec le Parlement de Bordeaux.
En 1793, l'église subit quelques mutilations du fait des violences révolutionnaires.
Sous le second Empire, en 1862, sous la férule du cardinal Donnet, l'ancien clocher est détruit et remplacé par le clocher actuel et la nef lambrissée. Les fresques qui ornent la nef ont été peintes en 1896.
Une croix de mission se trouve près de la mairie : La ferveur religieuse a été mise à mal lors de la Révolution française. Nombreux sont les fidèles à avoir déserté les offices. Au début du , l'Église a commencé une nouvelle évangélisation des campagnes françaises. Lors de la venue du missionnaire diocésain, sa mission était signalée par l'érection d'une croix dans le centre, ou à l'entrée du village. Celle de Bouliac date de 1834.
L'église a été classée au titre des monuments historiques en totalité en 1862.
L'iconographie romane
Les six chapiteaux du portail, par leurs techniques sculpturales, datent du . Les sculptures du chevet ont été tirées du répertoire de l'abbaye de La Sauve-Majeure établie à partir de 1130. Il y a donc eu au moins deux campagnes distinctes, la première pour la façade et la seconde, environ 100 ans plus tard, pour la partie orientale de l'église.
L'extérieur
Les sculptures romanes existantes à l'extérieur sont six chapiteaux au portail occidental, huit colonnes et chapiteaux autour du chevet, trois fenêtres et leurs colonnettes dans l'abside et les modillons qui soutiennent la corniche du chevet.
Le portail roman
Le portail occidental de l'église est abrité sous le clocher construit en 1862. Il est à voussures et constitué de trois archivoltes séparées par un bandeau sculpté d'entrelacs s'appuyant sur six colonnettes ornées de motifs géométriques. Seuls les six chapiteaux sont romans, les autres éléments du portail datent de la reconstruction du .
Dans l'ébrasure nord, les scènes sont une visite des rois Mages à Marie et Jésus, un piscifère portant son fardeau, le lavement, par Jésus, des mains et de la tête de saint Pierre.
Dans l'ébrasure sud, elles sont un archange tuant un dragon, le baptême de Jésus, Jésus guidant trois fidèles.
Ébrasement nord
Chapiteau nord 1 : Les Mages
Sur la face principale de la corbeille, Marie est assise en majesté, tenant Jésus dans son giron. Sur l'autre face, en arrière-plan, se présentent les trois Mages venus offrir leur cadeaux (de l'or, de l'encens et de la myrrhe) décrits par Mathieu II, 11. Le premier Mage, qui a perdu sa tête, est agenouillé.
Sur les deux faces du tailloir, deux anges volent. Celui qui porte un livre montre du doigt l'étoile qui a guidé les Mages depuis l'Orient et qui est un emprunt de la Nativité décrite par Luc II, 7-20 et dans laquelle Jésus est placé dans une crèche avec des bergers pour compagnie.
Quant à Marie, elle est parfaitement hiératique, encadrée par le guimpe des épouses et la houppelande qui recouvre ses épaules. Cependant, la représentation avec l'enfant, qui n'est pas un nouveau-né, semble venir de l'évangile apocryphe du pseudo-Mathieu du , où il est écrit que deux ans se sont passés après la naissance de Jésus avant que n'arrivent les mages avec leurs cadeaux, c'est-à-dire après la fuite en Égypte et le retour en Palestine. Ce chapiteau possède une particularité insolite : il n'y aucun signe, attribut ou geste désignant la divinité de Jésus aux Mages.
Chapiteau nord 2 Les porteurs de poissons
Le tailloir de ce chapiteau est décoré d'une onde de demi-palmettes issue de la bouche d'une tête de loup sur l'angle. Les masques « cracheur de rinceaux » sont toujours des symboles négatifs et le sujet de la corbeille est donc associé à un signe maléfique.
La corbeille, quant à elle, est dominée par un énorme poisson, arqué en une boucle presque parfaite. Il écrase, de son poids énorme, l'homme qui le porte. Un deuxième homme est venu en aide et a saisi la queue glissante. Les deux hommes sont chaussés et vêtus de bliauds luxueux.
Tobie or not Tobit : Certains guides affirment que le symbole du porteur de poisson fait référence à l'évangile apocryphe de Tobit, où Tobie a guéri la cécité de son père Tobit avec le fiel d'un énorme poisson. D'autres auteurs, comme Léo Drouyn, mettent en doute cette interprétation, car il y a des centaines de chapiteaux et modillons avec cette représentation et il n'y a jamais d'évocation d'une cécité. Par contre il y a souvent des symboles maléfiques, comme ici, ou signes des péchés associés au porteur de poisson. Les « porteurs de fardeaux » (poisson, bouc, cochon, tonnelet, etc.) ne sont qu'un procédé d'imagier pour figurer le pécheur accablé par sa conscience, comme si le poids des fautes l'écrasait physiquement. Le piscifère principal est l'emblème de l'intempérance et ici son compagnon est son complice dans le péché.
Chapiteau nord 3 : Le lavement des mains et de la tête de Pierre par Jésus
Quatre personnages de l'évangile selon Jean, XIII, v 4-12, sont représentés. Jésus, reconnaissable à son nimbe crucifère et Pierre, que l'usure à complètement effacé, mais il devait occuper cette position centrale. Jésus est présenté de trois quarts. Il a saisi le poignet droit de Pierre (c'est tout ce qu'il en reste) afin de lui laver les mains et la tête.
À gauche, un disciple barbu exhibe un tissu ou un vêtement. À droite, un troisième apôtre portait un objet, peut-être un bassin ou un essuie-mains, et observe la scène devant un arbuste à feuilles crossées.
Un détail a son importance, le fait que, tout à fait à droite du troisième disciple, on voit une palmette qui pousse dans un cénacle où elle n'a pas sa place . À l'époque romane, la question du lavement des pieds des apôtres par Jésus était complexe. Dans l'Aquitaine, comme en Espagne, on pratiquait, précisément le dimanche des Palmes, ce « lavement des mains et de la tête ». C'était une pratique totalement étrangère au rituel de l'Église romaine. La palmette n'était pas destinée à décorer un espace, elle évoquait ce temps de rameaux, caractérisé dans la France gallicane, par une liturgie exubérante dont le clergé bordelais défendait jalousement l'originalité et l'ancestralité.
Ébrasement sud
Chapiteau sud 1 : Archange tuant le dragon
Ce chapiteau est fortement endommagé. D'après Léo Drouyn, le tailloir aurait hébergé deux personnages allongés, comme le chapiteau des Mages. Son état de dégradation aujourd'hui ne permet pas une vérification. La petite face de la corbeille est quasi illisible. Sur la face principale, le corps d'un petit archange, nu-pieds, est sur un gros dragon disposé en bateau. L'ange n'a plus de tête, mais ses ailes sont bien visibles. On devine qu'il achève le dragon avec une lance (on discerne une partie du fût devant sa poitrine) tenue du bras gauche.
Chapiteau sud 2 : Baptême de Jésus
Le tailloir est conservé à moitié. Sur l'angle, se voit le protomé d'un loup. Sur la corbeille, Jésus est la figure centrale. Il est plongé aux trois quarts dans un dôme aplati, constitué de trois ondes concentriques poissonneuses, qui simule la rivière Jourdain. À sa droite, Jean-Baptiste, debout sur la rive, tient Jésus par l'épaule et la nuque. Il est habillé d'une tunique côtelée, différente du « pagne de cuir » et des « poils de chameau » du récit biblique.
Sur l'autre face de la corbeille, un ange, aux ailes ouvertes, tient la tunique du Christ. L'ange n'est pas mentionné dans le récit biblique non plus, mais sa présence était devenue coutumière au fil des siècles. Plus surprenante est l'absence d'une colombe représentant le Saint-esprit et de la « main de Dieu » sortant des cieux.
Chapiteau sud 3 : Jésus guide des fidèles
Cinq personnages sont représentés sur la corbeille : Jésus, identifiable par son nimbe crucifère et le seul à être nu-pieds. À la gauche de Jésus, progressent trois personnes, deux aux mains jointes dans le geste de la prière, précédées par un autre homme auquel Jésus donne sa main. Ils ont tous perdu leur tête et Jésus a été amputé de sa mâchoire. Tous les personnages portent des tuniques longues, sans ceinture et à manchettes évasées, ornées de bourrelets dessinant des motifs géométriques. À droite, mais derrière Jésus, se tient le cinquième homme. On note que son habit est ajusté par une agrafe tréflée, fréquente dans le haut clergé, et que l'homme serre dans sa main droite une barre fuselée, dont le sommet et le bas ont été brisés. Peut-être un bourdon de pèlerin ou une crosse pastorale ? Dans ce cas, l'homme serait probablement un prélat local.
Des indications montrent que l'ordre de ces six chapiteaux a été chamboulé lors d'un remontage à une époque indéterminée et la corbeille du Baptême de Jésus et celle des porteurs de poissons ont été échangées. En effet, la commémoration du baptême de Jésus ET la visite des rois Mages, qui s'appelaient Épiphanie ou Théophanie, étaient commémoré le six janvier. Il se peut que, pour associer les deux épiphanies, la corbeille du baptême ait primitivement occupé la place actuellement prise par les porteurs de poissons. Dans ce cas, les porteurs de poissons auraient dû être associés avec l'ange tuant le dragon. À l'intérieur, sur la même corbeille du chevet, on trouve précisément un piscifère et l'ange au dragon, sculpté par un atelier venu décorer l'église dans l'esprit de l'abbaye de La Sauve-Majeure, environ un siècle après la construction du portail.
Le chevet
Les baies du chevet
Le chevet est percé par cinq baies, dont les trois premières, commençant au nord du chevet, sont romanes.
Baie 1 : Décor végétal et baie 2 : Oiseaux et calice
Les corbeilles des colonnettes de la première baie ont un décor uniquement végétal.
Pour la deuxième baie, on voit, sur la corbeille du chapiteau est, deux oiseaux paisibles, à plumage de caille qui s'apprêtent à boire dans un calice situé sous leurs becs. Le contexte est neutre, ce doit donc être une évocation de l'Eucharistie. La corbeille du chapiteau ouest a un décor végétal.
Baie 3 : Ouroboros
Sur la corbeille du chapiteau sud, deux serpents, enlacés par le cou de façon à former un nœud, se mordent la queue.
Sur la corbeille du chapiteau nord, un décor de pignes de pin.
Les chapiteaux de la corniche
Six chapiteaux romans, dont deux sont historiés, soutiennent la corniche. Du sud au nord : feuillage d'acanthes, lions et dompteurs, acanthes, les quatre dogues, pignes, acanthes.
Les chapiteaux à décor végétal :
Chapiteau 2 : Les dompteurs de lion
En avant-plan, deux gros lions, tenus en laisse, s'affrontent avec une mimique d'intimidation mutuelle. En arrière-plan, se voient les bustes et les jambes de deux dompteurs. L'un est coiffé d'un bonnet à plissures, l'autre a les cheveux au vent. Les dompteurs serrent d'une main la chaîne de dressage et de l'autre main la queue du lion.
Dans la symbolique médiévale, les fauves étaient une métaphore des passions humaines. Elles subjuguent l'homme qui les flatte par la queue, alors que s'en libère celui qui sait les tenir en laisse.
Chapiteau 4 : Les dompteurs dévorés
Affrontés deux à deux, quatre dogues à collier clouté dévorent des têtes humaines.
L'illustration symbolique puisant dans le catalogue médiéval du mode à l'envers est celle du dompteur dompté c'est-à-dire, sur le plan moral, de celui qui s'est laissé vaincre par ses passions.
C'est encore une illustration du libre choix du chapiteau précédent.
Les modillons
L'église Saint-Siméon est très riche en modillons. On en trouve 21 supportant la corniche du chevet. Ce sont d'authentiques modillons romans. Sur les murs nord et sud de la nef, il y a une soixantaine de modillons, mais seulement une dizaine sont sculptés avec des figures géométriques ou des feuilles. Il y a également une dizaine de modillons autour de l'entrée du clocher. Ce sont des figures géométriques datant du .
Les modillons du chevet ne sont pas des fantaisies des sculpteurs ou des commanditaires, il y a une leçon de moralité, voulue par le clergé et destinée à la population locale : une mise en garde contre le péché de luxure. Par contre, les modillons des sont purement décoratifs, sans message de moralité.
Les modillons du chevet qui sont toujours lisibles :
L'intérieur
La partie romane de l'intérieur se limite aux deux chapiteaux de l'arc triomphal et aux deux chapiteaux du sanctuaire.
L'arc triomphal
Nord : Daniel dans la fosse aux lions et Habacuc
Le tailloir du chapiteau est tapissé d'une suite de palmettes. Sur la face principale de la corbeille, le prophète Daniel, sous les traits d'un jeune homme, nimbé, habillé d'une tunique à plis est assis entre deux fauves. Il prie à l'antique, les paumes étendues. En arrière-plan, sur la côté ouest, se trouve Habacuc, le petit prophète, chargé par l'« ange du Seigneur » d'apporter la nourriture à Daniel pendant son séjour dans la fosse.
Sud : La tentation et la chute d'Adam et Ève
La composition est un condensé du troisième chapitre du Livre de la Genèse. Elle se lit de droite à gauche, comme une bande dessinée. Sur le petit côté de la corbeille, le serpent glisse entre les feuilles de l'Arbre de la connaissance du bien et du mal et Ève, toute nue, les cheveux défaits, a pris le fruit défendu de sa main gauche.
Sur la face principale, la même Ève soutient de sa main droite celle gauche d'Adam et le fruit dont elle va le régaler.
Dans la scène suivante, Adam vient de découvrir l'angoisse et sa main droite s'est portée à sa gorge, un geste qui représente à lui seul le stéréotype de l'épisode. La nudité d'Adam est voilée par une feuille de figuier.
L'autre moitié de la corbeille montre Dieu, couvert d'un manteau et tenant dans sa main droite un sceptre fleurdelisé. Il pousse le couple hors du paradis en ayant pris soin de les vêtir de tuniques. Finalement, sur le petit côté, Adam doit travailler à retourner la terre pour se nourrir.
Les deux chapiteaux sont complémentaires : au nord on représente la vertu de l'obéissance, dont Daniel est le parangon et au sud, les effets de la désobéissance, dont les « premiers parents » sont l'archétype.
Le sanctuaire
Nord : Michel tuant un dragon et un porteur de poisson
Le tailloir du chapiteau est orné d'une frise d'acanthe sèche. Sur la face principale de la corbeille, un ange nimbé est grimpé sur le dos d'un dragon ailé symbolisant le Diable. L'ange aux pieds nus enfonce un épieu dans la gueule du dragon.
Sur le petit côté de la corbeille un homme, chaussé de bottines, porte sur son épaule gauche un énorme poisson.
L'enchaînement pécheur - ange tuant le Mal apparaît comme une iconographie positive : le protecteur céleste envoyé auprès de chaque homme en train de pécher, pour l'aider, s'il le veut bien, à anéantir le démon qui serpente au tréfonds de son âme.
Sud : Centaure tuant un démon
Le protagoniste principal est un Centaure se détournant pour tirer son arc sur un quadrupède qui grimace de frayeur. Ce dernier est une chimère curieuse : corps et pattes de lion, ailes déployées, buste et tête humaine avec oreilles en pointe et crête capillaire.
Une corbeille similaire existe à l'abbaye de la Sauve-Majeure.
C'est à partir du que le Centaure devient une figure centrale de l'imagerie de l'édification morale. Sa mission est d'exterminer le Mal (sirènes, et autres démons), mais la différence avec les anges est qu'il est un habitant de la terre et semblable en cela à n'importe quel homme, de sorte que le message implicite dans les images apparemment anodines était révolutionnaire, puisqu'il induit pour l'homme la possibilité de se libérer seul du Mal, en faisant appel aux seules forces terrestres.
La décoration et le mobilier
Les fresques
Les fresques sur les murs nord et sud de la nef sont l'œuvre des peintres Boudon et Vernay et datent de 1896. Elles sont longues de 20 mètres et hautes de 4 mètres. Sur chaque mur, se trouvent trois tableaux, séparés les uns des autres par les baies de la nef.
Au nord, le thème est l'amour dû à Dieu et, au sud, l'amour dû au prochain. On voit l'archevêque Pey Berland faisant construire à Bordeaux la tour qui porte encore son nom.
Les vitraux
Les vitraux du chevet :
Les vitraux de la nef
L'orgue
L'orgue a été conçu par le facteur d'orgues bordelais Gaston Maille en 1896. Le buffet qui l'habille est construit, ainsi que la tribune, par le menuisier Félix Boucher.
Construit sur une tribune à l'entrée de la nef, l'orgue a la particularité d'être scindé en deux parties distinctes, attachées l'une à l'autre par un clavier. Il comprend au total dix jeux répartis en deux claviers et un pédalier.
L'orgue a été restauré en 2012-2013.
Les statues
Vierge et enfant : Cette statue est une sculpture sur bois de la Renaissance, datant du et récemment restaurée ; elle fait l'objet d'un culte marial.
Saint Martin : statue en bois du .
Pey Berland : sous le porche du clocher, une statue du de Pey Berland (qui était curé à Saint-Siméon entre 1413 et 1422). Il tient à la main la tour qu'il a fait construire en 1440, complément de la cathédrale Saint-André de Bordeaux.
Toiles
La toile La Conversion de Madeleine (intitulée également Jésus chez Marthe et Marie), d'Agostino Scilla, date de 1678.
La Vierge aux donateurs est une copie, faite par Ernestine Froidure de Pelleport, d'un tableau d'Antoine van Dyck, actuellement au Louvre. La copie fut offerte par Napoléon III en 1860.
Sarcophages
Des sarcophages en pierre calcaire furent découverts à plusieurs reprises dans la nécropole qui remonte à l'époque gallo-romaine ; l'un d'eux était orné de stries et de chevrons. La plupart datent des périodes mérovingienne et carolingienne. Quelques-uns ont été réutilisés jusqu'au . Ils étaient disposés de telle sorte que le défunt ait les pieds tournés vers l'est. Les premiers ont été signalés à la Société Archéologique de Bordeaux en 1878 par A. Combes, puis en 1937 par T. Ricaud ; et en 1948, R. Cousté parle aussi de tombes en tegulae.
Voir aussi
Bibliographie
Michelle Gaborit et Marc Saboya, L'église Saint-Siméon de Bouliac, Édifices et Images du Moyen Âge, Centre de recherches Léo Drouyn Bouliac.
Articles connexes
Liste des monuments historiques de l'arrondissement de Bordeaux
Bouliac
Liens externes
L'église Saint-Siméon sur le site Visites en Aquitaine.
Références
Bouliac
Simeon Bouliac
Monument historique classé en 1862
Bouliac
Simeon Bouliac
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,221
|
Q: How to pass UlImage as Base64 String to Asp.Net Web Api I have following code for iOS 10 Swift 3
if let pic = info[UIImagePickerControllerOriginalImage] as? UIImage {
let sss = UIImagePNGRepresentation(pic)! as NSData
let s1 = sss.base64EncodedString(options: .lineLength64Characters)
// api call .....
}
in C# Web API I am doing following
byte[] imageBytes = Convert.FromBase64String(defultdr["@imageString"]);
MemoryStream ms = new MemoryStream(imageBytes, 0, imageBytes.Length);
// Convert byte[] to Image
ms.Write(imageBytes, 0, imageBytes.Length);
Image image = Image.FromStream(ms,true);
image.Save(ImageDirPath + defultdr["@imageName"]);
where defultdr["@imageString"] contains the image data as Base64 string from iOS Wwift 3
but I am getting following exception
Parameter is not valid.
at System.Drawing.Image.FromStream(Stream stream, Boolean
useEmbeddedColorManagement, Boolean validateImageData) at
System.Drawing.Image.FromStream(Stream stream) at
TWebApiSearch.Controllers.UploadController.ImageToDir(TRequest json)
in
C:\Users\tahmid\Downloads\APIWeb\TWebApiSearch\Controllers\UploadController.cs:line
53
on this line in Web API C#
Image image = Image.FromStream(ms,true);
What is the cause of this error?
A: You are loading the memory stream twice. Once when you initialized it
MemoryStream ms = new MemoryStream(imageBytes, 0, imageBytes.Length);
and second when you do the write.
ms.Write(imageBytes, 0, imageBytes.Length);
No need for the second one as you already loaded the memory stream buffer when initializing it with the byte array.
Try this
var imageBytes = Convert.FromBase64String(defultdr["@imageString"]);
var ms = new MemoryStream(imageBytes);
var image = Image.FromStream(ms,true);
image.Save(ImageDirPath + defultdr["@imageName"]);
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,381
|
Caminetto Moderno A Parete Con Focolare A Parete E images that posted in this website was uploaded by Drupaldistrowatch.com. Caminetto Moderno A Parete Con Focolare A Parete Eequipped with aHD resolution 800 x 600.You can save Caminetto Moderno A Parete Con Focolare A Parete E for free to your devices.
If you want to Save Caminetto Moderno A Parete Con Focolare A Parete Ewith original size you can click theDownload link.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,067
|
By AZBW
Traffic Delays Possible On Several Pinal Peak Roads
Forest roads 651 and 112 will be maintained Mon.-Thu., 6 am- 6 pm
Road maintenance activities will be conducted on Forest roads 651, and 112 over the next few weeks, Monday - Thursday, 6 am – 6 pm. There may be some traffic delays during this time.
Road repair and surface replacement work will be conducted four days weekly beginning on or before June 9, 2014 until the work is completed (estimated July 31). The roads will continue to be open to regular traffic, with extended periods of delay possible ranging from 30 minutes to 1 hour. The public is asked to practice caution due to the machinery, employees, and activities.
Forest Service road crews will be operating graders, dozers, backhoes, and water tenders making travel hazardous on the narrow and winding roads up the mountain.
"We ask for public understanding, support and cooperation," stated Rick Reitz, Globe District. "Fires in the area have removed soil-stabilizing vegetation which has contributed to erosion, loss of road surface, and the fill-in of drainage ditches."
Pinal Recreation residents and communication tower personnel should call the Globe Ranger District at 928-402-6200 if they have further questions about specific area maintenance activities.
'On The Road Again'
Tonto National Forest Closes Developed Recreation Sites Forestwide
Grand Canyon Extending Permit Time
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,172
|
'use strict';
var duplicateCreep = function (creep, priority, silence) {
var spawn;
priority = priority === true ? 'spawnPriorityQueue' : 'spawnQueue';
silence = silence === true;
// Find and add to queue
if (typeof creep.memory.spawn === "string") {
spawn = Game.getObjectById(creep.memory.spawn);
}
if (!(spawn instanceof Spawn)) {
Memory[priority].push({
role: creep.memory.role, memory: _.cloneDeep(creep.memory)
});
if (!silence) {
console.log("Added " + (creep.name || creep.memory.role) + " to global " + priority);
}
return;
}
if (!Memory.spawns[spawn.name]) Memory.spawns[spawn.name] = {};
if (!Memory.spawns[spawn.name][priority]) Memory.spawns[spawn.name][priority] = [];
Memory.spawns[spawn.name][priority].push({
role: creep.memory.role, memory: _.cloneDeep(creep.memory)
});
if (!silence) {
console.log("Added " + (creep.name || creep.memory.role) + " to " + priority + " at spawn " + spawn.name);
}
};
var command = function(flag, parameters) {
if (parameters.length < 2) {
console.log('Flag command creepClone: creepClone command has not enough parameters');
flag.remove();
return;
}
if (typeof Game.creeps[parameters[1]] !== 'object') {
console.log('Flag command creepClone: Creep ' + parameters[1] + ' not found');
flag.remove();
return;
}
var priority = false;
if (parameters.length > 2 && parameters[2].toLowerCase() === 'true') {
priority = true;
}
duplicateCreep(Game.creeps[parameters[1]], priority);
flag.remove();
};
var native = function(command, creep, priority, silence) {
return duplicateCreep(creep, priority, silence);
};
module.exports = {
exec: command,
command: "creepClone",
native: native,
help: 'Description:\n- Duplicates a creep\n\nUsage:\n- creepClone <creepName> [priority=false]',
};
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 518
|
{"url":"https:\/\/help.macrobond.com\/count-how-many-times-vix-has-been-lower-than-10\/","text":"# Count how many times VIX has been lower than 10\n\nIn other words, we need to sum the 1 and 0 of the previous logical expression, over time.\n\nIn the formula expression we can do this by using the function AggregateSum. It returns a result where the value at each observation is the sum of the values of all observations preceding it.\n\nAggregateSum(vix<10)\n\nUsing the vix<10 as the operator of AggregateSum, we can determine the number of occurrences, when VIX value has been lower than 10.\n\nThe blue line in the chart below is the final result.","date":"2022-12-05 03:31:13","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\": 0.7667273879051208, \"perplexity\": 359.095183716666}, \"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\/1669446711003.56\/warc\/CC-MAIN-20221205032447-20221205062447-00454.warc.gz\"}"}
| null | null |
Q: Loop through data object and ouput the object filename and message for each object How do i loop through my ajax request for each data object and pull out the filename, message for each file? Can anyone help?
Server side controller which is parsing back a List to the client side:
@RequestMapping(value = { "/fileUpload" }, method = RequestMethod.POST)
@ResponseBody
public List<FileUpload> uploadFile( @RequestParam("number") String number, @RequestParam("files[]") MultipartFile[] files, MultipartHttpServletRequest req, HttpServletResponse res)
{
WebUserSession session = (WebUserSession) req.getSession().getAttribute("webUserSession");
String windowsUsername = session.getUsername();
List<FileUpload> fileList = itsmService.uploadFile(files, windowsUsername, number);
int countTrue = 0;
int countFalse = 0;
for (FileUpload loopFile : fileList)
{
if (loopFile.getSuccess())
{
countTrue++;
}
else
{
countFalse++;
}
}
return fileList;
}
}
part of ajax request Client side data object:
success : function(data) {
FileUploadVisible(true);
console.log(data);
}
A: I'm not sure to understand your question, but if your server returns data like:
[{"name":"toto.txt", "message":"bla", "success":true},{"name":"titi.txt", "message":"blabla", "success":true}]
You can do:
success:function(fileList) {
fileList.forEach(function(file){
console.log("File name:" + file.name + ", msg:" + file.message + ", success: " + file.success);
})
}
or if you just want to display the files successfully uploaded:
success:function(fileList) {
var uploadedFiles = []
fileList.forEach(function(file){
if (file.success) uploadedFiles.push(file.name)
})
}
alert("Files uploaded: " + uploadedFiles.join(", ");
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,833
|
Q: I want to use two webcams on a single web page in html & javascript I want to capture images from both cameras and use them for some purposes is there anyone who can help me out? any tutorial? library? method? .
A: You can create two different streams, one for each camera, and show them simultaneously in two <video> tags.
The list of available devices is available using navigator.mediaDevices.enumerateDevices(). After filtering the resulting list for only videoinputs, you have access to the deviceIds without needing permission from the user.
With getUserMedia you can then request a stream from the camera with id camera1Id using
navigator.mediaDevices.getUserMedia({
video: {
deviceId: { exact: camera1Id }
}
});
The resulting stream can be fed into a <video> (referenced here by vid) by calling vid.srcObject = stream.
I have done this for two streams from two webcams simultaneously.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,450
|
Register for Events Events
Latest News News
My Northwestern Foundation & Alumni Community
Mark Zadorozny, '08
Mark Zadorozny '08
By Caitlin Hofen
"I came back to Northwestern for many reasons," Mark Zadorozny, '08, said. "I think it's a great place to work and I have many connections in this area. Knowing the University's atmosphere, I knew I'd be in good hands."
Zadorozny holds a Bachelor of Science degree in health and sports science education and a Master of Education degree in educational leadership, both from Northwestern. He is also the newest face of the health and sports science education faculty.
His experience includes middle school science teacher/coach at Woodward (Okla.) Public Schools; middle school science teacher at Yukon (Okla.) Middle School; and technical applications instructor at High Plains Technology Center in Woodward.
Zadorozny has coached softball, basketball, football and baseball. He is also currently an announcer at Northwestern for volleyball, basketball and baseball.
A native of Woodward, Okla., Zadorozny graduated from Cedarville, Kan., after moving there his freshman year of high school.
Both of Zadorozny's parents graduated from Northwestern, which helped to make his decision to return and teach future educators an easier choice.
"I grew up coming to Ranger football and basketball games. Ranger Red runs pretty deep in my family and having the opportunity to give back definitely helps.
"My grandmother, mother and sister are all teachers, so it felt right to go into that profession. I've always been around sports and education. I grew up playing sports and wanting to coach, but I hadn't ever thought about teaching in higher education. In the last few years, I realized that I wanted to go back and help future teachers."
Zadorozny is a very big family man. "I'm very close with my older brother and sister. They have families of their own and I love to see my nieces and nephew."
While the opportunity to attend more Ranger games is a plus, Zadorozny is most excited about getting to know the students.
"Even after I graduated, I would come back for games and such; this way, however, I really get to know the students and make connections with them. I'll get to guide them into becoming future coaches and educators. I take that as a challenge I'm willing to accept."
A huge Ranger supporter, Zadorozny is excited to give back to his alma mater and carry on the tradition of offering a quality Northwestern education. He believes alumni play a big role in the success of the University.
"I encourage alumni to start investing in the university. A little bit goes a long way and Northwestern wouldn't be what it is without the great support it receives. With the help of scholarships and other support, we attract even more students who we can turn into great teachers, doctors and professionals."
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,208
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Monthly Child Tax Credit Payments Will Start Going Out on July 15
Qualified families will receive a payment of up to $300 per month for each child under 6 and up to $250 per month for children between the ages of 6 and 17
By Josh Boak • Published May 17, 2021 • Updated on May 17, 2021 at 2:10 pm
The Treasury Department said Monday that 39 million families are set to receive monthly child payments beginning on July 15.
"This tax cut sends a clear and powerful message to American workers, working families with children: Help is here," Biden said in remarks at the White House.
HOW MUCH IS THE CHILD TAX CREDIT AND WHO IS ELIGIBLE?
Make It May 14, 2021
Tax Returns Are Due May 17. Here's How Much You Could Owe If You Don't File on Time
personal finance May 15, 2021
IRS Will Start Issuing Refunds on Unemployment Insurance Taxes—and These Taxpayers Will Get Theirs First
child tax credittaxesfinances
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,960
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Q: Visual Studio 2013 auto indent In Visual Studio 2013, under Tools > Options > Text Editor > File Extension, I have set three file extensions, namely .cginc, .compute, and .shader (these are Unity3D shader files) to use Microsoft Visual C++ for auto formatting.
Sometimes auto indent (for curly braces) works, and sometimes it doesn't...I can't find any rhyme or reason for this. When it doesn't work, it starts every new line flush to the left.
I Google around every couple weeks and can't find an answer.
Does anyone know any setting to make auto indent work consistently?
A: Try this:
Tools -> Options -> Select the language of your choice (Expand the menu) -> Select Formatting (Expand this sub menu) -> Select Indendation
Once you have selected Indentation, on the right are options displayed - Check/Tick the option: Indent Braces.
Note: You may also format the entire document/file using the shortcut Ctrl+K+D
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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module Fog
module Compute
class Ninefold
class Real
def create_ip_forwarding_rule(options = {})
request('createIpForwardingRule', options, :expects => [200],
:response_prefix => 'createipforwardingruleresponse', :response_type => Hash)
end
end
end
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,937
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<?xml version="1.0" encoding="utf-8"?>
<shape xmlns:android="http://schemas.android.com/apk/res/android">
<corners android:radius="5dp"/>
<solid
android:color="@color/retrieve_view_login_square_bg"/>
<stroke
android:width="1dp"
android:color="@color/retrieve_view_login_square_border_bg"
/>
</shape>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,747
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In loving memory
Madeleine 'Granny' Goad
1911–2005
ALSO BY PLUM SYKES
_Bergdorf Blondes_
_The Debutante Divorcée_
_Oxford Girl_
CONTENTS
1. Cover
2. Half-title Page
3. Dedication
4. Also by Plum Sykes
5. Title Page
6. Contents
7. One
8. Two
9. Three
10. Four
11. Five
12. Six
13. Seven
14. Eight
15. Nine
16. Ten
17. Eleven
18. Twelve
19. Thirteen
20. Fourteen
21. Fifteen
22. Sixteen
23. Seventeen
24. Eighteen
25. Nineteen
26. Twenty
27. Twenty-One
28. Twenty-Two
29. Twenty-Three
30. Twenty-Four
31. Twenty-Five
32. Twenty-Six
33. Twenty-Seven
34. Twenty-Eight
35. Twenty-Nine
36. Thirty
37. Thirty-One
38. Thirty-Two
39. Thirty-Three
40. Thirty-Four
41. Thirty-Five
42. Thirty-Six
43. Thirty-Seven
44. Thirty-Eight
45. Thirty-Nine
46. Acknowledgements
47. A Note on the Author
48. Copyright Page
1. One
2. Two
3. Three
4. Four
5. Five
6. Six
7. Seven
8. Eight
9. Nine
10. Ten
11. Eleven
12. Twelve
13. Thirteen
14. Fourteen
15. Fifteen
16. Sixteen
17. Seventeen
18. Eighteen
19. Nineteen
20. Twenty
21. Twenty-One
22. Twenty-Two
23. Twenty-Three
24. Twenty-Four
25. Twenty-Five
26. Twenty-Six
27. Twenty-Seven
28. Twenty-Eight
29. Twenty-Nine
30. Thirty
31. Thirty-One
32. Thirty-Two
33. Thirty-Three
34. Thirty-Four
35. Thirty-Five
36. Thirty-Six
37. Thirty-Seven
38. Thirty-Eight
39. Thirty-Nine
40. _Acknowledgements_
41. _A Note on the Author_
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1. Cover
2. Title Page
3. 1 Thursday, 14 October 1985, 0th week: morning
* * *
BY URSULA FLOWERBUTTON
* * *
The girl was lying on the chaise-longue when I found her that morning. She was still in her party dress. At first, I'd thought she was asleep. But she wasn't. She was dead. Even a girl like me – one who's read all seventy-five Agatha Christies – could never have imagined such a sinister start to her first term at Oxford. I was expecting books and ball-gowns, not a dead body...
Full story continues overleaf...
Thursday, 14 October 1985, 0th week*: morning
The thoughts going through Ursula Flowerbutton's mind as she gazed up at the gilded, gargoyled, turreted double gate tower of Christminster College, Oxford, were – mostly – of cucumber sandwiches. She might have studied Disraeli, Gladstone, Lincoln and De Gaulle to get through the impossible Oxbridge entrance exam, but in that moment, romantic nonsense of the fluffiest kind overpowered Ursula's intellectual capabilities. All she could think was that here she was, on her first day at Oxford University, about to commence three years of sepia-toned, _Brideshead Revisited_†-style bliss. She imagined herself poring over ancient historical manuscripts in hushed libraries by day; she'd sip hot Ovaltine while reading improving literature in her digs during the long winter evenings; there would be croquet on the college lawn on lazy summer afternoons, followed by tea and the aforementioned cucumber sandwiches—
'HEY!'
A sharp yell interrupted Ursula's reverie. She turned to see a rickety blue minibus, with the words J. Y. A. ORIENTATION EUROPE LTD emblazoned on the side, pulling away from the kerb opposite the college. The source of the voice was soon revealed to be a rather exotic specimen. A girl with a voluminous, puffed-up mound of dyed-blonde hair was waving desperately at Ursula. She was dressed in a green, bat-wing sweater, bubble gum-pink pedal-pushers, a trilby hat and white trainers. A heap of shiny orange suitcases surrounded her.
'Help!!!' came the voice again.
What a gorgeous American accent, thought Ursula, as light and frothy as an ice-cream soda. Ursula was in awe of the girl. She'd never met an American before, let alone one who looked like she'd walked straight off the set of _Sixteen Candles._ The girl's bright look made Ursula feel a little shabby. For her first day at Oxford she'd dressed in a Black Watch kilt, Fair Isle sweater, old tweed hacking jacket and a red woollen beret knitted by her grandmother.
'I'm coming!' Ursula called back, abandoning her beaten-up navy blue Globetrotter suitcase and trunk on the cobblestones. She darted across the road, dodging the students on bicycles whizzing along Christminster Lane.
'Hello,' she said to the girl when she reached her. 'Are you all right?'
'I'm totally confused,' replied the American. 'The traffic's all on the wrong side of the street and it's making me dizzy. I'm looking for the Christminster College campus.'
'Don't worry,' Ursula reassured her, indicating the gate tower across the street. 'You're here.'
' _That?!_ ' The girl's eyes widened as they travelled up-up-up the famous building. 'That is not a campus. _That_ is a museum.'
The American then grinned, flashing immaculately straight teeth. She had chestnut-coloured eyes, thick, dark eyebrows and tanned skin. With her streaky blonde mane, she looked like a cross between Madonna and Brooke Shields. The girl's nose was a pretty snub, very narrow, slightly turned up at the end, just as perfect as her teeth.
'You like my nose, huh?' she said.
'Sorry. I didn't mean to stare,' said Ursula, abashed.
'It's okay. I got it for my Sweet Sixteen. That's when everyone gets their nose jobs in New York. It's a rite of passage.'
Ursula liked this girl already. She'd never met anyone as refreshingly frank. The classmates with whom she had spent most of her teenage years were the repressed results of the institutionalised timidity encouraged at English all-girls' schools. The rite of passage for the majority of such creatures, Ursula included, was passing the Pony Club C Test.
'I'm Ursula, by the way,' she introduced herself.
'I'm so happy to meet you. I'm Nancy.' The other girl surprised Ursula by giving her a huge hug, as though they were long-lost best friends.
'Let's head over,' said Ursula, unfurling herself from Nancy and grabbing as many of her suitcases as she could manage.
Nancy loaded herself up with the remaining baggage and the girls zig-zagged across the street, chattering happily as they went.
'By the way, I love your outfit,' said Nancy.
'Really?' asked Ursula, amazed.
'Yeah. You look really cute, kinda like . . . Beatrix Potter.'
'Oh . . .' said Ursula, suddenly downcast. She had tried her absolute hardest this morning not to look like a country bumpkin. She'd clearly failed.
*
A wicket gate set into the vast, castle-like main door of the gate tower led Ursula and Nancy into an echoing, stone-flagged entrance. They dropped their luggage on the ground, gazing delightedly between the carved stone columns of the cloister just beyond into Christminster's famous Great Quad. The two girls drank in the scene before them: the immaculate half-acre of lawn, mown diagonally like a green checkerboard, flagged with a discreet _Please Keep Off The Grass_ sign; the late-morning sunshine glinting golden off the spires that topped the grand stone quadrangle buildings; a tutor, clad in a long black academic gown, scudding like a bat along the wide gravel paths round the edge of the grass.
'Wow! _Je_ adoring!' exclaimed Nancy.
'It's like a dream,' replied Ursula.
She felt as though she might burst with excitement. The college‡ was even more beautiful than she remembered from her interview a year ago.
Just then a glazed wooden door to the girls' right opened, and a boy appeared. He was tall and narrow-bodied, with a skull-like, pallid face. His gaunt look was not helped by the fact that his pale brown hair was prematurely thinning. He was wearing pebble-glasses with horn rims and was dressed in brown cords and a dark-green Loden jacket that Ursula recognised as _tracht_. The only thing indicating that he hadn't just stepped out of _Hansel and Gretel_ were the bright yellow Converse trainers on his feet. As he approached, Nancy nudged Ursula, declaring, 'He's not my type at all. Where are all those hot, floppy-haired Eton§ boys Oxford's so famous for?'
'Sshhh!' said Ursula, hoping the skinny boy hadn't heard.
She was surprised when he reached them and performed a little bow. Was everyone at Christminster going to be this formal? she wondered.
'Greetings, Freshers,' the boy said. He spoke with the faintest trace of a German accent. 'Allow me to introduce myself. I am Otto Schuffenecker, Prince of Carinthia, Austria. Second year. History.'
'Are you for real?' Nancy sounded gobsmacked.
'Absolutely.' Otto smiled proudly.
(Ursula sensed that this was not the moment to mention that her A-level History course had included the study of the official abolition of the Austrian nobility. There hadn't been a royal family in Austria since 1919, after the fall of Austria-Hungary.)
Otto went on to explain that, as one of the official Freshers' Liaison Officers, he was on hand to help new students navigate Christminster and Oxford. He seemed delighted that his first charges were to be Ursula and Nancy.
'Well, _Prince_ Otto,' said Nancy playfully, 'inspired by your full introduction, I am Nancy Feingold, of the Saddle River, New Jersey Feingolds, of Feingold's Gardening Tools, Inc. We live really close to Manhattan – I go all the time. Anyways, I'm a sophomore at Northwestern in Evanston, Illinois and I'm on my Junior Year Abroad here. Or should I say my _Party_ Year Abroad?'
'Are you planning on doing _any_ work while you're in Oxford?' Otto asked her with a smile.
'Sure . . . if I must,' replied Nancy. 'But honestly, I've been dreaming about Pimm's, punting and picnics all summer. My major's History. While I'm here, I plan on minoring in Earl-catching.'
Nancy exploded into giggles so contagious that Otto and Ursula soon found themselves overtaken by laughter. Finally, Otto managed to contain himself enough to tell Nancy, 'Well, you're in the right place for that. Oxford's stuffed with Earl-types.' He then turned to Ursula, asking, 'And you are?'
'Ursula Flowerbutton. Modern History,' was the extent of our heroine's modest introduction. Her ponies-dogs-muddy-walks-and-books upbringing in a farmhouse hidden in a country valley didn't seem exotic enough to elaborate on. But Otto raised his eyebrows quizzically, waiting for more.
'Miss,' stated Ursula firmly.
'Of?'
'Seldom Seen Farm, Dumbleton-under-Drybrook, Gloucestershire,' said Ursula. 'It's in the Cotswolds.'
'Lovely area,' he remarked. Then he helpfully explained to Nancy, 'It's the part of the English countryside that's on all the chocolate boxes. Right, girls, we'll sort out your luggage later. Follow me.'
*
Otto led them through the glazed door into the Porters' Lodge, a draughty little room that seemed far too small for the activity carrying on in it. From behind a worn wooden desk at the far end, the Head Porter, a stout, officious-looking man in a bowler hat, black three-piece suit and tie, was handing out keys to various students, one of whom had somehow managed to squeeze a bicycle into the lodge with him. Parcels and packages were piled up in heaps and notices were pinned on various boards. A tiny window looked out onto Christminster Lane. Ursula noticed that the north wall of the room was covered with mailboxes, labelled with students' names. Nancy pointed excitedly to a couple of aristocratic-sounding ones.
'"Lady India Brattenbury",' she read aloud. 'Sounds _very_ fancy . . . "Lord Wychwood". Isn't he out of _Pride and Prejudice_?'
'That's Wickham,' said Ursula. 'He was a terrible cad. Ran off with Elizabeth Bennet's sister Lydia.'
'Well, let's hope this Lord runs off with me!' giggled Nancy.
'He's taken, I'm afraid,' Otto told her. 'Right, these are your pigeonholes,' he explained, finding both girls' boxes. 'You should check them every day for messages. Your tutor will communicate with you here. Oh, and if you want to send a note to someone in another college, leave it with the porter by nine in the morning and Pigeon Post will collect and deliver it.'
'They deliver the mail by _pigeon_ here?' Nancy looked astonished. 'I feel like I'm in _The Wizard of Oz_.'
'Let me explain,' said Otto. 'Pigeon Post is one of many archaic Oxford traditions – it consists of college servants who ride round the city on mopeds delivering the students' mail. Usually it only takes two days to get a message back.'
' _Two days?_ ' Nancy was appalled. 'Can't we just phone from our dorm rooms?'
Nancy's astonished expression rapidly morphed into a highly traumatised one as Otto explained that Oxford undergraduates didn't have telephones in their rooms. There were, though, two payphones in college, which took two- or ten-pence coins.
'But no one really uses them, and if they ring, no one picks up,' he added, 'in case it's someone's parents.'
'But what if my mom needs to speak to me?' asked Nancy. 'Ursula, don't you ever want to talk to your parents?'
'I'd love to,' she replied. Then she added slowly, 'But they're both . . . gone.'
'I'm sorry, I didn't mean—' Nancy stammered, her face registering shock and sorrow.
'It's okay,' said Ursula. 'Really. My grandmothers raised me. They're wonderful.'
Otto curtailed the awkward moment by beckoning the girls to follow him. The trio squeezed through the crush of students in front of the porter's desk.
'Morning, Deddington,' Otto said to the bowler-hatted man when they reached the counter. 'I've got Miss Flowerbutton and Miss Feingold here. May we have their room keys, please?' He then told the girls, 'Not that you need to bother locking your rooms. No one ever does.'
Deddington handed Otto two large keys. 'Good morning, Flowerbutton, Feingold,' he said abruptly. (The girls would soon learn that Deddington had never grown used to the fact that Christminster had, finally, started accepting women. He didn't really approve, and could rarely bring himself to use the word 'Miss' when addressing the female minority in college.)
Just then, a hand clad in a bright yellow rubber washing-up glove plonked a cup of tea in front of Deddington. Ursula turned to find a shrivelled-looking woman standing beside her. She was wearing a brown polyester housecoat and old black lace-ups on her feet. Her greying hair was scraped into a tight bun.
'Thanks, love,' said Deddington, taking a gulp of the tea. Then he added, 'Feingold, Flowerbutton, this is my wife Mrs Deddington. She's the dons' scout.'
'The who's what?' asked a bewildered Nancy.
'I clean for the tutors,' explained Mrs Deddington in a thin voice.
She had a mouse-like face that looked as if it was permanently etched with worry. Ursula smiled sweetly at Mrs Deddington in an attempt to cheer her up but the scout's expression remained dour.
'Right, try not to lose your keys,' Deddington told the girls. 'If you get locked out at night, ring the bell. Our son Nicholas is one of the night porters here. You'll see him this evening.'
'Thanks, Deddington, I'll take the girls to their rooms now,' said Otto, heading out. 'You're on a Historians' staircase in the Gothic Buildings,' he told them.
'Catch you later.' Nancy waved goodbye to the porter and his wife.
'Thank you,' added Ursula as she left the lodge.
The girls followed Otto back out to the gate tower, where a group of returning Christminster students had started to gather. Their ebullient greetings echoed across the quad.
'This is a huge relief,' Nancy told Ursula, looking the male students up and down. ' _Tons_ of floppy-haired Eton boys.'
The boys were languid and tall, Ursula observed, and noticeably scruffily dressed with long fringes covering half their faces. Most of the girls had super-long, super-shiny hair that they swished around like show ponies' tails. Ursula and Nancy caught snatches of their conversation as they walked past.
'Jubie! Darling! _Ciao_!' squealed a girl who looked as if she had modelled herself on Princess Diana. She had fluffy, highlighted blonde hair and was wearing a lilac pleated skirt. Her pale pink striped blouse had a high, ruffled collar, which was tightly buttoned around her neck.
'Tiggy! Good hols?' Jubie yelled back.
'Yah. Absolutely schizo,' she replied. 'How was San Trop?'
' _Wild_ ,' said Jubie. 'Completely sick.'
'Is Bunter Up**?' asked one of the boys.
'Not sure. Apparently Teddy's arriving tomorrow,' replied another. 'He's bringing Ding Dong with him.'
'Yah? Great. I heard India's en route.'
'Who are they?' Nancy asked Otto curiously as they headed along the gravel path on the west side of Great Lawn towards the Gothic Buildings.
'The Yahs,' said Otto.
'Yahs?' said Nancy. 'What are Yahs?'
'Oh, sorry, I suppose you've never come across one. Let me explain. They're an easily identifiable English species – they all went to the same posh private schools, they still use their absurd childhood nicknames and they always say "yah" instead of yes. Hence, "Yahs".'
The Yahs, Otto said, dominated the Oxford scene. They ran the Oxford Union, the magazines, the balls, the dining societies and the drama clubs. They were Rowing Blues or drug addicts, sometimes both.
'Anyway, you two, you'll be straight in with the Yahs, you'll see. They love pretty girls,' said Otto.
He turned a sharp corner at the bottom of Great Quad and beckoned the girls to follow him along a narrow stone passage.
'All I can say is, thank god I'm not a Yah,' he went on. 'Ten per cent of them are dead before the end of Oxford.'
'What!' shrieked Nancy melodramatically.
'Okay, maybe not ten per cent. But there was that one in New College who ended up buried under her boyfriend's floorboards last term—'
'Ugghh!' shuddered Nancy.
'—and the third year who overdosed on heroin the day before his Classics finals.'
'How tragic,' said Ursula, feeling unnerved.
'You know what they had in common?'
'What?' asked Nancy, a note of terror in her voice.
'They were both,' concluded Otto, 'very posh.'
* Oxford terms begin with 1st Week, ending in 8th Week. Freshers arrive in 0th Week, before term officially starts. Undergraduates refer to dates by week, number and day, e.g. 'See you Tuesday of Third Week' makes total sense. 'See you on the twenty-ninth of October' does not.
† The TV version of _Brideshead Revisited_ , Evelyn Waugh's classic Oxford novel, was the _Downton Abbey_ of the 1980s, except everyone was secretly gay.
‡ There is no Oxford University 'campus'. Rather, the city of Oxford makes for an informal campus, with over thirty separate colleges dotted around the centre. Collectively these form the university. Each college has its own residential buildings, dining hall, chapel, libraries, grounds, etc., where students live and learn, although they can also study with tutors in other colleges. The oldest, richest, most famous colleges – Christ Church, New College, Magdalen College, All Souls – are as grand as palaces, occupying their own mini-estates within the town.
§ Notorious for its poshness, Eton College is now as well known for the hotness of its pupils. (Recent Old Etonian or 'O.E.' hotties include Princes William and Harry, Eddie Redmayne, Damian Lewis, Dominic West, Tom Hiddleston.)
** 'Going Up' = beginning term. 'Going Down' = ending term. Not to be confused with 'Sent Down' = expelled.
The Gothic Buildings were arranged around a rather spooky little courtyard dominated by a sprawling, gnarly-trunked yew in the centre. Very _Bleak House_ , thought Ursula as she looked around, noticing that the quad had four castellated turrets, one in each corner. The only dashes of colour were from a few pale pink autumn roses still open in the flowerbeds. Otto led the girls towards one of four stone archways. It had the letter 'C' carved above it.
'This is your staircase,' he informed Ursula and Nancy.
The girls peered curiously inside the archway. A flight of scrubbed wooden stairs led up towards a window on a high landing. Two girls were standing just inside the doorway, scrutinising a black-painted board attached to the wall on the left. The number of each room and the name of its occupant had been inscribed upon it in white italic lettering.
'Hi!' called one of the girls when she saw them. 'Felicia Evenlode-Sackville. Head Girl, Roedean. Lacrosse captain, Firsts. Everyone just calls me Moo.' She had the plummy, Sloane Ranger accent common to alumnae of one of England's most prestigious girls' boarding schools.
Moo was dressed in a navy blue regulation Roedean tracksuit and Green Flash plimsolls. The plump blonde pony-tail on top of her head seemed to bounce in time with her words. She had a sprinkling of freckles across her face and a healthy, sporty physique.
Moo's bumptious confidence put her in stark contrast to the other girl, who seemed too terrified to utter much more than her name: Claire Potter. Claire was a sad-faced girl, as plain as it was possible to be. She had acne-strewn skin that appeared to have the slippery consistency and pallid colour of lard. She was dressed in a drab, knee-length skirt, a scratchy-looking sweater and woolly tights. Her flat feet were clad in enormous ugly brogues. She was wearing clunky NHS spectacles and her mousy hair was chopped short.
Ursula felt sorry for Claire Potter. She'd try and be nice to her, but suspected they probably wouldn't become good friends. Although Claire had barely spoken, Ursula, as prone as any other eighteen-year-old girl to making snap judgements based on appearance, doubted they would have much in common. Ursula couldn't really relate to short-haired girls. She just didn't understand their philosophy of life.
'Your bathroom is through there,' said Otto, pointing to an open door on the left.
Ursula and Nancy stepped into the room and inspected the washing facilities. Four sinks were attached to the wall, and two whitewashed cubicles each contained a large bath. The room was icy, with the only possible relief offered by a meagre electric heater installed high on the wall above the door.
Nancy regarded the bathroom with a look of horror. 'Where are the showers?' she asked Otto.
He chuckled. 'As far as I know the only showers in Oxford are in the Randolph Hotel – oh!' He let out an excited cry. 'I don't _believe_ it. India!' he called out to a girl heading towards Staircase C. 'India!'
Ursula was immediately intrigued by the new arrival. Petite and fine-boned, India had a complexion as pale as a dove's wing. She wore her dark hair in a long bob, which had been left punkily unbrushed. Ray-Ban sunglasses revealed little more of her face than sky-high cheekbones. As she drew closer, Ursula could see that the girl's lips, which were stained a dark blackberry colour, were set in a full, dramatic pout. She was dressed in the Chelsea girl hipster uniform of the moment – black leather biker jacket, black Lycra mini skirt, black opaque tights and black suede pixie boots. Her hands were encased in black leather fingerless gloves. Her edgy look made Ursula regret her own tragic Beatrix Potter garb even more.
India greeted Otto with a kiss on each cheek when she reached him. She then started pinning something on one of the noticeboards just inside Staircase C.
'What on earth are _you_ doing Up in Freshers' Week, India?' Otto asked. 'Shame on you!'
'Rehearsing. It's _so_ annoying, I'm missing Beano's cocktail at Annabel's tomorrow night. Darling, don't tell a soul you've seen me here, yah?' she begged him.
'Wouldn't dream of it, sweetheart,' he agreed obligingly.
India stabbed a final drawing pin in a corner of the notice. It read 'OXFORD UNIVERSITY DRAMATIC SOCIETY – TREASURER REQUIRED. Apply to President, Dom Littleton, Magdalen College'.
'Anyway, we're desperate for a Treasurer and none of the acting lot would dream of doing anything so dull. So I had the brilliant idea of putting notices in the Freshers' staircases. Saddo first years love doing that kind of thing . . . but if no one takes this bait maybe we can recruit someone at the Freshers' Fair on Saturday.'
'I'll apply,' offered Claire Potter. She spoke in a soft Welsh accent, blushing furiously.
India pushed her sunglasses up on top of her head, revealing startling violet eyes and yard-long black eyelashes. She regarded Claire with the kind of disdain an eagle would reserve for an earthworm – an organism so insignificant it wasn't even worth contemplating as a meal.
'Sorry,' said Otto. 'Let me introduce you all. Freshers, this is Lady India Brattenbury.'
'Hi,' Lady India replied. In contrast to the friendly tone she'd used with Otto, she now spoke in a clipped, cold voice.
Ursula recognised the name from the pigeonholes. Lady India was clearly Someone Very Grand. As Otto introduced each Fresher by name, India's pout became ever-sulkier. Until, that is, she heard the words 'Nancy Feingold'.
'So _you're_ the Feingold girl,' she said to Nancy, her pout morphing into a welcoming smile. 'I heard on the grapevine you were coming Up. Read all about the Feingold gardening-tool dynasty in _Tatler._ It's my fave mag.'
'It's hardly a dynasty,' insisted Nancy, looking highly amused. 'It's only two generations, and that's including me and my brother!'
'Well . . . anyway, I've already thought of the most brilliant nickname for you. Lawnmower,' announced India, spluttering with laughter. 'Isn't that perfect?' She didn't register Nancy's rather surprised expression but carried on, 'You should come to Wenty's party on Sunday night. I'll get you an invitation.'
'Okay,' said Nancy. 'Why not?'
'It'll be wild. Otto's coming, aren't you?'
'Of course,' he declared happily.
The other three girls were, felt Ursula, rather pointedly excluded from this conversation. But finally, fixing Ursula with a bored expression, India said, 'Where were you at school?'
'St Swerford's,' said Ursula, hoping her local private school sounded a bit smarter than it really was. 'On a scholarship.'
When she saw India's smirk, she wished she'd never mentioned the scholarship.
'St Wherefords?' said India. 'M.P.S.D.* Never heard of it.'
She turned her attention to Moo, noticing her Roedean school tracksuit. 'We Wycombe Abbey girls always used to beat your lot at matches.'
Faster than shooting a lacrosse ball into goal, Moo retorted, 'And we beat Wycombe at chess.'
The uncomfortable pause that followed was eventually filled by Claire Potter, who stuttered, 'I-I-I went to the Grammar School in C-C-Cardiff.'
'Oh,' India replied, clearly underwhelmed.
Otto looked at his watch. 'Look, I must get these Freshers to their rooms or they'll never be in time for lunch in the Buttery. See you later, India?'
'Yah. Great to meet you, Lawnmower,' said India.
India departed. She didn't bother saying goodbye to the other girls.
* M.P.S.D. = Minor Public School Dear.
The first thing Ursula did after Otto had left her to unpack in her room was to find an old photograph of her parents, taken on their wedding day, and stand it on her desk. How beautiful her mother had been, with her auburn hair cascading around her shoulders. Everyone always told Ursula how alike they looked. She had been blessed with the same kind of hair as her mother – in the sunshine it glimmered like autumn leaves, and was so long and thick that it required almost the whole of her narrow wrist to flick it to one side. Ursula's grey eyes and china-white complexion had come from her father, though, and the freckles dotted over her cheekbones she blamed on the rare bursts of sunshine that occasionally reached the Gloucestershire hills.
Ursula's attic-like room, number four, was perched in the turret of Staircase C. (Nancy's, room three, stood opposite it, across the landing, with Claire and Moo on the floor below in one and two.) The décor was basic. The walls had long ago been painted an institutional yellow, which was peeling in places to reveal fragments of old floral wallpaper beneath. Apart from the desk and desk chair, the furniture consisted of a single bed, standard-issue chest of drawers, armchair, empty bookcase and narrow wardrobe. The original fireplace had been boxed in and the only heat source was a meagre, two-bar electric fire.
So what if her room was decorated like a sanatorium and colder than an igloo? It was her new home, and Ursula adored it. After putting her key away safely in the desk drawer, she unpacked her record player from her trunk, plugged it in, put on her favourite single – a-ha's 'Take On Me' _–_ and started decorating. She draped a favourite old bedspread printed with lilacs over the bed, and put an ancient chintz cushion of her grandmother's on the armchair. Then she set about organising her desk. Opposite the wedding photo, she laid out her stationery – pens, pencils, folders, writing paper, diary – on the right-hand side, before putting a reporter's notebook in the centre. She hoped she would fill it with ideas for articles for _Cherwell_ ,* the legendary Oxford student newspaper she longed to write for.
With a-ha's music to encourage her it didn't take Ursula long to unpack her clothes. In any case, she possessed an extremely limited wardrobe since, like most English girls, she had been dressed in the same ugly school uniform pretty much since she was eleven years old. She had a couple of tweed skirts, a few kilts, some Viyella blouses, several warm sweaters and a few pairs of jeans and cords. She had a smart velvet dress that she had made herself, but her pride and joy was her one proper black-tie 'ball-gown'. It was beautiful, even if it was a hand-me-down from her grandmother, and if she was _ever_ invited to an Oxford ball (please-please-please god, let me be invited to a ball one day! she prayed), she could go. The only 'trendy' things Ursula owned were her pair of Dr Marten's lace-ups, a stripy pair of fingerless gloves and a black satin bomber jacket, all precious birthday gifts from her groovy London godmother. Unsurprisingly, all of her clothing fitted easily into the narrow wardrobe.
She soon noticed a large white envelope addressed to her propped up on the mantelpiece. A mass of papers and notes fell out when she opened it. There were notices about Matriculation, the Freshers' photograph and Freshers' Fair, which were all scheduled to take place over the next two days, Friday and Saturday. Her 'Battels'* for the term were to be paid immediately. A photocopied note from the captain of the Christminster Women's Boat team invited her to try out for the college rowing eight. There were endless details about college mealtimes and a list of required dress for Formal Hall. An academic gown, Ursula realised, must be acquired that afternoon if she was going to be allowed to eat anything tonight. She also needed a mortarboard for the Freshers' photograph.
At the very bottom of the pile of information she found a small white card, embossed with the college coat of arms – a shield emblazoned with a stag at the bottom, a chevron in the middle, and three _fleurs de lis_ above – and handwritten with the words:
* * *
_Dr David Erskine_
_Professor Hugh Scarisbrick_
_At Home_
_Thursday, 14 th October_
_Room 3, Staircase B, Great Quad_
_7 p.m._ | _Sherry_
---|---
_Suits_ | _Gowns_
* * *
Sherry with her History tutors tonight! How thrilling, thought Ursula. Perhaps she could brainstorm with Nancy and Moo over lunch – they'd need something current to discuss with their dons. She guiltily hoped they wouldn't be obliged to hang out with Claire Potter too. There was something depressing about her.
By the time Ursula had finished unpacking and wandered across the landing to Nancy's digs to see if she wanted to go to lunch, the American student's room had undergone a dramatic transformation. Her bed was now fluffed up with a squillion-tog duvet encased in a leopard-print cover. There were matching cushions and even leopard-print pillowcases.
'I brought my own bedding,' said Nancy, flopping down on the newly plush bed.
The rest of the room was completely covered in clothes. As Ursula cast her eye over the sartorial chaos draped across the desk, chairs and floor, she couldn't help feeling slightly envious. There were party dresses and glittery shoes, skin-tight jeans and cropped tops, piles of sports gear, roller-skates, hot-pants and several varieties of plimsoll. There were jewellery cases bulging with teabag-sized diamanté earrings, ropes of _faux_ pearls and armfuls of bangles. There was even, Ursula noted, a pale, vanilla-coloured fur coat that glistened in the way only real mink would. Nancy had a white vanity case, now open and with crystal perfume bottles and makeup spilling out of it, that took up almost the entire surface of her desk.
Seeing Ursula looking longingly at the case, Nancy said, 'You can borrow any makeup you like.'
'Wow, amazing, thanks,' said Ursula, who had precisely one tube of mascara and one black eyeliner to her name.
'Even with no closets and that really unhygienic, twenty-thousand-year-old bathtub seven flights downstairs, I _love_ this attic,' sighed Nancy. 'The only thing I'm stressing about is how I'm gonna store my lipstick without a refrigerator.'
Ursula opened the window that overlooked Christminster Passage at the rear of the college. 'It's cold enough out here,' she said.
'Brrr! Freezing,' said Nancy, leaning out of the window and placing her most fashionable lipstick – Estée Lauder's Russian Red – on the sill.
The girls looked down. The narrow cobbled lane below was set between high stone walls. Two male students were walking along it, pushing bicycles. Their voices echoed up towards the girls' turret.
'. . . bloody good idea to get here in time for Freshers' Week. More opportunity. Before the O.E.s get in there with the Freshettes.'
'And the Freshers' photo will be out tomorrow and then we can really get started. Do you remember that _minger_ you snogged last time . . .'
For a split second the girls didn't know whether to be appalled or amused, but they chose the latter, collapsing into giggles.
'I hope all the boys here aren't that honest!' exclaimed Nancy, closing the window. Then she said, kindly, 'Hey, I'm really sorry for asking about your mom and dad earlier.'
'Don't be,' Ursula replied. 'I only know them from photographs. I was so young when they died. My grandmothers really are the best mum and mum. Anyway, come on, we need to get to lunch . . .'
Just then there was a rap on the door.
'A visitor! Maybe it's one of those Earls I'm planning on catching,' said Nancy with a wink. Then she called out, 'Hey, come in.'
A pert, bright-eyed woman wearing a floral cleaning tabard and bright yellow rubber gloves bustled into the room. Despite the fact that she was lugging an industrial-sized bucket of cleaning materials, she had an exceptionally cheerful expression on her face. She looked to be in her mid-thirties, and had her peroxided hair half pulled back from her very made-up face with a giant glittery butterfly clip. Beneath the apron, she seemed to have a voluptuous figure. Perhaps not all cleaning ladies suffered as much as Mrs Deddington, Ursula thought hopefully to herself.
'Good morning, Miss—?' she grabbed a list from her apron pocket and looked at it, then smiled gaily at Nancy '—Feingold. I'm Miss Blythe, your scout. But I let all the undergraduates call me Alice.'
'Awesome,' said Nancy.
'I'll be up here to clean your room and make your bed, miss,' Alice continued, smiling. She had traces of what sounded like a Northern accent. 'Every morning at ten-ish. And yours, Miss—?'
'Flowerbutton – I'm in the room across the landing,' said Ursula, thrilled. This was just too grown up for words, having a college housekeeper to look after her.
'Do you think,' said Nancy, reaching for her wallet and pulling out a crisp £1 note, 'you could do some extra tidying in here today? I just can't seem to get my closet sorted.'
Nancy beamed a persuasive smile at the scout. But Alice's hitherto jolly face suddenly took on a worried expression.
'I'm sorry, Miss Feingold. College servants aren't allowed to accept money from undergraduates,' she sighed, gazing longingly at the note.
'Gimme a break!' said Nancy, shoving the money into Alice's apron pocket.
The scout patted her pocket with a wink and said, 'I'm ever so grateful, Miss Feingold. It'll help with – well, things, you know, bills and suchlike. Your room will be perfect. But don't go saying a word to the High Provost, now will you?'
'Ssshhh,' said Nancy. 'I swear I won't.'
'Nor me,' said Ursula.
Suddenly there was another knock on the door. 'I feel _very_ popular this morning,' said Nancy as she pulled it open. There on the threshold stood India Brattenbury, holding a stiff white card in her hand.
'Your golden ticket, Lawnmower,' she said, handing it to Nancy. 'Wenty says he's dying to meet you.'
Nancy took the card and read aloud:
* * *
_The Earl of Wychwood_
_At home for his_
_Michaelmas Opening Jaunt_
_Sunday, 17 th October_
_The Old Drawing Room, Great Quad, Christminster_
_Dress: White Tie_ | _Pink Champagne_
---|---
_8 p.m. – Whenever_ | _Bonbons_
* * *
'A posh Oxford party, with a real Earl . . . oh my god!' she shrieked with delight. 'My mom would be going crazy. If I could actually call her. She's a real social climber – in a cute way, if you know what I mean. Just wants the best for me. Thank you _so much_ , India, this is literally my _dream_. Hey, come in.'
India stepped inside the room, looking straight through Ursula, but when she saw Alice her face brightened. 'Alice, I missed you all summer,' she cried, throwing her arms around the scout, much to Ursula's surprise.
'And I missed you, Lady India. Lovely to see you again.' The scout beamed at her. 'Righty-ho, I'll be going, don't want to disturb you ladies any longer.' As Alice hurried from the room with her bucket, she added, 'If you need anything, girls, you can always find me mid-morning in the Scouts' Mess by the Buttery having my tea break. Oh, and the washing machines are in the Monks' Cottages. You need ten-pence coins to operate them. I'll do your sheets.'
'Thank you!' Nancy and Ursula chorused as she left the room.
'That woman is the best scout in college,' said India. 'I tell her _everything._ She treats me like a daughter. You're so lucky, Nancy, to have her. She'll do _anything_ for a few quid. I mean last term for the ball she even—'
India whispered something into Nancy's ear so that Ursula couldn't hear. Nancy's face registered slight shock before she started giggling.
Ursula was starting to feel rather uncomfortable at this point in the proceedings. She liked Nancy so far, but the whispering had begun to make her feel like she was back in the Sixth Form Common Room at St Swerford's with the ultra-popular girls excluding her. She didn't need this at university too. She'd make her excuses and leave.
'I'm heading down to the Buttery for lunch, Nancy, if you want to join me.'
Before Nancy could answer, India informed her, 'I'm taking you to Brown's for lunch. They do a real American hamburger there.'
'Okay, maybe see you later,' said Ursula.
'Sure!' replied Nancy.
Ursula exited onto the landing, where she found Moo and Claire Potter. It looked like she would be having lunch with Claire, like it or not.
'We came to get you to go to the Buttery,' said Moo. 'Shall we get Nancy too?'
'She's going to a restaurant with India Brattenbury,' said Ursula, trying not to sound envious.
Just as the three girls headed down the stairs, India's voice rang out from behind Nancy's door.
'. . . and don't you dare hang out with those History beasts I met downstairs. A lezzy, a Gopper* and a scholarship girl. Your staircase is the saddest thing I've ever seen.'
* Launched in 1920, _Cherwell_ is the only student rag that can count Graham Greene, John le Carré and W. H. Auden among its undergrad contributors.
* 'Battels' = fees for food and board.
* 'Gopper', from 'Gopping Sloane', derogatory slang used by aristo teens to refer to their _slightly_ less posh upper-middle-class contemporaries – in this case, Moo.
After Ursula, Moo and Claire had forced down the tepid baked beans and rock-hard jam roly-poly on offer in the Buttery, they stopped by Walton Street Cycles to buy second-hand bicycles and then flew down the High on them. Their destination was the University outfitter.
'Shepherd and Woodward Est. 1852' read the gold lettering above the shopfront. Inside, the store was so crowded with Freshers all trying to buy their subfusc* in time for the Matriculation ceremony, that Ursula lost Moo and Claire almost immediately. It was rather a relief, she felt, to be without them for a moment. Moo had talked loudly and non-stop about lacrosse and skiing at lunch, and as for Claire, well, self-loathing radiated from her like a contagion. After one meal with her, Ursula felt as though she had been drowned in a cloud of melancholia.
It took forever to jostle her way to the counter at the back of the shop. A harassed-looking man stood behind it. He wore a white coat and had a small badge on his lapel reading 'Mr Hooker, Head Tailor'.
'Commoner?' he yelled at Ursula over the din being made by the other Freshers.
'Er, gosh, I'm not sure—' she said, wondering what on earth he meant.
Impatiently, Mr Hooker gestured towards two black academic gowns of different lengths hanging on the wall behind him. The longer one was labelled 'SCHOLAR', the shorter, 'COMMONER'.
Since Ursula hadn't had any special academic awards bestowed upon her with her entry to Oxford, she realised she would be wearing the lowliest gown on offer.
'I am definitely a Commoner . . .' she began.
But Mr Hooker didn't hear her. He'd already started talking to another customer.
'A _very_ good afternoon, Lord Wychwood. Nice to see you Up so early,' he said, his voice taking on an unctuous tone. 'How may I help you?'
Ursula glanced around to see who had taken her place. There was no other way to describe the person now standing next to her at the counter as anything other than absolutely the most perfect cucumber sandwich-type boy Ursula had ever seen. So tall he towered over the crowd, he had a mass of messy blond hair and properly azure eyes. He was dressed in faded jeans and a cricket sweater.
'Wychwood,' Ursula said to herself . . . wasn't he the Earl who was throwing the party Nancy was going to with India on Sunday night? Yes. That was it.
Well, _thank goodness_ she wasn't going to his party herself. The boy had dreadful manners, pushing in like that in front of a girl. There would be other cucumber sandwiches, Ursula told herself, ones of far superior quality.
'Good to see you again, Mr Hooker,' said the boy, shaking the tailor's hand and giving him a genuinely friendly smile. 'I need a couple of Blues* rowing shirts, please, and shorts,' he said. 'On my account.'
'Right you are, Lord Wychwood,' Mr Hooker replied. 'Congratulations on your Blue.'
The tailor trotted off to fetch the items, having clearly completely forgotten about Ursula's gown. She tapped Wychwood on the shoulder, saying, 'Excuse me, but I was in front of you. I'm in a rush.'
He glanced briefly at her. 'God, sorry. Mr Hooker's got a terrible crush on me, you see. It's very embarrassing, but no one else will be served in here 'til I leave.'
What a ridiculous excuse, thought Ursula to herself. Wychwood was clearly as vain as he was rude. She glared furiously at him but he didn't notice – he was already waving over her head at someone else.
There was nothing to do but await Mr Hooker's return. Ursula turned her back to Wychwood, consoling herself with the thought that she was not just glad, but _thrilled_ not to have been invited to his party on Sunday. The young Earl was probably a terrific snob with a dreadful character.
Mr Hooker eventually reappeared and started painstakingly wrapping Lord Wychwood's rowing shirts and shorts in crisp white tissue paper.
'I think you've forgotten about my gown,' Ursula told the tailor. 'And I'd like to buy a Christminster scarf, please,' she added, deciding to throw financial caution to the wind and break into her meagre term's allowance. The stacks of stripy woollen college scarves were irresistible. They were bound to be hideously expensive, but the chance to wear the Christminster colours, a scarlet stripe on a mint-green background, was worth it.
Clearly irritated to be interrupted while serving Wychwood, Mr Hooker grumpily snatched a gown, mortarboard and Christminster scarf from under the counter and chucked the lot in a bag.
'Seven quid for that,' he snapped.
Ursula handed over the cash, took the delicious scarf from the bag, wrapped it twice around her neck and picked up her shopping. She was just turning to leave when she noticed India Brattenbury, sunglasses covering half her face, dashing up to the counter.
'Wenty!' she called out. Wychwood turned. As his gaze landed on India, his eyes lit up. He enveloped her in his arms and the pair indulged in an extremely long embrace. They were obviously a couple. When they had finally unfurled themselves, India said, 'I just had the most hysterical lunch with that new American girl. She's a scream. I invited her shooting.'
'Good idea,' said Wychwood. 'There's nothing like a Yank to liven up a country weekend.'
'Exactly what I thought. I'm so bored of everyone.' India pouted then turned to Mr Hooker and said, 'Darling, can you get me a lacrosse stick? Left mine in the country.'
'I'll order one in, Lady India. Should be here by the middle of next week, all right?'
'Perfect,' said India, smiling at him. Then she turned to Wychwood and said, 'Walk me to rehearsals, baby?'
'Course, I will, bunny rabbit,' he replied fondly.
Hand in hand, the pair left the shop. The Fresher girls, Ursula included, mercilessly scrutinised India as she walked past them. What was so special, their expressions seemed to say, about that pout?
* Black gowns worn for formal occasions at Oxford.
* Blues are awarded for sporting prowess – the most prestigious for rowing, tennis, and cricket. Half-Blues are awarded for lesser sports, including gliding.
Thursday, 0th week: evening
In his new role as Freshers' Liaison Officer, Otto had offered to escort Ursula's staircase of Historians to the sherry party that night. He arranged to meet Ursula, Nancy, Claire and Moo in the Porters' Lodge a few minutes before seven. It was a chill, clear night and, walking from their rooms, the four girls could see desk lights glinting from the first floor of the Hawksmoor Library overlooking Great Quad.
Ursula felt oddly serious with her new academic gown draped over her clothes. Until this moment she had loved what she was wearing – her home-made maroon velvet dress with puffed sleeves and white Peter Pan collar – but next to Nancy's glitzy outfit, it seemed childishly twee. Nancy was dressed in a purple jumpsuit with a drawstring waist and such enormous sequined shoulder pads that her gown only just fitted over them. Her pile of blonde hair was held back on one side by an oversized silver bow, which matched her high-heeled pumps. A generous coating of shiny eyeshadow glistened from her eyelids. Moo was dressed Sloane-style in a pleated skirt and a pink and white spotted blouse with the collar turned up, while Claire Potter's sherry-party ensemble consisted of a calf-length navy frock that wouldn't have looked out of place on a Sunday School teacher.
Once inside the lodge, Ursula couldn't help but notice that Deddington's place at the porter's desk had been taken by a young man. He had a pile of accountancy textbooks stacked on the desk in front of him, the top one propped open.
'Evening, Freshers. I'm Nick Deddington, one of the night porters,' he said, getting up. Ursula thought Nick looked young, early twenties at the most.
'Your dad never said you were some kind of JFK Junior lookalike _,'_ said Nancy, flashing a flirtatious smile at him. _'_ May I call you Deddington Junior, as an _homage_?'
The American talent for saying whatever the hell you thought, Ursula mused to herself, was one of Nancy's many attractive qualities. She was right: Deddington Jnr was far dishier than his parents' ordinary looks would have led her to expect. Hair that was side-parted Kennedy-style, chocolate brown and thick, framed a chiselled, high-cheekboned face, and he was noticeably tall and athletic-looking. However, unlike Nancy, Ursula would have died before mentioning it to him. English girls just didn't do that kind of thing.
Taken aback, the young man said, 'Er . . . okay . . . thank you. Um, ladies, remember that the college gate is locked at midnight. If you're not back by then, ring the bell and—'
'Will _you_ be here at midnight?' asked Nancy, continuing to stare at Deddington Jnr with a decidedly lovelorn expression on her face. She wandered up to the desk and draped herself languidly across it.
The night porter coughed and reddened. He was thoroughly embarrassed.
'Yes. On duty, miss. As I was saying, if the door's locked, ring and I'll let you in.'
Deddington Jnr sat down and turned back to his books, studiously ignoring Nancy's presence. Finally, she removed herself from the porter's desk. She winked at Ursula and whispered, 'British boys are _so_ cute. I could literally have sex with the accent.'
Ursula stifled a giggle. Meanwhile, she noticed that Claire Potter had started industriously stuffing bright yellow flyers into each pigeonhole in the lodge. Seeing Ursula looking at her curiously, Claire handed a sheet of paper to her.
It stated, 'CROSSWORDS AND ICE CREAM SOCIETY. FIRST MEETING SUNDAY, 17TH OCTOBER, 7 P.M., JUNIOR COMMON ROOM, STAIRCASE B, GREAT QUAD'.
'I decided to start my own society. Will you join?' pleaded Claire, who seemed more talkative than she had been earlier. 'I'm going to have a table at the Freshers' Fair on Saturday where you can sign up.'
Ursula's main plan for the Freshers' Fair was to sign up for _Cherwell_. The last thing she wanted to do was join Claire's crossword club, but she knew she couldn't get out of it without seeming mean. And one thing Ursula was not was mean. She nodded a reluctant yes to poor Claire. After all, it wasn't as though there would be anything else to do on Sunday night. Nancy, peering over Ursula's shoulder at the flyer, was all enthusiasm.
'Crosswords and ice cream? How British! So eccentric! I'm there. I'll come on my way to the Earl's party.'
Claire managed a grateful smile, then handed Moo a flyer. 'Thanks,' said Moo, non-comittally.
Just then, Otto trotted into the lodge, exuding an air of brisk efficiency. He had swapped his Loden jacket for an immaculate tailor-made navy blue suit and navy and white spotted tie, over which his gown hung elegantly.
'Ready for a sherry hangover?' he said. The girls laughed and followed him out of the Porters' Lodge.
*
Dr David Erskine's set of rooms was located about halfway along the eastern side of Great Quad on Staircase B. To reach it, the four girls followed Otto through one of the many shadowy stone entrances that were dotted along each side of the quad, and up a steep flight of stairs to the first-floor landing. Dr Erskine's door was on the right-hand side, opposite one labelled 'Junior Common Room'.
Ursula couldn't have imagined a more heavenly place for her future tutorials than Erskine's set. The panelling had been lacquered in a glossy Chinese red. Persian miniatures had been hung densely around the mother-of-pearl-framed mirror above the carved stone fireplace, which, she noticed thankfully, was roaring with flames. On the opposite side of the room, situated in front of a huge Gothic window looking onto the Great Quad, was a large black lacquered desk, piled high with books and papers. A huge stuffed bird with a peacock-like tail sat proudly in the centre of it. A backstage pass to a Smiths concert hung around the bird's neck. Gosh, thought Ursula, Dr Erskine was terribly cool.
'It's a Great Argus,' Otto said, noticing Claire inspecting the creature on the desk. 'It came from Borneo. Dr Dave says it inspires his writing.'
'Weird,' said Claire, unimpressed.
'I think it's lovely,' said Moo.
Ursula meanwhile had noticed that the far wall, lined from floor to ceiling with books, had two 'secret' doors within it. One, which stood partially open, led into Dr Erskine's bedroom. If she craned her neck a little, she could just glimpse an unmade four-poster bed and a groovy ethnic rug on the floor.
'He _lives_ here?' Nancy asked Otto.
'Most of the dons live in College,' he explained.
'That is _so_ weird,' said Nancy.
Various comfy armchairs were dotted around the room, and a large chesterfield sofa, upholstered in worn tartan, was positioned to the right of the fireplace opposite a peacock-blue velvet-covered _chaise-longue_.
'Oh!' gasped Ursula, clutching Nancy's arm with a shudder. The _chaise-longue_ appeared to hold a motionless body. But suddenly it snored.
'That's Professor Scarisbrick. Started here in nineteen thirty-seven. He's the world's leading authority on Anglo-Saxon history. Secretly recruits for MI6,' Otto told the girls in a whisper. 'Gives all his tutorials lying down. Only rises if a student says something interesting. He's never got up in one of my tutes.'
The sleeping don, his glasses slightly skewiff on his snoozing face, had thinning white hair and a hearing aid in his right ear. He was dressed in a thick brown wool dressing gown beneath which there appeared to be a heavy tweed suit, checked woollen shirt and a maroon and cream dogtooth-check bow tie. The dressing gown was slightly grubby and clearly had food stains on the lapels.
Suddenly the secret door next to the bedroom opened, and Dr Erskine made his entrance. He was tall, with swishy, caramel-coloured hair and light brown eyes. Tonight he was wearing a pair of stonewashed Levi 501s with a trendy rip on the left knee, a navy velvet smoking jacket and a black cashmere turtleneck sweater. The finishing touch was his cream suede Gucci loafers, worn without socks, Continental-style.
'Freshers! _Salaam_!' he said smoothly. 'Now, drinks. There's sherry for Professor Scarisbrick, but I'm assuming you lot would prefer Kamikazes . . . I try to stick to historical themes with my cocktails. It inspires more . . . _reading_ ,' he continued with a mischievous grin.
'Awesome!' said Nancy gleefully.
Ursula was perplexed; so far, her only experience of Kamikazes had been the odd mention of suicidal Japanese pilots in books about World War II. But if a Kamikaze cocktail was some kind of educational beverage, she would, of course, try it. From a tiny refrigerator hidden in a bookcase the don produced ice, Triple Sec, vodka and lime juice. While Ursula watched Dr Erskine slosh everything into a cocktail shaker, Otto provided a running commentary on the Christminster History department.
Dr David Erskine, he explained, was so groovy and youthful that he insisted his students address him as Dr Dave. Only twenty-nine years old, he had climbed the greasy pole of academia with staggering speed, and was admired as much for his brains as for his belief that it was his duty to keep the myth of the bohemian, pyjama-wearing don alive. To that end, he attended many of the students' grander parties, and occasionally gave tutorials, dressed in paisley silk nightshirts bought on 'research' trips to Constantinople (as he referred to it).
Dr Dave's speciality – the East – was as trendy as his look and his first book, _From Constantinople to Jerusalem_ , had become a must-read in political circles. According to Otto, the history don adored being on television and discussing Palestine, mainly because the makeup was so flattering. His most beautiful female students – of whom he was considered to have rather too many – were nicknamed 'Dave's Babes', and it was rumoured that he had 'inspired' many of them with a good deal more than the partition of the Middle East.
'No one thinks he's creepy?' Nancy whispered to Otto.
'Quite the contrary,' Otto told her. 'He's considered one of the most glamorous dons in Oxford. Bit of a girl-magnet.'
As he handed round the full shot glasses, Dr Dave reminded himself of each girl's name. The group was soon joined by four male History Freshers, all in suits and gowns. Never having socialised with a tutor before, the students didn't have a clue what to talk about, and stood sipping their Kamikazes in silence.
'Guys, not like that – like this!' instructed Nancy, shooting her Kamikaze down her throat in one gulp.
Ursula followed suit. The drink was sharp and strong, and suddenly she started feeling floppy and giggly. Meanwhile, the other students glugged down their shots, resulting in a lot of hiccupping and spluttering. They looked rather shocked when Nancy held out her glass to Dr Dave for another drink, and he happily filled it – along with everyone else's.
'May I propose a toast, Dr Dave?' asked Nancy boldly.
The don nodded happily. 'You may.'
'Okay. To us!' she exclaimed, gulping down her next shot in one.
The other Freshers consumed their next drink enthusiastically. After her second shot, Ursula concluded to herself that she was absolutely, completely, definitely drunker than she had ever been in her life. It was lovely. The ice was broken, and the students were soon chattering away as though they'd known each other forever.
'GRRUUGGGGHHHH!' The party atmosphere was shattered by a loud grunt coming from the _chaise-longue._
'Sherry!' mumbled Professor Scarisbrick, sounding like an ancient growling dog. 'The vintage one, not the bloody cheap stuff we give the undergrads. Where's my wretched pipe?'
Without getting up, the Professor fumbled around on the Persian rug below the _chaise-longue._ He located and lit his pipe with ease considering that his eyes remained shut.
'Professor!' yelled Dave, as he took his drink over to him. 'Imbibe!'
With a great deal of clawing and moaning, Scarisbrick hauled himself into a sitting position and finally opened his eyes. He took the sherry and between sips glared at each student. 'SIDDOWWN!' he harrumphed eventually.
The slightly tipsy, rather alarmed Freshers promptly found seats and turned their attention to the esteemed Anglo-Saxon expert.
'There is only one rule in Oxford,' he barked, his deafness forcing him to speak far more loudly than he realised. 'And that is, to write your essay IN TIME FOR YOUR TUTORIAL, when you will read it OUT LOUD to me or Dr Dave, depending on which course you have selected this term. You will have one essay to write each week. You will all no doubt be wondering _how_ you will get this wonderful essay written on time.'
He stopped and pointedly eyed each of the eight students. As his pale watery gaze came to rest on Ursula, she had the sense that, were the weekly essay in question not achieved, she would be hung, drawn and quartered by the Professor, a torture about which, as a historian, she was well informed.
'Do you row, Mr . . .?' Professor Scarisbrick was scrutinising a meek-looking boy with curly black hair and a podgy face.
'Hunt,' replied the boy.
'You hunt! Gah!'
'No, I'm sorry, Professor, I mean, my name is Mr Hunt.'
'So you don't hunt?'
'No. I'm from Sidcup. They don't hunt in the suburbs.'
'Marvellous. Do you row?'
'Er . . .?'
'No? Good.'
The Professor turned his sights on Nancy next. 'Young lady, do you act?'
'I, well, sometimes . . .' Nancy started to explain.
'Sometimes should be never,' ordered Scarisbrick. 'All of you! The way to get your essays written in Oxford is simply to _work_. Don't do anything else. At the Freshers' Fair on Saturday, you will be seduced with invitations to scull, play hockey, write for magazines, act in plays, sing in choirs. Do not do any of it, or you will not have time to write your weekly essay. And whatever you do, DO NOT JOIN THE OXFORD UNION.'*
The students nodded, terrified.
'I have been known,' added the Professor, 'to Send Down students who arrive at my tutorials without an essay.'
Crumpets, thought Ursula. Being expelled for missing an essay seemed a harsh punishment, but the Professor clearly meant it. She vowed to herself that she would write her essay every week, on time, come what may, if that was what it took to stay here.
'Excuse me for asking, Dr Scarisbrick,' interjected Nancy politely.
'It's _Professor_ Scarisbrick,' he said crossly.
'Jeez, I'm sorry, Prof, but does that rule about being Sent Down apply to the American Year Abroad students?'
' _Professor_ Scarisbrick, please, Miss America. What is the point of having a title if no one uses it? To answer your query, yes, the rule includes _particularly_ the Year Abroad students, who have tended, in my forty-six-year tenure, to be statistically MOST likely to appear at a tutorial without an essay and are therefore MOST frequently Sent Down. Does that answer your question?'
Professor Scarisbrick did not appear to be expecting a response beyond a 'yes' but he got one anyway.
'So how does a student get an extension?' asked Nancy.
'Extension?' guffawed the Professor. 'There are no extensions in Oxford.'
' _No extensions!_ ' Nancy was horrified.
'You've hit the proverbial on the proverbial,' said Scarisbrick.
'Huh?'
'Read some bloody P. G. Wodehouse before your tutorial with me, my girl,' snorted Scarisbrick, before collapsing back onto the _chaise-longue_.
'Most informative, Professor Scarisbrick, thank you,' said Dr Dave. 'Now, let's get down to logistics. Tutorial times. Miss Flowerbutton?'
'Yes,' said Ursula.
'I'm afraid you've drawn the short straw. You're my first tutorial. I'll see you here at nine on Monday morning.'
'Excellent,' said Ursula, relieved she had been allocated Dr Dave as her first tutor rather than Scarisbrick.
Dr Dave assigned the other students their tutorial times. Half the intake – Moo, Claire and two of the boys – would study the Anglo-Saxon period with Professor Scarisbrick, the rest would take on the Reformation with Dr Dave. Nancy was thrilled to learn she was in his group.
He then added, 'And all of you, if you get to your tutorial and I don't answer when you knock on my door, just come straight into my rooms – you might have to get me out of bed. Whatever you do, please don't waste valuable tutorial time waiting outside like a lemon.'
* Ridiculously fabulous university debating society. Past Presidents include Benazir Bhutto (1977), Hilaire Belloc (1895), and Prime Minister William Gladstone (1830), whose Cabinet desk is in the Union library. _Everyone_ joins.
Sunday, 17 October, 1st week: evening
'There is _no way_ I can go without you.'
It was seven o'clock on Sunday evening, and Nancy had peered round the door of Ursula's room to plead with her to come along to Wentworth Wychwood's 'Opening Jaunt' that night.
'How would I know what to say to an Earl? I have no idea about posh British conversation. _Please_ come with me,' begged Nancy.
'But I haven't been invited,' Ursula protested. For her, as for many English girls brought up with good manners, there was no idea more dreadful, more humiliating, than turning up uninvited to a party from which one might be turned away – even a party to which one had decided one was thrilled not to have been invited. 'He doesn't even know me.'
'He doesn't know me either.'
'But you got an invitation.'
'Look, if Wentworth Wychwood doesn't know you, he won't know that he didn't invite you. Come on, Ursula, it'll be fun . . . if we're together.'
'Sorry, I can't. I promised Claire I'd go to her crosswords thing. Anyway, I don't like Wychwood much.'
'I thought you said you didn't know him,' Nancy said, looking confused.
'I don't.'
'Well, how do you know you don't like him then?'
'He's got bad manners,' said Ursula.
'Listen, no one's saying you have to be _friends_ with the guy. Just keep me company. India says it's going to be a beautiful party.'
Ursula couldn't help but be tempted by the thought of a white-tie party with pink champagne and bonbons. It did sound more exciting than Claire Potter's Crosswords and Ice Cream evening in the JCR. And Ursula did, after all, have Granny's lovely old ball-gown in her closet. If nothing else, she reasoned, the dress deserved an outing.
*
An hour and a half later, Ursula and Nancy hovered expectantly on the threshold of the Old Drawing Room. Between screams of laughter and groans of horror, snatches of gossip floated from the throng at Wentworth Wychwood's Opening Jaunt.
'. . . _Apparently Johnny Soames was so wasted he snogged a porter at Brasenose. When the provost asked him to explain himself, Johnny said he was terribly sorry but he'd mistaken the porter for a public telephone box_ . . .'
'. . . _I'm joining the Pooh Sticks Society. The only commitment you have to make is to show up once a term and toss a stick off Magdalen Bridge while drinking Fortnum's Afternoon Blend. Members are called Poohsers_ . . .'
'. . . _I can't possibly invite her to my twenty-first. She's too terribly Sloane_ . . .'
'. . . _Anyway he was wasted and the Assassins locked him in a Portaloo and rolled it down a hill and into a ditch and left him there all night_ . . .'
' _I'm starting term as I mean to go on – slightly drunk_.'
'This place looks more like Marie Antoinette's powder room than a party venue for two Oxford boys,' exclaimed Nancy, taking in the scene. 'I _love_ it!'
The Old Drawing Room, situated on the first floor of the Georgian wing on the west side of Great Quad, was famously the best undergraduate set in Oxford. Occupying it was a privilege traditionally accorded to the top Christminster Blue. This year, that was Wychwood – who then got to share the set with a room mate of his choice. The main room, a delicious confection of pistachio-painted panelling and swirls of gold leaf, was lit that night by candles and the promised bonbons – pale pink ones – were piled high on glass stands dotted around on side tables. Ursula noticed two doors at the far end of the room – she guessed these led into Wentworth's and his room mate's bedrooms. The view from the tall sash windows was spectacular, the full moon illuminating Great Quad as though it were an old black-and-white photograph.
The only thing that looked slightly out of place was a gloomy oil painting hanging above the elaborate marble fireplace. From a heavy mahogany frame, the stern visage of the college's founder, Thomas Paget, First Marquess of Anglesey, loomed ominously over the crowd.
'This makes the Freshers' Drinks look like a bad frat party,' sighed Nancy blissfully. 'Thank god we're out of Freshers' Week.'
Ursula agreed. 'Freshers' Week' – which actually only meant Thursday, Friday and Saturday – had been a social whirlwind ever since the History Drinks. There had been a formal Matriculation ceremony to enrol the Freshers into the University on Friday morning. This had been followed by an official Freshers' photograph on Great Lawn – at which they had been informed in no uncertain terms that this was the first and last time they would _ever_ set foot on the precious grass of Great Quad. There had been a Bop* on Friday night. A university-wide Freshers' Drinks, held in the gardens of the Oxford Union at eleven o'clock on the following morning, had mainly involved consuming copious amounts of a mysterious bright blue cocktail. The 'week' had culminated in the Freshers' Fair on Saturday, at which Nancy signed up for the drama clubs and Ursula, though nursing a headache, excitedly put her name down on a (long) list of wannabe-student journalists for _Cherwell_ , whose first meeting would be on Monday. Now here she was at the beginning of 1st Week and Ursula's Oxford life was starting in the most unexpected way – at an extraordinarily glamorous party. One that she was not invited to, she remembered suddenly, as she gazed upon the scene in the Old Drawing Room.
'British boys,' murmured Nancy, squeezing Ursula's arm. 'How come they're all so cute?'
'It must be the outfits,' said Ursula.
The clothes – well, Ursula had only seen such style in her grannies' old photo albums. Here in front of her were masses of young men – Oxford men, she reminded herself happily – dressed in 'white tie'. The dress code – which consisted of a high-cut tailcoat, white dress shirt, elaborate gold or even jewelled studs and cufflinks, a pastel-hued silk waistcoat, black trousers, shoes shiny enough to powder your nose in, a stiff white collar and a highly starched white piqué bow tie – made everyone look like a movie star according to Nancy.
'The ratio's at least four boys to one girl,' she declared. 'I predict a high probability of successful Earl-catching here.'
It was true, Ursula noted, that there were very few girls in the room. Most of the Oxford colleges had only seriously started admitting women in the last ten years. They were still only a small proportion of the student body. But then, she thought to herself, an eight-hundred-year-old boys' club was going to take longer than average to modernise.
Ursula was curious about the twenty or so girls she did see. They glittered like exotic birds among the sea of white tie and tails. Their party dresses, each one pouffier, shinier and more extravagant than the last, were made of brightly coloured taffeta, silk, velvet or lace, underpinned by generous sticky-out net underskirts. A few of the most glamorous girls even had thigh-high mini ball-gowns that looked like they might be from Christian Lacroix or Bruce Oldfield, famous designers whose dresses Ursula had seen in copies of _Vogue_. The young women were lavishly bedecked with diamanté bracelets, heavy paste earrings and reams of pearls at the neck. Gone were the swishy pony-tails – in their place was the curled, waved and crimped 'Big Hair' that was so fashionable now. Their eyes were circled with heavy eyeliner and their mouths painted with gleaming lip gloss.
Is my dress too old-fashioned? Ursula wondered to herself. The hand-me-down 1950s number that she had found in a trunk in the attic at Seldom Seen Farm suddenly seemed far less glamorous than it had at home. Her maternal grandmother, Violette Vernon-Hay, formerly a famous society beauty whom Ursula had nicknamed 'Vain Granny', had a few old ball-gowns and pieces of Paris couture in her attics, and had allowed Ursula to choose one gown to take with her to Oxford. (Her paternal grandmother, Jane Flowerbutton, 'Plain Granny', did not lend Ursula outfits. Her wardrobe mainly consisted of boiler suits in which to conduct farming activities.)
The dress was a three-quarter-length, hand-stitched copy of a Dior gown, in now-faded lilac silk grosgrain. The bodice was beautifully boned, and made Ursula's waist look minute. A generous ruffle of soft silk net edged the top of the corset and draped over the shoulders, and the skirt and its stiff petticoats stuck out. A huge velvet bow hooked over the back of the dress, and, thank goodness, covered a small tear that had been amateurishly darned. Even if the dress was old, Ursula loved the way it rustled as she walked.
'I feel super-duper underdressed,' said Nancy.
'No, you look cool,' Ursula reassured her.
Modern, thought Ursula, that's what she looks. Nancy's attire that night consisted of a teensy-weensy, skin-tight mini dress made of ruched, neon-yellow Lycra, fuchsia-pink suede Maud Frizon stilettos and an enormous silver down jacket that she had thrown over the ensemble. Her hair was backcombed and hair-sprayed into her signature towering mound, so fluffy it resembled candy floss, her eyes were boldly lined with cobalt-blue kohl and matching mascara, and her mouth glistened with the favourite Russian Red lipstick. Her tanned, bare legs added a sexy touch.
Ursula suddenly saw Wentworth Wychwood appear from the crowd. He glanced at them curiously. Oh, crumbs, thought Ursula to herself, he's coming over. The embarrassment of being found out as a gatecrasher, even at a party she hadn't wanted to be invited to, was too terrible to contemplate.
As he approached, Ursula could see that the letter 'W _'_ was elaborately engraved on the shiny brass buttons of his tailcoat. Even if he was a terrible snob, so grand that his buttons were initialled, Ursula couldn't deny that he was ridiculously good-looking.
' _You_ must be Lawnmower,' Wentworth said to Nancy, going straight up to her and kissing her on both cheeks. 'India says you're _very_ amusing.'
Ursula stood shyly behind Nancy. Hopefully Wychwood wouldn't notice her.
'Is everyone here really going to refer to me as a piece of gardening equipment?' grumbled Nancy, sounding miffed.
'Most likely. Although I must say, the name doesn't suit you at all – you're far more attractive than any lawnmower I've ever met. Mind you, everyone calls me Wenty, which is not an abbreviation of Wentworth. It's short for Went-Very-Wrong-Somewhere.'
He laughed jovially and then, laying eyes on Ursula, suddenly stopped. Oh, no! she thought. He's realised I'm a gatecrasher. She examined a splinter in the wooden floor.
'Hi. I'm Wenty,' he said finally.
Ursula slowly raised her eyes towards the boy, hoping her face didn't look as flushed as it felt.
To her surprise, he continued, 'You look . . . wonderful.'
'Oh . . . um . . .' Ursula didn't know what to say. At the occasional school dance she had attended while at Swerford's, the few boys present had barely noticed her, let alone paid her a compliment.
'I don't think we've met,' said Wenty.
'We have, actually,' she said curtly, reminding herself how rude he had been at the Shepherd and Woodward counter that day.
Wenty looked at her again, now with a flash of recognition in his eyes. 'Oh, that's it, I remember! I met you at Piggy's cocktail at Annabel's a couple of weeks ago in London.'
'No,' Ursula answered. 'I've never been to Annabel's.'
'You'll have to remind me.'
'You pushed in front of me at the counter at Shepherd and Woodward last week.'
Wenty looked completely blank. 'Did I? Really? I don't remember.'
'I'm sure you don't,' she replied. Wenty was even ruder than she had thought. Not only had he pushed in front of her, he couldn't even remember doing it.
'Look, sorry, okay,' he said, trying to appease her. 'What's your name?'
'I'm Ursula.'
'Ursula who, exactly?'
'Ursula Flowerbutton.'
'Flowerbutton. Lovely name. Unforgettable . . .'
Wenty gazed at Ursula for far longer than was strictly necessary. But she was determined not to be flattered. She imagined he probably called every girl he met 'wonderful' and 'unforgettable'. Finally, he peeled his eyes away from her, saying, 'Lawnmower, Unforgettable, I've already run out of champagne saucers. Come and help me find some more?'
He beckoned the girls to follow him across the landing, already swaying drunkenly as he walked. Nancy tailed him and Ursula followed reluctantly. They soon found themselves in a large, draughty bathroom with bleached wooden floors and a small window overlooking the courtyard at the back of the main quad.
A claw-footed bathtub that appeared to have been in situ since the twenties had been filled to the brim with icy water and at least twenty bottles of pink Bollinger were floating around in it. A tray of dirty glasses was perched on the loo seat and an abandoned bucket of cleaning items – washing-up gloves, J-cloths and the like – sat on the floor next to it.
'Where on earth have our dear washer-uppers got to?' Wenty said, glancing around the bathroom. 'Now . . . hmm, maybe there are some more glasses in here.'
He lurched over to peer inside a bathroom cabinet below the sink, and seemed shocked to find it contained his own toothbrush and toothpaste rather than clean glasses. He sighed, filled the sink with water, and attempted to wash one of the dirty champagne saucers from the tray, using a shrivelled-looking bar of Palmolive soap and a rather grubby face flannel. There was a crunch, an 'Ouch!', and his left hand came out of the washbasin covered in blood.
'Oh, bloody hell, forget it,' Wenty said, hurriedly wrapping his hand in a face towel that had 'W' embroidered in one corner in the same shade of pale blue as his lapels. He cleaned the blood off his hand, chucked the towel on the floor, and somehow found a bandage in the bathroom cabinet.
'Do you mind?' he asked, holding out the bandage to Ursula, who soon found herself wrapping the wound. 'Has anyone ever told you that you do that better than Florence Nightingale?' he said when she had finished.
'Do you flirt like this with everyone,' she replied, 'or just with girls who aren't your girlfriend?'
'I do adore a prude,' countered Wenty.
'I am _not_ a prude!' she protested.
'Okay, guys, chill,' Nancy interrupted. 'Let's get back to the party.'
With Nancy's help, Wenty managed to open a fresh bottle of champagne and slosh it into the used glasses on the tray, saying, 'Don't tell anyone!'
'That's so unhygienic,' Nancy told him. 'Haven't you heard of mono?'
But Wenty had tuned her out. 'Girls, bring more bottles of booze, will you?' he called, heading out of the bathroom.
Ursula and Nancy each grabbed a dripping champagne bottle, wiping them down with a bath towel so as not to spoil their dresses.
'He _really_ likes you,' Nancy whispered excitedly as they followed Wenty back across the landing.
'He's going out with India,' said Ursula in a hushed voice. 'He's only bothering to talk to me because I'm with you.'
'I think you're being mean about someone you hardly know,' said Nancy. 'I dig him.'
'Well, I don't,' said Ursula firmly.
She had decided, though, that she was not going to let her views on her host interfere with her enjoyment of the party. She made an executive decision to forget Wenty's ridiculous comments and have fun. As she passed through the gilded doorway of the Old Drawing Room and into the party, which was now packed, she felt as she imagined Alice must have when she fell down the rabbit hole. A huge smile crept across Ursula's face.
'Oh, sorry, Nancy, do you want to leave your jacket?' asked Wychwood, turning back to the girls.
'Are you kidding me?' she replied. 'This is a _Norma. Kamali. Sleeping. Bag. Coat._ '
Wenty looked blank.
Seeing his confusion, Nancy explained, 'It doesn't come off. It's part of the look. This is the most aspirational jacket in New York.'
'If you aspire to look like an astronaut,' their host replied mischievously. Then he carried on, 'Come and meet everyone – and grab a glass of champagne!'
Sipping glasses of delicious pink bubbles, the girls followed Wenty as he headed towards the centre of the room, carrying the tray. Guests were lounging on sofas or squashed three to a chair. Corks were popping, glasses were being filled, and tipsiness was being induced, all at a startling rate. A group had congregated around a white Steinway grand piano, on top of which a set of mixing decks had been temporarily installed. An olive-skinned boy in a white T-shirt, black jeans and black baseball cap with the word BOY* emblazoned in white letters across the front, was DJ-ing. As Culture Club's 'Do You Really Want to Hurt Me' came to an end, and Madonna's 'Like a Virgin' started up, he grinned at Wenty, making a thumbs-up gesture.
'That's Christian. Music scholar at New,' said Wenty over the music. 'He's far too London-trendy to wear white tie. The other weirdo over there is my good mate Horatio Bentley. Fondly known as The Man In Mauve. He's in rooms above the JCR. Studying Sanskrit. Come and meet him.'
Wenty waved at an eccentric-looking figure propped comfortably against the piano. He was a squat, tubby personage. His red hair was short apart from his fringe, which he wore in a long, asymmetric style he flicked dramatically off his forehead every now and again. Horatio was dressed in a full-length, lilac _djellaba_ , which, despite its voluminous folds, still strained across his considerable tummy, and his feet were clad in red velvet slippers. His neck was encased in a fat choker of tacky fake emeralds.
The girls followed Wenty over. 'Divine party, darling,' cooed Horatio at their host, then proceeded to kiss the young Earl on the lips.
'Horatio, can you stop acting _quite_ so gay, _all_ the time? It's getting boring,' protested Wenty, wiping his mouth on the back of his hand. 'How many times do I have to tell you, I am _not_ a homosexual and I will _never_ shag you. Darling.'
'Al–aaaa–as!' Horatio threw up his hands in a gesture of mock despair before turning his attention to Nancy and Ursula. His beady eyes darted quickly from one girl to the other, scrutinising them mercilessly.
Wenty said, 'Now, let me introduce you to Ursula Flowerbutton.'
'Hi,' said Ursula, smiling.
'And this is Nancy Feingold—'
'I know _all_ about _you_ already,' Horatio interjected.
'You do?' said Nancy, looking rather alarmed.
'I'm the gossip columnist on _Cherwell_. It's my job to know _everything_ about _everyone_. I read about you in _Tatler._ Your reputation as a beautiful American gardens heiress precedes you—'
'It's not actually gardens. It's gardening _tools_ ,' Nancy corrected him.
'Minor detail, darling, minor detail,' declared Horatio.
If he was a columnist on _Cherwell_ , thought Ursula to herself, she should get to know him.
'I signed up for the newspaper at the Freshers' Fair,' said Ursula. 'I'm coming to the meeting on Monday.'
'A girl like you will be a shining beacon of light amongst the blood-sucking cretins who staff it,' Horatio said dramatically. 'It'll be fun watching you fending off the tragic hacks.'
'Er . . . right,' replied Ursula, unnerved.
'I love your look, Horatio,' said Nancy.
'That's very flattering,' he said gratefully. He stroked the plastic gems at his throat. 'I'm channelling Talitha Getty in Morocco—'
'Horatio, cut the crap,' interrupted Wenty. 'Look after these beauties very carefully while I do some waitering.' He disappeared off into the crowd, filling glasses as he went.
'So, I'm curious,' Nancy asked Horatio. 'Who are all these people?'
'This lot?' he replied, gazing around the room. 'They're known as the Champagne Set – supposedly Oxford's most aristocratic, social, talented or beautiful undergraduates. But I think it's much _kinder_ –' Horatio smiled wickedly '– to describe them as immature twits still obsessed with their posh public-school peer groups.'
'Ouch!' gasped Nancy. 'That's mean.'
'But true,' said Horatio. 'My columns are notorious – for their honesty. If that means sometimes describing my subjects in the worst possible light, so be it.'
Just then, Ursula spotted a suave-looking boy crossing the room towards them. She had never seen such sophistication. His polished dark skin gleamed like a black diamond. A large gold signet ring on his left little finger flashed in the candlelight, and his white tie looked absolutely correct, as though it had been pressed and starched to within an inch of its life. He flashed a glossy smile at Horatio as he approached.
'Who's he?' Ursula asked, trying to sound super-casual.
'Eghosa Kolokoli. Wenty's room mate. He's from Nairobi. Very snappy dresser. But don't be taken in,' Horatio warned. 'Eg isn't _quite_ what he seems . . .'
But his warning faded into the din of the party. Ursula was mesmerised by Eg, who had suddenly transitioned to an expertly executed Michael Jackson-style moonwalk to reach Horatio and the girls.
'Eg, you're such a cliché of . . . _yourself_ ,' sighed Horatio wearily as Eg slapped him hello on the back. 'Have you met Nancy and Ursula?'
'Hello,' he said, turning to the girls. 'I'm Eghosa.' He had a deep voice and a very proper English accent that sounded like something out of the 1950s. He then turned to Ursula, took her hand and said, 'Dance?'
She couldn't quite believe that a boy like this could possibly want her as his dance partner. She couldn't say yes. Her dancing was far too amateurish. She shook her head. 'I'm okay, thanks,' she said. 'I'm sure Nancy would, though.'
'Great,' Nancy said, taking off her duvet jacket and flinging it on a sofa.
With that, Eg took Nancy's hand and proceeded to whirl her around the room. Thank goodness that's not me, Ursula told herself, as she watched the pair break into Ceroc. Luckily Nancy seemed to be some kind of Ginger Rogers. She and Eg twirled into any space they could find.
'Ah, Prince Shuffling Knickers doing his shuffling!' pronounced Horatio, pointing out Otto, who was dancing furiously by himself. 'Otto!' he called. 'Come over here, you lonely saddo!'
Otto stopped suddenly. When he turned and saw his audience he looked rather embarrassed, but still dashed over to Ursula, grabbed her hand and brushed his lips across it, Euro-style.
'India, _Liebling_!' he said, entwining his fingers in hers.
Ursula snatched her hand back. 'I'm Ursula!' How could he possibly mistake me for India? she wondered.
Otto rubbed his eyes and peered at her closely.
'Now I see. You are not India. You are Ursula. I am not wearing my glasses tonight. I never do for parties. Can't see a thing. Sincere apologies,' he said, bowing almost to the floor as he did so.
'Shuffling Pants, you are such a ponce,' Horatio admonished him. Ursula rather agreed. Otto's good manners, she was starting to think, were extreme, even for a minor Austrian princeling.
'Shut up, Horatio,' retorted Otto. 'A prince always bows when he apologises.'
'Otto, all this princeling twaddle is nonsense. No one cares that you rule over a pathetic pine forest on a craggy mountain somewhere in Carinthia—'
'Oh, but I do,' Nancy butted in, having just returned, breathless, from her dance. 'American girls just adore a prince, even one who lives someplace no one's _ever_ heard of and no one's _ever_ going to go.' She gave Otto a reassuring squeeze.
'Ooooh! I've spotted a gossip item,' said Horatio. He moved off towards a couple snogging violently on a window seat, half draped in curtains. 'See you all later.'
'Thank you for sticking up for me, Nancy. You're a sweet girl,' said Otto, bowing yet again.
'Oh my god, Otto,' blurted Nancy, 'you are so crazy posh.'
'Not as posh as he is, sadly,' he sighed, glancing wistfully at an extraordinarily beautiful boy standing by the fireplace.
' _Who_ is _he_?' asked Nancy, gazing at the gorgeous boy, who was immaculately turned out in a spanking-new tailcoat with black satin lapels. His feet were clad in black velvet evening slippers adorned with large black satin bows, while his white tie showed off his tanned face, brown eyes and dark, slicked-back hair to perfection. Lapis and diamond cufflinks and studs adorned his dress shirt. If Wenty was a cucumber sandwich, Ursula thought, then this specimen – well, he was absolutely a Bendick's Bittermint,* Ursula's favourite chocolate.
The Bendick's Bittermint was talking to a boy who was his exact opposite: a portly, ruddy-faced young man dressed in grubby-looking tartan trousers and a threadbare, claret-coloured velvet smoking jacket. He reminded Ursula of a worn-out teddy bear.
'The next Duke of Dudley,' said Otto.
Nancy's eyes lit up.
'Is that as good as an Earl?' she asked.
'Better,' replied Otto. 'Would you like me to introduce you?'
'No, I think I have a better strategy,' said Nancy, a determined look on her face.
'By the way, don't mention the Duke thing,' warned Otto. 'He's very embarrassed about it.'
'Watch me not mention it,' she said, dashing off towards the fireplace.
It only took a few seconds for Nancy to _faux_ -trip and accidentally-on-purpose throw her third glass of champagne all over the Next Duke of Dudley and his shabbily dressed friend. Shocked expressions turned to laughter as Nancy seductively patted the boys down with a napkin. Ursula looked on as Nancy flirted and joked with the Next Duke as though they were old friends already.
'Oh!' exclaimed Otto, observing the scene. It seemed he was about to say something, but then abruptly stopped himself. A sly smile spread across his face.
'What?' said Ursula. 'What is it?'
But Otto didn't answer. He had already moved on to his next social opportunity, and was waving at a stunning-looking girl, who stood chatting nonchalantly with Christian. Her beautiful, narrow face was cloaked by two curtains of long dark glossy hair, which came almost to her waist. She was wearing a starched man's white dress shirt that reached barely to mid-thigh, black tights, stilettos and not much else. An undone white bow tie hung around her neck.
'I like her take on the dress code,' said Ursula.
'Isobel Floyd. She's a scene-maker, she's always at the centre of everything. Her father's the Home Secretary. She was brought up in Belgravia. Everyone says she was the cleverest girl at St Paul's. She's dating Dom Littleton. He's the top theatre director among the students,' explained Otto, heading in Isobel's direction. 'Come on, I'll introduce you.'
'Okay,' said Ursula. She followed Otto, feeling somewhat intimidated after the write-up Otto had given this girl.
'I see you've taken the dress code literally,' Otto told Isobel as he went up to her. He kissed her on both cheeks. 'Very groovy.'
'It's fun, no?' she drawled. 'I can't face ball-gowns this early in the term. I've been barefoot in Koh Samui all summer.' She then drew back her curtain of hair and peered at Ursula, who had hung back shyly behind Otto. 'And you are?'
'Forgive me,' said Otto apologetically. 'Introductions. Ursula Flowerbutton, this is Isobel Floyd – the coolest girl in Oxford.'
'You're so embarrassing, Otto,' groaned Isobel. She then smiled smugly, as though to intimate that, yes, she probably was.
'Where's Dom? Isn't he coming?' asked Otto.
Isobel rolled her eyes, peeved. 'He said he'd meet me here ages ago. But he's late as usual. God knows where he is—'
She stopped talking, her gaze now fixed on the doorway, as the very late, rather dramatic arrival of India Brattenbury captured the attention of the entire room. Her look exuded glitzy modernity. Her long white silk-satin dress, beaded from head to toe with tiny seed pearls, was cut close to her body and puddled into a pool of glistening fabric, which kissed the floor. In sharp contrast, her hair had been teased into a messed-up, punky bob and she had perched an extraordinary pearl tiara with a diamond star at its centre on her head. It was probably a priceless family heirloom, Ursula guessed.
India was accompanied by a pony-tailed boy wearing torn jeans, a Rolling Stones T-shirt under a tailcoat, and pale blue John Lennon-style sunglasses. He zig-zagged through the throng, arm-in-arm with her. Meanwhile Horatio returned and joined Ursula and Otto, kissing Isobel hello.
'My best friend,' huffed Isobel, staring unhappily at India and her pony-tailed companion. 'And my boyfriend.'
'What _could_ they be doing together, I wonder?' said Horatio.
'Stop stirring, you prat,' spat Isobel. 'Dom's directing India in the play this term. They were probably . . . I don't know . . . rehearsing.'
'Well, she looks _rrrrr_ -adiant!' trilled Horatio. Ursula could see that he was enjoying winding up Isobel about India arriving with her boyfriend. 'Never have I seen a girl in pearls looking quite so groovilicious.'
'God, I can't believe she's got that Chanel bag,' said Isobel, enviously eyeing a miniature white-satin quilted handbag hanging by a chunky gold chain from India's shoulder.
'Now, now, Isobel, try and be nice,' tutted Horatio.
' _Liebling_ , over here,' interrupted Otto, waving enthusiastically at India.
'Hi darlings!' and air kisses were exchanged as India and Dom joined the group.
'Baby,' said Isobel to Dom before grabbing him by both shoulders and snogging him ravenously. No one seemed to be in the least bit embarrassed by the sight of her tongue darting its way in and out of Dom's mouth, except for Ursula.
'Don't wait, Dom, tell Isobel _now_ ,' said India, interrupting the PDA with a tap on Dom's shoulder. 'Good friends need to be . . . honest.'
'About what?' said Isobel, drawing away from her boyfriend.
Horatio nudged Ursula and chuckled, 'Oh, goodie! I do find a scream-up among the drama hacks terribly amusing.'
Meanwhile, Isobel was looking at India suspiciously.
'Shouldn't we tell Henry Forsyth first?' said Dom to India. 'I mean it _was_ his role—'
He stopped, seeing that a Goth-looking boy, with straggly, shoulder-length black hair and a month's worth of beard growth, had strolled up. In his tailcoat he looked like a teenage Dracula. Despite being only about five foot four tall, he had presence.
'The Rodent arriveth,' whispered Horatio mischievously to Ursula. 'Let the spat commence.'
'The Rodent?' repeated Ursula.
'My pet name for Henry Forsyth. Fitting, isn't it?'
'No!' But Ursula couldn't help giggling.
On tiptoes, Henry Forsyth craned his neck to peck Isobel on both cheeks.
'Henry, something's going on,' she hissed at the bearded boy. She couldn't hide the agitation in her voice. 'Something about your role.'
He stood back, regarded Isobel's distressed face, and said, somewhat melodramatically:
' _Isobel's angry; and the heavens themselves_
_Do strike at my injustice_.'
Otto looked quizzically at Henry.
' _The Winter's Tale_. I've been speaking in semi-intermittent iambic pentameter since the summer,' he explained. 'The Method. Preparation for taking on the Prince of Denmark.'
'Pretentious twat,' whispered Horatio to Ursula. She stifled a laugh.
'Please, Dom. Tell him.' India nudged Dom so hard he almost fell onto Henry.
'Dom?' was all Henry said.
'Er, well, yeah, Henry mate,' replied Dom nervously. 'Look, I want to be mega-radical with the Michaelmas production of _Hamlet_. I mean, yeah, any old public-school toff can play Hamlet. But Ophelia is a much bigger challenge for an Old Etonian like you. It will be so modern. India's completely right about that.'
'India! What the . . .?' Henry gagged. 'Am I to conclude that she is directing the director? Ugh. How pathetic.'
'Haven't you heard about gender-blind casting? It's so _moderne_ ,' India interjected. She smiled charmingly at Henry. 'I mean, you as Hamlet – Snoresville. You as Ophelia? Now _that_ will make waves.'
Henry Forsyth seethed. Gosh, he would have made a good Hamlet, Ursula thought. Finally, he announced, 'I have not invited every agent in London – contacts I have been building up in the theatre world since I played Julius Caesar in F Block* at school – to watch me nancy about in a dress in three weeks' time. _I_ am playing Hamlet, as we agreed last term.'
'Henry darling, listen, I _adore_ you. You're a _star,_ absolutely,' India cooed. 'But we've got to be relevant, otherwise everyone will carry on thinking Oxford drama is a stuffy old-boys' club. The problem is, everyone's seen Hamlet played by a man, it's not exactly _original_. So that's why, when Dom and I . . . _analysed_ the play . . . together a couple of days ago, he switched the casting and . . . suggested that _I_ play Hamlet now.'
'You?' Henry exploded.
'It's the in thing now at RADA to reverse the sexes in the classic roles,' she replied coolly.
'Sounds idiotic.'
By this point, some of the other guests at the party had realised that India, Dom and Henry were putting on a not-to-be-missed show, and a circle of spectators was forming around them.
'It's going to be thrilling for everyone,' India continued, 'particularly you, Isobel.'
'Really,' she deadpanned, looking about as thrilled as Anne Boleyn heading for the executioner's block.
'Isobel, sweetheart, you are my bestest friend _in the world._ I honestly couldn't wish for a better understudy, and you will _definitely_ get to play Hamlet as I'm _obviously_ going to be "ill" on the Saturday night of Third Week. It's my shooting party at Brattenbury Tower that weekend. So you get to be Hamlet.'
'Shooting weekend? What shooting weekend?' muttered Otto crossly to Ursula and Horatio. 'She hasn't invited _me_. Outrageous! After she came to our wild-boar shoot at the _Schloss_ and drank all our peach Schnapps and didn't leave any tips for the staff.'
'Are you surprised?' Horatio asked Otto, eyebrows raised. 'Rich girls are always _by far_ the meanest.'
Meanwhile, Ursula couldn't help but notice a radical change in Isobel's expression. A smug smile was suddenly dancing around her lips; she now resembled Marie Antoinette confronted by a huge slice of cake.
'. . . Oh, well,' she was saying. 'I mean, I agree, we have to modernise. Oxford can't carry on putting on _The Importance of Being Earnest_ every two minutes. It's so bourgeois.'
'Mate,' Dom said to his girlfriend, 'I'm so glad you're on side. Come on, Henry, how about it? How about Ophelia?'
Henry stared him down, unmoved. Finally he hissed, 'You've forgotten one tiny little detail, Dom. India is a bit of posh totty who couldn't act her way out of a local village hall.'
There were a few gasps from the group that had assembled and then a deathly silence fell, with all eyes on India. She smiled at Henry, rather seductively, and then, as if by magic, her face transformed. The bloom drained from her cheeks and her skin took on a ghastly pallor. Slowly, she raised one arm and pointed behind Henry's head, as though terrified of something. She took a deep breath and began, in a whisper:
_'Angels and ministers of grace defend us!_
_Be thou a spirit of health or goblin damned,_
_Bring with thee airs from heaven, or blasts from hell . . .'_
She was pointing, Ursula and the rest of the room now realised, at the gloomy portrait of Thomas Paget hanging over the fireplace. Her voice trembled, as though the ghost of Hamlet's father were standing before her.
_'Be thy intents wicked or charitable,_
_Thou comest in such a questionable shape_
_That I will speak to thee . . .'_
'I think this one can act her way out of a village hall and all the way to Broadway,' murmured Nancy, who had sidled up to Otto, Ursula and Horatio during India's speech, still flushed after her run-in with Next Duke.
'She _is_ Hamlet,' said Ursula, amazed.
At this point, India fell to her knees beneath the painting, and an utterly convincing tear rolled down her cheek as she finally ended Hamlet's soliloquy.
There was an astonished silence, then Wenty, rushing over, cried out, 'Bravo! Bravo!' and the room erupted into applause. India got up and smiled seductively at Wenty, who kissed her enthusiastically on the lips. Meanwhile, with a look of cold envy in his eyes, Henry Forsyth looked hard at India and said, 'You deserve _Macbeth_.'
India looked shocked. Henry stormed across the room, the champagne in his glass splish-sploshing angrily from side to side. In a far corner, a little group soon huddled round him to commiserate.
'Spoilsport,' concluded Lady India, regaining her composure. 'He's easily pretty enough to play Ophelia.'
*
Later that night, as Ursula, Horatio and Nancy hung out on the huge sofa beside the fireplace, the talk was of India.
' _Quel_ performance,' said Horatio drolly. 'Much better than the deathly drawing-room drivel Oxford University Dramatic Society usually bores everyone to tears with.'
'She's really a brilliant actress,' Nancy said. 'I'm surprised she hasn't got an agent already.'
'Oh, but she has, darling, she has,' replied Horatio. 'The same one as Rupert Everett, apparently.'
He got up to go.
'Well, toodle-pip, all,' he said. 'It's been fun. But I've got about six more drinks parties to get to tonight.'
''Bye!' said Nancy.
'See you at the _Cherwell_ meeting,' added Ursula.
As soon as Horatio had departed, Wenty took his place in an armchair opposite the girls and India perched on his lap, draping her arms languorously around his neck. Occasionally she stroked his cheek, her immaculately lacquered nails, painted as dark as ox blood, lingering around his lips. Ursula squirmed with embarrassment when Wenty started kissing India's fingers.
She was starting to feel her head swim. The drama of the party had exhausted her, and her 9 a.m. tutorial, followed by the all-important _Cherwell_ meeting, for which, it was dawning on her, she did not have one single idea for an interesting article, or even an idea for an uninteresting one, loomed at the back of her mind. Meanwhile, Isobel, Wenty, India, Nancy and everyone else were carrying on drinking as though they would be lying in until noon tomorrow.
'Enjoying your first Oxford party?' India asked Nancy.
'It's awesome,' she replied. 'Actually, I've even fallen in lust already.'
'What?' cried India. 'Who with?'
'The Next Duke of Dudley.'
India winced. 'Really?' she said, looking amazed. 'You do know the Dudley country place doesn't have central heating yet?'
Nancy's face lit up. 'He has a country house? With no central heating? How . . . eccentric! How romantic!' she said. 'On top of that, he's the hottest boy I've ever laid eyes on. Makes Rob Lowe look homely.'
'You _must_ be lovestruck,' laughed Wenty.
Ursula's room beckoned. She stood up, put down her glass and turned to go.
'Sneaking out?' said Wenty. He jumped up off the sofa and grabbed her by the wrist. 'Don't leave just yet, Unforgettable.'
Before she could say a word, India, fuming, had torn his hand from Ursula's wrist. 'Wenty. That's. _It_.'
With that, India released his hand, grabbed a glass of champagne and dashed towards the door of the old drawing room. Clutching a fresh bottle, Wenty followed in hot pursuit, calling out, 'My darling! Don't go! Champagne?'
'My star,' yelled Dom, chasing after them. 'Wait!'
'My boyfriend,' shrieked Isobel, scooting after Dom.
'My part!' raged Henry Forsyth, stamping furiously after everyone.
'Should we go after them too?' asked Nancy, wobbling dizzily as she got up from the sofa. She was very, very drunk.
'I think the party's over,' said Ursula. 'Let's go to our staircase.'
'What about stopping by Claire Potter's Crosswords and Ice Cream Society meeting?' giggled Nancy.
Oh, no, thought Ursula guiltily. They'd forgotten about poor Claire's club. 'I think it'll be finished by now,' she said, looking at her watch. It was a few minutes before midnight. 'Let's go.'
But Nancy had collapsed back on the sofa. Ursula tried to pull her up by her arm, but Nancy just giggled and asked for more champagne.
'Need a hand?' Eg had appeared out of nowhere.
'I think so!' said Ursula gratefully.
With his help, she guided Nancy through the party, grabbing her duvet coat _en route_. As they reached the door, Ursula noticed a boy lying across the threshold. He appeared to have fallen asleep, his head propped up against the doorframe, but as Ursula and Nancy tried to step over him, he jerked awake and grabbed at Ursula's ankle.
'Hey, gorgeousness, sleep with me,' he slurred.
'What? Ugh! No!' yelped Ursula, yanking her leg away from him.
'How about you, beautiful? Sleep with me?' the boy said to Nancy.
'That sounds _so_ romantic in your British accent,' she said woozily, stopping with one leg on either side of him.
'Just ignore Duncan,' said Eg, 'he propositions everyone when he's drunk. Even me.'
But Nancy was oblivious. She slumped down on top of the boy and started French-kissing him. Ursula turned to Eg, shocked.
'I think we're in for a long wait,' he said, shaking his head and smiling. Ursula couldn't help grinning back at him, despite her friend's predicament.
'Nancy . . . Nancy!' she said. 'What about the Next Duke of Dudley?'
Nancy managed to break off her snog for long enough to utter, 'Who?'
* Bops = puke-filled college discos.
* Mega-groovy King's Road boutique BOY was frequented by club kids and pop stars, including Madonna. It was so groovy in there that Billy Idol worked the till.
* Bendick's Bittermints, white fondant mints encased in dark chocolate, were purchased in yard-long boxes at Harrods. Pre-Diptique candles, they were the only acceptable pressie at grand English house parties.
* Year groups at Eton College are referred to by letter, from F to B. F Block is for the youngest boys, aged around thirteen.
Monday, 18 October, 1st week: morning
The possibility of being labelled a lemon by Dr Dave was, Ursula concluded, as she hovered uncertainly on the icy landing outside his rooms exactly ten minutes before the scheduled start of her tutorial, far more daunting than that of seeing her tutor in his pyjamas. If he did not answer the door at nine, she was determined, as per his instructions, to go straight inside, regardless of any pyjama-related embarrassment that might ensue.
The minutes ticked by painfully slowly. Ursula let out a yawn so long and rumbling that, had anyone heard it, they might have mistaken it for a wolf's howl. The truth was, she was shattered after Wenty's party last night. Despite her vow to be in bed by midnight, she had only been able to prise Nancy from Duncan's drunken clutches at around 1 a.m. Thank goodness Eg had hung around, chatting with her and keeping her entertained. By the time Ursula finally fell asleep – a state further delayed, she surmised, by the combination of the bone-chilling cold in her turret and the board-like sensation of her mattress – it was almost two in the morning.
God, it was cold on Dr Dave's landing. Even in one of her warm kilts, a duffel coat and her new college scarf, Ursula felt goose pimples appearing under her clothes. She checked her watch. Still only eight minutes to nine! She decided she would occupy herself by opening the surprisingly large number of envelopes she had found in her pigeonhole that morning. The first missive, a grand invitation printed on stiff cream card, was unexpected. It was as smart as the kind of thing a countess would send out for her debutante daughter's Coming Out ball, and read:
* * *
_The Perquisitors_
_request the pleasure of your company for_
_Buck's Fizz_
_Monday, First Week_
_at_
_The Monks' Undercroft, Magdalen College_
_R.S.V.P. The Hon. Rupert Bingham_ | _8 p.m._
---|---
_Magdalen_ | _Smart_
* * *
Who were the 'Perquisitors'? wondered Ursula. She didn't know anyone called the Hon. Rupert Bingham. She must have received the invitation by mistake. But then she noticed that her name was written in dark blue ink on the top right-hand corner of the invitation. It _was_ meant for her. Ursula wasn't sure whether to feel excited or creeped out. Were the Perquisitors one of Oxford's notorious secret societies? And how did they know her? It didn't matter, she thought, she couldn't possibly go anyway. It would be most odd to go to a party you'd been invited to by someone you'd never met. She'd had her ball-gown-wearing moment at Wenty's party. It had been wonderful, but tonight would be books and Ovaltine.
A feeling of increasing confusion enveloped Ursula as she opened the other envelopes. It seemed that she'd been invited to a different party – by boys she didn't know – almost every night that week. And not just any old party – these were strictly black-tie affairs. Wednesday night was to be reserved for a party being thrown by a society called the Gridiron Club.* Meanwhile four boys named Kit, Angus, Jamie and Tony had invited Ursula to join them on Friday for cocktails at a place called Vincent's. Who on earth were Kit, Angus, Jamie and Tony? And what was 'Vincent's'?
Just then, Ursula heard footsteps coming from the floor above her and looked round to see Horatio wandering down the stairs, a vision in mauves. He was dressed in a lavender-coloured velvet jacket, which was thrown over a violet gingham shirt and purple Prince of Wales check tweed trousers.
'Morning, Horatio,' Ursula called out.
' _Ciao_ , petal,' he replied, coming up to her and kissing her once on each cheek.
He tugged at her scarf. 'A tip. College scarves are _terribly_ passé. Burn it at once. They're for Japanese tourists.'
'Really?' said Ursula, contemplating the tragedy of burning such an item. Even if it was horribly gauche, she thought to herself, it was far too expensive to discard. And it was the cosiest thing in the world.
Horatio suddenly noticed the invitations in her hand. 'Good haul!' he congratulated her. 'Looks like you'll be in a ball-gown every night this week.'
'Horatio, I can't go to all these parties. I'll have far too much studying to do after my tutorial. In any case, I don't know any of these people, and I've no idea how they know me.'
'Let me explain. Remember the Freshers' photograph you did on Friday on Great Lawn?'
'What's that got to do with it?' asked Ursula.
'On the Friday night of every Noughth Week, as soon as the Freshers' photographs have been developed, Dave Brooks's dark room on Market Street is crammed with Hooray Henrys† going through each college photo looking for pretty Freshers. They circle the best-looking ones, get the girl's name off the photo and send them invitations to their societies and parties.'
'A beauty contest?' said Ursula, affronted.
'In polite British society, it's known as a Cattle Market!' hooted Horatio, bellowing with laughter.
'But what about the girls who aren't picked? They must feel so left out,' she said.
'It's very depressing for the Plain Janes,' he agreed, exhaling a long sigh. 'The selection process is so sexist it's probably illegal. But no one cares about all that crap. The parties are _great_ fun. You must go to the Perquisitors' tonight.'
Ursula couldn't help but feel her interest piqued. Despite the Hooray Henrys' truly wicked selection methods, she was secretly flattered to have been 'picked' by them. It was nice to be considered one of the pretty girls. Still, she told herself, she wouldn't go to the parties. How could you call yourself a modern girl if you didn't stand up to chauvinists? Ursula was going to stand up for feminism. She'd stand up for the Plain Janes. Perhaps she could write an article about that for _Cherwell_ – the inalienable right of the Plain Jane to attend _any_ Hooray Henry party she wished.
Ursula suddenly felt horrible about how she'd treated Claire Potter. She'd cruelly dismissed her as a potential friend because she was so dowdy. She'd written her off as a Plain Jane, just as cavalierly as the Hoorays would. Ursula vowed to make an effort with Claire. She was probably fascinating, regardless of her sad short hair.
'I'm definitely not going,' she told Horatio. 'I want to do some reading tonight.'
'Well, I'll be there, so if you change your mind – come!' he exclaimed. 'Don't forget, it's the _Cherwell_ meeting at twelve today. I'm going down to the offices now to sort everything out. Toodle-oo!'
As the clock on the gate tower chimed nine, Ursula stuffed the invitation cards into her satchel and knocked on Dr Dave's thick oak door.
No answer.
Two minutes passed, then three, four. No sign of Dr Dave. Not being a lemon, Ursula reminded herself, was crucial to making a good impression. The heavy door groaned as she gingerly pushed it open. Inside, Dr Dave's rooms were quiet as the grave, and the only light was a white sliver of sharp morning sunshine that curled around one of the curtains. Ursula could vaguely make out the familiar outlines of Dr Dave's armchairs, sofa and the _chaise-longue_ opposite. The silhouette of the Giant Argus loomed ominously in the half darkness.
'Oh!' Ursula gasped, suddenly noticing a large, person-shaped mound on the _chaise-longue_. Surely Dr Dave hadn't slept there? How embarrassing, she thought.
Perhaps if I noisily open the curtains, she decided, he'll wake up of his own accord. That would avoid an awkward moment. The thought of disturbing the author of _From Constantinople to Jerusalem_ from a snooze on his own sofa was simply too dreadful. She went over to the window and flapped the curtains unnecessarily wildly as she drew them back. Light flooded the room, picking up the gleaming red lacquer of the walls and glinting here and there off Dr Dave's Eastern trinkets. Ursula could now see that a thick Turkish blanket was pulled up over the sleeping figure.
But the mound was still motionless.
'Dr Dave?' she said brightly. 'Good morning?'
Silence.
Ursula sighed. In the daylight she could see that a pool of fabric was spilling onto the floor from beneath the Turkish blanket at one end of the _chaise-longue_. As she looked more closely, she realised that it was white satin, beaded with pearls. There was no mistaking the fabric or the gown – it had to be the extraordinary dress India had been wearing at Wenty's party last night. But why was _she_ asleep _here_ , in Dr Dave's rooms?
Ursula tiptoed closer to the sleeping girl. It must have been freezing in here last night – India had pulled the Turkish blanket up so high over her body that only the top part of the back of her head was visible. At the other end of the _chaise-longue_ , a hand was peeking out from beneath the blanket and grazing the floor. It appeared to be clutching a champagne saucer.
Ursula cleared her throat. 'Erm . . . hello?'
Nothing.
'Excuse me, Lady India?'
The girl didn't stir. Ursula stole closer to her. There wasn't a sound, a sigh, a movement coming from her.
Ursula could hardly breathe as she rushed around to the far side of the _chaise-longue._ Now she was behind it, she could see India's face. The Turkish blanket was drawn right up to her chin. Her eyes were closed, as though she were taking a nap. But this was not a peaceful sleep. India's cheeks were colourless, her complexion dulled to the chalky pallor of stale pastry.
At that moment Ursula was enveloped by fear as the sickening thought dawned that there might be no life in the girl before her. India's once-crimson lips, still in their customary haughty pout, were now the bluish hue of a nasty bruise.
_Stop being a lemon_ , Ursula chided herself. There was no time to waste. She must get help, and fast. As dire as the situation seemed, perhaps India could be revived. Ursula turned and rapped as violently as she could on Dr Dave's bedroom door, shouting his name. He must be in there, she thought, pulling the door open. But she was greeted by the sight of an empty, unmade bed. On the side table next to it was the Chanel bag India had been carrying last night. Ursula dashed out of the little bedroom and threw open the door to the bathroom. Empty. The room smelled noticeably lemony and fresh. She spotted a large bottle of Eau Sauvage cologne on the side of the basin. The top was off.
Ursula rushed from Dr Dave's rooms and out onto the landing. The JCR! she thought, seeing the door was ajar, and sped into the room opposite. Immediately she saw that there was someone in an armchair at the far end. She rushed over.
There was Otto Schuffenecker, sitting bolt upright in a worn wing-backed chair, still in his tails and white tie from the night before. He was fast asleep. Two champagne glasses and an empty bottle had been kicked over and were lying on the floor next to him.
'Otto!' Ursula yelled desperately.
He didn't stir. Please, god, don't let Otto be dead, she prayed. That would be too Agatha Christie for words. She shook the boy by his left arm, as violently as she dared, shouting, 'Otto. Otto! Wake up!'
'Ah, India darling, _mein Liebling_ , you are soooooggghhh sveee-ee-et,' he grunted, grabbing Ursula's waist and burying his head in her kilt.
'Ugh! Otto, GET OFF,' she screamed, whacking him with her notepad.
'Aggghh!' squealed Otto. 'Wenty! More champagne.'
'Otto, Wenty's party finished hours ago.'
Otto screwed up his eyes and squinted at her. 'Where's India?' He looked around the room. 'Where am I?'
'You're in the JCR.'
'NO!' He shuddered. 'How uncool. Please, never tell anyone you've seen me here, or I'll be sacked from the Gridiron Club . . .'
'Gridiron Club?' Ursula butted in. 'Was it you who—?' But before Otto could answer, she had refocused on the current emergency. 'I think something terrible has happened. It's India. She's . . . unconscious,' Ursula stammered, unable to articulate her worst fear.
'What?' cried Otto, virtually jumping out of his tailcoat. 'But . . . last night . . . we were so, she was so . . .' He seemed terribly confused.
'She was so what?' asked Ursula.
'Sexy,' he said, arching one eyebrow provocatively.
'Otto, what on earth are you talking about? Come with me, quick!'
Ursula grabbed his arm and heaved him from the chair, dragging him towards Dr Dave's rooms. As they left the JCR Ursula noticed a pile of blank crossword puzzles stacked neatly on a table near the windows, as well as a cluster of little round tubs filled with melted ice cream. It looked like only two pots had been eaten. Poor Claire. It didn't look as though the event had been a big success. She'd make it up to Claire for not going. Maybe she'd invite her for a chocolate digestive in her room or something.
'Ah, is so nice to hold your hand,' Otto cooed as he stumbled behind Ursula. 'I think I love you as much as I love India.'
'Otto, you're still drunk,' she reprimanded him, opening Dr Dave's door.
'Look,' she whispered desolately as they reached India. Otto walked round to the back of the _chaise-longue_. When he saw India's face, he finally appeared to sober up.
'Christ!' he said. 'I must feel the pulse on her neck. If there is anything, I will give her the kiss of life . . .'
He gingerly turned the blanket down from Lady India's chin. As he did so, he let out a blood-curdling screech, leaping away from the girl like a grasshopper who'd met up with a five-thousand-volt electric fence.
'What is it?' gasped Ursula, still on the opposite side of the _chaise-longue._ She rushed round it.
Ashen, speechless, Otto pointed at India's neck. The scene before them was ghastly.
India had a bloody slash wound extending from one side of her neck to the other. A crimson river had oozed down her collarbone and onto her décolletage where it had coagulated. The top part of the pearl-encrusted gown was soaked with blood. Ursula felt herself almost freeze with shock. The girl had been attacked with no mercy. A monstrous crime had been committed.
'Oh my god,' Ursula cried. 'No! It can't be! India! India!'
She started shaking the girl, hoping, praying, that she would miraculously take a breath. But India was cold, stiff with rigor, unresponsive.
'Let me try,' said Otto.
'Otto, there's no point. She's gone,' Ursula told him.
'I must try something – anything,' he insisted.
Otto leaned awkwardly over the _chaise-longue_ , put his lips on India's mouth and blew air into the girl's lungs as hard as he could, over and over, but to no avail. Eventually, he sat on the floor, head in hands. It was hopeless. India was gone.
Ursula sat down opposite him, took his hand and felt him squeeze it hard. The only thing to do now was to get help.
'Do you think you can manage to fetch a porter?' Ursula asked him gently.
'Er . . .' But Otto couldn't say more – his face turned a vile shade of green and he vomited violently on the Persian rug _._
'Ooops!' he said, looking at Ursula, mortified.
'Poor Otto,' she said, patting his back comfortingly. 'Go and get Deddington. I'll wait here.'
He nodded numbly, but got up and rushed down the stairs. When the thudding of his feet finally faded and Ursula was left alone with India's body, she felt totally dumbfounded. Poor India – killed. But by _whom_? _Why?_ Was there a murderer on the loose in Christminster? Ursula felt her spine crawl with fear.
The seconds seemed to tick by terribly slowly. Ursula carefully replaced the blanket over India's body, making sure to cover the terrible neck wound, as if restoring some semblance of peace to the dead girl. It was then that she noticed that the champagne saucer, still clutched in India's stiff left hand, was broken. Ursula crouched down on the floor to get a closer look: the stem of the glass was intact but half of the saucer part was missing.
Ursula's observations were suddenly interrupted by the sound of footsteps and a voice. She quickly jumped up from the floor and looked around to see Dr Dave breezily entering the room, holding a stack of books. He shut the door behind him and acknowledged Ursula, nodding his head to her. He then opened the book on the top of the pile and quoted from it dramatically.
'" _Blood defileth the land, and the land cannot be cleansed of the blood that is shed therein, but by the blood of him that shed it_ ,"' he read aloud.
'Blood?' she interjected, desperate to explain about the blood situation on the _chaise-longue_. 'Dr Dave—'
But he continued: "' _Blood indeed! Blood for blood!_ " That was how Ludlow – a Fifth Monarchy fanatic – saw the world. Now, your essay question for the week, Miss Flowerbutton . . .'
'Dr Dave, erm, I think, well – that's India Brattenbury,' said Ursula, gesturing at the body under the blanket.
'A not unusual occurrence,' he replied with a sly grin, and prodded India's leg. Ignoring the fact that she didn't respond, Dr Dave motioned to Ursula to sit in an armchair, impatient to begin the tutorial.
'But—' she started.
'Just listen, please, Miss Flowerbutton,' he said. 'Your moment to speak will come. Now, the Scottish Covenanters. After the Reformation, there is little doubt—'
Dr Dave stopped abruptly and wrinkled his nose. Like a dog following a scent, he soon found himself staring at Otto's pile of puke.
'If India parked that custard on my Persian rug . . .'
A rap at the door interrupted his flow. Dr Dave sighed. His eyes rolled to the top of their sockets and seemed to stick there, such was his annoyance. 'Come in,' he called.
Ursula jumped up from the armchair to see Deddington entering the room.
'Deddington? Something urgent? I'm in a tutorial with Miss Flowerbutton.'
'Yes, very sorry to interrupt you, Dr Erskine, but it has been brought to my attention that Lady India Brattenbury may be on your _chaise-longue_.'
'Standard procedure,' replied Dr Dave. 'Most mornings there's a student asleep on my _chaise-longue_. They're usually able to snooze through the first two tutorials of the morning if they got really drunk the night before. Anyway, since you've interrupted me, Deddington, can you make yourself useful and get my scout up here early to remove that?'
Dr Dave pointed at the yellowing, stinking pile of sick on the floor. It was too complicated at this point, Ursula felt, to explain that Otto Schuffenecker was to blame and not India.
'Absolutely, Dr Erskine,' agreed Deddington. 'But Lady India—'
Just then Otto burst in.
Dr Dave's face turned puce with annoyance. 'Otto, if you are here to tell me that your next essay is going to be late, come back never!' he snapped.
'It's India,' said Otto, looking shaken.
'I am well aware that India Brattenbury is asleep on my _chaise-longue_ ,' Dr Dave barked.
'But she's not asleep!' screeched Otto.
'Well, it doesn't take a doctorate in Aristotelian syllogistic logic to see the girl's plainly not awake. Which leads me to deduce that the only other thing she could possibly be is asleep,' said the tutor.
'Or . . . _dead_ ,' howled Otto, collapsing on the tartan sofa and shaking uncontrollably.
'Otto, you are clearly under the influence of a hallucinogenic drug!' barked Dr Dave. 'Right, Miss Flowerbutton,' he continued, ignoring Otto's shivering frame. 'After the Reformation the Scottish—'
'Dr Dave,' said Ursula, desperate. ' _Please_ listen!'
Profoundly fed up by this point, he growled, 'What is it now, Miss Flowerbutton?'
'I'm afraid Otto's right,' she said, her voice barely above a whisper. 'India's dead.'
Finally, Dr Dave directed his gaze back towards the _chaise-longue_ , walked over to the mound and put his hand on India's back, as though to check her breathing. When he got no response, he walked around the sofa and looked at her face. Slowly, he lifted the Turkish rug from the girl's neck. Shocked, he clutched at his forehead. 'My god! Deddington, go and inform the High Provost of the situation.'
The porter dashed out of the room while Ursula, Otto and Dr Dave stood in appalled silence.
Finally, Dr Dave said, 'Right. Try to remain calm. Ursula, despite the tragic circumstances of our tutorial today, your priority is writing your first essay this week. You may not be able to imagine it now, but work will take your mind off this frightful discovery you have made. So let's not waste these few minutes of uninterrupted tutorial time before the High Provost arrives. Why don't you start with S. A. Burrell's article in the _Scottish Historical Review_. Volume forty-three. In 1964 Burrell wrote a very famous essay called "The Apocalyptic Vision of the Early Covenanters". Tell me why his argument is wrong. I'll look forward to hearing you read out your essay this time next week.'
Ursula took a deep breath. She did need to calm down or she'd be no help to anyone, least of all herself. Deciding not to admit that she had no idea what an 'Early Covenanter' might be, she scribbled the title of the article on her notepad. As she did so, she heard Dr Dave utter the most awful retching sound. Ursula looked up. He had parked his own custard next to Otto's.
Alas, she thought wistfully, today hasn't been very _Brideshead Revisited_ at all.
* All-male dining society, started 1884. Past members include John le Carré and David Cameron. The rulebook runs to ten pages.
† Hooray Henrys = Sloane Ranger boys. Term often abbreviated to 'Hoorays'.
Ten minutes or so after Julius Scrope, the High Provost, had been alerted to the grisly situation, he appeared in Dr Dave's rooms with Deddington at his side. Ursula observed that High Provost Scrope, a former Christminster alumnus who had famously made his fortune as a corporate raider, was dressed today as though he were attending a City board meeting. His slicked-back hair, dapper pinstripe suit and silvery tie were far flashier attire than was required in the world of academia.
Eyeing Ursula and Otto suspiciously, he took in the scene and summed up the situation as a 'nasty, nasty . . . er . . . accident' before asking Deddington to call an ambulance.
'Right, sir. Please send my condolences to his lordship,' said Deddington, leaving the room.
'I will,' replied Scrope. He then turned to Dr Dave, saying regretfully, 'I'll have the unhappy task of informing the girl's father this morning.' Then as though thinking aloud, he added, 'God knows what this means for the legacy.'
'What?' asked Dr Dave suddenly.
'Oh, nothing. Nothing whatsoever,' replied Scrope curtly.
'How could anyone call this an accident?' Ursula whispered to Otto while Scrope ushered Dr Dave to a far corner of the room, where they consulted in low voices. 'I'm going to say something.' She steeled herself. 'Excuse me for interrupting, High Provost, but . . . perhaps we should call the police? I mean, in case it's not an accident.'
'Miss—'
'Flowerbutton,' Ursula informed him.
'As I was saying, Miss Flowerbutton. This is clearly a tragic . . . ahem . . . accident and that is the way the authorities will no doubt see it. Now, thank you for your help but I do suggest that this environment is not suitable for undergraduates and perhaps you and Mr . . .'
'Prince,' said Otto.
'Mr Prince, thank you, should be getting on with your academic work. And, don't say anything to anyone about the sad circumstances this morning, eh? There is a vast fortune at stake and the press mustn't get a whisper of this. We must respect the . . . er . . . privacy of the Brattenbury family at this tragic time.'
The two undergraduates nodded, as they were clearly expected to, and exited the room. At the bottom of the staircase they stood and looked at each other, both in some kind of shock.
'God,' said Otto. 'Mr!'
'Otto!' scolded Ursula. 'What are we going to do?'
The two of them stood in silence for a few moments, as though in mourning. Then Otto said, 'You know what you were asking me earlier?'
'What?' asked Ursula.
'About the Gridiron Club. I nominated you as a Potential Babe, when we were deciding which girls to invite to the party.'
'God, Otto, you are so shallow!' she admonished him. 'This is not the moment to be thinking about parties.'
He looked thoroughly embarrassed. 'Sorry, you're completely right. It must be the shock of everything.' Then he said, hopefully, 'You will come though?'
*
'CHERWELL, EST. 1920' read the shiny brass plaque positioned on the white-rendered wall of an old Tudor cottage in the city centre. Underneath, someone had stuck a Post-it note on which they had scrawled the words 'Editorial meeting – third floor'. Ursula rested her bicycle against a pile of others propped along the wall, locked it and pushed open a tiny, creaky medieval door studded with enormous iron nails. It led into a wonkily cobbled courtyard with a rickety staircase on the left. She headed up and soon found herself at _Cherwell_ HQ – an unnecessarily formal description for a couple of cramped attic rooms, a loo and a corridor piled with old copies of the student newspaper.
Ursula hadn't known what to do with herself that morning. On the one hand anxiety and fear overwhelmed her, as her mind kept flashing back to the sight of India's bloody, wounded neck. On the other, she found herself behaving in incredibly ordinary ways. It felt peculiar, almost surreal, to be buying a mid-morning Kit-Kat from the vending machine in Kitchen Quad. She watched with a sort of dazed amazement as other students chatted, argued and smoked as though it was a perfectly average Monday morning. Death, Ursula realised, did not interrupt the humdrum business of life, or the eating of Kit-Kats.
As she munched her chocolate, Ursula wondered what her grannies would do in this situation. Vain Granny would certainly have taken to her boudoir for at least a week. But Plain Granny would have remained unbowed. As she had told Ursula throughout her childhood, in difficult situations the only thing to do was Keep Going. (This advice was regularly interchanged with 'Kick On', a phrase Plain Granny would screech loudly at complete strangers on the hunting field.) Ursula _would_ Keep Going. However unsettled she felt, she must stick to her original plans for Monday, like the _Cherwell_ meeting she'd signed up for. She needed something, anything, to help distract her, even temporarily, from the horrific scene that morning.
Determined to be as professional as possible, she had brought her old school satchel, in which she had carefully arranged a yellow legal pad, a reporter's notebook and several pens and newly sharpened pencils. Her heart beating fast with anticipation, she squeezed her way along the corridor towards the sound of voices coming from one of the attic rooms. She peered shyly round the doorway, and took in her first glimpse of the _Cherwell_ offices. The main editorial room, with its low, sloping ceilings, cracked magnolia paint and institutional-looking, tatty blue carpet, was shabbier than Ursula had expected. But she could just glimpse the sparkling Oxford rooftops from two cobweb-covered dormer windows that looked out north over the city.
Along one side of the room a beaten-up collection of desks and tables, piled with the detritus of student journalism, had been pushed against the wall. Bliss, she sighed to herself, as she took in the sight of the surfaces overflowing with stacks of black-and-white photographs, old magazines, newspapers, rough layouts, pencil sharpenings, empty crisp packets, stained tea mugs and Coca-Cola cans. A couple of prehistoric-looking typewriters peeked out from under the mess. To Ursula, it was a paradise of creative chaos.
A gaggle of students – second- and third-year writers and editors, Ursula assumed, and mainly boys, she couldn't help noticing with a _frisson_ of excitement – were grouped round a large layout table in the centre of the room. They were leafing through back issues of the newspaper, gossiping like mad while consuming cigarettes and mugs of tea at a rate of knots. She spotted Horatio, legs crossed, perched on a high stool with a smouldering Sobranie cigarette (purple, naturally) in one hand and _Interview_ magazine in the other. As Ursula wandered in, he looked up and blew her an extravagant kiss. Oh, god, she thought, Horatio doesn't know yet. He's been here all morning and no one's told him what happened to India.
Over the next few minutes, a dozen or so wannabe Fresher journalists arrived and drifted hesitantly to the farthest corner of the room where they clustered in a nervous huddle. Ursula smiled shyly at the other students as she wandered over. No one smiled back: there was an atmosphere of unspoken but intense competition.
Eventually an auburn-headed boy with the lanky proportions of a stick insect stood up. He had a black hard-backed notebook in one hand and an unlit cigarette in the other. Dressed in drainpipe jeans and a maroon sweatshirt with the words MAGDALEN COLLEGE emblazoned across the front, he also wore clunky Clark Kent-style glasses and had a layer of soft, pale pink fluff on his chin, as though he had not quite developed real stubble yet. The students around the table quietened down as he began to speak.
'Hello. Right. Yup. Er. Well . . .' He scratched his head, as though he had forgotten what he was going to say. 'Er, yup, what I mean is, I'm Jago Summers and, erm . . . I'm the editor . . . of _Cherwell_ . . . this term. Yup.'
He lit the cigarette and inhaled deeply, as though pondering his next move. The group waited expectantly. Jago took another drag. By his third drag of the Marlboro Red, he was ready to continue.
'Right, yup, yup! Sorry. Freshers,' he said, aiming a long gaze at the group of wannabe writers in the corner. 'First of all, let me warn you, many of you will be disappointed today. Not for one reason, but for many. We are generous with our allocation of misery here.'
The Freshers giggled, but more out of fear than real amusement.
'Er, yup, so your first problem: there is a strict hierarchy here at _Cherwell_. Even though, well, I really hate hierarchies, they're completely anachronistic, but, er, I mean, that's sort of the way it is, right?'
'Apparently his nickname is Vague-o,' whispered a Fresher behind Ursula. The group stifled their chuckles.
' . . . so, it's like, each section has an editor, and each section has, like, a deputy section editor, and under them, feature writers, leader writers, sub-editors. And, I mean, it's really hard but no one gets to write for a section until they have written for News. By which I mean, forget about writing about anything glamorous like Arts or Movies until you have cut your teeth on local news issues, and local news issues include such intriguing matters as the allocation of, like, public toilets on Cornmarket or the fate of the rotting beams in the ceiling of Wadham's Great Hall. Woodward and Bernstein did not start out with Watergate as their first story.'
Ursula noticed two Fresher students pick up their bags and attempt to slip out of the meeting unnoticed. Jago waved one stick-insect arm at them. 'I totally get it if the whole hierarchy thing's getting to you,' he called after them as they left. Then to everyone else, he added, 'Sad, really sad. Right, the next disappointment I would like to announce is that not all of you will get to write something every issue. Anyone else want to leave?'
He glared at the group of deflated Freshers. No one moved a muscle.
'Okay. The next tragedy is that even if you are lucky enough to be commissioned, and actually write something, it will probably be killed.'
Ursula and the other Freshers, severely intimidated by now, didn't dare ask what he meant by 'killed'. But their mystified expressions prompted Jago to pick up a sheet of text off the layout table, scrumple it into a ball and aim it correctly at the bin.
'Oh, right,' gulped Ursula.
'In fact, it will take most of you an average of four or five attempts at stories before one gets run,' said Jago.
'Harsh,' murmured the Fresher standing beside Ursula.
'Right, let's allocate some stories,' the editor continued. 'Usual procedure is that anyone who's interested in writing something just puts their hand up.'
A scruffy young man, dressed in baggy shorts and a sweatshirt that read 'HERTFORD COLLEGE ULTIMATE FRISBEE' on the front and 'FEAR THE DEER' on the back, grabbed a piece of chalk and headed over to the large blackboard propped in one corner of the room.
'Everyone, this is Michael, our managing editor,' said Jago.
'Yeah,' said Michael, while frantically leafing through a notebook for article ideas. He then scribbled a list of upcoming stories on the blackboard. Ursula scanned the list for something she might be able to volunteer for. Among many other topics, she read:
_Trinity students steal boat club megaphone to shout abuse at Japanese tourists_
_Ancient Philosophy don at Pembroke resigns to become plumber_
_Emperor Hirohito's grandson seen doing own washing on first day at Merton College after bodyguard shows him how to use machine_
_First woman Fellow elected at All Souls after 545 years_
As Jago read each idea out loud, pretty much every hand in the room went up. I don't stand a chance, thought Ursula. Only one idea was allocated to a Fresher – the story of a cat who had given birth to three kittens in the Jesus College organ.
'Right, don't worry if you didn't get a story,' said Jago as he came to the end of the list. 'There'll be other chances. If you did get a story, congratulations. You may come and use the typewriters here in the office at any time. There's only one rule. The deadline for all stories is ten o'clock on Sunday mornings, we go to print on Sunday afternoon, publish on Mondays. Anyone who misses the deadline gets fired. Meanwhile, keep an eye out for stories. Sometimes things that don't seem like news turn into a great story. Do your trawls . . .'
'What are trawls?' asked a ballsy Fresher.
'Sorry,' said Jago. 'A trawl is fishing for information. Everyone needs to make at least three visits a week to someone important in their college or the university societies, to talk to them, get the gossip.'
'And speaking of gossip,' interrupted Horatio, 'proper gossip comes straight to me. I'm Horatio Bentley and I'm writing the gossip column – John Evelyn's Diary.* So if you hear any tales of backstabbing, debauchery, misdemeanours, drunkenness, inappropriate sex – well, you can find me in the Bodleian Library most mornings or in my rooms at Christminster College at teatime. Right now the hunt is on for Pushy Fresher of the Week. If you have any suggestions for candidates for the most odious and offensive first year in the university, be so kind as to let me know.'
'Thanks, Horatio,' said Jago. 'Right . . .'
He was interrupted by the arrival of a breathless girl wearing a CND T-shirt and cut-off denim shorts and tights. She had turquoise hair, and a ring through her nose.
'Late again, Karen,' huffed Jago. 'Everyone, this is Karen Porter, our news editor.'
'Jago, I've got something,' said Karen, ignoring his irritation. 'There's a rumour that a male undergraduate died this morning at Christminster. Some kind of accident.'
'What? Who?' cried Horatio.
'No names yet,' replied Karen.
'Shit,' said Jago. 'The story's yours.'
Should she say something? Ursula wondered. She knew the High Provost had forbidden her and Otto from talking about India's death, but clearly everyone was going to know about it sooner or later. If she was going to be a _Cherwell_ writer, she could hardly hold back information that might be crucial to a story. In the name of Truth, of Proper Journalism, of Full Disclosure (and a teensy bit for the sake of Personal Ambition, if she was being totally honest), she must make Jago aware of Karen Porter's mistake. If Woodward and Bernstein hadn't stuck up for The Truth, well . . . it didn't bear thinking about. Gingerly, Ursula raised her hand.
No one noticed.
'Excuse me,' she said.
Jago looked at her quizzically. 'Yup?'
'Well, it's just, I think I may have some information relevant to the Christminster story.'
Karen Porter frowned. She didn't need help from some Fresher, her expression seemed to say. If Jago sensed her displeasure, he chose to ignore it.
'Go on,' he said, his full attention now focused on Ursula.
'Well, the dead undergraduate . . . in fact, it was a girl.'
'How do you know?' asked Karen with a scowl.
'I found the body.'
There was a shocked silence. Ursula soon realised that twenty or so faces were staring at her expectantly. But she knew she shouldn't say much more now.
'Great scoop,' said Jago. 'Right, er, what's your name?'
'Ursula Flowerbutton.'
'I like it. Good name for a writer. Ursula, I'm reallocating the story to you.'
'But, it's Karen's story . . .' Ursula started to say. She'd only wanted to help the news editor, not snatch her scoop.
'You're at Christminster. It makes sense for you to do this story, Ursula. Karen's got plenty more to do, haven't you.' It wasn't a question.
Karen smiled a little too brightly. 'Yeah, loads,' she replied.
'Okay, thanks, everyone. I look forward to receiving your copy at the end of First Week,' said Jago, dismissing the meeting. 'Ursula, Horatio, wait behind, please.'
As the other students were leaving, Horatio dashed up to Ursula. 'Someone died in college? Who?'
'It's India.' Ursula could hardly bear to say what she knew she had to. 'She's dead.'
Horatio whitened. He gulped, unable to speak.
Jago shut the office door after the last of the students and editors had left. 'Did you say India?' he asked. 'India Brattenbury is dead? That pretty, posh, actressy girl? What happened to her?'
Ursula could barely hold back tears as she told Jago and Horatio about finding India's lifeless body on the _chaise-longue_ , the slit across her throat. Horatio shook his head in disbelief. 'Oh, god,' he whispered sadly.
'I don't think it was an accident,' said Ursula.
'You're saying this is a murder?' said Jago, almost hyperventilating with glee. 'Ursula, this could be _huge_ for _Cherwell_. You and I need to have a meeting. Why don't we get together tonight? Somewhere private. Somewhere like . . . my rooms. Come to Magdalen at six. I'm in the cloisters.'
Horatio lit another cigarette, took a long drag and regained his composure. He then pursed his lips, cocked his head to one side and stared dubiously at Jago, who ignored him, saying, 'We can discuss the piece. The tone. Your reporting skills. How to cover a police investigation . . . that sort of thing.'
'I'll be there,' said Ursula.
'I think you may well need a chaperone,' suggested Horatio, raising one eyebrow archly.
'Horatio, don't even think about nicking this story off Ursula,' Jago informed him. 'It's hers. Officially.'
'No need to worry about that, I'd much rather stick to the party reporting. Crime's not my style – not enough champagne or gossip on offer. I was just thinking that Ursula may need a companion at her meeting with you tonight,' said Horatio.
Jago crossed his arms. 'Sometimes I wish you'd naff off and stop interfering, you old queen.'
'I take enormous offence at that homophobic slur,' retorted Horatio. 'I am _not_ an old queen. I'm a young one.'
* _Cherwell_ 's longest-running column. Named after diarist John Evelyn (1620–1706), the column makes Page Six look kind.
Monday, 1st week: lunchtime
_Brideshead_ , bonbon _s,_ cucumber sandwiches – and now a murder. Ursula's sepia-hued fantasy of Oxford had been dramatically interrupted by India's death. After bidding farewell to Horatio and Jago, she sped back to Christminster on her bicycle, her brain buzzing with questions. For now, the _Cherwell_ assignment had done the trick – her fearful mood was, if not dissipated, edged to the back of her mind. Although she had intended to spend the afternoon in the library, she was soon diverted from her plan: the sight of an ambulance parked at an acute angle on the hard standing in front of Christminster called for immediate investigation. She threw her bicycle against the college railings, grabbed her satchel and dashed into the gate tower.
A small group of sombre-looking students stood gazing at the ambulance and talking to each other in low tones. Ursula caught snatches of their conversation.
'. . . Apparently it's a second year . . . Some kind of accident . . . Wentworth Wychwood's Opening Jaunt . . . Blood all over the walls . . .'
The group suddenly fell silent as Julius Scrope came into view accompanied by two paramedics. Head down, he was marching determinedly along the cloister towards the gatehouse, his heels clicking against the stone flags, the paramedics barely able to keep up.
' . . . I am afraid, High Provost,' Ursula heard one of them saying, as they came closer, 'we will need to alert the police _now_. We have delayed far too long this morning already.'
'There's no need for the police, surely?' Scrope looked agitated. 'The . . . ahem . . . disruption it will cause to the undergraduates. Police crawling all over the college – not conducive to academic studies. The dons would never allow it. I think it would be more . . . civilised to take her straight to hospital.'
'No can do, sir,' said the other man, rather too gleefully, thought Ursula, as he and his colleague followed Scrope into the cloister.
The High Provost stopped under the gate tower. It was only then that he noticed the small audience of undergraduates, who were clearly enjoying these proceedings immensely. He glowered at them.
'I presume you are all en route to lectures or the library,' said Scrope.
The group rapidly dispersed, while Ursula, from her spot in the shadows, slipped unnoticed into the Porters' Lodge, conveniently forgetting to completely close the door. Feigning an unnaturally keen interest in the hockey fixtures on the noticeboard, she listened as negotiations between the High Provost and the paramedics became more fraught.
'No. Can. Do?' huffed Scrope. 'Meaning what? Just get the girl to hospital.'
'Problem with that, sir,' Ursula heard one of the men say, 'is that we can only take a body to hospital if it's still alive . . .'
'. . . not that we can _officially_ say she's dead,' interrupted his partner.
'What?' snapped Scrope.
'Only the police can officially declare someone dead.'
'Well, if you can't declare her dead, surely that means she is, in theory, alive,' insisted Scrope, 'in which case you can take her to hospital.'
'Oh, no, sir, I'm not saying she's _alive_ , I'm just saying we can't formally declare her dead. But she clearly _is_ dead, to all intents and purposes. Which means we can't take her to hospital.'
Scrope snorted like a frustrated stallion.
'Even if the victim had been decapitated it's still the case that the police surgeon is the only one who can _officially_ declare her dead,' the man continued. 'Now if you'll excuse me, High Provost, I'm going to radio the police from the ambulance.'
The door to the Porters' Lodge received an irritated kick from Scrope, and its two-hundred-year-old hinges creaked in protest as he stormed inside.
'Bloody red tape! Deddington, in about ten minutes college is going to be crawling with coppers. Keep them under control, would you? And keep the press OUT,' he blustered, red-faced. 'Lord Brattenbury is expected to arrive fairly soon. Poor man. Send him straight to the mansion when he arrives.'
'Yes, High Provost,' Deddington replied.
As Scrope turned to go his eyes lighted on Ursula and he scrutinised her for a long, uncomfortable moment. Ursula prayed he wasn't going to accuse her of student journalism, punishable by some kind of ancient Oxford torture, and was relieved when he finally just said, 'You're the one who found her, aren't you?'
She nodded.
'Are you all right?' barked Scrope, not seeming particularly concerned.
There was a killer on the loose. Nothing was 'all right', Ursula thought to herself. Still, the last thing she was going to do was admit to Scrope – or herself – how terrified she really was. 'I'm fine,' she said.
'Jolly good. Don't disappear off anywhere.'
'I won't,' she promised.
*
Nothing could have prepared Ursula for the task that lay ahead. Not only had she never written a newspaper article, she'd never been mixed up in a murder either. She knew that if she were going to write something decent for _Cherwell_ , she'd need to get a scoop that no one else had. She would cover the police investigation from the minute it started, but in the meantime she needed to understand as much as she could about the girl India had been.
Ursula hoped desperately that Nancy was in her room. Perhaps her recollections of her lunch with India a few days ago might give Ursula a clue about where to start. As she dashed along the west side of the Great Lawn towards the Gothic Buildings, she glanced up towards the windows of the Old Drawing Room. The curtains were closed, even though it was past one o'clock already. Crumbs, she thought. Maybe Wenty didn't know yet. She prayed she wouldn't run into him now.
She took the steep stairs of the Gothic Buildings two at a time. Once she'd reached the top and was standing outside Nancy's door, she was completely out of breath.
'Nancy!' puffed Ursula, rapping on the door.
She waited a few seconds and knocked again. Finally, Nancy's voice could be heard uttering a muffled, 'Hey?'
'It's me, Ursula,' she said from the other side of the door.
'Come in, it's open.'
Ursula opened the door and stepped inside. Nancy's extensive evening wardrobe was still strewn around the room, and her desk was now submerged under an avalanche of writing materials. The surrounding floor was covered in scrunched-up balls of paper. Gosh, thought Ursula, American girls are much more studious than English ones. It looked as if Nancy had been working all night, even _before_ she'd had her first tutorial.
Nancy was still in bed, her hair distributed wildly across the leopard-print pillowcase upon which she was lying face down.
'Ow!' she croaked. 'My head.'
Gingerly, the American turned her face towards Ursula. Nancy's eyes were covered by a quilted, madras-check sleep mask on which her initials had been monogrammed in an elaborate copperplate script _._ The breast pocket of her light blue cotton Brooks Brothers nightshirt displayed the same monogram, as did, Ursula noticed, a pair of slippers that had been tossed beside the bed. (An infinitely more exciting way of identifying your clothes, thought Ursula, than the standard-issue Cash's nametapes that Plain Granny had sewn inside her uniform throughout her schooldays.)
Without removing the sleep mask Nancy sat up, wincing as she propped herself against the cold bars of her iron bedstead. 'I feel like an extra in _Annie_ ,' she complained, slowly pushing up the mask until it was perched like an Alice band on top of her head. With her mascara from the night before still smudged around her eyes, unkempt hair and several pillow marks indented on her cheek, she looked like an accidental version of Cyndi Lauper.* She rubbed her eyes, blinked a few times and yawned.
Suddenly she sat bolt upright and exclaimed, 'Oh my god, I overslept! What time is it? Have I missed my tutorial?'
'It's almost one-thirty,' said Ursula.
'Phew,' sighed Nancy. 'My tutorial's at two. How was yours? Utterly terrifying?'
'Well, um . . .' Unsure exactly how to explain just how frightening her tutorial and entire morning had been, Ursula let her voice trail off.
'Oh, Jesus,' said Nancy, rubbing her temples. 'I had way too much to drink last night. But, wow, what a party! I don't know how I'm going to cope with Dr Dave. My brain feels like one great big pink champagne bubble.'
Nancy tumbled out of bed and flopped down in the hard wooden chair at her desk. She started fumbling beneath the piles of papers there. Finally she retrieved a packet of Camel Lights and a glossy red matchbox with the word AREA splashed across it. Nancy offered Ursula a cigarette, but she shook her head. She was secretly desperate to try one, but St Swerford's punishment for smoking had been harsh enough to prevent her ever even contemplating it. Seeing Nancy blowing curls of smoke as proficiently as a 1940s movie star made Ursula feel unbelievably square.
'I've been writing all night,' Nancy said, casting her eyes over the chaotic desk. 'Letters of love. To Next Duke. Here,' she continued, handing Ursula a piece of fuchsia-pink paper. In extravagant, curly-whirly letters written in silver glitter pen, were the words:
Hi, Next Duke!
How are you? Can you ever forgive my totally gross American behavior last night? First off, I was so drunk that despite our awesome conversation, I cannot remember your Christian name. Secondly, sorry for spilling champagne all over your coat.
As all the JYAs were warned at our orientation last week in London, we must try to be 'civilised' now we are in Oxford. I guess I failed!!!!!!!!!!!!!!!!!!!!
The only way I can make it up to you is to come get your gorgeous coat and have it dry-cleaned.
R.S.V.P., please.
xxx
Nancy Feingold
(Room 3, Staircase C, Christminster)
'It's a very . . . _original_ love letter,' said Ursula, handing it back, though she couldn't quite see the romance in it.
Nancy winked as she folded the letter and put it in an envelope with 'F.A.O. The Next Duke of Dudley, Merton College', written on the front in the same glittery ink.
Discarding her nightshirt, Nancy pulled on a pair of drainpipe Gloria Vanderbilt jeans, added an oversized yellow sweatshirt, slipped her bare feet into high-top baseball boots and threw a stonewashed, shoulder-padded denim jacket over the whole ensemble.
'You look really fashionable,' Ursula complimented her, envying the casual look of the wonderful American outfit. Her own collection of kilts and cardis really did seem outdated, she thought, looking down at her own clothes.
Nancy must have sensed Ursula's sartorial discomfort. 'Hey, I had an idea for you,' she said sweetly. 'I mean, you're so cute, so pretty. You've got legs like Jerry Hall . . . but no one can see them. Why don't we cut your kilts short? Make them into mini-kilts. That would be _so cool_. Everything has to be mini now to look trendy.'
'Oooh . . .' Ursula smiled at the idea. Her kilts, shortened, would be much more fashionable than they were now. But it would take her forever to hem them, with all the pleats. 'But I don't have time for all that sewing.'
Ursula was interrupted by the unmistakable tones of Alice the scout calling from outside Nancy's room, 'Are you in there, girls?'
'Here!' replied Nancy.
Alice put her head round the door.
'I was just passing,' she said, 'and I wanted to check you were both all right. Dreadful news, isn't it?'
'What is?' asked Nancy curiously.
Alice's face fell.
'Hey, come in,' Nancy urged her.
'She doesn't know yet,' Ursula told the scout as she came into the room.
'Oh . . . oh, I see. Oh, dear.' Alice looked to be on the verge of tears.
'What's going on?' Nancy asked.
'Don't worry, I'll explain everything,' said Ursula.
'Thank you, Miss Flowerbutton,' Alice replied gratefully. 'I'll leave you now.' Just as she was closing the door, she added, 'Do take care, girls, won't you, round college? And I don't mean to seem nosy, but did I just hear one of you talking about kilts needing to be taken up? I take in sewing. I could have them done by tomorrow morning.'
'That would be great,' said Ursula. 'Thanks!'
'All right, I'll take them from your room,' said Alice, disappearing.
After the scout had gone, Nancy asked, 'What's happened?'
'Oh, Nancy, it's awful,' said Ursula, sitting down on the bed. 'I went into Dr Dave's rooms this morning for my tutorial. He wasn't there, but India was lying on his _chaise-longue_. She was dead.'
'Dead?' Nancy stared at Ursula, completely shocked.
'I think she was murdered.'
'Murdered?' Nancy gasped, her face turning pale. She took a long drag of her cigarette.
'Her neck . . . I don't know how to describe it . . .' The horror of the scene came back to Ursula. 'Her neck had been . . . slashed. The blood had dripped down her dress.'
'But . . . I can't believe it. And . . . I thought she was studying English, not History. Why would she be in Dr Dave's rooms anyway?'
'I don't know. But we've got to find out. I went to the _Cherwell_ meeting this morning and somehow . . . well . . . I've got to write an article about the murder. By Sunday morning. I don't know where to begin.'
'Come on,' said Nancy, grabbing a miniature Louis Vuitton rucksack into which she put a notebook, pens and the letter to Next Duke. 'We've got the perfect excuse to go back to Dr Dave's rooms. My tutorial's in twenty minutes.'
The girls clattered down the staircase. At the bottom they came upon Mrs Deddington and Alice with a mountain of sheets at their feet. Ursula noticed her kilts piled next to them. But work seemed to be the last thing on their minds. They were discussing something intently in whispers. When Mrs Deddington noticed the girls she gave them a wary glance and tapped Alice smartly on the shoulder. Both scouts abruptly stopped talking.
'Hello again, girls,' said Alice.
'Just helping out with a little bit of extra laundry this morning,' said Mrs Deddington. She started gathering up the dirty sheets. 'Are you okay?' she asked Ursula. 'I hear you were the one who found her.'
Ursula nodded gratefully. 'I'll be fine,' she said. 'Poor India.'
'It's a great shame,' said Alice. 'She was a lovely girl, wasn't she, Mrs D?'
Mrs Deddington didn't say anything. A sour expression momentarily clouded her face.
'You didn't like her?' said Nancy.
'I didn't really know her. Like Alice says, I am sure she was very nice,' protested Mrs Deddington. But her brittle smile was unconvincing. 'Better get on,' she said, and scuttled off with a large armful of laundry.
As Ursula and Nancy headed out into the courtyard of the Gothic Buildings, Ursula noticed the afternoon light was now starting to bathe the stone walls in mellow, amber rays.
'Can you tell me about your lunch last Thursday with India?' asked Ursula as they walked along.
'Sure,' Nancy said. 'So, it was like, totally awesome for me to be hanging out with this super-cool English girl. We walked to this cute little brasserie called Brown's on Woodstock Road. Anyway, we get there, she walks straight in past the line of people waiting – who all looked kinda irritated – and asks the maître d' to take us to her favourite table.'
Once Nancy and India had been seated at the best table in the restaurant, Nancy recalled, India had ordered a hamburger for her, saying, 'Wouldn't want you getting homesick.'
'It was like the smallest, most pathetic hamburger I'd ever seen. I could have eaten twelve, literally,' Nancy said. 'Anyway, then she asked me if I wanted to join her shooting party, in a few weekends' time, at her dad's country place.'
Ursula fished her reporter's notebook from her satchel and, as they walked, scribbled some notes:
_—Otto had mentioned shooting weekend. Annoyed he hadn't been invited._
_—Why would India invite Nancy, whom she only had met that day, and exclude old friend Otto?_
Nancy continued with her account of the lunch. 'So I said to India that I _loved_ to shoot, I used to spend hours at the range with my father in New Jersey. Anyway, India was totally freaked when I said this. She was like, "Girls don't shoot in England. We just load." So I asked her, hadn't she ever heard of Betty Friedan? She said, "Who?"'
The upshot was that, despite the thorough lecture Nancy had proceeded to give India on the subject of gender equality in the modern world, India deemed this information completely irrelevant to the politics of an English shooting weekend, telling Nancy that she hadn't been in England long enough to realise that even though the UK had Mrs Thatcher and punks, things weren't as modern as they seemed. Shooting weekends at 'Bratters', her ancestral home in the Derbyshire Dales, were run as they had been in 1925.
Still, Nancy had become extremely excited when India informed her that shooting parties had a dress code all of their own. The American girl loved the idea of dressing in tweed plus fours and green Wellington boots during the day and changing into a black-tie party dress for dinner.
'It sounded like _Falcon Crest_ ,* only classy,' Nancy told Ursula.
India had invited nine 'guns' – the boys – and nine girls. The tenth gun would be her father, Lord Brattenbury, the tenth girl India herself.
'What about her mother?' asked Ursula.
Nancy pondered for a moment. Then she said, 'You know, I don't think she mentioned her mom . . .' Nancy frowned, trying to recall the exact conversation. 'No, I'm sure she didn't.'
'So if she'd invited nine girls and only nine boys because her father was going to be the tenth gun, and India was the tenth girl,' interrupted Ursula, 'that must mean, for some reason, her mother wasn't going to be there. Did she say exactly who she'd invited?'
'Oh, yes, she was very clear that it was going to be a weekend for the Champagne Set. She said Wenty and Eg were definitely going, and Dom Littleton . . . and I _think_ she mentioned Dr Dave.'
While they headed towards the Porters' Lodge, Nancy peered over Ursula's shoulder while she scrawled:
_—India invited don to shoot. Why? Same don in whose rooms then found dead._
'That is _so_ creepola,' said Nancy.
'I know. And why invite an Oxford don and not Otto, who was her friend?'
Nancy shook her head. 'I wonder who's going to play Hamlet now?' she asked.
'Her understudy, I suppose . . .'
Ursula broke off. She and Nancy looked at each other as they walked. They seemed to be thinking the same thing.
'You don't think . . . Isobel? No _way_ ,' said Nancy slowly.
'She couldn't have,' Ursula agreed. 'They were best friends.'
Still, she found herself adding a grim note to her pad:
_—Could Isobel really have killed India over the Hamlet role?_
_—Or what about Henry Forsyth? Could he have done it?_
The girls had reached the gate tower. Deddington was standing underneath it talking to a smart-looking man dressed in a navy suit and a camel coat.
'. . . the entire college is saddened, Lord Brattenbury,' the porter was saying.
'Oh my god, India's dad!' gasped Nancy to Ursula.
'The High Provost awaits you in his mansion,' said Deddington, pointing to the far end of Great Quad.
'Thank you, Mr Deddington,' said Lord Brattenbury. 'Are the police here yet?'
'Just one junior officer so far. Apparently the senior detective's delayed. Something about a stolen Blues boat.'
'God almighty. There's a life lost and the police are tending to a boat. Oxford! Will you come and tell me the minute the detective arrives?'
Deddington nodded and went inside. Lord Brattenbury rushed past the girls with barely a glance at them. His face looked pinched, tragic.
They dashed into the Porters' Lodge.
'Ah, Feingold,' said Deddington. 'Delivery just arrived for you.'
'Oh?' said Nancy.
'You'll find it in front of college. Wouldn't fit into the lodge. Here,' he added, handing her a key.
'Ursula, quick!' said Nancy, glancing at her watch.
She put her letter for Next Duke into the Pigeon Post sack on Deddington's desk, and the girls grabbed their mail from their pigeonholes and sped out of the gate tower. The ambulance had now gone. In its place stood a glossy scarlet moped with a black leather seat and a white basket attached to the handlebars. The word SPREE was written in loopy white writing on the side. Nancy emitted a whoop of joy.
'My Spree ' she squealed, delighted, jumping astride the adorable little motorbike. Her mom, it transpired, had shipped Nancy's favoured mode of transport from Northwestern to Oxford. 'I never could really see myself on a bicycle. Look, Ursula, you can fit on the back.'
'It's gorgeous,' she said.
The huge college clock slowly chimed twice. 'I'm gonna be late for the tutorial,' shrieked Nancy, jumping off the moped. 'Come on!'
* Cyndi Lauper, of 'Girls Just Want to Have Fun' fame, was to Madonna what Meghan Trainor is to Taylor Swift.
* _Falcon Crest_ , which aired from 1981–1990, revolved around a feuding wine-growing family in the Napa Valley. Also known as 'Dallas with grapes'.
* Honda Spree = 'It' motorbike for 1980s American college kids.
Monday, 1st week: afternoon
As the girls walked speedily towards Dr Dave's staircase, Ursula opened a couple of the envelopes she had found in her pigeonhole. One of them contained an invitation to afternoon tea that day at four o'clock in the Junior Common Room. The host, Ben Braithwaite (Second Year, Engineering), 'President' of the JCR Committee, had invited the Freshers to meet their 'College Parents'. Ursula showed it to Nancy.
'An Oxford Mom and Dad? Cute! And "Afternoon Tea"! How retro,' she sighed. 'This place is more like the Ritz than school.'
Ursula had survived a very long time without parents, and wasn't sure now was the moment to acquire a mother and father. Still, an invitation to tea was hard to resist. Four o'clock tea by the fire at Seldom Seen Farm – hot buttered crumpets, Plain Granny's home-made scones with jam and clotted cream, Victoria sponge cake – had been Ursula's childhood treat, as cosy as it was scrumptious. Surely an Oxford tea would be just as comforting. And it would give her a good excuse to scope out the JCR unobserved. Ursula had a hunch that the room, which was right next to Dr Dave's set, might hold a clue to what had happened to India.
'Let's go to the tea together,' she said, following Nancy into Dr Dave's staircase. After that, she vowed to herself, she _would_ go to the library.
As the girls climbed the stairs, they could hear the sound of squabbling voices drifting down from the landing above them.
'I'm sorry, sir. You can't go in there.'
'But these are my rooms. I'm due to give a tutorial in about thirty seconds.'
'Your rooms are a crime scene. You'll have to wait for the Detective Inspector to arrive.'
On the first-floor landing, the girls found a uniformed police constable, who looked to be about their own age, barring entry to an irate Dr Dave.
'I shall be bereft without my Mont Blanc. May I at least get it from inside?' pleaded the don.
'Your what?'
'My fountain pen.'
'No one can be allowed into the crime scene. Might disturb evidence. I can lend you a Biro,' the constable offered, a film of nervous perspiration starting to dampen his face.
'Thank you, but . . . no. I have never written a word of my manuscripts with a Bic and I am not about to start now. I do hope that, despite the tragic circumstances of last night,' continued Dr Dave, 'the Inspector will understand I need access to my rooms as soon as is humanly possible. I need somewhere to give my tutorials, and volume two of _From Constantinople to Jerusalem_ , the sequel to my first bestselling tome on the Middle East, is due to be handed in to my publishers at the end of the month. I can only write sitting at my black lacquer desk with the Giant Argus watching over me . . .'
The constable maintained his position across the doorway, arms crossed. 'The D.I. will be here shortly,' he said, his expression set. Then, noticing Nancy and Ursula standing there, he blushed furiously, as though he had never seen such pretty girls in his life.
'Hello, officer. I'm here for my tutorial,' said Nancy. 'And good morning, Dr Dave.'
'Good _afternoon_ , Miss Feingold,' was his curt reply. Nancy reddened. 'I'm afraid we will be conducting our first tutorial . . .' he trailed off, looking around for a suitable perch, before his eyes alighted on the dust-laden window seat below the landing window '. . . here.'
He beckoned her to sit. 'Now,' he began. 'The Scottish Covenanters—'
But he got no further before the policeman, concerned by Ursula's presence, interrupted.
'And may I ask what reason you have to be here, miss?' he said to her. 'This is a potential crime scene.'
It wasn't a good idea, Ursula presumed, to admit she was writing about India's murder for _Cherwell_. So she said, 'I thought I could help the police with their enquiries.'
The constable looked unimpressed. 'And how would you be proposing to do that, miss?'
'I'm not sure . . . but I was the one who found the victim.'
The constable started.
'Did you say _you_ found her?' he asked, sounding amazed.
'Yes,' said Ursula.
' _You_ were the first person on the scene?'
'Yes.'
'You're sure of that?'
'Yes.'
The constable narrowed his eyes at her suspiciously. 'Miss—?'
'Flowerbutton,' Ursula informed him.
'Right, Miss Flowerbutton. You must remain here until the Detective Inspector arrives. You are a Person of Interest.'
'I am?' remarked Ursula hopefully. Perhaps this would give her unlimited licence to skulk around the police investigation – how perfect for her article. 'I'll stay as long as you need me,' she said, planting herself on the staircase leading up to the second floor. Bother, she thought, if only she'd been to the library already she could have checked out her book and started reading about the Early Covenanters while she was waiting for the D.I.
'May I continue, officer, with my tutorial now?' groaned Dr Dave from the window seat where he and Nancy sat, patiently waiting.
The constable nodded.
'Well, Miss Feingold. An eventful morning, soon to be made only more so by the introduction of Scottish religious fanaticism into your life. Your task this week is to familiarise yourself with an article entitled "The Apocalyptic Vision of the Early Covenanters" and tell me why the author is wrong.'
'Tell you . . . sorry . . . who's what?' said Nancy. She looked puzzled.
'Why the author is wrong,' repeated Dr Dave.
'Ok–aaaaay . . . I guess.' For the first time, Ursula detected a chink in Nancy's armour of self-confidence.
'You'll find the college library very useful. The librarian up there, Olive – I mean, Ms Brookethorpe – will help you find the literature. She's the prim-looking one, with a nice bottom . . .'
Dr Dave's description of Ms Brookethorpe was interrupted by the sound of footsteps clattering towards them. The constable rushed to the top of the staircase. When he saw to whom the footsteps belonged, he hurriedly stood to attention.
'Detective Inspector Trott!' he said, greeting the man who soon appeared on the landing.
In his mid-thirties, Ursula guessed, Trott was a beady-eyed, clean-shaven man with a muscular jaw and black hair cropped so close to his skull that you could see the veins pulsing in his temples. He was wearing a Barbour jacket over an unremarkable navy suit, white shirt and shiny red tie. His face conveyed only unemotional professionalism, giving nothing away.
'Show me,' was all D.I. Trott said in reply to the constable, flashing a cursory glance in the direction of Ursula, Nancy and their tutor.
'She's in here,' said the constable, opening the door to Dr Dave's rooms. He had barely put his toe over the threshold before Trott snapped, 'Wait outside, please, Constable! Only one person at a time until we have gathered all the evidence.'
A tense silence settled over the party on the landing. As the minutes ticked slowly by, Ursula dreaded to think what Trott was witnessing inside. When the D.I. eventually reappeared, he was stern-faced.
'Constable, radio the station and tell them I need Scene of Crime officers, forensic scientists, a police photographer, the exhibits officer, the police surgeon,' he ordered. 'They'd better tell the Home Office pathologist to bring his Murder Bag with him.'
Trott now turned his attention to Ursula and Nancy with a searing glare. 'Who are these two?'
'Miss Flowerbutton discovered the body,' replied the constable, pointing at Ursula. 'Miss Feingold is one of Dr Erskine's students. She's been having a tutorial.'
'Right. Keep Miss Flowerbutton here. I'll question her myself later.'
'I'll wait with you,' Nancy offered.
'Thanks,' said Ursula gratefully.
Trott then opened the door to the JCR, peered inside and said, 'This'll have to do for now as a temporary Incident Room. Constable, anyone who arrives comes straight in here, please. No one is to enter the crime scene until they've reported to me.'
*
That afternoon, Ursula and Nancy perched, seemingly unnoticed, at the top of the flight of stairs above Dr Dave's rooms as the full force of the Thames Valley Police swung into action. The landing and JCR were soon populated by a collection of investigative personnel, each of whom would disappear into Dr Dave's rooms one at a time, only to reappear a few minutes later carrying miscellaneous items, which would then be discussed with D.I. Trott, now installed in the JCR. Ursula surreptitiously scribbled notes constantly, while Nancy consumed cigarette after cigarette.
Dr Dave, who had somehow managed to continue with his tutorial schedule from the window seat, became more agitated as each object was transported. The sight of an officer emerging from his rooms while swaying beneath the weight of his beloved Giant Argus was too much.
'Young man, young man!' he called out. 'Is there really any compelling reason to remove poor Panoptes?'
'Eh?' mumbled the officer.
'Argos Panoptes was the all-seeing giant of Greek mythology,' said Dr Dave. 'I named the Giant Argus you have in your arms after him. Does he really need to be moved?'
'Evidence,' replied the officer curtly, before adding, 'Let's hope he really is all-seeing.'
Before Dr Dave could protest, the officer had staggered away down the stairs under the weight of the stuffed bird.
'God knows how I will finish the book without Panoptes,' sighed Dr Dave, bereft.
Every now and again a local officer of some description – traffic warden, social worker, county councillor, transport policeman – would appear on the landing. When the constable asked them the purpose of their visit, they universally offered a variation on the phrase 'just popping in to have a look'. They were, Ursula observed, behaving as though they were here to greet a newborn baby, not inspect a freshly murdered corpse.
By mid-afternoon, D.I. Trott reappeared from the JCR and consulted in a low voice with the constable, though Ursula could only catch snatches of the conversation.
'. . . why you kept _her_ hanging around directly outside the crime scene . . . did they teach you nothing in training, Constable?' His voice rose with irritation. Trott was giving the young policeman a serious ticking-off.
'But I thought you said to keep the prime suspect here?' whimpered the constable.
'Ssshhh! I didn't mean _right here . . ._ Anyway, it's too late now. I need to get on.'
_I'm_ the prime suspect? thought Ursula. She felt utterly confused – and petrified.
'I'm ready for you, Miss Flowerbutton,' said Trott.
'See you later, at the JCR tea?' said Nancy, giving Ursula a comforting hug before she left. 'And then let's go to that party tonight together, The Acquisitors—'
'Perquisitors,' Ursula corrected her as she followed Trott into the JCR. 'I can't.'
'You didn't mean it, did you?' asked Ursula, worried, as she walked alongside the D.I. into the JCR.
'Mean what, miss?' said Trott, striding towards the far end of the room.
'That thing you said about me being the prime suspect?'
'Standard procedure, miss. The person who finds the body is always the prime suspect.'
'But why?'
'It's your alibi. Let's just imagine you killed your friend . . .'
'She wasn't my friend.'
'I see,' said Trott. He looked at Ursula warily.
'I mean, I don't mean that I hated her – what I mean is she wasn't my friend because I didn't really know her.' Ursula felt panicked.
'Right. Well, to return to your earlier question, let's just imagine you _did_ kill this girl who you _say_ isn't your friend. The easiest thing is to say that you found her. It gives you an excuse to be in the room where the body is, and accounts for any of your fingerprints that might be found. So that's why you're officially the prime suspect. Only you're not meant to know that you are,' he grumbled. 'That young constable . . . I don't know. Now, if you could wait until I start my interview before you speak again, please?'
Ursula had been too frantic to notice much about the JCR this morning. But now, as she talked with Trott, she tried to take in every detail. She didn't want to miss a clue, if there was one.
It had probably been a very grand room once. Now, it had the gloomy air of an abandoned nursery. The oak panelling, once polished to a conker-like shine, no doubt, was scarred by years of student life. The two tall Gothic Revival stone windows at the front of the room had a pretty view of the immaculate Great Lawn, but cobwebs drooped grubbily in their arches. A fly buzzed hungrily over the remains of Claire Potter's Crosswords and Ice Cream party, which still filled the table beneath the windows. Ursula hadn't paid attention this morning, but she now saw that two chairs were pulled up to the far end of the table and that a pink plastic spoon had been left inside one of two empty ice-cream tubs there.
Trott and Ursula reached the far end of the room, where various officers were attempting to set up the temporary Incident Room the D.I. had demanded.
'Sir,' barked a woman officer as Trott and Ursula headed past her. She was in the process of turning the tatty billiard table in the centre of the room into a makeshift exhibits area.
The far end of the room was dominated by a huge, bulky television set positioned in one corner and surrounded by a semi-circle of ugly brown and beige sofas and armchairs with stuffing poking out here and there. They were now occupied by six or seven police employees. Behind the television, shelves were piled high with books, video-cassettes and old boxes of Scrabble and Monopoly.
'Right. Sit down, Miss Flowerbutton,' Trott finally said, gesturing towards two chairs drawn up either side of a red Formica-topped table in the corner opposite the television set.
They took their seats, and Trott opened a brand-new, blue-backed, A4-sized notebook, picked up a pen, and pressed PLAY on the tape recorder that had been set up on the table.
'Now, tell me, how did you come upon the body this morning?' he asked.
'Well, it all started because I didn't want to be a lemon,' said Ursula. 'You see, Dr Dave has very, very strict rules . . .'
Ursula tried to remember everything from the absolute beginning. She told Trott that Dr Dave encouraged students to enter his rooms if he didn't answer a knock; finding India on the _chaise-longue_ ; the dreadful realisation that she was not breathing; discovering Otto asleep in the JCR.
'He was in that chair,' added Ursula, pointing to the wing-backed chair in the middle of the room.
'Constable!' yelled Trott to no one in particular.
Four officers simultaneously sprang to their feet from the TV corner.
'That chair's evidence. Get it photographed, take fibre samples, and bag it up,' ordered Trott. 'And don't touch it.'
He turned his attention back to Ursula.
'Right, miss, you were saying a young man was in here?'
'Yes, Otto Schuffenecker. It was he who realised India was definitely dead.'
'Someone track down an Otto Schuffenecker and interview him today!' Trott barked at his team. 'How did he know she was dead?'
Ursula recounted to Trott the grim moment when Otto had turned back the blanket covering India, the sight of her neck, the blood on her white gown, his attempt to resuscitate her.
'Then I asked him to run and get the porter, Deddington. And I stayed with India . . .'
A single tear finally rolled down Ursula's cheek as she recounted the sight of the lifeless girl. Trott offered a tissue, but his expression remained unchanged, emotionless. He transcribed her story methodically, with the air of a man who was writing a polite thank-you note for a particularly uninspiring Christmas gift. Every few minutes another member of his team would come and interrupt him with a question, but he seemed to possess a remarkable ability to delegate detailed tasks without ever losing the thread of the story Ursula was telling him.
'Why are there so many people here?' she asked, noticing that the stream of police personnel coming in and out of the room was ever-increasing. She was going to need to fully understand the police investigation if she was to be able to report on it properly in her article.
'Murders aren't solved by two friendly bobbies these days, miss. It's not like you see on TV. There will be at least a hundred people on this over the next forty-eight hours. Before the evidence disappears.'
'Really?' said Ursula, her faith in Hercule Poirot brutally shattered.
'Yup. Although most of us in the force are well aware that it's more than likely to be an utter waste of police time and resources.'
'Why?' asked Ursula.
'Murders are often the least interesting crimes to solve. Most of them are straightforward domestics. It's usually The Husband or The Boyfriend. Look close to home, I always say. Look close to home. Now, you were saying that you stayed in the room while this –' Trott looked down at his notes '– Mr Schuffenecker . . .'
'Prince,' Ursula corrected him.
'Eh?' The D.I. frowned.
'It's Prince Otto Schuffenecker. He's an Austrian royal – at least he would have been if they hadn't abolished them after the First War. He gets very offended if anyone calls him Mr.'
'Right, er . . . thank you. So you were saying, this Prince fellow, he's gone down to the lodge to alert the porter?'
'Yes, and—'
But Trott's attention had been diverted. 'Hello, Doc!' he exclaimed, signalling to a very tall, white-haired gentleman who had just entered the JCR.
'Who's that?' asked Ursula.
'Home Office pathologist,' replied Trott. 'Dr Euan Rathdonnelly. Everyone just calls him Doc.'
Despite his advancing years, Doc, who was dressed in an immaculate Prince of Wales check suit, strode energetically across the room towards Ursula and Trott. He possessed the sort of pink-cheeked vigour and beaming demeanour of the jolly headmaster Ursula had so adored at pre-prep school. His elegance was marred only by an ungainly black duffel bag weighing down his left shoulder.
'Marvellous murder you've got here!' declared Doc gaily, sounding like a cheerfully excited father who's just watched his child in their school pantomime. 'I had a quick peek. Hope you don't mind.'
'All right to get on?' asked Trott, shaking Doc's hand warmly.
'Right away,' the pathologist replied.
He plonked the duffel bag on the floor, unzipped it and retrieved a pair of Wellington boots, a large leather apron and some black rubber gloves. By the time he had put the items on over his Savile Row suit and lit a cigar, he looked, thought Ursula, like an upmarket butcher. Next he rifled around in the duffel and retrieved swabs, tweezers and a few plastic bags, revealing the sharp blade of a scalpel glinting inside the holdall. Ursula gulped.
'Right, miss, you stayed in the room while your friend went down to the Porters' Lodge. Correct?' Trott continued. 'Yes,' replied Ursula. 'And while I was waiting, that's when Dr Dave arrived. He was due to give me my first tutorial on—'
She stopped suddenly, noticing from the corner of one eye that an officer was clearing away the remains of Claire Potter's tragic Crosswords and Ice Cream evening. Ursula jumped up from her chair and dashed across the room towards the table beneath the Gothic windows.
'Wait!' she gasped. 'Don't move a thing!'
A rather surprised-looking officer froze, a pot of melted ice cream in one hand and a black bin bag in the other. 'Sir?'
'Miss, I'd appreciate it if we could continue with your statement,' responded Trott, impatiently checking his watch.
'But – the ice cream – it's a _clue_ ,' said Ursula.
An agitated Trott harrumphed, 'Cool it, Nancy Drew.'
(Ursula was secretly flattered by Trott's remark. After all, when it came to girl detectives, Nancy Drew was her heroine.)
'Trott, just a minute,' Doc said, bounding over to where Ursula was standing. 'If there's a clue in the ice cream, I would like to know what it is.'
Trott's usually stony face twitched very slightly with irritation, and he looked at Ursula coldly. Then he turned to his tape recorder and stated for the record, 'Interview temporarily suspended. Witness believes there is something of significance in . . . ice cream.'
Frowning, he turned off the tape recorder and headed over to the table beneath the Gothic windows. He indicated to the officer that he should return the pot of melted ice cream to its former position.
Ursula seized the moment.
'It's not exactly _in_ the ice cream, this clue,' she stammered excitedly, 'but, look, if you sit down here . . .'
She slid into one of the two chairs that had been drawn up to the table, with the two empty tubs of ice cream in front of them. From the corner of her eye, she saw Trott raise an eyebrow sceptically.
'You can see that someone sat here while they were eating ice cream,' she continued, pointing out the two pots on the left-hand side of the table, which were indeed scraped clean. Each tub would have contained enough ice cream for one person. The pink plastic spoon that Ursula had noticed earlier was still resting inside one of them. 'But no one touched the pots at the other end of the table, they're full of melted ice cream.'
Trott rapped his fingers impatiently on the table-top, but, undaunted, Ursula continued.
'You see, Claire Potter – a Fresher – held her first Crosswords and Ice Cream Society party in here last night. But it looks like only one other person came, because Claire sat here, with that person, and, well, they ate one small pot of ice cream each.'
Trott sighed. 'Shall we get back to the interview now, miss?'
'Sorry. I'm not being clear,' said Ursula. 'If you were sitting here last night, eating ice cream, perhaps all alone, waiting for someone, or perhaps with someone . . . well, anyway, _look_!'
From her seat, she pointed through the tall Gothic windows and out towards the lawn of Great Quad below.
Trott reluctantly squatted beside her. 'Nice view,' he said curtly.
'Whoever was here could see exactly who was coming and going across Great Quad all evening.'
'In the dark?' said Trott.
'It was a full moon last night,' she said. 'Beautiful and bright. India was at a party in the Old Drawing Room – I know because I saw her there – which is almost directly across the quad from here. At some point she must have crossed it to get to Dr Dave's rooms. Whoever was sitting in these chairs might have seen her.'
Trott looked at Doc and jerked his head to him. They moved off for a moment and conferred in low voices.
Within minutes, an officer was despatched to track down and interview Claire Potter, and the police photographer was summoned from Dr Dave's rooms to the JCR. After the photographer had snapped photographs of the table, chairs and ice-cream tubs, Doc used a pair of tweezers to place the used tubs and single plastic spoon in clear bags and label them. Officers started delicately dusting the table and chairs with powder for fingerprints. A cigar wedged into the side of his mouth, Doc then headed off into Dr Dave's rooms to examine India's body.
'I think you'd better tell me all about this party,' Trott said to Ursula.
'Of course,' she said. 'It was _lovely_. It was Wenty's party. Lord Wentworth Wychwood. He's a second year. Rowing Blue. That's why he has the Old Drawing Room, it's a privilege for Blues. He's terribly pleased with himself, being an Earl and a Blue – you can imagine, Inspector.'
Trott concurred. 'I know the type.'
'Anyway, the party was, I suppose, a sort of first-night-of-term thing, everyone in white tie and ball-gowns.'
'Very nice,' commented Trott tartly.
'Yes, Detective Inspector, it _was_ heavenly. I mean, there were pink bonbons, and matching pink champagne. My American classmate Nancy thought it was _very_ unhygienic that Wenty – that's Lord Wychwood's nickname – used dirty champagne saucers because his washer-uppers had disappeared. Nancy's from New York so she's terrified of mono.'
'You say this, er –' Trott looked down at his notes '– Lord Wychwood served alcohol in champagne saucers?'
'Yes,' said Ursula.
' _Only_ saucers, not flutes or wine glasses?' asked Trott.
'Saucers. I'm sure of it. Wenty is far too much of a snob to use flutes.'
Trott beckoned an officer over. 'Track down a Lord Wentworth Wychwood for interview. He's a second year here. Person of Interest.'
'Yes, sir,' said the officer, and dashed off.
'Oh, you don't think – Wenty?' Ursula gasped, the horrible image of India's bloody neck zooming back into her mind. 'He was in love with India, he wouldn't want to . . .'
'So Wenty was Lady India's boyfriend, was he?' continued the D.I. 'Oh, dear me . . . that makes him a Person of Great Interest.'
Ursula's mind suddenly flashed back to Trott's earlier comment. What had he said? 'Look close to home. It's usually the husband or the boyfriend.' But Wenty? Ursula suddenly found herself feeling defensive.
'I don't think he'd ever harm her. Wenty genuinely cared about India.'
'How do you know?'
'When she got upset about something –' here Ursula paused. Instinct told her it wouldn't be wise to elaborate on India's reaction to hearing Wentworth call her 'Unforgettable.' '– and ran out of the party, he chased after her.'
As Trott scribbled his notes, Ursula noticed the door to the JCR being slowly pushed open. First, a very large bottom came into view, then an enormous tea trolley containing a huge, steaming tea urn and piles of cups and saucers. The owner of the significant bottom, a pudgy boy dressed in bleached jeans, an anorak and an old pair of grubby plimsolls, dragged the trolley all the way into the room. He was followed by a group of six students clutching clipboards, posters, drawing pins and various lists.
'. . . you see, this is why I am _not_ happy about first years holding events in here. What a mess!' the pudgy boy was complaining loudly, seeing the table still covered in pots of melted ice cream. 'And guess who gets to clear up? Yours truly, the JCR President.'
'Your committee will help,' volunteered a girl dressed in a tie-dyed tunic and clogs, with a ring through her nose and long, matted blonde dreadlocks. She grabbed a bin bag from the bottom of the trolley and started chucking the pots into it.
An officer dashed forward. 'Excuse me, miss, please don't touch those!'
Seeing his uniform, the girl backed away from the table and twisted a dreadlock nervously. Trott stood and strode over to the interlopers, saying, 'I'm afraid you can't come in here.'
The pudgy boy drew himself up to his full height, summoning all the courage he could muster.
'Sorry, I think there's been a misunderstanding. I'm Ben Braithwaite, JCR President,' he said, then paused, as though waiting for some kind of recognition of his authority from Trott. None was forthcoming.
'Anyway, I – well, we, the JCR committee – have this room booked from four till six for the Freshers' Tea,' Ben went on, trying to sound as official as he could.
'I'm Detective Inspector Trott, Thames Valley Police Criminal Investigation Department. Unfortunately, a homicide has taken place in college today. We're using this space as a temporary Incident Room.'
Ben and his committee looked appalled.
'So it's true? A _murder?_ In college? I thought it was just people starting silly rumours,' said Ben.
'I'm heading up the enquiry,' replied Trott.
'Oh, what a shock.' Ben seemed stunned for a moment, then looked at his watch. 'Problem is, eighty Freshers are due to arrive here in . . . gosh . . . less than five minutes, to meet their College Parents. I'm afraid there's not much I can do about that now.'
'Detective Inspector,' added the dreadlocked girl with a pleading look, 'it is _crucial_ for the _welfare_ of the Freshers that they each have a College Mother and Father.'
Trott raised his eyebrows and sighed.
'All right, get on with it,' he huffed. 'I'm going to clear our lot out of here. We need to get a proper Incident Room set up down at the station anyway.'
Ben nodded gratefully and reached down to the bottom tier of the tea trolley from where he procured a plate piled high with dry-looking biscuits.
'Custard Cream, Detective Inspector?'
For the first time, Ursula detected the glimpse of a smile on Trott's face.
'Don't mind if I do,' he said, gobbling one down. 'And I'll have a cup of tea while you're at it.'
Ben hurriedly produced one, and after a few gulps, looking satisfied, Trott told Ursula, 'Right, Miss Flowerbutton. I'm going to suspend my interview with you for now. I'll want to talk to you again, most likely. Don't leave Oxford without letting me know.'
Thank goodness, she thought to herself. The interview session had been surprisingly tiring. She still had so much to do today: she needed to get to the library after the tea and find the article Dr Dave had assigned, get to Jago's room by six for her editorial meeting with him, and then back to her room and spend the evening reading, as planned.
While the JCR committee prepared for the tea party, Ursula perched on one of the windowsills overlooking Great Quad. Her mind was whirring, and she hastily jotted a few notes on her pad:
_—India crossed Great Quad last night. Her white dress and tiara would have been hard to miss in the moonlight. If Claire Potter and her mysterious ice-cream-eating companion had seen her, could they have followed India into Dr Dave's rooms?_
_—But why would Claire Potter, or her unknown friend, have any interest in disposing of India Brattenbury—_
Ursula suddenly stopped writing. The tiara! What on earth had happened to the tiara?
Monday, 1st week: teatime
The clank of cheap teacups interrupted Ursula's cogitations. She looked up from her notebook. The police team had started to relocate from the JCR, but she must speak to Trott before he left. She speedily weaved her way through the bewildered-looking Freshers, who were gradually filling the room and helping themselves to cups of tea.
'Excuse me, Detective Inspector,' Ursula said, spotting him by the door. 'There's something I forgot to tell you. India was wearing a tiara at the party – pearls and diamonds, I think it was some sort of family heirloom – but it wasn't with her this morning when I found her.'
'Hmmmm,' he mused. 'We'll investigate.'
After he'd left, Ursula joined a group of Freshers clustered around a trestle table where the tea urn had been set up along with plates of Custard Creams and a few packets of Malted Milks. She grabbed a cup, sighing wistfully to herself as she sipped the over-stewed PG Tips. The JCR tea wasn't exactly the comforting _g _â_ teaux_-fest she'd been hoping for. Still, she hadn't eaten since breakfast and was starving. She devoured a biscuit.
While the tea got under way, Ben Braithwaite and the dreadlocked girl pinned an elaborate family tree on the JCR noticeboard. Maybe, Ursula mused, it would be fun to have a mother and a father for a change. Having said that, her friends at St Swerford's had only ever mentioned their own parents with an agonised roll of the eyes and pretence at vomiting.
Ursula wandered over to the family tree to learn her fate, and saw that the dreadlocked girl was now pinning up a garish orange poster on the board next to it. In large, black lettering it screamed,
_STRESS?! ANXIETY?! DEPRESSION?!_
_SPEAK TO YOUR WELFARE OFFICER_
Beneath the words were grainy black-and-white headshots of the girl along with Ben Braithwaite and the rest of the committee. Already, Ursula noticed, the Welfare Officers were hungrily prowling the room, as though each was hoping to be the first to land a Fresher suffering from Stress! Anxiety! or Depression!
The Freshers did not seem to have been afflicted by the aforementioned psychological conditions. Rather, they had been struck by a highly contagious case of Murder in the Dark. The news of India's death had clearly spread rapidly through the college already and the Freshers were jittery, but the death had also added a frisson of excitement to things. The possibility that any one of the students standing in the JCR at that moment could be next had brought a heightened sense of drama to their first week at Oxford. The questions that usually dominated these events – Where did you go to school? What subject are you studying? Did you take a gap year? – were replaced by darker gossip. As Ursula scanned the family tree for her name, she overheard ever-more hysterical outbursts among the Freshers around her.
'. . . _I heard a third year say she was Queen of the Yahs. Must have been a Lefty who did it_.'
'. . . _thirty-five stab wounds, she was unrecognisable apparently_.'
'. . . _strangled. A homeless drunk broke into the don's rooms looking for port and throttled her with baler twine_ . . .'
Ursula turned her attention back to the family tree, an extraordinarily detailed diagram of College Mothers, Fathers, Sons, Daughters, and even student Grandparents. It took Ursula ages to find her name, positioned next to those of her siblings. She had two 'brothers' – Paul Davies, Medicine, and Matt Owen, Geography. Her 'sister' was Claire Potter.
Ursula traced the line upwards to her parents. Her 'father', it turned out, was Ben Braithwaite. He seemed perfectly nice, if a bit officious. Finally her finger landed on the name of her 'mother' – someone named Jocasta Wright.
The dreadlocked girl, who had now finished putting up her Welfare sheet, looked over at Ursula and said, ' _Man!_ I'm your mum!'
Jocasta Wright then enveloped Ursula in an enormous hug, nearly suffocating her. She reeked so strongly of joss sticks that Ursula sneezed violently.
'Sorry!' she said, extricating herself from Jocasta and her tie-dyed tunic.
'No, don't be. I don't want you to be sorry for anything. That's, like, a total waste of head space. I'm here for you, man. Great, man,' Jocasta said in a low, languorous voice.
'Thank you,' said Ursula sincerely.
'No, man, don't thank me. It's my _duty_ to be here for you. My college mum . . .' Jocasta broke off and looked wistful '. . .she was, yeah, totally uninterested. I met her at the tea and never saw her again. But I'm going to be a _proper_ mother to you, I promise.'
What did that entail? Ursula wondered.
'So, right, any probs, I'm here. I'm Chairperson of the O-S-S-T-D-A-C – sorry! That stands for the Oxford Student Sexually Transmitted Diseases Advisory Committee – so any worries on that front, honestly, you can talk to me. In confidence. Herpes, VD, AIDS, contraception, abortion, family planning clinics . . . personally, I prefer the coil to the Pill . . .'
As Jocasta listed the various diseases and unwanted outcomes of sex that might afflict Ursula during her time at Christminster, she wondered what her student 'mother' would think if informed – which Ursula had no intention of doing – that she was a virgin, and a very innocent one at that.
The nearest Ursula had come to sex was a French kiss with her second cousin, Reggie Vernon-Hay, in the attic at Seldom Seen Farm one Christmas. The snog had occurred more out of Ursula's sense of desperation than any real attraction. After all, she had reached the age of fifteen and never kissed a boy. This was not out of reluctance, but rather a supply issue. Boys were thin on the ground if you attended an all-girls school and lived on a remote farm with elderly relatives. Snogging Reggie was, for Ursula, the only way she was going to pass the vital First Kiss milestone before she turned sixteen. Otherwise, she feared, she would be doomed to end up like one of Jane Austen's lonely spinster characters, crocheting doilies in an ivy-strangled cottage.
Despite Ursula concluding, post-snog, that the experience of touching tongues with a boy was only marginally less revolting than that of swallowing the lumpy school mashed potato that was served with most of Swerford's horrid school dinners, she was excited that she had _finally_ snogged someone. Even if it was her cousin, which was sort of incest-y, it still counted.
When it came to sex, Plain Granny (a Catholic, strict) had attempted to instil into her granddaughter the belief that sex before marriage was a sin. Vain Granny (Church of England, lapsed) espoused a different view. Her motto – 'Don't have sex before breakfast, you never know whom you might meet at lunch' – was oft-repeated in front of her granddaughter.
The result of this eccentric sex education was that Ursula felt free to settle herself somewhere between her grandmothers' extremes. If she fell in love, she'd have sex, but she wouldn't have sex just to rid herself of her virginity, and she certainly wouldn't get married just to have sex. Marriage was a very long way off – and would happen only after her career as a writer had been established. It was 1985, after all, not 1935. Ideally, the falling in love would happen in Oxford and Ursula's virginity would soon be happily in the past.
'. . . and we offer free packs of condoms to every student, male or female,' Jocasta was saying. 'Personally, I find Durex the most reliable, and pleasurable, with _loads_ of K-Y . . .'
Ursula felt herself reddening.
'It's nothing to be embarrassed about, just come and ask me for them . . . I could get some now for you?'
Luckily for Ursula, who felt as though her face was the colour of beetroot, Jocasta was stopped in her condom-sharing tracks by Claire Potter, now heading towards them with Ben Braithwaite. She pointed at Ursula and said, 'There she is.'
'My second daughter,' said Ben, greeting Ursula with a handshake. 'Hello.'
'Hi,' said Ursula, relieved to be rescued from Jocasta, and realising that now was the moment to make up for her stand-offish attitude to Claire. 'Look, I'm really sorry I didn't make it to your crossword event,' she told the other girl.
'That's okay. We all had a _really_ , _really_ great time, just like I told that policeman when he came to talk to me earlier,' she said with a wink.
What was that supposed to mean? Ursula wondered. She noticed that Claire had put on makeup for the tea: a thick layer of orangey foundation that only half-disguised her acne, and a violet-coloured metallic lip gloss. Something seemed different about her. She seemed almost cheerful today. Almost.
'Did you get your reading list yet?' Claire asked.
'Dr Dave didn't exactly give us a list. He only asked us to read one article.'
'Professor Scarisbrick read out a list of fifteen books on Anglo-Saxon history I have to get through by next week. Luckily I learned speed-reading at school,' Claire said with a slightly smug smile.
Ursula felt vaguely disconcerted. Why was one tutor assigning fifteen books and another just one article?
'If you have any questions about your tutorials, or any problems,' began Ben, 'think of talking to me as just like talking to one of your parents . . .'
Ursula nodded vaguely. She never told anyone her parents were gone unless it came up directly.
' . . . if you need a chat, _anytime_ , just come to my room . . .'
'I'd _love_ to,' said Claire, fluttering mascara-clogged eyelashes at him.
Jocasta pursed her lips disapprovingly at Ben, who coughed and cleared his throat.
'Sorry, not _my_ room – that wouldn't be appropriate – come and find me, er . . . here in the JCR. I'm always around.'
'Okay,' said Ursula, unable to imagine confiding in Ben about anything personal.
'And don't forget, you'll be allocated Moral Tutors by the end of the week,' added Jocasta. 'So there are plenty of people to talk to if, you know, you're suffering from Stress, Anxiety or Depression, Sexually Transmitted Diseases—'
'Hi, Moo,' interrupted Ursula, saved by the sight of her fellow historian bounding up to them. The pony-tailed blonde had traded her Roedean gear for a dark green tracksuit which bore the name of the college and a pair of crossed oars printed in scarlet on the front. In her left hand she had a clipboard and pen, and in her right an enormous wooden oar.
'Hi, all. I'm recruiting for the Freshers' Boat. Ursula, you'd make a good Seven with those long legs and arms of yours.'
'But I've never rowed,' protested Ursula.
Her Daddy Long Legs proportions had made sport a misery for her at school. She had liked riding and that was it. She had absolutely no intention of joining any kind of team whilst at Oxford.
'Hardly anyone's rowed before they come Up,' said Moo. 'The second years are going to teach us. Apparently it's really pretty down on the river in the mornings.'
An image of sunrise on the water popped into Ursula's head then, and suddenly the rowing thing seemed irresistibly romantic in a way that netball and lacrosse never had.
'I'll give it a try,' she said.
'If you're doing it, I'll do it too,' said Claire, grabbing the clipboard from Moo and scribbling her name at the top of the list. 'Since we're sisters now.'
Parents, sisters, Moral Tutors, condoms, boats – Ursula suddenly felt overwhelmed. Perhaps it was from being an only child and growing up in such a remote place, but she liked – no, needed – to be alone at times. The silence of the library beckoned alluringly. Maybe she could escape before her 'brothers' appeared.
Ursula managed to slip out of the JCR unnoticed and onto the landing. Dr Dave's pile of books was still on the dusty window seat, but he was gone. The constable was nowhere to be seen, and the door to Dave's set was closed, though semi-audible voices were coming from behind it. Ursula stood as close to the door as she could and put her ear to it.
'. . . looks like it's the jugular . . . but there isn't enough blood . . . ask the Coroner to order a post-mortem . . .'
Doc's Scottish accent was unmistakable. What had he been saying? Something about the jugular, and blood. No, 'there isn't enough blood'. What did that mean? Not enough blood for what, exactly?
So intent was Ursula's concentration that when Nancy's face appeared beside hers, she jumped, her heart pounding.
'You scared me!' gasped Ursula. India's murder had set her on edge far more than she had realised.
'What are you doing?' said Nancy. 'What about the tea party?'
'Ssshhh,' said Ursula. 'They're all in there, listen.'
Nancy put her ear to the door.
'. . . time of death?' they heard a male voice ask.
'That's Detective Inspector Trott's voice,' whispered Ursula.
'When was the victim last seen alive?' said another man.
'That sounds like Doc, he's the Home Office pathologist,' Ursula told Nancy.
'Maybe midnight,' said Trott. 'We need to speak to everyone who was at the party to establish who saw her last.'
'And what time was she found?' Doc asked.
'Nine in the morning.'
'Then the time of death was sometime between midnight and nine in the morning,' said Doc.
'Doesn't the hypostasis help?' asked Trott.
'Detective Inspector, you know as well as I do that too much has been claimed for the usefulness of hypostasis as an indicator of time of death. It's too variable.'
'What on earth are they talking about?' Ursula whispered to Nancy.
'Hypostasis,' she answered. 'When the blood stops circulating, gravity pulls it down to the lowest part of the body. The skin looks bluish-red.'
Ursula was impressed. 'How do you know that?'
'My older brother Frank's a med student, Yale. He's a super-geek. He's got some _horrible_ stories.'
Just then the doorknob turned, and the girls hastily retreated to the window seat and watched as D.I. Trott walked out together with a female officer. The two of them were talking so intently that they didn't notice Ursula and Nancy.
'W.P.C. Barwell, can you arrange for Lord Brattenbury to come down to the morgue later and formally identify the body?' Trott was saying in a low voice. 'I don't want the father seeing her like that.'
'Yes, sir, I'll make arrangements,' replied the other officer before they both disappeared down the stairs.
'This is so sad,' said Nancy.
'I know,' said Ursula.
She checked her watch. It was already half-past five. Where had the day gone?
'Nancy, I'm so sorry, I must go,' she said.
'What about the JCR tea?' asked Nancy. 'I need you with me for moral support.'
'I really have to get to Jago's.' Ursula was growing desperate about fitting everything in. 'I can't be late for the editor of _Cherwell_.'
Suddenly Nancy squealed. She and Ursula watched as two paramedics emerged from Dr Dave's rooms carrying a stretcher weighed down by a full body bag. Doc followed, closed the door behind him and then looked around. He immediately spotted Nancy and Ursula.
'Ah, Sherlock Holmes,' he said, eyeing Ursula. 'We meet again. Listen, you shouldn't be hanging around here. Nor should your friend.' Doc glowered at Nancy. 'This is a crime scene.'
'Sorry—' Ursula started to say.
'Oops!' The paramedic at the far end of the stretcher stumbled slightly as he reached the staircase.
To Ursula's horror, a mottled, bluish-looking foot suddenly lolled out from the bottom of the bag. A brown luggage tag was tied tightly around the big toe with a piece of string. India's name was scrawled on it in black capital letters.
'Ugghh—ggggh! I feel like we're in _Frankenstein_ ,' groaned Nancy, nonetheless transfixed by the ghoulish sight.
Doc hastily dashed to the end of the stretcher, stuffed India's foot back in the body bag and zipped it up. He said to the men, 'Can you accompany the victim to the mortuary? Then get hold of the Coroner and ask him to order an autopsy. I need to get her on the slab.' About to head down the stairs, he looked back at Ursula and Nancy and said, 'Girls, scarper!'
As soon as the men were out of sight, Nancy turned to Ursula and said, 'That toe. I think it's important.'
'How?' replied Ursula.
'It's got hypostasis. That suggests India's left foot was the lowest point of her body at the moment she died.'
'Right . . .' said Ursula, not quite sure of the significance of Nancy's conclusions. 'Look, I've got to go.' She made a move towards the stairs.
'Maybe I don't feel like your fancy British tea after all,' Nancy said, following her. 'I need a drink. Come to the Buttery Bar after your meeting with Jago?'
'I can't,' said Ursula. 'I'm sorry. I've got to go to the library as soon as I get back.'
'You, my friend,' complained Nancy, 'can be a real downer sometimes.'
'Ursula, yup, er . . . hi,' stuttered Jago as he opened the door to his room just after six that evening. 'Come in.'
Barefoot and dressed in jeans and a T-shirt, he held a poster for _Raging Bull_ and a packet of Blu-Tack in his hands. His room – a rather poky, north-facing space at the back of Magdalen's grand cloisters – was still in a state of disarray. 'Sorry about the mess,' he gulped. 'Haven't had time to unpack and sort anything yet.'
Jago stuck the Robert De Niro poster in the middle of the far wall. Ursula noticed the single bed was covered with more posters, from films like _American Gigolo_ , _Alien_ , _Scarface_ and _E.T._ A small sofa was heaped with several boxes of duty-free Marlboro Reds, a half-unpacked suitcase and that day's newspapers. Ursula glanced at the headlines, realising she hadn't looked at the news for days. The _Daily Mail_ headline screamed 'BRITISH TELECOM DITCHES RED PHONE BOX'. _The Times_ was more serious: 'REAGAN AND GORBACHEV TO END COLD WAR.'
'Hang on, let me move all this crap,' said Jago, clearing a space on the sofa so Ursula could sit.
'Thanks,' she said, taking her notebook and a pen from her satchel.
'Drink?' Jago offered.
'I'd love a cup of tea,' she said.
'My kettle's broken, sorry,' he replied. 'Glass of wine?'
'Okay,' said Ursula.
Jago somehow located a wine-box among the mess, and squirted white wine into two paper cups. 'Here,' he said, handing one to her. 'Now, let's get down to business.'
He planted himself on the sofa next to her, so close that his knee touched her left thigh. She moved a little to her right, to try and make space. Then he moved a little to his right, until his knee was squashed against her leg again.
'What's the latest?' Jago asked, polishing off his cup of wine in two swigs and helping himself to another.
'They're ordering an autopsy. They just took India's body from college.'
'How do you know?' he asked. 'You know we can't make any mistakes in this story. It's got to be spot on.'
'I saw them removing the body. It was horrible,' she told him. 'India's left foot fell out of the body bag. It had hypostasis in it.'
'Really?'
'I saw it with my own eyes,' she replied, taking a sip of the wine. It was warm and horribly vinegary.
'Didn't you say earlier that when you found India she was lying on Dr Dave's _chaise-longue_?' asked Jago.
'Yes. Why?'
'So, the hypostasis is in her left foot . . . meaning that was the lowest point of her body when she died.' Nancy had said exactly the same thing, thought Ursula. Jago continued hypothesising, 'The fact that the blood coagulated in her foot means that she died _exactly_ where you found her, because when you are sitting or lying on a _chaise-longue_ your feet are usually the lowest point of your body.'
Ursula noticed that Jago's right hand was now resting on her left knee. As subtly as she could, she slid out from underneath it. 'I don't quite follow.'
'What I mean is, we can conclude, from the foot situation, that India died _on_ the _chaise-longue_ ,' he said, sounding convinced and moving his hand back to her knee.
Ursula boldly lifted it off and put it back in Jago's lap. 'She was murdered lying down?'
'Maybe . . . If she died _on_ the _chaise-longue_ , it must mean that either she was already lying on it when her throat was cut, or she was standing close to it when she was attacked and fell back onto it.'
Ursula sensed that The Hand was now resting along the back of the sofa behind her head. She wished Jago wasn't quite so friendly as this.
'I think she was already on the _chaise-longue_ ,' said Ursula. 'It makes more sense.'
'How can you be so confident?'
'There was the stem of a broken champagne glass still in her hand when I found her,' Ursula replied. 'Surely she would have dropped it if she'd fallen backwards?'
'Which makes me ask, how did India end up with a broken champagne glass in her hand in the first place?' asked Jago.
Ursula winced. She felt The Hand touch the back of her head. She decided to ignore it, saying, 'It didn't look as though there'd been any violence or disturbance in the room. So, if there was no fight, if India was killed when she was lying on the _chaise-longue_ , maybe she knew her killer. Whoever it was came into the room – perhaps she was lying there waiting for them. She knew them so well and trusted them so much that she didn't even get up.'
'Perhaps,' Jago posited, 'she was expecting some kind of romantic rendezvous _.'_
'In Dr Dave's rooms? And that still doesn't explain the broken champagne glass.'
'Maybe she thought she was somewhere else, maybe she was too drunk to know that she'd gone to the wrong room,' he suggested.
'Wait!' said Ursula. 'What if it _wasn't_ the wrong room . . .'
Ursula stopped talking. From behind, she could feel The Hand twisting a lock of her hair.
'What is it?' asked Jago, looking surprised.
'I've got to go.' Ursula jumped up, yanking her hair from his hand at the same time. 'I've got someone to meet . . . um, _very_ urgently,' she said, trying to sound convincing. A drink in the bar with Nancy suddenly seemed like an excellent idea.
'But we haven't talked about police procedure yet. The protocol of covering a murder investigation.'
'Oh, yes, right,' stuttered Ursula. 'Any tips?'
'First off, when you write about the police investigation, get two or three verifications of every fact, every statement. Don't take anything the force says at face value. Having said that, my advice is pretty irrelevant – the police won't tell you anything useful anyway. You're going to have to figure out what happened to India by yourself.'
'You're saying I've got to _solve_ the murder?' Ursula said, feeling bewildered.
'Well, it would be a pretty boring piece if you didn't, wouldn't it? Ursula, unless you get the scoop and figure out who murdered India, your story gets killed.'
Jago might seem vague on the surface, but underneath the laid-back posturing, the boy was ruthless, she thought to herself.
'Won't be a problem,' she said, trying to sound as confident as possible while simultaneously wondering how on earth she was going to solve a murder, write a lead article for the newspaper _and_ complete her essay on the mysterious Early Covenanters, all in under a week. She really _did_ need to get to the library tonight.
Just as she was leaving, Jago said, 'Look . . . er . . . Ursula, I'd like to see you again.'
'Sure, at the _Cherwell_ office,' she replied in her firmest tones. Surely The Hand would stay where it should if they weren't alone for their next meeting.
'Well, no, actually.' Jago suddenly sounded awkward. 'Er . . . there's this cocktail thing at Vincent's on Friday. Why don't we go together?'
Was Jago asking her on a date? Ursula wondered, feeling perturbed.
'Er . . .' She hesitated. How awkward. It seemed a little inappropriate, professionally speaking, to go on some kind of date with her editor. But if she said no, would Jago be so humiliated he wouldn't even consider running her story?
'No strings attached,' he said, apparently sensing her reticence. 'Just, you know, as friends . . . colleagues.'
'Friends,' said Ursula, thoroughly relieved. 'Great.'
It would be interesting to have someone like Jago as a new friend, she thought. A cocktail at the mysterious Vincent's sounded fun. So long as The Hand wouldn't be revisiting The Knee.
Monday, 1st week: evening
Twenty minutes later, Ursula was relieved to find Nancy perched on a high wooden stool at the bar in the Buttery, a large, slightly dank half-cellar located in the old Kitchen Quad just to the east of the Porters' Lodge. The presence of two police officers, seated in a corner drinking orange juice and observing the scene like a couple of Stasi operatives, made for an edgy atmosphere.
'Hey, I thought you were going to the library,' said Nancy when she saw Ursula.
'I'm _en route_ ,' she said, taking the stool next to her friend. 'I'm going just as soon as I've had a shandy.'
'How was Jago?' Nancy enquired.
'Interesting.' Ursula decided not to say anything about The Hand for now.
Just then, Otto ambled in. He wandered up to the bar and leaned against it, slumping his head into his hands. Ursula noticed the policemen nudge each other and start whispering when they saw him.
'Otto, are you all right?' Ursula asked.
'Er . . . god . . . ugh!' Otto grunted. 'Bit hungover. I've spent the afternoon in the Eagle and Child trying to forget what happened last night. Didn't work.'
'Drink?' offered Nancy.
Otto shook his head. 'Maybe later.'
'Okay, so one jello shot for me, please,' Nancy asked the barman. 'And a shandy.'
'Thanks,' Ursula said.
Nancy's hands were shaking slightly. She seemed jittery.
'Are you okay?' asked Ursula.
'I've just never seen a real live dead toe before,' Nancy replied. 'And . . . the fact that it was India's toe. I mean, it's all so strange. I feel afraid, Ursula. There are police wandering all over college. They're everywhere.'
'They're here to protect us.' Ursula tried to sound reassuring.
'Maybe we should all carry rape whistles,' said Nancy nervously. 'If someone tries to attack us, at least we'd have that.'
The barman placed the girls' order on the bar. The fizzy shandy perked Ursula up and took away the taste of Jago's nasty white wine.
'No one else will be attacked,' said Otto suddenly. A film of perspiration was starting to appear on his face. He looked drained.
'You don't know that, Otto,' chided Nancy. 'Anything could happen. We need to protect ourselves.' She guzzled her jello shot in one. 'Another please?' she asked the barman.
'But – sorry . . . I mean . . .' Otto stuttered and wiped his forehead with a handkerchief.
'You mean what, Otto?' Ursula asked him firmly.
'I mean, yes, get the rape whistles. Keep safe.'
Just then, Ursula spotted Horatio sauntering into the bar. He had, naturally, changed for the evening and was now clad in a floor-length violet velvet silk kaftan.
'Great dress, Horatio,' said Nancy when she saw him.
'Thank you, darling,' he said, kissing the girls twice on each cheek and then looking curiously at Otto. 'God, you look like you've just been vomited from a sewer. Are you quite all right, Otto?'
'I have never known a day of such ghastliness,' the prince replied, looking ever-more pale.
'You'll get through this,' Horatio said kindly. 'I know you and India were great friends. You must be devastated.' He leaned against the bar and ordered two neat whiskies, handed one to Otto, who didn't touch it, and downed the other himself.
A few moments later, one of the officers in the corner walked purposefully over to the group at the bar. He immediately singled out Otto, saying, 'My colleague and I have reason to believe that you are one Otto Schuffenecker.'
Otto looked startled. 'I am,' he said in a whimper.
'We've been looking for you all afternoon. You're wanted down at the police station.'
'What! Why?' Otto was panicking.
'You're needed for interview and fingerprints.'
Otto reluctantly nodded his agreement. Without saying another word, the officer took him by the arm and marched him swiftly towards the door.
'Oh my _god_!' Nancy looked as though she were on the verge of weeping. 'Otto! _No!_ What about the Perquisitors' party tonight? I need you as my walker!'
But Otto didn't get a chance to reply. As he was taken up the stairs, Eghosa was descending them, dressed in fencing kit and carrying a foil. Eg looked askance at the police officer and his charge as they passed him.
'Ursula, now you've _got_ to come to the party with me tonight,' Nancy begged her. 'Next Duke might be there. Or some other Duke-type. Ursula, you _cannot_ let such a fabulous social opportunity pass me by.'
'I already told you, I can't go,' she replied. 'I'm sorry but we don't even know these Perquisitor people, and I've got to start my reading in the library.'
'Hey,' Nancy soothed her. 'Stop stressing. Dr Dave's only given us one little article. It'll take five minutes to read.'
'You really should go to the party,' Horatio urged Ursula. ' _Everyone_ will be there. You'll get some marvellous material for your article.'
This remark caught her attention. She knew that she desperately needed material, especially after her recent conversation with Jago. She felt her resolve softening.
'Maybe I could come for a _tiny_ bit, then go and study straight afterwards,' said Ursula, grateful that the Christminster Library stayed open all night. She'd still have plenty of time to start her reading.
'Told you you'd be in a ball-gown every night!' quipped Horatio.
'The same one, unfortunately,' laughed Ursula.
'I'll lend you a party dress,' Nancy said. 'I've got pretty much the entire Bloomingdale's eveningwear department in my room.'
'That's so kind of you,' said Ursula, suddenly delighted about the prospect of another ball-gown-wearing moment, particularly if the ball-gown was from a New York department store. Visions of a croissant-munching Audrey Hepburn in _Breakfast at Tiffany's_ , dressed in sunglasses and a Little Black Dress, came to mind. Perhaps Nancy could magically turn Ursula into a New York party girl for the night.
Eghosa came up to the bar and greeted Horatio and the girls. 'Mind if I join you?'
In his fencing regalia, he looked dazzling, Ursula thought. He put his mask and sharp-looking foil on the bar and took a stool next to her. She couldn't help staring at the sword – could it cut a throat?
'Just a Perrier, please,' said Eg to the barman.
Ursula could see that his mood today was very different from last night's. Gone was the suave, amusing disco-dancer, and in his place was a troubled young man.
'Are you all right?' asked Ursula shyly.
'It's not me I'm worried about. It's Wenty. He's in a bad way. The police have been in and out of our rooms all day. Wenty's freaking out.'
'Everyone in Oxford is terrified,' said Nancy.
Eg looked at the girls with concern in his eyes. 'Don't go anywhere alone, either of you. Always have someone with you, especially at night.'
'Maybe we should all walk together to the Perquisitors' party tonight?' suggested Ursula.
'I'd love to spend the evening with you,' Eg said. He caught Ursula's eye and she felt a flutter of excitement. 'I mean, I'd love to spend the evening with _all_ of you . . . but, actually, I'm going to give it a miss,' he continued. 'I said to Wenty I'd stay in tonight. Keep him company. We'll probably play Vingt-et-Un and get drunk.'
Eg finished his Perrier, settled his tab and got up to leave. As he departed, he said, 'Remember what I said. Go everywhere in a pair.'
Once he was out of earshot, Horatio said, 'No wonder everyone calls him Saint Eghosa.'
'Such a great guy,' agreed Nancy. 'Looking out for Wenty like that.'
'And us,' said Ursula. 'He's right. We should stick together. It's safer.'
Suddenly a voice from behind the girls bellowed, ' _Ciao!_ '
The trio turned to find Moo had entered the Buttery with her oar and a grinning Claire Potter in tow. Ursula wasn't sure why, but Claire's newly cheerful demeanour unnerved her. How could she be so uncharacteristically merry on such a dreadful day? Did she know something? What _had_ happened in the JCR last night? Ursula was determined to prise the information out of her.
'Heard you were in here! Hiding from your new parents already!' hooted Moo, jogging from one foot to the other. 'Nancy, I need you.'
'For?'
'The Freshers' Boat. We're recruiting a coxswain.'
'A who?' Nancy looked befuddled.
'The cox steers the boat. I need someone short and light. There's no rowing involved, you just have to have a loud voice to shout at everyone.'
'I can sure as hell yell,' she demonstrated accommodatingly.
'You're in,' said Moo. 'Training starts tomorrow morning. The crew is meeting in the Porters' Lodge to run down to the river.'
Moo turned and jogged away, Claire following. Just as Moo reached the door of the Buttery she looked back at them and added, 'See you at six a.m.!'
The Bloomingdale's eveningwear department, a.k.a. Nancy's wardrobe, was not, Ursula soon discovered, in the business of supplying simple, low-key, _Breakfast at Tiffany's_ -style cocktail frocks. Nancy, who had quickly swathed herself in a second skin of thigh-high, ruched gold lamé for the Perquisitors' party, offered Ursula a succession of ultra-trendy dresses to try.
'This would look _genius_ with your hair,' Nancy insisted, holding up a stretchy bright orange creation that looked more like a swimming costume than a party dress.
'A _bit_ bright for me maybe?' responded Ursula, dismissing the dress as kindly as she could.
'Or what about this?' Nancy went on, pulling out a green satin Vivienne Westwood-inspired puffball dress.
'I _love_ it,' Ursula gasped.
But before she could try it on, Nancy was stopping her, saying, 'No, wait, I have a _way_ better idea. This one.'
Ursula was soon happily adorned in a scarlet taffeta strapless mini dress printed with huge black polka dots. The tiny frock was held in place by what felt like suction around her chest, and frothed out into heavenly, miniature layers of thigh-high frills. Ursula had seen pictures of these super-fashionable 'ra-ra' dresses in Vain Granny's _Vogue_ s, but had never imagined she'd actually get to wear one.
'I think fur would work with that,' Nancy said, handing her a fake-fur crimson stole. Ursula wrapped the deliciously soft fabric around her shoulders as Nancy clipped glitzy _faux_ -ruby and diamond earrings onto her ears. A matching necklace and bracelet soon followed. They completed the outfit with silver fishnet tights and a pair of red suede shoes, which were far too small. Ursula dashed back to her room to find her old plain black court shoes, hoping no one would notice her feet.
'Okay, makeup,' said Nancy, when Ursula had returned. She opened her vanity case, which contained a Pantone of cheek and eye colours. Nancy applied emerald-green eyeshadow to Ursula's lids, matching mascara to her lashes and shimmery metallic blusher to her cheeks. She then insisted on sizzling Ursula's hair with a pair of boiling hot, high-tech American crimping irons. By the end of the terrifying process, Ursula's tresses resembled spun sugar.
Since there was no full-length mirror in either girls' room, Ursula could only review her look in thirds by standing on a chair and looking at various sections of her body in the mirror over Nancy's basin. She adored the dress portion of the outfit, but the makeup and jewellery were far more glitzy than she was used to.
'I look like David Bowie,' she protested.
'Stop!' ssh-ed Nancy. 'You look amazing. You look cooler than Sue Ellen in _Dallas_.'*
Finally, the girls were ready to go. As Ursula put her satchel over her shoulder, Nancy cried, ' _No!_ '
'What is it?'
'That bag. You _cannot_ accessorise a mini ra-ra dress with a high-school satchel. Here, borrow this,' Nancy said, offering Ursula a minuscule red velvet clutch bag.
'I can't fit my notebook in there.'
'You really need your work things?'
'Yes,' Ursula stated firmly.
Because, she vowed to herself, after the _teensiest_ stop at the Perquisitors' party, she was absolutely definitely, really and truly, honestly, going to the library.
*
'Is it normal in Oxford to throw a cocktail party in a dungeon?' Nancy asked a little later as they picked their way down the steep flight of stone steps beneath Magdalen College Chapel that accessed the Monks' Undercroft.
'It seems to be,' Ursula laughed, surveying the scene.
Despite being inhabited that night by a student DJ spinning poppy records by Wham! and Duran Duran, and sixty or so beautiful young Oxford undergrads in black tie, the place felt eerie to Ursula. Dripping church candles lit the vaulted, fifteenth-century crypt, and it was chilly enough down there that, even with the borrowed stole, Ursula's arms immediately prickled with goose pimples.
She and Nancy soon found the bar where they ordered two glasses of Buck's Fizz. Slightly desperately, they sipped their drinks as they scanned the room for a familiar face.
'Oh my god,' whispered Nancy suddenly. 'I cannot believe _she's_ here!'
'Who?'
'The "best friend" . . . I can't remember her name.' Nancy indicated someone on the other side of the room. 'I mean, shouldn't she be in mourning?'
Ursula spotted Isobel Floyd huddled in a distant corner surrounded by a group of boys. Looking like a groovier Little Lord Fauntleroy, she wore dark purple velvet knickerbockers with bows at the knees and a matching jacket with leg-o'-mutton sleeves. A lacy blouse peeked out from underneath the jacket.
'Oh, poor thing, I think she's crying,' Ursula said. 'Look, someone's just given her their handkerchief.'
'I guess she is really upset about India,' Nancy conceded.
'I think not.' The familiar voice came from behind them.
Horatio Bentley had arrived, to Ursula's and Nancy's relief.
'Horatio, help! We don't know anyone,' wailed Nancy.
'Don't worry, I'll introduce you to the Perquisitors,' he assured her.
'What the hell is a "Perquisitor" anyway?' Nancy asked.
'It means the original owner of an estate – from the Latin _perquisitum_ , for purchaser. My personal translation is White Tie Wankers . . . ha-ha-ha-ha-ha!' Horatio's belly jiggled beneath his kaftan as he laughed. 'Crikey . . . _brilliant_ idea for a weekly column. Jago would love it.' He retrieved a small Moleskine notebook from a pocket and scribbled WHITE TIE WANKERS on a blank page.
'So they're kind of like a high-school clique?' said Nancy.
'I suppose. The difference is that this clique, like most others in Oxford, is not bound by ties of friendship but by disdain for other people. You too, girls, will soon join a clique and spend your evenings laughing at the people you hung out with during Freshers' Week.'
'Which ones here are members?' asked Ursula.
Horatio pointed out four boys across the room, each one floppier-haired than the last.
'Rupert Bingham, Tom Higginbottom-Jones – fondly known as Wobbly Wobbly-Bottom to his friends – Alexander Fitzwilliam-Hughes and Lucian Peake. Between them their families own half the British Isles. Anyway, they ponce about in dinner jackets all term, providing reams of gossip for _Cherwell_. Among the mantelpiece-conscious, the Perquisitors' stiffie* is considered the ultimate invitation this week. Personally I say, the thicker the card, the more boring the party. Hmmm . . . that would be a _truly_ cruel opening for my article about tonight . . .'
Horatio frantically scribbled the line in his notebook, a mischievous grin on his face. 'By the way, Ursula, very fashionable outfit.'
'Thank you,' she said. The red polka-dot dress was turning out to be great fun – she had felt herself attracting admiring glances from a couple of boys as they sauntered up to the bar. 'It's Nancy's.'
'Looks like it,' replied Horatio. Ursula wasn't entirely sure whether to take this as a compliment.
Rupert Bingham, a sandy-haired boy with a smooth, shiny face, waved to Horatio and came over to greet him. Ursula noticed the boy's eyes hungrily devouring the sight of Nancy in gold lamé, before he directed an equally lascivious look at her in the polka-dot dress.
After Horatio had introduced the girls, Rupert pleaded, 'Now, Horatio, you _will_ give the party a nice write-up in your column, won't you?'
'I'll try, but I can't promise, Roo darling.' Horatio's tone was puckish. 'After all, that wouldn't be fair on all the parties I've slagged off, would it?'
While the others chatted, Ursula's gaze wandered and she soon spotted Dom Littleton mingling with friends. Despite the low lighting he was wearing his blue-lensed sunglasses. She watched, intrigued, as he sauntered up to Isobel, who pointedly turned her back on him and carried on talking to another boy. What was she so furious about? wondered Ursula. Horatio had been right when he'd encouraged her to come to the party earlier. She could definitely pick up some material here.
'Horatio, I think I need your help with the story,' she said as soon as Rupert Bingham had departed. 'Jago says the police won't tell student journalists anything. Have you got _any_ ideas about who might have wanted India out of the way?'
'Whodunnit?' Horatio raised one eyebrow. 'India was . . .' He paused, as though searching for the right words. 'Complicated. Spoiled. Difficult. Deceitful—'
'How?' asked Ursula.
'Well, I'm not suggesting that what she did was so deceitful that the Deceived-In-Question would have wished to cut her milky throat but, frankly, India didn't care who she trod on.'
At this point Horatio directed his gaze at Isobel, whose fury seemed to have miraculously dissipated. She was now talking intently with Dom Littleton, the hint of a smile on her face.
'Isobel? She did something to Isobel?' asked Ursula. 'But they were best friends.'
'So?' said Horatio. 'Don't you have best friends you are violently jealous of?'
Ursula shook her head. Plain Granny had seen to that. Jealousy was simply not allowed at Seldom Seen Farm.
'Hey, I know _exactly_ what you mean,' said Nancy. 'Literally _all_ my friends at Northwestern are totally jealous of me.'
'I can only imagine!' guffawed Horatio. Then he continued, 'Call me conceited, but I wrote a _terrifically_ witty column about India and Isobel last term. It was called _Les Soeurs Uglies_ , the thesis of which was that India and Isobel were so competitive and envious of each other that they were just like Cinderella's Ugly Sisters – cleverly in disguise as best friends and great beauties.'
'Horatio, are you saying that Isobel killed her bestie?' said Nancy, amazed.
'All I am saying is that Oxford is a very intense place. Rivalries get out of hand here. If you offend other people for too long, you generally exit this place with an assortment of meat cleavers through your back.'
'But what did India do to Isobel that was so awful?' Ursula was desperate to follow this lead. She knew she was onto something.
'All I know is that at the end of last term, Dom decided to direct _Hamlet._ Henry Forsyth was set to play the title role. Isobel was to be Ophelia. Then, the next minute, everything changed. Dom suddenly gave India the Ophelia role, and Isobel was demoted to her understudy. Everyone knows that Isobel is an even better actress than India was. It was bizarre.'
'So India stole her best friend's role last term?' Ursula clarified.
'Horrid, isn't it?' said Horatio. 'Then, at Wenty's party, we all find out that Dom has gone barmy and given India Henry Forsyth's role of Hamlet. The next thing we know, Girl-Hamlet is dead. Ah!' he said abruptly. 'We were _just_ talking about you!'
Isobel Floyd had arrived at the bar. She gave Horatio a smooch on each cheek.
'Dreadful about India,' he said.
'It's just so sad. Losing my greatest pal,' Isobel agreed. She visibly paled as she added, 'It's terrifying, to think there's some madman on the loose killing beautiful undergraduates. Makes me feel sick. I mean, I could be next.'
She shuddered. 'Anyway, I've got a scoop for you, Horatio. _Darling_.'
He raised his eyebrows. 'Pray tell.'
'I've agreed – with a very, very heavy heart, of course – to take on the Hamlet role.'
'You must be scared stiff!' Horatio blurted out. 'Your pretty throat could be next!'
Isobel whacked him hard on the arm. He looked sheepish.
'For _god's sake_ , Horatio,' she huffed. 'Anyway, as I was saying, I'll be taking on Hamlet. Obvs . . . in India's honour. We're going to dedicate the play to her. You won't forget to mention that in your article, will you, about the dedication? I know she'd want the show to go on, as they say on Broadway. She was such a great friend, truly my best friend in the world.'
Nancy nudged Ursula and hissed into her ear, 'The sub-plot thickens.'
'Of course I will, Isobel,' said Horatio with a coy smile. 'Any other Good Works you would like mentioned in my scoop?'
Isobel looked slightly confused. Then she said, 'Yah, you mean, like, the well I dug in Bangladesh on my gap year?
Horatio spluttered with laughter and Isobel reddened, realising she'd been played.
'You're hateful, Horatio,' she snarled. Then she suddenly burst into tears. 'I'm so-ooo-rrr-rreeeee-ggghhh,' she hiccupped, wiping her eyes on her frilly cuff. 'It's just, India, she was my soul . . . uggghhhh . . . mate. I just hope they find the murderer soon.'
'You may be able to help with that,' Horatio told her.
'What!?' she snapped, hurriedly snivelling her tears away. 'Why do you think _I_ know anything about the murder?'
'Ursula's trying to solve the case for _Cherwell_. Just tell her anything you know, it could really help.'
'Gosh, yes, of course, _anything_ I can do to help. Anything.' Isobel turned to Ursula and smiled sweetly. 'Let's go up into the quad so we can talk privately.'
*
Thank goodness she had her notebook and pen in her satchel, thought Ursula, as she followed Isobel up the stairs and out into Magdalen's main quad. Isobel led her towards the cloisters and sat on a stone bench there, huddling her knees to her chest.
'You know everyone calls him "Horrid Bentley"? Horrid by name, horrid by nature,' said Isobel, lighting a cigarette. She offered the packet to Ursula. 'Want one?'
Ursula was about to say she didn't smoke, but then thought better of it. She needed to bond with this girl. She took a cigarette from the packet of Marlboro Reds and twirled it in her fingers, delaying the moment she would have to attempt to light it.
'He's terribly cruel about people in his column. When he wrote about India he called her "The Hereditary Husky*". She wouldn't have been seen _dead_ in a Husky. Oh, god, I don't mean _dead_ dead. I meant, metaphorically dead. But – ugh! – now she _is_ dead!'
Isobel dropped her head into her hands. She seemed genuinely upset.
'Have you got any idea who might have done this to India?' asked Ursula gently.
'Why would _I_ know anything? I mean, of course I don't!' Isobel dragged needily on her cigarette.
'You were her best friend. Did she tell you about anything – _anything_ – that was worrying her?'
'Honestly, her life was great. I mean, she was going to play Hamlet. It was her dream.'
'Is it true,' Ursula ventured, somewhat timidly, 'that she stole the role of Ophelia from you last term?'
'Did Horrid tell you that? God, he's a stirrer. She got the part because she was . . .' Isobel hesitated '. . . better suited to the role. It was all very professional. I was fine with understudying. Really.'
Not very convincingly said, Ursula thought to herself. But she played along, asking, 'Do you mind if I write this down?'
'Of course not. I wouldn't want this to come out wrong in _Cherwell_. Scribble away.'
Ursula retrieved her notebook and pen and turned to a fresh page. 'Where did you go after Wenty's party?'
'Why?'
'India was killed sometime between leaving the party and nine in the morning when I found her. I'm trying to work out if anyone saw anything after the party, before she went to Dr Dave's rooms.'
'I went to bed. My room's in the Monks' Cottages. I'm right above the laundry. India and Otto are on a staircase near mine.'
'Were you alone?'
Isobel took a moment to answer. Eventually she said, 'I went to my room alone. I went to bed alone.'
'Dom didn't stay over?'
'He had tech rehearsals for _Hamlet_ with the lighting crew at eight the next morning. He decided to come back here to Magdalen, get a good night's sleep.'
'Did you see India between leaving the party and getting to your room?'
Isobel paused. She looked long and hard at Ursula. A sad expression clouded her face.
'I suppose there's no point in pretending. They were having a row. India and Wenty.'
'What about?' asked Ursula.
'You,' declared Isobel.
' _Me?_ 'exclaimed Ursula.
'India stormed out of the party, don't you remember?'
Ursula recalled India's abrupt exit from the room, and Wenty chasing after her.
'Well, Wenty ran out after her, and then Dom went after them both, and of course I went after Dom.'
'And Henry Forsyth went after you all, as far as I recall,' added Ursula.
'Henry was desperate. He said to me, when we were outside Wenty's room, that his "career" would never recover after what India had done. As if he has a career as an actor ahead of him! Horatio's nickname for him is The Rodent. Very fair, everyone thinks.' Isobel snickered. Then, suddenly looking frightened, she grabbed Ursula's arm. 'God! You don't think Henry did it?'
'Do you?'
'He's such a wimp, I can't imagine him murdering someone. But he had a motive, I suppose.'
'Where did Henry go when he left Wenty's staircase?'
Isobel released Ursula's arm and pursed her lips. 'I think he hung around, but I wasn't paying much attention,' she said. 'I was too distracted by the argument. We'd all chased down the staircase after Wenty and India, and there they were in the middle of Great Quad – on the _grass_ – and India was screaming at Wenty.'
'Why?' said Ursula.
'She was in a fury because she'd heard him call you "Unforgettable".' She was yelling at him that Unforgettable was his pet name for _her_. I heard her say, "There's only one 'Unforgettable' in your life and that's _me_."'
'What did Wenty say to that?'
'He told her, "Forgive me, my darling. It was only some random Fresher. It wasn't serious."'
'Oh,' said Ursula. She was rather offended to be labelled a 'random Fresher', but this just confirmed her doubts about Wenty's character: the boy clearly was as superficial as she'd suspected.
Isobel went on to explain that India, in a rage, had refused to accept Wenty's apology, and had accused him of being a terrible flirt.
'She was convinced that Wenty . . .' Isobel stopped abruptly. She seemed reluctant to go on.
'Convinced of what?' Ursula pressed her.
'That he was . . .' Isobel paused. She seemed uncharacteristically unsure of herself.
'That he was what?' urged Ursula.
'Sleeping around.'
'Who with?'
'Oh, um . . . God knows! Anyway, then Wenty started accusing _her_ of being the flirt. So she said, "I can't help it if everyone flirts with me!" Then she laughed. Wenty was fuming. He started talking about Dr Dave, accusing India of seducing him.'
'She seduced a _tutor_?' Ursula was properly shocked.
'Look, Dr Dave has affairs with lots of students. Everyone knows.'
Otto had mentioned something about her tutor having dalliances with his students, Ursula remembered, but she had thought he was exaggerating.
'But why did Wenty think Dr Dave and India were having an affair?' asked Ursula.
'They did have a _tiny_ fling, almost a year ago. Dr Dave spent a couple of weeks at Bratters over the Christmas holidays when India was a Fresher. _Supposedly_ he'd gone there to write. But there was more poetry going on in the rose bedroom with India than politics being written in the study. Dave's very . . . attractive, isn't he?'
'But he's a tutor!' exclaimed Ursula.
'I know. I mean, I'd never—' Isobel started. 'Anyway, Dave and India were infatuated with each other for a bit. But it was over in a few weeks. India told me all about it. They stayed friends. India saw Dr Dave as a _confidant_. Maybe Wenty was suspicious that there was still something going on. I mean, India was still always in and out of Dave's rooms, having endless long chats with him. Anyway, last night Wenty threatened to report Dr Dave to the High Provost, for sexual harassment.'
Maybe he should have, thought Ursula to herself. But she didn't say anything. Isobel was in full flow: 'India went mental. The only civilised thing about the scream-up was the way Wenty kept filling India's glass with champagne! Then the night porter came out and hollered at them to get off the grass, and I saw India run off towards Dr Dave's staircase, probably to warn him about Wenty's threat. She was very loyal to Dave.'
'It all sounds terribly complicated,' said Ursula.
'It is. When I heard that India had been found dead in Dr Dave's rooms, I couldn't believe it. There's _no way_ he did it.'
'How can you be so sure?' asked Ursula.
'Er . . . well, I don't know. I just don't think Dr Dave would do that kind of thing.'
Isobel looked away from Ursula suddenly. She lit another cigarette and inhaled deeply. Ursula dashed off some thoughts on her notepad:
_—Could Dr Dave have killed India? Had the affair with her been secretly reignited, as Wenty suspected? But . . . Dr Dave was sick when he saw India's corpse. Surely a cold-blooded killer wouldn't vomit at sight of own victim?_
'What about Wenty, where did he go when the night porter told everyone to leave?' Ursula asked.
'Must have gone back to the party, I imagine. I wasn't really paying attention,' Isobel told her.
Nor, unfortunately, was Ursula. She couldn't remember whether she'd seen Wenty at the party again or not.
'Anyway, this morning I woke up late,' Isobel went on. She faltered as she added, 'I couldn't believe it when my scout told me what had happened. Do you want a light?'
Ursula had completely forgotten that she still had an unlit cigarette in her left hand. She couldn't admit now that she didn't smoke so she put the cigarette between her lips and took a tiny, terrified drag to light it from Isobel's gold Zippo. It was disgusting. She coughed and spluttered. The cigarette went out.
If only I wasn't so square, she thought, I'd have learned to smoke at school like normal girls do. She made a note to herself to practise smoking in private before attempting it again in the presence of someone as cool as Isobel Floyd.
'It's a crap lighter,' lied Isobel, kindly saving her from further embarrassment.
'Look, I have to go, I've got rehearsals early tomorrow in the Burton rooms. If you need anything else, just come and find me there. By the way,' she added as she dashed off across the quad, 'that's a really cool dress.'
* The ultimate TV representation of eighties fashion, _Dallas_ (1978–1991) was a high-glam soap opera about the Ewings, a backstabbing Texan oil family. Typical looks included sequin-encrusted mohair sweaters, leopard-print, cleavage-revealing dresses, and silver knit batwing dresses. Having languished for decades in the depths of un-coolness, such items are now desperately trendy.
* Stiffie = Sloane speak for invitation, derived from the fact that smart invitations were once hand-engraved on exceptionally thick, stiff card.
* A shapeless, quilted green shooting coat, the Husky was key Sloane kit. The only person who looks good in one is the Queen.
Just as Ursula was about to return to Christminster – and the library she hadn't yet laid eyes on – she noticed a group of revellers entering the quad from the Monks' Undercroft. Nancy's gold dress shimmered in the moonlight.
'Yawn. Yawn. Ya-aaaa-awn!' Horatio's bellow was unmistakable. 'That party was as stiff as the invitation. Ursula!' he called. 'We're all going back to Dom's room.'
'Coming?' asked Nancy.
Perhaps there'd be more 'material' at Dom's gathering, Ursula reasoned. She'd go for half an hour and then, absolutely, definitely go to the library. 'Sure,' she said, following the group up a steep staircase to a smoke-filled room on the third floor of the cloisters.
Dom Littleton's room was easily large enough to accommodate a core group of ten or fifteen friends. Ursula was surprised to see Henry Forsyth in the room. He didn't seem pleased to be there – he looked as though he was about to punch someone. Dom's quarters weren't exactly luxurious: the only places to sit were the bed, a couple of hard chairs, the two window seats or a few grubby-looking cushions on the floor. His half-hearted attempt at decoration had consisted of draping the wall above his bed with an exotic hanging picturing the elephant-headed Indian god Ganesha. One corner was falling down. Ursula couldn't quite understand the attraction of Dom's room. Nancy, however, could.
'A bong!' she cried happily, kicking off her heels and plonking herself down on a cushion next to their host, who was already sitting cross-legged on the floor sucking on an enormous plastic tube full of smoke.
Ursula watched as Dom closed his eyes, exhaled a long plume of white smoke and then passed the peculiar contraption to Nancy. Ursula had never seen a bong before, but it was pretty clear what was going on. She had never touched drugs herself, and seeing Nancy giggling madly in a smoky haze didn't make Ursula any more inclined to try them.
She didn't really like the atmosphere in the room – everyone seemed focused on getting to the bong, the drugs. But she was here now and among India's friends. There might be clues to be uncovered, especially about the victim's last few hours.
Ursula sat down gingerly on the other side of Dom, very uncomfortably as it turned out. Her dress was so short that the only way to sit on the floor was with her knees pulled up awkwardly next to her.
'Sorry, man, the bong's going anti-clockwise,' Dom drawled to her.
'That's okay,' she replied, relieved.
'Did I see you talking to Isobel earlier?' he asked, filling a cigarette paper with loose tobacco.
'Yes, I'm writing an article about India for _Cherwell_. Isobel answered a few questions.'
'Turning detective?' Dom said, looking at her with surprise.
'Sort of,' she replied.
He fished a black substance out of a plastic bag and started crumbling it on top of the tobacco, rolled up the spliff and lit it.
'Smoke?' he offered. Ursula shook her head.
'So I guess you know all about India's infatuation with Dr Dave.'
'Isobel seems to think India and the tutor had a fling, but ages ago,' said Ursula.
'Yeah, man. Bet Isobel didn't tell you about _her_ one-night stand with Wenty, though!' Dom cackled, seeming completely stoned.
Ursula gasped. 'No, she didn't _._ What do you mean?'
'I followed India and Wenty out of the party. Isobel and Henry did too. We all saw the major slanging match between Wenty and India. First, India accuses him of being a terrible flirt—'
'—which he clearly is,' interjected Ursula.
'There's no argument there. But then, Wenty accuses India of still being involved with Dr Dave. She denies it, and then accuses Wenty of sleeping with her "best friend" – Isobel. My girlfriend! Wenty denied it, but India didn't believe him. She went nuts,' Dom recalled. 'That's when she ran off towards Dr Dave's rooms.'
'Dom, do _you_ think Wenty and Isobel ever—'
''Course they did. New College Ball behind the Pimm's tent this past summer. Everyone knew. Except India.'
'But, Dom, weren't _you_ upset?' Ursula couldn't quite believe how laid-back he seemed to be about his girlfriend's extra-curricular exploits.
'Man, do you dig _Siddhartha_?* I read it on my gap year in Nepal. I renounced all personal possessions, just like Siddhartha. Isobel doesn't belong to me. If she shags some idiot toff, that's her spiritual journey, man. When things go pear-shaped, I just say _Om._ Then nothing freaks me.
' _O-o-o-o-m-m-m-mmm_ . . .' chanted Dom, dragging deeply on the joint. He closed his eyes, crossed his legs, put his hands together in prayer, and sat there like a yogi, meditating.
His drug-induced trance didn't look like ending anytime soon. Ursula got up, amazed by how edited Isobel's version of the Great Quad row had been, and made her way over to Henry Forsyth, who was on the other side of the room nosing through Dom's record collection. Perhaps he could confirm this version of events.
'You a Run-D.M.C. fan?' said Henry when he saw Ursula. He was holding a single with the title 'It's Like That' emblazoned across the front. 'I dig hip hop.'
'Me too,' lied Ursula, deciding now was not the moment to admit that she was more of a sheet music kind of girl than a hip-hop fan.
Henry put the single on the record player and carefully lifted the needle onto the vinyl. Run started rapping, and an athletic-looking boy dressed all in black except for a white baseball cap, white sneakers and an enormous gold chain round his neck, suddenly broke into break-dancing mode. A little group, the most vocal of whom was Nancy, was soon cheering and clapping him on as he spun on his hips and shoulders.
'He should be on the stage,' declared Henry, admiring the boy's moves.
'I hear _you're_ a brilliant actor,' said Ursula.
Henry smiled for just a moment. 'To misquote Oscar Wilde, "To lose your Hamlet role to a girl once may be regarded as a misfortune, but to lose it twice looks like carelessness." No doubt you've heard about the scandal over my role?'
'Well, I was at Wenty's party last night,' Ursula replied. 'I saw the whole thing.'
'Quite a performance, wasn't it?' grumbled Henry. 'Although the drama in Great Quad after the party was truly worthy of an Oscar . . . and then a murder. God! Sunday night was like a Hitchcock film come to life.'
'Why do you think Dom gave India your role?' Ursula asked him.
Henry looked at her suspiciously. 'Why are you so interested?'
'I'm investigating what happened to India for _Cherwell_.'
'Press. I see,' he said sternly. Henry looked so fierce that Ursula wondered if she had blown her chance of getting him to open up. She should never have mentioned _Cherwell_.
'I wouldn't mention your name, I promise,' she said hurriedly. 'I'm just trying to find out a bit of background. That's all.'
'You misunderstand,' replied Henry. 'I'll only help you if you _do_ mention my name.'
'Er . . . okay.'
'I want _my_ Hamlet role back,' he declared, his face hardening. 'I wouldn't be in Dom Littleton's godforsaken room tonight unless I thought I could persuade him to get rid of Isobel. The publicity will help.'
'I _completely_ understand,' said Ursula, deciding she'd play along with him. 'It all sounds _very_ unfair,' she added with a sympathetic smile.
'It was absurd. What happened was _ab-_ solutely _ab_ -surd,' Henry complained. 'I've been building up to _Hamlet_ my entire life. Dom and I did some _awesome_ productions in our second year here. _Equus_ was the hottest ticket in Oxford last summer _._ Took all my clothes off – huge audiences! Dom and I had it all planned. We'd take on _Hamlet._ He'd direct. I'd star. We'd invite all the agents. Dom and I agreed on everything.
'Naturally, I wanted Isobel Floyd as my Ophelia. She's the best actress in Oxford. And the most beautiful. Only problem was, India Brattenbury thought that _she_ was the best actress in Oxford . . . It's such a pathetic cliché. They all want to be Ophelia, don't they?'
'I suppose so,' said Ursula.
'India didn't give a flying you-know-what about her best friend. All she cared about was herself. One minute Isobel was going to be Ophelia, the next minute India had the role. She seduced Dom—'
'What?' said Ursula. Had she heard Henry correctly? 'Did you say _India_ seduced _Dom_?'
''Course! That's how she got Isobel's role. The only talent involved was hers for shagging. Everyone in Oxford knew that India was pretty selfish, but stealing your best friend's role _and_ sleeping with her boyfriend – that's a double betrayal.'
Gosh, Ursula thought. This all made Louis XV's love life seem positively boring. 'Do you mind if I make some notes?' she asked Henry.
'Go ahead,' he replied.
Ursula jotted in her notebook:
_—Isobel: one-night stand with Wenty, while dating Dom_
_—Dom: infatuated with India, while dating Isobel_
_—India: dating Wenty, sleeping with Dom, confiding in Dr Dave_
_—Wenty: dating India, up to no good with Isobel_
'Anyway, I should have seen the whole Hamlet thing coming, I suppose,' continued Henry. 'When India arrived on Thursday night with Dom at Wenty's party, and they told me that _she_ was going to be Hamlet . . . God! I thought: He's really lost it. India had brainwashed him. The whole thing was utterly humiliating. I could have killed her. Tragically, I didn't get the chance.'
Henry laughed at his own horrid joke then his face lapsed into a cold, sullen expression.
'Anyway, I couldn't believe it when Wenty and India started rowing in Great Quad, and he suddenly accused her of shagging some tutor!'
'Apparently the affair with the tutor was over a while ago,' Ursula explained.
'Really? It was bad enough that India was stringing Wenty along as her boyfriend while she was bonking Dom. If she'd been at it with a tutor as well . . . Christ! No wonder Dom and Isobel started having a massive barney.'
Ursula was surprised. 'Isobel never mentioned that she argued with Dom that night,' she said. 'He didn't either. Isobel said that she and Dom agreed to spend the night apart because he had rehearsals early the next morning.'
Henry let out a hollow laugh. 'Isn't it amazing what people choose to forget when it suits them?'
'What happened with the two of them?' asked Ursula.
'Look, I'll try and get this right, but it's complicated. I was standing at the edge of Great Quad, with Isobel and Dom. When India accused Wenty of cheating on her with Isobel, Dom didn't really seem to react. It was as though he didn't care. But when Wenty accused India of sleeping with Dom, Isobel went bonkers on him.'
'Even though she'd slept with Wenty?' Ursula was incredulous at the story. From her virginal point of view, Oxford seemed to be in a state of permanent orgy.
'I know. Talk about double standards. Isobel whacked Dom on the head with her bag and announced she would be spending the night alone.'
'What did Dom do?' asked Ursula.
'He just said " _Om_ ", which really annoyed Isobel. She stomped off towards the Monks' Cottages, furious.'
Ursula couldn't believe that she'd been so naive. Isobel's version of events – that the row between Wenty and India had been over Dr Dave – was only a tiny portion of the truth. Horatio hadn't invented the rumour about India stealing her friend's role. Things were far messier than even he could have imagined.
No wonder Isobel had had her back turned to Dom at the beginning of the Perquisitors' party tonight – she was furious with him for cheating on her with India. Ursula wondered if giving Isobel the Hamlet role had been his way of trying to make up with her. Perhaps Isobel was even more ambitious than her late friend had been. She could conveniently forget her boyfriend's indiscretions if it meant being the star of the show. Jago was right: every story needed to be verified, two or three times over.
'So then Dom pushed off back to Magdalen—'
'Are you sure?' said Ursula.
Henry hesitated. Pondered for a moment. 'Well, I'm just _assuming_ he came back here . . . Maybe he did, maybe he didn't. You don't think . . . Dom would have wanted to hurt India, do you?'
Ursula shrugged her shoulders. 'Who knows? What happened after that?'
'It was all over pretty quickly. The night porter appeared from the gate tower and told everyone to leave. We all scattered, to god knows where . . . You will remember to mention my name in the article, won't you?'
'Absolutely.' Ursula smiled at him.
'And please do feel free to suggest that I'd make a far better Hamlet than Isobel,' added Henry.
'Sure,' said Ursula, crossing her fingers as they said goodbye.
She looked at her watch. It was almost midnight already. A perfect moment to depart for the library, _à la_ Cinderella. She found Horatio and Nancy sharing a cigarette in a corner. After updating them on the various love affairs India was embroiled in before she died, she told them she was going.
'No way! Why? It's early!' complained Nancy, puffing smoke into the air. 'And you're not allowed to go anywhere by yourself, remember?'
'I'm just . . . tired,' said Ursula. 'I'll be fine on my own.'
'I'll walk you back,' said Horatio. 'Can't have you being murdered while you're reporting on a murder, now can we? Although that would make a _marvellous_ diary item.'
Ursula poked him in the ribs. 'That is so mean!' she cried. 'But I'll take you up on the offer. Let's go.'
As they left the party he added, 'By the way, I do envy you this story. There's going to be enough revolting gossip in it to fill the entire men's public urinals on Cornmarket.'
* The holy word _Om_ saves Siddhartha from his own suicide in Hermann Hesse's 1922 Buddhist novel. Required reading among Oxford students in the 1980s, the book was seen as an antidote to the Thatcherite 'Yuppie' decade.
Monday, 1st week: midnight
'Excuse me,' Ursula whispered as quietly as she could. The midnight hush in the Hawksmoor Library was so still, so soft, she felt wretched marring it for even a moment. The only sound in the room was the occasional turning of a page.
A prim-looking woman in her late twenties was seated behind the librarian's counter, reading intently. 'Olive Brookethorpe, Ms, Librarian', read the badge pinned to her dress. The navy Laura Ashley number was printed with lilac flowers, with frills at the neck and cuffs, and Ms Brookethorpe's smooth, shoulder-length dark hair was held back by a matching navy velvet Alice band.
The librarian momentarily glanced up from her work, peered disapprovingly at Ursula's ra-ra dress, lifted her forefinger to her lips, and pointed to a large sign on the desk. It read SILENCE PLEASE. With that, she snapped her head back down and continued scribbling.
Ursula wasn't quite sure where to start. A lone graduate student, who looked Lilliputian against the vast room with its domed ceiling, was sitting at a desk poring over a pile of manuscripts. Perhaps she could help.
Ursula tiptoed, as quietly as she could, towards the student, who, she soon saw, was clad in a Tattersall nightie and huddled under a knitted blanket. Fuzzy pink slippers covered her feet and her hair, an uncontrollable frizz of mouse brown, tumbled out of a tortoiseshell comb on the side of her head. She looked up when she heard Ursula approach.
'Don't tell me you're trying to borrow a book,' she said in a low voice.
Ursula nodded.
'Alas!' whispered the girl.
Without raising her voice, she introduced herself as Flora White, a Classics scholar. The problem with book-borrowing, she explained, was that Ms Brookethorpe was notorious in college for her desire to create as hostile an atmosphere as possible in the Hawksmoor Library.
'As gatekeeper of some of the rarest manuscripts in the world, she sees it as her duty to be extremely unhelpful. After all, she can only protect the books from the _revolting_ , vandal-like paws of undergraduates if no one borrows any of them, and the way to ensure no one borrows the first editions or ancient folios is to make the – theoretical – borrowing process as arduous as possible,' explained Flora. 'Photo ID, signed letters from tutors, university seconders – there's always one more impossible-to-find document that Ms Brookethorpe requires.'
'Why is she like that?' asked Ursula.
'The less time Brookethorpe spends collecting books from the secure stacks in the tunnels underneath college, the more time she has to re-catalogue fragments of the thirteenth-century Bible that was written and illustrated by the Augustinian monks who first taught at Christminster,' Flora told her. 'She's here every night working on it. If she ever finishes the re-cataloguing, she'll be made a Fellow. But she never will. Everyone knows that. It's pathetic . . . Look, the protocol is, you fill out a request slip and leave it on the librarian's desk, and then she's obliged to serve you.'
Flora turned back to her work. 'Sorry, major thesis crisis,' she said apologetically. 'Two days to finish thirty thousand words on the moral significance – or not – of Virgil's _Aeneid_.'
Ursula thanked her and tiptoed back towards the librarian's desk.
'Sorry to interrupt,' she said, sotto voce. 'I'd like to fill out a request slip.'
'Bodleian card, please,' hissed Ms Brookethorpe.
She could barely hide her disappointment when Ursula produced the University library card with her photograph on it from her satchel. The librarian took it and disappeared into a small office behind the desk, from which Ursula could hear the sound of a photocopier whirring. After ten long minutes Ms Brookethorpe reappeared and handed the card back to Ursula, as well as a slip entitled HAWKSMOOR LIBRARY – REQUEST FORM.
She thanked the librarian and scribbled '"The Apocalyptic Vision of the Early Covenanters", _Scottish Historical Review_ , Volume 43, April 1964, S. A. Burrell' on the request form, and handed it back. Ms Brookethorpe scrutinised the request form, beady-eyed, and then declared, 'That doesn't mean anything to me.' Impatiently, she rolled her eyes heavenward.
'It's an article, from the _Scottish Historical Review_ ,' explained Ursula.
'I am well aware that you seek an article from the _Scottish Historical Review_ , Miss Flowerbutton.' In a deeply offended tone, she then addressed Ursula with the following monologue: 'If you are to borrow from this esteemed library, you must inform me of the shelf number and pressmark of the volume you require. The title and author are not enough. Then you must fill out each of these –' she handed Ursula a white form, a pink form, a green form and a blue form '– with all the information about the book, and leave it in the box on the librarian's desk. The pressmarks can be found in the catalogues at the other end of the library.' Ms Brookethorpe pointed towards the most distant part of the room.
Ursula set off. A monolithic statue of Thomas Paget, seated on a throne with a tower of books on one side and a sword and shield on the other, was prominently positioned in the centre of the room. As she traipsed the considerable length of the gallery, she allowed her gaze to drift towards the arched, double-height windows that overlooked Great Quad.
The north side of the room was lined with ornate bookcases. Over each section, the subject of the books below was indicated in Latin, inscribed in grand gilt lettering. Ursula passed ' _Theologia_ ', ' _Historia Antiqua_ ', ' _Historia Orientalis_ ', and ' _Philosophica_ ', all the while painfully aware of the click of her heels echoing on the black-and-white marble flags as she progressed along the gallery.
She finally found herself confronted by the three massive oak chests that contained the Hawksmoor Library catalogue. Within each were hundreds of cards containing the details of every book belonging to the college, organised alphabetically. Ursula quickly located the Bi–Bu section, wrote down the pressmark and shelf number of the article on the various forms and headed back to the librarian's desk, where she held out the forms to Ms Brookethorpe.
'All forms must be left in the librarian's box,' she ordered officiously, pointing to a completely empty brown leather box on her desk. Ursula put the forms inside.
'How long does it take to get a book?' she asked.
Ms Brookethorpe gave her a stern look and then regarded her watch.
'It's almost half-past midnight. As such, Night Time Lending Rules apply. We are _very_ understaffed. Depending on where your volume is located in the stacks, which contain over a hundred thousand items, well, there really is no predicting how long it could take to bring a book to you,' she concluded with a smile. She cocked her head to one side, as if to say, ' _Surely_ that will put you off.'
When Ursula replied, 'I'll wait,' Ms Brookethorpe huffed, looking irritated.
Ursula found a seat at the desk closest to the librarian's and started unpacking her satchel. She watched from the corner of her eye as Ms Brookethorpe continued working. Eventually, the librarian peered casually in the box, as though she had no recollection of Ursula's recent request. With a look of surprise on her face, she removed the slip and read through it, terribly slowly, before disappearing into the office again.
While Ursula waited, her head resting drowsily on one elbow propped on the desk, her gaze alighted on the window and thence the Great Quad below. However trying the acquisition of the _Scottish Historical Review_ article was proving to be, the moonlit scene more than made up for it. It looked so mysterious out there, the gables and turrets casting elongated shadows across the lawn. The college grounds were quiet, as though they had fallen into a deep sleep.*
Suddenly, Ursula sat bolt upright, alert, and rushed over to the window, pressing her face against the glass. A woman – she was sure it was a woman, she could make out the bottom of a knee-length skirt – was walking, head down, across the lawn. She appeared to have a bottle of champagne in one hand.
_Wait. No. It can't be_ , said Ursula to herself.
But it was. How could she not recognise the silhouette of that frumpy calf-length A-line skirt, the outline of hair cut unflatteringly short, the impression of glasses . . . Claire? Claire Potter was roaming the college at almost one in the morning, walking on the sacred, forbidden grass of Great Quad, with a bottle of champagne in her hand? She hadn't seemed that type at all. Ursula watched as she headed in the direction of the Monks' Cottages, soon disappearing into the passage leading to them. What was she doing going down there . . . wasn't that where India's rooms were? Who would want to go there alone in the dark, the day after a murder? Wasn't she scared?
Just then, Ursula heard the squeak of Ms Brookethorpe's pumps and looked round to see the librarian walking – as slowly as someone with two functioning legs could possibly manage – towards a locked bookcase nearby with the words _Historia Caledonia_ inscribed above it. She took a large brass key from her pocket, unlocked the door to the bookcase and retrieved a weighty leather-bound volume, which she brought over to Ursula's desk and wordlessly plonked in front of her with a heavy thud.
'Thank you,' whispered Ursula, picking up the book and rising to leave.
Ms Brookethorpe wrestled back the tome and slapped it on the desk. She tapped the spine of the book and pointed to a label, which read CONFINED TO THE LIBRARY in threatening black letters.
'You do realise this volume cannot be removed from the Hawksmoor Library?'
'Why?' asked Ursula.
'It is signed by the great historian S. A. Burrell, which makes it far too valuable to be taken to an undergraduate's room. It is priceless and cannot be replaced,' the librarian explained victoriously.
Ursula sat back down, took out her notepad and pen from her satchel and opened the _Scottish Historical Review_ at the Contents page. Her heart sank as she noted that the 'article' began on page 146 and ended on page 402.
'No pens!' Ms Brookethorpe rebuked her, snatching Ursula's from her hand. 'You may have this back when you leave. Readers are strictly forbidden from using ink in the Hawksmoor Library. Pencils only. 2B. You may collect one from the Porters' Lodge and the cost will be added to your Battels.'
This woman wanted her to lose the will to live, or at the very least the will to read, thought Ursula. She dug around and found a pencil in her satchel.
Ms Brookethorpe conceded that point with a wilted sneer and slunk back to her desk.
Ursula looked at her watch. It was already after one in the morning! She turned to page 146 and read the first sentence of the article:
_In the spring of 1639 Dr Walter Balcanquhal, soon to be Dean of Durham for his services as amanuensis and pamphleteer on behalf of King Charles I, remarked the astonishing arrogance of the Scottish Covenanting leaders in their various petitions and manifestos—_
Who was Dr Walter Balcanquhal? wondered Ursula. What was an 'amanuensis'?
She examined the footnote at the bottom of the page, hoping it might help. 'See "Queries of D. Balcanq. to the King as to the Declaration" (Wodrow MSS, Folio LXVI, No. 34 [1639?])', she read. Clarity did not envelop Ursula, even a tiny bit. She scanned the next lines:
_In 1643 while the Solemn League and Covenant was under negotiation with England, the poet Drummond of Hawthornden pointed out the seeming absurdity of Covenanting aspirations._
The lengthy footnote to this sentence read, 'William Drummond of Hawthornden, "SKIAMAXIA: or a Defence of a Petition Tendered to the Lords of the Council in Scotland", in _Works_ , 1711.'
It took her at least forty-five minutes to get through the first few paragraphs of the article. The reading for her essay was going to be much more involved than she'd imagined when Dr Dave had asked her to look at 'one short article'.
She stared dejectedly out of the window, exhaustion starting to overcome her. How bizarre: there was the ungainly silhouette of Claire Potter again. This time she was crossing the lawn of Great Quad in the other direction, from the Monks' Passage, the bottle of champagne still in her hand. When Claire reached the west side of the lawn, she disappeared towards the Gothic Buildings.
Ursula checked her watch again. Almost 2 a.m. As her tired eyes lingered on the moonlit quad, her brain suddenly sprang into life. What if . . .? she wondered. What if Ms Brookethorpe had been here late last night, working on her fragmented Bible project? Would she have seen anything? Something? Someone? Out there, either going to or leaving Dr Dave's staircase? Could she have seen the murderer?
Ursula took off her high heels and padded softly over to the librarian's desk.
'I was wondering . . .' she started.
Ms Brookethorpe's head shot up in irritation. 'What is it?'
Hmmm, thought Ursula, it might not be the best approach to dive straight in and ask if the librarian had seen Lady India's killer last night.
'Er . . . well, I'm kind of . . . stuck.'
Ms Brookethorpe regarded her suspiciously. 'Stuck?'
'Some of the language in the article . . . I mean, do you know what an amanuensis is?'
'It's the Latin for a scribe. Secretary.'
'Gosh, you're clever,' said Ursula.
For the first time, she saw Ms Brookethorpe look the tiniest bit pleased.
'Oh, well, no . . .' she started, embarrassed. 'I've got a doctorate in Middle English. University of Monmouthshire.'
'Do you really work here all night on the Bible fragments?' asked Ursula.
Ms Brookethorpe nodded. 'It's my passion.'
'So, you're here every night?' Ursula was getting closer.
'Mostly. Oh, except Sundays. I sing in the college choir every Sunday evening.'
Drat! thought Ursula. The one night Brookethorpe wasn't here was last night.
'But choir traditionally never takes place on the Sunday of First Week, so I came up here as usual.'
'Really?' said Ursula, trying not to let her excitement show. 'It's a terrific view from here over the quad at night.'
'I'm usually too busy working to notice it. Although last night . . . _frightful_ row out there. Couldn't concentrate at all. Terrible what happened to that poor girl. Unthinkable. In such a wonderful place . . .' Suddenly, Ms Brookethorpe seemed to want to chat.
'I know,' agreed Ursula. 'Did you see anyone out there you didn't recognise last night, after the row? Anything . . . unusual?'
'I've been racking my brains about that. Wondering, did I see something? Hear anything? Could I help the police? All I know is, well, there was a jolly good shouting match out there, _on the grass._ We all heard who was in love with whom. You know, usual student nonsense.'
'Well, thank you for your help with the Latin word,' said Ursula. 'I'm going to read now.'
'But what _was_ odd,' interjected Ms Brookethorpe, 'now I come to think of it . . .'
Ursula turned back to her. 'What?'
'Well, it was just . . . oh, it's probably nothing.'
'It might not be nothing.'
'I suppose at the time I didn't think anything of it. But a few minutes after the night porter had sent everyone home, I saw someone. Alone. Heading towards the Monks' Cottages. It must have been, oh, half-past midnight or so by then.'
Surely it hadn't been Claire Potter the librarian had seen? But then, it _could_ have been Claire. After all, she'd been right next to the scene of the crime for part of the evening. Claire could have seen India approaching Dr Dave's staircase, followed her into his rooms and attacked her. But why, after that, would she have gone to the Monks' Cottages, rather than the Gothic Buildings where her own room was located? What was there in the Monks' Cottages that so interested her? And why would she be going back tonight? Ursula wondered what Claire's motive for murder could have been anyway; even if she'd been upset that India had branded her a 'lezzy', was that a reason to slit someone's throat? This wasn't _Carrie._* Or was it? wondered Ursula, with a slight shiver.
'Was it a girl you saw?' she asked.
Ms Brookethorpe let out a bitter little laugh. 'A girl prowling the college grounds at night? I don't think so.'
'You think it was a man?'
'I _know_ it was a man. I know the cut of that jacket. The swoop of that . . .' here Ms Brookethorpe looked away, wistfully '. . . hair. In the full moon, I could see David – sorry, I mean, Dr Erskine – down there as though it were broad daylight.'
'Dr Dave?' gasped Ursula. 'How can you be sure it was him?'
'I just _know_. But don't say a thing to anyone,' the librarian begged her.
'No, of course I won't. But how peculiar that he was out there at that time,' said Ursula.
'Well . . . Perhaps it wasn't him. I don't want to cause any trouble. No, I must have made a mistake – I'm sure of it now. Please forget what I said about Dr Dave. Promise me,' she added, desperately clutching Ursula's hands, 'and I'll let you take a duplicate copy of the _Scottish Historical Review_ up to your room overnight.'
* Pubs and bars closed at 11 p.m. in England in the 1980s. It was impossible to buy a drink after that, except in a nightclub. By midnight, the grounds of Oxford colleges were deserted, the drinking carrying on in students' rooms.
* In _Carrie_ , Brian De Palma's 1976 high-school horror movie, the bullied outcast takes revenge on her class by killing them all at their Prom. _Glee_ , get lost!
Tuesday, 19 October, 1st week: 2.30 a.m.
How could she ever forget, Ursula wondered to herself as she wearily descended the library stairs, that Ms Brookethorpe had been so utterly convinced – and then so unconvincingly unconvinced – that she had seen Dr Dave in the Great Quad last night? Clutching the _Scottish Historical Review_ under one arm, Ursula felt vaguely guilty that she had accepted Brookethorpe's clear bribe to forget her tale, but getting the extra time to study the impenetrable text was an offer too advantageous to turn down.
'It's a bit late for the library, isn't it?' a male voice echoed up the stairwell.
Ursula started, afraid.
'Who's there?' she gasped, peering into the shadows at the bottom of the stairs.
'It's me, Eg.'
'What are _you_ doing here?' Ursula asked fearfully. Hadn't he said he was spending the night taking care of Wenty? If so, why on earth was he wandering around college at half-past two in the morning?
'Wenty's fallen asleep – finally. I wanted some air so I took a stroll round Great Quad. I saw you in the window of the library. I thought I'd wait and see you safely back to your room.'
'Thank you,' said Ursula uneasily, coming around the bend in the staircase to speak to him face to face. She wasn't sure whether to be grateful for or suspicious of Eg's gallantry. What if he was lying? What if _he_ was the killer, and had been waiting for her . . .
There was an awkward silence while Eg drank in the novel sight of Ursula in Nancy's party dress.
'Wow,' he said admiringly. 'You look more beautiful than . . . a Christmas tree.'
Ursula just smiled shyly, flattered.
'Look, I didn't mean to freak you out. Come on, I'll walk you back. We don't want another fatality tonight, do we?'
'You think the killer's here, in college? That's really horrible,' said Ursula as they set off.
'We just need to stay safe.'
As they walked under the gate tower, Ursula glanced through the window of the lodge and saw two uniformed police officers inside, both talking to the night porter.
'What are they doing here so late?' she asked Eg.
'I bet they'll be here twenty-four hours a day for a while. Hey – Ursula, where are you going?' he said, as she opened the door to the lodge.
Ursula turned and winked at him. 'Just checking my pigeonhole.'
'There won't be anything in it now.'
But Ursula ignored him and went inside, Eg following reluctantly. The pigeonhole was of course empty, but she grabbed a piece of paper and pretended to write a note while she listened to the police officers' conversation with the night porter.
'. . . you're saying you didn't see anyone unexpected that night?' asked one officer.
'I don't think so,' replied Nicholas Deddington. 'Everything was as usual.'
'When you say "as usual", sir, can you explain _exactly_ what you mean by that?' the other officer said.
'The main entrance is locked at midnight. Anyone arriving after that rings and I let them in.'
Just then Nick became aware of Ursula and Eg. 'I've got a parcel for you, Miss Flowerbutton,' he said. 'Wouldn't fit in your pigeonhole. Here.'
She thanked him and took the parcel, which had been wrapped in brown paper and tied with string, and stamped with the Dumbleton-under-Drybrook postmark. 'Grannies!' cried Ursula. 'A parcel from Granny and Granny,' she told Eg. She couldn't wait to get back to her room and open it.
A few minutes later, as they made their way along the west side of Great Quad towards the passage leading to the Gothic Buildings, Ursula said, 'You know, India and Wenty had the most dreadful row out here last night, before she died.'
'Wenty told me the whole story tonight over cards,' Eg recalled.
'Do you know if he came straight back to the Old Drawing Room after the argument?' asked Ursula. 'I don't remember seeing him again after he'd gone after India, do you?'
'Is it true you're writing this up for _Cherwell_?' Eg asked.
She nodded.
'I'm not going to be much help, I'm afraid. I was hanging out with Christian, DJ-ing for most of the night, so I didn't notice where Wenty was. There were so many people coming in and out.'
'True,' said Ursula. 'What time did you get to bed?'
'Two . . . three maybe . . .'
'Was Wenty there when you went to bed?'
'Wait – you don't think he did it? Come on!'
'I'm not saying that. I'm just trying to establish where everyone was last night.'
'Look, I was so knackered by the time I got to bed that I can barely remember my head touching the pillow. But when I woke up the next morning, very late, around midday, Wenty was still asleep in his room. We had a fag together in the Old Drawing Room. Of course, we had no idea that something terrible had happened to India.'
'What did you talk about?'
'Oh, mainly what an awesome party it had been. You know, the usual rubbish. Wenty bragging about how cool he was, that sort of thing.'
At this, Ursula sighed. Wenty was just as conceited as he seemed.
'Did he say anything about the argument he'd had with India?'
'He only mentioned it in passing. Wenty and India had so many rows that I didn't take much notice at the time. He seemed pretty chilled actually. He was extremely happy that the washer-uppers had come back and cleaned up. When I went to bed the Old Drawing Room was trashed, but when we got up it was immaculate.'
As Ursula's staircase came into view, she noticed someone emerging from the entrance, though she could only make out the silhouette of a caped figure. Eg must have seen the same person because he grabbed her by the waist and drew her swiftly into a flowerbed against the wall of the Gothic Buildings.
'Ouch!' yelped Ursula as a thorny rose bush pricked her legs.
The figure approached and peered at the flowerbed.
'Ursula! Daughter! I've been worried about you. Where have you been?'
It was Jocasta. Ursula's mind was playing tricks on her. She had mistaken the floaty tunic she was wearing for a far more sinister cloak.
'What are you _doing_?' Jocasta asked, squinting at her for a moment before spotting Eg's arm around Ursula's waist. 'Okay. Right. Sorry, man. I can see I'm interrupting something private,' she said apologetically. 'Am I an over-protective parent or what!'
'No,' protested Ursula, removing herself and her grannies' parcel from Eg and the flowerbed. 'You're not interrupting anything. Honestly.'
'I thought you were the murderer,' Eg explained as she followed Ursula back onto the path, 'so I grabbed Ursula and we hid in the rose bushes.'
Dubious of his explanation, Jocasta raised one eyebrow at the pair. 'Look, guys, it's _fine._ Just make sure you're being safe,' she said in an understanding tone. 'Have you got condoms?'
'It's not like that,' Eg tried to explain. 'We're not . . .'
But Jocasta was already forcing one into Ursula's hand. 'The instructions are on the packet,' she said, soon wafting away along the Gothic passage and leaving Eg and Ursula to whatever she imagined they wanted to be left to do. Mortified, Ursula shoved the condom into the front pocket of her satchel, hoping it was too dark for Eg to see.
'Right, I'll leave you here,' he said. 'Unless . . . would you like me to . . . um, well . . . escort you up the stairs?' He looked longingly at her, and she felt a flutter of excitement.
'No, honestly, I'll be fine from here.' She smiled, trying to regain her composure. 'Goodnight, and thank you for walking me back.'
Ursula had only taken a few paces when she heard Eg's voice again. She flinched, on edge. What did he want now? It was so late.
'Anytime. In fact, I was wondering . . . would you let me . . . well, erm, maybe . . . take you out to dinner?'
Phew, thought Ursula to herself. If Eg was asking her on a date, the indications were that he wasn't about to kill her now before she went to bed. Relief that she was not about to have her throat slit was soon replaced by the ecstatic realisation that Eghosa Kolokoli, the dishiest disco-dancer known, as far as she was concerned, to womankind, was asking _her_ to have dinner with him.
'Dinner,' Ursula repeated. The state of disbelief brought on by the word almost paralysed her with delight.
'At Chez Romain? On Turl Street?' asked Eg. 'It's French. Very nice.'
Before he could continue Ursula had blurted out, 'I'd love to.'
Ursula was desperate for bed, but the intense excitement of being asked on her very first date set her mind whirring. As she started to ascend the staircase, her brain bubbled over with delicious, dinner-date-type questions. What would she wear? What would she and Eg talk about? Would they kiss?
She had no idea how on earth she'd get to sleep tonight after so much drama, but she'd have to try. In less than four hours' time, she had pledged to be in the Porters' Lodge to jog down to the river for her first coaching session with the Christminster Freshers' rowing squad. She'd save her dear grandmothers' parcel as a treat for later.
Only . . . what was that? Ursula thought she heard a strange sound coming from the landing above her. She stood stock still and listened. The sound came again. Ursula's heart thudded. Silently, she removed her shoes and inched as quietly as she could up the last few steps. Now it was clear the sound was a human cry of something like 'Whhh—aaa-yyyy', followed by hiccupping and loud sobs.
Ursula tiptoed up onto the landing between her and Nancy's rooms. The sobs were becoming more intense, and were definitely coming from her friend's room. Was Nancy having some kind of crisis? Perhaps she was terribly homesick for New Jersey? Suddenly the sobbing turned into a gurgling, choking sound. Ursula froze. Was her friend being strangled? Was there a killer on the loose in college, as Eg thought?
Ursula didn't hesitate another moment. Heart pumping, breath suddenly short, she threw down her satchel, parcel and the _Scottish Historical Review_ and flung open the door to Nancy's room, ready to take on an attacker.
The sight that met Ursula was bizarre. The column of light coming from the landing behind her illuminated a shadowy figure writhing madly on the floor, while Nancy cowered on the bed, brandishing a hockey stick.
'Ursula!' she screamed. 'Thank god you're here. He's gone crazy.'
Now, you never know where things will lead in life. Being prep-school leapfrog champion had seemed, to Ursula, to be one of those achievements that was not going to add up to much. Until now. Reptile-style, she leaped upon the writhing form, squashed the flailing arms under her knees, and pushed its head to the floor.
The room was deathly quiet for a long moment. Then Nancy shrieked, 'I think you've killed him. Oh my god!'
The body beneath Ursula squirmed. Then it said, 'I am _not_ dead,' sounding irritated. 'But I wish I were. I deserve to die.'
Ursula instantly recognised the Germanic inflection. Otto. He started sobbing violently, almost toppling her. She stayed put, not sure whether it was safe to release her captive. What had he done to make Nancy so afraid?
'Nancy, what happened?' gasped Ursula.
'He just burst in here, like, going crazy, hysterical, saying, saying – that – that – oh, ggg-o-ugh-o-od!'
Now Nancy was blubbing. She wept as though the Trevi Fountain had sprung a leak.
'That he-he – _hiccup_ – he – _hiccup_ – thinks – said – but no!!! It can't be.'
She collapsed on her leopard-print pillowcase, mumbling unintelligibly, while Otto squirmed beneath Ursula. He managed to crick his neck about halfway round so that she could see a bloodshot left eye, and croaked, 'I know who killed India.'
Ursula slowly released her prisoner from her grip. After he'd wobbled uncertainly to his feet – he was definitely drunk, she thought – Otto brushed down his velvet smoking jacket and tuxedo trousers then stood sheepishly, hanging his head.
'Otto, did you just say that you know who killed India?' asked Ursula slowly.
'Yes,' he rasped.
'But how?'
'How?' wailed Otto, gulping back tears. 'How do I know? Because . . . i-i-i-it was . . . I think . . . it w-w-was . . . _me_.'
Ursula just looked at him, stunned, while Nancy elaborated.
'He's turned into some kind of lunatic. I thought he was going to kill me when he showed up in here, drunk and raging. Here, Ursula, tie him up,' she added, chucking her a yellow silk taffeta sash belonging to a ball-gown.
Otto offered Ursula his hands, which she tied in embarrassed silence, knotting the sash three times before finishing with a huge bow. He sank dejectedly into the chair by Nancy's desk, looking like a very strange birthday gift, buried his head in the yellow bow and awkwardly blew his nose into it.
'Noooo! That's Bill Blass demi-couture,' Nancy objected.
The misfortunes of Nancy's party frocks were not uppermost in Ursula's mind. Oddly calm, she looked hard at Otto and said, 'Try and remember exactly what happened.'
'I lied,' he told her. 'I lied to the police. When they questioned me today, they asked me when I last saw India. They were taking my fingerprints, my blood. How could I tell them the truth? I said I hadn't seen India after she left the party. I told them I'd fallen asleep in the JCR, that I hadn't seen her up there. None of it's true! I thought I could pretend. But I can't. I had to tell someone. That's the trouble with being a devout Roman Catholic. You need to confess. I thought Nancy was the right person to hear me out.'
'Why?' She looked amazed.
'You're from America. You have so many murderers there. You understand them.'
'I _so_ do not!' retorted Nancy crossly.
'Just tell us what _really_ happened, Otto,' Ursula pressed him. 'The truth.'
'It all started with Wenty. I knew that he wasn't serious about India, that she was just another conquest to him.'
'Wenty doesn't seem like your average douchebag,' protested Nancy. 'He was just charming at his party. And, _boy_ , is he hot.'
'The good-looking ones are _always_ the worst,' Ursula said. She wasn't at all surprised to hear that Otto thought Wenty was a player.
Otto continued, 'But _I_ was serious about India.'
' _You_ and India?' said Ursula, astonished.
'When she and I were Freshers, well . . . we . . . she was my First Love. There was this one fine autumn day, our first term. We took a punt out on the river.'
Otto's eyes had slightly glazed over, as if he were back on the waters of the Isis with India, not trapped in a college room in makeshift handcuffs.
'We drifted lazily along. We drank port and ate wild boar pâté that I'd brought back from the _Schloss_. I told her I'd shot the very boar we were eating. It wasn't true, but she was impressed – she loved to shoot – and that's when she invited me to Bratters. I felt . . . elated. An invitation to a prestigious English shoot! I knew my father would be proud.'
'Austrian parents sound _so weird_ ,' Nancy said in an aside to Ursula.
Oblivious to her comment, Otto continued, 'There was a romantic moment. We floated beneath a weeping willow tree and we kissed, touched tongues—'
'Eew! More than we need to know!' objected Nancy.
'That kiss,' said Otto, closing his eyes, 'was the most significant snog of my existence. It tasted _very_ much of the wild boar. Perfection.'
Nancy's face wrinkled with distaste. Ursula meanwhile hung on Otto's every word, searching for a clue.
'But that happiness I felt, our love—'
'India was in love with you as well?' asked Nancy sceptically.
Ignoring the question, he continued, 'She told me everything. I was always there for her. I always hoped, one day, that we'd kiss again, especially after I skilfully shot a woodcock that weekend at Bratters. But she grew distracted. There were many admirers. She was some kind of Zuleika Dobson.* Only . . . uuu-gguu-uuh,' he gulped, 'she is dead and her lovers are not!'
Tears squirted from his eyes.
'Otto, try and calm down. You need to tell us exactly what happened on Sunday night,' Ursula said.
'Something happened before that. It was last Thursday,' he snivelled. 'India came to my room – our rooms are both in the Monks' Cottages. Hers is – was – next door to mine. When we got our room numbers at the beginning of this term, I thought it was a sign . . . that we were meant to be together. Anyway, she was very upset. She said to me, did I know anything about Wenty and another girl? Had he cheated on her? Well, everyone had been gossiping that Wenty had got off with Isobel last term at the New College Ball. I couldn't _not_ tell her what I'd heard—'
'Did you think it was true?' Ursula interrupted.
Otto hesitated. 'I don't know why anyone would say something like that if it weren't true.'
'But, to be clear, you never _actually_ saw Isobel and Wenty make out?' Nancy pressed him.
'Er . . . I suppose . . . not . . . with my own eyes.'
'So how did India react when you told her this rumour about Wenty and Isobel?' said Ursula.
'She was infuriated. I don't know what she meant, but India said, "She'll see," and stormed out of my room.'
Nancy turned to Ursula and said, 'Do you think India showed up at Wenty's party with Dom just to spite Isobel?'
'Yes, and I think she'd been sleeping with him for ages,' replied Ursula. 'I think this all goes way back, to the summer, when India stole Isobel's role as Ophelia. But perhaps India had decided it was time Isobel knew about her involvement with Dom. Otto, go on, please.'
'From the moment she arrived at Wenty's party, looking so stunning, on Dom's arm, I could tell it was over with Wenty.'
'How did you know?' asked Ursula.
'It was in her eyes. Behind the sparkle, I don't know, there was something . . . hurt. I knew India very well, you see. And then, later, Wenty came back after she stormed out of the party and asked for my help.'
'Your help? How?' asked Ursula.
'He returned to the party in the Old Drawing Room sometime after midnight. He told me that he needed to speak to me "privately". We retreated to the bathroom across the landing from his set. It was just me, Wenty and a couple of scouts who were washing up. He was pretty drunk,' Otto went on. 'We all were that night. Wenty said that he and India had had an enormous row in the quad, and that he thought she had gone to find Dr Dave. He wanted me to go up and see, bring her back. So I told him I would. But I didn't mean it.'
'What?' said Ursula, confused.
'I sensed an . . . opportunity,' said Otto slowly, looking ashamed.
'What sort of opportunity?' she asked curiously. Surely he didn't mean he'd sensed an opportunity to kill India?
'It was my opportunity to have India. For myself-ugh-ugh-uggggg—' Otto's breathing speeded up and he started to sound hysterical, screeching, 'I double-crossed Wenty! He thought I was his friend! He _was_ my friend! But I wanted her too much!! I didn't care about him!!!'
For once, Ursula actually felt sorry for Wenty. His 'friend' wasn't exactly loyal.
'Otto, take a chill pill,' ordered Nancy. 'What happened next?'
'I took a bottle of champagne from the bathtub, two more glasses. I had one acid tab in my pocket. I swallowed half. Kept the other half for India,' Otto told them, gradually calming down. 'I was drunk, excited, tripping. I knew it was over between Wenty and India. It was my turn. When I reached Dr Dave's staircase, I called out for India. I heard a voice above me. Suddenly, there she was in front of me, a vision. A slightly blurry vision, I admit – I refuse to wear my glasses to black-tie parties. That white gown she was wearing, the pearls gleaming in her tiara . . . she looked marvellous. I think she had come out on to the landing, I can't remember exactly, but I said to her, " _Liebling_ , you are the most exquisite girl I have ever seen," and I took her hand. It was very romantic, gentle. I poured her a glass of champagne and put the half tab in it—'
'Did you tell her about the acid?' asked Ursula.
Otto looked hurt. 'Do I look like the kind of sleaze-bucket who spikes an innocent girl's drink?'
Ursula didn't say anything. Whether or not Otto had told India she was about to consume half an acid tab didn't really matter in the great scheme of things. The end result was that by this point in the chain of events, India was under the influence of drugs as well as the alcohol she had consumed at Wenty's party.
'Did you see anyone else on Dr Dave's staircase?' asked Nancy.
'I don't remember seeing anyone else at all,' said Otto. 'We went into the room almost immediately. We started kissing. We ended up on the sofa and at one point I fell off it, my passion was so great. But it didn't hurt. I felt as though I were floating, in a lilac cloud. I told her I loved her, that she was my one and only true _Liebling._ She said – even with my poor recall I can remember this as if it happened a minute ago – that she had never felt like this, ever. The joy I felt! I had waited so long! I sensed an invitation to her next shooting weekend would soon be forthcoming! And then I – _ach mein Gott_! I must have killed her after that.'
'Otto, no, you can't have—' Nancy started.
But he interrupted her, saying, 'I can't remember anything after that. My brain must have blocked it out. The act. The horror. I just remember waking up in the JCR and thinking Ursula was my _Liebling_.'
'Otto, are you sure you're not imagining things?' she protested. 'You were horrified when I showed you her body. You were so shocked you were sick.'
'Of course I was! I couldn't believe what I had done.'
'There was no blood on your clothes. How could you have killed her like that and not had any blood on you?'
'I have proof. Physical proof.'
'You do?' said Nancy, looking uneasy.
'Undo my trousers!' ordered Otto.
'Oh, Jesus, he's totally lost it,' Nancy bawled at Ursula.
Otto started fumbling awkwardly with his flies, despite his silken handcuffs, and his trousers fell to the floor, revealing a bony, pale pair of legs. He was wearing white Y-fronts.
'Look!' he said, gesturing to a huge, purple splodge of a bruise on his hip. 'You can't deny it. The fall from the sofa did this. It all happened. _I did it._ I was with her all night. I was the only person up there. I would have seen something if anyone else had done it. Or heard something. It was, maybe, a terrible accident.'
The girls stared at the bruise, speechless. There was no denying the physical evidence. And, when she really thought about it, Ursula felt that his behaviour on Monday morning, after seeing India's body, had been pretty peculiar, veering from one emotional extreme to the other.
'I need to tell the police the truth. And then they can hang me. With my _Liebling_ gone, I couldn't care less about dying. Bring me to the noose.'
'Otto, the death penalty was abolished here twenty years ago,' Ursula reminded him.
'How inconvenient,' he wailed. 'Regardless, first thing, I'm turning myself in.'
'Otto, don't do any such thing,' Ursula ordered him.
'You can help me?' he said, sounding desperate.
'I don't know yet. Just go to your room, get some sleep and don't say another word to anyone. _Especially_ not the police.'
*
It was around three in the morning by the time Ursula finally shut the door to her room, switched on the bedside lamp and sat on the bed. She was exhausted, but sleep seemed impossible after Otto's confession. Perhaps unwrapping the parcel from her grannies would calm her nerves. She smiled as she examined the contents: a bar of Kendal Mint Cake from Plain Granny, a bottle of Yves St Laurent Rive Gauche perfume from Vain Granny, and two letters, one from each of them. News from the farm! How comforting the idea of home seemed now. Ursula lay back on her pillows and began to read Plain Granny's letter:
Dearest Ursula—
In haste as sheep are out — miss you dreadfully. Without you house feels like a roly-poly pudding without the jam filling. Your old teddy bear Huggle very forlorn sitting on your bed all alone. Farm busy as ever, although chickens have stopped laying in protest at your departure. Making mountains of jam, blackberries this year are—
But Ursula was asleep, still in her party dress, dreaming of Seldom Seen Farm.
* To prove their love for the beauty of the title, every male undergraduate commits suicide in Max Beerbohm's novel _Zuleika Dobson_ (1911). Imagine _Gone Girl_ in Edwardian costume.
Tuesday, 1st week: morning
Someone was hammering on Ursula's door. It was a brutal awakening from the heavenly jam roly-poly and teddy-bears dream she was right in the middle of. Ursula sat bolt upright on her bed. The room was pitch black. The hammering intensified. It sounded as though someone was trying to bash her door down. There was nothing for it. Petrified, Ursula leaped from her bed and yanked her door open, simultaneously managing to hide behind it as she wondered who this could be.
Light from the landing illuminated something protruding into the room. Ursula heard the squeak of rubber soles crossing the threshold. As her eyes adjusted to the light, she realised that the object in silhouette was a large, wooden oar. It prodded the bed, as though searching for someone. Suddenly the most bizarre thing happened. The sound of singing filled the room:
_'Oh, what a beautiful morning,_
_Oh, what a beautiful day._
_Get yourself down to the ri–ver,_
_Or Christminster's boat rows away!'_
'Moo!' said Ursula crossly, recognising her voice. 'I thought I was about to get bonked on the head.'
'That's the idea. Got you out of bed pretty damn quick, didn't it? Fifteen minutes till the Freshers' Eight meets in the Porters' Lodge to run down to the river.' Moo, perky as hell and already in her college tracksuit and plimsolls, regarded the polka-dot dress Ursula was still wearing from last night. 'Super nightie.'
Ursula rubbed a sleepy bug from her left eye and looked at her alarm clock: 5.45 a.m. already. 'I'll be there,' she said.
'Good-o,' said Moo, jogging energetically across the landing to Nancy's digs where the door-bashing started all over again.
Ursula shut the door. As she dressed in her college tracksuit, she reviewed Otto's peculiar account of Sunday night. The facts certainly added up to a nasty outcome for him: Ursula had, after all, found him passed out in the room next to the one containing India's corpse, and he had freely admitted that he had spent most of the night in Dr Dave's rooms with India after he had left Wenty's party, and hadn't seen anyone else up there that night.
But, Ursula asked herself, how could Otto be so convinced that he had killed India if he couldn't remember actually doing it? If it was in fact possible to forget that you had just murdered a very close friend, which Ursula seriously doubted, there was the question of the influence of drugs and alcohol. Although Ursula had no personal experience of acid, she had heard that it made the user feel as though he or she was hallucinating, not homicidal. Was Otto's whole story in fact a nightmarish vision?
And what about India? Even if Otto had spiked her glass of champagne with half an acid tablet, would she really, after the famous row with Wenty, have suddenly become infatuated with someone else? Her love life was complicated enough surely.
Still, against all the odds, it appeared that Otto and India _had_ spent the night in each other's arms. Otto had the bruise to prove it. But that still didn't convince Ursula his tale of murder was completely reliable. That perfect slit she had witnessed in India's throat didn't quite gel with Otto's belief that he had killed her accidentally. It had been neat, clean, the work of someone who was careful and calculating, rather than the drunken mistake of a minor Austrian princeling. But why would Otto lie? If he was lying, was he protecting someone? If so, who?
Ursula wrapped her cosy college scarf tightly around her neck. It would be freezing outside this early, and even colder down on the water. She quickly glanced in the mirror. Her eyes were still daubed with the eyeshadow and mascara from last night, and her crimped hair had turned into a mad-looking bird's nest. With time running short, she hurrriedly wove it into two scruffy plaits and stepped out onto the landing. Despite having had only a couple of hours' sleep, excitement about her first Oxford rowing session energised her. Sunrise on the river would be magical.
'Hey!' Nancy was already outside Ursula's room waiting for her.
'Wow,' Ursula exclaimed as she took in her friend's rowing costume. There really was no other way to describe the outfit as anything other than a costume, she thought.
'Thank you,' said Nancy proudly. 'I was really worried about looking appropriate.'
Nancy's 'appropriate' rowing costume consisted of the following: a dark green blazer with red grosgrain trim and the Christminster coat of arms embroidered on the breast pocket, a starched white button-down shirt, striped college tie and a dark green pleated gym-slip skirt. Her knee-high socks were, of course, knitted in college colours, and her feet were clad in white puffy high-tops. She looked like a St Trinian's girl who'd got lost in a hip-hop video.
The girls clattered down the staircase and across Great Quad towards the Porters' Lodge. It was just starting to get light, but the day was disappointingly grey. Portentous, iron-black clouds were gathering above, and Ursula could already feel fat drops of rain plopping onto her cheeks.
'I _cannot_ believe what happened last night,' said Nancy as the girls walked. 'Otto seemed so nice when we met him that first morning. But on _America's Most Wanted_ it's always the cute, quiet ones who turn out to be homicidal maniacs.'
'We need to keep our options open. I think that Otto _thinks_ he killed India, but I'm not certain he did,' said Ursula, elaborating on her doubts about his tale.
Suddenly the droplets became a downpour, and they sprinted as fast as they could to the shelter of the gate tower. From the other direction, Mrs Deddington, head down against the weather, was dashing with a tray of tea from the Kitchen Quad. By the time she was under cover, her brown overall was limp and the two cups of tea and pile of Marmite on toast on the tray were spotted with rain and had lost any allure they might once have had.
'What a waste,' sighed the scout, heading into the Porters' Lodge.
'Oh, no, Marmite's _always_ delicious . . . Mmmm,' Ursula volunteered.
She and Nancy followed the Marmite's aroma, and Mrs Deddington, inside. There they found Alice, dusting industriously while chatting to Deddington and his son, who were swapping shifts. Mrs Deddington placed the tray on the porter's desk.
' _What_ is _that_ stuff?' asked Nancy, contorting her face at the sight of the dark brown substance on the toast.
'Only the most scrummy thing in the entire world,' said Ursula. 'You have to try it.'
'Miss Feingold, if you come down to the Scouts' Mess at five o'clock, I'm sure Mrs Deddington will make you some Marmite on toast to try, won't you, Linda? Miss Flowerbutton, come along as well,' said Alice.
Mrs Deddington put her hands on her hips and huffed.
'Come on, Linda,' Alice cajoled her. 'It's nice to have the students down for tea sometimes. We all need cheering up, don't we, after yesterday?'
'True,' she sighed. 'But only for five minutes, mind. I'm busy today.'
'See you later, Mum,' said Nick, gulping down his tea. He pecked his mother on the cheek, picked up his bag of textbooks in one hand, a wilted piece of toast in the other, and disappeared. Deddington, meanwhile, took up position behind the desk, nodded hello to the girls and tucked into his toast.
'I'll drop your kilts off in your room this morning,' Alice told Ursula. 'They look lovely now they're short.'
'Oh, thank you!' Ursula was delighted. 'I can wear one on my date tonight.'
'Date! _DATE_?' interrupted Nancy. 'What? Who?'
'Shhhh!' said Ursula. 'I'll tell you later.'
The Freshers' Eight – which, apart from Ursula, included Moo, Claire Potter, and five other young women who had arrived in dribs and drabs over the last few minutes – was finally gathered in its entirety. A second-year girl soon appeared and started barking at them officiously.
'Right! My name is Eleanor! Thompson! I am the President of the Christminster Boat Club!' shouted the girl, who was dressed in skin-tight cycling shorts and a college tracksuit top. Her hair was scraped into a tight bun, revealing a stern, plain face, and her muscular thighs, Ursula noted, bulged intimidatingly, like two prize marrows at a country fete.
'Remember! You're a team! From now on,' Eleanor continued, 'everyone needs to pull their weight. Literally. If one of you pulls her oar at the wrong time, you're _all_ in the water. It's down to your cox to tell you when to pull. Make sure you listen to her. Respect her.'
The team nodded obediently. Then Eleanor asked, 'Which one of you is the cox anyway?'
No one said anything. Ursula looked round. Nancy was staring dejectedly at her empty pigeonhole. Ursula nudged her.
'Nancy,' she hissed, 'you're needed.'
'Nothing from Next Duke,' her friend said, disconsolate. She turned from her pigeonhole, and then to Eleanor. 'I'm Nancy, the Freshers' cox. Hey.'
'I might have known,' said Eleanor, scrutinising Nancy's outfit disapprovingly. 'Right, Coach is meeting us down at the boathouse. Let's go.'
Just as the girls jogged out of the college gates, a bellowing crack of thunder delivered another torrent of rain. Pelted by the harsh weather, Ursula and Nancy jogged behind Claire Potter down Christminster Lane. Ursula hoped she could speak to Claire after the rowing session. She _must_ know something about that doomed Sunday night. Perhaps from her seat close to the window, maybe eating ice cream, she had seen India dashing from Great Lawn towards the JCR staircase. Or could she have heard Otto professing his undying devotion to his _Liebling_ on the landing outside? Might Claire have seen Dr Dave in the quad, as Ms Brookethorpe said she had? And what about Claire's companion? Ursula knew that two ice creams had been consumed that night, and, unless Claire had been extremely hungry, Ursula had to assume that someone else had eaten the second tub. But who? Had he or she seen anything? And, finally, what on earth had Claire been doing prowling Great Quad at 1 a.m. last night?
The girls reached the corner of Christminster Lane where they turned and jogged down the High Street, past the spires of Magdalen College Tower.
'So who's this mysterious date with?' panted Nancy as they jogged.
'Eg,' Ursula told her. 'Can you believe it, he's asked me out for dinner?' She still could not quite believe it herself.
'Oh my god, he is _so hot_. Do you have a crush on him?'
Before Ursula had a chance to answer, Eleanor Thompson shouted from behind them, 'Stop gabbing, you two, and get a move on!'
Ursula watched with awe as Nancy sprinted forward and easily overtook the rest of the Freshers' Eight. She was a superb runner and now seemed oblivious to the drenching she and the other girls were experiencing. Ursula's own tracksuit was soaked through and her plaits soon resembled two drowned worms. Her feet squelched in her Green Flash trainers. Her toes were becoming numb. But she refused to let her enthusiasm be dampened. As she followed the team along a track by the river, Ursula found herself captivated by its (very) damp charm. Rain droplets bounced brightly off the water. The finger-like fronds of drooping ash and weeping willow trees swept the surface. When the girls jogged past a herd of Highland cattle grazing the riverbank, Ursula felt as though she'd stepped into a scene from a picture postcard.
Thank goodness! she thought, winded, as the white-painted boathouses at last came into view. The girls trotted over a footbridge and finally caught their breath in front of the largest boathouse on the river's edge. On this section, the widest part, it was already rush hour. Sculls were powering smoothly up towards the Head of the River, coaches were yelling instructions at different teams, and rowers were warming up on the quayside.
The boathouse's huge wooden doors were already flung open, and a fit-looking group of male rowers expertly hoisted one of the many boats stored there above their heads. Ignoring the pelting rain, they marched out like a troop of soldiers, and effortlessly deposited the boat in the water. 'Your carriage awaits, ladies!' called one of them. Ursula had never seen so many dishy, well-built boys in her life.
Minutes later, she found herself seated at the front of the boat, the handle of a huge oar hovering above her knees. Moo was seated immediately behind her. The rest of the girls filled the other six rowing spots in the boat, and Nancy sat facing them from the bow. Her right hand rested on the rudder control. Her left hand gripped a loudspeaker. Even though her usually bouffant hair now resembled a wilted pancake, Nancy looked thrilled with her new role.
'Your coach today,' yelled Eleanor Thompson from the bank, 'is Wentworth Wychwood. He's a top rowing Blue. Please pay attention to his instructions. You'll learn a lot from him.'
Wenty? Oh no, thought Ursula. She couldn't imagine anyone she'd less like to be bossed around by in a boat than Wentworth Wychwood.
He soon appeared, sensibly dressed with a waterproof jacket over his Blues tracksuit, his hair covered by a thick black woollen hat. Though he looked worn and tired, Ursula couldn't help admitting to herself, grudgingly, that he was terribly handsome all the same.
'Morning, ladies,' said Wenty, flashing his beautiful smile at the Freshers, who responded, Ursula included, with a Group Swoon.
'Coach, before we start, I have a really important question,' Nancy said, her voice reverberating through the loudspeaker.
'Yes, Nancy,' replied Wychwood.
'It's about Next Duke. The Dudley one?'
'Nancy, please concentrate on coxing,' Wenty said, looking amused. 'Ladies, please lift your oars so they are flat above the water.'
The boat wobbled unevenly as the girls gingerly moved their oars.
'I'm just wondering,' continued Nancy, 'is he, you know . . . coming over to your rooms again soon?'
'Doubt it,' replied Wenty. 'He hardly ever leaves Merton. Except to go to chapel. He's a recluse. The only reason he came on Sunday was because he's had a crush on India since childhood . . .' Wenty suddenly stopped talking, and seemed desperately sad. Ursula couldn't help but feel sorry for him. After a few seconds he managed to pull himself together and said, 'Right, that's enough small talk. Ladies, turn your oar at a right angle to the water.'
Wenty expertly instructed the girls how to dip their oars in the water and pull them forward. Nancy was soon shouting 'PULL!' at the top of her lungs through the loudspeaker. Timidly at first, the Freshers' Eight dragged their oars through the water every time she made the command. Rowing was much more difficult than Ursula had imagined, and mostly the boat bobbed along like an ungainly bath toy. But the very occasional feeling of gliding that the girls achieved, even in the driving rain, was bliss. While she was out there on the water, Ursula felt energised and peaceful. But as soon as the crew had returned to the boathouse, she was back on her mission, and cornered Claire Potter immediately.
'Hi, Claire,' said Ursula. 'Did you enjoy that?'
'Not really,' the other girl answered as she dried her hair with an old towel. 'Ugh. I'm soaked.'
'Claire, may I talk to you? About what happened on Sunday night?'
'What about it?' she retorted, suddenly blushing bright red. She held the damp towel to her face as if to cool down her cheeks.
'I'm trying to find out what India did after she left the party.'
'What's it got to do with you? The police are all over college investigating. Why not leave it up to them?'
'Of course the police are investigating. But it's just . . . I'm supposed to be writing an article about it all for _Cherwell_.'
'Why would I know anything about what happened to India?' Claire said defensively.
Ursula was getting the distinct feeling that the girl had no interest in being even the slightest bit helpful to her, even after the apology. Still, she persisted.
'Claire, your Crosswords and Ice Cream party was almost next door to the room India died in. Did you see anything? Hear anyone?'
'I was very . . . er . . . occupied . . .' Claire hesitated '. . . with the . . . um . . . function that night.'
Why was she lying? Ursula wondered. After all, one of the few concrete facts she knew about the night in question was that only two tubs of ice cream had been eaten. It was most likely that Claire had had only one attendee, and that was only if she hadn't eaten two tubs of ice cream herself.
Claire now looked extraordinarily embarrassed. That's it! Ursula deduced. Claire was lying because she was ashamed her club had been such a flop. It would be cruel to confront the poor girl with her lies now, and anyway, Ursula might squirrel more information out of her by playing along.
'How long did all your friends stay in the JCR?' she asked, trying to sound casual.
'Er . . . I'm not sure,' Claire stammered.
'Did they leave before midnight? After?' Ursula went on.
'It's hard to say when you're . . . occupied.'
What on earth was Claire talking about? It was inconceivable that she hadn't heard anything from the JCR during Otto and India's tryst. After all, her virtually deserted party would have been extremely quiet. How could she have heard absolutely nothing?
'Unforgettable, hello,' Wenty suddenly interrupted them.
Ursula looked at him with amazement. How on earth could he call her 'Unforgettable' now, after everything that had happened? Perhaps sensing her discomfort, he cleared his throat and said, 'Sorry, I mean Ursula. I'm a mess . . . with everything that's happened.' He sounded sincere. 'Look, I'd like to talk to you. Privately. Would you allow me to . . .'
Wenty's voice trailed off and he stood looking at Ursula searchingly. A rather intense moment was abruptly curtailed by Claire Potter.
'I know when I'm not wanted,' she said, stamping off.
'Claire, wait . . .' Ursula called after her, unconvincingly. But her crewmate didn't look back.
'Allow you to . . .?' replied Ursula slowly to Wenty, returning her attention to him.
Wenty didn't answer her immediately. He just let his clear blue eyes linger on hers for what seemed like forever. Ursula felt slightly giddy. Eventually he said, 'To buy you a . . . fry-up?'
Ursula looked away from the boy and chided herself before she answered him. She did not need to start going all giddy on someone like Wenty. He was exactly the sort to be avoided in Oxford. She already had a date for dinner tonight. She didn't need a breakfast date as well.
'Sorry, no, I have plans,' she announced, feeling _very_ pleased with herself. Turning down Wenty was immensely satisfying.
'Plans for breakfast? Don't be ridiculous.'
'I am not being ridiculous!' Ursula retorted.
Wenty persisted, saying, 'Look, I just want to talk to you, about India, sooner rather than later.'
'Oh,' said Ursula, feeling humiliated. Wenty had not been asking her on a date after all.
'I've heard you're writing something for _Cherwell_ ,' he continued. 'I think we can help each other. Breakfast? Please?'
It was with some trepidation that Ursula followed Wenty under an awning displaying the words 'Since 1654' and through the tiny front door of the Queen's Lane Coffee House. Even if her rendezvous with him was strictly business, she had never experienced anything quite as intimate as an unaccompanied fry-up with a boy before.
The little café was a welcome, warm respite from the rainy street. The interior was spartan – dark brown chairs and tables, plain oak beams and ancient-looking plastered walls. Portraits of the café's previous undergraduate clientele – which apparently included Samuel Pepys and Lawrence of Arabia – were hung in nooks and corners, manna to tourists. The only clue that it wasn't still 1654 inside the coffee house was the speckled yellow linoleum floor tiles. The place was already crammed with students tucking into enormous cooked breakfasts of eggs, bacon, sausages, baked beans and dripping-drenched piles of fried white bread. Ursula was ravenous after the rowing session.
'Do you mind if we sit over there?' asked Wenty. 'Easier to talk about . . . well, everything.'
He led her towards a rickety round table in a far corner, away from the crowd. Ursula hesitated for a moment, thinking the situation looked far too secluded and romantic for a breakfast that was strictly business. Wenty noticed her reluctance.
'Don't worry, no one's going to think there's anything going on, Flowerbutton,' he said.
'Okay,' said Ursula, sitting down and noticing a pink plastic rose in a vase on the table. 'I don't want anyone thinking we're on, well, some kind of . . . breakfast date. It would be really unprofessional, for the article and everything.'
'Flowerbutton, I wouldn't ask you on a date if you paid me. You're adorable, but you're far too prim for me.' Wenty smiled and took a seat, looking very amused.
'Wenty,' Ursula exclaimed, 'you're so full of yourself! And I am _not_ prim.'
'All right – square.'
'Ughhhhhhh!' Ursula groaned. He was impossible. 'Can we change the subject, please?'
'Sorry. You must be freezing. Here.'
Wenty jumped up, took off his waterproof jacket, removed his dry sweatshirt and tucked it around Ursula's sodden shoulders. He was impossible, but perhaps the boy had a smidgen of gentlemanliness in his soul, she thought, hugging the sweatshirt round herself for warmth. A smidgen.
'You look like you need a hot chocolate,' he suggested.
'And that fry-up, please,' she said. 'I'm starving.'
'Flowerbutton,' declared Wenty, 'you're a one-off.'
'What?'
'Of all the beautiful girls I have invited to breakfast here, not _one_ has _ever_ ordered the fry-up.'
How predictable, thought Ursula. I am only one of _many_ girls to have spent a rainy morning in a cosy corner of Queen's Lane Coffee House with Wentworth Wychwood. The smidgen of goodness this boy possessed was almost negligible.
A waiter approached. After Wenty had ordered two Full English Breakfasts, he said to Ursula, 'India loathed coming here. She'd always order toast and marmalade, not touch it, and then complain to the waiter that it was cold. She was so contrary, but at the same time no one knew how kind she was underneath all her bravado. She used to visit her old nanny in her little cottage in Brattenbury every time she went home. Adored her. India even put up with my dreadful piano-playing. I'm hopeless, but she pretended I was good.'
'That's sweet,' said Ursula.
'She had her difficult side, of course. People thought she was spoiled, arrogant. But I understood why she behaved the way she did.'
At this, Ursula's ears pricked up. 'What do you mean?'
'Her mother ran off to a squat in Clapham to be a heroin addict with some cockney artist when India was three. Died a few years later. Tragic. India was raised by her nanny until Lord Brattenbury packed her off to boarding school when she was seven. India was the most insecure, most messed-up girl I ever met . . . I was crazy about her . . . she _made_ you crazy.'
Ursula nodded, but didn't feel any more enlightened. What did Wenty mean, India made him crazy? Crazy enough to kill her? Out of jealousy? If he was so infatuated with her, though, why had he slept with Isobel? And what about all the other girls everyone claimed Wenty was chasing behind India's back?
Just then, the waiter returned with a large tray and set two full platters on the table. Ursula and Wenty tucked in, both famished. Ursula sipped at the delicious hot chocolate. Wenty took a gulp of his coffee and then, looking sadly at her, said, 'Have you got any idea what happened to India? Do you know _anything_ , Flowerbutton?'
'Not much,' replied Ursula. She wasn't going to tell him about Otto's bizarre outburst the previous night. Or the information she'd gleaned from Isobel. 'But I think you might be able to help, Wenty. You're one of the last people who spoke to or saw India that night. I need you to tell me everything you know.'
He produced a sodden packet of Marlboro Reds and a plastic lighter from his back pocket.
'It's my fault that she's dead, you know,' he said, lighting the first of many cigarettes. 'I might as well have done it.'
'What?' said Ursula, trying not to look too shocked. Was she about to hear a second confession, of sorts?
Wenty rubbed one side of his head, the cigarette almost singeing his hair. 'Flowerbutton, if I hadn't called you "Unforgettable" that night, India would still be alive today. She was very insecure. Got easily jealous . . . you saw her. She was furious with me. Probably had every right to be.' Wenty stopped talking for a moment, an embarrassed expression on his face. 'I wasn't a very good boyfriend.'
'Really,' said Ursula, deadpan. His admission was hardly a surprise to her.
'Someone told her – god knows who, I could murder them – that I'd shagged Isobel Floyd. Of course, I told India I hadn't. I mean, Isobel was her best friend, for god's sake, what kind of person would do that?'
Had Otto been lying, wondered Ursula, when he had told India that Wenty had been up to no good with Isobel? Had he just used it to try and steal India away for himself?
'Why would someone say that about you and Isobel if it weren't true?' asked Ursula, before beginning to munch her fried bread.
Wenty looked sheepish. 'It wasn't _exactly_ not true.'
'I see.'
'Look, I shagged Isobel, okay. Behind the Pimm's tent, New College Ball. We were both blotto, so if you ask me it doesn't count. I couldn't tell India the truth, but I wasn't a very good liar. She knew. It was obvious. And then when she heard me call you "Unforgettable" at the party, that was it. She went nuts on me in Great Quad. I begged her to forgive me. You know, I loved her more than anything. I'd never have cheated on her if I'd been sober, never! But instead of forgiving me, she wanted to punish me. She was still at it with Dr Dave, I'm sure of that.'
'Really?' asked Ursula. Everyone had said the affair had been over for a long time, but perhaps Wenty was onto something.
'Revolting ponce. India was always up in his rooms having "long talks". I mean, please! If she wasn't still in love with him, why did she go to his rooms on Sunday night?'
'I don't know. No one does yet. What happened when you went back to the party after the row in Great Quad?'
'I was worried about India . . . angry . . . jealous. I went back into the Old Drawing Room and found Otto. We went to the bathroom for a private chat and I asked him to go after India and tell her I was sorry about Isobel. I couldn't bear to let a bit of extra-curricular activity part us forever. I thought I'd marry India one day.' A tear rolled slowly down Wenty's cheek. 'Sorry,' he said, brushing it away. 'Anyway, Otto said he knew India felt the same way about me, and then he went off to retrieve her. But he never came back. After an hour or so, I started getting concerned. I decided to go over to the staircase myself.'
That was odd, thought Ursula. Ms Brookethorpe hadn't said anything about seeing Wenty crossing Great Quad that night. She had only mentioned spotting Dr Dave walking towards the Monks' Cottages.
'Did you see anyone in Great Quad as you walked over?'
'No,' said Wenty. 'But I was so furious I probably wouldn't have noticed if the entire Household Cavalry had been on parade. I just wanted to get to India before . . .' He stopped speaking, face grim.
'Before what?' asked Ursula.
'Oh . . . nothing. It's pointless now. I was too late anyway.'
What did he mean? she wondered. Had he somehow known that India was in danger? Was he too late to save her life? Did this mean Otto had, in fact, done her in?
'Are you saying, Wenty, that India was already dead by the time you reached Dr Dave's rooms?'
'Dead? Hah! God, no.' He laughed bitterly. 'When I got to the staircase, India was alive and kicking . . . Or should I say, alive and shagging?'
'You saw her with—'
'I didn't need to,' he interrupted. 'I could hear them at it. Laughing, giggling, screaming. They were bonking so hard I could hear everything from the bottom of the staircase. That feeling I'd had for a while, that she was shagging someone behind my back – I was right.'
The whole thing was far worse than Wenty realised, thought Ursula. She dreaded to think what would happen when he discovered, which he inevitably would, that India had not been with Dr Dave that night, but with Wenty's 'loyal friend' Otto. 'What did you do then?'
'I . . . er . . . didn't want to go up. It would have been too humiliating,' Wenty said, then hesitated for a moment. 'So I went back to my rooms. Went to bed. Didn't wake up 'til midday on Monday.'
If Wenty hadn't gone up the stairs, as he said, he couldn't possibly have killed India. But was he telling the truth? A nasty sense of doubt crept through Ursula. Ms Brookethorpe didn't seem to have noticed Wenty going back across the quad to his rooms. Could she really have missed him twice?
'Look, sorry,' he continued, glancing at his watch. 'I've got to get to a lecture. We could carry this on later. Why don't you . . . come up to my rooms . . . after dinner?'
Ursula frowned at him. He seemed unable to resist flirting, whatever the circumstances. After her experience with Jago, the last thing she was going to do was go to Wenty's rooms alone, at night. She wasn't interested in becoming one of his many conquests.
'I can't,' she said. And couldn't resist adding, 'You see, I'm busy tonight. I've got a date. I'm going for dinner to Chez Romain.'
'Ooh, la-di-da! Chez Romance, don't you mean? You've only been in Oxford a few days and already you're going to the couples' restaurant.'
'The couples' restaurant?'
'It's so expensive that after one dinner there, you're officially a couple with whoever took you. It's much too extravagant for a casual date. Anyway, who are you going with?'
'Eg, as a matter of fact,' replied Ursula.
'Eg?' Wenty chortled.
The idea of Eg and Ursula on a date prompted tears, this time of laughter, to flow from Wenty's eyes. Ursula had no idea what was so funny.
By the time Ursula arrived back at college that morning, soaked and freezing, she was desperate to change into something warm and dry before starting her day properly. The freshly shortened mini-kilts beckoned. But her attention was soon distracted from her sartorial musings by the sight of Ben Braithwaite, looking like a frightened weasel, being shepherded out of college by two police officers. Ursula smiled at him as reassuringly as she could, but Ben, clearly paralysed with terror, did not respond. How on earth was he connected to India's death? she wondered.
En route to her room, she headed to the Porters' Lodge to check her pigeonhole. As she approached, she could hear Deddington's voice. He sounded irate.
'As I've already explained, college isn't open to the public this morning, sir.'
Ursula slipped into the lodge where she saw the porter talking to a man dressed in a long black overcoat and holding a scruffy briefcase. He was unshaven and had lank, greasy hair.
'Come on, just a quick walk around?' said the man, taking out a £10 note.
'Impossible.' Deddington pushed the note back into the man's pocket. 'The police are all over college. No one's allowed in unless they're a member of the University.'
Ursula took her time opening the one note she found in her pigeonhole while continuing to eavesdrop on the conversation between Deddington and the strange man.
'This was Lady India Brattenbury's college? You can confirm that at least?' the man persisted.
'No idea, sorry,' Deddington said, shaking his head. 'Now, do I need to show you out? Or should I ask the police to do that?'
'I'll find my own way.' With that, the man speedily exited, eyeing Ursula furiously as he did so.
'What was all that about?' she asked the porter.
' _Daily Mail._ Neil Thistleton, he said his name was. Snooping around. Don't speak to him if you see him round town. Gutter journalist,' grumbled Deddington.
'I won't,' Ursula promised.
She turned her attention back to the note in her hand. The photocopied sheet was from Eleanor Thompson and read:
_Freshers' Eight meet again 6 a.m. Saturday, Porters' Lodge_.
Alas, thought Ursula. Friday was the night of the Vincent's cocktail party. Getting up for 6 a.m. after an evening out wasn't exactly her idea of fun. But if she were ever to make the Freshers' Boat she didn't have much choice. Despite all his faults, Wenty's excellent coaching had inspired her to go for it, rowing-wise. She wanted to impress him and make the team next term.
Her plan for today was to spend the morning in the library, attacking the Apocalyptic Vision of the Early Covenanters with renewed vigour, and the afternoon gathering more material for her article. She needed to think how to approach Dr Dave. Was there a polite way to ask your don about the details of a love affair with his student?
As she made her way to her staircase from the lodge, Ursula suddenly heard agitated voices coming from the direction of the Provost's Lodgings, the Georgian mansion situated in the south-west corner of Great Quad. She drew close to the building and noticed the front door was just ajar.
'. . . I'm afraid it's not good news, High Provost.'
Ursula stopped outside the house. The dry tones of D.I. Trott were unmistakable. He continued, 'We believe the girl was murdered.'
'But you can't be sure?' Ursula now heard the High Provost's alarmed voice.
'We're ninety-nine per cent certain that Lady India died at someone else's hands. The autopsy is due to take place later this morning at the John Radcliffe Hospital. I have no doubt the procedure will confirm the suspicions of the police. Now, I have some questions for you, High Provost.'
'Rather busy this morning, I'm afraid. I'm chairing a meeting of the fundraising committee shortly. Could you come back another time? It would be so unfortunate if any members of the committee were to see a police officer here.'
'This won't take long,' she heard Trott say.
Suddenly Ursula noticed a woman peering at her through a large sash window on the ground floor of the house. Her face was powdered and lipsticked, and her hair coiffed into a twirl resembling a Walnut Whip. She was wearing a royal blue suit and a pale yellow blouse with a huge pussycat bow tied at the neck. She appeared to have modelled herself on Margaret Thatcher. She was sitting at an old-fashioned mahogany desk, with various documents neatly arranged on it. This must be Scrope's secretary, Mrs Gifford-Pennant, thought Ursula, noting the typewriter and telephone on the woman's desk.
'Wouldn't have thought a college as wealthy as Christminster needs a fundraising committee.' Trott sounded surprised.
'It does if we are to remain the richest college in Oxford,' Scrope told him curtly.
'Perhaps we could start with that,' Trott replied. 'I understand that generations of Brattenburys have attended Christminster and the family has pledged a considerable endowment to the college.'
'Mrs Gifford-Pennant!' called out Scrope. 'Please show this gentleman into the drawing room. And cancel the fundraising meeting.'
Ursula watched from outside as the secretary hurriedly jumped out of her seat and disappeared from the room. Next Ursula heard footsteps in the hall. Seconds later, Mrs Gifford-Pennant's meringue-like coiffure and powdered nose appeared around the slightly open front door of the Provost's house. She squinted at Ursula suspiciously and abruptly slammed the door.
*
What did the Brattenburys have to do with Christminster's finances? Was this endowment somehow connected to India's death? Ursula wondered as she clambered up the steep stairs to her room a few minutes later. When she reached it, Alice was inside, cleaning.
'You look like you need a mustard bath,' the scout said when she saw the girl's bedraggled state.
'No time,' Ursula told her. 'I've got to get to the library.'
'Well, I'll leave you to it then,' said Alice, ducking out of the room, which, Ursula noticed, was now immaculate.
As soon as she was alone, Ursula removed her damp tracksuit and changed into a black turtleneck sweater, a red tartan now mini-kilt and grey woolly tights. She laced up her Dr Marten's and threw on the bomber jacket that her groovy godmother had given her, hoping she looked much less Beatrix Potter-like than she had a few days before.
Ursula grabbed the duplicate volume of the _Scottish Historical Review_ and put it into her satchel, noticing that her desk had been neatly reorganised by Alice. A large red cardboard folder containing the hand-written notes for the _Cherwell_ article and her reporter's notebook had been placed in the centre. She opened the notebook and added her thoughts from her breakfast with Wenty:
_—If Wenty thought he heard India and Dr Dave 'shagging' (not knowing India was actually with Otto), could he have been angry enough to return to Dr Dave's staircase later that night and kill his girlfriend?_
She tore off her new notes and left them inside the folder. Then she put her notepad in her satchel and dashed out, but not before picking up Vain Granny's letter, too – she'd read it later when she needed a break from the Scottish Covenanters.
It was almost ten by the time she reached the library, where Ms Brookethorpe, who was restacking shelves of ancient manuscripts, acknowledged her with a faint nod of the head. Nancy was already seated, frantically chewing the end of her pencil and sighing loudly as she pored over the master volume of the _Scottish Historical Review._
'I need help. What does this mean?' she whispered as Ursula settled herself at the next desk. Nancy pointed at a passage on page one hundred and fifty-seven, which began:
_In 'Doctrine, Discipline, Regiment and Policie' the Scottish Kirk 'had the applause of forraine Divines' and was 'in all points agreeable unto the Word' until the crown had begun its policy of episcopal subversion._
'It's about the dispute,' Ursula started to say, 'between the English and Scottish over Rome—'
Before she had a chance to go on, Ursula realised the shadow of Ms Brookethorpe was looming over them.
'Silence in this reading room is non-negotiable,' said the librarian grandly.
'Sorry,' Nancy apologised.
Ms Brookethorpe nodded her acknowledgement and then, to Ursula's surprise, said to her in a low voice, 'Can you come into the office, please, I need to . . . check something . . . on your library card.'
Ursula followed her into the small office behind the librarian's desk, a room that was humble in comparison to the Hawksmoor Library. A large photocopier dominated it, and rows of box files lined the shelves on the walls. In one corner was a tiny, rather grubby butler's sink, next to which a kettle, a box of Yorkshire teabags and a pint of UHT milk stood on the Formica worktop. Ursula started to hand over her library card, but the librarian gestured for her to put it away. She flicked the kettle on and then stood stiffly facing Ursula, looking distinctly uncomfortable.
'I hear you're writing about the murder for _Cherwell_ ,' she said nervously. 'You mustn't mention me by name in your article. No one can know that I told you what I did. Dr Erskine has a lot of . . . well, I suppose you could say . . . he has a lot of influence over the History department in the University. I wouldn't want anything to jeopardise my work with the Bible fragments.'
'I completely understand,' said Ursula as sympathetically as she could. 'All my sources will remain anonymous.' (She imagined this must be how professional journalists reassured their sources.) 'I'll never mention your name, I swear.'
Ms Brookethorpe poured boiling water from the kettle on to a teabag in a mug and slowly stirred the brew as she spoke.
'Thank you, that's a relief. Now, I imagine you'll be wanting to get back to those Apocalyptic Covenanters.'
'Yes,' said Ursula. 'But there's just one thing I wanted to check first. About the night of the murder . . .'
'Oh?' The librarian looked anxious.
'Are you _sure_ you saw Dr Dave crossing Great Quad that night?'
'I've already told you what I saw. So far the police haven't asked to speak to me and I'm hoping it will stay that way. I don't want to be caught up in all this.'
'Did you see anyone else?'
'You really mean it, when you say no one will know that I have spoken to you?'
'I absolutely promise,' Ursula reassured her.
'All right then. I _suppose_ I did see someone else,' the librarian said finally.
'Why didn't you mention them before? When I spoke to you last night?' asked Ursula, a little annoyed.
'It was late . . . I . . . I don't know.' Ms Brookethorpe was hesitant. 'I must have forgotten. I didn't think a couple of undergraduates skulking around college was important.'
'Two?' asked Ursula. 'Together?'
'No, they weren't together. I can't be completely sure of the exact time, but about an hour after I noticed Dr Erskine walking towards the Monks' Cottages, someone ran across the lawn towards the JCR staircase from the direction of the Old Drawing Room.'
If everyone was telling her the truth, thought Ursula, it must have been Otto who had run past.
'Are you sure the person was running?'
'Yes. I remember thinking it quite odd that he was in such a hurry at that hour. It must have been almost one a.m.'
'He?'
'It was definitely a man – I could see the silhouette of a tailcoat in the moonlight. I don't know who it was, though, I'm afraid.'
'Did he return to the Old Drawing Room?'
'I don't think so. The only other person I saw that night was another man coming across from the Old Drawing Room to the JCR. Another undergraduate in tails, this one with fair hair. It was around two a.m. by then.'
'How do you know?'
'I remember thinking how sleepy I suddenly felt, and checking my watch. Bible fragments make you lose track of time, you know.'
It seemed as though Wenty had been telling the truth. Ursula felt relieved.
'How long do you think it was until the fair-haired boy returned to the Old Drawing Room?'
'Neither of them returned to the Old Drawing Room,' said Ms Brookethorpe.
'Are you sure?' Ursula pressed her.
'The only thing I saw after that was someone heading from the entrance of the JCR staircase towards the gate tower. It was definitely the same student.'
She must be mistaken, Ursula told herself. Wenty had returned to his room after visiting Dr Dave's staircase.
'How can you be sure?' she asked.
'When I left and went down to the lodge, I said goodnight to the porter – Deddington's boy – and left through the main entrance onto Christminster Lane. I saw someone riding away from college on a bicycle. I remember because I was amused to see him swaying from side to side along the lane. He must have been rather drunk.'
'And you say you know for sure that it was the same person who had crossed the lawn a few minutes earlier?'
'Oh, yes. I'm very good on hair, as you know. It was the blond one. Wentworth Wychwood.'
*
_. . . though there be new outcasts betwixt Christ and Scotland, I hope that the end of it will be, that Christ and Scotland shall yet weep in one another's arms . . ._
It was hopeless. The impact of the Scottish Reformation on the religious scene in the seventeenth century couldn't hold Ursula's attention in the way that India's murder could. She couldn't concentrate at all, so she took out Vain Granny's letter and read it.
Dearest Ursula,
I do hope you wear the Yves St Laurent scent every day. It's irresistible to young men – it smells of Paris. I am sure you are having a marvellous time, but please do not spend too much time in the library studying or you will turn into one of those ghastly bluestockings. Look at what's happened to that poor woman, Mrs Thatcher. Went to Oxford, became Prime Minister, closed the coal mines and is now the most hated person in the country! Do spare yourself that fate by going to as many parties as you can, flirting like mad, drinking masses of champagne and missing at least half of your tutorials. Meeting your first few husbands while young and beautiful is imperative.
Your most loving,
Grandmama Violette
P.S. More ball-gowns in attic if required.
Grandmama! Ursula sighed. She did miss her, a lot. What a character Violette was. She put the letter in her satchel and turned back to the Early Covenanters, re-reading the impenetrable paragraph, but her mind drifted. It was no good. She pulled out her reporter's notebook and jotted down more notes:
_—Either Brookethorpe is lying, or Wenty is. But why would Brookethorpe lie? Why would she make up a story about an undergraduate cycling out of college in the middle of the night? She wouldn't. But then, why would Wenty lie? Why would he have said he'd gone back to the Old Drawing Room when in fact he'd left college?_
_—Trott said, 'It's usually the husband or the boyfriend.' If Wenty is lying, is it because he was somehow involved in the murder? But if he had killed his girlfriend, would he really have cycled off out of college straight afterwards, presumably covered in blood?_
Ursula's analysis of her investigation was interrupted by Nancy's manicured hand depositing a note on the open page of her book. The scrawled writing read:
_I'm freaking out. I cannot understand any of this stuff. I don't know why they accepted me. I am not smart enough for Oxford. They're gonna find me out soon and send me back. I'm gonna be like a return at Saks._
_By the way, you look super-duper cute today in that mini._
Ursula scribbled a message back and handed it to her.
_Glad you approve of outfit. Let's visit Dr Dave later? He could help with Covenanters._
_P.S. I overheard Trott earlier. The autopsy's happening this morning._
Ursula saw Nancy's eyes widening as she read the last sentence. 'What are we waiting for?' she demanded, springing up from her desk. 'Come on!'
Tuesday, 1st week: midday
'This is a _really_ pretty place to be dead,' said Nancy, gazing admiringly at the classical stone façade of the Radcliffe Infirmary as they alighted from the Spree, having sped along the Woodstock Road to the hospital. Ursula had to agree that the hospital did look more like an ambassadorial residence than a medical centre. Elegant stone pillars and wrought-iron gates separated it from the road and a fountain featuring a stone figure of Triton bubbled in the centre of the front lawn. The ambulances parked outside looked almost incongruous.
'Nancy, we could get into _terrible_ trouble,' Ursula warned her as they made their way through the main gates. 'How on earth are we going to persuade them to let us inside?'
'Hey, I've talked my way into Danceteria every Thursday night in New York since I was fourteen. This'll be a breeze,' her friend replied.
Did they Send Down students for gatecrashing autopsies? Ursula wondered. And did she really have the stomach to witness a human dissection anyway? She started imagining all sorts of gruesome sights.
Nancy marched boldly past the fountain and straight up the steps to the main entrance. Ursula had no choice but to follow. Inside, a matron and a junior nurse dressed in matching blue uniforms, white aprons and stiff little white hats were sipping cups of tea at the front desk. Ursula was extremely surprised to see that they were talking to the creepy man who had been in the Porters' Lodge earlier.
'Oh my god,' she whispered to Nancy. 'That's the _Daily Mail_ journalist again. Don't say a thing to him, okay?'
Nancy nodded.
'I heard that Lady So-and-So was always going to come to a sorry end,' the younger nurse was telling him. 'The mother was a drug addict . . .'
Excited, Neil Thistleton scribbled violently on his notepad.
Suddenly spotting the two interlopers, Matron nudged the younger nurse. She scolded her, 'That'll do, Jean.' Turning to Thistleton she said, 'I'm sorry, sir, but you will have to leave now.'
'Look, I know the autopsy's going on in there—'
'Time to go,' Matron cut him off.
Ursula glanced around at the grim, institutional interior of the hospital. The long corridor beyond the nurses' station, painted a drab green, seemed endless. Stretchers and wheelchairs were abandoned at intervals along it. Scruffy signs hanging from the ceiling announced, with false jollity, the name of each ward – 'Primrose Ward', 'Daffodil Ward'.
Nancy and Ursula approached the desk. Before either girl could say a thing, the senior nurse barked, 'Visiting hours don't start 'til four!' She then turned back to Thistleton, saying, 'We can't have the general public wandering around the hospital. Please be on your way.'
The journalist turned on his heel, irritated. As he did so, his beady gaze landed on Ursula.
'You were there this morning, at Christminster,' he said, eyeing her coldly.
Ursula felt cornered, but managed to say, 'I think you've made a mistake.'
'No. I know about you. You're the one who found her, aren't you?'
How on earth did he know? Ursula was staggered.
'It's true, isn't it?' he badgered.
'I have no idea what you're talking about,' she replied, turning her attention to Matron. 'Excuse me . . .'
'Visiting hours don't start 'til four,' the nurse snapped again.
'We're not visiting a patient,' Nancy told her, leaning on the counter. 'We are here to observe Dr Rathdonnelly's autopsy today. For our . . . ahem . . . dissertations.' Nancy was a superb liar, thought Ursula to herself.
'Oh, you're medical students. Right, well, that's different,' said Matron, suddenly smiling. 'Go straight down to the end of the main corridor, turn left, third right. Look for the black double doors with opaque glass. There aren't any signs to the mortuary. There are surgical overalls hanging on the pegs before you go in. Use those.'
'Wait! _You two_ are going to Lady India's autopsy?' protested Thistleton. 'Why can't I observe it too?'
Matron smiled superciliously at the journalist. 'Observing a dissection is a privilege limited to members of the University,' she informed him.
'But . . .'
'Please leave now, sir.'
'Bloody privilege,' Thistleton cursed as he finally left.
*
The girls nervously approached the morgue via a long corridor. They could soon hear the sound of Doc's distinctive Scottish tones echoing from the other end of the passage.
'. . . are you writing this down? I know Trott wants my report ASAP. My first conclusion, on examination of the neck injury, is that the murder weapon was most likely the missing portion of the broken champagne saucer found clutched in the victim's hand . . .'
The double doors to the morgue swung back and forth as a stream of visitors – technicians, nurses and police officers – rushed in and out. The girls took green surgical overalls off the pegs outside the morgue and put them on over their clothes. To Ursula's amazement, no one took the least bit of notice of her or Nancy as they changed. They carried on listening to Doc's analysis while they did so.
'. . . On inspection, I have found a small shard of glass measuring three millimetres by one millimetre lodged inside the left edge of the slash wound. Its slightly curved character leads me to believe that it is a tiny piece of the aforementioned missing portion of the broken champagne saucer found in the victim's left hand at the scene.'
'Killed with her own champagne saucer,' said Nancy. 'That's horrible.'
'I know,' Ursula agreed.
Three junior doctors in white coats strolled into the morgue and Nancy quickly tagged along behind them, Ursula on her heels.* Once inside, the girls hung back close to the door, trying to be discreet.
The morgue glittered from every surface. Everything was white, shiny, glaring – white tiles on the floor, white tiles on the walls, dazzling steel benches, steel sinks and taps on hoses. A set of X-rays was illuminated on a lightbox on the far side of the room.
'Ugh!' Ursula gasped suddenly.
Close by, on a small dissection table, sat a pair of hands, _just_ hands, nothing else attached. A technician, wearing rubber gloves, picked one hand up and gently lifted one finger, pressed it onto a pad of ink, and then pressed the same finger onto a sheet of paper, producing a print. Ursula recognised the dark red polish on the nails. The hands belonged to India.†
Ursula's eyes soon came to rest on the porcelain slab in the centre of the room. There lay India, naked, bluish-white, only she didn't look much like India anymore. Her chest had been cut open in a Y-shape, from the side of each shoulder to her abdomen. Her organs had been placed in clear plastic tubs, arranged neatly on a metal trolley, and variously labelled 'STOMACH TUB', 'LIVER TUB' and so on. Her scalp and a section of her skull had been removed to retrieve her brain, which had been placed on another small table near her head. Of course, she had no hands. Had Ursula attempted to imagine a scene of such sterile horror, she couldn't have.
'Ee-ee-e-ew – this is terrible,' wailed Nancy into her ear.
'I just can't believe it,' Ursula said sadly.
She was amazed by how crowded the room was. There was a uniformed policeman with a Dictaphone, recording Doc's every word. A medical secretary stood next to him, scribbling down the report in shorthand. Various doctors and medical students in white coats looked on, transfixed. Lab technicians dashed around cleaning large knives and saws that wouldn't have looked out of place in a butcher's shop. Nurses wandered in and out, replenishing the staff with instant coffee. In death, as in life, India attracted quite an audience.
At the centre of the scene was Doc, rubber-booted, gloved and covered in a green operating gown and cap. In one hand he wielded an elongated pair of tweezers, in the other a steel ruler. He chomped on a cigar, jammed in one corner of his mouth, in the manner of a man relaxing at his club rather than dissecting another human being. Everyone listened intently as he went on, cigar in situ, saying, 'For the record, please state that I am defining the wound as a "slash" wound of the type typically produced by the exquisitely sharp edge of a piece of broken glass. The slash is twelve and a half centimetres long. It is longer than it is deep and the wound is deeper on the _left_ side of the neck than the _right_.'
'So the attacker was right-handed,' whispered Nancy.
'I suppose so,' Ursula whispered back.
'Which rules almost everyone in as a potential suspect,' Nancy said regretfully, her voice hushed.
Doc went on. 'The depth of the wound is twenty millimetres. The wound is straight-edged, which leads me to assume that the chin was raised during the attack. This stretching of the skin leads to a clean wound, rather than the jagged cut seen when a weapon is drawn over loose skin.'
'What sort of person lifts someone's chin to kill them?' Nancy asked quietly. 'That's _so_ weird.'
'I can't imagine.' Ursula shuddered.
'The jugular vein has been severed, causing the pooling of blood that was seen at the scene of the crime, and consistent with venous bleeding,' Doc continued. 'However, I must caution that the severing of the jugular vein was _not_ the cause of death. There was not enough blood loss for that.'
Ursula got out her reporter's notebook, which she hoped could pass for a medical student's, and started taking notes. She'd never get the details right otherwise.
'On _close_ examination, it is apparent that the jugular vein was divided during the attack. There is a hole – measuring half a centimetre – that left the wound open to the environment. This leads me to conclude that, since the victim could not and did not die as a result of blood loss, an air embolism is the official cause of death. An air embolism is caused by aspiration into a cut jugular vein while standing or sitting with the neck at a higher level than the thorax. It literally sucks air from the outside into the vein. The air remains in the right side of the heart. Death is usually immediate.'
'Do you think the killer just meant to wound her?' asked Nancy, her tone still low.
'Maybe,' Ursula replied.
'This is a peculiar case,' continued Doc. 'Often when someone is stabbed in the neck they don't die immediately. They're still capable of moving. They struggle. This girl didn't. There are no defence wounds on her arms or hands. No contusions. There was no blood anywhere else apart from on her neck and dress. The room was undisturbed with no sign of a fight. She was attacked in the exact position she was found, with her upper body slightly raised on the upper part of the _chaise-longue_. The hypostasis in her left foot confirms this.'
'I knew it!' said Nancy triumphantly.
A medic tapped her on the shoulder. 'Sssshhhhhh!' he scolded.
'Sorry,' Nancy apologised. Then, seemingly out of the blue, more quietly, she said, 'Hey, what are you gonna wear on your date tonight?'
'Er . . . I'm not really thinking about it right now,' murmured Ursula, trying to concentrate on her note-taking. 'Maybe I'll just wear this.' She looked down at her mini-kilt, which was definitely trendy enough for a date.
'You cannot wear a woollen skirt, however short it is, on a romantic dinner date!' Nancy told her, highly indignant. 'I'll lend you something. I've got the perfect dress. You'll die when you see it.'
Ursula smiled, looking forward to wearing one of Nancy's party dresses again. 'If you're sure,' she said gratefully.
Meanwhile, Doc was still pondering India's murder out loud. 'Now, why would a girl lie on a _chaise-longue_ and allow her throat to be slit without defending herself? Did she know her attacker extremely well? So well that when he came up behind her she had no reason to suspect any ill intent? How did the murderer manage to break a glass without alerting his victim to the fact that she was in imminent danger? Was the glass perhaps already broken when the murderer arrived on the scene? Did India break the glass herself? Luckily, I have no need to answer such questions. My job is purely to define the cause of death for the courts. My conclusion is that the girl died as a result of an air embolism.'
A technician rushed up with a note in his hand. 'Doc, the bloods are here,' he said. 'The victim had extremely high levels of alcohol in her blood. No indication of any narcotics. The stomach contents show that she hadn't eaten for at least ten or twelve hours before the attack took place. The alcohol would have been absorbed into her bloodstream much more quickly than if she had eaten. She would have felt highly intoxicated.'
Ursula nudged Nancy. 'No drugs. Wow. Looks like Otto didn't give her that half-tab of acid.'
'But why would he lie about that?'
'No idea,' said Ursula softly. 'I think he's covering something up. I don't know what, and I don't know why. But I just don't believe India would have spent the night in his arms without chemical help.'
'But she was "highly intoxicated".'
'True. But was she intoxicated _enough_ to feel as though she was in lust with Otto?'
'I see what you mean. I'd have to drink the entire Champagne region dry before I even kissed the guy on the cheek,' Nancy conceded.
Doc was finishing up, replacing his instruments and untying his gown. A nurse took it from him, and he handed her his used gloves. He suddenly stopped and jokingly whacked the side of his own head.
'Oops – forgot something,' he said. He removed the cigar from the corner of his mouth and addressed the medical secretary. 'Add this to the report: the fingerprints on the glass were smudged, so they're not much use. But there was a tiny trace of sodium dimethyldithiocarbamate on that shard of glass.'
'Sodium dimethyl-whatty?' repeated Ursula, attempting to scrawl down the name of the chemical as best she could.
'I haven't a clue,' said Nancy. 'But I know someone who would.'
'Who?' asked Ursula.
'My brother Frank. I told you, he's doing medicine. He knows all this kind of stuff. I'll call him as soon as we get back.'
'Call him!' Ursula was shocked. 'You can't _telephone_ America from a college payphone. You'd need at least forty or fifty ten-pence coins. Send him a telegram instead.'
'Okay, sure,' said Nancy. 'Fingers crossed he can help.'
'Right, inform the family that the funeral can go ahead on Thursday,' Doc told the uniformed policeman. Then he said to a junior doctor, 'Put everything back in and sew her up.'
'Yes, Doc,' said the young man, moving to retrieve the organs from the tubs.
Then Doc suddenly changed his mind, saying, 'On second thoughts, I'll keep her brain for the collection in my office. I've never had an aristocrat's brain before.* It'll be fascinating for lectures.'
With that, he plopped Lady India's brain into a large jar of formalin and gazed lovingly at the spongy-looking object as it sank to the bottom. 'Beautiful girl, beautiful brain,' he said cheerfully, before adding, 'ugly death.'
Clutching his trophy, Doc bade a jaunty farewell to the staff. Just as he was about to push open the swing doors, he caught sight of Ursula and Nancy. Oh, dear, thought Ursula to herself, we're in deep trouble now.
'Ah, Holmes and Watson, isn't it?' he said in a friendly tone.
'We prefer Cagney and Lacey,'† replied Nancy. 'More modern.'
'Quite right. Didn't realise you two were medics. Keep up the good work. We need more women doctors.'
'We sure do,' replied Nancy, flashing a confident smile at the pathologist.
'Well,' he said, 'I hope this morning kept you entertained.'
'It was awesome, Doc,' Nancy told him sincerely. 'But you've given me a phobia about champagne glasses. Almost makes me never want to go to another cocktail party again. _Almost._ '
* Before the advent of DNA-testing in the late 1980s, autopsies were more casual affairs than they are now. Since there were no DNA samples to contaminate, police, hospital staff and medical students would come and go fairly freely from the morgue.
† Until the late 1980s in Britain, it was accepted practice to sever a victim's hands in order to take a perfect fingerprint. The families never knew.
* As pathologists couldn't take a DNA sample from a body part in 1985, if they suspected that a case might be reopened during the next thirty years, they were legally obliged to preserve the relevant material. Until the late 1980s, pathologists kept many victims' brains and organs on their office shelves, 'just in case'.
† _Cagney and Lacey_ , 1982–1988, was the first 'feminist' buddy cop show. Worried that the female detective characters were too aggressive, CBS cancelled it in 1983. Furious fans staged a letter-writing campaign, forcing the network to reverse its decision. The show won fourteen Emmys.
The sight of Horatio Bentley holding court beneath the gate tower that afternoon was a soothing antidote to the bleak hours spent in the morgue. While Nancy popped into the lodge to check for a missive from Next Duke, Ursula listened with amusement as he assigned various gossipy stories to a couple of student hacks.
'. . . I heard the Corpus Christi sherry party was a _huge_ embarrassment,' said Horatio, grinning gleefully. 'One of the Latin scholars drank all the sherry before the guests arrived. A dry night was had by all. Gillian, you can write that up?'
'Sure,' a girl agreed, scribbling details on a notepad.
'Bobby, can you take on an item about Harry Bladon? He's been censured by the Keble JCR committee. The entire college is incensed by his hogging of the college payphone.'
'Who's he telephoning?' asked Bobby, an eager-looking boy.
'Seccies. The rumour is that he's so intellectually insecure he won't ask female undergraduates out. Thinks secretarial college students are easier to talk to. Pathetic twit.'
'No problem,' replied Bobby.
' _Au revoir_ hackettes!' cooed Horatio to the two writers as they dashed off after their stories. 'And _bonsoir_ beauty,' he added, noticing Ursula. 'How's your day going?'
'Rather gothic, actually,' she replied. Then dropping her voice, she continued, 'We've been at India's autopsy.'
'How grisly,' he gasped. 'What did you discover?'
Before Ursula could answer, she was interrupted by a shriek from Nancy.
'Guys, I don't believe it. A note from Next Duke!' Nancy was as pink and quivery as an over-whipped Angel Delight. 'Listen.
_'Dear Miss Feingold (if you must insist on using my title, I'll insist on using yours) . . ._
' _So_ witty!' she giggled. 'I dig funny guys. Oh my god, wait until you hear this.'
Her voice going up an octave every few lines, she read on:
_'. . . I was touched to receive your note and your extraordinarily generous offer to dry-clean my jacket. I cannot accept. Were the garment in question to be cleaned every time someone spilled a glass of champagne or vomited on it, it would spend so much time at Jeeves that I would never actually get to wear the tatty old thing . . ._
'Next Duke's _so_ self-deprecating. I mean, did you _ever_ see a better-dressed guy? How hot,' Nancy continued, enraptured.
But Ursula's mind was back at the hospital.
'If India didn't have any narcotics in her blood, that means Otto didn't give her any. Doesn't it?' she asked out loud.
'What are you on about?' Horatio looked thoroughly confused.
'Sorry. Otto told us he spiked India's drink with LSD, but the autopsy results contradict that.'
'Oh my god, listen to this!' squeaked Nancy, utterly engrossed in Next Duke's letter.
_'As a token of my appreciation of your offer, may I invite you to join me—'_
Ursula interrupted her friend. 'So if he didn't give the LSD to India, what did he do with it? Perhaps he gave it to someone else? But who? Or maybe he didn't give it to anyone at all. Perhaps it's still in the pocket of his tailcoat. And if it is, then—'
But Nancy wasn't listening. She went on,
_'—for a picnic breakfast at sunrise on Port Meadow this Saturday?'_
'Hang on a momento,' exclaimed Horatio. 'Am I to believe that Paddington is asking you on some kind of date?'
'Paddington?' Nancy looked puzzled.
'Everyone calls him that. He looks so like an orphaned teddy bear.'
'He does not!' Nancy retorted. 'Next Duke is the most beautiful boy I have ever wanted to make out with. And the poshest.'
'I feel a diary item coming on,' said Horatio. 'This is actual _news_. Algernon Dalkeith has _never_ been on a date since he started at Merton two years ago, despite the attentions of many wannabe-Duchesses.'
'Who?' asked Nancy, flummoxed.
'Sorry. Until he inherits the dukedom, the eldest son of a duke bears a different name and title to his father, the current Duke of Dudley. Next Duke's real name is Algernon, Marquess of Dalkeith,' Horatio explained.
'Next Duke's _way_ easier,' countered Nancy.
'What a shame you can't go on the sunrise date with him,' Ursula commented.
'What do you mean, I can't go?'
'Rowing squad. We meet on Saturday again at six, remember.'
Nancy's face fell. 'My parents would _literally_ stop paying for college if I turned down a date with the Next Duke of Dudley. This is a _major_ social opportunity for me.'
'Ursula, can't you make an excuse for her?' asked Horatio. 'For the sake of the Feingold clan. For the sake of _Cherwell_. This is going to be a marvellous John Evelyn story.'
'All right. Just this once,' she said reluctantly.
'Thank you, Ursula. I'll pay you back, I promise. Okay, I'm gonna go compose a poetic reply. See you at four-thirty – I'm psyched to try Marmite with Mrs Deddington.'
With that, Nancy dashed off in the direction of her room, clutching Next Duke's note tightly to her chest.
'I'm going to try and pin down Otto,' Ursula told Horatio. Her essay would just have to wait until after her dinner with Eg.
'Good luck, sweetie. I'm off to the cinema. _Another Country'_ s* showing again. It'll be even better the sixth time.'
Having bidden Horatio goodbye, Ursula set off towards Monks' Cottages. She'd started to feel convinced – she wasn't quite sure why, it was just a sense she had – that if she could learn the whereabouts of the missing half of the acid tablet, she'd be one step closer to working out what had really happened on Sunday night.
Monks' Cottages had a far cosier feel than any of the quadrangles Ursula had so far seen in Oxford. It could be accessed by a narrow passageway on the eastern side of Great Quad, and consisted of a collection of low, higgledy-piggledy stone dwellings built around a tiny grassy courtyard with a cobbled path running around the edge. There was an ancient well in the middle, overshadowed by an old apple tree, and the borders were filled with autumnal seed heads. She did hope she would be billeted here in her second year.
Ursula could hear the washing machines turning over on the ground floor of Staircase A, and noticed a couple of students wandering out of the laundry room lugging their washing. Otto's rooms, she knew, were in Staircase D, where India's had been as well.
A chill enveloped Ursula as soon as she stepped inside the arched entrance. India's name was, of course, still on the board listing the staircase's occupants – she had been in room three. Otto was in room four. Both rooms were on the first floor.
Ursula headed up the staircase towards Otto's room, praying that he would be in this afternoon. As she reached the top of the stairs, the first thing that caught her eye was the blue-and-white police tape across the front of India's room. CRIME SCENE DO NOT ENTER read the words on it. Next, she spotted a huddled heap on the floor outside the door to Otto's room. As she drew closer, it became apparent that the heap was human. Arms hugging its legs, head buried in its knees, the heap's face was not visible. Whoever it was, Ursula thought to herself, they were in a pathetic state.
'Erm . . . hello?' she said softly.
The heap responded by burying its head even deeper in the folds of its clothes. It was then that Ursula clocked the clunky brogues and thick maroon tights protruding from the bottom of a particularly unappealing grey flannel skirt. There was only one person in college who dressed like a geography teacher. Ursula knew instantly that the heap before her was none other than Claire Potter.
'Claire?' she asked. 'Are you okay?'
'FINE,' grunted the heap, sounding annoyed.
Claire Potter momentarily lifted her head and looked at Ursula. Her face was red and blotchy, her eyelids swollen. Submerged in the inflated puffiness of her complexion, her eyes appeared to have shrunk to the size of two currants.
Ursula was concerned. 'Claire, what's happened?'
'Nothing. That's what. NOTHING!' came the angry reply.
'Nothing?'
'He's ignoring me. Pretending nothing happened.'
'Who? Do you want to talk about it?' Ursula asked, sitting down next to her. She noticed the other girl was clutching a pen and paper in one hand.
'NO,' retorted Claire, and started frantically scribbling on the notepaper. Ursula looked over the girl's shoulder as she wrote.
Prince Otto Schuffenecker,
You are informed that your membership of the Christminster College Crosswords and Ice Cream Society has been revoked, nulled, voided and cancelled.
Claire Potter,
President
Christminster College Crosswords and Ice Cream Society
Tears spurting from her already swollen eyes, Claire heaved herself up from the floor and pinned her note to Otto's door.
'I never even invited him in the first place!'
'Invited him to what?'
'My Crosswords and Ice Cream evening.'
'Did he come?'
'Yes.'
'But that's _so_ strange,' Ursula said. Why would Otto have claimed he spent Sunday night locked in an embrace with India if he was at Claire's crossword club?
'What, that someone like _that_ would want to be with someone like _me_? Is that what you're saying is "so strange"?' Claire asked bitterly.
'No, of course not,' said Ursula, trying to calm the girl, although, if she was brutally honest, she did think it odd that Otto would have chosen to spend his time after Wenty's party with Claire rather than India. 'Just tell me what happened.'
Claire shook her head. 'I can't.'
'Why?' asked Ursula.
'It's too embarrassing. I just want to forget about it.' She turned and started to walk back downstairs.
'Claire, wait. This could be really important. I need to find out who killed India. It's scary, not knowing who did it. Having a murderer wandering around Oxford. Or even in college.'
Claire looked spooked and paused at the top of the stairs. 'I couldn't tell the police what really happened,' she croaked. 'I lied to them in my interview. What's going to happen to me?' Her voice rose higher with panic.
'Listen, I think you can help solve this case. No one will know what you say to me. But you've got to start being truthful with someone.'
'I know,' Claire said, hanging her head. 'But not here. I don't want to run into _him_.' She beckoned to Ursula to follow her.
A few minutes later she found herself in the student laundry, watching Claire angrily unload a large tumble dryer full of clothes.
'I'll fold,' offered Ursula when she saw Claire flick the iron on.
'Thanks. Look, you can't tell _anyone_ ,' the girl entreated, laying out an ugly, salmon-pink nightdress on the ironing board. It looked like the kind of thing a great-aunt living in a retirement home would wear.
'You have my word,' Ursula promised, shaking out a pair of jeans.
'I'd spent ages getting everything ready for the club. I was wearing a lovely white dress . . . my old Confirmation dress. I had photocopied tons of crosswords from _The Times_. I had quite a few little tubs of ice cream already dished out for the guests. I put everything on the big table in front of the window in the JCR. It looked really nice. I thought everyone was really going to enjoy themselves, only . . .' Claire bit her trembling lip. She hiccupped. 'Only . . . well, the thing is . . . no one c-c-c-a-aaaa-me.'
A fat tear rolled the length of her nose and perched there, wobbling as she spoke.
'I thought you said the club was really busy, when we talked at the boathouse yesterday,' said Ursula.
'That wasn't _exactly_ true,' Claire admitted, tapping a finger on the metal base of the iron to test its heat. 'I mean _I_ was busy but the party wasn't.'
Ursula frowned. 'How?'
'Like I said, no one showed up. I tried to stay cheerful, you know, trying some of the crosswords by myself, but after a couple of hours I just felt like such a loser,' Claire moaned, pressing the hot iron onto the nightdress. 'The ice cream was melting, it was such a waste. But I felt as though I had to stay all evening, in case someone showed up. So I sat at that big table in front of the window and watched to see if anyone would come. After ages and ages of nothing, there was that big argument in Great Quad at midnight. And then I saw India dashing towards the JCR staircase. I was so excited someone was finally coming to the party!
'I went to the door, opened it a crack, and listened. I heard someone coming up the stairs, peeked round the door, and sure enough, there was India. I think she was quite drunk because she was stumbling a bit. Her dress was so long that she kept tripping on the hem as she walked, and she was steadying herself against the wall of the staircase. She didn't even notice when her tiara fell off while she was on the landing . . .'
So that's what happened to the tiara, thought Ursula to herself. But that didn't tell her where it was now.
'India didn't notice me. But then no one ever does. When she got to the landing she didn't turn left to come into the JCR. She went and knocked on Dr Dave's door.'
'Did he answer?'
'No. She knocked a couple of times, but no one came. Then she turned the handle of the door and went in. I heard her call out "pudding" once she was inside the room.'
'"Pudding"? You're sure?'
'Yes. I was weirded out. What was India doing calling a don "pudding" in the middle of the night?'
'Did anyone answer her?'
'I don't think so. I mean, I didn't hear Dr Dave's voice. Just hers. Anyway, he'd already gone.'
'You saw him leave the room?'
'No, I didn't, but it was so strange. A few minutes after India had gone into his rooms, I sat back down at the table by the window, and I looked out over Great Quad, and there he was, walking along the path. It looked like he was heading from the Kitchen Quad towards Monks' Cottages.'
Ursula racked her brain. Hadn't Ms Brookethorpe said something similar about seeing Dr Dave out in the quad after the argument? Why had he gone outside at that particular moment?
'Did you notice him go down the staircase that night?' she asked Claire.
'That's the weirdest thing. I didn't hear anyone go _down_ that night, they only came _up_.'
'They?'
Claire looked away.
'This is the embarrassing bit,' she mumbled, sounding ashamed. 'I was sitting back in the JCR, and then, it must have been about half an hour or more after India had gone into Dr Dave's rooms, I heard footsteps on the staircase again. This time I was determined to recruit a club member. I put on a big smile and went and stood out on the landing. And then, this is really bad . . .' Claire's voice faltered.
'What happened?' Ursula asked her.
'It just looked so pretty, I couldn't help it.'
Ursula had no idea what Claire was talking about, but tried to look encouraging, hoping the girl would continue.
'I picked up India's tiara. I was going to give it back to her the next morning. But it was so lovely and I thought I'd just try it on for a moment. I'd put it on my head when Otto appeared. He was carrying a bottle of champagne and two glasses. He saw me! _He_ saw _me_. And he said all these really nice things.'
At this memory, Claire wept copiously, so much so that snot started dribbling from her nose onto her top lip. She wiped at it, unsuccessfully, with the back of her hand.
'Like what?' asked Ursula, wondering, again, where the tiara had ended up.
'He said I was the m-m-most – uu-ggg-hh – exquisite girl he had ever seen.'
'Are you _sure_?' Ursula pressed her. Hadn't Otto said he had told India she was 'exquisite'?
'Of course I'm sure! No one's ever said anything like that to me before. I'll never forget it. Then he took my hand, and we went into the JCR. We each ate a melted ice cream. He poured champagne for both of us, and once I'd finished it . . . I don't know, it was all so romantic . . . I fell in love with him! I thought he was in love with me. We kissed. It's the first time I've kissed a boy. It felt as though we were floating in a lilac cloud.'
'Claire, have you ever heard of LSD?'
'What does that stand for?'
'Don't worry about it,' said Ursula. Poor Claire. She didn't have a clue what had happened to her. 'What did you do with Otto?'
'Oh, it was marvellous. We didn't do any crosswords. But we ate ice cream together.'
'You're saying Otto ate a tub of ice cream?'
'Yes. He had the chocolate chip.'
Finally, the identity of the mystery ice-cream eater had been revealed – Otto – and the fate of the other half-tab of acid. Ursula knew Otto's story had been messy, he had been trying to cover something up – and it was this. He'd spent the night on Sunday not, as he must have wished, entwined with India, but tangled up with Claire Potter on the JCR sofa, and he was thoroughly embarrassed about it. Unless, of course, Claire was making the whole story up. But how would she know all the details Otto had mentioned if she were lying? She seemed sincere enough. Ursula had to believe that Claire was the honest Welsh lass she appeared to be.
'So how did the night end?'
'He fell off the sofa. I helped him get up and put him in the wing-backed chair. He said all these lovely German-y words to me. Like, "libb-ling" . . . something like that. Then he passed out. So I went to my room. Now he's acting like nothing happened. He walked straight past me in the Porters' Lodge the next day. Didn't even say hello, let alone "libb-ling".'
'I am so sorry,' said Ursula, sensing the other girl's deep pain and disappointment. 'Men can be awful.'
'I just want to die,' snivelled Claire. 'I've been a fool, haven't I? Why would someone like him want someone like me? I'm just some Welsh turnip.'
'Don't talk like that. Perhaps you both just . . . made a mistake,' Ursula told her, wondering how Otto would react when confronted with the error he had made. 'Claire, did you visit Otto late on Monday night?'
'How do you know?' she gasped, mortified.
'I saw you, from the library window.'
'I went to his rooms, but he wasn't there.'
That was certainly true, thought Ursula. He had been bawling his eyes out in Nancy's room at the time.
'Then he ignored me again this morning at breakfast. So I decided to take my revenge.'
'Revenge?' said Ursula, feeling alarmed.
'You saw the note. I'm banning him from the crossword club. Hopefully that'll hurt him as much as he's hurt me.'
With that, Claire flicked off the iron, grabbed her pile of clothes and put them in a basket, and started to head out of the laundry room.
'Claire, wait,' Ursula called out. 'What did you do with the tiara?'
'I was going to get it back to Lady India, honestly. But then, when it turned out she was dead . . . I was so terrified . . . I hid it. I couldn't tell the police I had the tiara, could I? They might have thought I'd killed her for it . . . and if my parents found out I'd stolen something, a tiara belonging to a murder victim – my dad would go mad. I wouldn't be allowed to stay at Oxford. I don't know what to do with the tiara now. It's still in my knicker drawer.'
'Oh, right.' Ursula was rather surprised by this news.
'Don't say a thing,' begged Claire. 'And thanks for listening.'
With that, she hurried out of the laundry. Ursula sat for a few moments, still astonished by the story she'd heard. She tried to order her thoughts by writing them down in her reporter's notebook.
_—It's not impossible to imagine that while on LSD Otto could have mistaken Claire in her white Confirmation dress for India, particularly as she had India's tiara on._
_—Cannot quite believe that Claire thought it was good idea to hide India's tiara with knickers._
Ursula decided to go back up to Otto's room and leave him a note asking him to come and find her this afternoon in the library. When she reached his door she noticed that Claire's note had gone. Otto must be inside. Before she knocked she couldn't resist trying the handle of India's room opposite, police tape or no police tape.
Unsurprisingly, the room was locked. Ursula crossed back to Otto's side of the landing, knocked on his door and called out his name.
'Enter,' a voice called out weakly from inside.
Ursula pushed open the door. Otto's room was small, untidy but cosy, white washed, with medieval oak beams in the ceiling. Two tiny dormer windows on the far side of the room overlooked the Monks' Garden. Otto was standing, face like a tomb effigy, in the middle of the room, dressed in black trousers and a black frockcoat embellished with elaborate gold frogging.
'For the funeral. It's on Thursday, at the church at Bratters. I just heard,' he said gloomily when he saw Ursula. 'I shall turn myself in to the police immediately afterwards. My only question is whether it is appropriate to wear the Austro-Hungarian Imperial sash at a wake.' He held up a ravishing strip of red and white brocade against the coat. 'What do you think?'
'I don't think you're going to be turning yourself in when I tell you my news. Or should I call it _your_ news?'
Otto threw down the sash on his unmade bed and grasped her by the shoulders. 'Tell me,' he said.
'Otto, you can't remember killing India because you didn't do it.'
'How can you be so sure? I have done many things I can't actually recall doing. When I was fourteen I was found drunk in the _Schnapps_ cellar at the _Schloss_ , dancing the polka with a chicken at three in the morning. I was wearing nothing but shooting socks. I don't remember a thing about it. That doesn't mean I didn't do it.'
'Otto, you weren't with India on Sunday night. You were with someone else.'
'What?'
'You were snogging a Fresher all night.'
Otto's face brightened.
'I knew it was you!' he cried. 'When I awoke that morning and there you were, standing before me. Your kilt felt so familiar against my cheek. _Liebling!_ '
He grabbed Ursula by the waist and started nuzzling her shoulder.
'Ugh!' she said, pulling away.
'You have gone off me already?' He looked terribly upset.
'I was never on you in the first place. I wasn't the Fresher you were with that night. It was someone much more . . . interesting.'
'Ah! You mean Lawnmower! The Dollar Princess. Yes, she would be wonderful for the _Schloss_. She could fund the re-thatching of the medieval barns.'
'Otto, the girl in question is a lovely Fresher named Claire Potter.'
'I know no one of that name.'
'But you do, Otto.'
'I do?' He looked blank.
'She was at the History drinks with us. She's on my staircase.'
'Describe her.'
'Okay. She's quite pale. Dark brown hair—'
'Sounds just like India,' he interrupted. 'No wonder I mistook her for my true _Liebling_.'
'Anyway, dark brown hair, cut very short. Glasses. She usually dresses in woollen skirts and brogues.'
Otto frowned. 'I hope you're not talking about who I think you're talking about.'
'She's very upset. Says you keep ignoring her.'
'Don't be absurd!' he exclaimed. 'Even without my glasses I would never exchange saliva with a girl like that one. I would rather have killed my darling India than kissed that toadlet. She is not rich. She is not from a great family.'
'Otto, how can you be such a terrible snob?'
'She is not beautiful,' he continued. 'In fact, she is the ugliest specimen of Fresher I've ever had the misfortune to set eyes on. I have never knowingly spoken to her so I can't tell you definitively that she is not even interesting but I suspect, were I to converse with her, which I hope never to, I would swiftly be proved right. I expect she is as dull as the colour of her hair.'
'Otto, I'm afraid you've done a whole lot more than converse with her. She's so upset about it all that she's banned you from her Crosswords and Ice Cream Society.'
'That note! On my door. I couldn't understand it.'
'Look, find your tailcoat. The one you were wearing on Sunday evening.'
'Why?' he asked, opening his wardrobe and retrieving it.
'You'll see.'
Ursula took the tailcoat from him and put her hand into the right-hand pocket. She was sure that what she was looking for would be there. She grasped something small and hard, took out her hand and showed Otto what was in it: a small, pink, plastic ice-cream spoon.
'You didn't do it, Otto. The night India was murdered, you were eating melted ice cream and snogging Claire Potter.'
* _Another Country_ (1984) starred eighties pretty boys Rupert Everett, Colin Firth and Cary Elwes as boarding-school pupils tangled in a web of homosexuality and spying. Shot like a Ralph Lauren ad, the film was as chic as it was camp.
Tuesday, 1st week: teatime
'Yeee-uuuuggg-cccchhhhh!'
Nancy had just bitten into her first-ever slice of hot buttered toast and Marmite.
Her expression was one of pure agony, though she somehow swallowed what was in her mouth.
'Sorry!' she said. 'People really eat this for fun here?'
'Most of the students wolf it down,' said Mrs Deddington sniffily.
'Honestly?' replied Nancy, sounding astonished. 'Give me peanut butter and jelly sandwiches any day.'
'I _adore_ Marmite on toast,' Ursula told them, taking a thick slice from the plate. 'Mmmmm!' she sighed, munching away. 'Delicious.'
'Nancy, I'll make you a cup of tea to wash away the taste,' said Mrs Deddington, flicking the kettle on.
The Scouts' Mess, located in the original college kitchen next to the Buttery Bar, was a rather dark, spartan room, imbued with a very British air of thrift. A pine dresser was stacked with basic blue-and-white-striped crockery and a few threadbare armchairs were arranged beside a mean-looking gas fire. Scouts took their meals on schoolroom chairs set around the two wooden tables arranged at one end of the room.
' _Iced_ tea?' Nancy asked, looking hopeful.
Pouring boiling water into the truly enormous sixteen-cup teapot, Mrs Deddington looked very confused.
'Okay, maybe not. Hot tea it is!' said Nancy, trying to sound pleased.
'Give it two minutes to brew,' said Mrs Deddington. 'Here, sit in the warm.' She beckoned the girls towards the armchairs, rubbing her hands together in front of the stove. 'Damp today, wasn't it? Autumn's really here now. But it'll be clear tonight. There'll be a frost in the morning.'
'I can't believe you guys think _that_ is a heater,' Nancy commented, looking at the stove's little flame. 'In America we'd call that a cigarette lighter.'
Just then, Alice bustled merrily in, dumped her cleaning stuff, took off her rubber gloves and said, 'How about a cuppa, Linda?'
'Just made,' said Mrs Deddington, pouring four cups of tea. She handed a steaming cup to each of the girls and another to Alice, who leaned against the sideboard as she sipped.
'Do you want to travel up to poor Lady India's funeral with us on Thursday?' Mrs Deddington asked her. 'We're driving.'
'I don't think I'm invited,' said Alice, looking heartbroken.
'Don't be silly, you must come,' insisted Mrs Deddington. 'Everyone knew you were Lady India's favourite scout. It would be remarked upon if you weren't there.'
'I suppose, yes, I would like to pay my respects. Say goodbye.' Ursula saw Alice's eyes well up a little. Poor thing, she thought, she must be just as devastated as India's friends. 'India was a really . . . _special_ girl.'
'Terrible, something like that happening here. In the grounds,' Mrs Deddington tutted. 'The police seem to be spending all their time in the High Provost's lodgings. Why they aren't speaking to _every_ student who was at that party of Lord Wychwood's, I'll never know.'
'Do you think the killer was at the party?' asked Ursula.
'I'm talking out of turn. I wouldn't know anything . . . about the party.' Mrs Deddington hesitated as she spoke, as though slightly unsure of herself. Ursula noticed her direct a particularly sharp look at Alice, who tried to avoid catching her eye. 'See, I wasn't here . . . in Christminster . . . on Sunday night. Can't miss my weekly dose of _Dallas_!'
That was an odd thing to say, thought Ursula – that she wasn't in college on Sunday night. Surely Mrs Deddington was never in college at night.
'Good episode this week, wasn't it?' said Alice.
'Er . . . yes, one of my favourites,' agreed Mrs Deddington. 'Anyway, Alice, let us take you with us to the funeral. I wouldn't have it any other way.'
'I suppose, yes, we all need to say goodbye to the dear girl. A proper farewell.' Alice looked downcast.
'At least that's settled,' said Mrs Deddington. 'Now, more tea, girls?'
'Otto and _Claire Potter?!!! No. Way_ ,' cried Nancy, amazed.
'I know – but don't tell anyone, I promised Claire I'd keep it all a secret,' said Ursula, doing up the pale pink satin buttons on the front of the navy velvet coatdress that Nancy had lent her for her date with Eg. It was slim-fitting, and Ursula loved its extravagant satin sailor collar. 'But Otto can't deny it. I found a plastic ice-cream spoon in his tailcoat pocket, exactly like the ones Claire had out that night. She feels like he's ignoring her, but the truth is he can't remember anything about it.'
'No wonder she's gutted,' said Nancy. Then, admiring Ursula's appearance, she added, 'This is _great_ for a date. You don't look slutty, but you don't look completely unavailable. That dress says, " _Maybe_ I'll make out with you, but no blow jobs tonight."'
Ursula tried her hardest not to look too shocked. Her goal was for tonight to conclude with a proper kiss that didn't resemble disgusting school food. She couldn't contemplate anything more complicated than that.
She had promised to accompany Nancy to see Dr Dave before heading off to meet Eg at the restaurant. As they made their way from the Gothic Buildings to his staircase, Nancy said, 'I'm seriously hoping Dr Dave's in right now. There is literally no way I am going to be able to produce a sentence, let alone an essay, if he doesn't explain that article to me. I read and re-read the first section again and again and I _still_ can't figure out one word of it.'
When they reached the tutor's landing, a male student emerged from his rooms clutching a heap of history texts.
'Looks like he's in,' said Nancy, relieved, and knocked on the door.
'Come in,' came the reply.
Ursula followed Nancy into Dr Dave's rooms, which were arranged almost exactly as they had been the first time they had visited for the sherry party. The only thing missing was the Giant Argus. As she took in the familiar sight of the _chaise-longue_ to the left of the fireplace, the memory of India's corpse lying there came back to Ursula vividly.
The evening was starting to close in, and the room seemed dark and slightly claustrophobic. Dr Dave was not in his usual ebullient form. He was gazing Byronically out of the window over Great Quad, as if lost in thought. Yes, thought Ursula, seeing him in profile, his hair _did_ have an undeniably languid swoop to it. Perhaps Ms Brookethorpe _had_ recognised him that night, very late, walking towards the Monks' Cottages.
The girls stood there awkwardly, not quite knowing what to do. Suddenly Dr Dave spun around.
'Forgive me, ladies,' he said. 'I'm shattered. I've spent half the afternoon being interrogated by that nincompoop Trott in a miserable room at the police station. The twit was trying to get me to confess to India's murder, based on nothing. It was like being persecuted by Oliver Cromwell. Anyway, what can I do for you?'
'It's about the article,' said Nancy hesitantly.
'Marvellous analysis of the Scottish Covenanters, isn't it?' Dr Dave brightened at the thought of it.
Suddenly, Nancy's face started to crumple. To Ursula's and Dr Dave's utter amazement, she began shedding tears by the gallon.
'Boyfriend problems?' the tutor asked tentatively.
'No, academic ones,' Nancy sniffed.
'Here.' Dr Dave offered her the blue silk handkerchief from his top pocket. 'Don't tell me bloody Brookethorpe wouldn't lend you the _Historical Review_?'
'It's not th-th-that,' stuttered Nancy, blowing her nose into the hanky. 'I-I-I c-can't understand why you guys let me into Oxford in the first place. I wanna drop out. I'm too dumb. Northwestern was so much easier. I can't understand a word of that article you want me to write about. I mean, what is a "forraine"?'
'There is a very useful book in the Hawksmoor Library. The _Dictionary of the Older Scottish Tongue to 1700_. If you were to look up the word "forraine" in it, you would discover that it means nothing more complicated than "foreigner". Someone like you, in fact.'
'Oh,' said Nancy, dabbing her eyes and looking relieved by the simplicity of the answer.
'Gin fizz, Miss Feingold?' offered the don. 'When a student comes to me with a crisis of confidence, I say, start on the booze. Restores the spirit far more quickly than anything else. Miss Flowerbutton, drink?'
The girls nodded. Dr Dave went to his fridge, put ice cubes, sloe gin, sugar syrup and soda water into a cocktail shaker and shook it violently. He then poured out the red, fizzy cocktail and handed each of the girls a glass. Ursula thought the drink was delicious – sharp and tangy and very alcoholic. The tutor drained his in one gulp and immediately poured himself another. Drawing himself up, he then pronounced, 'As the great Marcus Aurelius said, "If you are distressed by anything external, the pain is not due to the thing itself but to your own estimate of it, and this you have the power to revoke at any moment."'
'That's so deep,' said Nancy, cheering up a little.
'Miss Feingold, we dons have a tendency to suggest to students that the world is square. Christ, it's freezing in here!'
'Uh?' said Nancy, looking confused again.
Dr Dave took a match from the mantelpiece, struck it and lit the fire. Flames began to flicker about the kindling and newspaper underneath the logs.
'The onus is on you, the student,' he continued, 'to convince us that the world is round. Or hexagonal. Or triangular.'
'What? I don't get it!' Nancy said.
'Metaphorically speaking. I asked you to tell me in your essay why S. A. Burrell was wrong about the Scottish Covenanters. Have you asked yourself whether I am wrong to ask you if he is wrong?'
Nancy slowly shook her head. Meanwhile Dr Dave, having drained his second glass of fizz, was already most of the way through his third.
'Perhaps I am. The truth is, there is _nothing_ wrong with Mr Burrell's paper. It's a seminal work,' Dr Dave said wistfully. 'It's the kind of thing I hope to write myself one day.'
'Really?' Nancy looked incredulous.
The don nodded. 'Yes.' He regarded himself broodingly in the mirror, where he saw a lock of hair had fallen over his forehead, and rearranged his swoop to best advantage. 'No undergraduate ever understands the work here until their Final exams. The whole point of Oxford is to learn how to win an argument. To be so brilliant that you can convince me that the world is a rhombus, or that Cromwell was England's great liberator. In the meantime, just read, read, read. Start with that Scottish dictionary I mentioned.'
'Okay,' said Nancy.
'If you have any more problems, Miss Feingold, please let me know. What you are experiencing is a common occurrence. As a Moral Tutor I frequently find myself ministering to troubled undergraduates—'
'—like poor India?' interrupted Ursula, trying to sound as innocent as she could. It was the perfect moment to change the subject.
Dr Dave nodded and lit a cigarette.
'Ah, India. She had beauty, talent – and troubles,' he sighed, exhaling elaborate smoke rings. He plopped down on the sofa next to Nancy.
'Did she confide in you?' asked Ursula.
'Oh, yes, absolutely,' he said. 'We became quite . . . er . . .' He coughed, and looked faintly embarrassed. 'Yes . . . we were jolly . . . friendly! Mmm . . . I spent a lovely Christmas at Brattenbury Tower last year. Working on my new book.'
'Did India let you get _any_ work done?' said Nancy, directing a suggestive wink in his direction.
' _Much_ less than I had hoped.' Dr Dave smiled cheekily.
'I'm not surprised,' said Nancy.
He shrugged his shoulders casually, as if to say, _no one is_. He didn't seem the faintest bit perturbed by Nancy's hints that he'd been having an affair with a student. In fact, thought Ursula, he seemed pleased, almost flattered, by it. She decided the moment was right to dig a little deeper.
'Was that why India was here on Sunday night?' she asked.
'Rather curious, aren't you?' he said, topping up everyone's glass, including his own.
'Actually, I'm writing up the story for _Cherwell_ ,' she explained. 'It's so important to get the facts straight, isn't it?'
Dr Dave regarded her for a moment. Finally he said, 'I see. An amateur sleuth with journalistic ambitions. Hmmm . . . I like to encourage young writers. I'll try and help you.'
'Thanks. Can we go back to what happened on Sunday night?'
'God knows why she was here,' sighed Dr Dave, sounding frustrated. 'She had absolutely no reason to come. No reason at all. If she hadn't, she might still be alive.'
'But you were such _good friends_ ,' said Nancy suggestively.
'Ah, but the . . . _friendship_ , so to speak, was over,' he replied. 'I'd told her not to come up here. I couldn't allow her to visit me in college any more.'
'Why?' asked Ursula.
'My fiancée.' Dr Dave smiled gaily.
' _Fiancée?_ ' repeated Ursula, staggered by this new piece of information. 'You're engaged?'
'It's fairly new. We got engaged in Venice in September. Any close friendships with the girls here – well, all that had to stop. Naturally I told India about Fiammetta—'
'When?' interjected Ursula.
'As soon as India came up this term. She was upset. Blubbed a bit. But she didn't take much notice of what I'd said – she still kept appearing in my rooms last week, wanting advice, chats.'
It was no wonder, Ursula thought to herself, that Wenty thought India and Dr Dave were secretly still seeing each other. Dr Dave peered at the fire. The flames seemed to be disappearing. He picked up a pair of bellows and puffed at the kindling a few times until it relit.
'India was in love with you?' Nancy asked.
'I don't think so,' said Dr Dave. 'But she certainly liked to confide in me. I finally told her – this must have been on Saturday – that she must never, ever come to my rooms again.'
'So why _did_ she come up here then, on Sunday night?' asked Ursula.
'I've no idea.' Dr Dave shook his head. 'I wasn't here.'
That's not what Ms Brookethorpe had told her, thought Ursula. Or Claire Potter, for that matter. Both had said that they'd seen Dr Dave heading towards the Monks' Cottages just after India had entered the JCR staircase. Surely they couldn't both have been lying?
'Where were you?'
'I was at home.'
'Home?' Ursula was surprised. She'd always assumed that Dr Dave lived in college full-time – at least that's what Otto had said.
'Fiammetta and I bought a house on Cranham Terrace a few months ago. The pretty corner one with the wisteria. I only stay overnight in college when I absolutely have to these days. I was late in here the next morning, as you know, Ursula.'
She couldn't deny it. Dr Dave had certainly not been in his rooms when she had found the body. He hadn't arrived until well after nine in the morning, and so must have spent Sunday night elsewhere. What time had he actually left college? Ursula wondered. Long before India came to his rooms, as he said? Or after the argument took place on Great Quad, per Ms Brookethorpe and Claire's accounts? The existence of a fiancée further complicated things: could Fiammetta have been a motive for Dr Dave to have got rid of the girl who was almost stalking him? Or perhaps Fiammetta had found out about India and been driven to attack her fiancé's former fling herself?
The fire was starting to smoke slightly. Dr Dave puffed at it more fiercely with the bellows and more smoke swirled back into the room. The atmosphere was soon clogged enough for girls and don to be coughing. Dr Dave dashed over to the window and opened it.
'I think the chimney might be blocked,' Ursula volunteered.
'Oh, bother.' Dr Dave covered his nose and mouth with a handkerchief, stood next to the fireplace and somehow managed to reach his free arm up into the chimneybreast without catching fire or choking to death.
'It's definitely stuck up here,' he said, pulling at something.
He retrieved his hand, which was holding what looked like a scrunched-up towel.
'Odd,' he said, examining it. He unfolded the fabric and almost jumped out of his skin, letting out a yelp of fear at the same time.
Dr Dave flung the towel to the floor, where it landed in a heap. Ursula saw what had frightened him so much: the towel was caked in dark red dried blood. And on one of its corners, she spotted a familiar motif – an embroidered blue 'W'.
Tuesday, 1st week: evening
A swanky mixture of potted palms, mirrored panelling and Parisian toile-covered walls, the dining room at Chez Romain, the famous French restaurant on Turl Street, was the most glamorous – and romantic – place Ursula had ever been. She sat comfortably on a dark green leather banquette at a table laid with an immaculate white linen cloth opposite Eg, who was dressed in a crisp pale blue shirt and a sports jacket. He looked as striking as ever. A candle and a vase containing a red rose only upped the romantic ante. Ursula was in, she thought happily, for a major kissing session later on tonight.
'Wine list, sir?' asked a waiter wearing a white tuxedo.
'No, thank you,' said Eg. 'Just a Seven Up, please.'
'A glass of champagne for _mademoiselle_?' asked the waiter in a terrible fake-French accent.
Ursula smiled. 'Lovely,' she said.
Hopefully, she thought to herself, a glass of bubbly would settle her nerves after the discovery of Wenty's bloody towel in the chimney. After he contacted the police, Dr Dave had warned Ursula and Nancy not to mention the incident to anyone for the moment, and they'd agreed. That night, Ursula wanted to forget all about murder and enjoy herself with Eg. And anyway, he was so close to Wenty that Eg was the last person who needed to know about the latest discovery.
The drinks soon arrived on a little silver tray. Ursula took a long swig from her glass.
'Delicious!' she said.
'Good,' said Eg, glugging at his 7 Up.
'Why don't you have a glass of champagne?' she suggested. 'It's _so_ scrummy.'
'I don't drink,' he told her.
'Really?' Ursula asked, rather concerned. How on earth was Eg going to get around to the snogging marathon later without a little bit of alcohol to help things along?
'It's against my religion,' he explained.
'Right,' she said, trying not to sound too surprised.
'You look very pretty tonight, Ursula,' he told her.
Though flattered, she wasn't quite sure what to say. She tried to smile. Luckily, the moment was interrupted by the return of the _faux_ -French waiter.
'Menu?' he said, handing Eg and Ursula enormous leatherbound volumes.
Ursula opened hers and was so entranced by the array of dishes on offer that she had no idea how she was going to choose between the various delicacies. She noticed that her menu didn't include any prices and hoped she didn't choose anything expensive by mistake.
The waiter hovered impatiently over them. Finally he said, ' _Mademoiselle_?'
'Er . . . I'm not quite ready, sorry,' replied Ursula, hoping Eg would order first. If he ordered a starter, she would too. 'It _all_ looks so delicious.'
' _Monsieur_?'
'The melon balls to start, and then the duck _à l'orange_ ,' said Eg. He snapped the menu shut and handed it back to the waiter. 'Ready, Ursula?'
'Yes,' she said. 'May I please start with the prawn cocktail, followed by the salmon _en cro _û_ te_?' She hoped her choice would seem ultra-sophisticated.
' _Merci_ ,' said the waiter. He dashed off to place the order.
Now that the two of them were left alone, Ursula wondered what on earth they would talk about. They had at least two hours of dinner and chit-chat to get through before the kissing-fest would begin. It seemed a torturously long time to wait.
'I'm so glad you could come tonight, Ursula,' said Eg. 'Even with everything that's going on in college.'
'It's nice to forget about it all for a bit,' she replied. 'This is a lovely restaurant. Have you been here a lot?'
'No, actually. Never.' Eg sounded embarrassed.
'Why did you choose it then?' Ursula found herself giggling. How odd, she thought, to invite a girl on a date to a serious French restaurant that you'd never tried.
Eg looked tense. 'Er . . .well, I . . .' he stuttered. 'Apparently, this is where you take girls on dates.'
He looked round the room and Ursula followed his gaze, noticing that almost every table was occupied by a loved-up pair, most of them holding hands or staring longingly into each other's eyes.
'Where else have you taken girls on dates before, then?' she asked.
Eg looked deeply uncomfortable. It was most odd that the smoothest, most sophisticated boy Ursula had ever come across seemed to be allergic to any conversation related to going out with a girl.
'Well . . . the thing is . . . nowhere really,' he mumbled.
'Nowhere—' started Ursula.
'— _Mademoiselle. Monsieur_ ,' the head waiter interrupted.
He was trailed by two younger-looking waiters, each bearing a huge platter covered with an enormous, gleaming silver cloche. The platters were set down in front of Ursula and Eg and the silver domes simultaneously and flamboyantly removed.
Ursula could barely hide her disappointment when she saw her starter. The cloche had hidden an eggcup-sized glass containing no more than two teaspoonfuls of prawns and Marie Rose sauce. How on earth was she going to make this last?
'So this is what they call _nouvelle cuisine_?' Eg chuckled as he looked at his platter. Positioned in the centre were four melon balls no larger than marbles. 'Perhaps I should have taken you for a kebab.'
Ursula was just about to dig her spoon into the minuscule portion of prawn cocktail when she heard Eg clear his throat several times. She looked up to find that he had his eyes shut tight and his hands clasped in prayer over his plate.
'O Lord,' he began, 'who has multiplied loaves and fishes, and converted water to wine, O Lord, come to our table, as guest and giver to dine. Amen.'
He paused for a few moments, then opened his eyes. 'Sorry,' he said.
'Don't be,' replied Ursula. 'It's lovely to say grace sometimes.'
'It's just a bad habit really,' joked Eg, popping a melon ball into his mouth. 'Dad says grace at every meal.'
'Wow, he must be _very_ religious.'
'He has to be. He's the Bishop of Nairobi. Very strict Anglican.'
The last thing Ursula had expected was to discover that someone as suave as Eg was a clergyman's son.
'My mother was a Sunday School teacher,' he went on. 'She met my dad in church. What do your parents do, Ursula?'
Now it was her turn to feel anxious.
'My father, he was . . .' Ursula trailed off for a moment, remembering her father's broad smile in that treasured wedding photo. 'He was a doctor. Mummy was a musician.'
'Was?' said Eg kindly.
'They died when I was six. An accident.'
'I'm sorry,' he replied sympathetically. 'I'd never have imagined anything like that in your background. You're always so cheerful.'
'I have so much to be cheerful about,' Ursula reassured him. 'I was raised by my two grandmothers. They each lost a child, but they are the strongest women I know. They always told me, Mummy and Dad wouldn't want me to be sad, they'd want me to enjoy life. So that's what I try and do. I miss my parents every day, but I know they are watching over me, all the time, from Heaven.'
Ursula couldn't quite believe it when Eg teared up. He dabbed at his eyes with his napkin.
'Oh, Eg, don't be sorry for me,' she insisted. 'I'm lucky. My grannies are amazing. They love me so much.'
'Crikey, look at me,' he replied, blowing his nose. 'I am the biggest wimp. I cried all the way through _E.T._ '
The trio of waiters soon arrived with the next course. Ursula's salmon _en croûte_ was the size of a goldfish, she noted sadly when the next cloche came off, while Eg's duck _à l'orange_ was barely bigger than a chick.
'Do you miss Nairobi?' asked Ursula, taking a teeny-tiny bite in an attempt to make her food last.
'It's nice to have a bit of freedom, actually,' said Eg. 'In Nairobi, I certainly wouldn't be taking a nice English girl out for dinner.'
'Really? Why not?'
'Well, actually, I wouldn't be able to take _any_ girl for dinner. Kenya's really old-fashioned still. The Bishop's son is supposed to marry a good religious girl from our tribe.'
Ursula felt confused. If Eg's romantic destiny lay with a girl in Nairobi, why had he invited _her_ on a date?
After the main course had been cleared, a waiter wheeled the most extraordinary trolley full of puddings up to their table. There were meringues, _tarte tatin_ , _gâteaux_ , trifle, _crème brûlée_ – all in enormous quantities. Thank goodness, thought Ursula. Her tummy was still rumbling.
As they tucked into dishes heaped with two or three puddings each, Eg said, 'But – I'm not sure that's for me. When I saw you at Wenty's party – this exquisite English girl – I thought, I've _got_ to ask you out for dinner. Even if it's forbidden.'
'Forbidden?' She was starting to feel alarmed.
'Look, I'll admit something to you. This is the first date I've been on in Oxford,' he said, looking absolutely mortified.
Was this why Wenty had laughed so hard when she'd told him that Eg had invited her for dinner?
'Okay,' said Ursula. 'My turn to admit something. It's my first date, too!'
They burst out laughing together. Eg scooped up his last spoonful of pudding. 'Come on, let's get out of here.'
A little later, as they were walking back to college, he put his arm around Ursula's shoulders and pulled her in towards him. They wandered happily along Broad Street, past the cafés and tourist shops, and then turned into Christminster Lane. The only thing that dampened Ursula's mood was the sight of a police car departing from outside the college gates. She tried her hardest not to think about India's murder. She wanted to keep romance uppermost in her mind.
When they reached the landing outside her bedroom, Eg turned Ursula towards him, a hand on each of her shoulders, and said, 'You are . . .' he looked lingeringly at her lips '. . . very special.'
Finally, thought Ursula, kissing time. She closed her eyes expectantly. Absolutely nothing occurred. Patience, she told herself. She waited a little longer. She felt Eg remove his hands from her shoulders. Then she heard footsteps on the stairs.
Ursula opened her eyes. Eg was halfway down the staircase.
'Goodnight, lovely Ursula,' he said, half-turning back towards her. 'God bless you.'
Wednesday, 20 October, 1st week: morning
'The temperature in this bathroom is _literally_ a violation of my basic human rights. I still don't get why Northwestern has en-suites and Oxford basically has stables with bathtubs in them.'
Nancy, bellowing her complaints from the bath in the cubicle next to Ursula's, was not particularly enjoying the bracing draught circulating in the communal bathroom at the bottom of the girls' staircase in the Gothic Buildings at eight o'clock that Wednesday morning.
'Nancy, _no one_ turns the heating on in England in October. You just put on another sweater,' Ursula told her.
'In the bath?' replied Nancy. 'Anyway, how was it?'
'What?' Ursula replied, knowing perfectly well what Nancy was referring to.
'Your romantic dinner. Did you make out?'
This was _so_ embarrassing. 'I suppose . . . well . . . no,' she admitted.
'Why not?'
'I have no idea,' said Ursula. 'Instead of kissing me at the end of the night, he blessed me!'
'Oh my god, that is _so_ creepola.' Nancy sounded horrified. 'You need to be careful. He might be one of those gross Moonies.'*
Just then, Ursula heard the payphone ringing in the corridor outside for the first time since she had arrived in Oxford.
'It's way too cold to go answer it,' called Nancy from her cubicle.
'Otto told us never to answer the payphone anyway, in case it's someone's parents,' agreed Ursula.
Finally, the ringing stopped. But, a few seconds later, it began again.
'Maybe we should get it,' said Ursula.
'Rather you get fatal pneumonia than me, darling,' replied Nancy.
Ursula got out of her bath, wrapped a towel tightly around her and opened the door to her cubicle. Suddenly the ringing stopped.
'Drat,' said Ursula. Goose pimples were starting to appear on her forearms.
A few moments later, a bleary-eyed Claire Potter entered the bathroom, still dressed in her nightgown.
'There's someone wants to talk to you,' she said to Ursula. 'On the payphone. Says it's urgent.' Then she nudged Ursula and whispered into her ear, 'I can't go to the police about the tiara. I'm too scared. Anyway, no one is ever going to think of looking in _my_ knicker drawer for anything.'
'Claire,' said Ursula, her voice low, 'you can't get away with keeping it there forever. I'd better answer that call.'
Who on earth would be telephoning so early? she wondered, hoping Granny and Granny were all right. As she stepped from the bathroom into the gothic corridor, with its archway open to the quad, a brutal wind snapped against her naked legs and feet.
Ursula shivered as she picked up the handset that Claire had left hanging off the hook.
'Hello?' she said curiously.
'Unforgettable! Is that you?'
'Wenty?' said Ursula, surprised to hear his voice. Why on earth was he on the phone at this hour? 'I've told you a million times _not_ to call me that.'
'Okay, sorry, Flowerbutton. Look, there's no time to explain. I'm at Oxford police station. I was arrested late last night.'
'Oh, no,' Ursula gasped. Even though she'd known this was going to happen as soon as she'd seen the 'W' on that blood-stained towel, she still felt shocked. That must have been why the police car was leaving college when she and Eg returned from dinner.
'This is the last call I'm allowed today,' Wenty continued. 'I've spoken to my parents, and a lawyer, but I thought maybe you might know something – anything – that could help me get out of here. Because of that _Cherwell_ story you're writing.'
'What were you arrested for?'
'On suspicion of murder.'
Ursula was silent. She didn't know what to say. Even if Wenty was one of the most annoying people she had ever met, being arrested for murder was a fate Ursula wouldn't wish on anyone.
'Look, I didn't do it. You know that, don't you?' Wenty sounded desperate. 'But they're saying they've got new evidence against me since Monday when they first interviewed me in my rooms.'
'What kind of evidence?' she asked, although she knew exactly what he was going to say.
'It's a hand towel. Someone found it hidden in Dr Dave's rooms. It's one of the old Wychwood family ones. Got the bloody "W" on it. Bloody being the operative word.' He laughed weakly. 'It's covered in blood. The police are saying they tested it last night. Found my blood on it. And India's. They're saying it's proof I was in Dr Dave's rooms that night. They've shown it to me. It's my towel. There's no getting away from it.'
'I know,' said Ursula.
'What?' said Wenty. 'How do you know?'
'I was in Dr Dave's rooms when he found it. How did it get there?'
'You don't mean to say you think—'
'—I'm not saying I think anything,' said Ursula. 'Just tell me how it got there.'
'I don't know, do I? That's why I'm ringing you!' he wailed.
'But it's yours. How can you not know?'
'I told you, I didn't go into Dr Dave's rooms that night. God knows how it got there. You've got to believe me. Someone else put it there, and that someone is trying to frame me. I cut my hand when I was trying to wash up a champagne glass during the party. I wrapped my hand in the towel, remember?'
'Yes . . . yes, I do,' replied Ursula, thinking back to the moment when Wenty had tried to wash up the champagne saucers. It had seemed funny at the time.
'That's why my blood's on it. I think I left it on the floor in the bathroom. Someone used that towel when they killed India – to clean up. You've got to find out who hid the towel in Dr Dave's rooms. That person is India's killer, not me.'
The problem with Wenty's story, Ursula mused, was that he wasn't quite the plain, honest cucumber sandwich he would like to be taken for. If Ms Brookethorpe had been right about seeing him cycling away from college in the dead of night after she had left the library late on Sunday, it meant he had lied about his movements. Being a liar didn't necessarily make you a killer, but it didn't make you _not_ a killer either. He could easily have gone back to Dr Dave's rooms much later that night, long after Ms Brookethorpe had gone home, and killed India himself. Why else would Wychwood's towel be there, stained with his blood and India's?
'Wenty, I can't help you unless you are truthful with me,' Ursula said.
'Can you stop sounding like some kind of righteous headmistress, Flowerbutton? I _am_ telling the truth. I didn't kill India.'
'Why did you lie about Sunday night?' she said firmly. She didn't care if Wenty thought she sounded like a headmistress, if it was a way to get at the truth.
'What do you mean, lie?'
'Wenty, you didn't go back to your room after going to Dr Dave's staircase.'
'Of course I did.'
'Someone saw you leaving college. On your bicycle. Around two a.m.'
'Who?' he demanded.
'It doesn't matter who. Wenty, just be honest with me. Did you sleep in your own room that night, or were you somewhere else?'
There was silence at the other end of the line. Finally Wenty spoke. He sounded mortified.
'Well. Look . . . okay, I slept somewhere else.'
'Where?'
'That's not important.'
'Yes, it is, you twit.' Ursula was losing patience with him.
'Did you just call me a twit?'
'Yes. If you lie about your movements on the night of a murder, it does nothing except make you look like a deeply suspicious twit.'
'I am not a twit! Look, I can't say where I was.'
'I can't help you if you don't.'
'Flowerbutton, you're starting to sound like my mother. It's not very appealing.'
'Wenty, I'll take that as a compliment. I'm sure your mother's wonderful. Now, why didn't you tell me you'd left college that night?'
A long groan echoed down the phone.
'I was . . . embarrassed.'
'About what?'
'Look, after I heard India and Dr Dave _at it_ , so to speak, from the bottom of the staircase . . .'
Ursula decided not to tell Wenty yet that he had actually heard Otto and Claire Potter 'at it'. She didn't want to interrupt his flow, now he seemed prepared to talk.
'. . . I was jealous. Furious, actually. So I went off Hildebeest-hunting.'
'What sort of a sport is Hildebeest-hunting, Wenty?'
'You scramble over that garden wall at St Hilda's,* then go knocking on doors until a really top Hildebeest answers and, if you're lucky, you end up in bed with it.'
'You cannot go round calling female students "beasts"! That's horrible.'
'It's an in-joke,' he protested.
'Well, it's about the unfunniest, most sexist joke I've heard in my entire life.'
'Sorry.' Wenty actually sounded apologetic. For once.
'Anyway, did you end up in bed with a St Hilda's girl?'
' _Of course_ ,' he said, sounding faintly insulted at the suggestion that he might not have been successful. 'Geraldine Something-or-Other. Can't remember her surname.'
'Wenty!' Ursula gasped. 'If you were a character in a P. G. Wodehouse novel, you would have been labelled a hollow cad by now.'
'I know it makes me look awful. You can see why I didn't want anyone to know. Least of all a saintly prude like you, Flowerbutton.'
'I am not a saintly prude.'
'We'll agree to disagree on that. I mean, god, you know, there I was off with dear little Geraldine Something-or-Other, the night my girlfriend was killed. If that gets into _Dempster_† I'll never be accepted at White's.'‡
'Who cares about White's?' Ursula admonished him.
' _I_ do. And everyone I know, and everyone my parents know. Please don't mention Geraldine Something-or-Other in your article.'
'Can you stop worrying about yourself and focus on India? Just tell me anything else that happened that night.'
'Okay. Well, I must have left Geraldine's room at around six a.m., I suppose. That's the usual sort of time I escape those kind of, well . . . _scenarios_. I cycled back to college, left my bike at the gates and went straight to my staircase and up to my rooms. I was in bed fast asleep by six-forty-five. Didn't get up 'til midday.'
'Did anyone see you coming into college? Did you see anyone when you came in about six? Someone who could vouch for you?'
'God, er, can't remember really. I didn't have to ring. The gate was already unlocked. I suppose the night porter could have seen me.'
'If he happened to be looking out from the Porters' Lodge into the gate tower, at the very moment you came back, maybe. I can ask him tonight. And I could always interview Geraldine Something-or-Other. She'll vouch for you, won't she?'
'I'd rather you left Geraldine Thingummybob well alone, Flowerbutton.'
'Why?'
'Don't want to, you know, _encourage_ her . . . in _any_ way. Some of these girls they . . . they get the wrong idea, you know. They think they want a boyfriend, a "relationship", all that messy rubbish.'
'She might be able to give you a solid alibi.'
'But then I'd have to actually properly go out with her,' said Wenty reluctantly.
'Have you told the police all of this?' Ursula asked.
There were a few pip-pip-pip sounds while Wenty inserted money at the other end.
'. . . Running out of ten pees . . . I've told the police everything now. It's been _so_ embarrassing. But they're refusing to believe a word of it. They're insisting that the towel proves I was in the room at the time of India's death because my blood is on it as well as hers.'
'Have you told them about cutting your hand washing up?'
'Of course. But they're trying to get me to confess to India's murder. Flowerbutton, do you believe me?'
Ursula reflected that, logically speaking, if one were to invent a story about one's movements on a particular night, it would be unlikely to be one that painted as shabby a picture of one's moral character as Wenty's tale did. On that basis, she had to assume his was true.
'I do.'
'So you'll help?'
Wenty might be a hollow cad of a cucumber sandwich, Ursula thought to herself, but he was very possibly innocent. However insufferable his treatment of that poor St Hilda's girl was, it didn't make him a girlfriend-killer.
'Yes,' she replied. 'I'll try and help you.'
'Thank you, Flowerbutton. I'm worried that now the police have got me, they're going to stop looking for the real culprit. You've got to find the murderer, before they bang me up for good.'
'I must go,' said Ursula. She was freezing cold in her damp towel.
'Okay. Just make sure to go to the funeral tomorrow. I can't believe I'll be stuck in here when my poor India is buried. God,' Wenty said, sounding terribly saddened about his girlfriend. 'That was my last ten pee. Try and get in to see me here if you can and—'
_Pip-pip-pip. Pip-pip._
'And?' cried Ursula.
But she didn't hear Wenty finish his sentence. The line had gone dead.
* Rev. Sun Myung Moon's controversial Unification Church, nicknamed the Moonies, became a huge cult in the 1980s. It brainwashed thousands of virtual strangers into marrying each other at mass weddings.
* Founded in 1893 as a Hall for women, St Hilda's finally accepted men in 2008.
† The _Daily Mail_ 's Nigel Dempster was the most influential diarist of the 1980s and was nicknamed 'the Snake' for his love of gossip. London Zoo named a poisonous specimen after Dempster, much to his delight.
‡ White's = oldest, grandest gentlemen's club in London. The only woman ever allowed in was the Queen. Waiting list _at least_ seven years long.
The humble, red-brick workman's cottage on the corner of Cranham Terrace and Jericho Street, festooned with now-yellowing autumnal wisteria, looked like the kind of place Mrs Tiggy-Winkle might have lived. It had a pretty sash window looking onto the street, covered on the inside with a fine antique lace curtain, and the white-painted porch was shrouded in tumbling ivy.
'This has gotta be it,' said Nancy. 'Talk about a love nest.'
The girls propped their bicycles against an iron railing and made their way through the tiny front garden along a moss-covered path.
'I just hope _he_ doesn't answer the door,' Ursula said, realising she had no idea what she'd do if Dr Dave turned out to be at home. Nervously, she banged the brass knocker three times.
' _Pronto!_ ' came a sing-song Italian voice from inside the house.
The door opened to reveal a petite girl, barefoot and dressed in a lilac silk kimono edged with gold embroidery. Her hair was wrapped in a navy blue towel, and her enormous dark eyes were framed by ludicrously long eyelashes. She had tanned, gleaming skin and the delicious scent of tuberose perfume emanated from her. No wonder Dr Dave wanted to marry this enchanting creature, thought Ursula, spotting an engagement ring on her left hand. The girl smiled at them.
'Can I 'elp you?' she asked.
'We're students from Christminster,' said Ursula.
'You looking for Davide? 'E not here. Am sorry,' the girl said, shaking her head. She started to close the door.
'Wait,' said Nancy, stepping towards the threshold. 'We're looking for you, we think. Are you Fiammetta?'
She nodded.
'I'm writing a story for _Cherwell_ ,' explained Ursula. 'I wondered if you might be able to help . . . with background stuff. Have you got a few minutes to talk?'
'Sure. Come in.'
Fiammetta led the girls along a narrow passage towards the back of the house. They had to squeeze past a pile of suitcases with luggage tags attached, topped by a violin case.
'I'm sorry. I been travelling. Am in chaos!' she laughed. 'I 'ave not an idea even what day is.'
The threesome had now arrived in the modest, yellow-painted kitchen, whose French doors led out to a stretch of rambling cottage garden. Piles of sheet music, books and magazines covered most of the work surfaces.
'So, you writing about music?' asked Fiammetta, unfurling her turban to reveal her almost waist-length, damp dark hair, and gesturing for them to sit at the small café table in the middle of the room.
'Not _exactly_ ,' said Ursula.
'We're investigating a student's murder,' said Nancy boldly.
'In Oxford?! 'Ow awful,' exclaimed Fiammetta. She pronounced awful ' _ow-full_ '. 'What 'appened?'
'You haven't heard yet?' said Nancy, sounding astonished.
''Aven't 'eard what?'
'Lady India Brattenbury was murdered on Sunday night in Christminster College,' Nancy told her gravely.
'She was _studentessa_ at Christminster?'
'Yes,' said Ursula.
Fiammetta looked appalled. 'That is ' _orribile_ story. But I don't know 'ow I can 'elp. I've never met 'er. Or heard of 'er.'
Was she lying, wondered Ursula, or was she really as naive about her new fiancé as she claimed? There was no doubt that Dr Dave would have wanted to keep tales of his previous affairs with students from his future wife. But not to tell her about a murder in his own rooms – that was not just odd, it was, Ursula decided, deeply suspicious. Still, she didn't fancy being the one to break the news to Fiammetta that the most beautiful girl in Oxford had been found dead on her future husband's _chaise-longue_.
'I've only really got one question, Fiammetta,' Ursula said, her tone deathly serious.
'Yes?' replied the girl, now seeming jittery.
'Did your fiancé spend the night here on Sunday?'
'Please-mother-of-god, do not tell me you are suspecting 'im of this?'
'Of course not,' Ursula reassured her. 'We're just trying to figure out everyone's movements in college that night.'
'Oh, okay,' said Fiammetta, sounding slightly less alarmed. 'I was in Salzburg on Sunday night. We were playing Mozart's Violin Concerto Number 5 in A. I was First Violin.'
'Wow,' said Nancy.
'Thank you.' Fiammetta smiled graciously. 'Then another concert on Monday evening. I fly back yesterday, Tuesday evening. Is why we in chaos.'
'So, any idea where Dr Dave was on Sunday night?' Nancy persisted.
'I know for sure 'e wasn't 'ere,' replied Fiammetta decisively. ''E would have stayed in college. 'Ates it 'ere when I am gone.'
Why had Dr Dave lied about where he spent the night on Sunday? If he hadn't spent it at home, as he'd said, the only logical place to stay would have been in his rooms. But Ursula had seen him arrive the next morning well past 9 a.m., and after she had discovered India's body. Where on earth, she wondered, _had_ he spent the night, if he wasn't either at home or in his rooms? And why was he being so deceitful? Was he aware of rather more about the murder than he was letting on? Had he, in fact, been in his rooms much of the night? Perhaps, frustrated by India's refusal to stop harassing him, he had killed her himself, and 'arrived' late for Ursula's tutorial, to make it appear as though he had not been in the room overnight.
'You're _sure_ he wasn't here?' Nancy asked Fiammetta.
'I suppose 'e _could_ 'ave come 'ome on Sunday night. After all, I wasn't 'ere, but – _no_. It would be very unlike 'im.' The Italian girl shook her head. ''E likes company.'
'Thank you, Fiammetta,' said Ursula. 'You've been really helpful.'
'You're welcome. She wasn't 'istory student, this India, was she?' asked Fiammetta while showing the girls out.
'No,' said Nancy, shaking her head. 'English Literature.'
Fiammetta opened the front door, gasping with surprise as she did so. There on the front step stood two policemen.
'May we come in?' one of them asked.
'Well, I suppose,' she said resignedly, showing them in.
'Goodbye,' said Fiammetta as Nancy and Ursula stepped onto the path. 'Good luck with your article.' Then she added, as though to herself, 'I suppose that's why Davide didn't mention it to me. 'E probably didn't know 'er.'
'Probably not,' agreed Ursula.
*
'We're gonna have to go back to Dr Dave,' Nancy said as soon as she and Ursula were out of earshot.
'I know,' Ursula agreed, clambering onto her bike. 'But how do you accuse your own tutor of being a liar without being Sent Down?'
As soon as they were back in college, the girls zipped into the Porters' Lodge. Ursula found a hand-written note in her pigeonhole, which read:
Darling Ursula,
Your _Cherwell_ editors look forward to reading your copy on Sunday – have you discovered whodunnit yet?
No, I jolly well haven't discovered whodunnit, she said to herself, and scanned the rest of the note.
If you and Nancy-Doodle-Dandy would like a lift to the funeral on Thursday, I can pick you up outside college in my car at 7 a.m. Perhaps catch up at the Gridiron party tonight?
Mwah! Mwah!
Horatio
Ursula showed the note to her friend. 'Should we go to the funeral?'
'Sure we should. How else are we going to figure out "whodunnit"?' replied Nancy.
'But we haven't been invited,' protested Ursula.
'Horatio's invited us,' Nancy reasoned. 'And didn't you say Wenty said you must go, when he spoke to you on the phone this morning?'
'True.'
'And India invited me to her shooting party. There's no way she would have invited me to that and not wanted me at her funeral.'
'Okay,' agreed Ursula. 'We'll tell Horatio tonight.'
Just then Deddington beckoned the girls over to his desk.
'Good morning,' he said in an unusually quiet voice.
'Hey, Deddy,' chirped Nancy.
'Hi,' said Ursula.
'I've been hearing rumours about you, Flowerbutton,' he said. 'Rumours that you are writing about Lady India's death for _Cherwell_.'
'Yes. Can you help?'
'I'm not sure. But if you were to spend a few moments in the gallery of the Long Room you might find it very informative. I just saw that police bloke go in there.'
'You think there's a clue?' asked Nancy excitedly.
'Hush!' Deddington tried to calm her down. 'Go through the small attic door at the top of the staircase, two floors above the library. It's unlocked.'
'Thank you,' said Ursula. 'Oh, and one other thing, Deddington. Where were you on Sunday night?'
'Eagle and Child, the pub on St Giles.' Can't take that awful _Dallas_ Linda insists on watching. Much prefer a pint or two of ale.'
*
'Good afternoon, gentlemen,' High Provost Scrope was saying. 'Thank you very much for coming to this _postponed_ – and now emergency – meeting of the Christminster College Fundraising Committee. I know how busy you all are, so I'd like to get to the main business right away.'
Having clambered up the cramped, winding staircase to the attic door, Ursula and Nancy now found themselves crouching on a galleried landing. The Long Room beneath them was almost as large as the library one floor below. The walls were hung with ornate tapestries, and the windows looking onto Great Quad allowed a shaft of sunlight to illuminate the highly polished mahogany table in the centre of the room, groaning under the weight of the college silver arranged on top of it.
Ursula counted at least twenty men sitting around the table, most of them wearing suits. The absolute youngest was in late middle age, and they were in general so prosperously corpulent they looked as though they might burst. D.I. Trott, she noticed, was indiscreetly installed in an enormous tapestried throne at the farthest end of the room. White-gloved college servants were sloshing claret into the men's glasses, and while Scrope addressed them, the Fundraising Committee was becoming rather merrier than he seemed to realise.
'. . . the main business to be discussed today is the Brattenbury endowment. As you may know, Lord Brattenbury, who was a student here in the 1950s and went on to make a fortune in Africa from gold and diamond mining, has most generously already given almost a million pounds to the college, but now, with the death of his daughter – the _tragic, untimely_ death of his daughter India, who was an undergraduate here, and was the sole heir to the Brattenbury fortunes – there is no longer an immediate heir to the Brattenbury estate. Which means there may be the possibility of a much larger donation forthcoming to this college . . .'
Ursula could see that Trott's eyebrows were raised so high they were almost in lift-off mode, and Nancy whispered, 'How about if Scrope killed India for her inheritance?'
'Wouldn't it be too obvious?' replied Ursula.
'. . . funds which could secure the future of Christminster for the next hundred years or so. The key is to find a way to lobby Lord Brattenbury to rewrite his will in favour of Christminster. If the matter is _delicately_ handled with his lordship, I imagine we could be extremely successful. Although he is in his late forties, I have heard that the death of his only child has put his somewhat precarious health at risk. Time is of the essence.'
'Okay, perhaps we _should_ add Scrope to our list of suspects,' said Ursula, amazed by his blatant desire for India's fortune.
There was a hubbub while the committee members chatted amongst themselves. Then Scrope called out to the group, 'You are all distinguished alumni of the college. But is there anyone here who was an undergraduate with Brattenbury?'
A middle-aged gentleman dressed in a tweed suit raised his hand. In the upper-crust tones of the county squire Ursula assumed he was, he said, 'I seem to recall that I played a bit of rugger with 'im. Knocked 'im out cold once.'
'Marvellous!' squawked Scrope. 'The college would be most grateful if you could approach him about rewriting his will.'
'Certainly not,' replied the other man. 'I knew Brattenbury well enough to tackle 'im, not to discuss his personal affairs. Scrope, you'll have to get 'im to change his will yourself. Then knock 'im out and you'll get the money quicker!'
A chuckle ricocheted around the table but stopped dead with Scrope, who was not amused. His lips were puckered and white with frustration.
'Right, let's put the matter to a vote. Those in favour of my pursuing the Brattenbury fortune on behalf of the college, please raise your hand.'
Everyone at the table, except for the gent who had knocked out Lord Brattenbury in his youth, put up their hand.
'Right, that's settled. The next item on the agenda: funding for a Fellowship in Slavonic Linguistics . . .'
*
'Are you thinking what I'm thinking?' asked Nancy as the girls crept back down towards the Porters' Lodge.
'I don't know what you're thinking, Nancy. So I don't know if I'm thinking what you're thinking.'
'I'm thinking that Deddington sent us up there this morning because he suspects our dear High Provost cut India's throat himself, to steal her inheritance for Christminster. I mean, Scrope would have been in college that night.'
'Or,' said Ursula, 'Deddington sent us up there because he wants _us_ to think that _he_ thinks that Scrope had a motive to kill India.'
'You're not saying that Deddington killed India and is trying to frame Scrope? What possible motive could our nice porter have?'
'I don't know. But I feel as if there are a squillion sides to this,' Ursula concluded. 'There's a surfeit of suspects, each with a motive to kill.'
'Let's go back to Dr Dave right now,' said Nancy. 'We need to find out what he was _really_ up to on Sunday night.'
When the girls reached Dr Dave's rooms that Wednesday afternoon, they found the tutor hard at work at his desk. With the Giant Argus finally reinstated, the don, newly inspired, was typing ferociously with one finger at a brand-spanking-new, dark green Monica typewriter.
'Bourbon biscuit, girls? I was just about to take a break. Now that Panoptes is back, the words are flowing faster than the Euphrates.' He got up and handed them an open packet of Bourbon biscuits from his desk.
'Delicious, thank you,' said Ursula, munching away.
'Great,' said Nancy before taking a bite of hers. 'Eeooow!' she erupted. Her face crumpled with displeasure while she forced down the dry biscuit. 'Is it just me or do all British cookies taste like sidewalk?'
'I'm sorry our food isn't up to American standards, Miss Feingold. Dreadful business about the Wychwood towel, isn't it?' said their tutor, plopping into an armchair and taking a sip from a cup of tea. 'God knows why he'd want to dispose of his own girlfriend. He adored her.'
'I'm confused about something,' Nancy declared.
'Who isn't, my dear?' chuckled Dr Dave. 'Didn't the _Dictionary of the Older Scottish Tongue_ clear things up for you?'
'I'm not talking about the essay,' she replied. 'It's about Sunday night.'
'Oh,' said Dr Dave. The don's ebullience seemed to ebb rapidly.
'You see, you said you were at home that night. At your house in Cranham Terrace.'
'Indeed. That is where I was,' he agreed.
'But your fiancée says you weren't there.'
Dr Dave's face whitened, and he spluttered, 'Now look here, you two, you've got no right to go snooping around my fiancée for the sake of some article in a student rag.'
Ursula decided to step in before things got out of hand. 'We didn't tell her anything about India being found in your rooms.'
Dr Dave pulled a silk handkerchief from his top pocket and mopped his suddenly clammy brow.
'And nor should you have,' he retorted. 'There is no need for Fiammetta to know anything about the details of poor India's death. She doesn't know whether or not I was at home on Sunday night because she wasn't there. She was in Salzburg. Usually I never go home when she is away but on this occasion I did.'
He smiled, looking relieved. Ursula realised she was going to have to get to the nub of the matter in a less direct fashion.
'May I ask you something else?'
'If you _must_.'
'Is Ms Brookethorpe prone to telling fibs?'
'Not so far as I know,' he replied. 'What a peculiar question.'
'I think she's told a terrible lie,' said Ursula. She felt guilty as she spoke: she'd promised the librarian anonymity. But she had no choice in the matter if she were to get at the truth.
'It's about you,' added Nancy perkily.
'Well, that's absurd.' Dr Dave jumped up from his armchair and went and stared angrily out of the window.
'I don't know why,' said Ursula as innocently as she could, 'but she told me that she saw you walking along Great Quad towards Monks' Cottages after midnight on the night of India's murder.'
Dr Dave span around and looked at the girls. 'The woman is a complete fantasist. Admittedly she was attractive, in an unsophisticated sort of a way . . . a couple of years ago . . . but, no, with Olive – I mean, Ms Brookethorpe – you can't believe a word she says. Particularly about men.'
'I thought you said she wasn't the lying type,' Ursula reminded him.
'Did I?' asked Dr Dave. 'Then I was mistaken. The truth is, Brookethorpe has been obsessed with me since I was a Junior Fellow. We had a brief dalliance, a couple of years ago. Meaningless flingette, that sort of thing. After I dropped her, well, she's spread vile rumours about me ever since. She wants me out . . . anyway, even if she _had_ seen someone out on Great Quad at half-past midnight, how could she possibly have known it was me, in the dark?'
'It was a full moon,' Ursula pointed out. 'She mentioned the "swoop" of your hair as being particularly noticeable.'
'She did always have a thing about my hair . . . But this story is utter rubbish. She's a bitter, envious, lying old trout.'
'But someone else says they saw you there – a female Fresher,' said Ursula. She didn't want to reveal Claire Potter's identity to the don if she could avoid it. 'She said she saw you out there on Great Quad at _exactly_ the same time. She was holding an event in the JCR, across the landing. No one showed up and she was staring out of the window looking for guests.'
'Bloody hell,' Dr Dave growled, looking astonished. He glared at both girls then collapsed back in his armchair. 'Oh, all right, you've got me,' he conceded. 'I'll tell you where I was that night. I was in Monks' Cottages.'
'Why?' asked Nancy.
'I was doing my laundry,' he replied innocently. 'There, now that you know I wasn't at home, or here in my rooms, can we please end this uncomfortable interlude?'
'It took the _whole_ night to do your laundry?' asked Ursula.
Dr Dave groaned with irritation.
'Does it really matter? The important thing is that you can establish with utter certainty that I wasn't at home the night India died and I wasn't in my rooms when she came up here. Isn't that enough?'
The girls both looked at him, eyebrows raised. Finally, an expression of defeat came over his face.
'Look, none of this can get back to Fiammetta,' he insisted.
The girls assured him that it wouldn't.
'Or into your _Cherwell_ article, Flowerbutton.'
'I promise,' said Ursula.
'If it does you will _both_ be Sent Down for . . . I don't know . . . dabbling in crime writing.
'What transpired that night,' Dr Dave continued, 'was rather . . . sordid. I was planning to spend the night here on Sunday because Fiammetta was going to be away for a few days. I didn't want to be in the house without her cooking – and her, of course. I was doing some work, pretty late, here at my desk, when I heard this bloody great row from outside. I looked out of the window onto Great Quad and there were India and Wenty screaming at each other. The night porter eventually broke it up, and then I saw India run towards my staircase. I didn't want another scene, so I slipped out before she got to my landing.'
'But surely she saw you on the stairs?' said Ursula.
'Ah. Yes. I should explain.'
Dr Dave beckoned the girls to follow him, opened the door to his bedroom and strode in ahead of them.
'Come in,' he said.
'Hey,' Nancy stated firmly. 'We are not interested in some kind of weird threesome with you, Dr Dave.'
He smiled at her coolly. 'And, believe me, nor I with you. I am simply showing you the back stairs to my bedroom down which I fled on Sunday night.'
With that, the tutor pulled back the curtain on the far wall, revealing a small wooden door. He lifted the latch, ducked his head and went out through the low doorway. Nancy and Ursula followed and found themselves descending a cobweb-ridden spiral staircase, which led out to the south-west corner of the Kitchen Quad.
'So, you see,' said Dr Dave, after leading the girls back up the stairs to his rooms, 'while India was climbing the JCR staircase to my rooms, I skedaddled down here. I do wonder, if I hadn't been so keen on skedaddling, would she still be alive now?'
'Where did you skedaddle to for the night?' asked Ursula, sitting back down on Dr Dave's tartan-covered chesterfield.
'As I said, I spent the night in Monks' Cottages.'
'Where did you sleep? In a laundry basket?' queried Nancy in highly sceptical tones. Ursula could tell she wasn't going to let Dr Dave get away with this.
The don looked decidedly sheepish. 'Well, I was rather at a loose end. I couldn't go back to my rooms, knowing India was most likely in there, and the keys to Cranham Terrace were in there, too. So I . . . er . . . looked in . . . on . . . um . . . Isobel Floyd. Her room's above the laundry.'
'Did you "look in" for the whole night?' said Ursula, trying to phrase it as politely as possible.
'The looking in, yes, it took . . . erm . . . all night.' Dr Dave grimaced. 'And that's why, Ursula, I was a few minutes late for our tutorial.'
Nancy, who appeared to be far less concerned with maintaining British standards of politeness, just looked at the don and shrieked, 'Gross!'
Wednesday, 1st week: evening
'What do you think?'
Nancy appeared in Ursula's room that evening dressed in a completely see-through pair of Fiorucci 'jeans' fabricated from what looked like clear plastic. Beneath them, her athletic legs and a pair of silver, sequined hot-pants were clearly visible, and she had finished the look with a one-shouldered purple Lycra top and high heels.
'Really trendy,' said Ursula, looking up from her desk, where she'd been making notes about their meeting with Dr Dave. His behaviour perplexed her: she couldn't understand how someone could cheat so blatantly on their fiancée, let alone one as divine as Fiammetta.
'Hey, why aren't you ready?' asked Nancy, regarding Ursula's mini-kilt and sweater.
'I can't go to the Gridiron party tonight. I haven't done _any_ work on my essay. I think I need to spend the evening in the library instead.'
'What?' howled Nancy. 'You can't let me down at the last minute like this. I can't go by myself. I might get murdered on the way. Imagine how bad you'd feel if that happened.'
Was there really any getting out of tonight, knowing Nancy's persistence when it came to social events? With a long sigh, Ursula signalled her willingness to give in. 'I just need to finish this first.'
She added a final thought:
_—No wonder Isobel Floyd said that Dr Dave didn't do it. She was being 'dropped in on' by him when India was murdered._
Ursula left her article notes on the desk and hurriedly changed into Vain Granny's ball-gown – again. She couldn't help but feel slightly envious of her friend's endless supply of outrageous party dresses and New York clubbing outfits. Much as Ursula loved the lilac dress, it wasn't exactly of-the-moment. But it would have to do. There was no time to play dressing up in Nancy's room tonight.
*
The Gridiron Club dinner was a small, cliquey and mostly male affair. Ursula and Nancy arrived at the Golden Cross, a medieval, cobbled courtyard off Cornmarket Street, just before eight o'clock. They were directed past a Pizza Express restaurant on the ground floor and up a steep set of back stairs towards an attic dining room. As the girls reached the landing outside it, Ursula recognised the long sweep of a dark overcoat and saw Neil Thistleton trying to talk his way into the dinner.
'Look, mate,' one of the black-tied Gridiron members was saying to him in supremely polite tones while very effectively blocking the doorway, 'I'd love to help, really, but it's members only tonight. _Terrifically_ sorry and all that.'
Neil Thistleton turned and stared indignantly at the girls. Ursula looked at the floor, praying he wouldn't spot her.
'Members?' he said, turning back to the boy. 'Isn't this one of those snotty all- _male_ dining clubs?'
'We allow women as guests of members. No one else is allowed in, I'm afraid.'
'Sexist tosser,' said Neil Thistleton. He slunk reluctantly back down the stairs.
Beamed and whitewashed, the attic room that served as the headquarters of the secret society resembled a medieval refectory. A long table had been laid for dinner with starched white linen and pewter platters. There were about twenty members, mostly in black tie, and no more than seven or eight ravishingly pretty girls. Good, thought Ursula, spotting Isobel Floyd surrounded by an ever-adoring group of boys, I'll wait until she's tipsy before I mention Dr Dave. She spotted Eg standing by the bar, chatting with friends, looking as suave as ever in his tux, and wondered what they would say to each other after the peculiar end to their date last night.
'Ah, welcome!' Horatio Bentley greeted Ursula and Nancy, waddling towards them bearing two pewter goblets. 'Nancy, you look just like a disco ball.'
'Thank you,' she said, flattered by Horatio's attention and accepting a goblet.
'Huge apols in advance, sweetie. The Gridiron's wine doesn't quite live up to its pewter-ware,' he chuckled. 'It came out of a wine-box. If you drink enough, eventually you'll stop noticing the burning in your gut.'
Horatio led the girls to a trio of buttoned-leather chairs where he settled down, saying, 'I can't believe Wenty's been arrested. Dreadful news. You _are_ both going to come to the funeral tomorrow, aren't you?'
'Absolutely,' said Ursula.
'Good. You're bound to get some more material there,' said Horatio. Then, noticing a girl heading towards them, he said, ' _Ciao_ , Tiggy-Wiggy.'
Ursula recognised the Princess Diana wannabe that she'd seen in the gate tower on her first day at Christminster. Tiggy was now dressed in a navy silk dress with a high, pleated, stand-up collar over which she wore a gold choker. Her fluffy, streaky blonde hair grazed her enormous blue eyes but didn't conceal the determined look on her face.
'Sorry to interrupt, yah. Look, I'm up for Librarian-Elect at the Union. Are you two Freshers members yet?' she asked the girls.
'I'm thinking of joining,' said Nancy.
'You should,' replied Tiggy. 'It's a wonderful institution. And soon as you join, do register to vote in elections. Yah? _Great_.'
'By the way, Tiggy,' said Horatio, 'I voted in a Union election once.'
'Did you?' She looked suddenly excited by the prospect of another potential supporter.
'Yup. Can't remember who for, though.'
Deflated, Tiggy rushed off to canvass another group.
As soon as she was out of earshot Horatio said, 'Tiggy – ugh! She's one of those gauche, pushy, pony-club types. I hate to admit it, but that utterly talentless homuncule will most likely be President of the Onion one day.'
'Onion?' repeated Nancy.
'That's what the _Cherwell_ hacks call the Union because the group of twats who run it are so bitter – always making each other cry. Anyway, where were we before she interrupted us?'
'Wenty,' Ursula reminded him. 'He telephoned me this morning—'
'Hang on,' Horatio interrupted her. 'Wenty phoned _you_ , Ursula, from jail?'
'Yes, why not?'
'Nothing,' Horatio said with a coy grin. 'Nothing _at all_.'
'Look,' she went on, ignoring his innuendo, 'Wenty swears he didn't kill India. He's got an alibi for Sunday night that I need to check. He says he was at St Hilda's with a girl named Geraldine. Do you know her, by any chance, Horatio?'
'Geraldine who?'
'Something-or-Other.'
'Something-or-Other?' Horatio frowned.
'Wenty couldn't remember her second name. It sounded like he had a drunken one-night stand.'
'Oh, Wenty . . . So many girls to remember, so many names to forget!' laughed Horatio. 'There is _one_ Geraldine at St Hilda's. Geraldine Ormsby-Leigh. But it wouldn't be her. She's madly in love with her boyfriend, Hugo Pym. He's at Trinity.'
'Maybe she cheated on him,' suggested Nancy. 'I mean, it's not like everyone else isn't cheating on everyone else round here.'
'Let's ask her,' said Horatio suddenly. He waved across the room. 'Geraldine! Pumpkin! Over here!'
'No, wait—' Ursula tried to stop him, but it was too late.
The particular Geraldine in question swept over to Horatio and kissed him twice on each cheek. She certainly looked like she might be Wenty's type, thought Ursula to herself. Tall and slim with blonde locks almost to her waist, the girl was dressed in gold pedal-pushers, a sparkly green boob tube and lilac suede stilettos. Her bare shoulders gleamed with a dusting of golden glitter that also covered her cheekbones and eyelids, and her lips were slicked with an iridescent gloss.
Ursula examined this disco angel with some awe. Would her grannies ever approve of a garment as gorgeously minute as Geraldine's boob tube? Ursula was wondering to herself when she suddenly heard Horatio saying, 'Go on, ask her about Sunday night.'
'What about Sunday night?' interjected Geraldine.
'Well, I was just wondering . . .' Ursula trailed off, embarrassed. There wasn't a straightforward way to ask someone you'd just met about her love life. She muddled along, purposefully vague: '. . . if, yes . . . that's it, if you happened to, erm . . . _see_ Wenty on Sunday night?'
'Wenty?' said Geraldine.
'You know, Wentworth Wychwood. Toffy rower,' added Horatio.
'Why?'
'He says he – maybe – saw you,' said Ursula cautiously.
'Well, he can't have. I was on the sleeper train from Edinburgh. Coming back from a stalking weekend in Scotland with Hugo.'
'Oh, right,' said Ursula, as Geraldine drifted back to the bar.
'So, sounds like Wenty's got some explaining to do,' remarked Horatio as soon as she was out of earshot. 'Geraldine definitely wasn't with him on Sunday.'
Ursula felt exasperated. How did Wenty think he could get away with endless lies? And why had he lied about being with Geraldine? Did he think Ursula – or the police for that matter – wasn't going to bother to check his alibi? Could she trust what anyone said in Oxford?
'It looks pretty bad for Wenty,' said Nancy. 'I mean, his towel was at the crime scene covered in his and India's blood. He was jealous of her relationship with Dr Dave. He didn't spend the night in his own room, and he didn't spend it with Geraldine Whatever-Her-Name is either. If he wasn't at St Hilda's, maybe he _did_ follow India up to Dr Dave's rooms. Maybe Ms Brookethorpe saw him leaving college on his bicycle _after_ he'd killed her?'
'Or maybe it wasn't Wenty she saw at all. Maybe it was someone else,' suggested Ursula. 'Maybe Wenty was in Dr Dave's rooms all night and didn't leave until the early hours of the morning.'
'But how does anyone prove that?' asked Horatio.
Ursula was at a loss as to how to answer him. Why was it so impossible to find out what Wenty had really been up to that night?
Soon a rather ruddy-cheeked boy announced: 'Dinner is served!' The crowd found their place cards and sat down around the table while various tuxedoed members of the Gridiron Club distributed a white cardboard box onto each medieval platter. The logo across the top of the boxes read _Pizza Express_.
'This is literally the poshest take-out ever,' laughed Nancy, opening her box. She was sitting next to Horatio, and Ursula was seated on his other side, with Eg opposite her across the table.
He smiled shyly at her, saying, 'You've got that lovely dress on—'
'—again, sadly,' joked Ursula.
'No, you look so . . . pretty. Just like the first time I saw you.'
Ursula suddenly found herself being poked in the ribs. 'Heavens above,' Horatio whispered in her ear, 'the monk has fallen for the convent girl.'
'I am _not_ a convent girl,' Ursula whispered back.
'All right, the virgin,' he said.
Ursula rolled her eyes. She had nothing to say to that. After all, she _was_ a virgin, like it or not. Meanwhile, Horatio slid an enormous slice of Margherita pizza into his mouth and downed it at a gulp. Just then, Ursula noticed Isobel Floyd getting up from the table a few seats down from her.
'Just going to powder my nose, darling,' Isobel announced to the boy sitting next to her.
'I need to do a follow-up interview,' Ursula told her friends, getting up from her seat. 'Back in a moment.'
Isobel was retrieving a weighty golden YSL tube from her handbag when Ursula came upon her in the powder room a few moments later. She twisted it to reveal a dark purple lipstick, which she rolled expertly on her lips. She flashed a smile at Ursula.
'Very Goth, no?' she asked.
'It's lovely,' said Ursula, secretly asking herself why this exquisitely attractive girl wanted to look like Adam Ant.*
'I'm trying out looks for the play. My Hamlet's going to be a New Romantic.'†
'How are rehearsals going?' Ursula ventured, wondering how on earth she was going to change the subject to Dr Dave's night-time college rovings.
' _Really_ great, thanks. But then, Dom's a genius. I think he'll run the National Theatre one day. Why don't you come to a rehearsal? You could preview the play. Write it up for _Cherwell_.'
'I'm still quite busy with my article about India,' said Ursula. 'Actually, funnily enough, I wanted to ask you one more question.'
'Oh?' said Isobel, looking at Ursula out of one eye in the mirror as she dabbed emerald-green shadow onto her left eyelid.
'It was just, on Sunday night . . . you went back to your room after the argument—'
'I already told you that,' Isobel snapped.
'Yes. Did you by any chance forget to tell me that someone came to your room a little later that night?'
There was a clatter as Isobel's eyeshadow palette dropped from her hands and landed in the basin.
'Who told you that?' The colour of her face was rapidly darkening to match her purple lipstick.
'He did,' said Ursula.
Isobel sat down slowly on a tufted armchair and lit a cigarette.
'This can't get out. No one can know. Ever.'
Ursula nodded.
'At the time, it felt like revenge.'
'Revenge?' repeated Ursula.
'I wanted to get India back. Best friend? Ha! She took my part, my boyfriend – it was only fair for me to have a pop at Dr Dave.'
'But their affair was over months before,' Ursula pointed out.
'She was still obsessed with him. Saw him as her property. Anyway, he was in my room all night. We overslept. It seemed funny, at the time, when he dashed off saying he was late for a tutorial.'
'It was my tutorial,' said Ursula.
'And then, when I found out India was dead – god! I regretted what I had done so much. I couldn't tell a soul. I told the police what I told you – that I was alone all night. They didn't suspect I was lying for one minute,' Isobel said. Then she added with a smile, 'You see, I'm a pretty amazing actress when I need to be. Have you got any idea who killed India?'
'No,' admitted Ursula. 'But now I feel pretty sure it wasn't you or Dr Dave.'
'Good.' Isobel put her eyeshadow and lipstick back in her bag and she and Ursula returned to the dinner party.
At the table, Ursula found Nancy in mid-flow: 'I just don't get why they don't have sororities here. If you can have an all-male dining club, why not an all-female one?'
'None of the girls in Oxford would want to go to an all-female club,' a girl declared sharply from across the table. 'It was bad enough being at a girls' boarding school. Oxford's a release from that hell.'
'But what about feminism?' said Nancy. 'We're all equal now.'
'Saying you're a feminist,' said the girl across the table, 'is like saying you're a vegetarian or something – _weird_.'
Nancy rolled her eyes in despair as Ursula slipped back into her seat. She noticed, to her delight, that Eg was looking adoringly at her. As did Horatio. Like quicksilver, he whispered in her ear, 'He seems pretty glad Wenty's locked up. Eg was asking _all_ about you while you were gone.'
Ursula brushed him off, saying, 'Horatio, Eg's far too nice to be glad Wenty's in jail.'
Still, she couldn't help blushing slightly. Perhaps Eg _did_ really like her, despite the end to their evening. But other matters were more pressing. Ursula turned to Nancy. 'Do you mind if we go soon?'
'Why? I'm having a surprisingly good time at this anti-feminist dinner. It's like I'm witnessing some kind of anthropological experiment – from Victorian times.'
'Look, I've realised there's someone we've forgotten to speak to. He could be the key to India's death.'
*
It was almost 11 p.m. by the time the girls got back to college. The gate tower was dark and silent, but a warm light shone from inside the Porters' Lodge.
'Come on,' said Ursula, beckoning Nancy into the little room.
Deddington Jnr sat at the porter's desk, head buried in a pile of books as usual. He looked up when he heard the girls enter.
'Evening, ladies,' he said.
'Hey, sweetie,' said Nancy. She plonked her elbows on the desk, rested her chin on her hands and just stared at him. She seemed unable to resist flirting with him. 'Want me to go grab you a drink from the bar?'
'No – no, thank you,' responded Nick. 'I don't drink when I'm on duty.'
'You must be lonely here at night . . . I could hang with you, if you like?'
'Thank you, I can cope on my own,' he replied.
'Okay, be like that,' said Nancy, exhaling a long sigh.
'Nick, may I ask you something about your job?' said Ursula.
Deddington Jnr put down his pen and smiled at her. 'Sure.'
'Okay. What time are you here until in the mornings?'
'My dad takes over around seven. That is, unless it's his day off.'
'And what time do you start at night?'
'Usually seven p.m., unless it's my day off.'
'Did your father take over from you at seven a.m. this past Monday morning?'
Nick thought for a brief moment, then said, 'That was the day they found the girl dead, wasn't it? Yes. Everything was as usual that morning. I mean, usual except for the awful murder.'
'Did you see anyone coming in earlier than your father?'
'Only the scouts,' he replied. 'They all start between five and six in the morning. Mum always brings me a cuppa when she gets here. She brought me toast that morning. I remember, because she'd spread it with her homemade marmalade. Real treat.'
'Cute,' said Nancy.
'You don't remember seeing anyone else come in very early that morning, do you?' asked Ursula.
'Let me see.' Nick pursed his lips as he thought. 'Yes – yes, there was one other person I noticed.'
'Who?' asked Ursula, brimming with anticipation.
'The milkman. Always comes around six.'
'I see,' she said. 'Did you notice any undergraduates?'
'I don't think so, no . . . but I suppose someone could have slipped in without me noticing, once the gate was unlocked, if I was sorting mail or something. Why?'
'Nothing,' said Ursula regretfully.
Wenty was starting to look more and more guilty. Why didn't he ever seem able to tell the truth?
* Before Johnny Depp popularised pirate style, pop star Adam Ant (born plain Stuart Leslie Goddard) had the look down, only with way more makeup.
† A group of anti-punk London clubbers, the New Romantics dressed in Regency frills, vintage military jackets and highwayman garb. John Galliano's 1984 graduate show, _Les Incroyables_ , a direct reflection of the scene, was heavy on bows, pantaloons and tricorn hats.
Thursday, 21 October, 1st week: morning
'I feel like I'm in the _Thriller_ video,' said Nancy as they headed out the next morning at seven to meet Horatio. Oxford was shrouded in a fog so dense that the girls could barely see their way out of the gate tower.
Suddenly, from somewhere in the icy mist beyond, a voice called out, 'Wooo-oooo-hooooo!' As they edged closer to the sound, they could finally see that Horatio Bentley, the source of it, had parked outside college as promised. He waved at them from the window of a grubby, dented Reliant Robin whose engine was running. Ursula suspected the vehicle might, possibly, be green under the layer of grime covering it.
'Bentley by name, _not_ by nature!' Horatio declared. Despite the prospect of India's funeral ahead, his tone was as jovial as ever.
'Why does this thing only have three wheels?' asked Nancy, regarding the triangular-shaped car with a petrified expression on her face.
'I haven't got a driver's licence,' he explained. 'But this is classed as a motorbike, so I'm allowed to drive it.'
Nancy whitened.
'Stop dithering and get in,' he ordered.
Horatio, dressed in a black suit with a lilac bow tie and matching handkerchief, heaved himself out of the car and let Ursula clamber into the back seat, where she made a space for herself among the piles of old newspapers, crisp packets, and grimy piles of clothes. As she didn't own a black dress, she was wearing her homemade maroon velvet one under a duffel coat and her college scarf.
Nancy's funeral look was so elaborate that it took her some time to edge herself into the front passenger seat. She had chosen a tiny, skin-tight black 'bandage' dress, sheer black tights, high heels and a thigh-high black velvet swing coat with enormous puffed sleeves, a frilled collar and gold buttons down the front. Her blonde hair had been fluffed into the puffiest bun Ursula had ever seen.
'Darling,' said Horatio, 'you look just like Ivana Trump.'
'Oh my god, that is so cute of you to say,' said Nancy. 'She's my style icon.'
The fog in the car was almost as thick as outside. A Marlboro Red was smouldering in the ashtray – Horatio took a long drag and then said, 'Let's get going before the engine stalls. Might never start again. I call it my Unreliant Robin.'
He put his foot on the accelerator and jerked the car awkwardly into the road. Nancy clutched the dashboard and screamed with every lurch. The tiny vehicle felt, thought Ursula, no more substantial than a lawnmower on wheels.
'How far away is Brattenbury Tower?' Nancy gulped as they turned onto Broad Street.
'We'll be there in masses of time,' Horatio told her. 'Meanwhile there's hardboiled eggs and a flask of tea to keep us going. Made it all myself last night. Nancy, open that lunchbox at your feet and pass me an egg, please. I'm starving.'
The stench that filled the car when Nancy removed the lid from the plastic lunchbox took Ursula straight back to the revolting school meals she had suffered for so many years at St Swerford's.
'Here,' said Nancy, holding the egg as far from her nose as she possibly could.
'Help yourself,' said Horatio, swallowing his egg in two gulps.
'I'm not quite ready for one yet,' replied Nancy, trying to be polite.
Four hours and forty cigarettes later, Horatio's vehicle put-putted off the motorway and past a sign reading DERBYSHIRE DALES. The journey had taken well over an hour longer than it should have after Horatio had insisted on stopping at a greasy Little Chef in a motorway service station for a leisurely elevenses* of fish and chips.
'Brattenbury's only a few miles away now,' he said, taking a turn onto a stony track-like road, which wound down into a steep valley. The hills were streaked with the autumnal hues of dying bracken and lilac heather. Dry-stone walls divided the pastureland into rough fields on which wild-looking sheep were grazing.
'There's nothing, like, _here_ ,' declared Nancy bleakly, gazing out of the window at the stark landscape.
'In certain English circles – Sloaney ones – it's considered the height of glamour to live in the middle of absolutely bloody nowhere,' Horatio informed her.
A few minutes later, as they eventually motored through the tiny village of Brattenbury, Nancy's attitude abruptly changed. The village, which consisted of two or three stone-built farmhouses, a schoolhouse, a row of simple labourers' cottages, a village hall, a minuscule shop and a pub named the Brattenbury Arms, was simply charming. As they drove past a picturesque Norman church looming from the mist, Nancy cried excitedly, 'Oh my god, it's _so_ cute here. I wanna move in!'
The lane out of the village climbed steeply, becoming narrower and narrower as it wound around two or three sharp bends. About half a mile further on they finally arrived at a huge pair of stone gateposts topped with vast, intricately carved pineapples. An ornate gate lodge guarded the driveway, and miles of immaculately restored stone walling disappeared away from it in both directions. The property was clearly part of a huge estate, thought Ursula to herself. Horatio pulled up – taking care to leave the car's engine running – and a few seconds later an elderly man appeared from the gate lodge.
Horatio leaned out of the window to speak to him. 'Good morning, sir. We're here for the funeral.'
The old man looked at his watch and said in a strong Derbyshire accent, 'T'all went int'chapel half-hour gone. Go straight up t'drive, chapel's on t' right o' big 'ouse. Chop-chop, you're late!'
'I thought you said we'd have masses of time,' complained Nancy.
'We did,' retorted Horatio, pressing his foot down on the accelerator as hard as it would go. The car swayed precariously from side to side as they whizzed up the gravelled carriage drive.
Brattenbury Tower, which soon loomed into view, lived up to its name. Ursula had never seen such a romantic-looking house. An Elizabethan dream, the house had a stone faÇade so delicate it reminded her of an etching. Ancient stone mullioned windows soared up, four storeys high. At least twenty chimneys peeked above the balustraded roofline. A semi-circular flight of stone steps led up towards the impressive entrance. In front of the house the gravel drive swept around a circular lawn where various very smart cars, including a flashy gold Porsche, a Rolls-Royce and a couple of Range Rovers were parked.
'This is _so much_ cooler than Disneyland,' exclaimed Nancy, wide-eyed, as she drank in the sight. 'Can't you just imagine Rapunzel letting down her hair from up there?' she added, gazing upwards. 'And, look! I can't believe the Brattenburys even have their _own flag_ on their house . . . Hmmm, I wonder if Next Duke has one too.'
'Nancy darling, Duke-types have flags flying absolutely everywhere they can. Gives them a sense of identity,' Horatio explained.
'How . . . _sexy_ ,' sighed Nancy wistfully.
Ursula, meanwhile, noted sadly that the Brattenbury flag, decorated with the family coat of arms, was fluttering at half-mast. Horatio pulled up next to the Porsche and the three undergraduates alighted from their humble vehicle. There was no one in sight, but Ursula thought she heard the sound of an organ, and the faint echo of voices singing.
'I think it must be that way,' she said, pointing towards the east wing of the house.
The trio headed along a sheltered path before eventually coming round the corner of the building and spotting an ancient lych-gate at the far end of a vast lawn. Ursula, Nancy and Horatio dashed towards it, passed under the little archway and soon found themselves heading along a mossy path through a tunnel of yews leading to the Brattenbury family chapel. Horatio heaved open a heavy oak door and Nancy and Ursula slipped into the back of the chapel, with Horatio following, as quietly as they could.
Every pew of the tiny chapel was full, and the area behind the pews was crammed with standing guests. There was only room for them to squeeze in directly in front of the door. A lone choirboy, dressed in a frilled white collar and a red robe, stood in front of the altar singing the poignant verses of Psalm 23:
_. . . Yea, though I walk in death's dark vale,_
_Yet will I fear no ill:_
_For thou art with me, and thy rod_
_And staff me comfort still . . ._
'I wanna take that little skylark home,' whispered Nancy, as the boy's voice soared and echoed around the chapel.
As the last verses were sung, Ursula watched sorrowfully while India's coffin, beautifully adorned with heather, snowberries and wild brambles, was carried down the aisle. She found herself brushing a few tears from her cheek. Although she hadn't really known India, that didn't matter. When someone died so young, it was desperately sad. Her tears were not just for India – Ursula couldn't help thinking of her own mother and father, taken so early.
She took a deep breath and gathered herself. She remained standing with Nancy and Horatio at the back of the church, watching closely as the mourners who had been sitting at the front followed India's coffin from the chapel. The women – clad in enormous black hats and power suits with flashy gold buttons, coats of mink or sable, reams of pearls and generous sprinklings of diamonds – were accompanied by smart-looking husbands and boyfriends dressed in Savile Row suits and silk ties. These must, Ursula assumed, be the grander Brattenbury family relations and close friends. This group was followed at a respectful distance by a few humbler-looking types whom Ursula imagined must be estate workers or tenants. Finally, India's crowd of Oxford friends filed out of the chapel, the Yah girls striking in their black dresses and hats.
Ursula observed India's inner circle of friends intently as they passed in front of her. There were Dom and Isobel, holding hands, both looking suitably heartbroken. Eg and Otto followed behind them, Otto's Imperial red and white sash standing out among the sea of black. He gestured to Ursula, Nancy and Horatio to join them.
'I can't believe Wenty's not here,' said Eg miserably when he saw Ursula.
Once outside, she noticed Dr Dave and Fiammetta, fingers entwined, drifting along with the crowd of mourners making their way across the small family graveyard to the spot where India's coffin would be buried, beneath a huge old oak. Olive Brookethorpe walked behind them, chatting to Mr and Mrs Deddington, who were flanked by Alice the scout wearing a veiled hat. Ursula soon spotted High Provost Scrope and his secretary, Mrs Gifford-Pennant, talking to the grand-looking relations. Detective Inspector Trott and a constable were standing on the far side of the graveyard. The mourners watched, grimly silent, as India's coffin was lowered into the ground.
'That's peculiar,' said Horatio under his breath, glancing around.
'What is?' asked Ursula.
'Lord Brattenbury's not here.'
'You're right, I can't see him anywhere,' said Nancy, unable to spot India's father. 'Why would a dad not come to their own kid's funeral?' She paused, then added, 'Unless . . .'
'What?' asked Ursula.
'Unless he killed his own daughter.'
Ursula stared at her in disbelief.
'Be sensible, Nancy,' scolded Horatio. 'Why would Lord Brattenbury want to dispose of his only heir? Now's not the moment to kill off his daughter and reveal a love child in Australia.'
'I guess,' said Nancy.
Just then Ursula saw the elderly gentleman from the lodge house closing up the chapel doors. 'Excuse me,' she said politely, 'but where is Lord Brattenbury?'
'Got t'malaria back.'
'You can get malaria _here_?' Nancy shrank away from the old man.
'Don' worry y'self. Lord B got t'sickness in Africa. The fever comes back now and again. 'E's a'bed.'
'Horatio, we'll see you later,' said Ursula, turning on her heel. 'Come on, Nancy. We haven't got long.'
*
The Great Hall of Brattenbury Tower resembled an ancestral hunting lodge. Every spare inch of wall was decorated with old trophies – foxes' tails, stuffed game birds and huge sets of antlers hung among decorative swords, ancient shields and elaborate daggers.
'Forget _Rapunzel_ , this is totally _Jungle Book_ ,' Nancy quipped.
But Ursula's mind was already elsewhere, her attention focused on the dramatic staircase sweeping upstairs ahead of them. She and Nancy could easily sneak up there now and find Lord Brattenbury's bedroom, before the funeral party reached the house. Ursula glanced around the Great Hall – there was no one else about. 'Quick!' she said, beckoning Nancy to follow her upstairs.
The girls had only leaped up a couple of steps when a voice rang out behind them. They turned to find a severe-looking housekeeper dressed in black standing in the hall, trailed by several uniformed maids and butlers carrying silver trays laden with food. The housekeeper looked startled.
'Goodness, we're not ready for the wake yet,' she said, regarding her watch with a worried frown. 'I thought we had another half hour.'
'Actually, everyone's still outside the church,' said Ursula, thinking on her feet. 'We were just . . . erm . . . looking for the loo.'
The housekeeper sighed with relief. 'Go up to the first landing, turn left onto the main corridor upstairs, past the late Lady Brattenbury's old bedroom on the right, and you'll find a bathroom up there. And please keep your voices down as poor Lord Brattenbury is trying to rest. Now, if you'll excuse us, we'd better get the buffet laid out.'
'Oh my god, this is a Colefax and Fowler dream,' said Nancy, gazing admiringly at the décor as they headed along the first-floor corridor. 'Mom would _die_ to decorate like this.'
The corridor was painted a sunny yellow, and a frieze of bows and leaves had been hand-stencilled in crisp white beneath the cornicing. At each window, swagged, chintz curtains bulged onto the cream silk carpet, which had a yellow Greek key pattern embroidered on it. The look was, Ursula noted, extravagant, luxurious and of-the-moment.
'We're going to have to try every one,' said Ursula, looking along the vast corridor with doors all the way down it.
'Okay,' said Nancy, poking her head into an empty room on her right.
Suddenly Ursula noticed a door opening at the far end of the corridor. The girls froze for an instant then silently stepped behind a huge swagged curtain. From a tiny gap, they watched as a woman dressed in a green-and-white-striped nurse's uniform and an apron and cap walked past them towards the grand staircase.
'She must be Lord Brattenbury's nurse. He's got to be in that room,' whispered Ursula.
They heard the nurse's footsteps fade away. With the coast now clear, they tiptoed along until they reached the far end of the corridor.
The door to Lord Brattenbury's room was slightly ajar. With her heart thudding, Ursula pushed the door open a little more and took in the scene. The room was plain and masculine in style, the half-drawn curtains making for a gloomy atmosphere. The navy linen that lined the walls was barely visible, covered as it was with numerous sporting prints. A faded Aubusson rug covered the floor. A writing table had been taken over by a heap of tablet boxes, medicine bottles, thermometers, syringes and a pile of garish knitting. A mahogany-ended bed, flanked by two large side tables, was positioned in the middle of the far wall. The girls could see the outline of the sleeping Lord Brattenbury in it, hidden under mounds of blankets.
They walked as close as they dared to the bed, and Ursula noticed a glamorous black-and-white photograph on one side table. It showed a man dressed in white standing on the deck of a yacht. He looked gorgeously raffish and tanned, his thick hair ruffled by the wind, the sun catching his sharp cheekbones; he was holding a young child in his arms.
'That must be Lord B when he was young, with India. How sad,' whispered Nancy, pointing at it. 'But, boy, was he handsome. How come all these English guys look like JFK?'
'We just do, my dear,' said a thin voice suddenly.
There was a rustling among the bedcovers. Lord Brattenbury emerged from the layers of blankets and wearily propped himself up against the pillows. Despite the film of perspiration over his face, he still looked remarkably attractive, an older version of the astonishingly handsome man from the photograph. That thick head of hair, only slightly greying, was unmistakable, and his sharp cheekbones were instantly recognisable, if a little too prominent in his drawn face. He picked up a small hand towel from the other side table and patted down his forehead and cheeks.
Nancy and Ursula peered at him curiously. He managed a smile.
'You couldn't open a window, could you?' His voice was a rasp. Lord Brattenbury sounded very, very ill. 'That dreadful Nurse Ramsbottom is trying to kill me. All these blankets and heating too! Feels hotter than hell in here.'
'Of course,' said Ursula, pushing back a curtain and opening a window.
'Thank you. Air,' he said weakly. Then he asked, 'India's friends?'
The girls nodded.
'How lovely,' he said croakily, and asked for their names, which they gave.
'I remember,' said Lord Brattenbury, ever the gentleman, even as he sweated and sweated. 'Miss Feingold . . . I recall India had invited you to her shooting party. She did love Americans. Quite right. Did you go to the service?'
'Yes, it was really moving,' said Nancy.
'Poor, poor darling India. My heart's broken, you know. To lose a child before you die is the greatest punishment life can inflict on a mother or father . . .'
The girls watched as a tear left the corner of Lord Brattenbury's left eye and settled on his cheek. Suddenly, he started shivering violently. 'Sorry,' he stuttered. 'Can that window be shut now? Got the shaking chills again. My dear girl's death has brought the malaria back. I don't think I'm going to recover this time.'
'Don't say that!' exclaimed Ursula, closing the window.
Lord Brattenbury drew his blankets right up to his chin and lay back on his pillows. His teeth were chattering now, and it seemed he was weakening a little more with every word he spoke.
'The illegit . . .' he mumbled shakily. 'Can you find him?'
'The what?' asked Ursula.
'My son. The illegitimate boy . . . Mary. She was called Mary. Mary Crimshaw.'
'Who's Mary?' said Nancy.
'The girl. Her mother still lives in the village. Runs the shop . . .'
There was a light tap on the door. To their surprise, Nancy and Ursula suddenly found themselves face to face with High Provost Scrope and his secretary.
'Feingold? Flowerbutton? What on earth . . .?' demanded the Provost.
'We were just . . . er . . . chatting,' said Ursula with an innocent smile.
'Look, jolly sorry to interrupt the chat and all that, but I'm afraid I must speak to Lord Brattenbury urgently. Alone,' said Scrope.
'I think he's gone back to sleep,' said Nancy quietly. 'He seemed really sick.'
Lord Brattenbury was, indeed, lying on his pillows, eyes now closed.
'This is most inconvenient. I _suppose_ we'll have to wait until later. Mrs Gifford-Pennant, remind me to return here in an hour.' With that, Scrope and his secretary reluctantly left the room.
The door closed and Nancy and Ursula were left with the sleeping man.
'I can't believe what Lord Brattenbury just said,' whispered Ursula.
'There's another heir?' Nancy asked.
'I think so.'
Lord Brattenbury opened one eye and squinted at them both.
'Now get a move on,' he said, 'and find that boy before I croak!'
* Elevenses – old-fashioned British meal, usually consisting of tea or coffee, biscuits and cake, eaten at eleven in the morning.
By the time Ursula and Nancy got back down to the Great Hall, it had already filled with mourners. As the guests sipped more and more glasses of Lord Brattenbury's finest vintage champagne, served by an army of house staff, the funereal atmosphere was gradually replaced by that of a fabulous cocktail party. The girls soon located Horatio, who was, naturally, hogging the buffet, enthusiastically consuming large quantities of food and drink.
'Horatio, you gotta drive us down to the village,' said Nancy breathlessly.
'But I'm having a roaring time,' he protested, gulping down another forkful of Coronation chicken. 'This is a funeral of bacchanalian indulgence. I can't leave until I've tried absolutely everything.'
He whipped a devil-on-horseback from the sideboard, polishing it off in one bite. This was swiftly followed by a slice of fruit cake, a scone loaded with strawberry jam and clotted cream, and another glass of champagne. 'Mmmmmm!' Horatio groaned with pleasure. ' _Surely_ we don't need to leave yet.'
'Actually, we do. There's a new suspect in the case,' said Ursula.
Horatio's interest was piqued. 'Oh, who?'
'I don't know, exactly. But I'm _sure_ there is. Come on.'
'All right,' he agreed, sneaking some homemade shortbread into his pocket and following the girls outside.
By the time the three of them had made it back to the car, Ursula was starting to have serious doubts about Horatio's ability to drive. He had swayed his way out of the house, tripped over nothing at all on the gravelled drive and been unable to operate the handle of the driver's door to get inside the car. When he eventually got into it, he wrestled his way over the top of the front seat, collapsed onto the mess in the back and promptly passed out. His thumb soon found its way to his lips and he proceeded to make a noise like a blocked drain as he sucked it vigorously.
' _Now_ what are we going to do?' said Ursula, looking at the lumpen sleeping form.
Nancy stretched an arm into the back of the car, rifled through Horatio's pockets and found the car keys, which she dangled in front of Ursula.
'No self-respecting American high-schooler hits seventeen without getting her licence.'
A hair-raising twenty minutes later, during which Ursula tried to explain to her American friend the logic of driving on the left-hand side, Nancy parked the car opposite the village shop. The girls left Horatio snoring in the back seat.
A sign above the doorway of the shop read 'BRATTENBURY STORES'. The tiny window contained various parish notices and a few ancient-looking packets of crisps. As Ursula opened the door, a bell tinkled. Nancy tottered behind on her high heels.
The small shop was sparsely stocked with the kind of provisions that would last at least four or five years in a larder – or during a war. There were tinned peaches, baked beans, cans of corned beef and packets of 'squashed fly' biscuits. There was a confectionery shelf meagrely supplied with a few Walnut Whips, Caramac bars, Flumps, Black Jacks and foam bananas. Food-wise, the residents of Brattenbury were not exactly indulged.
Ursula spotted a neat pile of newspapers on the counter, on the top of which was the _Daily Mail._ She gulped when she saw the headline: WHO KILLED THE IT GIRL? Splashed under the words was a grainy black-and-white photograph of India dressed in a mini puffball dress at an Oxford ball.
'Nancy, look,' said Ursula. The girls read the article, which was of course by Neil Thistleton.
Brainy society beauty Lady India Brattenbury, only daughter of mining tycoon Lord Arthur Brattenbury, will be buried today at the lavish family estate in Derbyshire. Her body was discovered on Monday morning in the room of Oxford don Dr David Erskine, a rising star of the academic world. Lady India was last seen alive late on Sunday night. She had been partying at a wild bash with members of the so-called 'Champagne Set' in the rooms of her boyfriend, aristocratic rowing Blue and fellow Oxford student the Earl of Wychwood. Police sources told the Daily Mail, 'We have a young male suspect in custody,' but would not be drawn on his identity. However the Daily Mail has learned that Wentworth Wychwood was arrested late on Tuesday night.
'You know, I never thought I'd say this, but I actually feel sorry for Wenty,' Ursula said, returning the newspaper to the counter.
'Me too,' said Nancy.
There was no sign of anyone running the shop, but Ursula could see the door behind the counter was just ajar.
'Hello!' she called out, hoping there was someone inside.
'Two ticks!' came a locally accented voice from within.
Moments later an immaculately coiffed, overly made-up woman appeared. She was dressed in a cream-collared, neatly tailored red wool dress that strained over her ample bust. Her brightly dyed auburn hair had tell-tale grey streaks at the temples.
'Wow!' exclaimed Nancy when she saw her. 'You look like you should be running the cosmetics counter at Bloomingdale's.'
'Come again, love?' said the woman, looking confused. 'Right. What can I get you?'
'I'm starving,' said Nancy, picking up a Milky Bar and putting it on the counter. Ursula filled a paper bag with Flumps.
'Thirty pence, please.'
As Ursula handed over the coins, she said, 'Do you know where we could find Mrs Crimshaw?'
'You're looking at her,' replied the woman. 'Why?'
Ursula and Nancy regarded her nervously.
'What is it?' asked Mrs Crimshaw.
'Actually, we were wondering if you could tell us where your daughter Mary is?' said Nancy, unwrapping her Milky Bar. 'We need to speak to her. It's super urgent.'
A melancholy expression crossed the shopkeeper's face.
'She passed away,' said Mrs Crimshaw, in a desolate tone of voice. 'Died in childbirth. Twenty-two years ago now.'
'I'm so sorry,' said Nancy.
'Thank you,' said Mrs Crimshaw. Ursula noticed the woman's eyes welling up as she produced a handkerchief from her pocket and dabbed at her face. 'Dear me. Always get like this about poor Mary. Despite the trouble she caused when she was with us.'
Mrs Crimshaw took out a compact from a drawer beneath the counter and pressed powder as thickly as she could beneath her teary eyes.
'What kind of trouble?' said Ursula.
'So-fetching-she-were-pregnant-at-fourteen kind of trouble, that's what,' said Mrs Crimshaw.
'Fourteen?!' exclaimed Nancy. 'Oh my god. She was a child.'
'Unfortunately, she didn't look like one.' Mrs Crimshaw shook her head. 'She were always sneakin' off with boys. Got with one o' those lads in the village and that were it, she were pregnant.'
Clearly, thought Ursula, Mrs Crimshaw had absolutely no idea who the real father of Mary's baby was.
The woman went on, 'Poor lass. Poor bairn.'
'Barn?' Nancy looked puzzled.
'The baby. We've a saying in our family: "Those as thinks a bairn's a baby, call a sprog a child and a lass a lady". Mary's baby was a little boy. That's all I ever knew about it, really. I would have kept him, looked after him here even with the shame of it in the village, but Ian – that was Mary's father, my late husband – well, he'd arranged for the child to be adopted even before he was born. Said it wouldn't be fair on t'bairn to be brought up with everyone knowing he were illegitimate, but . . .' Mrs Crimshaw couldn't go on. Her face had clouded with regret.
'Do you have any idea who adopted Mary's baby?' asked Ursula, as gently as she could.
'Well, what a strange question,' said Mrs Crimshaw. 'Why come here asking that now, after all this time?'
'I'm trying to solve Lady India's murder. I think the whereabouts of your grandchild could help us work out who killed her.'
Mrs Crimshaw looked doubtfully at Ursula, then Nancy, then back at Ursula again.
'You don't look much like policewomen,' she said.
'I'm a student reporter,' explained Ursula, trying to sound as grown up and professional as she could. 'For _Cherwell_. It's a university newspaper. We really need your help.'
'All right then.'
Mrs Crimshaw walked to the door of the shop and turned the sign around so that CLOSED faced outwards, then returned to the counter and rested her ample bottom against it while she talked.
'As soon as Mary started showing, Ian packed her off to the Catholic Crusade of Rescue. Miles away, it was, Leeds somewhere. It was a home for unmarried mums and babies. If the mothers didn't have anywhere to go – and Ian wouldn't consider having Mary back home – the babies were adopted. When he told me that Mary had passed on, and the adoption was going ahead . . . well, there was nothing I could do.'
'It's a terrible story,' said Nancy.
'There's not a day goes by I don't regret not going and getting that bairn. But round here, it wouldn't be accepted. Not then, not now.'
'Do you think your husband had any idea who adopted the baby?' asked Ursula.
'He always said to me he didn't. But . . .' Mrs Crimshaw looked wistful '. . . sometimes, I wondered.'
'Why?'
'He used to go fishing for a day or two, a couple of times a year, with Brian Wood, the blacksmith, but there was this one time when he said he'd gone fishing and I saw Brian shoeing horses at the stables. I found an address in Ian's diary written next to that date.'
'Did you ever ask your husband why he had that address?'
'No . . . at the time I wondered if, you know, he was up to something, an affair . . . it seemed better to leave it alone. But looking back now, knowing Ian – he wouldn't have been up to anything. I think he went that day and saw where Mary's child lived.'
'Can you remember the address?' asked Nancy.
'Oh, yes. I'll never forget it,' said Mrs Crimshaw, 'because at the time I remember thinking it was so odd he would go all that way down south to Oxford without telling me. We didn't know anyone who lived there. But what has all this got to do with poor Lady India?'
Thursday, 1st week: evening
Number four Penny Farthing Place – the address that Mrs Crimshaw had scribbled on a piece of notepaper for the girls – was a two-up, two-down cottage located midway along a tiny, cobbled alley that ran along the back of St Ebbe's Street. A Victorian street lamp just outside the dwelling lit the narrow pavement.
It was almost eight o'clock by the time Nancy had parked the Reliant Robin opposite the house. While Horatio snored away in the back seat, she and Ursula sat, watching and waiting – for what, they weren't quite sure. But they were absolutely definitely waiting for something.
'You really think a ruthless, throat-slashing killer would have such a cute house?' whispered Nancy, peering at the cottage.
'Where am I?' whined a distressed voice from the back seat.
'Ssshhhhh, Horatio,' ordered Nancy. 'We don't want the murderer to know we're out here.'
'Murderer!' he whimpered, sounding petrified. 'What are you talking about?'
Ursula explained the state of play. The adoptive parents of Lord Brattenbury's illegitimate heir had, possibly, once lived in number four Penny Farthing Place. She was hoping against hope that they still did, perhaps with the adopted baby, who would now be twenty-two years old.
'Oh my god, someone's coming out,' Nancy said. 'Quick, hide!'
The three of them shrank as low as they possibly could in the car while still managing to see out.
'It can't be!' gasped Ursula.
'That is insane!' chimed in Nancy.
'Surely not,' added Horatio.
The trio watched, mouths agape, as the Christminster night porter, Nick, and his mother, Linda Deddington, spoke briefly on the doorstep of the house. He soon left and she retreated back indoors.
Ursula whispered, astonished, 'Mrs Deddington's still in her funeral clothes. They must have just got back.'
Horatio was completely flabbergasted. 'Am I to understand you believe, Ursula, that the Christminster College night porter is the illegitimate son of Lord Brattenbury? That Nicholas Deddington is the missing heir to the Brattenbury estate? Forty farms, two villages, ten thousand acres, and a garden square in Chelsea? Are you _sure_?'
'Well—' she began.
'—it's _definitely_ him,' Nancy interrupted her.
'How can you be so certain?' asked Horatio.
'Oh, that's super-easy,' she said confidently. 'It's all about hotness.'
'What?' guffawed Horatio.
'Look, the minute I met the night porter, I said he was literally as hot as JFK Junior. Remember, Ursula?'
'What on earth,' asked Horatio sceptically, 'has the undisputed hotness of JFK Junior got to do with all this?'
Nancy continued, 'Lord Brattenbury had this photo by his bed probably from when he was in his twenties. He was on a boat, all breezy and tan. He looked _exactly_ like JFK when he was young. I even said it out loud today. Lord B and Deddington Junior are _literally_ identical when you really think about it. They've got the same sexy cheekbones, same thick, Kennedy-ish hair. Gorgeous.'
'It's true,' said Ursula. 'Deddington Junior does look incredibly similar to the handsome, young Lord Brattenbury in that photograph.'
'So . . . are you saying, you think the night porter killed India?' asked Horatio. 'What a delicious twist that would be.'
'He certainly had motive – a huge inheritance, a grand title,' Ursula replied, 'and opportunity. Nick Deddington claims he was in college all night on Sunday, working. It's the perfect alibi.'
'It's true that if he was prowling around the grounds at the dead of night, no one would have suspected anything,' agreed Horatio. 'It's his job, after all.'
'But wait,' interjected Nancy. 'If he'd killed India, what would he have done with his bloody clothes when he had to go back to the lodge?'
'Easy enough to get rid of them,' said Ursula. 'If someone was helping him.'
'You mean . . .' Nancy's eyes drifted over to the Deddingtons' house.
'Maybe,' replied Ursula.
'Sounds like Wenty's off the hook,' said Horatio.
'I think it's a little more complicated than that,' said Ursula. 'He's still got an awful lot of explaining to do. Geraldine Ormsby-Leigh insists he wasn't with her. We still don't know exactly where he was that night.'
'Maybe we never will,' said Horatio. 'It's quite possible even Wenty doesn't know. He's too drunk to notice where he is most nights, after all.'
'Come on, Ursula,' said Nancy. 'Mrs Deddington's at home. Let's seize the moment.'
Before Ursula could do a thing about it, Nancy had sprung out of the car and was knocking on the door of number four.
'Go with her, Ursula,' said Horatio. 'This could be the key to the story. I'll keep a lookout.'
Ursula joined her friend on the Deddingtons' doorstep. Nancy knocked and a few moments later Mrs Deddington opened the door. She was in stockinged feet and looked tired, her black funeral dress only exacerbating the shadows beneath her eyes. When she saw the pair of them, she looked thoroughly taken aback.
'What you doing here, girls, in the dark?' she asked, her voice wavering.
'May we come in?' Nancy asked.
'Well, all right . . . but I've only just arrived back from the funeral.'
Frowning anxiously, Mrs Deddington ushered the girls inside. They squeezed along a narrow corridor and into a small, neat sitting room. There was a bulky TV set in one corner, and the room was furnished with a matching three-piece suite upholstered in flowery fabric. Ursula noticed various framed pictures of the family on the mantelpiece. There were snaps of Mr and Mrs Deddington with their son as a baby. There was Nicholas, smiling, gap-toothed, in school uniform. As a teenager, he was athletic-looking in football gear, his handsome features now apparent. The fact was, he didn't look the slightest bit like either of his parents, thought Ursula. There was no denying it. He _did_ have those Brattenbury cheekbones.
'I'm sorry to disturb you so late tonight, Mrs Deddington,' she said apologetically. 'But I want to talk to you about India.'
'I already told the police, I wasn't in college on Sunday night,' replied Mrs Deddington. Ursula thought she detected a guilty look come over the woman's face as she added, 'I was here all evening. _Dynasty_* was on.'
_Dynasty_? Hadn't Mrs Deddington said that she was watching _Dallas_ , not _Dynasty_ , when they were having Marmite on toast in the Scouts' Mess? And when Deddington had said he'd spent Sunday night at the Eagle and Child, hadn't he said it was because he hated _Dallas_? Ursula would have to check her notebooks when she got home tonight.
'It's not about you, Mrs Deddington,' Ursula told her.
'You might have said so,' she answered, looking suddenly relieved.
'It's about your son.'
'Nicholas? What's he got to do with any of this?'
'How old is Nick?'
'Twenty-two. Why?'
What an awkward thing to have to ask, Ursula thought to herself. She took a deep breath, then said, 'Does Nick know who his . . . it's just, I was sort of wondering . . . does he know who his real mother and father are?'
Mrs Deddington paled. She sank down onto the arm of one of the chairs.
'How do you know my boy is adopted?' she asked sharply.
'Long story . . .' said Nancy.
' _He_ doesn't even know he's adopted,' said Mrs Deddington.
'Really?' asked Nancy.
'We decided right from the start he'd never know. No point in complicating life for a young lad, is there?'
'Is there any way he could have found out?' asked Ursula.
Mrs Deddington paused for a long moment. Finally, she said, 'No one in our families ever knew. I lost my own baby, and Nicholas was . . .' her voice trailed off, sadly '. . . he came to us at the right moment. No one knew he wasn't mine. But if Nicholas had somehow discovered he was adopted, he'd have told us, I'm sure of it. We're a close family. I can't think how he'd ever have found out anyway. No. It's impossible, really.'
'So you don't think he's ever known who his real parents were?' asked Ursula.
'No. How could he? _We_ don't even know who they were. We weren't allowed any information about them when we adopted him. Now, what has this all got to do with poor Lady India anyway?'
'I'm not sure,' said Ursula. 'I think I've made a mistake. I'm sorry we bothered you tonight.'
'So am I,' huffed Mrs Deddington angrily. 'Remember, Nicholas must never know about any of this. Ever.'
*
'If Nick Deddington doesn't know that Lord Brattenbury is his real father, how could he have a serious motive for killing India?' asked Nancy.
They were sitting cross-legged on the floor of Ursula's room later that night, drinking tea and trying to warm up by the electric fire. Ursula had taken her notes about the murder out of her folder and spread them over the floor. Occasionally she wrote down a new thought.
'Maybe he does know,' she said.
'How?'
'I don't know . . . Or maybe Mrs Deddington does. Maybe _she's_ lying.'
'I can't see it,' said Nancy. 'She wasn't even here on Sunday night, we know that.'
'The thing is, I'm starting to wonder about Mrs Deddington's story about Sunday night,' said Ursula.
'Why?'
'When we first spoke to her about the night of the murder, she said she was at home that evening watching _Dallas._ Just now, she said she was watching _Dynasty. Dynasty_ 's not even on TV on Sunday nights.'
'That's weird,' Nancy agreed. 'I mean, how could anyone confuse _Dallas_ and _Dynasty_? The clothes are _completely_ different.'
'Exactly.'
'So, are you saying,' said Nancy slowly, 'that you think Mrs Deddington is lying about Sunday night? That she wasn't home after all? You think she _does_ know who Nicholas's parents were? That _she_ killed India?'
Just then, there was a rap on Ursula's door.
'Come in!' she called.
Both girls were startled to see Nick Deddington pop his head around the door. Ursula covered her notes with her arm as subtly as she could.
'Evening, Miss Flowerbutton,' he said, smiling at her. 'Actually, I'm looking for you, Miss Feingold – thought I might find you in here. There's a telegram for you.' He handed a brown envelope to Nancy.
'That's very kind of you,' she said.
'Goodnight, ladies,' he said.
The girls listened in silence as his footsteps faded away down the stairs.
'That was _really_ weird,' said Nancy. 'Do you think he could have heard what we were saying about his mom? Ugh! Spooky.'
'I know. Who's the telegram from?' asked Ursula.
'Hopefully it's Frank replying to mine,' Nancy said, tearing open the envelope.
She read the missive then handed it to Ursula. The telegram contained just two words:
RUBBER GLOVE
'I don't get it,' said Ursula.
'The sodium dimethyl-whatty,' replied Nancy. 'Frank's saying it's something to do with a rubber glove.'
'May I keep this?' said Ursula. Her mind was whirring. 'I think it might be useful.'
'Sure.'
She tucked it inside the folder. 'I think we need to go and see Wenty tomorrow. We can't properly rule him out unless we get the real truth from him about where he was late on Sunday night.'
'Do you think Detective Inspector Trott will let us see him?' said Nancy.
'I don't know, but it's worth a try.'
'It's already nearly eleven,' said Nancy, getting up to go to her room. 'I need to get some rest.'
After her friend had gone to bed, Ursula added to her notes, writing:
_—Is Mrs D lying to cover up for her son? Or her husband? Was Mr D really in Eagle and Child on Sunday night? Deddington Snr would, potentially, have had just as much motive as his wife to dispose of India._
_—And, as if I am not confused enough already, what on earth is significance of 'rubber glove'?_
* The Carringtons, the fictional family around which the TV show _Dynasty_ (1981–1989) revolved, were the Kardashians of the eighties. The whole world tuned in for the on-screen catfights between Alexis Carrington (Joan Collins) and Krystle Carrington (Linda Evans).
Friday, 22 October, 1st week: morning
Wentworth Wychwood managed a tired smile when Nancy and Ursula walked into the interview room to which he had been brought that morning. Having spent the last couple of days in a cell in Oxford police station, he was unshaven and looked dishevelled. The girls sat down on uncomfortable plastic chairs at a table opposite Wenty while a constable remained by the door.
'Flowerbutton,' Wenty said gratefully to Ursula, 'I _knew_ you'd sort everything out.'
Ursula shook her head sorrowfully. 'I have _not_ sorted everything out. Your story about Geraldine doesn't stand up.'
'What!' exclaimed Wenty. 'But—'
'Wenty, Trott's only allowing us twenty minutes with you,' Ursula interrupted. 'We need to go over the details of your story again, as quickly as possible.' Then she dropped her voice, hoping the constable couldn't hear. 'There's another lead.'
'What kind of lead?'
'There's an illegitimate heir to the Brattenbury fortune,' Nancy explained. 'Lord Brattenbury told us, when we were up in Derbyshire yesterday at the funeral. He called him "the illegit". Which I thought was really mean, by the way.'
'Are you saying that an illegitimate heir to Lord Brattenbury killed India so that he could inherit?' Wenty seemed dumbfounded.
'I wish it were that simple,' said Ursula.
'Trouble is,' added Nancy, 'the illegitimate heir doesn't _know_ he's the illegitimate heir. Or, at least, we don't think he knows. Which means he wouldn't have a motive for removing India.'
'So India had a brother?'
'Half-brother. Before he married India's mother, Lord Brattenbury got a local girl pregnant. Her name was Mary Crimshaw. She was the daughter of the village shopkeeper. But she died in childbirth and her baby boy was adopted. India never knew him,' explained Ursula.
'Actually,' pointed out Nancy, 'that's not _quite_ true. India did know her half-brother, but she didn't know that she knew her half-brother. In other words, she had no idea that the guy who was her half-brother was her half-brother.'
Wenty looked exasperated. 'You've lost me. Let me get this right. You're saying that the illegitimate heir didn't know he was the illegitimate heir, and that India knew the illegitimate heir but that she had no idea he was the illegitimate heir?'
'Exactly,' said Ursula.
Wenty shook his head. 'I don't see how I'm going to be any help in proving this theory of yours.'
'Let's just go over what happened on the night of the party,' said Ursula.
'If I must,' he sighed.
'Wenty, you haven't been telling the complete truth,' said Ursula.
'What do you mean? I've told you _everything_ , even the embarrassing stuff,' he pleaded.
'You lied about Geraldine Ormsby-Leigh. She was on the sleeper from Edinburgh on Sunday night,' Ursula continued.
'Geraldine . . . Geraldine Ormsby-Leigh? I didn't say Geraldine, I said _Gwen_ doline. Gwendoline Something-or-Other,' said Wenty. 'Yes. That was it. Gwendoline Orr-Little. They all sound the same.'
'I'm _sure_ it was a Geraldine you mentioned,' insisted Ursula.
Wenty looked sheepish. 'Look, Geraldine and I – okay. I _think_ we did it once. But not on Sunday. Sunday was Gwendoline. I swear to god, I didn't spend that night in college, murdering my girlfriend.'
Nancy and Ursula looked at each other doubtfully.
'Wenty, if this turns out not to be true . . .' warned Ursula. She paused for a moment, allowing her words to sink in. She couldn't help him unless she knew he was telling the truth. 'I think we need to go over the towel incident. I mean, the police are using it as their main piece of evidence against you, right?'
Wenty ran his hands through his hair. He looked exhausted.
'They're saying that I killed India, in a fit of jealousy, with a champagne glass from my own party, that I then returned to my rooms, collected a towel _with my initial on it_ , went back up to Dr Dave's rooms where I mopped the blood from India's neck and then stuffed the towel up the chimney, hoping it would be burned the next time Dr Dave lit his fire. I did explain to the police that, having been brought up in a freezing-cold country house heated _only_ by open fires and a pathetic broken Aga, I am well aware that if you want to burn something you put it _on_ the fire, not up the bloody chimney, where it has not a chance of burning to ash.'
'How did they take that?' asked Nancy.
'They said I was a fibbing toff,' he said despairingly. 'But, if I were going to kill my girlfriend, why would I do the deed and then go all the way back to my rooms – presumably covered in blood – to fetch a towel that identifies me? Someone put that towel there. Someone's trying to frame me.'
'Let's go back to the start of the party,' suggested Ursula, checking the clock. 'We've only got a few minutes left.'
Wenty sighed. 'Well, as you know, by the time you two arrived, at around nine, we'd started running out of champagne saucers. I met you two at the door of the Old Drawing Room, and asked you to come and help me get some more glasses from the bathroom.'
'True,' said Nancy. 'I remember being a little upset that my first experience of an Oxford ball was doing the dishes.'
'It wasn't a ball!' exclaimed Wenty. 'It was an Opening Jaunt. It's a completely different thing. Anyway, I remember that we three were in the bathroom, and I ended up trying to wash a champagne saucer in the basin because there were no clean ones and—'
'May I interrupt?' Ursula asked. 'Why were there no clean glasses?'
'The washer-uppers had disappeared,' replied Wenty. 'I think they must have been worried about being reported to the High Provost for moonlighting. They came back later, at around eleven.'
'Who were the washer-uppers?' asked Nancy.
'Just a couple of scouts. Anyway, while I was washing up the champagne glass, I cut my hand. So I used one of the monogrammed Wychwood hand towels to clean myself up. It was a nasty cut, lots of blood. But I was so pissed I barely noticed any pain.'
'Remind me where you put the towel,' said Ursula.
'I think I just chucked it on the floor,' Wenty recalled.
'Where anyone could have found it,' Nancy said.
'Not exactly _anyone_ ,' said Ursula. 'Only those who went into the bathroom that night.'
'Can you remember seeing anyone unusual in your bathroom?' asked Nancy.
'Not really.' Wenty replied. 'Masses of people must have gone in and out that night.'
'But, wait! What about anyone who was _not_ unusual?' said Ursula, suddenly excited.
'There's no way Eg killed India,' protested Wenty.
'I'm not talking about Eg,' she insisted. 'I'm talking about the washer-uppers. It wouldn't have been unusual, or particularly remarked upon, for them to have been in and out of the bathroom all night.'
'Except when I really needed them,' said Wenty.
'Which scouts were they?' asked Nancy.
'I managed to persuade Alice and Mrs Deddington to help out that night . . . Come to think of it, I still owe them a fiver each. Ursula, what are you doing?'
She was grabbing her notebook off the table.
'No wonder Mrs Deddington didn't know whether she watched _Dallas_ or _Dynasty_ on Sunday night. She didn't watch either of them! She wasn't at home. Come on, Nancy, we need to go.'
Nancy sprang out of her seat.
'But what about going over the details of Sunday night?' Wenty looked confused.
'Don't worry. I think we've got enough,' Ursula replied. 'You'll be out of here soon, I promise.'
As Ursula and Nancy rushed from the interview room, Wenty called, 'Try and get me out by Saturday night. I really want to go to Christian's night at the Playpen. It's such a brilliant club . . .'
'Wenty, I cannot believe you are worrying about your social life right now.' His priorities were, thought Ursula, amazingly superficial.
'Just trying to be positive,' said Wenty. 'Oh, and if you see Alice or Mrs Deddington, tell them I haven't forgotten about their fivers!'
Mrs Deddington was finishing off her morning tea break alone in the Scouts' Mess when Ursula and Nancy came upon her. She seemed jumpy.
'What is it now?' asked the scout, nervously putting her cup of tea aside.
'It's about Sunday night,' said Ursula.
'I told you. I was home. _Dallas_ was on. I told the police the same thing.' Mrs Deddington checked her watch. 'Now, I really must get on with the High Provost's ironing.'
She started gathering up a pile of shirts.
'The thing is, Wenty Wychwood says you were washing up at his party on Sunday night,' Nancy said, adding, 'He said to tell you he hasn't forgotten that he still owes you a fiver.'
Mrs Deddington turned a shade of crimson usually reserved for those unfortunate enough to have contracted scarlet fever.
'Girls, please, you _mustn't_ tell anyone I was working for an undergraduate,' she implored them. 'I could lose my job if anyone knows I've taken money from a student here.'
'Don't worry,' Nancy reassured her. 'We're not going to tell anyone. We just want to find out if you saw anything suspicious during the party.'
Mrs Deddington glanced anxiously around her. 'The other scouts will be in here soon for their break. I'll have to be quick.'
'Do you remember finding a blood-stained towel in the bathroom opposite the Old Drawing Room that night?' asked Ursula.
'The one with the "W" embroidered on it?' asked Mrs Deddington.
'So you did see it then?' Nancy said.
'Yes. Alice and I, we'd come down here for a bit – didn't want to get caught moonlighting. When we came back to carry on washing up, it must have been just after eleven, I saw the towel on the bathroom floor. I picked it up – of course I did, I like things spic-'n'-span.'
'What did you do with it?' asked Ursula.
Mrs Deddington looked surprised to be asked such a question.
'Put it in the laundry basket, of course. As I said, I do like things spic-'n'-span.'
At that moment the door to the mess opened and a couple of scouts appeared for their break.
'Cup of tea?' asked Mrs Deddington, filling the kettle again, before saying, rather louder than was necessary, 'Thank you for letting me know about the hot water breaking down, girls. Someone will be up to fix it today.'
*
Ursula was not quite sure what to do next. She had a non-negotiable deadline for her essay that coming Monday, and had made few inroads into the Apocalyptic – or not – Vision of the Scottish Covenanters, but the deadline for her article was sooner.
How on earth was she going to solve the riddle of India's murder and write it up before Sunday morning? After all, Mrs Deddington's story seemed to explain far less about the case than she had hoped. The only thing Ursula had ascertained from the brief interview with the scout was that she had lied about her whereabouts on Sunday night for fear of losing her job. The mystery of exactly how the bloody towel had made its way from Wenty's laundry basket, thence to India's garrotted neck and finally into Dr Dave's chimneybreast, was no nearer to being solved.
Ursula climbed the stairs to her room. Even if she didn't have the ending of her article yet, she had the grim beginning. She could at the very least add to the notes she'd started writing up last night. Maybe there was a crucial clue or fact she had overlooked that would be revealed while she was writing.
The door to her room was ajar when she arrived, and Ursula entered to find Alice beavering away, polishing the desk. Perhaps she could shed some light on the mysterious movements of the bloody towel.
'Hello, love,' said Alice. 'Am I all right to carry on?'
'Of course,' Ursula replied. 'I just need my desk.'
'I've got to do your bed and then I'll be gone.'
Alice bustled over to the bed and started making it. Ursula sat down at her desk where her work had been arranged in a neat pile, with the red folder containing her story notes on the top. She opened it and read the last note she had written on her pad:
_. . . what on earth is significance of 'rubber glove'?_
'Alice, can you help me with something?' said Ursula.
'I very much doubt I can help with your essay but go on, give me a try,' the scout chuckled.
'Actually it's about Sunday night.'
Alice stood stock still, a blanket in her hand. She turned slowly towards Ursula.
'Look, I know you were working at Wenty's party that night—'
'Miss Flowerbutton, please don't mention that to anyone,' Alice begged, sounding just as desperate as Mrs Deddington had. 'We tried to stay out of sight. The extra money's just so helpful.'
'Don't worry, no one's going to say anything. But I'm curious about something—'
A tap on the door, followed by Nancy's arrival in Ursula's room, interrupted them.
'Hey, Alice,' Nancy called to the scout. She was holding a blue airmail envelope in her hand. 'Mind if I hang out and read my mail here, Ursula?'
'Of course not,' she replied.
As Nancy sat in the armchair and started opening her letter, Ursula turned back to Alice and said, 'Anyway, I was curious about a towel that was left on the floor of Wenty's bathroom on Sunday night. It was one of his monogrammed hand towels.'
'I know the ones,' said Alice, arranging the blanket neatly on top of Ursula's bed. 'Very nice. But that one had a nasty stain on it.' She then addressed Nancy, saying in worried tones, 'Miss Feingold, you won't say anything about me washing up on Sunday night, will you?'
'No way,' said Nancy, scanning her letter.
'So, I was just wondering,' Ursula continued slowly, 'did you see Mrs Deddington put the hand towel into the laundry basket?'
'No.' Alice shook her head. 'She put it into the pocket of her apron.'
Ursula noticed Nancy stiffen in her chair. A startled expression appeared on her face.
'Mrs Deddington took it?' Nancy reiterated. 'You're sure?'
'Yes. We'd gone off for a break to the Scouts' Mess. When we came back later, there it was on the floor. She picked it up and said she was going to wash it herself,' replied Alice, starting to gather up her cleaning things now the bed was made. 'She wanted to get the stain out before it marked.'
'Oh, right,' said Ursula, trying to remain cool as she noted Alice's words in her book.
'Right, I'm off, girls,' said the scout, picking up her bucket.
Just as Alice was about to leave, Nancy suddenly emitted a long cooing sound. She was holding something in her hand.
'Oooooooh! Mom's sent the _cutest_ photograph!' she exclaimed, jumping up and holding out a photograph of a wrinkled new-born baby dressed in a smocked romper suit for Ursula to admire. 'My cousin has had her first kid. Winston. He's so adorable.'
'Sweet,' said Ursula, getting up from her desk to take a closer look.
'Lovely,' chimed in Alice, looking at the picture. 'Beautiful bairn.'
Suddenly, Nancy froze. What on earth was wrong with her? wondered Ursula.
'Did you say _barn_?' Nancy asked the scout.
'It means "baby",' replied Alice. 'That's what we say up where I come from. Now, I'll leave you ladies to it.'
The minute Alice had departed, Nancy shut the door to Ursula's room firmly, leaning against it as though to bar entry to any interlopers.
'Did you hear that?' she whispered dramatically.
'Hear what?' asked Ursula.
'She said _barn_.'
'It means a baby.'
'I know. She used the _exact_ same word for baby as Mrs Crimshaw did,' Nancy continued.
Suddenly Ursula twigged.
'Oh my god! Mrs Crimshaw said "bairn" yesterday, when she was talking about the illegitimate baby. Nancy, are you saying you think Mary Crimshaw might not be dead?'
'I'm saying I think she was standing right here two seconds ago. She uses the same local slang as Mrs Crimshaw. She could easily have been raised in Brattenbury.'
Ursula dashed back to her desk and turned to a fresh page in her notebook.
'Right, brainstorming time,' she said, hurriedly scribbling as she thought out loud. 'If Alice, who was in college on Sunday night, is in fact Mary Crimshaw, who is in fact the birth mother of Nicholas Deddington, who is in fact the illegitimate heir to Lord Brattenbury . . . then she had the motive _and_ the opportunity to murder India.'
Nancy nodded enthusiastically as she rifled around in her bag for a packet of cigarettes. She then retreated to the armchair, lit her cigarette and puffed madly on it while Ursula went on.
'But – hang on – how would Mary Crimshaw have found out where her son went when he was adopted?'
'Minor detail,' insisted Nancy between smoke rings. 'It probably wasn't that difficult to trace him. Perhaps her father secretly helped her. He knew where the baby had been taken.'
'But why would Mary's father have told his wife that their daughter was dead if she wasn't?' asked Ursula, halting her writing for a moment to ponder so many new twists.
'Look, we all know you British are pathetically uptight—'
'What?' Ursula protested.
'You heard Mrs Crimshaw. A pregnant, unmarried teenage daughter would have brought shame on Mary's family. Maybe her father forbade her from ever coming home. Maybe it was easiest to say she'd died,' Nancy suggested.
'Perhaps that's why Alice was wearing a veil at the funeral . . . she didn't want any of the locals to recognise her,' Ursula surmised. Suddenly, an extraordinary thought came to her: 'Do you think she just lied to us about seeing Mrs Deddington taking the towel?'
'That's it, Ursula!' Nancy sprang up from her chair. She was so excited she didn't notice the blizzard of ash falling from her cigarette to the floor as she did so. She came and perched on the corner of Ursula's desk. 'What if _Alice_ took the towel from the laundry basket, after Mrs Deddington had put it in there? What if _she_ did everything she's claiming Mrs Deddington did?'
How on earth could that be proved? Ursula wondered, opening the top drawer of her desk and retrieving the bar of Kendal Mint Cake Plain Granny had sent her. She broke off a large chunk and offered it to Nancy.
'You want me to eat Kryptonite now?' she said, shaking her head at the alien sight of the white bar.
'All the more for me,' Ursula declared happily, crunching on the sweet mint cake. It tasted sublime, and re-energised her to continue. 'Let's go back to the scene in the bathroom. Wenty said that after the row in Great Quad, he and Otto chatted in there – where, we now know, Alice and Mrs Deddington were washing up—'
'Wait! Washing up! That's it!!!' cried Nancy. 'It's what Frank said in that telegram. "Rubber glove". He meant a washing-up glove. You never see Alice without her yellow gloves. She must have been wearing them to hold the shard of glass when she killed India, so she didn't leave her fingerprints. That's why Doc found that trace of sodium diamithy-whatty on the tiny glass fragment that was found in India's neck.'
Ursula looked at Nancy with admiration. Her powers of deduction were razor-sharp. But there was a possible flaw in her theory.
'Lots of the other scouts wear washing-up gloves,' Ursula pointed out. 'Including Linda Deddington.'
'True.' Nancy looked deflated. She walked over to Ursula's basin, ran her cigarette stub under the cold tap and chucked it in the waste bin.
'But still, let's run with the hypothesis that Alice is the murderer,' Ursula went on.
'Okay,' said Nancy, taking a Milky Bar from her bag. She sat cross-legged on the floor munching chocolate.
'Alice, washing up in the bathroom at midnight, would have overheard Wenty asking Otto to go and get India back from Dr Dave's rooms.'
'Meaning she knew where India was that night,' added Nancy.
'Exactly. Now, I don't know why, but at some point that night Alice decided to go up there. No one would have suspected anything peculiar if they'd seen a scout go into Dr Dave's rooms at any time of the day or night, would they?'
'I guess not,' agreed Nancy.
'She had Wenty's bloody towel in her apron pocket, which she had taken from the laundry basket sometime that evening. She found India alone. Here was her chance to dispose of India, remove the obstacle to her son's possible inheritance and cast suspicion initially on Wenty by planting his bloody towel in Dr Dave's chimney. By later saying she'd seen Mrs Deddington take the towel, Alice could make it look as if Mrs Deddington had planted it in Dr Dave's chimney to frame Wenty, thereby making her the prime suspect.'
Ursula looked at Nancy questioningly, as if to say: Tell me if I'm completely mad.
'You've left something out,' Nancy said.
'What?'
'The flaw in Alice's plan – Mrs Deddington. If she doesn't know her adopted son is the offspring of Lord Brattenbury, then she had no motive to murder India. Mrs Deddington only lied about her whereabouts on Sunday night for fear of losing her job, not because she had killed someone.'
'The only problem now,' said Ursula, 'is that we don't know for sure Mary Crimshaw isn't dead.'
*
Ursula put a ten-pence coin in the slot in the payphone at the bottom of the staircase and dialled a Leeds number as fast as her fingers would allow. The line seemed to ring and ring forever before someone answered.
'Leeds Register Office,' a man's voice answered.
'Hello. I'm wondering if you can help me locate a death certificate,' said Ursula.
'I just need the name of the deceased and the year that they died.'
'Her name was Mary Crimshaw and I think she died some time in 1963 or 1964.'
'Right. You'll need to send a postal order for £2.50 made payable to Leeds Register Office. When we receive your payment we'll send you a copy of the death certificate.'
That was going to take weeks, thought Ursula.
'Can I ring back later to see if you've found it?' she begged. 'You see, this is really very urgent.'
'It's always "urgent",' came the deadpan reply.
'Oh.' There was nothing else for it. Ursula was going to have to tell a terrible lie. She crossed her fingers and said, in as sad a tone as she could, 'You see, the thing is, the dead woman . . . she was my mother.'
'Well . . . I suppose . . .' said the man, his tone softening '. . . in exceptional cases, I can elevate requests to "officially urgent".'
'That would be so kind,' she said, sniffing loudly.
'All right. Don't worry, love. Someone can try and find the certificate today. If you give me a telephone number, we'll ring you back later.'
Friday, 1st week: evening
The scene at Vincent's Club* that night was, declared Nancy, 'blazer heaven'. The sight of the 'Vinny's boys', the University's most elite sportsmen, lounging around on comfy-looking leather chesterfield sofas and swigging cocktails, dressed in the white-trimmed navy blazers and navy ties of the Oxford Blues, was, she concluded, 'better than a Ralph Lauren ad'. Thank goodness, Ursula thought to herself, she'd dressed up properly for the club's first cocktail party of term: Nancy had lent her a ruched, fitted, shocking-pink taffeta mini dress that only just covered her thighs. It wasn't exactly her style, but it was fun to wear. Nancy, meanwhile, was dressed in a frothy, off-the-shoulder dress that consisted of tumbling layers of pale yellow organdie frills held in place by tiny bows.
'Oh my god, Horatio!' Ursula exclaimed, suddenly spotting her friend across the room. This was the last place she'd expected to see him, especially wearing the navy blazer of the Full Blues.
'He's not exactly a jock,' observed Nancy.
Horatio saw the girls and beckoned them to join him.
'Horatio, I'm intrigued,' said Nancy. 'What sport did you get your Blue for?'
'Tennis,' he said, a contented smile on his face.
' _Really?_ ' Ursula couldn't help but sound amazed.
'If you are implying, dearest Ursula, as you seem to be, that I am far too rotund a personage to dash around a tennis court with any effectiveness—'
'No, no, I didn't mean that at all,' said Ursula, hoping she hadn't been rude. 'I wasn't saying—'
'—that I'm too fat to play tennis? Well, you're completely right. I'm far too podgy. I got my Blue for sitting on the umpire's chair. It's simply lovely up there, bossing everyone about. Everyone comes to my matches. Apparently I'm more entertaining than John McEnroe.'
Horatio ordered Snowball cocktails for the girls and then went on, 'Anyway, Ursula, I don't know what you're doing out enjoying yourself so close to your story deadline.'
'Actually, I said I'd meet Jago here for a drink.'
'Woooooh! Do be careful, Ursula. He can be dreadfully lecherous.'
'No, honestly, it's fine,' she insisted. 'It's an official No Strings Attached drink.'
'That old cliché,' chuckled Horatio.
'Really, I'm only staying for a bit tonight. I've got to get back to work on the story. I've got tons of notes to go through.'
Just then, Jago sauntered up to them. He was wearing the stripy jacket of a Half Blue.
'Ah, my star reporter!' he said, pecking Ursula on the cheek, throwing one arm around her shoulders and squeezing her to his chest.
Horatio whispered in her ear, ' _No Strings Attached._ Ha!'
'Solved the case yet?' said Jago. He sounded as though he was half-joking.
'Almost,' replied Ursula, removing his arm.
'You think you know who did it?' he said, suddenly serious. ' _Who?_ '
Ursula looked around the room, which was now bulging with guests. She didn't intend to go into the details of India's murder with so many people within earshot.
'I can't say yet.'
'You know how to drive a man crazy, don't you, Ursula?' said Jago, looking at her lasciviously.
Ursula said nothing, just looked coolly at him. She didn't want him getting any ideas about attaching any strings whatsoever to her. Finally, he said awkwardly, 'Anyway, yeah, I'm looking forward to reading your copy on Sunday morning. Do come and use the typewriters in the _Cherwell_ offices, won't you?'
'Sure,' she said non-committally.
Secretly, she was not convinced she would be anywhere near the _Cherwell_ typewriters by tomorrow afternoon. The final, crucial question in the mystery of India's death – whether or not Mary Crimshaw had died in childbirth – still remained unanswered. Ursula had stopped by the Porters' Lodge every half hour that afternoon, hoping that the Register Office in Leeds had called back and left a message – but there was nothing.
'Good,' said Jago. 'I do like a reporter who keeps to their deadline. Right, I'll get another round.'
He set off towards the bar and Ursula turned back to her friends, who were in the midst of an important conversation.
'Hey, what should I wear for my date tomorrow morning with Next Duke?' Nancy was asking Horatio.
'It might be very cold. A tweed suit would be ideal,' he suggested helpfully.
'Oh, god, that sounds _so_ old-school classy. Next Duke would like that, right? But where do I get one at this time in the evening?'
'I can lend you my old tweed hacking jacket,' offered Ursula. 'It's in my room.'
'Okay. That sounds totally authentic. What about the pants?'
'Sorry, I don't have tweed trousers. But I've got a pair of joddys you can borrow.'
'Joddys? What are those? They sound ugly,' Nancy declared.
'You know, jodhpurs . . . riding breeches. I've got Plain Granny's old ones, from when she was a girl. They look like wool pantaloons. They're beautiful.'
'Okay, I'll take your word for it,' said Nancy.
'The main thing is, you'll be warm. You just need some boots as well,' added Ursula.
'I'm good for boots. Can you leave the clothes in my room tonight?'
'All right,' said Ursula. 'I'm going back now to write. Horatio will walk me, won't you?'
When Horatio dropped Ursula back at the Porters' Lodge about thirty minutes later, Nick Deddington was at the desk. Nancy was right, Ursula thought to herself, he did look uncannily similar to the photograph of Lord Brattenbury as a young man that they had seen by his bedside yesterday. Ursula wondered what would happen when he discovered his true identity. She felt uneasy, knowing something so crucial about him that he didn't.
'Evening, Miss Flowerbutton,' he said when she entered. 'Someone telephoned, about half an hour ago. Said it was urgent.'
He handed her a note, which read:
_No certificate found. No need to send postal order. Leeds Register Office_.
She stared at it, then read the words over again. 'No certificate found?' No death certificate meant no death. Mary Crimshaw was not dead.
Ursula suddenly felt fearful, excited and nervous all at once. Had she discovered who the murderer was? _Was_ it Alice? Was a murderous scout, Ursula wondered as she walked hesitantly towards the Gothic Buildings, lurking somewhere between the lodge and Ursula's room, hiding in the shadows, waiting to kill again?
Perhaps, thought Ursula, she should tell Trott her suspicions, and let him deal with it. The situation was dangerous. But . . . could Alice _really_ be a cold-blooded killer? Alice seemed so nice, with her kind words, cosy cups of tea and jolly aprons. She'd been so friendly with India. If Ursula told Trott the latest developments, and it turned out that Alice was innocent, she could be landing her helpful scout in undeserved trouble.
In any case, there was nothing Ursula could do at this hour. She decided the best course of action would be to ask Alice some leading questions when she came to clean tomorrow morning after the 6 a.m. rowing session. Perhaps the scout would say something incriminating. Or else clear herself of suspicion completely.
In the meantime, back in her room, Ursula jotted down the new clues in her notebook, trying to prepare for writing the article on Saturday afternoon.
_—Mary Crimshaw is not dead. True._
_—Mary Crimshaw is real mother of Nicholas Deddington. True._
_—Is Alice Blythe Mary Crimshaw? Maybe._
_––Nicholas Deddington is illegitimate heir to Lord B. Very likely._
_—If Mary Crimshaw wanted her biological son to inherit, she had motive to kill India. Possible._
_—Am I wrong about it all? Possible. After all, how could someone who dusts with such dedication be a murderess?_
By ten o'clock, Ursula was done for the night. She put her notes carefully into the folder on her desk. It was only then that she noticed a brown paper bag sitting on her bed. What on earth was that, she wondered, and who had left it there? Ursula approached the bag with caution, picked it up and opened it. Inside, something was glistening. India's tiara! She reached her hand inside the bag and pulled out the jewelled headpiece. There was a note with it, from Claire Potter. It read,
_Knicker drawer no longer as safe as previously thought. CP._
No way! Did Claire Potter have a boyfriend? _Already?_ Ursula examined the tiara curiously. Inside, she noticed a stamp. This must be the hallmark, she thought, trying to read the tiny letters. She made out the words 'Butler & Wilson'. Butler & Wilson! Ursula couldn't help but laugh. She had heard of Butler & Wilson, an ultra-trendy costume jewellery shop in London where Chelsea girls shopped for gigantic diamanté earrings, golden chokers and glittery bangles. Claire needn't have worried about stealing a family heirloom. India had fooled everyone with her fake tiara. Ursula would hand it in to the police as soon as she could. She'd just say it had mysteriously appeared in her room, which was, after all, completely true.
Before bed, Ursula put out her old riding outfit for Nancy. The hacking jacket – dark brown tweed with a pale blue windowpane check – looked perfect. The jodhpurs, on the other hand, had an old grass stain on the left knee. She'd have to go and wash and tumble dry them in Monks' Cottages before she went to bed. Nancy wasn't the sort of girl to be seen on a first date in grubby clothing.
Twenty minutes later, as Ursula sat in the laundry waiting for the jodhpurs, doubts crept into her mind. What if Alice wasn't Mary Crimshaw? Ursula and Nancy had no real, concrete proof that she was, except for her Derby accent and the fact that she used the word 'bairn' for baby. Her rubber gloves _could_ place her at the scene of the murder, but no more so than any of the scouts in college. Maybe the sodium dimethyldithiocarbamate hadn't come from the killer at all. Maybe it had been on the glass before the killer touched it, transmitted when the glass had been washed or dried before Wenty's party.
Perhaps this whole journalism thing, the _Cherwell_ article, the dream of being a writer one day – well, maybe it wasn't going to happen. If Ursula didn't know the identity of India's murderer, she didn't have a story, and that was that. She consoled herself with the gruesome thought that maybe there'd be other murders while she was studying in Oxford with which she'd have better luck.
Ursula didn't get back to the Gothic Buildings with the dry joddys until close to midnight. Just as she was about to walk from the stone passage across the courtyard, she saw someone coming out of the archway of her staircase. Alarmed, Ursula scurried into the shadows of Staircase A and hid, watching.
There was no mistaking those yellow rubber gloves or that flowery apron. It was Alice. But what on earth was she doing in college at close to midnight, with her rubber gloves on? Surely it was much too late for cleaning, even for the most enthusiastic scout? Alice swiftly exited Staircase C, looking behind her as though to check no one had seen her. She then hurried straight past Staircase A, without seeing Ursula, and along the passage back towards Great Quad. She'd looked nervous, thought Ursula, who had never seen the usually jolly-faced scout in a fluster until now. Maybe she _was_ Mary Crimshaw.
Once upstairs, Ursula left the clean joddys, tweed jacket and a plaid shirt in Nancy's room, hoping her friend's sunrise breakfast would be as romantic as Nancy dreamed, before going back to her own quarters. As she walked into her room she noticed that two pages of her _Cherwell_ story notes had fallen onto the floor beneath her desk. She was sure she hadn't left them like that. She picked them up and put them back in the folder, carefully closing the cover.
Before she went to bed, she checked – several times – that she had locked the door to her room. She was properly scared. As she lay in bed, fretting, fearful, Ursula started to wonder if the idea of confronting Alice tomorrow was a sensible one. She resolved to ring Trott from the payphone downstairs the minute she woke up. In the end, the only thing that got her off to sleep was thinking about rowing practice. Counting oars, Ursula soon discovered, was as good a cure for sleeplessness as counting sheep.
* Membership of Vincent's Club on King Edward Street (started in 1863) requires possession of a Full or Half Blue, as well as intellectual and social standing to match. Women members were, finally, accepted in 2016. (No, that's not a typo.)
Saturday, 23 October, 1st week: sunrise
Ursula was having some kind of nightmare. She couldn't wake up from it. 'Uuughh-ggghhh!' she yelped. 'Uggrrrrhhhh.' She couldn't breathe in her dream. Air – there was no air! If only she could wake up, she could breathe. She wriggled. Struggled. But still she couldn't wake up. A pillow was being pressed over her face. Someone was suffocating her. She thought she heard an alarm clock ringing. But she still couldn't wake up. She was still in the nightmare. Not breathing. Choking.
Maybe Ursula had passed out for a few seconds. Maybe she'd dreamed that she'd fainted. Someone was banging on her door in her nightmare – rap-rap-rap!
'Ursula! You're late for rowing!'
But Ursula couldn't answer. She couldn't move her head. She realised she wasn't dreaming. She was awake and she was being smothered. Someone really did have a pillow over her face. Someone was trying to kill her. She tried to move her arms, to hit her attacker, but they were pinned to her chest by someone's body.
Suddenly there was a loud cracking sound. A body collapsed like lead on top of her, and Ursula could finally move her head. Gasping for breath, she pushed the body off her. It thudded to the ground. Ursula looked up. There, looming above her, oar in hand, was Moo.
'Didn't quite mean to, Ursula,' said Moo, 'but I think I've just bumped off our scout.' She sounded about as apologetic as a Girl Guide who'd just caught a mouse in a trap. 'I came in to wake you up for rowing and found Alice trying to suffocate you. So I decided to go for a massive bonk on the head.'
Ursula struggled up in bed and looked at the floor next to her. There lay Alice, slumped unconscious. Her head was oozing blood. Ursula couldn't help noticing the bright yellow rubber gloves on her hands.
'I can't believe she tried to kill me,' Ursula croaked, her voice almost gone after the shock of the attack.
'But why would Alice want to kill _you_?' gasped Moo.
Everything made sense to Ursula now. But, in her dazed state, her thoughts came out in a jumble.
'That's why she was always tidying my desk. She was secretly reading everything I wrote down about India's murder.'
'What are you talking about?' Moo looked concerned.
'I saw Alice coming out of our staircase late last night. She must have been in here, reading my notes. I found two pages on the floor – I never would have left them there.'
'Are you sure you're not concussed?' Moo asked.
'No, I'm fine. Once Alice knew that I knew she was actually Mary Crimshaw, mother of the illegitimate heir to the Brattenbury estate, she had to get rid of me.'
'Why would someone called Mary Crimshaw pretend she was Alice?' asked Moo. She looked completely and utterly confused.
'No, Alice was pretending she _wasn't_ Mary Crimshaw,' Ursula corrected her. 'She murdered India and tried to frame Mrs Deddington.'
Suddenly the scout stirred and gingerly lifted her bloodied head. 'I only wanted to get my son what was rightfully his,' she whimpered, before flopping to the floor again.
Moo then offered: 'I'm jolly good at boating knots. Do you have a belt?'
Without waiting for a response, Moo rifled in the wardrobe, dug out a couple of Ursula's belts and cheerfully tied them around Alice's hands and feet.
'Someone better call the police,' she said. 'Deddington Junior's in the Porters' Lodge. He can do it.'
'No. I'll ring from the payphone downstairs,' said Ursula.
Knowing what she did, she didn't think it was fair to ask Nick Deddington to turn in the woman who would soon be revealed as his birth mother to the police for murder. Ursula threw on her dressing gown. Just as she stepped onto the landing a loud scream came from Nancy's room. The door flew open and Nancy appeared, wearing Ursula's riding gear, which she had imaginatively teamed with a pair of very impractical high-heeled yellow suede boots. She looked absolutely terrified.
'Ursula!' shrieked Nancy. 'I've made a terrible mistake. Alice isn't the killer at all.'
'What?' cried Ursula.
'This weird street person showed up in my room this morning carrying champagne glasses and a bottle of Dom Perignon, claiming to be the next Duke of Dudley. But it's not him – it's an imposter! I think he wanted to kill me, just like he killed India, with the champagne saucer.'
'Where is he?' said Moo, rushing out of Ursula's room and into Nancy's, brandishing her bloodied oar.
Nancy pointed at the wardrobe. 'I shut him in there and locked it.'
A hammering came from inside the wardrobe.
'Could someone please let me out? I promise I _am_ the next Duke of Dudley and I have no interest in murdering the lovely American girl I met at Wenty's party, wailed a muffled voice.'
'What does he look like?' asked Ursula. She suddenly remembered the moment at Wenty's party when Nancy had spilled champagne over the Next Duke and Otto had laughed mischievously.
'His clothes are full of holes and he's got hardly any hair,' Nancy told her.
'Sounds just like a Duke-in-Waiting to me,' said Ursula, unlocking the door to the wardrobe.
'What are you doing?' screeched Nancy.
'He didn't kill India,' said Ursula as a scruffy-looking boy stumbled out, glasses and champagne bottle in hand. 'You were right about Alice. She _is_ Mary Crimshaw. She tried to suffocate me this morning.'
'No!' gasped Nancy.
'Don't worry,' said Moo. 'I saved the day. That's the whole point of being Head Girl.'
Ursula recognised the boy from the wardrobe immediately. He was the rather country-bumpkinish young man who'd been standing by the fireplace when Nancy had been flirting with the tall, dark, handsome undergrad she'd assumed was the Next Duke of Dudley. Ursula had wondered at the time whether the handsome guy had been a little too good to be true – no one was that good-looking _and_ the heir to a dukedom. How devilish of Otto to allow Nancy to carry on thinking he was!
'Hello,' said the scruffy boy. Horatio was right. Next Duke did have a Paddington Bear quality about him.
'Hi,' said Ursula. 'Are you okay?'
'I was expecting the Next Duke of Dudley,' said Nancy.
'Well, I _am_ the next Duke of Dudley,' replied the boy. 'Who did you think I was?'
Slowly, the truth started to dawn on Nancy.
'I'm so . . . sorry,' she stuttered, completely mortified. 'There's been a terrible mistake. I thought the Next Duke of Dudley was someone else.'
Next Duke looked bereft.
'I thought you were . . . _you_ ,' he said, downcast.
Nancy smiled at him, charmed. 'But your idea for a sunrise date in a meadow was _very_ cute.'
'It's going to be a beautiful dawn,' said Next Duke. 'Awful shame to miss it.'
Ursula saw Nancy flush a little.
'Yes . . .' she said.
'Would you like to see it . . . with-with me?' he mumbled.
'Actually, I would,' she replied, a dreamy smile coming over her face.
Next Duke looked overjoyed by Nancy's response. Then he said, taking in her equestrian garb, 'I'm afraid we'll be walking. We could ride another time.'
'I'm psyched!' said Nancy. Then she turned to Ursula and added, 'You will remember to make an excuse for me when you get to the river, won't you?'
'Actually, I need someone to make an excuse for me,' said Ursula, who now looked pleadingly at Moo.
'Just this once,' she agreed. 'I suppose being attacked by a bloodthirsty scout is an acceptable excuse for getting off games.'
'No, it's not that, I'm not hurt,' Ursula insisted.
'What is it then?' asked Moo.
'I've got an article to write. I've barely started it and I've got to hand it in first thing tomorrow. Everyone, will you guard Alice in my room while I go to the payphone? I need to call the police.'
As Ursula rushed down the stairs, she thought about India's tragic end. Murdered for nothing more than money. Killed for being a Yah. How sad that no one would ever see her perform Shakespeare. What were those famous words the ghost of Hamlet's father had said? ' _Murder most foul_ '. Indeed, thought Ursula as she dialled 999, India's murder was most foul. The phone rang only once before someone picked up.
'Hello,' she said, 'I'd like to report the capture of a murderess.'
The End
(Almost . . .)
Champagne Set Murder (continued from Page 1)
* * *
BY URSULA FLOWERBUTTON
* * *
After I telephoned the police on Saturday morning, I returned to my room where Algernon Dalkeith, a.k.a. the next Duke of Dudley, and Nancy Feingold (now holding hands), as well as Felicia Evenlode-Sackville were guarding Alice Blyth. The scout, whose real name is Mary Crimshaw, claimed that she had never intended to commit murder. Her plan had been to blackmail India Brattenbury. She said:
'I was going to tell India about Nicholas. I'd threaten to tell the college authorities about her affair with a famous don if she didn't promise to cut him into her inheritance.'
Crimshaw thought her moment had arrived on Sunday night. After overhearing Wentworth Wychwood telling Otto Schuffenecker that India had gone to the aforementioned don's rooms, the scout realised that if she caught India and the don together she could put her blackmailing plan into effect.
Having cleaned up Wychwood's party and bidden Mrs Deddington goodbye, Crimshaw removed Wychwood's bloody towel from his laundry basket and put it in her apron pocket. (She claims she was planning to wash it at home to prevent staining.)
Around 3 a.m. Crimshaw crept into the don's rooms, where she found India alone, passed out on the chaise-longue. Noticing a broken champagne saucer in her left hand, Crimshaw went to clear it up. As she picked up the sharp, broken piece of glass, she was convulsed by a terrible thought. Crimshaw says:
'I thought, wouldn't it be simpler for Nicholas if India was gone altogether?'
Wearing her rubber gloves, Crimshaw cut the girl's throat with the shard of glass. She claims that India was so drunk she didn't stir, despite the violence of the injury inflicted upon her.
Crimshaw then dabbed Wychwood's towel in the blood on India's neck and concealed it in the chimney, believing she could use it later to frame fellow scout Mrs Linda Deddington. When the towel was found, suspicion would initially fall on Wychwood. But Crimshaw would tell the police that Mrs Deddington was the last person seen with the towel, thereby placing her at the scene of the crime. As the adoptive mother of the illegitimate heir to the Brattenbury fortune, Mrs Deddington would appear to have motive and opportunity. Crimshaw planned to plant the piece of broken glass she had used as the murder weapon among Mrs Deddington's effects, but never found the opportunity.
'Nicholas would inherit one day. He'd never know what really happened,' insisted Crimshaw. 'I would disappear. Just like I had when I was fourteen. I was "dead" then. I could be dead again.'
Moments later, D.I. Trott and a constable arrived and arrested the scout.
As Crimshaw was led away by police Nancy Feingold said tearfully, 'So sad. I'm really gonna miss Alice. Those hospital corners she did on our beds were totally awesome.'
* * *
FROM JOHN EVELYN'S DIARY
by Horatio Bentley
Freshers – where did they come from? Why didn't they stay there? I am usually unsympathetic to Oxford's scarf-wearing, kettle-wielding newbies. But after an eventful First Week in which the University was shaken by the murder of Christminster beauty LADY INDIA BRATTENBURY, I can reveal that the delightful pair of Freshettes who unravelled the identity of the murderess have turned heads.
The crime-busting duo are as beautiful as they are brainy. American student NANCY FEINGOLD (affectionately known to her (new) friends as 'Lawnmower' – her cash comes from her parents' gardening-tools empire) was spotted dressed in saucy riding kit on Saturday morning cuddling up to one ALGERNON DALKEITH on a mossy knoll in Port Meadow. He is said to be infatuated with the New Jersey native, who has changed his nickname from 'Paddington' to plain old 'Next Duke'.
Nancy Feingold's bestie, URSULA FLOWERBUTTON, author of the riveting article on the preceding pages, also found time to dabble in romance on Saturday night. Despite the attempt made on her life earlier that day, this diarist noticed Miss Flowerbutton boogying enthusiastically to the romantic strains of 'The Love Cats' at the Playpen, the Saturday evening nightclub favoured by the CHAMPAGNE SET. She started the night dancing with Christminster disco-diva EGHOSA KOLOKOLI, but Cherwell editor JAGO SUMMERS cut in, stealing her away as soon as was politely possible.
When LORD WENTWORTH WYCHWOOD arrived at the club, just after being released from Oxford police station, he was overheard asking Ursula to dance. Flowerbutton refused the dishy Earl, saying she had to leave. When he asked her why, Ursula only answered, mysteriously, 'To write my essay'*.
* * *
*Ursula _did_ finish her essay on the Apocalyptic Vision of the Early Covenantors in time for her tutorial on the Monday morning of 2 Week. Dr Dave's only comment on nd her effort: 'This isn't high school, Miss Flowerbutton.' Devastated (until now she had been a straight-A student), Ursula promised herself that she really would, honestly, absolutely definitely, cross-her-heart-hope-to-die, stick-a-needle-in-my-eye, spend 2 Week in the library working much harder on her next essay.†
†Unless someone else was murdered.
ACKNOWLEDGEMENTS
Thank you so much to everyone who has helped me research, write, edit and publish this novel. I would like to thank especially: Detective Chief Inspector Gareth Bevan; John Smith, Chief Executive Officer of Police and Crime Commissioner Avon and Somerset; Tom Bailey; Lisa Wheden; Miranda Elvidge; Caitlyn Rainey; Miles Guilford; Dr Hugh White, retired forensic pathologist; Professor Peter Frankopan; Romain Reglade; Roberto Wheedon; the staff of the Oxford Union Library and the Codrington Library; Professor Marcus du Sautoy; Max Long; all at _Cherwell_ newspaper; Katie Bond; Christine Knight-Maunder; Kara Baker; Jonathan Burnham; Emily Griffin; Alexandra Pringle; Rebecca Carter; Luke Janklow; Jonathan Bate, Paula Byrne and all at Worcester College, Oxford; Eve MacSweeney; Crispin Jameson; Zac Posen; Jo Allinson; my brothers and sisters Lucy, Fred, Alice, Josh and Tom; my early readers – Victoria Elvidge, Catherine Ostler, my husband Toby Rowland, our daughters Ursula and Tess, and my mother Valerie.
A very special thanks to my Great Aunt Rosey Goad, who was the great P.D. James's editor. When I asked her advice about writing crime, she told me: 'Plum, don't have more than two bodies – it will confuse the reader. And never have more than one secret staircase.'
A NOTE ON THE AUTHOR
Plum Sykes was born in London and educated at Oxford. She has written two novels, _Bergdorf Blondes_ , a top ten _Sunday Times_ and _New York Times_ bestseller, and _The Debutante Divorcée_ , a _New York Times_ bestseller. She is a Contributing Editor at American _Vogue_ where she writes about fashion, celebrity and society. _Party Girls Die in Pearls_ is her third novel, the first in her Oxford Girl Mystery series. Plum Sykes lives on a Cotswolds farm with her husband and two children.
Find her on Instagram @therealplumsykes
First published in Great Britain 2017
This electronic edition published in 2017 by Bloomsbury Publishing Plc
© Victoria Sykes, 2017
Victoria Sykes has asserted her right under the Copyright, Designs and Patents Act, 1988, to be identified as Author of this work.
Image here © Getty Images.
This is a work of fiction. Names and characters are the product of the author's imagination and any resemblance to actual persons, living or dead, is entirely coincidental.
The moral right of the author has been asserted
All rights reserved. You may not copy, distribute, transmit, reproduce or otherwise make available this publication (or any part of it) in any form, or by any means (including without limitation electronic, digital, optical, mechanical, photocopying, printing, recording or otherwise), without the prior written permission of the publisher. Any person who does any unauthorised act in relation to this publication may be liable to criminal prosecution and civil claims for damages
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CONTENTS
1. Cover
2. Half-title Page
3. Dedication
4. Also by Plum Sykes
5. Title Page
6. Contents
7. One
8. Two
9. Three
10. Four
11. Five
12. Six
13. Seven
14. Eight
15. Nine
16. Ten
17. Eleven
18. Twelve
19. Thirteen
20. Fourteen
21. Fifteen
22. Sixteen
23. Seventeen
24. Eighteen
25. Nineteen
26. Twenty
27. Twenty-One
28. Twenty-Two
29. Twenty-Three
30. Twenty-Four
31. Twenty-Five
32. Twenty-Six
33. Twenty-Seven
34. Twenty-Eight
35. Twenty-Nine
36. Thirty
37. Thirty-One
38. Thirty-Two
39. Thirty-Three
40. Thirty-Four
41. Thirty-Five
42. Thirty-Six
43. Thirty-Seven
44. Thirty-Eight
45. Thirty-Nine
46. Acknowledgements
47. A Note on the Author
48. Copyright Page
1. One
2. Two
3. Three
4. Four
5. Five
6. Six
7. Seven
8. Eight
9. Nine
10. Ten
11. Eleven
12. Twelve
13. Thirteen
14. Fourteen
15. Fifteen
16. Sixteen
17. Seventeen
18. Eighteen
19. Nineteen
20. Twenty
21. Twenty-One
22. Twenty-Two
23. Twenty-Three
24. Twenty-Four
25. Twenty-Five
26. Twenty-Six
27. Twenty-Seven
28. Twenty-Eight
29. Twenty-Nine
30. Thirty
31. Thirty-One
32. Thirty-Two
33. Thirty-Three
34. Thirty-Four
35. Thirty-Five
36. Thirty-Six
37. Thirty-Seven
38. Thirty-Eight
39. Thirty-Nine
40. _Acknowledgements_
41. _A Note on the Author_
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5. v
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1. Cover
2. Title Page
3. 1 Thursday, 14 October 1985, 0th week: morning
|
{
"redpajama_set_name": "RedPajamaBook"
}
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************************************
Adobe Flash MX and Flash MX 2004
************************************
.. topic:: Introduction
This |ActionScript (TM)| 1.0 example can be used with
the Flash MX or Flash MX 2004 authoring tool.
.. contents::
Installation
============
First you need to install the free Flash Remoting
components for
`Flash MX <http://www.adobe.com/products/flashremoting/downloads/components/#flr_flash>`_
or `Flash MX 2004 <http://www.adobe.com/products/flashremoting/downloads/components/#flr_as1>`_.
The files are placed in your Flash MX (2004) folder under Configuration/Include.
You can leave them there and Flash will find them.
Source
======
- `helloworld.fla <../../examples/general/helloworld/flash/as1/src/helloworld.fla>`_
- `helloworld.as <../../examples/general/helloworld/flash/as1/src/helloworld.as>`_
- `helloworld.swf <../../examples/general/helloworld/flash/as1/deploy/helloworld.swf>`_
.. |ActionScript (TM)| unicode:: ActionScript U+2122
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 513
|
\section{INTRODUCTION}
The global market for unmanned aerial vehicles (UAVs) remains in the development stage with a potential market of \$45 billion in the civil sector alone \cite{kovalev2019analysis}, showing the enormous economic potential of drone swarms. Teams of robots and in particular quadcopters are found to be useful in many applications such as search-and-rescue missions \cite{karaca2018potential}, emergency communication \cite{camara2014cavalry} or package delivery \cite{shakhatreh2019unmanned}, the reason for which lies at least partially in their potential for low-level distributed processing applications such as decentralized object tracking and detection. At the same time, the usage of drone swarms in the real world adds various additional coordination challenges \cite{chmaj2015distributed}. As UAVs in swarms will often operate in close vicinity of each other, an important challenge for successful autonomous application of drone swarms is decentralized collision avoidance.
The study of collision avoidance has a long history and begins with traditional path-planning and sensor-based methods. Existing collision avoidance algorithms based on path-planning typically require expensive planning as well as full state information, complicating their usage in large drone swarms \cite{mellinger2012mixed}. On the other hand, although decentralized sensor-based methods can achieve very good performance -- e.g. ORCA \cite{alonso2013optimal} -- the disadvantage of such methods lies in their rigidity. While such algorithms may provide provable safety for the assumed model, they often lead to deadlock situations. In this context, end-to-end reinforcement learning (RL) may be able to provide better performance both in terms of deadlock occurrence and speed \cite{long2018towards}.
\begin{figure}
\centering
\vspace{0.2cm}
\includegraphics[width=0.9\linewidth]{fig/fig1.pdf}
\caption{Illustration of our proposed architecture (BICARL). A: The agent's own observations (orange block) are concatenated with observations from the $k$ nearest neighbors (blue block, $k = 1$) and passed to a feed-forward neural network to obtain an immediate action distribution. B: The corresponding situation where agents need to negotiate their way toward their target while avoiding other dynamical agents. Each agent conditions their actions on observations of its $k$-nearest neighbors and their own state.}
\label{fig:hlvl}
\end{figure}
In this work, we propose a scalable, biologically well-motivated design to learning collision avoidance using end-to-end RL, i.e. {B}iologically-{I}nspired {C}ollision {A}voidance using {RL} (BICARL). In our method, we apply the biological insight into flocks of starlings (Sturnus vulgaris) interacting only with a limited number of their nearest neighbors to quadcopters in order to achieve decentralized collision avoidance that avoids crashes or getting stuck while learning successful policies in 30 minutes on two commodity desktop computers using an Intel Core i7-8700K CPU, 16 GB of RAM and no GPUs. This provides a scalable end-to-end learning approach to decentralized collision avoidance, reaching the performance of other state-of-the-art algorithms. Even though our motion model is more complex than models in prior works \cite{chen2017socially, everett2021collision}, we find that the added complexity allows improved collision avoidance performance and at the same time direct application to real world quadrotors by combining with conventional low-level controllers. As a result, we obtain a very practical deep RL approach to decentralized collision avoidance that remains scalable and applicable to arbitrary task specifications while requiring minimal information about the swarm, i.e. only from the nearest neighbor. Finally, we validate our algorithm both in simulation and real world application, verifying real world applicability.
\section{RELATED WORK}
Traditional methods for collision avoidance include planning approaches with full state information \cite{hamer2018fast}, or hand-engineered sensor-based approaches using velocity obstacles \cite{fiorini1998motion} and potential fields \cite{sigurd2003uav}, see also \cite{hoy2015algorithms} for a survey.
Though the field of planning remains an active area of research \cite{honig2018trajectory}, typical disadvantages of planning-based approaches are the requirement of full state information and high computational cost, barring scalable application in decentralized drone swarms. On the other hand, sensor-based methods may be applied in decentralized swarms, but have the drawback of potential deadlocks. Amongst the sensor-based methods, Optimal reciprocal collision avoidance (ORCA) is a method based on velocity obstacles \cite{alonso2013optimal}, while force-based motion planning (FMP) is a state-of-the-art method based on potential fields \cite{semnani2019forcebased}.
\paragraph{Learning-based approaches}
Recently, there have been many attempts to use machine learning for collision avoidance \cite{long2018towards, willemsen2021mambpo}, typically outperforming traditional state-of-the-art methods in terms of success rate and speed by learning a collision avoidance rule via reinforcement learning \cite{chen2017decentralized}, though many prior works focus on simplified dynamics models and consider application only to small swarms or single drones \cite{kahn2017uncertainty}.
The GA3C-CADRL algorithm \cite{everett2021collision} applies LSTMs in a single integrator model to learn a reaction to the position and velocities of all other agents in order to jointly avoid collisions. From the imitation learning realm, GLAS \cite{riviere2020glas} focuses on imitating decisions of a global privileged planner with safety module. \cite{semnani2020multi} extend GA3C-CADRL to 3D dynamics and fuses FMP \cite{semnani2019forcebased} with RL in a hybrid algorithm. Finally, \cite{wang2020two} applies imitation learning on ORCA for initialization, and then refines using RL. All of the aforementioned approaches except for \cite{chen2017socially} remain conditioned on information of all agent's (relative) positions and velocities. Similar to our work, \cite{chen2017socially} proposes RL-based collision avoidance via nearest neighbor information in the single integrator model, though their approach remains limited to very small swarms and in \cite{everett2021collision} was found to become stuck in large swarms.
\paragraph{Biological inspiration}
In the field of robot navigation, there exist a great number of works inspired by biology. To name a few, one can find navigation algorithms inspired by bugs \cite{mcguire2019comparative}, optical flow navigation inspired by honey bees \cite{green2008optic} or rule-based swarming models inspired by collective motion \cite{vicsek2012collective}, see also \cite{hoy2015algorithms} for a review.
In our work, we give a biological motivation for the nearest neighbor method. To be precise, we take inspiration from the behavior of flocks of starlings following a topological nearest-neighbor interaction rule in order to achieve robust swarm flight behavior \cite{young2013starling}. In prior works, this type of biological insight has inspired flocking control design \cite{liu2020leader} and real-world deployment of drone swarms \cite{petravcek2020bio}. Somewhat more related, \cite{zhu2020multi} implements end-to-end-learned flocking control based on classical swarming models. However, their focus remains on flocking, and their observation model is fully observed and therefore not decentralized.
\section{MODEL}
In this work we consider partially observable stochastic games (POSG) as the typical setting of multi-agent reinforcement learning. Formally, in our setting a POSG is a tuple $(I, X, U, Z, T, r, p, \gamma)$. The index set $I = \{1, \ldots, N\}$ is the set of agents, $X$ is the state space, $U = \bigotimes_{i \in I} U_i$ is the joint action space and $Z = \bigotimes_{i \in I} Z_i$ is the joint observation space, where $U_i$ and $Z_i$ denote the action and observation spaces of each agent respectively. In our work, we use a deterministic transition function $T \colon X \times U \to X$ and a random observation emission density $p \colon Z \times X \to \mathbb R_{\geq 0}$. The reward function $r \colon X \times U \to \mathbb R^N$ and the discount factor $\gamma \in (0,1)$ give rise to the maximization objective
\begin{align}
J_i(\boldsymbol \pi) = \mathbb E_{\boldsymbol \pi} \left[ \sum_{t=0}^\infty \gamma^t r_i(x_t, u_t) \mid x_0 \sim \mu_0 \right]
\end{align}
of agent $i$ with initial state distribution $\mu_0$ over joint policies $\boldsymbol \pi = (\pi^1, \ldots, \pi^N)$ with $\pi^i \colon U \times Z_i \to \mathbb R_{\geq 0}$ and
\begin{align}
z_t \sim p(\cdot \mid x_t), \quad u_t^i \sim \pi^i(\cdot \mid z_t^i), \quad x_{t+1} = T(x_t, u_t)
\end{align}
for $t \geq 0, i \in I$ and $u_t \equiv (u_t^1, \ldots, u_t^N), z_t \equiv (z_t^1, \ldots, z_t^N)$.
\subsection{Dynamics}
In our work, we will consider both a 2D and a 3D case. The perhaps simplest model studied in multi-agent collision avoidance is the 2D single integrator model used in most prior works such as \cite{alonso2013optimal, everett2021collision}, where the states $x_t \equiv (\mathbf p^i_t, \mathbf p^{i,*}_t)_{i \in I}$ consist of the $\mathbb R^2$-valued position $\mathbf p^i_t = (x^i_t, y^i_t)$ as well as goal position $\mathbf p^{i,*}_t = (x^{i,*}_t, y^{i,*}_t)$ of agents $i$. As actions $u_t^i \equiv \mathbf v^i_t$, each agent may choose their $\mathbb R^2$-valued velocity $\mathbf v^i_t$ directly as an action under the constraint $\lVert \mathbf v^i_t \rVert_2 \leq v_{\mathrm{max}}$, leading to the deterministic transition function defined by
\begin{align}
\mathbf p^i_{t+1} = \mathbf p^i_t + \Delta t \cdot \mathbf v^i_t
\end{align}
for time step size $\Delta t \geq 0$.
We propose a more complex model with some degree of momentum by using the following modified double integrator model, where the state $x_t \equiv (\mathbf p^i_t, \mathbf v^i_t, \theta^i_t, \omega^i_t, \mathbf p^{i,*}_t)_{i \in I}$ is given not only by the positions, but also by the $\mathbb R^2$-valued velocities $\mathbf v^i_t$ as well as the $\mathbb R$-valued yaw angle and its associated angular rate $\theta^i_t, \omega^i_t$.
Note that this model can alternatively be understood as part of our algorithm on top of the single integrator model, i.e. the algorithm keeps track of any additional states and simulates the modified double integrator dynamics to choose velocities at every time step. Therefore, results for this more complex dynamics model are nonetheless applicable and comparable to the single integrator case.
An action $u_t^i \equiv (\tilde {\mathbf v}^i_t, \tilde {\omega}^i_t)$ of agent $i$ with chosen target velocity $\tilde {\mathbf v}^i_t \in \mathbb R^2$ and target angular velocity $\tilde {\omega}^i_t \in \mathbb R$ leads to the deterministic update
\begin{align}
\mathbf p^i_{t+1} &= \mathbf p^i_t + \mathbf v^i_t \cdot \Delta t , \\
\mathbf v^i_{t+1} &= \mathbf v^i_t - c_v (\mathbf R(\theta^i_t) \cdot \tilde {\mathbf v}^i_t - \mathbf v^i_t) \cdot \Delta t , \\
\theta^i_{t+1} &= \theta^i_t + \omega^i_t \cdot \Delta t , \\
\omega^i_{t+1} &= \omega^i_t - c_\omega (\tilde {\omega}^i_t - \omega^i_t) \cdot \Delta t
\end{align}
with $\lVert \tilde {\mathbf v}^i_t \rVert_2 \leq v_{\mathrm{max}}$, $\left| \tilde {\omega}^i_t \right| \leq \omega_{\mathrm{max}}$ and yaw rotation matrix
\begin{align}
\mathbf R(\theta^i_t) =
\begin{bmatrix}
\cos \theta^i_t & -\sin \theta^i_t \\
\sin \theta^i_t & \cos \theta^i_t
\end{bmatrix} \, .
\end{align}
Importantly, although the yaw angle is not strictly required, we add the yaw angle to empirically obtain significantly better performance, as the added model richness allows agents to implicitly save information without requiring e.g. recurrent policy architectures. This model of medium complexity will at the same time allow us to directly use the desired velocity output as a set point of more traditional low-level quadrotor controllers such as PID controllers.
Note that depending on the specific task, we add task-specific transitions for the goal position, see the section on experiments. For the 3D case, we simply add another $z$-coordinate to position and velocity that remains unaffected by the rotation matrix. It should be further noted that we observe no discernible difference when applying some small amount of noise to the dynamics.
\subsection{Observation model}
We let the observations of an agent $i$ be randomly given by the own position and velocity as well as the relative bearing, distance and velocity of other agents inside of the sensor range $K > 0$ and the goal, i.e.
\begin{multline}
z_t^i \equiv (\hat{\mathbf p}^i_t, \hat{\mathbf v}^i_t, d^{i,*}_t, \phi^{i,*}_t, \\
\{ (d^{i,j}_t, \phi^{i,j}_t, \mathbf v^{i,j}_t) \mid j \neq i \colon \lVert \mathbf p^j_t - \mathbf p^i_t \rVert_2 \leq K \})
\end{multline}
where the observations are Gaussian distributed according to
\begin{align}
\hat{\mathbf p}^i_t &\sim \mathcal N(\mathbf p^i_t, \sigma_p^2), \quad
\hat{\mathbf v}^i_t \sim \mathcal N(\mathbf v^i_t, \sigma_v^2), \\
d^{i,*}_t &\sim \mathcal N(\lVert \mathbf p^{i,*}_t - \mathbf p^i_t \rVert_2, \sigma_d^2), \\
\phi^{i,*}_t &\sim \mathcal N(\varphi^{i,*}_t \sigma_\phi^2), \\
d^{i,j}_t &\sim \mathcal N(\lVert \mathbf p^j_t - \mathbf p^i_t \rVert_2, \sigma_d^2), \\
\phi^{i,j}_t &\sim \mathcal N(\varphi^{i,j}_t, \sigma_\phi^2), \\
\mathbf v^{i,j}_t &\sim \mathcal N(\mathbf v^{j}_t - \mathbf v^{i}_t, \sigma_v^2)
\end{align}
with noise standard deviations $\sigma_p, \sigma_v, \sigma_d, \sigma_\phi > 0$ and bearing angles $\varphi^{i,*}_t = \arctantwo(y^{i,*}_t - y^i_t, x^{i,*}_t - x^i_t)$, $\varphi^{i,j}_t = \arctantwo(y^j_t - y^i_t, x^j_t - x^i_t)$ where $\arctantwo(y,x)$ is defined as the angle between the positive $x$-axis and the ray from $0$ to $(x,y)$. Note further that the observations allow application of e.g. ORCA \cite{alonso2013optimal} and FMP \cite{semnani2019forcebased}. In the 3D case, we additionally observe the $z$-coordinate difference to the target and other agents with Gaussian noise of standard deviation $\sigma_p$.
\subsection{Reward function}
As the reward function for all of our experiments, for each agent $i \in I$ we define a term for reaching the goal position and for avoiding collisions each by
\begin{multline}
r_i(x_t, u_t) = c_p \langle \mathbf v^i_t, \mathbf p^{i,*}_t - \mathbf p^i_t \rangle \\
- c_c \sum_{j \in I \setminus \{ i \}} \mathbf 1 \left( \lVert \mathbf p^j_t - \mathbf p^i_t \rVert_2 \leq C \right)
\end{multline}
with desired target position $\mathbf p^{i,*}_t \in \mathbb R^2$, avoidance radius $C \geq 0$ and reward / collision penalty coefficients $c_p, c_c \geq 0$, where $\langle \cdot, \cdot \rangle$ denotes the standard dot product.
Note that this objective is selfish, i.e. agents are only interested in their own collisions and reaching their own goal. This manual choice of reward function alleviates the well-known multi-agent credit assignment problem in multi-agent RL \cite{hernandez2019survey}, as shared, cooperative swarm reward functions are difficult to learn due to the increasing noise from other agent's behaviors as the number of agents rises.
\section{METHODOLOGY}
In this work, we propose a biologically-inspired design principle to learning collision avoidance algorithms. Recently, it was found that swarms of starlings observe only up to seven neighboring birds to obtain robust swarm flight behavior \cite{young2013starling}. This implies that it should be sufficient to use a similar observation reduction for collision avoidance. Further, it is well-known that the multi-agent RL domain suffers from the combinatorial nature of multi-agent problems \cite{hernandez2019survey}. Hence, the reduction of observations to e.g. the closest $k$ agents can greatly help learning effective behavior.
It is known that tractable exact solution methods and theoretical guarantees in the context of POSGs are sparse even in the fully cooperative case \cite{zhang2021multi}. Instead, we apply independent learning, i.e. we ignore the non-stationarity of other agents and solve assumed separate single-agent problems via RL \cite{tan1993multi}. Furthermore, we share a policy between all agents via parameter sharing \cite{gupta2017cooperative} and use the PPO algorithm to learn a single, shared policy \cite{schulman2017proximal}. For faster learning, we collect rollout trajectories from our simulation in parallel on two machines and use the RLlib \cite{liang2018rllib} implementation of PPO. We implemented our environment in Unity ML-Agents \cite{juliani2018unity}.
\begin{table}
\centering
\caption{Hyperparameters and parameters used in all experiments.}
\label{table:hyperparameters}
\begin{tabular}{@{}ccc@{}}
\toprule
Symbol & Function & Value \\ \midrule
$\Delta t$ & Time step size & \SI{0.02}{\second} \\
$c_v$ & Velocity coefficient & \SI{1}{\per \second} \\
$c_w$ & Angular rate coefficient & \SI{1}{\per \second} \\
$\sigma_p, \sigma_v$ & Noise standard deviations & \SI{1}{\milli\metre}, \SI{10}{\milli\metre}\\
$\sigma_d, \sigma_\phi$ & Noise standard deviations & \SI{1}{\milli\metre}, \SI{0.1}{\milli\metre}\\
$C$ & Avoidance radius & \SI{7}{\metre} \\
$c_p$ & Reward coefficient & \SI{0.3}{\second \per \square \metre} \\
$c_c$ & Penalty coefficient & $1$ \\
$v_{\mathrm{max}}$ & Maximum velocity & \SI{30}{\metre \per \second} \\
$\omega_{\mathrm{max}}$ & Maximum angular velocity & \SI{15}{\per \second} \\
$K$ & Sensor range & \SI{10}{\metre} \\
$k$ & Number of considered neighbors & 1 \\
$\gamma$ & Discount factor & $0.99$\\
\\ \midrule
& PPO & \\ \midrule
$l_{r}$ & Learning rate & \num{5e-5}\\
$\beta$ & KL coefficient & $0.2$ \\
$\epsilon$ & Clip parameter & $0.3$ \\
$B$ & Training batch size & $50000$ \\
$B_{m}$ & SGD Mini batch size & $2500$ \\
$M$ & SGD iterations & $20$ \\ \bottomrule
\end{tabular}
\end{table}
We parametrize our policy by a simple feedforward network consisting of two hidden layers with 64 nodes, and ReLU activations except for the output layer which outputs parameters of a diagonal Gaussian distribution over actions. Actions are clipped after sampling to fulfill constraints. We preprocess observations by reducing to the information of the nearest $k$ neighbors in $L_2$ norm on the positions $\mathbf p^i_t$. All results in this work are for $k=1$, i.e. we use observations $(d^{i,j}_t, \phi^{i,j}_t, \mathbf v^{i,j}_t)$ of the nearest neighbor $j$. Crucially, this observation reduction works well in our experiments, and it allows us to show that information of neighboring agents limited to the nearest neighbor information is indeed sufficient for collision avoidance. A resulting advantage of our design is that the resulting policy itself is very simple and memoryless.
Note that our policy architecture is very small and thus allows sufficiently fast computation on low-powered microcontrollers and UAVs such as the Crazyflie \cite{giernacki2017crazyflie} used in our real world experiments. In comparison, previous works use Deep Sets \cite{zaheer2017deep} or LSTMs \cite{everett2021collision} to process a variable number of other agents, which will scale worse in large swarms and is more computationally costly to learn and apply.
The scenarios considered in this work include formation change and package delivery. In the formation change scenario, all drones start on a circle and have their goal set to the opposite side. In the package delivery scenario, drones are continuously assigned new random package locations and destinations after picking and dropping off packages, where the locations and destinations are uniformly distributed e.g. on a line, circle or in a rectangle.
\section{EXPERIMENTS AND RESULTS}
\begin{table}
\centering
\caption{Performance of BICARL (ours), ORCA \cite{alonso2013optimal}, and FMP \cite{semnani2019forcebased} in the package delivery task, averaged over 4 runs of 100 seconds.}
\label{table:circle-comparison}
\begin{tabular}{@{}ccccccc@{}}
\toprule
Test setup & \multicolumn{3}{c}{Average collected packages} \\ \midrule
\# agents & BICARL & ORCA & FMP \\ \midrule
4 & \textbf{15.25} & 1.5 & 3 \\
6 & \textbf{8.66} & 1.16 & 2 \\
8 & \textbf{7.62} & 1.25 & 2.12 \\
10 & \textbf{5.8} & 0 & 1.2 \\
12 & \textbf{4.41} & 0 & 0.41 \\
14 & \textbf{4.14} & 0 & 0.25 \\ \bottomrule
\end{tabular}
\end{table}
\begin{figure}
\centering
\includegraphics[width=0.55\linewidth]{fig/stuck_fmp.pdf}
\caption{A constellation where FMP is stuck in package delivery: At this number of agents, the computed forces nullify each other, resulting in freezing agents. The red squares visualize the current goal of agents and the blue lines signify which agent is assigned which goal.}
\label{fig:fmp_stuck_circle}
\end{figure}
\begin{table*}
\centering
\caption{Performance analysis of BICARL (ours), ORCA \cite{alonso2013optimal}, and FMP \cite{semnani2019forcebased} in circle formation change, averaged over 5 runs.}
\label{table:comparison}
\renewcommand{\arraystretch}{1.21}
\begin{tabular}{@{}cllccccccccc@{}}
\toprule
\multicolumn{3}{c}{Test setup} &
\multicolumn{3}{c}{Extra time to goal (\si{\second})} &
\multicolumn{3}{c}{Minimal distance to other agents (\si{\metre})} &
\multicolumn{3}{c}{Extra travelled distance (\si{\metre})} \\ \midrule
\multicolumn{3}{c}{\# agents} &
BICARL &
ORCA &
FMP &
BICARL &
ORCA &
FMP &
BICARL &
ORCA &
FMP \\ \midrule
\multicolumn{3}{c}{5} & 4.02 & 14.52 & \textbf{1.86} & \textbf{12.38} & 4.94 & 6.18 & 1.28 & 1.17 & \textbf{1.11} \\
\multicolumn{3}{c}{10} & 4.51 & 15.31 & \textbf{1.95} & \textbf{7.01} & 4.78 & 3.78 & 1.26 & 1.65 & \textbf{1.15} \\
\multicolumn{3}{c}{15} & 4.87 & 14.98 & \textbf{2.15} & \textbf{6.14} & 3.42 & 4.44 & 1.27 & 1.82 & \textbf{1.19} \\
\multicolumn{3}{c}{20} & 6.51 & 18.68 & \textbf{2.34} & 3.14 & \textbf{4.74} & 3.87 & 1.45 & 2.7 & \textbf{1.26} \\
\multicolumn{3}{c}{25} & 7.52 & 20.13 & \textbf{2.28} & 3.3 & \textbf{4.7} & 3.34 & 1,60 & 3.56 & \textbf{1.23}\\
\multicolumn{3}{c}{30} & 7.42 & 31.51 & \textbf{3.66} & 3.52 & \textbf{4.7} & 2.94 & 1,61 & 4.68 & \textbf{1.38} \\
\multicolumn{3}{c}{35} & \textbf{7.81} & 41.15 & N/A & 2.52 & \textbf{4.72} & N/A & \textbf{1.60} & 5.71 & N/A \\
\multicolumn{3}{c}{40} & \textbf{9.18} & 45.64 & N/A & 2.35 & \textbf{2.75} & N/A & \textbf{1.79} & 10.17 & N/A \\
\multicolumn{3}{c}{45} & \textbf{8.91} & 76.25 & N/A & 1.94 & \textbf{3.21} & N/A & \textbf{1.76} & 8.14 & N/A \\
\multicolumn{3}{c}{50} & \textbf{10.1} & 81.03 & N/A & 1.67 & \textbf{2.66} & N/A & \textbf{1.87} & 17.95 & N/A \\ \bottomrule
\end{tabular}
\end{table*}
\begin{figure}
\centering
\includegraphics[width=0.95\linewidth]{fig/traj_drones_81012.png}
\caption{Simulated trajectories of agents for $1$-nearest neighbor BICARL with 2D dynamics. Left: 8 agents; Middle: 10 agents; Right: 12 agents.}
\label{fig:traj81012}
\end{figure}
For all of our experiments, we use the parameters and hyperparameters indicated in Table~\ref{table:hyperparameters}. We apply curriculum learning \cite{bengio2009curriculum} to improve learning performance, increasing the number of agents continuously while training. The single resulting policy is then evaluated for a varying number of agents to gauge generality and scalability. We consider a collision to occur when agents are closer than $\SI{1.5}{\metre}$ and accordingly tuned the penalty radius during training to $C=\SI{7}{\metre}$ to achieve reasonable behavior for up to $50$ agents.
\subsection{Simulation Results}
We have trained on two commodity desktop computers equipped with an Intel Core i7-8700K CPU, 16 GB RAM and no GPUs. We successfully and consistently learned good policies in 30 minutes upon reaching e.g. $1.4 \cdot 10^{6}$ time steps. The reason for the fast training lies in the parallelization of collecting experience in Unity, as we can run multiple simulations at the same time to collect data, and the simulation in Unity is fast. We first compare our results to the algorithms FMP \cite{semnani2019forcebased} and ORCA \cite{alonso2013optimal}. As mentioned earlier, for comparison we may also consider our dynamics as part of the algorithm for simpler single integrator dynamics.
\paragraph{Learned collision avoidance behavior}
In the package delivery scenario, we place the agents on a circle of fixed radius and gradually increase the number of agents. As soon as agents become closer than $\SI{3.5}{\metre}$ to the goal, we sample a new goal on the circle. In Table~\ref{table:circle-comparison}, it can be observed that the average packages collected per drone decrease as the number of drones increases, as the drones will increasingly be in each other's way. Further, for the package delivery task, FMP and ORCA eventually run into a deadlock. An example for FMP can be seen in Fig.~\ref{fig:fmp_stuck_circle}. In this case, our methodology provides robust, collision-free behavior that avoids getting stuck.
In the formation change scenario, during training we start with 4 diametrically opposed agents on a circle of radius $\SI{70}{\metre}$ and gradually increase the size of the swarm up to 40. It can be seen in Fig.~\ref{fig:traj81012} that rotation around a fixed direction emerges. Increasing the number of agents leads to situations where ORCA and FMP get stuck. \cite{trautman2010unfreezing} demonstrates that the solution is cooperative collision avoidance. In line with this finding, our learned policy is able to capture such cooperation, i.e. one drone gives way to another as can be seen in Fig.~\ref{fig:same_target} and supplementary videos.
Overall, from Table~\ref{table:comparison} we see that our solution is competitive, especially for many agents. Note that extra time to goal and travelled distance are measured relative to the baseline where drones fly in straight lines to the goal. Although FMP achieves very good results for small numbers of agents, at more than 35 agents FMP becomes stuck, while our method learns a robust behavior that works for large numbers of agents. Here, we tuned FMP parameters to obtain reasonably smooth flight. While improved results in FMP could be possible, additional parameter tuning would be required.
\begin{figure}
\centering
\includegraphics[width=0.95\linewidth]{fig/2A-Package-Collection.png}
\caption{Simulated flight behavior of two agents in the package delivery scenario with overlapping goals $(1^*, 2^*)$. In our approach, one drone yields and the other drone collects, obtaining a new goal. In FMP, the drones become stuck, while in BICARL one drone yields for the other drone. The red squares and blue lines have the same meaning as in Fig.~\ref{fig:fmp_stuck_circle}. Time progresses from left to right.}
\label{fig:same_target}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.95\linewidth]{fig/dis_drones_101520.png}
\caption{Simulated minimum inter-agent distances achieved in the circle formation change scenario. The red line indicates the radius considered as collision. Left: 10 agents; Middle: 15 agents; Right: 20 agents.}
\label{fig:dis_101520}
\end{figure}
In Fig.~\ref{fig:dis_101520}, we find that our method successfully avoids collisions with other agents while reaching the goals. During training, collisions may be caused by nearby other agents regardless of the own behavior. As a result, this often provides a negative feedback signal even if the drone itself is not responsible for the collision, resulting in behavior where agents avoid other agents even when the punishment for violating the avoidance radius (here $C=7$) is far off.
Analogously to the 2D case, we can consider formation change in the 3D case. In Fig.~\ref{fig:traj3D_81012} the trajectories of our learned policy are depicted. It can be seen that our methodology remains successful in guiding the agents to their goal destination. Furthermore, to make use of the additional space above, agents begin flying above and below each other.
\begin{figure}
\centering
\includegraphics[width=0.95\linewidth]{fig/traj_drones_3D81012.png}
\caption{Simulated trajectories of agents for $1$-nearest neighbor BICARL with 3D dynamics. Agents have free space above their starting position and use it to navigate past each other. Left: 8 agents; Middle: 10 agents; Right: 12 agents.}
\label{fig:traj3D_81012}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.95\linewidth]{fig/benchmark_LSTM_std.png}
\caption{Learning curve: Average achieved sum of rewards per episode over total time steps, plus variance as shaded region for three random seeds. It can be seen that our model learns significantly faster. Left: 2 agents; Middle: 4 agents; Right: 6 agents.}
\label{fig:lstm_bicarl}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{fig/3D-6A.png}
\caption{Real world circle formation exchange with the policy for the 3D case embedded on a swarm of 6 Crazyflie nano-quadcopters. The drones successfully switch their positions to the antipodal points on a circle by engaging in 3D collision avoidance. Time progresses from left to right.}
\label{fig:real}
\end{figure*}
\paragraph{Comparison to LSTM-based policies}
We compare our observation model with the LSTM variant proposed in \cite{everett2021collision}. We use an LSTM observation filter in conjunction with the two-layer architecture and run several tests with different parameters (e.g. number of hidden units) for 2, 4 and 6 agents, comparing the average achieved reward in each case.
The learning curves in Fig.~\ref{fig:lstm_bicarl} suggest that our observation model achieves comparable results and faster learning. As expected, training LSTMs is slow and our method speeds up learning greatly, especially for many agents. While this does not exclude that in general and for improved hyperparameter configuration, the LSTM observation model will be superior, our method regardless has less parameters to fine-tune, has a constant computation cost regardless of the number of agents and hence is cheaper to both train and apply in the real world.
Although in \cite{everett2021collision} it has been noted that learned collision avoidance algorithms such as \cite{chen2017socially} based on neighborhood information tend to get stuck in large swarms or cause collisions, we obtain a seemingly opposed result. The reason is that our double integrator model together with the added yaw angle allows agents to avoid getting stuck by implicitly saving information or communicating with each other via the yaw angle. Since the problem at hand is multi-agent, a general optimal solution should act depending on the whole history of observations and actions. If two drones are stuck, they can therefore remember past information and change their reaction, leading to better results. Indeed, in our experiments we found that the yaw angle is crucial to obtaining good performance, leading us to the conclusion that a sufficiently rich motion model may in fact ease the learning of good collision avoidance behavior.
\subsection{Real World Experiments}
We validate the real-world applicability of our policies by performing experiments on swarms of micro UAVs. We downscale lengths and speeds by $10$ (since the used indoor drones are only $\SI{0.1}{\metre}$ in length). We directly apply the desired velocity as a set point for the low-level controller instead of the acceleration as in conventional double integrator models, which is not easily regulated due to the non-linearity of typical quadrotor dynamics.
\paragraph{Hardware Setup}
For our real-world experiments, we use a fleet of Crazyflie nano-quadcopters \cite{giernacki2017crazyflie} with indoor positioning via the Lighthouse infrared positioning system and extended Kalman filter \cite{mueller2015fusing} with covariance correction \cite{mueller2017covariance}. Since the Crazyflies cannot sense each other, we simulate the inter-drone observations by exchanging information via TDMA-based peer-to-peer radio communication to broadcast the estimated position and velocity of each drone. Although the sensor range $K$ in the simulation can model a limited range of wireless communication, in our real indoor experiments there is no such limitation. Instead, drones ignore all information other than their supposed observations. As low-level controller, we use the default cascaded PID controller of the Crazyflie, setting desired velocities and yaw rates from the output of our policy directly as set point of the controller. Note that our code runs completely on-board, except for starting and stopping experiments.
\paragraph{Evaluation}
We find that the policies obtained in simulation work as expected for two to six drones in reality, and expect similar results for more real drones. In Fig.~\ref{fig:real}, it can be seen that the drones successfully complete formation changes. Using our modified double integrator model, we easily achieve stable results in real world tests, as the cascaded PID low-level-controller is able to achieve desired velocities while stabilizing the system. Due to the simulation-to-reality gap stemming from model simplifications, we can observe slight control artefacts (oscillations). Nonetheless, the combination of traditional controller with learned policy successfully realizes the desired collision avoidance behavior, and the on-board inference time of the policy remains below $\SI{1}{\milli \second}$, with $\SI{30}{\micro \second}$ on our PCs. As a result, by using our motion model of medium complexity, we obtain a practical approach to decentralized quadrotor collision avoidance.
\section{CONCLUSION}
To summarize, in this work we have demonstrated that information of $k$-nearest neighbors, non-recurrent policies and a sufficiently rich motion model are enough to find robust collision avoidance behavior. We have designed a high-level model that allows application of the learned policy directly to the real world in conjunction with e.g. a standard cascaded PID controller. In the large swarm case and for collecting packages, our scalable and decentralized methodology appears to show competitive performance. Furthermore, our method avoids getting stuck and is very fast to learn.
Interesting future work could be to consider external static or dynamic obstacles such as walls. One could combine our approach for $k>1$ with Deep Sets \cite{zaheer2017deep}, mean embeddings \cite{huttenrauch2019deep} or attention mechanisms \cite{manchin2019reinforcement} to further improve learning behavior. Finally, it may be of interest to further reduce the simulation-to-reality gap by using a more detailed model that models e.g. a 12-state quadrotor dynamics with simulated PID controller, motor lag, detailed drag models etc.
\bibliographystyle{IEEEtran}
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"redpajama_set_name": "RedPajamaArXiv"
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Q: How to get string after the last '/' character using a regular expression I would like to fetch last part of the string after character / which in our example is 'PAYLINK STALE CHECK ENTRI'.
Also, please note that, 'PAYLINK STALE CHECK ENTRI' is not a static string.
Example:
:61:1511171116CR00,10NMSC566666666/15139333333333333/CTC/MSC/PAYLINK STALE CHECK ENTRI
The output should be
PAYLINK STALE CHECK ENTRI
A: You don't need a regex to do this, just use lastIndexOf to find the last / and substr to get the substring after it.
var str = '103150800130001/CTC/MSC/PAYLINK STALE CHECK ENTRI';
console.log(str.substr(str.lastIndexOf('/')+1));
However, if you prefer a regex-based solution, this would work to strip off everything from the last / and before:
var str = '103150800130001/CTC/MSC/PAYLINK STALE CHECK ENTRI';
console.log(str.replace(/.*\//, ''));
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C2 - cubed
Karl Middlebrooks Rhythmic Noise Once again, I find myself reviewing an older release that, upon having listened to it a few times, I wonder how I missed it in the first place. Back in 2004, C2.control.org (a side project of control.org) released 'Cubed' on CD. I missed it then, but it came back as a digital release in 2011 on control.org's own record label, Control Freak Records.
I had to listen to this one a couple of times to get a good feel for it, because sometimes it's too early in the morning for me to listen to anything other than the pounding inside my own skull. The second time was the charm, though. On "cubed", C2 delivers an entire album full of driving, distorted beats and grooves. There's very little here in the way of samples, too; they're used pretty sparingly throughout the album, which I find to be a nice touch. It's raw, sparse distorted dance music that brings to mind acts like Winterkalte. Many of the tracks, like 'cubed' and 'c2', start off slow and take a bit to ramp up to full power, but once they do, they're good dancefloor stompers. 'Rockstar', is an out of control, noisy, pounder of a track that's one of the strongest on the album, and a new favorite of mine overall.
Not all the tracks are as compelling. A handful of tracks, like 'access' and 'poly', seem like they never quite go anywhere. In these moments, everything feels like it's bogged down a bit. I found myself wondering now and again when these tracks would end, so I could get back to the catchy tracks C2 have shown themselves capable of creating here and on their other releases. But despite the occasional miss, C2 bring several tracks that will end up in heavy rotation around here. April 1, 2012 http://www.brutalresonance.com/review/c2-cubed/ 4
Brutal Resonance
ReviewsInterviewsFeaturesNewsMovies & TVVideos
Released 2011 by Control Freak Records
Rhythmic Noise
Once again, I find myself reviewing an older release that, upon having listened to it a few times, I wonder how I missed it in the first place. Back in 2004, C2.control.org (a side project of control.org) released 'Cubed' on CD. I missed it then, but it came back as a digital release in 2011 on control.org's own record label, Control Freak Records.
Not all the tracks are as compelling. A handful of tracks, like 'access' and 'poly', seem like they never quite go anywhere. In these moments, everything feels like it's bogged down a bit. I found myself wondering now and again when these tracks would end, so I could get back to the catchy tracks C2 have shown themselves capable of creating here and on their other releases. But despite the occasional miss, C2 bring several tracks that will end up in heavy rotation around here.
Mentallo & The Fixer - 'A Collection'
I Just Got Insane - 'In Deiner Welt'
Karl Middlebrooks
info@brutalresonance.com
Writer and contributor on Brutal Resonance
Control.org - 'Degenerate'
Review, Apr 27 2011
C2 - 'QUADRANTS: the stories of four'
Review, May 08 2014
Embrionyc - 'Another Sleepless Night'
Review, Jul 20 2012
Arte Disfuncional - 'A History of Violence Sequence'
Review, Jan 24 2014
E.S.A.
Interview, Apr 02 2018
Shortly about us
Started in spring 2009, Brutal Resonance quickly grew from a Swedish based netzine into an established International zine of the highest standard.
We cover genres like Synthpop, EBM, Industrial, Dark Ambient, Neofolk, Darkwave, Noise and all their sub- and similar genres.
© Brutal Resonance 2009-2016
Designed by and developed by Head of Mímir 2016
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| null | null |
Q: Crash when running a command Code
MainClass: http://pastebin.com/U99LAQ6z
ListenerClass: http://pastebin.com/ZF5i8mw1
Explanation
I was working on my plugin and I am currently working on the portion that creates Lobbies. Basically what I have it do is when the player left-clicks with the position wand they set point 1 and when they right click they set point 2. What I have is a block selector that selects the first block then it selects all the blocks on the Y and all the blocks on Z and then on X till they equal the point2. It logs each of these blocks into a HashMap so I can call the blocks later when needed. My Problem is I think I have an infinite for loop or something like that because the console gives me an error. Sometimes it also gives me a longer stacktrace and im just confused about this.
So what I am trying to do is create a simple base that I am going to use for all of my future plugins that require areas defined. What this is suppose to do is when a player runs the command positionwand it gives them a tool that lets them select points. Left Clicking selects point1 and Right Clicking selects point2. I use getPoint1 and getPoint2 to get them because i store the points within a HashMap. Once the player has selected the points they run the command /lobbycreate (lobbyname) and when they run the command it is suppose get and store all of the blocks within the given points. To do this I use a series of for loops. To start off with I have the variable blockSelector. I want blockSelector to start off from getPoint1 and go to getPoint2 selecting all of the blocks in between the 2 points. Then we go to my first for Loop which while blockSelector.getBlockX() != getPoint2X, getPoint2X being a variable I used to get the getPoint2's X coordinate. The point of this is so all this loop and all of the loops within this loop will continue until both blockSelector and getPoint2X have the same value. Within this loop I have another loop that does this same thing except for with the Z-coordinate thus it selects all of the blocks on the Z coordinate but only after it does the loop within it that selects all of the blocks on the Y-coordinate. After all that goes through I add or subtract 1 depending on what replaceVar is and repeat the cycle till all 3 coordinates of blockSelector are equal to getPoint2 thus saving all of the blocks in the process. The reason for me having all of those if statements and the replaceVar is because When selecting coordinates players can select negative and postive coordinates and the getPoint1 could be greater or less than getPoint2 so I have to add or subtract depending on the 2 values.
Console output
jobisingh issued server command: /lobbycreate gulp
[09:46:50 ERROR]: The server has stopped responding!
[09:46:50 ERROR]: Please report this to http://www.spigotmc.org/
[09:46:50 ERROR]: Be sure to include ALL relevant console errors and Minecraft crash reports
[09:46:50 ERROR]: Spigot version: git-Spigot-b2c2c63-a3cb1bc (MC: 1.8.7)
[09:46:50 ERROR]: ------------------------------
[09:46:50 ERROR]: Server thread dump (Look for plugins here before reporting to Spigot!):
[09:46:50 ERROR]: ------------------------------
[09:46:50 ERROR]: Current Thread: Server thread
[09:46:50 ERROR]: PID: 17 | Suspended: false | Native: false | State: RUNNABLE
[09:46:50 ERROR]: Stack:
[09:46:50 ERROR]: java.util.HashMap.put(Unknown Source)
[09:46:50 ERROR]: me.jobisingh.MainClass.onCommand(MainClass.java:223)
[09:46:50 ERROR]: org.bukkit.command.PluginCommand.execute(PluginCommand.java:44)
[09:46:50 ERROR]: org.bukkit.command.SimpleCommandMap.dispatch(SimpleCommandMap.java:141)
[09:46:50 ERROR]: org.bukkit.craftbukkit.v1_8_R3.CraftServer.dispatchCommand(CraftServer.java:640)
[09:46:50 ERROR]: net.minecraft.server.v1_8_R3.PlayerConnection.handleCommand(PlayerConnection.java:1149)
[09:46:50 ERROR]: net.minecraft.server.v1_8_R3.PlayerConnection.a(PlayerConnection.java:984)
[09:46:50 ERROR]: net.minecraft.server.v1_8_R3.PacketPlayInChat.a(PacketPlayInChat.java:45)
[09:46:50 ERROR]: net.minecraft.server.v1_8_R3.PacketPlayInChat.a(PacketPlayInChat.java:1)
[09:46:50 ERROR]: net.minecraft.server.v1_8_R3.PlayerConnectionUtils$1.run(SourceFile:13)
[09:46:50 ERROR]: java.util.concurrent.Executors$RunnableAdapter.call(Unknown Source)
[09:46:50 ERROR]: java.util.concurrent.FutureTask.run(Unknown Source)
[09:46:50 ERROR]: net.minecraft.server.v1_8_R3.SystemUtils.a(SystemUtils.java:19)
[09:46:50 ERROR]: net.minecraft.server.v1_8_R3.MinecraftServer.B(MinecraftServer.java:714)
[09:46:50 ERROR]: net.minecraft.server.v1_8_R3.DedicatedServer.B(DedicatedServer.java:374)
[09:46:50 ERROR]: net.minecraft.server.v1_8_R3.MinecraftServer.A(MinecraftServer.java:653)
[09:46:50 ERROR]: net.minecraft.server.v1_8_R3.MinecraftServer.run(MinecraftServer.java:556)
[09:46:50 ERROR]: java.lang.Thread.run(Unknown Source)
[09:46:50 ERROR]: ------------------------------
[09:46:50 ERROR]: Entire Thread Dump:
[09:46:50 ERROR]: ------------------------------
[09:46:50 ERROR]: Current Thread: Chunk I/O Executor Thread-1
[09:46:50 ERROR]: PID: 38 | Suspended: false | Native: false | State: WAITING
[09:46:50 ERROR]: Stack:
> Press any key to continue . . .
A: I'm pretty sure these three loops are running indefinitely:
for(; blockSelector.getBlockX() != getPoint2X; blockSelector.setX(replaceVarX))
{
for(; blockSelector.getBlockZ() != getPoint2Z; blockSelector.setZ(replaceVarZ) )
{
for(; blockSelector.getBlockY() != getPoint2Y; blockSelector.setY(replaceVarY))
{
lobbies.put(blockSelector.getBlockY(), blockSelector);
}
lobbies.put(blockSelector.getBlockZ(), blockSelector);
}
lobbies.put(blockSelector.getBlockZ(), blockSelector);
}
I'm using X as example, but it is exactly the same for Y and Z.
Your loop runs as long as replaceVarX != getPoint2X.
However, replaceVarX never changes inside the loop, so there are two possibilities:
*
*replaceVarX == getPoint2X
The loop runs once.
*replaceVarX != getPoint2X
The loop runs indefinitely.
However, you make sure that the first one is never the case:
if(getPoint1X > 0 && getPoint2X > 0 && getPoint1X < getPoint2X)
{
replaceVarX = blockSelector.getBlockX() + 1;
}
else if(getPoint1X > 0 && getPoint2X > 0 && getPoint1X > getPoint2X)
{
replaceVarX = blockSelector.getBlockX() - 1;
}
else if (getPoint1X < 0 && getPoint2X < 0 && getPoint1X < getPoint2X)
{
replaceVarX = blockSelector.getBlockX() + 1;
}
else if (getPoint1X < 0 && getPoint2X < 0 && getPoint1X > getPoint2X)
{
replaceVarX = blockSelector.getBlockX() - 1;
}
else if (getPoint1X > 0 && getPoint2X < 0)
{
replaceVarX = blockSelector.getBlockX() - 1;
}
else if (getPoint1X < 0 && getPoint2X > 0)
{
replaceVarX = blockSelector.getBlockX() + 1;
}
else if (getPoint1Z < 0 && getPoint2Z < 0 && getPoint1Z < getPoint2Z)
{
replaceVarX = blockSelector.getBlockX() + 1;
}
To fix this, you'd have to move all those if / else inside the loops, so that the variables get updated accordingly.
You'd have to split them up over the loops though, so that replaceVarX gets updated in the outer loop, replaceVarZ in the middle one, and replaceVarY in the inner one, otherwise you'd break the outer two loops.
A (probably more efficient) alternative would be to replace replaceVarX with a variable outside the loop, which defines the direction (-1 or +1) for x, y and z each (I'll call them dx, dy and dz), and then to replace blockSelector.setX(replaceVarX) by blockSelector.setX(blockSelector.getBlockX() + dx).
This way you could avoid a lot of redundant checks (and thus computation time) inside those loops.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,487
|
Roulette is a game of numbers and chance. You can sit around studying the odds in roulette and the outcomes of trying to predict where that little ball will next end up but unfortunately you will be wasting a great deal of your time. Roulette is a game to be enjoyed. Sometimes you'll win and sometimes you'll lose. It's a game that has been played for centuries and no-one yet has found a way to truly beat the game. The game has been used to help develop statistical theories and processes such as Karl Pearson´s Goodness of Fit test. Could maths return the favour and help crack the code?
Countless people through the ages have tried to crack the roulette code using mathematics. Some of the more famous roulette systems are beased on mathematical sequences, strategies like the Fibonacci System, for example.
Mathematics seems the most obvious way in which you can attempt to beat the dealer but predicting any result is simply a matter of probability. The entire game is based upon independent events, which therefore means that any event that occurs when the wheel spins and the ball lands has no affect or influence on any previous or future spin. You should also check out our page that covers physics in roulette, by the way.
In European Roulette the payout on hitting any one of the 36 red or black numbers on the wheel is 35 to 1. It's the highest payout you can achieve with a single bet and so of course means that the probability is the lowest (at just 2.7% in fact). For an averagely better chance of winning many players will simply place an even-money bet on something like red or black, odds or evens or any one of the numbered groups. The probability is a massive 48.6% but the payout is just 1:1. But if the bet is pretty much a 50-50 chance then why is it only 48.6% and not 50%? Well, don't forget that the European table also includes the green zero pocket so this does mean that the odds will be slightly lower than 50/50. In fact, this little zero pocket is what gives the casino its roulette house edge, when combined with the payout structure.
British engineer Joseph Jagger did manage to use mathematics and predictability when he managed to win a significant sum of money on his visit to Monte Carlo in 1875. Jagger has hired 6 clerks to visit the casino Beaux-Arts Casino in Monaco to record all the outcomes of 6 different roulette wheels over a period of time. Using these findings he was able to discover a pattern of a group of numbers that kept coming in more frequently than others on just one of these wheels. He ended up placing his bets on this particular wheel using these numbers in question and was able to accumulate hundreds and thousands of pounds of winnings.
However, Jagger hadn't uncovered an incredible flaw in the game itself. He had simply exposed the fact a roulette wheel in the world's biggest gambling capital merely had a mechanical imbalance which gave the wheel a clear bias towards certain outcomes. And the problem with that discovery, was that the casino discovered it as well. And they fixed the wheel!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,981
|
package zmaster587.advancedRocketry.integration.jei.blastFurnace;
import mezz.jei.api.IJeiHelpers;
import zmaster587.libVulpes.interfaces.IRecipe;
import zmaster587.libVulpes.recipe.RecipesMachine;
import java.util.LinkedList;
import java.util.List;
public class BlastFurnaceRecipeMaker {
public static List<BlastFurnaceWrapper> getMachineRecipes(IJeiHelpers helpers, Class clazz) {
List<BlastFurnaceWrapper> list = new LinkedList<BlastFurnaceWrapper>();
for(IRecipe rec : RecipesMachine.getInstance().getRecipes(clazz)) {
list.add(new BlastFurnaceWrapper(rec));
}
return list;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,852
|
\section{Introduction}
The most familiar type of spontaneous symmetry breaking in quantum spin systems is magnetic long-range order. A paramount example is the N\'eel order in the ground state of the antiferromagnetic Heisenberg model on bipartite lattices \cite{tasaki2020physics}. While magnetic order breaks spin-rotation as well as time-reversal symmetry, chiral spin states (CSSs) \cite{wen1989chiral} epitomize the possibility of breaking the latter while keeping a vanishing expectation value for the spin operator. For SU(2)-invariant systems, the scalar spin chirality $\langle \mathbf S_i\cdot (\mathbf S_j\times \mathbf S_k)\rangle $ for three lattice sites defines an order parameter for CSSs. This large class of states includes, in particular, topologically nontrivial chiral spin liquids with anyonic excitations \cite{wen1989chiral,Kalmeyer1987,Laughlin1990,Greiter2009,Gong2014,Gong2015,Wietek2015,Cookmeyer2021}. For anisotropic exchange interactions, the generalizations of the scalar spin chirality are three-spin operators which remain invariant under discrete spin rotations. For instance, the operator $S_i^xS_j^yS_k^z$, invariant under global $\pi$ rotations about the $x$, $y$, and $z$ axes, appears in exactly solvable models with Kitaev interactions \cite{kitaev2006anyons,Yao2007,saket2010manipulating,Jianlong2019}.
Spontaneous chirality and magnetization are not necessarily competing orders, since they coexist in non-coplanar phases of frustrated magnets \cite{Messio2011,Batista2016,Hayami2021}. On the other hand, the order parameter of a collinear magnetic phase such as the N\'eel state vanishes in a CSS and vice versa. While a CSS preserves spin-rotation symmetries, a collinear magnetic state is usually invariant under a combination of time reversal and spatial or spin rotation that constitutes a broken symmetry in the CSS. As a result, in the absence of a coexistence region, the Landau-Ginzburg-Wilson (LGW) paradigm dictates that a generic phase transition from collinear magnetism to a CSS should be of first order \cite{landau2013statistical,Imry1975}.
Continuous phase transitions beyond the LGW paradigm have been discussed in the context of deconfined quantum criticality (DQC) \cite{senthil2004deconfined,senthil2004quantum,levin2004deconfined,Wang2017}. The most studied example is the continuous transition from the N\'eel state to a valence bond solid (VBS) on the square lattice \cite{Haldane1988,Read1989}. Such an exotic transition can be described by effective field theories with a rich phenomenology that includes order-parameter fractionalization, dualities, and emergent higher symmetries leading to non-compact gauge fields and deconfined spinons at the quantum critical point. Unambiguous numerical demonstrations of DQC in lattice models \cite{Sandvik2007,Melko2008} have been hindered by logarithmic corrections to finite-size scaling, which make it difficult to rule out a weakly first-order transition \cite{NahumPRX,Shao2016,Ma2020}. This challenge has motivated the study of Landau-forbidden transitions with analogies to DQC in one-dimensional (1D) models \cite{jiang2019ising,mudry2019quantum,Huang2019,Ogino2021,Roberts2021}, for which more controllable analytical and numerical methods are available. In fact, it has been known for a while that the same operator that gives rise to N\'eel order in the field theory for anisotropic spin-1/2 chains can also generate spontaneous dimerization \cite{Haldane1982}. The staggered magnetization and dimerization operators can be combined into a single order parameter, associated with the SO(4) symmetry of the SU(2)$_1$ Wess-Zumino-Witten conformal field theory (CFT) at the Heisenberg point \cite{Nahum2015}, and the continuous transition from the N\'eel to the dimerized phase in the frustrated XYZ chain has an emergent U(1) symmetry \cite{jiang2019ising,mudry2019quantum}.
In this work, we extend the set of Landau-forbidden phase transitions by constructing lattice models which exhibit a continuous transition from a CSS to a collinear magnetic phase. We start by unveiling a local mapping of the Hilbert space on a four-site unit cell that allows us to represent the chirality and the magnetic order parameters as two components of the same pseudospin. As the main example of our construction, we consider a zigzag spin chain with Kitaev interactions in addition to six-spin interactions that couple the chiralities on triangular plaquettes. For a particular choice of the parameters, our model reduces to the one proposed by Saket {\it et al.} \cite{saket2010manipulating} and becomes exactly solvable in terms of Majorana fermions and a static $\mathbb Z_2$ gauge field. The six-spin interaction stabilizes a chiral phase as it lifts the exponential ground state degeneracy of the model of Ref. \cite{saket2010manipulating}. In the regime of strong intercell Kitaev interactions in the chain direction, we find a collinear magnetic phase analogous to the stripe phase of the Kitaev-Heisenberg model on the triangular lattice \cite{kimchi2014kitaev,Becker2015,Li2015,Kazuya2016,Maksimov2019}. Thus, our work also fits in the context of recent studies of quasi-1D extended Kitaev models aimed at offering insight into two-dimensional phases \cite{LeHur2017,Yang2020,Nocera2020,Sorensen2021,Metavitsiadis2021}. We demonstrate the continuous transition between the CSS and the collinear magnetic phase using a combination of solvable effective Hamiltonians, bosonization of the low-energy theory, and numerical density matrix renormalization group (DMRG) simulations \cite{WhiteDMRG1992,white-dmrg-prb}. We show that the transition has an emergent U(1) symmetry and is described by a CFT with central charge $c=1$. We also analyze the transition from the point of view of a U(1) gauge theory with fermionic partons and discuss analogies with DQC in higher dimensions.
The paper is organized as follows. In Sec. \ref{sec:duality}, we present the pseudospin mapping and apply it to the zigzag chain model. In Sec. \ref{sec3}, we focus on the parameter regime in which the model is exactly solvable by a Majorana fermion representation. In this case, we obtain a chiral ground state and we classify the excitations in terms of fermionic modes and chirality domain walls. Section \ref{sec:transition} contains our analytical results for the transition between chiral and magnetic phases. We point out another exactly solvable limit of the model and use it as starting point for the effective field theory, uncovering some analogies with DQC. Our DMRG results which support the prediction of a $c=1$ CFT at criticality are presented in Sec. \ref{secDMRG}. Section \ref{sec2D} serves as an outlook, in which we offer some remarks about possible connections with 2D models that harbor chiral spin liquid ground states. Finally, we summarize our findings in Sec. \ref{secConclusion}.
\section{Chirality pseudospins and zigzag chain model\label{sec:duality}}
In this section, we present a pseudospin mapping that will prove useful in studying chiral phases of spin systems with bond-dependent anisotropic exchange interactions, as in quantum compass models \cite{Nussinov2015}. We note in passing that a remarkable duality between the scalar spin chirality and the staggered-dimer order parameter has been applied to interpret the chiral phase of the isotropic two-leg ladder with four-spin interactions \cite{Hikihara2003,Lauchli2003,Gritsev2004}.
Consider Pauli spin operators $\sigma_n^a$, with $a=x,y,z$, defined on four sites, $n=1,\dots,4$, represented as a square in Fig. \ref{fig1}(a). We choose the diagonal bond between sites $n=2$ and $n=3$ to divide the square into two triangles, and label the bonds as $x$, $y$, and $z$ so that each triangle contains one bond of each type. We then define the anisotropic spin chiralities on the triangles as the three-spin operators\begin{equation}
\tau_1^x=\sigma_2^x\sigma_3^y\sigma_1^z,\qquad \qquad \tau_2^x=\sigma_3^x\sigma_2^y\sigma_4^z.\label{taux}
\end{equation}
The spin indices in $\sigma_i^a\sigma_j^b\sigma_k^c$ obey the mnemonic rule that, in the triangle formed by sites $(i,j,k)$, site $i$ corresponds to the vertex opposite to an $a$ bond, site $j$ is opposite to a $b$ bond, and site $k$ is opposite to a $c$ bond. We define the $z$ components of the pseudospins as \begin{equation}
\tau_1^z=\sigma_1^x,\qquad \qquad \tau_2^z=\sigma_2^x.\label{tauz}
\end{equation}
The operators in Eqs. (\ref{taux}) and (\ref{tauz}) square to the identity and obey $[\tau_1^a,\tau_2^b]=0$, $\{\tau_1^x,\tau_1^z\}=\{\tau_2^x,\tau_2^z\}=0$. A duality transformation that exchanges the chiralities with one-spin operators can be obtained by applying the unitary $U=e^{-i\pi (\tau_1^y+\tau_2^y)/4}$, where $\tau_1^y=-\sigma_2^x\sigma_3^y\sigma_1^y$ and $\tau_2^y=\sigma_3^x\sigma_2^z\sigma_4^z$.
To complete the mapping, we define another pair of Pauli operators which commute with $\boldsymbol \tau_1$ and $\boldsymbol \tau_2$: \begin{eqnarray}
\rho_1^x=\sigma_2^x\sigma_3^y,\qquad \qquad \rho_2^x=\sigma_2^x\sigma_4^y,\\
\rho_1^z=\sigma_1^x\sigma_3^x,\qquad \qquad \rho_2^z=\sigma_2^x\sigma_4^x.\label{rhoz}
\end{eqnarray}
Note that these two-spin operators are invariant under time reversal, whereas all components of $\boldsymbol \tau_1$ and $\boldsymbol \tau_2$ are time-reversal odd.
\begin{figure}[t]
\centering{}\includegraphics[width=\columnwidth]{fig1.pdf}
\caption{Four-site unit cell and zigzag chain model. (a) The anisotropic chirality pseudospins on the triangular plaquettes are defined according to the bond labels $x$, $y$, and $z$ as written in Eq. (\ref{taux}). (b) In the zigzag chain, spins are coupled by Kitaev interactions. In addition, there is a six-spin interaction that couples the chiralities of triangles with the same orientation (up-pointing or down-pointing) on nearest-neighbor unit cells. }
\label{fig1}
\end{figure}
In order to illustrate the spin-chirality duality, we consider a spin model defined on a zigzag chain,
given by the Hamiltonian \begin{equation}
H = H_K+ H_Q,\label{HKQ}
\end{equation}
where
\begin{equation}
H_K= \sum_{a = x,y,z}K_a\sum_{\langle i,j\rangle_a} \sigma_i^a \sigma_j^a + K_x^\prime \sum_{\langle i,j\rangle_{x'}}\sigma^x_i\sigma^x_j
\label{zigzaghamiltonian}
\end{equation}
contains the bond-dependent Kitaev interactions. Here $y$ and $z$ bonds couple sites on different legs of the zigzag chain and $x$ bonds lie in the chain direction. We distinguish between two types of $x$ bonds. The bonds with Kitaev coupling $K_x$ lie within a unit cell and are represented by solid red lines in Fig. \ref{fig1}(b). The $x$ bonds between sites in neighboring unit cells have coupling $K_x^\prime$ and are represented by dashed red lines. We focus on the regime of antiferromagnetic Kitaev couplings, $K_a,K_x^\prime\geq 0$. The zigzag chain with $K_x=K_x'=K_y=K_z$ can be viewed as a strip of the Kitaev model on the triangular lattice, whose ground state has been controversial \cite{Becker2015,Li2015,Kazuya2016,Maksimov2019}. Besides the Kitaev interactions, we also add terms coupling the chiralities in different unit cells. The six-spin interaction preserves time-reversal symmetry and can be written in terms of the pseudospins in Eq. (\ref{taux}) as \begin{equation}
H_Q=-Q\sum_{r}(\tau^x_{1,r}\tau^x_{1,r+1}+\tau^x_{2,r}\tau^x_{2,r+1}),\label{HQ}
\end{equation}
where $\boldsymbol \tau_{1,r}$ and $\boldsymbol \tau_{2,r}$ denote the pseudospins for the unit cell at position $r$. This interaction is analogous to the coupling between the scalar spin chiralities on plaquettes of the square lattice proposed in Ref. \cite{wen1989chiral}. Physically, this type of multi-spin interaction can be associated with orbital currents in chiral magnets \cite{Bulaevskii2008,Grytsiuk2020}. We focus on $Q\geq 0$, favoring uniform chirality. The reason for the particular choice of coupling only triangles with the same orientation, see Fig. \ref{fig1}(b), will become clear in Sec. \ref{sec:transition}.
The relevant symmetries of the model are: time reversal $\mathcal{T}: i \mapsto -i$, $\boldsymbol{\sigma}_{n,r} \mapsto - \boldsymbol{\sigma}_{n,r}$ for $n=1,\dots,4$; a $C_2$ rotation symmetry about the center of a $z$ bond $\mathcal{R}: \boldsymbol{\sigma}_{1,r} \leftrightarrow \boldsymbol{\sigma}_{4,-r},\boldsymbol{\sigma}_{2,r} \leftrightarrow \boldsymbol{\sigma}_{3,-r} $;
and two discrete spin rotation symmetries $\mathcal{K}_1$ and $\mathcal K_2$ generated by $U_1 = \prod_r \tau^x_{1,r}$ and $U_2 = \prod_r \tau^x_{2,r}$, respectively, which can be understood as $\pi$ rotations about a different spin axis for each sublattice \cite{Chaloupka2015}. Their combined action is known as the Klein symmetry
because it is isomorphic to the Klein group, $\mathcal K = \mathcal K_1 \times \mathcal K_2 \simeq \mathbb Z_2 \times \mathbb Z_2$ \cite{kimchi2014kitaev}. The Klein symmetry is found in the Kitaev model on several lattices that obey a certain geometrical condition, including the triangular lattice, but is explicitly broken by more general interactions such as Heisenberg exchange. The product $U_1U_2=\prod_{r}\prod_{n=1}^4\sigma_{n,r}^z$ generates the usual global $\pi$ rotation about the $z$ spin axis.
In terms of the pseudospins, the Hamiltonian becomes\begin{eqnarray}
H&=&\sum_{r}\bigg[\sum_{l=1,2}\left(K_x \rho_{l,r}^z+K_x'\rho^z_{l,r}\tau^z_{l ,r}\tau^z_{l,r+1}-Q\tau^x_{l,r}\tau^x_{l,r+1}\right) \nonumber\\
&&+K_y(\rho_{1,r}^x\rho_{2,r}^x-\rho_{1,r}^y\rho_{2,r}^y\tau^x_{1,r}\tau^x_{2,r})\nonumber\\
&&+K_z(\rho^x_{1,r}\rho^y_{2,r}\tau^x_{2,r}+\rho^y_{2,r} \rho^x_{1,r+1} \tau^x_{1,r+1} ) \bigg].\label{newH}
\end{eqnarray}
This Hamiltonian is equivalent to a two-leg ladder with $l=1,2$ playing the role of a leg index. There are two pseudospins 1/2, namely $\boldsymbol \tau$ and $\boldsymbol \rho$, on each effective site specified by $(l,r)$. In the following sections, we will analyze special limits of the model to establish the existence of a continuous phase transition from a CSS in which $\langle \tau^x_{l, r}\rangle \neq 0$ to a magnetic phase in which $\langle \tau^z_{l, r}\rangle \neq 0$.
\section{Chiral Spin State in the exactly solvable model\label{sec3}}
In this section, we discuss the exact solution of the model with $K_x^\prime=0$. In this case, the local operators $\tau_{l,r}^x$ commute with the Hamiltonian in Eq. (\ref{newH}), generating an extensive number of conserved quantities. For $Q>0$, the ground state has the same eigenvalue for all $\tau_{l,r}^x$, corresponding to a uniform spin chirality that spontaneously breaks the $\mathcal T$ symmetry and preserves the $C_2$ rotation and Klein symmetries. Setting $\tau_{l,r}^x=1$, we obtain the effective Hamiltonian for the remaining pseudospins:\begin{eqnarray}
H_{\rho}&=&\sum_{r}\bigg[K_x\sum_{l=1,2} \rho_{l,r}^z +K_y(\rho_{1,r}^x\rho_{2,r}^x-\rho_{1,r}^y\rho_{2,r}^y)\nonumber\\
&&+K_z(\rho^x_{1,r}+ \rho^x_{1,r+1} ) \rho^y_{2,r}-2Q\bigg]\label{Hrho1}.
\end{eqnarray}
The model is now equivalent to a single chain with anisotropic exchange couplings in the $xy$ plane and an effective field in the $z$ direction. We apply the Jordan-Wigner transformation: \begin{eqnarray}
\rho_{l,r}^z&=&1-2d^\dagger_{l,r}d^{\phantom\dagger}_{l,r},\nonumber\\
\rho_{1,r}^+&=&d_{1,r}\prod_l\prod_{r'<r}(1-2d^\dagger_{l,r'}d^{\phantom\dagger}_{l,r'}),\\
\rho_{2,r}^+&=&d_{2,r}(1-2d^\dagger_{1,r}d^{\phantom\dagger}_{1,r})\prod_{r'<r}(1-2d^\dagger_{l,r'}d^{\phantom\dagger}_{l,r'}),
\nonumber
\end{eqnarray}
where $d_{l,r}$ are complex spinless fermions and $\rho^\pm_{l,r}=(\rho^x_{l,r}\pm i\rho^y_{l,r})/2$. The Hamiltonian becomes \begin{eqnarray}
H_{\rho}&=&\sum_{r}\bigg[-2K_x\sum_{l=1,2}d^\dagger_{l,r}d^{\phantom\dagger}_{l,r} +2K_y(d^\dagger_{1,r}d^\dagger_{2,r}+\text{h.c.})\nonumber\\
&&+iK_z(d^\dagger_{1,r}-d^{\phantom\dagger}_{1,r})(d^\dagger_{2,r}-d^{\phantom\dagger}_{2,r})\nonumber\\
&&-iK_z(d^\dagger_{1,r+1}+d^{\phantom\dagger}_{1,r+1})(d^\dagger_{2,r}+d^{\phantom\dagger}_{2,r}) \bigg]+\text{const}.\label{Hrho}
\end{eqnarray}
It is then clear that the model admits an exact solution in terms of free fermions.
The solution for $K_x^\prime=Q=0$ was discussed in Ref. \cite{saket2010manipulating} using a Majorana fermion representation. The key observation is that in this case the zigzag chain reduces to a tricoordinated 1D lattice, called the tetrahedral chain, and the model can be solved by the same methods as Kitaev's honeycomb model \cite{kitaev2006anyons}. Using Kitaev's representation, we write the original spin operator at site $j=(n,r)$ as $\sigma_{j}^a=ib^a_{j}c^{\phantom{a}}_{j}$, where $b^a_{j}$ and $c^{\phantom{a}}_{j}$ are Majorana fermions subjected to the local constraint $b^x_{j}b^y_{j}b^z_{j}c_j^{\phantom{x}}=1$. The tetrahedral-chain Hamiltonian can be written as\begin{equation}
H_0=\lim_{K_x^\prime\to0}H_K= \sum_{a = x,y,z}K_a\sum_{\langle j,l\rangle_a} iu^a_{jl} c_jc_l,\label{Majorana}
\end{equation}
where $u_{jl}^a=-ib_j^ab^a_l$ are locally conserved $\mathbb Z_2$ gauge fields defined on the $\langle j,l\rangle_a$ bonds, satisfying $u^a_{jl}=-u^a_{lj}$ and $[u^a_{jl},H_0]=[u^a_{jl},u^b_{j'l'}]=0$. The chirality operators can be identified with the $\mathbb Z_2$ fluxes in the triangles:\begin{eqnarray}
\tau^x_{1}=u^x_{13}u_{32}^zu_{21}^y,\qquad \tau^x_{2}=u^x_{42}u_{23}^zu_{34}^y.
\end{eqnarray}
Fixing a gauge with $u_{jl}^a=\pm1$ so that $\tau^x_{1,r}=\tau^x_{2,r}=1$ for all unit cells, we find that the remaining $c$-type Majorana fermions move freely in the background of the static gauge field.
Importantly, the spectrum also contains excitations which correspond to changing the $\mathbb Z_2$ flux configuration. Starting from the state with uniform chirality $\tau^x_{l,r}=1$ $\forall l,r $, we refer to an excitation with a single $\tau^x_{l,r}=-1$ as a type-$l$ vortex of the $\mathbb Z_2$ gauge field. In the solvable model, type-1 and type-2 vortices are localized in up-pointing and down-pointing triangles, respectively, and they can be created by changing the sign of a single $u_{jl}^x$. Note, however, that physical states must respect a global fermion parity constraint \cite{Pedrocchi2011}.
For $Q=0$, the exactly solvable model suffers from a ground state degeneracy that grows exponentially with system size \cite{saket2010manipulating}. The reason is that the energies of the exact eigenstates only depend on the total $\mathbb Z_2$ flux in each unit cell, {\it i.e.}, on the product $\tau_{1,r}^x\tau_{2,r}^x$, rather than the individual chiralities $\tau_{l,r}^x$. This degeneracy can be associated with a local symmetry as follows. For $Q=K_x^\prime=0$, flipping the sign of both chiralities in unit cell $r_0$ in Eq. (\ref{newH}) only affects the $K_z$ term that couples $d_{1,r_0}$ to $d_{2,r_0}$ and $d_{2,r_0-1}$ in the quadratic Hamiltonian Eq. (\ref{Hrho}). The sign change can be removed by applying a gauge transformation $d_{1,r_0}\mapsto -d_{1,r_0}$. Thus, all eigenstates with $\tau_{1,r_0}^x=\tau_{2,r_0}^x=-1$ have the same energy as the corresponding eigenstates with $\tau_{1,r_0}^x=\tau_{2,r_0}^x=1$. In the Kitaev representation, this means that pairs of vortices located in the same unit cell cost zero energy for $Q=0$. The role of the chirality-chirality coupling that we introduced in Eq. (\ref{HQ}) is therefore to introduce an energy cost proportional to $ Q$ for vortex pairs. For $Q>0$, there are only two ground states with uniform chirality $\tau_{l,r}^x=\pm1$.
Let us now discuss the excitations in the exactly solvable model. First, consider the vortex-free sector with uniform chirality $\tau_{l,r}^x=1$. Fixing the set of $u_{jl}^a$ in a translationally invariant ansatz and taking a Fourier transform in Eq. (\ref{Majorana}), we cast the Hamiltonian in the form $H=i\sum_{k>0}C^\dagger_{k}\mathcal A(k)C^{\phantom\dagger}_{k}+\text{const.}$, where $C_k^\dagger=(c^\dagger_{1,k},c^\dagger_{2,k},c^{ \dagger}_{3,k},c^{ \dagger}_{4,k})$ is a four-component spinor in the sublattice basis, with $c^\dagger_{n,k}=c^{\phantom\dagger}_{n,-k}$, and $\mathcal A(k)$ is a matrix obeying $\mathcal A^t(k)=-\mathcal A(-k)$.
The dispersion relations of the bands can be calculated analytically for arbitrary values of the Kitaev couplings. The single-fermion energy gap is given by $\Delta_c= \sqrt{K_x^2+K_y^2+K_z^2-2K_z\sqrt{K_x^2+K_y^2}}$ and vanishes for $K_z^2= K_x^2+K_y^2$ \cite{saket2010manipulating}. In the fermionic representation, the gap closing involves a change in the $\mathbb Z_2$ topological invariant $\nu=\text{sgn}\{\text{Pf}[\mathcal A(0)]\text{Pf}[\mathcal A(\pi)]\}$
for class D in one dimension \cite{Chiu2016}. The phase for $K_z^2> K_x^2+K_y^2$ corresponds to the nontrivial value $\nu=-1$.
However, in the spin representation this transition can be associated with local order parameters. According to the Hamiltonian in Eq. (\ref{Hrho1}), the limit of dominant $K_z$ corresponds to the $\boldsymbol \rho$ pseudospins ordering in the $xy$ plane. Importantly, nonzero expectation values of $\rho^x_{l,r}$ and $\rho^y_{l,r}$ break spin-rotation symmetries. Since time-reversal symmetry is already broken by the spontaneous spin chirality, we obtain $\langle \sigma^z_{1,r}\rangle\sim \langle \tau^x_{1,r}\rangle \langle \rho^x_{1,r}\rangle\neq 0$. Thus, for $K_z^2> K_x^2+K_y^2$ we encounter a phase with spontaneous magnetization in the $z$ spin direction coexisting with the spin chirality. Since this phase breaks more symmetries than the pure chiral phase for $K_z^2< K_x^2+K_y^2$, this is a conventional Ising transition and we do not discuss it further. Note that the gap for $c$ fermions closes at the critical point, but vortex excitations remain gapped across this transition.
\begin{figure}[t]
\centering{}\includegraphics[width=\columnwidth]{fig2.pdf}
\caption{Two types of chirality domain walls. The signs indicate the chirality $\tau_{l,r}^x$ for each triangle. (a) In the nonintegrable model with $K_x^\prime\ll K_a,Q$, an intracell domain wall can hop one unit cell to the right by applying the $K_x^\prime$ interaction on the $x'$ bond indicated by the dashed line. (b) An intercell domain wall moves two unit cells to the right by acting on the two bonds indicated by the dashed lines. }
\label{fig2}
\end{figure}
We now turn to vortex excitations. In this case, we compute the energies numerically by diagonalizing the Hamiltonian on a finite chain with open boundary conditions because the localized vortex breaks translational invariance. The energy of a single vortex behaves as $\Delta_{v}=\Delta^{(0)}_{v} +2Q$, where $\Delta^{(0)}_{v}$ denotes the vortex gap for $Q=0$. In particular, for $K_x=K_y=K_z$ we obtain $\Delta^{(0)}_{v} \approx 0.29K_x$. The vortex gap vanishes if any of the Kitaev couplings $K_a$ approach zero since the broken bonds make the $\mathbb Z_2$ fluxes ill defined. For $K_y=K_z\ll K_x$, we find $\Delta^{(0)}_{v}\approx K_y^2/(2K_x)$.
In the pseudospin representation, it may be more convenient to picture the elementary excitations in the flux sector as domain walls between regions of opposite chiralities. We distinguish between two types of domain walls as illustrated in Fig. \ref{fig2}. We call an intracell domain wall the situation in which the two domains are separated by a unit cell $r_0$ with $\tau^x_{1,r_0}\tau^x_{2,r_0}=-1$; see Fig. \ref{fig2}(a). By contrast, in an intercell domain wall the chirality switches between two adjacent unit cells so that $ \tau^x_{1,r}\tau^x_{2,r}=1$ for all unit cells in the vicinity of the domain wall; see Fig. \ref{fig2}(b). Starting from a ground state with uniform chirality, we create an intercell domain wall at position $r_0$ by applying the string operator $V_{r_0}= \prod_{r< r_0} \tau_{1,r}^z\tau_{2,r}^z$, with energy cost $2Q$. An intracell domain wall with energy $\Delta_{v}$ is created by $V_{r_0}\tau_{1,r_0}^z$ or $V_{r_0}\tau_{2,r_0}^z$.
The creation of chirality domain walls affects the fermionic spectrum. Here we restrict the parameters to the domain $K_y=K_z\leq K_x$. This condition puts the system in the pure chiral phase with $\nu=1$. We observe numerically that in the presence of an intercell domain wall the fermionic spectrum only comprises a continuum of extended states, as in the vortex-free case. On the other hand, the creation of an intracell domain wall gives rise to a midgap state in which a $c$ fermion is bound near the unit cell with $ \tau^x_{1,r}\tau^x_{2,r}=-1$. The energy of the bound state is not pinned at zero, but varies with the ratio $K_y/K_x$ as shown in Fig. \ref{fig3}. In particular, the bound state has zero energy for $K_x=K_y=K_z$. We expect the transition from the chiral phase to a non-chiral magnetic phase to be associated with the condensation of domain walls which carry the magnetization degree of freedom in the form of a bound state of a $c$-type matter fermion and a $b$-type flux fermion. This transition will be discussed in the next section.
\begin{figure}[t]
\centering{}\includegraphics[width=.95\columnwidth]{fig3.pdf}
\caption{Fermionic spectrum calculated from the Hamiltonian in Eq. (\ref{Majorana}) for an open chain containing a single intracell domain wall. Here we set $K_z=K_y$. The shaded region represents the continuum of extended states. The blue line represents the energy of the bound state. }
\label{fig3}
\end{figure}
\section{Transition to the magnetic state \label{sec:transition}}
For $K_x'>0$, the local chirality operators are no longer conserved because the operator $\tau^z_{l,r}\tau^z_{l,r+1}$ in Eq. (\ref{newH}) flips the chirality of two triangles with the same orientation in neighboring unit cells. Nevertheless, the parity of the number of type-$l$ vortices, encoded in the eigenvalues of $U_l=\prod_r \tau^x_{l,r}$, are still good quantum numbers. Clearly, the existence of two separately conserved parities is associated with the Klein symmetry. In the regime $K_x'\ll K_a,Q$, we can treat the $K_x'$ term as a perturbation to the solvable model discussed in Sec. \ref{sec3} and focus on a subspace with a fixed number of vortices or domain walls. Figure \ref{fig2} shows the processes that generate an effective hopping amplitude for the domain walls. While the intracell domain wall moves at first order in $K_x^\prime$, the intercell domain wall only moves at order $(K_x')^2$. As a result, both types acquire a dispersion, but the intracell domain wall has a larger bandwidth. We then expect that, as we increase $K_x^\prime$ in the regime $Q\gtrsim \Delta_v^{(0)}$, the gap for intracell domain walls will eventually close first, driving a phase transition. Below the critical value of $K_x'$, the mobile domain walls remain gapped and the system is in a CSS characterized by $\langle \tau^x_{l,r}\rangle \neq 0$. In the Kitaev representation, the gauge variables $u_{jl}^y$ and $u_{jl}^z$ still commute with the Hamiltonian, but $u_{jl}^x$ become fluctuating, and the chiral phase corresponds to $\langle u_{jl}^x\rangle\neq 0$.
To examine the phase transition, let us consider the limit of highly anisotropic Kitaev interactions. For $K_y,K_z\to0$, the $\boldsymbol \rho$ pseudospins are fully polarized with $ \rho^z_{l,r} = -1$ in the ground state. This condition imposes strong antiferromagnetic correlations in the $x$ spin direction between two spins on the same leg and in the same unit cell; see Eq. (\ref{rhoz}). In the regime $K_y,K_z\ll K_x$, we can safely assume that the $\boldsymbol \rho$ pseudospins are gapped out. We derive an effective Hamiltonian in the low-energy subspace by applying perturbation theory to second order in $K_y$ and $K_z$. We obtain \begin{eqnarray}
H_{\rm eff}&=&-\sum_{r} \sum_{l=1,2}\left( K_x' \tau^z_{l ,r}\tau^z_{l,r+1}+Q\tau^x_{l,r}\tau^x_{l,r+1}\right) \nonumber\\
&&-K_\perp\sum_r \tau^x_{1,r}\tau^x_{2,r}+\text{const.} ,\label{Heff}
\end{eqnarray}
where $K_\perp\approx K_y^2/(2K_x)$. This Hamiltonian describes a two-leg ladder with weakly coupled XY chains. Note that the interchain coupling only involves the $x$ components of the pseudospins. The interactions $ \tau^a_{1,r}\tau^a_{2,r}$ with $a=y,z$ are forbidden by the Klein symmetry.
\begin{figure}[t]
\centering{}\includegraphics[width=.95\columnwidth]{fig4.pdf}
\caption{Four ground states in the magnetic phase. The up and down arrows represent the magnetization in the $x$ spin direction. The symmetries that connect the different states are also indicated. }
\label{fig4}
\end{figure}
This analysis reveals that the model with $K_y=K_z=0$ and $K_x,K_x^\prime,Q>0$ is also exactly solvable. Setting $K_\perp=0$, we see that the Hamiltonian reduces to two identical and decoupled XY chains, even though spins on different legs of the original zigzag chain are coupled by the six-spin interaction. The exact excitation spectrum can be calculated by performing a Jordan-Wigner transformation. The critical point occurs at $K_x^\prime=Q$. The chirality vanishes continuously as we approach the critical point from $K_x'<Q$. For $K_x'>Q$, the system enters another ordered phase characterized by $\langle \tau^z_{l,r}\rangle\neq 0$. Since $\rho_{l,r}^z=-1$, this implies $\langle \sigma^x_{1,r}\rangle=-\langle \sigma^x_{3,r}\rangle\neq 0$ and $\langle \sigma^x_{2,r}\rangle=-\langle \sigma^x_{4,r}\rangle\neq 0$. Thus, we find a collinear magnetic phase with spontaneous magnetization in the $x$ spin direction. There are four ground states, labeled by $(\text{sgn}\langle \sigma^x_{1,r}\rangle,\text{sgn}\langle \sigma^x_{2,r}\rangle )$ and represented in Fig. \ref{fig4}. The magnetic order in the zigzag chain is analogous to the stripe phase of the antiferromagnetic Kitaev model on the triangular lattice as obtained in Ref. \cite{Maksimov2019}. The pair of states $(+,+)$ and $(-,-)$ are conjugated by $\mathcal T$ symmetry, and likewise for the pair $(+,-)$ and $(-,+)$. The $(+,+)$ and $(-,-)$ states break the $\mathcal R$ spatial rotation symmetry, but preserve $ \mathcal R\mathcal T$. The degeneracy between $(+,+)$ and $(+,-)$ is protected by the $\mathcal K_2$ symmetry; both $\mathcal K_1$ and $\mathcal K_2$ are spontaneously broken in this phase. All four ground states are invariant under the combination of time reversal and global $\pi$ rotation about the $z$ spin axis, a symmetry broken in the CSS. Remarkably, the $\mathcal T$ symmetry is broken on both sides of the transition, but restored at criticality. In the magnetic phase, the elementary excitations are kinks or domain wall in the staggered magnetization. The magnetization domain walls are mobile for any $Q>0$ and condense when we approach the phase transition from $K_x'>Q$.
We determine the universality class of the transition by taking the continuum limit in the effective Hamiltonian. Starting from $K_\perp=0$, we bosonize the pseudospins in the XY chains following the standard procedure \cite{giamarchi2003quantum} and then add the interchain coupling $K_\perp$ as a perturbation. We obtain the Hamiltonian density
\begin{eqnarray}
\mathcal H_{\rm bos}&=&\sum_l \left[\frac{v\kappa}{2}(\partial_x\theta_l)^2+\frac{v}{2\kappa}(\partial_x\phi_l)^2+\lambda \cos(\sqrt{4\pi}\theta_l)\right]\nonumber\\
&&-\lambda_\perp \cos(\sqrt\pi\theta_1) \cos(\sqrt\pi\theta_2),\label{Hbos}
\end{eqnarray}
where the bosonic fields satisfy $[\phi_l(x),\partial_{x'}\theta_{l'}(x')]=i\delta_{ll'}\delta{(x-x')}$, $v$ is the velocity of the pseudospin excitations, $\kappa$ is the Luttinger parameter, and $\lambda,\lambda_\perp>0$ are the coupling constants of the relevant operators with scaling dimensions $1/\kappa$ and $1/(2\kappa)$, respectively. In the vicinity of the transition, we have $v\sim K_x'$, $\lambda \sim K_x'-Q$, $\lambda_\perp \sim K_\perp$, and $|\kappa-1|\sim (K_\perp/K_x')^2$. The bosonized pseudospin operators can be written as $\tau^x_{l,r}\sim \cos(\sqrt\pi\theta_l)$ and $\tau^z_{l,r}\sim \sin(\sqrt\pi\theta_l)$.
For $\lambda_\perp=0$, the bosonic Hamiltonian reduces to two decoupled sine-Gordon models. The critical point at $\lambda=0$ contains two massless bosons and is described by a CFT with central charge $c=2$. The continuous symmetry $\theta_l\mapsto \theta_l +\text{const.}$ of the effective field theory at the critical point can be traced back to the lattice model. For $K_y=K_z=0$ and $Q=K_x'>0$, the operators $Y_l=\sum_r\tau^y_{l,r}$ with $l=1,2$ become conserved charges in the sector with $\rho^z_{l,r}=-1$. Away from the critical point, the relevant cosine perturbation flows to strong coupling under the renormalization group. The chiral phase corresponds to $\lambda<0$, with $\lambda\to-\infty $ at low energies pinning the $\theta$ field at $\theta_l=0$ or $\theta_l=\sqrt \pi$. For $\lambda>0$, the flow to $\lambda\to\infty$ pins the bosonic fields at $\theta_l= \sqrt \pi/2$ or $\theta_l= 3\sqrt \pi/2$, corresponding to the magnetic phase. For $\lambda_\perp=0$, both phases have a fourfold degenerate ground state. The extra ground state degeneracy of the chiral phase is due to the decoupling of the chiralities with different $l$ for $K_\perp=0$ in Eq. (\ref{Heff}).
When we switch on $\lambda_\perp >0$, the bosonic fields on different legs are coupled by a strongly relevant operator. According to the $c$ theorem \cite{Zamolodchikov1986}, the central charge of the CFT must decrease. Since the cosine operators commute with each other, we can proceed with a semiclassical analysis of the effective potential $V(\theta_1,\theta_2)=\lambda\sum_l \cos(\sqrt{4\pi}\theta_l)-\lambda_\perp \cos(\sqrt\pi\theta_1) \cos(\sqrt\pi\theta_2)$. For $\lambda_\perp>0$, the potential only becomes flat in the directions $\theta_2=\pm \theta_1$ (mod $2\sqrt \pi$) of the $(\theta_1,\theta_2)$ plane when $\lambda=\lambda_\perp/4>0$, as opposed to a completely flat potential when $\lambda=\lambda_\perp=0$. This means that the interchain coupling leaves out only one gapless boson at the transition, either $\theta_+=(\theta_1+\theta_2)/\sqrt2$ or $\theta_-=(\theta_1-\theta_2)/\sqrt2$. As a consequence, the generic transition for $\lambda_\perp>0$ has central charge $c=1$, associated with a single emergent U(1) symmetry. This result is similar to the N\'eel-VBS transition in frustrated spin chains \cite{jiang2019ising,mudry2019quantum}. At criticality, the correlation functions for both order parameters decay as power laws with the same exponent, $\langle \tau^a_{l,r}\tau^a_{l',r'}\rangle\sim |r-r'|^{-1/(2\kappa)} $ for $a=x,z$. Note that the transition at a critical value of $\lambda>0$ enlarges the region occupied by the chiral phase in comparison with the result for $\lambda_\perp=0$.
We can understand the ground state degeneracy by pinning the bosonic fields in the presence of the $\lambda_\perp$ interaction. Assuming that $\lambda_\perp$ gaps out $\theta_-$, we fix $\theta_2=\theta_1$. In the chiral phase, this condition implies $\theta_1=\theta_2=0,\sqrt\pi$; the two choices correspond to the ground states with either sign of $\langle \tau^x_{1,r}\rangle=\langle \tau^x_{2,r}\rangle\neq0$. If we assume instead that $\lambda_\perp$ gaps out $\theta_+$ and fix $\theta_2=-\theta_1$, we find precisely the same expectation values for the local physical operators. Thus, the ground state of the chiral phase is twofold degenerate for $\lambda_\perp>0$. On the other hand, the choice of gapping out $\theta_+$ or $\theta_-$ affects the expectation value of the magnetization when we pin the bosonic fields at $\theta_1=\pm\theta_2= \sqrt\pi/2, 3 \sqrt\pi/2$. In this case, there are still four possibilities labeled by the signs of $\langle \tau^z_{1,r}\rangle $ and $\langle \tau^z_{2,r}\rangle $. Provided that the Hamiltonian preserves the Klein symmetry, the ground state of the magnetic phase remains fourfold degenerate. In fact, the effective field theory allows us to analyze the effects of breaking the Klein symmetry, which in the bosonic representation acts as $\mathcal K_l:\phi_l\mapsto -\phi_l, \theta_l \mapsto -\theta_l $. Adding the perturbation $\lambda_\perp^\prime \sin(\sqrt\pi\theta_1)\sin(\sqrt\pi\theta_2)$ to the Hamiltonian density in Eq. (\ref{Hbos}), we find that the total potential still pins $\theta_2=\pm\theta_1$ and leaves out one gapless boson with $c=1$ at the transition. However, the ground state degeneracy of the magnetic phase is reduced to twofold, as the new interaction selects either $(+,+)$ and $(-,-)$ or $(+,-)$ and $(-,+)$, depending on the sign of $\lambda_\perp^\prime$.
In the bosonic Hamiltonian Eq. (\ref{Hbos}), we dropped all symmetry-allowed cosine operators involving the $\phi_l$ fields, such as $\cos(\sqrt{16\pi}\phi_l)$, because they are highly
irrelevant for $\kappa\approx 1$. Vertex operators of the form $\exp(i m\sqrt{\pi}\phi_l)$ with $m\in\mathbb Z$ create or annihilate domain walls, which in the bosonic theory correspond to kinks and antikinks in the field configuration, $\theta_l(x \to \infty)-\theta_l(x \to -\infty) = \pm\sqrt \pi$. In a semiclassical picture for the chiral phase, to go from the ground state with $\theta_l=0$ to $\theta_l=\pm \sqrt \pi$, the bosonic fields have to go through $\theta_l=\pm \sqrt\pi/2$, which can be interpreted as the magnetization $\langle \tau^z_{l,r} \rangle\sim\langle \sin (\sqrt\pi\theta_l)\rangle$ residing at the topological defect of the CSS. The same argument can be used to see how domain walls in the magnetic phase must carry spin chirality. Near the transition, the processes that change the number of domain walls in either picture become irrelevant.
Similar phenomenology is generally found in effective field theories for DQC \cite{senthil2004deconfined,senthil2004quantum,levin2004deconfined,Wang2017}:
topological defects in one phase nucleate the order parameter of the other phase. If the order parameters are
written in terms of fractionalized excitations, the resulting constraints lead to an emergent gauge field on which the topological defects are charged.
One can then understand both phases as distinct confined regimes, merging in a critical region corresponds to a gapless phase in the gauge theory.
In our model, the domain wall description can be obtained by a refermionization of the bosonic fields, defining the chiral fermions $\psi_{R/L,l} \sim \exp\left[-i\sqrt \pi (2 \phi_l \mp \theta_l /2)\right]$. The physical spin operators are then given by fermion bilinears. In terms of the two-component spinors $\Psi^\dagger_l = (\psi^\dagger_{L,l}, \psi^\dagger_{R,l})$, we have $\tau^x_{l,r} \sim \Psi^\dagger_l \sigma^x \Psi^{\phantom\dagger}_l$ and
$\tau^z_{l,r} \sim \Psi^\dagger_l \sigma^y \Psi^{\phantom\dagger}_l$, with $\sigma^a$ the Pauli matrices acting in the internal space. The effective Hamiltonian includes density-density interactions which arise from the cosine operators as well as quadratic terms in Eq. (\ref{Hbos}). The emergent symmetry at the critical point is manifested as Noether charges of the fermions, preventing pairing terms from appearing in the Hamiltonian. A mean-field decoupling of the quartic interactions generates mass terms for the chiral fermions in the ordered phases. Solitonic configurations in the mass terms support fermion bound states via the Jackiw-Rebbi mechanism \cite{jackiw1976solitons}, confirming the previous interpretation in terms of domain walls. Note that this mechanism applies to smooth domain walls in the low-energy theory for the transition, as opposed to the sharp domain walls deep in the chiral phase discussed in Sec. \ref{sec3}. For a smooth domain wall, a zero-energy bound state is formed even if the phase is topologically trivial \cite{robinson2019nontopological}.
To describe the transition in the fermionic picture, we start from the assumption of an emergent U(1)$\times$U(1) symmetry, which can then be gauged. The coupling to a U(1) gauge field can be obtained by noticing that the representation of the physical operators has a gauge redundancy, $\Psi_l(x)\mapsto e^{ie_l \Lambda_l(x)}\Psi_l(x)$, where $e_l$ play the role of gauge charges. We then impose a constraint on the fermion densities $\Psi^\dagger_l \Psi^{\phantom\dagger}_l\sim \partial_x \theta_l$, as usual in parton constructions \cite{Wenbook}. The resulting gauge-invariant lagrangian has the form \begin{equation}
\mathcal{L} = \sum_{l=1,2} \bar{ \Psi}_l i \gamma^\mu(\partial_\mu -ie_l a_\mu) \Psi_l +\frac{1}{4g^2}(f_{\mu\nu})^2 +\cdots,\label{fermiongauge}
\end{equation}
where we introduced the Maxwell tensor $f_{\mu\nu} = \partial_\mu a_\nu-\partial_\nu a_\mu$, the parameter $g$ in the Maxwell term controls the fluctuations of the gauge field, and we omitted quartic terms associated with short-range interactions. The terms highlighted in Eq. (\ref{fermiongauge}) comprise the $N_f = 2$ Schwinger model in $1+1$ dimensions, known to reduce to a single massless boson at low energies \cite{hosotani1997gauge, kim1999theory, sheng2008strong}. The gauge charges can be chosen arbitrarily as $e_l=\pm1$. The relative sign between $e_1$ and $e_2$ selects symmetric or antisymmetric modes with respect to exchanging the leg index, and is analogous to pinning either $\theta_+$ or $\theta_-$ in the bosonic theory. The critical point corresponds to fine tuning the quartic interactions so that the bosonic mode remains gapless, as described by Eq. (\ref{Hbos}) after we fix $\theta_2=\pm \theta_1$. Once again, we come to the conclusion that the transition between the chiral and magnetic phases is described by a $c=1$ CFT. At the fixed point, the two chiral sectors of the gapless boson decouple, and the CFT has an enlarged U(1)$\times$U(1) symmetry \cite{Affleck1985}.
\section{Numerical Results\label{secDMRG}}
In this section, we present our DMRG results for the phase transition between the CSS and the collinear magnetic state. To investigate the phases and the nature of the phase transition, we consider the zigzag chain with six-spin interactions described by the Hamiltonian in Eq. (\ref{HKQ}), equivalent to Eq. (\ref{newH}), and the two-leg XY ladder defined in Eq. (\ref{Heff}). In particular, we show results for the expectation values $\langle\tau^{x,z}_{l,r}\rangle$, the susceptibility of the ground-state energy density, and the entanglement entropy (EE).
\begin{figure}[t]
\centering{}\includegraphics[width=\columnwidth]{fig5.pdf}
\caption{Anisotropic spin chirality and local magnetization for the zigzag chain model, see Eqs. (\ref{HKQ})-(\ref{HQ}), as a function of $K_x'$ for $K_x=Q=1$ and $K_y=K_z=1$. The inset shows the same order parameters for $K_y=K_z=0.2$. }
\label{fig5}
\end{figure}
To compute the physical properties of interest, we have considered open chains with a maximum length of $L=400$. Keeping up to 400 states to represent the truncated DMRG blocks, we find that the largest truncation error acquired is smaller than $10^{-9}$ at the final sweep. As discussed in Secs. \ref{sec3} and \ref{sec:transition}, both chiral and magnetic phases exhibit degenerate ground states. Thus, to avoid linear combinations of the ground states in the numerical simulations, we have included weak and suitable perturbations that couple to the order parameters at the chain edges and select one ground state for a given phase. These small local perturbations do not affect the bulk properties, probed by observables computed near the middle of the chain.
Let us first focus on the zigzag chain model with six-spin interactions given by Eq. (\ref{HKQ}). In Fig. \ref{fig5}, we show the bulk values for the chiral and magnetic order parameters as a function of $ K'_x$ for $K_a=Q=1$, with $a=x,y,z$. While the CSS is characterized by a finite anisotropic spin chirality, the magnetically ordered phase displays an antiferromagnetic alignment along the $x$ spin direction (see Fig. \ref{fig4}). Note that there is a single phase transition at a critical value of $K_x'$. We have also considered the regime $K_y=K_z<K_x$ and found that the chiral phase becomes narrower as we decrease the ratio $K_y/K_x$, but the behavior is qualitatively the same as for $K_x=K_y=K_z$. No other transitions are observed as we vary $K_y/K_x$ for fixed $Q=K_x$; see the inset in Fig. \ref{fig5}.
\begin{figure}[t]
\centering{}\includegraphics[width=\columnwidth]{fig6.pdf}
\caption{Susceptibility of the ground-state energy density as a function of $K_x'$ for $K_x=K_y=K_z=Q=1$. The dashed lines represent the order parameters shown in the main plot of Fig. \ref{fig5}. The divergent behavior of $\chi_e$ at $K'_x\approx 0.312$ determines the critical point.}
\label{fig6}
\end{figure}
To pinpoint the location of the phase transition, we have analyzed the energy susceptibility, defined as \begin{equation}
\chi_e=-\frac{\partial^2 e_0}{\partial K_x'^2},
\end{equation}
where $e_0$ is the ground-state energy per site. In $d$ dimensions, the energy susceptibility diverges at the critical point as a power law with exponent $\alpha=(2/\nu)-(d+z)$, where $\nu$ and $z$ are the correlation and the dynamical critical exponents \cite{Albuquerque-scaling-susc,Sorensen2021}. In Fig. \ref{fig6}, we show $\chi_e$ as a function of $K_x'$ for $K_a=Q=1$. Note that $\chi_e$ exhibits a prominent peak, whose position in the $K_x'$ domain determines the critical point $K_x'=K'_{x,\text{crit}}$. For the set of couplings shown in Fig. \ref{fig6}, we obtain $K'_{x,\text{crit}}\approx0.312$. To verify the accuracy of the critical points extracted from $\chi_e$, we have also estimated $K'_{x,\text{crit}}$ from the analysis of the inflection point of the order parameters and the highest Schmidt eigenvalue. The latter was proposed in Ref. \cite{Sorensen2021} as a sensitive measure to detect phase transitions. Altogether, we found excellent agreement among the estimates obtained from these distinct procedures.
We now turn to the effective Hamiltonian in Eq. (\ref{Heff}), valid in the regime $K_y,K_z\ll K_x$. This model describes two weakly coupled XY chains with interchain coupling along the $x$ direction. In comparison with the original zigzag chain in Eq. (\ref{HKQ}), the dimension of the local Hilbert space in the effective ladder model is reduced by a factor of 2, providing a significant advantage for numerical simulations. Since we observed the same qualitative behavior for the original model with $K_x=K_y=K_z$ as for small $K_{y,z}/K_x$, see Fig. \ref{fig5}, we expect the effective XY ladder model in Eq. (\ref{Heff}) to capture the essential characteristics of the phase transition. Carrying out the same analysis as for the zigzag chain, we again find only one transition for fixed $Q$ and different values of $K_x'$. In agreement with the analysis in Sec. \ref{sec:transition}, the transition for $K_\perp>0$ shifts to larger values of $K_x'$ as compared to $K_x'=Q$ in the exactly solvable case $K_\perp=0$.
Setting $Q=1$, we determined the critical point $K_\perp=K_{\perp,\text{crit}}$ for $K_x'=1.2$ and $1.4$. The acquired values are $K_{\perp,\text{crit}}\approx0.187$ and $0.49$, respectively.
\begin{figure}[t]
\centering{}\includegraphics[width=\columnwidth]{fig7.pdf}
\caption{Entanglement entropy as a function of partition size $\ell$ for the critical two-leg XY ladder model, see Eq. (\ref{Heff}), with $(K_x',K_\perp)=(1.2,0.187)$ and $(1.4,0.49)$. The symbols represent the DMRG results for chain length $L=200$ and open boundary conditions. The solid red lines are fits to our numerical data using Eq. (\ref{entang}). The estimates for the central charge are indicated in the plot.}
\label{fig7}
\end{figure}
We investigate the universality class of the transition by extracting the central charge from the EE. Consider a chain composed of two partitions $\mathcal{A}$ and $\mathcal{B}$ with $\ell$ and $L-\ell$ sites, respectively. The EE is then defined as $S(\ell)=-\mathrm{Tr}(\rho_\mathcal{A}\ln\rho_\mathcal{A} )$,
where $\rho_\mathcal{A}$ is the reduced density matrix of partition $\mathcal{A}$. For critical 1D systems, the asymptotic behavior of $S(\ell)$ predicted from CFT is given by \cite{Calabrese2004}
\begin{equation}
S(\ell)=\frac{c}{3\eta}\ln\left[\frac{L}{\pi}\sin\left( \frac{\pi}{L}\ell\right)\right]+b,\label{entang}
\end{equation}
where $c$ is the central charge, $b$ is a nonuniversal constant, and $\eta=1(2)$ for periodic (open) chains. In Fig. \ref{fig7}, we show the EE as a function of $\ell$ for $(K_x',K_{\perp,\text{crit}})=(1.2,0.187)$ and $(1.4,0.49)$. We consider values of $\ell$ corresponding to partitions with an even number of rungs in an open chain. Fitting our DMRG results using Eq. (\ref{entang}), we obtain $c\approx 1.05$ and $1.03$, respectively. Considering different fitting intervals and system sizes, we have checked that our estimates are robust and the maximum deviation from $c=1$ is about $9\%$. The logarithmic scaling of the entanglement entropy with the subsystem size is clear evidence of critical behavior at the transition. Moreover, our results show remarkable agreement with the central charge predicted in Sec. \ref{sec:transition}. Finally, we have also investigated the effects of an explicit Klein-symmetry breaking by adding the interaction $K_\perp '\sum_r\tau^z_{r,1}\tau^z_{r,2}$ to the Hamiltonian in Eq. (\ref{Heff}). While the values of the critical couplings shift with the perturbation, no further transitions are observed and the central charge remains the same. Therefore, the Klein symmetry does not play any role in the universality class of the transition.
\section{Future directions in 2D\label{sec2D}}
The local pseudospin mapping of Eqs. (\ref{taux})-(\ref{rhoz}) can be used to fabricate Hamiltonians with six-spin interactions that reduce to known spin-1/2 models in two dimensions once we freeze out the $\boldsymbol \rho$ pseudospins. The phase with long-range order in $\tau^x$ would then correspond to a CSS. However, it is unclear if this approach can lead to deconfined transitions between chiral and magnetic phases in 2D compass models. The existence of a robust continuous transition between competing ordered phases is conjectured to be connected to the existence of non-trivial symmetry properties of topological defects \cite{senthil2004deconfined,senthil2004quantum,levin2004deconfined,Wang2017}. Therefore, if the $\boldsymbol \tau$ pseudospin is defined
envisioning the defects of $\tau^x$ and $\tau^z$ ordered phases on a given lattice, it may be possible to engineer a spin hamiltonian where gapping out the $\boldsymbol \rho$ pseudospin results in an effective model with a deconfined transition. An effective field theory description would then be described by a parton decomposition consistent with the defects \cite{Wang2017,jiang2019ising}.
A more interesting question is whether a continuous phase transition from a CSS to a collinear magnetic state or another ordered phase can be found in models that do not require six-spin interactions. Like the solvable model discussed in Sec. \ref{sec3}, the Yao-Kivelson model on the star lattice \cite{Yao2007} exhibits spontaneous time-reversal-symmetry breaking and two chiral phases separated by a phase transition at which the gap for dynamical matter fermions closes. In this case, the phases are topologically trivial and nontrivial chiral spin liquids distinguished by the Chern number. In the exact solution using the Kitaev representation, the topological excitations are vortices of the emergent $\mathbb Z_2$ gauge field, which bind Majorana zero modes in the nontrivial phase. One may then wonder if closing the vortex gap by adding integrability-breaking perturbations to the Yao-Kivelson model could drive an unconventional transition to a magnetic phase. In parallel, the dynamics of $\mathbb Z_2$ flux excitations and the relation to phase transitions in the extended Kitaev honeycomb model at zero magnetic field has been discussed based on parton mean-field theories \cite{Schaffer2012,Knolle2018} and a variational approach \cite{Zhang2021}.
Moving on to SU(2)-invariant models, the situation becomes less clear. Here new dualities involving the scalar spin chirality \cite{Hikihara2003} may prove instrumental. Numerically, a chiral spin liquid with spontaneous time-reversal-symmetry breaking has been found in the extended Heisenberg model on the kagome lattice \cite{Gong2014,Gong2015}. DMRG results on cylinder geometries suggest that the quantum phase transition from the chiral spin liquid to the $q=(0,0)$ N\'eel state is at least not strongly first order \cite{Gong2015}. The same can be said about transitions out of the chiral spin liquid phase in the triangular lattice Hubbard model \cite{Szasz2020}.
\section{Conclusions\label{secConclusion}}
In this work, we introduced a Kitaev-type model defined on the zigzag chain and showed the presence of two phases separated by a transition. On one side, we have a chiral spin state stabilized by coupling three-spin chiralities. On the other one, there is a collinear magnetic state, also found in the natural extension of our model to two dimensions, the Kitaev model on the triangular lattice \cite{Maksimov2019}. Numerical analysis and field theory arguments indicate a continuous phase transition, which would be forbidden by the traditional LGW paradigm due to the competing nature of the order parameters. Furthermore, a low-energy parton construction suggests an emergent symmetry along with the condensation of topological defects (domain walls) in the transition, similar to the phenomenology found in deconfined quantum criticality in two dimensions. Our work then provides an example of the recently found deconfined transitions in one dimension \cite{mudry2019quantum,jiang2019ising,Roberts2021}.
Further numerical investigation of this transition is also warranted. Critical exponents in correlation functions vary continuously for a $c=1$ (Gaussian) transition, and it would be interesting to see this behavior as the microscopic parameters are tuned. Moreover, the model in Eq. (\ref{newH}) may host other phases and transitions for a different range of parameters. We leave the complete mapping of the ground state phase diagram and the study of correlation functions at criticality for future work.
\begin{acknowledgments}
We thank J. C. Xavier for discussions on the DMRG implementation and the High-Performance Computing Center (NPAD) at UFRN for providing computational resources. We acknowledge funding by Brazilian agency CNPq (R.A.M. and R.G.P.). Research at IIP-UFRN is supported by Brazilian ministries MEC and MCTI. This work was also supported by a grant from the Simons Foundation (Grant Number 884966, AF).
\end{acknowledgments}
\bibliographystyle{apsrev4-2}
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Resumes are a crucial part of most organization's recruitment process. In many cases, it is the first point of contact that an applicant will have with an organization. Often, the information found in one's resume will result in them being placed in either the YES pile or the NO pile.
The desire to be placed in the YES pile can involve circumstances that cause applicants to take liberties with the truth on their resume or during the application process. An extreme example is Mike Ross (from the TV-show Suits) pretending to have a law degree in order to pay for his grandmother's medical bills. A more common scenario is that a job hunter may find that their search is beginning to drag on and hit a point of nervous desperation. Or, perhaps an individual might wish to avoid explaining an awkward employment gap or will embellish their experience because they really want the job. Some people don't feel that exaggeration is a big deal – "it's called selling yourself!" The fact is, it is a big deal.
Dishonesty on resumes can span from embellishments, to omissions, to outright lies. A simple embellishment might involve exaggerating the size of a leader's team or the scope of their role. I have seen applicants omit the end dates of their positions in an attempt to hide a 3-month employment gap or neglect to update the dates of their employment. In doing so, the candidate is attempting to mislead the hiring manager into believing they are still currently working. In a more extreme manner, there are times when people outright lie on their resumes.
Employers will find out. Recruiting processes have all sorts of checks in place. Interview questions are designed to give employers a clear understanding about career history and qualifications. Reference checks and other background checks are done to confirm career history and qualifications. Practical testing may also be done to ensure that a candidate has the skills and qualifications they claim to. Many organizations even use social media to qualify applicants. Even if you are a Mike-caliber liar and manage to get hired or if the recruitment process is not as thorough as it should be, there's a good chance that your performance will give you away resulting in further investigation and consequences.
The consequences. If an employee does get a job under false pretenses, an employer has the right to terminate the employment contract. This can happen early on or even years into your career with an employer. Leaving a job under such negative circumstances typically results in no references and puts candidates in an even more difficult position than they were prior to lying.
Loss of credibility. Some view a small embellishment or little white lie as a simple marketing spin. It may not seem like a big deal. However, in an employer's mind, if they find out that an applicant is willing to lie about something small, then why should the employer believe that they won't be dishonest about something large? Even if you are not terminated, your employer will not trust you 100%. Further to a damaged reputation, your career opportunities with that organization will likely be very limited.
It's wrong! Never mind all the consequences, dishonesty is simply wrong. Is it just me or is every "noble" thing that Mike does tainted by the fact that he's NOT A REAL LAWYER and that he lied to get to where he is? In addition to a damaged reputation and other consequences, dishonesty in the recruitment process also hurts the chances for all the honest jobseekers out there and may strip them of a job that they are actually qualified for.
We have all heard about situations where a little white lie can turn into a big green monster. Lying to get a job is not worth your credibility and reputation. It is far better to focus on honest strategies for getting ahead because once your reputation is tarnished it is difficult to build it up again.
For tips on how to positively impact your resume, take a look at our Resume Tips blog. For more information on Acuity HR Solutions and for a listing of our current opportunities, please visit our website at www.acuityhr.ca and sign up for our newsletter!
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{"url":"https:\/\/zbmath.org\/authors\/?q=ai%3Asolecki.slawomir","text":"# zbMATH \u2014 the first resource for mathematics\n\n## Solecki, S\u0142awomir\n\nCompute Distance To:\n Author ID: solecki.slawomir Published as: Solecki, S.; Solecki, Slawomir; Solecki, S\u0142awomir\n Documents Indexed: 69 Publications since 1990, including 1 Book\nall top 5\n\n#### Co-Authors\n\n 35 single-authored 3 Farah, Ilijas 3 Morayne, Micha\u0142 3 Todorcevic, Stevo B. 2 Irwin, Trevor L. 2 Kechris, Alexander S. 2 Moore, Justin Tatch 1 Arkhangel\u2019ski\u012d, Aleksandr Vladimirovich 1 Aviles Lopez, Antonio 1 Baltag, Alexandru 1 Bernett, H. R. 1 Cicho\u0144, Jacek 1 Dijkstra, Jan J. 1 Duncan, Jonathan 1 Godefroy, Gilles 1 Gruenhage, Gary F. 1 Hindman, Neil 1 Hjorth, Gregory 1 Kadets, Vladimir M. 1 Kawamura, Katsuyuki 1 Krupi\u0144ski, Krzysztof 1 Kubi\u015b, Wies\u0142aw 1 Kuchta, Ma\u0142gorzata 1 K\u00fcnzi, Hans-Peter A. 1 Kwiatkowska, Aleksandra 1 Leiderman, Arkady G. 1 Luschgy, Harald 1 Lutzer, David J. 1 Malicki, Maciej 1 Marciszewski, M. 1 Martin, Keye 1 Matheron, \u00c9tienne 1 Mislove, Michael W. 1 Moss, Lawrence S. 1 Nies, Andr\u00e9 Otfrid 1 Omiljanowski, Krzysztof 1 Pawlikowski, Janusz 1 P\u00e9rez Hern\u00e1ndez, Antonio 1 Pestov, Vladimir G. 1 Pillay, Anand 1 Pol, Roman 1 Reed, George Michael 1 Repov\u0161, Du\u0161an D. 1 Rosendal, Christian 1 Rubin, Matatyahu 1 Schneider, Friedrich Martin 1 Semenov, Pavel Vladimirovi\u010d 1 Shakhmatov, Dmitri B. 1 Siniora, Daoud 1 Spinas, Otmar 1 Srivastava, Shashi Mohan 1 Strauss, Dona 1 Tkachenko, Mikhail Gelievich 1 Toru\u0144czyk, Henryk 1 Uspenski\u012d, Vladimir Vladimirovich 1 van Mill, Jan 1 Zelen\u00fd, Miroslav 1 Zhao, Min 1 Zielinski, Jan M.\nall top 5\n\n#### Serials\n\n 6 The Journal of Symbolic Logic 5 Israel Journal of Mathematics 4 Advances in Mathematics 4 Journal of Functional Analysis 4 Proceedings of the American Mathematical Society 4 Topology and its Applications 3 Fundamenta Mathematicae 3 Real Analysis Exchange 3 Transactions of the American Mathematical Society 2 Colloquium Mathematicum 2 Journal of Combinatorial Theory. Series A 2 Ergodic Theory and Dynamical Systems 2 Annals of Pure and Applied Logic 1 Journal of Mathematical Analysis and Applications 1 Mathematical Proceedings of the Cambridge Philosophical Society 1 American Journal of Mathematics 1 Annales de l\u2019Institut Fourier 1 Bulletin of the London Mathematical Society 1 Proceedings of the London Mathematical Society. Third Series 1 European Journal of Combinatorics 1 Journal of the American Mathematical Society 1 Geometric and Functional Analysis. GAFA 1 The Bulletin of Symbolic Logic 1 Abstract and Applied Analysis 1 Journal of the European Mathematical Society (JEMS) 1 Journal of Mathematical Logic 1 Journal of the Institute of Mathematics of Jussieu 1 Wiadomo\u015bci Matematyczne 1 Forum of Mathematics, Sigma\nall top 5\n\n#### Fields\n\n 34 Mathematical logic and foundations\u00a0(03-XX) 31 General topology\u00a0(54-XX) 21 Measure and integration\u00a0(28-XX) 14 Topological groups, Lie groups\u00a0(22-XX) 11 Abstract harmonic analysis\u00a0(43-XX) 9 Combinatorics\u00a0(05-XX) 6 Group theory and generalizations\u00a0(20-XX) 6 Real functions\u00a0(26-XX) 5 Order, lattices, ordered algebraic structures\u00a0(06-XX) 5 Functional analysis\u00a0(46-XX) 4 Dynamical systems and ergodic theory\u00a0(37-XX) 3 Manifolds and cell complexes\u00a0(57-XX) 3 Probability theory and stochastic processes\u00a0(60-XX) 1 General and overarching topics; collections\u00a0(00-XX) 1 General algebraic systems\u00a0(08-XX) 1 Computer science\u00a0(68-XX) 1 Game theory, economics, finance, and other social and behavioral sciences\u00a0(91-XX)\n\n#### Citations contained in zbMATH\n\n55 Publications have been cited 729 times in 604 Documents Cited by Year\nThe logic of public announcements, common knowledge, and private suspicions.\u00a0Zbl\u00a01386.03019\nBaltag, Alexandru; Moss, Lawrence S.; Solecki, S\u0142awomir\n2016\nAnalytic ideals and their applications.\u00a0Zbl\u00a00932.03060\nSolecki, S\u0142awomir\n1999\nBorel chromatic numbers.\u00a0Zbl\u00a00918.05052\nKechris, A. S.; Solecki, S.; Todorcevic, S.\n1999\nExtending partial isometries.\u00a0Zbl\u00a01124.54012\nSolecki, S\u0142awomir\n2005\nAnalytic ideals.\u00a0Zbl\u00a00862.04002\nSolecki, S\u0142awomir\n1996\nA proof of uniqueness of the Gurari\u012d space.\u00a0Zbl\u00a01290.46010\nKubi\u015b, Wies\u0142aw; Solecki, S\u0142awomir\n2013\nAutomatic continuity of homomorphisms and fixed points on metric compacta.\u00a0Zbl\u00a01146.22003\nRosendal, Christian; Solecki, S\u0142awomir\n2007\nDecomposing Borel sets and functions and the structure of Baire class 1 functions.\u00a0Zbl\u00a00899.03034\nSolecki, S\u0142awomir\n1998\nCovering analytic sets by families of closed sets.\u00a0Zbl\u00a00808.03031\nSolecki, S\u0142awomir\n1994\nAmenability, free subgroups, and Haar null sets in non-locally compact groups.\u00a0Zbl\u00a01115.22002\nSolecki, Slawomir\n2006\nCofinal types of topological directed orders.\u00a0Zbl\u00a01071.03034\nSolecki, S\u0142awomir; Todor\u010devi\u0107, Stevo\n2004\nProjective Fra\u00efss\u00e9 limits and the pseudo-arc.\u00a0Zbl\u00a01085.03028\nIrwin, Trevor; Solecki, Slawomir\n2006\nFilters and sequences.\u00a0Zbl\u00a00976.03053\nSolecki, S\u0142awomir\n2000\nOn Haar null sets.\u00a0Zbl\u00a00887.28006\nSolecki, S\u0142awomir\n1996\nSize of subsets of groups and Haar null sets.\u00a0Zbl\u00a01079.43002\nSolecki, S.\n2005\nPolish group topologies.\u00a0Zbl\u00a00941.54034\nSolecki, S\u0142awomir\n1999\nDecomposing Baire functions.\u00a0Zbl\u00a00742.04003\nCicho\u0144, J.; Morayne, M.; Pawlikowski, J.; Solecki, S.\n1991\nExtreme amenability of $$L_0$$, a Ramsey theorem, and L\u00e9vy groups.\u00a0Zbl\u00a01172.22002\nFarah, Ilijas; Solecki, S\u0142awomir\n2008\nHaar null and non-dominating sets.\u00a0Zbl\u00a00994.28006\nSolecki, S\u0142awomir\n2001\nApproximation of analytic by Borel sets and definable countable chain conditions.\u00a0Zbl\u00a00827.54023\nKechris, A. S.; Solecki, S.\n1995\nClosed subgroups generated by generic measure automorphisms.\u00a0Zbl\u00a01350.37003\nSolecki, S\u0142awomir\n2014\nAbstract approach to finite Ramsey theory and a self-dual Ramsey theorem.\u00a0Zbl\u00a01283.05176\nSolecki, S\u0142awomir\n2013\n$$G_{\\delta }$$ ideals of compact sets.\u00a0Zbl\u00a01221.03046\nSolecki, S\u0142awomir\n2011\nA Fubini theorem.\u00a0Zbl\u00a01115.03065\nSolecki, S\u0142awomir\n2007\nBorel subgroups of Polish groups.\u00a0Zbl\u00a01100.03039\nFarah, Ilijas; Solecki, S\u0142awomir\n2006\nTwo $$F_{\\sigma\\delta}$$ ideals.\u00a0Zbl\u00a01022.06006\nFarah, Ilijas; Solecki, Slawomir\n2003\nAutomatic continuity of group operations.\u00a0Zbl\u00a00882.22001\nSolecki, S.; Srivastava, S. M.\n1997\nEquivalence relations induced by actions of Polish groups.\u00a0Zbl\u00a00852.04003\nSolecki, S\u0142awomir\n1995\nUnitary representations of the groups of measurable and continuous functions with values in the circle.\u00a0Zbl\u00a01306.22001\nSolecki, S\u0142awomir\n2014\nBorel equivalence relations and Lascar strong types.\u00a0Zbl\u00a01326.03042\nKrupi\u0144ski, Krzysztof; Pillay, Anand; Solecki, S\u0142awomir\n2013\nLocal compactness for computable Polish metric spaces is $$\\varPi ^1_1$$-complete.\u00a0Zbl\u00a006496626\nNies, Andr\u00e9; Solecki, Slawomir\n2015\nAvoiding families and Tukey functions on the nowhere-dense ideal.\u00a0Zbl\u00a01221.03043\nSolecki, S\u0142awomir; Todorcevic, Stevo\n2011\nIsometry groups of separable metric spaces.\u00a0Zbl\u00a01170.54013\nMalicki, Maciej; Solecki, S\u0142awomir\n2009\nA $$G_\\delta$$ ideal of compact sets strictly above the nowhere dense ideal in the Tukey order.\u00a0Zbl\u00a01155.03033\nMoore, Justin Tatch; Solecki, S\u0142awomir\n2008\nTranslation invariant ideals.\u00a0Zbl\u00a01053.28006\nSolecki, S\u0142awomir\n2003\nActions of non-compact and non-locally compact Polish groups.\u00a0Zbl\u00a00972.03043\nSolecki, S\u0142awomir\n2000\nVaught\u2019s conjecture and the Glimm-Effros property for Polish transformation groups.\u00a0Zbl\u00a00921.03049\nHjorth, Greg; Solecki, Slawomir\n1999\nSpatial models of Boolean actions and groups of isometries.\u00a0Zbl\u00a01230.37007\nKwiatkowska, Aleksandra; Solecki, S\u0142awomir\n2011\nA Ramsey theorem for structures with both relations and functions.\u00a0Zbl\u00a01247.05256\nSolecki, S\u0142awomir\n2010\nTrichotomies for ideals of compact sets.\u00a0Zbl\u00a01105.03040\nMatheron, \u00c9.; Solecki, S.; Zelen\u00fd, M.\n2006\nDominating and unbounded free sets.\u00a0Zbl\u00a00947.03067\nSolecki, Slawomir; Spinas, Otmar\n1999\nA Ramsey theorem for partial orders with linear extensions.\u00a0Zbl\u00a01348.05210\nSolecki, S\u0142awomir; Zhao, Min\n2017\nAbstract approach to Ramsey theory and Ramsey theorems for finite trees.\u00a0Zbl\u00a01267.05287\nSolecki, S\u0142awomir\n2013\nDirect Ramsey theorem for structures involving relations and functions.\u00a0Zbl\u00a01232.05226\nSolecki, S\u0142awomir\n2012\nSpecial issue: Based on lectures of the workshop on the Urysohn space, Ben-Gurion University of the Negev, Beer Sheva, Israel, May 21\u201324, 2006.\u00a0Zbl\u00a01151.54303\nLeiderman, A. (ed.); Pestov, V. (ed.); Rubin, M. (ed.); Solecki, S. (ed.); Uspenski\u012d, V.\n2008\nThe space of composants of an indecomposable continuum.\u00a0Zbl\u00a01014.54021\nSolecki, S\u0142awomir\n2002\nMeasurability properties of sets of Vitali\u2019s type.\u00a0Zbl\u00a00795.28010\nSolecki, S\u0142awomir\n1993\nOn sets nonmeasurable with respect to invariant measures.\u00a0Zbl\u00a00784.28006\nSolecki, S\u0142awomir\n1993\nBaire theorem for ideals of sets.\u00a0Zbl\u00a01348.54023\nAvil\u00e9s, A.; Kadets, V.; P\u00e9rez, A.; Solecki, S.\n2017\nTukey reduction among analytic directed orders.\u00a0Zbl\u00a006749579\nSolecki, S\u0142awomir\n2015\nRecent developments in finite Ramsey theory: foundational aspects and connections with dynamics.\u00a0Zbl\u00a01373.05199\nSolecki, S\u0142awomir\n2014\nA Boolean action of $$C(M, U(1))$$ without a spatial model and a re-examination of the Cameron-Martin theorem.\u00a0Zbl\u00a01253.37014\nMoore, Justin Tatch; Solecki, S\u0142awomir\n2012\nRecovering Baire one functions on ultrametric spaces.\u00a0Zbl\u00a01175.26003\nDuncan, Jonathan; Solecki, Slawomir\n2009\nLocal inverses of Borel homomorphisms and analytic p-ideals.\u00a0Zbl\u00a01096.22003\nSolecki, S\u0142awomir\n2005\nMartingale proof of the existence of Lebesgue points.\u00a0Zbl\u00a00701.26009\nMorayne, Micha\u0142; Solecki, S\u0142awomir\n1990\nA Ramsey theorem for partial orders with linear extensions.\u00a0Zbl\u00a01348.05210\nSolecki, S\u0142awomir; Zhao, Min\n2017\nBaire theorem for ideals of sets.\u00a0Zbl\u00a01348.54023\nAvil\u00e9s, A.; Kadets, V.; P\u00e9rez, A.; Solecki, S.\n2017\nThe logic of public announcements, common knowledge, and private suspicions.\u00a0Zbl\u00a01386.03019\nBaltag, Alexandru; Moss, Lawrence S.; Solecki, S\u0142awomir\n2016\nLocal compactness for computable Polish metric spaces is $$\\varPi ^1_1$$-complete.\u00a0Zbl\u00a006496626\nNies, Andr\u00e9; Solecki, Slawomir\n2015\nTukey reduction among analytic directed orders.\u00a0Zbl\u00a006749579\nSolecki, S\u0142awomir\n2015\nClosed subgroups generated by generic measure automorphisms.\u00a0Zbl\u00a01350.37003\nSolecki, S\u0142awomir\n2014\nUnitary representations of the groups of measurable and continuous functions with values in the circle.\u00a0Zbl\u00a01306.22001\nSolecki, S\u0142awomir\n2014\nRecent developments in finite Ramsey theory: foundational aspects and connections with dynamics.\u00a0Zbl\u00a01373.05199\nSolecki, S\u0142awomir\n2014\nA proof of uniqueness of the Gurari\u012d space.\u00a0Zbl\u00a01290.46010\nKubi\u015b, Wies\u0142aw; Solecki, S\u0142awomir\n2013\nAbstract approach to finite Ramsey theory and a self-dual Ramsey theorem.\u00a0Zbl\u00a01283.05176\nSolecki, S\u0142awomir\n2013\nBorel equivalence relations and Lascar strong types.\u00a0Zbl\u00a01326.03042\nKrupi\u0144ski, Krzysztof; Pillay, Anand; Solecki, S\u0142awomir\n2013\nAbstract approach to Ramsey theory and Ramsey theorems for finite trees.\u00a0Zbl\u00a01267.05287\nSolecki, S\u0142awomir\n2013\nDirect Ramsey theorem for structures involving relations and functions.\u00a0Zbl\u00a01232.05226\nSolecki, S\u0142awomir\n2012\nA Boolean action of $$C(M, U(1))$$ without a spatial model and a re-examination of the Cameron-Martin theorem.\u00a0Zbl\u00a01253.37014\nMoore, Justin Tatch; Solecki, S\u0142awomir\n2012\n$$G_{\\delta }$$ ideals of compact sets.\u00a0Zbl\u00a01221.03046\nSolecki, S\u0142awomir\n2011\nAvoiding families and Tukey functions on the nowhere-dense ideal.\u00a0Zbl\u00a01221.03043\nSolecki, S\u0142awomir; Todorcevic, Stevo\n2011\nSpatial models of Boolean actions and groups of isometries.\u00a0Zbl\u00a01230.37007\nKwiatkowska, Aleksandra; Solecki, S\u0142awomir\n2011\nA Ramsey theorem for structures with both relations and functions.\u00a0Zbl\u00a01247.05256\nSolecki, S\u0142awomir\n2010\nIsometry groups of separable metric spaces.\u00a0Zbl\u00a01170.54013\nMalicki, Maciej; Solecki, S\u0142awomir\n2009\nRecovering Baire one functions on ultrametric spaces.\u00a0Zbl\u00a01175.26003\nDuncan, Jonathan; Solecki, Slawomir\n2009\nExtreme amenability of $$L_0$$, a Ramsey theorem, and L\u00e9vy groups.\u00a0Zbl\u00a01172.22002\nFarah, Ilijas; Solecki, S\u0142awomir\n2008\nA $$G_\\delta$$ ideal of compact sets strictly above the nowhere dense ideal in the Tukey order.\u00a0Zbl\u00a01155.03033\nMoore, Justin Tatch; Solecki, S\u0142awomir\n2008\nSpecial issue: Based on lectures of the workshop on the Urysohn space, Ben-Gurion University of the Negev, Beer Sheva, Israel, May 21\u201324, 2006.\u00a0Zbl\u00a01151.54303\nLeiderman, A. (ed.); Pestov, V. (ed.); Rubin, M. (ed.); Solecki, S. (ed.); Uspenski\u012d, V.\n2008\nAutomatic continuity of homomorphisms and fixed points on metric compacta.\u00a0Zbl\u00a01146.22003\nRosendal, Christian; Solecki, S\u0142awomir\n2007\nA Fubini theorem.\u00a0Zbl\u00a01115.03065\nSolecki, S\u0142awomir\n2007\nAmenability, free subgroups, and Haar null sets in non-locally compact groups.\u00a0Zbl\u00a01115.22002\nSolecki, Slawomir\n2006\nProjective Fra\u00efss\u00e9 limits and the pseudo-arc.\u00a0Zbl\u00a01085.03028\nIrwin, Trevor; Solecki, Slawomir\n2006\nBorel subgroups of Polish groups.\u00a0Zbl\u00a01100.03039\nFarah, Ilijas; Solecki, S\u0142awomir\n2006\nTrichotomies for ideals of compact sets.\u00a0Zbl\u00a01105.03040\nMatheron, \u00c9.; Solecki, S.; Zelen\u00fd, M.\n2006\nExtending partial isometries.\u00a0Zbl\u00a01124.54012\nSolecki, S\u0142awomir\n2005\nSize of subsets of groups and Haar null sets.\u00a0Zbl\u00a01079.43002\nSolecki, S.\n2005\nLocal inverses of Borel homomorphisms and analytic p-ideals.\u00a0Zbl\u00a01096.22003\nSolecki, S\u0142awomir\n2005\nCofinal types of topological directed orders.\u00a0Zbl\u00a01071.03034\nSolecki, S\u0142awomir; Todor\u010devi\u0107, Stevo\n2004\nTwo $$F_{\\sigma\\delta}$$ ideals.\u00a0Zbl\u00a01022.06006\nFarah, Ilijas; Solecki, Slawomir\n2003\nTranslation invariant ideals.\u00a0Zbl\u00a01053.28006\nSolecki, S\u0142awomir\n2003\nThe space of composants of an indecomposable continuum.\u00a0Zbl\u00a01014.54021\nSolecki, S\u0142awomir\n2002\nHaar null and non-dominating sets.\u00a0Zbl\u00a00994.28006\nSolecki, S\u0142awomir\n2001\nFilters and sequences.\u00a0Zbl\u00a00976.03053\nSolecki, S\u0142awomir\n2000\nActions of non-compact and non-locally compact Polish groups.\u00a0Zbl\u00a00972.03043\nSolecki, S\u0142awomir\n2000\nAnalytic ideals and their applications.\u00a0Zbl\u00a00932.03060\nSolecki, S\u0142awomir\n1999\nBorel chromatic numbers.\u00a0Zbl\u00a00918.05052\nKechris, A. S.; Solecki, S.; Todorcevic, S.\n1999\nPolish group topologies.\u00a0Zbl\u00a00941.54034\nSolecki, S\u0142awomir\n1999\nVaught\u2019s conjecture and the Glimm-Effros property for Polish transformation groups.\u00a0Zbl\u00a00921.03049\nHjorth, Greg; Solecki, Slawomir\n1999\nDominating and unbounded free sets.\u00a0Zbl\u00a00947.03067\nSolecki, Slawomir; Spinas, Otmar\n1999\nDecomposing Borel sets and functions and the structure of Baire class 1 functions.\u00a0Zbl\u00a00899.03034\nSolecki, S\u0142awomir\n1998\nAutomatic continuity of group operations.\u00a0Zbl\u00a00882.22001\nSolecki, S.; Srivastava, S. M.\n1997\nAnalytic ideals.\u00a0Zbl\u00a00862.04002\nSolecki, S\u0142awomir\n1996\nOn Haar null sets.\u00a0Zbl\u00a00887.28006\nSolecki, S\u0142awomir\n1996\nApproximation of analytic by Borel sets and definable countable chain conditions.\u00a0Zbl\u00a00827.54023\nKechris, A. S.; Solecki, S.\n1995\nEquivalence relations induced by actions of Polish groups.\u00a0Zbl\u00a00852.04003\nSolecki, S\u0142awomir\n1995\nCovering analytic sets by families of closed sets.\u00a0Zbl\u00a00808.03031\nSolecki, S\u0142awomir\n1994\nMeasurability properties of sets of Vitali\u2019s type.\u00a0Zbl\u00a00795.28010\nSolecki, S\u0142awomir\n1993\nOn sets nonmeasurable with respect to invariant measures.\u00a0Zbl\u00a00784.28006\nSolecki, S\u0142awomir\n1993\nDecomposing Baire functions.\u00a0Zbl\u00a00742.04003\nCicho\u0144, J.; Morayne, M.; Pawlikowski, J.; Solecki, S.\n1991\nMartingale proof of the existence of Lebesgue points.\u00a0Zbl\u00a00701.26009\nMorayne, Micha\u0142; Solecki, S\u0142awomir\n1990\nall top 5\n\n#### Cited by 515 Authors\n\n 25 Solecki, S\u0142awomir 21 van Ditmarsch, Hans Pieter 17 Miller, Benjamin David 16 Ostaszewski, Adam J. 13 Todorcevic, Stevo B. 12 Lecomte, Dominique 12 Rosendal, Christian 10 Bingham, Nicholas Hugh 10 Farah, Ilijas 10 Kubi\u015b, Wies\u0142aw 10 van Benthem, Johan F. A. K. 9 Banakh, Taras Onufrievich 9 Kechris, Alexander S. 9 Marks, Andrew S. 9 Smets, Sonja J. L. 8 Balcerzak, Marek 8 Baltag, Alexandru 8 Conley, Clinton Taylor 8 G\u0142\u0105b, Szymon 8 Ne\u0161et\u0159il, Jaroslav 8 Sabok, Marcin 8 Shelah, Saharon 8 van Eijck, Jan 8 Vidny\u00e1nszky, Zolt\u00e1n 7 \u00c5gotnes, Thomas 7 Darji, Udayan B. 7 Doucha, Michal 7 Filip\u00f3w, Rafa\u0142 7 Hru\u0161\u00e1k, Michael 7 Kooi, Barteld Pieter 7 Kwiatkowska, Aleksandra 7 Malicki, Maciej 7 Zakrzewski, Piotr 6 Carroy, Rapha\u00ebl 6 Elekes, M\u00e1rton 6 Farkas, Barnab\u00e1s 6 Gao, Su 6 Herzig, Andreas 6 Hoshi, Tomohiro 6 Kwela, Adam 6 Lupini, Martino 6 Pol, Roman 6 Schwarzentruber, Fran\u00e7ois 6 Seward, Brandon 6 Tsankov, Todor 6 van der Hoek, Wiebe 5 Aucher, Guillaume 5 Ben-Yaacov, Ita\u00ef 5 Bolander, Thomas 5 Hjorth, Gregory 5 Krupi\u0144ski, Krzysztof 5 Melleray, Julien 5 Motto Ros, Luca 5 Mro\u017cek, Nikodem 5 Rec\u0142aw, Ireneusz 5 Szuca, Piotr 5 Tucker-Drob, Robin D. 5 Vel\u00e1zquez-Quesada, Fernando Raymundo 5 Wang, Yanjing 5 Zapletal, Jind\u0159ich 4 Barto\u0161ov\u00e1, Dana 4 Borodulin-Nadzieja, Piotr 4 Das, Pratulananda 4 D\u00e9gremont, C\u00e9dric 4 Di Prisco, Carlos Augusto 4 Ding, Longyun 4 Ditmarsch, Hans P.van 4 Drewnowski, Lech 4 Garbuli\u0144ska-W\u0119grzyn, Joanna 4 Hubi\u010dka, Jan 4 Lorini, Emiliano 4 Ma, Minghui 4 Ma\u0161ulovi\u0107, Dragan 4 Matheron, \u00c9tienne 4 Nguyen Van Th\u00e9, Lionel 4 Pestov, Vladimir G. 4 Protasov, Igor Volodymyrovych 4 Schneider, Friedrich Martin 4 Tryba, Jacek 4 Uzcategui, Carlos Enrique 4 Zelen\u00fd, Miroslav 3 Asher, Nicholas M. 3 Becker, Howard S. 3 Bowen, Lewis Phylip 3 Camerlo, Riccardo 3 Chodounsk\u00fd, David 3 Dobrinen, Natasha L. 3 Elek, G\u00e1bor 3 French, Tim 3 Ghasemi, Saeed 3 Gierasimczuk, Nina 3 Guzm\u00e1n Gonz\u00e1lez, Osvaldo 3 Kallman, Robert R. 3 Kihara, Takayuki 3 Kiss, Viktor 3 Kone\u010dn\u00fd, Mat\u011bj 3 Labuda, Iwo 3 Leonetti, Paolo 3 M\u00e1trai, Tam\u00e1s 3 Pacuit, Eric ...and 415 more Authors\nall top 5\n\n#### Cited in 100 Serials\n\n 50 Topology and its Applications 40 The Journal of Symbolic Logic 35 Synthese 31 Israel Journal of Mathematics 31 Journal of Mathematical Analysis and Applications 27 Advances in Mathematics 22 Annals of Pure and Applied Logic 21 Transactions of the American Mathematical Society 16 Proceedings of the American Mathematical Society 16 Journal of Logic, Language and Information 14 Ergodic Theory and Dynamical Systems 14 Archive for Mathematical Logic 13 Studia Logica 12 Journal of Philosophical Logic 12 The Bulletin of Symbolic Logic 10 Fundamenta Mathematicae 10 Journal of Functional Analysis 8 Artificial Intelligence 8 Journal of Applied Non-Classical Logics 7 Journal of Combinatorial Theory. Series A 7 Acta Mathematica Hungarica 7 Forum of Mathematics, Sigma 6 Information and Computation 6 Mathematical Logic Quarterly (MLQ) 6 Journal of Mathematical Logic 5 European Journal of Combinatorics 5 Journal of the American Mathematical Society 5 Forum Mathematicum 5 Revista de la Real Academia de Ciencias Exactas, F\u00edsicas y Naturales. Serie A: Matem\u00e1ticas. RACSAM 4 Discrete Mathematics 4 Inventiones Mathematicae 4 Indagationes Mathematicae. New Series 4 Journal of Applied Logic 3 Algebra Universalis 3 Annales de l\u2019Institut Fourier 3 Czechoslovak Mathematical Journal 3 Illinois Journal of Mathematics 3 Memoirs of the American Mathematical Society 3 Monatshefte f\u00fcr Mathematik 3 Order 3 Geometric and Functional Analysis. GAFA 3 Aequationes Mathematicae 3 Journal of the European Mathematical Society (JEMS) 3 Journal of the Institute of Mathematics of Jussieu 3 Logical Methods in Computer Science 2 Mathematical Proceedings of the Cambridge Philosophical Society 2 Archiv der Mathematik 2 Mathematische Annalen 2 Journal of Mathematical Sciences (New York) 2 Sarajevo Journal of Mathematics 2 Groups, Geometry, and Dynamics 2 European Journal of Mathematics 1 Bulletin of the Australian Mathematical Society 1 Journal d\u2019Analyse Math\u00e9matique 1 Russian Mathematical Surveys 1 Ukrainian Mathematical Journal 1 Applied Mathematics and Computation 1 Collectanea Mathematica 1 Commentationes Mathematicae Universitatis Carolinae 1 Dissertationes Mathematicae 1 Fuzzy Sets and Systems 1 Journal of the London Mathematical Society. Second Series 1 Journal of Mathematical Economics 1 Journal of Pure and Applied Algebra 1 Notre Dame Journal of Formal Logic 1 Proceedings of the London Mathematical Society. Third Series 1 Quaestiones Mathematicae 1 Real Analysis Exchange 1 Rendiconti del Circolo Matem\u00e0tico di Palermo. Serie II 1 Results in Mathematics 1 Ricerche di Matematica 1 Theoretical Computer Science 1 International Journal of Algebra and Computation 1 MSCS. Mathematical Structures in Computer Science 1 Linear Algebra and its Applications 1 Bulletin of the Polish Academy of Sciences, Mathematics 1 Applied Categorical Structures 1 St. Petersburg Mathematical Journal 1 Electronic Research Announcements of the American Mathematical Society 1 Topoi 1 Geometry & Topology 1 Journal of Group Theory 1 Revista Matem\u00e1tica Complutense 1 Annals of Mathematics. Second Series 1 Acta Mathematica Sinica. English Series 1 Comptes Rendus. Math\u00e9matique. Acad\u00e9mie des Sciences, Paris 1 Central European Journal of Mathematics 1 Oberwolfach Reports 1 Sibirskie \u00c8lektronnye Matematicheskie Izvestiya 1 Logica Universalis 1 Journal of Modern Dynamics 1 Banach Journal of Mathematical Analysis 1 Confluentes Mathematici 1 Journal of Logic and Analysis 1 The Review of Symbolic Logic 1 Forum of Mathematics, Pi 1 Axioms 1 Journal de l\u2019\u00c9cole Polytechnique \u2013 Math\u00e9matiques 1 Transactions of A. Razmadze Mathematical Institute 1 Annales Henri Lebesgue\nall top 5\n\n#### Cited in 33 Fields\n\n 401 Mathematical logic and foundations\u00a0(03-XX) 190 General topology\u00a0(54-XX) 94 Measure and integration\u00a0(28-XX) 72 Topological groups, Lie groups\u00a0(22-XX) 69 Combinatorics\u00a0(05-XX) 68 Computer science\u00a0(68-XX) 60 Dynamical systems and ergodic theory\u00a0(37-XX) 57 Functional analysis\u00a0(46-XX) 37 Group theory and generalizations\u00a0(20-XX) 37 Real functions\u00a0(26-XX) 26 Abstract harmonic analysis\u00a0(43-XX) 24 Order, lattices, ordered algebraic structures\u00a0(06-XX) 22 Sequences, series, summability\u00a0(40-XX) 18 Game theory, economics, finance, and other social and behavioral sciences\u00a0(91-XX) 9 Number theory\u00a0(11-XX) 9 Manifolds and cell complexes\u00a0(57-XX) 8 Probability theory and stochastic processes\u00a0(60-XX) 7 Difference and functional equations\u00a0(39-XX) 6 Category theory; homological algebra\u00a0(18-XX) 5 General and overarching topics; collections\u00a0(00-XX) 5 General algebraic systems\u00a0(08-XX) 4 Geometry\u00a0(51-XX) 4 Information and communication theory, circuits\u00a0(94-XX) 3 History and biography\u00a0(01-XX) 3 Operator theory\u00a0(47-XX) 2 Convex and discrete geometry\u00a0(52-XX) 2 Differential geometry\u00a0(53-XX) 2 Global analysis, analysis on manifolds\u00a0(58-XX) 1 Field theory and polynomials\u00a0(12-XX) 1 Linear and multilinear algebra; matrix theory\u00a0(15-XX) 1 Functions of a complex variable\u00a0(30-XX) 1 Ordinary differential equations\u00a0(34-XX) 1 Numerical analysis\u00a0(65-XX)","date":"2021-04-18 20:50:16","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\": 0.5270798802375793, \"perplexity\": 6429.024862742062}, \"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\/1618038860318.63\/warc\/CC-MAIN-20210418194009-20210418224009-00376.warc.gz\"}"}
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Sabia ovalifolia är en tvåhjärtbladig växtart som beskrevs av S. Y. Liu. Sabia ovalifolia ingår i släktet Sabia och familjen Sabiaceae. Inga underarter finns listade.
Källor
Tvåhjärtbladiga blomväxter
ovalifolia
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{
"redpajama_set_name": "RedPajamaWikipedia"
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Il Polo della Democrazia Sociale di Romania (in rumeno Polul Democrat-Social din România) è stata una coalizione politica romena di centro-sinistra, formatasi in vista delle Elezioni parlamentari in Romania del 2000.
Storia
Contesto e formazione della coalizione
In seguito alla sconfitta alle elezioni parlamentari e presidenziali del 1996, il Partito della Democrazia Sociale di Romania (PDSR), che sotto diverse denominazioni (FSN, FDSN) aveva dominato la scena politica dalla rivoluzione del 1989 in poi, andò incontro ad una ristrutturazione interna per fronteggiare il governo di centro-destra della Convenzione Democratica Romena (CDR), con l'obiettivo di tornare al potere nel 2000. Il PDSR del suo presidente Ion Iliescu e del vicepresidente Adrian Năstase, nello stesso anno siglò importanti accordi pre-elettorali con altre forze minori: il gruppo del centro-destra liberale del Partito Umanista Romeno (PUR, fondato nel 1991 e mai entrato in parlamento) e il Partito Social Democratico Romeno (PSDR, dal 1996 facente parte di un'instabile alleanza con i partiti di governo di centro-destra).
Il 25 febbraio il PDSR gettò le basi per un'alleanza con il Partito Umanista Romeno, che era finalizzata alla partecipazione su liste comuni alle elezioni parlamentari del 26 novembre e al sostegno congiunto ad Iliescu per l'incarico di presidente della repubblica. Fu, quindi, l'atto fondamentale che poi portò alla nascita del Polo della Democrazia Sociale di Romania (rumeno: Polul Democrat-Social din România). Dopo le elezioni amministrative di giugno, il 7 settembre il partito concluse due fondamentali accordi con il Partito Social Democratico Romeno di Alexandru Athanasiu. Grazie al primo il PSDR aderì alla coalizione del Polo della Democrazia Sociale di Romania e, con il secondo, si impegnò ad accettare la creazione di un gruppo parlamentare comune nel caso di vittoria alle elezioni, che avrebbe portato la fusione tra i due partiti nella prima parte del 2001.
Successo elettorale del 2000
Il 26 novembre 2000 in occasione delle elezioni legislative la coalizione conseguì 155 deputati e 65 senatori, mentre il secondo partito più votato fu l'ultranazionalista Partito Grande Romania (PRM) di Corneliu Vadim Tudor. Alle elezioni presidenziali, infatti, Iliescu si ritrovò ad affrontare il leader del PRM, personaggio estremista e giustizialista che rappresentava il voto di protesta dell'elettorato. Di fronte al pericolo rappresentato dall'estremismo nazionalista persino gli altri gruppi antagonisti del PDSR, cioè PNL, PD e UDMR, considerandolo come l'alternativa politicamente più credibile, appoggiarono Iliescu in occasione del ballottaggio presidenziale del 10 dicembre 2000. Iliescu ottenne il 66,83% dei voti e Vadim Tudor si fermò al 33,17%.
Mentre Iliescu divenne nuovamente presidente della repubblica, Năstase fu designato per il ruolo di primo ministro. La coalizione, che non aveva conseguito la maggioranza assoluta, per ottenere l'investitura e garantire la sopravvivenza del governo, fu costretta a richiedere l'appoggio parlamentare di Partito Nazionale Liberale (PNL) e Unione Democratica Magiara di Romania (UDMR). Sulla base di interessi comuni, quali lo sviluppo economico della Romania e l'integrazione euro-atlantica del paese, il 27 dicembre fu firmato il protocollo di intesa tra la coalizione di governo e gli altri due partiti. L'accordo con il PNL, tuttavia, sarebbe saltato il 18 aprile 2001.
In applicazione dell'accordo siglato tra PDSR e PSDR nel 2000, in occasione della conferenza nazionale del 16 giugno 2001, si concretizzò la fusione tra le due formazioni, che già partecipavano ad un gruppo parlamentare comune alla camera dei deputati e al senato. Si realizzò, quindi, l'unificazione di due tra i più importanti gruppi socialdemocratici del paese, che si unirono intorno all'insegna di Partito Social Democratico (PSD).
Nel maggio del 2001 anche il PUR cambiò nome, passando alla nuova dicitura di Partito Umanista di Romania - Social Liberale (rumeno: Partidul Umanist din România – Social Liberal, PUR-SL).
La coalizione ottenne 155 seggi alla camera e 65 al senato, così ripartiti:
Fine della coalizione
Nel giugno del 2003 il primo ministro Năstase, nel quadro di una più ampia risistemazione della squadra di governo, decretò l'abolizione del ministero per le piccole e medie imprese, l'unico ministero assegnato al PUR e condotto da Silvia Ciornei. Ritenendo scandalosa tale decisione, il leader del PUR Dan Voiculescu accusò il PDSR di aver rotto gli accordi stretti precedentemente, di aver mostrato una costante arroganza nei confronti del partito e di aver continuamente cercato di forzare l'identità dottrinaria del PUR. Questo decise di lasciare il governo e, nel mese di settembre, tutti i suoi deputati abbandonarono il gruppo parlamentare del PSD, passando al gruppo degli indipendenti. Rassegnarono le proprie dimissioni, inoltre, il ministro Ciornei, un viceministro, un segretario di stato al ministero delle comunicazioni, un prefetto e tre viceprefetti. I tre senatori del PUR (Ioan Pop de Popa, Dinu Marin e Aureliu Leca), invece, preferirono passare al PSD.
Risultati elettorali
Note
Voci correlate
Partito della Democrazia Sociale di Romania
Partito Social Democratico (Romania)
Politica della Romania
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Q: Ajax call getting CORS errors/conflicts What i'm trying to do is as follows, we have different external content pages that have one or more of our products and need a simple "add to cart" button.
To accomplish this i want to use a ajax call that talks to the magento api with json following the dev site of magento:
https://devdocs.magento.com/guides/v2.1/get-started/order-tutorial/order-add-items.html
Before i can add a product i need to create a cart (Quote id). So first i need to create a guest-cart and then add the products to cart, i have been able to do this in Postman but when i try this in a simple html page i'm getting errors...
I'm getting the following error:
Failed to load https:///rest/V1/guest-carts: No 'Access-Control-Allow-Origin' header is present on the requested resource. Origin 'http://localhost' is therefore not allowed access. The response had HTTP status code 404.
My code is as follows:
<script>
$(document).ready(function(){
var settings = {
"async": true,
"crossDomain": true,
"url": "https://<domain>/rest/V1/guest-carts",
"method": "POST",
"contentType": "application/json",
"headers": {
// "Content-Type": "application/json",
"Accept": "application/json",
"Cache-Control": "no-cache"
}
}
$("button").click(function(){
$.ajax(settings).done(function (result) {
console.log(result);
alert("Awesome, it worked!");
})
.fail(function( status, errorThrown ) {
alert( "Sorry, there was a problem!" );
console.log( "Error: " + errorThrown );
console.log( "Status: " + status );
})
});
});
</script>
I found that CORS was giving issues with chrome and the server because it sends first a method OPTIONS to check if the server allows the method POST. So i added the code from this site: https://gist.github.com/michiel/1064640 to the nginx config to allow OPTIONS, GET and POST. When i check in chrome at the Network tab i see that OPTIONS gives status code 204 back but the POST gets a 404 status code.
This is the part where i'm running out of options since i don't know what to do next. Hopefully you guys can point me in the right direction.
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"redpajama_set_name": "RedPajamaStackExchange"
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Senior Executives, Directors and Advisory Partners
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Ms. Gitelson joined Oaktree in 2014. Prior to Oaktree, she worked for the global head of Blackstone Advisory on business development and communications matters, and as an associate in Blackstone's Advisory and Restructuring team in London. Prior to this role, Ms. Gitelson spent four years in fixed income institutional sales at Deutsche Bank in New York where she was a vice president in their Structured Product Sales and Structured Private Placements groups. Ms. Gitelson began her career as an attorney at Davis Polk & Wardwell in New York where she worked on cross-border M&A and securities transactions as part of their Latin America and Spain group. Ms. Gitelson graduated cum laude with a B.A. in intellectual history and minors in French and art history from the University of Pennsylvania, holds a J.D. from Stanford Law School and a Master's in Finance from London Business School. She is fluent in Spanish and proficient in French.
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"redpajama_set_name": "RedPajamaCommonCrawl"
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\section{Introduction}
\label{s:intro}
With recent advances in data collection technologies, it is increasingly common to have images as a part of collected data. For instance, in clinical trials, we can collect both fMRI images and clinical information from the same group of patients. However, it is challenging to analyze such data because variables represent different forms; we have both matrix (or tensor) images and vector covariates. The traditional statistical approaches face inferential issues when dealing with input variables in the form of images due to the high dimensionality and multicollinearity issues. Convolutional neural networks (CNN) \citep[cf.][]{krizhevsky2012imagenet,he2016deep} are popular approaches for image processing without facing inferential challenges. However, CNNs cannot quantify the prediction uncertainty for the response variable and the estimation uncertainty for the covariates, which are often of primary interest in statistical inference. Furthermore, the complex structure of CNN can lead to nonconvex optimization \citep{ge2015escaping,kleinberg2018alternative,daneshmand2018escaping,alzubaidi2021review}, which cannot guarantee the convergence of weight parameters. In this manuscript, we propose a computationally efficient Bayesian approach by embedding CNNs within the generalized linear model (GLM). Our method can take advantage of both statistical models and advanced neural networks. We observe that our two-stage approach can provide a more accurate parameter estimate compared to the standard CNN.
There is a growing literature on Bayesian deep learning methods \citep[cf.][]{neal2012bayesian, blundell2015weight} to quantify the uncertainty by regarding weight parameters in deep neural networks (DNN) as random variables. With an appropriate choice of priors, we have posterior distributions for the standard Bayesian inference. Furthermore, Bayesian alternatives can be robust for a small sample size compared to the standard DNN \citep{gal2016bayesian, gal2016dropout}. Despite its apparent advantages, however, Bayesian DNNs have been much less used than the classical one due to the difficulty in exploring the full posterior distributions. To circumvent the difficulty, variational Bayes (VB) methods have been proposed to approximate complex high-dimensional posteriors; examples include VB for feed-forward neural networks \citep{graves2011practical, tran2020bayesian} and CNNs \citep{shridhar2019comprehensive}. Although VB can alleviate computational costs in Bayesian networks, to the best of our knowledge, no existing approach provides a general Bayesian framework to study image and vector type variables simultaneously. Furthermore, most previous works have focused on prediction uncertainty but not on estimation uncertainty of model parameters; interpretations of regression coefficients have not been studied much in the Bayesian deep learning context.
In this manuscript, we propose a Bayesian convolutional generalized linear model (BayesCGLM) that can study different types of variables simultaneously (e.g., regress a scalar response on vector covariates and array type images) and easily be implemented in various applications. A BayesCGLM consists of the following steps: (1) train Bayes CNN \citep{gal2016bayesian} with the correlated input (images or correlated variables) and output (response variable); (2) extract Monte Carlo samples of features from the last layer of the fitted Bayes CNN; and (3) fit a GLM by regressing a response variable on augmented covariates with the features obtained from each Monte Carlo sample. For each Monte Carlo sample of the feature, we fit an individual GLM and aggregate its posterior, which can be implemented in parallel. These steps allow us to fully account for uncertainties in the estimation procedure. We also study an often overlooked challenge in Bayesian deep learning: practical issues in obtaining credible intervals of parameters and assessing whether they provide nominal coverage.
In our method, we model Bayes CNN via a deep Gaussian process \citep{damianou2013deep}, which is a nested Gaussian process. Gaussian process \citep{rasmussen2003gaussian} are widely used to represent complex function relationships due to their flexibility. Especially, \citet{gal2016bayesian, gal2016dropout} show that neural networks with dropout applied before the last layer is mathematically equivalent to the approximation of the posteriors of the deep Gaussian process \citep{damianou2013deep}. This procedure is referred to as Bernoulli approximating variational distributions \citep{gal2016bayesian,gal2016dropout}. We build upon their approach to sample features from the approximated posterior of the last layer in Bayes CNN. Training Bayes CNN with dropout approximation does not require additional computing costs compared to the frequentist one. There have been several recent proposals to extract features from the hidden layer of neural network models \citep{tran2020bayesian,bhatnagar2020computer,fong2021forward}, and our method is motivated by them. The features can be regarded as reduced-dimensional summary statistics that measure complex non-linear relationships between the high-dimensional input and output; the dimension of the extracted features is much lower than that of the original input. We observe that CNN is especially useful for extracting information from correlated inputs such as images or spatial basis functions.
The outline of the remainder of this paper is as follows. In Section 2, we introduce a Bayesian DNN based on Gaussian process approximation and its estimation procedure. In Section 3, we propose a computationally efficient BayesCGLM and provide implementation details. In Section 4, we study the performance of BayesCGLM through simulated data examples. In Section 5, we apply BayesCGLM to several real-world data sets. We conclude with a discussion and summary in Section 6.
\section{A Bayesian Deep Neural Network: Gaussian Process Approximation} \label{BayesDeep}
In this section, we first give a brief review of Bayesian DNNs and their approximations to deep Gaussian processes. Let $\mathbf{D}=\lbrace (\mathbf{x}_n,\mathbf{y}_n), n=1,\cdots, N\rbrace$ be the observed dataset with input $\mathbf{x}_n \in {\mathbb R}^{k_0}$ and response $\mathbf{y}_n \in {\mathbb R}^{d}$. A standard DNN is used to model the relationship between the input $\mathbf{x}_n$ and the output $\mathbf{o}_n \in {\mathbb R}^{d}$, which can be regarded as an expectation of $\mathbf y_n$ given $\mathbf x_n$, in terms of a statistical viewpoint. Suppose we consider a DNN model with $L$ layers, where the $l$th layer has $k_l$ nodes for $l=1,\cdots, L$. Then we can define the weight matrix $\mathbf{W}_l \in {\mathbb R}^{k_{l} \times k_{l-1}}$ that connects the $(l-1)$th layer to $l$th hidden layer and the bias vector $\mathbf{b}_l \in {\mathbb R}^{k_{l}}$. Then $\bm{\theta}= \lbrace (\mathbf{W}_l,\mathbf{b}_l), l=1,\cdots, L\rbrace$ is a set of parameters in the neural network that belongs to a parameter space $\bm{\Theta}$.
To model complex data, the number of neural network parameters grows tremendously with the increasing number of layers and nodes. Such large networks are computationally intensive to use and can lead to overfitting issues. Dropout \citep{srivastava2014dropout} has been suggested as a simple but effective strategy to address both problems. The term "dropout" refers to randomly removing the connections from nodes during the training step of the network. While training, the model randomly chooses the units to drop out with some probability. The structure of neural networks with dropout is
{\small
\begin{equation}
\mathbf{o}_{n}=\sigma_L \Big(\mathbf{W}_{L}\sigma_{L-1}\Big(\mathbf{W}_{L-1} \cdots \sigma_3 \Big(\mathbf{W}_{3}\sigma_2 \Big(\mathbf{W}_{2} \sigma_1 \Big(\mathbf{W}_{1}\mathbf{x}_{n} +\mathbf{b}_1\Big)\circ\mathbf{d}_{2} +\mathbf{b}_2\Big)\circ\mathbf{d}_{3} +\mathbf{b}_3\Big) \cdots \Big)\circ\mathbf{d}_{L-1} + \mathbf{b}_{L-1} \Big)\circ\mathbf{d}_{L} +\mathbf{b}_{L} \Big)
\label{eq1drop}
\end{equation}}
where $\sigma_l(\cdot)$ is an activation function; for instance, the rectified linear (ReLU) or the hyperbolic tangent function (TanH) can be used. Here $\mathbf{d}_{l} \in {\mathbb R}^{k_l}, l=2,\cdots,L$, are dropout vectors whose components follow Bernoulli distribution with pre-specified success probability $p_l$ independently. The notation ``$\circ$" indicates the elementwise multiplication of vectors. Although dropout can alleviate overfitting issues, quantifying uncertainties in predictions or parameter estimates remains challenging for the frequentist approach.
To address this, Bayesian alternatives \citep{friedman1997bayesian, friedman2003being, chen2012good} have been proposed by considering probabilistic modeling of neural networks. The Bayesian approach considers an arbitrary function that is likely to generate data $\mathbf{D}$. Especially, a deep Gaussian process (deep GP) has been widely used to represent complex neural network structures for both supervised and unsupervised learning problems \citep{gal2016dropout, damianou2013deep}. By modeling data generating mechanism as a deep GP, a hierarchical Bayesian neural network \citep{gal2016dropout} can be constructed. For $l\geq 2$, let $\mathbf{f}_{n,l} =\mathbf{W}_l \bm{\phi}_{n,l-1} + \mathbf{b}_l \in {\mathbb R}^{k_l}$ be the linear feature which is a random variable in the Bayesian framework; we set independent normal priors for weight and bias parameters as $p(\mathbf{W}_{l})$ and $ p(\mathbf{b}_l)$ respectively. Then we can also define $\bm{\phi}_{n,l}=\sigma_{l}(\mathbf{f}_{n,l}) \in {\mathbb R}^{k_l}$ as the nonlinear output feature from the $l$th layer after applying an activation function. For the first layer (i.e., $l=1$), we have $\mathbf{f}_{n,1}=\mathbf{W}_1 \mathbf{x}_{n} + \mathbf{b}_1$. Then the deep GP can be written as
\begin{equation}
\begin{split}
\mathbf{f}_{n,2}|\mathbf{x}_n &\sim N(0, \mathbf{\Sigma}_{nn,1})\\
\mathbf{f}_{n,3}|\mathbf{f}_{n,2}&\sim N(0, \mathbf{\Sigma}_{nn,2})\\
\vdots \\
\mathbf{f}_{n,L}|\mathbf{f}_{n,L-1} &\sim N(0, \mathbf{\Sigma}_{nn,L-1})\\
\mathbf{y}_n|\mathbf{f}_{n,L} &\sim p(\mathbf{y}_n|\mathbf{f}_{n,L}),
\label{multilayers}
\end{split}
\end{equation}
where the covariance $\mathbf{\Sigma}_{nn,l} \in {\mathbb R}^{k_l \times k_l}$ is
\begin{equation}
\begin{split}
\mathbf{\Sigma}_{nn,l} &= \int \int \sigma_l(\mathbf{W}_l \bm{\phi}_{n,l-1} + \mathbf{b}_l)
\sigma_l(\mathbf{W}_l \bm{\phi}_{n,l-1} + \mathbf{b}_l)^{\top}p(\mathbf{W}_l)p(\mathbf{b}_l)d\mathbf{W}_l d\mathbf{b}_l.
\label{covfuncsmultilayer}
\end{split}
\end{equation}
This implies that an arbitrary data generating function is modeled through a nested GP with the covariance of the extracted features from the previous hidden layer. Since the integral calculation in \eqref{covfuncsmultilayer} is intractable, we can replace it with a finite covariance rank approximation in practice; with increasing $k_l$ (number of hidden nodes) the finite rank approximation becomes close to \eqref{covfuncsmultilayer} \citep{gal2016dropout}. In \eqref{multilayers}, $\mathbf{y}_n$ is a response variable that has a distribution $p(\cdot)$ belonging to the exponential family with $g(E[\mathbf{y}_n|\bm{\phi}_{n,L-1}])=\mathbf{W}_{L}\bm{\phi}_{n,L-1}+\mathbf{b}_{L}$; here $g(\cdot)$ is a one-to-one continuously differential link function. Note that $g^{-1}(\cdot)$ can be regarded as $\sigma_{L}(\cdot)$ in \eqref{eq1drop} in terms of traditional DNN notation. To carry out Bayesian inference, we need to approximate the posterior distributions of deep GP.
However, Bayesian inference for deep GP is not straightforward to implement. With increasing hidden layers and nodes, the number of parameters $\bm{\theta}$ grows exponentially, resulting in a high-dimensional posterior distribution. Constructing Markov chain Monte Carlo (MCMC) samplers \citep{neal2012bayesian} is impractical due to the slow mixing of high-dimensional parameters and the complex sequential structure of the networks. Instead, variational approximation methods can be practical and computationally feasible for Bayesian inference \citep{bishop2006pattern, blei2017variational}. Variational Bayes approximates the true $\pi(\bm{\theta}|\mathbf{D})$ via the approximate distribution (variational distribution) $q(\bm{\theta})$, which is a class of distribution that can be easily evaluated. Given a class of distributions, the optimal variational parameters are set by minimizing the Kullback–Leibler (KL) divergence between the variational distribution $q(\bm{\theta})$ and the true posterior $\pi(\bm{\theta}|\mathbf{D})$ where minimizing the KL divergence is equivalent to maximizing $\mbox{E}_q[{\log(\pi(\bm{\theta},\mathbf{D}))}]-\mbox{E}_q[\log q({\bm{\theta}})]$, the evidence lower bound (ELBO).
\citet{gal2016dropout} shows that a deep neural network in \eqref{eq1drop} is mathematically equivalent to a variational approximation of deep Gaussian process \citep{damianou2013deep}. That is the log ELBO of deep GP converges to the frequentist loss function of deep neural network with dropout layers. With the independent variational distribution $q(\bm{\theta}):=\prod_{l=1}^{L}q(\mathbf{W}_l)q(\mathbf{b}_l)$, the log ELBO of the deep GP is
\begin{equation} \begin{split}
\mathcal{L}_{\text{GP-VI}} := \sum_{n=1}^{N}\int \cdots \int \prod_{l=1}^{L}q(\mathbf{W}_{l})q(\mathbf{b}_{l}) \log p(\mathbf{y}_n| \mathbf{x}_n,\lbrace \mathbf{W}_{l}, \mathbf{b}_{l} \rbrace_{l=1}^{L}) d\mathbf{W}_1\cdots d\mathbf{b}_{1}\cdots d\mathbf{W}_L d\mathbf{b}_{L} \\
- \text{KL}\Big (\prod_{l=1}^{L}q(\mathbf{W}_{l})q(\mathbf{b}_{l})\Big |\Big |p( \lbrace\mathbf{W}_{l},\mathbf{b}_{l}\rbrace_{l=1}^{L}) \Big ).
\end{split}
\label{gaussianVI}
\end{equation}
Here, $p(\mathbf{y}_n|\mathbf{x}_n,\lbrace \mathbf{W}_{l}, \mathbf{b}_{l} \rbrace_{l=1}^{L})$ is a conditional distribution, which can be obtained by integrating the deep GP \eqref{multilayers} with respect to $\mathbf{f}_{n,l}$ for all $l$.
\citet{gal2016dropout} used a normal mixture distribution as a variational distribution for model parameters as follows:
\begin{equation}\begin{split}
q(\mathbf{W}_{l}) & = \prod_{\forall i,j} q(w_{l,ij}),~~~q(\mathbf{b}_{l}) = \prod_{\forall i} q(b_{l,i})\\
q(w_{l,ij}) &= p_lN(\mu^{w}_{l,ij},\sigma^2)+(1-p_l)N(0,\sigma^2) \\
q(b_{l,i}) &= N(\mu^{b}_{l,i},\sigma^2),
\end{split}
\label{variationaldist}
\end{equation}
where $w_{l,ij}$ is the $i,j$th element of the weight matrix $\mathbf{W}_{l}\in {\mathbb R}^{k_l\times k_{l-1}}$ and $b_{l,i}$ is the $i$th element of the bias vector $\mathbf{b}_{l} \in {\mathbb R}^{k_l}$. For the weight parameters, $\mu^{w}_{l,ij}$ and $\sigma^2$ are variational parameters that control the mean and spread of the distributions, respectively. As the inclusion probability $p_l \in [0,1]$ becomes close to 0, $q({w}_{l,ij})$ becomes $N(0,\sigma^2)$, indicating that it is likely to drop the weight parameters (i.e., $w_{l,ij}=0$). Similarly, the variational distribution for the bias parameters is modeled with normal distribution. Our goal is to obtain the variational parameters that maximize \eqref{gaussianVI}. Since the direct maximization of \eqref{gaussianVI} is challenging due to the intractable integration, \citet{gal2016dropout} replace it with approximation with distinct Monte Carlo samples for each observation as
\begin{equation}\small
\mathcal{L}_{\text{GP-MC}} :=\frac{1}{M}\sum_{m=1}^{M}\sum_{n=1}^{N} \log p(\mathbf{y}_n|\mathbf{x}_n,\lbrace \mathbf{W}_{l}^{(n,m)}, \mathbf{b}_{l}^{(n,m)} \rbrace_{l=1}^{L})-\text{KL}\Big(\prod_{l=1}^{L}q(\mathbf{W}_{l})q(\mathbf{b}_{l})\Big|\Big|\prod_{l=1}^{L}p(\mathbf{W}_{l})p(\mathbf{b}_{l})\Big),
\label{GPMC}
\end{equation} \normalsize
where $\lbrace\lbrace \mathbf{W}_{l}^{(n,m)}, \mathbf{b}_{l}^{(n,m)} \rbrace_{l=1}^{L}\rbrace_{m=1}^{M}$ is Monte Carlo samples from the variational distribution $\prod_{l=1}^{L}q(\mathbf{W}_l)q(\mathbf{b}_l)$ for $n$th observation. Note that the Monte Carlo samples are generated for each stochastic gradient descent updates, and the estimates from $\mathcal{L}_{\text{GP-MC}}$ would converge to those obtained from $\mathcal{L}_{\text{GP-VI}}$ \citep{bottou1991stochastic,paisley2012variational,rezende2014stochastic}. In Section 4, we observe that even $M=1$ can provide reasonable approximations though the results become accurate with increasing $M$. Furthermore, \citet{gal2016dropout} shows that \eqref{GPMC} converges to the frequentist loss function of a deep neural network with dropout layers when $\sigma$ in \ref{variationaldist} becomes zero and $k_l$ becomes large. This implies that the frequentist neural network with dropout layers is mathematically equivalent to the approximation of the posteriors of deep GP. We provide details in the Web Appendix A of the Supporting Information. Given this result, we can quantify uncertainty in the Bayesian network without requiring additional costs compared to the frequentist network.
In many applications, prediction of unobserved responses is of great interest. Based on the variational distributions obtained from \eqref{GPMC}, \citep{gal2016dropout} provide a Bayesian framework for predictions in deep neural networks, which is called {\it{Monte Carlo dropout}}. Consider an unobserved response $\mathbf{y}^{\ast}\in {\mathbb R}^{d}$ with an input $\mathbf{x}^{\ast} \in {\mathbb R}^{k_0}$. For given $\mathbf{x}^{\ast}$, we can obtain $\bm{\phi}^{\ast}_{l} \in {\mathbb R}^{k_l}$ from the posterior predictive distribution of the deep GP in \eqref{multilayers}.
Although \citet{gal2016dropout} focuses on predicting response variable $\mathbf{y}^*$, this procedure allows us to construct informative and low-dimensional summary statistics for unobserved $\mathbf{x}^{\ast}$. Especially, we can sample an output feature vector $\bm{\phi}^{\ast}_{n, L-1} \in {\mathbb R}^{k_{L-1}}$ which summarizes the complex nonlinear dependence relationships between input and output variables up to the last layer of the neural network.
In Section 3, we use these extracted features as additional covariates in the GLM framework.
\section{Flexible Generalized Linear Models with Convolutional Neural Networks}
We propose a Bayesian convolutional generalized linear model (BayesCGLM) based on two appealing methodologies: (1) CNN \citep{krizhevsky2012imagenet} have been widely used to utilize correlated predictors (or input variables) with spatial structures such as images or geospatial data. (2) GLM can study the relationship between covariates and non-Gaussian response by extending a linear regression framework. Let $\mathbf{X}=\lbrace \mathbf{X}_n \rbrace_{n=1}^{N}$ be the collection of correlated input (i.e., each $\mathbf{X}_n$ is a matrix or tensor) and $\mathbf{Y}=\lbrace \mathbf{y}_n \rbrace_{n=1}^{N}$ be the corresponding output. We also let $\mathbf{Z}=\lbrace \mathbf{z}_n \rbrace_{n=1}^{N}$ be a collection of $p$-dimensional vector covariates.
We use CNN to extract features from $\mathbf{X}_n$ and employ it in a GLM with the covariates $\mathbf{z}_n$ to predict $\mathbf{y}_n$. The uncertainties in estimating the features are quantified using Monte Carlo dropout. Now let $\bm{\Phi}^{(m)} =\lbrace \bm{\phi}^{(m)\ast}_{n, L-1}\rbrace_{n=1}^{N} \in {\mathbb R}^{N \times k_{L-1}} $ be the extracted features, which is the last hidden layer of CNN for $\mathbf{X}_n$, in the $m$th iteration of Monte Carlo dropout ($m=1,\cdots,M$). We begin with an outline of our framework: \\
\noindent {\it{Step 1.}} Train a Bayes CNN with the correlated data $\mathbf{X}$ (images or spatial variables) and covariate $\mathbf{Z}$ as input and $\mathbf{Y}$ as output. Here, the covariates $\mathbf{Z}$ are concatenated right after the last hidden layer. \\
\noindent {\it{Step 2.}} Using Monte Carlo dropout, extract features $\bm{\Phi}^{(m)}$ from the last hidden layer of the trained Bayes CNN for $m=1,\cdots M$.\\
{\it{Step 3.}} Fit GLMs by regressing $\mathbf{Y}$ on $[\mathbf{Z}, \bm{\Phi}^{(m)}]$ for $m=1,\cdots,M$ in parallel. Posteriors from different $m$ (the ``$m$th feature-posterior") are aggregated to construct an ensemble-posterior distribution. \\
A graphical description of \noindent BayesCGLM is presented in Figure~\ref{BayesCGLM}. We provide the details in the following subsections.
\begin{figure}
\begin{center}
\includegraphics[width = 0.8\textwidth]{figure1.png}
\end{center}
\caption[]{Illustration for the BayesCGLM.}
\label{BayesCGLM}
\end{figure}
\subsection{Bayesian Convolutional Neural Network}\label{Sec:BayesCNN}
As a variant of deep neural networks, convolutional neural networks (CNN) have been widely used in computer vision and image classification \citet{krizhevsky2012imagenet, shridhar2019comprehensive}. CNN can effectively extract important features of images or correlated datasets with non-linear dependencies. A Bayesian perspective, which regards weight parameters as random variables, can be useful because it enables uncertainty quantification in a straightforward manner through posterior densities. Especially, \citet{gal2016bayesian} proposes an efficient Bayesian alternative by regarding weight parameters in CNN's kernels as random variables. By extending a result in \citet{gal2016dropout}, \citet{gal2016bayesian} shows that applying dropout after every convolution layer can approximate intractable posterior distribution of deep GP.
Let $\mathbf{X}_n\in {\mathbb R}^{M_0 \times R_0 \times C_0}$ be an input with $M_0$ height, $R_0$ width, and $C_0$ channels. For example, $M_0$ and $R_0$ are determined based on the resolution of images and $C_0=3$ for color images ($C_0=1$ for black and white images). There are three types of layers in CNN: the convolution, pooling, and fully coresulting innnected layers. Convolutional layers move kernels (filters) on the input image and extract information, resulting in a ``feature map". Each cell in the feature map is obtained from element-wise multiplication between the scanned input image and the filter matrix. The weight values in filter matrices are parameters to be estimated in CNN. The higher the output values, the more the corresponding portion of the image has high signals on the feature. This can account for spatial structures in input data; we can extract the important neighborhood features from the input by shifting the kernels over all pixel locations with a certain step size (stride) \citep{ravi2016deep}. Then the pooling layers are applied to downsample the feature maps. The convolution and pooling layers are applied repeatedly to reduce the dimension of an input image. Then the extracted features from the convolution/pooling layers are flattened and connected to the output layer through a fully connected layer, which can learn non-linear relationships between the features and the output.
Consider a CNN with $L$ layers, where the last two are fully connected and output layers; we have convolutional operations up to $(L-2)$th layers. For the $l$th convolution layer, there are $C_l$ number of kernels $\mathbf{K}_{l,c} \in {\mathbb R}^{H_{l} \times D_{l} \times C_{l-1}}$ with $H_l$ height, $D_l$ width, and $C_{l-1}$ channels for $c=1,\cdots,C_l$. Note the dimension of channel is determined by the number of kernels in the previous convolution layer. After applying the $l$th convolution layer, the $(m,r)$th element of the feature matrix $\bm{\eta}_{n,(l,c)}\in {\mathbb R}^{M_{l} \times R_{l}}$ becomes
\begin{equation}
[\bm{\eta}_{n,(l,c)}]_{m,r}= \sigma \Big(\sum_{i=1}^{H_{l}}\sum_{j=1}^{D_{l}}\sum_{k=1}^{C_{l-1}} ([\mathbf{K}_{l,c}]_{i,j,k} [\bm{\eta}_{n,(l-1,c)}]_{m+i-1,r+j-1,k}) + b_{l,c}\Big),
\label{convolutionweight}
\end{equation}
where $b_{l,c}$ is the bias parameter. For the first convolution layer, $\bm{\eta}_{n,(l-1,c)}$ is replaced with an input image $\mathbf{X}_n$. (i.e., $\bm{\eta}_{n,(0,c)}=\mathbf{X}_n$). For computational efficiency and stability, ReLU functions are widely used as activation functions $\sigma(\cdot)$. By shifting kernels, the convolution layer can capture neighborhood information, and the neighborhood structure is determined by the size of the kernels.
\citet{gal2016bayesian} points out that extracting the features from the convolution process is equivalent to matrix multiplication in standard DNN (Figure~\ref{CNNmultiplication}). Without loss of generality, consider we have a single feature matrix $\bm{\eta}_{n,(l,1)}\in {\mathbb R}^{M_{l} \times R_{l}}$ with a kernel $\mathbf{K}_{l,1} \in {\mathbb R}^{H_l \times D_l}$ (i.e., $c=1$). As described in Figure~\ref{CNNmultiplication}, we can rearrange the $(l-1)$th input $\bm\eta_{n,(l-1,1)}$ as $\Tilde{\bm\eta}_{n,(l-1,1)} \in {\mathbb R}^{ M_lR_l \times H_{l}D_{l}}$ by vectorizing the $H_{l} \times D_{l}$ dimensional features from $\bm\eta_{n,(l-1,1)}$. Similarly, we can vectorize the kernel $\mathbf{K}_{l,1}$ and represent it as a weight matrix $\Tilde{\mathbf{W}}_{l,1} \in {\mathbb R}^{ H_{l} D_{l}}$. Then, we can perform matrix multiplication $\Tilde{\bm\eta}_{n,(l-1,1)}\Tilde{\mathbf{W}}_{l,1}\in {\mathbb R}^{M_lR_l}$ and rearrange it to obtain a matrix $\bm\eta_{n,(l,1)} \in {\mathbb R}^{M_{l} \times R_{l}}$, which is equivalent to output from the $l$th convolution layer. The above procedure can be generalized to the tensor input $\bm\eta_{n,l-1}=\lbrace \bm\eta_{n,(l-1,c)} \rbrace_{c=1}^{C_{l-1}} \in {\mathbb R}^{M_{l-1} \times R_{l-1} \times C_{l-1}}$ with $C_l$ number of kernels $\mathbf{K}_{l,c} \in {\mathbb R}^{H_{l} \times D_{l} \times C_{l-1}}$; therefore, the convolution process is mathematically equivalent to standard DNN. Similar to DNN, we can construct posterior distributions for weights and bias parameters.
Then the posterior distributions are approximated through variational distributions in \eqref{variationaldist}, which are normal mixtures. As described in Section 2, applying dropout after every convolution layer before pooling can approximate deep GP \citep{damianou2013deep}.
\begin{figure}
\begin{center}
\includegraphics[width = 0.8 \textwidth]{figure2.png}
\end{center}
\caption[]{Illustration for the convolution operation. The convolution process is mathematically equivalent to the matrix product in standard DNN.}
\label{CNNmultiplication}
\end{figure}
After the $L-2$th convolution layers, the extracted feature tensor $\bm\eta_{n,L-2} \in {\mathbb R}^{M_{L-2} \times R_{L-2} \times C_{L-2}}$ is flattened in the fully connected layer. By vectorizing $\bm\eta_{n,L-2}$, we can represent the feature tensor as a feature vector $\bm\phi_{n,L-2}=\vect(\bm\eta_{n,L-2}) \in {\mathbb R}^{k_{L-2}}$, where $k_{L-2} = M_{L-2} R_{L-2} C_{L-2}$. Finally, we can extract a feature vector $\bm\phi_{n,L-1} \in {\mathbb R}^{k_{L-1}}$ from the fully connected layer; the collection of feature vectors $\bm{\Phi}=\lbrace \bm\phi_{n,L-1}\rbrace_{n=1}^{N} \in {\mathbb R}^{N \times k_{L-1}}$ summarizes high-dimensional input into a lower dimensional space.
\subsection{Bayesian Convolutional Generalized Linear Model}
We propose BayesCGLM by regressing $\mathbf{Y}$ on $\bm{\Phi}$ and additional covariates $\mathbf{Z}$. Here, we use $\bm{\Phi}$ as a basis design matrix that encapsulates information from high-dimensional and correlated input variables in $\mathbf{X}$. The proposed method can provide interpretation of the regression coefficients and quantify uncertainties in prediction and estimation. As the first step, we train Bayes CNN with the $\mathbf{X}$ and $\mathbf{Z}$ as input and $\mathbf{Y}$ as output. We concatenate the covariate $\mathbf{Z}$ right after the last hidden layer to model the effects of $\mathbf{X}$ on $\mathbf{Y}$ for a given fixed $\mathbf{Z}$. Compared to the traditional CNN, the training step does not require additional computational costs. Then we can extract the feature $\boldsymbol{\Phi} \in {\mathbb R}^{N\times k_{L-1}}$ from the last hidden layer. We can regard this feature as an informative summary statistic that can measure potentially non-linear effects of the input $\mathbf{X}$ on $\mathbf{Y}$. Since we place a posterior distribution over the kernels of the fitted CNN (Section~\ref{Sec:BayesCNN}), we can generate Monte Carlo samples of the features $\bm{\Phi}^{(m)}$, $m=1,\cdots,M$ from the variational distribution. To account for the uncertainties in estimating the features $\bm{\Phi}$, we use the entire Monte Carlo samples of the features rather than the point estimates of them. In Section~\ref{Sec:Simul}, we demonstrate that this allows us to fully account for uncertainties in the feature extraction step through various simulated examples.
With the covariate matrix $\mathbf{Z} \in {\mathbb R}^{N \times p}$, we fit $M$ different GLMs by regressing $\mathbf{Y}$ on $\mathbf{A}^{(m)}=[\mathbf{Z}, \bm{\Phi}^{(m)}] \in {\mathbb R}^{N \times (p+k_{L-1})}$. Then, our model is
\begin{equation} \begin{split}
g(E[\mathbf{Y}|\mathbf{Z},\bm{\Phi}^{(m)}]) &= \mathbf{Z}\bm{\gamma}_m + \bm{\Phi}^{(m)}\bm{\delta}_m = \mathbf{A}^{(m)}\bm{\beta}_m
\end{split}
\label{model}
\end{equation}
where $\bm{\beta}_m=(\bm{\gamma}_m^{\top}, \bm{\delta}_m^{\top})^{\top} \in {\mathbb R}^{p+k_{L-1}}$ is the corresponding regression coefficients and $g(\cdot)$ is a one-to-one continuously differential link function.
For each Monte Carlo sample $m$, we can fit individual GLM in parallel; we have $m=1,\cdots, M$ number of posterior distributions, and each of them can be called the ``$m$th feature-posterior." This step is heavily parallelizable, so computational walltimes tend to decrease with an increasing number of available cores. Then, we construct an aggregated distribution (the ``ensemble-posterior") through the mixture of feature posteriors. In the following section, we provide details about constructing the ensemble-posterior distribution. The fitting procedure of BayesCGLM is summarized in Algorithm~\ref{algo}.
\begin{algorithm}
\caption{BayesCGLM algorithm}\label{MCGLMalgo}
\textbf{Input}: $\mathbf{X}$ (image or correlated input), $\mathbf{Y}$ (response), $\mathbf{Z}$ (scalar covariates)
\textbf{Output}: Posterior distribution of model parameters.
\begin{algorithmic}[1]
\State Fit Bayes CNN by using $\mathbf{X}$ and $\mathbf{Z}$ as input and $\mathbf{Y}$ as output. Then for each $m$
\For {$m=1,2,\ldots,M$}
\State Extract features $\bm{\Phi}^{(m)}$ from the last hidden layer.
\State Fit GLM by regressing $\mathbf{Y}$ on $[\mathbf{Z}, \bm{\Phi}^{(m)}]$ through the Laplace approximation
\State Obtain the $m$th feature-posterior distribution of model parameter.
\EndFor
\State Construct an ensemble-poseterior by aggregating $M$ feature-posteriors.
\end{algorithmic}
\label{algo}
\end{algorithm}
Since BayesCGLM in \eqref{model} can capture non-linear effects of $\mathbf{X}$ on $\mathbf{Y}$ through the extracted features $\bm{\Phi}$, we can improve prediction accuracies compared to the standard GLM. In general, inference for the regression coefficients $\bm{\gamma}$ is challenging for DNN. Although one can regard the weight parameters at the last hidden layer as $\bm{\gamma}$, we cannot obtain uncertainties of estimates. Note that even Bayes DNN \citep{gal2016dropout} only provides point estimates of weight parameters, while they can quantify uncertainties in response prediction. On the other hand, our method directly provides the posterior distribution of $\bm{\gamma}$, which is useful for standard inference. In addition, the complex structure of DNN often leads to nonconvex optimization \citep{ge2015escaping, kleinberg2018alternative, daneshmand2018escaping}, resulting in inaccurate estimate of $\bm{\gamma}$. Stochastic gradient descent algorithm requires simultaneous updates of high-dimensional weight parameters. Instead, our two-stage approach can alleviate this issue by using extracted features $\bm{\Phi}$ as a fixed basis matrix in GLM.
Figure~\ref{loglikelihood_contour} illustrates the profile log-likelihood surfaces over $\bm{\gamma}$ based on a single replicate from our simulation studies in Section 4. Note that we use the same CNN structure to make the profile log-likelihood surfaces comparable. We observe that our two-stage approach provides more accurate estimates compared to BayesCNN. Since the stopping rule of the stochastic gradient descent algorithm is based on prediction accuracy (not based on convergence in parameter estimation), the convergence of individual weight parameters, including $\bm{\gamma}$ cannot be guaranteed.
The problem gets further complicated by a large number of weight parameters that need to be simultaneously updated in CNN \citep{alzubaidi2021review}.
On the other hand, BayesCGLM converts the complex optimization into simple nonparametric regression problems by utilizing the extracted features $\bm{\Phi}$ as a fixed basis function. This allows application of more accurate optimization algorithms based on convergence in parameter estimates. In Section 4, we repeat the simulation 300 times under different model configurations and show that BayesCGLM can recover the true $\bm{\gamma}$ values well, while estimates from Bayes DNN can be biased.
\begin{figure}[htbp]
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.99\linewidth]{likelihood_binary_contour_BayesCNN_final_update_v2.pdf}
\caption{The profile log-likelihood in BayesCNN}
\label{BayesCNN_loglike}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.99\linewidth]{likelihood_binary_contour_GLM_final_update_v2.pdf}
\caption{The profile log-likelihood in BayeCGLM}
\label{BayesCGLM_loglike}
\end{subfigure}
\caption{The yellow circles show the true coefficient $\bm{\gamma}=(1,1)$, the green x-crosses indicate the maximum likelihood points, and the red triangles represent the estimates obtained from BayesCNN and BayesCGLM, respectively. BayesCGLM provides a closer estimate to the true $\bm{\gamma}$ than BayesCNN.}
\label{loglikelihood_contour}
\end{figure}
\subsection{Bayesian Inference}
Our framework requires Bayesian inference on ensemble GLMs in \eqref{model}, i.e., fitting $M$ different GLM models in a Bayesian way. One may choose the standard MCMC algorithm, which can provide exact Bayesian inference. However, this may require a long run of the chain to guarantee a good mixing, resulting in excessive computational cost in our framework. To fit ensemble GLMs quickly, we use the Laplace approximation method \citep{tierney1986accurate}. In our preliminary studies, we observe that there are little benefits from the more exact method; for enough sample size, the Laplace approximation method provides reasonably accurate approximations to the posteriors within a much shorter time.
For each dropout realization $m$, we can obtain the maximum likelihood estimate (MLE) $\widehat{\bm{\beta}}_{m}$ of \eqref{model} using the iteratively reweighted least square algorithm, which can be implemented through {\tt{statmodels}} function in {\tt{Python}}. Then we can approximate the posterior of $\bm{\beta}_{m}$ through $\mathcal{N}(\widehat{\bm{\beta}}_{m}, \widehat{\mathbf{B}}^{-1}_{m})$, where $\widehat{\mathbf{B}}_{m} \in {\mathbb R}^{(p+k_{L-1}) \times (p+k_{L-1})}$ is the observed Fisher information matrix from the $m$th Monte Carlo samples. We can obtain such approximated posteriors for $m=1,\cdots,M$ from Monte Carlo dropout realizations. We can aggregate these feature-posteriors with the mixture distribution as
\begin{equation}
\widehat{\pi}(\bm{\beta}|\mathbf{D}) = \sum_{m=1}^{M} \omega_m \varphi(\bm{\beta}; \widehat{\bm{\beta}}_{m},\widehat{\mathbf{B}}^{-1}_{m})
\label{posterirdist}
\end{equation}
where $\varphi(\bm{x}; \bm{\mu},\bm{\Sigma})$ is a multivariate normal density with mean $\bm{\mu}$ and covariance $\bm{\Sigma}$.
Here, we used the equal weight for $\omega_m$ across different Monte Carlo dropout realizations (i.e., $\omega_m=1/M$). Especially, we focus on describing inference for the posterior distribution of the regression coefficient $\bm{\gamma}$, which can quantify the impact of $\mathbf{Z}$ on $\mathbf{Y}$. Since we have an ensemble-posterior distribution \eqref{posterirdist}, we can conduct standard Bayesian inference naturally.
Our Bayesian framework also quantifies uncertainties in predictions which is another advantages over existing neural networks. For unobserved response $\mathbf{Y}^{\ast} \in {\mathbb R}^{N_{cv}}$, we have $\mathbf{A}^{\ast}_{m}=[\mathbf{Z}^{\ast}, \bm{\Phi}^{(m)}]\in {\mathbb R}^{N_{cv}\times (p+k_{L-1})}$ for $m=1,\cdots,M$. Then from \eqref{posterirdist}, the distribution of the linear predictor is
\begin{equation}
\widehat{\pi}(\mathbf{A}^{\ast}\bm{\beta}|\mathbf{D}) = \sum_{m=1}^{M} \omega_m \varphi(\mathbf{A}^{\ast}\bm{\beta}; \mathbf{A}_{m}^{\ast}\widehat{\bm{\beta}}_{m},\mathbf{A}_{m}^{\ast}\widehat{\mathbf{B}}^{-1}_{m}\mathbf{A}^{\ast\top}_{m}).
\label{linearpredictor}
\end{equation}
From the linear predictor above, we can construct the posterior predictive distribution. For given a posterior mean $\widehat{\bm{\beta}}$ from \eqref{posterirdist}, we can generate $\mathbf{Y}^{\ast}$ from the exponential family distribution with mean $\mathbf{A}^{\ast}\bm{\hat{\beta}}$. For example, when we have a Gaussian response we have $\mathbf{Y}^{\ast} \sim \mathcal{N}( \mathbf{A}^{\ast}\bm{\hat{\beta}},\hat{\sigma}^2)$ where $\hat{\sigma}^2 = \frac{\sum_{n=1}^{N}(\mathbf{A}^{\ast}\bm{\hat{\beta}}-\mathbf{Y})}{N}$ for the Gaussian response. For the count response, we can simulate $\mathbf{Y}^{\ast}$ from the Poisson distribution with intensity $\mathbf{A}^{\ast}\bm{\hat{\beta}}$.
\section{Simulated Data Examples}\label{Sec:Simul}
In this section, we apply BayesCGLM to three different simulated examples, with a Gaussian response (described in the Web Appendix D), binary response (Section \ref{binary_case}), and Poisson response (Section \ref{Poisson_case}). We implement our approach in {\tt{TensorFlow}}, an open-source platform for machine learning. Parallel computation is implemented through the {\tt{multiprocessing}} library in {\tt{Python}}. The computation times are based on 8 core AMD Radeon Pro 5500 XT processors. The configuration of the CNN structures and tuning details used in our experiments are provided in the Web Appendix B.1 of the Supporting Information.
We generated 1,000 lattice datasets as images, which have a spatially correlated structure given by the Gaussian process. The responses are created by the canonical link function $g^{-1}$ and the natural parameter $\bm{\lambda}=g^{-1}(\bm\Phi + \mathbf{Z}\bm{\gamma})$ with $\bm{\gamma}=(1,1)$. Here, $\bm\Phi$ indicates the vectorized local features of the images, $\mathbf{Z}$ represents covariates generated from a standard normal distribution, and $\bm{\gamma}$ are the true coefficients. To demonstrate the performance of our model, we compared it with two existing models: Bayesian CNN and GLM. The details of simulation experiments are described in the Web Appendix C of the Supporting Information.
\subsection{Binary Case} \label{binary_case}
We now describe the simulation study results when $\mathbf{Y}$ is in the form of binary response, i.e., $\mathbf{Y} \sim \text{Bernoulli}(\bm{\lambda})$. Figure \ref{simul_pred_binary} compares the true response and the estimated probabilities (for the response to be 1), which show a good agreement. We observe that the predicted probability shows higher values for the binary responses with the true value of 1. The area under the curve (AUC) from the receiver operating characteristic curve (ROC) is about 0.85, which is reasonably close to 1 (Figure \ref{simul_pred_binary2}).
\begin{figure}[htbp]
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.8\linewidth]{simulation_binary_result1.pdf}
\caption{Predicted probability surface}
\label{simul_pred_binary1}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.99\linewidth]{simulation_binary_result2.png}
\caption{ROC plot}
\label{simul_pred_binary2}
\end{subfigure}
\caption{ (a) The scatter plot of estimated probability of $Y=1$ and true binary responses. Blue line indicates the estimated probability function. (b) The ROC curve for binary prediction. Blue line denotes AUC of the ROC curve. }
\label{simul_pred_binary}
\end{figure}
In Table~\ref{simul_binary} we observe that parameter estimates from BayesCGLM are accurate, and the coverages are close to the 95\% nominal rate even with a single Monte Carlo dropout ($M=1$). On the other hand, the estimates from Bayes CNN \citep{gal2016bayesian}, and GLM are biased.
Deep learning methods (BayesCGLM, Bayes CNN) show much better prediction performance than GLM because they can extract features from image observations.
\begin{table}[tt]
\centering
\caption[]{Inference results for the simulated binary datasets. For all methods, mean of $\bm{\gamma}$, estimation coverage, accuracy, recall, precision, and average computing time (sec) are calculated from 300 simulations. The numbers in the parentheses indicate standard deviations obtained from the repeated simulations.}
\label{simul_binary}
\begin{tabular}{c c c c c}
& \textbf{BayesCGLM}(M$=300$) &\textbf{BayesCGLM}(M$=1$) & \textbf{Bayes CNN} & \textbf{GLM} \\ \hline
$\gamma_1$ & 1.029 (0.154) & 1.030 (0.154) & 0.848 (0.117) & 0.389 (0.086) \\
Coverage & 0.956 & 0.966 &- & 0 \\
\hline
$\gamma_2$ & 1.030 (0.151) & 1.024 (0.155) & 0.851 (0.120) & 0.388 (0.081) \\
Coverage & 0.954 & 0.952 & - & 0 \\
\hline Accuracy & 0.869 (0.019) & 0.857 (0.022) & 0.850 (0.022) & 0.600 (0.028) \\
Recall & 0.869 (0.031) & 0.855 (0.033) & 0.851 (0.036) & 0.600(0.066) \\
Precision & 0.868 (0.030) & 0.858 (0.033) & 0.863 (0.029) & 0.602 (0.043) \\
\hline Time & 37.028 & 17.661 & 15.022 & 0.003\\
\end{tabular}
\end{table}
\subsection{Poisson Case} \label{Poisson_case}
We describe the simulation study results when $\mathbf{Y}$ is generated from the Poisson distribution, i.e., $\mathbf{Y} \sim \text{Poisson}(\bm{\lambda})$.
Figure \ref{simul_pred_poisson} shows that the true and predicted responses are well aligned for a simulated dataset except for some extreme values. We observe that HPD prediction intervals include the true count response well.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width = 0.5\textwidth]{simulation_poisson_result.pdf}
\end{center}
\caption[]{Comparison between the true and predicted Poisson responses from BayesCGLM. The vertical lines indicate 95\% highest posterior density (HPD) from the predictive distribution. }
\label{simul_pred_poisson}
\end{figure}
Table~\ref{simul_poisson} indicates that BayesCGLM shows comparable (or better) prediction performance compared to Bayes CNN while providing uncertainties of estimates. We observe that the coverage of the credible and prediction intervals becomes closer to the 95\% nominal rate with the repeated Monte Carlo dropout sampling ($M=300$), compared to a single Monte Carlo dropout sampling ($M=1$). Compared to Bayes CNN, BayesCGLM with $M=300$ results in better point estimates for the parameters $\bm{\gamma}$ and point predictions for the response $\mathbf{Y}$. Compared to GLM, BayesCGLM with $M=300$ results in better empirical coverage for the credible intervals for $\bm{\gamma}$ as well as better prediction accuracy and empirical coverage for the prediction intervals for $\mathbf{Y}$.
\begin{table}[tt]
\centering
\caption[]{Inference results for the simulated Poisson datasets. For all methods, mean of $\bm{\gamma}$, estimation coverage, RMSPE, prediction coverage, and average computing time (min) are calculated from 300 simulations. The numbers in the parentheses indicate standard deviations obtained from the repeated simulations. }
\label{simul_poisson}
\begin{tabular}{c c c c c}
& \textbf{BayesCGLM}(M$=300$) &\textbf{BayesCGLM}(M$=1$) & \textbf{Bayes CNN} & \textbf{GLM} \\ \hline
$\gamma_1$ & 0.987 (0.027) & 0.989 (0.031) & 0.910 (0.106) & 0.997 (0.058) \\
Coverage & 0.936 & 0.862 & - & 0.542 \\
\hline
$\gamma_2$ & 0.986 (0.026) & 0.989 (0.031) & 0.911 (0.101) & 0.997 (0.059) \\
Coverage & 0.958 & 0.868 & - & 0.550 \\
\hline RMSPE & 2.305 (1.610) & 2.687 (2.190) & 2.737 (2.537) & 4.364 (3.739) \\
Coverage & 0.963 & 0.890 & 0.967 & 0.924 \\
\hline Time & 22.73 & 15.285 & 13.753 & 0.004 \\
\end{tabular}
\end{table}
\section{Applications}
We apply our method to three real data examples: (1) malaria incidence data, (2) brain tumor images, and (3) fMRI data for anxiety scores. We attached fMRI data for anxiety score result in the Web Appendix E of the Supporting Information. BayesCGLM shows comparable prediction with the standard Bayes CNN while enabling uncertainty quantification for regression coefficients. Details about the CNN structure are provided in the Web Appendix B.2 of the Supporting Information.
\subsection{Malaria Cases in the African Great Lakes Region}
Malaria is a parasitic disease that can lead to severe illnesses and even death. Predicting occurrences at unknown locations is of significant interest for effective control interventions. We compiled malaria incidence data from the Demographic and Health Surveys of 2015 \citep{ICF2017}. The dataset contains malaria incidence (counts) from 4,741 GPS clusters in nine contiguous countries in the African Great Lakes region.
We use average annual rainfall, vegetation index of the region, and the proximity to water as spatial covariates. Under a spatial regression framework, \citet{gopal2019characterizing} analyzes malaria incidence in Kenya using these environmental variables. In this study, we extend this approach to multiple countries in the African Great Lakes region. We use $N = 3,500$ observations to fit the model and save $N_{\text{cv}} = 1,241$ observations for cross-validation.
To create rectangular images out of irregularly shaped spatial data, we use a spatial basis function matrix as the correlated input $\mathbf{X}$ in Algorithm~\ref{algo} (line 1). Specifically, we use thin plate splines basis defined as $ \mathbf{X}_{j}(\mathbf{s}) = ||\mathbf{s}-\mathbf{u}_{j}||^{2}\log(||\mathbf{s}-\mathbf{u}_{j}||)$, where $s$ is an arbitrary spatial location and $\lbrace \mathbf{u}_{j} \rbrace_{j=1}^{239}$ is a set of knots placed regularly over the spatial domain. From this we can construct basis functions for each observation, resulting in $\mathbf{X} \in {\mathbb R}^{3,500 \times 239}$ input design matrix. We first train Bayes CNN with input $\mathbf{X}, \mathbf{Z}$ and the count response $\mathbf{Y}$ while applying Monte Carlo dropout, with a dropout probability of 0.2. We set the batch size and epochs as 10 and 500, respectively. We use ReLU and linear activation functions in the hidden layers and use the exponential activation function in the output layer to fit count response. From this, we extract a feature design matrix $\bm{\Phi} \in {\mathbb R}^{3,500 \times 16}$ from the last layer of the fitted Bayes CNN. With covariate matrix $\mathbf{Z} \in {\mathbb R}^{3,500 \times 3}$, we fit Poisson GLMs by regressing $\mathbf{Y}$ on $[\mathbf{Z}, \bm{\Phi}^{(m)}] \in {\mathbb R}^{3,500 \times 19}$ for each $m$; here, we have used $M=500$.
We compare our method with a spatial regression model using basis expansions \citep[cf.][]{sengupta2013hierarchical, lee2020scalable} as
\[
g(E[\mathbf{Y}|\mathbf{Z},\mathbf{X}]) = \mathbf{Z}\bm{\gamma} + \mathbf{X}\bm{\delta},
\]
where $\mathbf{X} \in {\mathbb R}^{3,500 \times 239}$ is the same thin plate basis matrix that we used in BayesCGLM. To fit a hierarchical spatial regression model, we use {\tt{nimble}} \citep{nimble2017}, which is a convenient programming language for Bayesian inference. To guarantee convergence, an MCMC algorithm is run for 1,000,000 iterations with 500,000 discarded for burn-in. Furthermore, we also implement the Bayes CNN as in the simulated examples. Table~\ref{malaria_table} indicates that vegetation, water, and rainfall variables have positive relationships with malaria incidence in deep learning methods. BayesCGLM shows comparable prediction performance compared to Bayes CNN. Although Bayes CNN can provide prediction uncertainties, it cannot quantify uncertainties of the estimates. Although, we can obtain HPD intervals from spatial regression, it provides higher RMSPE and lower coverage than other deep learning methods.
\begin{table}[tt]
\centering
\caption[]{Inference results for the malaria dataset from different methods. For all methods, posterior mean of $\bm{\gamma}$, 95\% HPD interval, RMSPE, prediction coverage, and computing time (min) are reported in the table. }
\label{malaria_table}
\begin{tabular}{c c c c}
& \textbf{BayesCGLM(M=500)} & \textbf{Bayes CNN} &\textbf{Spatial Model} \\ \hline
$\gamma_1$ (vegetation index) & 0.099 & 0.103 & 0.115 \\
& (0.092 0.107) & - & (0.111 0.118) \\
$\gamma_2$ (proximity to water) & 0.074 & 0.058 & -0.269 \\
& (0.068 0.080) & - & (-0.272 -0.266) \\
$\gamma_3$ (rainfall) & 0.036 & 0.027 & -0.122 \\
& (0.027 0.045)& - &(-0.126 -0.117) \\
\hline RMSPE & 27.438 & 28.462 & 42.393 \\
Coverage & 0.950 & 0.947 & 0.545 \\
\hline Time & 57.518 & 30.580 & 41.285 \\
\end{tabular}
\end{table}
Figure \ref{fig1_malaria} shows that the true and predicted malaria incidence from BayesCGLM have similar spatial patterns. We observe the prediction standard errors are reasonably small across the entire spatial domain. Similar to the simulation studies, we illustrate the 95\% HPD prediction intervals, which include the true count response well.
\begin{figure}[htbp]
\centering
\begin{subfigure}[b]{1.1\textwidth}
\includegraphics[width=\linewidth]{malaria_result_1_update.png}
\caption{Prediction map}
\label{fig1_malaria}
\end{subfigure}
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\linewidth]{malaria_result_2.pdf}
\caption{Prediction plot with confidence interval}
\label{fig2_malaira}
\end{subfigure}
\caption[]{(a): The true (left panel) and predicted (middle panel) malaria incidence from BayesCGLM. The right panel shows the corresponding prediction standard errors. (b): Comparison between the true and predicted malaria incidence from BayesCGLM. The vertical lines indicate 95\% highest posterior density (HPD) from the predictive distribution.}
\end{figure}
\subsection{Brain Tumor Image Data}
We apply our method to brain tumor image data. The dataset is collected from Brain Tumor Image Segmentation Challenge (BRATS) \citep{menze2014multimodal}. The binary response indicates whether a patient has a fatal brain tumor. For each patient, we have an MRI image with its two summary variables (first and second-order features of MRI images). Here, we use $N = 2,508$ observations to fit our model and reserve $N_{\text{cv}} = 2,007$ observations for validation.
We first fit Bayes CNN using $240 \times 240$ pixel gray images $\mathbf{X}$, scalar covariates $\mathbf{Z}$ as input, and the binary response $\mathbf{Y}$ as output with Monte Carlo dropout, with the dropout probability of 0.25. We set the batch size and the number of epochs as 3 and 5, respectively. As before, we use ReLU and linear activation functions in the hidden layers and use the sigmoid activation function in the output layer to fit binary response. From the fitted CNN, we can extract $\bm\Phi \in {\mathbb R}^{2,508 \times 16}$ from the last hidden layer. In this study, we use two summary variables (first and second-order features of MRI images) as covariates. With the extracted features from the CNN, we fit logistic GLMs by regressing $\mathbf{Y}$ on $[\mathbf{Z},\bm\Phi^{(m)}] \in {\mathbb R}^{2,508 \times 18}$ for $m=1,\cdots, 500$.
We compare our method with GLM and Bayes CNN as in the simulated examples. Table \ref{table_tumor} shows inference results from different methods. Our method shows that the first-order feature has a negative relationship, while the second-order feature has a positive relationship with the brain tumor risk. We observe that the sign of $\bm{\gamma}$ estimates from BayesCGLM are aligned with those from GLM. For this example, BayesCGLM shows the most accurate prediction performance
while providing credible intervals of the estimates.
\begin{table}
\centering
\caption[]{Inference results for the brain tumor dataset from different methods. For all methods, the posterior mean of $\bm{\gamma}$, 95\% HPD interval, accuracy, recall, precision, and computing time (min) are reported in the table. }
\label{table_tumor}
\begin{tabular}{c c c c c}
& \textbf{BayesCGLM(M=500)} & \textbf{Bayes CNN} & \textbf{GLM} \\ \hline
$\gamma_1$ (first order feature) & -5.332 & 0.248 & -2.591 \\
& (-7.049,-3.704)& - & (-2.769,-2.412) \\
$\gamma_2$ (second order feature) & 4.894 & 0.160 & 2.950 \\
& (3.303, 6.564) & - & (2.755, 3.144) \\
\hline Accuracy & 0.924 & 0.867 & 0.784 \\
Recall & 0.929 & 0.787 & 0.783 \\
Precision & 0.901 & 0.907 & 0.715 \\
\hline Time & 293.533 & 103.924 & 0.004\\
\end{tabular}
\end{table}
Figure~\ref{tumoraccuracy1} shows that the predicted probability surface becomes larger for the binary responses with a value of 1. Furthermore, the true and predicted binary responses are well aligned. The AUC for the ROC plot is about 0.96, which shows an outstanding prediction performance (Figure~\ref{tumoraccuracy2}).
\begin{figure}[htbp]
\centering
\begin{subfigure}{0.5\textwidth}
\includegraphics[width=0.9\linewidth]{tumor_1.pdf}
\caption{Predicted probability surface}
\label{tumoraccuracy1}
\end{subfigure}
\begin{subfigure}{0.5\textwidth}
\includegraphics[width=0.9\linewidth]{tumor_2.png}
\caption{ROC plot}
\label{tumoraccuracy2}
\end{subfigure}
\label{tumoraccuracy}
\caption[]{(a): The scatter plot of estimated probability of $Y=1$ and true binary responses in the brain tumor example. Blue line indicates the estimated probability function. (b): The ROC curve for binary prediction. Blue line denotes AUC of the ROC curve.}
\end{figure}
\section{Discussion}
We propose a flexible regression approach, which can utilize image data as predictor variables via feature extraction through convolutional layers. The proposed method can simultaneously utilize predictors with different data structures such as vector covariates and image data.
Our study of real and simulated data examples shows that the proposed method results in comparable prediction performance to existing deep learning algorithms in terms of prediction accuracy while enabling uncertainty quantification for estimation and prediction. By constructing an ensemble-posterior through a mixture of feature-posteriors, we can fully account for the uncertainties in feature extraction; we observe that BayesCGLM can construct accurate HPD intervals that achieve the nominal coverage. BayesCGLM is efficient, in that multiple GLMs are fitted in parallel with fast Laplace approximations.
Our work is motivated by recently developed feature extraction approaches from DNN. These features can improve the performance of existing statistical approaches by accounting for complex dependencies among observations. For instance, \citet{bhatnagar2020computer} extract features from the long-short term memory (LSTM) networks \citep{gers2001lstm,graves2005framewise} and use them to capture complex inverse relationship in calibration problems. Similar to our approach, \citet{tran2020bayesian} use the extracted features from DNN as additional covariates in generalized linear mixed effect models. \citet{fong2021forward} propose a nonlinear dimension reduction method based on deep autoencoders, which can extract interpretable features from high-dimensional data.
The proposed framework can be extended to a wider range of statistical models such as cox regression \citep{cox1972regression} in survival analysis or regression models in causal inference. Extracting features from other types of DNN is also available; for example, one can use CNN-LSTM \citep{wang2016dimensional} to capture spatio-temporal dependencies. Our ensemble approaches can significantly improve prediction and provide interpretable parameter estimates.
\section*{Acknowledgement}
JP and YJ were partially supported by the National Research Foundation of Korea (NRF-2020R1C1C1A0100386812). SH was partially supported by a National Research Foundation of Korea grant funded by the Korean government (NRF-2019R1A2C1007399). The authors are grateful to anonymous reviewers for their careful reading and valuable comments.
\section{Dropout as a Bayesian Approximation}
In this section, we describe results in \cite{gal2016dropout} that a deep neural network with dropout layers is mathematically equivalent to the approximation of the posteriors of the deep GP. Especially we focus on a Gaussian response $\mathbf{y}$ case, but it can be easily extended to non-Gaussian responses. Consider a neural network with dropout layers as
\begin{equation}
\mathbf{y}=\sigma_L \Big(\mathbf{W}_{L}\sigma_{L-1}\Big(\mathbf{W}_{L-1} \cdots \sigma_3 \Big(\mathbf{W}_{3}\sigma_2 \Big(\mathbf{W}_{2} \sigma_1 \Big(\mathbf{W}_{1}\mathbf{x} +\mathbf{b}_1\Big)\circ\mathbf{d}_{2} +\mathbf{b}_2\Big)\circ\mathbf{d}_{3} +\mathbf{b}_3\Big) \cdots \Big)\circ\mathbf{d}_{L-1} + \mathbf{b}_{L-1} \Big)\circ\mathbf{d}_{L} +\mathbf{b}_{L} \Big)
\label{eq1dropSuppl}
\end{equation}
where $\mathbf{d}_{l}$ for $l=2,\cdots,L$ is defined as $k_l$-dimensional vectors, each component follows Bernoulli distribution with success probability $p$ independently.
In a traditional approach, we can train the above model by minimizing the following loss function with $L_2$ regularization terms:
\begin{equation}
\mathcal{L}_{dropout} := -\frac{1}{2N}\sum_{n=1}^{N} ||\mathbf{y}_n - \mathbf{\hat{y}}_n||_2^2+\sum_{l=1}^{L}\lambda^{\mathbf{w}}_{l}||\mathbf{W}_l||_2^2+\sum_{l=1}^{L}\lambda^{\mathbf{b}}_{l}||\mathbf{b}_l||_2^2.
\label{dropoutloss}
\end{equation}
Here $\lambda^{\mathbf{w}}_{l}$, $\lambda^{\mathbf{b}}_{l}$ are the shrinkage parameters for weight and bias parameters respectively. Note that we can replace \eqref{dropoutloss} with other types of loss functions for non-Gaussian responses; for instance, we can use a sigmoid loss function for a binary classification problem.
In a Bayesian alternative, we can represent \eqref{eq1dropSuppl} as a deep GP. As we described in the main manuscript, the posterior distribution of the deep GP is approximated through the variational distributions as
\begin{equation}\begin{split}
q(\mathbf{W}_{l}) & = \prod_{\forall i,j} q(w_{l,ij}),~~~q(\mathbf{b}_{l}) = \prod_{\forall i} q(b_{l,i})\\
q(w_{l,ij}) &= p_lN(\mu^{w}_{l,ij},\sigma^2)+(1-p_l)N(0,\sigma^2) \\
q(b_{l,i}) &= N(\mu^{b}_{l,i},\sigma^2),
\end{split}
\label{variationaldistsuppl}
\end{equation}
where $w_{l,ij}$ is the $i,j$th element of the weight matrix $\mathbf{W}_{l}\in {\mathbb R}^{k_l\times k_{l-1}}$ and $b_{l,i}$ is the $i$th element of the bias vector $\mathbf{b}_{l} \in {\mathbb R}^{k_l}$.
Then the Monte Carlo version of log evidence lower bound is
\begin{equation}
\mathcal{L}_{\text{GP-MC}} =\sum_{n=1}^{N} \log p(\mathbf{y}_n|\mathbf{x}_n,\lbrace \mathbf{W}_{l}^{(n)}, \mathbf{b}_{l}^{(n)} \rbrace_{l=1}^{L})-\text{KL}(\prod_{l=1}^{L}q(\mathbf{W}_{l})q(\mathbf{b}_{l})||p(\lbrace \mathbf{W}_{l}, \mathbf{b}_{l} \rbrace_{l=1}^{L})),
\label{GPMCsuppl}
\end{equation}
where $\lbrace \mathbf{W}_{l}^{(n)}, \mathbf{b}_{l}^{(n)} \rbrace_{l=1}^{L}$ is Monte Carlo samples from the variational distribution $\prod_{l=1}^{L}q(\mathbf{W}_l)q(\mathbf{b}_l)$ for $n$th observation; for notational convenience, we describe the Monte Carlo approximation with a distinct single sample as in \cite{gal2016dropout}. Note that $\mathcal{L}_{\text{GP-MC}}$ is the stochastic objective function, and estimates from $\mathcal{L}_{\text{GP-MC}}$ would converge to those obtained from $\mathcal{L}_{\text{GP-VI}}$ \citep[cf.][]{bottou1991stochastic,paisley2012variational,kingma2014autoencoding,rezende2014stochastic}.
According to \cite{gal2016dropout}[Proposition 1], with large $k_l$ and a small value of constant $\sigma$ (hyper parameter), $\text{KL}( q(\mathbf{W}_l)|| p(\mathbf{W}_l))$ can be approximated as
\begin{equation}\begin{split}
\text{KL}( q(\mathbf{W}_l)|| p(\mathbf{W}_l)) &\approx \sum_{\forall i,j \in l} \frac{p_l}{2}((\mu^{w}_{l,ij})^{2}+(\sigma^{2} -(1+\log2\pi)-\log\sigma^{2})+C),
\end{split}
\label{KLdiversuppl1}
\end{equation}
with some constant $C$. Similarly, $\text{KL}( q(\mathbf{b}_l)|| p(\mathbf{b}_l))$ can be approximated as
\begin{equation}\begin{split}
\text{KL}( q(\mathbf{b}_l)|| p(\mathbf{b}_l)) &\approx \sum_{\forall i,j \in l} \frac{p_l}{2}((\mu^{b}_{l,ij})^{2}+(\sigma^{2} -(1+\log2\pi)-\log\sigma^{2})+C).
\end{split}
\label{KLdiversuppl2}
\end{equation}
By plugging in these approximations in \eqref{GPMCsuppl}, we have
\begin{equation} \begin{split}
\mathcal{L}_{\text{GP-MC}} &\approx \sum_{n=1}^{N} \log N(y_n; \mathbf{W}_{n,L}\phi_{n,L-1}+\mathbf{b}_L,\tau^{-1}\mathbf{I}_N)\\ &-\sum_{l=1}^{L} \frac{p_l}{2}(||\bm{\mu}^{\mathbf{w}}_{l}||^{2}-k_l k_{l-1}(\sigma^{2} -(1+\log2\pi)-\log\sigma^{2}))
-\sum_{l=1}^{L} \frac{1}{2}(||\bm{\mu}^{\mathbf{b}}_{l}||^{2}-k_l(\sigma^{2} -(1+\log2\pi)-\log\sigma^{2}))
\label{GPMCKLsuppl}
\end{split}
\end{equation}
with a constant precision $\tau$ (hyper parameter).
By ignoring constant hyper parameter terms ($\tau,\sigma$), the approximated version of the log evidence lower bound scaled by a positive constant $\frac{1}{\tau N}$ becomes
\begin{equation} \begin{split}
\mathcal{L}_{\text{GP-MC}} \approx -\frac{1}{2N}\sum_{n=1}^{N} ||\mathbf{y}_n - \mathbf{\hat{y}}_n||_2^2 - \sum_{l=1}^{L}\frac{p_l}{2\tau N} ||\bm{\mu}^{\mathbf{w}}_{l}||_2^2 - \sum_{l=1}^{L}\frac{1}{2\tau N}||\bm{\mu}^{\mathbf{b}}_{l}||_2^2.
\label{GPMC1suppl}
\end{split}
\end{equation}
This implies that the posterior distribution of the weight parameter $w_{l,ij}$ is approximated with the mixtures of the spike distributions; one is centered around $\mu^{w}_{l,ij}$ and the other is centered around 0. Similarly, the posterior of bias parameter $b_{l}$ is approximated through the spike distribution centered around $\mu^{b}_{l,i}$. The loss function \eqref{GPMC1suppl} becomes equivalent to \eqref{dropoutloss} by setting $\lambda^{\mathbf{w}}_l=\frac{p_l}{2\tau N}$ and $\lambda^{\mathbf{b}}_l=\frac{1}{2\tau N}$.
\clearpage
\section{CNN Structures}
\subsection{Simulations}
\paragraph{Gaussian Data}
\begin{itemize}
\item Optimizer: Adam optimizer
\item Learning rate: $1e-4$
\item Loss function: mean squared error
\item Batch size: 32
\item Epoch size: 300
\end{itemize}
\title{\textbf{Summary of 2D-CNN Configurations:}}
\begin{table}[htt]
\centering
\begin{threeparttable}
\begin{tabular}{cc@{\qquad}ccc}
Layer Type & Dimension & Kernel Size & Strides & Activation Function \\ \midrule
\makecell{Convolution Layer} & $8\times 8$ & $4\times4$ & $2\times2$ & relu\\ \cmidrule(l r){1-5}
\makecell{Dropout(p=0.2)} & - & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Max Pooling} & $2\times 2$ & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Convolution Layer} & $16\times 16$ & $3\times3$ & $2\times2$ & softmax \\ \cmidrule(l r){1-5}
\makecell{Dropout(p=0.2)} & - & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Max Pooling} & 2 & - & - & -\\ \cmidrule(l r){1-5}
\makecell{Flatten} & - & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Dense} & $1\times 32$ & - & - & relu \\ \cmidrule(l r){1-5}
\makecell{Dropout(p=0.2)} & - & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Dense} & $1\times 16$ & - & - & relu \\ \cmidrule(l r){1-5}
\makecell{Dropout(p=0.2)} & - & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Dense} & $1\times 16$ & - & - & softplus \\ \cmidrule(l r){1-5}
\makecell{Dropout(p=0.2)} & - & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Concatenate} & $\mathbf{Z}$ \\ \cmidrule(l r){1-5}
\makecell{Dense} & $1\times 1$ & - & - & mse \\ \cmidrule(l r){1-5}
\end{tabular}
\end{threeparttable}
\caption{Summary of 2D-CNN configurations for Gaussian simulation}
\end{table}
\clearpage
\paragraph{Binary Data}
\begin{itemize}
\item Optimizer: Adam optimizer
\item Learning rate: $1e-4$
\item Loss function: binary cross entropy
\item Batch size: 3
\item Epoch size: 2,000
\end{itemize}
\title{\textbf{Summary of 2D-CNN Configurations:}}
\begin{table}[htt]
\centering
\begin{threeparttable}
\begin{tabular}{cc@{\qquad}ccc}
Layer Type & Dimension & Kernel Size & Strides & Activation Function \\ \midrule
\makecell{Convolution Layer} & $16\times 16$ & $3\times3$ & $1\times1$ & softmax\\ \cmidrule(l r){1-5}
\makecell{Dropout(p=0.25)} & - & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Max Pooling} & $2\times 2$ & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Convolution Layer} & $32\times 32$ & $3\times3$ & $1\times1$& softmax \\ \cmidrule(l r){1-5}
\makecell{Dropout(p=0.25)} & - & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Max Pooling} & 2 & - & - & -\\ \cmidrule(l r){1-5}
\makecell{Flatten} & - & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Dense} & $1\times 16$ & - & - & Relu \\ \cmidrule(l r){1-5}
\makecell{Dropout(p=0.25)} & - & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Dense} & $1\times 8$ & - & - & linear \\ \cmidrule(l r){1-5}
\makecell{Dropout(p=0.25)} & - & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Concatenate} & $\mathbf{Z}$ \\ \cmidrule(l r){1-5}
\makecell{Dense} & $1\times 1$ & - & - & sigmoid \\ \cmidrule(l r){1-5}
\end{tabular}
\end{threeparttable}
\caption{Summary of 2D-CNN configurations for binary simulation}
\end{table}
\clearpage
\paragraph{Poisson Data}
\begin{itemize}
\item Optimizer: Adam optimizer
\item Learning rate: $1e-3$
\item Loss function: Poisson
\item Batch size: 3
\item Epoch size: 2,000
\end{itemize}
\title{\textbf{Summary of 2D-CNN Configurations:}}
\begin{table}[htt]
\centering
\begin{threeparttable}
\begin{tabular}{cc@{\qquad}ccc}
Layer Type & Dimension & Kernel Size & Strides & Activation Function \\ \midrule
\makecell{Convolution Layer} & $8\times 8$ & $4\times4$ & $2\times2$ & softmax\\ \cmidrule(l r){1-5}
\makecell{Dropout(p=0.2)} & - & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Max Pooling} & $2\times 2$ & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Convolution Layer} & $32\times 32$ & $3\times3$ & $1\times1$ & softmax \\ \cmidrule(l r){1-5}
\makecell{Dropout(p=0.2)} & - & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Max Pooling} & 2 & - & - & -\\ \cmidrule(l r){1-5}
\makecell{Flatten} & - & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Dense} & $1\times 32$ & - & - & softplus \\ \cmidrule(l r){1-5}
\makecell{Dropout(p=0.2)} & - & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Dense} & $1\times 16$ & - & - & linear \\ \cmidrule(l r){1-5}
\makecell{Dropout(p=0.2)} & - & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Concatenate} & $\mathbf{Z}$ \\ \cmidrule(l r){1-5}
\makecell{Dense} & $1\times 1$ & - & - & exponenetial \\ \cmidrule(l r){1-5}
\end{tabular}
\end{threeparttable}
\caption{Summary of 2D-CNN configurations for Poisson simulation}
\end{table}
\clearpage
\subsection{Real data applications}
\paragraph{Malaria Incidence}
\begin{itemize}
\item Optimizer: Adam optimizer
\item Learning rate: $1e-4$
\item Loss function: Poisson
\item Batch size: 10
\item Epoch size: 2500
\end{itemize}
\title{\textbf{Summary of 1D-CNN Configurations:}}
\begin{table}[htt]
\centering
\begin{threeparttable}
\begin{tabular}{cc@{\qquad}ccc}
Layer Type & Dimension & Kernel Size & Strides & Activation Function \\ \midrule
\makecell{Convolution Layer} & $1\times 32$ & $3$ & $1$ & tanh\\ \cmidrule(l r){1-5}
\makecell{Dropout(p=0.25)} & - & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Max Pooling} & $1\times 1$ & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Convolution Layer} & $1\times64$ & $3$ & $1$ & tanh \\ \cmidrule(l r){1-5}
\makecell{Dropout(p=0.25)} & - & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Max Pooling} & 2 & - & - & -\\ \cmidrule(l r){1-5}
\makecell{Flatten} & - & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Dense} & $1\times 32$ & - & - & relu \\ \cmidrule(l r){1-5}
\makecell{Dropout(p=0.2)} & - & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Dense} & $1\times 16$ & - & - & linear \\ \cmidrule(l r){1-5}
\makecell{Dropout(p=0.2)} & - & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Concatenate} & $\mathbf{Z}$ \\ \cmidrule(l r){1-5}
\makecell{Dense} & $1\times 1$ & - & - & exponenetial \\ \cmidrule(l r){1-5}
\end{tabular}
\end{threeparttable}
\caption{Summary of 1D-CNN configurations for malarial incidence}
\end{table}
\clearpage
\paragraph{Brain tumor MRI images}
\begin{itemize}
\item Optimizer: Adam optimizer
\item Learning rate: $1e-4$
\item Loss function: binary cross entropy
\item Batch size: 3
\item Epoch size: 5
\end{itemize}
\title{\textbf{Summary of 2D-CNN Configurations:}}
\begin{table}[htt]
\centering
\begin{threeparttable}
\begin{tabular}{cc@{\qquad}ccc}
Layer Type & Dimension & Kernel Size & Strides & Activation Function \\ \midrule
\makecell{Convolution Layer} & $64\times 64$ & $3\times3$ & $1\times1$ & relu\\ \cmidrule(l r){1-5}
\makecell{Dropout(p=0.25)} & - & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Max Pooling} & $2\times 2$ & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Convolution Layer} & $32\times 32$ & $3\times3$ & $1\times1$& relu \\ \cmidrule(l r){1-5}
\makecell{Dropout(p=0.25)} & - & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Max Pooling} & 2 & - & - & -\\ \cmidrule(l r){1-5}
\makecell{Flatten} & - & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Dense} & $1\times 16$ & - & - & linear \\ \cmidrule(l r){1-5}
\makecell{Dropout(p=0.25)} & - & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Concatenate} & $\mathbf{Z}$ \\ \cmidrule(l r){1-5}
\makecell{Dense} & $1\times 1$ & - & - & sigmoid \\ \cmidrule(l r){1-5}
\end{tabular}
\end{threeparttable}
\caption{Summary of 2D-CNN configurations for brain tumor MRI images}
\end{table}
\clearpage
\paragraph{fMRI Data for Anxiety Score}
\begin{itemize}
\item Optimizer: Adam optimizer
\item Learning rate: $1e-3$
\item Loss function: mean squared error
\item Batch size: 3
\item Epoch size: 500
\end{itemize}
\title{\textbf{Summary of 2D-CNN Configurations:}}
\begin{table}[htt]
\centering
\begin{threeparttable}
\begin{tabular}{cc@{\qquad}ccc}
Layer Type & Dimension & Kernel Size & Strides & Activation Function \\ \midrule
\makecell{Convolution Layer} & $8\times 8$ & $3\times3$ & $2\times2$ & relu\\ \cmidrule(l r){1-5}
\makecell{Dropout(p=0.25)} & - & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Max Pooling} & $2\times 2$ & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Convolution Layer} & $16\times 16$ & $3\times3$ & $2\times2$ & relu \\ \cmidrule(l r){1-5}
\makecell{Dropout(p=0.25)} & - & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Max Pooling} & 2 & - & - & -\\ \cmidrule(l r){1-5}
\makecell{Flatten} & - & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Dense} & $1\times 16$ & - & - & relu \\ \cmidrule(l r){1-5}
\makecell{Dropout(p=0.25)} & - & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Dense} & $1\times 8$ & - & - & softplus \\ \cmidrule(l r){1-5}
\makecell{Dropout(p=0.25)} & - & - & - & - \\ \cmidrule(l r){1-5}
\makecell{Concatenate} & [$\mathbf{Z}$] \\ \cmidrule(l r){1-5}
\makecell{Dense} & $1\times 1$ & - & - & linear \\ \cmidrule(l r){1-5}
\end{tabular}
\end{threeparttable}
\caption{Summary of 2D-CNN configurations for fMRI data}
\end{table}
\clearpage
\section{Introduction}
With recent advances in data collection technologies, it is increasingly common to have images as a part of collected data. For instance, in clinical trials, we can collect both fMRI images and clinical information from the same group of patients. However, it is challenging to analyze such data because variables represent different forms; we have both matrix (or tensor) images and vector covariates. The traditional statistical approaches face inferential issues when dealing with input variables in the form of images due to the high dimensionality and multicollinearity issues. Convolutional neural networks (CNN) \citep{krizhevsky2012imagenet, szegedy2015going, he2016deep} are popular approaches for image processing without facing inferential challenges. \textcolor{red}{However, CNNs cannot quantify the prediction uncertainty for the response variable and the estimation uncertainty for the covariates, which are often of primary interest in statistical inference. Furthermore, the complex structure of CNN can lead to nonconvex optimization \citep{ge2015escaping,kleinberg2018alternative,daneshmand2018escaping,alzubaidi2021review}, which cannot guarantee the convergence of weight parameters. } In this manuscript, we propose a computationally efficient Bayesian approach by embedding CNNs within the generalized linear model (GLM). Our method can take advantage of both statistical models and advanced neural networks.
There is a growing literature on Bayesian deep learning methods \citep[cf.][]{neal2012bayesian, blundell2015weight} to quantify the uncertainty by regarding weight parameters in deep neural networks (DNN) as random variables. With an appropriate choice of priors, we have posterior distributions for the standard Bayesian inference. Furthermore, Bayesian alternatives can be robust for a small sample size compared to the standard DNN \citep{gal2016bayesian, gal2016dropout}. Despite its apparent advantages, however, Bayesian DNNs have been much less used than the classical one due to the difficulty in exploring the full posterior distributions. To circumvent the difficulty, variational Bayes (VB) methods have been proposed to approximate complex high-dimensional posteriors; examples include VB for feed-forward neural networks \citep{graves2011practical, tran2020bayesian} and CNNs \citep{shridhar2019comprehensive}. Although VB can alleviate computational costs in Bayesian networks, to the best of our knowledge, no existing approach provides a general Bayesian framework to study image and vector type variables simultaneously. Furthermore, most previous works have focused on prediction uncertainty but not on estimation uncertainty of model parameters; interpretations of regression coefficients have not been studied much in the Bayesian deep learning context.
In this manuscript, we propose a Bayesian convolutional generalized linear model (BayesCGLM) that can study different types of variables simultaneously (e.g., regress a scalar response on vector covariates and array type images) and easily be implemented in various applications. A BayesCGLM consists of the following steps: (1) train Bayes CNN \citep{gal2016bayesian} with the correlated input (images or correlated variables) and output (response variable); (2) extract Monte Carlo samples of features from the last layer of the fitted Bayes CNN; and (3) fit a GLM by regressing a response variable on augmented covariates with the features obtained from each Monte Carlo sample. For each Monte Carlo sample of the feature, we fit an individual GLM and aggregate its posterior, which can be implemented in parallel. These steps allow us to fully account for uncertainties in the estimation procedure. We also study an often overlooked challenge in Bayesian deep learning: practical issues in obtaining credible intervals of parameters and assessing whether they provide nominal coverage.
In our method, we model Bayes CNN via a deep Gaussian process \citep{damianou2013deep}, which is a nested Gaussian process. Gaussian process \citep{rasmussen2003gaussian} are widely used to represent complex function relationships due to their flexibility. Especially, \cite{gal2016bayesian, gal2016dropout} show that neural networks with dropout applied before the last layer is mathematically equivalent to the approximation of the posteriors of the deep Gaussian process \citep{damianou2013deep}. This procedure is referred to as Bernoulli approximating variational distributions \citep{gal2016bayesian, gal2016dropout}. We build upon their approach to sample features from the approximated posterior of the last layer in Bayes CNN. Training Bayes CNN with dropout approximation does not require additional computing costs compared to the frequentist one.
There have been several recent proposals to extract features from the hidden layer of neural network models \citep{zhu2017deep,tran2020bayesian,bhatnagar2020computer}, and our method is motivated by them. The features can be regarded as reduced-dimensional summary statistics that measure complex non-linear relationships between the high-dimensional input and output; the dimension of the extracted features is much lower than that of the original input. We observe that CNN is especially useful for extracting information from correlated inputs such as images or spatial basis functions.
The outline of the remainder of this paper is as follows. In Section 2, we introduce a Bayesian DNN based on Gaussian process approximation and its estimation procedure. In Section 3, we propose a computationally efficient BayesCGLM and provide implementation details. In Section 4, we study the performance of BayesCGLM through simulated data examples. In Section 5, we apply BayesCGLM to several real-world applications such as malaria incidence data, brain tumor image data, and functional magnetic resonance imaging data (fMRI) data for anxiety scores. We conclude with a discussion and summary in Section 6.
\section{A Bayesian Deep Neural Network: Gaussian Process Approximation}
In this section, we first give a brief review of Bayesian DNNs and their approximations to deep Gaussian processes. Let $\mathbf{D}=\lbrace (\mathbf{x}_n,\mathbf{y}_n), n=1,\cdots, N\rbrace$ be the observed dataset with input $\mathbf{x}_n \in {\mathbb R}^{k_0}$ and response $\mathbf{y}_n \in {\mathbb R}^{d}$. A standard DNN is used to model the relationship between the input $\mathbf{x}_n$ and the output $\mathbf{o}_n \in {\mathbb R}^{d}$, which can be regarded as an expectation of $\mathbf y_n$ given $\mathbf x_n$, in terms of a statistical viewpoint.
Suppose we consider a DNN model with $L$ layers, where the $l$th layer has $k_l$ nodes for $l=1,\cdots, L$. Then we can define the weight matrix $\mathbf{W}_l \in {\mathbb R}^{k_{l} \times k_{l-1}}$ that connects the $(l-1)$th layer to $l$th hidden layer and the bias vector $\mathbf{b}_l \in {\mathbb R}^{k_{l}}$. Then $\bm{\theta}= \lbrace (\mathbf{W}_l,\mathbf{b}_l), l=1,\cdots, L\rbrace$ is a set of parameters in the neural network that belongs to a parameter space $\bm{\Theta}$.
To model complex data, the number of neural network parameters grows tremendously with the increasing number of layers and nodes. Such large networks are computationally intensive to use and can lead to overfitting issues. Dropout \citep{srivastava2014dropout} has been suggested as a simple but effective strategy to address both problems. The term "dropout" refers to randomly removing the connections from nodes during the training step of the network. While training, the model randomly chooses the units to drop out with some probability. The structure of neural networks with dropout is
{\small
\begin{equation}
\mathbf{o}_{n}=\sigma_L \Big(\mathbf{W}_{L}\sigma_{L-1}\Big(\mathbf{W}_{L-1} \cdots \sigma_3 \Big(\mathbf{W}_{3}\sigma_2 \Big(\mathbf{W}_{2} \sigma_1 \Big(\mathbf{W}_{1}\mathbf{x}_{n} +\mathbf{b}_1\Big)\circ\mathbf{d}_{2} +\mathbf{b}_2\Big)\circ\mathbf{d}_{3} +\mathbf{b}_3\Big) \cdots \Big)\circ\mathbf{d}_{L-1} + \mathbf{b}_{L-1} \Big)\circ\mathbf{d}_{L} +\mathbf{b}_{L} \Big)
\label{eq1drop}
\end{equation}}
where $\sigma_l(\cdot)$ is an activation function; for instance, the rectified linear (ReLU) or the hyperbolic tangent function (TanH) can be used. Here $\mathbf{d}_{l} \in {\mathbb R}^{k_l}, l=2,\cdots,L$, are dropout vectors whose components follow Bernoulli distribution with pre-specified success probability $p_l$ independently. The notation "$\circ$" indicates the elementwise multiplication of vectors. Although dropout can alleviate overfitting issues, quantifying uncertainties in predictions or parameter estimates remains challenging for the frequentist approach.
To address this, Bayesian alternatives \citep{friedman1997bayesian, friedman2003being, chen2012good} have been proposed by considering probabilistic modeling of neural networks. The Bayesian approach considers an arbitrary function that is likely to generate data $\mathbf{D}$. Especially, a deep Gaussian process (deep GP) has been widely used to represent complex neural network structures for both supervised and unsupervised learning problems \citep[cf.][]{titsias2010bayesian, gal2016dropout, damianou2013deep}. By modeling data generating mechanism as a deep GP, a hierarchical Bayesian neural network \citep{gal2016dropout} can be constructed. For $l\geq 2$, let $\mathbf{f}_{n,l} =\mathbf{W}_l \bm{\phi}_{n,l-1} + \mathbf{b}_l \in {\mathbb R}^{k_l}$ be the linear feature which is a random variable in the Bayesian framework; we set independent normal priors for weight and bias parameters as $p(\mathbf{W}_{l}), p(\mathbf{b}_l)$ respectively. Then we can also define $\bm{\phi}_{n,l}=\sigma_{l}(\mathbf{f}_{n,l}) \in {\mathbb R}^{k_l}$ as the nonlinear output feature from the $l$th layer after applying an activation function. For the first layer (i.e., $l=1$), we have $\mathbf{f}_{n,1}=\mathbf{W}_1 \mathbf{x}_{n} + \mathbf{b}_1$. Then the deep GP can be written as
\begin{equation}
\begin{split}
\mathbf{f}_{n,2}|\mathbf{x}_n &\sim N(0, \mathbf{\Sigma}_{nn,1})\\
\mathbf{f}_{n,3}|\mathbf{f}_{n,2}&\sim N(0, \mathbf{\Sigma}_{nn,2})\\
\vdots \\
\mathbf{f}_{n,L}|\mathbf{f}_{n,L-1} &\sim N(0, \mathbf{\Sigma}_{nn,L-1})\\
\mathbf{y}_n|\mathbf{f}_{n,L} &\sim p(\mathbf{y}_n|\mathbf{f}_{n,L}),
\label{multilayers}
\end{split}
\end{equation}
where the covariance $\mathbf{\Sigma}_{nn,l} \in {\mathbb R}^{k_l \times k_l}$ is
\begin{equation}
\begin{split}
\mathbf{\Sigma}_{nn,l} &= \int \int \sigma_l(\mathbf{W}_l \bm{\phi}_{n,l-1} + \mathbf{b}_l)
\sigma_l(\mathbf{W}_l \bm{\phi}_{n,l-1} + \mathbf{b}_l)^{\top}p(\mathbf{W}_l)p(\mathbf{b}_l)d\mathbf{W}_l d\mathbf{b}_l.
\label{covfuncsmultilayer}
\end{split}
\end{equation}
This implies that an arbitrary data generating function is modeled through a nested Gaussian process with the covariance of the extracted features from the previous hidden layer. Since the integral calculation in \eqref{covfuncsmultilayer} is intractable, we can replace it with a finite covariance rank approximation in practice; with increasing $k_l$ (number of hidden nodes) the finite rank approximation becomes close to \eqref{covfuncsmultilayer} \citep{gal2016dropout}. In \eqref{multilayers}, $\mathbf{y}_n$ is a response variable that has a distribution $p(\cdot)$ belonging to the exponential family with $g(E[\mathbf{y}_n|\bm{\phi}_{n,L-1}])=\mathbf{W}_{L}\bm{\phi}_{n,L-1}+\mathbf{b}_{L}$; here $g(\cdot)$ is a one-to-one continuously differential link function. Note that $g^{-1}(\cdot)$ can be regarded as $\sigma_{L}(\cdot)$ in \eqref{eq1drop} in terms of traditional DNN notation. To carry out Bayesian inference, we need to approximate the posterior distributions of deep GP.
However, Bayesian inference for deep GP is not straightforward to implement. With increasing hidden layers and nodes, the number of parameters $\bm{\theta}$ grows exponentially, which results in the high-dimensional posterior distribution. Constructing Markov chain Monte Carlo (MCMC) samplers \citep{neal2012bayesian, mcdermott2019bayesian} is impractical due to the slow mixing of high-dimensional parameters and the complex sequential structure of the networks. Instead, variational approximation methods can be practical and computationally feasible for Bayesian inference \citep{bishop2006pattern, blei2017variational}. Variational Bayes approximates the true $\pi(\bm{\theta}|\mathbf{D})$ via the approximate distribution (variational distribution) $q(\bm{\theta})$, which is a class of distribution that can be easily evaluated. Given a class of distributions, the optimal variational parameters are set by minimizing the Kullback–Leibler (KL) divergence between the variational distribution $q(\bm{\theta})$ and the true posterior $\pi(\bm{\theta}|\mathbf{D})$ where minimizing the KL divergence is equivalent to maximizing $\mbox{E}_q[{\log(\pi(\bm{\theta},\mathbf{D}))}]-\mbox{E}_q[\log q({\bm{\theta}})]$, the evidence lower bound (ELBO).
\subsection{Dropout as an Approximation to the Gaussian Process}
\cite{gal2016dropout} shows that a deep neural network in \eqref{eq1drop} is mathematically equivalent to a variational approximation of deep Gaussian process \citep{damianou2013deep}. That is the log ELBO of deep GP converges to the frequentist loss function of deep neural network with dropout layers. With the independent variational distribution $q(\bm{\theta}):=\prod_{l=1}^{L}q(\mathbf{W}_l)q(\mathbf{b}_l)$, the log ELBO of the deep GP is
\begin{equation} \begin{split}
\mathcal{L}_{\text{GP-VI}} := \sum_{n=1}^{N}\int \cdots \int \prod_{l=1}^{L}q(\mathbf{W}_{l})q(\mathbf{b}_{l}) \log p(\mathbf{y}_n| \mathbf{x}_n,\lbrace \mathbf{W}_{l}, \mathbf{b}_{l} \rbrace_{l=1}^{L}) d\mathbf{W}_1\cdots d\mathbf{b}_{1}\cdots d\mathbf{W}_L d\mathbf{b}_{L} \\
- \text{KL}\Big (\prod_{l=1}^{L}q(\mathbf{W}_{l})q(\mathbf{b}_{l})\Big |\Big |p( \lbrace\mathbf{W}_{l},\mathbf{b}_{l}\rbrace_{l=1}^{L}) \Big ).
\end{split}
\label{gaussianVI}
\end{equation}
Here, $p(\mathbf{y}_n|\mathbf{x}_n,\lbrace \mathbf{W}_{l}, \mathbf{b}_{l} \rbrace_{l=1}^{L})$ is a conditional distribution, which can be obtained by integrating the deep GP \eqref{multilayers} with respect to $\mathbf{f}_{n,l}$ for all $l$.
\cite{gal2016dropout} used a normal mixture distribution as a variational distribution for model parameters as follows:
\begin{equation}\begin{split}
q(\mathbf{W}_{l}) & = \prod_{\forall i,j} q(w_{l,ij}),~~~q(\mathbf{b}_{l}) = \prod_{\forall i} q(b_{l,i})\\
q(w_{l,ij}) &= p_lN(\mu^{w}_{l,ij},\sigma^2)+(1-p_l)N(0,\sigma^2) \\
q(b_{l,i}) &= N(\mu^{b}_{l,i},\sigma^2),
\end{split}
\label{variationaldist}
\end{equation}
where $w_{l,ij}$ is the $i,j$th element of the weight matrix $\mathbf{W}_{l}\in {\mathbb R}^{k_l\times k_{l-1}}$ and $b_{l,i}$ is the $i$th element of the bias vector $\mathbf{b}_{l} \in {\mathbb R}^{k_l}$. For the weight parameters, $\mu^{w}_{l,ij}$ and $\sigma^2$ are variational parameters that control the mean and spread of the distributions, respectively. As the inclusion probability $p_l \in [0,1]$ becomes close to 0, $q({w}_{l,ij})$ becomes $N(0,\sigma^2)$, indicating that it is likely to drop the weight parameters (i.e., $w_{l,ij}=0$). Similarly, the variational distribution for the bias parameters is modeled with normal distribution. Our goal is to obtain the variational parameters that maximize \eqref{gaussianVI}. Since the direct maximization of \eqref{gaussianVI} is challenging due to the intractable integration, \cite{gal2016dropout} replace it with approximation with distinct Monte Carlo samples for each observation as
\begin{equation}
\mathcal{L}_{\text{GP-MC}} :=\frac{1}{M}\sum_{m=1}^{M}\sum_{n=1}^{N} \log p(\mathbf{y}_n|\mathbf{x}_n,\lbrace \mathbf{W}_{l}^{(n,m)}, \mathbf{b}_{l}^{(n,m)} \rbrace_{l=1}^{L})-\text{KL}\Big(\prod_{l=1}^{L}q(\mathbf{W}_{l})q(\mathbf{b}_{l})\Big|\Big|\prod_{l=1}^{L}p(\mathbf{W}_{l})p(\mathbf{b}_{l})\Big),
\label{GPMC}
\end{equation}
where $\lbrace\lbrace \mathbf{W}_{l}^{(n,m)}, \mathbf{b}_{l}^{(n,m)} \rbrace_{l=1}^{L}\rbrace_{m=1}^{M}$ is Monte Carlo samples from the variational distribution $\prod_{l=1}^{L}q(\mathbf{W}_l)q(\mathbf{b}_l)$ for $n$th observation. Note that the Monte Carlo samples are generated for each stochastic gradient descent updates, and the estimates from $\mathcal{L}_{\text{GP-MC}}$ would converge to those obtained from $\mathcal{L}_{\text{GP-VI}}$ \citep[cf.][]{bottou1991stochastic,paisley2012variational,kingma2014autoencoding,rezende2014stochastic}. In Section 4, we observe that even $M=1$ can provide reasonable approximations though the results become accurate with increasing $M$. Furthermore, \cite{gal2016dropout} shows that \eqref{GPMC} converges to the frequentist loss function of a deep neural network with dropout layers when $\sigma$ in \eqref{variationaldist} becomes zero and $k_l$ becomes large. This implies that the frequentist neural network with dropout layers is mathematically equivalent to the approximation of the posteriors of deep GP. We provide details in the supplementary material. Given the result in this theorem, we can quantify uncertainty in the Bayesian network without requiring additional computing costs compared to the frequentist network.
In many applications, prediction of unobserved responses is of great interest. Based on the variational distributions obtained from \eqref{GPMC}, \cite{gal2016dropout} provide a Bayesian framework for predictions in deep neural networks, which is called {\it{Monte Carlo dropout}}. Consider an unobserved response $\mathbf{y}^{\ast}\in {\mathbb R}^{d}$ with an input $\mathbf{x}^{\ast} \in {\mathbb R}^{k_0}$. For given $\mathbf{x}^{\ast}$, we can obtain $\bm{\phi}^{\ast}_{l} \in {\mathbb R}^{k_l}$ from the posterior predictive distribution of the deep GP \eqref{multilayers}.
Although \cite{gal2016dropout} focuses on predicting response variable $\mathbf{y}^*$, this procedure allows us to construct informative and low-dimensional summary statistics for unobserved $\mathbf{x}^{\ast}$. Especially, we can sample an output feature vector $\bm{\phi}^{\ast}_{n, L-1} \in {\mathbb R}^{k_{L-1}}$ which summarizes the complex nonlinear dependence relationships between input and output variables up to the last layer of the neural network. It can be used as an informative variable in statistical inference. In Section 3, we use these extracted features as additional covariates in the GLM framework.
\section{Flexible Generalized Linear Models with Convolutional Neural Networks}
In this section, we propose a Bayesian convolutional generalized linear model (BayesCGLM) based on two appealing methodologies: (1) Convolutional neural networks (CNN) \citep{krizhevsky2012imagenet} have been widely used to utilize correlated predictors (or input variables) with spatial structures such as images or geospatial data. (2) Generalized linear models (GLM) can study the relationship between covariates and non-Gaussian response by extending a linear regression framework. Let $\mathbf{X}=\lbrace \mathbf{X}_n \rbrace_{n=1}^{N}$ be the collection of correlated input (i.e., each $\mathbf{X}_n$ is a matrix or tensor) and $\mathbf{Y}=\lbrace \mathbf{y}_n \rbrace_{n=1}^{N}$ be the corresponding output. We also let $\mathbf{Z}=\lbrace \mathbf{z}_n \rbrace_{n=1}^{N}$ be a collection of $p$-dimensional vector covariates.
We use CNN to extract features from $\mathbf{X}_n$ and employ it in a GLM with the covariates $\mathbf{z}_n$ to predict $\mathbf{y}_n$. The uncertainties in estimating the features are quantified using Monte Carlo dropout. Now let $\bm{\Phi}^{(m)} =\lbrace \bm{\phi}^{(m)\ast}_{n, L-1}\rbrace_{n=1}^{N} \in {\mathbb R}^{N \times k_{L-1}} $ be the extracted features, which the last hidden layer of CNN for $\mathbf{X}_n$, in the $m$th iteration of Monte Carlo dropout ($m=1,\cdots,M$). We begin with an outline of our framework: \\
\noindent {\it{Step 1.}} Train a Bayes CNN with the correlated data $\mathbf{X}$ (images or spatial variables) and covariate $\mathbf{Z}$ as input and $\mathbf{Y}$ as output. Here, the covariates $\mathbf{Z}$ are concatenated right after the last hidden layer. \\
\noindent {\it{Step 2.}} Using Monte Carlo dropout, extract features $\bm{\Phi}^{(m)}$ from the last hidden layer of the trained Bayes CNN for $m=1,\cdots M$.\\
{\it{Step 3.}} Fit GLMs by regressing $\mathbf{Y}$ on $[\mathbf{Z}, \bm{\Phi}^{(m)}]$ for $m=1,\cdots,M$ in parallel. Posteriors from different $m$ ("$m$th feature-posterior") are aggregated to construct an ensemble-posterior distribution. \\
A graphical description of \noindent BayesCGLM is presented in Figure~\ref{BayesCGLM}. We provide the details in the following subsections.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width = 0.8\textwidth]{figure1.png}
\end{center}
\caption[]{Illustration for the BayesCGLM.}
\label{BayesCGLM}
\end{figure}
\subsection{Bayesian Convolutional Neural Network}\label{Sec:BayesCNN}
As a variant of deep neural networks, convolutional neural networks (CNN) have been widely used in computer vision and image classification \citep[cf.][]{krizhevsky2012imagenet, he2016deep, shridhar2019comprehensive}. CNN can effectively extract important features of images or correlated datasets with non-linear dependencies. A Bayesian perspective, which regards weight parameters as a random variable, can be useful because it enables uncertainty quantification in a straightforward manner through posterior densities. Especially, \cite{gal2016bayesian} proposes an efficient Bayesian alternative by regarding weight parameters in CNN's kernels as random variables. By extending a result in \cite{gal2016dropout}, \cite{gal2016bayesian} shows that applying dropout after every convolution layer can approximate intractable posterior distribution of deep GP.
Let $\mathbf{X}_n\in {\mathbb R}^{M_0 \times R_0 \times C_0}$ be an input with $M_0$ height, $R_0$ width, and $C_0$ channels. For example, $M_0, R_0$ are determined based on the resolution of images and $C_0=3$ for color images ($C_0=1$ for black and white images). There are three types of layers in CNN: the convolution, pooling, and fully connected layers. Convolutional layers move kernels (filters) on the input image and extract information, resulting in a "feature map." Each cell in the feature map is obtained from element-wise multiplication between the scanned input image and the filter matrix. Here, the weight values in filter matrices are parameters to be estimated in CNN. The higher the output values, the more the corresponding portion of the image has high signals on the feature. This procedure can account for spatial structures in input data; we can extract the important neighborhood features from the input by shifting the kernels over all pixel locations with a certain step size (stride) \citep{ravi2016deep,goodfellow2016deep,khan2020survey}. Then the pooling layers are applied to downsample the feature maps. The convolution and pooling layers are applied repeatedly to reduce the dimension of an input image. Then the extracted features from the convolution/pooling layers are flattened and connected to the output layer through a fully connected layer, which can learn non-linear relationships between the features and the output.
Consider a CNN with $L$ layers, where the last two are fully connected and output layers; we have convolutional operations up to $(L-2)$th layers. For the $l$th convolution layer, there are $C_l$ number of kernels $\mathbf{K}_{l,c} \in {\mathbb R}^{H_{l} \times D_{l} \times C_{l-1}}$ with $H_l$ height, $D_l$ width, and $C_{l-1}$ channels for $c=1,\cdots,C_l$. Note the dimension of channel is determined by the number of kernels in the previous convolution layer. After applying the $l$th convolution layer, the $(m,r)$th element of the feature matrix $\bm{\eta}_{n,(l,c)}\in {\mathbb R}^{M_{l} \times R_{l}}$ becomes
\begin{equation}
[\bm{\eta}_{n,(l,c)}]_{m,r}= \sigma \Big(\sum_{i=1}^{H_{l}}\sum_{j=1}^{D_{l}}\sum_{k=1}^{C_{l-1}} ([\mathbf{K}_{l,c}]_{i,j,k} [\bm{\eta}_{n,(l-1,c)}]_{m+i-1,r+j-1,k}) + b_{l,c}\Big),
\label{convolutionweight}
\end{equation}
where $b_{l,c}$ is the bias parameter. For the first convolution layer, $\bm{\eta}_{n,(l-1,c)}$ is replaced with an input image $\mathbf{X}_n$. (i.e., $\bm{\eta}_{n,(0,c)}=\mathbf{X}_n$). For computational efficiency and stability, ReLU functions are widely used as activation functions $\sigma(\cdot)$. By shifting kernels, the convolution layer can capture neighborhood information, and the neighborhood structure is determined by the size of the kernels.
\cite{gal2016bayesian} points out that extracting the features from the convolution process is equivalent to matrix multiplication in standard DNN (Figure~\ref{CNNmultiplication}). Without loss of generality, consider we have a single feature matrix $\bm{\eta}_{n,(l,1)}\in {\mathbb R}^{M_{l} \times R_{l}}$ with a kernel $\mathbf{K}_{l,1} \in {\mathbb R}^{H_l \times D_l}$ (i.e., $c=1$). As described in Figure~\ref{CNNmultiplication}, we can rearrange the $(l-1)$th input $\bm\eta_{n,(l-1,1)}$ as $\Tilde{\bm\eta}_{n,(l-1,1)} \in {\mathbb R}^{ M_lR_l \times H_{l}D_{l}}$ by vectorizing the $H_{l} \times D_{l}$ dimensional features from $\bm\eta_{n,(l-1,1)}$. Similarly, we can vectorize the kernel $\mathbf{K}_{l,1}$ and represent it as a weight matrix $\Tilde{\mathbf{W}}_{l,1} \in {\mathbb R}^{ H_{l} D_{l}}$. Then, we can perform matrix multiplication $\Tilde{\bm\eta}_{n,(l-1,1)}\Tilde{\mathbf{W}}_{l,1}\in {\mathbb R}^{M_lR_l}$ and rearrange it to obtain a matrix $\bm\eta_{n,(l,1)} \in {\mathbb R}^{M_{l} \times R_{l}}$, which is equivalent to output from the $l$th convolution layer. The above procedure can be generalized to the tensor input $\bm\eta_{n,l-1}=\lbrace \bm\eta_{n,(l-1,c)} \rbrace_{c=1}^{C_{l-1}} \in {\mathbb R}^{M_{l-1} \times R_{l-1} \times C_{l-1}}$ with $C_l$ number of kernels $\mathbf{K}_{l,c} \in {\mathbb R}^{H_{l} \times D_{l} \times C_{l-1}}$; therefore, the convolution process is mathematically equivalent to standard DNN. Similar to DNN, we can construct posterior distributions for weights and bias parameters.
Then the posterior distributions are approximated through variational distributions in \eqref{variationaldist}, which are normal mixtures. As described in Section 2, applying dropout after every convolution layer before pooling can approximate deep GP \citep{damianou2013deep}.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width = 0.8\textwidth]{figure2.png}
\end{center}
\caption[]{Illustration for the convolution operation. The convolution process is mathematically equivalent to the matrix product in standard DNN. }
\label{CNNmultiplication}
\end{figure}
After the $L-2$th convolution layers, the extracted feature tensor $\bm\eta_{n,L-2} \in {\mathbb R}^{M_{L-2} \times R_{L-2} \times C_{L-2}}$ is flattened in the fully connected layer. By vectorizing $\bm\eta_{n,L-2}$, we can represent the feature tensor as a feature vector $\bm\phi_{n,L-2}=\vect(\bm\eta_{n,L-2}) \in {\mathbb R}^{k_{L-2}}$, where $k_{L-2} = M_{L-2} R_{L-2} C_{L-2}$. Finally, we can extract a feature vector $\bm\phi_{n,L-1} \in {\mathbb R}^{k_{L-1}}$ from the fully connected layer; the collection of feature vectors $\bm{\Phi}=\lbrace \bm\phi_{n,L-1}\rbrace_{n=1}^{N} \in {\mathbb R}^{N \times k_{L-1}}$ summarizes high-dimensional input into a lower dimensional space.
\subsection{Bayesian Convolutional Generalized Linear Model}
In this section, we propose BayesCGLM by regressing $\mathbf{Y}$ on $\bm{\Phi}$ and additional covariates $\mathbf{Z}$. Here, we use $\bm{\Phi}$ as a basis design matrix that encapsulates information from high-dimensional and correlated input variables in $\mathbf{X}$. The proposed method can provide interpretation of the regression coefficients and quantify uncertainties in prediction and estimation.
As the first step, we train Bayes CNN with the $\mathbf{X}$ and $\mathbf{Z}$ as input and $\mathbf{Y}$ as output. We concatenate the covariate $\mathbf{Z}$ right after the last hidden layer to model the effects of $\mathbf{X}$ on $\mathbf{Y}$ for a given fixed $\mathbf{Z}$. Compared to the traditional CNN, the training step does not require additional computational costs. Then we can extract the feature $\boldsymbol{\Phi} \in {\mathbb R}^{N\times k_{L-1}}$ from the last hidden layer. We can regard this feature as an informative summary statistic that can measure potentially non-linear effects of the input $\mathbf{X}$ on $\mathbf{Y}$. Since we place a posterior distribution over the kernels of the fitted CNN (Section~\ref{Sec:BayesCNN}), we can generate Monte Carlo samples of the features $\bm{\Phi}^{(m)}$, $m=1,\cdots,M$ from the variational distribution. To account for the uncertainties in estimating the features $\bm{\Phi}$, we use the entire Monte Carlo samples of the features rather than the point estimates of them. In Section~\ref{Sec:Simul}, we demonstrate that this allows us to fully account for uncertainties in the feature extraction step through various simulated examples.
With the covariate matrix $\mathbf{Z} \in {\mathbb R}^{N \times p}$, we fit $M$ different GLMs by regressing $\mathbf{Y}$ on $\mathbf{A}^{(m)}=[\mathbf{Z}, \bm{\Phi}^{(m)}] \in {\mathbb R}^{N \times (p+k_{L-1})}$. Then, our model is
\begin{equation} \begin{split}
g(E[\mathbf{Y}|\mathbf{Z},\bm{\Phi}^{(m)}]) &= \mathbf{Z}\bm{\gamma}_m + \bm{\Phi}^{(m)}\bm{\delta}_m = \mathbf{A}^{(m)}\bm{\beta}_m
\end{split}
\label{model}
\end{equation}
where $\bm{\beta}_m=(\bm{\gamma}_m^{\top}, \bm{\delta}_m^{\top})^{\top} \in {\mathbb R}^{p+k_{L-1}}$ is the corresponding regression coefficients and $g(\cdot)$ is a one-to-one continuously differential link function.
For each Monte Carlo sample $m$, we can fit individual GLM in parallel; we have $m=1,\cdots, M$ number of posterior distributions, and each of them can be called "$m$th feature-posterior." This step is heavily parallelizable, so computational walltimes tend to decrease with an increasing number of available cores. Then, we construct an aggregated distribution (the "ensemble-posterior") through the mixture of feature posteriors. In the following section, we provide details about constructing the ensemble-posterior distribution. The fitting procedure of BayesCGLM is summarized in Algorithm~\ref{algo}.
\begin{algorithm}
\caption{BayesCGLM algorithm}\label{MCGLMalgo}
\textbf{Input}: $\mathbf{X}$ (image or correlated input), $\mathbf{Y}$ (response), $\mathbf{Z}$ (scalar covariates)
\textbf{Output}: Posterior distribution of model parameters.
\begin{algorithmic}[1]
\State Fit Bayes CNN by using $\mathbf{X}$ and $\mathbf{Z}$ as input and $\mathbf{Y}$ as output. Then for each $m$
\For {$m=1,2,\ldots,M$}
\State Extract features $\bm{\Phi}^{(m)}$ from the last hidden layer.
\State Fit GLM by regressing $\mathbf{Y}$ on $[\mathbf{Z}, \bm{\Phi}^{(m)}]$ through the Laplace approximation
\State Obtain the $m$th feature-posterior distribution of model parameter.
\EndFor
\State Construct an ensemble-poseterior by aggregating $M$ feature-posteriors.
\end{algorithmic}
\label{algo}
\end{algorithm}
Since BayesCGLM in \eqref{model} can capture non-linear effects of $\mathbf{X}$ on $\mathbf{Y}$ through the extracted features $\bm{\Phi}$, we can improve prediction accuracies compared to the traditional GLM. In general, inference for the regression coefficients $\bm{\gamma}$ is challenging for DNN. Although one can regard the weight parameters at the last hidden layer as $\bm{\gamma}$, we cannot obtain uncertainties of estimates. Note that even Bayes DNN \citep{gal2016dropout} only provides point estimates of weight parameters, while they can quantify uncertainties in response prediction. On the other hand, our method directly provides the posterior distribution of $\bm{\gamma}$, which is useful for standard inference. In addition, the complex structure of DNN often leads to nonconvex optimization \citep{ge2015escaping, kleinberg2018alternative, daneshmand2018escaping,alzubaidi2021review}, resulting in inaccurate estimate of $\bm{\gamma}$. Stochastic gradient descent algorithm requires simultaneous updates of high-dimensional weight parameters. Instead, our two-stage approach can alleviate this issue by using extracted features $\bm{\Phi}$ as a fixed basis matrix in GLM.
\textcolor{red}{Figure~\ref{loglikelihood_contour} illustrates the profile log-likelihood surfaces over $\bm{\gamma}$ based on a single replicate from our simulation studies in Section 4. We observe that our two-stage approach provides more accurate estimates compared to BayesCNN. Since the stopping rule of the stochastic gradient descent algorithm is based on prediction accuracy (not based on estimation accuracy), the convergence of individual weight parameters, including $\bm{\gamma}$ cannot be guaranteed. The simultaneous optimization becomes more challenging, especially when there are many weight parameters to be updated \citep{alzubaidi2021review}. On the other hand, BayesCGLM coverts the complex optimization into simple nonparametric regression problems by utilizing the extracted features $\bm{\Phi}$ as a fixed basis function. In Section 4, we repeat the simulation 300 times under the different model configuration. We show that BayesCGLM can recover the true $\bm{\gamma}$ values well, while estimates from Bayes DNN can be biased.}
\begin{figure*}[htbp]
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.99\linewidth]{likelihood_binary_contour_BayesCNN_final.pdf}
\caption{The profile log-likelihood in BayesCNN}
\label{BayesCNN_loglike}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.99\linewidth]{likelihood_binary_contour_GLM_final.pdf}
\caption{The profile log-likelihood in BayeCGLM}
\label{BayesCGLM_loglike}
\end{subfigure}
\caption{\textcolor{red}{The green x-crosses indicate the true $\bm{\gamma}$, and the red triangles represent the estimates obtained from BayesCNN and BayesCGLM, respectively. BayesCGLM provides a closer estimate to the true $\bm{\gamma}$ than BayesCNN.}}
\label{loglikelihood_contour}
\end{figure*}
\subsection{Bayesian Inference}
Our framework requires Bayesian inference on ensemble GLMs in \eqref{model}, i.e., fitting $M$ different GLM models in a Bayesian way. One may choose the standard Markov chain Monte Carlo (MCMC) algorithm, which can provide exact Bayesian inference. However, this may require a long run of the chain to guarantee a good mixing, resulting in excessive computational cost in our framework. To fit ensemble GLMs quickly, we use the Laplace approximation method \citep{tierney1986accurate, rue2009approximate}. In our preliminary studies, we observe that there are little benefits from the more exact method; for enough sample size, the Laplace approximation method provides reasonably accurate approximations to the posteriors within a much shorter time.
For each dropout realization $m$, we can obtain the maximum likelihood estimate (MLE) $\widehat{\bm{\beta}}_{m}$ of \eqref{model} using the iteratively reweighted least square algorithm, which can be implemented through {\tt{statmodels}} function in {\tt{Python}}. Then we can approximate the posterior of $\bm{\beta}_{m}$ through $\mathcal{N}(\widehat{\bm{\beta}}_{m}, \widehat{\mathbf{B}}^{-1}_{m})$, where $\widehat{\mathbf{B}}_{m} \in {\mathbb R}^{(p+k_{L-1}) \times (p+k_{L-1})}$ is the observed Fisher information matrix from the $m$th Monte Carlo samples. We can obtain such approximated posteriors for $m=1,\cdots,M$ from Monte Carlo dropout realizations. We can aggregate these feature-posteriors with the mixture distribution as
\begin{equation}
\widehat{\pi}(\bm{\beta}|\mathbf{D}) = \sum_{m=1}^{M} \omega_m \varphi(\bm{\beta}; \widehat{\bm{\beta}}_{m},\widehat{\mathbf{B}}^{-1}_{m})
\label{posterirdist}
\end{equation}
where $\varphi(\bm{x}; \bm{\mu},\bm{\Sigma})$ is a multivariate normal density with mean $\bm{\mu}$ and covariance $\bm{\Sigma}$.
Here, we used the equal weight for $\omega_m$ across different Monte Carlo dropout realizations (i.e., $\omega_m=1/M$). Especially, we focus on describing inference for the posterior distribution of the regression coefficient $\bm{\gamma}$, which can quantify the impact of $\mathbf{Z}$ on $\mathbf{Y}$. Since we have an ensemble-posterior distribution \eqref{posterirdist}, we can conduct standard Bayesian inference naturally.
Our Bayesian framework also quantifies uncertainties in predictions which is another advantages over existing neural networks. For unobserved response $\mathbf{Y}^{\ast} \in {\mathbb R}^{N_{cv}}$, we have $\mathbf{A}^{\ast}_{m}=[\mathbf{Z}^{\ast}, \bm{\Phi}^{(m)}]\in {\mathbb R}^{N_{cv}\times (p+k_{L-1})}$ for $m=1,\cdots,M$. Then from \eqref{posterirdist}, the distribution of the linear predictor is
\begin{equation}
\widehat{\pi}(\mathbf{A}^{\ast}\bm{\beta}|\mathbf{D}) = \sum_{m=1}^{M} \omega_m \varphi(\mathbf{A}^{\ast}\bm{\beta}; \mathbf{A}_{m}^{\ast}\widehat{\bm{\beta}}_{m},\mathbf{A}_{m}^{\ast}\widehat{\mathbf{B}}^{-1}_{m}\mathbf{A}^{\ast\top}_{m}).
\label{linearpredictor}
\end{equation}
From the linear predictor above, we can construct the posterior predictive distribution. For given a posterior mean $\widehat{\bm{\beta}}$ from \eqref{posterirdist}, we can generate $\mathbf{Y}^{\ast}$ from the exponential family distribution with mean $\mathbf{A}^{\ast}\bm{\hat{\beta}}$. For example, when we have a Gaussian response we have $\mathbf{Y}^{\ast} \sim \mathcal{N}( \mathbf{A}^{\ast}\bm{\hat{\beta}},\hat{\sigma}^2)$ where $\hat{\sigma}^2 = \frac{\sum_{n=1}^{N}(\mathbf{A}^{\ast}\bm{\hat{\beta}}-\mathbf{Y})}{N}$ for the Gaussian response. For the count response, we can simulate $\mathbf{Y}^{\ast}$ from the Poisson distribution with intensity $\mathbf{A}^{\ast}\bm{\hat{\beta}}$.
\section{Simulated Data Examples}\label{Sec:Simul}
In this section, we apply BayesCGLM to simulated Gaussian, binary, and Poisson data. We implement our approach in {\tt{TensorFlow}}, an open-source platform for machine learning. Parallel computation is implemented through the {\tt{multiprocessing}} library in {\tt{Python}}. The computation times are based on 8 core AMD Radeon Pro 5500 XT processors. The source code can be downloaded from https://github.com/jeon9677/BayesCGLM. The configuration of the CNN structures and tuning details used in our experiments are provided in the supplementary material.
\paragraph{Simulation Design}
We first set up spatial locations $\mathbf{s}_1,\cdots,\mathbf{s}_{900}$ on a $30\times 30$ regular lattice with a spacing of one. We generate a spatially correlated data from a Gaussian process with mean 0 and the Mat\'{e}rn class covariance function \citep{stein2012interpolation}
\[ \varphi(d) = \frac{\sigma^2}{2^{\nu-1}\Gamma(\nu)} (d/\rho)^{\nu}K_\nu(d/\rho),
\]
where $d$ is the Euclidean distance between two points. Here $\sigma$, $\nu$, and $\rho$ denote the variance, smoothness, and range parameters, respectively. In our simulation studies, we set $\sigma=1,\nu=0.5$ and $\rho=15$. We repeat the above procedure for 1,000 times to generate $1,000$ number of image observations $\mathbf{X} = \lbrace\mathbf{X}_n \rbrace_{n=1}^{1000}$. Figure \ref{exampleimage} illustrates a single realization of the simulated image.
\begin{figure*}[htbp]
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[width=.9\linewidth]{Sample_simulation.png}
\caption{Illustration for a simulated image}
\label{exampleimage}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.9\linewidth]{Sample_simulation_filter.png}
\caption{Filter images}
\label{filters}
\end{subfigure}
\caption{(a) An example of generated $30 \times 30$ size image, $\mathbf{X}_n$. (b) Images of four filters used in our simulation studies. Each of the filters highlights top-left ($\mathbf{K}_1$), bottom-right ($\mathbf{K}_2$), top-right ($\mathbf{K}_3$), and center ($\mathbf{K}_4$) respectively. For all figures, the darker gray region corresponds to small pixel values.}
\label{Simulated_images}
\end{figure*}
Then we extract features from simulated images through $C$ number of different filters $\mathbf{K}_c\in {\mathbb R}^{30 \times 30}$ for $c=1,\cdots,C$ (Figure \ref{filters}). In our study, we used $C=4$ for Gaussian and binary cases and $C=2$ for a Poisson case. The weights in each filter are designed to capture local characteristics in different parts of the input images. Especially, we use inverse quadratic basis functions $\varphi(d) = 1/(1+(\delta d)^2)$, where $d$ is the Euclidean distance between pixels in $\mathbf{K}_c$ and a focal point $\mathbf{u}_c$, and $\delta$ is the tuning parameter that controls the decay of the kernel as the distance grows. Here, we set $\mathbf{u}_1=(0,0)$, $\mathbf{u}_2=(10,20)$, $\mathbf{u}_3=(15,15)$, $\mathbf{u}_4=(30,30)$ to extract local features in different areas, and $\delta=0.1$ for all $c=1,\dots,4$. Based on the filters, we can extract features $\Phi_{n,c}$ as
\[
\Phi_{n,c} = \frac{1}{30}\frac{1}{30}\sum_{i=1}^{30} \sum_{j=1}^{30} \mathbf{K}_{c,ij} \mathbf{X}_{n,ij},
\]
for $c=1,\cdots,C$. Each $\mathbf{K}_c$ extracts a local feature of the images in different areas such as bottom-left, bottom-right, center, or top-right. Therefore, the whole feature vector is $\bm\Phi_n=\lbrace \Phi_{n,c} \rbrace_{c=1}^{C} \in {\mathbb R}^{C}$.
Finally, we generate covariates $\mathbf{Z} =\lbrace \mathbf{z}_{n} \rbrace_{n=1}^{1000} \in {\mathbb R}^{1000\times 2}$, where individual elements are sampled from a standard normal distribution. For given the generated features $\bm\Phi=\lbrace \bm\Phi_{n} \rbrace_{n=1}^{1000} \in {\mathbb R}^{1000\times C}$ above, we calculate $\bm{\lambda}=g^{-1}(\bm\Phi + \mathbf{Z}\bm{\gamma})$ with $\bm{\gamma}=(1,1)$, i.e., the true values of the regression coefficients for the covariates are 1's. We simulate $\mathbf{Y} \sim N_{1000}(\bm{\lambda},\mathbf{I})$ for normal, $\mathbf{Y} \sim \text{Poisson}(\bm{\lambda})$ for count, and $\mathbf{Y} \sim \text{Bernoulli}(\bm{\lambda})$ for binary cases. Note that the features $\bm\Phi_{n}$'s used to generate $\mathbf{Y}$ are not used for model fitting. Our BayesCGLM uses the generated images $\mathbf{X}_n$ and $\mathbf{z}_n$ as input variables and conducts feature extraction on $\mathbf{X}_n$ by itself. We use the first $N=700$ observations for model fitting and the remaining $N_{cv}=300$ for performance testing.
To measure the prediction accuracy, we calculate the root mean square prediction error (RMSPE) and the empirical coverage of prediction intervals. We also report the average computing time from simulations. For each simulation setting, we repeat the simulation 500 times.
\paragraph{Comparative Analysis}
To demonstrate the performance of our approach, we compare the parameter estimation and response prediction performance of proposed BayesCGLM with Bayesian CNN \citep{gal2016bayesian} and GLM. In the training step, we use the standard early stopping rule based on prediction accuracy to avoid overfitting issues. We train CNN by using both images $\mathbf{X}$ and covariates $\mathbf{Z}$ as input and response $\mathbf{Y}$ as output. By placing $\mathbf{Z}$ at the fully connected layer, we can obtain the weight parameters that correspond to the regression coefficients $\bm{\gamma}$ in \eqref{model}; note that this can only provide point estimates of the weights. Since it is challenging to use images as predictors in the GLM, we fit GLM by regressing $\mathbf{Y}$ on $\mathbf{Z}$ without using images.
\subsection{Gaussian Case}
We first describe the simulation study results when the response $\mathbf{Y}$ is generated from a Gaussian distribution, i.e., $\mathbf{Y} \sim N_{1000}(\bm{\lambda},\mathbf{I})$. For Monte Carlo dropout sampling, we set $M=300$, which seems to result in good uncertainty quantification performance while requiring computing resources. We also compare the results with a single dropout sample (i.e., $M=1$) to examine how much improvement in inference performance can be attributed to the repeated dropout sampling.
Figure \ref{simul_pred_normal} illustrates agreement between the true and predicted responses in a simulated dataset. Since BayesCGLM can quantify uncertainties in predictions, we also visualize the 95\% highest posterior density (HPD) prediction intervals, which include the true $\mathbf{Y}$ well.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width = 0.5\textwidth]{simulation_normal_result.pdf}
\end{center}
\caption[]{Comparison between the true and predicted responses from BayesCGLM. The vertical lines indicate 95\% highest posterior density (HPD) from the predictive distribution. }
\label{simul_pred_normal}
\end{figure}
Table \ref{simul_normal} reports the inference results from different methods. We observe that the parameter estimates obtained from BayesCGLM and GLM are similar to the simulated truth on average, while the estimates from Bayes CNN \citep{gal2016bayesian} are biased. As we have pointed out in Section 3.2, simultaneous updates of high-dimensional weight parameters through a stochastic gradient descent algorithm can lead to inaccurate estimates of $\bm{\gamma}$. Our BayesCGLM method with $M=300$.
Our BayesCGLM with $M=300$ results in accurate uncertainty quantification, yielding credible intervals for the regression coefficients $\bm{\gamma}$ whose empirical coverage is closer to the nominal coverage (95\%) than the BayesCGLM with $M=1$ and GLM. This shows that both the repeated Monte Carlo dropout sampling and the use of extracted features improve uncertainty quantification performance for $\bm{\gamma}$.
Furthermore, the prediction accuracies (RMSPE, prediction coverage) from BayesCGLM are comparable to the Bayes CNN. On the other hand, RMSPE obtained from GLM is much higher than the other deep learning approaches.
\begin{table}[tt]
\centering
\begin{tabular}{c c c c c}
\toprule
& \textbf{BayesCGLM}(M$=300$) &\textbf{BayesCGLM}(M$=1$) & \textbf{Bayes CNN} & \textbf{GLM} \\ \hline
$\gamma_1$ & 0.993 (0.041) & 0.954 (0.058) & 0.888 (0.093) & 1.003 (0.057) \\
Coverage & 0.944 & 0.918 & - & 0.930 \\
\hline
$\gamma_2$ & 0.992 (0.041) & 0.946 (0.060) & 0.883 (0.104) & 1.000 (0.055) \\
Coverage & 0.954 & 0.924 & - & 0.912 \\
\hline RMSPE & 1.048 (0.045) & 1.044 (0.045) & 1.076 (0.053) & 1.416 (0.058) \\
Coverage & 0.940 & 0.930 & 0.947 & 0.975 \\
\hline Time & 35.725 & 19.037 & 18.894 & 0.003 \\
\bottomrule
\end{tabular}
\caption{Inference results for the simulated Gaussian datasets. For all methods, mean of $\bm{\gamma}$, estimation coverage, RMSPE, prediction coverage, and average computing time (sec) are calculated from 300 simulations. The numbers in the parentheses indicate standard deviations obtained from the repeated simulations.}
\label{simul_normal}
\end{table}
\subsection{Binary Case}
We now describe the simulation study results when $\mathbf{Y}$ is in the form of binary response, i.e., $\mathbf{Y} \sim \text{Bernoulli}(\bm{\lambda})$. Figure \ref{simul_pred_binary} compares the true response and the estimated probabilities (for the response to be 1), which show a good agreement. We observe that the predicted probability shows higher values for the binary responses with the true value of 1. The area under the curve (AUC) from the receiver operating characteristic curve (ROC) is about 0.85, which is reasonably close to 1 (Figure \ref{simul_pred_binary2}).
\begin{figure*}[htbp]
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.8\linewidth]{simulation_binary_result1.pdf}
\caption{Predicted probability surface}
\label{simul_pred_binary1}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.99\linewidth]{simulation_binary_result2.png}
\caption{ROC plot}
\label{simul_pred_binary2}
\end{subfigure}
\caption{ (a) The scatter plot of estimated probability of $Y=1$ and true binary responses. Blue line indicates the estimated probability function. (b) The ROC curve for binary prediction. Blue line denotes AUC of the ROC curve. }
\label{simul_pred_binary}
\end{figure*}
In Table~\ref{simul_binary} we observe that parameter estimates from BayesCGLM are accurate, and the coverages are close to the 95\% nominal rate even with a single Monte Carlo dropout ($M=1$). On the other hand, the estimates from Bayes CNN \citep{gal2016bayesian}, and GLM are biased.
Similar to the Gaussian case, deep learning methods (BayesCGLM, Bayes CNN) show much better prediction performance than GLM because they can extract features from image observations.
\begin{table}[tt]
\centering
\begin{tabular}{c c c c c}
\toprule
& \textbf{BayesCGLM}(M$=300$) &\textbf{BayesCGLM}(M$=1$) & \textbf{Bayes CNN} & \textbf{GLM} \\ \hline
$\gamma_1$ & 1.029 (0.154) & 1.030 (0.154) & 0.848 (0.117) & 0.389 (0.086) \\
Coverage & 0.956 & 0.966 &- & 0 \\
\hline
$\gamma_2$ & 1.030 (0.151) & 1.024 (0.155) & 0.851 (0.120) & 0.388 (0.081) \\
Coverage & 0.954 & 0.952 & - & 0 \\
\hline Accuracy & 0.869 (0.019) & 0.857 (0.022) & 0.850 (0.022) & 0.600 (0.028) \\
Recall & 0.869 (0.031) & 0.855 (0.033) & 0.851 (0.036) & 0.600(0.066) \\
Precision & 0.868 (0.030) & 0.858 (0.033) & 0.863 (0.029) & 0.602 (0.043) \\
\hline Time & 37.028 & 17.661 & 15.022 & 0.003\\
\bottomrule
\end{tabular}
\caption{Inference results for the simulated binary datasets. For all methods, mean of $\bm{\gamma}$, estimation coverage, accuracy, recall, precision, and average computing time (sec) are calculated from 300 simulations. The numbers in the parentheses indicate standard deviations obtained from the repeated simulations.}
\label{simul_binary}
\end{table}
\subsection{Poisson Case}
Finally, we describe the simulation study results when $\mathbf{Y}$ is generated from the Poisson distribution, i.e., $\mathbf{Y} \sim \text{Poisson}(\bm{\lambda})$. For the Poisson case, we use only two filters ($\mathbf{K}_1$ and $\mathbf{K}_2$, which extract features in the top-left area and the bottom-right area respectively) to extract features. This is because using all four filters leads to too large $\bm\lambda$ values, resulting in a large dispersion of Poisson responses $\mathbf{Y}$. Figure \ref{simul_pred_poisson} shows that the true and predicted responses are well aligned for a simulated dataset except for some extreme values. We observe that HPD prediction intervals include the true count response well.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width = 0.5\textwidth]{simulation_poisson_result.pdf}
\end{center}
\caption[]{Comparison between the true and predicted Poisson responses from BayesCGLM. The vertical lines indicate 95\% highest posterior density (HPD) from the predictive distribution. }
\label{simul_pred_poisson}
\end{figure}
Table~\ref{simul_poisson} indicates that BayesCGLM shows comparable (or better) prediction performance compared to Bayes CNN while providing uncertainties of estimates. We observe that the coverage of the credible and prediction intervals becomes closer to the 95\% nominal rate with the repeated Monte Carlo dropout sampling ($M=300$), compared to a single Monte Carlo dropout sampling ($M=1$). Compared to Bayes CNN, BayesCGLM with $M=300$ results in better point estimates for the parameters $\bm{\gamma}$ and point predictions for the response $\mathbf{Y}$. Compared to GLM, BayesCGLM with $M=300$ results in better empirical coverage for the credible intervals for $\bm{\gamma}$ as well as better prediction accuracy and empirical coverage for the prediction intervals for $\mathbf{Y}$.
\begin{table}[tt]\centering
\begin{tabular}{c c c c c}
\toprule
& \textbf{BayesCGLM}(M$=300$) &\textbf{BayesCGLM}(M$=1$) & \textbf{Bayes CNN} & \textbf{GLM} \\ \hline
$\gamma_1$ & 0.987 (0.027) & 0.989 (0.031) & 0.910 (0.106) & 0.997 (0.058) \\
Coverage & 0.936 & 0.862 & - & 0.542 \\
\hline
$\gamma_2$ & 0.986 (0.026) & 0.989 (0.031) & 0.911 (0.101) & 0.997 (0.059) \\
Coverage & 0.958 & 0.868 & - & 0.550 \\
\hline RMSPE & 2.305 (1.610) & 2.687 (2.190) & 2.737 (2.537) & 4.364 (3.739) \\
Coverage & 0.963 & 0.890 & 0.967 & 0.924 \\
\hline Time & 22.73 & 15.285 & 13.753 & 0.004 \\
\bottomrule
\end{tabular}
\caption{Inference results for the simulated Poisson datasets. For all methods, mean of $\bm{\gamma}$, estimation coverage, RMSPE, prediction coverage, and average computing time (min) are calculated from 300 simulations. The numbers in the parentheses indicate standard deviations obtained from the repeated simulations. }
\label{simul_poisson}
\end{table}
\section{Applications}
We apply our method to three real data examples: (1) malaria incidence data, (2) brain tumor images, and (3) fMRI data for anxiety scores. BayesCGLM shows comparable prediction with the standard Bayes CNN while enabling uncertainty quantification for regression coefficients.
\subsection{Malaria Cases in the African Great Lakes Region}
Malaria is a parasitic disease that can lead to severe illnesses and even death. Predicting occurrences at unknown locations is of significant interest for effective control interventions. We compiled malaria incidence data from the Demographic and Health Surveys of 2015 \citep{ICF2017}. The dataset contains malaria incidence (counts) from 4,741 GPS clusters in nine contiguous countries in the African Great Lakes region: Burundi, the Democratic Republic of Congo, Malawi, Mozambique, Rwanda, Tanzania, Uganda, Zambia, and Zimbabwe. We use average annual rainfall, vegetation index of the region, and the proximity to water as spatial covariates. Under a spatial regression framework, \citet{gopal2019characterizing} analyzes malaria incidence in Kenya using these environmental variables. In this study, we extend this approach to multiple countries in the African Great Lakes region. We use $N = 3,500$ observations to fit the model and save $N_{\text{cv}} = 1,241$ observations for cross-validation.
To create rectangular images out of irregularly shaped spatial data, we use a spatial basis function matrix as the correlated input $\mathbf{X}$ in Algorithm~\ref{algo} (line 1). Specifically, we use thin plate splines basis \citep{wahba1990spline} defined as $ \mathbf{X}_{j}(\mathbf{s}) = ||\mathbf{s}-\mathbf{u}_{j}||^{2}\log(||\mathbf{s}-\mathbf{u}_{j}||)$, where $s$ is an arbitrary spatial location and $\lbrace \mathbf{u}_{j} \rbrace_{j=1}^{239}$ is a set of knots placed regularly over the spatial domain. From this we can construct basis functions for each observation, resulting in $\mathbf{X} \in {\mathbb R}^{3,500 \times 239}$ input design matrix. We first train Bayes CNN with input $\mathbf{X}, \mathbf{Z}$ and the count response $\mathbf{Y}$ while applying Monte Carlo dropout, with a dropout probability of 0.2. We set the batch size and epochs as 10 and 500, respectively. We use ReLU and linear activation functions in the hidden layers and use the exponential activation function in the output layer to fit count response. From this, we extract a feature design matrix $\bm{\Phi} \in {\mathbb R}^{3,500 \times 16}$ from the last layer of the fitted Bayes CNN. We provide details about the CNN structure in the supplementary material. With covariate matrix $\mathbf{Z} \in {\mathbb R}^{3,500 \times 3}$, we fit Poisson GLMs by regressing $\mathbf{Y}$ on $[\mathbf{Z}, \bm{\Phi}^{(m)}] \in {\mathbb R}^{3,500 \times 19}$ for each $m$; here, we have used $M=500$ Monte Carlo samples.
We compare our method with a spatial regression model using basis expansions \citep[cf.][]{sengupta2013hierarchical, shi2017spatial, lee2020scalable} as
\[
g(E[\mathbf{Y}|\mathbf{Z},\mathbf{X}]) = \mathbf{Z}\bm{\gamma} + \mathbf{X}\bm{\delta},
\]
where $\mathbf{X} \in {\mathbb R}^{3,500 \times 239}$ is the same thin plate basis matrix that we used in BayesCGLM. To fit a hierarchical spatial regression model, we use {\tt{nimble}} \citep{nimble2017}, which is a convenient programming language for Bayesian inference. To guarantee convergence, an MCMC algorithm is run for 1,000,000 iterations with 500,000 discarded for burn-in. Furthermore, we also implement the Bayes CNN as in the simulated examples. Table~\ref{malaria_table} indicates that vegetation, water, and rainfall variables have positive relationships with malaria incidence in deep learning methods. BayesCGLM shows comparable prediction performance compared to Bayes CNN. Although Bayes CNN can provide prediction uncertainties, it cannot quantify uncertainties of the estimates. Although, we can obtain HPD intervals from spatial regression, it provides higher RMSPE and lower coverage than other deep learning methods.
\begin{table}[tt]\centering
\begin{tabular}{c c c c}
\toprule
& \textbf{BayesCGLM(M=500)} & \textbf{Bayes CNN} &\textbf{Spatial Model} \\ \hline
$\gamma_1$ (vegetation index) & 0.099 & 0.103 & 0.115 \\
& (0.092 0.107) & - & (0.111 0.118) \\
$\gamma_2$ (proximity to water) & 0.074 & 0.058 & -0.269 \\
& (0.068 0.080) & - & (-0.272 -0.266) \\
$\gamma_3$ (rainfall) & 0.036 & 0.027 & -0.122 \\
& (0.027 0.045)& - &(-0.126 -0.117) \\
\hline RMSPE & 27.438 & 28.462 & 42.393 \\
Coverage & 0.950 & 0.947 & 0.545 \\
\hline Time & 57.518 & 30.580 & 41.285 \\
\bottomrule
\end{tabular}
\caption{Inference results for the malaria dataset from different methods. For all methods, posterior mean of $\bm{\gamma}$, 95\% HPD interval, RMSPE, prediction coverage, and computing time (min) are reported in the table. }
\label{malaria_table}
\end{table}
Figure \ref{fig1_malaria} shows that the true and predicted malaria incidence from BayesCGLM have similar spatial patterns. We observe the prediction standard errors are reasonably small across the entire spatial domain. Similar to the simulation studies, we illustrate the 95\% HPD prediction intervals, which include the true count response well.
\begin{figure}[htbp]
\centering
\begin{subfigure}[b]{1.1\textwidth}
\includegraphics[width=\linewidth]{malaria_result_1_update.png}
\caption{Prediction map}
\label{fig1_malaria}
\end{subfigure}
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\linewidth]{malaria_result_2.pdf}
\caption{Prediction plot with confidence interval}
\label{fig2_malaira}
\end{subfigure}
\caption[]{(a): The true (left panel) and predicted (middle panel) malaria incidence from BayesCGLM. The right panel shows the corresponding prediction standard errors. (b): Comparison between the true and predicted malaria incidence from BayesCGLM. The vertical lines indicate 95\% highest posterior density (HPD) from the predictive distribution.}
\end{figure}
\subsection{Brain Tumor Image Data}
Here, we apply our method to brain tumor image data. The dataset is collected from Brain Tumor Image Segmentation Challenge (BRATS) \citep{menze2014multimodal}. The binary response indicates whether a patient has a fatal brain tumor. For each patient, we have an MRI image with its two summary variables (first and second-order features of MRI images). In this study, we use $N = 2,508$ observations to fit our model and reserve $N_{\text{cv}} = 2,007$ observations for validation.
We first fit Bayes CNN using $240 \times 240$ pixel gray images $\mathbf{X}$, scalar covariates $\mathbf{Z}$ as input, and the binary response $\mathbf{Y}$ as output with Monte Carlo dropout, with the dropout probability of 0.25. We set the batch size and the number of epochs as 3 and 5, respectively. As before, we use ReLU and linear activation functions in the hidden layers and use the sigmoid activation function in the output layer to fit binary response. From the fitted CNN, we can extract $\bm\Phi \in {\mathbb R}^{2,508 \times 16}$ from the last hidden layer. We provide details about the CNN structure in the supplementary material. In this study, we use two summary variables (first and second-order features of MRI images) as covariates. With the extracted features from the CNN, we fit logistic GLMs by regressing $\mathbf{Y}$ on $[\mathbf{Z},\bm\Phi^{(m)}] \in {\mathbb R}^{2,508 \times 18}$ for $m=1,\cdots, 500$.
We compare our method with GLM and Bayes CNN as in the simulated examples. Table \ref{table_tumor} shows inference results from different methods. Our method shows that the first-order feature has a negative relationship, while the second-order feature has a positive relationship with the brain tumor risk. We observe that the sign of $\bm{\gamma}$ estimates from BayesCGLM are aligned with those from GLM. For this example, BayesCGLM shows the most accurate prediction performance
while providing credible intervals of the estimates.
\begin{table}[tt]
\centering
\begin{tabular}{c c c c c}
\toprule
& \textbf{BayesCGLM(M=500)} & \textbf{Bayes CNN} & \textbf{GLM} \\ \hline
$\gamma_1$ (first order feature) & -5.332 & 0.248 & -2.591 \\
& (-7.049,-3.704)& - & (-2.769,-2.412) \\
$\gamma_2$ (second order feature) & 4.894 & 0.160 & 2.950 \\
& (3.303, 6.564) & - & (2.755, 3.144) \\
\hline Accuracy & 0.924 & 0.867 & 0.784 \\
Recall & 0.929 & 0.787 & 0.783 \\
Precision & 0.901 & 0.907 & 0.715 \\
\hline Time & 293.533 & 103.924 & 0.004\\
\bottomrule
\end{tabular}
\caption{Inference results for the brain tumor dataset from different methods. For all methods, the posterior mean of $\bm{\gamma}$, 95\% HPD interval, accuracy, recall, precision, and computing time (min) are reported in the table. }
\label{table_tumor}
\end{table}
Figure~\ref{tumoraccuracy1} shows that the predicted probability surface becomes larger for the binary responses with a value of 1. Furthermore, the true and predicted binary responses are well aligned. The AUC for the ROC plot is about 0.96, which shows an outstanding prediction performance (Figure~\ref{tumoraccuracy2}).
\begin{figure*}[htbp]
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.9\linewidth]{tumor_1.pdf}
\caption{Predicted probability surface}
\label{tumoraccuracy1}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.9\linewidth]{tumor_2.png}
\caption{ROC plot}
\label{tumoraccuracy2}
\end{subfigure}
\caption{(a) The scatter plot of estimated probability of $Y=1$ and true binary responses in the brain tumor example. Blue line indicates the estimated probability function. (b) The ROC curve for binary prediction. Blue line denotes AUC of the ROC curve.}
\label{tumoraccuracy}
\end{figure*}
\subsection{fMRI Data for Anxiety Score}
The resting-state fMRI data is collected by the enhanced Nathan Kline Institute-Rockland Sample (NKI-RS)\citep{nooner2012nki} to create a large-scale community sample of participants across the lifespan. For 156 patients, resting-state fMRI signals are scanned for 394 seconds. Then we transform fMRI images to 278 brain functional brain Regions-of-Interest (ROIs) following the parcellation scheme of \cite{shen2013groupwise}. From this, we extracted time series of 278 brain functions and regressed out head motion, cerebrospinal fluid (CSF), white matter signals, and then band-pass filtered the time course data between 0.01-0.10 Hz to obtain valid signals . Functional network was computed using pairwise Pearson correlation coefficient constructing 278 $times$ 278 square symmetric matrix, which is called the ROI-signal matrix. Correlation matrix of 278 brain functions have been widely used to study functional connectivity of the brain \citep{worsley2002general,friman2003adaptive,smith2013functional}. Therefore, we also use the correlation matrix as input in Algorithm~\ref{algo} (line 1). Each participant is asked to respond to the survey data, which measure a wide array of physiological and psychological assessments. In this study, we use $N=156$ observations to fit our model and reserve $N_{\text{cv}}=52$ observations for validation. We use conscientiousness, extraversion, agreeableness, and neuroticism as scalar covariates $\mathbf{Z}$ and the averaged anxiety score as a response $\mathbf{Y}$.
We first train Bayes CNN with $\mathbf{X}$, $\mathbf{Z}$ as input and the continuous response $\mathbf{Y}$ as output, with a Monte Carlo dropout probability of 0.25. We set the batch size as 3 and epochs as 5. Except for the last hidden layers and the output layer, we use "ReLU" as an activation function. We use the "softplus" activation function in the last hidden layers. For the output layer, we use the "linear" activation function to model continuous responses. From the trained Bayes CNN, we extract a feature matrix $\bm\Phi \in {\mathbb R}^{104 \times 8}$ from the last hidden layer. We provide the details of the CNN structure in the supplementary material. Then we regress $\mathbf{Y}$ on $[\mathbf{Z},\bm\Phi^{(m)}]$ for $m=1,\cdots,500$.
Here, we compare our method with GLM and Bayes CNN. Table~\ref{NKI_table} indicates that BayesCGLM has comparable prediction performance compared to Bayes CNN and can quantify uncertainties of regression coefficients. We observe that all the methods provide similar coefficient estimates; the direction of the relationship between the response and covariates are identical. We can infer that the person who has emotional instability and vulnerability to unpleasant emotions (neuroticism) and who is less likely to rebel against others (agreeableness) has a high anxiety and depression score which gives the comparable result from \citet{Kaplan2015}. The person who is more inclined to seek sociability and is eager to attain goals, on the other hand, has lower anxiety and depression scores.
\begin{table}[tt]\centering
\begin{tabular}{c c c c }
\toprule
& \textbf{BayesCGLM(M=500)} & \textbf{Bayes CNN} & \textbf{GLM} \\ \hline
$\gamma_1$ (neuroticism) & 3.496 & 3.480 & 4.318 \\
& (2.629,4.29) & - & (2.572,6.064)\\
$\gamma_2$ (extraversion) & -1.123 & -1.149 & -1.653\\
& (-1.829,-0.386) & - & (-3.198,-0.108) \\
$\gamma_3$ (agreeableness) & 1.113 & 1.204 & 1.196\\
& (0.480,1.760)& - &(-0.229,2.621) \\
$\gamma_4$ (conscientiousness) & -1.394 & -1.416 &-1.283\\
& (-2.226,-0.507)& - &(-3.056,0.491) \\
\hline RMSPE & 9.03 & 9.79& 10.53 \\
Coverage & 0.923 & 0.981 & 0.923 \\
\hline Time & 3.459 & 0.491 &0.001 \\
\bottomrule
\end{tabular}
\caption{Inference results for the fMRI dataset from different methods. For all methods, posterior mean of $\bm{\gamma}$, 95\% HPD interval, RMSPE, prediction coverage, and computing time (min) are reported in the table. }
\label{NKI_table}
\end{table}
Figure \ref{NKI_results1} shows that there is some agreement between the true and predicted responses. We observe that HPD prediction intervals include the true response well (92.3\% prediction coverage). Due to the small sample size, prediction intervals are wider than previous examples, which is natural.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width = 0.5\textwidth]{NKI_result.pdf}
\end{center}
\caption[]{Comparison between the true and predicted anxiety score from BayesCGLM. The vertical lines indicate 95\% highest posterior density (HPD) from the predictive distribution.}
\label{NKI_results1}
\end{figure}
\section{Discussion}
In this manuscript, we propose a flexible regression approach, which can utilize image data as predictor variables via feature extraction through convolutional layers. The proposed method can simultaneously utilize predictors with different data structures such as vector covariates and image data.
Our study of real and simulated data examples shows that the proposed method results in comparable prediction performance to existing deep learning algorithms in terms of prediction accuracy while enabling uncertainty quantification for estimation and prediction. By constructing an ensemble-posterior through a mixture of feature-posteriors, we can fully account for the uncertainties in feature extraction; we observe that BayesCGLM can construct accurate HPD intervals that achieve the nominal coverage. BayesCGLM is computationally efficient, in that multiple GLMs are fitted in parallel with fast Laplace approximations.
Our work is motivated by recently developed feature extraction approaches from DNN. These features can improve the performance of existing statistical approaches by accounting for complex dependencies among observations. For instance, \cite{bhatnagar2020computer} extract features from the long-short term memory (LSTM) networks \citep{gers2001lstm,graves2005framewise} and use them to capture complex inverse relationship in calibration problems. Similar to our approach, \cite{tran2020bayesian} use the extracted features from DNN as additional covariates in generalized linear mixed effect models. \cite{fong2021forward} propose a nonlinear dimension reduction method based on deep autoencoders, which can extract interpretable features from high-dimensional data.
The proposed framework can be extended to a wider range of statistical models such as cox regression \citep{cox1972regression} in survival analysis or regression models in causal inference. Furthermore, extracting features from other types of DNN is available; for example, one can use CNN-LSTM \citep{wang2016dimensional} to capture spatio-temporal dependencies. Our ensemble GLM approaches can significantly improve prediction and provide interpretable parameter estimates.
\section*{Acknowledgement}
Jaewoo Park and Yeseul Jeon were supported by the National Research Foundation of Korea (NRF-2020R1C1C1A0100386812). Sanghoon Han was partially supported by a National Research Foundation of Korea grant funded by the Korean government (NRF-2019R1A2C1007399). The authors are grateful to Seokjun Choi for the helpful discussion. The authors are grateful to anonymous reviewers for their careful reading and valuable comments.
\section{Dropout as a Bayesian Approximation}
In this section, we describe results in \cite{gal2016dropout} that a deep neural network with dropout layers is mathematically equivalent to the approximation of the posteriors of the deep GP. Especially we focus on a Gaussian response $\mathbf{y}$ case, but it can be easily extended to non-Gaussian responses. Consider a neural network with dropout layers as
\begin{equation}
\small\mathbf{y}=\sigma_L \Big(\mathbf{W}_{L}\sigma_{L-1}\Big(\mathbf{W}_{L-1} \cdots \sigma_3 \Big(\mathbf{W}_{3}\sigma_2 \Big(\mathbf{W}_{2} \sigma_1 \Big(\mathbf{W}_{1}\mathbf{x} +\mathbf{b}_1\Big)\circ\mathbf{d}_{2} +\mathbf{b}_2\Big)\circ\mathbf{d}_{3} +\mathbf{b}_3\Big) \cdots \Big)\circ\mathbf{d}_{L-1} + \mathbf{b}_{L-1} \Big)\circ\mathbf{d}_{L} +\mathbf{b}_{L} \Big)
\label{eq1dropSuppl}
\end{equation}
where $\mathbf{d}_{l}$ for $l=2,\cdots,L$ is defined as $k_l$-dimensional vectors, each component follows Bernoulli distribution with success probability $p$ independently.
In a traditional approach, we can train the above model by minimizing the following loss function with $L_2$ regularization terms:
\begin{equation}
\mathcal{L}_{dropout} := -\frac{1}{2N}\sum_{n=1}^{N} ||\mathbf{y}_n - \mathbf{\hat{y}}_n||_2^2+\sum_{l=1}^{L}\lambda^{\mathbf{w}}_{l}||\mathbf{W}_l||_2^2+\sum_{l=1}^{L}\lambda^{\mathbf{b}}_{l}||\mathbf{b}_l||_2^2.
\label{dropoutloss}
\end{equation}
Here $\lambda^{\mathbf{w}}_{l}$, $\lambda^{\mathbf{b}}_{l}$ are the shrinkage parameters for weight and bias parameters respectively. Note that we can replace \eqref{dropoutloss} with other types of loss functions for non-Gaussian responses; for instance, we can use a sigmoid loss function for a binary classification problem.
In a Bayesian alternative, we can represent \eqref{eq1dropSuppl} as a deep GP. As we described in the main manuscript, the posterior distribution of the deep GP is approximated through the variational distributions as
\begin{equation}\begin{split}
q(\mathbf{W}_{l}) & = \prod_{\forall i,j} q(w_{l,ij}),~~~q(\mathbf{b}_{l}) = \prod_{\forall i} q(b_{l,i})\\
q(w_{l,ij}) &= p_lN(\mu^{w}_{l,ij},\sigma^2)+(1-p_l)N(0,\sigma^2) \\
q(b_{l,i}) &= N(\mu^{b}_{l,i},\sigma^2),
\end{split}
\label{variationaldistsuppl}
\end{equation}
where $w_{l,ij}$ is the $i,j$th element of the weight matrix $\mathbf{W}_{l}\in {\mathbb R}^{k_l\times k_{l-1}}$ and $b_{l,i}$ is the $i$th element of the bias vector $\mathbf{b}_{l} \in {\mathbb R}^{k_l}$.
Then the Monte Carlo version of log evidence lower bound is
\begin{equation}
\mathcal{L}_{\text{GP-MC}} =\sum_{n=1}^{N} \log p(\mathbf{y}_n|\mathbf{x}_n,\lbrace \mathbf{W}_{l}^{(n)}, \mathbf{b}_{l}^{(n)} \rbrace_{l=1}^{L})-\text{KL}(\prod_{l=1}^{L}q(\mathbf{W}_{l})q(\mathbf{b}_{l})||p(\lbrace \mathbf{W}_{l}, \mathbf{b}_{l} \rbrace_{l=1}^{L})),
\label{GPMCsuppl}
\end{equation}
where $\lbrace \mathbf{W}_{l}^{(n)}, \mathbf{b}_{l}^{(n)} \rbrace_{l=1}^{L}$ is Monte Carlo samples from the variational distribution $\prod_{l=1}^{L}q(\mathbf{W}_l)q(\mathbf{b}_l)$ for $n$th observation; for notational convenience, we describe the Monte Carlo approximation with a distinct single sample as in \cite{gal2016dropout}. Note that $\mathcal{L}_{\text{GP-MC}}$ is the stochastic objective function, and estimates from $\mathcal{L}_{\text{GP-MC}}$ would converge to those obtained from $\mathcal{L}_{\text{GP-VI}}$ \citep[cf.][]{bottou1991stochastic,paisley2012variational,rezende2014stochastic}.
According to \cite{gal2016dropout}[Proposition 1], with large $k_l$ and a small value of constant $\sigma$ (hyper parameter), $\text{KL}( q(\mathbf{W}_l)|| p(\mathbf{W}_l))$ can be approximated as
\begin{equation}\begin{split}
\text{KL}( q(\mathbf{W}_l)|| p(\mathbf{W}_l)) &\approx \sum_{\forall i,j \in l} \frac{p_l}{2}((\mu^{w}_{l,ij})^{2}+(\sigma^{2} -(1+\log2\pi)-\log\sigma^{2})+C),
\end{split}
\label{KLdiversuppl1}
\end{equation}
with some constant $C$. Similarly, $\text{KL}( q(\mathbf{b}_l)|| p(\mathbf{b}_l))$ can be approximated as
\begin{equation}\begin{split}
\text{KL}( q(\mathbf{b}_l)|| p(\mathbf{b}_l)) &\approx \sum_{\forall i,j \in l} \frac{p_l}{2}((\mu^{b}_{l,ij})^{2}+(\sigma^{2} -(1+\log2\pi)-\log\sigma^{2})+C).
\end{split}
\label{KLdiversuppl2}
\end{equation}
By plugging in these approximations in \eqref{GPMCsuppl}, we have
\begin{equation} \small \begin{split}
\mathcal{L}_{\text{GP-MC}} &\approx \sum_{n=1}^{N} \log N(y_n; \mathbf{W}_{n,L}\phi_{n,L-1}+\mathbf{b}_L,\tau^{-1}\mathbf{I}_N)\\ &-\sum_{l=1}^{L} \frac{p_l}{2}(||\bm{\mu}^{\mathbf{w}}_{l}||^{2}-k_l k_{l-1}(\sigma^{2} -(1+\log2\pi)-\log\sigma^{2}))
-\sum_{l=1}^{L} \frac{1}{2}(||\bm{\mu}^{\mathbf{b}}_{l}||^{2}-k_l(\sigma^{2} -(1+\log2\pi)-\log\sigma^{2}))
\label{GPMCKLsuppl}
\end{split}
\end{equation}
with a constant precision $\tau$ (hyper parameter).
By ignoring constant hyper parameter terms ($\tau,\sigma$), the approximated version of the log evidence lower bound scaled by a positive constant $\frac{1}{\tau N}$ becomes
\begin{equation} \begin{split}
\mathcal{L}_{\text{GP-MC}} \approx -\frac{1}{2N}\sum_{n=1}^{N} ||\mathbf{y}_n - \mathbf{\hat{y}}_n||_2^2 - \sum_{l=1}^{L}\frac{p_l}{2\tau N} ||\bm{\mu}^{\mathbf{w}}_{l}||_2^2 - \sum_{l=1}^{L}\frac{1}{2\tau N}||\bm{\mu}^{\mathbf{b}}_{l}||_2^2.
\label{GPMC1suppl}
\end{split}
\end{equation}
This implies that the posterior distribution of the weight parameter $w_{l,ij}$ is approximated with the mixtures of the spike distributions; one is centered around $\mu^{w}_{l,ij}$ and the other is centered around 0. Similarly, the posterior of bias parameter $b_{l}$ is approximated through the spike distribution centered around $\mu^{b}_{l,i}$. The loss function \eqref{GPMC1suppl} becomes equivalent to \eqref{dropoutloss} by setting $\lambda^{\mathbf{w}}_l=\frac{p_l}{2\tau N}$ and $\lambda^{\mathbf{b}}_l=\frac{1}{2\tau N}$.
\section{CNN Structures}
\subsection{Simulations}
\paragraph{Gaussian Data}
\begin{itemize}
\item Optimizer: Adam optimizer
\item Learning rate: $1e-4$
\item Loss function: mean squared error
\item Batch size: 32
\item Epoch size: 300
\end{itemize}
\begin{table}[tt]
\centering
\caption{Summary of 2D-CNN configurations for Gaussian simulation}
\begin{tabular}{c c c c c c}
\textbf{Layer Type} &\textbf{Dimension}& \textbf{Kernel Size} & \textbf{Strides} & \textbf{Activation Function} \\ \Hline
Convolution Layer& $8\times 8$ & $4\times4$ & $2\times2$ & Relu \\
Dropout(p=0.2) & - & - & - & - \\
Max Pooling & $2\times 2$ & - & - & - \\
\hline
Convolution Layer & $16\times 16$ & $3\times 3$ & $2\times2$ & softmax \\
Dropout(p=0.2) & - & - & - & - \\
Max Pooling & 2 & - & - & -\\
\hline
Flatten & - & - & - & - \\
Dense & $1\times 32$ & - & - & Relu \\
Dropout(p=0.2) & - & - & - & - \\
Dense & $1\times 16$ & - & - & Relu \\
Dropout(p=0.2) & - & - & - & - \\
Dense & $1\times 16$ & - & - & softplus \\
Dropout(p=0.2) & - & - & - & - \\
\hline
Concatenate & $\mathbf{Z}$ \\
Dense & $1\times 1$ & - & - & mse \\
\hline
\end{tabular}
\end{table}
\paragraph{Binary Data}
\begin{itemize}
\item Optimizer: Adam optimizer
\item Learning rate: $1e-4$
\item Loss function: binary cross entropy
\item Batch size: 3
\item Epoch size: 2,000
\end{itemize}
\begin{table}[tt]
\centering
\caption{Summary of 2D-CNN configurations for binary simulation}
\begin{tabular}{c c c c c c}
\textbf{Layer Type} &\textbf{Dimension}& \textbf{Kernel Size} & \textbf{Strides} & \textbf{Activation Function} \\ \Hline
Convolution Layer& $16\times 16$ & $3\times3$ & $1\times 1$ & softmax \\
Dropout(p=0.25) & - & - & - & - \\
Max Pooling & $2\times 2$ & - & - & - \\
\hline
Convolution Layer & $32\times 32$ & $3\times 3$ & $1\times 1$ & softmax \\
Dropout(p=0.25) & - & - & - & - \\
Max Pooling & 2 & - & - & -\\
\hline
Flatten & - & - & - & - \\
Dense & $1\times 16$ & - & - & Relu \\
Dropout(p=0.25) & - & - & - & - \\
Dense & $1\times 8$ & - & - & linear \\
Dropout(p=0.25) & - & - & - & - \\
\hline
Concatenate & $\mathbf{Z}$ \\
Dense & $1\times 1$ & - & - & sigmoid \\
\hline
\end{tabular}
\end{table}
\paragraph{Poisson Data}
\begin{itemize}
\item Optimizer: Adam optimizer
\item Learning rate: $1e-3$
\item Loss function: Poisson
\item Batch size: 3
\item Epoch size: 2,000
\end{itemize}
\begin{table}[tt]
\centering
\caption{Summary of 2D-CNN configurations for Poisson simulation}
\begin{tabular}{c c c c c c}
\textbf{Layer Type} &\textbf{Dimension}& \textbf{Kernel Size} & \textbf{Strides} & \textbf{Activation Function} \\ \Hline
Convolution Layer& $8\times 8$ & $4\times 4$ & $2\times 2$ & softmax \\
Dropout(p=0.2) & - & - & - & - \\
Max Pooling & $2\times 2$ & - & - & - \\
\hline
Convolution Layer & $32\times 32$ & $3\times 3$ & $1\times 1$ & softmax \\
Dropout(p=0.2) & - & - & - & - \\
Max Pooling & 2 & - & - & -\\
\hline
Flatten & - & - & - & - \\
Dense & $1\times 32$ & - & - & softplus \\
Dropout(p=0.2) & - & - & - & - \\
Dense & $1\times 16$ & - & - & linear \\
Dropout(p=0.2) & - & - & - & - \\
\hline
Concatenate & $\mathbf{Z}$ \\
Dense & $1\times 1$ & - & - & exponential \\
\hline
\end{tabular}
\end{table}
\subsection{Real data applications}
\paragraph{Malaria Incidence}
\begin{itemize}
\item Optimizer: Adam optimizer
\item Learning rate: $1e-4$
\item Loss function: Poisson
\item Batch size: 10
\item Epoch size: 2500
\end{itemize}
\begin{table}[tt]
\centering
\caption{Summary of 1D-CNN configurations for malarial incidence}
\begin{tabular}{c c c c c c}
\textbf{Layer Type} &\textbf{Dimension}& \textbf{Kernel Size} & \textbf{Strides} & \textbf{Activation Function} \\ \Hline
Convolution Layer& $1\times 32$ & $3$ & $1$ & tanh \\
Dropout(p=0.25) & - & - & - & - \\
Max Pooling & $1\times 1$ & - & - & - \\
\hline
Convolution Layer & $1\times 64$ & $3$ & $1$ & tanh \\
Dropout(p=0.25) & - & - & - & - \\
Max Pooling & 2 & - & - & -\\
\hline
Flatten & - & - & - & - \\
Dense & $1\times 32$ & - & - & Relu \\
Dropout(p=0.25) & - & - & - & - \\
Dense & $1\times 16$ & - & - & linear \\
Dropout(p=0.25) & - & - & - & - \\
\hline
Concatenate & $\mathbf{Z}$ \\
Dense & $1\times 1$ & - & - & exponential \\
\hline
\end{tabular}
\end{table}
\paragraph{Brain tumor MRI images}
\begin{itemize}
\item Optimizer: Adam optimizer
\item Learning rate: $1e-4$
\item Loss function: binary cross entropy
\item Batch size: 3
\item Epoch size: 5
\end{itemize}
\begin{table}[tt]
\centering
\caption{Summary of 2D-CNN configurations for brain tumor MRI images}
\begin{tabular}{c c c c c c}
\textbf{Layer Type} &\textbf{Dimension}& \textbf{Kernel Size} & \textbf{Strides} & \textbf{Activation Function} \\ \Hline
Convolution Layer& $64\times 64$ & $3 \times 3 $ & $1 \times 1$ & Relu \\
Dropout(p=0.25) & - & - & - & - \\
Max Pooling & $2\times 2$ & - & - & - \\
\hline
Convolution Layer & $32\times 32$ & $3 \times 3$ & $1\times 1$ & Relu \\
Dropout(p=0.25) & - & - & - & - \\
Max Pooling & 2 & - & - & -\\
\hline
Flatten & - & - & - & - \\
Dense & $1\times 16$ & - & - & linear \\
Dropout(p=0.25) & - & - & - & - \\
\hline
Concatenate & $\mathbf{Z}$ \\
Dense & $1\times 1$ & - & - & sigmoid \\
\hline
\end{tabular}
\end{table}
\paragraph{fMRI Data for Anxiety Score}
\begin{itemize}
\item Optimizer: Adam optimizer
\item Learning rate: $1e-3$
\item Loss function: mean squared error
\item Batch size: 3
\item Epoch size: 500
\end{itemize}
\begin{table}[tt]
\centering
\caption{Summary of 2D-CNN configurations for fMRI data}
\begin{tabular}{c c c c c c}
\textbf{Layer Type} &\textbf{Dimension}& \textbf{Kernel Size} & \textbf{Strides} & \textbf{Activation Function} \\ \Hline
Convolution Layer& $8\times 8$ & $3 \times 3 $ & $2 \times 2$ & Relu \\
Dropout(p=0.25) & - & - & - & - \\
Max Pooling & $2\times 2$ & - & - & - \\
\hline
Convolution Layer & $16\times 16$ & $3 \times 3$ & $2\times 2$ & Relu \\
Dropout(p=0.25) & - & - & - & - \\
Max Pooling & 2 & - & - & -\\
\hline
Flatten & - & - & - & - \\
Dense & $1\times 16$ & - & - & Relu \\
Dropout(p=0.25) & - & - & - & - \\
Dense & $1\times 8$ & - & - & softplus \\
Dropout(p=0.25) & - & - & - & - \\
\hline
Concatenate & $\mathbf{Z}$ \\
Dense & $1\times 1$ & - & - & linear \\
\hline
\end{tabular}
\end{table}
\section{Simulation Experiment}
\paragraph{Simulation Design}
We first set up spatial locations $\mathbf{s}_1,\cdots,\mathbf{s}_{900}$ on a $30\times 30$ regular lattice with a spacing of one. We generate a spatially correlated data from a Gaussian process with mean 0 and the Mat\'{e}rn class covariance function \citep{stein2012interpolation}
\[ \varphi(d) = \frac{\sigma^2}{2^{\nu-1}\Gamma(\nu)} (d/\rho)^{\nu}K_\nu(d/\rho),
\]
where $d$ is the Euclidean distance between two points. Here $\sigma$, $\nu$, and $\rho$ denote the variance, smoothness, and range parameters, respectively. In our simulation studies, we set $\sigma=1,\nu=0.5$ and $\rho=15$. We repeat the above procedure for 1,000 times to generate $1,000$ number of image observations $\mathbf{X} = \lbrace\mathbf{X}_n \rbrace_{n=1}^{1000}$. Figure \ref{exampleimage} illustrates a single realization of the simulated image.
\begin{figure}[htbp]
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[width=.9\linewidth]{Sample_simulation.png}
\caption{Illustration for a simulated image}
\label{exampleimage}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.9\linewidth]{Sample_simulation_filter.png}
\caption{Filter images}
\label{filters}
\end{subfigure}
\caption{(a) An example of generated $30 \times 30$ size image, $\mathbf{X}_n$. (b) Images of four filters used in our simulation studies. Each of the filters highlights top-left ($\mathbf{K}_1$), bottom-right ($\mathbf{K}_2$), top-right ($\mathbf{K}_3$), and center ($\mathbf{K}_4$) respectively. For all figures, the darker gray region corresponds to small pixel values.}
\label{Simulated_images}
\end{figure}
Then we extract features from simulated images through $C$ number of different filters $\mathbf{K}_c\in {\mathbb R}^{30 \times 30}$ for $c=1,\cdots,C$ (Figure \ref{filters}). In our study, we used $C=4$ for Gaussian and binary cases. For the Poisson case, we use only two filters ($\mathbf{K}_1$ and $\mathbf{K}_2$, which extract features in the top-left area and the bottom-right area respectively) to extract features. This is because using all four filters leads to too large dispersion of Poisson responses. The weights in each filter are designed to capture local characteristics in different parts of the input images. Especially, we use inverse quadratic basis functions $\varphi(d) = 1/(1+(\delta d)^2)$, where $d$ is the Euclidean distance between pixels in $\mathbf{K}_c$ and a focal point $\mathbf{u}_c$, and $\delta$ is the tuning parameter that controls the decay of the kernel as the distance grows. Here, we set $\mathbf{u}_1=(0,0)$, $\mathbf{u}_2=(10,20)$, $\mathbf{u}_3=(15,15)$, $\mathbf{u}_4=(30,30)$ to extract local features in different areas, and $\delta=0.1$ for all $c=1,\dots,4$. Based on the filters, we can extract features $\Phi_{n,c}$ as
\[
\Phi_{n,c} = \frac{1}{30}\frac{1}{30}\sum_{i=1}^{30} \sum_{j=1}^{30} \mathbf{K}_{c,ij} \mathbf{X}_{n,ij},
\]
for $c=1,\cdots,C$. Each $\mathbf{K}_c$ extracts a local feature of the images in different areas such as bottom-left, bottom-right, center, or top-right. Therefore, the whole feature vector is $\bm\Phi_n=\lbrace \Phi_{n,c} \rbrace_{c=1}^{C} \in {\mathbb R}^{C}$.
Finally, we generate covariates $\mathbf{Z} =\lbrace \mathbf{z}_{n} \rbrace_{n=1}^{1000} \in {\mathbb R}^{1000\times 2}$, where individual elements are sampled from a standard normal distribution. For given the generated features $\bm\Phi=\lbrace \bm\Phi_{n} \rbrace_{n=1}^{1000} \in {\mathbb R}^{1000\times C}$ above, we calculate $\bm{\lambda}=g^{-1}(\bm\Phi + \mathbf{Z}\bm{\gamma})$ with $\bm{\gamma}=(1,1)$, i.e., the true values of the regression coefficients for the covariates are 1's. We simulate $\mathbf{Y} \sim N(\bm{\lambda},\mathbf{I})$ for normal, $\mathbf{Y} \sim \text{Poisson}(\bm{\lambda})$ for count, and $\mathbf{Y} \sim \text{Bernoulli}(\bm{\lambda})$ for binary cases. Note that the features $\bm\Phi_{n}$'s used to generate $\mathbf{Y}$ are not used for model fitting. Our BayesCGLM uses the generated images $\mathbf{X}_n$ and $\mathbf{z}_n$ as input variables and conducts feature extraction on $\mathbf{X}_n$ by itself. We use the first $N=700$ observations for model fitting and the remaining $N_{cv}=300$ for performance testing.
To measure the prediction accuracy, we calculate the root mean square prediction error (RMSPE) and the empirical coverage of prediction intervals. We also report the average computing time from simulations. For each simulation setting, we repeat the simulation 500 times.
\paragraph{Comparative Analysis}
To demonstrate the performance of our approach, we compare the parameter estimation and response prediction performance of proposed BayesCGLM with Bayesian CNN \citep{gal2016bayesian} and GLM. In the training step, we use the standard early stopping rule based on prediction accuracy to avoid overfitting issues. We train CNN by using both images $\mathbf{X}$ and covariates $\mathbf{Z}$ as input and response $\mathbf{Y}$ as output. By placing $\mathbf{Z}$ at the fully connected layer, we can obtain the weight parameters that correspond to the regression coefficients $\bm{\gamma}$ in $g(E[\mathbf{Y}|\mathbf{Z},\bm{\Phi}^{(m)}]) = \mathbf{Z}\bm{\gamma}_m + \bm{\Phi}^{(m)}\bm{\delta}_m$; note that this can only provide point estimates of the weights. Since it is challenging to use images as predictors in the GLM, we fit GLM by regressing $\mathbf{Y}$ on $\mathbf{Z}$ without using images.
\section{Simulation study: Gaussian case}
We first describe the simulation study results when the response $\mathbf{Y}$ is generated from a Gaussian distribution, i.e., $\mathbf{Y} \sim N_{1000}(\bm{\lambda},\mathbf{I})$. For Monte Carlo dropout sampling, we set $M=300$, which seems to result in good uncertainty quantification performance while requiring computing resources. We also compare the results with a single dropout sample (i.e., $M=1$) to examine how much improvement in inference performance can be attributed to the repeated dropout sampling.
Figure \ref{simul_pred_normal} illustrates agreement between the true and predicted responses in a simulated dataset. Since BayesCGLM can quantify uncertainties in predictions, we also visualize the 95\% highest posterior density (HPD) prediction intervals, which include the true $\mathbf{Y}$ well.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width = 0.5\textwidth]{simulation_normal_result.pdf}
\end{center}
\caption[]{Comparison between the true and predicted responses from BayesCGLM. The vertical lines indicate 95\% highest posterior density (HPD) from the predictive distribution. }
\label{simul_pred_normal}
\end{figure}
Table \ref{simul_normal} reports the inference results from different methods. We observe that the parameter estimates obtained from BayesCGLM and GLM are similar to the simulated truth on average, while the estimates from Bayes CNN \citep{gal2016bayesian} are biased. As we have pointed out in Section 3.2, simultaneous updates of high-dimensional weight parameters through a stochastic gradient descent algorithm can lead to inaccurate estimates of $\bm{\gamma}$. Our BayesCGLM method with $M=300$.
Our BayesCGLM with $M=300$ results in accurate uncertainty quantification, yielding credible intervals for the regression coefficients $\bm{\gamma}$ whose empirical coverage is closer to the nominal coverage (95\%) than the BayesCGLM with $M=1$ and GLM. This shows that both the repeated Monte Carlo dropout sampling and the use of extracted features improve uncertainty quantification performance for $\bm{\gamma}$.
Furthermore, the prediction accuracies (RMSPE, prediction coverage) from BayesCGLM are comparable to the Bayes CNN. On the other hand, RMSPE obtained from GLM is much higher than the other deep learning approaches.
\begin{table}[tt]
\centering
\caption{Inference results for the simulated Gaussian datasets. For all methods, mean of $\bm{\gamma}$, estimation coverage, RMSPE, prediction coverage, and average computing time (sec) are calculated from 300 simulations. The numbers in the parentheses indicate standard deviations obtained from the repeated simulations.}
\label{simul_normal}
\begin{tabular}{c c c c c}
& \textbf{BayesCGLM}(M$=300$) &\textbf{BayesCGLM}(M$=1$) & \textbf{Bayes CNN} & \textbf{GLM} \\ \hline
$\gamma_1$ & 0.993 (0.041) & 0.954 (0.058) & 0.888 (0.093) & 1.003 (0.057) \\
Coverage & 0.944 & 0.918 & - & 0.930 \\
\hline
$\gamma_2$ & 0.992 (0.041) & 0.946 (0.060) & 0.883 (0.104) & 1.000 (0.055) \\
Coverage & 0.954 & 0.924 & - & 0.912 \\
\hline RMSPE & 1.048 (0.045) & 1.044 (0.045) & 1.076 (0.053) & 1.416 (0.058) \\
Coverage & 0.940 & 0.930 & 0.947 & 0.975 \\
\hline Time & 35.725 & 19.037 & 18.894 & 0.003 \\
\end{tabular}
\end{table}
\section{fMRI Data for Anxiety Score}
The resting-state fMRI data is collected by the enhanced Nathan Kline Institute-Rockland Sample (NKI-RS)\citep{nooner2012nki} to create a large-scale community sample of participants across the lifespan. For 156 patients, resting-state fMRI signals are scanned for 394 seconds. Then we transform fMRI images to 278 brain functional brain Regions-of-Interest (ROIs) following the parcellation scheme of \citet{shen2013groupwise}. From this, we extracted time series of 278 brain functions and regressed out head motion, cerebrospinal fluid (CSF), white matter signals, and then band-pass filtered the time course data between 0.01-0.10 Hz to obtain valid signals . Functional network was computed using pairwise Pearson correlation coefficient constructing 278 $times$ 278 square symmetric matrix, which is called the ROI-signal matrix. Correlation matrix of 278 brain functions have been widely used to study functional connectivity of the brain \citep{friman2003adaptive,smith2013functional}. Therefore, we also use the correlation matrix as input $\mathbf{X}$. Each participant is asked to respond to the survey data, which measure a wide array of physiological and psychological assessments. In this study, we use $N=156$ observations to fit our model and reserve $N_{\text{cv}}=52$ observations for validation. We use conscientiousness, extraversion, agreeableness, and neuroticism as scalar covariates $\mathbf{Z}$ and the averaged anxiety score as a response $\mathbf{Y}$.
We first train Bayes CNN with $\mathbf{X}$, $\mathbf{Z}$ as input and the continuous response $\mathbf{Y}$ as output, with a Monte Carlo dropout probability of 0.25. We set the batch size as 3 and epochs as 5. Except for the last hidden layers and the output layer, we use "ReLU" as an activation function. We use the "softplus" activation function in the last hidden layers. For the output layer, we use the "linear" activation function to model continuous responses. From the trained Bayes CNN, we extract a feature matrix $\bm\Phi \in {\mathbb R}^{104 \times 8}$ from the last hidden layer. We provide the details of the CNN structure in the Web Appendix B.2 of the Supporting Information. Then we regress $\mathbf{Y}$ on $[\mathbf{Z},\bm\Phi^{(m)}]$ for $m=1,\cdots,500$.
Here, we compare our method with GLM and Bayes CNN. Table~\ref{NKI_table} indicates that BayesCGLM has comparable prediction performance compared to Bayes CNN and can quantify uncertainties of regression coefficients. We observe that all the methods provide similar coefficient estimates; the direction of the relationship between the response and covariates are identical. We can infer that the person who has emotional instability and vulnerability to unpleasant emotions (neuroticism) and who is less likely to rebel against others (agreeableness) has a high anxiety and depression score which gives the comparable result from \citet{Kaplan2015}. The person who is more inclined to seek sociability and is eager to attain goals, on the other hand, has lower anxiety and depression scores.
\begin{table}[tt]
\centering
\caption[]{Inference results for the fMRI dataset from different methods. For all methods, posterior mean of $\bm{\gamma}$, 95\% HPD interval, RMSPE, prediction coverage, and computing time (min) are reported in the table. }
\label{NKI_table}
\begin{tabular}{c c c c }
& \textbf{BayesCGLM(M=500)} & \textbf{Bayes CNN} & \textbf{GLM} \\ \hline
$\gamma_1$ (neuroticism) & 3.496 & 3.480 & 4.318 \\
& (2.629,4.29) & - & (2.572,6.064)\\
$\gamma_2$ (extraversion) & -1.123 & -1.149 & -1.653\\
& (-1.829,-0.386) & - & (-3.198,-0.108) \\
$\gamma_3$ (agreeableness) & 1.113 & 1.204 & 1.196\\
& (0.480,1.760)& - &(-0.229,2.621) \\
$\gamma_4$ (conscientiousness) & -1.394 & -1.416 &-1.283\\
& (-2.226,-0.507)& - &(-3.056,0.491) \\
\hline RMSPE & 9.03 & 9.79& 10.53 \\
Coverage & 0.923 & 0.981 & 0.923 \\
\hline Time & 3.459 & 0.491 &0.001 \\
\end{tabular}
\end{table}
Figure \ref{NKI_results1} shows that there is some agreement between the true and predicted responses. We observe that HPD prediction intervals include the true response well (92.3\% prediction coverage). Due to the small sample size, prediction intervals are wider than malaria incidence and brain tumor examples, which is natural.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width = 0.5\textwidth]{NKI_result.pdf}
\end{center}
\caption[]{Comparison between the true and predicted anxiety score from BayesCGLM. The vertical lines indicate 95\% highest posterior density (HPD) from the predictive distribution.}
\label{NKI_results1}
\end{figure}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,730
|
<?php
Route::controllers([
'administrator' => 'Cristabel\Administrator\Http\Controllers\AdministratorController'
]);
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,360
|
Lucky Clover Shake is Back!
RXBAR identified the potential for undeclared peanut in two varieties in December and initiated a recall of those varieties. They are now expanding the recall out of an abundance of caution, after recently receiving consumer contacts regarding allergic reactions to additional varieties.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,790
|
{"url":"http:\/\/tex.stackexchange.com\/tags\/ieeetran\/hot","text":"# Tag Info\n\n3\n\nYou need to change some parameters used internally by the document class. If you are submitting an article to an IEEE journal then this is not a good idea. It would be better to provide a photo of the appropriate size. Still, since you asked how to do it... \\documentclass{IEEEtran} \\usepackage{lipsum} \\makeatletter % Edit the following values as ...\n\n3\n\nFirst, I agree with @vonbrand on the advice to adhere to the specific journal rules and not to change settings by yourself; its their responsibility not yours and they will end up editing the whole paper again prior to publication. But, if you have to, you can use the etoolbox package to patch the relevant command in the .cls file to force it behave the way ...\n\n2\n\nNowadays You can load enumitem and use its additional parameters. In older versions of the ieeetran-class it was not possible due to incompatibility issues, but it got fixed with the most recent version. Changelog: 10) Removed support for legacy IED list commands, legacy QED and proof commands and the legacy biography and biographynophoto ...\n\n2\n\nTo make BibTeX insert a \"tie\" (unbreakable space) instead of an ordinary (breakable) space after \"pp.\" when using the IEEEtran bibliography style, you could proceed with the following solution: Locate the file IEEEtran.bst in your TeX distribution, and make a copy of the file; name the copy, say, IEEEtrantie.bst. (Do not edit the file IEEEtran.bst ...\n\n1\n\nYou have modified the ref to the author an delete the IEEE: \\documentclass[conference,a4paper]{IEEEtran} \\ifCLASSINFOpdf \\usepackage[pdftex]{graphicx} \\else \\usepackage[dvips]{graphicx} \\fi \\usepackage{caption} \\usepackage{subfig} \\usepackage{amsmath} \\usepackage{array} \\begin{document} \\title{Conference title.} \\author{ \\IEEEauthorblockN{Author ...\n\n1\n\nYou have to add the necessary options to the itemize environment. It might be easier with enumitem, but I don't think the copy editors would be happy if you load it. \\documentclass{IEEEtran} \\usepackage{calc} \\usepackage{showframe} % just for the example \\usepackage{blindtext} % just for the example \\begin{document} \\blindtext \\begin{itemize}[% ...\n\nOnly top voted, non community-wiki answers of a minimum length are eligible","date":"2015-11-26 07:04:59","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8919789791107178, \"perplexity\": 3628.3812832968374}, \"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-2015-48\/segments\/1448398446535.72\/warc\/CC-MAIN-20151124205406-00262-ip-10-71-132-137.ec2.internal.warc.gz\"}"}
| null | null |
module Demoable
class ApplicationController < ActionController::Base
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,710
|
Levant kan syfta på:
Levant – vind vid västra Medelhavet, se Levant (vind)
Île du Levant – en ö utanför den franska medelhavskusten
Oscar Levant (1906–1972), en amerikansk pianist och filmskådespelare
Levanten – en region vid östra Medelhavet
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,633
|
package com.baomidou.mybatisplus.plugins.pagination;
import static com.baomidou.mybatisplus.enums.DBType.getDBType;
import org.apache.ibatis.session.RowBounds;
import com.baomidou.mybatisplus.exceptions.MybatisPlusException;
import com.baomidou.mybatisplus.plugins.pagination.dialects.DB2Dialect;
import com.baomidou.mybatisplus.plugins.pagination.dialects.H2Dialect;
import com.baomidou.mybatisplus.plugins.pagination.dialects.HSQLDialect;
import com.baomidou.mybatisplus.plugins.pagination.dialects.MySqlDialect;
import com.baomidou.mybatisplus.plugins.pagination.dialects.OracleDialect;
import com.baomidou.mybatisplus.plugins.pagination.dialects.PostgreDialect;
import com.baomidou.mybatisplus.plugins.pagination.dialects.SQLServer2005Dialect;
import com.baomidou.mybatisplus.plugins.pagination.dialects.SQLServerDialect;
import com.baomidou.mybatisplus.plugins.pagination.dialects.SQLiteDialect;
import com.baomidou.mybatisplus.toolkit.StringUtils;
/**
* <p>
* 分页方言工厂类
* </p>
*
* @author hubin
* @Date 2016-01-23
*/
public class DialectFactory {
/**
* <p>
* 生成翻页执行 SQL
* </p>
*
* @param page
* 翻页对象
* @param buildSql
* 执行 SQL
* @param dialectType
* 方言类型
* @param dialectClazz
* 自定义方言实现类
* @return
* @throws Exception
*/
public static String buildPaginationSql(Pagination page, String buildSql, String dialectType, String dialectClazz)
throws Exception {
// fix #172, 196
return getiDialect(dialectType, dialectClazz).buildPaginationSql(buildSql, page.getOffsetCurrent(), page.getSize());
}
/**
* Physical Pagination Interceptor for all the queries with parameter
* {@link org.apache.ibatis.session.RowBounds}
*
* @param rowBounds
* @param buildSql
* @param dialectType
* @param dialectClazz
* @return
* @throws Exception
*/
public static String buildPaginationSql(RowBounds rowBounds, String buildSql, String dialectType, String dialectClazz)
throws Exception {
// fix #196
return getiDialect(dialectType, dialectClazz).buildPaginationSql(buildSql, rowBounds.getOffset(), rowBounds.getLimit());
}
/**
* <p>
* 获取数据库方言
* </p>
*
* @param dialectType
* 方言类型
* @param dialectClazz
* 自定义方言实现类
* @return
* @throws Exception
*/
private static IDialect getiDialect(String dialectType, String dialectClazz) throws Exception {
IDialect dialect = null;
if (StringUtils.isNotEmpty(dialectType)) {
dialect = getDialectByDbtype(dialectType);
} else {
if (StringUtils.isNotEmpty(dialectClazz)) {
try {
Class<?> clazz = Class.forName(dialectClazz);
if (IDialect.class.isAssignableFrom(clazz)) {
dialect = (IDialect) clazz.newInstance();
}
} catch (ClassNotFoundException e) {
throw new MybatisPlusException("Class :" + dialectClazz + " is not found");
}
}
}
/* 未配置方言则抛出异常 */
if (dialect == null) {
throw new MybatisPlusException("The value of the dialect property in mybatis configuration.xml is not defined.");
}
return dialect;
}
/**
* <p>
* 根据数据库类型选择不同分页方言
* </p>
*
* @param dbType
* 数据库类型
* @return
* @throws Exception
*/
private static IDialect getDialectByDbtype(String dbType) {
IDialect dialect;
switch (getDBType(dbType)) {
case MYSQL:
dialect = MySqlDialect.INSTANCE;
break;
case ORACLE:
dialect = OracleDialect.INSTANCE;
break;
case DB2:
dialect = DB2Dialect.INSTANCE;
break;
case H2:
dialect = H2Dialect.INSTANCE;
break;
case SQLSERVER:
dialect = SQLServerDialect.INSTANCE;
break;
case SQLSERVER2005:
dialect = SQLServer2005Dialect.INSTANCE;
break;
case POSTGRE:
dialect = PostgreDialect.INSTANCE;
break;
case HSQL:
dialect = HSQLDialect.INSTANCE;
break;
case SQLITE:
dialect = SQLiteDialect.INSTANCE;
break;
default:
throw new MybatisPlusException("The Database's Not Supported! DBType:" + dbType);
}
return dialect;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,910
|
A month ago Ford presented a revamped Ford Focus, however Blue Oval doesn't prepare to quit keeping that. First spy pictures of 2020 Ford Focus ST are a testimony that Detroit-based firm has huge plans for this model. So, all you speed up fans must be thrilled that a brand-new flashy Focus variation is on the way, right? New, 2020 Focus ST was spied at Nurburgring track and the roadways that border it. The ST version will look quite much like the normal car in terms of design, however mechanical parts will certainly be a different tale.
Under the hood, a 1.5-liter turbocharged inline-4 will probably change the 2.0-liter turbo-4 readily available on the existing ST. The brand-new system will certainly have the ability to generate near to 275 horsepower. Just for comparison the present version loads 'only' 252 hp. This amount of power means that we will certainly be looking at an improved turbocharging system.
The changes will certainly continue with a new eight-speed transmission which will be an optional attribute. At the moment just readily available gearbox is a six-speed manual. It was unexpected that this model had only manual transmission, but that's around to alter. Just what we anticipate to stay the exact same is the front-wheel-drive which will certainly continue to be the only choice.
2020 Ford Focus ST will certainly be underpinned by firm's next-gen modular platform planned for tiny cars. The lately launched standard variation comes with upgraded torsional strength thanks to the new chassis and suspensions design. The distinction in stiffness in between brand-new and old Focus is 20%, and the new setup will certainly move ride and managing capabilities of ST to the next level.
New Focus comes geared up with Ford's Co-Pilot360 Suite which teems with electronic chauffeur aids. It additionally gets a new shade touchscreen which is mounted on top of the control panel. The 2020 Ford Focus ST will certainly have added tools such as Recaro seats, ST badges around the cabin, and probably brings couple of performance choices, and driving settings.
The new generation of Focus will certainly strike the dealers later this year after it was showcased a few months ago. However, all you performance fans will have to wait practically a full year before ST variation concerns the marketplaces. As the screenings of mechanical parts are only in onset, 2020 ST Focus won't be out before 2019. Unlike the basic car which will certainly be marketed as 2019 MY, its performance variation will probably be sold as 2020 MY. Look at summer of next year for its release.
After the ST variation Blue Oval will most likely start work with RS model, but also for that, we'll need to wait beyond 2019.
There's no word on the price as of yet, however it will exceed the price of the typical variation, as we are accustomed. The price of the ongoing car stars from $18,825 for the base version, and all as much as $25,145 for completely filled Focus. The ongoing ST is valued at $26,000 which is much above the normal variation. Same can be expected from a new design which will have a price closer to $27,000.
When it hits the marketplace, new 2020 Ford Focus ST will go against automobiles such as Hyundai Veloster N, Volkswagen Golf GTI, and Honda Civic Type R.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,336
|
\section{Introduction}
\label{intro}
Stars form in molecular clouds that are both turbulent and magnetized.
The relative importance of the turbulence and magnetic field in
controlling star formation is a matter of debate (McKee \& Ostriker
2007). Early quantitative studies have concentrated on the formation
and evolution of individual (low-mass) cores out of quiescent,
magnetically supported clouds (Nakano 1984; Shu et al. 1987;
Mouschovias \& Ciolek 1999). Recent numerical simulations have
focused more on the role of turbulence in the dynamics of larger
clouds and core formation in them (as reviewed in, e.g., Mac Low
\& Klessen 2004). Ultimately, the debate must be settled by
observations, especially direct measurements of the magnetic
field strength in the bulk of the molecular gas, which are
generally difficult to do (e.g., Heiles \& Crutcher 2005). In the
absence of such measurements, we have to rely on indirect evidence.
The best observed region of active star formation is arguably
the nearby Taurus molecular cloud complex. Although it is not
clear whether the complex is magnetically supported or not as
a whole, the more diffuse regions are probably magnetically
dominated. The strongest evidence comes from thin strands of
$^{12}$CO emission that are aligned with the directions of
the local magnetic field (Heyer et al. 2008), as traced by
polarization of background
star light. The magnetic field in the strands is apparently
strong enough to induce a measurable difference between the
turbulent velocities along and perpendicular to the field
direction. Heyer et al. concluded that the field strength
is probably high enough to render the relatively diffuse
striated region subcritical, with a magnetic flux-to-mass
ratio greater than the critical value $2\pi G^{1/2}$ (Nakano
\& Nakamura 1978). The CO striations are strikingly
similar to those observed in the nearby Riegel-Crutcher HI
cloud, mapped recently by McClure-Griffiths et al. (2006),
using 21cm absorption against the strong continuum emission
towards the galactic center region. Its hair-like strands
are also along the directions of the local magnetic field,
again traced by the polarization vectors of the background
star light. The authors estimated a field strength of $\sim
30$~$\mu$G inside the strands. Kazes \& Crutcher (1986)
measured the line-of-sight field strength at a nearby
location, and found $B_{los}\sim 18~\mu$G, which is
consistent with the above estimate if a correction of $\sim
2$ is applied for projection effect. These are clear examples
of magnetic domination in, respectively, cold HI
and relatively diffuse molecular clouds.
It is unclear, however, how representative these two clouds are.
In particular, the degree of magnetization is uncertain in
giant molecular clouds (GMCs), where the majority of stars
form. Elmegreen (2007) argued that the envelopes of GMCs, where
most of the molecular gas resides, are probably magnetically
critical or even subcritical, and can be long-lived if not for
disruption by rapid star formation in GMC cores, which are
probably supercritical, with internal dynamics dominated by
turbulence; this supposition needs to be tested by magnetic
field observations on GMC scales (e.g., Novak, et al. 2007).
Nevertheless, the
Taurus cloud complex and the Riegel-Crutcher cloud are probably
best mapped cloud in CO and HI, respectively. How such relatively
diffuse, apparently magnetically dominated clouds proceed to
condense and collapse to form stars is the focus of our paper.
Diffuse clouds are expected to condense more easily along than
across the field lines as long as they are dominated by ordered
large-scale magnetic fields. Such an anisotropic condensation
can increase the density greatly with little corresponding
increase in the magnetic field strength. This expectation is
consistent with the available Zeeman measurements of field
strength, which is more or less constant below a number density
of order $10^3$~cm$^{-3}$ (Heiles \& Crutcher 2005). Above this
density, the field strength tends to increase with density, indicating
that contraction perpendicular to the field lines becomes
significant. We interpret the break in the observed field
strength-density relation as marking the point where the
self-gravity of the cloud becomes dynamically important in the
cross-field direction. Before this point is reached, one expects
the self-gravity to first become significant along the field
lines (where there is no magnetic support), pulling matter into
condensations, which can be either a sheet, a filament,
or even a knot, depending on the degree of anisotropy in the
initial mass distribution.
In the case of the Taurus molecular cloud complex, most of the
dense molecular gas traced by $^{13}$CO and especially C$^{18}$O
are distributed
in structures elongated more or less perpendicular to the
large-scale magnetic field (e.g., Onishi et
al. 1996, 2002; Goldsmith et al. 2008). It has long been suspected
that they have condensed along the field lines (e.g., Heyer et
al. 1987; Tamura et al. 1987). Cross-field contraction must have
already taken place in the condensed structures, at least locally,
since stars have been forming in these structures for at least
a few million years (Kenyon \& Hartmann 1995). Palla \& Stahler
(2002) examined the star formation pattern in the Taurus
clouds in both space and time, concluding that stars began
to form at a low level at least 10 million years ago, in
a spatially dispersed fashion. The majority of the stars are
formed, however, in the last 3 million years, near where the
dense gas is observed today. This pattern of accelerating
star formation is precisely what is expected of a magnetically
dominated cloud that condenses in two stages: first along the
field line, when little star formation is expected, except
perhaps in some pockets that have an exceptionally weak local
magnetic field to begin with or have their magnetic fluxes
reduced by an exceptionally strong compression-enhanced ambipolar
diffusion, and then across the field, when more active star
formation can take place throughout the dense sheets and
filaments that have condensed along the field lines. A strong
support for this supposition comes from the fact that star
formation is slow even in dense gas: the rate of star
formation for the last 3 million years is about ${\dot M}_*
\sim 5\times 10^{-5}~M_\odot$yr$^{-1}$ (Goldsmith et al. 2008),
two orders of magnitude below the free-fall rate for the dense
($\sim 10^4$~cm$^{-3}$) C$^{18}$O gas (Onishi et al. 2002; see
\S~\ref{taurus} for actual numbers); Krumholz \& Tan (2006) found
that a similar result holds for objects of a wide range of density.
Although the observed moderately supersonic motions (${\cal M}\sim
2$) can provide some support for the dense gas, we believe that
the star formation is slow mainly because the bulk of the dense
gas remains magnetically supported even after cross-field
contraction has begun in localized regions in the condensed
structures.
This requirement can naturally be satisfied if such structures
are marginally magnetically critical (Basu \& Ciolek 2000),
where individual stars can form on a relatively short time
scale, but the overall rate of star formation remains well
below the free-fall value, as we have demonstrated using
simulations in 2D sheet-like geometry (Li \& Nakamura 2004
and Nakamura \& Li 2005; LN04 and NL05 hereafter; see also
Kudoh et al. 2007 and Kudoh \& Basu 2008). The Taurus cloud
complex is discussed further in \S~\ref{taurus}, along
with the Pipe Nebula (\S~\ref{pipe}), which may represent an
earlier phase of star formation in a magnetically dominated
cloud (Lada et al. 2008; F. Alves et al., in preparation).
The rest of the paper is organized as follows. Section~\ref{setup}
describes the model formulation, including governing equations,
initial and boundary conditions, and numerical code. It is followed
by three sections on numerical results. Section~\ref{result} concentrates
on a standard model with a particular combination of initial turbulence
and outflow feedback. It illustrates the essential features of the
magnetically regulated star formation in three dimensions, including
strong stratifications in density and turbulent speed, broad
probability distribution functions (PDFs) of volume and column
densities and, most importantly, low rate of star formation. This
model is contrasted, in \S~\ref{other}, with others that have
different levels of initial turbulence and outflow feedback. We
devote \S~\ref{core} to a detailed analysis of dense cores, including
their shapes, mass spectrum, velocity dispersions, flux-to-mass
ratios, angular momenta, and whether they are bound or not. In
\S~\ref{discussion}, we outline a general scenario of magnetically
regulated star formation in quiescent condensations of relatively
diffuse, turbulent clouds, and make connection to observations
of the Taurus molecular cloud complex and Pipe Nebula. The main
results of the paper are summarized in \S~\ref{conclude}. Readers
interested only in the general scenario and their connection to
observations can skip to \S~\ref{discussion}.
\section{Model Formulation}
\label{setup}
\subsection{Governing Equations}
The basic equations that govern the evolution of isothermal, weakly
ionized, magnetized molecular gas are (see, e.g., Shu 1991)
\begin{equation}
{\partial \rho_n \over \partial t} + \nabla \cdot (\rho_n
\mbox{\boldmath{$V$}}_n) = 0 \ ,
\end{equation}
\begin{equation}
\rho_n {\partial \mbox{\boldmath{$V$}}_n \over \partial t} +
\rho_n (\mbox{\boldmath{$V$}}_n \cdot \nabla \mbox{\boldmath{$V$}}_n)
= -\rho_n \nabla \Psi - c_s^2 \nabla \rho_n + {1 \over 4 \pi}
(\nabla \times \mbox{\boldmath{$B$}}) \times \mbox{\boldmath{$B$}} \ ,
\end{equation}
\begin{equation}
{\partial \mbox{\boldmath{$B$}} \over \partial t}
=\nabla \times (\mbox{\boldmath{$V$}}_i \times \mbox{\boldmath{$B$}}) \ ,
\end{equation}
\begin{equation}
\nabla ^2 \Psi = 4 \pi G \rho_n \ ,
\end{equation}
where
$\rho$, $\mbox{\boldmath{$V$}}$, $\mbox{\boldmath{$B$}}$, $c_s$,
and $\Psi$ are, respectively,
density, velocity, magnetic field, isothermal sound speed,
and gravitational potential, with the subscripts $n$ and $i$
denoting neutrals and ions. The drift velocity between ion
and neutral is
\begin{equation}
\mbox{\boldmath{$V$}}_i - \mbox{\boldmath{$V$}}_n = {t_c \over 4 \pi
\rho_n} (\nabla \times \mbox{\boldmath{$B$}}) \times
\mbox{\boldmath{$B$}} \ ,
\end{equation}
where the coupling time between the magnetic field and neutral matter,
$t_c$, is given by
\begin{equation}
t_c = {1.4 \over \gamma C \rho_n^{1/2}},
\end{equation}
in the simplest case where the coupling is provided by ions
that are well tied to the field lines and the ion density $\rho_i$
is related to the neutral density by the canonical expression
$\rho_i = C \rho_n^{1/2}$. We adopt an ion-neutral drag coefficient
$\gamma = 3.5\times 10^{13}$~cm$^3$~g$^{-1}$~s$^{-1}$ and $C = 3
\times 10^{-16}$~cm$^{-3/2}$~g$^{1/2}$ (Shu 1991). The factor 1.4
in the above equation comes from the fact that the cross section
for ion-helium collision is small compared to that of ion-hydrogen
collision (Mouschovias \& Morton 1991). These governing equations
are solved numerically subject to a set of initial and boundary
conditions.
\subsection{Initial and Boundary Conditions}
Our simulations are carried out in a cubic box (of size $L$), with
standard periodic conditions imposed at the boundaries. The cloud
is assumed to have a uniform density $\rho_0$ initially. The
corresponding Jeans length is
\begin{equation}
L_J= \left({\pi c_s^2/G \rho_0}\right)^{1/2},
\end{equation}
where the isothermal sound speed $c_s=1.88\times 10^4(T/10
K)^{1/2}$~cm/s, with $T$ being the cloud temperature. The
initial density $\rho_0 = 4.68\times 10^{-24} n_{H_2,0}$~g~cm$^{-3}$,
where $n_{H_2,0}$ is the initial number density of molecular
hydrogen, assuming 1 He for every 10 H atoms.
We consider relatively diffuse molecular clouds, with a fiducial
H$_2$ number density of $250$~cm$^{-3}$. It is the value adopted
by Heyer et al. (2008) for the sub-region in Taurus that shows
magnetically aligned $^{12}$CO striations. Scaling the density by
this value, we have
\begin{equation}
L_J=1.22 \left(\frac{T}{10 K}\right)^{1/2}
\left(\frac{n_{H_2,0}}{250\ {\rm cm}^{-3}}\right)^{-1/2} {\rm pc},
\end{equation}
and a Jeans mass
\begin{equation}
M_J= \rho_0 L_J^3 = 31.6 M_\odot \left(\frac{T}{10 K}\right)^{3/2}
\left(\frac{n_{H_2,0}}{250\ {\rm cm}^{-3}}\right)^{-1/2}.
\end{equation}
The Jeans length is smaller than the dimensions of molecular cloud
complexes such as the Taurus clouds, which are typically tens of
parsecs. Although global simulations on the complex scale are
desirable, they are prohibitively expensive if individual star
formation events are to be resolved at the same time. After some
experimentation, we settled on a moderate size for the simulation
box, with an initial Jeans number $n_J=L/L_J=2$. It represents
a relatively small piece of a larger complex. Scaling the Jeans
number by 2, we have a box size
\begin{equation}
L=2.44 \left(\frac{n_J}{2}\right) \left(\frac{T}{10 K}\right)^{1/2}
\left(\frac{n_{H_2,0}}{250\ {\rm cm}^{-3}}\right)^{-1/2} {\rm pc},
\end{equation}
and a total mass
\begin{equation}
M_{tot}= 253\ M_\odot \left(\frac{n_J}{2}\right)^3 \left(\frac{T}{10
K}\right)^{3/2}\left(\frac{n_{H_2,0}}{250\ {\rm cm}^{-3}}\right)^{-1/2}
\end{equation}
in the computation domain. Even though the initial Jeans number
adopted is relatively small, there is enough material inside
the box to form many low-mass stars, especially in condensations
where the Jeans mass is much reduced.
We impose a uniform magnetic field along the $x$ axis at the beginning
of simulation. The field strength $B_0$ is specified by the parameter
$\alpha$, the ratio of magnetic to thermal pressure, through
\begin{equation}
B_0=3.22~\alpha^{1/2} \left({T\over 10 K}\right)^{1/2}
\left(\frac{n_{H_2,0}}{250\ {\rm cm}^{-3}}\right)^{1/2} (\mu~G).
\label{fieldstrength}
\end{equation}
In units of the critical value $2\pi G^{1/2}$ (Nakano \& Nakamura
1978), the flux-to-mass ratio for the material inside the box is
\begin{equation}
\Gamma_0 = 0.45 n_J^{-1} \alpha^{1/2}.
\label{centralgamma}
\end{equation}
Since we are interested in magnetic regulation of star formation,
we choose a relatively strong magnetic field with $\alpha=24$,
corresponding to a mildly magnetically subcritical region with
$\Gamma_0=1.1$ (or a dimensionless mass-to-flux ratio $\lambda
=0.91$) for $n_J=2$, which is close to the value of 1.2
used in the 2D simulations of LN04 and NL05. For the fiducial
values for temperature and density, the initial field strength
is $B_0=15.8\ \mu$G according to equation~(\ref{fieldstrength});
this value is not unreasonably high (see \S~\ref{taurus} for
a discussion of the magnetic field in the Taurus region). We
postpone an investigation of initially supercritical clouds to
a future publication.
Molecular clouds are highly turbulent, particularly for the
relatively diffuse gas that we have assumed for the initial
state. It is established that supersonic turbulence decays
rapidly, with or without a strong magnetic field (e.g., Mac
Low \& Klessen 2004). How the turbulence is maintained is
uncertain (e.g., McKee \& Ostriker 2007). For the relatively
small, parsec-scale region that is modeled here, a potential
source is energy cascade from larger scales, which can
in principle keep the turbulence in the sub-region for a
time longer than the local crossing time. Ideally, this
(and may other) form of driving should be included in a
model, although it is unclear how this can be done in
practice. We will take the limit that the turbulence decays
freely, except for the feedback from protostellar outflows
that are associated with forming stars. Other forms of
driving will be considered elsewhere. Following the standard
practice (e.g., Ostriker et al. 2001), at the beginning of
the simulation ($t=0$) we stir the cloud with a turbulent
velocity field of power spectrum $v_k^2 \propto k^{-3}$ in
Fourier space.
A useful time scale for cloud evolution is the global gravitational
collapse time
(Ostriker et al. 2001)
\begin{equation}
t_g= {L_J \over c_s} = 6.36 \times 10^6 \left(\frac{n_{H_2,0}}
{250\ {\rm cm}^{-3}}\right)^{-1/2} ({\rm years}),
\label{gravtime}
\end{equation}
which is longer than the free-fall time at the initial density,
$t_{ff,0}=[3\pi/(32 G \rho_0)]^{1/2}$, by a factor of 3.27. In
the presence of a strong magnetic field, as is the case in our
simulation, gravitational condensation is expected to occur
preferentially along field lines, creating a flattened, sheet-like,
structure in which most star formation activities take place.
Inside the sheet, the characteristic Jeans length is
\begin{equation}
L_{s} \equiv {c_s^2 \over G \Sigma_0} = {1 \over \pi n_J} L_J
= 0.195 \left(\frac{2}{n_J}\right) \left(\frac{T}{10 K}\right)^{1/2}
\left(\frac{n_{H_2,0}}{250\ {\rm cm}^{-3}}\right)^{-1/2} {\rm pc}\ ,
\end{equation}
corresponding to a local gravitational collapse time
\begin{equation}
t_{s} \equiv {L_{s} \over c_s} = {c_s \over G \Sigma_0}
={1 \over \pi n_J} t_g \ = 1.01 \times 10^6 \left(\frac{2}{n_J}\right)
\left(\frac{n_{H_2,0}} {250\ {\rm cm}^{-3}}\right)^{-1/2} ({\rm years}) ,
\end{equation}
where the column density $\Sigma_0$ is measured along the initial
magnetic field lines, $\Sigma_0 = \rho_0 L = n_J \rho_0 L_J$.
The sheet has a Jeans number $n_{s}\equiv L/L_{s} = \pi n_J^2$,
corresponding to $12.6$ for $n_J=2$, which is close to the value
of 10 adopted in the 2D simulations of LN04 and NL05. Note that
the gravitational collapse time $t_s$ is longer than the free-fall
time of the condensed sheet
\begin{equation}
t_{\rm ff,s}=\left({3\pi \over 32 G \rho_s}\right)^{1/2},
\label{freefalltime}
\end{equation}
by a factor of 2.31, where the characteristic sheet density is
given by
\begin{equation}
\rho_s = \frac{\pi G \Sigma_0^2}{2 c_s^2} = \frac{\pi^2 n_J^2}{2}
\rho_0 \ .
\end{equation}
\subsection{Numerical Code}
We solve the equations that govern the cloud evolution using a 3D MHD
code based on an upwind TVD scheme, with the ${\nabla\cdot {\bf B}}=0$
condition enforced through divergence cleaning after each time step.
The base code is the same as the one
used in the cluster formation simulations of Li \& Nakamura (2006)
and Nakamura \& Li (2007), except that we have now included ambipolar
diffusion in the induction equation. The ambipolar diffusion term is
treated explicitly, which puts a stringent requirement on the time
step (since it is proportional to the square of the grid size, Mac
Low et al. 1995), especially in the lowest density regions. To
alleviate the problem, we set a density threshold, $\rho_{AD}=0.1
\rho_0$, below which the rate of ambipolar diffusion is reduced to
zero. A similar approach was taken by Kudoh et at. (2007), and can
be justified to some extent by increased ionization of low (column)
density gas from UV background (McKee 1989). Even with this measure,
it still takes more than a month to run a standard $128^3$ simulation
for two global gravitational collapse time $t_g$ (or $4\pi$ times the
sheet collapse time $t_s$) on the vector machine available to us.
Since our focus is on the evolution of, and star formation in, the
condensed gas sheet, to speed up the simulations, we follow the
initial phase of cloud evolution on a coarser grid of $64^3$ up to
a maximum density of $\sim 40~\rho_0$, before switching to the
$128^3$ grid. To gauge the numerical diffusion of magnetic field on
such a relatively coarse grid, we have followed the evolution of a
model cloud in the ideal MHD limit (with the ambipolar diffusion
term turned off; Model I0 in Table \ref{tab:model parameter}) up
to $2 t_g (=12.6t_s)$, and found that the mass-to-flux ratio is
conserved to within $1.5\%$,
which seems to be much better than the two-dimensional
higher resolution runs with ZEUS ($512^3$, Krasnopolsky \& Gammie 2005).
A difficulty with grid-based numerical codes such as ours in
simulating star formation in turbulent clouds is that, once
the collapse of a piece of the cloud runs away to form the
first star, the calculation grinds to a halt without special
treatment. Because the bulk of the stellar mass is believed
to be assembled over a rather brief (Class 0) phase of $\sim
3\times 10^4$~yrs (Andre et al. 1993) and the outflows are
most active during this phase (much shorter than the fiducial
characteristic time $t_s\sim 10^6$~yrs), we idealize the
processes of individual star formation and outflow ejection
as instantaneous, and treat them using the following simple
prescriptions. When the density in a cell reaches a threshold
value $\rho_{th}=10^3\rho_0$, we define around it a critical
core using a version of CLUMPFIND algorithm
(Williams, de Geus, \& Blitz 1994;
see \S~\ref{core} for detail). The core identification
routine is specified by a minimum threshold density $\rho_{\rm
th, min}$ and the number of density levels $N$. They are set
to $200~\rho_0$ and 5 respectively. The mean density of the
core so identified is estimated at $\sim 10^5$~cm$^{-3}$,
comparable to those of H$^{13}$CO$^+$ cores in Taurus studied
by Onishi et al. (2002). We extract $30\%$ of the mass from
the core (if its mass exceeds the local Jeans mass), and put
it in a Lagrangian particle located at the cell center; its
subsequent evolution is followed using the standard leap-frog
method. The extracted mass fraction is
motivated by the observations of J. Alves et al. (2007),
who found a mass spectrum for dense cores that is similar to
the stellar IMF in shape but more massive by a factor of $\sim 3$
(see also Ikeda, Sunada, \& Kitamura 2007).
It is within the range of $25-75\%$ for the star formation
efficiency per dense core estimated theoretically by Matzner \&
Mckee (2000).
The material remaining in the core after mass extraction is
assumed to be blown away in a two-component outflow: a bipolar
jet of $30^\circ$ half-opening angle around the local magnetic
field
direction that carries a fraction $\eta$ of the total momentum,
and a spherical component outside the jet that carries the
remaining fraction. The total outflow momentum is assumed to
be proportional to the stellar mass, with a proportionality
coefficient of $P_*=35$~km/s.
The prescriptions for star formation
and outflow are similar to those used in Nakamura \& Li (2007).
One difference is the use of CLUMPFIND here to define the critical
core. Another is that, when an outflow reenters the computation
box because of periodic boundary condition, we reduce its strength
by a factor of ten if its speed exceeds $50~c_s$, to prevent
it from strongly perturbing or even disrupting the condensed
sheet.
\section{Standard Model}
\label{result}
We have run four models that include ambipolar diffusion, with
parameters listed in Table \ref{tab:model parameter}. They
differ in random realization of the initial velocity field,
relative strength of the jet and spherical outflow components,
and level of initial turbulence. An ideal MHD run was also
performed for comparison. In this section, we focus on the
model with initial turbulent Mach number ${\cal M}=3$ and
jet momentum fraction $\eta=75\%$. It serves as the standard
against which other models, to be discussed in the next
section, are compared.
The initial phase of cloud evolution is similar to those already
presented in Krasnopolsky \& Gammie (2005), who carried out
high-resolution, ideal MHD simulations of magnetically subcritical
clouds in two dimensions, including self-gravity and an initially
supersonic turbulence that decays freely, as in our simulation.
The initial turbulence creates many shocklets that are oriented
more or less perpendicular to the field lines; they appear as
filaments of enhanced density and are the main sites of turbulence
dissipation. As the turbulence decays, the fragments start to
coalesce, eventually forming a single, approximate Spitzer sheet.
In the ideal MHD limit, the sheet formation marks the end of
cloud evolution. With ambipolar diffusion, it signals the beginning
of a new phase---star formation\footnote{For a strong enough
turbulence and/or initially inhomogeneous distribution of
mass-to-flux ratio, stars can form before the formation of a
well-defined sheet (see \S~\ref{discussion} for further
discussion).}.
This latter phase is the focus of our discussion next.
\subsection{Star Formation Efficiency}
\label{SFE}
We begin the discussion with a global measure of star formation in the
standard model: the efficiency of star formation, defined as the ratio
of the mass of all stars
to the total mass of stars and gas. The star formation efficiency (SFE
hereafter) is plotted in Fig.~\ref{fig:sfe} as a function of time in
units of the gravitational collapse time of the sheet $t_s$. Sheet
formation
is finished by the end of the first global collapse time $t_g=2\pi\
t_s$. Soon after, the first star appears, around $t \approx 7~t_s$.
By the end of simulation at $t=11.4~t_{s}$ (nearly $2~t_g$), 37 stars
have formed, yielding an accumulative SFE of 6.5\%. The average mass
of a star is close to $0.45 M_\odot \left(T/10 K\right)^{3/2}\left(n_{H_2,0}
/250 {\rm cm}^{-3} \right)^{-1/2}$, typical of low-mass stars. The
rate of star formation is given by the slope of the SFE curve, which
is estimated at $\sim 1.5\%$ per $t_s$. At this
rate, it would take $\sim 65~t_s$ to deplete the gas due to star
formation. The gas depletion time corresponds to $\sim 10~t_g$ or
$\sim 33$ times the free-fall time $t_{{\rm ff},0}$ at the {\it
initial} (uniform) density $\rho_0$. In other words,
the star formation efficiency per {\it initial} free fall time
(Krumholz \& McKee 2005) is SFE$_{{\rm ff},0}\sim 3\%$. Interestingly,
this rate is comparable to that of cluster formation in
protostellar outflow-driven turbulence (Nakamura \& Li 2007).
However, a large fraction of the gas is at a much higher density
than the initial density $\rho_0$ after the formation of the
condensed sheet. It is more appropriate to measure the star
formation rate using the free fall time at the characteristic
sheet density $\rho_s=2 \pi^2\ \rho_0$, which is $t_{{\rm ff},s}=
0.43\ t_s$ (see equation~[\ref{freefalltime}]), yielding
SFE$_{{\rm ff},s}\sim 0.7\%$. To help understand the rather slow
and inefficient star formation, we next examine in some detail
the cloud structure and dynamics at $t=11~t_s$, representative of
the post-sheet formation phase of cloud evolution when stars
are forming steadily, starting with overall mass distribution
as revealed by column density.
\subsection{Column Density Distribution}
The cloud material is expected to condense along the magnetic field
lines into a flattened structure. This expectation is borne out by
Figs.~\ref{colden_y} and \ref{colden_critical}, which show the
distributions of column density along, respectively, the $y$- and
$x$-axis (the direction of the initial magnetic field). Viewed
edge-on, the structure resembles a knotty, wiggly filament
(Fig.~\ref{colden_y}). The dense filament is sandwiched by a
low-density ``halo,'' which contains a number of spurs more or
less perpendicular to the filament. The spurs are roughly aligned
with the global magnetic field. They resemble the diffuse, fuzzy
$^{12}$CO streaks that are frequently observed to stream away
from dense filaments (Goldsmith et al. 2008). Similar spurs are
seen in the recent SPH-MHD simulations of Price \& Bate (2008).
The face-on view of the condensed structure is quite different
(Fig.~\ref{colden_critical}). It appears highly fragmented, with
low-density voids as well as high-density peaks. Some
over-dense regions cluster together to form a short filament;
others are more scattered. The actual appearance of the cloud
to outside observers will depend, of course, on the viewing
angle. It can be filament-like (as in Fig.~\ref{colden_y}) or,
more likely, patchy (as in Fig.~\ref{colden_critical}). Superposed
on the face-on column density map is the distribution of the
33 stars that have formed up to the time plotted. With few
exceptions, they follow the dense gas closely. Most of the stars
are found in small groups, which is broadly consistent with the
distribution of young stars in Taurus (Gomez et al. 1993). Since
the clouds is initially
magnetically subcritical everywhere, in order for any localized
region to collapse and form stars, it must become supercritical
first.
A rough measure of magnetic criticality is provided by the
average mass-to-flux ratio $\bar{\lambda}$ normalized to
the critical value,
\begin{equation}
\bar{\lambda}(y,z) = \int_{-L/2}^{+L/2} 2\pi G^{1/2}
\rho/B_x dx.
\label{AverageLambda}
\end{equation}
Contours of $\bar{\lambda}=1$ are plotted in Fig.~\ref{colden_critical},
showing that most of the dense regions are indeed magnetically
supercritical, and thus capable of forming stars in principle.
The supercritical regions enclosed within the contours contain
a significant fraction ($34\%$) of the total mass.
The mass fraction of supercritical gas increases gradually with
time initially, and remains more or less constant at later times
($t\gtrsim 6~t_s$), as shown in Fig. \ref{fig:supercritical}.
This value provides an upper limit to the efficiency of
star formation.
If the supercritical regions collapse to form stars within a local
free fall time, the star formation rate would be much higher
than the actual value. An
implication is that the bulk of the supercritical material
remains magnetically supported to a large extent and is rather
long lived. Only its densest parts (i.e., dense cores),
which contain a small fraction of the supercritical mass,
directly participate in star formation at any given time.
The longevity is characteristic of the (mildly) supercritical
regions that are produced through turbulence-accelerated
ambipolar diffusion in nearly magnetically critical clouds.
It is in contrast with the highly supercritical material
in weakly magnetized clouds, which is expected to collapse
promptly unless supported by turbulence. This same behavior
was found in 2D calculations (NL05).
To quantify the mass distribution further, we plot in Fig.~\ref{colden_PDF}
the probability distribution function (PDF) of the column density
along the $x-$axis, in the direction of initial magnetic field. The
PDF peaks around $\Sigma \sim 1.7~\rho_0 L_J$ (where $\rho_0$ and
$L_J$ are the density and Jeans length of the initial, uniform state),
close to the average value of $2~\rho_0 L_J$. The distribution can be
described reasonably well by a lognormal distribution, although there
is significant deviation at both low and high density end. The
considerable spread in column density may be surprising at the
first sight, since one may expect little structure to develop in
a subcritical sheet dominated by a strong, more or less uniform
magnetic field, especially since the initial mass distribution is
also chosen to be uniform. This expectation is consistent with the
column density PDF of the ideal MHD counterpart of the standard
model (plotted in the same figure for comparison), which is much
narrower. The key difference is that ambipolar diffusion turns an
increasingly large fraction of the initially subcritical cloud into
supercritical material (until saturation sets in at late times,
see Fig.~\ref{fig:supercritical}). Once produced, the supercritical
material behaves drastically differently from the subcritical
background. It is easier to compress by an external flow because
its magnetic field is weaker. The compressed supercritical region
also rebounds less strongly, because the magnetic forces driving
the rebound are cancelled to a larger extent by the self-gravity
of the region. More importantly, the supercritical regions can
further condense without external compression, via self-gravity.
The accumulation of supercritical material and subsequent
(magnetically-diluted) gravitational fragmentation in such material
are responsible for the broad column density PDF.
These key aspects of the problem are {\it not} captured by the
ideal MHD simulation, but are well illustrated in 2D simulations
that include ambipolar diffusion (e.g., NL05; Kudoh \& Basu
2006). Other aspects are better discussed in 3D, however. These
include the dependence of various physical quantities on volume
density, to which we turn our attention next.
\subsection{Density Stratification}
Our model cloud is strongly stratified in density. It contains dense
cores embedded in a condensed sheet which, in turn, is surrounded
by a more diffuse halo. The relative amount of mass distributed at
different densities is displayed in Fig.~\ref{lognorm_density},
along with that for the ideal MHD case and for a Spitzer sheet.
Whereas the mass distribution for the ideal MHD case resembles
that of the Spitzer sheet closely, the same is not true for
the standard model with ambipolar diffusion: its mass distribution
is considerably broader, with more material at both the low and
high density end. The former indicates that the diffuse material
outside the sheet is dynamically active, as we show below. The
excess at the high density end is produced, on the other hand,
primarily by ambipolar diffusion, which enables pockets of
the sheet material to condense across magnetic field lines
gravitationally. The bulk ($62.5\%$) of the cloud material
resides in the condensed sheet, which we define somewhat
arbitrarily as the material in the density range between
$5~\rho_0$ and $50~\rho_0$, corresponding to a range in
H$_2$ number density between $1.25\times 10^3$~cm$^{-3}$ and
$1.25\times 10^4$~cm$^{-3}$ (assuming $n_{H_2,0}=250$~cm$^{-3}$).
The remaining cloud material is shared between the diffuse halo
($30.0\%$), defined
loosely as the material with density below $5\ \rho_0$, and
dense cores ($7.5\%$), defined as the material denser than
$50 \rho_0$ (see \S~\ref{core} below).
Our model cloud is also strongly stratified in turbulent speed.
Fig.~\ref{rms_vel} shows the mass weighted rms velocity as a
function of density. At densities corresponding to the
condensed sheet ($5-50~\rho_0$), the turbulent Mach number is
about 2. The moderately supersonic turbulence persists in the
condensed sheet for many local free fall times, despite the
absence of external driving. It is driven by a combination
of outflows (through either direct impact or MHD waves) and
ambipolar diffusion-induced gravitational acceleration.
The modest increase in rms velocity toward the high density
end is because the denser gas is typically closer to star
forming sites and is thus more strongly perturbed by outflows.
The much higher level of turbulence in the diffuse halo is
more intriguing. One may expect little turbulence to remain at
the representative time $t=11~t_s$
shown in the figure, which is well beyond the time ($t\sim 6\ t_s$)
of sheet formation --- a product of turbulence dissipation. The
expectation is consistent with the ideal
MHD simulation, which shows that, in the absence of external
driving, the turbulent motions die down quickly everywhere.
The turbulence decay is changed by ambipolar diffusion
which, accelerated by the initial strong turbulence, enables
stars to form soon after sheet formation. The stars stir
up the halo via protostellar outflows in at least two ways.
First, the outflows are powerful enough to disrupt the
star-forming cores, break out from the condensed sheet, and
inject energy and momentum directly into the halo. Second, a
fraction of the outflow energy and momentum is also
deposited in the sheet, where gas motions can shake the
field lines and send into the halo MHD waves whose amplitudes
grow as the density decreases (Kudoh \& Basu 2003); the waves
help maintain the turbulence in the diffuse region, particularly
in the cross-field direction.
Even though the turbulence speed is relatively high in the halo,
it is still sub-Alfv\'enic. This is shown in
Fig.~\ref{ave_MA}, where the average Alfv\'en Mach
number $M_A$ is plotted as a function of density. The flow
is moderately sub-Alfv\'enic at densities between $\sim 0.3$
and $\sim 10~\rho_0$, with $M_A$ within $20\%$ of $0.5$. The
Alfv\'en Mach number
drops slowly towards lower densities, indicating that the
lower-density gas becomes increasingly more magnetically
dominated in our model cloud. A caveat is that, in real
clouds such as the Taurus complex, the dynamics of the
low-density halo on the parsec-scale of our modeled region
are expected to be tied to that of the larger complex. It
is likely for the diffuse gas to receive energy cascaded
from larger scales and become more turbulent than suggested
by Fig.~\ref{ave_MA}. Some of this energy may even be fed
through Alfv\'en waves into the condensed sheet and increase
its velocity dispersion as well. The increase is not expected
to be large, however, since the Alfv\'en speed becomes
transonic at high densities (see Fig.~\ref{ave_MA}), where
highly supersonic motions should damp out quickly through
shock formation. In future refinements, it will be
desirable to include external driving of the halo material,
which may modify the way stars form, especially at early
times (see discussion in \S~\ref{discussion}). Nevertheless,
we believe that the decrease of turbulent speed with increasing
density found in our simulation (and earlier in Kudoh \&
Basu 2003) is a generic feature of the magnetically
regulated star formation in three dimensions. It is broadly
consistent with the observed trend (e.g., Myers 1995).
Another generic feature is the distribution of magnetic field
strength as a function of density. The distribution is shown
in Fig.~\ref{ave_B}. There is a clear trend for the average
field strength to remain more or less constant at relatively
low densities, before increasing gradually towards the high
density end. A similar trend is inferred
in Zeeman measurements of field strength over a wide range of
densities, and has been interpreted to mean that the initial
stage of cloud condensation leading to star formation is
primarily along the field lines, which increases the density
without increasing the field strength (Heiles \& Crutcher
2005). This is precisely what happens in our model,
where the initial accumulation of diffuse material into the
condensed sheet is mostly along the field lines, and further
condensation inside the sheet is primarily in the cross-field
direction. The cross-field condensation leads to an increase
in field strength with density, although by a factor that is
less than that expected from flux-freezing because of ambipolar
diffusion.
\subsection{Sheet Fragmentation: Turbulent or Gravitational?}
The star formation activities in our model are confined mostly to
the condensed sheet. Its dynamics hold the key to understanding the
magnetically regulated star formation. An overall impression of
the sheet can be gained from Figs.~\ref{colden_y} and
\ref{colden_critical},
which showed an edge-on and face-on view of the sheet in column
density. The sheet has a highly inhomogeneous
mass distribution which is, of course, a pre-requisite for active
star formation. To illustrate the clumpy mass distribution further,
we plot in Fig.~\ref{den_pot_vel} the distribution of the density
in the ``mid-plane'' of the sheet, defined as the $yz$ plane
(perpendicular to the initial magnetic field along the $x$ axis)
that passes through the minimum of gravitational potential. The
density distribution spans seven orders of magnitude, from
$10^{-4}~\rho_0$ (the density floor adopted in the computation) to
$10^3~\rho_0$ (the threshold for Lagrangian particle generation).
The question is: how is the density inhomogeneity created and
maintained?
A widely discussed possibility is turbulent fragmentation, where
regions of high densities are created by converging flows (e.g.,
Padoan \& Nordlund 2002; Mac Low \& Klessen 2004). There is indeed
a random velocity field in the mid-plane, as shown by arrows in
the figure. However, the flow speed is typically small, comparable
to the sound speed in most places.
It is unlikely for such a weak turbulence to create the high density
contrast shown in Fig.~\ref{den_pot_vel} by itself. This conclusion
is strengthened by the presence of a strong magnetic field, which
cushions the flow compression. Indeed, the turbulent speeds in the
plane of the sheet are significantly below the approximate
magnetosonic speed $v_{ms}=[B_x^2/(4\pi\rho)+c_s^2]^{1/2}$ in
most places, and are thus not expected to create large density
fluctuations; they are essentially a collection of magnetosonic
waves. The low level of turbulence points to gravity rather than
flow motions as the primary driver of fragmentation in the sheet.
Gravity is expected to be dynamically important in the sheet, a structure
formed by gravitational condensation along field lines in the
first place. The fact that the sheet has a dimension much larger than
the thickness indicates that it contains many thermal Jeans masses.
If the thermal pressure gradient was the main force that opposes
the gravity, one would expect the bulk of the sheet to collapse
in one gravitational time $t_s$. And yet, the vast majority of
the sheet material remains uncollapsed after at least $\sim 5~t_s$.
The longevity indicates that the
gravitational fragmentation proceeds at a much reduced
rate. The reduction is due, of course, to magnetic forces,
which cancel out the gravitational forces in the plane of
the sheet to a large extent (Shu \& Li 1997, Nakamura et al.
1999). The near cancellation
can be inferred from Fig.~\ref{den_pot_vel}, where contours of
gravitational potential are plotted along with the density
and velocity vectors. If the gravitational forces were
unopposed, one would expect the material to slide down the
potential well, and picking up speed towards the bottom,
with the infall speed reaching highly supersonic values.
For example, for a potential drop of 8~$c_s^2$ (say between
contours labeled ``$-8$'' and ``$-16$''), an increase in speed
by 4~$c_s$ is expected. These expectations are not met. The
actual velocity field appears quite random, and generally
transonic. The slow random motions in a rather deep
gravitational potential well provide a clear indication
for magnetic regulation of the sheet dynamics. Active
outflows also play an important role in generating the density
inhomogeneities observed in the condensed sheet. Their effects
are discussed in \S~\ref{outflow} below.
\subsection{Comparison to 2D Calculations}
\label{comparison}
There are a number of differences between the 3D simulation and the
previous 2D calculations. The 2D treatment does not account for the
mass stratification along the field lines, and thus cannot
address the important issue of dynamical coupling between the
condensed material and the diffuse halo. A related difference is
that, in 3D, there is an initial phase of mass settling to form the
sheet, which is is absent in 2D. Furthermore, the mass of each star
in the 2D models is held fixed (typically at $0.5~M_\odot$); in 3D,
it is fixed at $30\%$ of the mass of the criticl core that forms
the star. In addition, the prescription for outflow
\footnote{We note that the spherical component of the outflow
in the standard 3D model carries a momentum close to
(within $15\%$ of) that of the 2D outflow in the plane of
mass distribution in the standard model of NL05.}
and the degree
of initial cloud magnetization are also somewhat different. Despite
these differences, the results of the 2D and 3D models are in
remarkably good agreement.
The most important qualitative agreement is that, once settled
along the magnetic field lines, the condensed sheet material
produces stars at a slow rate. In the standard 2D model of NL05,
about $5.3\%$ of the total mass is converted into stars within
a time interval of $3.35~t_s$ (see their Fig.~1), corresponding
to a star formation rate of $\sim 1.6\%$ per sheet gravitational
collapse time $t_s$. This rate is nearly identical to the value
of $\sim 1.5\%$ estimated in \S~\ref{SFE} for the standard 3D
model (which is also close to that of the 3D ${\cal M}=10$ model,
see \S~\ref{other} below). The (face-on) mass distributions
are also similar in 2D and 3D models. In both cases, dense,
star-forming cores are embedded in moderately supercritical
filaments which, in turn, are surrounded by more diffuse,
magnetically subcritical material (compare
Fig.~\ref{colden_critical} with the lower panels of
Fig.~2 of NL05). More quantitatively, the mass fraction of
supercritical material increases gradually initially and
reaches a plateau of $\sim 30\%$ in both 2D and 3D models. The
similarity extends to the turbulent speed of the condensed
material, which is about twice the sound speed in both cases
at late times (see Fig.~\ref{rms_vel} and Fig.~11 of NL05).
The similarities indicate that 2D calculations capture the
essence of the slow, magnetically regulated star formation
in moderately subcritical clouds. The same conclusion was
drawn by Kudoh et al. (2007) based on 3D simulations of
weakly perturbed magnetically subcritical sheets. The
2D calculations have the advantage of being much faster to
perform. They can be done with higher spatial resolution
($512^2$ as compared to the $128^3$ used in this paper)
and in larger numbers. The number of 3D simulations is
necessarily more limited, especially because of the
small time step required by our treatment of ambipolar
diffusion using an explicit method. Nevertheless, we are
able to perform several additional simulations, to check
the robustness of the standard model, and to help elucidate
the effects of two key ingredients: initial turbulence and
outflow feedback.
\section{Other Models}
\label{other}
\subsection{Initial Turbulence}
\label{turbulence}
Even though the initial turbulence dissipates quickly, it seeds
inhomogeneities in mass distribution and mass-to-flux ratio
that are expected to affect star formation later on. To illustrate
the effects of initial turbulence, we focus on a model identical
to the standard model, except for a higher initial turbulent Mach
number ${\cal M}=10$ (instead of $3$). The overall pattern of
cloud evolution follows that seen
in the standard model: dissipation of turbulence leads to
gravitational settling of cloud material onto a condensed sheet,
the densest parts of which collapse to form stars; the stars,
in turn, stir up the sheet and its surrounding halo through
protostellar outflows. One difference is that the stronger
turbulence initiates star formation at an earlier time, as seen
in Fig.~\ref{sfetwo}. The first star forms around $\sim 4.5~t_s$
in the ${\cal M}=10$ case, compared to $\sim 6.8~t_s$ for
${\cal M}=3$. Nevertheless, their rates of star formation are
comparable, with values between $\sim 0.5\%$ and $\sim 1.0\%$
per free fall
time $t_{ff,s}=0.43\ t_s$ in both cases. The earlier start, but
still low rate, of star formation in the more turbulent case
is consistent with the turbulence-accelerated, magnetically
regulated scenario that we advocate based on previous 2D
calculations (LN04, NL05; see also recent 3D simulations of
Kudoh \& Basu 2008).
The basic structure and kinematics of the initially more turbulent
cloud are also similar to those of the standard model cloud. In
both cases, the condensed sheet is highly fragmented but dynamically
``cold,'' with moderately supersonic random motions. It is surrounded
by a halo moving at highly supersonic, but sub-Alfv\'enic, speeds.
An interesting difference is that the sheet in the ${\cal M}=10$
case appears thicker, as illustrated in Fig.~\ref{colden_y_M10},
where an edge-on view of the column density distribution (along
the $y$-axis, perpendicular to the initial magnetic field
direction) is plotted for a representative time $t=8~t_s$. It is
to be compared with Fig.~\ref{colden_y} for the standard model.
The main reason for the larger apparent thickness is that the
condensed sheet, while still thin intrinsically, is more warped.
An example of warps is shown in Fig.~\ref{den_Bfield_M10}, where
the density distribution in a plane perpendicular to the $y$-axis
is plotted; superposition of such warps in different parts of
the sheet along the line of sight gives rise to the more
puffed-up appearance.
Associated with the sharp warp of condensed material in
Fig.~\ref{den_Bfield_M10} is a strong
distortion of magnetic field lines. Both are likely driven by
protostellar outflows, which are the main source of energy
that drives the cloud evolution at late times. In particular,
outflows ejected at an angle to the large-scale magnetic field
can excite large-amplitude Alfv\'en waves, which perturb the
field lines in the halo that thread other parts of the condensed
sheet. The large Alfv\'en speed in the diffuse halo allows
different parts of the sheet to communicate with one another
quickly, as emphasized recently by Shu et al. (2007) in the
context of poloidally magnetized accretion disks. The
communication raises the interesting possibility of global,
magnetically mediated, oscillations for the condensed material
that may have observational implications. This issue warrants
a closer examination in the future.
We conclude by stressing that condensations produced by gravitational
settling along field lines do not have to appear highly elongated,
even when viewed edge-on. They can be further distorted by outflows
from stars formed in neighboring condensations or other external
perturbations that are not included in our current calculations. We
have experimented with simulations that allow active outflows to
re-enter the simulation box, and found that the condensed sheet
became more puffed up and, sometimes, even completely disrupted.
\subsection{Outflow Feedback}
\label{outflow}
Protostellar outflow plays an important role in our model. It
limits the amount of core mass that goes into a forming star
and injects energy and momentum back to the cloud. In our
standard model, we have assumed that $75\%$ of the outflow
momentum is carried away in a bipolar jet (of $30^\circ$ in
half opening angle), and the
remaining $25\%$ in a spherical component. Here, we contrast
the standard model with one that has a stronger spherical
component (carrying $75\%$ of outflow momentum). The star
formation efficiencies (SFEs) of both models are plotted
in Fig.~\ref{sfedependm3}. It is clear that the stronger
spherical component of outflow has reduced the rate (and
efficiency) of star formation significantly, by a factor of
$\sim 2.3$. This trend is the opposite of what we found in
the case of cluster formation in dense clumps that are
globally magnetically supercritical (Nakamura \& Li 2007):
for a given total outflow momentum, more spherical outflows
tend to drive turbulence on a smaller scale that dissipates
more quickly, leading to a higher rate of star formation.
There are a couple of reasons for the seemingly contradictory
result. First, whereas the supercritical cluster-forming
clumps are supported mainly by turbulent motions, the bulk
of the star-forming condensations in subcritical or nearly
critical clouds are magnetically supported (in the cross-field
direction). As a result, faster dissipation of
turbulence leads to more rapid star formation in the former,
but not necessarily in the latter. Second, the jet component,
while crucial for supporting the more massive, rounder
cluster-forming clump, has a more limited effect on the flattened
condensation: it escapes easily from the sheet (and quickly
out of the simulation box) and its energy is mostly ``wasted''
as far as driving turbulence in the star-forming, condensed
material is concerned. The spherical component, in contrast,
impacts the condensed material more directly, especially in
the vicinity of a newly formed star. The dispersal of the
residual core material after star
formation to a wider region appears to be the main reason
for the lower star formation rate in the case of stronger
spherical component: it takes longer for the more widely
dispersed material to recondense and form stars. The stars
in the stronger spherical component model are less clustered
than those in the standard model, consistent with the above
interpretation.
\section{Dense Cores}
\label{core}
\subsection{Identification}
A variant of the CLUMPFIND algorithm of Williams et al. (1994) is
ussed for core identification. We apply their ``friends-of-friends''
algorithm to the density data cube in regions above a threshold
density, $\rho_{\rm th, min}$, and divide the density distribution
between $\rho_{\rm th, min}$ and $\rho_{\rm th, max}$ into 15 bins
equally spaced logarithmically. The maximum is set to $\rho_{\rm th,
max}=10^3\ \rho_0$, the density above which a star (Lagrangian
particle) is created if the core mass exceeds the local Jeans
mass. The minimum is set to $\rho_{\rm th, min}=50\ \rho_0$,
corresponding to $n_{H_2}=1.25 \times 10^4$~cm$^{-3}$ for the
fiducial value of initial density. We include only those
``spatially-resolved'' cores containing more than 30 cells in
the analysis, to ensure that their properties are determined with
reasonable accuracy. The minimum core mass is thus $M_{c, \rm min}
\equiv \rho_{\rm th, min} \times 30\Delta x\Delta y\Delta z=7.19
\times 10^{-2} M_{J,s}$, where $M_{J,s}\equiv \rho_s L_s^3
\simeq2.5M_\odot\left(T/10 K\right)^{3/2}\left(n_{H_2,0}/250
{\rm cm}^{-3} \right)^{-1/2}$ is the Jeans mass of the Spitzer
sheet.
In our standard simulation, the cloud material settles into a large
scale gas sheet by the time $t\sim 6\ t_s$. Soon after, a dense core
collapses in a run-away fashion to form the first star, signaling
the beginning of active star formation. We will limit our analysis
of dense cores to this star-forming phase, which complements the
analyses of, e.g., Gammie et al. (2003) and Tilley \& Pudritz (2007),
which focus on the phase prior to the runaway collapse.
There are typically a handful of dense cores in our standard model at
any given time (see Fig.~\ref{colden_critical}), which is too small
for a statistical analysis. To enlarge the core sample, we rerun the
standard model with a different realization of the initial turbulent
velocity field, and include all ``resolved'' cores at six selected
times $t=6.0$, 7.0, 8.0, 9.0, 10.0, and 11.0~$t_s$ from both runs.
The total number of cores is 72. Since the majority of them do not
survive for more than 1~$t_s$ (they do last for several local
free-fall times typically), there is relatively little overlap in
dense cores from one time to another. We treat all cores independently
regardless whether they appear at more than one selected time or
not. They have densities in the range 10$^4\sim 10^5$ cm$^{-3}$
and radii of $0.04 \sim 0.12$ pc, corresponding roughly to
H$^{13}$CO$^+$ or N$_2$H$^+$ cores.
\subsection{Core Shape}
We compute the shape of a core using the eigenvalues of the
moment-of-inertia tensor:
\begin{equation}
I_{ij} \equiv \int \rho x_i x_j dV \ ,
\end{equation}
where the components of position vector, $x_i$ and $x_j$,
are measured relative to the center of mass.
We compute the lengths of the three principal axes, $a, b,$ and $c$
($a\ge b \ge c$) by dividing the eigenvalues of the tensor,
$M_ca^2$, $M_cb^2$, and $M_c c^2$, by the core mass $M_c$.
The core shape is then characterized by the axis ratios
$\xi\equiv b/a$ and $\eta \equiv c/a$.
Figure \ref{fig:coreshapenew} shows the distribution of axis ratios
of the dense cores in our sample. It is clear that the cores
are generally triaxial; they show little clustering around either the
prolate ($b/a=c/a<1$) or oblate ($b/a=1, c<1$) line in the figure.
This result is perhaps to be expected, since the cores are distorted
by external turbulent flows and occasionally outflows, neither of
which are symmetric.
Following Gammie et al. (2003), we divide the $\xi$-$\eta$ plane
into three parts: a prolate group which lies above the line
connecting $(\xi, \eta)=(1.0, 1.0)$ and (0.33, 0.0), an oblate group
which lies below the line connecting $(\xi, \eta)=(1.0,1.0)$ and
(0.67, 0.0), and a triaxial group which is everything else.
Of the 72 identified cores, 43\% (31) are prolate, 44\% (32) are
triaxial, and 13\% (9) are oblate. These fractions are in general
agreement with Gammie et al. (2003). One
difference is that the ratio $\eta=c/a$ tends to be smaller for
our cores [see Figure 10 of Gammie et al. (2003)]. It is likely
because dense cores in our model are embedded in a condensed sheet,
which has already settled
gravitationally along the field lines, although part of the
difference may also come from the different methods used in
core identification. Note that more massive cores tend to be more
flattened along their shortest axis. This trend lends support to
the idea that gravitational squeezing plays a role in core
flattening.
Another trend is that more massive cores tend to be more oblate.
For example, 9 out of 15 (60\%) cores having masses more than 1.5
times the average core mass
($\left<M_c\right>=0.93~M_{J,s}$)
lie
below the line connecting $(\xi, \eta)=(1.0,1.0)$ and (0.5, 0.0)
(the dotted line in Fig.~\ref{fig:coreshapenew}). Less massive cores
tend to be more prolate. Their shapes may be more susceptible to
external influences, such as ambient flows.
\subsection{Virial Analysis}
Virial theorem is useful for analyzing the dynamical states of dense
cores. We follow Tilley \& Pudritz (2007) and adopt the virial
equation in Eulerian coordinates (McKee \& Zweibel 1992):
\begin{equation}
{1 \over 2} \ddot{I} + {1 \over 2} \int _S \rho r^2
\mbox{\boldmath $v$} \cdot d\mbox{\boldmath $S$}
= U+K+W+S+M+F \ ,
\label{eq:virial}
\end{equation}
where
$$
I=\int _V \rho r^2 dV\ ,
U= 3 \int _V P dV \ ,
K= \int _V \rho v^2 dV \ ,
W= -\int _V \rho \mbox{\boldmath $r$} \cdot \nabla \Psi dV \ ,
$$
$$
S= -\int _S \left[P\mbox{\boldmath $r$}+
\mbox{\boldmath $r$}\cdot (\rho \mbox{\boldmath $v$}
\mbox{\boldmath $v$})\right] \cdot d\mbox{\boldmath $S$} \ ,
M= {1 \over 4\pi}\int _V B^2 dV \ ,
F= \int _S \mbox{\boldmath $r$} \cdot \mbox{\boldmath $T$}_M \cdot d
\mbox{\boldmath $S$} \ .
$$
Here, $\mbox{\boldmath $T$}_M$ is the Maxwell stress-energy tensor
\begin{equation}
\mbox{\boldmath $T$}_M={1 \over 4\pi}\left(
\mbox{\boldmath $B$}\mbox{\boldmath $B$}-{1 \over 2} B^2
\mbox{\boldmath $I$}\right) \ ,
\end{equation}
where $\mbox{\boldmath $I$}$ is the unit tensor.
The terms, $I$, $U$, $K$, $W$, $S$, $M$, and $F$ denote, respectively,
the moment of inertia, internal thermal energy, internal kinetic
energy, gravitational energy, the sum of the thermal surface
pressure and dynamical surface pressure, internal magnetic energy,
and the magnetic surface pressure.
The internal thermal, kinetic, and magnetic energies are always
positive. The gravitational term is negative for all the
cores in our sample, although it can in principle be positive,
especially in crowded environments. Other terms can be either
positive or negative.
The second term on the left hand side of equation (\ref{eq:virial})
denotes the time derivative of the flux of moment of inertia
through the boundary of the core.
As is the standard practice (e.g., Tilley \& Pudritz 2007), we
ignore the left hand side of equation (\ref{eq:virial}) in our
discussion, and consider a core to be in virial equilibrium if
the sum $U+K+W+S+M+F=0$.
The equilibrium line is shown in Fig.~\ref{fig:virialparamnew}, where
the sum of the surface terms ($S+F$) is plotted against
the gravitational term ($W$),
both normalized to the sum of the internal terms ($U+K+M$).
The surface terms (dominated by the external kinetic term)
are generally important; they are more so for lower mass cores
than higher mass cores.
The majority of the cores more massive than 1.5 times
the average core mass lie below the equilibrium line, and
are thus bound; they are expected to collapse and form stars.
The majority of lower mass cores lie, in contrast, above the line,
and may disperse away, if they do not gain more mass through
accretion and/or merging with other cores, or further reduce
their turbulence support through dissipation and/or magnetic
support through ambipolar diffusion.
Tilley \& Pudritz (2007) found in their ideal MHD simulations
that as the mean magnetic field becomes stronger, the surface
terms contribute more to the virial equation.
Indeed, for their most strongly magnetized cloud, a good
fraction of cores have surface terms that are greater than
the sum of the internal terms ($U+K+M$). This is not the
case for our non-ideal MHD model, even though our clouds are
more strongly magnetized than theirs. The reason is that the
cloud dynamics is modified by ambipolar diffusion, especially
in dense cores, where the diffusion rate is the highest. A
related difference is that there are more bound cores in our
models than in their strong field models, where most of the
cores are magnetically subcritical (see also Dib et al. 2007).
Ambipolar diffusion enables these cores to become supercritical,
and thus more strongly bound. It is a crucial ingredient that
cannot be ignored for core formation, even for relatively
weakly magnetized clouds (of dimensionless mass-to-flux ratio
$\lambda$ of a few). For such clouds, the cores tend to be more
strongly magnetized (relative to their masses) than the cloud
as a whole, especially in driven turbulence (Dib et al/ 2007),
since only a fraction of the cloud mass along any given flux
tube is compressed into a core.
The boundedness of a core is often estimated using the virial
parameter, the ratio of the virial mass to core mass,
particularly in observational studies.
In Figure \ref{fig:virialrationew}, we plot against core mass
the virial parameter, defined as
\begin{equation}
\alpha_{\rm vir}={5 \Delta V_{1D}^2 R_c \over GM_c},
\end{equation}
where $\Delta V_{1D}$ is the one-dimensional velocity dispersion
including the contribution from thermal motion.
There is a clear trend for more massive cores to have smaller
virial parameters. Indeed, nearly all cores
more massive than 1.5 times the average
($M_c \gtrsim 1.5 \left<M_c\right>\simeq 1.4 M_{J,s}$)
have virial parameters smaller than unity.
It is consistent with the fact that most of these
cores lie below the line of the virial equilibrium in
Fig.~\ref{fig:virialparamnew}, indicating that they are bound.
The reason for the agreement is that the surface terms are not
as significant as the gravitational term for massive cores, and
that their magnetic support is greatly reduced by ambipolar
diffusion (consistent with the trend for cores with smaller
flux-to-mass ratios to have smaller virial parameters, see
Fig.~\ref{fig:virialparamnew}).
The reduction of surface terms makes the internal kinetic energy,
which is directly related to the virial parameter, more important.
We conclude that the virial parameter can be used to gauge the
importance of self-gravity for dense cores formed in strongly
magnetized clouds, as long as ambipolar diffusion is treated
properly. The virial parameter can be fitted by a power law
$\alpha_{\rm vir} \propto M_c^{-2/3}$, in good agreement
with the scaling found by Bertoldi \& McKee (1992) for nearby
GMCs.
\subsection{Velocity Dispersion-Radius Relation}
The kinematics of molecular clouds and dense cores embedded in them
are probed with molecular lines. Since the pioneering work of Larson
(1981), many observational studies have shown that the linewidth of
a region correlates with the size of the region. The linewidth-size
relation is often approximated by a power law $\Delta V\propto
R^{0.5}$. However, this relation may not hold universally, especially
on small scales. There are many cases where no clear correlation is
found between the two (e.g., Caselli \& Myers 1995; Lada et al. 1991;
Onishi et al. 2002).
In Fig.~\ref{fig:velocitysizenew}, we plot the
three-dimensional
nonthermal velocity
dispersion against radius (equivalent to the linewidth-size relation)
for the cores in our sample. No correlation
is apparent between the two, consistent with the observations of
Onishi et al. (2002) for the dense cores in the Taurus molecular
clouds. There is a large scattering in the velocity dispersion-radius
plot, especially for cores with small masses. The reason may be that
some cores (especially the less massive ones) are not in virial
equilibrium and that their formation is strongly influenced by
external flows.
\subsection{Angular Momentum Distribution}
Angular momentum is one of the crucial parameters that determine the
characteristics of formed stars or stellar systems. In Fig.~\ref{am1},
we plot the specific angular momenta of the cores against their radii.
There is no tight correlation between the specific angular momentum
and radius, although there is a slight tendency for more massive
cores to have somewhat larger specific angular momenta. For our
sample cores, the specific angular momentum ranges from 0.01 to
0.4 $c_s L_s$ (where $c_s$ is the isothermal sound speed and $L_s$ the
Jeans length of the Spitzer sheet). The distribution peaks around
0.08 $c_s L$, corresponding
to $0.9 \times 10^{21}$ cm$^2$ s$^{-1}$ (for the fiducial cloud
parameters). For comparison, we
superpose in the figure the specific angular momenta of 7 starless
N$_2$H$^+$ cores deduced by Caselli et al. (2002). They are broadly
consistent with our results. The average for the N$_2$H$^+$ cores
is estimated at $\sim 1.6 \times 10^{21}$ cm$^2$ s$^{-1}$, somewhat
larger than the peak value of our distribution.
An oft-used quantity for characterizing the rotation of a dense core
is the ratio of rotational energy to gravitational energy. For a
uniform sphere of density $\rho$ rotating rigidly at an angular
speed $\Omega$, it is given by $\beta=\Omega^2/(4\pi G\rho)$.
Even though our cores are not uniform, and do not rotate strictly
as a solid body, we have computed the value of $\beta$ using the
average angular speed and mean density. The distribution of
$\beta$ peaks around $\sim $ 0.02, in good agreement with
the average value of 0.018 for the starless N$_2$H$^+$ cores of
Caselli et al. (2002). It is also consistent with the well-known
result that rotation is generally not important as far as core
support against self-gravity is concerned (Goodman et al. 1993).
\subsection{Flux-to-Mass Ratio}
Ambipolar diffusion enables dense cores to be less magnetized relative
to their masses. In Fig.~\ref{fig:mass2fluxnew}, we plot the magnetic
flux-to-mass ratios of the cores against their column densities.
The ratio is normalized to the critical value $2\pi G^{1/2}$, and the
column density to that of the initially uniform state,
$4\pi\rho_0 L_s$.
The magnetic flux-to-mass ratio is
approximated as $\Gamma_c\equiv \pi R_{c}^2 \left< B\right>/M_c$, where
$\left<B\right>$ is the mean magnetic field strength inside a core
(e.g., Dib et al. 2007) and the mean core radius $R_c$ is
defined as the radius of a sphere enclosing the same volume as
the core. The column density is calculated from the core mass
divided by the area of a circle whose radius is equal to $R_{c}$.
There is a clear trend for cores with higher column densities to have
lower flux-to-mass ratios. Indeed, all cores with masses greater
than $1.5\left<M_c\right>$ are magnetically supercritical, with
flux-to-mass ratios as low as $\sim 0.35$, indicating
that they have all experienced significant ambipolar diffusion.
The weakening of magnetic support may have enabled these cores to
condense to a high column density in the first place.
\subsection{Core Mass Spectrum}
In Figure \ref{fig:coremf} we plot the mass spectrum for our sample
of dense cores. The core mass is normalized to the Jeans mass of
the Spitzer sheet, $M_{J,s}$.
The lowest mass at $\sim 0.1~M_{J,s}$ corresponds to the minimum
core mass determined from the criteria for core identification. There
is a prominent break around $1~M_{J,s}$ in the mass spectrum. Above
the break, the spectrum can be fitted roughly by a power law $dN/dM
\propto M^{-2.5}$, not far from the Salpeter stellar initial mass
function. We note that cores above the break contain more than
$\sim 100$ cells; their structures should be reasonably well resolved.
The core spectrum flattens below the break, which is also
broadly consistent with observations.
Similar core spectra are also
obtained in the turbulent fragmentation model of core formation that
involves much weaker magnetic field
and much stronger turbulence
(e.g., Padoan \& Nordlund 2002).
The characteristic mass below which the core mass spectrum flattens
apparently coincides with the Jeans mass of the condensed sheet,
which may result from magnetically regulated gravitational fragmentation.
Caution must be exercised in trying to differentiate various
scenarios of core formation, such as the magnetically regulated
gravitational fragmentation and the turbulent fragmentation, based
on core mass spectrum and, perhaps by extension, stellar mass
spectrum. The stellar mass spectrum for the 52 stars formed in the
standard model and its close variant are plotted in Fig.~\ref{fig:coremf}
for comparison. The stellar mass spectrum also resembles the
Salpeter IMF.
\section{Discussion}
\label{discussion}
\subsection{Magnetically-Regulated Star Formation in Quiescent
Condensations}
\label{scenario}
We have simulated star formation in relatively diffuse, mildly
magnetically subcritical clouds that are strongly turbulent
initially. The decay of initial turbulence leads to gravitational
condensation of the diffuse material along field lines into
a highly fragmented sheet (see Fig.~\ref{3D}), which produces
stars in small groups. The bulk of the condensed material
remains magnetically supported in the cross-field direction,
however, with typical motions that are moderately supersonic
and approximately magnetosonic. Only a small fraction of the
condensed material collapses gravitationally to form stars.
The calculations lead us to a picture of slow, inefficient,
star formation in a relatively quiescent, large-scale
condensation (which can be a filament if the cloud is flattened
in a direction perpendicular to the field lines to begin with)
that is embedded in a much more turbulent halo of diffuse gas.
The above picture is an unavoidable consequence of the anisotropy
intrinsic to the large-scale magnetic support. Before the self-gravity
becomes strong enough to overwhelm the magnetic support in the
cross-field direction (to initiate widespread star formation), it
should be able to condense material along the field lines (where
there is no magnetic resistance) first. The condensed material
is expected to be relatively quiescent. Its field-aligned velocity
component is damped during the condensation process, whereas its
cross-field motions are constrained by the field lines. The
lateral magnetic support ensures that the bulk of the condensed
material is relatively long-lived, allowing more time for the
turbulence to dissipate. The dissipation is hastened by the high
density in the condensation, which makes it easier for shock
formation, because of low Alfv\'en speed. Indeed, when the condensed
layer accumulates enough material for its mass-to-flux
ratio to approach the critical value (a condition for widespread
star formation), its Alfv\'en speed drops automatically to a value
comparable to the sound speed (NL05, Kudoh et al. 2007). Rapid
dissipation of super-magnetosonic motions ensures that the bulk of
the condensed material moves at no more than moderately supersonic
speeds, as we find here and previously in 2D calculations (NL05).
It is in this relatively quiescent environment that most star
formation takes place.
The quiescent, star-forming, condensed material is surrounded by
a highly turbulent, fluffy halo. The stratification in turbulent
speed is a consequence of the stratification in density (Kudoh \&
Basu 2003): the amplitude of a wave increases as it propagates
from high density to low density, and faster motions can survive
for longer in the lower density region because of higher Alfv\'en
speed. We believe that this is a generic result independent of
the source of turbulence. In our particular simulations, the
saturated, highly supersonic turbulence in the halo at late times
is driven mainly by protostellar outflows internally. It is
conceivable, perhaps even likely, that most of the turbulence
in the diffuse regions of star forming clouds comes from energy
cascade from larger-scale molecular gas (perhaps even HI envelope).
Depending on the details of energy injection, the external driving
of turbulence can slow down the
gravitational settling along field lines and may even suppress
it altogether. Even when the large-scale gravitational settling
is suppressed, star formation can still occur, perhaps in a less
coherent fashion: stars can form in localized pockets of a
magnetically subcritical cloud directly from strong compression
through shock-enhanced ambipolar diffusion (NL05), especially
when the shock is highly super-Alfv\'enic (Kudoh \& Basu 2008).
Vigorous driving may favor a mode of turbulence-accelerated star
formation at isolated locations, whereas decay of turbulence may
allow the gravity to pull the diffuse gas into large-scale
condensations, in which stars form in a more coherent fashion.
The first mode is now under investigation, and the results will
be reported elsewhere. The second corresponds to the picture
outlined above.
The dichotomy is not unique to star formation in strongly magnetized
clouds. It is also present when the magnetic field is weak or
non-existent: whereas stars form rapidly in a clustered mode in a
decaying non-magnetic turbulence, they are produced more slowly, in
a more dispersed fashion, if the turbulence is driven constantly,
especially on small scales (e.g., Mac Low \& Klessen 2004). Different
modes of star formation may operate in different environments, and
there is a urgent need for sorting out which mode dominates in which
environment. In the next subsection, we will concentrate on the Taurus
molecular clouds, the prototype of distributed low-mass star forming
regions (see Kenyon et al. 2008 for a recent review), and the Pipe
nebula, which appears to be a younger version of the Taurus complex
(Lada et al. 2008).
\subsection{Connection to Observations}
\subsubsection{The Taurus Molecular Cloud Complex}
\label{taurus}
The Taurus molecular cloud complex may represent the best case for
the picture of magnetically regulated star formation investigated
in this paper. One line of evidence supporting this assertion
comes from the studies of young stars by Palla \& Stahler (2002).
They inferred, using pre-main sequence tracks, that the star formation
in the complex has two phases. It began at least 10 Myrs ago, at
a relatively low level and in widely dispersed locations (Phase
I hereafter). In the last three million years or so, stars are
produced at a much higher rate (Phase II). The majority of the
stars produced in Phase II are confined within the dense,
striated, C$^{18}$O filaments (Onishi et al. 2002). Palla \&
Stahler interpret these results to mean that the Taurus clouds
evolved from an initial, relatively diffuse, state
quasi-statically. They envision that, during the early epoch
of quasi-static contraction, few places have contracted to
the point of forming dense cores and protostars, and hence only
scattered star formation activity is seen at isolated locations
(Phase I).
Eventually, further contraction of the cloud leads to the formation
of nearly contiguous, dense filaments where multiple cores can
form and collapse simultaneously, leading to elevated star formation
(Phase II).
In view of the discussion
in the last subsection, it is natural to identify Phase I with the
shock-accelerated mode of star formation in a magnetically
subcritical or nearly critical cloud (or sub-region), when the
turbulence is still
strong enough to prevent large-scale gravitational condensation
along the field lines. In this picture, Phase II started when the
turbulence had decayed to a low enough level that the gravity was
able to collect enough mass along the field lines into condensed
structures for a faster, more organized, mode of star formation
to take over.
A salient feature of our picture is that the rate of star formation
in Phase II should remain well below the limiting free-fall rate
of the condensed material. There is strong evidence that the
conversion of dense material in the Taurus clouds into stars is
indeed slow. The dense, filamentary material is well traced by
C$^{18}$O, which Onishi et al. (1996) has studied in great detail.
These authors estimated a mass and average density for the C$^{18}$O
gas of $M_{{\rm C}^{18}{\rm O}}\sim 2900~M_\odot$ and $n_{H_2}\sim
4000$~cm$^{-3}$ (comparable to that in the condensed sheet in
our simulations). The latter corresponds a free fall time
$t_{{\rm C}^{18}{\rm O},ff}=4.86 \times 10^5$~yrs. The maximum, free-fall,
rate of turning C$^{18}$O material into stars is therefore
${\dot M}_{*,{\rm C}^{18}{\rm O},ff}\sim 6\times
10^{-3}~M_\odot$~yr$^{-1}$. This
rate is to be compared with the actual rate of star formation in
the complex, ${\dot M}_*\sim 5\times 10^{-5}~M_\odot$yr$^{-1}$,
estimated by Goldsmith et al. (2008) based on an assumed average
mass of $0.6~M_\odot$ for the 230 stars observed in their
mapped region (Kenyon et al. 2008) and a duration of 3~Myrs for
Phase II of star formation where most stars are produced (Palla
\& Stahler 2002). Therefore, the rate of converting the even
dense (relative to the diffuse $^{12}$CO gas) C$^{18}$O material
into stars is extremely slow, amounting to $\sim 1\%$ per local
free fall time. This rate is comparable to those obtained in our
simulations.
One may argue that the slow rate of star formation in C$^{18}$O
gas is due to turbulence support. However, the C$^{18}$O gas
in the Taurus clouds is observed to be unusually quiescent.
Onishi et al. (1996) estimated a mean equivalent line-width
(defined as the integrated intensity divided by the peak
temperature of the line) of 0.65~km~s$^{-1}$, corresponding
to a 3D velocity dispersion of 0.45~km~s$^{-1}$ for a Gaussian
profile. The velocity dispersion is $\sim 2.4$ times the
sound speed for a gas temperature of 10~K, indicating that
the moderately dense gas traced by C$^{18}$O is not
highly supersonic in general. Indeed, the velocity
dispersion is quite comparable to that of the condensed
sheet material in our standard model at similar densities
(see Fig.~\ref{rms_vel}). Nearly 1/3 ($937~M_\odot$ out
of $2900~M_\odot$) of the C$^{18}$O gas is classified
into 40 cores, which have a mean line-width (FWHM) of
0.49~km~s$^{-1}$, comparable to that estimated above
for the C$^{18}$O gas as a whole. Onishi et al. concluded
that most of the cores are roughly gravitationally bound,
based on their masses, sizes and line-widths. These authors
argued that the general alignment of the minor axes of the
cores with the overall field direction indicates that the
magnetic field plays a role in their formation. We agree
with this assessment.
The relatively low velocity dispersion of the C$^{18}$O gas
and the morphology of dense cores provide indirect support
for the picture of magnetically regulated star formation.
A more direct test of the picture would come from a
determination of the mass-to-flux ratio.
For a given field strength $B$, the critical column density
along the field line is
\begin{equation}
\Sigma_{\vert\vert,c}=58.7~B_{20}~M_\odot~{\rm pc}^{-2},
\label{critical}
\end{equation}
corresponding to an $A_V \approx 2.5~B_{20}$, where $B_{20}$
is the field strength in units of 20~$\mu$G.
Goldsmith et al. (2008) estimated a total mass of
$2.4\times 10^4 M_\odot$ in their
mapped area of $28$~pc$\times 21$~pc, yielding an average
column density of 40.8~M$_\odot$~pc$^{-2}$ along the line
of sight. Only $\sim 2/3$ of the surveyed area have detectable
$^{12}$CO emission in individual spectra, however (their
Masks 1 and 2). For these regions, the total mass and area
are $1.95\times 10^4 M_\odot$ and $\sim 392$~pc$^2$, yielding
a somewhat higher average column density of 49.7~M$_\odot$~pc$^{-2}$
along the line of sight. If the column density along the field
lines is comparable to this value, the corresponding critical
field strength would be $16.9~\mu$G, according to
equation~(\ref{critical}). About half of the mass (9807~M$_\odot$)
resides in eight ``high-density'' regions (such as L1495 and
Heiles Cloud 2) that together occupy $\sim 31\%$ of the area,
with an average line-of-sight column density of
79.5~M$_\odot$~pc$^{-2}$. This value would correspond to
a critical field strength of $27.1~\mu$G if the magnetic field
is exactly along the line of sight. There is, however, evidence
for a large-scale magnetic field in the plane of sky in the
Taurus cloud complex from the polarization of background
star light. Indeed, the velocity anisotropy measured by Heyer
et al. (2008) in a relatively diffuse sub-region points to
a magnetic field that is mainly in the plane of sky. If this
magnetic orientation is correct, and if matter condenses along the
field lines into flattened structures, then projection effects
may be considerable. Applying a moderate correction factor
of $\sqrt{2}$ (corresponding to a viewing angle of $45^\circ$)
would bring the average field-aligned column density of the
``high-density'' regions down to 56.2~M$_\odot$~pc$^{-2}$, with
a corresponding
critical field strength of $19.2~\mu$~G. It is therefore likely
that a magnetic field of order 20~$\mu$G is strong enough to
regulate star formation in the Taurus cloud complex.
The magnetic field strength has been measured for three dense
cores in Taurus through OH Zeeman measurements using Arecibo
telescope (Troland \& Crutcher 2008). They are B217-2, TMC-1
and L1544, with a line-of-sight field strength of $B_{\rm los}
\approx 13.5$, $9.1$, and $10.8$~$\mu$G respectively. Although
the true orientation of the magnetic field is unknown, a
statistically most probable correction factor of 2 would bring
the total field strength to $\sim 18.2 - 27.0~\mu$G. Given that
OH samples moderately dense gas approximately as C$^{18}$O
does, and that the field strength changes relatively little at
low to moderate densities (see Fig.~\ref{ave_B}), it is plausible
such a field permeates most of the space in the complex. A
worry is that such a strong field is not detected in several
other cores in the Taurus in the same Arecibo survey. Whether
these non-detections can be reconciled with our picture of
magnetically regulated star formation through, e.g., unfavorable
field orientation, remains to be seen.
Indirect evidence for a relatively strong magnetic field
comes from the observation that the magnetic field as
traced by the polarization vectors of the background star
light is well ordered over many degrees in the sky,
indicating that the magnetic energy is larger than the
turbulent energy.
If we use conservative estimates for H$_2$ number density
of $10^2$~cm$^{-3}$ (see Table 2 of Goldsmith et al. 2008),
and turbulent speed of 2~km~s$^{-1}$,
we can put a lower limit to the field strength at $\sim
15~\mu$G. The limit agrees with the value of $\sim 14~\mu$G
estimated by Heyer et al. (2008) based on comparing the
observed velocity anisotropy in a striated, relatively
diffuse region with MHD turbulence simulations; they point
out that
the value is to be increased if the magnetic field has
a substantial component along the line of sight. A
strong line-of-sight magnetic field of $\sim 20~\mu$G
or more is deduced by Wolleben \& Reich (2004) at the
boundaries of molecular clouds a few degrees away from
the sub-region studied by Heyer et al. (2008), from
modeling the enhanced rotation measures inferred from
polarization observations at 21 and 18~cm. Subsequent
HI Zeeman observations did not confirm the deduced
field strength, however (C. Heiles, priv. comm.).
Although none of the pieces of evidence discussed above is
conclusive individually, taken together, they make a
reasonably strong case for a global magnetic field of order
$\sim 20\mu$G in strength. The existence of such a field
may not be too surprising from an evolutionary point of view.
It is now established that the cold neutral medium (CNM) of
HI gas (likely precursor of Taurus-like molecular clouds)
is fairly strongly magnetized in general, with a median
field strength of $\sim 6~\mu$G (Heiles \& Troland 2005).
For denser HI clouds, it is not difficult to imagine a
somewhat higher field strength.
A case in point is the nearby Riegel-Crutcher HI cloud
that we mentioned in the introduction. It was mapped
recently by McClure-Griffiths et al. (2006) using 21~cm
absorption against strong continuum emission towards the
Galactic center. They found long strands of cold neutral
hydrogen that are remarkably similar to the CO striations
in Taurus studied by Heyer et al. (2008): both are aligned
with the local magnetic field. For the magnetic energy to
dominate the turbulent energy inside the filaments, the
field strength must be of order $30~\mu$G or larger. This
value is consistent with $B_{los}\sim 18~\mu$G obtained
by Kazes \& Crutcher (1986) through Zeeman measurements
at a nearby location, if a correction of $\sim 2$ is
applied for projection effects. The physical conditions
inside the HI filaments of the Riegel-Crutcher cloud are
not dissimilar to those of diffuse CO clouds: number
density $n_H\sim 460$~cm$^{-3}$,
velocity dispersion $\sigma \sim 3.5$~km~s$^{-1}$, and
temperature $T\sim 40$~K. The main difference is that the
column density is $\sim 10^{20}$~cm$^{-2}$, which is not
sufficient for shielding the CO from photodissociation
(van Dishoeck \& Black 1988).
The above discussion leads us to the following scenario for
cloud evolution and star formation in the Taurus molecular
clouds: we envision the cloud complex to have evolved from a
turbulent, strongly magnetized, non-self-gravitating HI cloud
similar to the Riegel-Crutcher cloud. Once enough material
has been accumulated, perhaps by converging flows
(Ballesteros-Paredes et al. 1999) or some other means
(such as differential turbulence dissipation, which may
lead to a condensation through ``cooling flow'' along the
field lines, e.g., Myers \& Lazarian 1998), for CO
shielding, it becomes visible as a molecular cloud.
Depending on the degree of magnetization, level of
inhomogeneity, and strength of turbulence, star formation
can start quickly in some pockets (perhaps even during the
process of molecular cloud formation), despite of a strong
global magnetic field, either because they happen to be
less magnetized to begin with or because their magnetic
fluxes are force-reduced by shock-enhanced ambipolar
diffusion. This corresponds to the low-level, Phase I of
star formation deduced by Palla \& Stahler (2002). Once
enough material has settled along field lines for the
mass-to-flux ratio of the condensed material to approach
the critical value, the rate of star formation increases
drastically. We identify this phase of more organized star
formation in condensed structures as the more active,
Phase II of star formation deduced by Palla \& Stahler
(2002). Even during this relatively active phase, the star
formation rate is still well below the free-fall rate of
the condensed gas, because of magnetic regulation.
The above picture can be contrasted with that of Ballesteros-Paredes
et al. (1999) and Hartmann et al. (2001). These authors advocated
a picture of rapid cloud formation and star formation for the
Taurus-Auriga complex, motivated mainly by a lack of ``post-T Tauri
stars'' older than $\sim 5$~Myrs in the region; indeed, the
majority of stars are younger than $\sim 3$~Myrs, as mentioned
earlier. They argue that it is difficult to understand how star
formation can occur simultaneously over a spatial extent of $\sim
20$~pc (or more) over a timescale as short as a few million years,
given that the observed turbulent speed is $\sim 2$~km~s$^{-1}$,
corresponding to a crossing time of $\sim 10$~Myrs (or more),
unless the star formation is coordinated by some agent other than
the currently observed turbulence. They suggest that this agent is
external: a large-scale converging flow with speeds of 5-10~km~s$^{-1}$.
In our picture, it is the global gravity and a strong, large-scale,
internal magnetic field that coordinate the star formation: the
strong field suppresses vigorous star formation until enough
material has collected along field lines into condensed structures
for the gravity to start overwhelm magnetic support in the
cross-field directions. The degree to which the star formation is
actually synchronized will, of course, depend on how the mass is
initially distributed along the field lines and rate of turbulence
dissipation and replenishment, neither of which is, unfortunately,
well constrained at the present. Indeed, a hybrid scenario may be
possible, with the mass accumulation along the field lines sped
up by a large-scale converging flow. Nevertheless, we believe
that, at least in the particular case of the Taurus molecular cloud
complex, for the large amount of dense, relatively quiescent,
C$^{18}$O gas to form stars at $\sim 1\%$ of the free-fall rate,
magnetic regulation is needed. A similar case can be made for the
Pipe Nebula.
\subsubsection{Pipe Nebula}
\label{pipe}
Although less well studied than the Taurus cloud complex, the Pipe
nebula is an important testing ground for theories of low-mass
star formation in relatively diffuse dark clouds. It has $\sim
10^4~M_\odot$ of molecular material as traced by $^{12}$CO (Onishi
et al. 1999), not much different from the Taurus complex. The higher
density gas traced by $^{13}$CO and C$^{18}$O is concentrated in
a long filament that spans more than 10~pc, again similar to the
Taurus case. The main difference is that there is no evidence
for star formation activity except in the most massive core, B59.
The lower rate of star formation indicates that the Pipe nebula is
at an earlier stage of evolution compared to the Taurus complex, as
pointed out by Muench et al. (2007). The relative inactivity is
not due to a lack of dense, quiescent material that is a
pre-requisite of (low-mass) star formation. J. Alves et al. (2007)
identified 159 dense cores from extinction maps, many of which are
formally gravitationally unstable, with masses above the critical
Bonner-Ebert mass (Lada et al. 2008). In the absence of additional
support, such cores should collapse and form stars on the free-fall
time scale, which is $t_{ff}=3.7\times 10^5$~yrs for the inferred
median core density $n(H_2)=7\times 10^3$~cm$^{-3}$. This time
scale is much shorter than the turbulence crossing time $t_c=L/v
\sim 5\times 10^6$~yrs, over a region of $L \sim 15$~pc, and for
a typical turbulent speed of $v\sim 3$~km~s$^{-1}$. The mismatch
between the turbulent crossing time and star formation time is
therefore much more severe than that noted above for the Taurus
region.
There are two possible resolutions to the above conundrum. If
the dense cores (at least those massive enough to be formally
gravitationally unstable) are indeed short-lived objects, with
a lifetime of order the average free-fall time ($\sim 3\times
10^5$~yrs), their formation must be rapid and well coordinated.
It is tempting to
attribute the coordination to a large-scale converging flow, as
Ballesteros-Paredes et al. (1999) and Hartmann et al. (2001)
did for the Taurus case. However, to cover a distance of $15$~pc
(the size of the cloud, which is comparable to the length of
the chain of dense cores) in $\sim 3\times 10^5$ years, the
flow needs to converge at a speed of order $\sim 50$~km~$s^{-1}$,
which is uncomfortably high. A more likely alternative is that
the cores live much longer
than their free-fall times would indicate. While the longevity
is not necessarily a problem for those non-self-gravitating,
lower mass cores that can be confined by external pressure,
it is problematic for the more massive cores that should have
collapsed on the free-fall time scale, unless they are supported
by some (so-far invisible) agent in addition to the observed
thermal and turbulent pressures. The most likely candidate is
a magnetic field. There is some evidence for the existence of
an ordered magnetic field that is more or less perpendicular
to the long filament that contains the dense cores from stellar
polarization observations (F. Alves \& Franco 2007; F. Alves et
al., in preparation), although much more work is needed, particularly
for the diffuse region outside the filament, to firm up or reject
this possibility. In any case, it is plausible that
the Pipe nebula represents an earlier stage of the magnetically
regulated cloud evolution and star formation than the Taurus
clouds: material has started to collect along the field lines
into dense, relatively quiescent, structures, but their
mass-to-flux ratios are yet to reach the critical value,
except perhaps in the most massive core, B59. In this case,
the ages of the cores are set by the rate of mass condensation
along the field lines, which may depend on the ill-understood
process of turbulence replenishment in the diffuse gas, and
may be accelerated by large-scale converging flows (Ballesteros
et al. 1999; Hartmann et al. 2001).
\subsection{Modes of Star Formation in Strongly Magnetized Clouds}
\label{mode}
Strong magnetic fields tend to resist cloud condensation and
star formation. This resistance can be overcome by either
turbulence or gravity, leading to two distinct modes of star
formation. Which mode dominates depends on the relative
importance of gravity and turbulence. When the turbulence is
strong enough to prevent gravitational settling along field
lines, star formation is still possible. Turbulence can
overcome magnetic resistance in localized regions that are
compressed by converging flows to high densities; such regions
are expected to have large magnetic field gradients and low
fractional ionization, both of which enhance the rate of
ambipolar diffusion. The
turbulence-accelerated ambipolar diffusion enables dense
pockets to become self-gravitating
and collapse into stars (NL05, Kudoh \& Basu 2008), even when
the background cloud
remains well supported. In this mode, the crucial step of
star formation---core formation---is driven by turbulent
converging flows but regulated by magnetic fields, with
self-gravity playing a secondary role. After a bound core
has formed, its further evolution is driven primarily by
the self-gravity, but remains regulated by the magnetic
field.
The second mode corresponds to a turbulence that is not strong
enough to prevent gravitational settling, either because it
is weak to begin with or because it has decayed without being
adequately replenished. Condensation would occur along field
lines, which would lead to further increase in self-gravity
and more rapid dissipation of turbulence. In the condensed
structure, turbulence becomes less important dynamically, and
star formation is driven mainly by gravity in the cross-field
direction. The gravitational contraction is resisted primarily
by magnetic fields, although protostellar outflows play an
important role in dispersing away the dense gas that remains
after the formation of individual stars; the dispersal slows
down the global star formation. In this mode, the gravity and
magnetic fields are of primary importance, with turbulence
playing a secondary role.
The above two modes of star formation may be interconnected: it
is natural to expect the second to follow the first, as a result
of turbulence dissipation. The transition from one to the
other should occur when the mass-to-flux ratios in the condensed
structures approach the critical value. It may be possible for
the mass-to-flux ratio to stay near the critical value due to
outflow feedback: if too much mass is added to the structure
so that its mass-to-flux ratio increases well above the critical
value, it would form stars at a higher rate, producing more
outflows that stop further mass accumulation, similar to the
outflow-regulated scenario of cluster formation envisioned
in Matzner \& McKee (2000). They have shown that regions of
outflow-regulated star formation tend to be overstable,
oscillating with increasingly large amplitudes. Indeed, the
outflows may unbind the condensed structure completely,
terminating its star formation quickly. An abrupt termination
is implied by the dearth of the well-observed star forming
regions in the declining phase of star formation (Palla \&
Stahler 2000).
If, on the other hand, outflows fail to retard mass accumulation
along field lines, the star formation may change over to
yet another, more active, mode --- cluster formation. The
transition is expected to take place when the mass-to-flux ratios
of the condensations become substantially greater than the critical
value (say by a factor of two), making global (not just local)
contraction perpendicular to the field lines possible. The
higher rate of star formation should lead to a stronger
outflow feedback, which could increase the turbulence
to a level that balances the global collapse (e.g., Nakamura
\& Li 2007; Matzner 2007). In this mode, protostellar
outflow-driven turbulence becomes the primary agent that
resists the global gravitational collapse, with the magnetic
field playing a secondary role.
In summary, we envision three, perhaps sequential, modes of star
formation in strongly magnetized clouds, dominated (a) first by
turbulence and magnetic fields, (b) then by magnetic fields and
gravity, and (c) finally by gravity and turbulence (again). In
this sequence, stars form first directly out of turbulent diffuse
material in a dispersed fashion, then more coherently in
relatively quiescent, condensed structures, and finally as
clusters in dense turbulent clumps. The strong turbulence
is primordial initially. It is transformed into a protostellar
turbulence by outflows in regions of active cluster formation.
\section{Summary and Conclusions}
\label{conclude}
We have performed 3D simulations of star formation in turbulent,
mildly magnetically subcritical clouds including protostellar
outflows and ambipolar diffusion. The simulations are motivated
by observations of molecular gas and young stars in the Taurus
cloud complex and the realization that cold neutral HI gas, a
likely progenitor of molecular gas, is magnetically subcritical
in general. They are natural extensions of our previous 2D
calculations. The main results are:
1. The decay of turbulence leads to gravitational condensation
of material along the field lines. In the ideal MHD limit, the
resulting structure closely resembles a smooth Spitzer sheet,
as found previously (Krasnopolsky \& Gammie 2005). Ambipolar
diffusion changes the structure drastically. It creates
supercritical material that behaves differently from the
subcritical background in a fundamental way: it can be condensed
further by self-gravity. As a result, the distributions of mass
with respect to volume and column densities are greatly broadened.
Indeed, the probability distribution function (PDF) of the (more
easily observable) column density can be fitted approximately
by a broad, lognormal distribution (see Fig.~\ref{colden_PDF}),
as is the case for weakly magnetized clouds fragmented by strong
turbulence (e.g., Padoan \& Nordlund 2002; P. S. Li et al. 2004).
It may be difficult to distinguish the turbulent fragmentation
in weakly magnetized clouds and the ambipolar diffusion-enabled
gravitational fragmentation in strongly magnetized clouds based
on mass distribution.
2. The magnetized clouds evolve into a configuration strongly
stratified in density at late times, with dense cores embedded
in fragmentary condensations which, in turn, are surrounded
by a more diffuse halo (see Fig.~\ref{3D}).
The condensed material can be significantly warped and may appear
thicker than its intrinsic width, even when viewed edge-on,
because of superposition of warps along the line of sight. While
the diffuse halo is highly turbulent, the condensations are
more quiescent, with moderately supersonic velocity dispersions
in general. One reason is that the bulk of the condensed material
is magnetically supported, which allows more time for the
turbulence to decay. Another is their relatively low Alfv\'en
speed (comparable to the sound speed), which forces highly
supersonic motions to dissipate quickly, through shock
formation. The moderately supersonic turbulence is maintained,
however, for many local free-fall times, through a combination
of outflow feedback, MHD waves, and gravitational motions
induced by ambipolar diffusion.
3. Stars form at a low rate. Only one percent or less of the cloud
material is converted into stars in a free-fall time at the
characteristic density of the condensed material. The slow star
formation cannot be due to support by turbulence because the
star-forming, condensed material is rather quiescent. It is
regulated by magnetic fields. The low rate of star formation is
not sensitive to the level of initial turbulence, although more
turbulent clouds produce stars early, consistent with the picture
of turbulence-accelerated star formation in strongly magnetized
clouds. The rate of star formation is further reduced by
protostellar outflow, particularly the component in the plane
of mass condensation.
4. Dense cores formed in our simulations are typically triaxial.
More massive cores tend to have lower magnetic flux-to-mass
ratios, be more tightly bound gravitationally, and be more
oblate. Virial analysis indicates that the surface term due to
external kinetic motions contributes significantly in general,
especially for less massive cores. We find a larger number
of bound cores than in previous ideal MHD simulations even
though our clouds are more strongly magnetized; they are made
possible by ambipolar diffusion. The cores have specific
angular momenta comparable to the observed values, and a mass
spectrum that resembles the Salpeter stellar IMF towards the
high mass end. The spectrum flattens near the characteristic
Jeans mass of the condensed material, which is a few solar
masses for typical parameters. The stellar mass spectrum also
resembles the Salpeter IMF.
5. The Taurus molecular cloud complex may represent the best case
for magnetically regulated star formation. The strongest evidence
comes from the existence of a large amount of dense, relatively
quiescent, C$^{18}$O gas, which produces stars at a rate two orders
of magnitude below the maximum free-fall rate. Most likely, the
star formation is regulated by a strong, ordered magnetic field.
Several lines of evidence suggest that a magnetic field of the
required strength (of order $\sim 20 \mu G$) is indeed present
in the region, although direct Zeeman measurements are available
for only three cores, and their interpretations are complicated
by projection effects.
6. We speculate that, depending on the (uncertain) rates of
turbulence replenishment and outflow feedback, there may be
three distinct modes of star formation in strongly magnetized
clouds: (a) inefficient star formation through
turbulence-accelerated ambipolar diffusion at dispersed
locations in relatively diffuse, magnetically subcritical
clouds, (b) more coherent, but still inefficient,
magnetically regulated star formation in relatively
quiescent, nearly magnetically critical condensations,
(c) more efficient cluster formation in magnetically
supercritical clumps that may be regulated by outflow-driven
protostellar turbulence. Much work remains to be done to
understand these modes and their possible interconnection.
\acknowledgments
This work is supported in part by a Grant-in-Aid for Scientific Research
of Japan (18540234, 20540228),
and NSF and NASA grants (AST-0307368 and NAG5-12102).
Parts of this work were performed while the authors were in residence
at the Kavli Institute for Theoretical Physics at UCSB.
The numerical calculations were carried out mainly
on NEC SX8 at YITP in Kyoto University,
on Fujitsu VPP5000 at the Center for Computational Astrophysics, CfCA,
of the National Astronomical Observatory of Japan,
and on Hitachi SR8000 at Center for Computational Science
in University of Tsukuba.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,444
|
Q: Windows 7 x64 not booting properly I'm not really sure "what" happened to my newly mounted pc.
The only thing I know is that it could be either a REGISTRY problem or a HDD failure.
The "why" is because I accidentally removed almost all of HKEY_LOCAL_MACHINE\Software\ Wow6432Node directory while uninstalling a program completely, OR because someone(...) disconnected it from the power source when it was hibernating.
And the conclusion is that it's stuck on that start screen below.
Which do you think is more likely to have happened, and, if possible, how to repair it?
Thanks.
A: If you had system protection turned on you can press F8 during boot and go to either "repair your computer" or "last known good configuration". Registry should be backed up in the system restore point, so one way or another you should be able to restore it.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,168
|
Connecting Mogadishu: Somalia, 2012
At The Village Restaurant, a popular open-air hangout for Mogadishu's returning diaspora community, a charcoal-powered Italian espresso machine brews Somalia's best cappuccino.
Wi-fi internet beams throughout the cafe, as patrons check email, download music videos, and keep tabs on Somalia's latest news.
As Mogadishu shifts from two decades of civil war to a quivering democracy, opportunities for business - from hotels to off-grid espresso makers to cafes like the Village - are flourishing. And so too are the opportunities for bringing them online.
Perched between the tattered ruins of a flattened landscape, the glow of wireless receiver antennas has gradually replaced the orange glow of stray bullets, bringing a new era of global connectivity and freedom of information to the city's estimated one million residents.
In 2000, Somalia was one of the last African nations to get online. Since then, the internet industry here has seen as much turbulence and turnover as the country itself.
According to Abdulkadir Hassan Ahmed, general manager of Global Internet Company, Somalia's largest internet provider, at least 17 internet companies in Somalia have gone under in the past decade.
It's a tough job. All the time, companies are coming in and going out"
Global Internet Company, founded in 2003 by a consortium of Somalia's leading telecom companies including Hormuud and NationLink, provides dial-up, DSL and some point-to-point wireless.
Yet even Mr Ahmed admits his own company's connections can be slow and expensive. After nearly 10 years in business, Global Internet is almost profitable, he says, but is more of a loss leader for telecoms.
Unlike Somalia's thriving telecoms sector, where two decades of lawlessness, lack of regulation, and cut-throat competition for an increasingly mobile market have driven services up and prices to rock bottom (less than one cent per minute), internet in Mogadishu has been archaic.
Dial-up is the cheapest option, at around $30 (£18) a month per computer, but is painfully slow - less than 56kbs - and highly oversubscribed, according to many.
Read on at http://www.bbc.co.uk/news/business-19961266
Connecting Mogadishu-1.jpg
Connecting Mogadishu-10.jpg
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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Jonas Magni Livin, född april 1663 i Norrköping, död 26 september 1733 i Linderås församling, Jönköpings län, var en svensk präst.
Biografi
Jonas Livin föddes 1663 i Norrköping. Han var son till kyrkoherden Magnus Livin och Margareta Prytz i Säby församling. Livin blev 1680 student vid Uppsala universitet och prästvigdes 1688. Han blev pastorsadjunkt i Linderås församling och 1694 kyrkoherde i församlingen. År 1720 utnämndes han till prost. Livin avled 1733 i Linderås församling.
Familj
Livin gifte sig första gången 1686 med Gunilla Bjugg. Hon var dotter till kyrkoherden Petrus Johannis Bjugg och Anna Olsdotter Berenfelt i Linderås församling.
Livin gifte sig andra gången med Catharina Elisabeth Wong. Hon var dotter till domprosten Olavus Johannis Wong och Margareta Laurinus i Linköping. De fick tillsammans barnen häradshövdingen Magnus Livin, kaptenslöjtnanten Olof Livin vid Smålands kavalleriregemente, kyrkoherden Jonas Livin i Vinnerstads församling, kapellan Claes Livin i Kungsholms församling och Margaretha Livin som var gift med kyrkoherden Samuel Samuelis Mörtling i Linderås församling.
Bibliografi
De Livia Columba Pr. L. Norrman, Uppsala 1686.
Referenser
Noter
Födda 1663
Avlidna 1733
Män
Svenska präster under 1600-talet
Svenska teologer under 1600-talet
Svenska präster under 1700-talet
Svenska teologer under 1700-talet
Präster från Norrköping
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,840
|
Iceland's most famous landmarks, mountains, glaciers, caves and much more. Here below you will find general information about Iceland and how to plan your visit to Iceland, whether you visit during summer or winter season.
Do you want to explore the underground world by yourself? Arnarker Cave is the perfect cave to do so.
Useful information with video on how to drive in Iceland. Road conditions, insurance, filling stations and mountain tracks.
List over ten of Iceland's largest rivers by volume. Iceland is home to large glacier rivers and other natural phenomenons.
List of the ten longest rivers in Iceland. General information about Iceland, at Extreme Iceland.
List of Iceland's 10 Biggest Glaciers. Iceland has the largest glacier in Europe, Vatnajokull.
Ten of Icelands largest lava fields. The largest lava field in Iceland is Storavitishraun lava field, 30-50 cubic kilometers.
List of the ten oldest churches in Iceland. The oldest church in Iceland is Baenhusid in Nupsstadur, from around 1650 (the base could be even older).
List of the Ten longest lava tubes in Iceland. The longest lava tube cave in Iceland is Laufbalavatn lava tube system, around 5012 meters long.
In this blog you will find some of my personal experiences as well as some practical information which could be useful when planning to hitchhike in Iceland.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 251
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By Diane Rommel / June 30, 2017
Delta Is Having a Fire Sale on Flights to Canada. Here's Where to Go.
From lakes to waterfalls to drinking till sunrise
In honor of Canada's 150th birthday, Delta is offering terrific fares to several Canadian airports, including Toronto, Montreal, Calgary and Vancouver. Many of the country's most amazing destinations are within easy (enough) access, which means it's time to take advantage of the brief, wondrous Canadian summer before it (and the Delta fare sale) conclude.
Here are our picks for the best destinations.
You've seen the uncannily blue waters of Lake Louise in a million Instagram photos. It's as beautiful as it looks — and worth summertime crowds for summertime temps.
Nearest airport: Calgary
Kayakers will want to make a beeline for this park criss-crossed by six rivers and 400 lakes; everyone else might want to head skyward — to the park's via ferrata path.
Nearest airport: Montreal
The cluster of three waterfalls at Niagara is viewed as tourist kitsch sometimes, but the fact is, they're absolutely beautiful — and well worth the drive from Toronto.
Nearest airport: Toronto
FATHOM FIVE NATIONAL MARINE PARK
This freshwater marine park presents a smorgasbord for divers — it's filled with the remains of turn-of-the-century schooners, with nearly two dozen wrecks in all.
MONTREAL BAR-HOPPING
Like Paris without the attitude, Montreal offers a master class in laidback loafing. We say start with smoked meat at Schwartz's — an appropriate way to kick off the night for any newbie — and then follow the crowd.
BANFF HOT SPRINGS
One hundred percent natural hot spring water is currently flowing at Banff's Upper Hot Springs, where visitors have been taking the waters for over a century. At under ten bucks an entrance, these calming pools, set in gorgeous surroundings, may be the best deal in the country.
PARC NATIONAL DE LA GASPESIE
Go for a hike in this incredible sub-arctic wilderness, with herds of caribou and other northern animals making their home on the tundra. There's also kayaking, canoeing, salmon fishing and even stand-up paddling.
You don't even need to leave Vancouver to get outside in Stanley Park, a regular feature on World's Best Park lists thanks to features like 12 miles of hiking paths, heated outdoor pools, picnic spots and more. Bonus: All of the city's top-notch restaurants, hotels and shops are still within easy reach.
Nearest airport: Vancouver
See the world from your inbox.
Sign up for The Journey, our Travel newsletter.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,071
|
In recent years, Spafford has witnessed a truly meteoric rise in the jam scene. Coming together serendipitously in 2009, the band gained a dedicated fan base in their home state of Arizona and quickly earned a reputation as the Southwest's go-to jam act. Through word of mouth, buzz about the group spread, and due to unprecedented demand, their headlining tours consistently sold out venues across the country. Most recently, Spafford has been selected to play high-profile events like Bonnaroo, The Peach Music Festival, Summer Camp, Firefly Music Festival, and Suwanee Hulaween.
Coming from New Orleans, Tank and the Bangas are surrounded by plenty of grand musical traditions. And the five-piece group has a rare knack for combining various musical styles — fiery soul, deft hip-hop, deep-groove R&B and subtle jazz — into one dazzling, cohesive whole that evokes the scope of New Orleans music while retaining a distinctive feel all its own.
Sponsored by Wells Fargo, Down By Downtown (DxDT) takes place April 11 to 13, 2019, in downtown Roanoke, Virginia. For more information, visit https://downbydowntown.com/.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 245
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Lavora alla 81 Produce ed è nota soprattutto per essere la doppiatrice originale del personaggio Delia Ketchum nell'anime Pokémon
Doppiaggio
Anime
Pokémon (Delia Ketchum)
Ristorante Paradiso (Angela)
Guerriere Sailor Venus Five (Afrodite)
Kindaichi shōnen no jikenbo
Bakuretsu Hunter (Butter)
Akira
Silent Möbius
Cooking Papa (Hanaoka-sensei)
Generator Gawl (Keiko)
Alien 9 (Mamm di Kumi)
Let's & Go - Sulle ali di un turbo (Lizhi)
Detective Conan (Mika Mitsui)
Arjuna - La ragazza terra (Newscaster)
100% Fragola (di Nishino)
Red Garden (mamma di Rose)
Seikai no monshō (Rurune)
DNA² (Insegnante)
Moominland, un mondo di serenità
Ranma ½ (Yuka)
Doppiatrici italiane
Cinzia Massironi: Delia Ketchum
Collegamenti esterni
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,310
|
def reverseList(list):
# get the last index
lastIndex = len(list) - 1
# iterate through the index
for item in list:
# assign last item value to temp
temp = list[lastIndex]
# exchange elements
list[lastIndex] = list[list.index(item)]
# restore the value
list[list.index(item)] = temp
# decrement index
lastIndex -= 1
if (lastIndex >= list.index(item)):
break
# print the reversed list
for item in list:
print item
# Test list 1
list1 = [1, 2, 3, 4, 5]
reverseList(list1)
# Test list 2
list2 = [72, 272, 920, 111]
reverseList(list2)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,480
|
FULL TIME COURSES
Master of Arts in Scoring for Film, TV and Interactive Media
BA (Hons) Degree in Audio and Music Technology
BA (Hons) Degree in Music Production
BA (Hons) Degree in Music Production with Composition
PART TIME COURSES
Diploma in Live Music and Performance Technology
Music Production With Apple Logic Pro X
Music Production With Ableton Live
Diploma in Film Production Full Time
Diploma in Film Production Part Time
Scriptwriting and Screenwriting Course
Adobe After Effects Level One
Adobe Premiere Pro CC2023
Video Editing With Final Cut-Pro X
BA (Hons) in Creative Technologies and Digital Art
Diploma in Game Development
Diploma in 3D Game Art
3D Environmental Design
BA Degree in Animation
Certificate in Animation
Springboard+ in Animation
Character Design with Gary Erskine
Introduction to 3D Animation
International Students Information
So, fellow gamers, we're half way through the year and the game industry isn't showing any sign of slowing down! In fact, it's continuing to grow and evolve as game developers and designers keep pushing the boundaries, trying to create the next big thing.
It's been an exciting 6 months with a bunch of top new game releases, a new console from Nintendo and plenty of chatter in the rumour mill!
Let's take a look into the next few months to see what's coming down the line…
Consoles – The End of an Era?
We're pretty used to seeing big leaps every few years in the console arena – PlayStation 1, 2, 3, 4…Are we seeing a shift in this approach fuelled by the Smartphone manufacturers like Apple? The world's largest smartphone producers, Apple and Samsung, have become experts at updating their hardware, creating a buzz with consumers and in turn a big demand for their latest devices.
iPhone 6, iPhone 6S, iPhone 6SPlus… Samsung take a similar approach – Galaxy S6, Galaxy S7, each version including incremental hardware upgrades. The older versions then fall in price making them more affordable for consumers. Now console manufacturers are getting in on this act. Sony are starting to adopt this approach adding the 4K enabled PlayStation 4 Pro which dropped right before last Christmas.
Microsoft are also trying this out and have been teasing gamers across the globe with the Xbox One Scorpio with their "Feel True Power" slogan. Whilst some might be sad to think there's a change coming it should mean we start to see a fall in prices for some of the latest consoles. Ain't nothing wrong with that!
Oh and for the retro game fans out there – we hope you got your pre-orders in for the SNES Classic Mini launching the end of September!
Physical Sales
Digital sales of games first overtook physical sales back in 2013 and hasn't stopped since! It is estimated that almost 75% of all game sales in 2016 were digital. Online shops have become very popular with gamers because faster internet speeds allow for quick downloading of the latest titles.
It's not all about downloading games though. A growing trend, and big revenue stream, is downloading in game add-ons such as expansion packs, subscriptions etc. Defo something to think about when in the game design phase.
Mobile gaming is booming and driving a lot of the trends we're talking about in this article, including downloads of course. Games like Pokémon Go have raked in millions (possibly billions!) and use all the available tech such as geo location and the smartphone camera to create augmented reality.
Smartphone graphics have come on in leaps and bounds and gamers can sit back anywhere and dip in and out of a game with ease helping mobile gaming thrive. Strategy games such as Clash of Clans are making a big impact in multiplayer and we are seeing more and more games designed to be played online with friends.
Multiplayer Leading the Way
Essential Facts found that in 2016, 53% of the most frequent games played were multiplayer games and over half of gamers said that online games helped them to connect with their friends.
We think this trend is going to continue as new ways of interacting and playing games online with people come on stream. Game developers are going to need to know all the latest techniques to keep up with multiplayer design in the coming years.
Develop Your GamesDev Knowledge
If you're interested in learning more about game development, game design, game testing or any other element of creating great games we offer several courses where you can learn the necessary skills to become part of an exciting and fast growing industry.
Drop along to an open event to learn more.
[openday]
Music | Audio | Film Campus
Windmill Lane Studios
20 Ringsend Road
Games | Animation Campus
The Market Studios
Halston St. & Mary's Lane
© 2023 Pulse College
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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\section{Introduction}
\begin{figure}
\centering
\includegraphics[width=0.75\textwidth]{img/Breathing_interval.pdf}
\caption{
Breathing cycle length versus number of observations. The lengths vary
from a minimum of 0.2 seconds to a maximum of 16 seconds.
}
\label{fig:breathing_interval}
\end{figure}
The human respiratory system may be affected by pathological conditions which
typically alter the patterns of the emitted sound events
\cite{Zak2020}. Recently, due to the COVID-19 pandemic, such kind of
diseases became of primary importance for the public health, being a novel
cause of death for large part of the population \cite{9555564}.
Interestingly, medical acoustics, i.e. the scientific field using audio signal
processing and pattern recognition methods for health diagnosis
\cite{Beach2014}, can play a fundamental role in the development of
user-friendly tools to prevent and limit the spread of respiratory diseases
\cite{Ntalampiras2019}. The ability to create automatic methods to detect
anomalies is useful for both patients and physicians. If the first ones can
take advantage from automatic diagnosis methods, physicians can save time and
minimize potential errors in the performed diagnoses, improving the treatment
path for the patients. Importantly, such solutions may also reduce the overall
pressure on public health systems.
Respiratory/breathing cycle is the result of the combined operation of the
diaphragm and rib muscles enabling inhalation and exhalation, i.e. breathing in
and out. Breathing cycles are evaluated by physicians with the aid of a
stethoscope to spot abnormalities that can represent the onset of a disease.
Unfortunately, stethoscopes can suffer from external noises, sounds emitted by a variety of organs, etc.
A respiratory cycle can be classified as normal or abnormal. Abnormal breathing
cycles can mainly contain two types of abnormal sounds: \textit{crackles}
and \textit{wheezes}. \textit{Crackles} are mostly found in the
inhalation phase, although in some cases, they appear in the exhalation phase~\cite{bohadana2014fundamentals}. They are characterized by a short duration and they
are explosive. \textit{Wheezes} are longer than crackles, around 80-100
milliseconds, and the frequency can be below 100 Hz. They are usually between
100 Hz and 1000 Hz and rarely exceed 1000 Hz~\cite{bohadana2014fundamentals}.
The literature includes various works focusing on the creation of automatic systems that
can detect abnormalities in respiratory cycles. Usually, feature extraction is
performed to capture significant properties of the signals (audio features) in
order to train machine learning models, such as Hidden Markov
Model~\cite{jakovljevic2017hidden}, Support Vector Machine~\cite{serbes2017automated}
or boosted decision tree~\cite{chambres2018automatic}. Recently, deep
learning has been introduced in the audio signal processing community, bringing
advances to medical acoustics as well. In this case, typical features are
matrix-like representations of the signal in a frequency-time space. Usually,
FFT-based representations are extracted and used as an input for training deep
learning networks, such as MLP~\cite{do2021classification},
CNN~\cite{perna2018convolutional,demir2019convolutional,do2021classification}, RNN~\cite{perna2019deep}, and other neural
architectures~\cite{pham2022ensemble,pham2020robust}. In some works, multiple
type of features are used with exceptional results~\cite{do2021classification,
tariq2022featurebased}.
Most of the existing works dealing with the classification of respiratory
cycles assume availability of data representing every potential class, thus modeling the specific task as a multi-class classification problem. In this paper,
instead, we propose an anomaly detection approach. The underlying idea is that
abnormalities in respiratory cycles are difficult to record and can vary
between different patients, due to different ages and/or sexes. To account for
such variability, a sophisticated modeling would be required, which is a hard
requirement given the little amount of available data. Therefore, we decided to
opt for an anomaly detection task that requires fewer data for training.
Interestingly, Variational Autoencoders have been successfully employed for
anomaly/change/novelty detection tasks, such as bird species
\cite{Ntalampiras2021bird}, ultrasounds \cite{9552041}, time series \cite{Matias2021RobustAD}, computer networks
\cite{9527980}, etc.
The main contribution of this article is to investigate how Variational
Autoencoders (VAE) can be effectively employed for detecting anomalies existing
in respiratory sounds. The proposed method achieves state-of-art results, while
only requiring recordings of normal cycles for training, and as such improving
considerably the model's generalization abilities. To distinguish the
abnormalities, a threshold must be set, which can be optimally computed from a
set of abnormalities. The following sections describe the
\begin{inparaenum}[a)]
\item employed dataset of respiratory sounds, \item audio signal preprocessing,
\item VAE's architecture and training procedure, \item experimental set-up and results, as well as \item our conclusions.
\end{inparaenum}
\section{The Dataset of Respiratory Sounds}
\label{sec:dataset}
We used the Respiratory Sound database provided by the 2017 International
Conference on Biomedical and Health Informatics
(ICBHI)~\cite{rocha2019open}. The database consists of 5.5 hours of audio:
6898 breathing cycles from 920 audio recordings of 126 patients. Audio
recordings last from 10 to 90 seconds and the audios include different types of
noises typical of real-world conditions. Moreover, the audio samples were
acquired using various equipments, sampling rates and detection methods. Each
respiratory cycle is categorized into four classes: \textit{crackles},
\textit{wheezes}, \textit{crackles and wheezes}, and \textit{normal}.
The creators of the challenge predefined train and test subsets of data which
permits the reliable comparison between different approaches. Such division is
performed at the level of recordings -- i.e.\ the 920 records from which the
respiratory cycles are extracted. In this way, a patient's observations can be
either in the train set or in the test set, but not in both of them. As such,
any patient-specific bias is eliminated. The direct consequence is that using
the predefined division, the generalization abilities of the models are better
assessed.
The predefined division is shown in the Table~\ref{tab:ICBHI_60-40}. Overall,
the train set accounts for the 60\% of the whole dataset, while the test part
consists of the remaining 40\%. In addition to the predefined division, several
works in literature used random splits defined at the breathing cycle levels.
To compare the proposed method to existing works, we tested our method in this
setting as well. Namely, we used 80\% of the cycles for training and the
remaining 20\% for testing -- see Table~\ref{tab:ICBHI_80-20}.
\begin{table}
\centering
\footnotesize
\caption{ICBHI Dataset 60\%/40\% splitting (statistics at the cycle level). Note that in
the train set abnormal cycles are excluded.}
\begin{tabular}{|c|c|c|c|c|c|} \cline{2-6}\multicolumn{1}{c|}{} &
\textbf{Crackles} & \textbf{Wheezes} &
\textbf{Both} & \textbf{Normal} &
\textbf{Total} \\ \hline \textit{Training} & 962 & 401 & 281 & 1669
& \textbf{3313} \\ \hline \textit{Validation} & 253 & 100 &
89 & 394 & \textbf{829}
\\ \hline \textit{Testing} & 649 & 385 &
143 & 1579 & \textbf{2756} \\ \hline
\textbf{Total} & \textbf{1864} &
\textbf{886} & \textbf{506} &
\textbf{3642} & \textbf{6898} \\
\hline
\end{tabular}
\label{tab:ICBHI_60-40}
\end{table}
\begin{table}
\centering
\footnotesize
\caption{ICBHI Dataset 80\%/20\% splitting. Note that in the train set, abnormal
classes are excluded.}
\begin{tabular}{|c|c|c|c|c|c|} \cline{2-6}\multicolumn{1}{c|}{} &
\textbf{Crackles} & \textbf{Wheezes} &
\textbf{Both} & \textbf{Normal} &
\textbf{Total} \\ \hline \textit{Training} & 1192 & 572 & 325 & 2325
& \textbf{4414} \\ \hline \textit{Validation} & 290 & 144 &
93 & 577 & \textbf{1104}
\\ \hline \textit{Testing} & 382 & 170 &
88 & 740 & \textbf{1380} \\ \hline
\textbf{Total} & \textbf{1864} &
\textbf{886} & \textbf{506} &
\textbf{3642} & \textbf{6898} \\
\hline
\end{tabular}
\label{tab:ICBHI_80-20}
\end{table}
Finally, since the proposed method performs anomaly detection, we grouped all
the anomalies in one class ($anomalies = \{crackles, wheezes, both\}$) and the normal observations
in another class ($normal = \{normal\}$). Moreover, for the sake of evaluating
the generalization abilities of the proposed model, we used a validation set
randomly drawn from the train set. Since the proposed method is weakly
supervised, in the training set we disregarded the abnormal respiratory cycles
and only employed the normal ones. We still used the abnormal cycles in the
validation and test sets following both of the above-mentioned divisions.
\section{Preprocessing of Respiratory Sounds}
All recordings were resampled at 4 KHz since, according to the
literature~\cite{jakovljevic2017hidden, serbes2017automated}, most of the relevant information is below 2
KHz. As such, according to the Nyquist Theorem, 4 KHz is the minimum-sample
rate that allows to reconstruct the important information content.
Subsequently, we extracted each breathing cycle from each audio file using
the annotations available in the dataset; in the example shown in
Table~\ref{tab:annotationsample}, we extracted 9 audio excerpts, disregarding the
remaining non-useful portions of audio. The available audio excerpts varied in
duration, with the average being 2.7 seconds. Fig.~\ref{fig:breathing_interval}
shows the distribution of the obtained audio duration.
\begin{table}[b]
\centering
\footnotesize
\caption{Example of breathing cycles recorded in one audio file. 0 means
``absence'', 1 means ``presence''.}
\begin{tabular}{|c|c|c|c|c|}
\hline
\textbf{ID} & \textbf{Start} & \textbf{End} &
\textbf{Crackles} & \textbf{Wheezes}
\\ \hline
1 & 2.3806 & 5.3323 & 1 & 0 \\ \hline 2 &
5.3323 & 8.2548 & 0 & 0 \\ \hline 3 &
8.2548 & 11.081 & 1 & 1 \\ \hline 4 &
11.081 & 14.226 & 1 & 1 \\ \hline 5 &
14.226 & 17.415 & 1 & 1 \\ \hline 6 &
17.415 & 20.565 & 1 & 1 \\ \hline 7 &
20.565 & 23.681 & 0 & 1 \\ \hline 8 &
23.681 & 26.874 & 0 & 1 \\ \hline 9 &
26.874 & 30 & 0 & 1 \\
\hline
\end{tabular}
\label{tab:annotationsample}
\end{table}
To ease the processing of the audio excerpts, and similarly to previous works,
we used a fixed-length for each excerpt \cite{9412226}. Specifically, respiratory cycles
lasting less than 3 seconds were wrap-padded while those that lasted more 3
seconds were truncated. To represent the audio characteristics of each excerpt and their evolution in
time, we extracted Mel-Frequency Cepstral Coefficients (MFCCs) on
moving windows. MFCCs are widely used in audio signal processing, including
speech emotion recognition~\cite{9373397}, acoustic scene
analysis~\cite{8721379}, music analysis~\cite{simonetta2022context}, and
medical acoustics~\cite{Ntalampiras2020}. First, the audio signal was
converted in a log-amplitude spectrogram using windows lasting 40ms and
overlapping by 50\%; Hamming windowing function was applied before of
computing the FFT-based log-amplitude spectrum. Then, the Mel filter banks are
applied to each spectrogram column in order to extract perceptually relevant
audio characteristics. Finally, the Discrete Cosine Transform (DCT) is used to
compress the extracted information; the first component, which is proven to be
highly correlated with the energy of the signal~\cite{zheng2001comparison}, was
removed to prevent the model of learning information strongly correlated to the recording conditions. As such, 12 MFCCs were considered.
\section{Anomaly Detection Model}
\begin{figure*}[t]
\label{fig:schema}
\includegraphics[width=\textwidth]{img/APPROACH_.pdf}
\caption{Block diagram of the proposed anomaly detection approach. Starting
from the database, a pre-processing phase is performed which is followed by
a feature extraction step. The model is then trained with samples belonging to
the normal class learning how to reconstruct the input, the optimal
hyper-parameters are determined and tested. Finally, the optimal
thresholds
for the validation and test sets are determined.}
\end{figure*}
Variational Autoencoders (VAE) are neural networks that unify variational
inference approaches with autoencoders. Autoencoders are neural networks composed by two parts: the first part, named
\emph{encoder}, learns a mapping from the input sample
$x$ to a latent representation $z$; the
second part, instead, is named \emph{decoder} and learns to map a
point from $z$ to $x$.
In variational inference, instead, a distribution is inferred via
point-estimation methods to approximate Bayesian inference when the number of
parameters or the size of datasets makes the problem intractable with Monte
Carlo methods~\cite{blei2017variational}.
In the case of VAEs, the latent representation $z$ must
satisfy two important properties, i.e. it must be
\begin{inparaenum}[a)]
\item a continuous distribution, and
\item easily samplable.
\end{inparaenum}
Usually, $z$ is modeled as a Gaussian distribution. In this
case, for each sample, the encoder predicts a mean and a variance, defining the
corresponding latent Gaussian distribution. Then, a single point from the
predicted distribution is sampled and passed to the decoder.
Figure~\ref{fig:VAE_structure} depicts a VAE with Gaussian latent distribution.
Compared to basic autoencoders, VAEs allow to draw random samples at inference
time, making them suitable for generation tasks, such as creativity in music
applications~\cite{Yamshchikov2020}. Moreover, they allow to mimic Bayesian
models, which by construction predict distributions. Indeed, when multiple
samples are drawn from $z$, one can analyze a set of outputs
that constitute a distribution by themselves, allowing to analyze the epistemic
certainty measure of the model regarding its own prediction. In other words,
VAEs can be used to help the final user with an estimation of the certainty of
the prediction; for instance, a medical expert could decide if the probability
score predicted by the model should be trusted or not.
As regards to the loss function, we used the sum of Mean Squared Error between the reconstructed matrix and the input, and the Kullback-Leibler Divergence between the unit Gaussian and the latent distribution.
We defined each layer as the sequence of a convolutional layer, a
batch-normalization layer, a ReLU function, and a dropout layer while training.
The encoder is then built from a sequence of such layers, while the decoder is
composed by the corresponding layers with transposed convolutions instead of
simple convolutions. The latent means and standard deviations are computed with
pure convolutional layers. Figure~\ref{fig:schema} illustrates the whole anomaly detection pipeline.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\textwidth]{img/VariationalAutoencoder.drawio.png}
\caption{Structure of the proposed VAE consisting of 4 convolutional layers.}
\label{fig:VAE_structure}
\end{figure}
\subsection{VAE Training}
\begin{table}
\centering
\footnotesize
\caption{Hyper-parameters and the optimal value found for each of them. BH indicates the best hyperparameters while HS indicates the hyperparameter space that has been considered.}
\begin{tabular}{|c|c|c|}
\cline{2-3}\multicolumn{1}{c|}{} & \textbf{BH} & \textbf{HS} \\ \hline
\textbf{Loss function} & MSE + $D_{KL}$ & \{MAE,MAPE,MSE, MSLE\} \\ \hline
\textbf{Optimizer} & Adam & \{SGD, RMSprop, Adam, Adadelta\} \\ \hline
\textbf{Learning rate} & $1 \times 10^{-4}$ & \{$1 \times 10^{-5}$,$1 \times 10^{-4}$,$1 \times 10^{-3}$\}\\ \hline
\textbf{Epochs} & 1000 & - \\ \hline
\textbf{Patience} & 10 epochs & \{5,10,15\} \\ \hline
\textbf{Batch size} & 32 & \{32,64,128\} \\ \hline
\textbf{Activation Functions hidden layer} & ReLU & \{LeakyReLU, ReLU, sigmoid, tanh\} \\ \hline
\textbf{Activation Functions output layer} & Linear & \{linear, sigmoid\} \\ \hline
\textbf{Dropout rate} & 0.3 & \{0.1,0.2,0.3,0.4,0.5\} \\ \hline
\end{tabular}
\label{tab:HP}
\end{table}
During training, we first searched for the optimal hyper-parameters using Bayesian
Optimization method~\cite{brochu2010tutorial} with a Gaussian Process as
surrogate model, 2.6 as exploration-exploitation factor $k$,
and Lower Confidence Bound (LCB) as acquisition function. The hyper-parameters
and their optimal values are shown in Table~\ref{tab:HP}. We then trained the model using the Adam optimization algorithm with a learning
rate equal to $1 \times 10^{-4}$.
Training is performed in a weakly-supervised fashion by using the anomaly
labels of the validation set only. Specifically, the model is trained to
reconstruct samples from the normal class, thus when the input is an excerpt
from the abnormal class, we expect that the model will not be able to
efficiently reconstruct the input. Since the network is trained to minimize the
Mean Squared Error (MSE) between the input and the output, we expect a small
MSE for normal respiratory cycles and larger MSEs for abnormal ones.
Consequently, the MSE computed on the training set observations can be used as
a threshold to spot anomalies.
Since the dataset is not fully balanced, the threshold is chosen so that it
maximizes the balanced accuracy on the validation set. Balanced accuracy,
compared to Matthews Correlation Coefficient \cite{Matthews1975}, allows
for an easy interpretation, being the average between true-positive-rate (TPR)
and true-negative-rate (TNR). In our case, TPR is the rate of correctly
identified anomalies, while TNR is the rate of correctly identified normal
observations~\cite{chicco2021matthews,bekkar2013evaluation}.
\section{Experimental Set-Up and Results}
\begin{table}[t]
\footnotesize
\caption{Results obtained on the testing set using two different thresholds,
one computed on the validation and one computed on the testing set.}
\label{tab:thresholds}
\centering
\begin{tabular}{cc|ccc|}
\cline{3-5}
\multirow{2}{*}{} & & \multicolumn{3}{c|}{Validation threshold} \\ \cline{3-5}
& & \multicolumn{1}{c|}{TPR} & \multicolumn{1}{c|}{TNR} & ACC \\ \hline
\multicolumn{1}{|c|}{\multirow{2}{*}{60\%/40\% split}} &
\begin{tabular}[c]{@{}c@{}}Validation\\ threshold\end{tabular} &
\multicolumn{1}{c|}{0.33} & \multicolumn{1}{c|}{0.80} & 0.57 \\ \cline{2-5}
\multicolumn{1}{|c|}{} &
\begin{tabular}[c]{@{}c@{}}Test\\ threshold\end{tabular} &
\multicolumn{1}{c|}{0.44} & \multicolumn{1}{c|}{0.70} & 0.57 \\ \hline
\multicolumn{1}{|c|}{\multirow{2}{*}{80\%/20\% split}} & \begin{tabular}[c]{@{}c@{}}Validation\\ threshold\end{tabular} &
\multicolumn{1}{c|}{0.48} & \multicolumn{1}{c|}{0.71} & 0.60 \\ \cline{2-5}
\multicolumn{1}{|c|}{} &
\begin{tabular}[c]{@{}c@{}}Test\\ threshold\end{tabular} &
\multicolumn{1}{c|}{0.58} & \multicolumn{1}{c|}{0.61} & 0.60 \\ \hline
\end{tabular}
\end{table}
In the context of the ICBHI challenge, the multi-class accuracy is computed as the
average of TPR for each class. Having combined all the anomalous
classes into one, we calculated TPR for two classes (\emph{anomalies}
and \emph{normals}) and from there, the ICBHI score, which corresponds
to the balanced accuracy. Moreover, we assessed the performance of
the model using ROC curves and AUC values which are well-established figures of
merit in the related literature.
We first observed the distribution of the MSE values in the validation set --
see Fig.~\ref{fig:Validation} -- finding that MSE allowed to partially
separate excerpts coming from the two classes. We also observed the ROC and AUC while
training, discovering that an optimal threshold could successfully separate the
two classes. Fig.~\ref{fig:ROC_VAL} shows the performance of the trained
model on the validation set.
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{img/Validation.pdf}
\caption{
\label{fig:Validation}
Distributions of normal and anomalous samples existing in the validation set. The figure on the
left is related to the division in 60\%/40\%, while the one on the right in 80\%/20\%.
}
\centering
\includegraphics[width=\textwidth]{img/ROC_validation.pdf}
\caption{
\label{fig:ROC_VAL}
ROC and AUC computed on the validation set. The left image is related to the 60\%/40\% division, while the one on the right to the 80\%/20\% one.
}
\end{figure}
We performed the same evaluation on the testing set to assess the
generalization abilities of the model -- see
Fig.~\ref{fig:Testing}~and~\ref{fig:ROC_Testing}. It is particularly
interesting comparing the optimal threshold computed on the validation set and
the corresponding one computed on the testing set. Using the 80\%/20\% division, the
proposed method identified a slightly smaller threshold than using the
60\%/40\% division.
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{img/Testing.pdf}
\caption{
\label{fig:Testing}
Distributions
of normal and anomalous respiratory sounds as regards to the testing set. The left image is related to
the division in 60\%/40\%, while the one on the right to the 80\%/20\% one.
}
\includegraphics[width=\textwidth]{img/ROC_Testing.pdf}
\caption{
\label{fig:ROC_Testing}
ROC and AUC computed on the testing set. The left image is related to the division in 60\%/40\%, while the one on the right in the 80\%/20\% one.
}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{img/Threshold.pdf}
\caption{
MSE thresholds when processing the testing set.
The left image is related to the division in 60\%/40\%, while the one on the right in the 80\%/20\% one.
}
\label{fig:Threshold}
\end{figure}
However, by comparing the results obtained using the two thresholds -- see
Table~\ref{tab:thresholds} -- we observe that the validation threshold is an
effective approximation of the testing one, proving the generalization
abilities of the model.
A fundamental aspect that must be considered when comparing different works is
the data division into training and testing sets. As explained in
Section~\ref{sec:dataset}, the creators of the database offer a
subdivision (60\%/40\%) at the level of audio recordings, so that observations associated with a given patient can be either in train or test set.
The division into 80\%/20\% used by some authors is instead extracted randomly
at the level of the excerpts in which each audio recording is segmented
(respiratory cycles). Unfortunately, such a division may be biased due to
patient dependency. Using such a division, the performance measure increases
essentially for two reasons:
\begin{itemize}
\item there is a greater amount of data in the training phase, and
\item observations of the same patient can be both in training
and in testing.
\end{itemize}
Moreover, the usage of random seeds can generate dataset
division that are particularly successful, thus hiding a pre-bias -- i.e.\
searching good random splits to boost the final scores.
\begin{table}[t]
\footnotesize
\centering
\caption{
ICBHI Challenge results on the detection of crackles
and wheezes (four-class anomaly detection-driven prediction) with the proposed method.
$TPR$ is the true-positive-rate, $TNR$ is the true-negative-rate and
$ACC$ is the balanced accuracy (corresponding to the ICBHI score).
}
\begin{tabular}{|c|c|c|c|c|c|} \hline \textbf{Method} &
$\mathbf{TPR}$ & $\mathbf{TNR}$ &
$\mathbf{ACC}$ & \textbf{Split} &
\textbf{Task}
\\ \hline HMM~\cite{jakovljevic2017hidden} & 0.38 & 0.41
& 0.39 &
\multirow{6}{*}{\textit{60/40}} & \multirow{5}{*}{\textit{4 classes}}
\\ STFT+Wavelet~\cite{serbes2017automated} & 0.78
& 0.20 & 0.49 &
&
\\ Boosted Tree~\cite{chambres2018automatic} & 0.78 & 0.21
& 0.49 &
&
\\ Ensemble DL~\cite{pham2022ensemble} & 0.86
& 0.30 & 0.57
& &
\\ \cline{6-6} \textbf{Proposed Method} & 0.33
& 0.80 & $ 0.57^*$ &
& \textit{2 classes}
\\ \hline LSTM~\cite{perna2019deep} & 0.85 & 0.62
& 0.74 & \multirow{7}{*}{\textit{80/20}} &
\multirow{3}{*}{\textit{4 classes}}
\\ CNN-MoE \& C-RNN~\cite{pham2020robust} & 0.86 & 0.73 & 0.80
& &
\\ \cline{6-6} LSTM~\cite{perna2019deep} & -
& -
& 0.81 &
& \multirow{4}{*}{\textit{2 classes}}
\\ CNN-MoE \&
C-RNN~\cite{pham2020robust} & 0.86 & 0.85 & 0.86 & & \\
\textbf{Proposed Method} & 0.58 & 0.61 &
$0.60^*$ & &
\\ \hline
\end{tabular}
\label{tab:ICBHI_competitors}
\end{table}
Overall, the proposed approach offers performance which is in line with the
state of the art even though it employs respiratory sounds representing only
healthy conditions. Interestingly, such a line of thought addresses the problem
in a more realistic way since it is unreasonable to assume availability of
abnormal respiratory sounds representing the entire gamut of such diseases.
\section{Conclusions}
In this work we presented a framework modeling the MFCCs extracted from
healthy respiratory sounds using a fully convolutional Variational Autoencoder.
To the best of our knowledge, this is the first time that the detection of
respiratory diseases is faced from an anomaly detection perspective.
Interestingly, the proposed model achieved state of the art results in a
patient-independent experimental protocol even though it is only
weakly-supervised.
The small size of the available dataset poses the problem of overfitting and
model generalization abilities. To this end, we employed Variational Inference,
which allows the model to estimate its own epistemic confidence, in addition to
the estimation of the anomaly probability.
In order to improve the performance of the presented anomaly detection system,
data augmentation could be a fundamental addition, as it will provide
additional training samples. Moreover, different architectures could be
tested including networks able to take into account time dependencies, such as
attention-based nets and Recurrent Neural Networks. Finally, multiple features
could be used to improve the reconstruction abilities~\cite{do2021classification,tariq2022featurebased}.
\balance
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\bibliographystyle{splncs04}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section*{SUPPLEMENTAL MATERIAL}
\setcounter{figure}{0}
\renewcommand{\figurename}{Fig.\,S}
\section{Two-dimensional interactions and theoretical predictions}
In our experiment we create an effectively 2D system by means of a tight axial confinement along the $z$-direction. The system is then characterized by the 3D scattering length $a_{3D}$ and, as an additional length scale, the oscillator length $\ell_z=\sqrt{\hbar/m\omega_z}$. Both parameters uniquely determine the effective 2D scattering length $a_{2D}$ of the system. In our measurements $\ell_z$ is constant and we tune $a_{2D}$ by changing $a_{3D}$ via a magnetic Feshbach resonance.
Unlike in 3D, there is a two-body bound state for all interaction strengths in two dimensions. In the BCS limit ($a_{2D} \gg \ell_z$), it is weakly bound and the size of the dimers, which is on the order of $a_{2D}$, is much larger than $\ell_z$. Thus, the dimers
have a 2D character. On the contrary, in the BEC limit the dimers are much smaller than $\ell_z$. They are thus not influenced by the axial confinement, and internally have a 3D character. However, the behavior of the dimers can be described in the 2D framework regardless of their internal structure. In particular, the scattering processes in this system
can be mapped onto those of purely two-dimensional interactions.
We review here the concepts of 2D scattering length and confinement induced two-body bound state.
\textbf{2D scattering length.} The 2D scattering length $a_{2D}$ is defined from low-energy scattering of particles \cite{Bloch2008,Makhalov2014,Levinsen2014}. For this purpose we consider the scattering amplitude of two atoms with relative momentum $\hbar k$ and energy $E_k=\hbar \omega_z/2+\hbar^2k^2/m$ colliding in the continuum above the ground state of the harmonic oscillator in $z$-direction. For sufficiently low $k$ we have $|E_k-\hbar\omega_z/2|\ll \hbar\omega_z$ and higher excitations in the $z$-direction do not affect the collision. This regime is experimentally relevant here for all magnetic fields since $\mu,k_B T\lesssim \hbar\omega_z$. The scattering amplitude for atoms then reads
\begin{align}
\label{a2d1} f(k) = \frac{4\pi}{\sqrt{2\pi} \ell_z/a_{3D}-\ln(\pi k^2\ell_z^2/A)+\mathrm{i} \pi}
\end{align}
with $A=0.905$ \cite{Petrov2001,Bloch2008}. This formula is valid for all values of $\ell_z/a_{3D}$ as long as $k$ is sufficiently low \cite{Bloch2008,Levinsen2014}. In particular, from the definition of the 2D scattering length $a_{2D}$ according to $f(k) \to 4\pi/[-\ln(k^2a_{2D}^2)+\mathrm{i} \pi]$ for $k\to 0$ we deduce
\begin{align}
\label{a2d2} a_{2D} = \ell_z \sqrt{\frac{\pi}{A}} \exp\Bigl(-\sqrt{\frac{\pi}{2}}\frac{\ell_z}{a_{3D}}\Bigr)
\end{align}
for all magnetic field values.
This definition includes the underlying three-dimensionality of the scattering events up to the two-body sector.
In the weak confinement limit ($\ell_z \gg a_{3D}$) realized far on the BEC side, we obtain
\begin{align}
\label{a2d3} f(k) \simeq \tilde{g} := \sqrt{8\pi} \ \frac{a_{3D}}{\ell_z},
\end{align}
which constitutes an energy-independent effective coupling constant. This is the expected low-energy limiting behavior for a short ranged potential whose range $r_e$ is much smaller than $\ell_z$ \cite{Bloch2008}. Expression (\ref{a2d3}) can be used to compare the molecules on the BEC side to two-dimensional bosons with coupling constant $\tilde{g}_b$ by replacing $a_{3D} \to 0.6 a_{3D}$ \cite{Petrov2005} and $m\to 2m$. We then obtain $\tilde{g}_b=0.60$ for $692$G, demonstrating that the fermionic system realizes strongly coupled two-dimensional bosons in this limit. Eq. (\ref{a2d2}) for the 2D scattering length has been employed for the whole crossover in Ref. \cite{Makhalov2014}, where the limit (\ref{a2d3}) provides for the correct bosonic weak-coupling limit.
\textbf{Confinement induced bound state.} The confinement along the $z$-direction induces a two-body bound state with binding energy $\tilde{E}_B$ for all values of the 3D scattering length $a_{3D}$. The corresponding binding energy $\tilde{E}_B$ is found from \cite{Petrov2001,Bloch2008}
\begin{align}
\frac{\ell_z}{a_{3D}} =\int_0^\infty \frac{\mbox{d}u}{\sqrt{4\pi u^3}} \Bigl(1-\frac{e^{-\tilde{E}_Bu/\hbar\omega_z}}{\sqrt{(1-e^{-2u})/2u}}\Bigr).
\label{EqTrans}
\end{align}
From the binding energy an associated length scale $\tilde{a}_{2D}$ can be defined according to $\tilde{E}_B=\hbar^2/m\tilde{a}_{2D}^2$. This length scale can be used to quantify the interactions of the system. However, only far on the BCS side of the crossover, where ${a}_{2D}\gg\ell_z$ and the internal structure of the dimer is two-dimensional, this length scale coincides with the 2D scattering length.
This limit is reached for magnetic fields $\gtrsim 852$G. Far on the BEC side, in contrast, $\tilde{E}_B$ approaches the binding energy of the 3D system since the size of dimer is much smaller than $\ell_z$ and thus the effect of the strong confinement on the dimer vanishes. The difference between $a_{2D}$ and $\tilde{a}_{2D}$ reaches almost three orders of magnitude at $692$G. The interaction strengths at the critical temperature using both definitions are listed in Table I.
Far on the BEC side ($\ln(k_Fa_{2D})\ll-1$), the critical temperature can be computed from the theory of weakly coupled two-dimensional bosons. Based on the corresponding Monte Carlo calculations presented in \cite{Prokofev2001,Prokofev2002} the BKT transition temperature is found \cite{Petrov2003} to be
\begin{align}
\frac{T_c}{T_F} = \frac{1}{2}\Bigl[\ln\Bigl(\frac{C}{4\pi}\ln\Bigl(\frac{4\pi}{k_F^2a_{2D}^2}\Bigr)\Bigr)\Bigr]^{-1}
\end{align}
with $C=380(3)$.
\section{Preparation of the sample}
We start our experimental sequence by transferring a 3D Fermi gas of $^6$Li atoms in states $\ket{1}$ and $\ket{2}$ \cite{Zuern2013} from a magneto-optical trap into an optical dipole trap (ODT). This surfboard-shaped trap has an aspect ratio of $\omega_x$:$\omega_y$:$\omega_z = 1$:$8$:$44$ and is far red detuned ($\lambda = 1064\,$nm) from the optical transition.
The gas is then evaporatively cooled into degeneracy at a magnetic offset field of $795\,$G on the BEC side of the broad Feshbach resonance at $832.2\,$G \cite{Zuern2013}. We therefore obtain a 3D molecular Bose-Einstein condensate (mBEC) consisting of about $10^5$ molecules with negligible thermal fraction. This sample is finally transferred into a standing-wave optical dipole trap (SWT) as illustrated in Fig. 1A in the main text.
The SWT is created by two elliptical focused $1064\,$nm Gaussian beams, which intersect under an angle of $\simeq 14^\circ$. This leads to a standing wave interference pattern where the maxima have a distance of $\simeq4.4\,\mu$m. The ellipticity of the beams is chosen such that the interference maxima have a circular symmetry in the xy-plane. At the position of the SWT, the magnetic offset field has a saddle point. It leads to an additional weak magnetic confinement (anti-confinement) in radial (axial) direction. The measured trap frequency of the magnetic confinement in radial direction is $\omega_{mag}(B) \approx 2\pi \times 0.39 \text{Hz} \, \sqrt{B [\text{G}]}$.
At a magnetic offset field of $795\,$G, the combined trapping frequencies for the central layers of the SWT are given in the main text and lead to an aspect ratio of $\omega_x:\omega_y:\omega_z = 1:0.997:309$.
In order to align the relative position of the atoms in the ODT with one layer of the SWT, we apply a magnetic field gradient
in z-direction, which can shift the atoms up or down in the ODT.
To optimize the fraction of atoms transferred into this single layer, we furthermore decrease the vertical size of the atoms in the ODT by modulating the position of the ODT in the transverse x-direction on a time scale much faster than all trapping frequencies. This creates a time averaged potential where the width of the trap in x-direction is increased by a factor of approximately 5.
In order to further reduce the extension in z-direction of the sample, we ramp to a magnetic offset field of $730\,$G over $600\,$ms. This reduces the repulsive interaction and thus the size of the sample. After the transfer into the SWT, we ramp back to $795\,$G, where we further evaporatively cool the sample by simultaneously applying a magnetic field gradient and lowering the trap depth. This also allows us to control the total number of particles.
In order to access higher temperatures in a controlled fashion, we can then apply a heating procedure. For the lowest three temperatures, we hold the sample in the SWT at $795\,$G for a variable time ($0 \dots 1\,$s) during which it is heated by technical noise. For higher temperatures, we parametrically heat the sample by modulating the depth of the SWT with variable amplitude. After letting the sample equilibrate for $300\,$ms, we ramp the magnetic offset field to the value we want to investigate, where we wait for an additional $20\,$ms before probing the system. All magnetic field ramps are performed with ramp speeds $\lesssim 1.9\,$G/ms.
To ensure adiabaticity of the magnetic field ramps, we compare the temperature of a sample held at an offset field of $732\,$G to the temperature of a sample which was ramped across the Feshbach resonance to $900\,$G and back in the same time. We find that for this ramp speed the two temperatures agree within their uncertainties. All magnetic field ramps are thus adiabatic, and we probe the crossover in an isentropic way.
\section{Distribution of particles in the standing-wave trap}
\label{subsec:tomography}
In order to probe the distribution in the layers of the SWT, we use a radio-frequency tomography technique. We apply a magnetic field gradient along the z-axis to make the
transition frequency $\nu_{\ket{2}\ket{3}}$ between states $\ket{2}$ and $\ket{3}$ spatially dependent on z. The dependence of $\nu_{\ket{2}\ket{3}}$ on the magnetic field is given by $d\nu_{\ket{2}\ket{3}}/dB\simeq 6.3 \,$kHz/G.
We can thus visualize the density distribution by counting the number of transferred atoms as a function of the transition frequency. To minimize the line width of the transition, we need to exclude interaction effects and three-body losses. Hence, we first remove the particles in state $\ket{1}$ by applying a resonant laser pulse for about $10\,\mu$s. To minimize heating and losses, this is done at a magnetic offset field of $1000\,$G, where the atoms are not bound into molecules at our temperatures and interactions are comparatively weak. Although we still observe significant heating, the thermal energy is small compared to the trap depth and it is therefore not expected that particles get transferred between the individual layers.
After a ramp back to $795\,$G, we apply a magnetic field gradient of approximately $70\,$G/cm along the z-axis. We thus achieve a difference of approximately $200\,$Hz in transition frequency between atoms in neighboring layers. We then drive the $\ket{2}$-$\ket{3}$ transition and record the number of transferred particles using state-selective absorption imaging along the y-axis as a function of frequency (see Fig. S\ref{fig:figures1}).
\begin{figure} [!htb]
\centering
\includegraphics [width= 7 cm] {somfig1_004.pdf}
\caption{\textbf{Tomographic measurement of the particle distribution in the standing-wave trap.} Data points represent the number of particles transferred to state $\ket{3}$ as a function of the transition frequency $\nu_{\ket{2}\ket{3}}$.
The central maximum at $81.9591\,$MHz corresponds to atoms in the central layer, the neighboring layers are only slightly populated. The sum of three Gaussian profiles (solid line) is fitted to the data and yields a population of the central layer with approximately $89$\% of the particles.}
\label{fig:figures1}
\end{figure}
The large central maximum at $81.9591\,$MHz in Fig. S\ref{fig:figures1} corresponds to atoms in the central layer. Note that only a fraction of atoms is transferred to state $\ket{3}$, and thus the displayed atom number is considerably lower than the total atom number in the trap. The neighboring layers lie at roughly $81.9589\,$MHz and $81.9593\,$MHz which was confirmed in previous measurements where several layers were filled. By repeating the tomographic measurement, we can assure that the position of the layers is stable within a range of $\pi/8$ over time scales of more than a week.
We fit the distribution shown in Fig. S\ref{fig:figures1} with three Gaussian profiles of the same width and thus estimate the fraction of atoms in the non-central peaks to be $11$\%. Note that this value is a conservative upper bound and overestimates the number of atoms in the non-central layer for two reasons: the magnetic field gradient applied during the measurement tilts the trap and thus removes a large percentage (approximately $25$\%) of all atoms before the transition is driven. Since the central layer is filled with more atoms and to higher energies than the surrounding layers, a greater fraction of atoms will be lost from the central layer. In addition, the atom numbers detected in the non-central peaks are at the detection limit, and are thus influenced by phenomena such as dispersive non-resonant interactions between the imaging light and atoms in state $\ket{2}$.
The phase space density of atoms in the non-central layers is low, and we therefore expect them to follow a thermal distribution. Hence, their influence on the measured condensate fraction, peak condensate density and temperature is negligible. However, the in situ density distribution may be influenced. This is discussed in section \ref{subsec:systematic_errors}.
\section{Influence of the finite aspect ratio}
In contrast to theory, where the dimensionality of a system can easily be set, experimental realizations of low dimensionality will always remain an approximation. For instance, a two-dimensional system can be realized by strongly confining particles in one of the three spatial dimensions. However, there will always be a residual influence of the third dimension. Its magnitude can be determined by comparing the relevant energy scales of the system (the temperature $T$, the chemical potential $\mu$ and dimer binding energy $E_B$) to the axial oscillator energy $\hbar \omega_z$.
For $T, \mu \gtrsim \hbar \omega_z$, particles populate the axially excited trap levels. We ensure the absence of a significant population of these excited levels by performing a measurement which is explained below. This includes the center-of-mass motion of atom pairs. However, depending on the dimer binding energy $E_B$ the internal structure of atom pairs can be three-dimensional. For $E_B\ll \hbar \omega_z$, the internal structure of the pairs is 2D. For $E_B\gg \hbar \omega_z$, the pairs are deeply bound and their internal 3D structure is not resolved. Hence, only for $E_B \approx \hbar \omega_z$ which in our system occurs at an interaction strength $\ln(k_Fa_{2D})\approx 0.5$, the internal structure of the pairs might affect the behavior of the system.
Estimating this effect is complicated: a theoretical treatment beyond the two-body sector is extremely difficult due to strong interactions, and experimental studies would require even larger trap aspect ratios or smaller atom numbers, both of which are currently unfeasible.
We estimate the population of axially excited states due to finite $T$ and $\mu$ by investigating the system's momentum distribution in the axial direction. This measurement is performed similar to the technique described in \cite{Dyke2011} and it relies on the same principles as the previously used band-mapping technique \cite{froehlich2011}.
\begin{figure} [!htb]
\centering
\includegraphics [width= 7 cm] {somfig2_006.pdf}
\caption{\textbf{Measurement of axially excited population.} The axial width $\sigma_z$ is obtained from a Gaussian fit to the density distribution after $3\,$ms time-of-flight. For atom numbers up to approximately $60\,000$,
only the axial ground state of the trap is occupied and $\sigma_z$ is constant. For higher atom numbers, axially excited trap levels become populated and $\sigma_z$ increases. Lines are linear fits to the data, the fit range is indicated by the solid part of each line.}
\label{fig:figures2}
\end{figure}
We release the sample from the SWT, let it expand for a time-of-flight of $3\,$ms, and measure its vertical extension. In the non-interacting limit, atoms in the axial ground state of the trap have a Gaussian wave function in the axial direction. Their axial expansion can then be described by the dispersion of a Gaussian wave packet, which is independent of the number of atoms in the axial ground state. Fig. S\ref{fig:figures2} shows the axial width $\sigma_z$, which is determined from a Gaussian fit to the density distribution,
as a function of the number of prepared atoms per spin state $N$ in the weakly interacting Fermi regime at $1400\,$G and at the coldest attainable temperature. One observes that the axial width is independent of the number of atoms up to approximately $N_{2D}=60\,000$
atoms per spin state. For $N>N_{2D}$ the axial width starts to increase with growing $N$. This change in behavior indicates population of the axially excited states, where the atoms have additional momentum in axial direction. The obtained critical atom number $N_{2D}$ is in agreement with the expectations for a trap with the given aspect ratio and anharmonicities. By keeping the atom number below $N_{2D}$, we can thus ensure that for the investigated temperature only a negligible fraction of atoms populates the axial excited state.
This measurement is performed in the fermionic regime, where due to the Pauli principle multiple occupation of trap levels is suppressed.
All other measurements presented here are performed at lower magnetic fields, closer to the bosonic limit ($\ln(k_F a_{2D}) \rightarrow -\infty$).
The measurement performed at $1400\,$G ($\ln(k_F a_{2D})\gtrsim6$) thus represents an upper bound on the fraction of particles in the axially excited states as for lower fields the atoms tend to form molecules which occupy lower energy states.
To ensure the absence of a significant amount of axial excitations also for lower magnetic fields and higher temperatures, we make use of the relation between the radial size of a harmonically trapped gas and the energy of the highest occupied oscillator level.
We compare the radial size of each sample to that of the measurement at $1400\,$G, where we have excluded significant population of axially excited states. For this comparison, we estimate the radial Fermi radius $r_F$ by the radius where the particle density reaches the noise floor for a particle number $N_{2D}$ at a magnetic field of $1400\,$G. Then we integrate the number of particles outside this radius. In this way, we obtain $N_{ex,r}$, an estimate for the number of particles whose energy is larger than $\hbar \omega_z$. Assuming that all degrees of freedom are equally occupied, the number of particles in axially excited states is given by $N_{ex,z} \simeq N_{ex,r}/2$. We find that $N_{ex,z} \lesssim 1.5\%$ for all interactions strengths below $\ln(k_F a_{2D}) = 3$ and temperatures below $T/T_F \simeq 0.3$. Furthermore, for all $T/T_F \lesssim 0.2$, the fraction of axially excited particles is $N_{ex,z} < 1\%$.
Note that this estimate for the axially excited fraction is conservative, as it assumes constant trap frequencies. It does not take into account that the radial trap frequency decreases by up to about $16$\% when the magnetic field is decreased to $692\,$G.
This leads to an increase in the corresponding Fermi radius by up to approximately $8$\%, and to a corresponding overestimation of $N_{ex,z}$.
We have thus measured the phase diagram of a quasi-2D system with small but finite influence of the third dimension. This influence has to be considered when comparing the experimental data to true 2D predictions.
Recent theoretical work shows that these effects influence the system and can lead to a higher critical temperature \cite{Fischer2014}.
\section{Obtaining the pair momentum distribution}
\label{subsec:momentumdistribution}
In order to probe the momentum distribution of the sample, we use the combination of an interaction quench and a matter wave focusing technique described in detail in the main text and in \cite{Murthy2014}. We turn off the optical SWT, and let the sample expand ballistically in the weak magnetic potential, which is confining in the radial direction. Due to the harmonic shape of this potential, the in situ momentum distribution of the 2D sample is mapped to a spatial distribution after an expansion time of $t_{\text{exp}}=T/4=\pi/(2\omega_{\text{exp}})$. In our case, $\omega_{\text{exp}} = \omega_{\text{mag}} \approx 2\pi \times 10\,$Hz, which leads to $t_{\text{exp}} = 25\,$ms.
To obtain the actual in situ momentum distribution, it is fundamentally important that interactions are negligible while the gas is expanding, since they would result in a redistribution of momentum. Due to its large aspect ratio, our sample expands rapidly in z-direction. Thus, its density drops rapidly and interactions between the expanding particles are quenched. However, for large interactions strengths there is still residual scattering, which can affect the obtained momentum distribution.
We thus minimize the interactions by quickly ramping to the lowest accessible interaction strength on the BEC side ($692\,$G) on a timescale shorter than $125\,\mu$s just before releasing the sample from the SWT. This procedure leads to a negligible scattering rate during the expansion \cite{Murthy2014} and projects correlated pairs of atoms onto tightly bound molecules. The measured momentum distribution thus does not contain the relative momentum of the atoms in a pair, but only the center-of-mass momentum of the pair. Thus, fermionic Cooper pairs and bosonic molecules yield the same signature of enhanced low-momentum density in the pair momentum distribution. As a consequence, information about the Tan contact \cite{Tan2008} cannot be obtained from the pair momentum distribution.
We validate that this method does not alter the temperature of the system by comparing the momentum distributions obtained with and without the interaction quench both below ($732\,$G) and above ($872\,$G) the resonance. In both cases the observed temperature are consistent within the experimental uncertainties.
Furthermore, we confirm that the measured non-Gaussian fraction $N_q/N$ remains unchanged for low magnetic fields where pairs are deeply bound. This is achieved by comparing data with and without the magnetic field ramp. For high magnetic fields however, we cannot directly access $N_q/N$ without projecting correlated pairs into molecules. We thus need to make sure that we probe the properties of the interacting system at the original magnetic field, i.e. that the sample does not adapt to the interaction strengths at lower magnetic fields during and after the ramp.
To estimate this effect, we prepare a sample at $900\,$G at a high temperature where we expect $N_q/N = 0$ and perform the rapid ramp without releasing the sample from the trap. We find that it takes more than $11\,$ms for the momentum distribution to adapt to the interaction strength at the new magnetic field value of $692\,$G and develop a non-Gaussian fraction.
This is two orders of magnitude larger than the timescale of the rapid ramp ($< 125 \,\mu$s). Hence, the influence of the rapid ramp technique on the measured quantities can be neglected.
\section{Absorption imaging parameters and calibrations}
\label{subsec:imaging}
We use absorption imaging along the z-axis (see Fig.1, main text) to determine the integrated column density $n_{2D}\left(x,y\right)$. To obtain a reasonable signal-to-noise ratio we set the imaging intensity to $I \simeq I_{sat}$. Thus, for zero detuning one obtains \cite{Reinaudi2007,Yefsah2011}
\begin{align}
n_{2D}\left(x,y\right) \sigma_0^* &\,=\, - \ln{\frac{I_t\left(x,y\right)}{I_0\left(x,y\right)}}+\frac{I_0\left(x,y\right)-I_t\left(x,y\right)}{I_{sat}^*} \label{eq:imaging}\\
&\,=\, OD\left(x,y\right)+\frac{I_0\left(x,y\right)}{I_{sat}^*}(1-e^{-OD\left(x,y\right)}), \label{eq:imagingrewrite}
\end{align}
where $I_t$ is the transmitted intensity after the atomic cloud, $I_0$ is the initial intensity before the atomic cloud, $I_{sat}^*$ is the effective saturation intensity, $\sigma_0^*$ is the effective scattering cross section and the optical density $OD$ is defined as $OD=-\ln\frac{I_t}{I_0}$.
Due to the uniform intensity distribution of the imaging beam at the position of the atoms, $I_0\left(x,y\right)$ is independent of $x$ and $y$ to a good approximation. In order to calibrate $I_0/I_{sat}^*$, we take several subsequent data sets of a pure atomic sample at $\unit[1400]{G}$ both with our regular imaging settings and with a $\unit[10]{dB}$ attenuated imaging intensity. We then use equation (\ref{eq:imagingrewrite}) and adjust $I_0/I_{sat}^*$ such that the RHS yields the same result both for the regular and the low-intensity setting. Averaging over the data sets then results in $I_0/I_{sat}^*=0.97_{-0.08}^{+0.13}$. The systematic uncertainties are estimated by the minimum and maximum $I_0/I_{sat}^*$ obtained for the individual data sets. This leads to a systematic uncertainty of $_{-4\%}^{+7\%}$ for the atom number $N$ and the peak density $n_{0}$, and a negligible uncertainty for $T$ and $N_q/N$. In addition, we independently measure the power of the imaging beam and thus determine the imaging intensity to be $I \approx I_{sat}$. This justifies using the literature value $\sigma_0$ \cite{gehm2003} for $\sigma_0^*$.
On the BEC-side, the binding energy of the molecules shifts the resonance frequency which leads to a decreased detection efficiency. Using in situ images at different fields, we calibrate this factor for our imaging settings. For magnetic fields below $782\,$G, it deviates from $1$ and reaches $N_{at}/N_{mol}=1.33_{-0.07}^{+0.10}$ at $692\,$G. This leads to a systematic uncertainty of up to $8$\% in atom number $N$ and peak density $n_{0}$ for the affected magnetic fields. More details about the systematic uncertainties can be found in section \ref{subsec:systematic_errors}.
The duration of our imaging pulse is $\tau = \unit[8]{\mu s}$. Due to the small mass of $^6$Li, the atoms are accelerated during the imaging pulse. This results in a Doppler shift of approximately $\unit[10]{MHz}$ at the end of the imaging pulse. To compensate for this effect, we linearly sweep the imaging laser frequency during the pulse.
In order to reduce the shot noise in the absorption images, we use a ten times longer reference pulse. To further improve the quality of the absorption images, we apply a fringe removal algorithm \cite{Ockeloen2010}.
\section{Temperature determination}
\label{subsec:temp_det}
We obtain the temperature $T$ of each sample by fitting a Boltzmann distribution given by
\begin{align}
\tilde{n}(p,t=0) = n(x,t_{\text{exp}}) = A_0 \exp\left(- \frac{M \omega^2_{\text{exp}} x^2}{2 k_B T} \right)
\label{eq:boltzmann}
\end{align}
to the wing of the radial momentum distribution \cite{Murthy2014}. Here, $M$ is the mass of the expanding particle, $k_B$ is Boltzmann's constant, $A_0$ is the amplitude of the fit function, and $\omega_{\text{exp}}$ is the trapping frequency of the radial magnetic confinement the particles expand in. As evident from Fig. 3A, this function describes the data well over a range of more than $50$ pixels. The temperatures used in the main text are the average of approximately $30$ realizations. In order to obtain the degeneracy temperature $T/T_F$, we obtain $T_F$ from the in situ peak density of the sample as described in the main text.
For magnetic fields $\leq 782\,$G, the thermal part of the sample has the momentum distribution of molecules. We verify this in a measurement where we prepare the sample at different magnetic fields and let it evolve in time-of-flight for $3\,$ms before ramping the magnetic field to $527\,$G, where molecules are deeply bound and are thus not detected in absorption imaging resonant with free atoms. We observe that all atoms are bound in molecules after this expansion experiment at all investigated temperatures for magnetic fields $\leq 782\,$G. Thus, they are also bound in the trap. For these magnetic fields,
we thus use the molecule mass in equation (\ref{eq:boltzmann}). For magnetic fields $\geq 892\,$G, the binding energy of the quasi-2D dimer is significantly smaller than the thermal energy in our sample.
The thermal wing thus has the momentum distribution of atoms and we use the atom mass. For intermediate fields, the thermal part crosses over from the molecular to the atomic momentum distribution. Thus, using the atom and molecule mass one obtains an upper and lower bound on the temperature.
We determine the degeneracy temperature at the intermediate fields from a linear interpolation of $T/T_F$ versus $\ln(k_Fa_{2D})$ between $782\,$G and $892\,$G for samples where we applied the same heating parameter. This interpolation is depicted in Fig. S\ref{fig:figures3} for the lowest attainable temperature. The behavior of the interpolated temperature and of the temperatures obtained using the molecular (red) and atomic (green) mass justifies the interpolation procedure. The interpolated temperature always lies between the molecular and the atomic limit. It is close to the molecular limit on the BEC side, and crosses over to the atomic limit as $\ln(k_Fa_{2D})$ increases.
We estimate the systematic error of the interpolated temperature using two assumptions: $T/T_F$ has to be monotonous in $\ln(k_Fa_{2D})$, which yields an interval between the $T/T_F$ of the two points between which we interpolate, and $T/T_F$ has to lie between the values obtained from a fit with the molecule mass and the atom mass (red and green data in Fig. S\ref{fig:figures3}). The overlap of these two intervals is indicated by the gray area in Fig. S\ref{fig:figures3}. It gives an upper bound for the systematic uncertainty of the interpolation result. The statistical errors of the interpolated $T/T_F$ are obtained from the statistical errors at $782\,$G and $892\,$G.
\begin{figure} [!htb]
\centering
\includegraphics [width= 7 cm] {somfig3_002.pdf}
\caption{\textbf{Temperature interpolation in the strongly interacting regime} at the lowest attainable temperature. We obtain the temperature at magnetic fields $782\,\mathrm{G} < B < 892\,\mathrm{G}$, where the thermal part of the gas does not exclusively consist of molecules or atoms, from a linear interpolation between the points at $782\,$G and $892\,$G (solid line). The temperatures obtained with the molecule (atom) mass are depicted as red squares (green triangles). The systematic uncertainty of the interpolated temperatures are indicated by the gray area. Dashed lines are guides to the eye. Each data point is the average of approximately $30$ individual measurements, the error bars denote the standard error of the mean.}
\label{fig:figures3}
\end{figure}
In addition to the temperature determination from the momentum distribution, we also extract the temperature from the in situ data. Applying the local density approximation to the whole cloud, we plot the in situ density as a function of the trapping potential $V(r)$ and fit its wing with a Boltzmann distribution, which in this case is given by
$n(V) = B_0 \exp\left(- \frac{\alpha V}{k_B T} \right)$, where $1 \leq \alpha \leq 2$ takes into account whether the thermal wing consists of atoms or molecules. For the intermediate magnetic fields, we perform an interpolation similar to the one mentioned above to determine $\alpha$.
For magnetic fields up to $812\,$G, the temperatures obtained from both methods agree for low temperatures. For low fields and the highest temperatures, the in situ fit yields larger temperatures. For larger fields, the temperature from the in situ data is systematically lower than the temperature obtained from the momentum distribution.
The reasons for this deviation are still unclear.
Nevertheless, within their errors, the extracted critical temperatures from both methods are still compatible with each other. The values for $T_c/T_F$ obtained with both methods are listed in table \ref{tab:tc}.
\begin{table*} [htb]
\begin{center}
\small
\begin{tabular}{|c| c | c | c |c | c |} \hline
\: $B$ [Gauss] \: &\: $\ell_z/a_{3D}$ \:&\: $\ln(k_Fa_{2D})_{T_c}$ (stat.)(sys.)\:& \: $\ln(k_F\tilde{a}_{2D})_{T_c}$ (stat.)(sys.)\:&\: $T_c/T_F$ (stat.)(sys.) \:&\: $(T_c/T_F)_{\text{in situ}}$ (stat.)\: \\
\hline
& & & & & \\
692 & \,\,7.11 &-\,7.30 \,(4) $\left(_{-5}^{+4}\right)$ &-\,0.96 \,(4)$\left(_{-3}^{+2}\right)$ & 0.089 \, (15) $\left(_{-13}^{+14}\right)$ & 0.090 \, (13) \\ [5pt]
732 & \,\,3.98 &-\,3.42 \,(2) $\left(_{-6}^{+4}\right)$ &-\,0.45 \,(2)$\left(_{-5}^{+3}\right)$ & 0.100 \, (22) $\left(_{-15}^{+17}\right)$ & 0.099 \, (27) \\ [5pt]
782 & \,\,1.55 &-\,0.59 \,(1) $\left(_{-7}^{+4}\right)$ &\,\,\,0.20 \,(1)$\left(_{-6}^{+4}\right)$ & 0.129 \, (35) $\left(_{-18}^{+24}\right)$ & 0.112 \, (44) \\ [5pt]
812 & \,\,0.55 &\,\,\,0.57 \,(1) $\left(_{-7}^{+2}\right)$ &\,\,\,0.79 \,(2)$\left(_{-7}^{+2}\right)$ & 0.146 \, (25) $\left(_{-23}^{+50}\right)$ & 0.146 \, (21) \\ [5pt]
832 & \,\,0 &\,\,\,1.23 \,(1) $\left(_{-8}^{+2}\right)$ &\,\,\,1.33 \,(1)$\left(_{-8}^{+2}\right)$ & 0.167 \, (39) $\left( _{-34}^{+48}\right)$ & 0.122 (103) \\ [5pt]
852 & -0.46 &\,\,\,1.72 \,(1) $\left(_{-9}^{+2}\right)$ &\,\,\,1.76 \,(1)$\left(_{-9}^{+2}\right)$ & 0.167 \, (27) $\left( _{-22}^{+42}\right)$ & 0.122 \, (60) \\ [5pt]
\hline
\end{tabular}
\caption{\textbf{Measured critical temperatures.}
The measured critical temperatures $T_c/T_F$ are given with their respective statistical and systematic errors as a function of the magnetic offset field $B$, $\ell_z/a_{3D}$, and the 2D interaction parameter $\ln(k_Fa_{2D})$ at $T_c/T_F$. In addition, also the alternative 2D interaction parameter $\ln(k_F\tilde{a}_{2D})_{T_c}$ obtained with equation (\ref{EqTrans}) and the critical temperature $(T_c/T_F)_{\text{in situ}}$ obtained from the in situ temperature fit are given.}
\label{tab:tc}
\end{center}
\vskip -0.5cm
\end {table*}
\section{Systematic uncertainties}
\label{subsec:systematic_errors}
The errors given in the main text are the statistical errors of our measurements. In addition, systematic uncertainties arise due to uncertainties in the following quantities:
\textbf{Imaging intensity $\boldsymbol{I_0/I_{sat}}$:} The measured atom density depends on the intensity $I_0$ of the imaging beam (see equation (\ref{eq:imagingrewrite})). In our experiments we use an imaging intensity of $I_0/I_{sat}^*=0.97_{-0.08}^{+0.13}$ (see section \ref{subsec:imaging}). This leads to an uncertainty in the peak density $n_{0}$ and thus we obtain $T_F$ $_{-4\%}^{+7\%}$ and $k_F$ $_{-2\%}^{+3.5\%}$.
\textbf{Atoms in non-central pancakes:} We obtain an upper bound of $11\%$ for the fraction of atoms in the non-central layer of the SWT (see section \ref{subsec:tomography}). To estimate their influence on the measured peak density $n_{0}$, we assume that their temperature is not affected by the evaporative cooling in the SWT because of the small atom number. However, it is affected by the heating procedure where the depth of the trap is modulated. The minimum temperature of these atoms is thus about $100\,$nK, the temperature at the transfer into the SWT. Assuming a thermal Boltzmann gas, we calculate the density of $5500$ atoms in the two non-central layers to be $n_{0,\text{nonc.}} = 0.14\,$atoms$/\mu$m$^2$ \cite{Ketterle2008}. This leads to an overestimation of $T_F$ by $5\%$ for the lowest magnetic fields, and $19\%$ at the highest magnetic fields, where $n_{0}$ is smaller due to the fermionic character of the sample. Analogously, $k_F$ is overestimated by $2.5\%$ to $9.5\%$.
\textbf{Reduced absorption cross section of molecules:} A finite molecular binding energy leads to a shift of the optical transition frequency of the dimers and results in a reduced absorption cross section. By rescaling the obtained images for the lowest three magnetic fields ($692\,$G, $732\,$G and $782\,$G), we compensate for this effect (see section \ref{subsec:imaging}). The uncertainty in the rescaling factor leads to an uncertainty in the peak density $n_{0}$ and thus $T_F$ which is smaller than $8\%$. The corresponding uncertainty in $k_F$ is smaller than $4\%$.
\textbf{Magnification of the imaging system:} We calibrate the magnification of the imaging system using Kapitza-Dirac scattering of atoms on an optical standing wave potential with known periodicity. The resulting uncertainty in the magnification is approximately $3\%$. This uncertainty quadratically enters the temperature obtained from the fit with equation (\ref{eq:boltzmann}). Thus, $T$ and $T/T_F$ have a relative systematic uncertainty of approximately $6\%$.
\textbf{Trap frequency of the expansion potential:} We measure the trap frequency of the expansion potential $\omega_{\text{exp}}$ with a relative uncertainty of approximately $3\%$. This leads to an uncertainty of $6\%$ in $T$ (see equation (\ref*{eq:boltzmann})).
\textbf{Expansion time:} The measured momentum distribution depends on the expansion time. The extracted temperature has a maximum for $t_{\text{exp}}=T/4=\pi/(2\omega_{\text{exp}})$. Deviations from the ideal expansion time lead to a systematic underestimation of $T$ by approximately $5$\%.
\textbf{Fitted region for temperature determination:} The temperature determined from the fit to the tail of the momentum distribution weakly depends on the region included in the fit. This leads to a relative uncertainty of $7$\% in $T$ for low temperatures and up to $13$\% at the highest investigated temperatures. The same effect also leads to a systematic uncertainty in the determined nonthermal fraction $N_q/N$. The absolute uncertainty in $N_q/N$ ranges from $0.06$ at low temperatures to about $0.02$ at high temperatures.
\textbf{Temperature interpolation ($\boldsymbol{812}\,$G - $\boldsymbol{852}\,$G):} For the magnetic fields $812\,$G, $832\,$G and $852\,$G, the reduced temperature $T/T_F$ is determined by interpolation because here the sample consists of a mixture of molecules and atoms (see section \ref{subsec:temp_det}). The systematic uncertainties are estimated as detailed in section \ref{subsec:temp_det}.
The obtained relative uncertainties are largest for low temperatures. They are usually on the order of $10$\%-$25$\% and range up to $50$\% for few individual values. For the critical temperature $T_c$, the total systematic uncertainties including those from temperature interpolation are listed in table \ref{tab:tc}.
\textbf{Resulting total systematic uncertainties:} Assuming a Gaussian distribution of the previously mentioned $8$ independent error sources, we calculate the total relative systematic uncertainty of $T/T_F$.
For magnetic fields where the temperature is not interpolated, one obtains $T/T_F$$_{-13\%}^{+15\%}$ for low temperatures and low magnetic fields to $T/T_F$$_{-15\%}^{+28\%}$ at high temperatures and high magnetic fields. At the critical temperature, one obtains approximately $T_c/T_F$$_{-15\%}^{+17\%}$.
At the magnetic fields where the temperature is determined by interpolation, the systematic uncertainties are significantly bigger and one obtains approximately $T_c/T_F$$_{-20\%}^{+30\%}$ at the critical temperature.
The systematic uncertainty in $\ln(k_F a_{2D})$ is dominated by the uncertainties in $k_F$. We thus neglect uncertainties in $a_{2D}$. The total absolute systematic uncertainty then lies between $\ln(k_F a_{2D})$$_{-0.05}^{+0.04}$ for low magnetic fields and $\ln(k_F a_{2D})$$_{-0.10}^{+0.02}$ for high magnetic fields.
The systematic uncertainties of $T/T_F$ and $\ln(k_F a_{2D})$ at the critical temperature $T_c$ are listed in table \ref{tab:tc}.
\textbf{Temperature determination from in situ profiles:} As mentioned in section \ref{subsec:temp_det}, we additionally determine the temperature from the in situ profiles for comparison. For magnetic fields $\ge 832\,$G, these temperatures are systematically smaller than those determined from the momentum distribution. The corresponding values for $(T_c/T_F)_{\text{in situ}}$ are listed in table \ref{tab:tc}.
\end{document}
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\section{Introduction\label{sec:intro}}
It is always a challenge to accurately estimate the column density of the galactic foreground interstellar medium in the direction of extragalactic sources. It is also one of the important parameters when calculating the physical parameters of gamma-ray burst (GRB) host galaxies. We started an investigation of the infrared sky brightness towards GRBs using \textit{AKARI} Far-Infrared Surveyor (\textit{AKARI} FIS) of \cite{kawada2007} all-sky maps
\footnote{The sky maps can be retrieved from the data archive web site: \textit{http://www.ir.isas.jaxa.jp/ASTRO-F/Observation/}}.
\cite{doi2015}.
GRBs are the most energetic explosions in the Universe. A massive star undergoes core collapse, or a double neutron star or a neutron star and a black hole binary merges \cite{woosley2006}. X-ray and optical afterglows can outshine the brightest quasars. The redshift distribution of Swift GRBs shows that these objects may provide information up to high z values on: galaxy evolution, star formation history, intergalactic medium, see eg. \citet{gomboc2012}.
Most of the known physical parameters of the GRB and the GRB host galaxy are calculated from the afterglow. An estimate on the galactic foreground hydrogen column density towards the GRB is part of the calculations. It is based on galaxy counts and HI \citep{burstein1982}, HI surveys eg. the LAB survey \citet{kalberla2005}, extinction maps calculated from infrared surveys \citet{schlegel1998} and \citet{schlafly2011}, or from spectroscopic measurements and colors of nearby Galactic stars.
\section{Analysis of the \textit{AKARI} FIS All Sky Survey images \label{sec:2}}
\citet{doi2012}, and recently \citet{doi2015} have processed full sky images of the \textit{AKARI} FIS at 65$\mu$m, 90$\mu$m, 140$\mu$m and 160$\mu $m. The images achieve a detection limit of $< 10$\,MJysr$^{-1}$ with absolute and relative photometric accuracies of $< 20$\%. The spatial resolution of the survey is 1$'$. We substracted 30 by 30$\square '$ images centered on 283 GRBs with known redshifts used by \citet{horvath2014} in their analysis (see references therein). We selected 30 images for a test of foreground FIR emission, these were GRBs with associated FIR extragalactic sources \citet{toth2016} and GRBs associated to the large-scale Universal structure Hercules Corona Borealis Great Wall at a redshift of $z \approx 2$ by \citet{horvath2014}.
The color temperature maps of the large grain emission were estimated using the 90$\mu$m, 140$\mu$m and 160$\mu$m images. The maps were convolved to a 2$'$ resolution and, for each pixel, the spectral energy distribution (SED) was fitted with B$_\nu$(T$_{dust}$)$\nu^\beta$ with a fixed $\beta$=2.0 spectral index. The column densities averaged over a 2$'$ beam were calculated using the following equation with the intensity and temperature values from the SED fits:
\begin{equation}
N(H)=\frac{2I_\nu}{B_\nu(T)\kappa\mu m_H}
\end{equation}
We used $\mu$=2.33 for the particle mass per hydrogen molecule and a dust opacity $\kappa$ obtained from the formula 0.1cm$^2$/g ($\nu$/1000 GHz)$^\beta$.
\section{FIR foreground\label{sec:3}}
\subsection{The foreground galaxy of GRB\,060117\label{sec:31}}
ISM rich galaxies may have a size 2 times larger than their apparent optical size when measured from HI 21 or FIR data. We looked for GRBs with associated \textit{AKARI} galaxies from \citet{toth2016} and selected the 5 closest associations.
The 90 $\mu$m images have both a high spatial resolution $93\times 64 \square "$ and a high enough sensitivity to detect galaxies. One of the fields, the one centered on GRB\,0601175 showed a foreground galaxy 2-3 times more extended in FIR than it's NIR size of 0.8$'$ by 0.2$'$ \citet{skrutskie2006}.
GRB\,060117 is a "long" type GRB with duration of 25\,s \citep{campana2006} at a photometric redshift of $z=0.98\pm0.24$ \citep{xiao2011}.
The bright FIR object is 2MFGC\,16496 a flat galaxy as appears in 2MASS images \citep{mitronova2004} at $z\approx0.0042$ \citep{jones2009}, i.e. it is clearly a foreground object. In optical and NIR images the foreground galaxy is relatively far from the GRB. Its galactic disk however is rather extended and apparently increases the foreground FIR sky brightness towards the GRB by approximately 1\,MJysr$^{-1}$. That emission by 2MFGC\,16496 may mislead us estimating the galactic FIR foreground, unless it is carefully subtracted. See Figure~\ref{fig:pkasfig1} for the \textit{AKARI} FIS 90\,$\mu$m image of the $30\times 30 \square '$ surroundings of GRB\,060117. The lowest contour is set at 3 times the standard deviation over the minimum surface brightness in the field.
\begin{figure}[h]
\centering
\includegraphics[width=80mm]{toth_fig1.eps}
\caption{\textit{AKARI} FIS 90\,$\mu$m image of GRB\,060117. Black cross indicates the GRB's position. The contour levels are at 2, 3.1 and 4\,MJysr$^{-1}$. The foreground galaxy west of the position of GRB\,060117 is 2MFGC\,16496. The \textit{AKARI} 90\,$\mu$m beam size is indicated in the lower right corner as a white ellipse.\label{fig:pkasfig1}}
\vspace{5mm}
\end{figure}
\subsection{Structure of the Galactic foreground of GRBs in the Hercules Corona Borealis Great Wall\label{sec:32}}
\citet{horvath2013} discovered a concentration of $1.6<z<2.1$ GRBs in the Hercules-Corona Borealis region. Detailed statistical tests by \citet{horvath2014} indicate a significant clustering of those GRBs, that is also called as "the Hercules Corona Borealis Great Wall". This huge structure lies ten times farther away than the Sloan Great Wall \citep{gott2005}. The size of the structure defined by these GRBs is about 2000-3000\,Mpc, or more than six times the size of the Sloan Great Wall or more than twice the size of the Huge Large Quasar Group \citep{cloves2013}.
We investigated the structure of galactic foreground ISM of 24 GRBs all belong to the Hercules Corona Borealis Great Wall. A constant color temperature in the line of sight was estimated pixel-by-pixel using \textit{AKARI} FIS all sky survey 90, 140, 160\,$\mu$m images. We assumed an emissivity of $\beta =2$. The distribution of the hydrogen column density $N($H$)$ was derived, as described in Section~\ref{sec:2}. The galactic foreground cirrus structures show a fluctuation on 3-4$'$ scale, sometimes with small chains of knots in the whole column density range.
In order to test the accuracy of our column density estimates based on lower angular resolution data, we calculated the averages $\bar{N}($H$)_6$ in a radius of 3$'$ and $\bar{N}($H$)_{30}$ for the $0.5 \times 0.5 \square '$ surroundings of the GRBs.
We compared the calculated column density averages with the central column density ($N($H$)_c$) value towards the GRB. A linear correlation was found for the 24 tested directions with a relatively large scatter. The linear correlation coefficients were 0.57 and 0.28 for the $\bar{N}($H$)_6$ vs. $N($H$)_c$ and the $N($H$)_{30}$ vs. $N($H$)_c$, respectively.
In as many as 50\% of the directions the $\bar{N}($H$)_6 - N($H$)_c$ difference was over $\pm$30\% of $N($H$)_c$ GRBs, and for 40\% the $\bar{N}($H$)_{30}$ average was more then 100\% off.
The \textit{Planck Space Telescope} \citep{tauber2010} observed the sky in 9 frequency bands covering 30 - 857\,GHz. The \textit{Planck} images at 545 and 857\,GHz (550\,$\mu $m and 350\,$\mu $m respectively) have a spatial resolution of approximately 5$'$ \citep{planck2011}, similarly to IRIS \citep{miville2005}, the 100\,$\mu $m calibrated \textit{IRAS} images.
We compared the \textit{AKARI} based column density estimates with estimates derived from IRIS 100$\,\mu m$, \textit{Planck} 857\,GHz and 575\,GHz images, and in general a good correlatio nwas found. A more detailed analysis including the use of the DustEM model \citep{compiegne2011} will be given elsewhere.
\subsection{Galactic foreground and the hydrogen column density of the GRB host galaxies\label{sec:33}}
We estimated the effect of the galactic foreground correction on the calculated hydrogen column density of the GRB host galaxy $N($H$)_{host}$.
We downloaded spectra from the \textit{Swift-XRT} GRB Catalogue\footnote{\url{http://www.swift.ac.uk/xrt_live_cat/} maintained by the UK Swift Science Data Centre (UKSSDC)}, and analyzed those with Xspec\footnote{Xspec is part of the HEASOFT Software package of NASA's High Energy Astrophysics Science Archive Research Center (HEASARC), available at \url{http://heasarc.gsfc.nasa.gov/lheasoft/download.html}} \citep{arnaud1996}. The \textit{Swift-XRT} spectral data provided by the UKSSDC is calibrated and has the appropriate format for Xspec. We used exactly the same model as in the automatic analysis of the UKSSDC \citep{evans2009}. Each spectra was fitted with an absorbed power law with two absorbing components. The first component takes the Galactic foreground into consideration, and it is held fixed during a fit, while the second component gives the absorption due to the excess hydrogen column which is determined by the fitting.
We selected test GRBs with a range of X-ray flux, at different galactic latitudes (that means varying $N($H$)_{foregr}$), and with a range of the initial values of $N($H$)_{host}$. We altered the $N($H$)_{foregr}$ foreground column density ($\pm 50$\%) and recalculated $N($H$)_{host}$. A 50\% increase or decrease of the assumed $N($H$)_{foregr}$ resulted in 15 to 35\% change of $N($H$)_{host}$. We consider that as a non-negligible difference.
\section{Conclusions}
Our tests indicate that a careful examination of the FIR foreground may in one hand reveal foreground FIR objects, on the other hand a high resolution mapping of the galactic cirrus foreground may significantly increase the accuracy of the estimation of foreground extinction. \textit{AKARI} FIS sky survey images are the proper data for that foreground analysis, that may serve as a basis for a recalculation of GRB host parameters.
\acknowledgments
This research is based on observations with AKARI, a JAXA project with the participation of ESA. This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester. This research was supported by OTKA grants NN111016 and K101393 and JSPS KAKENHI Grant 25247016.
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Holly James es una cantante de música Dance-Pop-House nacida en Londres, Reino Unido.
Biografía
Nació en Londres en el 1981 y desde muy pequeña deseó actuar en musicales o en el mundo de la música. Fue bailarina y cantante, y ahora triunfa en el mundo de la música Pop en UK. Fundó su propio grupo con tres amigas más, llamado Tymes 4, con el que viajaron por toda Inglaterra cantando en eventos y fiestas. Después, estuvo trabajando en el Mcdonald's de Oxford Circus, durante 3 años.
En el año 2001 le surgió la oportunidad de su vida. Conoció al DJ Jason Nevins, con quién grabó un sencillo titulado "I'm In Heaven" en el 2001. El sencillo fue publicado, a través de Polydor en el verano de aquel mismo año, llegando al Número 1 en el UK Dance/Club Charts, y al Número 8 en el UK Singles Chart. El sencillo fue un éxito en las Listas de Ventas en UK.
Su primer Hit-Single "I'm In Heaven" fue publicado en el verano del 2003 en Europa, disfrutando de gran éxito. Más tarde saldría en Australia, China y Japón, con gran éxito.
Después de este Hit, en el año 2003 sacaría al mercado su segundo Maxi-Single, titulado "Touch It", junto con el grupo de música House "Lee Cabrera". El sencillo fue publicado a finales del 2003 por la discográfica independiente inglesa Tommy Boy Records UK. El sencillo coincidió con el mes de las fechas de salida de los sencillos de Sophie Ellis-Bextor - "Mixed Up World", Kylie Minogue - "Slow" o Sugababes - "Hole In The Head", entre otros, situándose en el Top 10 del Reino Unido. Pero, el sencillo de Holly no alcanzó el Top 10, quedándose en el #58 en el UK Singles Chart, y en el #24 en el UK Dance/Club Chart.
Su primer álbum, Miscellaneous, iba a ser publicado en diciembre del 2003 a través de la discográfica Tommy Boy Records UK, pero debido al fracaso de su segundo sencillo en el Reino Unido y Europa, nunca publicaron el primer álbum de Holly James.
Actualmente interpreta a Carmen en Fame, The Musical en UK.
Discografía
Álbumes
Miscellaneous (2003) - Álbum no publicado.
Sencillos
I'm In Heaven Feat. Jason Nevins & U.K.N.Y. - #8 UK (2001).
Touch It Feat. Lee Cabrera - #58 UK (2003).
Enlaces externos
Página Web Oficial de la Cantante: - Actualmente Cerrada.
Cantantes femeninas del Reino Unido
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\section{Introduction}
The idea of dark photon \cite{okun1982,holdom1986,fh1991} has been studied by many theorists and experimentalists. Usually, the interactions of the dark photon to the visible matter are assumed from an abelian kinetic mixing between the standard model (SM) $U(1)_Y^{}$ gauge group and a dark $U(1)_X^{}$ gauge group. This $U(1)$ kinetic mixing can not be sizable due to the constraints from different experiments including low energy colliders \cite{Merkel:2014avp,Lees:2014xha,Kou:2018nap}, meson decays \cite{Pospelov:2008zw,Batley:2015lha,Banerjee:2018vgk}, beam dump experiments \cite{Riordan:1987aw,Blumlein:2013cua,Blumlein:2011mv} and high-energy colliders \cite{Curtin:2014cca,Anelli:2015pba,Ilten:2016tkc,Blinov:2017dtk,Aaij:2019bvg,Sirunyan:2019wqq}. However, it is not obvious to explain the smallness of a renormalizable $U(1)$ kinetic mixing.
In this paper we shall consider the non-abelian kinetic mixing to realize another possibility that the dark photon originates from a dark $SU(2)_X^{}$ gauge group and hence its couplings to the matter do not appear at renormalizable level \cite{ccf2009,bcn2016,ahopr2017}. In the presence of the non-abelian kinetic mixing between the dark $SU(2)_X^{}$ gauge group and the SM $SU(2)_L^{}\times U(1)_Y^{}$ gauge groups, one of the dark gauge bosons becomes a dark photon, while the others keep stable to form a dark matter particle. This scenario predicts a nearly degenerate mass spectrum of dark photon and dark matter.
\section{Dark sector}
Besides the dark gauge fields $X^{1,2,3}_\mu$, the $SU(2)_X^{}$ dark sector only contains a dark Higgs doublet $\chi=(\chi_1^{},\chi_2^{})^T_{}$. The full Lagrangian of the dark sector is
\begin{eqnarray}
\mathcal{L}_{\textrm{Dark}}^{} &=& -\frac{1}{4} X_{\mu\nu}^{a} X_{}^{a\mu\nu} + \left(D_\mu^{}\chi \right)^\dagger_{} \left(D^\mu_{}\chi \right) - \mu_\chi^2 \chi^\dagger_{}\chi \nonumber\\
&&- \lambda_\chi^{} \left(\chi^\dagger_{}\chi\right)^2_{} ~~\textrm{with}\nonumber\\
&& X_{\mu\nu}^a = \partial_\mu^{} X_\nu^{a} - \partial_\nu^{} X_\mu^{a} + i g_X^{} \varepsilon^{abc}_{} X^{b}_{\mu} X^{c}_{\nu}\,,\nonumber\\
&&D_\mu^{} \chi = \left(\partial_\mu^{} - i g_X^{} \frac{\tau_a^{}}{2} X^a_\mu\right) \chi\,,
\end{eqnarray}
where $g_X^{}$ is the $SU(2)_X^{}$ gauge coupling. At the renormalizable level, the dark sector can interact with the SM only through the Higgs portal as below,
\begin{eqnarray}
\mathcal{L}_{\textrm{int}}^{} &=&-\lambda_{\phi\chi}^{} \phi^\dagger_{}\phi \chi^\dagger_{}\chi\,.
\end{eqnarray}
Here $\phi=(\phi_{}^{0},\phi_{}^{-})^T_{}$ is the SM Higgs doublet for the spontaneous electroweak symmetry breaking, i.e.
\begin{eqnarray}
\phi \Rightarrow \phi =\left[\begin{array}{c} \frac{1}{\sqrt{2}}\left(v_\phi^{} + h_\phi^{}\right)\\
[2mm]
0
\end{array}\right] \,.
\end{eqnarray}
When the dark Higgs doublet $\chi$ develops its vacuum expectation value (VEV) for spontaneously breaking the $SU(2)_X^{}$ symmetry, i.e.
\begin{eqnarray}
\chi \Rightarrow \chi=\left[\begin{array}{c} \frac{1}{\sqrt{2}}\left(v_\chi^{} + h_\chi^{}\right)\\
[2mm]
0
\end{array}\right] \,,
\end{eqnarray}
the dark gauge fields $X^{1,2,3}_\mu$ can obtain their masses,
\begin{eqnarray}
\mathcal{L}_{\textrm{Dark}}^{} &\supset&\frac{1}{8} g_X^2 v_\chi^2 \sum_{a=1,2,3}^{}X^{a}_{\mu} X^{a\mu}_{} \nonumber\\
&=&m_X^2 X^{+}_\mu X^{-\mu}_{} + \frac{1}{2} m_X^2 X^{3}_{\mu} X^{3\mu}_{}~~\textrm{with}\nonumber\\
&&X^{\pm}_\mu = \frac{1}{\sqrt{2}}\left(X^{1}_\mu \mp i X^{2}_\mu\right)\,,~~m_X^2 = \frac{g_X^2 v_\chi^2 }{4} \,.
\end{eqnarray}
In the following, we shall conveniently refer to $X^{\pm}_\mu$ as the charged dark gauge bosons and $X^{3}_\mu$ as the neutral dark gauge boson although these dark gauge bosons do not carry the ordinary electric charge.
\section{Non--abelian kinetic mixing}
So far the dark gauge bosons $X^{\pm}_\mu$ and $X^{3}_\mu$ are exactly degenerate. This feature can be modified if the dark $SU(2)_X^{}$ gauge fields $X^{1,2,3}_{\mu}$ have a non-abelian kinetic mixing with the SM $SU(2)_L^{}\times U(1)_Y^{}$ gauge fields $W^{1,2,3}_\mu, B^{}_{\mu}$, i.e.
\begin{eqnarray}
\label{effective}
\mathcal{L} &\supset& -\frac{1}{\Lambda^2_{6}} \chi^\dagger_{} \frac{\tau_a^{}}{2} \chi X^{a}_{\mu\nu} B^{\mu\nu}_{}-\frac{1}{\Lambda^4_{8}} \chi^\dagger_{} \frac{\tau_a^{}}{2} \chi X^{a}_{\mu\nu} \phi^\dagger_{} \frac{\tau_b^{}}{2} \phi W^{b\mu\nu}_{}\nonumber\\
&&\textrm{with}~~ B^{}_{\mu\nu}=\partial_\mu^{} B_\nu^{} - \partial_\nu^{} B_\mu^{}\,,\nonumber\\
&&\quad\quad \quad \!W^{a}_{\mu\nu}=\partial_\mu^{} W_\nu^{a}
- \partial_\nu^{} W_\mu^{a} + i g \varepsilon^{abc}_{} W^{b}_\mu W^{c}_\nu\,.
\end{eqnarray}
Here $g$ is the $SU(2)_L^{}$ gauge coupling. As for the other non-abelian kinetic mixing, they can be forbidden by an additional $U(1)_D^{}$ global symmetry under which the dark Higgs doublet $\chi$ is non-trivial, i.e.
\begin{eqnarray}
\mathcal{L} ~/ \!\!\!\!\!\supset -\frac{1}{\tilde{\Lambda}^2_{6}} \chi^\dagger_{} \frac{\tau_a^{}}{2} \tilde{\chi} X^{a}_{\mu\nu} B^{\mu\nu}_{}-\frac{1}{\tilde{\Lambda}^4_{8}} \chi^\dagger_{} \frac{\tau_a^{}}{2} \tilde{\chi} X^{a}_{\mu\nu} \phi^\dagger_{} \frac{\tau_b^{}}{2} \phi W^{b\mu\nu}_{}\,.
\end{eqnarray}
The non-abelian kinetic mixing (\ref{effective}) can lead to the kinetic mixing between the dark $X^{3}_\mu$ and the SM $W^{3}_\mu,B_\mu^{}$, i.e.
\begin{eqnarray}
\mathcal{L}&\supset & -\frac{\epsilon_B^{}}{2} \tilde{X}^{3}_{\mu\nu} B^{\mu\nu}_{}-\frac{\epsilon_W^{}}{2} \tilde{X}^{3}_{\mu\nu} \tilde{W}^{3\mu\nu}_{}~~\textrm{with}\nonumber\\
&&\epsilon_B^{} \equiv \frac{v_\chi^2 }{2\Lambda^2_{6}}\,,~~ \tilde{X}^{3}_{\mu\nu}=\partial_\mu^{} X_\nu^{3} - \partial_\nu^{} X_\mu^{3}\,,\nonumber\\
&&\epsilon_W^{} \equiv \frac{v_\chi^2 v_\phi^2}{8\Lambda^4_{8}}\,,~~ \tilde{W}^{3}_{\mu\nu}=\partial_\mu^{} W_\nu^{3} - \partial_\nu^{} W_\mu^{3}\,,
\end{eqnarray}
after the dark and electroweak symmetry breaking, i.e.
\begin{eqnarray}
SU(2)_X^{}\times U(1)_D^{} &\stackrel{\langle\chi\rangle}{\longrightarrow}& U(1)_{DE}^{}\,,\\
SU(2)_L^{}\times U(1)_Y^{}&\stackrel{\langle\phi\rangle}{\longrightarrow}& U(1)_{em}^{}\,.
\end{eqnarray}
Here $U(1)_{DE}^{}$ is an unbroken global symmetry to define a dark electric charge. Therefore, the neutral dark gauge boson $X^{3}_\mu$ indeed becomes a dark photon so that it can be distinguished from the charged dark gauge bosons $X^{\pm}_{\mu}$. More details will be given shortly.
The effective operators in Eq. (\ref{effective}) can be induced by integrating out certain scalar(s) crossing the dark and SM sectors. For example, an $[SU(2)_X^{}]$-doublet scalar with $U(1)_Y^{}$ charge can mediate the $SU(2)_X^{} \times U(1)_Y^{}$ mixing, an $[SU(2)_X^{}]$-doublet and $[SU(2)_L^{}]$-triplet scalar without $U(1)_Y^{}$ charge can mediate the $SU(2)_X^{} \times SU(2)_L^{}$ mixing, while an $[SU(2)_X^{}\times SU(2)_L^{}]$-bidoublet scalar with $U(1)_Y^{}$ charge or an $[SU(2)_X^{}]$-doublet and $[SU(2)_L^{}]$-triplet with $U(1)_Y^{}$ charge can mediate both of the $SU(2)_X^{} \times U(1)_Y^{}$ and $SU(2)_X^{} \times SU(2)_L^{}$ mixing. It should be noted these crossing scalars are also non-trivial under the $U(1)_D^{}$ global symmetry to forbid some unexpected couplings.
\subsection{The case only with $SU(2)_X^{}\times U(1)_Y^{}$ kinetic mixing }
In this case, the crossing scalar could be an $[SU(2)_X^{}]$-doublet scalar with a non-trivial $U(1)_Y^{}$ hypercharge,
\begin{eqnarray}
\eta=\left[\begin{array}{c} \eta_1^{Y_\eta^{}}\\
[3mm]
\eta_2^{Y_\eta^{}}
\end{array}\right] \,,
\end{eqnarray}
with $Y_\eta^{}$ being the $U(1)_Y^{}$ hypercharge. The crossing scalar $\eta$ also carries a $U(1)_D^{}$ charge as the same with the dark Higgs doublet $\chi$. The following $\chi-\eta$ couplings,
\begin{eqnarray}
\mathcal{L}\supset -\lambda^{}_{1} \chi^\dagger_{}\eta \eta^\dagger_{}\chi - \lambda^{}_{2} \tilde{\chi}^\dagger_{}\eta \eta^\dagger_{}\tilde{\chi} \,,
\end{eqnarray}
then can break the mass degeneracy between the $\eta_{1,2}^{}$ components, i.e.
\begin{eqnarray}
\mathcal{L}&\supset & -m_{\eta_1^{}}^2 \eta^{\ast}_{1}\eta^{}_{1} -m_{\eta_2^{}}^2 \eta^{\ast}_{2}\eta^{}_{2} ~~\textrm{with}\nonumber\\
&&
m_{\eta_1^{}}^2 - m_{\eta_2^{}}^2 = \frac{1}{2}\left(\lambda_{1}^{} - \lambda_{2}^{}\right) v_\chi^2\,.
\end{eqnarray}
We then can compute the $X^{3}_{}-B$ kinetic mixing at one-loop level \cite{fltw2011},
\begin{eqnarray}
\label{kinetic}
\epsilon_B^{} = \frac{g_X^{} g' Y_\eta^{} }{96\pi^2_{}} \ln \left(\frac{m_{\eta_1^{}}^2}{m_{\eta_2^{}}^2} \right)\,,
\end{eqnarray}
with $g'$ being the $U(1)_Y^{}$ gauge coupling.
We should keep in mind the crossing scalar $\eta$ carries a non-zero electric charge so that it should be heavy enough and decay before the BBN. Otherwise, it should have been ruled out experimentally. This means we need other $[SU(2)_X^{}]$-singlet mediator scalar(s) to make the crossing scalar $\eta$ unstable. For example, the mediator scalar can be a singly charged dilepton scalar $\xi$ with the following couplings,
\begin{eqnarray}
\mathcal{L} \supset - f_\xi^{}\xi \bar{l}_L^c i\tau_2^{} l_L^{} -\lambda_{\xi\eta\chi}^{} \xi^2_{}\eta^\dagger_{}\chi +\textrm{H.c.}\,.
\end{eqnarray}
We can also consider a doubly charged dilepton scalar $\zeta$ to be the mediator scalar. The related couplings include
\begin{eqnarray}
\mathcal{L} \supset - f_\zeta^{} \zeta \bar{e}_R^c e_R^{} -\mu_{\zeta\eta\chi}^{} \zeta \eta^\dagger_{}\chi +\textrm{H.c.}\,.
\end{eqnarray}
In this case, the formula (\ref{kinetic}) should be modified because of the $\eta_{1,2}^{}-\zeta$ mixing. Alternatively, the crossing scalar $\eta$ could be colored when the mediator is composed of certain diquark or leptoquark scalar(s). These mediator scalars may be useful in the generation of radiative neutrino masses \cite{zee1986,babu1988,bl2001}.
We would like to emphasise that the crossing scalars do not have the following gauge-invariant terms because of the $U(1)_D^{}$ global symmetry,
\begin{eqnarray}
\label{stable}
\mathcal{L} ~/\!\!\!\!\!\supset - \lambda^{}_{12} \chi^\dagger_{}\eta \eta^\dagger_{}\tilde{\chi} - \tilde{\lambda}_{\xi\eta\chi}^{} \xi^2_{}\eta^\dagger_{}\tilde{\chi}- \tilde{\lambda}_{\zeta\eta\chi}^{} \zeta \eta^\dagger_{}\tilde{\chi} +\textrm{H.c.}\,.
\end{eqnarray}
In the absence of the above couplings, the $\eta^{}_{2}$ component of the crossing scalar $\eta$ can only decay into a charged dark gauge boson $X^{\pm}_{\mu}$ with a real or virtual $\eta^{}_{1}$ component. Therefore, the charged dark gauge bosons $X^{\pm}_{\mu}$ can keep stable.
\subsection{The case only with $SU(2)_X^{} \times SU(2)_L^{}$ kinetic mixing}
In this case, the crossing scalar could be an $[SU(2)_X^{}]$-doublet and $[SU(2)_L^{}]$-triplet scalar without $U(1)_Y^{}$ charge,
\begin{eqnarray}
\Delta =\left[\begin{array}{c}\delta_1^{}\\
[2mm]
\delta_2^{} \end{array}\right]~~\textrm{with}~~\delta_{i}^{}=\left[\begin{array}{cc}\frac{1}{\sqrt{2}}\delta^{0}_{i}&\delta^{+}_{i2}\\
[2mm]
\delta^{-}_{i1}& -\frac{1}{\sqrt{2}}\delta^{0}_{i} \end{array}\right]\,.
\end{eqnarray}
For simplicity we do not explicitly calculate the $X^3_{}-W^3_{}$ kinetic mixing, which should depend on the mass difference among the components of the crossing scalar $\Delta$, like the formula (\ref{kinetic}).
Note the crossing scalar $\Delta$ can decay through the following quartic coupling,
\begin{eqnarray}
\mathcal{L}\supset -\lambda_{\chi \phi \Delta}^{}\left[\tilde{\phi}^T_{} i\tau_2^{} \left( \chi^\dagger_{} \Delta\right) \phi +\textrm{H.c.}\right]\,,
\end{eqnarray}
without resorting to additional mediator scalars. Clearly, this crossing scalar can pick up a VEV $\langle\Delta\rangle$ after the dark and electroweak symmetry breaking.
\begin{eqnarray}
\langle\Delta\rangle =\left[\begin{array}{c}\langle\delta_1^{}\rangle\\
[2mm]
0 \end{array}\right]~~\textrm{with}~~\langle\delta_{1}^{}\rangle=\left[\begin{array}{cc}\frac{1}{\sqrt{2}}\langle\delta^{0}_{1}\rangle&0\\
[2mm]
0& -\frac{1}{\sqrt{2}}\langle\delta^{0}_{1}\rangle \end{array}\right]\,.
\end{eqnarray}
\subsection{The case with both $SU(2)_X^{} \times SU(2)_L^{}$ and $SU(2)_X^{} \times U(1)_Y^{}$ kinetic mixing}
In this case, the crossing scalar could be an $[SU(2)_X^{}\times SU(2)_L^{}]$-bidoublet scalar with $U(1)_Y^{}$ charge,
\begin{eqnarray}
\Sigma =\left[\begin{array}{c}\sigma_1^{}\\
[2mm]
\sigma_2^{} \end{array}\right]~~\textrm{with}~~\sigma_{i}^{}=\left[\begin{array}{c} \sigma^{0}_{i}\\
[2mm]
\sigma^{-}_{i} \end{array}\right]\,.
\end{eqnarray}
or an $[SU(2)_X^{}]$-doublet and $[SU(2)_L^{}]$-triplet with $U(1)_Y^{}$ charge,
\begin{eqnarray}
\Omega =\left[\begin{array}{c}\omega_1^{}\\
[2mm]
\omega_2^{} \end{array}\right]~~\textrm{with}~~\omega_{i}^{}=\left[\begin{array}{cc}\frac{1}{\sqrt{2}}\omega^{+}_{i}&\omega^{++}_{i}\\
[2mm]
\omega^{0}_{i}& -\frac{1}{\sqrt{2}}\omega^{+}_{i} \end{array}\right]\,.
\end{eqnarray}
The formula on the $X^3_{}-B$ and $X^3_{}-W^3_{}$ kinetic mixing should be similar to Eq. (\ref{kinetic}). For simplicity we do not show the calculations.
The crossing scalars $\Sigma$ and $\Omega$ have the following couplings to the dark Higgs doublet $\chi$ and the SM Higgs doublet $\phi$, i.e.
\begin{eqnarray}
\mathcal{L}&\supset& - \mu_{\chi \phi \Sigma}^{} \left( \chi^\dagger_{} \Sigma \phi +\textrm{H.c.} \right) \nonumber\\
&&- \lambda_{\chi \phi \Sigma}^{} \left[\phi^T_{}i \tau_2^{} \left(\chi^\dagger_{} \Omega\right) \phi +\textrm{H.c.} \right] \,.
\end{eqnarray}
Therefore, we need not introduce additional mediator scalars in this case.
\section{Dark photon}
We can remove the loop-induced kinetic mixing between the neutral dark gauge field $X^{3}_\mu$ and the SM gauge fields $B_\mu^{},W^{3}_\mu$ by a non-orthogonal rotation as below,
\begin{eqnarray}
X_\mu^{3}&=&\frac{1}{\sqrt{1-\epsilon^2_{B}- \epsilon^2_{W}}} X'^{3}_\mu\,,\nonumber\\
B_\mu^{}&=&B'^{}_\mu - \frac{\epsilon_B^{}}{\sqrt{1-\epsilon^2_{B} -\epsilon^2_{W} }} X'^{3}_\mu\,,\nonumber\\
W_\mu^{3}&=&W'^{3}_\mu - \frac{\epsilon_W^{}}{\sqrt{1 - \epsilon^2_{B} -\epsilon^2_{W}}} X'^{3}_\mu\,.
\end{eqnarray}
In the basis of $B'^{}_\mu, W'^{3}_\mu, X'^{3}_\mu$, we can obtain the massless photon and the $Z$ boson by
\begin{eqnarray}
A_\mu^{} &=& W'^3_\mu \sin \theta_W^{} + B'^{}_\mu \cos \theta_W^{}\,,\nonumber\\
Z_\mu^{} & =& W'^3_\mu \cos\theta_W^{} - B'^{}_\mu \sin\theta_W^{}\,,\end{eqnarray}
where $\theta_W^{}$ is the Weinberg angle as usual, i.e. $\tan \theta_W^{} = g'/g $. For the following demonstration, we would like to conveniently denote the parameters,
\begin{eqnarray}
\epsilon_Z^{}&=&\epsilon_W^{} \cos\theta_W^{} - \epsilon_B^{} \sin\theta_W^{} \,,\nonumber\\
\epsilon_A^{} &=& \epsilon_W^{} \sin\theta_W^{} + \epsilon_B^{} \cos\theta_W^{} \,,\nonumber\\
\bar{\epsilon}_Z^{} &=& \frac{\epsilon_Z^{}}{\sqrt{1 - \epsilon^2_{Z} -\epsilon^2_{A}}}\,,~~ \bar{\epsilon}_A^{} =\frac{\epsilon_A^{}}{\sqrt{1 - \epsilon^2_{Z} -\epsilon^2_{A}}} \,.
\end{eqnarray}
Now the $Z$ boson has a mass mixing with the $X'^3_{}$ boson, i.e.
\begin{eqnarray}
\mathcal{L} &\supset& \frac{g^2_{} v_\phi^2 }{8 \cos^2_{}\theta_W^{}}\left(Z_\mu^{} - \bar{\epsilon}_Z^{} A'^{}_\mu \right) \left(Z^\mu_{} - \bar{\epsilon}_Z^{} A'^\mu_{}\right) \nonumber\\
&&+ \frac{g_X^2 v_\chi^2}{8(1-\epsilon^2_{Z}- \epsilon^2_{A})} A'^{}_{\mu} A'^{\mu}_{} \nonumber\\
&=&\frac{1}{2}m_Z^2 Z_\mu^{} Z^\mu_{} + \frac{1}{2} m_{A'}^2 A'^{}_\mu A'^\mu_{} + m_{ZA'}^2 Z_\mu^{} A'^\mu_{}~~\textrm{with}\nonumber\\
&&m_Z^2 = \frac{g^2_{} v_\phi^2 }{4 \cos^2_{}\theta_W^{}} \,,\nonumber\\
&&m_{A'}^2 = \frac{g_X^2 v_\chi^2}{4(1-\epsilon^2_{Z}- \epsilon^2_{A})}+ \frac{ g^2_{} v_\phi^2 \epsilon_Z^2}{4 \cos^2_{}\theta_W^{} \left(1 - \epsilon^2_{Z} -\epsilon^2_{A}\right)}\,,\nonumber\\
&&m_{ZA'}^2= - \frac{ g^2_{} v_\phi^2 \epsilon_Z^{} }{4 \cos^2_{}\theta_W^{} \sqrt{1 - \epsilon^2_{Z} -\epsilon^2_{A}}} \,.
\end{eqnarray}
Here we have identified the $X'^3$ boson as the $A'$ boson. The mass eigenstates of the $Z$ and $A'$ bosons should be
\begin{eqnarray}
\hat{Z}_\mu^{} &=&Z_\mu^{} \cos \zeta - A'^{}_\mu \sin \zeta ~~\textrm{with}\nonumber\\
&&m^2_{\hat{Z}}= \frac{m_{Z}^2 + m_{A'}^2 - \sqrt{(m_Z^2 - m_{A'}^2)^2_{} + 4 m_{ZA'}^4 }}{2}\,,\nonumber\\
\hat{A}'^{}_\mu &=& Z_\mu^{} \sin \zeta + A'^{}_\mu \cos \zeta ~~\textrm{with}\nonumber\\
&&m^2_{\hat{A}'}= \frac{m_{Z}^2 + m_{A'}^2 + \sqrt{(m_Z^2 - m_{A'}^2)^2_{} + 4 m_{ZA'}^4 }}{2}\,,\nonumber\\
&&
\end{eqnarray}
where the rotation angle is determined by
\begin{eqnarray}
\tan 2 \zeta = \frac{2 m_{ZA'}^2 }{m_{A'}^2 - m_{Z}^2 }\,.
\end{eqnarray}
The $A^{}_\mu$, $\hat{Z}_\mu^{}$ and $\hat{A}'^{}_\mu$ bosons couple to the SM fermions,
\begin{eqnarray}
\mathcal{L}&\supset& \frac{g}{c_W^{}}J_{NC}^\mu \left(Z_\mu^{} - \bar{\epsilon}_Z^{} A'^{}_\mu \right) + eJ_{em}^{\mu} \left(A_\mu^{} - \bar{\epsilon}_A^{} A'^{}_\mu\right) \nonumber\\
&=& \left[\frac{g \left(\sin\zeta - \bar{\epsilon}_Z^{}\cos\zeta \right) }{c_W^{}}J_{NC}^\mu - e \bar{\epsilon}_A^{} \cos\zeta J_{em}^\mu \right]\hat{A}'^{}_\mu\nonumber\\
&&+\left[\frac{g \left(\cos\zeta + \bar{\epsilon}_Z^{}\sin\zeta \right) }{c_W^{}}J_{NC}^\mu + e \bar{\epsilon}_A^{} \sin\zeta J_{em}^\mu \right]\hat{Z}_\mu^{}\nonumber\\
&&+eJ_{em}^{\mu} A_\mu^{}\,, \end{eqnarray}
with $J_{em,NC}^\mu$ being the SM electromagnetic and neutral currents,
\begin{eqnarray}
J_{em}^\mu &=& -\frac{1}{3}\bar{d}\gamma^\mu_{} d + \frac{2}{3} \bar{u}\gamma^\mu_{} u - \bar{e} \gamma^\mu_{} e \,,\nonumber\\
J_{NC}^\mu &=& \frac{1}{4}\bar{d}\gamma^\mu_{} \left[\left(-1+\frac{4}{3}\sin^2_{}\theta_W^{} \right)+ \gamma_5^{}\right]d \nonumber\\
&&+ \frac{1}{4}\bar{u}\gamma^\mu_{} \left[\left(1-\frac{8}{3}\sin^2_{}\theta_W^{} \right)- \gamma_5^{}\right]u \nonumber\\
&&+ \frac{1}{4}\bar{e} \gamma^\mu_{}\left[\left(-1+4 \sin^2_{}\theta_W^{}\right) + \gamma_5^{}\right] e \nonumber\\
&&+ \frac{1}{4}\bar{\nu}\gamma^\mu_{} \left[ 1-\gamma_5^{}\right]\nu \,.
\end{eqnarray}
For a small $\epsilon_Z^{}$, the mass eigenstates $\hat{Z},\hat{A}'$ can well approximate to the $Z,A'$ bosons, and their mass eigenvalues $m_{\hat{Z}, \hat{A}'}^2$ can also be simplified, i.e.
\begin{eqnarray}
\hat{Z}_\mu^{} &\simeq &Z_\mu^{} ~~\textrm{with}~~m^2_{\hat{Z}}\simeq \frac{g^2_{} v_\phi^2 }{4 \cos\theta_W^2}\,,\nonumber\\
\hat{A}'^{}_\mu &\simeq & A'^{}_\mu ~~\textrm{with}~~m^2_{\hat{A}'}\simeq \frac{g_X^2 v_\chi^2}{4(1-\epsilon_Z^2- \epsilon_A^2)}\,.
\end{eqnarray}
In this limiting case, the $\hat{A}'$ boson only interacts with the neutral currents $J_{NC}^\mu$ at the second order in $\epsilon_Z^{}$. The fashion is similar to the interaction between the $\hat{Z}$ boson and the electromagnetic currents $J^\mu_{em}$. We hence can simply take
\begin{eqnarray}
\mathcal{L}\supset eJ_{em}^{\mu} A_\mu^{} + \frac{g}{c_W^{}}J_{NC}^\mu \hat{Z}_\mu^{} -\bar{\epsilon}_A^{} e J_{em}^\mu \hat{A}'^{}_\mu + \mathcal{O}(\epsilon^2_{Z})\,.
\end{eqnarray}
In this sense, we can name the $A'$ boson as a dark photon. The experimental constraints and implications on the dark photon $A'$ have been studied in a lot of literatures. (For a recent review, see for example \cite{Fabbrichesi:2020wbt})
Actually, the present model could provide a purely dark photon without any couplings to the neutral currents if the $SU(2)_X^{}\times U(1)_Y^{}$ mixing parameter $\epsilon_B^{}$ and the $SU(2)_X^{}\times SU(2)_L^{}$ mixing parameter $\epsilon_W^{}$ are chosen to be
\begin{eqnarray}
\frac{\epsilon_W^{}}{\epsilon_B^{}}=\tan \theta_W^{} \Rightarrow \epsilon_Z^{} =0\,.
\end{eqnarray}
Since the parameters $\epsilon_{B,W}^{}$ are both induced at loop level, they should be quite small. Therefore, the dark photon $A'$ can be only slightly heavier than the charged dark gauge bosons $X^\pm_\mu$, i.e.
\begin{eqnarray}
m_{A'}^2 - m_{X^\pm_{}}^2 &\simeq &m_X^2 \left( \frac{1}{1-\epsilon_Z^2- \epsilon_A^2}-1\right)\nonumber\\
& \simeq & m_X^2\left(\epsilon_Z^2 + \epsilon_A^2 \right) \ll m_X^2\,.
\end{eqnarray}
\section{Dark matter}
Although the neutral dark gauge boson $X^3_\mu$ now has become the dark photon coupling to the SM electromagnetic and neutral currents, its charged partners $X^{\pm}_{\mu}$ can still keep stable because of the $U(1)_D^{}$ global symmetry. We expect the $X^{\pm}_{\mu}$ bosons to account for the dark matter relic. For this purpose, the annihilations of the $X^{\pm}_{\mu}$ bosons into some light species should arrive at a desired strength. This can be achieved, thanks to the dark Higgs boson $h_\chi^{}$, i.e.
\begin{eqnarray}
\mathcal{L}_{\textrm{Dark}}^{}\supset \frac{m_X^2}{v_\chi^{}} h_\chi^{} X^{+}_\mu X^{-\mu} + \frac{m_X^2}{v_\chi^2}h_\chi^{2} X^{+}_\mu X^{-\mu}\,.
\end{eqnarray}
For example, if the dark Higgs boson $h_\chi^{}$ is lighter than the charged dark gauge bosons $X^\pm_\mu$, we can expect the annihilations $X^{+}_{} X^{-}_{} \rightarrow h_\chi^{} h_\chi^{} $. Subsequently, the dark Higgs boson can mostly decay into the mediator scalars including dileptons, diquarks and/or leptoquarks. In the case the dark Higgs boson is too heavy to appear in the final states, it can mediate a $s$-channel annihilation of the charged dark gauge bosons into the mediator scalars. In any of these cases, the mediator scalars eventually can decay into the SM fermion pairs. This means we even can expect a leptophilic dark matter if the mediator is dominated by certain dilepton scalar(s) \cite{gos2009,ghsz2009}. Alternatively, we can resort to the coupling between the dark Higgs boson and the SM Higgs boson for the required annihilations.
The dark $SU(2)_X^{}$ gauge bosons as the dark matter have been studied in other scenario. For example, all of three dark $SU(2)_X^{}$ gauge bosons can keep stable if we do not consider the high dimensional operators (\ref{effective}). The dark $SU(2)_X^{}$ gauge bosons thus can provide three degenerate dark matter particles \cite{hambye2009}. Alternatively, like the 't Hooft-Polyakov monopole model \cite{thooft1974,polyakov1974}, the $SU(2)_X^{}$ gauge symmetry can be spontaneously broken down to a dark $U(1)_X^{}$ gauge symmetry by a real dark Higgs triplet. In consequence, the charged dark gauge boson $X^{\pm}_\mu$ can be a stable dark matter particle while the neutral dark gauge boson $X^3_\mu$ does keep massless and does not couple to the SM electromagnetic and neutral currents \cite{bkp2014}.
\section{Conclusion}
In this paper we have demonstrated an interesting scenario where the dark photon and the dark matter particle can have a nearly degenerate mass spectrum. Specifically we consider the non-abelian kinetic mixing between the dark $SU(2)_X^{}$ gauge group and the SM $SU(2)_L^{}\times U(1)_Y^{}$ gauge groups. Because of these non-abelian kinetic mixing, one of the dark gauge bosons becomes the dark photon, meanwhile, the others keep stable to serve as the dark matter particle. The non-abelian kinetic mixing also makes the dark photon slightly heavier than the dark matter. The quasi-degenerate dark photon and dark matter could be tested if the dark photon and the dark matter are both observed in the future.
\textbf{Acknowledgement}: This work is supported by the Natural Science Foundation of the Higher Education Institutions of Jiangsu Province under grant No.\,22KJB140007.
|
{
"redpajama_set_name": "RedPajamaArXiv"
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{"url":"http:\/\/devmaster.net\/posts\/16342\/get-list-of-files-with-dds","text":"0\n101 May 04, 2009 at 09:37\n\nHey, I\u2019ve been looking into populating an array of LPCSTR\u2019s with the file paths of all the .dds files in a particular folder (Media\/LightProbes\/). I\u2019ve seen a number of methods that use the boost libraries file system but I\u2019d rather stay away from boost for now.\nThe basic process would go like this\n\nscan the folder\nint numDDSFiles = 0;\nfor (each file, check dds)\nnumDDSFiles++;\n\nLPCSTR DirectoryArrays[numDDSFiles];\nfor(int i = 0; i < numDDSFiles; i++)\n{\nDirectoryArrays = \u201cMedia\/LightProbes\/\u201d + fileName + \u201cdds\u201d;\n}\n\ntrouble is I don\u2019t know how to check how many files there are in a particular file that have a certain file extension, or how to get what that file name is. I\u2019d be happy to use STL or Win32 library and am open to other methods, but as stated above would rather not use boost.\n\nThanks =)\n\n#### 24 Replies\n\n0\n102 May 04, 2009 at 14:47\n0\n101 May 05, 2009 at 04:45\n\nhey monjardin, thanks for the reply. I\u2019m having some trouble understanding how it works, and how to implement, I understand that under his given framework, I would add a filter for *.dds files where he says this:\n\nYou can implement simple filtering by filename (or extension) like this:\n\nbool CheckUseFile(LPCTSTR, WIN32_FIND_DATA* pwfd)\n{\nreturn ::PathMatchSpec(pwfd->cFileName, _T(\u201c*.jpg\u201d));\n}\n\nalthough much of the code that is presented in the download file seems very complicated and I\u2019m finding it difficult to understand, is there not an easier way to go about it?\n\nThanks\n\n0\n139 May 05, 2009 at 05:05\n\nDo you just want to find files in a single folder (no sub-folders), or an entire tree (folders, its sub-folders, and their sub-folders, etc\u2026)? For an entire tree, monjardin\u2019s link may be useful, but if you just want to look at a single folder, you can just use FindFirstFile and FindNextFile. You\u2019d pass FindFirstFile a search string like \u201cc:\\myproject\\*.dds\u201d, then run a loop calling FindNextFile until it returns false.\n\n0\n101 May 05, 2009 at 05:20\n\nI\u2019m just looking to find the files in a single folder, \u201cMedia\/LightProbes\/\u201d and get just the names of the files with a .dds file extension.\n\n0\n139 May 05, 2009 at 05:26\n\nThen FindFirstFile and FindNextFile should be effective and pretty simple to use.\n\n0\n101 May 05, 2009 at 06:10\n\nk thanks :)\n\n0\n101 May 05, 2009 at 07:35\n\nOk, I\u2019ve tried implementing a simple one that just gets all the files in a directory, here\u2019s the code\n\nHANDLE hHandle;\nWIN32_FIND_DATA* pData = NULL;\nstd::vector<LPCSTR> Directories;\n\nhHandle = FindFirstFile(\"Media\/LightProbes\/\", pData);\n\nBOOL bMoreFiles = TRUE;\nwhile(bMoreFiles == TRUE)\n{\nbMoreFiles = FindNextFile(hHandle, pData);\nif(bMoreFiles == TRUE)\n{\nDirectories.push_back(LPCSTR(pData->cFileName));\n}\n}\n\n\nbut when I check the size of Directories and it is 0, I have checked in the file Media\/LightProbes and there are 9 files there, what\u2019s happening?\n\n0\n101 May 05, 2009 at 08:14\n\nTry this \u2026\n\nHANDLE hHandle;\nWIN32_FIND_DATA* pData = NULL;\nstd::vector<LPCSTR> Directories;\n\nhHandle = FindFirstFile(\"Media\/LightProbes\/*\", pData);\n\nBOOL bMoreFiles = TRUE;\nwhile(bMoreFiles == TRUE)\n{\nbMoreFiles = FindNextFile(hHandle, pData);\nif(bMoreFiles == TRUE)\n{\nDirectories.push_back(LPCSTR(pData->cFileName));\n}\n}\n\n\nEdit: Heh \u2026 the code block appears to get proper confused by a \/* in a string \u2026\n\n0\n101 May 05, 2009 at 08:55\n\nHey, thanks for the reply, I\u2019m getting a compile error that directs me to this in vector\n\n{ \/\/ allocate array with _Capacity elements\n_Myfirst = 0, _Mylast = 0, _Myend = 0;\nif (_Capacity == 0)\nreturn (false);\nelse if (max_size() < _Capacity)\n_Xlen(); \/\/ result too long\nelse\n{ \/\/ nonempty array, allocate storage\n_Myfirst = this->_Alval.allocate(_Capacity);\n_Mylast = _Myfirst;\n_Myend = _Myfirst + _Capacity;\n}\nreturn (true);\n}\n\n0\n101 May 05, 2009 at 09:34\n\nAnd what does the error tell you?\n\nThere is another bug though. An LPCSTR is a Long Pointer to a Constant STRing. Basically after you call FindNextFile the pData->cFileName is most probably de-allocated and freed. To fill your structure you should be copying the string to your own allocation (Which you, of course, need to free after you\u2019ve used it).\n\nSo try this code \u2026\n\nHANDLE hHandle;\nWIN32_FIND_DATA* pData = NULL;\nstd::vector<TCHAR*> Directories;\n\nhHandle = FindFirstFile(\"Media\/LightProbes\/*\", pData);\n\nBOOL bMoreFiles = TRUE;\nwhile(bMoreFiles == TRUE)\n{\nbMoreFiles = FindNextFile(hHandle, pData);\nif(bMoreFiles == TRUE)\n{\nconst size_t strLen = _tcslen( pData->cFileName ) + 1;\nTCHAR* pNewStr = new TCHAR[strLen];\n_tcscpy_s( pNewStr, strLen, pData->cFileName );\nDirectories.push_back( pNewStr );\n}\n}\n\n0\n101 May 05, 2009 at 09:41\n\nHey, what includes and lib\u2019s do I need for _tcslen and _tcscpy_s?\n\n0\n101 May 05, 2009 at 09:46\n\nyou need to #include <tchar.h>\n\n0\n101 May 05, 2009 at 09:47\n\nIts worth noting that all its doing is modifying the code so that it will work with UTF-8 or UTF-16 strings. You can easily change TCHAR to char and change _tcs* to str*\n\n0\n101 May 05, 2009 at 09:49\n\nGot the same thing as before, with that section from vector and this:\n\nFirst-chance exception at 0x7c902128 in RTHDRIBLEngine.exe: 0xC0000005: Access violation writing location 0x00000000.\nUnhandled exception at 0x7c902128 in RTHDRIBLEngine.exe: 0xC0000005: Access violation writing location 0x00000000.\n\n0\n101 May 05, 2009 at 09:54\n\nOh yeah don\u2019t use a WIN32_FIND_DATA pointer unless you are prepared to allocate the memory for it. Basically you are telling FindFirstFile to fill memory at address 0 (NULL) and that raises an access violation (0xc0000005) from Windows.\n\n0\n101 May 05, 2009 at 09:58\n\nbecause i\u2019m feeling nice (Even though the link above has a perfect explanation of how to use FindFirstFile) \u2026 this code should work. If it doesn\u2019t i suggest you try to figure out why (because I haven\u2019t compiled or run the code, myself) using the documentation.\n\nHANDLE hHandle;\nWIN32_FIND_DATA data;\nstd::vector<TCHAR*> Directories;\n\nhHandle = FindFirstFile(\"Media\/LightProbes\/*\", &data );\n\nBOOL bMoreFiles = TRUE;\nwhile(bMoreFiles == TRUE)\n{\nconst size_t strLen = _tcslen( data.cFileName ) + 1;\nTCHAR* pNewStr = new TCHAR[strLen];\n_tcscpy_s( pNewStr, strLen, data.cFileName );\nDirectories.push_back( pNewStr );\n\nbMoreFiles = FindNextFile( hHandle, &data );\n}\n\n0\n103 May 05, 2009 at 10:02\n\nWhy dont you learn the windows file explorer.\n\n0\n101 May 05, 2009 at 10:06\n\n@rouncer\n\nWhy dont you learn the windows file explorer.\n\nHow does that help you to programmatically enumerate files in a directory?\n\n0\n101 May 05, 2009 at 10:22\n\nCool, it seems to work prefectly now, here\u2019s the code for getting all of a certain type of file ina folder (I find all .dds files obviously, just change the file extension on the FindFirstFile function);\n\n HANDLE hHandle;\nWIN32_FIND_DATA pData;\nstd::vector<TCHAR*> Directories;\n\nhHandle = FindFirstFile(\"Media\/LightProbes\/*.dds\", &pData);\n\nconst size_t strLen = _tcslen( pData.cFileName ) + 1;\nTCHAR* pNewStr = new TCHAR[strLen];\n_tcscpy_s( pNewStr, strLen, pData.cFileName );\nDirectories.push_back( pNewStr );\n\nBOOL bMoreFiles = TRUE;\nwhile(bMoreFiles == TRUE)\n{\nbMoreFiles = FindNextFile(hHandle, &pData);\nif(bMoreFiles == TRUE)\n{\nconst size_t strLen = _tcslen( pData.cFileName ) + 1;\nTCHAR* pNewStr = new TCHAR[strLen];\n_tcscpy_s( pNewStr, strLen, pData.cFileName );\nDirectories.push_back( pNewStr );\n}\n}\n\n\nAs a note to others, make sure you save the name of whatever cFileName is after FindFirstFile is called otherwise you will miss it out as FindNextFile will not go back to the start of the file, thanks heaps Goz for your help, just another couple of questions if you don\u2019t mind?\n\n1. ok, so I\u2019ve got the name and file extension, how can I effectively go\n\nLPCSTR caPaths[Directories.size()];\n\nfor(int i = 0; i < Directories.size(); i++)\n{\ncaPaths = LPCSTR(\u201cMedia\/LightProbes\/\u201d + Directory);\n}\n\n??\n\n1. What exactly is an \u201caccess violation\u201d? I\u2019ve seen them so many times but never actually known what they are.\n\nAgain thank you for your help.\n\n0\n101 May 05, 2009 at 10:37\n\n@Phlex\n\n1. ok, so I\u2019ve got the name and file extension, how can I effectively go\n\nLPCSTR caPaths[Directories.size()];\n\nfor(int i = 0; i < Directories.size(); i++)\n{\ncaPaths = LPCSTR(\u201cMedia\/LightProbes\/\u201d + Directory);\n}\n\n??\n\nFirstly you should be using LPSTRs not LPCSTRs as the const will play havoc with you. Secondly you don\u2019t seem to have an idea as to how pointers work. You need to allocate the space for the string before you assign to it.\n\nFailing that you could just use the STL string (std::string).\n\nUsing that you could re-write that code as\n\nstd::string caPaths[Directories.size()];\n\nfor(int i = 0; i < Directories.size(); i++)\n{\ncaPaths[i] = std::string( \"Media\/LightProbes\/\" ) + std::string( Directory[i] );\n}\n\n\nThis way you are letting C++ do all the work. Microsoft also provides a similar string class called CString. Its available under MFC and ATL. Failnig that I\u2019m sure there are several string classes you could use. I wrote myself one recently for some cross platform fun.\n\n1. What exactly is an \u201caccess violation\u201d? I\u2019ve seen them so many times but never actually known what they are.\n\nAn access violation is an exception raised by the operating system. It informs you that you have tried to access some memory you aren\u2019t allowed to access (hence access violation). You can easily recreate them by writing to a NULL pointer, writing to a pointer higher than 0x80000000 (in a standard win32 setup) or, simply, by writing to a block of memory that hasn\u2019t been assigned to your process (ie allocated).\n\nPrimarily I think you need to do more research into what pointers are and how to use them as this does appear to be quite a large hole in your current knowledge.\n\n0\n101 May 05, 2009 at 10:43\n\ndon\u2019t forget to call FindClose(Handle) when you finshed searching.\n\n0\n103 May 05, 2009 at 16:55\n\nOh i didnt understand, i thought youy were just writing a file loader, sorry my mistake.\n\n0\n101 May 05, 2009 at 19:53\n\nI quite agree, my knowledge on pointers is rather terrible, thanks for the help everyone, your comments are all taken into consideration.\n\n@Adrian, thanks for the tip :)\n\n@rouncer, no trouble, thanks for helping\n\n0\n101 May 05, 2009 at 21:29\n\n@Phlex\n\nI quite agree, my knowledge on pointers is rather terrible, thanks for the help everyone, your comments are all taken into consideration.\n\nI\u2019d strongly recommend looking into a bit of assembly (Simply debugging your code using the disasembler is a great way to learn if you are prepared to figure out what is going on in memory and the registers). It will give you a great idea of how memory works.\n\nAddressable memory on a standard win 32 process can be thought of as a HUGE array of 2\\^31 bytes (or chars). A memory address is just the array index into that array. Its obviously gets a lot more complicated than that because you cannot guarantee exactly where memory will be allocated. However if you allocated a struct that has 4 DWORDs in it you know that there are 4 DWORDS each DWORD is made of 4 bytes. Therefore the allocate takes 16 bytes. The memory address of the structure is the first \u201cindex\u201d of the first byte. So to get the 3rd DWORD in the structure you know it is the base index PLUS 12 bytes. The following 4 bytes then make up the DWORD.\n\nNot sure that will help at all .. but well worth spending some serious time playing with pointers :)","date":"2013-12-06 00:14:05","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\": 0.3272120952606201, \"perplexity\": 3968.4037567310584}, \"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-2013-48\/segments\/1386163048663\/warc\/CC-MAIN-20131204131728-00065-ip-10-33-133-15.ec2.internal.warc.gz\"}"}
| null | null |
{"url":"https:\/\/en.wikipedia.org\/wiki\/Unconditionally_convergent","text":"# Unconditional convergence\n\n(Redirected from Unconditionally convergent)\n\nUnconditional convergence is a topological property (convergence) related to an algebraical object (sum). It is an extension of the notion of convergence for series of countably many elements to series of arbitrarily many. It has been mostly studied in Banach spaces.\n\n## Definition\n\nLet ${\\displaystyle X}$ be a topological vector space. Let ${\\displaystyle I}$ be an index set and ${\\displaystyle x_{i}\\in X}$ for all ${\\displaystyle i\\in I}$.\n\nThe series ${\\displaystyle \\textstyle \\sum _{i\\in I}x_{i}}$ is called unconditionally convergent to ${\\displaystyle x\\in X}$, if\n\n\u2022 the indexing set ${\\displaystyle I_{0}:=\\{i\\in I:x_{i}\\neq 0\\}}$ is countable and\n\u2022 for every permutation of ${\\displaystyle I_{0}:=\\{i\\in I:x_{i}\\neq 0\\}}$ the relation holds:${\\displaystyle \\sum _{i=1}^{\\infty }x_{i}=x}$\n\n## Alternative definition\n\nUnconditional convergence is often defined in an equivalent way: A series is unconditionally convergent if for every sequence ${\\displaystyle (\\varepsilon _{n})_{n=1}^{\\infty }}$, with ${\\displaystyle \\varepsilon _{n}\\in \\{-1,+1\\}}$, the series\n\n${\\displaystyle \\sum _{n=1}^{\\infty }\\varepsilon _{n}x_{n}}$\n\nconverges.\n\nEvery absolutely convergent series is unconditionally convergent, but the converse implication does not hold in general: if X is an infinite dimensional Banach space, then by Dvoretzky\u2013Rogers theorem there always exists an unconditionally convergent series in this space that is not absolutely convergent. However, when X\u00a0=\u00a0Rn, then, by the Riemann series theorem, the series ${\\displaystyle \\sum x_{n}}$ is unconditionally convergent if and only if it is absolutely convergent.","date":"2017-03-24 16:22:00","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 13, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9778698682785034, \"perplexity\": 184.47847094529973}, \"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-2017-13\/segments\/1490218188213.41\/warc\/CC-MAIN-20170322212948-00222-ip-10-233-31-227.ec2.internal.warc.gz\"}"}
| null | null |
Q: Underscore _.template Uncaught Syntax error token I'm working on Addy Osmani's Backbone Fundamentals tutorial. http://addyosmani.github.io/backbone-fundamentals/#application-view
and am getting an "Uncaught SyntaxError: Unexpected token % " in Chrome.
It's pointing to a line in Underscore.js and also a line in my views/app.js.
In views/app.js, the line it's pointing to is:
statsTemplate: _.template( $('#stats-template').html() ),
It says "anonymous function" in the error message. This was copied from the tutorial, so I'm not sure why it's throwing an error. Thanks
Template markup:
<script type="text/template" id="stats-template">
<span id="todo-count">
<strong>
<%= remaining %>
</strong>
<%= remaining === 1 ? 'item':'items'%> left
</span>
<ul id="filters">
<li>
<a class="selected" href="#/">All</a>
</li>
<li>
<a href="#/active">Active</a>
</li>
<li>
<a href="#/completed">Completed</a>
</li>
</ul>
<% if(completed) {% >
<button id="clear-completed">Clear completed (<%= completed %>)</button>
<% } %>
</script>
A: The template markup has a space between % and > which is causing Underscore to balk. This:
<% if(completed) {% >
Should be this:
<% if(completed) { %>
Fiddle
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 2,698
|
Enjoy your planning and revel even more with All 27 Promocode.club's great Forever 21 Canada promotion codes!
Choose a Forever 21 Canada coupon you would like to use from Promocode.club. Press use Forever 21 Canada code box, which is located just below. Now your web browser should copy the All 27 Forever 21 Canada code for you, although it is best to make sure and copy it manually. The https://www.forever21.com web page will open in a new window for you. Go to their Forever 21 Canada checkout section and find the Promotional Keycode box. Paste your Forever 21 Canada coupon code there and click apply.
|
{
"redpajama_set_name": "RedPajamaC4"
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| 1,562
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Con il nome Filippo Acciaiuoli (o Filippo Acciaioli) si identificano:
Filippo Acciaiuoli, podestà di Ferrara nel 1365;
Filippo Acciaiuoli (1637-1700) poeta e musicista italiano;
Filippo Acciaiuoli (1700-1766), cardinale dal 1759.
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 5,638
|
Deloitte Deloitte Consulting - Summer Scholar (Strategy) in Chicago, Illinois
Deloitte Consulting - Summer Scholar (Strategy)
Same job available in 20 locations
Cincinnati, Ohio, United States
Cleveland, Ohio, United States
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McLean, Virginia, United States
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This summer, jump start your career and take your business skills to the next level when you join Deloitte Consulting LLP. We're looking for a sharp, critical thinker with a gift for analysis to help us develop and implement actionable recommendations to our client's largest business issues. If you're an outside-the-box thinker who enjoys exploring how strategy, business processes, and technology help shape the future of our clients' organizations, then you may be the right fit for the role.
Work You'll Do
As a Strategy Summer Scholar, you'll work in a fast-paced environment that exposes you to a variety of experiences across industries, disciplines, and geographies, while you develop your business skills. You will work alongside industry-leading clients to help them make decisions and implement enhancements to their organizations' strategy, productivity, performance, and long-term profitability. You'll also assist in determining the requirements of clients' projects and programs by defining, analyzing, and managing requirements to help fulfill clients' needs. While the specific job deliverables will vary according to clients' needs, our Strategy Summer Scholars have the broad skills needed to tackle a wide variety of business initiatives. You will have the opportunity to exercise strong analytical and critical thinking skills with the chance to solve complex problems and communicate your findings. You will work independently and collaboratively with a team.
As a Strategy Summer Scholar, you are comfortable with ambiguity and creating your own path. You are driven to lead and make an impact no matter how big or small the problem. You leverage a 'business first lens' across functional and industry topics to drive strategic transformation and deliver solutions to clients. You will be successful by demonstrating a breadth and depth of capabilities and will be positioned as a 'best athlete' who is prepared to take on a variety of business, technological and organizational challenges. Examples of these may include innovating and improving processes and experiences; implementing tools, new offerings and data powered advancements; shaping lifecycle activities; identifying efficiencies to enhance organizational performance; and architecting innovative operating models. This individual will help some of the largest and most innovative organizations address their most complex and pressing business challenges.
Our consultants offer insightful recommendations and data-fueled strategies to help our clients address some of their most complex business issues. You will work alongside and learn from cross-functional teams with leaders and peers with varying backgrounds, education, and experiences. We are a diverse and talented team that supports a collaborative culture and delivers value through the application of strategy and process improvement. We tackle our clients' strategy, operational, financial management, supply management, innovation, and growth issues and help to enhance their business performance, productivity, and long-term profitability. Join us and help in designing our clients' roadmap to the future.
Junior standing in a full-time academic program with a target graduation date between December 2021 and August 2022
Bachelor's degree in progress in the following or related majors:
Business (e.g., Finance, Marketing, Org Behavior)
Liberal Arts (e.g., Psychology, Industrial and Labor Relations)
Must be legally authorized to work in the United States without the need for employer sponsorship, now or at any time in the future
Strong academic track record (minimum GPA of 3.5)
Availability to participate full-time for a 10-week internship June to August 2021
Ability to travel up to 80-100% (While 80-100% of travel is a requirement of the role, due to COVID-19, non-essential travel has been suspended until further notice)
During this challenging and inspiring learning experience, you'll gain first-hand insights into technical skills, critical professional behaviors, standards, and mindsets needed to be effective as a strategy consultant. You'll work with a wide variety of talented strategic leaders and learn how to develop, implement, and measure innovative strategies, tackle real-world business issues head-on, and experience first-hand the day-to-day challenges and opportunities of this exciting career. Summer Scholars will also be invited to participate in the Consulting Summer Scholar Internship Conference hosted during the summer at Deloitte University. Explore Deloitte University, The Leadership Center.
From developing a stand out resume to putting your best foot forward in the interview, we want you to feel prepared and confident as you explore opportunities at Deloitte.Check out recruiting tips from Deloitte recruiters.
At Deloitte, we know that great people make a great organization. We value our people and offer employees a broad range of benefits.Learn more about what working at Deloitte can mean for you.
Our people and culture
Our inclusive culture empowers our people to be who they are, contribute their unique perspectives, and make a difference individually and collectively. It makes Deloitte one of the most rewarding places to work. But don't take our word for it:Be inspired by the stories of our people.
From entry-level employees to senior leaders, we believe there's always room to learn. We offer opportunities to build new skills, take on leadership opportunities and connect and grow through mentorship. From on-the-job learning experiences to formal development programs, our professionals have a variety of opportunities to continue to grow throughout their career.Learn more about our commitment to developing our people.
As used in this posting, "Deloitte" means Deloitte Consulting LLP, a subsidiary of Deloitte LLP. Please seewww.deloitte.com/us/aboutfor a detailed description of the legal structure of Deloitte LLP and its subsidiaries. Certain services may not be available to attest clients under the rules and regulations of public accounting.
All qualified applicants will receive consideration for employment without regard to race, color, religion, sex, sexual orientation, gender identity, national origin, age, disability or protected veteran status, or any other legally protected basis, in accordance with applicable law.
Deloitte will consider for employment all qualified applicants, including those with criminal histories, in a manner consistent with the requirements of applicable state and local laws.See notices of various ban-the-box laws where available.
Requisition code: 2688
© 2021. SeeTerms of Usefor more information.
Deloitte refers to one or more of Deloitte Touche Tohmatsu Limited, a UK private company limited by guarantee ("DTTL"), its network of member firms, and their related entities. DTTL and each of its member firms are legally separate and independent entities. DTTL (also referred to as "Deloitte Global") does not provide services to clients. In the United States, Deloitte refers to one or more of the US member firms of DTTL, their related entities that operate using the "Deloitte" name in the United States and their respective affiliates. Certain services may not be available to attest clients under the rules and regulations of public accounting. Please seewww.deloitte.com/aboutto learn more about our global network of member firms.
Deloitte Consulting - Summer Sch...
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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function optionsFromArray(array) {
var options = [];
for(var i = 0; i < array.length; i++)
options.push($j('<option/>').val(array[i][1]).text(array[i][0])[0]);
return options;
}
function nestedOptionsFromJSONArray(array,prompt_option_text) {
var options = [];
options.push($j('<option/>').val(0).text(prompt_option_text));
//gather together by parent id
var parents = {};
for(var i = 0; i < array.length; i++) {
var item = array[i];
if (parents[item.parent_id]) {
var parent = parents[item.parent_id];
parent.children.push({id:item.id,title:item.title});
}
else {
var parent = {title:item.parent_title,id:item.parent_id,children:[]};
parent.children.push({id:item.id,title:item.title});
parents[item['parent_id']]=parent;
}
}
//build into optgroups, with options clustered according to parent
for (parent_id in parents) {
var parent=parents[parent_id];
console.log(parent);
var group = $j('<optgroup/>').attr('label',parent.title);
for(var i=0;i<parent.children.length;i++) {
var child=parent.children[i];
group.append($j('<option/>').val(child.id).text(child.title));
}
options.push(group);
}
return options;
}
var Associations = {};
// Object to control a list of existing associations
Associations.List = function (template, element) {
this.template = template;
this.element = element;
this.element.data('associationList', this);
this.listElement = $j('ul', this.element);
this.items = [];
};
Associations.List.prototype.toggleEmptyListText = function () {
var noText = $j('.no-item-text', this.element);
if(this.items.length === 0)
noText.show();
else
noText.hide();
};
Associations.List.prototype.add = function (association) {
this.items.push(new Associations.ListItem(this, association));
this.toggleEmptyListText();
};
Associations.List.prototype.remove = function (listItem) {
var index = this.items.indexOf(listItem);
if (index > -1)
this.items.splice(index, 1);
this.toggleEmptyListText();
};
Associations.List.prototype.removeAll = function () {
this.items = [];
this.listElement.html('');
this.toggleEmptyListText();
};
// Object to control an element in the association list
Associations.ListItem = function (list, data) {
this.list = list;
this.data = data;
// Create and append element to list
this.element = $j(HandlebarsTemplates[this.list.template](data));
this.list.listElement.append(this.element);
// Bind remove event
var listItem = this;
this.element.on('click', '.remove-association', function () {
listItem.remove();
});
};
Associations.ListItem.prototype.remove = function () {
this.element.remove();
this.list.remove(this);
};
// A collection of lists, for example if you want to split associations into different lists depending on a certain
// attribute.
Associations.MultiList = function (element, groupingAttribute) {
this.element = element;
this.element.data('associationList', this);
this.groupingAttribute = groupingAttribute;
this.lists = {};
};
Associations.MultiList.prototype.addList = function (attributeValue, list) {
this.lists[attributeValue.toString()] = list;
};
Associations.MultiList.prototype.add = function (association) {
var value = digValue(association, this.groupingAttribute);
var list = this.lists[value.toString()];
list.add(association);
};
Associations.MultiList.prototype.removeAll = function () {
for(var key in this.lists) {
if(this.lists.hasOwnProperty(key))
this.lists[key].removeAll();
}
};
// Object to control the association selection form.
Associations.Form = function (list, element) {
this.list = list;
this.element = element;
this.element.data('associationForm', this);
this.selectedItems = [];
this.commonFieldElements = [];
};
Associations.Form.prototype.reset = function () {
this.selectedItems = [];
};
Associations.Form.prototype.submit = function () {
var commonFields = {};
this.commonFieldElements.forEach(function (element) {
var name = element.data('attributeName');
if(element.is('select')) { // <select> tags store both the value and the selected option's text
commonFields[name] = { value: element.val(),
text: $j('option:selected', element).text() };
} else {
commonFields[name] = element.val();
}
});
var list = this.list;
this.selectedItems.forEach(function (selectedItem) {
// Merge the common fields with the selected item's attributes
var associationObject = $j.extend({}, commonFields, selectedItem);
list.add(associationObject);
});
if(this.afterSubmit)
this.afterSubmit(this.selectedItems.length);
this.reset();
};
Associations.filter = function (filterField) {
$j.ajax($j(filterField).data('filterUrl'), {
data: { filter: $j(filterField).val() },
success: function (data) {
$j(filterField).siblings('[data-role="seek-association-candidate-list"]').html(data);
}
}
);
};
$j(document).ready(function () {
// Markup
$j('[data-role="seek-associations-list"]').each(function () {
var list = new Associations.List($j(this).data('templateName'), $j(this));
var self = $j(this);
var existingValues = $j('script[data-role="seek-existing-associations"]', self).html();
if(existingValues) {
JSON.parse(existingValues).forEach(function (value) {
list.add(value);
});
}
});
$j('[data-role="seek-associations-list-group"]').each(function () {
var multilist = new Associations.MultiList($j(this), $j(this).data('groupingAttribute'));
var self = $j(this);
$j('[data-role="seek-associations-list"]', self).each(function () {
multilist.addList($j(this).data('multilistGroupValue'), $j(this).data('associationList'));
});
var existingValues = $j('script[data-role="seek-existing-associations"]', self).html();
if(existingValues) {
JSON.parse(existingValues).forEach(function (value) {
multilist.add(value);
});
}
});
$j('[data-role="seek-association-form"]').each(function () {
var element = this;
var listId = $j(element).data('associationsListId');
var list = $j('#' + listId).data('associationList'); // Get the List object from the DOM element
var form = new Associations.Form(list, $j(this));
// Strip the name of the element and store it as a data attribute, to stop it being submitted as a field in the
// main form
$j(':input[data-role="seek-association-common-field"]', $j(element)).each(function () {
$j(this).data('attributeName', this.name);
this.name = '';
form.commonFieldElements.push($j(this));
});
$j(element).on('click', '.selectable[data-role="seek-association-candidate"]', function () {
$j(this).toggleClass('selected');
if(!$j(this).parents('[data-role="seek-association-candidate-list"]').data('multiple')) {
$j(this).siblings().removeClass('selected');
}
form.selectedItems = [];
$j(this).parents('[data-role="seek-association-candidate-list"]').find('[data-role="seek-association-candidate"].selected').each(function () {
// Merge common fields and association-specific fields into single object
form.selectedItems.push({
id: $j(this).data('associationId'),
title: $j(this).data('associationTitle')
});
});
return false;
});
$j('[data-role="seek-association-confirm-button"]', $j(element)).click(function (e) {
e.preventDefault();
$j('.selectable[data-role="seek-association-candidate"]', $j(element)).removeClass('selected');
form.submit();
});
});
$j('[data-role="seek-association-filter"]').keypress(function (e) {
// If more than two characters were entered, or the input was cleared, or the ENTER key was pressed..
var filterField = this;
if($j(filterField).val().length == 0 || $j(filterField).val().length >= 2 || e.keyCode == 13) {
Associations.filter(filterField);
if(e.keyCode == 13)
e.preventDefault();
}
});
// If no initial association candidates provided, make a filter call to get some.
$j('[data-role="seek-association-filter"]').each(function () {
var list = $j(this).siblings('[data-role="seek-association-candidate-list"]');
if($j.trim(list.html()) == '')
Associations.filter(this);
});
});
|
{
"redpajama_set_name": "RedPajamaGithub"
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| 8,960
|
Thakur may refer to:
Thakur (title), a feudal title and surname used by erstwhile nobility of India
Rajput, an Indian caste also known by the name "Thakur"
Thakur village, a residential locality in Mumbai, India
See also
Thaker, an Indian family name
Thakkar, an Indian family name
Thakar (tribe), an Adivasi tribe of Maharashtra, India
Thakor, a Hindu Koli caste in Gujarat
Thakura (disambiguation)
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,407
|
Der Langenprozeltener Forst ist ein 8,66 km² großes gemeindefreies Gebiet im Landkreis Main-Spessart im bayerischen Spessart. Es ist komplett bewaldet.
Geographie
Lage
Das Gebiet liegt westlich von Gemünden am Main mit dem namensgebenden Ortsteil Langenprozelten. Die höchste Erhebung ist die Kuppe mit .
Nachbargemeinden
Siehe auch
Liste der gemeindefreien Gebiete in Bayern
Weblinks
Gemeindefreies Gebiet Langenprozeltener Forst in OpenStreetMap (Abgerufen am 27. August 2017)
Gemeindefreies Gebiet in Bayern
Geographie (Spessart)
Waldgebiet im Landkreis Main-Spessart
Waldgebiet in Europa
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,522
|
\section{Introduction}
High precision measurements at the $Z$ pole at LEP
combined with polarized forward backward asymmetries at SLC
and other measurements of electroweak observables at lower energies
have been used to place stringent limits on new physics beyond the standard
model \cite{altanew,langanew}.
Under the assumption that the dominant effects of the new physics
would show up as corrections to the gauge boson self-energies, the
LEP measurements have been used to parameterize the possible new
physics in terms of three observables $S$, $T$, $U$
\cite{pestak}; or equivalently
$\epsilon_1$, $\epsilon_2$, $\epsilon_3$ \cite{alta}. The difference
between the two
parameterizations is the reference point which corresponds to the
standard model predictions. A fourth observable
corresponding to the partial width $Z \rightarrow b \overline{b}$ has
been analyzed in terms of the parameter $\delta_{bb}$ \cite{langanew} or
$\epsilon_b$ \cite{alta}.
In view of the extraordinary agreement between the standard model
predictions and the observations, it seems reasonable to assume
that the $SU(2)_L \times U(1)_Y$ gauge theory of electroweak interactions
is essentially correct, and that the only sector of the theory
lacking experimental support is the symmetry breaking sector. There
are many extensions of the minimal standard model that incorporate
different symmetry breaking possibilities. One large class of models
is that in which the interactions responsible for the symmetry breaking
are strongly coupled. For this class of models one expects that there
will be no new particles with masses below 1~TeV or so, and that their
effects would show up in experiments as deviations from the minimal
standard model couplings.
In this paper we use the latest measurements of partial decay widths
of the $Z$ boson to place bounds on anomalous gauge boson couplings.
Our paper is organized as follows. In Section~2 we discuss our formalism
and the assumptions that go into the relations between the partial
widths of the $Z$ boson and the anomalous couplings. In Section~3
we present our results. Finally, in Section~4 we discuss the
difference between our calculation and others that can be found in
the literature, and assess the significance of our results by comparing
them to other existing limits. Detailed analytical
formulae for our results are relegated to an appendix.
\section{Formalism}
We assume that the electroweak interactions are given
by an $SU(2)_L \times U(1)_Y$ gauge
theory with spontaneous symmetry breaking to $U(1)_{EM}$, and
that we do not have any information on the symmetry breaking sector
except that it is strongly interacting and that any new particles have
masses higher than several hundred GeV. It is well known that this
scenario can be described with an effective Lagrangian with operators
organized according to the number of derivatives or gauge fields
they have. If we call $\Lambda$ the scale at which the symmetry breaking
physics comes in, this organization of operators corresponds to
an expansion of amplitudes in powers of $(E^2~{\rm or}~v^2)/\Lambda^2$.
For energies $E \leq v$ this is just an expansion in powers of the
coupling constant $g$ or $g^\prime$, and for energies $E \geq v$ it
becomes an energy expansion.
The lowest order effective Lagrangian for the symmetry breaking
sector of the theory is \cite{longo}:
\begin{equation}
{\cal L}^{(2)}={v^2 \over 4}{\rm Tr}\biggl[D^\mu \Sigma^\dagger D_\mu
\Sigma \biggr].
\label{lagt}
\end{equation}
In our notation $W_{\mu}$ and $B_{\mu}$ are
the $SU(2)_L$ and $U(1)_Y$ gauge fields with
$W_\mu \equiv W^i_\mu \tau_i$.\footnote{$Tr(\tau_i \tau_j)=2\delta_{ij}$.}
The matrix $\Sigma \equiv \exp(i\vec{\omega}\cdot \vec{\tau} /v)$, contains the
would-be Goldstone bosons $\omega_i$ that give the $W$ and $Z$ their
masses via the Higgs mechanism and the $SU(2)_L \times U(1)_Y$
covariant derivative is given by:
\begin{equation}
D_\mu \Sigma = \partial_\mu \Sigma +{i \over 2}g W_\mu^i \tau^i\Sigma
-{i \over 2}g^\prime B_\mu \Sigma \tau_3.
\label{covd}
\end{equation}
Eq.~\ref{lagt} is thus the $SU(2)_L \times U(1)_Y$ gauge
invariant mass term for the $W$ and $Z$. The physical masses
are obtained with $v \approx 246$~GeV. This non-renormalizable Lagrangian
is interpreted as an effective field theory, valid below
some scale $\Lambda \leq 3$~TeV. The lowest order interactions between
the gauge bosons and fermions, as well as the kinetic energy
terms for all fields, are the same as those in the minimal
standard model.
For LEP observables, the operators that can appear at tree-level
are those that modify the gauge boson self-energies. To order
${\cal O}(1/\Lambda^2)$ there are only three \cite{longo,appel}:
\begin{equation}
{\cal L}^{(2GB)}=\beta_1{v^2\over 4} \biggl({\rm Tr} \biggl[
\tau_3 \Sigma^\dagger D_\mu \Sigma\biggr]\biggr)^2
+ \alpha_8g^2 \biggl({\rm Tr}\biggl[\Sigma \tau_3 \Sigma^\dagger
W_{\mu \nu}\biggr]\biggr)^2
+ g g^{\prime}{v^2 \over \Lambda^2} L_{10}\, {\rm Tr}\biggl[ \Sigma
B^{\mu \nu}
\Sigma^\dagger W_{\mu \nu}\biggr],
\label{oblique}
\end{equation}
which contribute respectively to $T$, $U$ and $S$. Notice that for the
two operators that break the custodial $SU(2)_C$ symmetry we have
used the notation of Ref.~\cite{longo,appel}.
In this paper we will consider operators that affect the $Z$ partial
widths at the one-loop level. We will restrict our study to only those
operators that appear at order ${\cal O}(1/\Lambda^2)$ in the gauge-boson
sector and that respect the custodial symmetry in the limit
$g^\prime \rightarrow 0$. They are:
\begin{eqnarray}
{\cal L}^{(4)}\ &=&\ {v^2 \over \Lambda^2} \biggl\{ L_1 \,
\biggl( {\rm Tr}\biggl[
D^\mu\Sigma^\dagger D_\mu \Sigma \biggr]\biggl)^2
\ +\ L_2 \,\biggl({\rm Tr}\biggl[
D_\mu\Sigma^\dagger D_\nu \Sigma\biggr]\biggl)^2
\nonumber \\
&& - i g L_{9L} \,{\rm Tr}\biggl[W^{\mu\nu} D_\mu
\Sigma D_\nu \Sigma^\dagger\biggr]
\ -\ i g^{\prime} L_{9R} \,
{\rm Tr}\biggl[ B^{\mu \nu}
D_\mu \Sigma^\dagger D_\nu\Sigma\biggr]\biggr\},
\label{lfour}
\end{eqnarray}
where the field strength tensors are given by:
\begin{eqnarray}
W_{\mu\nu}&=&{1 \over 2}\biggl(\partial_\mu W_\nu -
\partial_\nu W_\mu + {i \over 2}g[W_\mu, W_\nu]\biggr)
\nonumber \\
B_{\mu\nu}&=&{1\over 2}\biggl(\partial_\mu B_\nu-\partial_\nu B_\mu\biggr)
\tau_3.
\label{fsten}
\end{eqnarray}
Although this is not a complete list of all the operators that can arise
at this order, we will be able to present a consistent picture in
the sense that our calculation will not require additional counterterms
to render the one-loop results finite. Our choice of this subset of operators
is motivated by the theoretical prejudice that violation of custodial
symmetry must be ``small'' in some sense in the full theory \cite{siki}.
We want to restrict our attention to a small subset of all the operators that
appear at this order because there are only a few observables that have been
measured.
The operators in Eq.~\ref{oblique} and Eq.~\ref{lfour} would
arise when considering the effects
of those in Eq.~\ref{lagt} at the one-loop level, or
from the new physics responsible for symmetry
breaking at a scale $\Lambda$ at order $1/\Lambda^2$. We, therefore,
explicitly introduce the factor $v^2/\Lambda^2$ in our definition
of ${\cal L}^{(4)}$ so that the coefficients
$L_i$ are naturally of ${\cal O}(1)$.\footnote{
We do not introduce this factor in
the $SU(2)_C$ violating couplings $\beta_1$ and $\alpha_8$ since we
do not concern ourselves with them in this paper. They are simply
used as counterterms for our one-loop calculation.}
The anomalous couplings that we consider would have tree-level effects
on some observables that can be studied in future colliders. They have been
studied at length in the literature \cite{boud}.
In the present paper we will compute their
contribution to the $Z$ partial widths that are measured at LEP. These
operators contribute to the $Z$ partial widths
at the one-loop level. Since we
are dealing with a non-renormalizable effective Lagrangian, we will
interpret our one-loop results in the usual way of an effective field theory.
We will first perform a complete calculation to order ${\cal O}
(1/\Lambda^2)$. That is, we will include the one-loop contributions from
the operator in Eq.~\ref{lagt} (and gauge boson kinetic energies).
The divergences generated in this calculation are absorbed by
renormalization of the couplings in Eq.~\ref{oblique}. This calculation
will illustrate our method, and as an example, we use it to place
bounds on $L_{10}$.
We will then place bounds on the couplings of Eq.~\ref{lfour} by considering
their one-loop effects. The divergences generated in this one-loop calculation
would be removed in general by renormalization of the couplings in the
${\cal O}(1/\Lambda^4)$ Lagrangian of those operators that modify the
gauge boson self-energies at tree-level; and perhaps by additional
renormalization of the couplings in Eq.~\ref{oblique}. This would occur
in a manner analogous to our ${\cal O}(1/\Lambda^2)$ calculation.
Interestingly,
we find that we can obtain a completely finite result for the $Z \rightarrow
\overline{f} f$ partial widths using only the operators in Eq.~\ref{oblique}
as counterterms.
However, our interest is to place bounds on the couplings
of Eq.~\ref{lfour} so we proceed as follows. We first regularize the
integrals in $n$ space-time dimensions and remove all the poles in
$n-4$ as well as the finite analytic terms by a suitable definition of
the renormalized couplings. We then base our
analysis on the leading non-analytic
terms proportional to $L_i \log \mu$. These terms determine the
running of the $1/\Lambda^4$ couplings and cannot be generated by
tree-level terms at that order. It has been argued in the literature
\cite{georgi}, that with a carefully chosen renormalization scale $\mu$
(in such a way that the logarithm is of order one), these terms give us
the correct order of magnitude for the size of the $1/\Lambda^4$
coefficients. We thus choose some value for the renormalization
scale between the $Z$ mass and $\Lambda$ and require that this
logarithmic contribution to the renormalized couplings falls in the
experimentally allowed range. Clearly, the LEP observables do not
measure the couplings in Eq.~\ref{lfour}, and it is only from
naturalness arguments like the one above, that we can place bounds
on the anomalous gauge-boson couplings.
{}From this perspective, it is clear that these bounds are not a substitute
for direct measurements in future high energy machines. They should,
however, give us an indication for the level at which we can expect
something new to show up in those future machines.
We will perform our calculations in unitary gauge, so we set $\Sigma=1$
in Eqs.~\ref{lagt}, \ref{oblique} and \ref{lfour}. This results in interactions
involving three, and four gauge boson couplings, some of which we
present in Appendix~A. Those
coming from Eq.~\ref{lagt} are equivalent to those in the minimal
standard model with an infinitely heavy Higgs boson, and those coming
from Eq.~\ref{lfour} correspond to the ``anomalous'' couplings.
For the lowest order operators we use the conventional
input parameters: $G_F$ as measured in muon decay;
the physical $Z$ mass: $M_Z$; and $\alpha(M_Z)=1/128.8$. Other lowest
order parameters are derived quantities and we adopt one of the
usual definitions for the mixing angle:
\begin{equation}
s_Z^2 c_Z^2 \equiv{\pi \alpha(M_Z)\over \sqrt{2} G_F M_Z^2}.
\label{szdef}
\end{equation}
We neglect the mass and momentum of the external fermions
compared to the $Z$ mass. In particular, we do not include the $b$-quark
mass since it would simply introduce corrections of order
$5\%$ and our results are only order of magnitude estimates.
The only fermion mass that is kept in our calculation is
the mass of the top-quark when it appears as an intermediate state.
With this formalism we proceed to compute the $Z\rightarrow f \overline{f}$
partial width from the following ingredients.
\begin{itemize}
\item The $Z\rightarrow f {\overline f}$ vertex, which we write as:
\begin{equation}
i {\Gamma}_\mu = -i{e\over 4 s_Z c_Z}
\gamma_\mu\biggl[
(r_f +\delta r_f)(1+\gamma_5) +(l_f+\delta l_f)
(1-\gamma_5)\biggr ]
\label{vertex}
\end{equation}
where $r_f=-2Q_f s_Z^2$ and $l_f=r_f+T_{3f}$.
The terms $\delta l_f$ and $\delta r_f$ occur at one-loop both at order
$1/\Lambda^2$ and at order $1/\Lambda^4$ and are given in Appendix~B.
\item The renormalization of the lowest order input parameters.
At order $1/\Lambda^2$ it is induced by tree-level anomalous
couplings and one-loop diagrams with lowest order vertices. At
order $1/\Lambda^4$ it is induced by one-loop diagrams with an
anomalous coupling in one vertex. We present analytic formulae
for the self-energies, vertex corrections and boxes in Appendix~B.
The changes induced in the lowest order input parameters are:
\begin{eqnarray}
{\Delta \alpha\over \alpha} &=& {A_{\gamma \gamma}(q^2)\over q^2}
\mid _{q^2=0}
\nonumber \\
{\Delta M_Z^2\over M_Z^2}&=&{A_{ZZ}(M_Z^2)\over M_Z^2}
\nonumber \\
{\Delta G_F\over G_F} & = & {2 \Gamma_{W e \nu}\over {\cal A}_0}
-{A_{WW}(0)\over M_W^2}+(Z_f-1)+B_{\rm box}
\label{shifts}
\end{eqnarray}
The self-energies $A_{VV}$ receive tree-level contributions from the
operators with $L_{10}$, $\beta_1$ and $\alpha_8$. They also
receive one-loop contributions from the lowest
order Lagrangian Eq.~\ref{lagt} and from the operators with
$L_1$, $L_2$, $L_{9L}$ and $L_{9R}$. The effective $W e \nu$ vertex,
$\Gamma_{W e \nu}$, receives one-loop contributions from all
the operators in Eq.~\ref{lfour}. The fermion wave function
renormalization factors
$Z_f$ and the box contribution to $\mu \rightarrow e \nu
{\overline \nu}$, $B_{\rm box}$, are due only to one-loop
effects from the lowest order effective Lagrangian and are thus
independent of the anomalous couplings. Notice that $B_{\rm box}$ enters the
renormalization of $G_F$ because we work in unitary gauge where box
diagrams also contain divergences.
\item Tree-level and one-loop contributions to $\gamma Z$ mixing.
Instead of diagonalizing the neutral gauge boson
sector, we include this mixing as an additional
contribution to $\delta l_f$ and $\delta r_f$ in Eq.~\ref{vertex}:
\begin{equation}
\delta l_f^{\prime} = \delta r_f^{\prime}
= -{ c_Z \over s_Z} r_f { A_{\gamma Z}(M_Z^2) \over M_Z^2}
\label{mix}
\end{equation}
\item Wave function renormalization. For the external fermions we include it
as additional contributions to $\delta l_f$ and $\delta r_f$ as shown in
Appendix~B. For the $Z$ we include it explicitly.
\end{itemize}
With all these ingredients we can collect the results from Appendix~B into
our final expression for the physical partial width. We find:
\begin{eqnarray}
\Gamma(Z\rightarrow f {\overline f}) &
=& \Gamma_0 Z_Z \biggl[ 1
-{\Delta G_f \over G_f} - {\Delta M_Z^2 \over M^2_Z}
+{2(l_f \delta l_f + r_f \delta r_f)\over l_f^2 + r_f^2}
\nonumber \\ && -
{2 r_f(l_f + r_f)\over l_f^2 + r_f^2}{c^2_Z \over s^2_Z -c^2_Z}
\biggl({\Delta G_f \over G_f} + {\Delta M_Z^2 \over M^2_Z}
-{\Delta \alpha \over \alpha} \biggr)\biggr],
\label{full}
\end{eqnarray}
where $\Gamma_0$ is the lowest order tree level result,
\begin{equation}
\Gamma_0 (Z\rightarrow f {\overline f})=
N_{cf} (l_f^2 + r_f^2) {G_F M_Z^3 \over 12 \pi \sqrt{2}},
\label{widthl}
\end{equation}
and $N_{cf}$ is 3 for quarks and 1 for leptons.
We write the contributions of the different anomalous couplings
to the $Z$ partial widths in the form:
\begin{equation}
\Gamma(Z\rightarrow f \overline{f}) \equiv \Gamma_{SM}(Z\rightarrow f\overline{f})
\biggl( 1 + {\delta \Gamma_f^{L_i} \over \Gamma_0(Z\rightarrow f\overline{f})}\biggr).
\label{defw}
\end{equation}
We use this form because we want to place bounds on the anomalous
couplings by comparing the measured widths with the one-loop standard
model prediction $\Gamma_{SM}$. Using Eq.~\ref{defw} we introduce
additional terms proportional to products of standard model one-loop
corrections and corrections due to anomalous couplings. These are
small effects that do not affect our results.
We will not attempt to obtain a global fit to the parameters in our
formalism from all possible observables. Instead we use the
partial $Z$ widths. We believe this approach to be adequate
given the fact that the results rely on naturalness assumptions.
Specifically we consider the observables:
\begin{eqnarray}
\Gamma_e &=& 83.98\pm 0.18 ~{\rm MeV}~{\rm~Ref.~\cite{altanew}}
\nonumber \\
\Gamma_\nu &=& 499.8 \pm 3.5~{\rm MeV}~{\rm~Ref.~\cite{glas}}
\nonumber \\
\Gamma_Z &=&2497.4 \pm 3.8~{\rm MeV}~{\rm~Ref.~\cite{glas}}
\nonumber \\
R_h &=& 20.795\pm 0.040~{\rm~Ref.~\cite{glas}}
\nonumber \\
R_b &=& 0.2202\pm 0.0020~{\rm~Ref.~\cite{glas}}
\label{data}
\end{eqnarray}
The bounds on new physics are obtained by subtracting the standard
model predictions at one-loop from the measured partial widths
as in Eq.~\ref{defw}. We
use the numbers of Langacker \cite{langanew} which use the global
best fit values for $M_t$ and $\alpha_s$ with $M_H$ in the range
$60-1000$~GeV. The first error is from the uncertainty in $M_Z$ and
$\Delta r$, the second is from $M_t$ and $M_H$, and the one in
brackets is from the uncertainty in $\alpha_s$.
\begin{eqnarray}
\Gamma_e &=& 83.87~\pm 0.02 \pm 0.10~{\rm MeV}~{\rm~Ref.~\cite{langap}}
\nonumber \\
\Gamma_\nu &=& 501.9 \pm 0.1 \pm 0.9~{\rm MeV}~{\rm~Ref.~\cite{bernabeu}}
\nonumber \\
\Gamma_Z &=& 2496 \pm 1 \pm 3 \pm [3]~{\rm MeV}~{\rm~Ref.~\cite{langanew}}
\nonumber \\
R_h &=& 20.782 \pm 0.006 \pm 0.004 \pm[0.03]~{\rm~Ref.~\cite{langanew}}
\nonumber \\
\delta_{bb}^{new} &=& 0.022 \pm 0.011~{\rm~Ref.~\cite{langap}}
\label{theory}
\end{eqnarray}
where $\delta_{bb}^{new}\equiv [\Gamma(Z \rightarrow b\overline{b})
-\Gamma(Z \rightarrow b\overline{b})^{(SM)}]/\Gamma(Z \rightarrow b\overline{b})^{(SM)}$.
We add all errors in quadrature.
\section{Results}
In this section we compute the corrections to the $Z \rightarrow f \overline{f}$
partial widths from the couplings of Eq.~\ref{lfour}, and compare them
to recent values measured at LEP. We treat each coupling constant
independently, and compute only its {\it lowest} order contribution to the
decay widths. We first present the complete ${\cal O}(1/\Lambda^2)$
results. They illustrate our method and serve as a check of our calculation.
We then look at the effect of the couplings $L_{1,2}$ which affect only
the gauge-boson self-energies. We then study the more complicated case of the
couplings $L_{9L,9R}$. Finally we isolate the non-universal effects
proportional to $M_t^2$. As explained in the previous section, we do not
include in our analysis the operators that appear at ${\cal O}(1/\Lambda^2)$
that break the custodial symmetry. As long as one is interested in
bounding the anomalous couplings one at a time, it is straightforward
to include these operators. For example, we discussed the parity violating
one in Ref.~\cite{dvplep}.
\subsection{Bounds on $L_{10}$ at order $1/\Lambda^2$.}
The operators in Eq.~\ref{oblique} are the only ones that induce a
tree-level correction to the gauge boson self-energies to
order ${\cal O}(1/\Lambda^2)$. This can be
seen most easily by working in a physical basis in which the neutral gauge
boson self-energies are diagonalized to order ${\cal O}(1/\Lambda^2)$.
This is accomplished with renormalizations described in the literature
\cite{holdom,cdhv}, and results in modifications to
the $W {\overline f}^{\prime} f$ and the $Z f \overline{f}$ couplings.
This tree-level effect on the $Z \rightarrow f \overline{f}$ partial
width is, of course, well known. It corresponds,
{\it at leading order}, to the new physics
contributions to $S,~T,~U$ or $\epsilon_{1,2,3}$
discussed in the literature \cite{alls}.
In this section we do not perform the diagonalization mentioned above, but
rather work in the original basis for the fields. This will serve two
purposes. It will allow us to present a complete ${\cal O}(1/\Lambda^2)$
calculation as an illustration of the method we use to bound the other
couplings. Also, because the gauge boson interactions that appear at
this order have the same tensor structure as those induced by
$L_{9L}$ and $L_{9R}$, we will be able to carry out the calculation
involving those two couplings simultaneously. In this way, even though
the terms with $L_{9L,9R}$ are order $1/\Lambda^4$,
the calculation to order $1/\Lambda^2$ will serve as a check of
our answer for $L_{9L,9R}$.
To recover the $1/\Lambda^2$ result we set $L_{9L,9R}=0$ (and also
$L_{1,2}=0$ but these terms are clearly different) in the results
of Appendix~B. As explained in Section~2, we have regularized our
one-loop integrals in $n$ dimensions and isolated the ultraviolet
poles $1/\epsilon = 2/(4-n)$. We find that we obtain a finite answer
to order $1/\Lambda^2$ if we adopt the following renormalization
scheme:\footnote{That is, a finite answer for the physical observables
$Z \rightarrow \overline{f} f$, not for quantities like the self-energies.}
\footnote{
In these expressions we have simply dropped any finite constants
arising from the loop calculation. These constants would only
be of interest in a complete effective field theory analysis
applied to a problem where all unknown constants can be measured
and additional predictions made. For example, that is the status of
the ${\cal O}(p^4)$ $\chi$PT theory for low energy strong interactions.}
\begin{eqnarray}
{v^2 \over \Lambda^2} L^r_{10}(\mu) &=& {v^2 \over \Lambda^2}
L_{10}-{1 \over 16 \pi^2} {1 \over 12}
\biggl( {1 \over \epsilon} + \log{\mu^2 \over M_Z^2} \biggr)
\nonumber \\
\beta_1^r(\mu) &=& \beta_1 -
{e^2 \over 16 \pi^2}{3 \over 2 c^2_Z}
\biggl( {1 \over \epsilon} + \log{\mu^2 \over M_Z^2} \biggr).
\label{renorm}
\end{eqnarray}
We thus replace the bare parameters $L_{10}$ and $\beta_1$ with the
scale dependent ones above. As a check of our answer, it is interesting
to note that we would also obtain a finite answer by adding to the
results of Appendix~B, the one-loop contributions to the self-energies
obtained in unitary gauge in the minimal standard model with one Higgs
boson in the loop. Equivalently, the expressions in Eq.~\ref{renorm}
correspond to the value of $L_{10}$ and $\beta_1$ at one-loop in the
minimal standard model within our renormalization scheme.\footnote{Our
result agrees with that of Refs.~\cite{longo,herrero}.}
Our result for $L_{10}$ at order $1/\Lambda^2$ is then:
\begin{equation}
{\delta \Gamma_f^{L_{10}} \over \Gamma_0(Z\rightarrow f\overline{f})}=
{e^2 \over c_Z^2 s_Z^2}L^r_{10}(\mu)
{v^2 \over \Lambda^2}{2r_f(l_f +r_f) \over
l_f^2+r_f^2}{c_Z^2 \over s_Z^2-c_Z^2}
\label{shten}
\end{equation}
Once again we point out that, at this order, the contribution of
$L_{10}$ to the
LEP observables occurs only through modifications to the self-energies
that are proportional to $q^2$. At this order it is therefore
possible to identify the effect of $L_{10}$ with the oblique
parameter $S$ or $\epsilon_3$. If we were to compute the effects
of $L_{10}$ at one-loop (as we do for the $L_{9L,9R}$), comparison with
$S$ would not be appropriate. Bounding $L_{10}$ from existing analyses
of $S$ or $\epsilon_3$ is complicated by the fact that the
same one-loop definitions must be used. For example, $L_{10}(\mu)$ receives
contributions from the standard model Higgs boson that are usually
included in the minimal standard model calculation. We will simply
associate our definition of $L_{10}(\mu)$ with {\it new} contributions
to $S$, beyond those coming from the minimal stardard
model.\footnote{We are being sloppy by not matching our $L_{10}$ at
one loop with the precise definitions used to renormalize the
standard model at one-loop. This does not matter for our present
purpose.}
Numerically we find the following $90\%$ confidence level bounds on $L_{10}$
when we take the scale $\Lambda=2$~TeV:
\begin{eqnarray}
\Gamma_e &\rightarrow & -1.7 \leq L^r_{10}(M_Z)_{new} \leq 3.3 \nonumber \\
R_h &\rightarrow & -1.5 \leq L^r_{10}(M_Z)_{new} \leq 2.0 \nonumber \\
\Gamma_Z &\rightarrow & -1.1 \leq L^r_{10}(M_Z)_{new} \leq 1.5
\label{tennum}
\end{eqnarray}
We can also bound the {\it leading order} effects of
$L_{10}$ from Altarelli's latest global fit
$\epsilon_3 = (3.4 \pm 1.8) \times 10^{-3}$ \cite{altanew}. To do this,
we subtract the standard model value obtained with $160\leq M_t \leq
190$~GeV and $65 \leq M_H \leq 1000$~GeV as read from Fig.~8 in
Ref.~\cite{altanew}. We obtain the $90\%$ confidence level interval:
\begin{equation}
-0.14 \leq L^r_{10}(M_Z)_{new} \leq 0.86
\label{sfalta}
\end{equation}
We can also compare directly with the result of Langacker
$S_{new}=-0.15\pm 0.25^{-0.08}_{+0.17}$\cite{langanew}, to obtain
$90\%$ confidence level limits:
\begin{equation}
-0.46 \leq L^r_{10}(M_Z)_{new} \leq 0.77
\label{sflanga}
\end{equation}
The results Eqs.\ref{sfalta}, \ref{sflanga} are better than our result
Eq.~\ref{tennum} because they correspond to global fits that include
all observables.
\subsection{Bounds on $L_{1,2}$ at order $1/\Lambda^4$.}
The couplings $L_{1,2}$ enter the one-loop
calculation of the $Z \rightarrow f \overline{f}$
width through four gauge boson couplings as depicted schematically
in Figure~1.
\begin{figure}[htb]
\centerline{\hfil\epsffile{f1.ps}\hfil}
\caption[]{Gauge boson self-energy diagrams involving the couplings $L_{1,2}$.}
\end{figure}
Our prescription
calls for using only the leading non-analytic contribution
to the process $Z \rightarrow f \overline{f}$. This contribution can be extracted
from the coefficient of the pole in $n-4$.
Care must be taken to isolate the poles of
ultraviolet origin (which are the only ones that interest us) from those
of infrared origin that appear in intermediate steps of the calculation
but that cancel as usual when one includes real emission processes as well.
We thus use the results of Appendix~B with the replacement:
\begin{equation}
{1 \over \epsilon} = {2 \over 4-n} \rightarrow \log\biggl({\mu^2 \over M_Z^2} \biggr)
\label{prescrip}
\end{equation}
to compute the contributions to the partial widths using Eq.~\ref{full}.
Since in unitary gauge $L_{1,2}$ modify only the four-gauge
boson couplings at the one-loop level, they
enter the calculation of the $Z$ partial widths only through the
self-energy corrections and Eq.~\ref{shifts}.
These operators induce a non-zero value for $\Delta \rho \equiv
\Pi_{WW}(0) / M_W^2 - \Pi_{ZZ}(0)/ M_Z^2$. For the observables
we are discussing, this is the {\it only} effect of $L_{1,2}$.
We do not place bounds on them from global fits of the oblique
parameter $T$ or $\epsilon_1$, because we have not shown that
this is the only effect of $L_{1,2}$ for the other observables
that enter the global fits. It is curious to see that even
though the operators with $L_1$ and $L_2$ violate the custodial
$SU(2)_C$ symmetry {\it only} through the hypercharge coupling,
their one-loop effect on the partial $Z$ widths is equivalent to
a $g^4$ contribution to $\Delta\rho$, on the same footing as two-loop
electroweak contributions to $\Delta\rho$ in the minimal standard
model. The calculation to ${\cal O}(1/\Lambda^4)$
can be made finite with the following renormalization of $\beta_1$:
\begin{equation}
\beta_1^r(\mu) = \beta_1
+{3 \over 4}{\alpha^2 (1 + c_Z^2)\over s_Z^2 c_Z^4}
\biggl(L_1 + {5 \over 2} L_2\biggr){v^2 \over \Lambda^2}
\biggl({1 \over \epsilon}+\log{\mu^2 \over M_Z^2}\biggr).
\label{shone}
\end{equation}
Using our prescription to bound the anomalous couplings, Eq.~\ref{prescrip},
we obtain for the $Z$ partial widths:
\begin{equation}
{\delta\Gamma_f^{L_{1,2}} \over \Gamma_0(Z\rightarrow f \overline{f})}=
-{3 \over 2}{\alpha^2 (1 + c_Z^2)\over s_Z^2 c_Z^4}
\biggl(L_1 + {5 \over 2} L_2\biggr){v^2 \over \Lambda^2}\log\biggl(
{\mu^2 \over M_Z^2}\biggr)
\biggl(1 +{2 r_f (l_f + r_f) \over l_f^2 + r_f^2}{c_Z^2 \over s_Z^2 - c_Z^2}
\biggr).
\label{resonetwo}
\end{equation}
Using $\Lambda=2$~TeV, and $\mu=1$~TeV we find $90\%$ confidence level bounds:
\begin{eqnarray}
\Gamma_e &\rightarrow & -50 \leq L_1 + {5 \over 2}L_2 \leq 26 \nonumber \\
\Gamma_\nu &\rightarrow & -28 \leq L_1 + {5 \over 2}L_2 \leq 59 \nonumber \\
R_h &\rightarrow & -190 \leq L_1 + {5 \over 2}L_2 \leq 130 \nonumber \\
\Gamma_Z &\rightarrow & -36 \leq L_1 + {5 \over 2}L_2 \leq 27
\label{rhoana}
\end{eqnarray}
Combined, they yield the result:
\begin{equation}
-28 \leq L_1 + {5 \over 2} L_2 \leq 26
\label{combineonetwo}
\end{equation}
shown in Figure~2.
\begin{figure}[htb]
\centerline{\hfil\epsffile{f2.ps}\hfil}
\caption[]{$90\%$ confidence level bounds on $L_{1,2}$ from the $Z\rightarrow
f {\overline f}$ partial widths, (Eq. 24). The allowed region is shaded.}
\end{figure}
As mentioned before, the effect of $L_{1,2}$ in
other observables is very different from that of $\beta_1$.\footnote{
An example are the observables discussed by us in Ref.\cite{bdv}.} It is
only for the $Z$ partial widths that we can make the ${\cal O}
(1/\Lambda^4)$ calculation finite with Eq.~\ref{shone}.
\subsection{Bounds on $L_{9L,9R}$ at order $1/\Lambda^4$.}
The operators with $L_{9L}$ and $L_{9R}$ affect the
$Z$ partial widths through Eqs.~\ref{vertex}, \ref{shifts}, and
\ref{mix}. We find it convenient to carry out this
calculation simultaneously with the one-loop effects of the lowest
order effective Lagrangian, Eq.~\ref{lagt}, because
the form of the three and four gauge
boson vertices induced by these two couplings is the same as that
arising from Eq.~\ref{lagt}. This can be seen from Eqs.~\ref{conv},~\ref{unot}
in Appendix~A. Performing the calculation in this way, we obtain a result
that contains terms of order $1/\Lambda^2$ (those independent of $L_{9L,9R}$),
terms of order $1/\Lambda^4$ proportional to $L_{9L,9R}$, and terms
of order $1/\Lambda^4$ proportional to $L_{10}$ and $\beta_1$.
As mentioned before, we keep these terms together
to check our answer by taking the limit $L_{9L}=L_{9R}=0$.
This also allows us to cast our answer
in terms of $g_1^Z$, $\kappa_\gamma$ and $\kappa_Z$ which is convenient
for comparison with other papers in the literature.
It is amusing to note that the divergences generated by the operators
$L_{9L,9R}$ in the one-loop (order $1/\Lambda^4$) calculation of the
$Z \rightarrow \overline{f} f$ widths can all be removed by the following
renormalization of the couplings in Eq.~\ref{oblique} (in the $M_t=0$ limit):
\begin{eqnarray}
\beta_1^r(\mu) &=& \beta_1 -
{\alpha \over \pi} {e^2\over 96 s_Z^4 c_Z^4}
{v^2 \over \Lambda^2}
\biggl[c_Z^2(1-20s_Z^2)L_{9L}+s_z^2(10-29c_Z^2)L_{9R}\biggr]
\biggl({1 \over \epsilon} + \log{\mu^2 \over M_Z^2}\biggr)
\nonumber \\
L^r_{10}(\mu)&=& L_{10} -
{\alpha \over \pi}{1\over 96 s_Z^2 c_Z^2}
\biggl[(1-24c_Z^2)L_{9L}+(32c_Z^2-1)L_{9R}\biggr]
\biggl({1 \over \epsilon} + \log{\mu^2 \over M_Z^2}\biggr)
\label{amusing}
\end{eqnarray}
This proves our assertion that our calculation to order ${\cal O}
(1/\Lambda^4)$ can be made finite by suitable renormalizations
of the parameters in Eq.~\ref{oblique}. However, we do not expect
this result to be true in general. That is, we expect that a
calculation of the one-loop contributions of the operators in
Eq.~\ref{lfour} to other observables will require counterterms
of order $1/\Lambda^4$. Thus, Eq.~\ref{amusing} does {\it not}
mean that we can place bounds on $L_{9L,9R}$ from global fits
to the parameters $S$ and $T$. Without performing a complete
analysis of the effective Lagrangian at order $1/\Lambda^4$
it is not possible to identify the renormalized parameters of
Eq.~\ref{amusing} with the ones corresponding to $S$ and $T$
that are used for global fits.
Combining all the results of Appendix~B into Eq.~\ref{full},
and keeping only terms
linear in $L_{9L,9R}$, we find after using Eq.~\ref{renorm}, and
our prescription Eq.~\ref{prescrip}:
\begin{eqnarray}
{\delta\Gamma^{L_9} \over \Gamma_0}&=&{\alpha^2 \over 24}
{1\over c_Z^4 s_Z^4}{v^2 \over \Lambda^2}
\log\biggl({\mu^2 \over M_Z^2}\biggr) \nonumber \\
&&\biggl\{ \biggl[L_{9L}(1-24c_Z^2)+L_{9R}(-1+32c_Z^2)\biggr]
{2r_f(l_f+r_f)\over l_f^2 +r_f^2}{c_Z^2 \over s_Z^2-c_Z^2}
\nonumber \\
&&+2\biggl[L_{9L}c_Z^2(1-20s_Z^2)+L_{9R}s_Z^2(10-29c_Z^2)\biggr]
\biggl(1+{2r_f(l_f+r_f)\over l_f^2 +r_f^2}
{c_Z^2 \over s_Z^2-c_Z^2}\biggr)\biggr\}
\nonumber \\
&&+{\alpha^2 \over 12}
\log\biggl({\mu^2 \over M_Z^2}\biggr)
{1+2c_Z^2 \over c_Z^4 s_Z^4 (l_b^2 +r_b^2)}
{v^2 \over \Lambda^2}
{M_t^2 \over M_Z^2}\biggl[L_{9R}s_Z^2-7L_{9L}c_Z^2\biggr]\delta_{fb}
\label{finalres}
\end{eqnarray}
The last term in Eq.~\ref{finalres} corresponds to the non-universal
corrections proportional to $M_t^2$ that are relevant only
for the decay $Z \rightarrow \overline{b} b$.
Using, as before,
$\Lambda=2$~TeV, $\mu=1$~TeV we find $90\%$ confidence level bounds:
\begin{eqnarray}
\Gamma_e &\rightarrow & -92 \leq L_{9L}+0.22L_{9R} \leq 47 \nonumber \\
\Gamma_\nu &\rightarrow & -79 \leq L_{9L}+1.02L_{9R} \leq 170 \nonumber \\
R_h &\rightarrow & -22 \leq L_{9L}-0.17L_{9R} \leq 16 \nonumber \\
\Gamma_Z &\rightarrow & -22 \leq L_{9L}-0.04L_{9R} \leq 17
\label{rhonum}
\end{eqnarray}
We show these inequalities in Figure~3.
\begin{figure}[htb]
\centerline{\hfil\epsffile{f3.ps}\hfil}
\caption[]{$90\%$ confidence level bounds on $L_{9L,9R}$ from the
$Z\rightarrow f {\overline f}$ partial widths, (Eq. 27). The allowed
region is shaded. The solid, dotted, dashed, and dot-dashed lines
are the bounds from $\Gamma_e$, $\Gamma_\nu$, $R_h$, and $\Gamma_Z$,
respectively.}
\end{figure}
If we bound one coupling at a time
we can read from Figure~3 that:
\begin{eqnarray}
-22 \leq &L_{9L}& \leq 16 \nonumber \\
-77 \leq &L_{9R}& \leq 94
\label{oneattime}
\end{eqnarray}
In a vector like model with $L_{9L}=L_{9R}$ we have
the $90\%$ confidence level bound:
\begin{equation}
-22<L_{9L}=L_{9R}< 18 .
\label{vectorbound}
\end{equation}
We can relate our couplings of Eq.~\ref{lfour} to the conventional
$g_1^Z$, $\kappa_\gamma$ and $\kappa_Z$ by identifying our unitary gauge
three gauge boson couplings with the conventional parameterization of
Ref.~\cite{hagi} as we do in Appendix~A. However, we must emphasize that
there is no unique correspondence between the two. Our framework assumes,
for example, $SU(2)_L\times U(1)_Y$ gauge invariance and this results in
specific relations between the three and four gauge boson couplings that are
different from those of Ref.~\cite{burgess} which assumes only electromagnetic
gauge invariance. Furthermore,
if one starts with the conventional parameterization of the three-gauge-boson
coupling and imposes $SU(2)_L\times U(1)_Y$ gauge invariance one does not
generate any additional two-gauge-boson couplings.
It is interesting to point out that within our formalism there are only
two independent couplings that contribute to three-gauge-boson couplings
($L_{9L,9R}$) but not to two-gauge-boson couplings (as $L_{10}$ does).
{}From this it follows that the equations for $g_1^Z$, $\kappa_\gamma$
and $\kappa_Z$ in terms of $L_{9L.9R,10}$ are not independent. In fact,
within our framework we have:
\begin{equation}
\kappa_Z = g_1^Z + {s_Z^2 \over c_Z^2}\biggl( 1 -\kappa_\gamma \biggr).
\label{depen}
\end{equation}
The same result holds in the formalism of Ref.~\cite{zeppenfeld}.
For the sake of comparison with the literature we translate the bounds
on $L_{9L}$ and $L_{9R}$ into bounds on $\Delta g_1^Z$, $\Delta \kappa_Z$
and $\Delta \kappa_\gamma$. For this exercise we set $L_{10}=0$.
We use $L_{9R}=0$ to obtain the bound
on $\Delta g_1^Z$. We then solve for $L_{9L}$ and $L_{9R}$ in terms of
$\Delta \kappa_Z$ and $\Delta \kappa_\gamma$, and bound each one of these
assuming the other one is zero. We obtain the $90\%$ confidence level
intervals:
\begin{eqnarray}
-0.08 < & \Delta g_1^Z& < 0.1\nonumber \\
-0.3 < & \Delta \kappa_Z & < 0.3\nonumber \\
-0.3 < & \Delta \kappa_\gamma & < 0.4 .
\label{ourdg}
\end{eqnarray}
Similarly, if there is a non-zero $L_{10}$, these couplings receive
contributions from it. Setting $L_{9L,9R}=0$, we find from Eq.~\ref{tennum}
the bounds: $-0.004 < \Delta g_1^Z < 0.005$,
$-0.003 < \Delta \kappa_Z < 0.004$, and
$-0.009 < \Delta \kappa_\gamma < 0.007$.
These bounds are stronger by a factor of about 20, just as the bounds on
$L_{10}$, Eq.~\ref{tennum} are stronger by about a factor of 20 than the
bounds on $L_{9L,9R}$, Eq.~\ref{oneattime}. However, these really are bounds
on the oblique corrections introduced by $L_{10}$ (which also contributes
to three gauge boson couplings due to $SU(2)_L\times U(1)_Y$ gauge invariance).
It is perhaps more relevant to consider the couplings of operators without
tree-level self-energy corrections. This results in Eq.~\ref{ourdg}.
\subsection{Effects proportional to $M_t^2$}
As can be seen from Eq.~\ref{finalres}, the $Z\rightarrow \overline{b} b$
partial width receives non-universal contributions proportional to
$M_t^2$. Within our renormalization scheme, the effects that correspond to
the minimal standard model do not occur. Our result corresponds entirely to
a new physics contribution of order $1/\Lambda^4$ proportional to
$L_{9L,9R}$. These effects have already been included to some extent in
the previous section when we compared the hadronic and total widths of the
$Z$ boson with their experimental values. In this section we isolate
the effect of the $M_t^2$ terms and concentrate on the $Z\rightarrow \overline{b} b$
width. Keeping the leading non-analytic contribution, as usual, we find:
\begin{equation}
{\Gamma(Z\rightarrow \overline{b} b) \over \Gamma_0}-1 =
{\alpha^2 \over 12}{1+2c_Z^2 \over c_Z^4 s_Z^4 (l_b^2 +r_b^2)}
{v^2 \over \Lambda^2}
{M_t^2 \over M_Z^2}\biggl[L_{9R}s_Z^2-7L_{9L}c_Z^2\biggr]
\log\biggl({\mu^2 \over M_Z^2}\biggr)
\label{nonuni}
\end{equation}
We use as before $\mu=1$~TeV, and we neglect the contributions
to the $Z \rightarrow \overline{b} b$ width that are not proportional to
$M_t^2$. We can then place bounds on the anomalous couplings by
comparing with Langacker's result $\delta_{bb}^{new}=0.022 \pm
0.011$ for $M_t = 175 \pm 16$~GeV \cite{langap}. Bounding the
couplings one at a time we find the
$90\%$ confidence level intervals:
\begin{eqnarray}
-50 \leq &L_{9L}& \leq -4 \nonumber \\
90 \leq & L_{9R}& \leq 1200 .
\label{dbblim}
\end{eqnarray}
Once again we find that there is much more sensitivity to
$L_{9L}$ than to $L_{9R}$. The fact that the $Z \rightarrow \overline{b}b$
vertex places asymmetric bounds on the couplings is, of course,
due to the present inconsistency between the measured value and
the minimal standard model result. Clearly, the implication that
the couplings $L_{9L,9R}$ have a definite sign cannot be taken
seriously. A better way to read Eq.~\ref{dbblim} is thus:
$|L_{9L}|\leq 50$ and $|L_{9R}|\leq 1200$.
\section{Discussion}
Several studies that bound these ``anomalous couplings'' using the
LEP observables can be found in the literature. Our present study
differs from those in two ways: we have included bounds on some
couplings that have not been previously considered, $L_{1,2}$ and
we discuss the other couplings, $L_{9L,9R}$ within an $SU(2)_C
\times U(1)_Y$ gauge invariant formalism. We now
discuss specific differences with some of the papers found in the
literature.
The authors of Ref.~\cite{hervegas} obtain their bounds by regularizing the
one-loop integrals in $n$ dimensions, isolating the poles in $n-2$ and
identifying these with quadratic divergences. This differs from our
approach where we keep only the (finite) terms proportional to the
logarithm of the renormalization scale $\log\mu$. To find bounds, the
authors of Ref.~\cite{hervegas} replace the poles in $n-2$ with
factors of $\Lambda^2/M_W^2$. We believe that this leads to the
artificially tight constraints \cite{quaddiv}
on the anomalous couplings quoted
in Ref.~\cite{hervegas} ($2\sigma$ limits):
$-9.4\times 10^{-3} \leq \beta_2 \leq 2.2\times 10^{-2}$ and
$-1.5\times 10^{-2} \leq \beta_3 \leq 3.9\times 10^{-2}$. We translate
these into $90\%$ confidence level intervals:
\begin{eqnarray}
-1.0 \leq &L_{9R} & \leq 2.4 \nonumber \\
-1.6 \leq &L_{9L} & \leq 4.2
\label{vegas}
\end{eqnarray}
which are an order of magnitude tighter than our bounds. Conceptually,
we see the divergences as being absorbed by renormalization of other
anomalous couplings. As shown in this paper, the calculation of the
$Z\rightarrow \overline{f}f$ can be rendered finite at order ${\cal O}(1/
\Lambda^4)$ by renormalization of $\beta_1$ and $L_{10}$. Thus, the
bounds obtained by Ref.~\cite{hervegas}, Eq.~\ref{vegas}, are really bounds on
$\beta_1$ and $L_{10}$. They embody the naturalness assumption
that all the coefficients that appear in the effective Lagrangian at
a given order are of the same size. Our formalism effectively
allows $L_{9L,9R}$ to be different from $L_{10}$.
The authors of Ref.~\cite{burgess} do not require that their effective
Lagrangian be $SU(2)_L\times U(1)_Y$ gauge invariant, and instead they are
satisfied with electromagnetic gauge invariance. At the technical level
this means that we differ in the four gauge boson vertices associated with the
anomalous couplings we study. It also means that we consider different
operators. In terms of the conventional anomalous three gauge boson couplings,
these authors quote $1\sigma$ results
$\Delta g_1^Z = -0.040 \pm 0.046$,
$\Delta\kappa_\gamma = 0.056 \pm 0.056$, and
$\Delta\kappa_Z = 0.004 \pm 0.042$.
These constraints are tighter than what we obtain from the contribution
of $L_{9L,9R}$ to $\Delta g_1^Z$, $\Delta\kappa_\gamma$ and
$\Delta\kappa_Z$, Eq.~\ref{ourdg}; they are weaker than what we obtain from the
contribution of $L_{10}$.
The authors of Ref.~\cite{zeppenfeld} require their effective Lagrangian to
be $SU(2)_L \times U(1)_Y$ gauge invariant, but they implement the symmetry
breaking linearly, with a Higgs boson field. The resulting power counting
is thus different from ours, as are the anomalous coupling constants. Their
study would be appropriate for a scenario in which the symmetry breaking
sector contains a relatively light Higgs boson. Their anomalous couplings
would parameterize the effects of the new physics not directly attributable
to the Higgs particle. Nevertheless, we can roughly compare our results to
theirs by using their bounds for the heavy Higgs case (case (d) in Figure~3
of Ref.~\cite{zeppenfeld}). To obtain their bounds they consider the case
where their couplings $f_B=f_W$ and $f_{WB}=0$ which corresponds to our
$L_{10}=0$, and $L_{9L}=L_{9R}$. For $M_t=170$~GeV they find the following
$90\%$ confidence level interval
$-0.05 \leq \Delta\kappa_\gamma \leq 0.12$, which we translate into:
\begin{equation}
-7.8 \leq L_{9L}=L_{9R} \leq 18.8
\label{zepp}
\end{equation}
This compares well with our bound
\begin{equation}
-22 \leq L_{9L}=L_{9R} \leq 18
\label{uscomp}
\end{equation}
Finally, if we look {\it only} at those corrections that are proportional
to $M_t^2 /M_W^2$ and that would dominate in the $M_t \rightarrow \infty$ limit, we
find that they only occur in the $Z \rightarrow \overline{b} b$ vertex. This means
that they can be studied in terms of the parameter $\epsilon_b$ of
Ref.~\cite{alta} or $\delta_{bb}$ of Ref.~\cite{langanew}. Converting our
result of Eq.~\ref{dbblim} to the usual anomalous couplings and
recalling that only two of them are independent at this order, we
find, for example:
\begin{eqnarray}
-0.28 \leq &\Delta g_1^Z& \leq -0.03 \nonumber \\
0.6 \leq &\Delta \kappa_\gamma& \leq 5.2
\label{compdbb}
\end{eqnarray}
This result is very similar to that obtained in Ref.~\cite{eboli}.
We now compare our results\footnote{
Our normalization of the $L_i$ is different from that
of Ref. \cite{boud,bdv,fls}. We have translated their results into
our notation.}
with bounds that future colliders are
expected to place on the anomalous couplings.
\begin{figure}[htb]
\centerline{\hfil\epsffile{f4.ps}\hfil}
\caption[]{Comparison of the $95\%$ confidence level bounds from the
$Z$ partial widths (shaded region) with that obtainable at LEPII
with $\sqrt{s}=196~GeV$ and $\int {\cal L}=500~pb^{-1}$,
(dotted contour) \cite{boud}.}
\end{figure}
In Fig. 4, we compare
our $95\%$ confidence level bounds on $L_{9L}$ and $L_{9R}$ with those which
can be obtained at LEPII with $\sqrt{s}=196$~GeV and an integrated
luminosity of $500~pb^{-1}$ \cite{boud,bm2}.
We find that LEP and LEPII are
sensitive to slightly different regions of the $L_{9L}$ and $L_{9R}$
parameter space, with the bounds from the two machines being of
the same order of magnitude. The authors of Ref.~\cite{fls} find that
the LHC would place bounds of order $L_{9L} < {\cal O}(30)$ and a factor of
two or three worse for $L_{9R}$. We find, Eq.~\ref{oneattime}, that
precision LEP measurements
already provide constraints at that level.
We again emphasize our caveat that
the bounds from LEP rely on naturalness arguments and are no substitute
for measurements in future colliders.
The limits presented here on the four point couplings
$L_{1}$ and $L_{2}$ are the first available for these couplings.
They will be measured directly at the LHC. Assuming a coupling
is observable if it induces a $50\%$ change in the high momentum
integrated cross section, Ref.~\cite{bdv} estimated that the
LHC will be sensitive to $\mid L_{1}, L_{2}\mid \sim {\cal O}(1)$, which is
considerably stronger that the bound obtained from the $Z$
partial widths.
\section{Conclusions}
We have used an effective field theory formalism to place
bounds on some non-standard model gauge boson couplings.
We have assumed that the electroweak interactions are an
$SU(2)_L \times U(1)_Y$ gauge theory with an unknown,
but strongly interacting, scalar
sector responsible for spontaneous symmetry breaking.
Computing the leading contribution of each operator, and allowing
only one non-zero coefficient at a time, our $90~\%$ confidence level
bounds are:
\begin{eqnarray}
-1.1 < &L_{10}^r(M_Z)_{new}& < 1.5 \nonumber \\
-28 < &L_1& < 26 \nonumber \\
-11 < &L_2& < 11 \nonumber \\
-22 < &L_{9L}& < 16 \nonumber \\
-77 < &L_{9R}& < 94.
\label{rescon}
\end{eqnarray}
Two parameter bounds on ($L_1,L_2$) and ($L_{9L},L_{9R}$) are
given in the text. The bounds on $L_1,L_2$ are the first
experimental bounds on these couplings. The bounds on $L_{9L}$
and $L_{9R}$ are of the same order of magnitude as those which
will be obtained at LEPII and the LHC.
\vspace {1.in}
\noindent{\bf Acknowledgements} The work of G. V. was supported in part
by a DOE OJI award. G.V. thanks the theory group at BNL for their
hospitality while part of this work was performed.
We are grateful to W. Bardeen, J.~F.~Donoghue, E. Laenen,
W. Marciano, A.~Sopczak, and A. Sirlin for useful discussions.
We thank P.~Langacker for providing us with his latest numbers.
We thank F.~Boudjema for providing us with the data file for
the LEPII bounds in Figure~4.
|
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| 3,455
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{"url":"http:\/\/math.stackexchange.com\/questions\/59092\/neighborhood-and-open-sets","text":"# Neighborhood and open sets\n\nShow that every neighborhood is an open set.\n\nLet $E = N_{r}(p)$. Then I want to show that for every $q \\in E$, $q$ is an interior point. This means I just have to find a neighborhood of $q$ which is in $E$?\n\n-\nNot every neighborhood is an open set. Wkipedia states this en.wikipedia.org\/wiki\/Neighbourhood_%28mathematics%29 (\"Note that the neighbourhood V need not be an open set itself\"). Also, what's $N_r(p)$? \u2013\u00a0George Lowther Aug 22 '11 at 23:19\nMaybe you're trying to show that every ball (something of the form $N_r(p)$, in your notation) in a metric space is open? A neighborhood of a point $p$ is usually an open set containing $p$, or a set containing an open set containing $p$, depending on the author. \u2013\u00a0Dylan Moreland Aug 22 '11 at 23:22\nAnyway, if I'm right about the nature of your actual question then I think it would be good to draw a picture (so we're in the setting of $\\mathbf R^2$ with the usual metric). \u2013\u00a0Dylan Moreland Aug 22 '11 at 23:25\nHow do you define neighborhood? \u2013\u00a0lhf Aug 22 '11 at 23:32\n\nDylan's suggestion of drawing a picture is good. Here we have our point $p$, the open ball $E=N_r(p)$, and an arbitrary point $q\\in E$. I have labeled the distance from $p$ to $q$ as $d$.\nIf Dylan and I understand the question correctly, you want to show that there is some $s>0$ for which the open ball $N_s(q)$ is contained in $E$ (i.e. $N_s(q)\\subseteq E$)? If that's the case, then can you think of an $s$, depending on $r$ and $d$, for which this is true? Think about drawing the circle of radius $s$ around the point $q$ in the picture.\n $s = r-d$ would work? \u2013\u00a0James Aug 23 '11 at 0:05 @James: Yup! This follows from the triangle inequality, which is part of the axioms of a metric space. \u2013\u00a0Zev Chonoles Aug 23 '11 at 0:08","date":"2013-05-22 16:31:34","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\": 0.9100916385650635, \"perplexity\": 156.12452972090634}, \"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-2013-20\/segments\/1368702019913\/warc\/CC-MAIN-20130516110019-00073-ip-10-60-113-184.ec2.internal.warc.gz\"}"}
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{"url":"https:\/\/curriculum.illustrativemathematics.org\/k5\/teachers\/grade-3\/unit-7\/lesson-6\/lesson.html","text":"# Lesson 6\n\nDistance Around Shapes\n\n## Warm-up: Notice and Wonder: Paper Clips & Shapes (10 minutes)\n\n### Narrative\n\nThe purpose of this warm-up is for students to visualize the idea of perimeter and elicit observations about distances around a shape. It also familiarizes students with the context\u00a0and materials they will be working with in the next activity, where they will use paper clips to form the boundary of shapes and compare or quantify their lengths.\n\n### Launch\n\n\u2022 Groups of 2\n\u2022 Display the image.\n\u2022 \u201cWhat do you notice? What do you wonder?\u201d\n\u2022 1 minute: quiet think time\n\n### Activity\n\n\u2022 1 minute: partner discussion\n\u2022 Share and record responses.\n\n### Student Facing\n\nWhat do you notice? What do you wonder?\n\n### Student Response\n\nFor access, consult one of our IM Certified Partners.\n\n### Activity Synthesis\n\n\u2022 \u201cCan you predict how many paper clips it might take to go around the whole shape?\u201d\n\u2022 Consider asking: \u201cThe paper clips in the picture are standard paper clips. If we use jumbo paper clips, which are larger, would we need more or fewer paper clips to build the shape?\u201d\n\n## Activity 1: What Does It Take to Build the Shapes? (15 minutes)\n\n### Narrative\n\nThe purpose of this activity is to give students a concrete experience of building the boundary of shapes and quantifying that length of the boundary, allowing them to conceptualize perimeter as a measurable geometric attribute. Students use $$1\\frac{1}{4}$$-inch\u00a0paper clips as the units for measuring the distances around four shapes. The reasoning here prepares them to reason about equal-size intervals that can be marked on the sides of a shape to measure its length (as students will see in the next activity).\n\n### Required Materials\n\nMaterials to Gather\n\nMaterials to Copy\n\n\u2022 What Does It Take to Build the Shapes?\n\n### Required Preparation\n\n\u2022 Each group of 4 needs 25-50 paper clips that are $$1\\frac{1}{4}$$-inch long each.\n\u2022 If using 1-inch paper clips, use 80% scale when making copies of the\u00a0blackline masters.\n\n### Launch\n\n\u2022 Groups of 4\n\u2022 Give each group a copy of the blackline master and 25\u201350 paper clips.\n\u2022 \u201cMake a prediction: Which shape do you think will take the most paper clips to build?\u201d\n\u2022 30 seconds: quiet think time\n\u2022 Poll the class on whether they think shape A, B, C, or D would take the most paper clips to build.\n\n### Activity\n\n\u2022 \u201cWork with your group to find out which shape takes the most paper clips to build. You may need to take turns with the paper clips.\u201d\n\u2022 5\u20137 minutes: small-group work time\n\u2022 Monitor for students who:\n\u2022 place paper clips one at a time and then count the total\n\u2022 place paper clips one at a time and then add up the numbers on all sides\n\u2022 use multiplication ($$3 \\times 3$$ for B, $$(2 \\times 4) + (2 \\times 2)$$ for C, or $$(2 \\times 6) + 2$$ for D)\n\u2022 use estimation\n\n### Student Facing\n\nYour teacher will give you four shapes on paper and some paper clips.\n\nWork with your group to find out which shape takes the most paper clips to build. Explain or show how you know. Record your findings here. Draw sketches if they are helpful.\n\nA\n\nB\n\nC\n\nD\n\n### Student Response\n\nFor access, consult one of our IM Certified Partners.\n\n### Activity Synthesis\n\n\u2022 Select previously identified students to share their responses and strategies for finding the number of paper clips. Record the different strategies for all to see.\n\u2022 \u201cWhat we have done in this activity is to find the length of the boundary of shapes. The boundary of a flat shape is called its perimeter.\u201d\n\u2022 \u201cWhen we find the distance all the way around a shape, like we did with paper clips, we are measuring its perimeter.\u201d\n\u2022 \u201cTo find the length of the perimeter of a shape, we can find the sum of its side lengths.\u201d\n\u2022 \u201cWe can say that the perimeter of shape A is 13 paper clips long.\u201d\n\u2022 \u201cWhen we say \u2018find the perimeter,\u2019 we mean find the length of the perimeter.\u201d\n\n## Activity 2: Distance Around (20 minutes)\n\n### Narrative\n\nIn this activity, students find the perimeter of shapes\u2014first on dot paper, and then using the tick marks on the sides of the shapes. Students may\u00a0need a reminder that when we measure length, we count the number of length-units, not the number of endpoints. While students may count the tick marks on all sides and add them, they may also observe that some side lengths are the same, especially on shapes A and B, and use this structure and multiplication by 2 to find the perimeter\u00a0efficiently (MP8).\n\nMLR8 Discussion Supports. Synthesis: To support the transfer of new vocabulary to long-term memory, invite students to chorally repeat these phrases in unison 1\u20132 times: perimeter and distance around a shape.\nAction and Expression: Develop Expression and Communication. Synthesis: Identify connections between strategies that result in the same outcomes but use differing approaches.\nSupports accessibility for: Visual-Spatial Processing\n\n### Launch\n\n\u2022 Groups of 2\n\u2022 \u201cEarlier, we used paper clips to measure the distance around shapes. What are some other units we could use to measure distances or lengths?\u201d (The side length of a square on grid paper, the distance between points on dot paper, centimeter, inch, foot)\n\u2022 \u201cLet\u2019s find the length of the perimeter of some shapes on dot paper and some shapes whose side lengths are shown with tick marks.\u201d\n\n### Activity\n\n\u2022 \u201cWork independently to find the perimeter of each shape. Afterwards, share your responses with your partner.\u201d\n\u2022 5 minutes: independent work time\n\u2022 3\u20135 minutes: partner discussion\n\u2022 Monitor for the different strategies students use to find the perimeter of each shape, such as counting one unit at a time, adding the number of length units of all sides, and using multiplication.\n\n### Student Facing\n\nFind the perimeter of each shape. Explain or show your reasoning.\n\n### Student Response\n\nFor access, consult one of our IM Certified Partners.\n\nIf students lose track of the side lengths they are counting, consider asking:\n\n\u2022 \u201cHow are you finding the distance around the shape?\u201d\n\u2022 \u201cHow could you keep track of the side lengths as you work?\u201d\n\n### Activity Synthesis\n\n\u2022 Select students to share their responses and strategies. Start with those who counted individual units and move toward those who use operations.\n\u2022 Record the strategies for all to see.\n\u2022 \u201cWhich shape, C, D, or E,\u00a0has the greatest perimeter?\u201d (C. Its perimeter is 33 units. The other two are 30 and 32 units.)\n\u2022 Consider asking: \u201cWhich shape has a greater perimeter: the rectangle in the first problem or the triangle?\u201d (The triangle has 33 units while the rectangle has 22 units, but the units aren\u2019t the same length, so we don\u2019t know which perimeter is greater.)\n\n## Lesson Synthesis\n\n### Lesson Synthesis\n\n\u201cToday we learned what perimeter is. How would you describe perimeter to a friend?\u201d (Perimeter is the distance around a shape or the length around a shape. It\u2019s the length of all the sides added together.)\n\n\u201cHow do you find the perimeter of a shape?\u201d (We can count the number of units all the way around a shape or add up the number of units on each side.)\n\n\u201cOne situation where we might find the perimeter of a shape is when putting a frame around a piece of artwork. The perimeter of the artwork can tell us how much framing material we need.\u201d\n\n\u201cCan you think of other situations where it might be helpful to find the perimeter of a shape?\u201d (Enclosing a yard with a fence. Decorating the edges of a piece of paper with ribbon.)\n\n## Cool-down: What is the Perimeter? (5 minutes)\n\n### Cool-Down\n\nFor access, consult one of our IM Certified Partners.","date":"2021-12-08 21:59:40","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 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\": 0.4570992588996887, \"perplexity\": 1914.0357451290201}, \"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-2021-49\/segments\/1637964363598.57\/warc\/CC-MAIN-20211208205849-20211208235849-00419.warc.gz\"}"}
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Recently in Webcomics Category
The Busiest Damn Week of These People's Lives
By Zach on May 4, 2007 12:05 AM
Project for when I am bored: Go through the Questionable Content archives and figure out exactly how much time has passed within the comic since the first strip. The strip is great, but the pacing is very, very slow and the author, Jeph Jacques, is unusually consistent about having each series of strips occur on the same day, then moving on to the next day, and so on. The trick is, though, that the plot moves along in real-time, even as time moves slowly within the strip. It's jarring when a new day dawns in QC's Boston and you look back through the archives and realize that the last three months of plot developments all occurred in one day. Seriously, I'll bet the events of full four years of the strip thus far have transpired over the course of two weeks in the characters's lives. A month at the outside.
"I try not to encourage people visiting me. It sometimes causes them to think I have interest in their company."
By Zach on April 19, 2006 11:30 PM | 6 Comments
While we're recommending webcomics, I thought I should put in a word for my current personal favorite, Something Positive.
Something Positive's interesting and unique, as webcomics go. It started out as a fairly simple gag-a-day strip, but has gradually become a lot more serious and dramatic. That's actually not too uncommon; the same trajectory from funny-to-teh-drama happens in a significant portion of webcomics. The difference in Something Positive is that R.K. Milholland has managed to successfully transition from a pure gag-a-day strip to a strip with well-developed characters and an ongoing plot without sacrificing humor or changing the fundamental feel of the comic. The first strip is cynical and witty and it maintains that same essential tone throughout the whole run. It's a pretty rare strip that doesn't have some kind of joke, and the few that are pure drama are handled so well that it's hard to complain about them.
The strip started out with a fairly small cast of real characters and a coterie of cardboard cut-outs. Since then the main cast has grown and evolved, but done so in a very gradual way. Davan now is different than Davan four years ago, but it's not the result of any hammer-on-the-head life-changing after-school-special moments. And that's happened with all of the main characters. Meanwhile, the background characters have been developed and fleshed out into genuine people. It's weird to go back and read Monette in the early strips, where she's played as just a big dumb pseudo-lesbian, because now she's a full-on main character with emotions and depths. She's (literally) become a part of the family.
The plot's also well-done. Milholland does an excellent job of developing the story in a way that you don't necessarily expect, but he does it without cheating. The movement of the plot makes complete sense when you go back and re-read the strips, even if you didn't see where things were going at the time. Another unique thing about S*P is the way that a lot of the plot is told in a very subtle way. Milholland respects his audience enough not to hit them over the head explaining things. You're expected to pick up on certain plot points that happen in the background while the strip is looking elsewhere.
Reading through the archives, the strip starts out with a pretty vicious tone. The first strip is representative of the nastiness that was S*P early on. As the comic has evolved the humor's gotten a bit softer at the edges, but no less cynical. The surprising thing about Something Positive is that a strip that started off with a joke in astoundingly poor taste about abortion could evolve into what is probably the most emotional and affecting webcomic on the scene today. The characters are cynical bastards, but they're cynical bastards you come to really care about. So even if the early strips turn you off, I'd recommend plunging through and seeing if it doesn't grow on you. I'd recommend going through at least April 21, 2002. I also would advise against starting at the current strip and working backwards; there are a lot of spoilers in the more recent strips, and it'd be a shame to ruin some of the surprises. Also, don't read the cast page. It's up to date, which is nice, but that means it's full of spoilers.
Oh, also, Choo-Choo Bear. Choo-Choo Bear is impossible not to love. And that's good, because chemo-kitties need extra love!
Four Posts! One Day! No Real Content!
By Zach on March 21, 2006 11:59 PM | 5 Comments
I see that Questionable Content is touching on the sexy librarian thing today. Having said that, today might not be the greatest Questionable Content for the first-time reader. As the drama has ramped up lately, the funny has gotten a tad stale. The artwork has gotten pretty amazing lately, though, and I quite enjoy the strip now, just in a somewhat different way than I did before. I think Jeph Jacques's doing a good job of keeping QC a gag-a-day strip, even as he moves it in a more serious direction and develops the characters. Things are sort of in transition now for the characters, and once things get settled again I'm confident the funniness will return to prior levels. In the meantime, I recommend going through the archives.
Though, having said that, my heart belongs to R.K. Milholland. If you only read one webcomic on my recommendation, read Something Positive. That is all.
Hello, Something Positive readers!
By Zach on November 4, 2005 3:39 PM
Man, I'm getting a hell of a lot of google hits from people searching for Something Positive Nancy. Which just goes to show that I'm not the only one who was confused by Milholland's pulling an obscure background character out of his hat. In any case, sit and stay for a while! This isn't a Something Positive blog, or even primarily a webcomics blog, but maybe you'll find something interesting. Feel free to comment.
Also, Nancy is one of Aubrey's girls from Nerdrotica. She first showed up in the Class of Ninety-Bore storyline, when she was one of the girls Jason brought down to help Davan show up his classmates. You'll note that in that storyline they hit it off, but it didn't go anywhere until now. Since then there have been a few mentions of her, for instance in the series around when Episode 3 came out Aubrey mentions her being the first one to get her alien request form in. Further, if you look at the post-its on Davan's desk from when he was working at the insurance company, they all say "Nancy Called," indicating that Nancy's been trying to contact him a lot and, until now, he hasn't reciprocated. Hope this helps!
R.K. Milholland needs to update his character page
By Zach on October 27, 2005 10:31 AM
Does anyone out there read Something Positive? Does anyone know who "Nancy," the girl who showed up in the most recent strip, is? Davan acts as though he knows her, and they have banter right out the gate, but I can't tell if this is one of Davan's friends who left Boston before the strip started or if this is just one of the many characters who have come and gone in the strip since the last time Milholland updated his character page. If she has appeared in prior strips, she probably had a different hair color. Any ideas?
UPDATE: Ah, found it. She was mentioned in this strip. For those curious, she's one of Aubrey's Nerdrotica phone sex girls. It's possible that she's made a physical appearance elsewhere in the strip, but she's only just mentioned in that strip. In any case, I think I can be forgiven for forgetting about her.
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-Mob Justice – You call Me a Bad Name?
It is said that "If you want to kill a dog, give it a bad name". Someone who does not appreciate you would always have something negative to say about you, real or unreal. As humans, in our inter-personal interactions and relations, things happen and we find ourselves speaking about another in a negative light. This may be normal, but often not the best of circumstance, especially where it completely denies the person of any positive qualities, or castigated as though he or she has no atom of goodness. A person may have some weak points, but he or she has some strengths and good qualities too. Though, it does not mean that we deny when something goes wrong or we colour 'black' situations as 'white'. It is still important to call a spade a spade, especially to one's self.
However, the emphasis in this piece is when we colour a person "bad", not from an objective point of view. There is something about this person that you just don't like; not necessarily that the person has done any wrong thing to you – "You colour me bad." This could be out of envy and jealousy, which may even be unconscious. In fact, when asked if you envy this person, you would easily say that you have nothing to envy about this person.
Could it be that this person is generally doing well, or s/he is generally friendly and easily liked by others? This becomes a reason to "Colour me bad". However, remember that giving me a bad name does not add to your good name.
Such happens in everyday life. The bad name calling happens in families, communities, social groups, work place settings, in politics and in many human engagements.
As parents, if we make a habit of always colouring our kids bad, a lot of that tends to stick with them through life. Surely, no parent wants that!
In our public sphere today, we hear of and may have seen mob justice take place. Have you ever witnessed a situation of mob-justice? At such moments, you see human beings let loose and behave like animals. They descend on their victim as though their lives depended on it. All they craze for at that moment is to see the blood of the victim flow. A quick look tells you that the crowd would hurt, injure and would kill the victim so easily. "You colour me bad and kill me." Even persons who have no idea if this is an innocent victim or not, hold on to the name called and pick up stones to end that life.
Mind you, the criminal guys who are mobbed are no less cruel in calling their victims a bad name. They steal, attack, destroy, loot and/or kill as though other humans were just inanimate objects.
Once I had gone to the market to plait my hair, and suddenly, there was an uproar. A crowd of market persons began crowding and following a young man. What was his offence? He had just snatched a phone from a lady sitting in a parked vehicle. My! The place was in chaos. Even the young man attending to me suddenly left and picked up his own item to club the alleged thief.
In no time, the young man was already bleeding from the head. He had to return the phone before he was left to walk away, nursing his bleeding head, as he held it with his shirt. As he walked past alone, I thought to myself, "This guy would have simply lost his precious life to mob justice for a phone theft!".
When you speak to persons involved in mob justice, they give you multiple reasons why they take the law into their hands rather than take the person or persons to the police. One reason they would tell you is that it serves as an immediate warning to others to desist from similar acts or crimes. They people in the locality may not have enough confidence in the local police to deal with the issue. No reason justifies mob justice as it could also become a tool for revenge for even simple offences done against someone. Besides, the law on Presumption of Innocence protects a person as being innocent until proven guilty.
Back to calling me bad, if someone comes to you venting negatively about another person, don't be quick to add to the bitterness, hurt and pain. Perhaps, this person simply needs a listening ear – just hear me out. Or perhaps, you could point out a different perspective or way of looking at the issue, to the person. However, it may be that you too have suffered some negative experience from that person, and this complaint prompts you to also begin venting your anger.
All in all, this person you speak about can't be all bad. Now that you know his/her weak points, perhaps you can now see how best to relate with him/her without digging the negative experience deeper, by allowing it affect your peace and goodwill towards the person.
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| 8,562
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{"url":"https:\/\/dspace.kaist.ac.kr\/handle\/10203\/31649","text":"#### Molecular reaction dynamics study using high resolution and high sensitivity spectroscopy = \uace0\ubd84\ud574\ub2a5 \uace0\uac10\ub3c4 \ub808\uc774\uc800 \ubd84\uad11\ud559\uc744 \uc774\uc6a9\ud55c \ub2e4\uc6d0\uc790\ubd84\uc790\uc758 \ub3d9\ub825\ud559 \uc5f0\uad6c\n\nCited 0 time in Cited 0 time in\n\u2022 Hit :\u00a0304\n$\\emph{Absolute primary H atom quantum yield measurements in the 193.3 nm and 121.6 nm photodissociation of acetylene}$ Absolute quantum yields for primary H atom formation ($\u03c6_H$) were measured under collision-free conditions for the room-temperature gas-phase dissociation of acetylene ($C_2H_2$) after photoexcitation at 193.3 nm and at the H-atom Lyman-\u03b1 wavelength (121.6 nm) by a pulsed laser photolysis (LP)-laser-induced fluorescence (LIF) \u2018pump-and-probe\u2019 technique. Using HCl and $CH_4$ photolysis at 193.3 and 121.6 nm, respectively, as a reference, values of $\u03c6_H$ (193:3 nm) = 0.94\u00b10.12 and $\u03c6_H$ (121.6 nm) = 1.04\u00b10.16 were obtained which demonstrate that for both photolysis wavelengths the H + $C_2H$ product channel dominates the primary $C_2H_2$ photochemistry. $\\emph{Absolute quantum yield measurements for the formation of oxygen atoms after UV laser excitation of$SO_2$at 222.4 nm}$ The dynamics of formation of oxygen atoms after UV photoexcitation of $SO_2$ in the gas-phase was studied by pulsed laser photolysis (LP)-laser-induced fluorescence \u2018pump-and-probe\u2019 technique in a flow reactor. $SO_2$ at room-temperature was excited at the KrCl excimer laser wavelength (222.4 nm) and $O(^3P_j)$ photofragments were detected under collision-free conditions by vacuum ultraviolet (VUV) laser-induced fluorescence (LIF). The use of narrow-band probe laser radiation, generated via resonant third-order sum-difference frequency conversion of dye laser radiation in Krypton, allowed the measurement of the nascent $O(^3P_{j=2,1,0})$ fine-structure state distribution: $n_{j=2}\/n_{j=1}\/n_{j=0}$ = (0.88 \u00b1 0.02)\/(0.10 \u00b1 0.01)\/(0.02 \u00b1 0.01). Employing $NO_2$ photolysis as a reference, a value of $\u03c6_{o(^3p)}$=0.13 \u00b1 0.05 for the absolute $O(^3P)$ atom quantum yield was determined. The measured $O(^3P)$ quantum yield is compared with the results of earlier fluorescence quantum yield measurements. A suitable mechanism is suggested in which the dissociation proceeds via in...\nRyoo, RyongresearcherJung, Kyung-Hoonresearcher\uc720\ub8e1researcher\uc815\uacbd\ud6c8researcher\nDescription\n\ud55c\uad6d\uacfc\ud559\uae30\uc220\uc6d0 : \ud654\ud559\uacfc,\nPublisher\n\ud55c\uad6d\uacfc\ud559\uae30\uc220\uc6d0\nIssue Date\n2005\nIdentifier\n244765\/325007 \u00a0\/\u00a0020015184\nLanguage\neng\nDescription\n\n\ud559\uc704\ub17c\ubb38(\ubc15\uc0ac) - \ud55c\uad6d\uacfc\ud559\uae30\uc220\uc6d0 : \ud654\ud559\uacfc, 2005.2, [ xi, 133 p. ]\n\nKeywords\n\nPhotodissociation;\u00a0vinyl bromide;\u00a0acetylene;\u00a0\uad11\ubd84\ud574;\u00a0\ubc18\uc751 \ub3d9\ub825\ud559;\u00a0\ubd84\uad11\ud559;\u00a0\ube44\ub2d0 \ube0c\ub85c\ub9c8\uc774\ub4dc;\u00a0\uc544\uc138\ud2f8\ub80c;\u00a0reaction dynamics;\u00a0spectroscopy\n\nURI\nhttp:\/\/hdl.handle.net\/10203\/31649","date":"2020-11-25 03:49:39","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 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\": 0.3548648953437805, \"perplexity\": 9999.276181691786}, \"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-2020-50\/segments\/1606141180636.17\/warc\/CC-MAIN-20201125012933-20201125042933-00495.warc.gz\"}"}
| null | null |
'use strict';
import 'vs/css!./hover';
import * as nls from 'vs/nls';
import {ListenerUnbind} from 'vs/base/common/eventEmitter';
import {KeyCode, KeyMod} from 'vs/base/common/keyCodes';
import * as platform from 'vs/base/common/platform';
import {TPromise} from 'vs/base/common/winjs.base';
import {IKeyboardEvent} from 'vs/base/browser/keyboardEvent';
import {IOpenerService} from 'vs/platform/opener/common/opener';
import {KbExpr} from 'vs/platform/keybinding/common/keybindingService';
import {Range} from 'vs/editor/common/core/range';
import {EditorAction} from 'vs/editor/common/editorAction';
import {Behaviour} from 'vs/editor/common/editorActionEnablement';
import * as editorCommon from 'vs/editor/common/editorCommon';
import {CommonEditorRegistry, ContextKey, EditorActionDescriptor} from 'vs/editor/common/editorCommonExtensions';
import {ICodeEditor, IEditorMouseEvent} from 'vs/editor/browser/editorBrowser';
import {EditorBrowserRegistry} from 'vs/editor/browser/editorBrowserExtensions';
import {ModesContentHoverWidget} from './modesContentHover';
import {ModesGlyphHoverWidget} from './modesGlyphHover';
class ModesHoverController implements editorCommon.IEditorContribution {
static ID = 'editor.contrib.hover';
private _editor: ICodeEditor;
private _toUnhook:ListenerUnbind[];
private _contentWidget: ModesContentHoverWidget;
private _glyphWidget: ModesGlyphHoverWidget;
static getModesHoverController(editor: editorCommon.ICommonCodeEditor): ModesHoverController {
return <ModesHoverController>editor.getContribution(ModesHoverController.ID);
}
constructor(editor: ICodeEditor,
@IOpenerService openerService: IOpenerService
) {
this._editor = editor;
this._toUnhook = [];
if (editor.getConfiguration().hover) {
this._toUnhook.push(this._editor.addListener(editorCommon.EventType.MouseDown, (e: IEditorMouseEvent) => this._onEditorMouseDown(e)));
this._toUnhook.push(this._editor.addListener(editorCommon.EventType.MouseMove, (e: IEditorMouseEvent) => this._onEditorMouseMove(e)));
this._toUnhook.push(this._editor.addListener(editorCommon.EventType.MouseLeave, (e: IEditorMouseEvent) => this._hideWidgets()));
this._toUnhook.push(this._editor.addListener(editorCommon.EventType.KeyDown, (e:IKeyboardEvent) => this._onKeyDown(e)));
this._toUnhook.push(this._editor.addListener(editorCommon.EventType.ModelChanged, () => this._hideWidgets()));
this._toUnhook.push(this._editor.addListener(editorCommon.EventType.ModelDecorationsChanged, () => this._onModelDecorationsChanged()));
this._toUnhook.push(this._editor.addListener('scroll', () => this._hideWidgets()));
this._contentWidget = new ModesContentHoverWidget(editor, openerService);
this._glyphWidget = new ModesGlyphHoverWidget(editor);
}
}
private _onModelDecorationsChanged(): void {
this._contentWidget.onModelDecorationsChanged();
this._glyphWidget.onModelDecorationsChanged();
}
private _onEditorMouseDown(mouseEvent: IEditorMouseEvent): void {
var targetType = mouseEvent.target.type;
if (targetType === editorCommon.MouseTargetType.CONTENT_WIDGET && mouseEvent.target.detail === ModesContentHoverWidget.ID) {
// mouse down on top of content hover widget
return;
}
if (targetType === editorCommon.MouseTargetType.OVERLAY_WIDGET && mouseEvent.target.detail === ModesGlyphHoverWidget.ID) {
// mouse down on top of overlay hover widget
return;
}
this._hideWidgets();
}
private _onEditorMouseMove(mouseEvent: IEditorMouseEvent): void {
var targetType = mouseEvent.target.type;
var stopKey = platform.isMacintosh ? 'metaKey' : 'ctrlKey';
if (targetType === editorCommon.MouseTargetType.CONTENT_WIDGET && mouseEvent.target.detail === ModesContentHoverWidget.ID && !mouseEvent.event[stopKey]) {
// mouse moved on top of content hover widget
return;
}
if (targetType === editorCommon.MouseTargetType.OVERLAY_WIDGET && mouseEvent.target.detail === ModesGlyphHoverWidget.ID && !mouseEvent.event[stopKey]) {
// mouse moved on top of overlay hover widget
return;
}
if (this._editor.getConfiguration().hover && targetType === editorCommon.MouseTargetType.CONTENT_TEXT) {
this._glyphWidget.hide();
this._contentWidget.startShowingAt(mouseEvent.target.range, false);
} else if (targetType === editorCommon.MouseTargetType.GUTTER_GLYPH_MARGIN) {
this._contentWidget.hide();
this._glyphWidget.startShowingAt(mouseEvent.target.position.lineNumber);
} else {
this._hideWidgets();
}
}
private _onKeyDown(e: IKeyboardEvent): void {
var stopKey = platform.isMacintosh ? KeyCode.Meta : KeyCode.Ctrl;
if (e.keyCode !== stopKey) {
// Do not hide hover when Ctrl/Meta is pressed
this._hideWidgets();
}
}
private _hideWidgets(): void {
this._glyphWidget.hide();
this._contentWidget.hide();
}
public showContentHover(range: editorCommon.IEditorRange, focus: boolean): void {
this._contentWidget.startShowingAt(range, focus);
}
public getId(): string {
return ModesHoverController.ID;
}
public dispose(): void {
while(this._toUnhook.length > 0) {
this._toUnhook.pop()();
}
if (this._glyphWidget) {
this._glyphWidget.dispose();
this._glyphWidget = null;
}
if (this._contentWidget) {
this._contentWidget.dispose();
this._contentWidget = null;
}
}
}
class ShowHoverAction extends EditorAction {
static ID = 'editor.action.showHover';
constructor(descriptor: editorCommon.IEditorActionDescriptorData, editor: editorCommon.ICommonCodeEditor) {
super(descriptor, editor, Behaviour.TextFocus);
}
public run(): TPromise<any> {
const position = this.editor.getPosition();
const range = new Range(position.lineNumber, position.column, position.lineNumber, position.column);
(<ModesHoverController>this.editor.getContribution(ModesHoverController.ID)).showContentHover(range, true);
return TPromise.as(null);
}
}
EditorBrowserRegistry.registerEditorContribution(ModesHoverController);
CommonEditorRegistry.registerEditorAction(new EditorActionDescriptor(ShowHoverAction, ShowHoverAction.ID, nls.localize('showHover', "Show Hover"), {
context: ContextKey.EditorTextFocus,
kbExpr: KbExpr.has(editorCommon.KEYBINDING_CONTEXT_EDITOR_TEXT_FOCUS),
primary: KeyMod.chord(KeyMod.CtrlCmd | KeyCode.KEY_K, KeyMod.CtrlCmd | KeyCode.KEY_I)
}, 'Show Hover'));
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 765
|
\section{\label{sect1}Introduction}
Lithium isotope $^7$Li is a target material nucleus that plays an important role in the fields of nuclear technology and nuclear engineering, such as the compact accelerator-driven neutron source and International Fusion Materials Irradiation Facility (IFMIF) \cite {Garin2011}. The neutron beam produced by compact accelerator-driven neutron sources can be applied to nondestructive testing and medical treatment \cite{M.Paul2015}. For improving the reliability in design of target system for compact accelerator-driven neutron sources, accurate nuclear data of proton induced reactions on $^7$Li are required for Monte Carlo calculation code. The reaction data of $p + ^7$Li are important due to not only the value of the applications but also the basic research interest in the field of nuclear reaction. There are many open reaction channels at incident proton energy even below $20$ MeV for $p + ^7$Li reaction, so the reaction mechanism is very complex. The cluster separations, such as $^5$He $\rightarrow n+\alpha$, $^5$Li $\rightarrow p+\alpha$ and $^8$Be $\rightarrow \alpha+\alpha$, are involved besides the sequential particle emission processes. The calculation of the nuclear reaction is sensitive to the spin, excited energy and parity of the nuclear levels, especially the elastic and inelatic scattering angular distribution, double-differential cross section for neutron and proton induced nuclear reactions with $1p-$shell light nuclei involved. Therefore, the detailed study on $p + ^7$Li reaction will extend the knowledge of light charged particle induced nuclear reactions with $1p$-shell light nuclei involved, as well as the accurate structural information of some unstable light nuclei. Furthermore, this study will provide abundant proofs to test the reliability of statistical theory of light nucleus reactions (STLN), which has been applied successfully to calculate the double differential cross sections of outgoing neutrons both for neutron and proton induced nuclear reactions with $1p$-shell light nuclei involved \cite{X.J.Sun2017,Zhang2001Li6,Zhang2002Li7,Sun2009Be9,Duan2010Be9,Zhang2003B10,Zhang2003B11,Sun2008kerma,Zhang1999C12,Sun2007C12,Yan2005N14,Zhang2001O16,Duan2005O16,Duan2007,Zhang2015,Yan 5He}.
The double differential cross sections can give the most detailed information of the emission particles \cite{Trkov2011}. Therefore double differential cross sections of the emitted particles are very important both for the theoretical calculations and the applications. Due to the lack of the appropriate theoretical method for light nucleus reaction, the evaluation or model calculation of outgoing neutron and light charged particles double differential cross section for $p + ^7$Li reaction is not satisfactory. The complete nuclear reaction data for $p + ^7$Li reaction are scarce both for theory and experiment. There are a few data on partial cross sections for $p + ^7$Li reaction, such as $^7$Li($p,p$), $^7$Li($p,n$), $^7$Li($p,d$) and $^7$Li($p,\alpha$) in ENDF/B-VII.$1$ \cite{M.B.Chadwick2011} and ENDF/B-VIII.$0$ \cite{A.D.Carlson2018}. The nuclear reaction data in ENDF/B-VII.1 and ENDF/B-VIII.0 were given based on R-matrix analysis \cite{Philip2005} and fitting to experimental data. In the JENDL-$4.0$/HE \cite{K.Shibata2011}, evaluation was performed by the multi-channel R-matrix fitting the experimental data at incident proton energy below $10$ MeV, and the CCONE code \cite{Iwamoto.O2013} was used to fit ($p,xn$) spectra in incident proton energy range from $10$ to $200$ MeV. The double differential cross sections of some outgoing charged particles for $p + ^7$Li reaction are not given in those three nuclear reaction databases \cite{M.B.Chadwick2011,A.D.Carlson2018,K.Shibata2011}, and cluster separation processes are not also considered in the evaluations.
In addition, $n + ^6$Li reaction \cite{T.Matsumoto2011} and $n + ^7$Li reaction \cite{D.Ichinkhorloo2011} had been calculated with continuum discretized coupled channels (CDCC) method in $2011$ and $2012$, respectively. After that, proton induced reactions on $^{6,7}$Li \cite{H.Guo2013} are also calculated with CCDC method in $2013$. Based on this method, the double differential cross sections of proton and triton from $p + ^7$Li reaction \cite{H.R.Guo2019} were given recently by one of our collaborators. The calculated double differential cross sections of outgoing proton agree well with the experimental data, but the results of outgoing triton are overestimated at large outgoing angles comparing with the measurements. The same method was applied to calculate the double differential cross sections of outgoing neutron for $n+^{6,7}$Li reactions, and the results are underestimated in some low-emission-energy region comparing with the experimental data \cite{H.R.Guo2019}. Moreover, the double differential cross sections of outgoing deuteron are not given in references \cite{T.Matsumoto2011,D.Ichinkhorloo2011,H.Guo2013,H.R.Guo2019}. One of reasons is that the sequential secondary particle emission processes are not considered in CDCC model. Obviously, for high incident proton energies, the effects of secondary particle emission processes between the discret levels and cluster separations in light nuclear reactions are very important.
The reaction cross sections for $p + ^7$Li reaction have been successfully analyzed by R-matrix theory in previous studies, however the analysis of energy-angular spectra of the outgoing particles is still an open problem. There are two main problems in calculating double differential cross sections for neutron and proton induced light nucleus reactions. One is the theoretical description of the emission process from a compound nucleus to the discrete levels of the residual nuclei and from the discrete levels of the first residual nuclei to the discrete levels of the secondary residual nucleus with angular momentum and parity conservation through pre-equilibrium process. Another is the recoil effect, which is very important for light nucleus reaction, must be taken into account exactly to keep energy conservation in different reaction processes. Fortunately, these two problems have been solved by STLN \cite{Zhang2015,X.J.Sun2016}.
Based on the unified Hauser-Feshbach and exciton model \cite{Zhang1993} , which can describe the particle emission processes between the discrete levels with energy, angular momentum, and parity conservation, STLN is proposed to calculate the double differential cross sections of outgoing neutron and light charged particles for neutron induced reactions with the $1p$-shell nuclei involved. STLN has been applied successfully to calculate the double differential cross sections of outgoing neutron for neutron induced reaction on $^6$Li \cite{Zhang2001Li6},$^7$Li \cite{Zhang2002Li7},$^9$Be \cite{Sun2009Be9,Duan2010Be9},$^{10}$B \cite{Zhang2003B10},$^{11}$B \cite{Zhang2003B11},$^{12}$C \cite{Sun2008kerma,Zhang1999C12,Sun2007C12},$^{14}$N \cite{Yan2005N14},$^{16}$O \cite{Zhang2001O16,Duan2005O16} and $^{19}$F \cite{Duan2007}, and the calculated results very well reproduced the measurements. Furthermore, STLN has been improved to apply to the light charged particle induced nuclear reaction with the $1p$-shell nuclei involved. For example, the double differential cross section of outgoing neutron for $p + ^9$Be reaction had been calculated and analyzed in $2016$, and the calculated results successfully reproduced the measurements \cite{X.J.Sun2016}. In this paper, we will further improve STLN, such as analyzing the probable open reaction channels, considering the Coulomb barrier and the emission ratio of the composite particle, to obtain the double differential cross sections of outgoing charged particles for $p + ^7$Li reaction.
In Sec. \ref{sect2}, the theoretical model used in this work is briefly introduced. The reaction channels of $p + ^7$Li reaction below $20$ MeV are analyzed in detail in Sec. \ref{sect3}. The comparisons between calculated results with experimental data and analysis are given in Sec IV, and summary is given in the last section.
\section{ THEORETICAL MODEL}\label{sect2}
\subsection{ Theoretical Frame}\label{sect2.1}
Since the approach of decribing neutron induced light nucleus reactions with 1$p$-shell light nuclei involved had been proposed in $1999$ \cite{Zhang1999C12}, many experimental data, especially the double differential cross sections, had been analyzed. In dynamics, the angular momentum coupling and parity effect in pre-equilibrium emission process from discrete levels of the first residual nuclei to the discrete levels of the secondary residual nuclei were proposed to accurately keep the angular momentum and parity conservations \cite{Zhang2001O16}, so the double differential cross sections of secondary emitted particles can be calculated by this theoretical model. In kinematics, the recoil effect is strictly taken into account, and the energy balance of particle emission during the different reaction processes could be held accurately. Recently, this approach has been improved to describe proton induced light nucleus reactions with 1$p$-shell light nuclei involved, and to calculate the double differential cross sections of neutron and light charged particles. For illustrating the physical picture, the fundermental formulas are simply given in this paper. The detailed description of STLN can be found in Refs. \cite{Zhang1999C12,Zhang2015,X.J.Sun2016}.
Based on the unified Hauser-Feshbach and exciton model \cite{Zhang1993}, the cross sections of the first emitted particles from compound nucleus to the discrete energy levels of the first residual nuclei can be expressed as
\begin{eqnarray}\label{eq1}
\sigma_{m_1,k_1}(E_L)=\sum_{j\pi}\sigma_a^{j\pi}(E_L)\{\sum_{n=3}^{n_{max}}P^{j\pi}(n)
\frac{W_{m_1,k_1}^{j\pi}(n,E^*,\varepsilon_{m_1}^c)}{W_T^{j\pi}(n,E^*)}
+Q^{j\pi}(n)\frac{W_{m_1,k_1}^{j\pi}(E^*,\varepsilon_{m_1}^c)}{W_T^{j\pi}(E^*)}\}. \nonumber\\
\end{eqnarray}
Where $P^{j\pi}(n)$ is the occupation probability of the $n$-th exciton state in the $j\pi$ channel ($j$ and $\pi$ denote the angular momentum and parity in final state, respectively). $P^{j\pi}(n)$ can be obtained by solving the $j$-dependent exciton master equation under the conservation of angular momentum in pre-equilibrium reaction processes \cite{Zhang1994}. $Q^{j\pi}(n)$ is the occupation probability of the equilibrium state in $j\pi$ channel. $W_{m_1,k_1}^{j\pi}(n,E^*,\varepsilon_{m_1}^c)$ is emission rate of the first emitted particle $m_1$ at the $n$-th exciton state with outgoing kinetic energy $\varepsilon_{m_1}^c$ in center-of-mass system (CMS), and $W_T^{j\pi}(n,E^*)$ is total emission rate at the $n$-th exciton state. $W_{m_1,k_1}^{j\pi}(E^*,\varepsilon_{m_1}^c)$ is emission rate of the first emitted particle $m_1$ at the equilibrium state with outgoing kinetic energy $\varepsilon_{m_1}^c$ in CMS, and $W_T^{j\pi}(E^*)$ is total emission rate at the equilibrium state. $E^*$ is excited energy of compound nucleus, $\sigma_a^{j\pi}(E_L)$ is absorption cross section in $j\pi$ channel. In Eq. (\ref{eq1}), the first term in the brace denotes the contribution of the pre-equilibrium process, which dominates the light nucleus reactions with 1$p$-shell light nuclei involved. And the second term in the brace denotes the contribution of the equilibrium process.
The cross section of the secondary outgoing particle from discrete energy level of first residual nucleus to the discrete energy level of the secondary residual nucleus can be expressed as
\begin{eqnarray}\label{eq2}
\sigma_{k_1 \rightarrow k_2}(n, m_1, m_2)=\sigma_{k_1}(n, m_1)\cdot R_{m_2}^{k_1\rightarrow k_2}(E_{k_1}),
\end{eqnarray}
where $\sigma_{k_1}(n, m_1)$ is cross section of the first emitted particle $m_1$ expressed in Eq. (\ref{eq1}), and $R_{m_2}^{k_1\rightarrow k_2}(E_{k_1})$ is the branching ratio of the secondary outgoing particle $m_2$ from energy level $E_{k_1}$ of first residual nucleus $M_1$ to the energy level $E_{k_2}$ of secondary residual nucleus $M_2$.
The formulas (\ref{eq1}) and (\ref{eq2}) describe the particle emission from compound nucleus to discrete energy levels of first residual nuclei, and from the discrete energy levels of the first residual nuclei to the discrete levels of the secondary residual nuclei with angular momentum and parity conservation through the pre-equilibrium and equilibrium reaction process. Our previous researches indicate that the contributions of the total double differential cross sections of outgoing particle for light nucleus reactions are mainly from the pre-equilibrium emission process \cite{X.J.Sun2017,Zhang2001Li6,Zhang2002Li7,Sun2009Be9,Duan2010Be9,Zhang2003B10,Zhang2003B11,Sun2008kerma,Zhang1999C12,Sun2007C12,Yan2005N14,Zhang2001O16,Duan2005O16,Duan2007,Zhang2015,Yan 5He,X.J.Sun2015,X.J.Sun2016}. Only the equilibrium reaction process does not reproduce the double differential cross sections of the light nucleus reactions.
The linear momentum-dependent exciton state density model \cite{M.B.Chadwick1991} is used to obtain the Legendre expansion coefficients of the first outgoing particle and its residual nucleus. The double differential cross sections of the first outgoing deuteron, triton, $^3$He, and $\alpha$ are calculated with the improved Iwamoto-Harada model \cite{A.Iwamoto1982,J.S.Zhang93,J.S.Zhang1996}, which describes the light composite particle emissions. The representation of the double differential cross sections of secondary outgoing particle had been obtained by the accurate kinematics in Refs.\cite{Zhang2003B10,Zhang1999C12}. And the representation of the double differential cross sections of cluster separation and three-body breakup process can be found in Refs.\cite{Zhang2001Li6,Zhang2002Li7,Zhang2003B11}. Energy conservation is held strictly during nuclear reaction process in laboratory system (LS) for different reaction processes. A new integral formula \cite{X.J.Sun2015}, which is not compiled in any integral tables or any mathematical softwares, had been employed to describe the double differential cross sections of outgoing particles.
According to the Heisenberg's uncertainty relation, the level widths and energy resolution could be considered in fitting experimental data. The fitting procedure for double differential cross sections of outgoing particles are performed with Gaussian expansion form. And the transformation formulas from CMS to LS have been given in Ref. \cite{Zhang1999C12}. All of the energy level widths is derived from the experimental measurements \cite{Tilley1992,Tilley2002,Tilley2004} as fixed input parameters.
The optical model is very important to calculate the reduced penetration factor, which determines the emission rate of the first emitted particle. The phenomenological spherical optical model potential is employed in the model calculations. The potential parameters of the incident and ejected channels are determined by various reaction cross sections, and the angular distributions of the elastic and inelastic scattering.
\subsection{ Coulomb Barrier}\label{sect2.2}
Since Coulomb barrier has significant effect for open reaction channels of charged particles, it must be reasonably considered in the calculation for incident channel and outgoing channels.
Considering the energy-momentum conservation in CMS for outgoing channel, the definitive kinetic energy $\varepsilon _{{m_1}}^c$ of the first emitted particle can be easily derived as
\begin{eqnarray}\label{eq3}
\varepsilon _{{m_1}}^c = \frac{{{M_1}}}{{{M_C}}}\left( {{E^*} - {B_1} - {E_{{k_1}}}} \right).
\end{eqnarray}
Where $M_1$ is mass of the first residual nucleus after emitting the first particle $m_1$, and $M_C$ is mass of compound nucleus. For convenience, $m_1$ and $M_1$ also denote the first outgoing particle and the first residual nucleus, expectively. $E^*$ is excited energy of compound nucleus. $B_1$ is binding energy of the first emitted particle in compound nucleus. $E_{k_1}$ is excited energy of the $k$-th discrete level of the first residual nucleus.
Considering the energy-momentum conservation in CMS for incident channel, the excited energy of compound nucleus can be expressed as
\begin{eqnarray}\label{eq4}
{E^*} = \frac{{{M_T}}}{{{M_C}}}{E_p} + {B_p},
\end{eqnarray}
where $M_T$ is mass of target nucleus. $E_p$ is kinetic energy of incident particle. $B_p$ is binding energy of incident particle in compound nucleus. According to Eqs. (\ref{eq3}) and (\ref{eq4}), the threshold energy $E_{th}$ can be calculated.
Due to effect of Coulomb barrier \cite{G.R.Satchler1991,Peter W1970}, the kinetic energy of first outgoing charged particle must be higher than the Coulomb barrier $V_{Coul}$ , namely $ \varepsilon _{{m_1}}^c > {V_{Coul}}$. According to the assumption of the spherical nucleus \cite{Zhang2015}, the Coulomb barrier ${V_{Coul}}$ can be approximatively expressed as
\begin{eqnarray}\label{eq5}
{V_{Coul}} = \frac{{{e^2}{Z_{{M_1}}}{Z_{{m_1}}}}}{{{r_C}(A_{{M_1}}^{\frac{1}{3}} + A_{{m_1}}^{\frac{1}{3}})}},
\end{eqnarray}
where $Z_{M_1}$ and $Z_{m_1}$ is charge number of residual nucleus and first outgoing charged particle, respectively. $r_C(=1.2\sim1.5$fm) is charge radii parameter. For proton, deuteron, triton, $^3$He, $\alpha $ and $^5$He, their charge radii $r_C$A$^\frac{1}{3}$ will be substituted by the measurements compiled in Ref. \cite{I.Angeli2013}.
Therefore, the incident energy $E_p$ must meet Eq. (\ref{eq6}) to open reaction channels, i.e.
\begin{eqnarray}\label{eq6}
{E_p} > \frac{{{M_C}}}{{{M_T}}}(\frac{{{M_C}}}{{{M_1}}}{V_{Coul}} + {E_{{k_1}}} + {B_1} - {B_p}).
\end{eqnarray}
Obviously, the Coulomb barrier can affect significantly the open reaction channels. It is necessary that the reduced penetration factor calculated by optical model potential is $0$, if $ \varepsilon _{{m_1}}^c <{V_{Coul}}$.
\subsection{ Double Differential Cross Section of Light Composite Particle}\label{sect2.3}
The double differential cross sections of the emitted neutron and proton can be calculated using the linear momentum-dependent exciton state density model \cite{M.B.Chadwick1991}. The formulas for double differential cross sections of outgoing light composite particles ( deuteron, triton, $^3$He, $\alpha $ and $^5$He) can be expressed as \cite{Zhang2015}
\begin{eqnarray}\label{eq7}
\frac{{{d^2}\sigma }}{{d\varepsilon d\Omega }} = \sum\limits_n {\frac{{d\sigma (n)}}{{d\varepsilon }}A(n,\varepsilon ,\Omega )} .
\end{eqnarray}
Where $\frac{{d\sigma (n)}}{d\varepsilon }$ is energy spectrum in $n$-th exciton state, and can be calculated with angular momentum and parity conservation. A$(n,\varepsilon ,\Omega )$ is angle factor satisfying the normalization condition, expressed as
\begin{eqnarray}\label{eq8}
A(n,\varepsilon ,\Omega ) = \frac{1}{{4\pi }}\sum\limits_l {(2l + 1)\frac{{{G_l}(\varepsilon ,n)}}{{{G_0}(\varepsilon ,n)}}} \frac{{{\tau _l}(n,\varepsilon )}}{{{\tau _0}(n,\varepsilon )}}{P_l}(\cos \theta ).
\end{eqnarray}
Where $\Omega$ is solid angle of outgoing particle. ${\tau _l}(n,\varepsilon )$ is lifetime of the $l$-th partial wave with outgoing particle energy $\varepsilon$ emitted from $n$-th exciton state, and can be derived from the exciton model with angular momentum and parity conservation. ${G_l}(\varepsilon ,n)$ is geometric factor in $n$-th exciton state with outgoing particle energy $\varepsilon$, expressed as
\begin{eqnarray}\label{eq9}
G_l^b({\varepsilon _b}) = \frac{1}{{{x_b}}}\int\limits_{\max \left\{ {1,{x_b} - {A_b} + 1} \right\}}^{\sqrt {1 + \frac{E}{{{\varepsilon _F}}}} } {{x_1}d{x_1}} \int\limits_{{x_b} - {x_1}}^{{A_b} - 1} {dy{Z_b}(y){P_l}(\cos \Theta )}.
\end{eqnarray}
Where $\varepsilon _b$ is kinetic energy of the outgoing composite particle, and $\varepsilon _F$ is Fermi energy. $E^*$ is excited energy of compound nucleus. $A_b$ is mass number of outgoing particle $b$. Here $x_1=p_1/p_F$, $p_1$ is momentum of the first nucleon in the outgoing composite particle $b$, and $p_F$ is Fermi momentum. And $x_b=p_b/p_F$, $p_b$ is momentum of the outgoing composite particle. $y=p_y/p_F$, and $p_y$ is total momentum of nucleons except the first nucleon in the outgoing composite particle $b$. ${Z_b}(y)$ is a factor related to emitted composite particle, expressed as
\begin{eqnarray}\label{eq10}
{Z_b}(y) = \left\{ {\begin{array}{*{20}{l}}
{y, ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~~~~~~~ ~~ ~~~~~~~~~b = \textmd{deuteron}}\\
{y{{(y - 2)}^2}(y + 4), ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~~~~~~~ ~~ ~ ~~~~~~~~ ~~b = \textmd{triton}, \textmd{$^3$He}}\\
{{{(y - 3)}^4}({y^3} + 12{y^2} + 27y - 6),~~~~~~~~~~~~~~~~~~~~b = {\alpha} }\\
{{{(y - 4)}^6}({y^4} + 24{y^3} + 156{y^2} + 224y - 144),~~~~~b = \textmd{$^5$He}}.
\end{array}} \right.
\end{eqnarray}
Cosinoidal function $ \cos \Theta$ can be expressed as
\begin{eqnarray}\label{eq11}
\cos\Theta = \frac{{x_b^2 + x_1^2 - {y^2}}}{{2{x_b}{x_1}}}.
\end{eqnarray}
The formulas mentioned above can be used to calculate the double differential cross sections of the outgoing neutron, proton, deuteron, triton, $^3$He, $\alpha$, and $^5$He in this work. The calculated results are given in Sec IV.
\section{ANALYSIS OF REACTION CHANNELS FOR $p + ^7$Li REACTION}\label{sect3}
For proton induced $^7$Li reaction, reaction channels exist theoretically at incident energy $E_p \leq 20$ MeV in terms of the reaction threshold energy $E_{th}$ as follows:
\begin{eqnarray}\label{eq12}
p+^7\textmd{Li}\rightarrow ^{8}\textmd{Be}^* \rightarrow \left\{
\begin{array}{llr}
(p, \gamma)^{8}\textmd{Be}, ~~~~~Q=+17.255 \textmd{MeV}, ~~~~~E_{th}=0.000 \textmd{MeV}\\
(p, n)^{7}\textmd{Be},~~~~~Q=-1.643 \textmd{MeV}, ~~~~~~E_{th}=1.879 \textmd{MeV}\\
(p, p)^{7}\textmd{Li},~~~~~~~Q=~0.000 \textmd{MeV}, ~~~~~~E_{th}=0.000 \textmd{MeV}\\
(p, \alpha)\alpha, ~~~~~~~~Q=+17.348 \textmd{MeV}, ~~~~E_{th}=0.000 \textmd{MeV}\\
(p, ^3\textmd{He})^{5}\textmd{He}, ~~~Q=-4.125 \textmd{MeV}, ~~~~~E_{th}=4.7175 \textmd{MeV}\\
(p, d)^{6}\textmd{Li}, ~~~~~~~Q=-5.025 \textmd{MeV}, ~~~~~E_{th}=5.7468 \textmd{MeV}\\
(p, t)^{5}\textmd{Li}, ~~~~~~~Q=-4.434 \textmd{MeV}, ~~~~~E_{th}=5.0709 \textmd{MeV}\\
(p, 2n)^{6}\textmd{Be}, ~~~~Q=-12.320 \textmd{MeV}, ~~~~E_{th}=14.0897 \textmd{MeV}\\
(p, np)^{6}\textmd{Li}, ~~~~~~Q=-7.249 \textmd{MeV}, ~~~~E_{th}=8.2903 \textmd{MeV}\\
(p, pn)^{6}\textmd{Li}, ~~~~~~Q=-7.249 \textmd{MeV}, ~~~~E_{th}=8.2903 \textmd{MeV}\\
(p, n\alpha)^{3}\textmd{He}, ~~~~~Q=-3.230 \textmd{MeV}, ~~~~E_{th}=3.694 \textmd{MeV}\\
(p, nd)^{5}\textmd{Li}, ~~~~~~Q=-10.691 \textmd{MeV}, ~~~E_{th}=12.2267 \textmd{MeV}\\
(p, 2p)^{6}\textmd{He}, ~~~~~~Q=-9.974 \textmd{MeV}, ~~~~E_{th}=11.4067 \textmd{MeV}\\
(p, pt)^{4}\textmd{He}, ~~~~~~Q=-2.467 \textmd{MeV}, ~~~~E_{th}=2.8214 \textmd{MeV}\\
(p, tp)^{4}\textmd{He}, ~~~~~~Q=-2.467 \textmd{MeV}, ~~~~E_{th}=2.8214 \textmd{MeV}\\
(p, pd)^{5}\textmd{He}, ~~~~~~Q=-9.619 \textmd{MeV}, ~~~E_{th}=11.0007 \textmd{MeV}\\
(p, dp)^{5}\textmd{He}, ~~~~~~Q=-9.619 \textmd{MeV}, ~~~E_{th}=11.0007 \textmd{MeV}.
\end{array}
\right.
\end{eqnarray}
Considering the conservations of the energy, angular momentum, and parity in the particle emission processes, the reaction channels of the first particle emission are listed as follows:
\begin{eqnarray}\label{eq13}
p+^7\textmd{Li}\rightarrow ^{8}\textmd{Be}^* \rightarrow \left\{
\begin{array}{l}
n+ ^{7}\textmd{Be}^* ~~(k_1=gs, 1, 2, ...,7),\\
p+ ^{7}\textmd{Li}^* ~~(k_1=gs, 1, 2, ..., 10),\\
\alpha+{\alpha}^* ~~(k_1=gs, 1, 2, ..., 14),\\
^3\textmd{He}+ ^{5}\textmd{He}^* ~~(k_1=gs, 1),\\
d+ ^6\textmd{Li}^* ~~(k_1=gs, 1, 2, ..., 5),\\
t+ ^5\textmd{Li}^* ~~(k_1=gs,1,2).
\end{array}
\right.
\end{eqnarray}
Where $gs$ and $k_1$ denote the ground state and the $k_1$-th energy level of the first residual nuclei $M_1$ taken from measurements \cite{Tilley1992,Tilley2002,Tilley2004}, respectively.
For the first particle emission channel $^7$Li($p, n$)$^7$Be$^*$, the first residual nucleus $^7$Be$^*$, which attains the seventh energy level, can still emit a proton with the residual nucleus $^6$Li or a alpha particle with the residual nucleus $^3$He. Furthermore, the secondary residual nucleus $^6$Li can break up into deuteron and alpha particle \cite{Zhang2001Li6} if $^6$Li is in the first, third and fourth discrete energy levels. Therefore, the first particle emission channel $^7$Li($p, n$)$^7$Be$^*$ can further open ($p, np$)$^6$Li, ($p, npd\alpha$) and ($p, n\alpha$)$^3$He reaction channels in the final state.
For the first particle emission channel $^7$Li($p, p$)$^7$Li$^*$, the first excited level of residual nucleus $^7$Li cannot emit any particle, so it purely contributes to the inelastic scattering channel. The second and the third excited levels of $^7$Li can emit triton, so they contribute to the ($p, pt\alpha$) reaction channel. If the first residual nucleus $^7$Li$^*$ is at the $k_1$-th ($k_1 \geq 4$) excited energy level, some
energy levels will emit a neutron, so they contribute to the $(p, pn)^6$Li reaction channel. Furthermore, the secondary residual nucleus $^6$Li with high excited energy can break up into $d + \alpha$, thus this reaction process belongs to ($p, pnd\alpha$) reaction channel. If the first residual nucleus $^7$Li$^*$ is at the $k_1$-th ($k_1 \geq 6$) excited energy level, some energy levels will emit proton and deuteron with the corresponding secondary residual nuclei as $^6$He$_{gs}$ and $^5$He, respectively. Considering the two cluster separation reaction, i.e. $^5$He $\rightarrow$ $n + \alpha$ \cite{Yan 5He}, so these reaction processes belong to ($p,2p$)$^6$He$_{gs}$ and ($p,pnd\alpha$) reaction channels, respectively. Therefore, first particle emission channel $^7$Li($p, p$)$^7$Li$^*$ will contribute to ($p, pn$)$^6$Li, ($p, pt\alpha$), ($p, npd\alpha$) and ($p, 2p$)$^6$He$_{gs}$ reaction channels in the final state, besides the elastic and inelastic scattering.
For the first particle emission channel $^7$Li($p,d$)$^6$Li$^*$, besides reaction process $^6$Li$^*$ $\rightarrow$ $d + \alpha$ as mentioned above belongs to ($p, 2d\alpha$) reaction channel in the final state, some excited energy levels ($k_1 > 3$) of the first residual nucleus $^6$Li$^*$ can emit proton with the secondary residual nucleus $^5$He. As mentioned above, $^5$He is unstable and can be separated into a neutron and an alpha particle spontaneously \cite{Yan 5He}, so ($p, dp$)$^5$He reaction channel belongs to ($p, pnd\alpha$) reaction channel in the final state.
Considering the two-cluster separation processes, i.e. $^5$Li $\rightarrow$ $p + \alpha$ and $^5$He$ \rightarrow$ $n + \alpha$, so the first particle emission channels such as ($p, t$)$^5$Li and ($p, $$^3$He)$^5$He belong to ($p, pt\alpha$) and ($p, n\alpha$)$^3$He reaction channels in the final state, respectively.
For proton induced $^7$Li reaction at $E_p=14$ MeV, the compound nucleus $^8$Be can even reach the twenty-seventh discrete energy level with $28.6$ MeV in term of Eq. (\ref{eq4}), so it can emit neutron, proton, deuteron, triton, $^3$He, and break up into two alpha particles. Because of the high excited energy of the compound nucleus $^8$Be, alpha particles through two body breakup process are also at high excited energy in term of the energy conservation. So $^4$He can emit a proton at $k_1$-th ($k_1\geq 1$) energy level, a neutron at $k_1$-th ($k_1\geq 2$) energy level, and break up into two deuterons at $k_1$-th ($k_1\geq 9$) energy level, respectively. These reaction processes belong to ($p, 2\alpha$), ($p, pt\alpha$), ($p,n\alpha$)$^3$He and ($p, 2d\alpha$) reaction channels. Certainly, the gamma decay obviously competes with the particle emission, and the branching ratios can be obtained by means of the model calculation in STLN.
According to the analysis of reaction channels mentioned above, the total spectra could be produced by adding all of the partial spectra of the same outgoing particle yielded from every reaction channel. The contributions of the double differential cross sections of total emitted proton are from elastic scattering, inelastic scattering, ($p,np$)$^6$Li, ($p,2p$)$^6$He$_{gs}$, ($p, npd\alpha$) and ($p, pt\alpha$) reaction channels. The contributions of the double differential cross sections of total emitted deuteron are from ($p,d$)$^6$Li, ($p, npd\alpha$) and ($p, 2d\alpha$) reaction channels. The contributions of the double differential cross sections of total emitted triton are just from $(p,t)^5$Li and ($p, pt\alpha$) reaction channels. The contributions of the double differential cross sections of total emitted neutron are just from ($p,n$)$^7$Be, ($p, n^3$He)$\alpha$, $(p,npd\alpha)$ and $(p,np)^6$Li reaction channels. The contribution of the double differential cross sections of total emitted $^3$He is only from ($p, n^3$He)$\alpha$ reaction channel. In conclusion, for the proton induced $^7$Li reaction, reaction channels
exist at incident energy $E_p \leq 20$ MeV as follows:
\begin{eqnarray}\label{eq14}
p + {}^7\textmd{Li} \to {}^8\textmd{Be}^* \to \left\{ {\begin{array}{*{20}{l}}
{n + {}^7\textmd{Be}^*\left\{ {\begin{array}{*{20}{l}}
{k_1 = gs,1~~~~~~~~~~~~~~~~~~~~~~~~~(p,n){}^7\textmd{Be}}\\
{k_1 = 2,7~~~~~~~~~~~~~~~~~~~~~~~~~(p,n{}^3\textmd{He})\alpha }\\
{k_1 = 4,7~~~~~~~~~~~~~~~~~~~~~~~~(p,np){}^6\textmd{Li}_{gs}}
\end{array}} \right.}\\
{p + {}^7\textmd{Li}^*\left\{ {\begin{array}{*{20}{l}}
{k_1 = gs ~~~~~~~~~~~~~~~~Compound~elastic}\\
{k_1 = 1 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(p,{p'})}\\
{k_1 = 2,...,10~~(t + \alpha )~~~~~~~~~~~~~~(p,pt\alpha )}\\
{k_1 \ge 6(d + {}^5\textmd{He})~~~~~~~~~~~~~~~~~~(p,npd\alpha )}\\
{k_1 \ge 6(p + {}^6\textmd{He}_{gs})~~~~~~~~~~~~(p,2p){}^6\textmd{He}_{gs}}\\
{k_1 \ge 4(n + {}^6\textmd{Li})~~~~~~~~~~~~~~~~~(p,np){}^6\textmd{Li}}
\end{array}} \right.}\\
{\alpha + {\alpha ^*}\left\{ {\begin{array}{*{20}{l}}
{k_1 = gs~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(p,2\alpha )}\\
{k_1 = 1 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(p,pt\alpha )}\\
{k_1 \ge 2~~~~~~~~~~~~~~~~~~~(p,pt\alpha ),(p,n{}^3\textmd{He}\alpha )}\\
{k_1 \ge 9~~~~~~~~(p,pt\alpha ),(p,n{}^3\textmd{He}\alpha ),(p,2d\alpha )}
\end{array}} \right.}\\
{{}^3\textmd{He} + {}^5\textmd{He}~(k_1 = gs,1) \to (n + \alpha )~~~~~~~~(p,n{}^3\textmd{He}\alpha )}\\
{d + {}^6\textmd{Li}^*\left\{ {\begin{array}{*{20}{l}}
{k_1 = gs,2~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(p,d)}\\
{k_1 = 1,3,4,5~~~~~~~~~~~~~~~~~~~~~~~~~(p,2d\alpha )}\\
{k_1 = 4,5(p+^5\textmd{He} \to n + \alpha)~~~~(p,npd\alpha)}
\end{array}} \right.}\\
{t + {}^5\textmd{Li}~(k_1 = gs,1,2) \to (p + \alpha )~~~~~~~~~~~~~~(p,pt\alpha )}.
\end{array}} \right.
\end{eqnarray}
\section{THE CALCULATED RESULTS AND ANALYSIS}\label{sect4}
The experimental double differential cross sections of proton for $p + ^7$Li reaction had been measured only at incident proton energy $E_p = 14$ MeV in 1989 \cite{N. Koori1989}. The experimental double differential cross sections of deuteron and triton for $p + ^7$Li reaction had also been given in 1991 \cite{N. Koori1991}. The PUNF code based on STLN is developed for calculating the cross sections, elastic angular distributions and the double differential cross sections of outgoing neutrons, proton and light charged particles. In this paper, the comparisons between the calculated results with the measurements of double differential cross sections of total outgoing proton, deuteron and triton for $p + ^7$Li reaction will be performed.
The comparisons of the calculated double differential cross sections of total outgoing proton with the measured data are shown in Figs. \ref{Fig1} - \ref{Fig3} at the incident proton energy $14$ MeV for outgoing angles $20^\circ, 30^\circ, 40^\circ, 50^\circ, 60^\circ, 70^\circ, 80^\circ, 90^\circ, 100^\circ, 110^\circ, 120^\circ, 130^\circ, 140^\circ, 150^\circ, 160^\circ $ and $165^\circ$, respectively. The black points denote the experimental data derived from Ref. \cite{N. Koori1989}, and the red solid lines denote the calculated total double differential cross sections of outgoing proton. The calculated results agree well with the measurements except some peaks which are contaminated by the scattering from $^1$H, $^{12}$C and $^{16}$O as illuminated in Ref. \cite{H.R.Guo2019,N. Koori1989}. Fig. \ref{Fig_Par_P1} shows the partial double differential cross sections of outgoing proton from reaction channel $^7$Li($p,{p'}$)$^7$Li with outgoing angle $60^\circ$ at $E_p = 14$ MeV in LS. The black solid lines denote the partial spectra of the first outgoing proton from the compound nucleus to the ground state, up to the eighth excited energy levels of the first residual nucleus $^7$Li as labeled in Fig. \ref{Fig_Par_P1}. In this paper, only the cross sections with values larger than 10$^{-3}$ mb are given. Fig. \ref{Fig_Par_P2} shows the partial spectra of secondary outgoing proton from the third to sixth excited energy levels of $^7$Be to the ground state of $^6$Li, and from the fifth excited energy level of $^7$Be to the first excited energy level of $^6$Li for $^7$Li($p, np$)$^6$Li reaction channel labeled by the blue dashed lines. The green dotted lines denote the partial spectra of secondary outgoing proton from the fourth and fifth excited energy levels of $^6$Li to the ground state of $^5$He for $^7$Li($p, dp$)$^5$He reaction channel. The orange dash-dotted lines denote the partial spectra of secondary outgoing proton from ground state and the first excited energy level of $^5$Li to ground state of $^4$He for $^7$Li($p, tp$)$^4$He reaction channel. Fig. \ref{Fig_Par_P3} shows the partial spectra of secondary outgoing proton from the first to 10th excited energy levels of $^4$He which breaks into $p$ and $t$ for $^7$Li($p, \alpha p)t$ reaction channel labeled by the black solid lines. From Figs. \ref{Fig_Par_P1} - \ref{Fig_Par_P3}, one can see that the contributions of the secondary outgoing proton is far smaller than that of the first outgoing proton, as shown in Ref. \cite{H.R.Guo2019}.
The calculated double differential cross sections of total outgoing deuteron for $p + ^7$Li reaction at $14$ MeV are compared with the experimental data with outgoing angles of $10^\circ, 20^\circ, 30^\circ, 40^\circ, 50^\circ, 60^\circ, 70^\circ, 80^\circ, 90^\circ, 100^\circ, 110^\circ, 120^\circ, 130^\circ, 140^\circ, 150^\circ, 160^\circ$ and $165^\circ$, as shown in Figs. \ref{Fig4} - \ref{Fig6}. The black points denote the experimental data derived from Ref. \cite{N. Koori1991}, and the red solid lines denote the calculated total double-differential cross sections of outgoing deuteron. One can see that the calculated results agree well with the measurements. Fig. \ref{Fig_Par_D1} shows the partial double differential cross sections of outgoing deuteron from reaction channels $^7$Li($p,d$)$^6$Li, $^7$Li($p,pd$)$^5$He, $^7$Li($p,dd$)$^4$He and $^7$Li($p,npd\alpha$) with outgoing angle $60^\circ$ at $E_p = 14$ MeV in LS. In Fig. \ref{Fig_Par_D1}, the black solid lines denote the partial spectra of first outgoing deuteron from compound nucleus to the ground state, up to the fifth excited energy level of the first residual nucleus $^6$Li for $^7$Li$(p, d)^6$Li reaction channel. The blue dashed line denotes the contribution of secondary outgoing deteron from the eighth excited energy level of $^7$Li to the ground state of $^5$He for $^7$Li$(p, pd)^5$He reaction channel. The orange dotted lines denote the contributions of secondary outgoing deteron from the first, third, fourth and fifth excited energy levels of $^6$Li to the ground state of $^4$He for $^7$Li$(p, dd)^4$He reaction channel. The magenta dash-dotted line denotes the contribution of secondary outgoing deteron for reaction channel ($p, np$)$^6$Li $\rightarrow$ ($p, np+d\alpha$) from the fifth excited energy level of $^7$Be to the first excited energy level of $^6$Li, which can break up into $d + \alpha$. The green dash-dotted lines denote the contributions of secondary outgoing deteron for reaction channel ($p, pn$)$^6$Li $\rightarrow$ ($p, pn+d\alpha$) from the seventh and eighth excited energy levels of $^7$Li to the first excited energy level of $^6$Li. Fig. \ref{Fig_Par_D2} shows the partial double differential cross sections of outgoing deuteron from reaction channel $^7$Li($p,\alpha d$)$d$ with outgoing angle $60^\circ$ at $E_p = 14$ MeV in LS. In Fig. \ref{Fig_Par_D2}, the black solid lines denote the secondary outgoing deuteron from the ninth to $13$th excited energy levels of $^4$He, which can further break up into 2$d$.
The calculated double differential cross sections of total outgoing triton for $p + ^7$Li reaction at $14$ MeV are compared with the experimental data with outgoing angles of $10^\circ, 20^\circ, 30^\circ, 40^\circ, 50^\circ, 60^\circ, 70^\circ, 80^\circ, 90^\circ, 100^\circ, 110^\circ, 120^\circ, 130^\circ, 140^\circ, 150^\circ, 160^\circ$ and $165^\circ$, as shown in Figs. \ref{Fig7} - \ref{Fig9}. The black points denote the experimental data derived from Ref. \cite{N. Koori1991}, and the red solid lines denote the calculated total double-differential cross sections of outgoing triton. One can see that the calculated results agree well with the measurements. Fig. \ref{Fig_Par_T1} shows the partial double differential cross sections of outgoing triton from reaction channels $^7$Li($p,t$)$^5$Li and $^7$Li($p,pt$)$^4$He with outgoing angle $60^\circ$ at $E_p = 14$ MeV in LS.
In Fig. \ref{Fig_Par_T1}, the blue dashed lines denote the partial spectra of the first outgoing triton from the compound nucleus to the ground state and the first excited energy level of $^5$Li for $^7$Li$(p, t)^5$Li reaction channel. The black solid lines denote the contributions of secondary outgoing triton from second to eighth excited energy levels of $^7$Li to the ground state of $^4$He for $^7$Li$(p, pt)^4$He reaction channel. Fig. \ref{Fig_Par_T2} shows the partial double differential cross sections of secondary outgoing triton from reaction channel $^7$Li($p,\alpha p$)$t$ with outgoing angle $60^\circ$ at $E_p = 14$ MeV in LS. In Fig. \ref{Fig_Par_T2}, the black solid lines denote the partial spectra of secondary outgoing triton from the first to 10th excited energy levels of $^4$He, which can break up into $p$ and $t$. As shown in Figs. \ref{Fig_Par_P3} and \ref{Fig_Par_T2}, there are some wave-form partial spectra because of the too small Gaussian expansion coefficients no more than.
\section{SUMMARY AND CONCLUSION}
\label{sect4}
Based on the unified Hauser-Feshbach and exciton model \cite{Zhang1993}, which can describe the nuclear reaction emission processes between the discrete levels with energy, angular momentum, and parity conservation, STLN has been applied successfully to calculate the double differential cross sections of outgoing neutrons for neutron and proton induced reactions with the $1p$-shell nuclei involved. In this paper, the STLN has been improved to calculate the double differential cross sections of outgoing neutron, proton, deuteron, triton, $^3$He, and alpha particle for proton induced $^7$Li nuclear reaction. The calculated results very well reproduce the existed measurements of outgoing proton, deuteron and triton. The model calculation for $p + ^7$Li reaction indicates that the pre-equilibrium emission process is the dominant reaction mechanism in 1$p$-shell light nucleus reactions. Due to the light mass of $1p$-shell nuclei, the recoiling effects in various emission processes are strictly taken into account. And the calculated results indicate that the double differential cross sections of outgoing particles are very sensitive to energy, spin and parity of the discrete energy levels both for the target nucleus and the corresponding residual nuclei. Furthermore, the complete nuclear reaction data with ENDF-6 format for $p + ^7$Li can be obtained by PUNF code on the basis of STLN.
\textbf{Acknowledgements}
This work is supported by Natural Science Foundation of Guangxi (No. 2019GXNSFDA185011); National Natural Science Foundation of China (No. 11465005); and Foundation of Guangxi innovative team and distinguished scholar in institutions of higher education (No. 2017GXNSFGA198001).
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,923
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2 Word Domains With Keyword "Mind"
Labeled as budget: above $50 in Domain Buyer Requests started by ADDomain, Sep 24, 2018.
I'm looking for 2-word names with the keyword "mind".
Budget: $25-$100 per domain. Up to $300 for exceptional names.
Please, don't spam me with poor-quality names.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,671
|
Q: Delete button in table, jQuery I wrote the code, using JS and jQuery. The code adds new rows to my table. Every row contains the delete button and it do not work. I'm going to implement also edit button.
My table:
<div class="row">
<div class="col-md-12">
<table class="table table-striped" id = "my-table">
<thead>
<tr>
<th>Date</th>
<th>Example</th>
</tr>
</thead>
<tbody>
</tbody>
</table>
</div>
</div>
</div>
JS code:
$(document).ready(function(){
var today = new Date();
var dd = today.getDate();
var mm = today.getMonth()+1; //January is 0!
var yyyy = today.getFullYear();
if(dd<10) {
dd='0'+dd
}
if(mm<10) {
mm='0'+mm
}
today = mm+'/'+dd+'/'+yyyy;
//var counter = 1;
$('button#sending_button').click(function(){
var text = $('textarea#mytextarea').val();
$('#my-table').append('<tr><td>'+today+'</td><td>'+text+'</td><td><button type="button" id="delete-button" class="btn btn-danger">Usuń</button></td></tr>');
$('textarea#mytextarea').val('');
//counter++;
});
$('#my-table').on('click', 'button.btn btn-danger', function() {
$(this).parents('tr').remove();
});
});
I don't know what more I should explain about this problem.
A: Your problem is here:
$('#my-table').on('click', 'button.btn btn-danger', function() {
It should be button.btn.btn-danger.
Some random suggestions:
*
*Don't repeat the id of each button ("delete-button"): id's should be unique
*Use a semantically meaningful class instead (delete button would also be good)
*Use said class when defining the delegate event listener, instead of button.btn.btn-danger (which is too bound to the DOM and its style definition)
*Use parent instead of parents to avoid surprises. They're two different methods with different goals
*Event better, use closest.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,133
|
\section{Introduction}\label{sect:Intro}
This paper concerns the `Sterrenkundig Laboratorium
Kapteyn'\footnote{\normalsize Because it concerns names I will as much as possible
keep the formal names of institutions and observatories in the language
of their location, so Sterrenkundig Laboratorium Kapteyn or for short
Kapteyn Laboratorium, the Sterrewacht
Leiden, Bergedorfer Sternwarte, Helsingfors Observatorion, Mount Wilson
Observatory, etc. But, since it would be impractical, I will
refrain from applying this to universities, but I will not translate
`Rijks-universitei Groningen' into the sometimes used `State University of
Groningen' (a correct translation would be `National University',
since the term
`State University' refers to universities funded by states
in the USA rather than by a nation) but use the official University of
Groningen.}
in Groningen, the Netherlands, during the directorship of
Pieter Johannes van Rhijn (1886--1960)\footnote{\normalsize I add in general
for persons when they first appear
in this text their full set of first names and the years of birth an death,
but for those still alive I refrain from quoting their year of birth.},
which lasted from 1921 to 1957.
At the start of the academic year in 1921 Pieter van Rhijn, see
Fig.~\ref{fig:1921}, was
appointed professor of astronomy at the University of Groningen and director
of its astronomical laboratory. This laboratory had been founded by Jacobus
Kapteyn, see Fig.~\ref{fig:JCK},
and on the occasion the curators of the University and with
consent of the Minister had renamed it
the `Sterrenkundig Laboratorium Kapteyn'. When Kapteyn's
attempts -- starting not long after he was appointed
professor of astronomy in 1878 without any facility to speak of for research
and teaching in astronomy -- to obtain a real observatory had failed,
he had reverted to this 'observatory without a telescope'. Under Kapteyn it
had risen to become one of the most prominent
institutions worldwide, overshadowing the Sterrewacht Leiden, which was in a
deep decline. Under van Rhijn the Groningen Laboratorium slipped in turn to a
lower rank on the ladder of prominence, status and prestige.
I will look into the circumstances, context and background to this.
\bigskip
I will start with context and briefly consider astronomy
in the Netherlands up to roughy the end
of this period, i.e. the 1950s.
A first reference would be David Baneke's history of astronomy
in the Netherlands in the twentieth century \cite{DB2015}, but this still awaits
being published in English translation. In recent years a
few further studies on particular aspects have been published. But also
a number of biographies of Dutch astronomers of this period
have appeared (a few unfortunately also in Dutch only).
\begin{figure}[t]
\sidecaption[t]
\captionsetup{width=.36\textwidth}
\includegraphics[width=0.64\textwidth]{fig1921.jpg}
\caption{\normalsize Pieter Johannes van Rhijn in his late twenties around the time
he obtained his PhD in 1915. Kapteyn Astronomical Institute.}
\label{fig:1921}
\end{figure}
\begin{figure}[t]
\sidecaption[t]
\includegraphics[width=0.64\textwidth]{figJCK.jpg}
\caption{\normalsize Jacobus Cornelius Kapteyn. Painting by Jan Veth, now
located in the Senate Room of the Academy Building of the University of
Groningen. It was originally intended as a present to his wife at the 40-th
anniversary of his professorship, but she disliked it and a new one was
painted. Veth later over-painted it with gown, barret and jabot for its present
purpose. For the full story see \cite{JCKbiog} or \cite{JCKEng}.
University of Groningen.}
\label{fig:JCK}
\end{figure}
Although astronomy in the Netherlands is generally considered to have started
with Christiaan Huygens (1629--1695)
and his 'discovery' of Saturn's rings and satellite Titan (biography in
\cite{CDAn}), modern astronomy in the
Netherlands goes back to Frederik Kaiser (biography in \cite{RGent}).
Kaiser was the founder of the Sterrewacht
Leiden, the building modeled after Pulkovo Observatorium and opened in 1861.
Kaiser had made quite a name for
himself when Halley's comet was due to return in 1835. He revised calculations
made earlier by renowned scientists
and predicted the time of passage of the perihelion to an astounding accuracy
of 1.5 hours, where many celebrated
astronomers had been wrong by much larger time intervals, sometimes
as much as 9 days.
Under Kaiser the Sterrewacht became well-known for its extremely accurate
astrometry. However, under his successor Hendricus Gerardus van de
Sande Bakhuyzen (1838--1923), it lost much of its influence because he decided
not to embrace photography as the new tool while in fact it
turned out superior to visual
observing. Under his younger brother Ernst Frederik (1848--1918), who was
appointed director in 1908, Leiden lost even more of its
status of excellence.
Instead Groningen took over, where Jacobus Kapteyn became one of the most
prominent astronomers worldwide with his studies of the
distributions of stars in space and stellar kinematics (biography in
\cite{JCKbiog} and \cite{JCKEng}).
The turning point for Leiden came in 1918, when after the sudden death of
Ernst van de Sande Bakhuyzen, Willem de Sitter (1872--1934) became director,
having been appointed professor of astronomy in Leiden
already in 1908. De Sitter
(biography in \cite{Guich}) was Kapteyn's first student working on the
structure of the system of Galilean satellites
around Jupiter. In the reorganization of the Sterrewacht of 1918, in which
Kapteyn had had an important voice as well, a new department on stellar
astronomy was created next to de Sitter's theoretical one (in this Einstein's
General Relativity became a second focus). The new department was headed by
Ejnar Hertzsprung (1873--1967), who already was a world renowned astronomer
(no recent biography, but see \cite{Herrmann}), but the third
department on astrometry remained without a head, when the Prime Minister
refused to appoint Antonie Pannekoek (1873--1960) to the post because of his
communist sympathies. Pannekoek was appointed subsequently at the
municipal University of Amsterdam, where he founded an
Astronomical Institute and made a
name by mapping the Milky Way and more impressively, founding astrophysics of
stellar atmospheres, exploring ways to apply the Saha ionization
formula to study the physical conditions in these outer envelopes (see
\cite{Pannekoek}). In 1925 Jan Hendrik Oort was appointed at the
vacant Leiden position.
Utrecht under Marcel Gilles Jozef Minnaert (1893--1970) rose to prominence
particularly for producing a detailed solar spectrum and founding a line of
research in solar physics and astrophysics
(biography in \cite{Minnaert}). Later additions to the staffs
until 1957 (the end of van Rhijn's term in office)
were Hendrik Christoffel (Henk) van de Hulst (1918--2000)
(biography in \cite{Delft}), Pieter Theodorus Oosterhoff (1904--1978)
and Adriaan Blaauw (1916--2010) in
Leiden, Herman Zanstra (1894--1972) in
Amsterdam and Cornelis (Kees) de Jager (1921--2021) in Utrecht. Comprehensive
biographies of these last four remain to be written.
In the twentieth century, Dutch astronomy had become a major player worldwide
in the field of astronomy. Consider the large number of Dutch
astronomers that were born before WWII and rose to leading positions (usually
directors) abroad, as listed by Jan Oort \cite{JHO1982}: Jan Schilt, Willem
van den Bos, Dirk Brouwer, Peter van de Kamp. Willem Luyten, Gerard Kuiper,
Bart Bok, Leendert Binnendijk,
Maarten Schmidt, Lo Woltjer, Gart Westerhout, Tom Gehrels and Sydney
van den Bergh. To illustrate this point further I note that almost all
astronomers in the Netherlands mentioned in the previous paragraph
were awarded what was probably
the most prestigious honor, the Gold Medal of the Royal
Astronomical Society: Kapteyn 1902, Hertzsprung 1929, de Sitter 1931, Oort,
1946, Minnaert, 1947, Pannekoek, 1951, Zanstra 1961, de Jager 1981.
Most of these also received the Bruce Medal of the Astronomical Society of the Pacific.
Van de Hulst, Blaauw and Oosterhoff are missing in the Gold Medal list, but
the first two of them received the Bruce Medal in 1978
and 1988.
In this success story van Rhijn is conspicuously missing.
\bigskip
In his presentation during a symposium in 1999 in Groningen on the {\it Legacy
of Jacobus C. Kapteyn} \cite{Legacy}, Woodruff T. Sullivan put it as follows
(\cite{WTS99}, p.230):
\par \begingroup \leftskip2.5em \rightskip2em \large
\noindent
[...]. at Groningen after the death of Kapteyn in 1922, his eponymous
Laboratory went into a period of decline coinciding with the long directorship
[...] of his prot\'eg\'e Pieter van Rhijn. Van Rhijn did some excellent work
in the 1920s [...], but
both his research and leadership for the rest of his career were unimaginative
and stuck in a rut (Blaauw, 1984).[...]
Kapteyn's Plan of Selected Areas, the study and organization of which
occupied a large portion of his successor van Rhijn's career [...] , this
work was done in a routine and unimaginative manner, and contributed to
Groningen's period of decline.
\par \endgroup
In the 1984-paper of Adriaan Blaauw
that Sullivan refers to to support his case, Blaauw actually is much milder
(\cite{Blaauw1983}, p.56; my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
[...] the foundation of an astronomical laboratory. But the complete dependence
on the cooperation of foreign observatories that went along with that
was a very heavy burden on the future of the institute.
Could van Rhijn reasonably be expected to continue the special
fame that the institute had acquired through Kapteyn, and which was a
necessary condition for this kind of cooperation to maintain? [...]
[Van Rhijn] was a thorough scientific researcher [...].
His approach to research, however, was highly schematic,
strongly focused on further evaluation of previously
designated properties or quantities. This benefited the thoroughness of
the intended result, but came at the expense of flexibility and
reduced the opportunity for unexpected perspectives; as a result, it did not
lead to surprising discoveries.
\par \endgroup
Or in Blaauw's interview for the American Institute of Physics Oral History
Interviews Project \cite{Blaauw1978}:
\par \begingroup \leftskip2.5em \rightskip2em \large
[...] the main lines at Groningen by van Rhijn, who followed very much in the
course set by Kapteyn, [...]. And in that time, he
accomplished a very large amount of, you might call it routine works. But it
is a very thorough compilation of photometry and proper motions in the
Selected Areas and also work on such problems as luminosity functions.
\par \endgroup
\bigskip
It may not be surprising that no comprehensive biography of van Rhijn
has ever been written, but it remains remarkable that shortly after his
death no extensive obituary concerning van
Rhijn has appeared in any major international
astronomical journal. The only somewhat extensive one
was written by Adriaan Blaauw and Jean Jacques Raimond (1903--1961)
-- both having obtained
their PhD's with van Rhijn as `promotor' (supervisor) -- in Dutch in the
periodical of the association of amateur astronomers and meteorologists
\cite{BlaRaim} (the contribution by Adriaan Blaauw in the {\it Biographical
Encyclopedia of Astronomers} \cite{Bea}, is modeled on this
article by him and Raimond). But they did not think it worthwhile,
necessary or appropriate to publish an English language article
to honor the life of an important astronomer.
Bartholomeus Jan (Bart) Bok (1906--1983),
who also obtained his PhD under van Rhijn,
published obituaries in {\it Sky \&\ Telescope} on Pannekoek
and van Rhijn \cite{BokST}. And
in 1969 Ernst Julius \"Opic (1893--1985), catching up on a suspension of the
{\it Irish Astronomical Journal}, listed van Rhijn among a set of over
twenty astronomers that had died in previous years \cite{IrAJ}.
But that is all.
In fact, there is not a single article on van Rhijn and Groningen astronomy
under him in the English literature other than
Blaauw's article in the {\it Biographical Encyclopedia}, and
in Dutch there is only the one by Blaauw \cite{Blaauw1983}
that Sullivan referred to.
At van Rhijn's seventieth birthday in 1954 a kind of festschrift appeared
(the Dutch title translates into English as `in
Kapteyn's footsteps'; see below) with articles by his colleagues and students,
but the articles are mostly in Dutch and refer to
then current research by the authors themselves and do not address van Rhijn's
contributions in any detail.
\bigskip
In this contribution I will critically examine the Kapteyn
Laboratorium during the period of van Rhijn's directorate 1921-1957.
Questions addressed include: Was van Rhijn the successor Kapteyn had wished
for or had he preferred someone else? What future might Kapteyn have had
in mind for his Laboratorium? Was van Rhijn's work really so unimaginative
and how important has it turned out to be?
Was Kapteyn's concept of an astronomical laboratory,
an observatory without a telescope, a viable one in the long run?
What precisely caused the Kapteyn Laboratorium to loose its prominent place
in the international context? How important was the
{\it Plan of Selected Areas} for the progress of astronomy and did
it ever reach the goals Kapteyn had in mind for it? Did van Rhijn get the
support he deserved from his university and the government? How influential
was van Rhijn and what was his role in the prominence of Dutch astronomy
that followed during the twentieth century? What made Adriaan Blaauw accept his appointment as van Rhijn's successor?
I rely heavily on Adriaan Blaauw's 1983 chapter \cite{Blaauw1983}, and
I will when appropriate quote verbatim from this in translation.
\bigskip
A few words on archival sources. The van Rhijn Archives are now part of the
Regionaal Historisch Centrum `Groningen Archieven', founded in 2001 by the
Ministry of Education, Culture and Science and the Municipality of Groningen,
where the University of Groningen has deposited its Archives. The van Rhijn
Archives were transferred from the Kapteyn Astronomical Institute through the
`Archief Depot' of the University. Much of the correspondence is missing in
the Groninger Archieven under the corresponding heading, including that
between van Rhijn and the Curators of the university. Why these letters have
been culled from the collection is unknown, but probably to avoid duplication.
This, unfortunately, has made this very much less accessible.
Van Rhijs corresponded quite extensively with Jan Hendrik Oort (1900--1992)
in Leiden. These letters are missing in Groningen
also, but quite a collection is available in the Oort Archives,
which contains over 400 letters between van Rhijn and Oort. These are mostly
handwritten letters, of which usually no copies were kept,
so most are from van Rhijn
to Oort. Also a few letters of Adriaan Blaauw and Lukas Plaut to Oort
proved to be relevant to this article. These letters are almost entirely in
Dutch of course. An extensive inventory of the Oort
Archives has been published by Jet Katgert-Merkelijn \cite{JKM97}. No
professional scans are available for the Oort Archives, but over 27,000
reasonable to good quality photographs, taken by me, of mainly letters and
notes are available on my special Oort homepage accompanying my biography of
him \cite{JHObiog}.
Much information on the University of Groningen during van Rhijn's days
is available in part II of Klaas van
Berkel's monumental history of the university \cite{KvB2017}. Again this is in
Dutch unfortunately, and of not much use to English speaking researchers. An
extensive English summary has been announced.
\section{Pieter Johannes van Rhijn and his early research}\label{sect:Early}
I start this section with some biographical notes on van Rhijn. As mentioned,
at present no comprehensive biography on him has been written.
Pieter Johannes van Rhijn was born on March 24, 1886 in the city of Gouda, some 20
km northeast of Rotterdam. His father, Cornelis Hendrikus van Rhijn (1849--1913),
was a clergyman and not long after Pieter's birth he was appointed professor of
theology at the University of Groningen. In \cite{Nijen} we read about C.H. van
Rhijn (my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
Van R. had a hard time fitting into the directional scheme of his time. The
liberal, tolerant upbringing, received in the parental home, continued to
influence him throughout his life. He wanted to be an unapologetic biblical
theologian.
\par \endgroup
Pieter's upbringing therefore was undoubtedly relatively liberal as well. Cornelis
van Rhijn had had a first marriage, which however lasted only three years after
which his wife died. This marriage produced no children. Cornelis married again
in 1882 with Aletta Jacoba Francina Kruijt (1859--1945), and Pieter Johannes was
their second child. He had one older sister and three younger brothers, of which
one, Maarten van Rhijn (1888--1966), eventually like his father became a professor
of theology, but then at the University of Utrecht. On the Web there is a long and
detailed genealogy of the family van Rhijn (earlier often spelled van Rijn),
\cite{PJgenea}. The keeper of this, Gert Jan van Rhijn, is a grandson of van
Rhijn's brother Maarten, has provided some very useful biographical material
to me.
The genealogy goes back to the year 1510 and to ancestors who lived very close to
Leiden, but the famous seventeenth century painter from Leiden, Rembrandt with
the same surname, turns out to be not part of his ancestry. Pieter grew up in
Groningen, where eventually he studied astronomy with Kapteyn. In 1915
obtained his PhD with Kapteyn as supervisor (`promotor').
\begin{figure}[t]
\sidecaption[t]
\includegraphics[width=0.49\textwidth]{figwedding.jpg}
\includegraphics[width=0.49\textwidth]{figchild.jpg}
\caption{\normalsize Left: Pieter J. van Rhijn and Regnera (`Rein') L.G.C. de
Bie at their wedding in 1932. Right: Van Rhijn with his son, who was born in
1934. From the Website of the `Stichting Het geslacht van
Rhijn' \cite{PJgenea}, with permission.}
\label{fig:wedding}
\end{figure}
In 1932, at the age of 46, he married Regnera Louise Geertruid Constantia de Bie
(1906--1997), twenty years his junior (see Fig.~\ref{fig:wedding}, left).
In his letters
van Rhijn referred to her as `Rein' (interestingly pronounced the same as 'Rhijn').
She had a degree in Law. The marriage certificate says she was 'without occupation'.
Her father was a juvenile judge and later president of a district court
(Arrondissementsrechtbank) in Rotterdam. Van Rhijn and his wife eventually would
have two children, a son and a daughter (see Fig.~\ref{fig:wedding}, right).
Before his marriage van Rhijn was well occupied by the fact that his elder
sister Adriana Josephine van Rhijn (1884--1976),
whose nickname was Joos, lived in Groningen after being divorced in 1921.
She had four children, at the time of the divorce between 2 and 10
years of age and van Rhijn helped her extensively in raising the children.
At the time of his own marriage these children were
eleven years older and his involvement with them must slowly have become
less time consuming.
During the Second World War van Rhijn contracted tuberculosis and his
recovery took a fair number of years. More about this is covered in the later
sections of this paper.
It did have a major impact on the rest of his life, for his vigor never fully
returned.
Not too much has been written about van Rhijn as a private person. Through
Gert Jan van Rhijn, the keeper of
the van Rhijn genealogy \cite{PJgenea}, I have been able to get answers to some
questions about Pieter van Rhijn from the latter's daughter, who is still alive
(contrary to his son). With his background of being the son of a clergyman and
theology professor, the question arises how religious Pieter van Rhijn really was.
According to his daughter
he went to church but not every week and the church he
went to was that of the Remonstrants, because `he could not believe the creation
story'.
The Remonstrant movement had started with the liberal Leiden professor Jacobus
Arminius ($\pm$1555--1609). The view of Calvinism was that humans were predestined
for Heaven or Hell and had no free will to choose to live either in virtue or in
sin. The followers of Arminius (sometimes called the `Rekkelijken' or Flexibles)
opposed this and a few related points of view, which were collected in the
`Five Articles of Remonstrance' (objection or statement or defence)
in 1610, which caused them to be designated
Remonstrants. Their views were particularly disputed by Franciscus Gomarus
(1563--1641), whose followers were called `Preciezen' (Strict Ones) or
Counter-Remonstrants. Gomarus actually already taught in Leiden when Arminius
arrived there. During the Synod of Dordrecht (1618--1619) the Remonstrants were
expelled by the Protestant Church and the movement separated from the
Calvinists.
The movement still exists in the form of the Remonstrant Brotherhood
and has somewhat over 5000 members and `friends' (sympathizers) in the Netherlands.
It stands for a liberal interpretation of Christianity; their faith is characterized
by a non-dogmatic attitude, in which freedom of thought and belief based on one's
own insights is central and developments in modern science and insights in general
are taken into consideration. There was therefore room for van Rhijn to adhere
modern scientific developments and to adopt evolution over biblical creation.
Van Rhijn's daughter (who like het aunt was called `Joos' in the family)
stressed that (my translation)
\par \begingroup \leftskip2.5em \rightskip2em \large
van Rhijn did not like dogmatic beliefs, but his wife had a more orthodox
upbringing. At home they read from the Bible, but Pieter Johannes never led in
prayer. He had a broad view and was averse to strict rules. His wife had had a
strict upbringing. If something had to be done this way or that, Pieter Johannes
van Rhijn would say: So, who stipulates that [`Wie stelt dat vast', stressing
that no other authority than himself would decide on his behavior, action or
options]?
They [he and his wife] went to church together. The faith was also passed on, so
daughter Joos went with them to catechism. Mother found it liberating that Pieter
Johannes thought more freely about religion.
\par \endgroup
Van Rhijn had no special functions in the church and did not participate in
activities other than the Sunday service. In the Netherlands, people had (and have)
attached small signs to their front doors, traditionally with the name of the man,
as head of the family, imprinted on it. In van Rhijn's case his wife's maiden name
was added to it.
Van Rhijn was deeply infuriated about the German occupation during Worl War II.
He was hurt and outraged by the supposed German superiority and breach of Dutch
sovereignty. In a
in a letter of December 1945 to his sister-in-law\footnote{\normalsize This is
Lies de Bie, a
younger sister of his wife, who and her preacher husband Foppe Oberman at the time were still in Java. A copy this letter was kindly provided by Gert
Jan van Rhijn.} he expressed this as follows (my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
What has struck me most in the Germans is their unsparing arrogance (`das
Herrenvolk') and their virtuosity in lying. They can lie brilliantly. Therein lies
a certain merit. The Krauts only did it stupidly. And sometimes I almost burst at
the pharisaism of e.g. Seys Inquart (the German chief in the Netherlands), who
said he had come to the Netherlands to protect us from Bolshevism. But those are
echos from a distant past. The gentlemen Nazi leaders including Seys Inquart are
now on trial in Nuremberg and within a few months they will probably no longer be
among the living.
\par \endgroup
\noindent
Arthur Seyss-Inquart (born Artur Zajtich; 1892--1946) actually was not German, but Austrian.
Pieter van Rhijn played the viola, but there was no frequent playing together in
the household. Adriaan Blaauw has recorded \cite{Blaauw1983} his fond recollection
of (my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
\noindent
the warmth of his domestic environment, where playing music together -- van Rhijn
was an excellent viola player -- was a bright spot in the dark War years.
\par \endgroup
\noindent
Blaauw himself played the flute. Van Rhijn's daughter has remarked that
\par \begingroup \leftskip2.5em \rightskip2em \large
\noindent
as a younger man in his student days he had played in ensembles. He was not
good enough to play first violin. [...] Eventually he found it more
interesting to play the viola. Later in life he did not have the strength [to
play].
\par \endgroup
She also recorded that she went to performances of Bach's Matth\"aus Passion
together with her father. Mrs. van Rhijn sang in a choir. Van Rhijn was interested
in Mozart and Schubert quartets. When recovering from tuberculosis after the War
he noted in the letter to his sister-in-law referred to above that he was allowed
out of bed eight hours a day and walking 30 minutes, and added that
in addition to reading he
listened to concerts on the radio, where Mozart was an obvious favorite. He
wrote (my translation)
\par \begingroup \leftskip2.5em \rightskip2em \large
Mozart is a musician who really doesn't belong on this Earth. When hearing
his music, it is as if the heavens break open over this cursed
world. My colleague
van der Leeuw professor of theology (now a Minister) says that after hearing a
Mozart concert he no longer believes in the Original Sin for three days.
\par \endgroup
Gerardus van der Leeuw (1890--1950), a good friend of van Rhijn,
was historian and philosopher of religion, and for a short period
Minister of education immediately after the War.
Van Rhijn
read a lot throughout his life, but particularly when he was recovering from
tuberculosis, especially English novels. British authors that he liked
included Howard Spring, John Galsworthy, Nevil Shute, J.B. Priestley, Graham
Greene, W. Somerset Maugham, Duff Cooper, C.L. Morgan, Jane Austin and the
Bronte sisters. American writers were fewer, but included J.P. Marquand, Louis
Bromfield and Pearl Buck. He mentioned in the letter quoted that he also had read
some excellent poetry.
As for outdoors activities, he also often went hiking to south of Groningen, one
would think with his family.
Pieter Johannes van Rhijn died in Groningen on May 9, 1960 at age 74.
\bigskip
Van Rhijn obtained his PhD in 1915, on a thesis consisting of two quite
separate studies. For this he had spent an extended period at Mount Wilson
Observatory (1912--1913), which of course had been arranged
by Kapteyn with George Ellery Hale (1868--1938).
The title of the thesis was {\it Derivation of the
change of colour with distance and apparent magnitude together with a new
determination of the mean parallaxes of the stars with given magnitude and
proper motion}. The thesis was never fully published
in a journal; a shorter paper appeared with the same title as the thesis itself
\cite{PJvR1916}, although it treated really
only the first part, which concerned the question
of reddening of starlight with distance from the Sun. In 1909, Kapteyn
had hypothesized that extinction of starlight would be primarily
scattering rather than absorption, and that this could be expected to be
stronger at shorter wavelengths, giving rise to reddening of starlight with
distance \cite{JCK1909}. Van Rhijn addressed this issue using a study
called the {\it Yerkes Actinometry}, which was published in John
Adelbert Parkhurst (1861--1925) in 1912.
With this data-set, together with proper motions from other places, van Rhijn
determined the change in color of stars with distance. Kapteyn has assumed
(as we now know incorrectly)
that the wavelength dependence was like that of Rayleigh
scattering in the Earth's atmosphere, so that reddening could be related to
total extinction. Van Rhijn thus found that it
amounted to '{\it ceteris paribus} [all other things being equal] only
$0.000195 \pm 0.00003$ magnitudes per parsec', only about half that what Kapteyn
had found in 1909. Interestingly -- and disappointingly --
there was absolutely no
discussion of this difference, neither in the paper, nor in the thesis itself.
We will see in what follows that this is the first instance of an emerging pattern
of van Rhijn never
adding a discussion in a paper to put things in a larger context.
Van Rhijn's value is much lower than the currently adopted roughly one
magnitude per kiloparsec. However, this region of {\it Parkhurst Actinometry}
is centered on the celestial pole and that means almost $30^{\circ}$ Galactic
latitude. So in hindsight
it is not surprising that the inferred extinction was smaller
than in all-sky determinations or regions exclusively in the Milky Way.
The second part of the thesis
formed an important step in the preparation to derive the
distribution of stars in space, the ultimate aim of Kapteyn's scientific work,
and was published much later after more work and additional data.
Because it is relevant to the discussion of van Rhijn's work for the rest
of this paper, I summarize
first the procedure Kapteyn had designed for this very briefly. It begins with
two fundamental equations of statistical astronomy.
They link the counts of stars and the average parallax as a function of apparent
magnitude to the distribution of the total density of stars as a function of
the position in space and to the luminosity function. I will
not present these here; if interested see the full, online version of Appendix A
\cite{AppA} of \cite{JCKEng}\footnote{\normalsize Note that in Box 15.3 or
\cite{JCKbiog} the function
$D(r)$ has been erroneoously omitted from the integrand of both equations.}.
Distances of stars can be measured directly from trigonometry using as base the
diameter of the Earth's orbit. But Kapteyn and others had concluded that it
was not possible to do this in a wholesale manner. So he decided to use the
motion of the Sun through space as baseline. Already in one year this amounts
to four times larger a baseline and furthermore only increases with time.
This method of so-called secular parallaxes uses proper motions which contain
a reflection of the motion of the Sun through space in addition to the
peculiar motion of each star itself. His assumption was that these latter
components were random and isotropic. However because of these random
motions these secular parallaxes can only be derived statistically.
Kapteyn developed the method to solve these
fundamental equations basically in two papers,
\cite{JCK1900} and \cite{JCK1902}. The first step is to determine, with the
secular parallax method, the average parallax of the stars of any apparent
magnitude $m$ and proper motion $\mu$. We can also estimate the average absolute
magnitude when we know the average parallax. Of course for an individual
star of an observed $m$ and $\mu$ the parallax will be different from the mean.
But for the problem of the distribution of the stars in space it is not
necessary to know the distance of each individual star, only how many stars
there are at a certain distance. Furthermore, we need to know in principle
the distribution law of the parallaxes around the average value.
However, Kapteyn showed that widely
differing assumptions of this law led to results that differ but little. Now,
since it thus becomes possible to find, for any given distance, the number of
stars of each absolute magnitude, we can evidently derive the two fundamental
laws which determine the arrangement of the stars in space, viz. the
distribution of the stars over absolute magnitude, which Kapteyn called the
'luminosity curve' and for each line of sight
the law of total stellar density with distance from the Sun.
Kapteyn developed a
convenient numerical procedure of conducting these computations, after he had
introduced two assumptions: (1) there is no appreciable extinction of light in
space; (2) the frequency of the absolute magnitudes (the luminosity
curve) does not change with the distance from the Sun.
The point then is that this procedure
gives indeed both the luminosity curve in the solar
neighborhood {\it and} the density law, but the latter
up to only small distances, since stars
with measured proper motions are involved. So this really
only constitutes a first step, namely determining the local luminosity
function. The next is to use deep star counts to find the
star density out to larger distances, but that is a separate problem of
`inverting' these two fundamental equations. This summary
illustrates how the second part of van Rhijn's thesis, {\it a new determination
of the mean parallaxes of the stars with given magnitude and proper motion},
fitted into the grand scheme of things, and that it constituted the
important preliminary step before one could derive a model for the distribution
of stars in space.
\bigskip
After his return from Mount Wilson, van Rhijn had been appointed assistant at
the Groningen Laboratorium in 1914.
In addition to completing his PhD thesis, van Rhijn was heavily involved
in Kapteyn's activities to prepare for his first attack on the problem of the
structure of the Sidereal System. Much of this work had been done
done with the assistants, first Willem de Sitter,
Herman Albertus Weersma (1877--1961) and Frits Zernike (1886--1966),
and later van Rhijn. The program led to major contributions in the
{\it Publications of the Astronomical Laboratory at Groningen}:
van Rhijn in 1917
\cite{PJvR1917}, Kapteyn, van Rhijn \&\ Weersma in 1918 \cite{KRW1918},
Kapteyn \&\ van Rhijn in 1920 \cite{KvR1920a}. All this work was part of the
necessary preparatory effort to bring
together information and data etc. available into
a form that could be used to study the 'Construction of the Heavens'. This was
finally done for the first time by Kapteyn and van Rhijn in 1920 \cite{KvR1920b}, in
which they presented a crosscut through the Stellar System, which
came to be known as the Kapteyn
Universe. Since the analysis was done with count
averages over all longitudes, this
schematic model had the Sun by definition
in the center. It was the basis for the {\it First
attempt at a theory of the arrangement and motion of the Sidereal System} by
Kapteyn \cite{JCK1922}, which was the
start of observational galactic dynamics and moved
the Sun out from the center by 0.65 kpc.
Under and in collaboration with Kapteyn, van Rhijn had been extremely
productive. He even found time to complete another fundamental study {\it On
the brightness of the sky at night and the total amount of starlight}
\cite{PJvR1921}. This was another old favorite of Kapteyn, who wanted to find a
constraint on the distribution of stars in space by determining an observational
value for the amount of integrated starlight.
It had been attempted before in a PhD thesis under Kapteyn by a
student with the name Lambertus Yntema (1879--1932),
who tried to measure this from a dark
site not too far from Groningen by two ingenious methods developed by Kapteyn,
comparing an illuminated small surface to the sky viewed next to it and by
photographing the sky through a small hole
and comparing that to an exposure of the full moon
\cite{LY1909}. He found that there was a
significant amount of 'Earth light', a sort of 'permanent aurora', but he
speculated that part of his observed light would be diffuse starlight.
Yntema left astronomy ending up as rector of a lyceum or grammar school in Bussum.
Van Rhijn, during his stay at Mount Wilson, had repeated this study from the
mountain top. Zodiacal light is another factor; it is in fact comparable in
brightness to integrated starlight. Actual direct measurements
of integrated starlight have only been obtained in recent times, particularly
by the Pioneer 10 spacecraft on its way to
Jupiter, when in 1973 it had passed the asteroid belt. The zodiacal light
there was negligible and the spacecraft
produced maps of the integrated star light that could be used to study the
properties of the stellar distribution in the
Galaxy; for details see \cite{PCK86}.
Both Yntema and van Rhijn claimed values that compared to
our current
knowledge are not bad at all, but this must be fortuitous. For a more
thorough and detailed discussion see \cite{JCKbiog} or \cite{JCKEng}.
All of this goes to show that van Rhijn
produced quite a lot of high
quality work, which however was initiated by Kapteyn and executed under
his supervision.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.98\textwidth]{figlab.jpg}
\end{center}
\caption{\normalsize The `Sterrenkundig Laboratorium', when it had been
named after Kapteyn on the occasion of his retirement.
Kapteyn Astronomical Institute.}
\label{fig:lab}
\end{figure}
\section{Kapteyn's Plan of Selected Areas}\label{sect:PSA}
Kapteyn launched the {\it Plan of Selected Areas} in 1906 \cite{SA},
after a few years of planning and
discussions. A few relevant pieces of the text are important to cite here for
background to what is discussed below.
The Plan was designed to address a particular problem:
What is the distribution of stars in space and what is the structure of the
Sidereal System? But there is more to it, as we can read in Kapteyn's
Introduction in \cite{SA} (p.3):
\par \begingroup \leftskip2.5em \rightskip2em \large
[...] to make at last definite plans for the future work of the laboratory.
\par \endgroup
So, it was more than an endeavor for the interest of answering a scientific
question, it was also Kapteyn's way of providing a future for the laboratory
he had founded (Fig.~\ref{fig:lab}). The text continues:
\par \begingroup \leftskip2.5em \rightskip2em \large
After much consideration the most promising plan seems to me to
be some plan analogous to that of the gauges of the Herschels.
In accordance with the progress of science, however, such a plan
will not only have to include the numbers of the stars, but all the data
which it will be possible to obtain in a reasonable time. My intention had
been all along to give to this plan such an extension that, with the
exception of the work at the telescope, the whole of it could be undertaken
by our laboratory.
In working out details, however, I soon found out that, with a plan
any ways on a scale meeting the requirements of the case, such would be
impossible unless the funds of our institution were materially increased.
\par \endgroup
\begin{figure}[t]
\sidecaption[t]
\includegraphics[width=0.64\textwidth]{figSANorth.jpg}
\caption{\normalsize The \textit{Selected Areas}
in the \textit{Systematic Plan} of the northern hemisphere; the
numbers run from Area no. 1 in the center (the North Pole) to No. 115 on the
equator. The scale on the border is Right Ascension in hours or 15$^{\circ}$\
on the equator.
Declinations are indicated next to the circles, which are spaced by 15$^{\circ}$.
The dashed line indicates the center line of the Milky Way,.
Year numbers indicate when the volume of the
\textit{Bergedorf(-Groningen-Harvard)
Spektral-Durchmusterung} for the relevant declination range has been published.
Kapteyn Astronomical Institute.}
\label{fig:SANorth}
\end{figure}
Kapteyn presented the {\it Plan of Selected Areas}
as a refinement of the star gauges of William
Herschel (1738--1822) and his son John Frederick William Herschel (1792--1871).
Kapteyn quoted that they covered 145 square degrees (of about 41000 for the
full $4\pi$ steradians of sky) and counted about 117,600 stars. But this work
constituted only counts, no magnitudes or other properties. Kapteyn set an aim
of 200,000 stars, or almost twice the number of stars in the Herschel gauges.
This number for stars down to magnitude 14 required 400 square degrees, for
which Kapteyn then defined 252 areas, of dimensions either rectangular with
sides of order 75 arcmin or circular with radius 42 arcmin
(in both cases an area of about 1.5 square degrees).
This in regard of a reasonable field of view
for plates in photographic telescopes. The size eventually was not a fixed
property, areas could be smaller in crowded parts of the sky or larger in
sparse ones. The program envisaged counts to as faint a level as possible,
photographic and visual magnitudes (and thus colors),
and as far as possible parallaxes,
proper motions, spectral types and radial velocities. Kapteyn estimated that this
(not counting `additional short exposures') would require in total 9710
exposures on 2620 plates plus an unknown number for (hopefully) some 6300
radial velocities. The Plan also provided for determinations of the
total sky brightness to set a further constraint.
At the insistence of Edward Charles Pickering (1846--1919), director of the
Harvard College Observatory, the Plan consisted of two parts,
a {\it Systematic Plan} of 206 areas distributed evenly across the sky,
as envisaged by Kapteyn, and an
additional 46 for a {\it Special Plan} chosen to be particularly
suitable for more
detailed studies of the structure of the Stellar System.
The {\it Systematic Plan}
aimed for a mean separation of the areas of order 15$^{\circ}$. The distribution
of the regions 1 to 115 in the northern hemisphere
from the pole to the equator is shown in
Fig.~\ref{fig:SANorth}. This is a projection of a sphere onto a flat plane and
distortions resulting from this hide the approximate regularity
on the sky. The
declinations are as required for the desired spacing in rings of 15$^{\circ}$\
difference. The 24 regions on the equator are spaced by one hour or
15\degspt0. The rings at declinations 15$^{\circ}$, 30$^{\circ}$, and 45$^{\circ}$\ also
contain 24 regions that have spacings of respectively 14\degspt5, 13\degspt0,
and 10\degspt6. For declination 60$^{\circ}$\ the 12 regions are spaced by
15\degspt0, while for the 6 at declination 75$^{\circ}$\ it is 15\degspt5. This is
indeed a rather uniform distribution, except that it is not evident
why Kapteyn did not restrict
the declination zone 45$^{\circ}$\ to 18 regions spaced by 14\degspt1
instead of 24 of 10\degspt6.
The distribution is not {\it precisely} uniform because for calibration
purposes they were required, if at all possible, to be centered on a star of
magnitude 8.0 to 9.0. But there also should be no star brighter than this --
as Pickering had stressed --, since its light would scatter
over much of the photographic emulsion and affect the determination of
magnitudes of the faint stars.
This sounds like a straightforward selection procedure,
but in fact is less trivial than it may seem, as
a quick calculation shows. The density of stars between 8-th or 9-th magnitude
in photometric band V (roughly visual) averages about 2 per square degree (in
round numbers 90,000 stars for the full sky of 41,000 square degrees). A
Selected Area covers about 1.5 square degrees. The density varies of course
substantially with Galactic latitude, ranging from somewhat less than one in
the poles to up to ten or more stars
per square degree at low latitude (for these
numbers see Fig.~1 in \cite{PCK86}, which is produced with the export version
of the Galaxy model computer code of \cite{BS80}). At high latitude this is a
serious limitation for the choice of fields, since fluctuations around an
average of one star per field yields a significant number of field with none.
The further condition that there should be no bright star -- of, say,
magnitude 7 or brighter -- in the Area excludes a significant part of the sky;
there are about 16,000 of such stars over the full sky or one star per 2.6
square degrees. This would imply that a randomly chosen field of 1.5 square
degrees has a chance of one in two or so to have a star brighter than magnitude
7 in it. Kapteyn did not comment on this but his combined conditions must have
been a serious constraint and have made the choice of the positions of his Areas
far from simple.
The selection of the Areas for the {\it Special Plan} was done along a
number of lines, that I will summarize as follows with excerpts from Kapteyn's
original 1906 proposal \cite{SA}:
\par \begingroup \leftskip2.5em \rightskip2em \large
A. Quite a number of areas (19) in which […] maps show a black opening
surrounded by rich parts [...]; or a rich part, or rich parts, between dark
spaces [...]; or where there is at least a sudden change in star density
[...]. \\
B. A series of areas on the great branches of the Milky way and the rift
between them. [...] Moreover several plates of the series A will very
usefully serve the study of this division of the Milky way.\\
C. Further: very rich or very poor parts of the Galaxy [...]: Here too, many
of the plates of series A will be useful, so for instance [the one] on the
border of the southern Coalsack.\\
D. Two Milky way areas [...] for which de Sitter finds a marked difference,
in opposite direction, between photographic and visual magnitude.\\
E. For the rest I have included, of {\it extra}-galactic regions: two
areas
[..] coinciding with the richest parts of the {\it Nubecula minor};
{\it two} areas
[...] coinciding with the parts richest in stars and nebulae of the {\it
Nubecula major}; {\it one} area [...] coinciding with the part of the Orion nebula
with strongest contrast in star density; one plate [...] covering a part of
the sky, near the North Pole of the Milky way, exceptionally rich in small
nebulae. [Italics from Kapteyn]
\par \endgroup
It is interesting to consider the last category (E.) in somewhat more detail.
The term {\it extra-}galactic means of course away from the Milky Way on the
sky, while eventually it became to mean outside our Galaxy.
The {\it Nubecula major} and {\it Nubecula minor} are the Large and Small
Magellanic Clouds. At the time the Magellanic Clouds were generally considered
to be detached parts of the Milky Way, although the suggestion that they were
separate stellar systems had been made \cite{CA1867}. The brightest stars in
them are of apparent magnitude 9 or 10 and this by
itself would also suggest relatively large distances. Kapteyn noted in a
footnote that there is no field in the {\it Systemstic Plan} that covers
either of the Clouds.
The exceptional density of nebulae in the final Area refers to the Virgo
Cluster. It is one of the interesting coincidences in galactic and extragalactic
astronomy that the relatively nearby (some 20 Mpc) Virgo Cluster of galaxies
is on the sky located close to the Galactic north pole and consequently that
our Galaxy would be seen close to face-on for astronomers there. The dense
center of the Virgo Cluster is some 15$^{\circ}$\ from the pole,
and there is an only about 3\%\ chance to see another spiral
galaxies that close to face-on
\cite{PCKMWG}\footnote{\normalsize
See this reference also for some more remarkable
coincidences related to the orientation of nearby galaxies and the position of
the Sun in the disk of the Galaxy.}. The region around the Galactic
North Pole is indeed
rich in nearby galaxies belonging to the Virgo Cluster and its extensions into
the Local Supercluster. Kapteyn's Area No. 32 in the {\it Special Plan} is
about 1$^{\circ}$\ away from the actual Galactic north
pole as defined now. Kapteyn has not specified what he proposed to about
the small nebulae. In the catalogue
resulting from the {\it Special Plan} no special attention has been given to
the nebulae in this Area.
\bigskip
There are two aspects that I want to stress. Although Kapteyn
may have considered his plan a work of service to posterity,
much like the {\it Carte du Ciel}, and the great star catalogues {\it Bonner
Durchmusterung}, his own and David Gill's (1843--1914)
{\it Cape Photographic Durchmusterung}, and
the {\it Astronomische Gesellschaft Katalogs}, he had in the first place, and in
clear distinction to these works, a well-defined scientific goal in mind.
He might have had some hope in
1906 that he himself would be able to use at least a major part of the data
to construct a model of the Sidereal System. That
hope turned out to be unrealistic,
and he was forced to replace this desire with his {\it First attempt at a
theory of the arrangement and motion of the Sidereal System} \cite{JCK1922},
based on
data before the bulk of the Selected Areas had been surveyed.
But he would lay the foundations for future
astronomers to do so. He expected though that his
successor would surely complete what he had been unable to accomplish.
Secondly, he saw it as a program for his Groningen Laboratorium to carry out
completely by itself except for the data collection at the telescopes.
And that this would
require extra investment. However, it turned out that he should not have
counted for this on his university, and certainly not on his government.
Just as when in 1886 he set out
to measure up the plates for the CPD and reduce the date to positions and
magnitudes, he had to rely for the largest part on private funds. He now had
his own Laboratorium with, since 1911, adequate housing, contrary to the 1880s
when physiology professor Dirk Huizinga (1840--1903)
generously provided some working space (as it happened in the same building
that later housed the Laboratorium). Luckily,
he was saved then by George Hale, when in 1908 he adopted the {\it Plan of
Selected Areas} as the prime observing program for his brand new giant 60-inch
telescope on Mount Wilson. Kapteyn was appointed Research Associate by the
Carnegie Institution of Washington, which enabled him to visit California
annually (at least until the first World War broke out) and take charge of the
execution of the program at the big telescope.
But is turned out unrealistic that all work could be done in Groningen.
For example, the following was an unforeseen complication. It turned out
much less straightforward to derive stellar magnitudes from photographic plates
taken with a large reflector than a small photographic telescope.
To illustrate this the following. The field of view of the 60-inch telescope was
only about 23 arc minutes and many of the fields had no stars brighter than
12-th magnitude.
Using the same Fig.~1 of \cite{PCK86} as above it can be deduced that
averaged over the
whole sky the number of stars of 12-th magnitude or brighter is
about 70 per square degree, which means on average 10 or
so per 23 arcmin field of the 60-inch. But at high latitudes it is on
average one or a few, so that fluctuations will assure that a
non-negligible fraction has no stars brighter than magnitude 12.
This complicated matters as calibration of the zero-point of the magnitude
scale relative to the standards around the celestial North Pole had to be
permormed in such Areas without making use of stars brighter than magnitude 12.
Furthermore stars away from the center of the field had
distorted images from optical aberration's such as coma
due to the parabolic shape of the mirror.
All this required
special attention and much work was taken off Kapteyn's hands
by Mount Wilson astronomer Frederick Hanley Seares (1871--1964), who took
this upon himself to solve. This was an enormous effort and it lasten until
1930 until the project was concluded with the publication of the
catalogue \cite{MWilson}. For manpower in Groningen Kapteyn
relied on funding from various sources rather than the university and
significant parts of the manpower was actually paid for by the Carnegie
Institution. The university and government provided only one
computer (`rekenaar', a person whose primary task was to perform calculations)
and one clerk in addition to the professorship and assistant. When
Kapteyn retired the pay-roll of the university staff consisted of the
professor, one assistant, three computers and a clerk (see Table 1 in section
\ref{sect:staff} below), but that increase had only occurred in 1919.
The University of Groningen had committed to the {\it Plan of Selected Areas}
when it agreed with Kapteyn's appointment by the Carnegie Institution (it was
less generous than this suggests, since
he had to make up in teaching that he failed to provide during his Mount Wilson
visits over the remaining months). To some extent the lack of financial
support is understandable as the funding for the universities in general was
tight. However, due to its remoteness from `Den Haag'
(the government seat in the Hague) and the difficulty
of effective lobbying, and due to low student numbers in Groningen, it was
particularly tight there.With the appointment of Willem de Sitter
as director in 1918 a
major reorganization had taken place at the Sterrewacht Leiden, and its staff
was three or four times larger than at the Groningen Laboratorium.
\bigskip
The execution of the {\it Plan of Selected Areas} during the remaining years
of Kapteyn was restricted,
as far as the work in Groningen is concerned, to the cataloging
of stars in the Areas from the plates it received from Harvard and
Mount Wilson with the vital support of
the directors Edward Pickering and George Hale.
The plates of all 206 Areas Pickering had committed to
were taken at the Harvard College Observatory in the north and at
its Boyden Station in Arequipa, Peru, in the south, respectively
between 1910 and 1912 and between 1906 and 1916, and from all of these plates
stellar positions and magnitudes had to be derived.
All those measurements were performed in Groningen.
The catalogue resulting from the Harvard plates, which went down to about
magnitude 16, was published in three installments by Pickering \&\ Kapteyn in
1918 and by Pickering, Kapteyn \&\ van Rhijn in 1923 and 1924 in the
{\it Annals of the Astronomical Observatory of Harvard College}
\cite{Harvard1}\cite{Harvard2}\cite{Harvard3}.
The work had been a joint undertaking between Harvard and Groningen. As
Pickering wrote in the Preface to the first volume:
\par \begingroup \leftskip2.5em \rightskip2em \large
It was, therefore, a pleasure to aid the plans of Professor Kapteyn in his
work on the Selected Areas. We have accordingly taken the photographs needed
to show the faint stars, determined the magnitudes of a sequence of standard
stars in each area, and provided the means of publication of the results.
Professor Kapteyn has undertaken the laborious work of measuring the plates,
reducing the measures, and determining the positions and magnitudes of large
numbers of stars.
\par \endgroup
The Durchmusterung became known as the {\it Harvard Durchmusterung} in spite of
the fact that while the plates were indeed taken at Harvard, the bulk of the
work was done in Groningen. I will therefore refer to it as the {\it
Harvard(-Groningen) Durchmusterung}.
According to the {\it Third Report} of the progress of the Plan
\cite{PJvRrep3}, the number of stars in the
{\it Harvard(-Groningen) Durchmusterung} was some 250,000, for all of which
positions and magnitudes had been measured in Groningen. An examination of the
three volumes shows accurate quotes for the total number in the first two
volumes, while adding up numbers in Table 1 of the third volume is easily
performed. This results then is a total of 84,002 + 65,753 + 82,226 = 231,981
stars, so it is actually somewhat less than what had been advertised.
The work in Groningen was quite extensive and, since great care had to be taken,
very time consuming. Most of it was done by dedicated staff,
but had to be overseen by
an astronomer. In Kapteyn's Introduction to the first installment in 1918
on the northern hemisphere, he stressed that his own contribution was minor,
but that the supervision of the work was left to the assistants, which he
listed as Willem de Sitter, Herman Weersma, Frits Zernike and Pieter van Rhijn.
The second and third volumes covering the south were authored
by Pickering, Kapteyn (posthumously) and van Rhijn, and the latter added Jan
Oort to this list of assistants.
\bigskip
The primary plates at the Mount Wilson 60-inch of the 139 Selected Areas
north
of declination -1$^{\circ}$\ were taken between 1910 and 1912, plates with shorter
exposure time and/or reduced aperture were taken from 1913 to 1919, and a set
of exposures of the Selected Areas and a field with stars in the {\it
North Polar Sequence} with a 10-inch telescope in the same years.
Overall more than one
thousand plates were taken. The project was executed at Mount Wilson under
the leadership of Frederick Seares.
For the {\it Mount Wilson(-Groningen) Catalogue} the work was more evenly
divided than for the {\it Harvard(-Groningen) Durchmusterung}. In addition to
the 'long-exposure plates' together with short exposure plates, that were all
sent to Groningen, to facilitate the transition from faint to brighter stars,
a long and thorough investigation had been conducted at Mount Wilson to learn
how to do photographic photometry with large reflectors
and to define a magnitude
scale and zero-point in each of the Selected Areas. For this purpose
another set of exposures, using the 60-inch telescope stopped to
smaller apertures was obtained. As a consequence two sets of stars were
catalogued: the long-exposure plates were only studied over 15 arcmin areas at
low Galactic latitudes (below 40$^{\circ}$) and 20 arc\-min at high latitude. The
plates measured for magnitude scales and zero points were measured over a
larger area and this yielded 'supplementary' stars outside the 'Groningen
areas'. The {\it Mount Wilson Catalogue of photographic magnitudes in Selected
Areas 1–139} by Seares, Kapteyn, van Rhijn, Joyner \&\ Richmond appeared in
1930. It reached magnitude 18.5 or so. The final catalogue contains 67,941
stars, of which 45,448 had been measured in Groningen.
\bigskip
The measurements and reductions for the plates for the {\it Special Plan}
\cite{vRK1952} were seriously delayed by the economic depression and associated
shortage of manpower and then the Second World War and in the end were only
published in 1952. It was no longer a joint publication between Groningen and
Harvard, but solely by the Kapteyn Laboratorium. The work concerned all the
46 Areas of the {\it Special Plan} and the total number of stars measured
was 140,082 stars down to 15 to 16 photographic magnitude. Measurements and
reductions were supervised by van Rhijn's assistants starting with Willem
Jan Klein Wassink, then Bart Bok, Petrus Bruna, Jean-Jacques Raimond, Adriaan
Blaauw and ending with Lukas Plaut (and van Rhijn himself did some of this
also of course). The plates were taken decades before the
publication, most of them in 1911 and 1912, but some as long ago as in 1898
(so before the {\it Plan of Selected Areas} was even defined and at that time
for some other purpose). The measurements extended only over a restricted part
of the plates, in squares of roughly 50 to 100 arcmin on a side,
in general depending on the structure in the star field in the
particular Area or the richness.
The production of this catalogue constituted a very large
expenditure in terms of time and manpower. The use that has been made of
it is very limited it
seems, to some extent of course since its appearance was long overdue, decades
after the plates had been taken. The NASA Astrophysics Data System
ADS\footnote{NASA/ADS (ui.adsabs.harvard.edu) provides some information in
terms of numbers of citations
and of reads as measures of usefulness or at least one aspect of this.
Citations to other publications are reasonably complete in ADS
since the mid-fifties, when journals started to
collect references at the end of papers
rather than in footnotes in the text, so that it was practical for
ADS to collect them (see section 14.6 of \cite{JHObiog} for
some more details). Reads, when the full record in the ADS has
been downloaded by a user, are available since roughly the year 2000.}
records no
(zero!) citations after the publication of the {\it Durchmusterung},
so it is unlikely it has ever been used as source material
for a study as envisaged initially. The number of
reads is only a few (except in 2021, which are due to me for this study).
But then finding it on ADS merely yields the information
that the publication is not available online. As we will see below
(section~\ref{sect:extinction}), the results have actually been used
in a preliminary form in the 1938 PhD thesis of Broer Hiemstra \cite{BH38},
who refers to the unpublished counts of the {\it Special Plan}. This thesis
concerned distances and total extinction of dark clouds, and contained
extensive determinations of proper motions from other plates
than the Harvard ones for the {\it Special Plan}. As far as I am aware no
extensive use is made of the counts in the fields in the Magellanic Clouds;
these amount tot 9777 stars in the richest part of the SMC and 7818 in that
of the LMC.
It would not be surprising if more than maybe a few researchers of the Clouds
are or were actually aware of the existence of these uniform data sets of
magnitudes and star counts.
\begin{figure}[t]
\sidecaption[t]
\includegraphics[width=0.51\textwidth]{figdeSitter.jpg}
\includegraphics[width=0.47\textwidth]{figWeersma.jpg}
\caption{\normalsize Willem de Sitter and Herman A. Weersma.
These pictures come from the {\it Album Amicorum}, presented to H.G. van de
Sande Bakhuyzen on the occasion of his retirement as professor of astronomy
and director of the the Sterrewacht Leiden in 1908 \cite{Album}.}
\label{fig:dSW}
\end{figure}
\section{The succession of Kapteyn}\label{sect:Succession}
In 1908 there had been a vacancy in the directorship of the Sterrewacht Leiden
after the retirement of Hendrikus van de Sande Bakhuyzen.
It had of course been offered to Kapteyn, but -- as actually
he had done earlier in the case of
Utrecht -- he had declined. Ernst van de Sande Bakhuyzen
(1848--1918) had been appointed director and de Sitter (see Fig.~\ref{fig:dSW})
professor. Herman Weersma (Fig.~\ref{fig:dSW})
in Groningen in 1908 took de Sitter's place and became Kapteyn's
assistant. Weersma had obtained his PhD in that year, the thesis title was
{\it A determination of the apex of the solar motion according to the methods
of Bravais}.
Kapteyn considered Weersma his probable successor, but in 1912
Weersma left to take up a position as secondary school teacher, because he
wanted to have more time to devote to his `real interest', philosophy.
Kapteyn was at a loss. There was no suitable candidate in the Netherlands.
After I had written my biography of Kapteyn \cite{JCKbiog}, Prof. Kalevi
Mattila of Helsinki University informed me of a development. The story is as
follows:
On November 3, 1912, not long after Weersma's departure, Kapteyn wrote to his
longtime friend and collaborator Anders Severin Donner (1854--1938) of
Helsingfors Observatorion (Helsingfors is Swedish for Helsinki) as follows
(provided by Mattila, my translation from the German):
\par \begingroup \leftskip2.5em \rightskip2em \large
At present my greatest worry is to find a successor that will keep the
laboratory from going into a decline. My assistant leaves me. Do you know
anyone -- assiduous, practical, reliable -- who without too much pay (he will
get immediately 2000 German Mark, but I see a possibility, if he is really good,
that next year he might be paid and extra 1200 Mk)
could develop in the next 9 years to
become my successor?
\par \endgroup
\noindent
Prof. Matilla continued his email to me:
\par \begingroup \leftskip2.5em \rightskip2em \large
Donner's pupil Yrj\"o V\"ais\"al\"a had just completed his Masters degree the same
month and Donner approached him with this proposal. Prof. Olli Lehto published
in 2005 a triple biography (in Finnish) of the three V\"ais\"al\"a
brothers, who all became professors: Vilho (meteorologist), Yrj\"o
(astronomer, geodesist), and Kalle (mathematician). Having heard from me
of this letter by Kapteyn, Vilho got access also to some private
correspondence of Yrj\"o V\"ais\"al\"a to his bride Martta. Yrj\"o V\"ais\"al\"a
was very seriously considering the great scientific opportunities offered by
this position and how pleasent it would be to live among the nice and decent
people of Holland [the Netherlands]\footnote{\normalsize Holland is sometimes used as the
name of the nation, although it really is only two of the western provinces
of the Netherlands bordering the North Sea, where Amsterdam, Rotterdam
and other cities thrived as centers
of commerce and culture in the seventeenth century. The use of Holland instead
of the Netherlands is much the same as the often heard incorrect referring to
Great Britain as England, and both should be avoided.}.
However, as he wrote to Martta: `Away from Finland for many
years, perhaps forever! I think that we would loose much when doing that. And
what about my little Martta in that foreign country! What if we would become
parents of even smaller Marttas and Yrj\"os! Would they become Finns or Dutch?
Or neither of them.' After such deliberations he finally asked Donner to
answer with `no' to Kapteyn. Donner motivated V\"ais\"al\"a's refusal in a
somewhat different, perhaps more diplomatic way: `We Fins are passionate
patriots'. In fact, this was a period in the
history of Finland with strong national feelings while the Russian tsar tried
to suppress the autonomous status of Finland. Five years later, in 1917,
Finland declared its independence.
\par \endgroup
Yrj\"o V\"ais\"al\"a (1891--1971) went on to found the astronomy department
at Turku University, where he designed a Schmidt-like telescope with which he
and his collaborators discovered 807 asteroids and 7 comets.
\bigskip
Kapteyn's next
hope had been Frits Zernike,
who was appointed assistant, but he also left
already in 1914 to pursue a career in physics, which earned him eventually (in
1953) a Nobel Prize for his invention of the phase-contrast microscope.
In \cite{Blaauw1983}(p.61),
Blaauw confirmed this. He quoted Zernike in a speech
in 1961 at
the occasion of the donation of the well-know Veth painting of Kapteyn to the
university that Kapteyn had told him he would have liked to designate him as
the next director of the Laboratorium.\footnote{\normalsize
I have not been able to locate
a copy of this speech that Adriaan Blaauw must have seen, to confirm this.}
In 1916 Kapteyn arranged for van
Rhijn to be appointed his assistant, who became then the probable candidate to
succeed him as professor of astronomy and director of the
Laboratorium. We may have a brief look at other young astronomers in the
Netherlands at the time as possible candidates,
say those who obtained a PhD between 1910 and 1920. Kapteyn
had four students in that period: Etine Imke Smid (1883--1960, PhD thesis
{\it Determination of the proper motion in right ascension and
declination for 119 stars} in 1914),
Samual Cornelis Meijering (1882--1943;
PhD thesis {\it On the systematic motions of the K-stars} in 1916), Willem
Johannes Adriaan Schouten (1893--1971; PhD thesis {\it On the
determination of the principal laws of statistical astronomy} in 1918) and
Gerrit Hendrik ten Brugge Cate (1901--1961; PhD thesis {\it Determination and
discussion of the spectral classes of 700 stars mostly near the north pole}
in 1920). These were not obviously extraordinary theses, so they were
apparently not considered suitable candidates.
Etine Schmid was the first female astronomy PhD in the Netherlands and
may not even have been eligible for a professorship or the directorate.
The thesis was written in Dutch; I translated the title.
She went to Leiden where she worked with among others
physicist and later Nobel laureate Heike Kamerlingh Onnes (1853--1926),
but she left scientific research in 1916, when she got married.
Meijering became a
teacher at a high-school, Schouten had alienated Kapteyn because of his
unnecessary rude remarks on von Seeliger's work and ten Brugge Cate (who later
changed his name into ten Bruggecate) also became a teacher at a high-school.
In Leiden new PhD's in this period were Christiaan de Jong
(1893--1944; PhD thesis {\it Investigations on the
constant of precession and the systematic motions of stars} in
1917), Jan Woltjer (1891--1946; PhD thesis {\it Investigations in the theory of
Hyperion} in 1918) or August Juliaan Leckie (1891--19??; PhD thesis
{\it Analytical and numerical theory of the motions of the orbital planes of
Jupiter's satellites} in 1919). De Jong, whose supervisor
was Ernst van de Sande Bakhuyzen, already became a gymnasium teacher of
mathematics in Leiden before completing the thesis, which was
written in Dutch (title above is my translation). He may not have been
interested in a career in astronomy in the first place. De Jong was executed
by the Germans as retaliation of an act of sabotage he had not been concerned
with. Woltjer became a Leiden staff member and worked on stellar pulsation
theory until his death in 1946 as a result of undernourishment in the last
year of the War. His interests were remote from the ongoing investigations in
Groningenand anyway. Leckie moved to the Dutch East Indies, where be worked at
the Bandung Institute of Technology. The last two were students of Willem de
Sitter and worked on theoretical subjects involving planetary satellites.
In Utrecht Adriaan van Maanen
(1884--1946) had defended his thesis {\it The proper motions of 1418 stars in
and near the clusters h and $\chi $ Persei} in 1911 under Albertus Antonie
Nijland (1868--1936) and Kapteyn; he might have been a potential candidate,
but had moved permanently to Mount Wilson Observatory.
This list is not very long, so candidates were difficult to find.
Among these persons that
obtained their PhD's before or in 1916, there is no obvious other
candidate that could
have been considered as being preferred over van Rhijn for the position of
assistant in Groningen. After van Rhijn had
been appointed, a further appointment would
have been difficut to make so especially after that only exceptionally promising
candidate could have been considered. There is no such person obvious in
the list Nor is there anyone among these that would be so
outstanding to be appointed as Kapteyn's
successor in 1921 in place of van Rhijn.
Before 1921 van Rhijn had done relevant and excellent
research. He had extensively worked on
proper motion observations and with Kapteyn had used simulations to show that
with counts down to magnitude 17, which the plates taken with the 60-inch
would certainly be able to provide,
`the densities should become pretty reliable for the whole of
the domain within which the density exceeds 0.1 of that near the Sun'.
In Kapteyn's last paper before retirement that van Rhijn co-authored, they
examined the matter of the Cepheid distances and the consequences for those of
globular clusters, concluding that Shapley's distances were much
too large; Shapley's result was indeed
incorrect due to the now known concept of Stellar
Populations and the unknown differences between pulsating variables in these
two Populations (for more see \cite{JCKbiog}, p.592).
Van Rhijn before he assumed
the directorship definitely had been quite productive and had
produced excellent astronomy, but under the direction of
and on research programs initiated by Kapteyn.
The conclusion must be that van Rhijn with his dedication to the work of
Kapteyn and the Laboratorium
effectively by default became Kapteyn's successor. He had done
very good research, although he never seemed to have had a
chance to define his own projects. This discussion serves to illustrate the
problem of finding suitable candidates for leading positions in astronomy at
the time. There simply was no other option.
Van Rhijn had certainly
shown himself to be a competent researcher, so he was a man
Kapteyn could expect to dedicate himself to the continuation of his life's work.
There is no evidence that Kapteyn looked for other possibilities
abroad, being confident apparently that van Rhijn was the person to whom he
could entrust the future of his Laboratorium and the further execution of his
{\it Plan of Selected Areas}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.98\textwidth]{figdrawing.jpg}
\end{center}
\caption{\normalsize Pieter Johannes van Rhijn. Reproduction of a crayon
drawing donated by his relatives to the Kapteyn Astronomical Institute.
It dates from 1926. It has been produced by Eduard Gerdes (1887--1945),
who was a Dutch painter and art teacher. Later, in the 1930s, he would sympathize
with the national socialist party, and hold important functions in art policies
during the War. Gerdes died soon after the War ended, but the cause of his
mysterious death has never been
determined. He was posthumously found guilty for collaborating with the German
occupation. This drawing decorates the Kapteyn Room.}
\label{fig:draw}
\end{figure}
\section{Van Rhijn's astronomical work up to 1930}\label{sect:1920s}
When Kapteyn retired from his positions of professor and director at the end
of the academic year in 1921, he
actually left Groningen, never to return, and completely terminated all
involvement in astronomy in Groningen.
During the first decade of van Rhijn's directorship, roughly the 1920s, the
Laboratorium still flourished.
A very important part of the {\it Plan of Selected
Areas}, the star counts in the {\it Harvard(-Groningen) Durchmusterung} and the
{\it Mount Wilson(-Groningen) Catalogue} progressed vigorously.
The publication of the
first was completed in 1924, encompassing the {\it Systematic Plan} of the 206
all-sky Areas. This had been a major investment of effort on behalf of
the Laboratorium, which in the mean time had been named
`Sterrenkundig Laboratorium Kapteyn' or just `Kapteyn Laboratorium'.
The {\it Special Plan} of 46 areas in the Milky Way on which Pickering had
insisted so much and which Kapteyn had been forced to accept to ensure
his collaboration, went along at a very slow pace
and was published only in 1952. The Groningen part of the work on the Mount
Wilson program was also completed expeditiously, but the problem of deriving
accurate magnitudes using large reflectors took much longer to solve and
as we have seen, was a
responsibility of Seares at Mount Wilson; still it was completed and published
in 1930.
In 1923 van Rhijn had followed up the first and second reports,
published by
Kapteyn \cite{JCK1911}
on the progress of the {\it Plan of Selected Ares}, by a third
report \cite{PJvRrep3}.
It was an almost one hundred page booklet, published by the
Laboratorium. It was followed up with a fourth report in 1930 in the {\it
Bulletin of the Astronomical Institutes of the Netherlands} \cite{PJvRrep4}.
So van Rhijn had picked up his responsibilities with vigor.
But he did also some very significant research, some in providing additional
observational material to the Plan, some in analysis relevant to the
aim of deriving a model for the Stellar System. In this he was assisted by
young astronomers. Upon his appointment the then vacant position of assistant
had for a short period been filled by Jan Oort, who soon after this
appointment had been promised a staff position in Leiden by de Sitter, and
had left for a period of two years to work with Frank Schlesinger (1871--1943)
at Yale Observatory in New Haven to learn astrometry.
It had been filled subsequently by students preparing a PhD
thesis: one year by Peter van de Kamp (1901--1995)
who defended a thesis on {\it De zonsbeweging met
betrekking tot apparent zwakke sterren} (The solar motion with respect to
apparent faint stars) in 1924, and subsequently by Willem Jan
Klein Wassink (1902--1994),
who submitted a thesis on {\it The proper motion and the
distance of the Praesepe cluster} in 1927. Klein Wassink left astronomy and
like so many young PhDs in astronomy became a physics and mathematics teacher,
in this case in Zierikzee in the province of Zeeland in the southwest.
After him the
position was filled by students that would obtain their doctorate after 1930,
Bart Bok and Jean Jacques Raimond (for details see below).
\begin{figure}[t]
\sidecaption[t]
\includegraphics[width=0.64\textwidth]{fig1926JHO.jpg}
\caption{\normalsize Pieter van Rhijn in 1926 at the dinner after the PhD thesis
defense of Jan Hendrik Oort, for which he acted as supervisor. Sitting at the
table are Willem de Sitter and his wife. Standing next to van Rhijn is Oort's
(then future) mother-in-law and on the left Oort's mother. From the Oort
Archives, see \cite{JHObiog}.}
\label{fig:JHOdiner}
\end{figure}
In addition to this van Rhijn (see Fig.~\ref{fig:JHOdiner})
had been `promotor' (thesis supervisor) of
three students, that had actually started under Kapteyn. The
first is Egbert Adriaan Kreiken (1896--1964; for an obituary see \cite{EAK}),
who completed a PhD thesis {\it On the color of the faint
stars in the Milky Way and the distance to the Scutum group}, in 1923.
Kreiken had been born in exactly the same house as Kapteyn in Barneveld,
where his parents had taken over the boarding school of Kapteyn's
parents after the Kapteyn Sr. had died. After many peregrinations, Kreiken
ended up in Ankara, Turkey were he founded the observatory which is now named
after him. Jan Schilt (1894--1982; for an obituary by Oort see \cite{JHO1982}),
who had defended his thesis
{\it On a thermo-electric method of measuring photographic magnitudes}
in 1924, had moved to Leiden. Next
moved to the USA, where eventually he became chair of
the Columbia astronomy department. Jan Oort had already moved to Leiden when
he defended his thesis in 1926 on {\it The stars of
high velocity}.
\bigskip
In addition to work on the {\it Plan of Selected Areas} catalogues and
associated matters, the coordination of the Plan's execution and writing
reports, and in addition to his teaching and supervision of students, van
Rhijn did some very important and fundamental work. There are actually three
papers in this category that I will very briefly introduce. The first concerns
what we have seen was the first step to derive a model for the distribution of
stars in space an that is the mean parallax of stars as a function of proper
motion and magnitude in various latitude zones. In 1923 van Rhijn published an
improved version of what he had published with Kapteyn in 1920, but now for
spectral classes separately \cite{PJvR1923}. Secondly, after first having
discussed the relative merits and comparisons of trigonometric, statistical
and spectroscopic parallaxes \cite{PJvR1925a}\footnote{\normalsize
Van Rhijn used 'mean
statistical parallax' in the title (note that ADS has omitted the 'mean').
Statistical parallax is different from secular parallax and this may be
confusing to some readers. I use the following. Usually
statistical parallax is that for a group or cluster
of stars for which both proper motions and radial velocities are known, where
then the average parallax is derived from comparing the scatter in the
radial velocities to that in the proper motions. It is therefore fundamentally
different from secular parallax from proper motions relative to the solar apex.
Van Rhijn has in the titles of the (sub-)sections in
the paper been careful to avoid confusion,}, van Rhijn
produced in 1925 an improved version of the luminosity curve of stars
\cite{PJvR1925}.
\begin{figure}[t]
\sidecaption[t]
\includegraphics[width=0.64\textwidth]{figRhijn2.jpg}
\caption{\normalsize Pieter van Rhijn around 1930. Kapteyn
Astronomical Institute.}
\label{fig:vRhijn2}
\end{figure}
This was a very profound paper cited for a long time. It presented
what became known as the
'van Rhijn luminosity function'. Its importance today lies with studies of
the Initial Mass Function (IMF) of star formation, where for stars below about
one solar mass (that live longer than the age of the Galaxy)
it directly follows from this local luminosity function. ADS,
the NASA Astrophysics Data System, still shows of order 3 reads per year over
the last 15 or so. It has been of major importance. The ADS is highly
incomplete in citations before the mid-1950s, so we cannot illustrate its
impact using these.
And thirdly, in 1929 van Rhijn (Fig.~\ref{fig:vRhijn2})
presented a comprehensive set of star counts
\cite{PJvR1929}. It was the final product of the {\it Harvard-(Groningen)
Durchmusterung} and the {\it Mount Wilson(-Groningen) Catalogue},
the latter before it was
actually published. It was presented as a table with positions at intervals of
$10^{\circ}$ in Galactic longitude and latitude with the number of stars per
square degree at integer values of photographic magnitude down to magnitude 18.
Alternatively it was presented as a table of coefficients as a function of
position for an analytical fit to the counts. The counts in the {\it
Selected Areas} in the Harvard and Mount Wilson data were for magnitudes
larger than 10, between 6 and 10 van Rhijn used counts derived by Antonie
Pannekoek using the {\it Bonner} and {\it Cordoba Durchmusterungs} and still
brighter the {\it Boss Preliminary General Catalogue}. This together
constituted a major milestone in Kapteyn's Plan.
This would have been a point in time for van Rhijn to redo the analysis in the
paper of Kapteyn and van Rhijn in 1920 \cite{KvR1920b} and derive an improved
version of the distribution of stars in space.
But in the mean
time it had become a general feeling among astronomers that indeed interstellar
extinction did exist, see for example Oort (1927, 1931),
waiting only for the final blow to the notion of the
absence of absorption to be delivered by Robert Julius Trumpler
(1886--1956) in 1930. In any case, van Rhijn postponed
such an exercise, at least felt it was premature to do so.
That the matter of interstellar extinction was definitely on
his mind is clear from his paper in 1928, in which he studied the matter
using diameters of globular clusters \cite{PJvR1928}. This was a clever and
timely exercise. Remember that the notion of an absence of absorption was
primarily due to Harlow Shapley's (1885--1972)
study of colors of stars in globular clusters. Shapley
had found that, if Kapteyn's value were correct, the bluest
stars in globular clusters had
to be intrinsically bluer by of order two magnitudes than those in the solar
neighborhood and therefore had concluded that interstellar extinction was
negligible. Now, Arthur Stanley Eddington (1882--1944)
had proposed a mechanism in which no
reddening would accompany extinction \cite{ASE26}. Eddington considered
`molecular absorption', by which he meant dissociation of
molecules by starlight, which are continuously `disrupted and reformed'.
This is now known to
be much too inefficient. Van Rhijn decided to use diameters of
globular clusters to investigate this. The formal solution for the amount of
extinction he obtained was $.036 \pm .021$ magnitudes per kiloparsec,
which means very little if any absorption. Van
Rhijn noted that this formal value would change
the distance of Shapley's most distant globular
cluster from 67 to 35 kpc, and then mentioned but did not discuss the large
uncertainty of this. With modern eyes this result would mean that he had
found that indeed space was probably mostly transparent towards the globular
clusters. It is interesting to note that this approach of diameters of globular
clusters foreshadows Trumpler's seminal analysis of open clusters two years
later.
\bigskip
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.98\textwidth]{fig4studs.jpg}
\end{center}
\caption{\normalsize Four students of Pieter van Rhijn in 1939. From left to
right Bart Bok, Jan Oort, Peter van de Kamp and Jan Schilt.
Kapteyn Astronomical Institute.}
\label{fig:4studs}
\end{figure}
At the close of the decade of the twenties van Rhijn could look back onto a
productive period. The important Harvard and Mount Wilson
catalogs in Kapteyn's {\it Plan of Selected Areas} had successfully been
completed and important first steps for an analysis towards a model of the
Stellar System, namely
star counts, mean parallaxes for the different spectral types
and a local luminosity function had been derived. Under his coordination the
Plan was progressing well.
While during Kapteyn's directorship 27 numbers of the {\it Publications of the
Astronomical Laboratory at Groningen} had appeared (in 22 years, but he
had started in 1900
with backlog of a few), during the first nine years of van Rhijn's
directorate (1921-1939) fourteen had been published.
In addition to this and the
Harvard and Mount Wilson Durchmusterungs, he had (co-)authored eight papers
in refereed journals, a progress report on the Plan (and prepared another one
to appear in 1930), and an obituary
about Kapteyn. In addition to the three PhD students he inherited from Kapteyn,
three PhD students had produced theses entirely under his supervision
and two more were on their way:
Bart Bok would finish in 1932 on {\it A
study of the $\eta$ Carinae region} and Jean Jacques Raimond on {\it The
coefficient of differential Galactic absorption} in 1933 (see
Fig.~\ref{fig:4studs}).
There might have been other students as well that had an interest in astronomy
but did not pursue that, at least not in Groningen.
One was Helena Aletta (Heleen) Kluyver (1909--2001).
On March 15, 1929, van Rhijn wrote to
Jan Oort in Leiden (Oort Archives, website accompanying \cite{JHObiog},
Nr. 149; my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
During the Easter vacations Helena Kluyver, a very interested astronomy
student, will visit
Leiden. She wishes to see the instruments and to get an idea of the work
done at the Sterrewacht.
Can you help her a little?
\par \endgroup
Heleen Kluyver ended up moving to Leiden, where she remained in various
functions at the Sterrewacht, eventually obtaining a PhD (see \cite{JHObiog}
or \cite{JHOEng} for more on her).
And on January 12, 1931 Oort wrote (same source):
\par \begingroup \leftskip2.5em \rightskip2em \large
You will have heard that Heleen Kluyver has become half-assistant at the
Sterrewacht. She did obtain her Candidaats
[Bachelor nowadays] here. She is quiet and not very accessible, but when you
get to know her she is really a very nice
girl. In the beginning she may have been lonely in Leiden, so if the
occasion arises you may make some contact with her.
\par \endgroup
During the 1930s van Rhijn's productivity in terms of number of
publications took a downward
turn and no new initiatives were taken. To examine the background and the
reasons for this more closely we have
to look at the program van Rhijn started soon after assuming the directorate
with the Bergedorfer Sternwarte at Hamburg and his attempts to obtain his own
telescope, an undertaking that Kapteyn had failed to successfully complete.
\bigskip
But before that I close this section by noting a remark Adriaan Blaauw made in
his 1983 chapter \cite{Blaauw1983}, where he points out that
(p.51; my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
\noindent
[...] in these days the Groningen laboratory still internationally was held in high
regard, is evident from the 1933 book {\it Makers of astronomy} by Hector
MacPherson, in which a prominent place was reserved for van Rhijn and his
institute.
\par \endgroup
Hector Carsewell MacPherson (1888--1956), in his day, was
a highly respected astronomy teacher and
writer, especially of biographies of astronomers, see \cite{BG2011}. In the
final chapter of the book \cite{Makers}, {\it Explorers of the Universe},
the persons presented by MacPherson are David Gill, Jacobus Kapteyn,
Willem de Sitter, Vesto Melvin Slipher (1875--1969), Arthur Eddington,
Harlow Shapley and Pieter van Rhijn.
The latter, in MacPherson's view,
is in the same class as Gill, Kapteyn, Eddington or Shapley. He
discussed in detail and praised the work leading up to the 'Kapteyn Universe',
giving part of the credit for it to van Rhijn, even designating it the
'Kapteyn-van Rhijn Universe'. The Laboratorium in Groningen was staged as
'the world-famed institution'. Maybe van Rhijn here gets more credit than he
deserves, but the high regard of Groningen astronomy is obvious.
\section{Bergedorf(-Groningen-Harvard) Spektral- Durchmusterung}\label{sect:Spektral-D}
The work in the preceding section is quite substantial. But another major
project had been started by van Rhijn, which however would turn out a major
investment of time with in comparison very little return. This is called
in full the {\it Bergedorfer Spektral-Durchmusterung der 115 n\"ordlichen
Kapteynschen Eichfelder enthaltend f\"ur die Sterne bis zur 13. photographischen
Gr\"osse die Spektren nach Aufnahmen mit dem Lippert-Astrographen der Hamburger
Sternwarte in Bergedorf und die Gr\"o{\ss}en nach Aufnahmen des Harvard College
Observatory bestimmt durch das Sterrenkundig Laboratorium Kapteyn}.
At the start of the {\it Plan of Selected
Areas} it was foreseen that spectral types would be determined
for as many stars as possible. Now, stars brighter than magnitude 8.5 were
classified as part of the {\it Henri Draper Catalogue}, published by Annie Jump
Cannon (1863--1941) and Edward Pickering in nine volumes of the
{\it Annals of Harvard
College Observatory} between 1918 and 1924, containing about 250,000 stars
(and later extended with the {\it Henri Draper Extension}). As van Rhijn
remarked in the {\it Third progress report} on the
{\it Plan of Selected Areas} \cite{PJvRrep3}:
\par \begingroup \leftskip2.5em \rightskip2em \large
We thus have only to provide for the
classification of selected areas stars fainter than 8\magspt5.
\par \endgroup
The enormous
tasks of classifying a quarter of a million stars had been accomplished almost
entirely by Annie Cannon, which van Rhijn had used in \cite{PJvR1923} and which
had earned Cannon at Kapteyn's proposal a
doctorate {\it honoris causa} from the University of Groningen in 1921
\cite{PCK2021}. In addition to this Milton Lasell Humason (1891--1972)
at Mount Wilson Observatory obtained
spectral types of at least ten stars per Area between magnitudes 11 and 12
with the 60-inch telescope of all 115 northern Areas, in total 4066 stars
(published in 1932, \cite{Hum32}). The aim, however, was to obtain many more spectral types.
Soon after he had assumed the directorship of the Laboratorium, van Rhijn had
approached Richard Reinhard Emil Schorr (1867--1951),
the director of the Bergedorfer Sternwarte near Hamburg,
and proposed to set up a joint project to fill the gap. Schorr accepted and
assigned Friedrich Karl Arnold Schwassmann (1870--1964) to carry out the
program. Arnold Schwassmann had, after a PhD from G\"ottingen in 1893,
worked at a number of places, until he was appointed in 1902 on the staff at
Hamburg and was very much involved in moving the observatory out of Hamburg to
Bergedorf (completed 1912). Schorr had convinced
local businessman Eduard Lippert to finance an astrograph for the new site
rather than setting up a private observatory. It was a triple
telescope, consisting of a double astrograph with
in addition a telescope according to the {\it Carte du Ciel} specifications on
the same mount. Schwassmann, later together with Arno Arthur Wachmann
(1902--1990), used it for
hunting comets and asteroids. The instrument included an object prism that
could be fitted to any of the three astrographs (see Fig.~\ref{fig:Lippert}).
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.98\textwidth]{figLippert.jpg}
\end{center}
\caption{\normalsize Arthur Arno Wachmann at the Lippert Astrograph, with
which the objective prism (and accompanying direct) plates for the
\textit{Bergedorf(-Groningen-Harvard)
Spektral-Durchmusterung} were taken. From \cite{Hamhist2016}, Hamburger Sternwarte, reproduced with permission.}
\label{fig:Lippert}
\end{figure}
This survey aimed at determining spectral classes of stars in the magnitude
range 8.5 to 13. The method was to take photographic images of the sky with the
objective prism in front of one of the astrographs, so that each stellar image
was replaced with a spectrum on the photographic plate. The other half
of the double astrograph was used to take a direct plate
of the same area at the same time so that the
spectra could be directly related to the stars that corresponded to it. The
method was tested by taking plates in the same manner of stars with known
spectral type. The limit of the spectroscopic survey was magnitude 13.5 or so.
The fields obtained at the Lippert astrograph measured 1.5 degrees on a side,
much larger than the 20 arcmin in the {\it Harvard(-Groningen) Durchmusterung},
so that it was necessary to determine apparent magnitudes
over much larger fields.
The Groningen part of the project involved the determination of accurate
positions and
photographic magnitudes. For this, extensive new plate material was collected at
Harvard with the cooperation of director Harlow Shapley. These were measured
in Groningen using the photometer that Jan Schilt had developed, first in
Groningen as part of his PhD thesis work and improved while he worked at
Leiden Observatory.
Schilt developed this, as he noted in his thesis, following a suggestion by
Kapteyn. The practice after the start of the use of
photographic plates to determine magnitudes was to estimate the
diameter of the image. Schilt wrote
\cite{Schilt1922}:
\par \begingroup \leftskip2.5em \rightskip2em \large
At the request of Professor J.C. Kapteyn
I have made lately some experiments at the Groningen Astronomical Laboratory
concerning the determination of photographic
magnitudes by means of a thermopile and a galvanometer. The principle is this:
the image of a small circular lens is formed on
the photographic plate. The image of a star on the plate is brought to the
center of the circular bright spot projected on the plate
by the illuminated lens; the darker the image of the star, the smaller will
be the fraction of the light transmitted by the
photographic plate. This fraction is measured by means of the thermocurrent.
The reading of the galvanometer, which indicates
the intensity of this current, is thus a measure of the photographic magnitude
of the star.
\par \endgroup
A thermopile is a set of thermocouples operated usually (as in this case)
in series, a thermocouple being two different
materials that are electric conductors. When these have a different
temperature the energy difference gives rise to an
electric current (the Seebeck effect). The temperature difference results from
illumination through the photographic plate.
This current then is measured with a galvanometer and the strength of the
current can then be related to the amount of light transmitted by the
photographic emulsion. This requires a very good
thermopile, and Schilt had obtained an excellent one, produced by Frits
Zernike, who after obtaining his PhD in Amsterdam had returned to Groningen in
the physics department.
It is
illuminating to read Blaauw's description in 1983 (\cite{Blaauw1983}, p.54 and
further; my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
The measurement program carried out [...] at the Kapteyn Laboratorium [...]
has greatly absorbed the capacity of the laboratory over a period of more than
two decades, probably much longer than van Rhijn had expected. The German
partner has discharged its task with full responsibility and German
`gr\"undlichkeit', but the cooperation with the Harvard College Observatory was a bit
more difficult. However, Shapley and his collaborators cannot actually be
blamed for this. The difficulty lay rather in the nature of the cooperation.
While in Bergedorf and Groningen the staff worked for one's own institute, it
was difficult for those at the Harvard College Observatory who were charged with the
recording of the photographic plates, always devoting themselves to it with
the necessary altruism. It went well as long as this was in the hands of pure
observers who, in the patient, careful nocturnal work with the telescope, find
full satisfaction, a mentality one finds more readily among the best of the
astronomical amateurs than among the `professionals'. The
work at the Harvard College Observatory required more and more effort from advanced
students who were difficult to motivate and moreover, were quickly replaced
by new, inexperienced ones. No wonder, then, that the work, as I sometimes
heard afterwards, gradually got the name of `the damned van Rhijn program'.
That it turned out ultimately satisfactory, must be partly due to the
circumstance that van Rhijn's assistant Bok became a staff member of the
Harvard College Observatory and thus was able to keep an eye on things.
\par \endgroup
What we see here is a first crack (or one of the first) in the model by
Kapteyn of an astronomical
laboratory relying entirely on observatories elsewhere for in this case plate
material. In part this may be the changing 'zeitgeist', a growing reluctance
to work predominantly for the benefit and in the
service of others. What probably played a rol as well is that van Rhijn did
not quite have the unquestionable authority Kapteyn had had. Blaauw continued
(my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
[...] Bergedorfer Spektral Durchmusterung.
This is not a fortunate name, for the large Groningen share is not expressed
in it; in fact, the Groningen effort, counted in man-years, must have been
considerably greater than the German. But it was not in van Rhijn's
character to give
much attention to this external aspect; the main thing
was for him that the job got done.
\par \endgroup
In an article on the history of the Hamburger Sternwarte
\cite{Hamhist2016} it is stated (p.197):
\par \begingroup \leftskip2.5em \rightskip2em \large
In the 1920s multiple survey programmes began, among them the Bergedorfer
Spektral-Durchmusterung by Schwassmann and
Wachmann, who used the Lippert
Astrograph to obtain spectra of more than 160,000 stars,
\par \endgroup
\noindent
ignoring the positional part, the co-authorship of van Rhijn and the
collaborative nature with the Kapteyn Laboratorium and Harvard College Observatory
even though this is manifestly clear from the full title of the Durchmusterung.
Completing Blaauw's remarks on the subject (my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
Many generations of van Rhijn's temporary
assistants we find back in his introductions to the five issues mentioned as
co-workers: in the first, of 1935, B.J. Bok, J.J. Raimond en P.P. Bruna; in
the one of 1938 P.P. Bruna; in 1947 me; in 1950 and 1953 L. Plaut. To the
measurement and calculation work participated G.H. M\"uller, W. Ebels, P.
Brouwer, P.G.J. Hazeveld, H.J. Smith, J.B. van Wijk, M. Seldenthuis, D.
Huisman, H. Schurer, M.L. Wagenaar and W.L. Gr\"ummer. During the War years,
cooperation with Bergedorf naturally became increasingly difficult.
Until it stopped completely. It must have been around 1943 that I myself, then
an assistant, was summoned to come to the office of a high German official in
Groningen to justify why Bergedorf no longer received any response from us
(van Rhijn was staying in a sanatorium with a
serious illness), but it was not difficult for me to point out to the
gentlemen that for some well-known reason the delivery of the plates from
Harvard had been interrupted. At that time in Bergedorf they had very little
idea of the conditions prevailing here under the occupation.
\par \endgroup
\bigskip
Bruna is Petrus Paulus, born in Utrecht in 1906, who was assistant at
the Laboratorium from 1934 to 1939. He did publish some work in the
{\it Monthly Notices of the Royal Astronomical Society}
on the determination of the motion of the
Sun in space. The only thing in public records I could find
about him is that he got married in in 1940 and he must have left
astronomy probably to take up a job as high school teacher or so.
In the correspondence in the Oort Archives
there is one reference to him. On 15 January
1941, van Rhijn wrote to Oort (Oort Archives, website accompanying
\cite{JHObiog}, Nr. 149; my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
With Bruna it is a difficult case. He has done the routine work of the
Laboratorium as an assistant quite well, but he has no gifts for
scientific research. Added to this is the fact that he and his wife are ill
quite often and he has a busy job as a teacher. At the
moment he is busy writing down the work. But that is also going slowly because
the pieces he sends are poorly written. He has
been working on the research for about seven years. It is indeed high time
that the work is finished. I will write him again.
\par \endgroup
Bruna never completed the thesis.
A number of
the computers can be found in Fig.~\ref{fig:staff}. The HBS mentioned in the
caption is the 'Higher Civic School', instituted in 1863 by liberal Prime
Minister Johan Rudolph Thorbecke (1798-1872) as part of a major overhaul of the
Dutch educational system. It was designed to provide
education preparing upper or middle class boys for a career in industry or
commerce, but which as a result of the excellent training in science by
teachers with PhD's quickly became the preferred secondary school leading to
academic studies in mathematics and natural sciences.
\bigskip
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.98\textwidth]{figstaff.jpg}
\end{center}
\caption{\normalsize Computers and observators of the Kapteyn Laboratorium;
photograph taken in 1926 or 1927. From left to right (caption taken from
Adriaan Blaauw in \cite{Blaauw1983}; my translation):
\textbf{J.B. van Wijk} -- appointed 1-11-1924 after HBS training and
military service, celebrated 1-11-1964
his 40th anniversary in office and in 1966 wrote his memoirs of the first
years.
\textbf{J.M. de Zoute} -- worked under Kapteyn and van Rhijn, initially
studied theology,
but had to terminate his studies to earn a living. Was a computer and wrote
letters for Kapteyn and van Rhijn in the final `neat' version
before sending. He was always
computing or writing standing at a high lectern (which in 1938 was still in the
`computer's room'). Was released
in the crisis years due to university budget cuts.
\textbf{W. Ebels} -- appointed in the early 1920s, worked like van Wijk and
M\"uller until his retirement.
\textbf{T.W. de Vries} -- worked for 36 years under Kapteyn, then 6 years
under Van Rhijn, did in particular much measuring work.
\textbf{H.J. Smith} -- computer under van Rhijn, died in the War years.
\textbf{G.H. M\"uller} -- amanuensis, instrument maker, already worked under
Kapteyn, did measurement work and
was van Rhijn's right hand in organizing the installation of the
telescope on top of the laboratory building.
\textbf{P. Brouwer} -- affiliated to the laboratory from Sept. 1925 to Dec.
1935, afterwards career with the police,
helped A. Blaauw (March 1983) to identify the persons on this photograph.
Kapteyn Astronomical Institute.}
\label{fig:staff}
\end{figure}
Eventually, the results were published in five volumes with the long title
quoted at the beginning of this section
by the Hamburger Sternwarte at Bergedorf between 1935 and 1953
\cite{Bergedorf}. Schwassmann and van Rhijn were the authors. The magnitude
range came out as between 8.5 and 13 photographic and the total number of
stars was 173,599. As Blaauw noted \cite{Blaauw1983}, the daily supervision of
the work had been the duty of the assistants; this was a full-time job, so the
preparation of PhD theses was restricted to the evenings and the weekends.
In this article I wish to honor Blaauw, van Rhijn and others in Groningen
and Harvard by
referring to it as the {\it Bergedorf(-Groningen-Harvard)
Spektral-Durchmusterung}.
It should be mentioned
here that a similar exercise was performed at the Potsdam Sternwarte near
Berlin as the {\it Spektral-Durchmusterung der Kapteyn-Eichfelder des
S\"udhimmels}, also in five volumes, published by Wilhelm Becker (1907--1996)
between 1929 and 1938,
partly in collaboration with Hermann Alexander Br\"uck (1905--2000).
\section{Research based on the Selected Areas and the Spektral-Durchmusterung}\label{sect:PSAreserach}
One might ask whether these spectral surveys have been of much use to others
and constituted a worthwhile investment of funds, effort and manpower. I would
think to some extent yes, but also that in the case of Groningen it went at
the expense of momentum in research. The five volumes of the {\it
Bergedorf(-Groningen-Harvard)
Spektral-Durchmusterung} together have collected in ADS of order one
citation every two years at a rather constant rate since the 1950s.
And since the start of the current millennium it
had and still has some ten reads per year. It certainly was not a major
advance, although it definitely not a
complete waste, but whether it was a wise decision in view of the future of the
laboratory to make such a long term commitment of most of its resources is
another matter. It is likely that the more useful part of the {\it
Durchmusterung} was the spectral type information provided by Bergedorf rather
than the magnitudes from Groningen.
\bigskip
I stressed above that the {\it Plan of Selected Areas} was more than a provision
of fundamental data to the community at large. It was first and for all
designed to solve for a detailed model of the distribution of stars and their
kinematics and to study the dynamics of the Stellar System. After the `first
attempt' by Kapteyn in 1922 \cite{JCK1922} no further attempts had been made.
Van Rhijn had taken the
three next steps by refining the mean parallaxes for separate
spectral types \cite{PJvR1923}, determined a state of the art luminosity
function \cite{PJvR1925} and he had derived improved star counts in 1929 based
on the Harvard and Mount Wilson Durchmusterungs \cite{PJvR1929}, but
he stopped here and no
new model for the stellar distribution had been attempted. Undoubtedly this
had to do with the problem of how to allow for interstellar extinction.
\begin{figure}[t]
\sidecaption[t]
\includegraphics[width=0.64\textwidth]{figRS1935.jpg}
\caption{\normalsize Density distributions (`Dichte $\Delta$')
of stars of separate spectral class
with distance from the Sun at latitude $+50^{\circ}$ as determined by van Rhijn
\&\ Schwassmann \cite{vRS35}, based on 22 \textit{Selected Areas} at roughly
this latitude. The values have been normalized at
the solar position. No distinction is
made between dwarfs and giants here, but this is done in tabular form
in the paper for G, K (and M)-stars. Kapteyn Astronomical Institute.}
\label{fig:RhijnSchwas}
\end{figure}
Since the aim of the {\it Plan of Selected Areas} was to elucidate the
structure of the Stellar System, it is important to stress that van Rhijn did
use the {\it Bergedorf(-Groningen-Harvard) Spektral-Durchmusterung} for this purpose. Probably since the
effects of absorption could not very reliably be eliminated he and Schwassmann
choose to study distributions of various spectral types at a Galactic latitude
of $50^{\circ}$. Their paper \cite{vRS35} is in German, it collected only 11
citations in ADS, of which 9 after 1955 (the last one in 1985), and attracted
about one read per year in the last two decades. It used determinations by van
Rhijn of the luminosity functions per spectral type B, A, F, G, K and M,
distinguished dwarfs and giants (and supergiants) by assuming reasonable
ratios between these two for each apparent magnitude. The result is quite
significant and has been illustrated in Fig.~\ref{fig:RhijnSchwas}. They
found that the layer of B-stars is thinnest, increasing with later spectral
type.
This was not entirely new, earlier studies and in particular Oort's
well-known $K_z$ study of 1932 \cite{JHO1932} had shown this as well.
But it is an important confirmation. In the 1930s it was not
understood at all how this came about. The explanation was
found only in the 1960s. Stars are born in the thin gas layer and have therefore initially the same distribution and mean
vertical velocity (velocity dispersion) as the gas clouds.
For any generation of stars this dispersion increases with time
due to scattering of stars by massive molecular clouds, spiral
structure or in-falling satellites and the average distance
from the symmetry plane will increase with time for each
generation. The sequence of stars corresponds to an increase
of mean average age and the effect reflects this so-called `secular evolution'.
I will elaborate on this study to show
the similarities and the differences
in approach and in scope with van Rhijn's work.
A more detailed analysis of the paper is given in my
Oort biography \cite{JHObiog}. Oort had used stars at high latitudes with
known radial velocities, absolute magnitudes and spectral types (mostly from
the {\it Henry Draper Catalogue}) and he first determined the
dispersions of vertical velocities (the mean random motion). This showed no
significant trend with distance from the plane, especially for K- and M-giants,
on which most of his final analysis rested. He then used space distributions,
particularly from papers by van Rhijn and others, and used dynamics to find the
vertical force $K_z$ up to 0.5 kpc. Oort then turned the problem around: if
for a group of stars (range of absolute magnitudes or of spectral types) one
knows the velocity dispersion, the force field can be used to calculate what
the vertical density distribution for that group of stars would be. By comparing
that to actual counts he arrived at an improved form for the luminosity function.
But that was only the beginning. First he simulated star counts for fainter
stars, both apparently and intrinsically and compared
that to van Rhijn's counts. When that was satisfactory he constructed mass
models of the Galaxy using a spherical mass around the center and a homogeneous
ellipsoid, satisfying constraints as the rotation parameters (rotation velocity and Oort
constants).
He then derived the gravitational forces, using these forces,
the corresponding star counts for
different assumptions of the velocity dispersions. He did (rather had his
computers do) this enormous amount of work first as a function of latitude
(concluding that above 15$^{\circ}$\ no absorption effects were evident) and then
as a function of longitude. This resulted in two important deductions. The
first is the often quoted Oort limit, the total amount of mass in a column
perpendicular to the plane of the Galaxy at the position of the Sun,
since he now knew the asymptotic
value for the vertical force. The second was a first approximation of the
stellar density in a vertical crosscut through the disk and the Galactic
center, at least away from the plane and extinction. This is reproduced in
Fig.~\ref{fig:crosscut1}.
\bigskip
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.98\textwidth]{figCrosscut1.jpg}
\end{center}
\caption{\normalsize Crosscuts through the Galactic System from Oort in 1932.
Surfaces of equal density of
starlight, as ellipse fits centered on the Galactic center at
1/25th (dots) and 1/100th (crosses) of that near the Sun. The Sun is at
10000 pc, the center of the Galaxy at zero to the right. The dashed
lines indicate latitude $20^{\circ}$. From \cite{JHO1932}. From the Oort
Archives.}
\label{fig:crosscut1}
\end{figure}
It is true that Oort in this work used many results of van Rhijn's earlier
investigations and in fact the analysis would not have been possible without
it. Van Rhijn \&\ Schwassmann's conclusions
did constitute a significant strengthening of Oort's conclusion as it
was based on many more spectral types of fainter stars.
On the other hand, Oort's work was far more imaginative and went much further.
He aimed at an
understanding of the structure of the local Galaxy in terms of distributions,
kinematics and dynamics, and the possible
presence of unseen matter, while van Rhijn and
Schwassmann merely mapped distributions without addressing the deeper issue of
understanding the structure of the Galaxy. Van Rhijn followed on the program of
Kapteyn to collect the data, while Oort
followed up on Kapteyn's final analysis to use this to map the Stellar System.
\section{Interstellar extinction}\label{sect:extinction}
In addition to the distribution of stars in space, van Rhijn had a second main
line of research in studying interstellar extinction. The first half of his
thesis was devoted to this \cite{PJvR1916} on the basis of reddening of stars
with distance, and he had, as we have seen, in 1928 unsuccessfully tried
to determine this effect on apparent magnitudes on the
basis of diameters of globular clusters \cite{PJvR1928}.
I will not review in detail the history of the discovery of interstellar
extinction. For a general listing of the chronology see \cite{Li}, and for
the argument why the discovery should be credited to Robert Trumpler, as it
usually is, see \cite{SeelBe}.
After William Herschel had noted apparent holes
in the distribution of stars in the Milky Way, and Wilhelm Struve suggested
this to mean that there was an apparent decline in the number of stars with
distance from the Sun, which could be due to absorption, it was Edward
Pickering's consideration of the star ratio \cite{Pick1903},
that sparkled early in the
twentieth century the debate about extinction. The star ratio is the relative
number of stars between subsequent apparent magnitudes, which with fainter
magnitudes deviated more and more from the theoretical value of log~0.6 = 3.981
for a uniform distribution. George Cary Comstock (1855--1934)
noted in 1904 that the average proper motion decreased
much less rapidly with apparent magnitude then expected (at least at low
Galactic latitudes) and proposed that this was due to extinction
\cite{Com1904}. Kapteyn
argued from detailed modeling that the matter was more complicated and the effects
of changes in extinction and in space density were degenerate and
considerations of analysis of star counts could not resolve this.
He proposed that the observed effects were
a mix of the two. It was also Kapteyn who in the second of a set of two papers
{\it On the absorption of light in space} \cite{JCK1909}
suggested that it could be predominantly scattering and
that therefore the effects would be more severe at bluer wavelengths. He
subsequently actually found these effects of increasing reddening
with distance. So in addition to star
counts, proper motions and studies of the distribution of stars there now was a
new sign of extinction, namely interstellar reddening.
Extinction had been established beyond doubt by Trumpler using reddening
of stars in open clusters, comparing the distances of about one hundred open
star clusters determined from apparent magnitudes and spectral types with
those from angular diameters, assuming these are on average the same
\cite{Trump}. He derived an extinction of 0\magspt7
photographic magnitudes per kpc. This was
improved in several ways by astronomers, of which I take two examples without
wanting to suggest all the progress came from these two studies. Former van Rhijn
student Peter van de Kamp -- in the mean time having moved to Leander
McCormick Observatory -- used an interesting method; he assumed it was zero at
high latitudes and then used galaxy counts as a function of latitude to
estimate average values of 0.8 to 1.4 photographic magnitudes per kpc
\cite{vdKamp1932}. Jean Jacques Raimond (for an obituary see \cite{JHO1961b})
wrote a thesis in 1933, {\it The
coefficient of differential Galactic absorption}, under van Rhijn in which he
compared colors of stars at high latitudes with those of stars of the same
spectral type in the plane. His result \cite{JJR1934} can be summarized as
follows. He used magnitudes (photographic on the Mount Wilson scale and visual
on the Harvard scale) and spectral types from various sources and estimated
distances from van Rhijn's \cite{PJvR1923}
mean parallaxes as a function of magnitudes, proper
motions, spectral types and Galactic latitude or from spectroscopic parallaxes.
This gave him absolute magnitudes. From stars at high latitude he calibrated
the relation between colors and absolute magnitudes and then could find the
reddening for stars at low latitude. He assumed a layer of absorbing material
with a thickness of 200 pc uniformly filled, and then found a reddening of
(converted to units used here) $0.50\pm 0.03$ magnitudes per kpc.
The problem remained what the corresponding amount of extinction would be.
This requires
knowing the wavelength dependence of the scattering. Kapteyn had been the first
to hypothesize that the signature of interstellar extinction would be a
reddening of starlight with distance and he had assumed that the effect was
similar to that in the Earth's atmosphere giving rise to the blue color of the
daylight sky. This is, however,
the result of scattering of light on very small particles such as {\it molecules}.
Most of this
is Rayleigh scattering first described by Lord Rayleigh (John William
Strutt; 1842--1919) around 1870 (it is now known that Raman scattering
also plays a role,
which is inelastic while Rayleigh scattering is elastic,
and results in exchange of energy). These processes occur
from scattering by particles {\it smaller} than the wavelength, while it became
clear that interstellar scattering really is from {\it dust grains} with sizes
{\it comparable} to the wavelength.
Consequently Kapteyn's initial guess that the dependence was
as for Rayleigh scattering, that is inversely proportional to the fourth
power of the wavelength, was incorrect; it actually is
in first approximation linear with the inverse wavelength.
It was Alfred Harrison Joy (1882--1973), who cleverly derived the
relation between reddening and extinction from Cepheids, which are bright,
can be seen over large distances, have relatively small random motions, and
whose absolute magnitudes follow from their period of the brightness
variations. He used Jan Oort's formalism in which on scales
significantly smaller than the size of the Galaxy the radial velocity should
increase linearly with distance as a result of differential rotation (see Sect.
4.5 in \cite{JHObiog} or Sect. 5.4 in \cite{JHOEng}). Joy published this
result in 1939 \cite{Joy}, but it was reported already in the 1932 annual
report of the director of Mount Wilson Observatory.
In 1936, van Rhijn \cite{PJvR1936}
followed this up by applying the method to open star clusters (with
distances from the Hertzsprung-Russell diagram) and to F to M stars
(in addition
to Galactic rotation using spectroscopic and secular parallaxes). This
together resulted in average extinctions of 1.1 and 0.55 magnitudes
photographic and visual respectively per kpc.
Van Rhijn, in the same paper,
used this to do a somewhat general mapping in the plane of the
Galaxy, using the northern Selected Areas at roughly latitude zero.
Remarkably, beyond a kpc from the Sun there is a strong increase
in density towards the center of the Galaxy, which is not accompanied by a
rapid decrease in the anti-center direction. This would be consistent with the
Sun being located just outside a spiral arm, but that would have been rather
speculative and was not offered as an explanation by van Rhijn.
In fact, there is no discussion at all
on the broader view of our Galactic System or comparison to external galaxies.
The two studies by van Rhijn -- the one with Schwassmann on the distributions
for different spectral types along a line of sight at high latitude \cite{vRS35}
and the one on
distributions in the Galactic plane corrected with an assumed uniform
extinction \cite{PJvR1936} --
provide no profound new insight into the structure of the
Stellar System as a second step following and improving the 1920 analysis
with Kapteyn \cite{KvR1920b}. It was not the case that this was impossible, or
too early, van Rhijn simply did not address it.
\bigskip
There was one more thesis completed under van Rhijn during the 1930s and
that person would have been a better choice for the assistant position
than Bruna.
This was Broer Hiemstra (1911–??), who defended a PhD thesis in 1938
\cite{BH38}, but had not been appointed assistant.
He was born in Pingjum (a village south of Harlingen and close to where the
Afsluitdijk joins Friesland) as a son of an elementary school teacher.
He is not listed
among the staff of the Laboratorium in the 'Jaarboeken', the annual reports of
the University of Groningen, so he must have been supported by his parents or
been supporting himself. Hiemstra had obtained his doctoral exam in November
1936, but since 1934 the assistant position had been occupied by Bruna.
In the certificate of his marriage in Dokkum (Friesland)
in 1938 Hiemstra is listed as having the profession 'teacher' (according to a
newspaper clipping they were engaged to marry already in 1931!, also in Dokkum),
so a possibility is that, while working on his thesis, he supported himself as
part-time teacher, which turned into full-time after his thesis defense
and made the marriage possible (he was 28 years by then and his bride
Arendina Maria Vlietstra, an elementary school teacher, 29). They settled in
Amsterdam where the young Mrs. Hiemstra tragically
died in 1940, one would think probably
in child birth. According to a photograph in the Den Haag Municipal Archives
(Nr. 1.21111), Hiemstra remarried and in 1976 at the age of 65 retired as
director of a large secondary school conglomerate there (the Hague).
So he pursued a
successful career in secondary education.
No more can be found on Hiemstra in official records or
newspapers, except the announcement of the death of his father in 1944.
The thesis was defended in 1938 and concerned {\it Dark clouds in Kapteyn's
Special Areas 2, 5, 9 and 24 and the proper motions of the stars in these
regions}, and this study is rather ingenious and
very much worthy of review here. It concerned
determinations of distances and amounts of extinction in dark clouds.
These Areas are four of the ones especially selected by Kapteyn for
this purpose
These properties had been
estimated before, usually from counts of stars projected
onto the dark cloud compared to those in the immediate vicinity. The way to do
this is to determine from star counts and the luminosity function what the
apparent density along a line of sight is towards the cloud and one
or more just next to
it on the sky. The problem
using star counts is that this involves solving integral equations. Counts of
stars in a particular direction as a function of apparent magnitude, follow
from a summation. For each element of distance from the Sun along a line of
sight each apparent magnitude corresponds to a certain absolute magnitude. The
number of stars seen at that apparent magnitude from this element then is the
total density of stars there multiplied by the the fraction of stars of the
required absolute magnitude, which is the value of the luminosity curve at
that absolute magnitude. All such contributions from elements at other
distances along the line of sight, corresponding to different {\it absolute}
magnitudes, then have to be added up to find the total count of stars of that
particular {\it apparent} magnitude.
Mathematically this is an integral, which can
be evaluated in principle in a straightforward manner. But the solution
required is the inverse. Given the form of the luminosity curve and the star
counts at apparent magnitudes, the problem is to determine the total density
as a function of distance. That means having to
'invert' the integral and inversion of integral equations is notoriously
difficult. For the solution for the 'Kapteyn Universe'
\cite{KvR1920b}, Kapteyn and van Rhijn had used Schwarzschild's analytic method
for the special case that the luminosity curve could be approximated by a
Gaussian.
This method, which German astronomer Karl Schwarzschild (1873--1916), then from
the Sternwate G\"ottingen,
had actually described in a paper in 1912
\cite{KS1912}, would have played a role in
Kapteyn proposing Schwarzschild for an honorary doctorate from the
University of Groningen on the occasion of its tricentennial in 1914 (see
\cite{PCK2021}).
In the early 1920, Anton Pannekoek in Amsterdam had also addressed the
problem of the distance and extinction of a dark cloud \cite{AP1921}. He used
Kapteyn's density distribution and the luminosity function to compute the
theoretical decrease of the number of stars of different magnitudes for
different assumptions for the distance and extinction in the cloud and
compared those to actual counts towards the cloud. In other words he solved
the inversion problem by trial and error.
Other major contributions to this field had come from Maximilian Franz Joseph
Cornelius Wolf (1863--1932) at the Landessternwarte Heidelberg-K\"onigstuhl
(\cite{Wolf1923} and subsequent papers in the {\it Astronomische Nachrichten}).
Wolf introduced what became known as Wolf diagrams, which compare star counts
for the obscured and adjacent regions. This shows immediately at what
magnitude the curves begin to deviate, from which the distance may be
estimated, and the amount of absorption from at what magnitude the
curves become parallel again. For an interesting review of the history of
this subject, see Pannekoek (1942) \cite{AP1942}.
All of this was based on counts of stars as a function of apparent magnitude
and solving the corresponding integral equation of statistical astronomy. The
innovative idea in Hiemstra's thesis was to use counts of proper motions and
solve the corresponding equation, in which the luminosity law is replaced by
the distribution law of space velocities. Hiemstra attempted to determine
distances and amounts of extinction of the dark clouds in these areas using
proper motions of stars in front of and next to these clouds. For this he
measured proper motions from plates obtained at Radcliffe Observatory and made
available by its director Harold Knox-Shaw (1885--1970).
The method he used is a very interesting one, so I will explain its principles.
But first I will, for those unfamiliar with this, explain the notions of velocity
ellipsoid and vertex deviation. Kapteyn used the effects of the solar velocity
on the distribution of proper motions to derive statistical distances (secular
parallaxes), which required random, isotropic motions. When he had found that
in addition
there was a pattern of two opposite Stellar Streams, Karl Schwarzschild quickly
realized that this could also be explained as an asymmetry in the velocities,
which would then be Gaussian with three unequal principal axes. Galactic
dynamics required the long axis of this velocity ellipsoid to be directed
towards the Galactic center, but Kapteyn's Star Streams were directed some
20$^{\circ}$\ away from this. This deviation of the vertex was not a fluke in
Kapteyn's analysis, but convincingly confirmed and eventually attributed to
gravitational effects of spiral structure by Oort \cite{JHO1940}.
Hiemstra chose components of the proper motions
that line up as much as possible
with the long axis of the velocity ellipsoid as given by Oort (so not assuming
this points at the Galactic center, but taking the vertex deviation into
account) and at at least 66$^{\circ}$\ from the solar motion in space, both
conditions ensuring a minimal effect of the parallactic motion. The
distribution of proper motions is a combination of the distribution of
parallaxes and that of the linear velocities and comparable integral equations
can be written down for counts as a function of proper motions as for those
of apparent magnitude. This function follows from a study
of radial velocities and is assumed uniform with position in space.
Hiemstra inverted the relevant integral equation by trial and error
and in this way arrived at a distribution of parallaxes for the stars next to
the dark nebula. From this he determined what star counts would look like
taking only stars up to various maximum distances.
Assuming that the density of stars and their luminosities is
the same in the obscured region and towards the dark nebula he then arrived
at an estimate of the distance of the cloud and the amount of extinction it
gives rise to. The four dark clouds were at distances ranging from 300 to
1000 parsecs and
their total extinctions at least 0\magspt5 to 2$^{\rm m}$ in the photographic band.
This provided excellent, independent confirmation of results from other studies.
Hiemstra's paper has no citations in ADS but still averages 2 reads per year. It
seems to me the conclusions are still valid according to current insight and
constituted a very significant result, that probably did not at the time
attract the attention it deserved. For example Pannekoek in his review in
\cite{AP1942} mentioned Hiemstra's publication but only in passing, noting
it as an illustration that his own
formalism can easily be adapted for proper motions. At
about the same time (1937) a study appeared by Freeman Devold Miller
(1909--2000) of Swasey Observatory in Ohio, who used the 'Pannekoek equation'
(the integral equation of star counts in the case of an intervening dark cloud)
\cite{Miller37}. Hiemstra compared his work to Miller's and the results were
similar. Yet, the Miller paper currently has on average only one read per year.
The Wolf 1923-paper has become a classic with some 1 to 2 citations and 15 or
so reads per year, while the 1921-paper by Pannekoek
was cited for the last time in 1993. But it is not easily accessible;
it has about one read per year, probably only for the person involved
to find out that the text is not on the Web. Hiemstra's thesis,
it seems, still is significant.
\section{Oort's second approximation}\label{sect:Oort1938}
Actually the time was ripe for a new attempt to synthesize a model for the
distribution of the stars in space and the structure of the Stellar
System, and it was Oort who took this step in
\cite{JHO1938}! I quote from Oort's historical discussion on {\it The
development of our insight into the structure of the Galaxy between 1920 and
1940} (\cite{JHO1972}, p.263):
\par \begingroup \leftskip2.5em \rightskip2em \large
I wanted to determine the general density distribution outside the disk as a
function of Galactic latitude as well as of longitude, and thereby to extend
Kapteyn and van Rhijn's initial investigation to a second approximation. It
was found that beyond about 500 pc on either side of the Galactic plane, the
star density increased rather smoothly toward the center. It showed a
logarithmic density gradient of 0.14 per kpc at levels between about 800 and
1,600 pc above and below the Galactic plane, and the equidensity surfaces
made an angle of $10^{\circ}$ with this plane.
\par \endgroup
This paper had been summarized and discussed extensively in section 8.4 of
\cite{JHObiog} (or 6.6 of \cite{JHOEng}),
so I will only give a brief summary here in order to
demonstrate the ingenuity of the approach. When Oort started this investigation
(well before the publication year of 1938) the {\it Plan of Selected Areas}
had progressed quite well. At this point it is relevant to summarize the state
of progress in the Plan at that time, say the middle of the 1930s.
As for star counts, the {\it Harvard(-Groningen) Durchmusterung} down to
magnitude 16 had been completed as well as the {\it Mount
Wilson(-Groningen) Catalogue}
of the northern Areas to magnitude 19.
The next step would be to determine
spectral types and most importantly, proper motions. The spectral types were
not fully published but mostly available from the Bergedorfer Sternwarte
for the northern fields and the Potsdam Sternwarte near Berlin for
the south. Oort must have had access to all or most unpublished material from
these two Durchmusterungs as far as available at any particular time, which
meant for stars brighter than magnitude 12 or so. As mentioned above, for the
northern Areas Humason at Mount Wilson used the 60-inch
telescope to determine the spectral type of at least 10 stars per
Area down to magnitude 13.5 or so.
Another major advance was the publication in 1934 of {\it The Radcliffe
catalogue of proper motions in the Selected Areas 1 to 115} by Harold Knox-Shaw
(1885–1970) and his assistant
H.G. Scott Barrett. Radcliffe Observatory, which was part of
Oxford University, had a 24-inch telescope; it was first used to try and
measure parallaxes in the Selected Areas, but when that was unsuccessful,
attention was re-directed towards proper motions. This had been started in
1909 under Arthur Alcock Rambaut (1859--1923), taking photographs in duplicate,
one being developed immediately, the other stored for re-exposure at least ten
years later. The second stage was started in 1924 under Knox-Shaw. The
catalogue, published in 1934, contained proper motions of some 32,000 stars.
The Oxford Radcliffe site was actually vacated in 1935 and the Observatory
moved to a site near Pretoria, South Africa. Another important source for Oort
to use was a study by Peter van de Kamp and Alexander Nikolayevich Vyssotsky
(1888--1973) at Leander McCormick Observatory near Charlottesville, Virginia,
which had a 26-inch refracting telescope. Van de Kamp and Vyssotsky had
measured the proper motions of 18,000 stars with this telescope.
All these data together already constituted a substantial fraction of what
Kapteyn must have envisioned necessary for an attack on the problem
when he conceived the {\it Plan of Selected Areas}.
But before embarking on an analysis of the available material with the aim to
derive a new picture of Galactic structure, Oort went through some important
preparatory exercises improving estimates for the constant of precession
(vital for determining proper motions),
the motion of the equinox and the Oort constants of differential Galactic
rotation\footnote{Oort of course did not call them that, but in his lectures
introduced them as `two constants of Galactic rotation', to which he
would add `,which we will call $A$ and $B$', hesitating a
few moments as if to make up his mind what he would decide to call them.}.
And from radial velocities
and proper motions combined with estimated distances of spectroscopic and
other parallaxes, he derived an improved estimate of the shape of the velocity
ellipsoid (the distribution of velocity vectors).
This left the problem of the extinction. For this Oort used galaxy
counts by Edwin Powell Hubble's (1889--1953)
mapping the `Zone of Avoidance', with which he was
provided by Hubble before publication. The problem was to turn the distribution
of galaxy counts into one of magnitudes of extinction. Oort did that by
determining the colors of fainter stars in Selected Areas and comparing
that to those of nearby, unreddened stars of the same spectral type, and
found the reddening. This related galaxy counts to reddening, but he still
needed to transform reddening into the corresponding actual amount
of absorption. For this one would have to measure the brightness as a function
of wavelength for the reddened and the un-reddened
stars and in this way the wavelength dependence of the
extinction. Such a study had just been published by by Jesse
Leonard Greenstein (1909--2002), who had calibrated spectra in terms of the
brightness as a function of wavelength using photo-electric techniques, and
had found the dependence of the absorption to be inversely proportional to the
wavelength. Greenstein determined also the actual amount of absorption by
comparing stars still visible behind dark clouds to stars just next to it,
which also gave him the factor to turn color excess into actual magnitudes of
extinction.
From studies of faint stars Oort concluded that
\par \begingroup \leftskip2.5em \rightskip2em \large
\noindent
in all probability a
considerable fraction, if not all, of the absorption derived from the counts
of nebulae must take place within a relatively small distance from the Galactic
plane.
\par \endgroup
Oort then felt justified to approximate the situation by assuming that
the entire absorption indicated by the nebulae took place in front of the
stars considered. He ignored the Selected Areas at too low Galactic latitude,
generally below 10$^{\circ}$\ or so, but without any strict criterion.
So with that assumption and using van Rhijn's luminosity
function, for each Kapteyn Selected Area
he proceeded to calculate the star density along the line of sight at three
distances from the plane. He found two important things. First a general and
large-scale increase in stellar density towards the center of the Galaxy,
and secondly, that there is symmetry between both sides of the Galactic plane.
\begin{figure}[t]
\sidecaption[t]
\includegraphics[width=0.64\textwidth]{figCrosscut2.jpg}
\caption{\normalsize Crosscut through the Galactic System from Oort in 1938.
Surfaces of equal density of
starlight, at various light densities. The Sun is at zero,
the Galactic center to the right at 10 kpc. From the Oort Archives.}
\label{fig:crosscut2}
\end{figure}
Oort continued by sketching from his data isodensity curves in a cross-cut
through the Galaxy at the solar position in a plane at right angles to that of
the Milky Way and along the direction towards the center. This has been
reproduced in Fig.~\ref{fig:crosscut2}.
It looks very much like what would be expected if the
Galaxy had a spiral structure like many external systems. The precise form of
the 'arms' could not be determined, but Oort was relatively certain about one
thing and that is that 'the Sun should be between two spiral arms'. At larger
distances from the plane the spiral structure is replaced with a general
density gradient towards the center, amounting to a factor 1.398 in the stellar
density per kpc. In modern terms this corresponds to a scalelength (e-folding)
of about 3 kpc; my preferred value is more like 4.5 kpc (see \cite{vdKF2011},
section 3.1.1), but this is open to different opinions.
This result is a rough, first approximation which quite well
fits current understanding. In
Oort's analysis, hot and bright O- and B-stars play a significant role and
these are now known to be concentrated heavily to the spiral arms, where in a
spiral galaxy star formation is preferentially taking place. So an analysis
relying heavily on short-lived OB-stars probably
enhances the real contrast between arm
and interarm regions. Modern studies have revealed the spiral structure in our
Galaxy in much more detail. The Sun then is
located near a local arc or
branch between the major spiral arms, as found also in modern
studies (see e.g. \cite{Gon21}). Oort's analysis is along the lines that
Kapteyn must have had in mind when he defined the {\it Plan of Selected Areas}
and Oort's paper is therefore a major outcome of this Plan.
\bigskip
I entered into this somewhat detailed discussion to illustrate that
in the course of the 1930s the work on
the Selected Areas actually had progressed to the point that an analysis
as originally envisaged by Kapteyn was possible and that Oort actually had
performed this. Of course more work needed to be completed and published, but
I would argue that with this study by Oort the
primary goal of the {\it Plan of Selected Areas}
had been accomplished. What followed was to a large extent
provision of more catalogue data. After all,
in his presentation of the Plan
(\cite{SA}, p.1), Kapteyn wrote:
\par \begingroup \leftskip2.5em \rightskip2em \large
The aim of it is to bring together, as far as possible with such an effort,
all the elements which at the present time must seem most necessary for a
successful solution to the sidereal problem, that is: the problem of the
structure of the sidereal world.
\par \endgroup
It would seem that with adding more data no substantial improvement could be
achieved. A main reason for that is that the distribution of the dust that
gives rise to Galactic extinction is {\it highly} irregular.
Van Rhijn's 1936 study
\cite{PJvR1936} had failed to show anything fundamentally new and indeed Oort
\cite{JHO1938} does not even mention the outcome of this work. He refers to
this paper twice, in both cases related to B-stars and Cepheids being bright and
therefore more readily included
in samples selected at random. So, Oort's paper and the
crosscut in Fig.~\ref{fig:crosscut2} is the final result of the {\it Plan of
Selected Areas} in terms of the goal set by Kapteyn. Significant progress
in the mapping of the stellar component of the Galaxy
became only feasible with the recent astrometric satellites Hipparcos and Gaia.
\bigskip
Although, as I stated, Oort's 1938-study constituted to a large extent what
Kapteyn had defined as the aim of the {\it Plan of Selected Areas}, it was
not at all the end of the undertaking. In 1930 van Rhijn
had published the fourth
progress report in the {\it Bulletin of the Astronomical Institutes of the
Netherlands}, and after that it was taken up by the International Astronomical
Union as a dedicated Commission 32. Van Rhijn chaired this commission
this until 1958 and
edited progress reports in the tri-annual {\it Transactions of the IAU}
\cite{IAU2307}. In 1958 at a meeting of Commission 33 (Structure of the
Galactic System), chaired by Adriaan Blaauw, Nancy Grace Roman (1925--2018)
moved a proposal to incorporate it as a sub-commission to Commission 33, which
after being seconded by Bart Bok was carried. The new chair was Tord
Elvius (1915--1992); his last report to the IAU appeared in 1970. At the
corresponding General Assembly in Brighton, Commission 33 President George
Contopoulos proposed to abandon the separate sub-commission and incorporate the
work in that of the
Commission itself. This was in fact the {\it formal} end of the {\it
Plan of Selected Areas}.
In addition to the IAU reports, two overview articles have been written on the
1960s that should be mentioned here. The first of
these is a very nice overview by Beverly Turner Lynds, discussing developments
up to 1963 \cite{L1963}, and the second a presentation in 1965 in the
famous {\it Galactic Structure} volume V of the {\it Stars and Stellar Systems}
compendium by Blaauw and Elvius \cite{ABTE1965}.
In a sense the {\it informal} end of the Plan was marked
in 1953 at what became known as the Vosbergen
conference, as I submitted in my Kapteyn biography \cite{JCKbiog}.
This meeting, which as number 1 marks the beginning of the IAU-Symposia
series, was a tribute to van Rhijn and therefore was held near Groningen.
I return to this below.
As I noted, significant progress
in the mapping of the stellar component of the Galaxy
became only feasible with the recent astrometric satellites.
Still, the {\it Plan of Selected
Areas} had a significant impact on Galactic studies, at least through the
1930s. This is not too obvious from the authoritative discussion of the
development of the field between 1900 and 1952 by Robert Smith \cite{RHS2006}.
He mentions the {\it Plan of Selected Areas} only by its inception by Kapteyn
in 1906 and the latter's first progress report in 1924, and does not cover
further studies of the distributions of stellar properties in its context
during the 1920s and 1930s (such as e.g. \cite{JHO1932} or \cite{JHO1938}).
\bigskip
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.98\textwidth]{figOortMW.jpg}
\end{center}
\caption{\normalsize The structure of the Galaxy as it emerged towards the end
of the 1920s as summarized by Frederick Seares \cite{FHS1928}. This drawing has
been published first by Willem de Sitter in his book \textit{Kosmos} of 1932
\cite{WdS1932}, and attributed there to `Dr. Oort'. He probably drew it earlier
than this. It was used and published by Jan Oort himself in his inaugural
lecture as Extraordinary Professor in 1936 \cite{JHO1936}. The scale is in
parsecs with the Galaxy extending from -15 to +40 kpc from the Sun.
From the Oort Archives, see \cite{JHObiog}.}
\label{fig:OortMW}
\end{figure}
\begin{figure}[t]
\sidecaption[t]
\includegraphics[width=0.64\textwidth]{figSeares.jpg}
\caption{\normalsize Frederick Seares at the time of his involvement with
the \textit{Plan of Selected Areas} commenced. This picture comes from the
\textit{Album Amicorum}, presented to H.G. van de Sande Bakhuyzen on the
occasion of his retirement as professor of astronomy and director of the
Sterrewacht Leiden in 1908 \cite{Album}.}
\label{fig:Seares}
\end{figure}
The canonical picture prevalent in the 1930s and maybe some
years before was that the Galaxy consists of a flattened disk extending much
further then extinction allows us to see and a system of globular systems
of which the centers coincide and that it resembles extragalactic spiral nebulae.
Oort's well-known sketch for Willem de Sitter's book in 1932 in
Fig.~\ref{fig:OortMW} sums this up well.
This picture was preluded by various astronomers, such as Henri Norris
Russell (1877--1957) in an important review on the state of affairs at the
end of the 1910s on {\it Some problems of sideral astronomy} \cite{HNR18},
or somewhat later Oort in his
inaugural lecture as privaat-docent in 1926 (Appendix A. in \cite{Legacy} for
an English translation). In an important paper from 1928 mostly coming out
of his involvement in the {\it Mount Wilson(-Groningen)
Catalogue} as part of the {\it Plan of
Selected Areas}, Frederick Seares \cite{FHS1928}, see Fig.~\ref{fig:Seares},
put it as follows (p.176 and
further):
\par \begingroup \leftskip2.5em \rightskip2em \large
As a directive aid to further research [...] lead to the following picture of
the probable structural relations in the stellar system: The galactic system
is a vast organization resembling Messier 33, although probably larger [...] It
includes a central condensation, scattered stars, and aggregations of stars
distributed over the galactic plane, [...] Like the spirals, the system also
includes diffuse nebulous material concentrated near the galactic plane, dark
and obscuring, or luminous if stimulated to shine by the radiation of
neighboring stars of high temperature. The central condensation, in longitude
325$^{\circ}$, is hidden behind the dark nebulosity which marks the great rift in
the galaxy. Stellar aggregations, seen above and below the obscuring clouds in
the manner familiar in edge-on spirals, form the two branches of the Milky
Way, extending from Cygnus to Circinus. The diameter of the system is large --
80,000 to 90,000 parsecs, if it may be regarded as coextensive with the system
of globular clusters; [...] The sun is situated almost exactly in the galactic
plane defined by faint stars, but a little to the north of that determined by
Cepheids, faint B stars, and some other special classes of objects. Its
distance from the central condensation is perhaps 20,000 parsecs, [...]
Obscuring clouds conceal the great surface density which otherwise might be
expected in the direction of the center of the larger system.
\par \endgroup
Note that this precedes Trumpler's 'discovery' of interstellar absorption.
Oort's 1932 and 1938 papers significantly strengthened the understanding of
Galactic structure; van Rhijn's work on the luminosity function was important,
but in terms of a fundamental understanding of
the overall structure of the Galaxy his work was not groundbreaking.
For completeness: By 1939 the distance to the Galactic center had shrunk to
10 kpc (but is still over 10 kpc in Fig.~\ref{fig:OortMW}),
e.g. on the basis of the review by John Stanley Plaskett (1865--1941)
\cite{JSP1939}.
As Smith also noted in the article cited above \cite{RHS2006},
by the end of the 1930s the
hopes that further analysis of star counts, and the {\it Plan of Selected
Areas} would help further understanding of the structure of the Galaxy were
diminishing. Studies of light distributions and kinematics in external
galaxies, as Oort's seminal, pioneering study of NGC 3115 \cite{JHO1940},
presented at the 1939 McDonald dedication, were promising in that he showed
the feasibility of photographic surface photometry, but the field
awaited larger telescopes and better
observing techniques to measure stellar motions in these systems.
Star counts were not dead yet, though. Bart Bok still saw great promise in
them, especially when he and his Irish student Eric Mervyn Lindsay (1907--1974)
discovered and studied apparent windows in the obscuring layer along the
Galactic plane \cite{LB1937}. They wrote (\cite{BL1938}, p.9):
\par \begingroup \leftskip2.5em \rightskip2em \large
Maybe the study in or near windows in the distribution of obscuring material
can help, if studies on the distribution of faint external galaxies go hand in
hand with studies on the distribution of colors and variable stars, spectral
surveys and star counts to faint magnitude limits on a definite photometric
system. For several years our ignorance concerning galactic absorption has
appeared to be the chief obstacle in the way of effective analysis of galactic
structure; this obstacle can now be removed. Our knowledge of galactic
structure can be further advanced if detailed studies of the distribution of
stars and nebulae are made for more or less obscured fields near galactic
windows as well as for the windows themselves.
\par \endgroup
But this program never produced much fundamental new insight.
\section{Van Rhijn's plans for his own observatory}\label{sect:Observatory}
Already in the 1920s it had become clear that Kapteyn's model of an
astronomical laboratory, an observatory without a telescope, had started to
show cracks. Blaauw (\cite{Blaauw1983}, p.57)
attributed this among others to the
difference in personalities between Kapteyn and van Rhijn (my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
Two character traits had made Kapteyn succeed: his exceptional gifts as a
scientific researcher and his engaging, optimistic personality. In these
respects, what may we say of van Rhijn? He was a thorough scientific researcher
and had a wide interest in the developments in astronomy, also outside his own
field of research. [...] His approach to research, however, was highly
schematic, strongly focused on further evaluation of previously determined
properties or quantities. This benefited the thoroughness of the intended
result, but came at the expense of flexibility and
reduced the opportunity for unexpected perspectives; as a result,
it did not lead to surprising discoveries.
Also, van Rhijn's personality was so different from what has been
handed down to us about Kapteyn; meetings with foreign colleagues, offering
the opportunity for casual discussion and the development of friendships
hardly occurred, because he was not very keen on travel; he liked organizing
and conducting much of the work from behind his desk, admittedly very well and
carefully, but in an almost impersonal way that could unintentionally come
across as `pushing'. All in all, circumstances that made working with the
outside world not always run smoothly [...].
\par \endgroup
At the time of Kapteyn's death and the birth of the IAU, or
to be precise the first
General Assembly, the {\it Carte du Ciel} and the {\it Plan of Selected Areas}
were integrated into the Union as dedicated Commissions. No such new
initiatives on this scale of participation were attempted after that;
in the 1950s the {\it National
Geographic Society -- Palomar Observatory Sky Survey} was performed at a
single observatory with an especially built telescope. Kapteyn had profited
from the existence of the Helsingfors Observatorion and Anders Donner, who was
interested primarily in supplying data for others: participating in the {\it
Carte du Ciel}, but in addition providing data for Kapteyn
to use for the latter's
research. No such opportunity presented itself to van Rhijn. Major
observatories were no longer ready to act as service institutions merely to
supply large masses of observational material for others
(if they ever were). The story
of the grumbling astronomers at Harvard to take plates for the 'damned van
Rhijn program' illustrate this. The {\it Bergedorf(-Groningen-Harvard)
Spektral-Durchmusterung} was performed with a clear division of tasks.
Bergedorf did not supply observational material to Groningen, the observatory
obtained the spectra and classified them,
while Groningen relied on Harvard plates to
determine positions and magnitudes.
Blaauw attributed the success of Kapteyn to both his extraordinary talent for
scientific research and his outgoing personality, a combination of
attributions van Rhijn did not
possess. This would explain both the prominence of the Laboratorium in
Kapteyn's days as well as the decline under van Rhijn. I actually would go a
bit further: the model of an astronomical laboratory relying on others to
supply observational material, simply was not a viable one and was bound to
fail not only in the long run, but already as soon as Kapteyn had stepped down.
Kapteyn will have realized this when he presented his
{\it Plan of Selected Areas}, where in addition to the scientific goals quoted
above he put it forward as a means to secure funding, actually increased
funding, for his laboratory. As I put forward in \cite{JCKbiog} and
\cite{JCKEng}, Kapteyn
must have realized that the future of Dutch astronomy lay in
Leiden with a real observatory, which after the reorganization of 1918
consisted of three
departments lead by renowned astronomers. Groningen had no observatory, nor
the support the Sterrewacht had from its university and particularly from the
Ministry and was bound to become second rate. It could survive only with a
long term commitment by a large group of observatories to his Plan,
concluded under Kapteyn's leadership.
Van Rhijn must have realized this also. His experience with the Groningen part
of the {\it Spektral-Durchmusterung} and the reliance on
Harvard made abundantly clear that the model of a laboratory
would fail. No wonder
that van Rhijn picked up what Kapteyn had failed to accomplish:
obtain his own telescope.
\bigskip
I briefly recount the story of Kapteyn's failure to obtain an observatory.
He was appointed the first professor in Groningen
with the teaching of astronomy as his
primary assignment as a result of a new law on higher education that
stipulated that the curricula at the three
nationally funded universities should be
the same. This roughly doubled the number of professors in the country and
this and the stronger emphasis on research resulted in a comparable increase
in the funding of the universities. However, to a significant extent due to
negative advice by the directors of the observatories in Leiden and Utrecht,
Hendricus van de Sande Bakhuyzen and Jean Oudemans, who did not wish to share
the resources for astronomy with a third partner, Kapteyn's proposals to build
a small observatory in Groningen were rejected. This development made Kapteyn
give up hopes for an observatory and instead establish his astronomical
laboratory. In the 1890s he requested funding for a photographic telescope for
Groningen, proposing Leiden would concentrate on astrometry and Groningen on
astrophotography. However, van de Sande Bakhuyzen reversed his earlier
position, in which he questioned the usefulness of photography, and
refrained from Leiden participation in the {\it Carte du Ciel}.
He must have seen Kapteyn's proposal as a threat to Leiden losing its
dominant position in the country, and submitted a competing
proposal. This was significantly cheaper than Kapteyn's since he only added
on to existing infrastructure and the choice was not difficult for the
Minister.
Another possibility to secure regular access for Groningen to a telescope occurred
during the late 1910s, when rich tea-planter and amateur astronomer Karel Albert Rudolf Bosscha
(1865--1928) from Bandung, Dutch East-Indies, assisted by Rudolf Eduard
Kerkhoven (1848--1918), made plans to establish an observatory there \cite{LKBF}.
This was stimulated by
Joan George Erardus Gijsbertus Vo\^ute (1879–1963), who was
originally a civil engineer, but who had turned to astronomy and, without ever
defending an PhD thesis\footnote{\normalsize On September 21, 1922, Joan Vo\^ute acted as
leader of a joint German-Dutch solar eclipse expedition to Christmas Island to
the south of Java, for which the Friedrich-Wilhelms Universit\"at in Berlin --
now the von Humboldt-Universit\"at -- awarded him an honorary doctorate in
1923.},
worked as observer at Leiden and Cape Town observatories before joining the
meteorological office in the Dutch East-Indies.
Kapteyn felt this observatory should be managed by a national committee,
while de Sitter took the position that the Sterrewacht director (so he himself)
should be responsible for defining the research program and organizing the
interface with Dutch astronomy. De Sitter argued that the Minister's attempts
at specialization and concentration could be diverted by agreeing that 'Leiden
specializes in everything except the specializations of Groningen and Utrecht'
(from a letter to Kapteyn of May 24, 1920, see W.R. de Sitter in \cite{Legacy},
p.99), making the latter two in fact subsidiaries to the first.
In the end de Sitter's coup failed and Kerkhoven and Bosscha established the
observatory with the authority for the scientific program in the hands of
{\it its} director, the first of whom of course was Vo\^ute.
De Sitter secured access to a telescope in South-Africa,
first through an agreement with the Unie Sterrewag in Johannesburg and later by
erecting a telescope there through funding by the Rockefeller Foundation in
order to lure Ejnar Hertzsprung to Leiden,
but this was not offered as a national
facility. Only after World War II would this situation be replaced by
establishment of national and international facilities. But until then Leiden
kept a firm grip on access to the telescope. And, as we will see, the Ministry
actively supported for many years Leiden's dominance through its funding
decisions.
That this gradually changed after Word War II can be appreciated from the story
of the related `Leids Kerkhoven-Bosscha Fund'
(LKBF). In 1950, upon the death of his widow, it became known that Rudolf
Albert Kerkhoven (1879–1940), son of Rudolf E. Kerkhoven and amateur
astronomer, had left a considerable sum of money to support astronomy in the
Dutch East-Indies. In the mean time this had become Indonesia and the result
eventually (in 1954) was the founding of the LKBF \cite{LKBF}.
A reasonable fraction of
the allocations that the Fund provided went, and still goes, to the Bosscha
Observatory. But the majority of the proceeds of the investments is spent on
grants to support Dutch astronomers; and, in spite of the 'Leids' in the Fund's
name, astronomers from all Dutch institutes are eligible equally. Actually,
astronomers from Dutch universities other than Leiden, like myself, have
chaired the Board.
\bigskip
Van Rhijn revived the efforts to install a telescope in Groningen, but ran into
the same problems Kapteyn had had before:
although having support from Curators,
being rejected funding by the government. This was not an isolated development.
The university of Groningen was systematically disadvantaged across the board
by the government.
In 1928 it induced Rector Magnificus Johannes Lindeboom (1882–1958), theologian
and historian of religion, to state in the Rector's annual report on thr
`Lotgevallen' (literally the report on the fates) of the university
(my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
It is a fact that almost every year the vacancies caused by departures to other
universities have to be listed among the
misfortunes. [...] The fact that Groningen is a bit far —
actually really not that far — from the center cannot be changed. The
authorities in Groningen are doing their best to make Groningen a very pleasant
city to live in, and when the authorities in Den Haag [the Hague]
would be willing to do the same, there is nothing else [...] to worry about.
But they could, for example,
ensure that Groningen gets its fourth postal delivery back, and that it
as the third commercial city of the country and a scientific and
cultural center, no
longer is, as it is at the moment, put behind almost every national
establishment. All of this may — this in passing — be an explanation of the
fact that your Rector believed to do the right thing by, in a letter to Z.E.
[His Excellency] the Minister of Waterways and Public Works, plead the
interests of the University in this matter.
\par \endgroup
In the beginning of the 1920s significant budget cuts had been introduced when
the economy, dependent as it was on trade with Germany that suffered from the
aftermath of the World War I, stagnated, although stabilized later in the decade
\cite{KvB2017}. The situation worsened with the Wall Street crash of 1929. It
would be logical for the Minister to look for savings in the expenses on
astronomy, and with the existence of extensive infrastructure in Leiden, which
had been thriving after de Sitter's reorganization of 1918, and that in Utrecht,
it would be understandable that any extra expenses on the small laboratory in
distant Groningen would be seen as an extravagance. However, the remoteness of
Groningen affected all of the university. In 1934/35 there was a (not for the
first time) seriously considered option to close one of the universities, and
that would then be the one in Groningen. An immediate extensive and intense
lobby prevented this \cite{KvB2017}, but it underlined the vulnerability of
the university.
In view of all this it may come as no surprise that van Rhijn failed to receive
any support or funds for a telescope to be put on the roof of the
Kapteyn Laboratorium. I quote from Adriaan Blaauw's article
(\cite{Blaauw1983}, p.58 and further; my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
[…] van Rhijn wanted to have a little more control over the future of the work
himself by means of his own equipment. His wish was to install a telescope of
modest dimensions on the laboratory building.
Who would think that the fame which
the Kapteyn Laboratorium had acquired internationally — few scholars, in fact,
have contributed so much to the Dutch scientific reputation at the beginning
of this century as Kapteyn — would be rewarded by this low-cost plan is
mistaken, and can only look in astonishment at the spirit of the ministry's
responses. After van Rhijn first had to acquire the telescope with the help of
private funds, even his attempts at ministerial level to obtain the funds
needed for the construction of the dome were rejected. In spite of the
intercession of the Curators of the university […], the ministry persisted in
its chosen position: there should be nothing from which it would appear that
Groningen astronomy could pretend to have a claim for its own equipment. […]
In 1929, in response to a modern reader rather haughty letter from the
Ministry to the Groningen Curators, van Rhijn patiently and extensively
refuted the arguments put forward therein, but the university still got a
negative response, so he decided to look for private funds for the construction
of the dome as well.\footnote{\normalsize See the letter from Curators of the University to
van Rhijn dated April 27, 1929 (no. 617) with copy of letter from Curators to
the Minister of 10 April (no. 539) and of the reply of the Minister of 22 April
(no. 1592, department H.O.); Van Rhijn's letter of 4 May 1929 to Curators and
their message of December 16, 1929 (no. 2719) regarding the Minister's
dismissive answer; Van Rhijn's letter of 3 March 1930 to the Curators and their
reply of 17 July 1930 with copy of the Minister's letter of 14 July (no. 31621
dept. H.G.); all this correspondence in the van Rhijn Archives at the Kapteyn
Laboratorium.}
In March 1930 he could report to the Curators that he had succeeded. In 1931
the telescope, a reflecting telescope of 55 cm diameter, was installed.
The state of affairs described above must have been a bitter experience for
van Rhijn, and it was not the only one: also in attempts around that time to
bring the remuneration of his computers and observers and their future
prospects to the same level as elsewhere, especially in Leiden, he also
encountered this unreasonable rejection by the higher
authorities. What, for example, to think of the following introductory
remarks in a letter from the Minister to the Curators of 2 October
1929\footnote{\normalsize No.1498 Department of H.O.}:
'With reference to the letter mentioned, I have the
honor to inform your Board that I am prepared to consider a further proposal
from your Board, on the understanding that, in my opinion, in applying the
salary regulation to the personnel of the Groningen Laboratorium, the
University should take into account the difference between the other
institutions and the Sterrewacht Leiden, also with regard to the grades to
which the personnel can be promoted.'\footnote{\normalsize From a copy attached to the letter from
the Curators to van Rhijn of October 24, 1929, no. 2429.} But he was too much
of a man of character to speak out about these matters to his co-workers at
the laboratory.
\par \endgroup
Blaauw's reference to letters in the footnotes
in the van Rhijn archives no longer
is a valid one. As explained in the Introduction, these
archives (at the initiative of Blaauw himself) have been transferred to the
central archival depot of the university and from
there, after inventories and removal of duplicates and irrelevant parts, to
the municipal Groningen Archives for permanent
keeping. In this process these letters have been lost.
\section{Realization of van Rhijn's telescope}\label{sect:telescope}
So who funded the telescope? All I can find is the remark 'private sources'.
There is no correspondence on the acquisition of the telescope at all in the
van Rhijn Archives. Whether
private sources were individuals or funds is not known, so it could
in fact have been a kind
of crowd-funding {\it avant la lettre}. The only remark on the subject in the Oort
correspondence is a remark van Rhijn made in a letter to Oort, dated November 24,
1952 (more of this letter is cited in section 20; from the Oort Archives Nr. 149,
my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
Kapteyn has tried to establish a proper possibility for scientific research in
Groningen, which he has been refused. The only
possibility, which I saw to change this, was the purchase of a second-hand mirror,
which happened to be offered to me at the
time.
\par \endgroup
How and by whom this offer was made is not mentioned.
What is known is that in the end the {\it Kapteyn Fund} and the {\it Groningen
University Fund} bore most of the cost of the construction of the dome.
These funds might have contributed to the purcvhase of the telescope as well.
I digress a bit to describe very briefly the histories of these funds. The
Kapteyn Fund was formally founded on February 21, 1925, by the signing of a
deed at the office of a notary in Groningen. The first board encompassed the
directors of the astronomical institutes in the Netherlands: Pieter van Rhijn,
Willem de Sitter, Albert Nijland and Anton Pannekoek, but also a person called
W. Dekking. This was Rotterdam merchant Willem Dekking (1873--1923), who was a
friend of Kapteyn's elder daughter Jacoba Cornelia (1880--1960). The latter had
studied medicine and married a fellow-student in medicine Willem Cornelis
Noordenbos (1875--1954) and lived for a short period in Rotterdam. Decking had
been very much impressed by Kapteyn and offered his help managing the Fund and
soliciting donations. The deed mentioned a starting capital of 'eleven hundred
sixty eight guilders and sixty-eight cents, brought together by Professor
Kapteyn during his life', which corresponds to a current purchasing power of
18,650\textgreek{\euro}\footnote{\normalsize A useful currency conversion site is provided
by the International Institute of Social History \cite{IISG}.}.
This sentence survived in the latest update, of 2010, which I signed in my
capacity of the chairman
of the Board at that time. The current capital is of
order 70,000\textgreek{\euro}. The Fund is called the {\it
Sterrenkundig Studiefonds J.C.
Kapteyn} and aims to support studies of students or research by astronomers
from all over the Netherlands. It was and still
is not a particularly large fund and the
contribution to the cost of the dome must have been modest.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.468\textwidth]{figteles.jpg}
\includegraphics[width=0.512\textwidth]{figtel2.jpg}
\end{center}
\caption{{\normalsize Left: The 55-cm
telescope on top of the Kapteyn Astronomical
Institute, pointing at the tower of the central Academy Building of the
University of Groningen. The Laboratorium was located in the center of
Groningen just behind this main university building with its 66 meter high
tower, which is positioned to the southeast of the telescope at only some
20 meter distance (see Fig.~\ref{fig:air})}.
{\normalsize Right: View of the telescope and mount. Note that
the spectrograph and plate holder at the Newtonian focus. Kapteyn Astronomical Institute.}}
\label{fig:tel}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.98\textwidth]{figlabtel.jpg}
\end{center}
\caption{\normalsize The Kapteyn Astronomical Institute with the telescope
`dome' on the roof. Kapteyn Astronomical Institute.}
\label{fig:labtel2}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.98\textwidth]{figair.jpg}
\end{center}
\caption{\normalsize The Academy Building of the University of Groningen
(center) with the Kapteyn Laboratorium to the left with the
cylindrical telescope dome on the roof. The photograph dates from 1952
and has been taken from the southwest.
The telescope has been installed in 1931 and removed in 1959 (but the dome left
intact) and the building was demolished in 1988. The church on the right
opposite the
Academy is the Roman Catholic Sint-Martinus Church, not to be
confused with the Martini Church and Tower near the Grote Markt.
It was demolished in 1982 and replaced with the Central University Library.
Aviodrome Lelystad.}
\label{fig:air}
\end{figure}
The {\it Groningen University Fund} has a history \cite{GUF},
which is relevant to discuss here in the context of the support
it gave to
van Rhijn where the Minister had failed to do so. The 'GUF' was founded on
March 4, 1893 by the professors of Groningen University, following examples
at the other universities, where in chronological order at Utrecht,
Am\-sterdam and Leiden such funds had established.
It grew out of a much older fund,
the 'Professor's Fund', providing grants for promising students that had no
other means to enroll in universities. It had been established in 1843 and in
addition to voluntary contributions from professors and parts of the tuitions
they received,
levied fines on professors that for no good reason had failed to turn up at
meetings of the Senate. A direct reason for replacing this fund with the GUF
was that the law on higher education of 1876, the same that was behind the
creation of Kapteyn's professorship, had severely reduced the freedom to choose
where to spend the finances provided by the Minister. The GUF would, among
other things, provide a buffer to offset this.
The new Fund hoped to increase its capital from private donations by professors
and other supporters of Groningen university. The original statutes state as
the aim (my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
The University of Groningen will use the funds for the promotion of study at
the University and of the flourishing of the University in the broadest sense.
To this end, the following will be considered: assisting with teaching materials
and financial support for talented students with limited means; expanding the
University library and the academic collections; promoting scientific research
and trips, to be undertaken by professors, lecturers, private tutors or students
at the Groningen University; temporarily employing private tutors to the
University in scientific subjects, which are not taught at a national
university, and promoting actions and measures, which may increase the prestige
of the University at home and abroad.
\par \endgroup
The starting capital was 4000 Guilders, equivalent to nowadays about
55,000\textgreek{\euro}. In 1930 the capital had grown to about 123,000
Guilders (in current purchasing power a little over a million
\textgreek{\euro}). In more recent years (I was on the board between 1997 and
2015, seven years as treasurer and eleven as chairman) the capital varied due
to economic ups and downs and new donations between some 7 and 9 million
\textgreek{\euro},
while the returns on the capital were used mostly to co-fund foreign travel
related to studies of several hundred students per year and scientific
symposia, etc. in Groningen. The need to be dependent as little as possible
upon funding for research from the government was well justified, as the story
of the financing of van Rhijn's telescope and dome
clearly shows. I continue now with the story of the telescope.
\bigskip
The 'Lotgevallen' for 1931 mention (my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
The Kapteyn Sterrenkundig Laboratorium has undergone a metamorphosis with the
construction of an observatory dome. The scientific work in this renowned
laboratory is based on determining the position of stars with the help of
photographs provided by domestic and foreign observatories. The director,
colleague van Rhijn, has purchased an astronomical reflector in order not to be
depending on others, being able to take photographs
of the sky himself. The observer,
Mr. G.H. M\"uller, has shown himself capable not only of observation, but
also of designing what is required for that purpose. The costs of the dome
have been covered by private funds, mainly from the J.C. Kapteyn Fund and
the University Fund.
\par \endgroup
The telescope (Figs.~\ref{fig:tel}, \ref{fig:labtel2} and \ref{fig:air})
was a reflecting one
with an aperture (mirror diameter) of 55 cm and a focal length of 2.75 m,
and was originally planned apparently to be used for astrometric work.
Later it had a spectrograph in the Newtonian focus, that means
near the focus of the primary mirror the light was
reflected at a right angle towards the
telescope tube near its top, where at that location a spectrograph was mounted.
The van Rhijn Archives contain extensive correspondence with the Karl Zeiss
Company in Germany about auxiliary equipment such as a slitless spectrograph
(1936), and an 'Astro-spectrograph for one or three prisms' (1952), but the
latter file also contains a letter from Dr. J.H. Bannier, the director of the
Netherlands Organization for Pure Research, stating that it will not honor a
request for funding. There are also quotes for other extensions from Zeiss.
The idea to use it spectroscopically seems to stem from the 1930s.
The only formal description of the telescope is in a few sentences in
a paper in 1953 \cite{PJvR1953}, not even mentioning the constructor,
but it is very likely to
be the Karl Zeiss Company as well.
For this paper \cite{PJvR1953}, the first one
based on material obtained with the telescope, spectra had been taken with a
slitless spectrograph.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.57\textwidth]{figfiles.jpg}
\includegraphics[width=0.41\textwidth]{figpermit.jpg}
\end{center}
\caption{\normalsize Left:
The stack of copies of the allocation of permits issued
by the Municipality of Groningen for outdoor lighting for advertising.
Right: One example of a permit. From the Archives of the Kapteyn
Astronomical Institute.}
\label{fig:permits}
\end{figure}
The location in the center of the city of Groningen posed a major problem of
disturbing city lights, and when van Rhijn was intending to work
spectroscopically in particular those of advertising installations on the
outside of buildings, would be a serious problem. Such observations
would be seriously affected by strong neon spectral lines. Van Rhijn
arranged that special conditions were attached to the granting of permits by
the municipal authorities for such installations. The van Rhijn Archives
contain a large folder of copies of letters from the City of Groningen
concerning such permits (Fig.~\ref{fig:permits}).
These run from the early 1930s to well into the 1950s.
I estimate that this concerns some two hundred such permits. They
invariably contain the following passage (with sometimes minor variations on
this) (my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
... to grant permission [...] to install a lighting installation in accordance
with the attached drawing on the understanding that the light of the
installation can be extinguished during those evenings and nights when
observations are made by Kapteyn Sterrenkundig Laboratorium, and that for this
purpose, at a location to be determined by the Municipal Works Department, a
switch shall be installed which can be operated either by or on behalf of the
Director of the aforementioned laboratory and ...
\par \endgroup
According to Blaauw very little use has been made of this. It would be
something to see on a clear winter night van Rhijn or his assistant
taking their bikes and going
around the city to switch off all advertising before opening up their dome.
Maarten Schmidt, who was an undergraduate student in Groningen in the
1940s after the war, in his interview for the American Institute of Physics
Oral History Interviews \cite{Schmidt75} recollected:
\par \begingroup \leftskip2.5em \rightskip2em \large
It was a reflector of about fifty centimeters in diameter and it stood on the
roof of the observatory building which was next to the University in the
center of the city. I took spectra of blue stars, spectra of stars to study
the effects of dust in between the stars and us, [...]
That observatory, being in the middle of the city, they
had a funny system that was hardly ever used, I think, but they had gotten the
municipality to require that every building that had roof top advertisements
in lights would take care that there was a switch at ground level that could
be handled with a particular key so that if any observer found it necessary
to observe low in the sky in the direction, he would get a bike, go to that
part of the city, go to that particular part and turn off that darn light.
[laugh] I think we once used it or never did it. I have the inkling that we
once did it and I felt very guilty, you know, these advertising lights.
\par \endgroup
There even was the following addition to the formal Regulations:
\par \begingroup \leftskip2.5em \rightskip2em \large
\noindent
23 August 1949:
Regulation amending the Police Regulations for the Municipality of Groningen
[...] After article 113, a new article is inserted, numbered 114, and reading
as follows:\\
'It is forbidden to have a illuminated advertisement, facade illumination or
any other light source on or in a building or in a yard, the light of which
radiating upwards could be a hindrance to astronomical observation by the
University or one of its institutions. This prohibition does not apply
to light sources for which a permit has been granted by the Mayor and Aldermen,
as long as the conditions attached to this permit are complied with.'
\par \endgroup
\bigskip
The question is what van Rhijn had in mind using the telescope for.
Let me first quote Adriaan Blaauw \cite{Blaauw1983} (p.58 \&\ 59;
my translation)
and then fill in the details in the next section.
\par \begingroup \leftskip2.5em \rightskip2em \large
Van Rhijn imagined to attack with the telescope especially the problem of the
reddening of the stars by the interstellar medium, and collected the necessary
spectrophotometric equipment in the course of the following years. However,
the work got off to a slow start. There was little technical assistance among
the staff, which had been reduced in the crisis years, and the scientific
assistant had to give attention to the main measurement programs.
After the war, the work got somewhat under way and results of this color
dependence research were published by van Rhijn himself and by Borgman.
However, it never came to a substantial contribution, in the first place
because new photoelectric techniques quickly superseded the photographic
ones. Moreover, the situation of the telescope, located in the middle of the
city, where city lights and smoke obstructed the sky, became an impossible
one. When, during my directorate, around 1959, the new wing of the university
building was being built and the walls of the laboratory that carried the heavy
telescope began to show cracks, the presence of a tall construction crane was
taken advantage of to lift the instrument out of the dome.
\par \endgroup
As Adriaan Blaauw told those interested on a few occasions (among
which myself)
he bribed the crane operator by presenting him with a box of cigars.
My colleague Jan Willem Pel, who worked at the Kapteyn Sterrenwacht
(which was established in
the 1960s as part of the Kapteyn Laboratorium) in Roden near Groningen informs
me: '... this mirror lay in a chest under the dome in Roden; after that it
ended up in the optical lab in Dwingeloo, where I
think it still is.' The staff and workshop of the Kapteyn Sterrenwacht has
been moved to the Radio-sterrenwacht in Dwingeloo together
with a similar workshop and staff from the Sterrewacht Leiden to
form the 'Optical IR Instrumentation group' there. We will
meet Jan Borgman, mentioned here, extensively below.
\section{Adriaan Blaauw and Lukas Plaut}\label{sect:ABLP}
Adriaan Blaauw and Lukas Plaut joined the Kapteyn Laboratorium in 1938 and
1940 respectively;
Blaauw when a position as assistant became available, Plaut as a result of the
antisemitism brought along by the Nazi party and the German occupation of the
Netherlands. Blaauw was appointed assistant, which was normally
(and indeed in this case) associated
with the preparation of a PhD thesis. In order to put things in context I
have to backtrack a little.
In the years up to 1940 much of the resources and manpower was used to carry
the work on the {\it Spektral-Durchmusterung} with the Hamburg and Harvard
collaborators along.
As documented above, some other very significant and notable work was performed
as well, although no breakthroughs or major new insight had resulted from this.
That is not to say that nothing of significance happened. I have already
mentioned the PhD theses by Bart Bok and Jean-Jacques Raimond in 1932 and 1934
and Broer Hiemstra in 1938.
The position of assistant, occupied by Bok and Raimond when they had completed
their theses was filled by P.P. Bruna, who, as mentioned above, left without
completing a PhD thesis, and this opened up a position for Blaauw.
\bigskip
On October 1, 1938 Adriaan Blaauw (see Fig.~\ref{fig:NAC1941})
joined the Kapteyn Laboratorium as the new
assistant. Blaauw has himself summarized his biographical
details in his prefatory, autobiographical
chapter in {\it Annual Reviews of Astronomy \&\
Astrophysics} \cite{Blaauw2004}. Blaauw was born in Amsterdam, the son of a
bank auditor and a governess. He entered Leiden university in 1932 to study
astronomy and had obtained in 1938 his 'candidate' degree (now called
Bachelor), but when he was appointed in Groningen, not yet the 'doctorandus'
degree (equivalent to the current Master). In
his AIP interview \cite{Blaauw1978}, Blaauw stated that he obtained the degree
in 1941 in Leiden before the university closed. The University of Leiden was
closed in the fall of 1940 after the famous lecture by Rudolph Pabus
Cleveringa (1894--1980), protesting the dismissal of Jewish
professor of Law Eduard Maurits Meijers (1880--1954) and others.
However, exams were allowed again after April 30,
1941, but forbidden again after November 30, 1941.
Blaauw must have obtained his degree in this period.
For more on Leiden University during World War II,
see \cite{Ottersp}. The Germans gave Leiden students
permission to continue their studies at another university
and do their exams there.
Blaauw, as we saw, at first was involved extensively in supervising the ongoing
Groningen contribution to the {\it Bergedorf(-Groningen-Harvard)
Spektral-Durchmusterung}, but
this slowed down after the outbreak of World War II and stopped completely
when the USA entered the war, because of course no more
new plate material arrived
from Harvard. In 1942 he married Alida Henderika van Muijlwijk (1924--2007).
Because the work at the Laboratorium had essentially come to a standstill he had
ample time to work on research for a PhD thesis, which would concern the
structure of the Scorpius-Centaurus Association. This is a star cluster
which contains many hot, young stars (of spectral types O and B) and
in fact is a very young cluster at about
130 parsec distance. Blaauw used improved proper motions of the B-stars in
order to derive a better parallax for the association. He found that the group
was expanding and dissolving, but, as he stated later, he failed to see the
obvious fact that the association had to be have formed {\it very}
recently. I will return to Blaauw further on.
\bigskip
\begin{figure}[t]
\sidecaption[t]
\includegraphics[width=0.64\textwidth]{figNAC41.jpg}
\caption{\normalsize Detail of the group photograph of the Nederlandse
Astronomen Conferentie in Doorn in 1941. The person on the
left is Adriaan Blaauw. In the front we see Lukas Plaut. From the
Website of the Koninklijke Nederlandse Astronomen Club, the Dutch Society
of professional astronomers \cite{KNAC}.}
\label{fig:NAC1941}
\end{figure}
Lukas Plaut (1910--1984) joined the Kapteyn Laboratorium in 1940 after the
outbreak of the war. Since no autobiographical notes from him exist I will
spend some more space here on Plaut than on Blaauw. A biographical article,
written by Barbara Henkes recently in 2020, focusing on his suffering from
antisemitism and based on extensive interviews
with and material provided by his widow in the 1990s, is very much worth
reading \cite{Henkes}.
Adriaan Blaauw has written a short obituary of Plaut in {\it Zenit},
the successor
of the amateur periodical {\it Hemel \&\ Dampkring} \cite{Blaauw1978}. His
contribution to the {\it Biographical Encyclopedia of Astronomers} \cite{Bea}
is modeled on this. He wrote:
\par \begingroup \leftskip2.5em \rightskip2em \large
Occupation of the Netherlands by the German army in 1940 forced Plaut to move
first to Groningen (the Kapteyn Laboratorium, followed by a modest teaching
job), then to a labor camp, and finally to the concentration camp in
F\"urstenau, Germany.
\par \endgroup
This is a bit misleading, at least it gives the impression
of having been imprisoned for some extensive time in an extermination
camp.\footnote{\normalsize In an interview remininiscing on Plaut in 2010, a few months before his death,
Blaauw gives a much more balanced, but in details not fully accurate,
account (vimeo.com/16073067).}
I will detail the story here somewhat extensively. Plaut's experience certainly
has been traumatic and this is not intended to take away from
this aspect.
Lukas Plaut (see Fig.~\ref{fig:NAC1941}) and his identical twin brother
Ulrich Hermann were born in Japan from Jewish, German parents. Their
father was a teacher of arts
and an art dealer; the parents gave the children (there was
also a younger sister Eva) a liberal upbringing, hardly following Jewish
rituals and only attending the synagogue exceptionally.
In fact Lukas was not circumcised, which
may have saved his life later. When the children were teenagers, the parents
sent them to their native Germany. Lukas went on to study mathematics and
physics at the Friedrich-Wilhelm Universit\"at in Berlin, after which in 1931 he
obtained a position at the Neubabelsberg Sternwarte in nearby Potsdam. The
boys must have continued their lives separately at some stage; the brother
became an architect, married in Shanghai in 1935 and moved to Los Angeles.
Lukas had in the mean time denounced Judaism, but this would not save him
from measures against Jews in Germany.
In April 1933 he was expelled from the observatory in the wake of the
growing antisemitism, resulting in a boycott by the Nazi's of Jewish businesses
and suppression of Jews. With the
help of his mother, who, feeling he should move to the Netherlands,
came from Japan to visit the Sterrewacht Leiden with him,
as a result of which he took up a study in astronomy in Leiden,
financially supported by his parents. He studied with Jan Oort and Ejnar
Hertzsprung, eventually earning a PhD in 1939 under the latter on {\it
Photogra\-phi\-sche photometrie der veranderlijke sterren CV Carinae en WW
Draconis} (Photographic photometry of the variables CV Carinae and WW Draconis). In
1938 he had married a (non-Jewish) Dutch woman, Stien Witte (1911--2007).
This was not straightforward, as
it was forbidden by German law for a (German) Jew to marry an Aryan without
permission. Dutch authorities had adopted this measure (to avoid tensions
with Germany), but with the support and intervention of Jan Oort and a friendly
and helpful layer this was performed with a exceptional permit, but as
quietly (and quickly) as possible. When the permit came they were called away
from their work to marry within hours lest it
would be repealed, which robbed Stien Witte, much to her chagrin, the
opportunity to ever wear a wedding dress.
After the outbreak of WWII Plaut remained at first in Leiden, where
Hertzsprung had arranged some modest allowance when transfer of funds from his
parents in Japan became impossible. But increasingly measures against Jews
were introduced, resulting in Plaut being ordered in September 1940 to leave
Leiden and the neighboring Leiderdorp where they lived. On the long list of
places that were also forbidden for him, Groningen was lacking, so that is where
he went. His wife and the daughter they had by then came a few days later. This
whole affair turned Lukas Plaut from being a supporting husband in to one being
protected by his wife, as she now, being `Aryan', had to take initiatives and
deal with authorities. They rang the doorbell of the van Rhijn residence, who
declined to provide them shelter (probably apprehensive of the possible
consequences), but showed them a list of places to rent modest rooms.
But van Rhijn gave Plaut the opportunity to enter the Laboratorium.
From Blaauw's biographical notes it would follow he did this with some
regularity, even being paid for a little bit of (illegal?) teaching.
But things got worse and worse. They apparently kept a stipend Hertzsprung
had arranged, which was however
barely sufficient to survive. At the insistence of his
brother in Los Angeles the Plauts filed for emigration to the USA at the
American Consulate in November 1940,
providing written guarantees from the brother and positive
recommendations from Oort and Hertzsprung. Yet the application was turned down
(the German authorities would still have had to agree as well, which also
happened rarely). In 1941 Plaut had to register as non-Dutch Jew, which could
mean deportation. Being married to an Aryan wife saved him from this however.
On the other hand,
because of their marriage Lukas lost his German citizenship and his wife
her Dutch, so they had become stateless! Of course as of May 1942 Lukas had
to wear the yellow star, identifying him as a Jew; in principle he could
be picked up any time in the street for deportation. So he and his wife lived
as inconspicuously as possible, always terrified he might be
deported. In 1943 they had a second daughter.
However, the inevitable happened. In March of 1944 he was called upon to enter a
forced labor camp to work on the construction of an new airport near the
village of Havelte, not far from Meppel, some 60 km to the southwest of
Groningen. The Germans had decided to build a new airport, `Fliegerhorst
Havelte', to relieve Amsterdam Airport Schiphol, which was the major airfield
in the country, and have a second airfield further to the east
for military purposes. The construction had
started in 1942. On satellite photographs one can still easily discern the
outline of the more than one kilometer long, east-west runway. He was allowed
to write home regularly, reporting that the food was reasonable and sufficient
but complaining of fatigue and discomfort like flu, and actually was paid a
small salary. His wife was allowed to send him food, clothes and books, etc.
In fact the treatment of mixed Aryan-Jew marriages offered some protection.
But the strenuous physical effort took its toll, just like the lack of privacy
and the strain of the sharing of the sleeping quarters with
twentyfive noisy men which
he found difficult to endure. On September 5, 1944, 'dolle dinsdag' (mad
Tuesday), when incorrect news spread throughout the country that the
Allied forces were about to liberate the Netherlands, guards flew and Plaut
returned to Groningen. Although the emerging airfield was well camouflaged,
the Allies of course knew about it pretty well and when
it was almost completed the airport
was destroyed on March 25, 1945 in a massive bombing attack, leaving over
2000 bomb craters.
Plaut went into hiding in Groningen in the home of Johannes Gualtherus van
der Corput (1890--1975), a professor of mathematics. Van de Corput has been
active in the resistance movement throughout the War.
He survived and after the War he first
moved to Amsterdam and later the USA. Plaut was given a new non-Jewish,
Aryan identity; he was supposed to be an agricultural engineer from the
School of Higher Education in Agriculture,
the predecessor of the current Wageningen
University \&\ Research. However, he no longer could send
letters to his wife, who even was not allowed to know where he was, although a
courier picked up his dirty laundry, brought it to her and after cleaning
returned it, and she provided the
necessary financial support. On February 23, 1945, a mere eleven weeks before
the end of the war, Plaut, van de Corput and a third person
were arrested following a tip the Germans had received
that there were persons in hiding at the address where he lived and
taken to the prisoner camp F\"urstenau. Not as a Jew, but as an Aryan male
hiding to avoid the {\it Arbeitseinsatz} of Dutch men in Germany, the forced
labor to keep German (war) industry and economy going. His liberal Jewish
upbringing, resulting in him not being circumcised, supported his forged
identity and very likely saved him from being transported to an extermination
camp. As I noted above, Blaauw, in his contribution to the {\it
Biographical Encyclopedia of Astronomers} \cite{Bea} described F\"urstenau as
a concentration camp, and those citing him have copied that, but it really was
a prisoner camp and not an
extermination facility for Jews as this word would suggest.
F\"urstenau lies some distance into Germany from the Netherlands at the
latitude of Zwolle, at about 130 km from Groningen. After the liberation of
the camp some time in April 1945,
Plaut was released and he walked the distance to Groningen, appearing
unannounced at his wife's doorstep. She had not heard from
him since his arrest. In a letter of Plaut to Oort on May 19 (see
Section~\ref{sect:recover} below) he mentioned that he worked on a farm for a
few weeks and returned back in Groningen 'three weeks ago,'
so this must have been before the German
surrender. Throughout the war
he had not been in immediate extermination threat in a concentration camp but
the danger of arrest and deportation had been hovering over him and his wife
the full five years of the war.
He survived this ordeal but never was able to free
himself from the trauma of the injustice, humiliation and indignation of being
subjected to all of this only because of his Jewish descent. I remember him as
a kind, soft-spoken man with an obvious all-consuming, but untold and
unresolved grief, suffering from the post-traumatic stres syndrome that
held him in its grip. His mental well-being deteriorated more and more when
he grew older. He died in 1984 from an heart attack likely not unrelated to
the terror and distress he had been subjected to. His twin brother had died
already in 1971; his parents survived the War also, they moved the the USA,
probably before Japan entered the war.
As a sideline:
when Lukas Plaut retired in 1975 I was offered (and I accepted) the vacancy at
the staff he left at the Laboratorium. I took over from him the introductory
astronomy lectures he had been giving for many years to first-year students
in physics, mathematics and astronomy.
He had obviously enjoyed doing that; the book he used to accompany his course,
{\it Abri\ss\ der Astronomie} by Hans–Heinrich Voigt, he translated with Nancy
Houk into an English edition in two paperback volumes \cite{Abriss},
of which he presented me with a copy with obvious pride.
\section{Van Rhijn and the Laboratorium during the War}\label{sect:WWII}
When the war
started, van Rhijn had been in the middle of his one-year term as
Rector Magnificus of Groningen University (see Fig.~\ref{fig:RM}). He
completed his term before all the troubles with Jewish staff started.
In September 1940, when he was scheduled to present the Rector's departing
lecture, he was expressly ordered
by the German authorities to avoid any reference to the political situation.
This lecture concerned {\it The continuity in the development
of research in the natural sciences} (September 16, 1940). He mentioned that
this subject was chosen `in connection with a lecture course [...], given by
Professor Zernike on 'Principle, direction and purpose of physics'.' He asked
the following questions (my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
But why should I believe what science teaches today when I know with a fair
degree of certainty that after only a few decades natural scientists will
reject our present views and perhaps teach the opposite of what we now believe
to be true? What value can one place on a science that has repeatedly
contradicted itself in the course of time and continues to do so to this very
day?
\par \endgroup
\noindent
His answer is continuity. Van Rhijn sums the characteristics of this up as
\par \begingroup \leftskip2.5em \rightskip2em \large
First, the present solution of a problem is built on the previous one
and forms the basis of the next solution.
Secondly, the older conception is understood in the newer theory as a special
case, so that for this special case the older conceptions are also considered
admissible according to the newer theory.
\par \endgroup
He then proceeded to illustrate this with a description of the development of
our understanding of the motions of Sun and planets, from Ptolemy via
Copernicus, Brahe, Kepler, Galilei, Newton to Einstein.
\bigskip
\begin{figure}[t]
\sidecaption[t]
\includegraphics[width=0.64\textwidth]{figRM.jpg}
\caption{\normalsize Pieter Johannes van Rhijn as Rector Magnificus
(1939--1940) of the University of Groningen. He wears the chain of office,
decorated with the university's coat of arms, which includes (in abbreviation)
the motto \textit{ Verbum Domini Lucerna Pedibus Nostris} (the
word of the Lord is a light for our feet). From the
Website of the Stichting `Het geslacht van
Rhijn' \cite{PJgenea}, with permission.}
\label{fig:RM}
\end{figure}
As successor of van Rhijn as Rector Magnificus, the Germans appointed a
sympathizer of the Nazi regime by the name of Johannes Marie Neele Kapteyn
(1879--1949) -- no known relative of Jacobus Cornelius. He had been professor
of German language and literature in Groningen since 1924 (later also Frisian).
In his traditional report on the `Lotgevallen' of the University in the past
year, van Rhijn actually did refer -- in spite of the instructions -- to
the political situation. He apparently got away with it. It started as follows
(my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
\noindent
Ladies and Gentlemen,\\
Looking back on the past Academic year, one single coherent series of
events dominates our thoughts: mobilization of our armed forces to protect our
neutrality, war with Germany, capitulation of the Dutch army and the
occupation of the Netherlands by the German Wehrmacht. We have all suffered
bitterly under these battles and are suffering indiscriminately from the
consequences these events have had for our country and our people.
\par \endgroup
And he finished before transferring the office of Rector Magnificus to his
successor with:
\par \begingroup \leftskip2.5em \rightskip2em \large
As Dutchmen, we have a reputation to sustain in the field of science
and the practice of science according to our best traditions is of the utmost
importance for the preservation of our independent national existence, which
we so ardently wish to maintain.
And finally, ladies and gentlemen students, if the courage to continue your
studies fails you because the future is too dark and uncertain, let me remind
you of the motto of our University: an open Bible with the words: {\it Verbum
domini lucerna pedibus nostris} [The word of the Lord is a light for our feet].
It is this Word that has been the support and strength of our leaders in very
difficult times.
{\it Verbum domini}. The Word that leaves us in no doubt about the situation
of this world that has turned away from God, but also opens a view far beyond
this world to the Kingdom of Heaven that has been and is to come, the Word
that calls us to work in the service of that Kingdom.
And what remains, with all the events of the world in which we find ourselves
today, is the classic phrase: {\it ora et labora} [pray and work].
My rectorate has come to an end and I now hand over the office of Rector
Magnificus of this University to my successor, Dr. J.M.N. Kapteyn.
Dear Kapteyn. You are assuming the rectorate of the University in an eminently
difficult period of its existence. I express the wish that you may be granted
the wisdom to lead the University in such a way that it will fulfill its
mission also in these turbulent times. I ask you to come forward and while I
decorate you with the sign of your dignity, I greet you with the eternal wish
of salvation.
\noindent
{\it Salve, Rector Magnifice, iterumque salve!} [Hail, Rector Magnificus, and
hail again!].
\par \endgroup
After van Rhijn's speech mentioned above and the transfer of the rectorate
van Rhijn and all, or at least most, of the
professors stayed away from the usual reception to congratulate
the new Rector and wish him well (see \cite{Berkel05}).
In his
correspondence with Oort, van Rhijn wrote (my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
\noindent
18 November 1940.\\
After the transfer of the rectorate Rein and I spent two weeks in Zuidlaren
and we very much enjoyed that. Beautiful weather and the surrounding
countryside is also very nice. It is easier to forget all the misery outside
than inside the city. Furthermore we were a week at the academic summer
conference in ter Apel. Approximately 200 students and professors were
gathered there. It was a very successful attempt to bring more unity among the
students and also between the students and the professors. It is good that
professors and students do not only meet in the lecture room but also go on a
fox hunt together like in ter Apel.
\par \endgroup
This summer conference, which actually took place in July 1940 was intended
to be a, what in our current
parlance would be, team building exercise. As Rector van Rhijn had spent the
full week there, underlining the
need for good relations and unity among the academic community.
Vossenjacht or fox hunt is a type of scavenger hunt in which participants
must search for
foxes within a certain area.
The foxes are game leaders who walk around dressed in
distinctive clothing, for example of a certain profession. In this case the
foxes might have been the professors and students the hunters.
During the War the work at the Kapteyn Laboratorium came to an almost complete
standstill. Arrival of new plate material from
Harvard ceased, and the staff, mostly male computers, went
into hiding to escape internment as prisoner of war when having served in the
army or to avoid transportation to Germany in the context of the
{\it Arbeitseinsatz} (labor deployment), etc.
In a letter of November 24, 1941, Pieter van Rhijn wrote to Jan Oort in Leiden
(my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
There is so much misery and so much injustice that it is almost impossible to
bear. But we will not lose courage. One can tyrannize us and destroy all that
is valuable in our country. But if we remain courageous, they cannot change
our disposition and thus be victorious over us. The strong language and the
compassion and the trust in God and in the Wilhelmus van Nassouwe can renew our
strength in these days.
\par \endgroup
The 'Wilhelmus' is the Dutch national anthem. It would have been forbidden
even to mention this, but obviously both Oort and van Rhijn trusted each other
enough to do so anyway. Van Rhijn referred to God; he was religious
and a church-goer but Oort was not.
At first teaching at the universities in the Netherlands went on as usual. In
November 1940 the German occupation ruled that Jewish professors had to
resign, which resulted in a well-known, strong protest in Leiden
mentioned above by Rudolph Cleveringa, at that time Dean of the Faculty of Law,
supporting his mentor, Jewish professor also of law, Eduard Meijers.
Students distributed transcripts of this lecture. As a result of
this Leiden University was closed and the possession of such a transcript was
declared illegal. Many students went to other universities, mostly Amsterdam.
Unlike Leiden University, the University of Groningen did not close during
the war.
In a letter of November 7, 1941, van Rhijn asked Jan Oort whether there were
any students in Leiden that would
want to do their exams in Groningen. A week later he wrote that a student van
Tulder has visited him. Johannes Jacobus Maria van Tulder (1917--1992) wanted
to obtain his doctoral (Masters) degree with van Rhijn in Groningen. There is
some more correspondence from the next few weeks on this in the Oort Archives,
in which Oort recommends him for it. However, van Rhijn is worried that van
Tulder sympathized with the Nazi's, since he had been seen by the personnel of
the Laboratorium with a well-known police officer that was a member of the NSB,
the Dutch Nazi party (NSB is the Dutch acronym of the National Socialist
Movement). Van Rhijn wrote to Oort on this because this person (identified
only by his initial K) had betrayed various persons that had been transported
to Germany, and cautioned Oort to be careful in not mentioning this to van
Tulder. Fortunately, van Tulder is eventually cleared when it turned out that
the person showing him to the Laboratorium was another person with the same
surname (van Rhijn now identifying these persons as K$_1$ and K$_2$) living in
the same street, but this K$_2$ definitely was not a 'colborateur' of the
Germans. He had been identified incorrectly by the suspicious personnel of the
Laboratorium.
The {\it Yearbooks} \cite{Yearbooks}
of Groningen University lists van Tulder as having passed
this doctoral
exam in April of 1942. Oort and van Tulder published three papers in 1942
in the {\it Bulletin of the Astronomical Institutes of the Netherlands}, but
the latter dropped out of astronomy and eventually obtained a PhD in sociology
and became a professor in that discipline in Leiden. In Groningen no
other exams in astronomy were taken during the War.
The Germans initially left the university alone. But in the fall of 1940
-- as already alluded to -- Jewish
professors were fired. Contrary to Leiden this was accepted resignedly. Next,
Jewish students were told to leave the university, which again was accepted
without major resistance.
Teaching continued until early 1943, when students were rounded up for
transport to Germany as workforce to keep the German war economy going. The
students protested this time and stayed away from lectures. To turn he tide,
the Germans promised to leave them alone if they would return attending
lectures, but only after signing a loyalty statement promising no further
resistance to the occupying authorities. Only a small fraction (of order 10\%)
signed. On April 18, 1943, van Rhijn wrote to Oort about this:
\par \begingroup \leftskip2.5em \rightskip2em \large
The situation at the universities is uncertain and changes with the day,
sometimes with the hour of the day. In Groningen not even 10 percent of the
students have signed, elsewhere a little more. But in any case the declaration
is a failure. The Germans don't want to withdraw the declaration. The case is
hopelessly deadlocked and nobody can see a way out. It will be the end of
higher education in the Netherlands for the time being.
\par \endgroup
\noindent
And on May 8:
\par \begingroup \leftskip2.5em \rightskip2em \large
A few hundred students were deported to Germany. Only the NSB-ers, the girls
and the good Dutchmen who went into hiding, remain. I have no idea what will
happen to the universities now.
\par \endgroup
The university remained officially open until the liberation in April 1945.
But teaching practically came to a standstill. There were no astronomy students
in the first place, but it should be remembered that for van Rhijn teaching
involved mostly general
astronomy lectures for physics and mathematics majors. And in the last years
of the War he was away being nursed for tuberculosis and formal replacement
was not really necessary.
\bigskip
\begin{figure}[t]
\sidecaption[t]
\includegraphics[width=0.64\textwidth]{figNAC.jpg}
\caption{\normalsize
Pieter van Rhijn lecturing in the open air during the
second Netherlands Astronomers Conference in 1942. From the Website of the
Royal Netherlands Astronomical Society \cite{KNAC}.}
\label{fig:NAC}
\end{figure}
In the first years of the War there were still some national activities in
astronomy, such as one-day meetings of the society of professional
astronomers, the Netherlands Astronomers Club NAC. These were
supplemented by a new kind of meeting, in the beginning independent from the
NAC. The first Netherlands Astronomers Conference had been organized
in Doorn, a village not too far from Utrecht, in June
1941 by Marcel Minnaert, see Fig.~\ref{fig:NAC1941}.
These conferences were meant to bring together the Dutch astronomical
community, including students, for longer than the single half-day of meetings
of the Netherlands Astronomers Club (see my more
extensive discussion in chapter 9.3 of \cite{JHObiog} or ch. 7.4 in
\cite{JHOEng}).\footnote{\normalsize On the occasion of its centennial the predicate Royal
has been awarded. Contrary to my optimistic and hopeful expectation in
\cite{JHObiog} the change in the name from a 'Club' to the more professional
'Society' unfortunately did not happen, at least not in the Dutch name.
However the abbreviation NAC was changed into KNA, at
least removing the confusion with the Astronomers Conference.\label{note:KNA}}
A second meeting
in the same place was organized in July 1942. Adriaan Blaauw attended both
meetings, Lukas Plaut only the first (see Fig.~\ref{fig:NAC1941}),
and Pieter van Rhijn only
the second one (Fig.~\ref{fig:NAC}). Some more information and photographs
are available on the Website of the NAC \cite{KNAC}. After 1942 even such
conferences ceased to be possible,
but that was picked up again after the War and they are
still being organized annually, but now under the auspices of the KNA
(see footnote~\ref{note:KNA}).
During the war, meetings of the Club were being held every now and then,
sometimes preceded by special colloquia such as the famous one in 1944
where the possibility of using the 21-cm line of neutral hydrogen in astronomy
was announced by Henk van de Hulst.
\bigskip
During the War, in November of 1943 according to his letter to his
sister-in-law in 1945 referred to in section~\ref{sect:Early},
van Rhijn started suffering from tuberculosis for which he was
hospitalized for a long time in a sanatorium. Well after the War, in 1948 or
1949, he returned
to his job (part-time before that), but his health remained fragile, and this
probably played a role throughout
the rest of his career. In relation to van Rhijn's attitude after the War to
start radio astronomy in the Netherlands, Adriaan Blaauw has said in an
unpublished interview with Jet Katgert-Merkelijn (conducted in preparation
for what came to be my Oort biography \cite{JHObiog}; my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
So the last year of the War he [van Rhijn] was in a sanatorium. He was in
Hellendoorn and maybe in another place too. But when
he was admitted to that sanatorium, he forced himself to continue to provide some
leadership at the institute. I think
he actually could have retired quite well, because he had had a severe form of
tuberculosis after all.
But I always felt that -- because he had a very young family, he
married very late, so those children were younger even than the children of
Oort. In any case, that he forced himself to continue anyway, to put it
crudely, to keep his salary for that family. Well, so it was a man who really
couldn't handle much more than he did at the time anyway. And starting such an
adventure in radio astronomy, which was something very new, I think his mental
and physical condition prevented him from saying, 'I'm going to participate
in that enthusiastically.'
\par \endgroup
Hellendoorn was one of the larger sanatoria in the Netherlands; it now is a
nursing home for persons suffering from dementia and comparable ailments.
It is the name of a nearby village 90 km directly to the south from Groningen,
15 km to the west of the city of Almelo, in a forest area with conditions
favorable for treatment of tbc. Van Rhijn was nursed there from December 1943
to December 1944, when he was transported back to Groningen (see below).
\bigskip
Through Gert Jan van Rhijn, keeper o the van Rhijn genealogy,
I obtained some more information on this period from
van Rhijn's daughter. She mentioned that in the first months he was
much too ill to read, just was lying in bed in a small room. This
must have been very difficult. His wife was allowed to visit him
twice per month, but later during the War this had become
much too dangerous. He did have a few other people visiting,
like his brother Maarten, but this was exceptional. When this
became possible he wrote letters to his wife, about four per week on size A5
paper. By the time he was able to read he asked his wife repeatedly for two
books: {\it Geloof en openbaring} (Faith and
revelation) by G.J. Heering, who was a remonstrant clergyman, and
{\it Physik der Sternatmosph\"aren mit besonderer
Ber\"ucksichtigung der Sonne} by A. Uns\"old. These could not
be sent for danger of getting lost, so only a visitor could
deliver them and visiting was difficult and rare. It shows
that he divided his interest between religion and science.
The Oort Archives contain a number of letters from van Rhijn to Oort during
the first's treatment, that contain some more information on
his illness (Oort Archives, website accompanying \cite{JHObiog}, Nr. 149). On
March 18, 1944 (last letter before this is dated July 1943 before he fell ill)
he wrote from the sanatorium to Hulshorst, where Oort and his family were in
hiding (see \cite{JHObiog} or \cite{JHOEng} for more background to this;
my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
I am also doing well, according to the circumstances. My temperature has been
normal for weeks, the coughing is much better, and the sick spot in my lungs
has shrunk a bit. I can write letters and read very
simple reading material (Ivans for example) only a few hours a day.
But that is progress already.
Some months ago I was too tired to write or read anything.
\par \endgroup
Van Rhijn refers here to Jakob van Schevichaven (1866--1935), author of
detective stories, acclaimed as the first professional writer in this genre in
the Netherlands. His pseudonym, Ivans, was derived from
abbreviating his name to J. van S.
And on 18 October 1944 (same source; my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
Our expectations have since been disappointed and we have to be patient. I
would very much appreciate it if you could visit me sometime. But there can be
no question of that for the time being. Of course Rein cannot come either. It
is quiet lying in bed day after day. The sanatarium has no shortage yet. The
electricity is produced here. There still appears to be some supply of fuel.
Recently my blood was tested, the result was positive. I also feel better than
I did in the summer and less tired. Sometimes I even have the urge to get up!
The last few weeks I have been reading through Blaauw's dissertation. Every
now and then patients who lie here ask me astronomical questions, such as: How
do you weigh the Earth?
\par \endgroup
On December 18, 1944 van Rhijn wrote from one of the three Groningen
hospitals, called the Diaconessenhuis (Deaconess Hospital). In addition to
this hospital for Protestants there was a Roman-Catholic and an academic
hospital in Groningen. Van Rhijn would choose for a hospital
for people of his religion if at all possible. Hellendoorn on
the other hand was a `volkssanatorium', `volks' (meaning people's)
indicating open to all
religions. Part of the letter reads (same source again; my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
\noindent
Dear Jan,\\
Thank you for your letter of 3 December, which I received in Hellendoorn. I am
now back in Groningen in the Diaconessenhuis. The matter is that in the last
four weeks V2 flying bombs have been launched near the sanatorium. Several of
these failed; one of the failed bombs landed a few kilometers from the
sanatorium and killed 18 workers who were working in the fields. It became too
unsafe in Hellendoorn. With great difficulty Rein rented a car that
transported me to Groningen. It gives me peace that I am here now. It is a
wonderful feeling to be so close to Rein and the children. It was difficult in
Hellendoorn, especially the last few months when no more visitors came.., [...]
My illness is healing. At the end of November I was examined with favorable
results. The blood is completely normal again and the sick spots in my lungs
have become smaller. The doctor who treats me here is also satisfied. When I
traveled in the car from Hellendoorn to here I had to get dressed for the first
time in 1944. You can understand my dismay when it turned out that my top
pants could not be closed at all anymore! There was a gap of 4 to 5 centimeters!
You won't recognize me when you see me again. I have become a fat puffy guy
with a double chin. This is a punishment from heaven because I used to look
down on fat people. It gives something smug about the whole person. And now I
am becoming such a fat person myself.
\par \endgroup
Traveling by car was rather dangerous as very often they were fired
upon from Allied airplanes, but
van Rhijn fortunately made it. It was only during the last few months
that his children were allowed
to visit him, but -- as his daughter recollects --
sitting in the window sill with the windows opened.
\bigskip
During the last two or three years of the War in effect only Blaauw and
observer M\"uller (and whenever possible Plaut)
kept the Kapteyn Laboratorium going. Blaauw
spent all his time on his thesis subject, but very little other research was
being done. In 1946 van Rhijn wrote a short article summarizing the
astronomical work performed in the Netherlands during the War \cite{PJvR1946}.
He mentioned his own work on using interstellar H
and K line intensities (from singly ionized calcium) as a distance criterion
and derived the density and luminosity functions of O and early B stars.
This was published after the War in the {\it Groningen Publications} and I will
return to it. Then there was Blaauw's work already noted and a paragraph
on Lucas Plaut:
\par \begingroup \leftskip2.5em \rightskip2em \large
Dr. L. Plaut has worked at the Kapteyn Laboratorium since 1940. He has been
imprisoned for some time in a camp in Germany but returned safely after the
liberation. He has investigated a field of 10$^{\circ}$ x 10$^{\circ}$\ for variable
stars with the blink comparator of the Kapteyn Laboratorium. The plates have
been taken by Dr. van Gent with the Franklin-Adams camera at Johannesburg.
\par \endgroup
So, Plaut was regarded as working at the Laboratorium since his arrival in
Groningen.
\bigskip
In his autobiographical article in 2004 \cite{Blaauw2004}, Adriaan Blaauw wrote:
\par \begingroup \leftskip2.5em \rightskip2em \large
A curious incident occurred toward the end of the war. One day in February
1945, my doorbell rang and two lads of high school age, amateur astronomers,
asked to see me. With all the bad things the War brought us, the forced
blackouts at least provided amateur astronomers with a unique opportunity to
watch the sky. These lads now also wanted to observe the sky and speak to a real
astronomer, and of course they were welcome to enter. The lads were Maarten
Schmidt and Jan Borgman; both later had excellent careers in astronomy. As one
of them reported years later, my wife met them with the words, 'You are lucky,
he has just been released from prison!' Why had I been in prison? I had been
accused of listening to and spreading news broadcast from London -- something
strictly forbidden. Luckily, the War was approaching its conclusion, and
although nobody could tell how long it would continue, some lower-ranking
German officials were ripe for some gentle bribery. My relatives and friends
collected a sufficiently persuasive sum and succeeded in getting the prison
doors opened for me.
Two things I recall of that memorable release on February 8,
1945 -- it was my little son's second birthday, and I could at last satisfy my
hunger, as far as circumstances allowed.
\par \endgroup
\section{Recovering from the War}\label{sect:recover}
The Second World War ended in the Netherlands on May 5, 1945. Soon after that
astronomers started looking forward, and in renewing contacts they did relay
to each other their experiences and circumstances.
The following comes from the
Oort Archives. Van Rhijn wrote Oort immediately on May, 5 (Oort Archives,
website accompanying \cite{JHObiog}, Nr. 149; my translation).
For clarity I remind the reader, that Joos is his elder sister, who lived with
her four children in Groningen (see section`\ref{sect:Early}).
\par \begingroup \leftskip2.5em \rightskip2em \large
Last night the news that the German army in
[the Netherlands] has capitulated; it is almost unbelievable
that for us the mistreatment by that execrable people has come to an end. There
was quite some fighting here from April 13 to 16. Relatively few casualties
(about 100). But many houses burned, [...] Rein and the children and Joos and
her family came out of the fight unharmed; the houses only have glass damage.
[...] Blaauw and his family got off well. Plaut has miraculously returned from
Germany on foot, where he was imprisoned for a few weeks; he only had to walk.\\
\noindent
[long list of family and friends that did not survive]
I am doing well, the diet has been now so good that I have to eat a little less
to keep from getting too heavy! At the end of March there were no more bacilli
in my sputum, indicating recovery.
The laboratory has not suffered any damage, [...].
There was hard fighting in the vicinity of the Diaconessenhuis. The Krauts
shot from the garden of the house. It was dangerous for us though. We were in
the basement of the building.
\par \endgroup
\noindent
Adriaan Blaauw wrote Oort on May 15 (Oort Archives, website accompanying
\cite{JHObiog}, Nr. 161; my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
First, that Plaut did not fall prey to persecution, but resumed his work at
the Laboratory. It was a close call, however: [details the hiding and capture
of Plaut and his return and his own short stay in prison without mentioning
why he was arrested].
We went through very nervous days during the fighting around Groningen. Much
has been destroyed in the city, and with more unfavorable winds the Laboratory
would also have been affected by the fires. Now we got away with a few bullet
holes. The staff that had been in hiding is all but one present again, so that
the programs have been resumed. I myself am busy preparing my article on the
Scorpio-Centaurus cluster for the printer. Unfortunately Hoitsema's entire
printing facility has burned down, so printing is suspended waiting for attempts
to get help. Nevertheless, I hope that the work can be started within a few
days. Also burned in the fire are the 150 pages of the Special Areas
and the two publications of Prof. van Rhijn, completely corrected.
\par \endgroup
\noindent
And Lukas Plaut on May, 19 (Oort Archives, website accompanying \cite{JHObiog},
Nr. 163; my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
\noindent
Dear Professor Oort,\\
Now that the postal connection seems to be getting a little better I would
like to write to you in a little more detail. Since last year
March it has not been
possible for me to do anything for astronomy. I had to go to Havelte to
work on the airfield. In September I was discharged and went home again. Many
were rounded up at that time and taken to Westerbork, moreover, there was
compulsory digging for all men, so I preferred to go into hiding. However, in
February I was arrested and after a few days taken to Germany. Compared to
many I had a good time there, mainly because it was not
discovered that my identity card was false. When the Allied armies arrived, I was released and
worked on a farm for a few weeks before returning home. That was three weeks
ago now. My wife and the children have fortunately come through the last few
months well. The house did not suffer from the fighting either.
Since September my wife has received financial support from an illegal fund.
The fund has now been absorbed into the `Nationaal Steunfonds' [National
Support Fund], of which I do not
dare knock at the door, as so many need to be supported who have lost
everything. Now I would like to ask you about the possibilities of getting a
job as an assistant or another scholarship. If at all possible, I would like
to continue working at an observatory. I do not believe there are any other
possibilities for me.
\par \endgroup
The `illegal' fund was in fact illegal in the sense that the underground
resistance movement had arranged this to support persons that had no income.
The National Support Fund had been set up by the Dutch government upon its
return to the Netherlands.
\bigskip
When the War ended in 1945 slowly business was being picked up again.
In 1956 van Rhijn would turn seventy, so he still had a bit more than a
decade to go. But he had not yet recovered from
tuberculosis and did not fully return on his job. The other personnel did
get back to the Laboratorium. For many years there were in addition to the
professor and assistant four persons on the staff, of which one
was amanuensis (to copy letters or manuscripts from drafts), one observer and
two computers. Three of these were already at the Laboratorium before the
partial closure and returned. Already before the liberation Oort had offered
Adriaan Blaauw a position in Leiden when things would be back to normal.
Blaauw accepted and started his appointment in October 1945. He finished his
thesis {\it A study of the Scorpio-Centaurus cluster} and defended it in
Groningen in July 1946.
What about Lukas Plaut? Henkes \cite{Henkes} noted:
\par \begingroup \leftskip2.5em \rightskip2em \large
The much-praised 'modesty', which Lukas Plaut demonstrated immediately after
his flight to the Netherlands, had not diminished in the years of his
persecution. Making himself invisible had become second nature to him,
reinforced by the advice to the few Jews who had survived not to draw
attention, 'to show modesty, and to be grateful'.
\par \endgroup
Lukas Plaut had asked Oort in his letter in May for support with finding some
kind of employment. Oort had taken this op by recommending him to van Rhijn as
assistant, now that the position had been vacated by Blaauw.
In the Oort Archives there is
an envelope on the back of which Oort has made a note for himself that he had
replied on June 14, 1945 to the personal letter from van Rhijn of May 5 (see
above) and after some personal matters he wrote (Oort Archives, website
accompanying \cite{JHObiog}, Nr. 149):
\par \begingroup \leftskip2.5em \rightskip2em \large
As quickly as possible proposal to appoint Blaauw. Appointment will take
some
time and he can stay in Groningen for a while if necessary. Need him to help
build up the new spirit.
What does he think of Plaut as his successor? Recommended him highly. Also
mentioned van de Hulst, Objection: mainly theoretical.
\par \endgroup
Oort had taken the initiative to see if Plaut could be appointed as assistant
in Groningen. Van Rhijn will
have hesitated, since Plaut worked on variable stars, and with himself still
suffering from tuberculosis he would have preferred someone more suited to
take the supervision of work related to the {\it Plan
of Selected Areas} upon himself.
Variable stars had been a major line of research in Leiden under Hertzsprung.
The latter had stayed on as director after reaching the age of seventy in 1943
and now was moving back to Denmark. It would seem reasonable to expect that
Pieter Theodorus Oosterhoff (1904--1978) would continue this (see
Fig.~\ref{fig:NAC1952}). The formal appointment of Oort as professor at Leiden
University and director of the Sterrewacht was made on November 8, 1945. And
indeed on the same date, Pieter Oosterhoff was appointed lecturer and
adjunct-director (Oosterhoff would be appointed extra-ordinary (honorary)
professor in 1948; for an obituary see \cite{JHO1978} ).
So, why would van Rhijn use his single staff position for
someone working in a field that was already a major line of research in Leiden?
Furthermore, the assistant had for many years been a person supervising work
on the {\it Plan of Selected Areas} and the {\it
Spektral-Durchmusterung}, preparing at the same time a PhD theses and then
vacating the position. No wonder van Rhijn hesitated to propose Plaut for the
assistantship.
This left Plaut uncertain about his future and his wife decided to write Oort
without telling her husband. She did so on August 1, because, she explained,
she no longer could handle the situation. She did not directly ask
him to approach van Rhijn about the Groningen assistantship, but asked for
support in a more indirect manner (Oort Archives, website accompanying
\cite{JHObiog}, Nr. 163; my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
My husband is seriously ill (jaundice). This is otherwise a nasty disease, but
harmless. However, my husband is mentally totally exhausted and no longer
believes that he can get better. For a long time he has been pondering about
his future, but saw no light. Now he is so exhausted physically and mentally
that he is no longer able to think about it. I would like to ask you, isn't
there anyone in the whole astronomical world who can help this man?
\par \endgroup
She proposed that Oort may write to Plaut to cheer him up. At the top of the
letter Oort has scribbled that he had written a note to Plaut on August 15
and had 'tried to give him some
encouragement'. And also that he hoped Plaut and Blaauw could visit Leiden after
his trip to England. However, it took a while for Oort could make the trip.
Jan Oort was General Secretary of the International Astronomical Union. The
president Arthur Eddington had died in 1944, and through vice-president Walter
Sydney Adams (1876--1956) of Mount Wilson Observatory it had been arranged that
Harold Spencer Jones (1890--1960), director of the Royal Greenwich Observatory
and Astronomer Royal, would be interim president. Oort was planning to fly to
England to discuss the course of action with Spencer Jones. The trip was
arranged by Gerrit Pieter (later Gerard Peter) Kuiper (1905–1973), a graduate
from Leiden, but by then naturalized American, who had come to Europe as part
of the ALSOS Mission, which investigated and collected intelligence on the
German atomic, biological and chemical weapons research. He has been reported
to have staged a daring rescue mission into eastern parts of Germany, during
which he took famous physicist Max Planck into Allied care hours before the
Soviets arrived. Due to visa and other formalities Oort's trip kept being
postponed but eventually took place from September 23 to October 7. Obviously
Oort had many things on his mind, busy as he was with putting the Sterrewacht
and the IAU back on track.
In the meantime Lukas Plaut had writen
to Oort again on August 30 (Oort Archives, website
accompanying \cite{JHObiog}, Nr. 163; my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
Yesterday morning I had a meeting with Prof. van Rhijn about the possibility
of becoming an assistant here. Prof. van Rhijn, however, would have liked to
have some certainty about my future after the assistantship ended and only
wanted to take further steps with the trustees or the Board of Rehabilitation
when, for example, I could show that there was a possibility for me to go to
America. So I wrote to my brother in Los Angeles and asked him to take care of
the official papers again for obtaining a visa after a few years. It is
unfortunate that the decision has been delayed, so I am still in limbo for
some time.
\par \endgroup
So, van Rhijn was still hesitant to allocate his assistant position to Plaut
and wanted some assurance that the position could become available again after
a few years and that Plaut would have a future when it had ended.
There is no further
mention of this situation in the Oort Archives. In the end van Rhijn did
arrange for Plaut to be appointed assistant, filling the vacancy created by
Blaauw's departure.
The appointment started on October 1, 1945, the date on which Blaauw's had
ended, so van Rhijn's hesitations would not have lasted long after August 30.
Plaut had to take over part of van Rhijn's teaching because of the latter's
illness and for this he was given (by Ministerial decree as required) a
teaching assignment in 1946 in prodaedeutic (first year introductory) astronomy.
He eventually remained on the staff until his retirement. Later Plaut's
presence in Groningen played a major role in defining the {\it
Palomar-Groningen Survey}, which concerned mapping of a particular type of
stars, RR Lyrae variables, combining the study of variable stars, on which
Plaut was an expert, with Galactic structure, the research field of van Rhijn
and the Laboratorium. But that was only in 1953 and cannot
possibly have played a role in Plaut's appointment in 1945.
\section{Staff and workers at the Laboratorium}\label{sect:staff}
At this point it may be appropriate to spend some time on the development of the
actual manpower (no women yet) of the Laboratorium over the years. Table 1
shows the personnel between 1898 and 1960.
This is compiled from the lists published annually in the
{\it Yearbook} \cite{Yearbooks}
of the university of Groningen and since 1946 partly in a supplement to
this, the {\it University Guide} \cite{UGuide}.
From 1900 onward Kapteyn had had an assistant. The first appointment
had been done at the beginning of the academic year
1899-1900, in the 'Lotgevallen' in the {\it Yearbook 1898-1899}
\cite{Yearbooks} it is mentioned (my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
An increase in personnel is to be recorded at the `Astronomische Laboratorium',
where for the first time an assistant was appointed, namely Mr. L.S. Veenstra.
\par \endgroup
\begin{table*}[t]
\centering
\begin{threeparttable}[b]
\caption{\normalsize Table of personnel at the Kapteyn Laboratorium from 1898
to 1960 according to the {\it Yearbook} \cite{Yearbooks}
(Jaarboek der Rijks-Universiteit
Groningen) and since 1946 in part to the {\it University Guide}
\cite{UGuide} (Groninger
Universiteitsgids). Years in which the personnel is the same as in
the preceding one, haven been omitted.}\label{table:first}
\begin{tabular}{rlll}
\toprule
Year\tnote{a} \ \ \ & \ \ \ Professor & \ \ \ Assistant & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Other personnel\tnote{b} \\
\midrule
1898\ \ \ & \ J.C. Kapteyn\tnote{c} & \ \ \ \ \ \ \ \ -- & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -- \\
1900\ \ \ & \ J.C. Kapteyn & L.S. Veenstra & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -- \\
1901\ \ \ & \ J.C. Kapteyn & W. de Sitter\tnote{d} & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -- \\
1904\ \ \ & \ J.C. Kapteyn & W. de Sitter & T.W. de Vries\\
1908\ \ \ & \ J.C. Kapteyn & H.A. Weersma\tnote{e} & T.W. de Vries\\
1910\ \ \ & \ J.C. Kapteyn & H.A. Weersma & T.W. de Vries, J. Jansen \\
1913\ \ \ & \ J.C. Kapteyn & F. Zernike & T.W. de Vries, J. Jansen\\
1915\ \ \ & \ J.C. Kapteyn & P.J. van Rhijn\tnote{f} & T.W. de Vries, J. Jansen \\
1918\ \ \ & \ J.C. Kapteyn & P.J. van Rhijn\tnote{g} & T.W. de Vries, J. Jansen \\
1919\ \ \ & \ J.C. Kapteyn & P.J. van Rhijn & T.W. de Vries, G.H. M\"uller, J.M. de Zoute, \\
&&& D. van der Laan, J. Jansen\\
1921\ \ \ & \ P.J. van Rhijn\tnote{i} & J.H. Oort & T.W. de Vries, G.H. M\"uller, J.M. de Zoute, \\
&&& D. van der Laan\\
1922\ \ \ & \ P.J. van Rhijn\ \ \ \ \ & P. van de Kamp\ \ & T.W. de Vries, G.H. M\"uller, J.M. de Zoute,\\
&&& D. van der Laan\\
1923\ \ \ & \ P.J. van Rhijn & W.J. Klein Wassink & T.W. de Vries, G.H. M\"uller, J.M. de Zoute,\\
&&& D. van der Laan, W. Ebels \\
1927\ \ \ & \ P.J. van Rhijn & W.J. Klein Wassink & W. Ebels, G.H. M\"uller, J.M. de Zoute,\\
&&& D. van der Laan\\
\bottomrule
\end{tabular}
\begin{tablenotes}
\normalsize
\item[a]{Situation at the start of the academic year.}
\item[b]{Functions were described as amanuensis, observator, computer or clerk.}
\item[c]{Professor of astronomy, probability theory and mechanics since 1878.}
\item[d]{Also privaat docent (additional appointment for teaching) in astronomical perturbation theory and related subjects.}
\item[e]{Also privaat docent in orbital determination and perturbation theory of celestial bodies.}
\item[f]{Also privaat docent in theoretical and stellar astronomy, later astronomy, probability theory.}
\item[g]{In addition between 1918 and 1922 A.E. Kreiken was listed as assistant b.b. ('without burden to the country's treasury').}
\item[i]{Professor of astronomy and probability theory.}
\end{tablenotes}
\end{threeparttable}
\end{table*}
\addtocounter{table}{-1}
\begin{table*}[t]
\centering
\begin{threeparttable}[b]
\caption{\normalsize {\it (continued from previous page)} The last line of the first part has been repeated.}
\begin{tabular}{rlll}
\toprule
Year \ \ \ & \ \ \ Professor & \ \ \ Assistant\tnote{a} & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Other personnel \\
\midrule
1927\ \ \ & P.J. van Rhijn & W.J. Klein Wassink & W. Ebels, G.H. M\"uller, J.M. de Zoute,\\
&&& D. van der Laan\\
1928\ \ \ & P.J. van Rhijn & B.J. Bok & W. Ebels, G.H. M\"uller, J.M. de Zoute,\\
&&& D. van der Laan, H.J. Smith \\
1929\ \ \ & P.J. van Rhijn & J.J. Raimond & W. Ebels, G.H. M\"uller, J.M. de Zoute, \\
&&& D. van der Laan, H.J. Smith\\
1934\ \ \ & P.J. van Rhijn & P.P. Bruna & W. Ebels, G.H. M\"uller, J.M. de Zoute, \\
&&& D. van der Laan, H.J. Smith\\
1939\ \ \ & P.J. van Rhijn & A. Blaauw & W. Ebels, G.H. M\"uller, D. van der Laan, H.J. Smith\\
1941\ \ \ & P.J. van Rhijn & A. Blaauw & W. Ebels, G.H. M\"uller, J.B van Wijk, H.J. Smith\\
1945\ \ \ & P.J. van Rhijn & A. Blaauw & D. Huisman, G.H. M\"uller, W. Ebels, J.B. van Wijk\\
1946\ \ \ & P.J. van Rhijn & L. Plaut\tnote{b} & D. Huisman, G.H. M\"uller, W. Ebels, J.B. van Wijk\\
1956\ \ \ & P.J. van Rhijn & L. Plaut, J. Borgman & D. Huisman, G.H. M\"uller, W. Ebels, J.B. van Wijk \\
1957\ \ \ & A. Blaauw\tnote{c} & L. Plaut, J. Borgman & D. Huisman, W. Ebels, J.B. van Wijk, H.R. Deen, \\
&&& E.R.R. Timmerman \\
1958\ \ \ & A. Blaauw & L. Plaut, J. Borgman & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ --\tnote{d} \\
&& H. van Woerden & \\
1960\ \ \ & A. Blaauw & L. Plaut, J. Borgman, & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -- \\
&& H. van Woerden, & \\
&& A.B. Muller & \\
\bottomrule
\end{tabular}
\begin{tablenotes}
\item[a]{In 1947 this changed into conservator, and from 1948 onward scientific staff member.}
\item[b] Also a teaching assignment between 1946-1949 in propaedeutic (first year introductory) astronomy.
\item[c] Professor of astronomy.
\item[d] After 1957 no longer available in \textit{Universiteitsgids}.
\end{tablenotes}
\end{threeparttable}
\end{table*}
But in the same {\it Yearbook} \cite{Yearbooks}
Veenstra is not listed as the personnel of the
Laboratorium (Table 1 is based on these listings and presents this such that
for example
the line 1900 refers to the situation at the start of the academic year
1900-1901). Schelto Leonard Veenstra (1874--1951) is listed in the 1898-1899
{\it Yearbook}
as a fifth-year student in mathematics and natural sciences, but I
have not found him in other {\it Yearbooks}
around that time in lists of students or
persons that passed an exam. So, who was this Mr. Veenstra?
The information I have been able to uncover in municipal
archives is the following. In March 1901 Veenstra (26 years of age) married in
Groningen 19-year old Doetje Heidstra and on the certificate he is described
as having the position observator. They seem to have moved to Utrecht in 1908.
The `Canon Autisme Nederland', an organization concerning persons suffering from
autism, notes in the description of a {\it Cannon on the history of autism},
that in 1910 (my translation)
\par \begingroup \leftskip2.5em \rightskip2em \large
by the Ministry [of Law] an Inspector of Probation was appointed to supervise
the government support of probation. According to this organization the
first inspector was `the energetic
Schelto Leonard Veenstra (1874–1951), former captain of the Salvation Army,
characterized [...] as `deeply religious, broad-minded'.
\par \endgroup
Veenstra probably terminated his studies and academia. It is likely he never
contemplated a career in astronomy in the first place, but took the position
for a short period to earn a living.
Willem de Sitter came next (see \cite{Guich} for an informative biography).
He had passed his Doctorandus (Masters) degree in 1900, after having his
studies interrupted 1897–1899 for a period of observational experience with
David Gill at the Cape Observatory. While he was away Kapteyn had arranged
the assistantship for him when he returned, but on a temporary basis
had appointed
Veenstra on it. In 1901 de Sitter
defended a thesis on the orbits of Jupiter's Galilean satellites -- a
subject that remained his principal interest throughout his career --,
and his appointment continued until he moved to Leiden in 1908. Like some of
his successors, de Sitter also held an appointment as privaatdocent
to give lectures. His assignment (and that of his successors) is given
in footnotes to Table 1. When de Sitter left for Leiden, Herman Weersma took
over, but he left astronomy after a few years. Frits Zernike came next, but he
left to do a PhD in Amsterdam in physics.
Van Rhijn was given the position in 1915 (like de Sitter
and Weersma, but unlike Zernike, this included also
one as privaat docent, but only after he had defended his PhD thesis).
The position of assistant apparently could be filled either as a temporary
appointment of a student working on a PhD thesis, or on a longer or indefinite
basis by a person holding a PhD. De Sitter and Weersma
resigned rather than having come to the end of some term.
So from Oort up to and including Blaauw, who resigned in 1945 to accept an
appointment in Leiden, van Rhijn used the assistant position for a person,
who had obtained a Doctorandus (Master) degree and was now preparing a PhD
thesis.
Starting with Kapteyn's share in the {\it Cape Photograph Durchmusterung} in
the 1880s the measurement of photographic plates and the associated reductions
was performed by persons employed from grants obtained especially for that
purpose. Kapteyn started with T.W. de Vries paid from grants from a number of
funds, and more personnel followed. This soft money did not provide any
long-term security to the employees and Kapteyn over many years sought ways to
provide that to de Vries (see \cite{JHObiog}). As table 1 shows this finally
succeeded only in 1904. When the {\it Plan of Selected Areas} had
been started, George Hale had adopted it as the primary project for Mount
Wilson and his 60-inch telescope, and Kapteyn had been appointed Research
Associate to the Carnegie Institution of Washington. But that was not all; the
Carnegie Institution also provided funding to hire personnel in Groningen
to measure up the 60-inch plates. The situation did change in
1919, when the University of Groningen appointed three more computers on the
university staff in addition to the amanuensis de Vries and the clerk Janssen.
This was still the situation at the time van Rhijn took over from Kapteyn.
So at the start of his term as director van Rhijn had one assistant and four
positions for technical support. Various astronomers were
subsequently employed on the
assistant position. Jan Oort filled it for a short period.
Then followed Peter van de Kamp, Willem Klein Wassink, Bart Bok,
Jean-Jacques Raimond, Petrus Bruna and Adriaan Blaauw. Except Bruna all
these assistants finished and defended a PhD thesis under van Rhijn
(see Fig.~\ref{fig:vRhijn3}).
Not long after van Rhijn took office a fifth position was added to the
Laboratorium, but at the end of the thirties during the a Great Depression
this disappeared again, not to come back before van Rhijn's retirement. So,
the net result was that as far positions on the university pay-roll are
concerned, the number of staff had remained the same as it was at the start
of van Rhijn's directorship.
\bigskip
So, how does this compare to the university as a whole? For this I refer to
Table~\ref{table:Univstaff},
which covers the period 1920 to 1960 in steps of a decade, so roughly
encompassing van Rhijn's tenure of the directorship of the Kapteyn
Laboratorium. The growth of the university is understandably modest
during the Great Depression and the
Second World War, but picks up after this,
especially during the 1950s.
The number of institutes is somewhat more than half the number of professors.
The column professors, however, includes extraordinary professors
(appointments on an individual basis in addition to the normal contingent),
honorary
professors and so-called ecclesiastical professors (who in the Theological
Faculty teach particular subjects on behalf of churches or religious
organization), so ignoring these it follows that as a rule the majority of
ordinary professors were also directors of an institute.
The number of professors
increased by a factor 2.4 and of institutes by 2.1, so this situation did not
change much. Not surprisingly, astronomy remained having one institute with
one professor. Roughly speaking one professor out of four was supported by a
lecturer, a senior scientist involved in research and teaching. Although the
total number of lecturers
increased, it remained relatively constant at only three or four
for the natural sciences and -- maybe a not unreasonably --
none for the Kapteyn
Laboratorium. The growth university-wide was in assistants, as we have seen
junior scientists, in Kapteyn's day as long term employment (de Sitter to van
Rhijn), but later as temporary appointments and usually to prepare a PhD thesis
(under van Rhijn from Oort to Blaauw). From the end of the War to the end of
van Rhijn's term this was a junior permanent position occupied by Plaut. There
were no new positions for junior staff in astronomy,
while this increased by a factor three
in the university and Faculty of Sciences. The increase to four assistants
occurred after Blaauw took office and was part of the conditions he bargained
for as bonus for his acceptance of the directorate. Whereas de Sitter, Weersma
and van Rhijn were appointed privaat docent in addition, this was not the case
of the 'graduate students' under van Rhijn. Plaut was given a teaching
assignment, but only temporarily during van Rhijn's illness. While the number
of such appointments increased, none was available structurally to astronomy.
In this respect van Rhijn was worse off than his predecessor and contemporaries
\begin{table*}[t]
\caption{\normalsize University personnel employed at the University of Groningen}
\centering
\begin{threeparttable}[b]
\begin{tabular}{r|rrrr|rrrr|rrrr|rrrr|rrrr|rrrr}
\toprule
Year\tnote{a}\ \ & \multicolumn{4}{|c|}{Institutes\tnote{b}}
& \multicolumn{4}{|c|}{Professors} & \multicolumn{4}{|c|}{Lecturers} & \multicolumn{4}{|c|}{p.d./l.o.\tnote{c}} & \multicolumn{4}{|c|}{Assistants, etc.} & \multicolumn{4}{|c}{Further personnel} \\
& u & n & k & \ & u & n & k &\ & u & n & k &\ & u & n & k &\ & u & n & k &\ & u & n & k &\ \\
\hline
1920\ \ & \ \ 25 & \ \ \ \ 9 & \ \ \ \ 1 &\ & \ \ 46 & \ \ \ \ 11 & \ \ \ \ 1 &\ & \ \ 9 & \ \ \ \ 2 & \ \ \ \ 0&\ & \ \ 7 & \ \ \ \ 4 & \ \ \ \ 1 &\ & \ \ \ 52 & \ \ \ \ 15 & \ \ \ \ 1 &\ & \ \ \ 83 & \ \ \ \ 35 & \ \ \ \ 5&\ \\
1930\ \ & 31 & 11 & 1 &\ & 54 & 12 & 1 &\ & 9 & 1 & 0 &\ & 13 & 5 & 0 &\ & 81 & 32 & 1 &\ & 101 & 45 & 5 &\ \\
1940\ \ & 36 & 15 & 1 &\ & 57 & 14 & 1 &\ & 13 & 3 & 0 &\ & 21 & 9 & 0 &\ & 116 & 39 & 1 &\ & 107 & 52 & 4 &\ \\
1950\ \ & 53 & 14 & 1 &\ & 76 & 18 & 1 &\ & 18 & 4 & 0 &\ & 35 & 12 & 0 &\ & 171 & 47 & 1 &\ & 135 & 60 & 4 &\ \\
1960\ \ & 60 & 16 & 1 &\ & 97 & 25 & 1 &\ & 28 & 4 & 0 &\ & 21 & 8 & 0 &\ & 172 & 41 & 4 &\ & --\tnote{d} & -- & -- &\ \\
\bottomrule
\end{tabular}
\begin{tablenotes}
\item{a} Situation at the start of the academic year.
\item{b} Here and in the further columns: u = University, n = Faculty of Natural Science, k = Kapteyn Laboratorium.
\item{c} p.d. = privaat docent (additional appointment for teaching), l.o. = teaching assignment.
\item{d} No longer listed in the Universiteitsgids.
\end{tablenotes}
\end{threeparttable}
\label{table:Univstaff}
\end{table*}
In terms of scientific staff astronomy did not profit at all from the
national and local growth of university budgets,
student numbers and staff during van Rhijn's
directorship. In further personnel the situation also showed a negative trend;
while there was a significant increase (by a factor 1.6 university-wide and 1.7
in the Faculty) for astronomy it went went from 5 to 4 and stayed at that level
for a long time. In this category
there was a modest increase as well under Blaauw
as part of the conditions under which he accepted his appointment.
\begin{figure}[t]
\sidecaption[t]
\includegraphics[width=0.64\textwidth]{figRhijn3.jpg}
\caption{\normalsize Pieter Johannes van Rhijn, probably in the late 1940s.
From the collection of
Adriaan Blaauw and Kapteyn Astronomical Institute.}
\label{fig:vRhijn3}
\end{figure}
The Kapteyn Laboratorium remained a marginal effort in Groningen; while
the university grew by a factor of about three or so between 1921 and 1957, it
remained at the same level, probably below critical mass in the long run. On
a national level this falling behind of Groningen astronomy was even more
evident. For nation-wide comparison the
available literature in English is limited.
David Baneke's book {\it De ontdekkers van de
hemel} on the history of Dutch astronomy in the twentieth century \cite{DB2015}
is not available in English, as is also the case for a number of his other
informative papers on related issues except for a few, e.g. \cite{DB2010}.
From all of this I compile the following numbers. In
the 1930s there were eleven staff positions (professors, assistants, etc.) in
astronomy at Dutch universities: six in Leiden, two in Utrecht and Groningen
and one in Amsterdam. Compared with the Sterrewacht Leiden the Groningen
Laboratorium was insignificant. In 1933, Willem de Sitter wrote a history of
the Sterrewacht on the occasion of its tricentennial \cite{WdS1933} and listed
the staff at that time: two professors (of which one extraordinary), two
conservators (senior staff comparable to lecturer), two observators
(scientific staff), four assistants (plus six voluntary, so not paid from the
regular budget), ten computers, two technical staff, a typist and a carpenter.
The observatory itself had various telescopes and a station in Johannesburg,
South-Africa, which included one telescope and access to at least two more.
The teaching
load (astronomy staff taught astronomy courses to physics and mathematics
majors, and some times physics or mathematics courses) was divided between at
least four persons. This is no comparison to the other universities, where all,
or at best almost all, the teaching had to be done by a single person.
Utrecht had an observatory as well, concentrating on solar research
\cite{Utrecht370}, but expanding into astrophysics in particular hot, massive
stars. The Amsterdam Institute of Pannekoek was not significantly
different from the Groningen establishment \cite{Pannekoek}. Anton Pannekoek
had modeled his institute along Kapteyn's lines in the sense of relying on
observational material from elsewhere. Pannekoek concentrated furthermore on
theoretical work, pioneering the field of stellar atmospheres. The institute
remained small under his leadership, and also under his successor Herman
Zanstra (1894-–1972), who took over after Pannekoek's retirement in 1946.
Not only did Leiden dominate Dutch astronomy and therefore making it more
difficult for other universities to strengthen their astronomy
departments, more importantly for the Kapteyn Laboratorium is that Groningen
astronomy had under van Rhijn lost ground by failing to increase its size in
step with the rest of the university.
\bigskip
As far as students contemplating a possible career in astronomy are concerned,
the situation in Groningen was also minimal. Maarten Schmidt had enrolled as
a student with a strong interest in astronomy in 1946, and obtained his
Candidaats (Bachelor) in 1949. But on invitation of Jan Oort he moved to
Leiden at that stage. The single promising student left, Jan Borgman, although
of almost exactly Schmidt's age, entered Groningen university only in 1950. He
progressed in a satisfactory way, obtaining his 'Doctoraal' (Masters) in 1955
and a PhD in 1956. However the PhD degree was granted with physics professor
Hendrik Brinkman (1909--1994) as supervisor, and concerned electronic variable
star recognition on plate pairs.
No further PhD students came forward during the rest of van Rhijn's
professorship.
Up until the first years of WWII (the period
1921--1942) the number of PhD theses under
van Rhijn had been significant compared to the other universities, namely eight (Groningen)
versus fourteen (Leiden), eleven (Utrecht) and two (Amsterdam). But between 1945 and 1957
van Rhijn had only one, compared to eight (Leiden), four (Utrecht) and four (Amsterdam).
This one PhD was Adriaan Blaauw in 1946, but after him there were none for
almost two decades, unless one counts the physics thesis of Jan Borgman. Of the
total of nine Groningen PhDs under van Rhijn, however, 5 had been recruited as PhD students
from elsewhere: Jan Schilt and Peter van de Kamp from Utrecht, and Jean
Jacques Raimond, Bart Bok and Adriaan Blaauw from Leiden. Egbert Kreiken,
Jan Oort, Willem Klein Wassink and Broer Hiemstra had obtained their Doctoraal
(Masters) in Groningen, but the first two were in a sense students of Kapteyn.
Students in physics and mathematics do attend astronomy lecture courses in
their first years and inspiring astronomy professors might ignite an interest
in their field. Van Rhijn did not recruit many new astronomy students, although
it must be said that after the War his tuberculosis would have made this
difficult. Klein Wassink and Hiemstra started as Groningen undergraduates
and went on to a PhD; they are the only ones that can be seen as having been
pure students of van Rhijn. The small number of students would not help
persuading Curators or the Minister to create or assign new staff
positions at the astronomy department.
\section{The late forties and early fifties}\label{sect:forfif}
During the first years after the War work at the Kapteyn Laboratorium was slow
in picking up.
The work on variables was painstakingly slow, but very competently executed
by Plaut
and van Rhijn was out of action a large part of the time. Work on the telescope
was performed but no results appeared. Van Rhijn and Plaut made up the full
scientific staff, which had never grown compared to Kapteyn's days
(see Fig.~\ref{fig:NAC1952}). Teaching went
on as normal, lectures to a large extent being given by Plaut. There was one
astronomy major student, Maarten Schmidt, but as already mentioned,
he left for Leiden halfway through his studies.
\bigskip
During the War the work on the {\it Bergedorf(-Groningen-Harvard)
Spektral-Durchmus\-te\-rung}
had first slowed down and then essentially halted.
Van Rhijn had been able to do some research, part
of it probably while being nursed for tuberculosis. In any case he did publish
two papers in the {\it
Publications of the Astronomical Laboratory at Groningen}.
In his review of Dutch astronomy during the Second World War \cite{PJvR1946}
he had noted:
\par \begingroup \leftskip2.5em \rightskip2em \large
The H and K lines are absorption lines due to calcium and can only be
distinguished from stellar lines of
these have not such lines in their spectra, which means they have to be
spectral types O or B.
During the fighting in Groningen in April 1945 two publications, written during
the War by the director, have been burned, but a copy of the manuscripts is
undamaged.
Dr. P.J. van Rhijn investigated the accuracy of the interstellar line
intensities as a criterion of distance and reduced the known equivalent widths
of the H and K lines, which appeared to be most suited to the purpose, to
distances. The density and luminosity functions of the O and early B stars were
derived; contrary to former results it appears that the density does not
decrease with increasing distance from the Sun.
\par \endgroup
After the War van Rhijn had not sit
still in spite of his tuberculosis, from which he recoverd only very slowly.
A large research project that he completed in 1949, was a mapping of
extinction on the basis
of color excesses measured at various places all over the sky \cite{PJvR1949}.
According to the Introduction:
\par \begingroup \leftskip2.5em \rightskip2em \large
It is the purpose of the present paper to derive the mean differential
absorption and the distribution of the amount of the differential absorption
for stars at a specified distance and Galactic latitude. If, as seems probable,
the photographic and differential absorption are in a constant ratio, the mean
photographic absorption and the distribution of this absorption for stars at a
specified distance and latitude can be found. These functions can be used
advantageously in the solution of the equation of stellar statistics, [...]
\par \endgroup
For clarity I remind the reader that differential absorption is what we
would now call reddening, in those days
usually between photographic (blue) and photovisual (yellow) wavelength bands.
The distribution of actual values in a particular direction was
found to be approximately Gaussian and could be
explained by a chance distribution of absorbing clouds in regions near the
Milky Way. The number of clouds on a line of sight near the Milky Way was
6 per kpc on average, and the
photographic absorption of a single cloud van Rhijn estimated as 0\magspt27.
The differential absorption of such a cloud turned out to be
0\magspt035. The
distribution of cloud sizes was not addressed. Interesting as this may be the
use of this to solve the equation of stellar statistics is limited, because of
the severe irregularities in the distribution of extinction on the sky and
along sight lines. As far as I am aware these results have never been used for
such a purpose. In ADS van Rhijn's paper
has a single citation in 1963 and only an occasional
read (the peak in 2021 is me preparing this paper).
It should be recorded that van Rhijn after the war
faithfully fulfilled his task as
coordinator of the {\it Plan of Selected Areas}; he wrote a comprehensive
report for Commission 32 for the IAU General Assembly in Z\"urich, Switzerland,
in 1948. But he did not attend and during the meeting Jan Oort took the lead as
acting President of that Commission together with Adriaan Blaauw as Secretary.
\bigskip
Plaut did vigorously pursue his research in variable stars. Some of it during
the War was probably completing work he had started in Leiden before he
moved to Groningen. This means extremely labor intensive and time consuming
work of which the yield was small and progress slow. But is shows the kind of
work Plaut did and how very patient a person he was. In 1946 he published
a paper on an eclipsing binary from 341 plates taken between February 1930 and
August 1936 \cite{Plaut1946}. The plates had been obtained at the Leiden
southern station using the 10-inch Franklin-Adams Telescope of the Unie
Sterrewag in Johannesburg, which had been donated by John Franklin-Adams
(1843--1912), the maker of the first all-sky photographic survey. In the deal
de Sitter had negotiated, Leiden astronomers had access to this telescope.
Hertzsprung and collaborators used it primarily to study variable stars from
repeated exposures. Plaut had measured these plates with the Schilt photometer
(see above) to construct a detailed light curve, the form of which gave among
others information on the eccentricity of the orbit.
This is not all.
As noted above, van a Rhijn \cite{PJvR1946} had noted in his article on Dutch
astronomy during the war, that Plaut
had been searching for variable stars on plates
taken at the Leiden southern station with the Franklin-Adams telescope.
Obviously, this involved plates taken
by the Leiden representative there, which in this case was Hendrik van
Gent (1900--1947), who had worked on variable stars with Hertzsprung
and obtained a PhD under him in 1932.
Plaut used the blink comparator of the Kapteyn Laboratorium
(rapidly switching the view between two plates of the same area)
and published a
paper in 1948 [56], in which he reported 91 variables by comparing
12 pairs of plates, subsequently estimating the magnitudes on a full set of 300
plates. These estimates were performed by eye. This was a matter of patience
and extreme care. And in
1950 Plaut published a review collecting the available data on all 117 known
eclipsing binaries brighter than magnitude 8.5 at maximum and determining their
orbital parameters and updated that in 1953 (\cite{Plaut1953}
and earlier references therein). Plaut had established himself as an authority
on variable stars.
\begin{figure}[t]
\sidecaption[t]
\includegraphics[width=0.64\textwidth]{figNAC52.jpg}
\caption{\normalsize Detail of the group photograph of the Nederlandse
Astronomen Conferentie in 1952. The full delegation from Groningen consisted
of only two persons: on the lower-left Lukas Plaut, on the lower right
Jan Borgman. The latter was a second year student. In the
top we see Maarten Schmidt, who by then had moved to Leiden.
Other notable persons are Cees Zwaan from Utrecht top-right and Pieter
Oosterhoff from Leiden between Schmidt and Plaut.
From the Website of the Koninklijke Nederlandse Astronomen Club \cite{KNAC}.}
\label{fig:NAC1952}
\end{figure}
As we have seen, van Rhijn was out of action for most of the rest of the 1940s
after the War and the Laboratorium relied mostly on Lukas Plaut for teaching.
In the 'Lotgevallen' for 1949, covering the previous academic year, Rector
Magnificus Pieter-Jan Enk (1885--1963), Professor of Latin, noted (my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
It fills us with great joy that Prof. Dr. P.J. van Rhijn, after an illness that
lasted more than four years, has now completely recovered and is again going
to devote himself to his work with full vigor. We wish our colleague all the
best in this respect.
\par \endgroup
\noindent
And
\par \begingroup \leftskip2.5em \rightskip2em \large
The teaching assignment given to Dr. L. Plaut, in connection with Prof. van
Rhijn's illness, to teach Propadeutic Astronomy was withdrawn by Ministerial
Decree of November 18, 1948.
\par \endgroup
\bigskip
We have seen in section~\ref{sect:extinction}
that van Rhijn had been very much involved in research
into interstellar extinction, in fact it was part of his PhD thesis.
Summarizing the background very briefly,
following Kapteyn it had become clear that it was due to
some kind of scattering which caused reddening and part of the problem was to
turn amount of reddening into amount of extinction. This is given by what is
usually referred to as the ratio total-to-selective absorption. This is a
dimensionless property (in fact magnitudes per magnitude) and once known can be
applied without any other information: measure the reddening and with the ratio
gives the extinction. The ratio depends of course on the wavelength
dependence
of the scattering and has a different numerical value for different photometric
bands. The total-to-selective absorption depends not only on the physics of the
scattering,
but also on the size distribution of dust particles and cannot simply
be calculated from first principles of physics.
Van Rhijn's telescope was originally intended for stellar photometry,
but it soon became clear that the
sky conditions in the center of Groningen were too poor for this.
He therefore resorted to spectroscopy
for which condition are less critical. We will seen that he
corresponded in 1936 with the Zeiss
Company about the purchase of a slitless spectrograph,
which he probably obtained in that year or
shortly after that. At that time the selective-to-total absorption
was still very poorly known and he wished to
improve this with his telescope taking spectra of obscured stars and determine
the energy distribution over wavelength and do the same for unreddened stars of
the same spectral type. Eventually he was aiming at using stellar spectra to
more fully determine the distribution of dust in the Galaxy, and address
questions on what the size distribution of the particles was and whether or not
this changed substantially with position in space.
Work on relatively bright stars in the sky was the only kind of
observational astronomy that could
conceivably be done as relevant research from the center of a city like
Groningen. The reason is that a good
fraction of the bright background in a city is concentrated
in small wavelength regions of bright spectral lines. Outside these
wavelength regions the city sky is more amenable to astronomical research.
This project took a very long time getting started. In the 1930s getting the
telescope ready for operation was a tedious and slow process, to a large extent
due to the economic depression, which prevented the hiring of additional staff,
but also as a result of the enormous effort required for the {\it
Bergedorf(-Groningen-Harvard) Spektral-Durchmusterung}.
The telescope required a
spectrograph, but the – what looks like final – quote in the correspondence
with the Zeiss Company in the van Rhijn Archives on a 'Spaltlose
Einprismen-Spectrograph' (slitless single prism spectrograph) stems from 1936.
Where the required 4100 RM (ReichsMark), corresponding to an estimated 70000
\textgreek{\euro} at present, came from
is not explained. It may have taken a few years
to raise that amount. Some observing did take place during the war, but
whether useful material for this project was obtained then is doubtful. The
collecting of spectra only proceeded successfully apparently after the war, as
noted by Blaauw in the quote at the end of section~\ref{sect:telescope} and
Maarten Schmidt described his experience with learning observing from Lukas
Plaut as a student in the late forties in the citation preceding that of
Blaauw. The first results on the wavelengths dependence of
interstellar extinction were published by van Rhijn in 1953 in a single
authored paper, \cite{PJvR1953}, in which he acknowledged Plaut for taking
most of the plates. His technical and computational staff, M\"uller and Huisman,
are noted for help with the reduction (and M\"uller also with the observing).
It seems that the work at the telescope was mainly done by Lukas Plaut.
This would have been a major effort, as it also involved obtaining experience
with the equipment and the problem of the poor observing conditions in the
center of the city of Groningen. Plaut has not published any paper on data
obtained with the Groningen telescope and did not use the telescope for studies
of variable stars; for useful research purposes it has been used only with
the spectrograph.
Since the design and installation of the telescope astronomy had changed
considerably. Would the idea of measuring brightness distributions of reddened
stars as a function of wavelength using photographic recordings of spectra have
been timely in the late 1920s or the 1930s, it was definitely not state of the
art after the war. The field of stellar photometry had been revolutionized by
the introduction of photoelectric photometry (and spectophotometry),
and indeed applied to the problem
of the wavelength dependence of interstellar extinction by astronomers as
Jesse Greenstein, Joel Stebbins (1878--1966), Albert Edward Whitford
(1905--2002) and others, and these studies had established that it was
approximately inversely proportional to the wavelength. In fact, van Rhijn did
acknowledge in the first line of his paper [98] that the detailed nature of
the wavelength dependence had been addressed by Stebbins and Whitford. These
had published a series of six papers entitled {\it Six-color photometry of
stars}, starting in 1943 with \cite{SW1943}. They used the Mount Wilson 60-
and 100-inch telescopes to photoelectrically measure the intensity distribution
over the wide wavelength range of 3530 to 10,300 \AA\ (they included
measurements on galaxies) and had found relatively small, but
significant deviations from a simple inverse-$\lambda$ law.
In \cite{PJvR1953} van Rhijn described how the photographic spectra
were measured. For this he used 'the thermoelectric photometer of the Kapteyn
Laboratory', which must be still the 'Schilt photometer' built by Jan
Schilt in his thesis research and among others used for the
{\it Spektral-Durchmusterung}
(see section~\ref{sect:Spektral-D} and
\cite{Schilt1922}). The method van Rhijn had designed was to use a pair of stars
of the same spectral type, one significantly reddened and one unreddened, so
respectively at low and high Galactic latitude.
The stars used are of spectral type late O or
early B, since these are stars with the weakest spectral features, the
spectral lines present predominantly come from hydrogen and helium but
with no significant lines of other elements (early O-type stars are unsuitable
because of strong helium lines, and in later B-stars the hydrogen lines become
stronger). Their apparent magnitudes were between 5.5 and 8.0. These were
exposed various (usually four) times on a single plate together with exposures
of a late-B or early-A star. The latter are stars with spectra with strong and
sharp hydrogen lines that could be used for wavelength calibration. The spectra
of the two program stars were measured at a set of seven wavelengths ranging
from about 3900 to 6300 \AA, or a factor of about 1.6. The research reported
concerned 14 pairs and it confirmed the Stebbins and Whitford result.
This was followed up soon by Jan Borgman.
He had spent some years working at the Philips Company,
developing his technical skills and gaining
experience in building electronic instrumentation. He obtained his doctorandus
degree (Masters) in 1955.
His PhD thesis work (defended in 1956) will be described below, but as a student
he published in 1954 and 1956 two more studies (with the same title as van
Rhijn's paper), extending van Rhijn's work with another 10 and 27 late O- and
early B-stars, \cite{JB1954}\cite{JB1956}.
Again the agreement with the Stebbins and Whitford curve was very good.
The work suffered from all problems associated with quantitative photographic
work in that is was tedious and had to be carried out with extreme care and
perseverance. The photographic emulsion is a very a-linear detector: it suffers
from under- and over-exposure and the relation between the photographic density
-- how opaque the emulsion is, zero when the transmission is 100\%, unity when
this is 10\%, etc. --, known as the characteristic curve, varies
from plate to plate and sometimes even in different parts of the
same plate. On top of this there is
low-intensity reciprocity failure -- at faint light levels the exposure of,
say twice, fainter light source takes more than twice the exposure time to
reach the same photographic density. Van Rhijn, Borgman and Plaut did overcome
all these problems and produced reliable results.
The program was conceived by van Rhijn twenty
or so years earlier and indeed worked
out excellently, and could in the 1930s in all likelyhood have resulted in
papers that would have made a major impact, rivaling others worked on this
subject at that time (see e.g. the 1934 review by Trumpler for more details),
\cite{Trump}. But due to very long delays van Rhijn was not breaking any new
ground, merely confirming what had been determined in the mean time in
particular by photoelectric techniques that held more promise for the future.
But the fact that the photographic work in Groningen actually worked and could
produce results that were comparable in accuracy to the photoelectric data is
a tribute to the quality of the work of van Rhijn and Borgman.
\section{The Vosbergen conference}\label{sect:Vosberen}
In June 1953, the International Astronomical Union held its Symposium Nr. 1 on
{\it Co-ordination of Galactic Research}. The initiative for this meeting had
been taken by Jan Oort, who was President of Commission 33 of the IAU on
'Stellar Statistics' (at the IAU General Assembly of 1958 in Dublin this was
changed into 'Structure and Dynamics of the Galactic System'). The purpose and
character of the conference, quoted from a circular letter sent to all
participants -- and obviously written by Oort -- was described as follows:
\par \begingroup \leftskip2.5em \rightskip2em \large
During the first third of this century an important concentration of work on
Galactic structure and motions has been promoted by the Plan of the Selected
Areas, initiated in 1906 by Kapteyn. Although the scheme outlined in 1906
has not lost its significance, it is widely felt that further research into
structure
and dynamics of the Galaxy should be extended beyond the original Plan.
At the same time it is felt by many that some kind of coordination of effort
remains highly desirable, if only because the value of many observations is
greatly enhanced if the data can be combined with other data for the same
stars.
Several observatories have asked for suggestions with regard to future work
on Galactic structure. The recent advent of several large Schmidt telescopes
furnished with objective prisms, red-sensitive plates, etc., brings forward with
some urgency for each of the observatories concerned the problem of how to
organize and restrict the work with telescopes that can produce much more
than can be measured and discussed by the existing observatories. The question
arises, whether it is desirable to formulate a new plan of attack, and whether
such a plan should also include recommendations for concentrated work on
particular regions. If the answer to the latter is affirmative, which regions
should be selected and what data would be the most urgent.
\par \endgroup
The meeting was hosted by Pieter van Rhijn and the Kapteyn Laboratorium; the
venue was the estate Vosbergen, which was used by he University of Groningen as
a conference center. It is located near the village of Eelde some 5 km south of
Groningen. Lukas Plaut had done the local organizing. Adriaan Blaauw edited
the proceedings, which was unusual according to current standards, in that
it did not consist of papers or articles of the presentations made,
but instead Blaauw
had summarized all presentations and the long discussions (Blaauw would
move to Yerkes before finishing the proceedings). The format
of the meeting was that
there were introductory presentations (some of which have been published
separately in full), followed by discussion. The Scientific Organizing Committee
(SOC), of which Oort was chairman and Blaauw secretary, consisted of in total
eight persons. Van Rhijn was not part of this (nor Plaut); it was dominated by
Oort and Blaauw from Leiden and Whilhelm Heinrich Walter Baade (1893--1960)
(see \cite{OstBaade} or \cite{JHO1961a}) with whom Oort had been
conducting extensive correspondence and who visited Leiden in the months
preceding the meeting. This meeting also marked the first discussions on the
European Southern Observatory, taking place first in Leiden the Sunday before
the meeting. Oort had invited a few leading astronomers from
European countries to travel to the Netherlands a day earlier for this purpose.
More disucssions took place during a boat trip on the IJsselmeer (the dyked off
former Zuiderzee) as part of the symposium.
If the meeting was intended to be at Groningen as a tribute to van Rhijn's
dedication to the {\it Plan of Selected Areas} and guardian of the Kapteyn
legacy, it would seem a bit of an affront not to invite him on the SOC. Even
as host this would have been appropriate. After all, van Rhijn was only three
years away from retirement by now and this would be a excellent occasion to
honor him on an international stage. It is somewhat reminiscent of
the story of the Board of the Foundation Radio Radiation of Sun and Milky Way
that Oort had founded after the war
to realize the construction of radio telescopes (see \cite{JHObiog}
or \cite{JHOEng}), where
for a national effort it would have seemed appropriate that the directors
of the institutes interested in using the facilities would be board members.
But van Rhijn was not invited.
Although this was later corrected (upon protests by van Rhijn), it remained a
remarkable move. In addition to the eight SOC members of the Vosbergen meeting,
twenty more astronomers were invited to attend
\par \begingroup \leftskip2.5em \rightskip2em \large
because they represented institutions which might take part in
future galactic research, or because of the character of their research.
\par \endgroup
In
total ten introductory papers were presented in seven sections. The members of
the SOC all held introductory reviews (and a few of the other persons invited
to attend). Van Rhijn was not invited for this.
From the proceedings the impression is that van Rhijn
remained somewhat in the background. He is mentioned in the proceedings only
when the need for understanding the wavelength dependence of interstellar
extinction was discussed as an example of one having addressed that issue and
for his stressing (with Oort), that
\par \begingroup \leftskip2.5em \rightskip2em \large
\noindent
a very important task for the
observatories equipped with Astrographic Catalogue refractors will be the
determination of proper motions in regions centered on the Kapteyn Selected
Areas, but much larger than those given in the Pulkovo and Radcliffe
Catalogues.
\par \endgroup
\bigskip
The desiderata arrived at during the Vosbergen symposium showed a change in
emphasis on collecting as much data of various kinds as possible on stars in
the Selected Areas to large programs concerning each a particular set of
objects to elucidate well-defined issues. For example the inventory of stars
in the areas defined by Kapteyn
was replaced by especially designed surveys to determine
the distribution of stars in the halo or the disk using tracer objects, an
inventory of the distributions of stars in the local volume as a function of
age, surveys for proper motions and radial velocities of various types of
objects, or programs designed to address special problems. For example the
formation and evolution of the disk could be traced by studying the
distribution of stars of different spectral types on the Main Sequence or
late-type giants. B-stars could be used to determine from the distribution of
their radial velocities a better estimate of Oort constant $A$ of
Galactic rotation.
And two more actions would be made effective.
First the establishment of two IAU sub-committees, one within IAU Commission
33 on the coordination of observational programs and one within Commission 27
(Variable Stars) to organize further work on variable-star surveys. Memberships
were rather limited; for the first it was Baade, Blaauw, Lindblad, Oort,
Parenago, and van Rhijn, for the second Baade, Blaauw, Kukarkin, Oosterhoff.
These persons have been identified in Fig.~\ref{fig:Vosbergen}. Note that the total
attendance was only twenty five or so. I will go into
this list of members in some detail since it is remarkable,
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.98\textwidth]{figVosbergen.jpg}
\end{center}
\caption{\normalsize Participants at the 1953 Vosbergen Conference,
\textit{Coordination of Galactic Research}, with some of their coats, hats,
suitcases, etc. stored not really out of sight in the front.
Persons referred to in the
text can be identified as follows:
Second from the left Bertil Lindblad, next to him Walter Baade, four persons to
the right Jan Oort, in front of Oort Lukas Plaut and
standing next to him Bill Morgan, between and behind them Adriaan Blaauw,
and with papers in his hand Pieter van Rhijn, to the left of him, partly
hidden by the lady directly to his left, Jason Nassau, and behind Nassau
Pieter Oosterhoff. The
two persons on the right in the second row (numbers 2 and 4 from the right)
are respectively P.P. Parenago and B.V. Kukarkin. Kapteyn Astronomical
Institute.}
\label{fig:Vosbergen}
\end{figure}
One may argue the membership of these sub-commissions within the IAU was
chosen to optimize the interface with the IAU: Lindblad had just finished as IAU
President and was adviser to the Executive Committee, Oort and Boris
Vasilyevich Kukarkin (1909–1977) were at the time Presidents of Commissions 27
and 33, while Baade was President of Commission 28 (Galaxies),
Pieter van Rhijn of Commission 32 (Selected Areas) and Pieter Oosterhoff was
IAU General Secretary. Pavel Petrovich Parenago (1906--1960) and
Kukarkin were both from Moscow (the first specializing
in stellar structure of the Galaxy and the latter in variable stars) and were
to cover the Eastern countries; both were very influential in the Soviet Union.
This makes sense, but the rest of the membership involved persons
that were all special to
Oort and the Dutch dominance is both obvious and out of balance. Oosterhoff,
Blaauw and van Rhijn were all Dutch (Blaauw was appointed Secretary of both
sub-committees). Bertil Lindblad (1895–1965) had for a long time
been a leading figure in the field of structure of the Galaxy,
Oort had special relationships with him and with Baade, and although
they were clearly
leaders in the field, for example some American prominent astronomers that
were in the audience would have been fitting to include, such as William
Wilson (Bill) Morgan (1906--1994) or Jason John Nassau (1893--1965), who both have
contributed very important studies involving extensive surveys of respectively
early and late type stars. I have also identified Morgan and Nassau in
Fig.~\ref{fig:Vosbergen}. As a sideline: Donald Edward Osterbrock (1924--2007),
in his biography of Walter Baade \cite{OstBaade}, noted about Baade and Nassau:
\par \begingroup \leftskip2.5em \rightskip2em \large
The two men had the same general build, height, hair style and color, prominent
hawk nose, and liked to dress alike at scientific meetings which they both
attended. Frequently in group pictures they stood at opposite ends of the front
row, often striking very similar poses, which they must have rehearsed or at
least discussed in advance.
\par \endgroup
This picture would be a notable exception to this.
The second desideratum was to hold another meeting soon. It was
thought to possibly take place during the 1955 General Assembly in Dublin, but
was eventually deferred to 1957 as IAU Symposium Nr. {\bf 7}, {\it Second
Conference on Co-ordination of Galactic Research}, held near Stockholm. There
was reasonable, but no considerable overlap with the first meeting among the
participants. Van Rhijn was absent. By now he was beyond retirement age, but had
taken the directorship upon himself as long as no successor had arrived.
The format was the same, but Adriaan Blaauw this time coordinated a
small group of editors, who shared the burden of this work. It is beyond the
scope of this article to discuss its proceedings in more detail.
\bigskip
The change from emphasis on completion of Kapteyn's {\it Plan of Selected Areas}
that had been the focus of van Rhijn's efforts was replaced by directing the
efforts to specific surveys related to specific questions concerning aspects
of the quest to determine the structure, kinematic, dynamics and eventually
formation and evolution of the Milky Way Galaxy. In this sense it was the
unofficial end of the {\it Plan of Selected Areas}, since -- as noted above --
IAU Commission 32 remained operational until 1958, continued as a sub-commission
within Commission 33 and was formally disbanded only at the General
Assembly in Brighton in 1970. This last action then
should be seen as the official end. In the end 28 observatories from 11
different countries had contributed to the Kapteyn's {\it Plan of Selected
Areas}.
For the Kapteyn Laboratorium the outcome of the first meeting was profound in
that it provided a focus on an extensive, long-term undertaking that would
occupy Lukas Plaut and a large part of the Laboratorium's manpower
for may years. I will turn to this next.
\section{Groningen-Palomar Variable-Star Survey}
One of the major desiderata of the Vosbergen meeting was to undertake what
became known as the {\it Palomar-Groningen Variable-Star Survey}, which
concerned variable stars in the Galactic bulge and halo. It was proposed at
Vosbergen by Walter Baade as a means of finding out what the stellar
distribution in the bulge and halo of the Galaxy was and to investigate the
differences and similarities between the Milky Way Galaxy and the Andromeda
Nebula. The halo was defined primarily by the globular clusters as tracers of
Baade's Population II, introduced in 1944 upon resolving stars in the Andromeda
Nebula and companion dwarf galaxies. Baade had noted that the brightest stars
in the halo Population II were red (giant) stars, while in the disk Population
I these were blue OB-stars. This resulted in the realization that there was a
fundamental difference between stars in the disk and halo. At the time of the
Vosbergen meeting the situation was (see \cite{Vosbergen}, p.4-5):
\par \begingroup \leftskip2.5em \rightskip2em \large
From all the evidence available at present it seems that the distribution of
the population II objects is symmetrical with respect to the axis of rotation
of the Galaxy. (Distribution of globular clusters in the Galaxy, distribution
of population II objects in extra-galactic nebulae.) The first task of future
work, therefore, will be to determine the way in which the density varies with
the distance from the Galactic Centre, […] The following proposal was made by
Baade.
For the determination of the surfaces of equal density in the halo, we can
restrict ourselves to determining the density distribution in a cross-section,
perpendicular to the Galactic plane and going through the Sun and the axis of
rotation. […] It is proposed that, to begin with, three fields be chosen,
centered at approximately
$l$=327$^{\circ}$,$b$=about 20$^{\circ}$ (latitude as close to the nucleus as
interstellar absorption permits);
$l$=327$^{\circ}$,$b$=45$^{\circ}$;
$l$=147$^{\circ}$, $b$=somewhere about 10$^{\circ}$.\\
The ideal instrument for such a survey would be the 48-inch Palomar Schmidt,
which may become free in the next few years after the sky survey is concluded.
It gives fields of 7$^{\circ}$$\times$7$^{\circ}$, free from vignetting and can
easily reach the 20th photographic magnitude in 10-min. exposures. This
provides an ample margin to make sure that the survey will be complete up to
17.5 median magnitude corrected for absorption, i.e. distances up to 30
kiloparsecs for the RR Lyrae variables — even if the absorption in one of the
fields should amount to one or one and a half magnitudes. In the case of the
two latter fields it is very probable that regions of heavy or irregular
absorption can be avoided. As to the first, it is a matter of searching the
sky survey plates in this general direction before deciding on the most
suitable co-ordinates.
\par \endgroup
These coordinates are of course 'old' Galactic coordinates with the zero-point
of longitude is at the crossing of the Galactic and celestial equators.
Longitude 327$^{\circ}$\ then
is that of the center of the Galaxy. The fields were as close as possible
above the Galactic center, 45$^{\circ}$\ up
from the center and towards the anticenter close to the Galactic plane.
The sky survey referred to is
the {\it National Geographic Society – Palomar Observatory Sky
Survey (NGS-POSS)}, which was substantially completed in the middle of 1956.
\begin{figure}[t]
\sidecaption[t]
\includegraphics[width=0.64\textwidth]{figPlaut.jpg}
\caption{\normalsize Lukas Plaut at later age. From the collection of
Adriaan Blaauw, Kapteyn Astronomical Institute.}
\label{fig:LPlaut}
\end{figure}
At the Vosbergen meeting at had become clear that Lukas Plaut
(Fig.~\ref{fig:LPlaut}) and the Kapteyn
Laboratorium would be prepared to take this project, the
{\it Palomar-Groningen Variable-Star Survey}, upon them. For the precise choice
of the fields Plaut relied on Baade, who had surveyed the region around the
center extensively both with the Palomar 18-inch Schmidt and the Mount Wilson
100-inch telescopes. The 18-inch Schmidt had been an initiative of Fritz Zwicky
(1898--1974). He and Walter Baade had proposed in 1934 that supernovae were
the source of cosmic rays, and Zwicky wanted a small, wide-angle telescope to
search external galaxies for supernovae. The choice for a Schmidt telescope is
undoubtedly due to Baade. The design of a wide-angle telescope with
a spherical mirror and objective corrector plate to correct for spherical
aberration was first made by Bernhard Woldemar Schmidt (1879--1935)
at Hamburg Observatory where Baade worked
before joining Mount Wilson Observatory in 1931.
The 18-inch was the first telescope on
Palomar Mountain, becoming operational in 1936, well before the 48-inch (1948)
and 200-inch (1949) telescopes. Zwicky was a stubborn and ill-tempered person,
and regarded the 18-inch as his telescope. In the beginning he and Baade were
on good terms and published together, but later they avoided each other. In
the late 1930s Baade used the 18-inch, which recorded images of the sky on
photographic film over a circular area of almost 9 degrees diameter, to map a
large region around the center of the Galaxy. He did this in red light (with a
panchromatic film and a red filter), identifying areas of little extinction.
At the Vosbergen meeting, Plaut estimated the time required to perform the
blinking of plates and determination of light curves of RR Lyrae stars and
other variables as `something of the order of 15,000 hours or six man-years'.
At the start of the project four different fields were selected, three near
the center in new coordinates ($l$,$b$) at (0$^{\circ}$,+29$^{\circ}$),
(4$^{\circ}$,+12$^{\circ}$), and (0$^{\circ}$,-10$^{\circ}$) plus a field in a direction
more or less
perpendicular to this at (82$^{\circ}$,+11$^{\circ}$). Most observing was done for the
first three fields during two extended periods in California by Plaut in
1956 and 1959, and some plates were taken in between by other observers. The
three-year time-span allowed discovery of long-period variables. The fourth
field was never completed.
The organization of the survey took much time and also irritation on the part
of van Rhijn and Plaut. Much depended on Walter Baade, who was responsible for
choosing the exact location of the fields and
organizing the access to the Palomar 48-inch Telescope with Ira Sprague Bowen
(1898--1973), the director
of the Mount Wilson and Palomar Observatories.
In a letter to Jan Oort of May 4, 1954 (in the Oort
Archives, Nr. 149), Pieter van Rhijn complained (my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
The correspondence between Baade and me has been as follows: Baade wrote me
Oct. 13, 1953, that he himself was in favor of the plan to search for the
variable stars in some fields in the plane perpendicular to the Milky Way
with the Schmidt Palomar, but that he still had to consult with Bowen about
the execution of the plan, Also the possibility, that Plaut would cooperate
in the execution of the plan at Mount Palomar, would be considered. In a short
letter of Jan. 8, 1954, I asked Baade if he had already spoken to Bowen about
the matter and if there was a reasonable chance, that the matter would go
ahead. I repeated the same question on Feb. 9, `54. To date I have received
no reply from Baade. The way Baade has handled this case has annoyed me.
Surely the least he could have done was to answer me to my reasonable question
of whether there was a good chance that the plan would be realized. Instead,
he makes me wait six months and then asks you to tell me that he cannot give
accurate dates for the work. Dates that I, you should note, had not even asked
for. The sentence in which he says he was 'much amused by my writing' I will
leave unmentioned. Such remarks will not promote good cooperation.
\par \endgroup
\noindent
Oort wrote back two days later (same source, again my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
\noindent
Dear Piet,\\
You should not take such statements from Baade too seriously, nor his
non-answers to letters. I do nor know at all what he was referring to in the
phrase in question, but I find it hard to imagine that it would be worthwhile
to get angry or annoyed about. After all, people often have strange manners of
correspondence. If I wanted to be annoyed with Baade I would have jumped out
of my skin last year before his visit here. He did not let me hear anything
about his arrival for months, despite my repeated questions, and then finally
wrote, after the date that had been firmly agreed upon had already passed,
that he had come into conflict with other appointments and promises that he
had made before and had forgotten again. And so on and so forth. We also have
had more experiences like this. When he was here, it was most pleasant. We
benefited enormously from his stay, also the students did, with whom he was
very kind.
This is to help you get over your annoyance, which is never pleasant or helpful.
\par \endgroup
\bigskip
It must have been quite difficult also to find finances for Plaut's necessary
travel to Pasadena and Palomar Observatory. According to Plaut's acknowledgments
in the publications resulting from the survey it involved originally the
United States Educational Foundation in the Netherlands (Fulbright Travel
Grant), Commission 38 of the International Astronomical Union, and the
Netherlands Organization for the Advancement of Pure Research (ZWO).
Later the Leids Kerkhoven-Bosscha Fund was added. The Kapteyn Laboratorium would
have contributed also, but that is (understandably)
not mentioned in the acknowledgments.
Almost 400 plates were taken in the photographic wavelength region plus about
90 in the photovisual to measure colors.
In the course of the project
a field called {\it Baade's Window} was added as a final part of the
{\it Variable-Star Survey}. This field had been discovered
also by Baade as part of his survey of the region around the Galactic center
and is a field very close (only four degrees) from the actual direction of the
center, ($l$,$b$) = (1$^{\circ}$,-4$^{\circ}$). It is in the brightest part of the Milky
Way in this general direction
and very uniform, suggesting very little extinction. It is centered on the
globular cluster NGC 6522, which is at about 7.5 kpc along the line of sight, so
in space close the center of the
Galaxy; the extinction is not zero, but about 'only' 2.5 magnitudes in
photographic magnitudes according to Baade \cite{Baade1946}.
Baade had studied this window extensively with the
Mount Wilson 100-inch telescope and discovered some 159 RR Lyraes (also known
as cluster-type variables, because they occur in large numbers in globular
clusters). Plaut reanalyzed these data, redetermining their
periods and the photometric zero-points, using the 1-meter telescope by then
available at the La Silla site of the European Southern Observatory. The
results of the {\it Palomar-Groningen Variable-Star Survey} were published by
Plaut in six papers, the last one appearing in 1973 (\cite{Plaut1973}, which
has references to the earlier papers). It was concluded in 1975 by Jan Oort and
Lukas Plaut in a discussion of the whole project in terms of the distribution
of RR Lyrae variables in the inner region and halo of the Galaxy and its
implication for the distance to its center and the rotation constants,
\cite{OP1975}.
\bigskip
The main effort in the program was to find the variable stars. The problem was
the fact that blinking plates was tedious, tiring, slow and difficult to do
consistently for anything more than a few hours at most. And therefore time
consuming and requiring an experienced astronomer. It was known very well that
completeness levels were poor except for the largest amplitudes. In general
three methods had been in use that provided useful results, always involving
two photographic plates recorded at different epochs with the same telescope
and the same exposure time. These were the following:
The first was direct blinking in a comparator or
microscope, where the two plates were examined by rapidly switching the view
between the two. The second method involved superimposing of a
negative of one of the plates on a
positive of the other, so that the images of the non-variable stars were
canceling. A third option was to examine
the plates in a stereo-comparator, which means
viewing two plates at the same time each with one eye. All these methods
produced rather low completenesses, as could be judged from repeating the
process later by the same or a different person. New variables then
turned up, but
also some were not rediscovered. Of course the chance of discovery depended
also on the average magnitude of the star and magnitude difference between
the two observations. Whichever method was used, searching for variable stars
remained a tedious business.
The classical study in this issue is that by Hendrik van Gent in 1933,
who worked for a
long time as the Leiden observer at the Unie Sterrewag in Johannesburg,
\cite{vGent33}.
Using a blink microscope, he found the chance of discovery on a single pair
of Franklin-Adams plates around magnitude 13 or 14 and variations of 1 to 2
magnitudes to be usually below 10\%. The actual value depended
of course on the amplitude of the variation and the phase in the light curve.
Astronomers used often ten or more pairs of plates, but even then the
incompleteness can be still quite significant. Plaut in the famous volume V of
the compendium {\it Stars and Stellar Systems}, \cite{Plaut1965},
his Appendix I, summarized
the completeness issue, and provided further details such as the dependence
on type of variable. Obviously completeness in detecting variables was a
major concern.
This had worried Plaut for a long time. At the suggestion of a young
physicist in Groningen (Herman de Lang) he tried a new method by
observing one plate in the blink
comparator through blue and one through a red filter, such that
non-variable stars were black while variable stars because of their differing
dimension on the plates had a blue or red ring around them. He involved student
Jan Borgman in this, which resulted in a short note \cite{PB1954}, reporting
that the method worked well, but also announcing they (this really was Borgman)
were developing an electronic device for the purpose of identifying variable
objects more completely and conveniently. Indeed Jan Borgman designed and
built an instrument for electronic discovery of variables in the framework of
his PhD research. The thesis that resulted from this was defended in 1956
under Hendrik Brinkman and was entitled {\it Electronic scanning
for variable stars}, \cite{Borgthesis}. Brinkman was a professor of
experimental physics in Groningen. He worked in many areas of physics, and is
particularly noted for the founding of the nuclear physics accelerator
institute in Groningen.
\begin{figure}[t]
\sidecaption[t]
\includegraphics[width=0.64\textwidth]{figBorgmanapparaat.jpg}
\caption{\normalsize The electronic instrument designed and constructed by
Jan Borgman to identify variable stars
by comparing a pair of photographic plates. See
text for explanation. This work constituted his PhD
thesis under physics professor
Hendrik Brinkman. From Borgman's PhD thesis \cite{Borgthesis}.}
\label{fig:Borgmanapp}
\end{figure}
The design was in principle straightforward. It made use of the principle of
so-called flying-spot scanning developed for television. On a cathode ray tube,
sitting on the floor in Fig.~\ref{fig:Borgmanapp}, a rectangular grid or raster
is formed on the screen, and a bright spot is formed moving across that grid.
This spot is focused by an optical system from below onto the
two photographic plates lying on the table. The optical system divides the light
in two by a semi-transparent/semi-reflecting mirror. This system is positioned
such that the same portions of the plates are covered by this grid. The light
spot on the screen on the cathode ray tube results in two beams of light on
the photographic emulsions, at any time illuminating the same position. As the
spot moves across the raster on the screen of the cathode ray tube the
corresponding spots on the photographic plates are illuminated simultaneously.
Two photomultiplier tubes above the plates convert the light transmitted by the
photographic emulsions into electrical impulses (a video signal). These signals
are then subtracted electronically and the resulting signal is amplified and
transferred to the grid of another cathode ray tube, whose dot is synchronous
with the dot on the first tube. This then is displayed on the television screen
on top. The image shows noise (resulting chiefly from the grains in the
photographic emulsions) as a gray background level in positions where the two
plates are the same, but show a white or black dot where a variable star is
present. Fig.~\ref{fig:JB} shows the results.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.88\textwidth]{figJBpictures.jpg}
\end{center}
\caption{\normalsize The left panels show images of parts of plates obtained
with the Palomar 48-inch Schmidt telescope. The middle panels show the same
areas from a different plate but now with the video signal's phase reversed.
The right panels show the difference signals. The variable star in the top
panels changed from magnitude 12.1 to 13.5, and at the bottom from 17.9 to
below the detection limit of 19. Adapted from Borgman's contribution to
\cite{Voetspoor}. Kapteyn Astronomical Institute.}
\label{fig:JB}
\end{figure}
Plaut used the Borgman machine to measure up the plates for the {\it
Palomar-Groningen Variable-Star Survey}. The Borgman machine was essential
for the project because it speeded up the blinking
process substantially; with classical methods it would have been
impossible within any
reasonable timescale. This was the first time the device
was used extensively. He reported that it took 40 hours to search a plate pair
(square of 14-inch sides or about 6\degspt5). The work was still strenuous and
could not be done for more than 4 hours a day, so searching each pair had
to be spread over at least 10 days. The detection probability for
amplitudes in excess of 1\magspt5 was over 80\%, and still of order 50\%\ for
1 magnitude amplitude, at least for the first field. For the more crowded field
it was worse. As mentioned, Plaut blinked ten pairs of plates for each of the
three fields. Altogether the three fields yielded about 1100 RR Lyraes,
with 110 more from the reanalysis of Baade's Window, and on top of that
determined magnitudes for all objects found on the total set of almost
500 plates. In addition the survey
yielded similar numbers of long-period and other types of variable stars,
including eclipsing binaries.
The estimation of brightnesses was done visually using the technique introduced
by Friedrich Wilhelm August Argelander (1799--1875) and designated the
`Argelander step estimation method', in which a brighter and a fainter
comparison star are used and it is estimated where the variable's brightness
is positioned as a step value of the fraction
of the magnitude difference between these stars.
The fascinating history of these magnitude estimations from the Herschels to
Argelander and beyond is summarized by John B. Hearshaw (\cite{JBH}, chapter 2).
The {\it Palomar-Groningen Variable-Star Survey} was a major, long term
undertaking, extending well beyond van Rhijn's directorate, actually even
beyond that of his successor Adriaan Blaauw. It dominated Plaut's research
efforts for the rest of his academic career. The most important
results announced in the paper by Oort and Plaut in 1975 is that the
distance to the Galactic center is 8.7 kpc, with an uncertainty of 0.6 kpc,
and that the density distribution in the bulge and halo
to about 5 kpc is almost spherical and
varies approximately as $R^{-3}$ \cite{OP1975}. This constitutes fundamental,
very important results of a very long-term investment of
time and effort of Lukas Plaut, but of course does not generate a stream of
press releases announcing breakthroughs.
\section{Van Rhijn's final research and seventieth birthday}
The final installment of the {\it Bergedorf(-Groningen-Harvard)
Spektral-Durchmusterung} appeared at the time of the Vosbergen Symposium
(in the same year 1953), as did van Rhijn's paper with the first results of the
spectrographic work with his
telescope addressing the issue of the wavelength dependence of interstellar
extinction. In the year before, 1952, finally the measurement of the Harvard
plates for positions and magnitudes of stars in the {\it Selected Areas Special
Plan} had been completed and published. It was presented as a separate
publication by the Kapteyn's Laboratorium with van Rhijn as first author and
Kapteyn posthumously as the only
other author \cite{vRK1952}. But other work was going on in Groningen -- other
than the programs by Lukas Plaut on variable stars and by Jan Borgman on
spectral energy distributions -- in the framework of the {\it Plan of Selected
Areas}, as can be seen from the IAU Commission 32 reports.
In the 1955-report there is a paragraph on collaborations between `Alger and
Groningen'. This involved proper motions near the equator. The measurements of
these had been performed at the Kapteyn Laboratorium and the plates had been
taken at Alger. This institution had been established by l'Observatoire de
Paris as l'Observatoire astronomique de Bouzareah, so named after the suburb
where it had been located.
The observatory had been involved in the {\it Carte du Ciel}
project, in which it had been assigned the equatorial zone. These plates served
as first epoch plates. Their useful extent covered 30-80\%\ of the
3\degspt5$\times$3\degspt5 area of those of the {\it Spektral Durchmusterung},
which served as second epoch plates. At Alger some special plates were taken
to investigate possible systematic errors.
The project has been described in some detail by Plaut in \cite{Voetspoor}.
First relative proper motions were derived for 10,960 stars.
These were made absolute using various fundamental
catalogues for which there was overlap of 850 stars. The results were published
in 1955 and 1959 in two volumes of the {\it Publications and
of the Kapteyn Astronomical Laboratory at Groningen}, the first part
on the relative proper motions by van Rhijn and Plaut \cite{vRP1955} and the
part on the absolute motions by Plaut by himself \cite{Plaut1959}. This enormous
volume of proper motions constituted another major investment of time. It is
not clear what use has been made of it, since until recently these publications
had not been included in ADS (see below).
The second project concerned also proper motions and was a collaboration with
the Yale-Columbia Southern station. It was aimed at measuring proper motions of
the stars in the zone of declination -15$^{\circ}$. The first epoch plates date
from around 1927 and the interval was twenty-four years, which meant that the
probable error of the proper-motion components was $\pm$0\secspt0022. The
telescope was the 26-inch refractor originally erected in 1924 in Johannesburg
under Frank Schlesinger (1871--1943), director of Yale Observatory. Actually,
when he was at Yale in 1922-1924, Jan Oort was briefly considered to be
appointed the astronomer responsible for the setting up of the telescope.
Oort went to Leiden instead, but Schlesinger's intervention had meant
for Oort an
earlier and shorter military service (see \cite{JHObiog}).
In 1952 the telescope was
moved to Mount Stromlo near Canberra, Australia, where it was destroyed
in a bush fire in 2003. The proper motion survey was set up by Dirk Brouwer
and Jan Schilt after they had as directors of respectively Yale and Columbia
initiated the collaboration and orchestrated the move from Johannesburg to
Canberra. The plates were measured in Groningen, but as far as I am aware
the results were never published.
These two programs did constitute the final contribution from Groningen to
the {\it Plan of Selected Areas}.
\bigskip
As indicated above, Borgman took over the program of determining the wavelength
dependence of interstellar extinction with the Groningen telescope. As already
described above he published two more papers on this subject \cite{JB1954} and
\cite{JB1956}. As he stated in the
Introduction of the second of these two papers :
\par \begingroup \leftskip2.5em \rightskip2em \large
The publication of these results concludes the present investigation; the
material available for our instrument and method is almost exhausted and,
moreover, it is considered that the present state of photo-electric techniques
does no longer justify a photographic investigation of the absorption law.
\par \endgroup
So the paper by Borgman in 1956 marked the end of research with the Groningen
telescope. The total harvest was three papers on wavelength dependence of
extinction, when the technique was actually no longer state of
the art. Most likely the telescope
had also been used for student training (as far as there were students).
I will briefly summarize Borgman's further career.
Borgman after completing the work with the Groningen telescope first developed
the instrument to identify variable stars for Plaut and the {\it
Palomar-Groningen Survey} and then moved on to photoelectric photometry. He was
introduced to the Yerkes and McDonald Observatories, undoubtedly through
Adriaan Blaauw, who was adjunct-director at Yerkes under Gerard Kuiper as
director, and in 1957 the new director in Groningen. Borgman collaborated with
Bill Morgan and Bengt Georg Daniel Str\"omgren
(1908--1987) at Yerkes and McDonald Observatories and with Harold Lester
Johnson (1921--1980) of Lowell Observatory and observed extensively at McDonald
and Lowell. Borgman developed a seven color narrow-band (width 5 to 10\%\ of
the wavelength) photometric system and designed a new d.c. amplifier (which
is used to amplify the output photocurrent of the 1p21 photomultiplier used
in photometers). With this seven color system he analyzed the spectra of
OB-stars and attacked again the problem of the wavelength dependence
of extinction.
Eventually he became director of the new
Kapteyn Observatory in Roden (15 km SW from Groningen) and his career
moved to infrared, ultraviolet, and balloon and space astronomy and
in the 1970s on to
science administration. He became Dean of the Faculty of
Mathematics and Natural Sciences, Rector Magnificus and Chairman of the
Governing Board, all of Groningen University, before moving to Den Haag
(the Hague) as
chairman of the Netherlands Organization for Scientific Research NWO and
finally Brussels as first chairman of the European Science and Technology
Assembly of the European Commission.
\bigskip
In 1956 van Rhijn turned seventy, the age of retirement. There was a
small event celebrating this milestone, honoring him for his accomplishments
and major contributions. A special booklet was produced entitled
in English translation {\it In
Kapteyn's footsteps: Sketches of the work program of Dutch astronomers,
offered by friends and colleagues to Prof. Dr P.J. van Rhijn on the occasion of
his seventieth birthday}, published by The Dutch Astronomers Club and dated
Groningen, April 7, 1956. Van Rhijn's birthday was March 24. The date April 7,
1956 was a Saturday, so it was offered at a
special meeting of the NAC on that date held in Groningen.
The contributions to the booklet do not look like written accounts of oral
presentations, but rather short contributions written for the occasion.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.465\textwidth]{figVoetsp2.jpg}
\includegraphics[width=0.525\textwidth]{figVoetsp1.jpg}
\end{center}
\caption{\normalsize Title page and table of contents of the booklet
\textit{In het voetspoor van Kapteyn} presented to van Rhijn at his 70th
birthday. Kapteyn Astronomical Institute.}
\label{fig:voetspoor}
\end{figure}
\begin{figure}[t]
\sidecaption[t]
\includegraphics[width=0.64\textwidth]{figPJvRVoet.jpg}
\caption{\normalsize Photograph of van Rhijn as frontispiece of the
`Voetspoor' booklet of Fig.~\ref{fig:voetspoor}. There is no caption or
accompanying text. Kapteyn Astronomical Institute.}
\label{fig:frontispiece}
\end{figure}
It gives the impression of a sloppy, hastily prepared document (see
Fig.~\ref{fig:voetspoor}). It is produced using offset printing, bound
cheaply with a soft cover without any illustration. On the Contents page the
entry has Bart Bok's name typed as 'B.J. Kok', and then corrected by
over-typing the 'K' with a 'B'. There is no introduction, and the editor
or coordinator is not identified . There is a picture of
van Rhijn in the front of the booklet (Fig.~\ref{fig:frontispiece}),
but without a caption.
Lukas Plaut has, as follows from the correspondence below, organized
the whole and collected the contributions into a single booklet.
The title refers to an expression quite similar in Dutch and in English,
meaning 'following someone's example' or 'going to do something similar to what
someone else has done before'. Maybe unintended this points at a lack of
originality; after all in a literal sense putting your feet on the same spot
as another person means you are not breaking any new ground. This choice for the
title of what could have been a festschrift is a rather minimal token of
appreciation, just like the fact noted in the Introduction of the lack of any
prominent obituary in international journals after van Rhijn's death.
There is a little bit more background in the Oort-Blaauw correspondence.
Jan Oort mentioned on February 21, 1955, that he had received a letter from
Lukas Plaut (not present in the Oort Archives), and noted he is
uncertain what to do about this. It is a typed letter in Dutch, so what follows
is my translation.
\par \begingroup \leftskip2.5em \rightskip2em \large
From Plaut I received a letter these days, a copy of which you will
find enclosed. I do not really know what to make of it. I think it is very
difficult for the people he mentions to write something in 2 or 3 pages that
would be of use to van Rhijn. That is why I think it would be more appropriate
to offer his complete work in beautifully bound volumes, and perhaps also an
album with photos and signatures of all the people who have just collaborated
with him. Let me or Plaut hear what you think about this.
\par \endgroup
On 27 February 1955 Blaauw wrote to Oort from Yerkes
(handwritten in Dutch, my translation).
\par \begingroup \leftskip2.5em \rightskip2em \large
I am as unenthusiastic as you are about Plaut's proposal concerning the tribute
for van Rhijn. I seriously doubt that van Rhijn would care much for a
collection of essays as proposed by Plaut, especially if we apparently do not
think it is worthwhile to print them. I also doubt that I could get people
like Ambartsumian and Baade to write such an unpublishable article.
I believe that the tribute should in any case have a rather intimate character
and should be limited to those who have worked with van Rhijn. I would feel
very much for an album with signatures and photographs and perhaps letters
addressed to van Rhijn from these fellow workers. One could then indeed expand
it with a nicely bound collection of van Rhijn's works, but I do not find that
solution ideal. The collection cannot be complete, for how can the work on the
Bergedorfer Spektral Durchmusterung be represented in it. And would van Rhijn
really care enough? He will probably rarely look in it, since he has his notes
in his much-used hand copies anyway. What do you think of the idea of having
all those who received their doctorates from him write a short article about
the current view of the subject of their dissertation? Such articles, which at
the same time give a picture of the development of astronomy, might be of
interest to a wider public and could be printed somewhere in Dutch or English.
Writing this would not be any problem for most participants. Would the articles
not be suitable material for Hemel en Dampkring, e.g. a special issue. That
would reduce the costs.
\par \endgroup
Interestingly they refer to him as 'van Rhijn' and not 'Piet' or 'Pieter',
which if talking about a mutual good friend would be more logical
(however, they refer to 'Plaut' rather than 'Lukas').
They used first names throughout their correspondence for most other colleagues.
On October 12 Blaauw wrote (typed and in English):
\par \begingroup \leftskip2.5em \rightskip2em \large
I have been very busy lately and not given much thought to the Groningen
succession. I just received Plaut's letter for a short article, which I shall
write.
\par \endgroup
No further mention of the whole business occurs in subsequent letters
between Oort and Blaauw. From this correspondence it appears that Lukas
Plaut played a definite
role in setting up the tribute on the occasion of van Rhijn's imminent
retirement and inviting persons
to contribute to a small publication with collected contributions by
colleagues and friends. Plaut edited the booklet and in his modesty
felt it inappropriate for
him to add an introduction, or even mention his role.
The 13 contributors comprise van Rhijn's students
Jan Schilt, Jan Oort, Bart Bok,
Jean-Jacques Raimond, Peter van de Kamp, and Adriaan Blaauw, and also
Jan Borgman. Dirk Brouwer was a Leiden PhD but
collaborated in the {\it Plan of Selected Areas}. Lukas Plaut was a colleague
and collaborator, Henk van de Hulst and Piet Oosterhoff were important
colleagues from Leiden, while Herman Zanstra and Marcel Minnaert
were directors of the astronomy institutes in Amsterdam and Utrecht. All of them
described their {\it own} work or that of their Institutions, usually, but not
even always, concluding with a sentence or two of personal appreciation of van
Rhijn, but nowhere is anything included like a summary of van Rhijn's
contributions or a statement on the importance of his work. Those persons that
had emigrated to the USA and established themselves there permanently (Bok,
Brouwer and van de Kamp) wrote in English, the rest in Dutch. It seems
altogether not a volume intended for a wide audience.
The Oort Archives (No. 149) contain a text of the speech Oort gave at
the special NAC meeting, presenting the booklet. He acknowledged Plaut's
editing work, regretting he was not present. He described the set-up as
letting each contributor address what interested them at the time. Oort
does review van Rhijn's work in some detail and noted (my translation)
\par \begingroup \leftskip2.5em \rightskip2em \large
Prof. van Rhijn saw, as it were, a great task before him when he was appointed
to the Groningen Laboratory, and he has been completely faithful to this task
during the long period he has been attached to it. This commands admiration.
It is clear from the scope of this work, but also from its nature. It is a
task that only someone capable of self-sacrifice can accomplish.
\par \endgroup
\noindent
And Oort finished by saying
\par \begingroup \leftskip2.5em \rightskip2em \large
Both you and I will be reminded on this occasion of a day when, I believe
almost ten years ago, in more intimate circle, we also faced each other in
much the same way. What I remember most about that day, Piet, is how
beautifully your friend van der Leeuw, who died so much too soon, spoke to you
then.
Van der Leeuw then spoke to you. He more or less said that the main reason why
he felt such a warm friendship for you was your aversion to making too much
fuss and your simplicity, which springs from a deep-rooted sense of truth.
It is not only van der Leeuw who saw this in you. We all appreciate you for
this.
It is also because of this that today, instead of the frills of an official
tribute, we only wanted to give you something of
that which ultimately affects us most deeply: our own work.
\par \endgroup
\noindent
It would have been fitting had Oort provided an introduction along these lines
or an advance summary of this speech to include in the booklet.
\bigskip
In his final years before retirement and actually also beyond van Rhijn
published three more papers in the {\it Publications of the Astronomical
Laboratory at Groningen}, the final one even the last volume of that series.
The institute publications in the Netherlands existed alongside the {\it
Bulletin of the Astronomical Institutes of the Netherlands}, which had been
founded as a national journal in 1921. The {\it Supplement Series} of the
BAN was instituted in the 1960s, the first volume starting in 1966 and the
institute series were already in anticipation phased out then. Between 1960
and 1966 there had been no need for a Supplement publication from Groningen,
but eventually Plaut used the new series and published his measurements for
the {\it Palomar-Groningen Survey} in the Supplements of the BAN.
Van Rhijn's final research was quite original and timely. It actually resulted
also from the Vosbergen Conference, where the local structure in the Galaxy was
seen as an important and urgent subject of research. He started
determining the space distribution of A0-stars near the Sun from low latitude
Selected Areas and for higher latitudes for A0- to A5-stars, also in
Selected Areas, but other sources as well as in cases, where
the number of A0-stars in
the Areas is too small, see \cite{PJvR1955}. It was suggested at
Vosbergen to use A0-stars because of their small spread in intrinsic luminosity.
The spectral types were
taken from the {\it Henri Draper Catalogue}, {\it Bergedorf(-Groningen-Harvard)
Spectral-Durchmusterung} and the Potsdam southern equivalent, and the magnitudes
were corrected for absorption from the color excesses, which could be determined
since the spectral types were known. The densities then follow from the
distribution of corrected apparent magnitudes and the luminosity function,
which he assumed to be a Gaussian around a mean value with 0.5 magnitudes
dispersion. Within 200 pc from the plane of the Galaxy he found that the
A-star density correlated with the hydrogen density derived from the 21-cm
surveys at Kootwijk by Leiden observers.
Next van Rhijn turned to K-giants, \cite{PJvR1956}, doing essentially the same
thing. The percentages of giants among K-stars was taken to be the same as in
various detailed studies in the literature. Using new determinations for the
hydrogen densities derived in Leiden, he found, while for the A-stars that
were closer than
50 pc from the Galactic plane their density correlated with that of
hydrogen, for K-giants there was no evidence for such a correlation. Above
50 pc neither the A-star nor the K-star distributions correlated with that of
the hydrogen. Typically for van Rhijn, there is no discussion on what this
would mean for understanding Galactic structure or the presence of spiral arms.
Van Rhijn's final research paper appeared after his retirement, and concerned
{\it The ellipsoidal velocity distribution of the A stars as a function of the
distance from the Galactic plane}, \cite{PJvR1960}. He extended his study of
A0 stars to A5 at low latitudes
using proper motions to derive the distribution of their
random motions, and from this the properties of the Schwarzschild distribution,
the dispersions in the principal directions and the vertex deviation (see
Section~\ref{sect:extinction}), at various distances from the plane of the
Galaxy. Solving for the axes of the velocity ellipsoid for a sample of stars
can be done from their distribution of proper motions (the components parallel
and perpendicular to the Galactic equator) using a method developed in 1948 in
Leiden by Coert Hendrik Hins (1890--1951) and Adriaan Blaauw. Van Rhijn found
the dispersion of the motions to increase with vertical distance, while their
ratio's remained more or less constant.
These and the vertex deviation of 26$^{\circ}$\
were not unreasonable. He attributed the change in mean vertical velocities with
height above the plane to a mixture of two components with a different velocity
dispersion. Although he investigated the possibility of a difference in spectral
types among the two distributions and presented for this a table, but he
failed to discuss this table and to draw any conclusions from it, nor did he
discuss the implication for our understanding of the structure of the Galaxy.
Note that this was published in 1960,
when the concept of Stellar Populations was widely
accepted and a discussion of the results in this context would have been highly
appropriate. The two components had dispersions of about 10 and 25 km/sec (the
smaller one contributing some 90\%), so could in terms
of the scheme of the Vatican Symposium of 1957
be interpreted as young population I and the (older) disk population
\cite{AB1965}. In that context it would have been no
surprise that A-stars, being relatively young, would overwhelmingly
be part of the young Population I.
These three investigations, comprising van Rhijn's final research, stemmed
from the Vosbergen meeting. The results were interesting and provided new
insight; yet, van Rhijn presented them without any real discussion on what the
impact on our understanding of
Galactic structure at a fundamental level would be. Unfortunately,
the impact as revealed by citations cannot be evaluated with the ADS. During
this research I discovered that the last six volumes of the {\it Publications
of the Astronomical Laboratory at Groningen} were not included in ADS. It
turned out that the scans were in the system, but for some reason the
corresponding records were not created. I was promised this would be corrected.
\section{Van Rhijn's succession}\label{sect:PJvRsuccession}
Van Rhijn's retirement would be at the end of of the academic year 1955-1956.
The University of Groningen had the
choice between on the one hand continuing the Laboratorium and try and raise
its prominence to what it had been in Kapteyn's days by appointing a prominent
astronomer or on the other close it and arrange for a physics professor to take
care of astronomy teaching as a task secondary to his main activities or
appoint an astronomer from Leiden (Utrecht and Amstardam were
possibly even too small in staff for this)
to come on a part-time basis to teach. In the
end they chose to pursue the first option. But it was clear that there was a
need for not only a change in leadership but in addition a modernization of
the way the laboratory was
organized and associated with that new investments and
an increase in staff were required.
The complete reliance on other observatories providing
the plate material, to be measured and reduced in Groningen, was no longer a
guarantee for success.
Van Rhijn was very much concerned with the issue of his succession and the
future of his Laboratorium. Already on November 24, 1952, he wrote to Oort
(from the Oort Archives Nr. 149, my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
\noindent
Dear Jan,\\
I would like to write to you about a matter which has been occupying me for
some time now and which also gives me concern, namely the future development
of the Kapteyn Laboratorium.
As you know, the possibilities for work here are very limited, Kapteyn has
tried to establish a proper possibility for scientific research in Groningen,
which he has been refused. The only possibility, which I saw to change this,
was the purchase of a second-hand mirror, which happened to be offered to me
at the time. I have had the greatest difficulty in obtaining permission to
install the instrument on the roof of the laboratory from money from private
funds, that is how strongly `Den Haag' [the Minister] was opposed to expansion.
The instrument is very useful […] But the weather in Groningen is too
bad for photometric work. A few years ago I applied to Z.W.O. for a
spectrograph, which of course can be used even if the sky is not too clear.
The application was turned down. And so we are still stuck with very limited
possibilities of research. Many times I have received a refusal on a request
to take plates and even if they were granted, the work of the Groningen
laboratory was subordinated to that of the particular
Observatory itself. It is not my
intention to reproach anyone for this. But it does make research in Groningen
very difficult.
In a few years time I will have to resign and of course I am thinking
about the future of the Kapteyn Laboratorium. It has been suggested a few
times to close the laboratory as a financial cut-back measure. I would regret
this very much, not only because the University of Groningen is already
disadvantaged in several respects compared to Leiden and Utrecht, but also
because a concentration of astronomy does not seem to me to be in the best
interest of the enterprise. Just think of the time of Bakhuyzen in Leiden,
Nijland in Utrecht and Kapteyn in Groningen. Leiden and Utrecht were of little
importance for the development of astronomy at that time. Especially in such
times the importance of an independent institute, which goes its own way,
becomes apparent.
If, however, Groningen is to continue to exist, there will have to be other
possibilities for research here in the future than there have been so far.
But I have given this matter some thought and one obvious possibility is radio
research. In the decisions to be made concerning this research I would ask you
to consider the possibility that in the future the Groningen astronomers will
wish to participate in this research. I would also be glad to attend the
meetings of the Foundation, since the development of radio research in the
Netherlands is, in my opinion, also very important for the future of the
Kapteyn Laboratorium. What do you think of this?
\par \endgroup
Oort replied on 29 Noyember, 1952, saying among others (Oort Archives Nr. 149; my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
The starting point is who you will be able to get as your successor upon
retirement. Should Blaauw by then be available for that, I would not be the
least bit worried about the future prosperity of the lab.
\par \endgroup
He then went into a long argument why it is difficult to add another astronomer
to the Board of the Foundation for Radioastronomy, in the end suggesting
\par \begingroup \leftskip2.5em \rightskip2em \large
[...] that you attend a meeting occasionally, when an issue in which you have
a special interest is on the agenda? Another possibility would be to have
somewhat alternating meetings to discuss the technical issues
of the instruments and
the like and meetings to discuss the general scientific side, which could then
include a few more people.
\par \endgroup
The issue of Oort keeping van Rhijn out of the Board of the radio astronomy
foundation is outside the present scope, but has been treated in detail in
Astrid Elbers' thorough study of the history of Dutch radio astronomy
\cite{Elbers} and need not be covered here. In the end the outcome was
that van Rhijn joined the board. On September 14, 1956, according to him a few
days before his formal retirement, van Rhijn wrote to Oort, saying he was
grateful the latter considered including others than Leiden astronomers in
the board of the Kerkhoven-Bosscha Fund, but complained rather bitterly that he
(and Kapteyn) had been denied an observatory with the argument that two in
Leiden and Utrecht was enough, but now a very expensive third one was being
built (in Dwingeloo) without involving the director in Groningen in this.
Returning to the future of astronomy in Groningen, I quote from a letter of
van Rhijn to Oort of December 15, 1952 (Oort Archives, Nr. 149; my
translation). On the issue of appointing a successor or closing the
Laboratorium, he described the position of the Faculty:
\par \begingroup \leftskip2.5em \rightskip2em \large
In such a Faculty [covering natural sciences] astronomy
should, in the opinion of our Faculty, be represented. Astronomy is too
important to leave the teaching of that subject to a lecturer who comes to
Groningen once a fortnight from the west of the Netherlands,
\par \endgroup
This shows that there was a firm decision in Groningen to continue with a
professor and an institute in astronomy. He continued:
\par \begingroup \leftskip2.5em \rightskip2em \large
If one develops the practice of astronomy at an institute to a very high degree
compared to others, then there is a danger that the smaller institutes will be
closed down as prosperity declines in the Netherlands. Thus in 1932 the plan
was to close down the Observatory in Utrecht and the Kapteyn Laboratory was in
those days also on a proposal to disappear. All the more reason not to make
the difference in size and importance of the various institutes too great.
It is undoubtedly true that a competent astronomer, appointed as my successor,
will be able to bring the Kapteyn Laboratory to a favorable development. But
it is precisely the question of how to make the appointment to Groningen
plausible for this man, and to this end new possibilities for scientific work
will have to be created here. Otherwise he will be grateful for the honor, but
look for a job elsewhere.
Sending a Groningen astronomer abroad to collect material could indeed
partially solve the difficulties. The Kapteyn Laboratorium would have to have
a workshop for the production of instruments and tools, which are not available
at the observatory. If one does not have access to a workshop, it will be very
difficult to undertake something new and one is dependent upon work of the same
nature as is already done at that observatory.
In this way one remains dependent upon the directors of other institutes. This
remains a major obstacle, as I have painfully experienced on several occasions.
\par \endgroup
Van Rhijn added to the letter:
\par \begingroup \leftskip2.5em \rightskip2em \large
\noindent
P.S. I am convinced of Blaauw's abilities. His articles on the O-associations
I have read. It is excellent work. It seems unlikely to me that he would be
willing to come from Yerkes to Groningen in a few years time. If he feels
somewhat at home in America he will stay there because of the greater
possibilities for research.
\par \endgroup
So, Jan Oort had been involved from an early stage on. It was obvious that his
favorite was Adriaan Blaauw. The problem was, of course, the latter's move to
Yerkes Observatory.
In the Oort Archives we can find some correspondence between Blaauw and Oort on
this issue. On March 8, 1956, Blaauw wrote (handwritten, in Dutch) to Oort
(my translation).
\par \begingroup \leftskip2.5em \rightskip2em \large
\noindent
Dear Jan,\\
As you will of course know, van Rhijn has indeed asked me, on behalf of the
Groningen Faculty, to succeed him. I have already replied to him that I am
seriously considering the offer, and have exchanged views with him about
certain conditions which I believe must be met in any case. However, I have
also made it clear that my decision will also depend on the future character
of my position here, which, now that the first three-year period ends soon,
will have to be reviewed in any case. On this last point I have not yet heard
a decision. Nevertheless, I am resolved to make the decision within a few weeks
whether or not I desire the Groningen position, at least in principle. Van
Rhijn would like a speedy decision.
I have already informed van Rhijn, among other things, that I believe that the
special character of the laboratory as a place for processing material obtained
elsewhere could be fruitfully maintained if the necessary arrangements were made
so that regularly Groningen observers could personally collect the material
elsewhere. This requires, among other things, a guarantee of a relatively high
annual travel fund, and an expansion of the staff with at least one young and
energetic astronomer.
A subject to which van Rhijn alluded but which I do not yet clearly understand
concerns the part that Groningen should play in radio astronomy. Van Rhijn did
mention that Groningen has a seat on the board of the Radio Foundation, but I
would like to be clear first of all of the view that you and Henk van de Hulst
have on this. You may remember that we touched on this point during our
conversation in Hamburg last summer, and my impression was that you were rather
reserved about the participation of other institutes in the 21 cm work.
[...] Would you write me what you
think now of a possible Groningen share in Dutch research? Of course I would
also like to hear your views on the Groningen research work in general.
\par \endgroup
He added that he will drop the plan to visit the Netherlands that
summer partly in view of his possibly going to Groningen,
but mainly because of a serious backlog of work.
This apparently crossed a letter by Oort to Blaauw on March 12, in which Oort
mentioned that he had already
heard from Maarten Schmidt that Blaauw would not visit the
Netherlands that summer, but invited him to come in the fall for a few months
as lecturer on a temporary position.
When Oort received Blaauw's letter he answered `immediately' (March 14, 1956;
my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
\noindent
Dear Adriaan,\\
\noindent
[...]
I am glad that now Groningen has taken the first step and congratulate you with
the fact đat you have been placed as Number 1 on the list. I have not seen van
Rhijn for a long time, it seems as if his already very limited appetite for
travel has become even less lately.
I don't have to tell you how much I would enjoy your coming to
Groningen. There is not the slightest doubt in my mind that we would work
together pleasantly and fruitfully in all the areas you would like. I hope
that you feel the same way and that you would never feel that the larger
Sterrewacht in Leiden is drawing too much in for itself. This is certainly not
our ideal. It is our ideal to build a useful collaboration for all concerned.
Groningen, Leiden, Dwingeloo, Hartebeestpoort and Pretoria could form a
wonderful set of institutes for research of the Milky Way system, interstellar
media and evolution.
I meanwhile agree with you that making ample funds available for travel to
America and South Africa and expanding the staff with a scientific staff member
for Groningen are the necessary conditions for healthy development. In most
cases this can at present be done incidentally through applications to Z.W.O.
Although the acceptance of such requests will depend on the resources available
at Z.W.O. and the number of other requests, in my experience if the request is
well-founded we can almost count on it being accepted [...]
But now to the actual question. I can answer without reservation that there
will certainly be very much room for Groningen to participate in the work of
the radio observatory, at least if someone like you were to come and put it
in good shape, [...]
\par \endgroup
As remarked above already,
it is remarkable that both Blaauw and Oort kept referring to `van Rhijn'
without using his first name, while they used first names addressing each
other, and in their correspondence used first names when referring to e.g.
Henk van de Hulst or Pieter Oosterhoff (although in correspondence above they
refer to 'Plaut' without using his first name).
Both Oort and Blaauw had obtained
their PhD's with van Rhijn as their 'promotor', but were senior now
themselves. Oort addressed van Rhijn as `Piet' when
writing him, and Blaauw probably did the same.
The reference to `Dwingeloo' of course is to the 25-meter Dwingeloo radio
telescope, about to become operational, and `Hartebeespoortdam' to the Leiden
Southern Station. This latter facility resulted from the condition for
Hertzsprung to accept his appointment in 1918 that there should be a southern
telescope for his research. Willem de Sitter then had arranged access for
Leiden to the facilities of the Unie Sterrewag in Johannesburg, including the
Franklin-Adams Telescope (referred to as F.A. below). Later this was
extended with the Rockefeller Telescope, a twin 40-cm astrograph, that Leiden
had built from funds provided by the Rockefeller Foundation. After the war
Oort had furthermore arranged funding from Leiden University for an additional
90-cm `light-collector', which would become operational in Hartebeespoortdam
near Pretoria in 1958. It was to be
outfitted with a five-channel photometer built by Th\'eodore (Fjeda) Walraven
(1916--2008). The Leiden station (and the Unie Sterrewag telescopes) had been
moved from Johannesburg to Hartebeespoortdam in 1957.
\bigskip
Oort followed his letter up on April 16, 1956 (my translation;
`Pieter' is Pieter Oosterhoff):
\par \begingroup \leftskip2.5em \rightskip2em \large
In this connection I would like to write to you that during the NAC meeting in
Groningen, where van Rhijn was honored, Borgman gave a lecture with
demonstration on the electronic blink-machine. I was very impressed by the
skill and complete mastery of the subject which he demonstrated. He has thought
the problems through exceptionally well and also has great skill in putting
things together so that they work effectively. He reminded me very much of
Walraven in this respect, only he is much calmer and more balanced, If you
could keep him in Groningen you would, I believe, have a worker of the very
highest quality.
Indeed I have written to the Groningen Faculty that we would also like to
foster a collaboration at our Southern Station. Pieter and I have reviewed
this after receiving your letter. We are of the opinion that there will be
room enough to also have Groningen observers there at times, when we have both
the light-collector and the Rockefeller there as well as part of the F.A.
Moreover, Walraven is going to build another telescope for continuous
observation of special variable stars [...]
Fee for the use of our instruments we would rather not want to ask. Only if
you occasionally want to take over part of our time with the Radcliffe
telescope would it be reasonable if Groningen would bear the cost.
\par \endgroup
However, on April 20, 1956 Adriaan Blaauw wrote (typed, in English):
\par \begingroup \leftskip2.5em \rightskip2em \large
\noindent
Dear Jan,\\
I have, after long hesitation, decided to decline the offered position at
Groningen, It was a most difficult decision to make, and several times during
the past two months I felt strongly inclined to go back to [the Netherlands].
But I was, after all, not really convinced that it would be worth discontinuing
my present position and work, and transferring my family again, especially not
because I believe there are other astronomers who may be interested in this
position and do as good a job -- and who at present work under less favorable
circumstances than I do.
\par \endgroup
He must have
had a particular person in mind, but did not specify who it was. Oort urged
him to reconsider, or at least postpone a final decision, until after the summer
when he might spend some time in Leiden and could visit Groningen.
In a handwritten PS to an (otherwise missing) letter to Oort on
May 29, 1956, Blaauw mentioned that van Rhijn had asked him who he had had in
mind and that he had told him it was Bruno van Albada.
Gale Bruno van Albada (1912--1972; for an obituary see \cite{JHO1973}), uncle
of at that time student Tjeerd van Albada in Groningen, had studied in
Amsterdam and obtained a PhD under Pannekoek in 1945. The thesis was in
Dutch but concerned pioneering methods of analysis of spectral lines to
obtain chemical element
abundances that would come to fruition only after electronic computer were
used, and he worked on theories of the origin of the chemical elements. After
a short stay in the USA van Albada moved to Indonesia in 1949 to become
director of the Bosscha Sterrenwacht, which he rebuilt into a working
observatory under extremely difficult (`less favorable' in Adriaan Blaauw's
words is a euphemism) circumstances.
From the correspondence between Oort and van Rhijn it can be learned that
Oort suggested all activities in Groningen related to van Rhijn's succession
should be postponed. Blaauw was to
visit Leiden in the fall and Oort seemed hopeful he could persuade Adriaan
Blaauw to visit Groningen and reconsider. It turned out, according to this exchange,
that in addition to
van Albada also Peter van de Kamp was reconsidered an option. He was a student
of van Rhijn and worked in the field he had always worked in, astrometry. Van
Rhijn asked Oort on his opinion on these two persons
without giving his own. He did apparently not
know van Albada very well. He wrote to Oort (May 7, 1956, mt translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
We have heard that he is stubborn and finds it difficult to accept the opinions of
others. These traits would make it difficult to work with the Faculty.
\par \endgroup
Oort replied that he never had any problems with van Albada and if he had a
difficult character that was 'more appearance than reality'. He did think
van de Kamp would certainly do good work, but Groningen needed someone with
the 'enthusiasm and talent to start something new'. Of the two he
definitely preferred van Albada.
Blaauw in the mean time had also himself
suggested to postpone any decision until September
when he was to visit the Netherlands. As far as I am aware, van Albada and van
de Kamp were never approached.
Through Oort's intervention (as seems the case according to the exchamge of
letters) Groningen University postponed the decision and
decided not to approach van Albada, and await
Blaauw's reconsideration. Eventually, but a few years later,
Bruno van Albada felt forced to leave
Indonesia in view of the growing discontent towards Dutchmen as former
colonizing nationals. He moved to Amsterdam in 1958, where he became director
in 1959.
Blaauw indeed spent a few months in Leiden in the fall of 1956, eventually
October and November, on a temporary appointment as lecturer arranged by Oort.
The visit had the effect Oort had
hoped for. On December 10, 1956, Adriaan Blaauw's wife Atie wrote to Oort that
the day after his return Adriaan had fallen ill with what had been diagnosed
in the meantime as jaundice and was unable to write himself (my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
\noindent
In principle A. has decided to accept the position, but he needs to write to
Groningen about further details. [...]\\
Prof. van Rhijn does not yet know all this; A. has not talked to him about this.
What do your think of this in principle?
\par \endgroup
Oort responded on December 17 that 'Mieke and I are very much delighted'.
Blaauw accepted the position in Groningen with some special agreed conditions.
These conditions were a new staff position, travel fund for
observing and provision to build an observatory with a workshop to build
instrumentation outside Groningen. Hugo van
Woerden was appointed at the new staff position to start radio astronomy as part
of the research effort. Jan Borgman and student Tjeerd van Albada did site
surveying in the area around Groningen and found the village of Roden, some
20 km to the southwest of Groningen, to be best in terms of darkness at night
and accessibility. The Kapteyn Sterrenwacht has been inaugurated there in 1965
and operated a 62-cm telescope.
\bigskip
Blaauw had on September 1, 1956, become adjunct-director of Yerkes and McDonald
Observatories, so his future in the USA seemed very bright. Yet, he decided
to accept the Groningen directorship. In 1984 Adriaan Blaauw celebrated his
70-th birthday.
At this occasion in September 1984 the Dutch astronomical community organized
a symposium to honor him. At this meeting Hugo van Woerden, who was
the first appointment Blaauw had made
in Groningen and his first PhD student, gave an extensive
overview of his career \cite{HvW1985} and I quote from this as follows:
\par \begingroup \leftskip2.5em \rightskip2em \large
Groningen had acquired world fame under Kapteyn in 1900-1920. But
while van Rhijn, as successor to Kapteyn (1921–1956), had continued the
latter's work with distinction [...], the Kapteyn Laboratory had remained small
and poorly equipped, and its research no longer drew much attention.
Within the Netherlands, astronomy was dominated by Leiden under Oort's
directorship. Leiden Observatory was 'big' and famous; it had 3 full
professors (Oort, Oosterhoff and van de Hulst), a tenured academic staff of
about 10 plus a dozen or more graduate students; its personnel totaled about
40. Its pre-war fame due to de Sitter, Hertzsprung and Oort had been strongly
enhanced by post-war developments, notably the 21-cm line studies of rotation
and spiral structure of our Galaxy, but also research on interstellar matter,
stellar associations, comets and variable stars. Utrecht, with Minnaert as
professor, and Houtgast and de Jager as associate professors, was smaller but
growing rapidly; it was famous for its solar work and developing in stellar
astrophysics. Am\-ster\-dam, under Zanstra, was small but had many good students.
Groningen, with van Rhijn retiring, had one staff member (Plaut), one
scientific assistant (Borgman), 3 computing assistants, one draftsman, one
junior technician, no secretary, and ... no students. Borgman's doctorate in
1956 had been the first in 10 years.
\par \endgroup
That is to say, no {\it graduate} students. In 1957 Tjeerd van Albada and Harm
Habing entered their fourth and third year respectively as undergraduate
students.
\par \begingroup \leftskip2.5em \rightskip2em \large
\noindent
[...] By
early 1968, when Blaauw accepted the Scientific Directorship of ESO, the total
number of people working in the Department amounted to about 40 — a growth by
a factor of five in ten years.
\par \endgroup
\bigskip
Now the question is, what made Adriaan Blaauw accept the appointment at the
Kapteyn Laboratorium? And why he did he go to Yerkes in the
first place? For this it is important to find out what kind of person
he was. First note that Blaauw was a person that liked adventures,
challenges and exploring new grounds. But he was also unconventional and could
be annoyed and amused at the same time
by bureaucracy. To illustrate this consider
the following anecdote. In the central building of the
University of Groningen (the Academy Building) many rooms are decorated with
paintings of well-known (emeritus) professors, dressed in gown, jabot and
sometimes barret
(as Kapteyn in Fig.~2). Now Blaauw was not regarded an emeritus professor, as
he had {\it resigned} in 1975 and therefore never {\it retired} from a Groningen
professorship. And the administrators adored bureaucracy,
as they were cautious not to set any precedents.
After his retirement from Leiden Blaauw had been given the status of guest
researcher in Groningen, because he lived nearby, but that would not qualify
him to have his portrait added. It meant he also
did not receive announcements and invitations other emeriti did. As others I
was annoyed by this and thought it was thankless and unfair, and I appealed a
few times to the upper officials, trying to use my weight as director of the
Kapteyn Institute or dean of the Faculty of Mathematics and Natural Science,
but to no avail. Blaauw was on the one hand a bit piqued by this bureaucratic
nitpicking, but foremost amused. We were unable to break this deadlock for a
long time. With Adriaan's upcoming 90-th birthday in 2004 we wanted to add a
painting of him to the gallery in the Academy Building.
In the end Rector Magnificus Simon Kuipers intervened and Blaauw
was appointed for one year as honorary professor so that he would qualify for
the status of Groningen emeritus professor after that and a painting of him was added.
Typically for Adriaan Blaauw he selected without hesitation the by far most
unconventional painter of the lot he could choose from. This was Adriana Engelina
Maria (Janneke) Vieger. It was displayed on
the cover of the Institute's 2004 Annual Report \cite{AnnRep2004}, see
Fig.~\ref{fig:AB2004}.
\begin{figure}[t]
\sidecaption[t]
\includegraphics[width=0.64\textwidth]{fig2004.jpg}
\caption{\normalsize The cover of the 2004 Annual Report of the Kapteyn
Astronomical Institute \cite{AnnRep2004} with Adriaan Blaauw's painting
by Janneke Viegens
for the gallery in the Academy Building. University of Groningen.}
\label{fig:AB2004}
\end{figure}
That Blaauw was also adventurous can be illustrated with his travels when a
young staff member in Leiden. He has told on many occasions that he very much
enjoyed his participation of the second Kenya expedition of the Sterrewacht
Leiden. These two expeditions (in 1931-1933 and 1947-1951) aimed at measuring
absolute declinations, that is to say free from corrections for flexure in the
telescope and atmospheric refraction. The principle is that at the equator
declinations can be measured as the angle along the horizon (azimuth)
between the positions where a star did rise and set or
between one of these and that of the north (see \cite{JKM91} or
\cite{JHObiog}). Blaauw was at the
Kenya site for five months in 1949-1950. His description of the extensive
travel involved, full of detours through Africa,
in his autobiographical contribution to the {\it Annual Review},
\cite{Blaauw2004}, illustrates very well his adventurous character.
With this in mind we can
understand that Blaauw gave up a permanent
position in Leiden to accept a temporary one at Yerkes and McDonald with less
pay. Of this he said in an unpublished interview with Jet Katgert-Merkelijn
some time in the 1990s (my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
But then you see, I did decide to take that job in America. Of course, that was
also a decision of, um.... Well, on the one hand it was quite nice in Leiden.
On the other hand, I also had the feeling that others in Leiden also had,
of .... a certain stabilized situation. Oort was the boss and there was, eh
.... Oosterhoff, who directed things in that other building which we then
called the `Astro'. And then you had the feeling of `Yes, you could sit
here for another 20 or 25 years, but the world is so big and there's so much
more interesting to look at, so, eh .... let's do that, so I just resigned as
a lecturer then. And I accepted a temporary job at the University of Chicago
at the Yerkes Observatory. In retrospect, actually a curious gamble. And I
think few people at present would still do that, that you give up a
permanent job for ..... And that was also a job in America in which I was, I
think, paid less than I was in Leiden, but there was apparently so much that
feeling of either you stay here and then you're a bit walled in, although,
walled in in a nice community, or you go and have another adventure. Well,
that's what I did. Look, others have done it too of course, Gart Westerhout
with the same feeling. And there have been others like that.
\par \endgroup
Adriaan Blaauw saw a lack of room to develop and start new initiatives in
Leiden where Oort dominated and everything was set and determined by him.
Yerkes Observatory of
the University of Chicago had been founded just across the state border
from Illinois in Williamsbay, Wisconsin, by George Hale in 1897, who erected
there the largest refracting telescope with an aperture lens of 40 inches.
Yerkes had become one of the leading centers in astronomy in the United States,
especially under the directorship of Otto Struve (1897–1963), descendent of a
long von Struve line of famous Russian astronomers from St. Petersburg (Otto
had dropped the 'von' when he became a USA citizen). Blaauw twice spent
extensive periods at Yerkes of nine and six months (1947-1948 and 1952) --
as was
the case for Kenya without his family. Fig.~\ref{fig:Yerkes} shows him with
Jan Oort, director Struve and later director Gerard Kuiper in 1947.
\begin{figure}[t]
\sidecaption[t]
\includegraphics[width=0.64\textwidth]{figYerkes.jpg}
\caption{\normalsize Adriaan Blaauw (left)
at Yerkes Observatory with from left to
right Gerard Kuiper, Jan Oort, and Otto Struve. These frames are cut
from a group photograph of staff and visitors at Yerkes Observatory in 1947.
Oort Archives.}
\label{fig:Yerkes}
\end{figure}
Yerkes employed some first rate astronomers, including in addition to Struve,
Jesse Greenstein (after his work referred to above
on the wavelength dependence of interstellar extinction he moved to the
California Institute of Technology), Surahmanyan Chandrasekhar (1920--1995;
known e.g.
from his work on white dwarfs and degeneration pressure), William (Bill)
Morgan, William Albert Hiltner (1914--1991; another pioneer of photo-electric
photometry), Bengt Str\"omgren (wel-known for his work chemical composition of
stars and physics of the interstellar medium), and others.
Yerkes was responsible for the
operation of the 84-inch telescope on McDonald Observatory in Texas, one of
the most productive American telescopes, and Yerkes astronomers had easy access
to it. In 1947 Struve left for Berkeley and then Kuiper had been director until
Str\"omgren was appointed in 1950; the latter stayed until 1957 when he went to
Princeton and Kuiper took over again. Blaauw had published important papers
with Struve and Morgan.
\bigskip
Coming back to the Groningen appointment the question is: why then did he
elect to leave the USA and become director of a place that was in poor
shape with little promise to flourish again? By that time he was 43 years of
age. This question was not covered in the
Katgert-Merkelijn interview (which concerned Jan Oort primarily), but it
was in the AIP interview. Here are some excerpts:
\par \begingroup \leftskip2.5em \rightskip2em \large
So I said, 'Well, certainly I want to stay at Yerkes till somewhere in 1957.'
\noindent
[...] It was to complete things I had been working on, and also because we were
not quite sure that we wanted to go back to [the Netherlands]. We had gone to
the United States, as I said, more or less as emigrants from [the Netherlands],
and we liked it very much there. Where we lived at William's Bay was a very
pleasant place and we had very nice friends there. The educational situation
for the children was not bad. [...]
I think we all felt that the possibilities for the future of children in [the
Netherlands] were rather limited. [...]
On the other hand, I felt attracted to the job in [the Netherlands], and there
was a circumstance -- the general situation at Yerkes that had developed at
that time. [...]
And while people have sometimes said that you can have at an opera only one
prima donna, and a lot of good players, but you cannot have six prima donnas.
It doesn't last for long, unless you have one person who very strongly keeps
it together. And maybe one should say that there were all these prima donnas,
but not one who kept the whole thing together strongly enough, by one common
motivation, you see. I think one should look at it in that way. Maybe not the
fact that there were frictions. There are frictions almost everywhere. I don't
know of any observatory where there have not been frictions. […]
[...], the catalogue work [at the Kapteyn Laboratorium] had been practically
finished by that time. On the other hand, radio astronomy was developing in
[the Netherlands]. There was the Dwingeloo Radio Observatory that was dedicated
in 1956. I was still away then. It really opened the future for participation
in that work. So then when I went back to [the Netherlands], I said of course,
`Now, this place has to be, in a way, rebuilt, modernized, and we want to
participate in the radio work.'
\par \endgroup
Blaauw saw it as a challenge and an opportunity to bring the laboratory in
Groningen back to prominence, while at the same offering him the possibility
to gather his own research group around him and lead it to become a significant
factor in astronomy. The complex composition and balance between personalities
on the Yerkes staff with many `prima donna's' was not the environment he was
looking for. You either strive to be a prima donna too or settle with being one
of the smaller players. This was to some extent the same as working in Leiden
under the dominance of Oort had been, which made him leave Leiden.
Groningen meant working next to Oort,
not under him. And, of course a challenge and an adventure were opportunities
never to let pass.
\begin{figure}[t]
\sidecaption[t]
\includegraphics[width=0.64\textwidth]{figportret.jpg}
\caption{\normalsize Pieter Johannes van Rhijn in 1957. From the collection of
Adriaan Blaauw and Kapteyn Astronomical Institute.}
\label{fig:portret}
\end{figure}
\section{The Laboratorium after van Rhijn}
At the end of the academic year 1955-56 van Rhijn had been given honorable
discharge since he had become 70 years of age earlier during that year. Like the
appointment of professors this was by Royal Decree. However, since he had no
successor he was appointed, also by Royal Decree and on the same date of July
30, on a temporary basis to be responsible for education in astronomy. He
remained listed in the {\it Yearbook} \cite{Yearbooks} of the University of
1956 as the director of the Kapteyn Laboratorium. The next {\it Yearbook}
mentions the appointment of Blaauw, but no discharge of van Rhijn.
When Blaauw had taken over, van Rhijn prepared the publishing of his final
research as detailed above: his last publication \cite{PJvR1960} constituted the
final volume of the {\it Groningen Publications}, number 61 in 1960. He also
died that same year, on May 9, 1960 at the age of 74. In the {\it Yearbook}
of the University \cite{Yearbooks} for 1960 he is remembered in an {\it In
Memoriam} by Adriaan Blaauw, of which I will quote a part (my translation).
\par \begingroup \leftskip2.5em \rightskip2em \large
When van Rhijn accepted his position in 1921 as Kapteyn's successor with
a lecture on `The Gravitation Problem', he consciously followed in his teacher's
footsteps. He considered it his task to carry out the grand plan designed by
Kapteyn. This plan envisaged an extensive investigation of the structure of
the Galaxy. At the time of Kapteyn's resignation, important preparatory research
had been completed; some highly significant results had already been obtained.
However, only the foundations had been laid for the broad scope actually
intended. A start had been made on international cooperation, with the aim of
systematically collecting all kinds of observation data. Throughout his career,
van Rhijn has been the driving force behind this `Plan of Selected Areas'.
With enthusiasm and unyielding perseverance he managed to obtain the
cooperation of observatories all over the world and, what was certainly not
the easiest thing, he managed to persuade almost all collaborators to complete
their often time-consuming work. The arsenal of data thus created is a lasting
monument to van Rhijn's efforts.[...]
\noindent
Van Rhijn had a, for astronomy, large number of students. They can be found in
important posts at observatories abroad — in Australia and in the United States
-- and in the Netherlands. In all their work the imprint of their teacher can
be detected. Many of them -- both students and colleagues-- are now mourning
the passing of a friend whose exemplary rectitude they admired. Because his
thinking and his actions, both in his work and in his daily life, were
dominated by sincerity and modesty, those who came into contact with him on a
superficial level were under the impression that he barely cared about what
was going on in the world around him. But those who knew him better know that
he had a profound interest in and a sharp judgment of what was going on in the
world.
The War years were very difficult for van Rhijn: his rectorate ended
under the German occupation and then he suffered
from extended illness. With great willpower
he resigned himself to the therapy that demanded so much patience in
sanatorium and hospital. Fortunately, after the war, he was given many more
years of fruitful scientific work.
Van Rhijn was averse to insincere honors. He could speak with glowing
indignation about speeches at special occasions in which merits were
exaggerated and shortcomings concealed. He lived in the deep awareness that
there is a judgment of man and his work, higher and purer than that which can
be pronounced by fellow men.
\par \endgroup
Van Rhijn left behind his twenty years younger wife after 28 years of marriage
and their two children, who by now had grown up and were in their twenties.
Regnera L.G.C. de Bie survived him by 37 years. After she died in 1997 she
was buried next to her husband in a cemetery in the north of the city of
Groningen (see Fig.~\ref{fig:graf}).
\bigskip
\begin{figure}[t]
\sidecaption[t]
\includegraphics[width=0.64\textwidth]{figgraf.jpg}
\caption{\normalsize Tombstones on the graves of van Rhijn and his wife on the
cemetery `Selwerderhof' in the city of Groningen. From
`Online begraafplaatsen' \cite{Graven}.}
\label{fig:graf}
\end{figure}
It would carry us beyond the scope of this article to go into the development
of the Kapteyn Laboratorium under the directorship of Blaauw in the same
detail as done here for the long period of van Rhijn's leadership. Yet in
order to contrast how it to some extent regained its prominence, a short
summary of the first decade or so is appropriate.
Adriaan Blaauw assumed his position
in Groningen on September 1, 1957. He immediately
started to expand the Laboratorium by hiring a young radio-astronomer from
Leiden, Hugo van Woerden (1926--2020). Van Woerden had studied astronomy in
Leiden and had been involved in the research with the Kootwijk radiotelescope,
a refurbished German radar antenna of the Atlantikwall, a 7.5 meter
`W\"urzburg Riese'. In 1955 he had started a PhD protect that involved the
25 meter Dwingeloo radiotelescope that subsequently had become operational in
1956. This work on the structure and motion in the interstellar gas in the
Orion region was transferred to Groningen and resulted in the first PhD
thesis completed and defended under Blaauw in 1962.
Blaauw's next hire was Andreas Bernardus (Andr\'e) Muller (1918--2006). He also
came from Leiden after having obtained his PhD in 1953 with Oosterhoff as
supervisor on a thesis on the variable star XZ Cygni, an RR Lyrae type variable
star. He spent a few years at the Leiden Southern Station, as main observer
in long-term photometric programs and oversaw the move
of the Rockefeller Telescope from Johannesburg to Hartebeespoortdam. At the
new site Walraven was busy erecting the Light-collector. Two Leiden astronomers
in South-Africa was too much of a luxury, and Muller and Walraven did not get
along very well, so one of the two had to come back to the Netherlands. In his
interview with Jet Katgert-Merkelijn, Blaauw recalls:
\par \begingroup \leftskip2.5em \rightskip2em \large
Then there are persons like Andr\'e [Muller], who were not treated very nicely
by Jan [Oort]. Andr\'e had been in South Africa with his family. It must have
been around '57, when Andr\'e came back with his wife and he said to me that
he had been told to look for a job elsewhere. That was very painful and had
to do with Oort feeling that Walraven was the better man to be there. Andr\'e
and his wife Louise had a large family. They had 6 children, although I don't
know they had already 6 then. [...] In any case André had been in Africa for
quite some time and looked after things very well, but he was simply dismissed
and I felt that was very unjust.
\par \endgroup
Now, the developments in the project to found a large European Observatory in
the south, for which Oort and Baade had taken the initiative had developed to
the stage that there had been an `ESO Committee' of which by then Oort was
permanent President. With the limited funding that was provided by the
members of the informal organization, site testing campaigns had been organized.
Blaauw was very much involved in all this and wanted to hire Muller to
contribute (my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
Well, Andr\'e then was given a job in Groningen. I had convinced people to
accept him, since the situation in Leiden was very poor. And then the
developments in the framework of ESO demanded that someone would supervise
things that developed and Andr\'e with his experience in South Africa was the
right person. And slowly he shifted from a job in Groningen to one at ESO. It
all worked out well for him, but he and his family sacrificed a lot when he
worked in Chile. That has been very difficult.'
\par \endgroup
The sequence of events was as follows:
In the same year 1957, Oort in this capacity as President of the ESO Committee
had approached the Ford Foundation for funding to proceed to found the
European Southern Observatory ESO. This was successful in 1959.
In that year Blaauw took up the role of Secretary of the ESO Committee and
was keen on playing a major role in the setting up of this organization.
In January 1964, four of the five countries (the Netherlands, Sweden, Germany and
France) had signed and ratified a convention, the Ford funds
became available. Belgium ratified only in 1967, a few
months after Denmark had joined as the sixth member state.
The appointment of Andr\'e Muller in 1959 ensured a major involvement of the
Kapteyn Laboratorium in ESO \cite{ABESO}.
Muller turned out instrumental for ESO; a
well-written article was published by Richard West at the time of Muller's
death, which describes his work and personality very well \cite{WestAM}.
In 1961 a young student Martien de Vries was added to the personel
of the laboratorium to work with
Muller and Blaauw on stellar photometry using the Leiden Southern Station.
Unfortunately de Vries never finished his PhD thesis and after he
participated setting up the Kapteyn Observatory in
Roden under Borgman, he went on long
term sick-leave and eventually was dismissed from the staff late in the 1970s.
In 1968 Tjeerd van Albada and Harm Habing obtained their Candidaats
degrees (Bachelor). In
respectively February 1962 and March 1963 they passed their Doctoraal exams
(Masters) and started PhD research.
The radio group under van Woerden was extended in 1962 with the appointment of
Swiss born Ulrich J. Schwarz (1932--2018), who joined the Kapteyn Laboratory in 1962 to become its second radio astronomer. For an {\it in memoriam} see
\cite{UJS}. He had studied physics in Bern, but did spend a year (1957--1958)
in Leiden working on radio astronomy and receivers (with American Charles
Seeger preparing for the Benelux Cross Antenna projects, see e.g.
\cite{JHObiog} or \cite{JHOEng}), where he had met Hugo van Woerden's sister
Elisabeth, whom he married in 1961. Van Woerden left after obtaining his
PhD for a two-year fellowship of the Carnegie Institution at the Mount Wilson
and Palomar Observatories and Schwarz took over the coordination of radio
astronomical work in Groningen
And in 1960 a professorship of astrophysics had been allocated by Groningen
University, which after a long search was filled in 1963 by appointing
Stuart Robert Pottasch (1932--2018) \cite{SRP}. He studied at Harvard and
Colorado and had been a postgraduate fellow at Leiden and postdoctoral fellow
at Utrecht and Paris before joining the Kapteyn Laboratorium. His field was
stellar atmospheres and interstellar physics.
So within six years the Kapteyn Laboratorium had grown from three staff
members (Blaauw, Plaut and Borgman) and no students to seven staff members
(plus van Woerden, Muller, Schwarz and Pottasch) and (as it turned out)
three students!
\bigskip
\begin{table}[t]
\begin{center}
\caption{\normalsize
Number of articles in the \textit{Bulletin of the Astronomical Institutes
in the Netherlands} and its \textit{Supplement Series} (BAN) and the share
of the Kapteyn Laboratorium Groningen (RUG).}
\begin{tabular}{crcccr|crccr}
\toprule
Years & BAN & RUG & \% &\ \ \ &\ \ \ & Years & BAN & RUG & \% \\
\toprule
1921-1925 &163\ \ & 7 & 4 &&& 1946-1950 & 97 & \ \ 2 & 2 \\
1926-1930 & 215\ \ & 7 & 3 &&& 1951-1955 & 59 & \ \ 2 & 3 \\
1931-1935 & 104\ \ & 0 & 0 &&& 1956-1960 & 85 & \ \ 5 & 6 \\
1936-1940 & 142\ \ & 2 & 1 &&& 1961-1965 & 97 & 12 & 12 \\
1941-1945 & 82\ \ & 1 & 1 &&& 1966-1969 & 99 & 21 & 21 \\
\midrule
Main 1921-1969 &1104 & 59 &5 &&& Suppl. 1966-1969 & 34\ \ & 5 & 15 \\
\bottomrule
\end{tabular}
\label{table:BAN}
\end{center}
\end{table}
The growth was also reflected in the publications output. Most, but not all
papers appeared in the {\it Bulletin of the Astronomical Institutes of the
Netherlands} and extensive data in tabular form in the the {\it Publications of
the Astronomical Laboratory at Groningen},
the latter having been terminated with
van Rhijn's final publication and was replaced by the {\it Supplement Series}
of the {\it BAN}, of which the first volume started in 1966.
In 1969 both the main and supplements editions were terminated
and incorporated in {\it Astronomy \&\ Astrophysics}.
Table~\ref{table:BAN} shows the number of papers in the {\it BAN} over the
full period it appeared in five-year totals. The {\it Supplement Series} was
published only during the final interval (1966-1969)
and since of these papers an abstract
was published in the main journal and in listings counted as an article, the
numbers for the main journal have been corrected for this. The overall
Groningen share is 5.3\%, but there is a pronounced change from a few percent
before 1955 under van Rhijn
to 20\%\ towards the end. It should be noted that up to the Second
World War the numbers are significantly distorted by Ejnar Hertzsprung's
publication record, who published large numbers of short,
(often single page,) single author
papers often on 'elements' (periods, amplitudes, etc.) of variable stars
and other objects. In the period 1921-1942, he published 123 papers of the
total number of 638 articles in the journal, which amounts to 19\%.
Keeping this in mind, even then the effect of the change of the directorship
from van Rhijn to Blaauw is very large.
There was of course also the budget increase for travel.
According to van Woerden
\cite{HvW1985} it amounted to the sizable sum of 10,000 guilders per year,
which in current times would be around 35,000 \textgreek{\euro}. It was
probably used among other things to pay for part of
Plaut's long stays at Palomar Observatory
for the {\it Palomar-Groningen Variable-Star Survey} and Borgman's observing and
research trips to McDonald, Lowell and Yerkes Observatories.
It is very likely that this budget has been used also for
some of the expenses related to the sight-testing activities for ESO in South
Africa under Andr\'e Muller.
The Groningen Laboratory did not contribute any further data sets to
the {\it Plan of Selected Areas} after van Rhijn finished his research.
With van Rhijn no longer present
as the driving force it also disappeared from the research program of
Kapteyn's laboratory, which badly needed a new focus and new inspiration.
\bigskip
It would carry too far to go into details of the new research projects that
were initiated under Blaauw. In addition to his own research, which concerned
young stars, stellar associations, interstellar medium and star formation, the
fields opened were Galactic structure using pulsating stars under Lukas
Plaut, Galactic radio astronomy with the large Dwingeloo dish under Hugo van
Woerden \cite{HvWS} and under Jan Borgman the founding of the Kapteyn
Observatory in Roden with a 62-cm telescope and a well-equipped workshop to
develop instrumentation, infrared photometry from balloons
and ultraviolet photometry in the first Astronomical Netherlands Satellite ANS.
It is fair to say that the prominent position the Kapteyn Laboratorium had
under Kapteyn, and had to a major extent lost under van Rhijn, was largely
recovered in Adriaan Blaauw's term as director.
\section{Discussion}
I want to start this discussion with a few numbers that illustrate the enormous
amount of work that the Kapteyn Astronomical Laboratorium has devoted to
provide fundamental data. Here are the numbers of stars of the Groningen shares
for which positions and magnitudes have been derived in catalogues produced as
part of the {\it Plan of Selected Areas}:
\begin{tabular}{lr}
{\it Harvard(-Groningen) Durchmusterung}: & 231,981,\\
{\it Mt. Wilson(-Groningen) Catalogue}: & 45,448,\\
{\it Durchmusterung of the Special Plan}: & 140,082,\\
{\it Bergedorf(-Groningen-Harvard) Spektral Durchmusterung}: & 173,559.\\
\end{tabular}
These in total 591,070 position and brightness determinations were performed
roughly between 1910 and 1950, so 40 years of work
(not allowing for termination and slowing down of work during the Depression
and World War II). Now the
workweek usually was 6 days and since 1919 the workweek was by law set at
45 hours.
Allowing for closure during public holidays, we may take a year as 51 workweeks
or 2295 working hours. With these assumptions this means that during
working hours another star with measured coordinates and apparent magnitude
was added to the data base on average every 9 to 10 minutes. And even
less if allowance is made for times of fewer operating hours. An incredible
accomplishment! Of course, at any time a number of persons were working on
this.
Although remarkable it should be said that it is not unprecedented.
For example the initial version of the {\it Bonner
Durchmusterung} was produced by Friedrich Argelander and his assistants
between 1846 and 1863 (observations
ran from 1853 to 1859, see \cite{Charlier21}) and this resulted in
a catalogue of the positions and apparent
magnitudes of approximately 325,000 stars. And Annie J. Cannon
classified some 350,000 stellar spectra for the {\it
Henri Draper Catalogue} and extensions between about 1900 and
1940, with a reported peak frequency of about
three per minute \cite{Sheis}!
The production of positions and magnitudes started even earlier
in Groningen with
the {\it Cape Photographic Durchmusterung} or {\it CPD}, which also concerned
positions and magnitudes. This involved a total of 454,875 stars and was
achieved in 12 years, so repeating the calculation with these numbers yields a
rate of adding another star with measured and reduced coordinates and apparent
magnitude to the set of completed measurements on average every 4 or 5 minutes
for this period.
This is even faster, but it should be realized that the {\it CPD} plates were
measured with Kapteyn's very clever 'parallactic method'. In this method a
small telescope was placed at a distance from the plate exactly the same as
the focal distance of the telescope, so that in a correct orientation the two
perpendicular angular distances of a star from the plate center that were read
from dials on the axes of the small telescope were direct measurements of the
differences in right ascension and declination of the star and the plate center.
The Kapteyn Laboratorium between 1888 and 1957 (the year of van Rhijn's
retirement) measured 1.045,945 positions and magnitudes of stars in the
context of the Cape and Selected Areas Durchmusterungs. A quick scan
of the {\it Groningen Publications} shows that in addition, over this period,
there have been measurements of over 27,500 parallaxes (often of course upper
limits) and/or proper motions from plates provided by Anders Donner at
Helsingfors Observatorion, Ejnar Hertzsprung at Potsdam Observatorium, the
Cape Royal Observatory, Radcliffe Observatory, l'Observatoire Bouzareah
(Algers), and a number of other places. Of these about 7800 were obtained in
Kapteyn's time and the Algers collaboration accounted for
the largest faction
(41\%). So between 1888 and 1957, the Kapteyn Laboratorium produced a new
position/magnitude combination every 9 minutes during working hours and a
proper motion and/or parallax every $5{1\over2}$ hours.
After the War there have been
identifications and magnitude measurements of thousands of variable stars in
Lukas Plaut's research. This of course continued further in the latter's
variable star survey on Palomar plates up to about 1970. When Kapteyn in 1885
wrote David Gill to take on the measurements for the CPD, which he estimated
would be an effort of six or seven years, and a little later developed his
concept of an astronomical laboratory, he could not have foreseen the
enormous production this would lead to.
\bigskip
Before addressing the questions I formulated in the Introduction I quote
Adriaan Blaauw, who has expressed himself more than
others on van Rhijn's personality
and style of research. In his interview with Jet Katgert-Merkelijn
he said about van Rhijn (my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
Yes, he must have regretted seeing that the great development in the field of
Galaxy research went a little bit past him. But at the same time, if you were
to say, that was also kind of his own fault, you're right about that too. And
I think he also sensed that a bit. But in his last years he had, I think,
enough problems to deal with: his health, his family, and yes... the cuts from
the University and the staff. He also wasn't a.... He was a man who very
conscientiously carried out a program, but with little.... what can I say...?
Not very inspired, I would say.
\par \endgroup
I now return to the questions posed in the Introduction:
\bigskip
\noindent
{\it Was van Rhijn the successor Kapteyn had wished for or had he preferred someone else?}
We have no written statement or account by Kapteyn, but we can say a few things
about this. It should be seen in a
national context. He had been offered the directorship of the Sterrewacht
Leiden at the retirement of Gerardus van de Sande Bakhuyzen in 1908, which was
untimely as at that time he started his Research Associate appointment with the
Carnegie Institution. So he had pushed for his student Willem de Sitter for
this. Against his advice
Leiden had appointed Ernst van de
Sande Bakhuyzen as director and de Sitter in the professorship.
This first action was indeed a failure due to the
first's conservative nature and when in 1918 he unexpectedly died before
retirement, de Sitter and Kapteyn had already considered a course of action,
which resulted in a new structure with three departments with renowned leaders
(like Hertzsprung). From this it is clear that Kapteyn must have viewed the
future of Dutch astronomy to be the much better staffed and equipped
Sterrewacht in Leiden and not his much smaller Groningen establishment. It is
also important to realize that in the proposal for his {\it Plan of Selected
Areas}, Kapteyn had presented it in addition to the science as a means to
secure the near future of the Laboratorium.
So, Kapteyn would have looked for someone that would be dedicated to executing
the Plan, rather than someone to start new initiatives before
this Plan had
been completed. Someone to follow 'in his footsteps', we may say. Van Rhijn
was just that and must have been seen also by Kapteyn as the right person. He
obviously expected a great future for Jan Oort, who studied with him, and
recommended him strongly to de Sitter for a future job in Leiden (along with
Jan Schilt, but Oort with preference). The conclusion would be that Kapteyn
might have been content with van Rhijn taking over his position in Groningen.
Van Rhijn had carried out, one
would think to Kapteyn's satisfaction, the preparatory work for the star count
analysis in 1920 \cite{KvR1920a}, which would lead to the Kapteyn Universe and
was the basis for his 'first attempt' \cite{JCK1922}.
\bigskip
\noindent
{\it What future might Kapteyn have had in mind for his Laboratorium?}
At the opening of his Astronomical Laboratorium, still on a temporary location,
on January 16, 1896, Kapteyn said in his lecture (see \cite{JCKbiog}, p.228; my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
[...] we need to look at the work that needs to be carried out at a
photographic observatory. It consists of two parts: $1^{\rm st}$. The work
of the photographer. His part ends with the completed photographic plate.
$2^{\rm nd}$. The work of the astronomer. His part starts with the completed
photographic plate. […]
Eliminate the photographer and all you need is a building with some six rooms,
for which no very special requirements exist. Still we cut out what seems to
us the least vital, the lack of which would present an eminent advantage to
most astronomers if they prefer research to teaching, […] No laborious efforts
on the part of the photographer, only concentration on the real astronomical
work! What pleasure, what alleviation!
\par \endgroup
In his presentation of the {\it Plan of Selected Areas} he wrote in the
Introduction:
\par \begingroup \leftskip2.5em \rightskip2em \large
\noindent
That such a plan was evolved at the Laboratory of Groningen is only natural.
The nature of our astronomical institution makes our work dependent on that of
other observatories. Every work of a practical nature undertaken here is of
necessity a work of cooperation with another astronomer who possesses the
means of having photographs of the sky taken. [...]
At the same time, in pursuance of the same plan, I devoted all my leisure left,
to the investigation of the structure of the stellar system by means of the
data already available. […] These labors have now come to that point, that
they ought to enable me to make at last definite plans for the future work of
the laboratory. [...]
\par \endgroup
Not only was the Plan a means of securing a future for the institution,
it also illustrated that to secure the collaboration of others required a
vision in terms of a plan, the usefulness of which had to convince directors
of observatories. In the 1896 lecture above Kapteyn's assumption was that the
observatories produced abundantly more plates than they were able to measure,
reduce and analyze. The case of Anders Donner at Helsingfors was a clear
example. Presenting a good proposal was sufficient to obtain the plates. As
long as this situation prevailed the concept of an astronomical laboratory
could survive.
However, in van Rhijn's days these assumptions started to fail. The work at the
Mount Wilson telescopes by astronomers like Harlow Shapley, Edwin Hubble,
Walter Baade, etc. was manifestly not intended to provide research material
for others. Even in the collaboration for the {\it Mount Wilson(-Groningen)
Catalogue}, Frederick Seares pursued his own part independently and published
the catalogue without Groningen in its name (but with Kapteyn as co-author).
In the days of Kapteyn,
observatories were often collecting data to produced catalogues for the general
benefit, but these times were the past in van Rhijn's time and
would never return.
\bigskip
\noindent
{\it Was van Rhijn's work really so unimaginative and how important has it
turned out to be?}
I start with the second part. This concerns his scientific research, not the
efforts for the {\it Plan of Selected Areas}. It can be maintained, I believe,
that van Rhijn's work on the Luminosity Function, the provision of tables of
mean parallaxes of stars as a function of position, apparent magnitude, proper
motion and spectral type, etc. were no more than continuing Kapteyn's approach.
His research work on the distribution of stars in the plane of the Galaxy after
corrections for average extinction, and that of distributions of stars at a
line of sight at higher Galactic latitude for various spectral types in the
1930s were in itself good pieces of research,
but the papers failed to put the results in a
larger context. The PhD theses produced under him were of good or high quality,
especially the one by Broer Hiemstra was ingenious and original. So my
assessment is that van Rhijn's scientific work in general was solid, but in
general it was straightforward and not following a new,
innovative strategy.
A useful illustration would be to compare van Rhijn's 1936 study, in which he
studied the distribution of stars in the plane of the Galaxy \cite{PJvR1936},
with Oort's 1938 study of the distribution of stars in the Galaxy in
directions out of that plane \cite{JHO1938}. In van Rhijn's paper average
extinctions per kiloparsec were derived and applied to Selected Areas near
latitude zero. He ignored the effects of irregularities in the dust
distribution and in the end found only a local result (of which he seemed not
really convinced) that the Sun is located in a local cluster with a radius of
order 1 kpc and a drop in density of a factor almost two lower at the borders.
Results from larger distances on
the structure of the `larger system' are ambiguous, uncertain and inconclusive.
As usually the case with van Rhijn papers, he failed to provide a discussion of
the repercussions of his work on the larger picture of the structure of the
Galaxy.
Oort's 1938 study \cite{JHO1938} contrast with this in that he is interested
in the larger view on the Sidereal System and attempts to supersede Kapteyn's
`first attempt' with an improved version with the inclusion of what had become
known subsequently. In particular his approach is to limit himself to high
latitudes and at the lowest latitudes only to include those for which Hubble's
galaxy counts allow an estimate of the corrections involved. Furthermore, he
stays away from very low latitudes, because he then no longer can simplify the
work by assuming the absorption takes place in front of the bulk of the stars.
This leads to a first glimpse at how our Stellar
System compares to external spiral galaxies (from the Summary, p.233):
\par \begingroup \leftskip2.5em \rightskip2em \large
The structural features indicated are large-scale phenomena, extending on both
sides of the Galactic plane to distances of 500 or 700 parsecs from this plane.
The analogy between this structure and the spiral structure of strongly
flattened extra-galactic systems is discussed, but the data are still too
inaccurate and incomplete to permit any conclusion about the course of the
eventual `arms', except that the Sun should be between two arms.
\par \endgroup
Van Rhijn's work is not unimaginative and certainly not unimportant. But he
does not take the wider view of how his work fits in with the emerging picture
of the Galaxy we live in. Or at least he does not address this when presenting
results of his investigations.
As it turns out, Adriaan Blaauw also took these same two papers to make a
comparison between van Rhijn and Oort in his interview with Jet
Kargert-Merkelijn (quoted earlier). He took a somewhat different approach and
stressed another fundamental difference in conducting research (my translation):
\par \begingroup \leftskip2.5em \rightskip2em \large
Look, van Rhijn's working method was, in a way, that of Kapteyn. You make a
plan, you know those and those observational data are all here, this is the
problem we want to solve and we are going to do that step and that step and
you finish that way. But of course when you do that, it happens that at a
certain moment you say: `Hey, there is something I never thought of. Now I
will just have to look at the matter again.' Well, that flexibility, he had
very little of. A `serendepity'-development did not fit in van Rhijn's
schedule. [...]
Van Rhijn's style of research can be characterized well by comparing two pieces
of research that appeared in 1936 and 1938. […] Here you can see the different
ways of approach: the somewhat rigid, systematic, programmatic one of van Rhijn
and a more flexible one of Oort. [...]
Well, there were two people, both of them had the task of `we must do Milky Way
research and take into account the interstellar matter.' They had the same data
and then you see van Rhijn working according to a certain scheme, he says in
advance, `I am going to go with those reddening measurements, I am going to
calculate the absorption per kiloparsec and then I will apply that [...] .'
And Oort leaves things more open in the approach he will take. And at a certain
moment he says: `Well, I will leave the low latitudes for what they are, I can
at least say something about the higher latitudes. Because I know how thick
that layer is and I know that if I look up by a certain amount, I can trust the
star counts.
\par \endgroup
\bigskip
\noindent
{\it Was Kapteyn's concept of an astronomical laboratory, an observatory
without a telescope, a viable one?}
It should be noted that this concept was not Kapteyn's preferred option; he was
forced to accept it as the only way to get himself involved in astronomical
research at the forefront. I have described this development of Kapteyn not
obtaining his own observatory and telescope earlier, \cite{JCKbiog} or
\cite{JCKEng}, as a blessing in disguise. Had he been running his own
observatory, he might have been prevented from conducting research, measuring
plates or interpreting the results thereof to the extent he now has been able
to. Add to this the fact that the material he did receive to measure was far
superior to what he could have obtained from Groningen.
The success formula worked for Kapteyn's lifetime, but would it hold up in the
long run? In the first place it required a personality like him to succeed him
and also that the world would not change.
As mentioned above, the eminent astronomers drawn to
Pasadena to exploit the new Mount Wilson Observatory were not the kind that
were satisfied to provide others with unique observational material; they
wanted to push astronomy forward themselves and take the credit for the
progress. And this was not limited to Mount Wilson, but probably part of a
much more fundamental development. Blaauw's story of the `damned van
Rhijn program', as Harvard astronomers described the obligatory taking of
plates only to be sent to Groningen, illustrates this well.
Van Rhijn was aware of this and took steps to obtain his own telescope and did
in fact have a viable observing plan for it. The delay due to economic
depression and World War does not invalidate this point. But after the War the
real world moved to a situation where observing facilities of top quality (or
simply superior size) were not possible for the average university institute.
Blaauw realized this and accepted the position of successor of van Rhijn only
with guarantees for access to facilities (Dwingeloo, Hartebeespoortdam, foreign
observatories) and
funds to actually use these.
So, yes the `observatory without a telescope' concept worked for the person
Kapteyn in his times. The commitment to the {\it Plan of Selected Areas} worked
for van Rhijn profitability as a work plan to pursue, but without a road-map
how to exploit the data and synthesize a model for the structure of the
Galaxy it produced little more than catalogues and data bases. Van Rhijn seemed
to have been satisfied with that, but the future of the astronomical laboratory
needed a change in tactics. Having funds to go observe elsewhere solved at least
part of the issue under Blaauw. What remained was the problem of how to be
awarded the necessary observing time, but collaborations helped to solve that
(Plaut at Palomar, Borgman at Yerkes, McDonald, etc.) for the moment. The
existence of a nationally funded foundation to run the Dwingeloo dish solved
the radio astronomical needs. The initiative to build ESO would do the same in
the optical.
So, is the concept of an astronomical laboratory still a viable one?
Maybe paradoxically I would say: Yes, I would
think so. After all, in a sense the current Kapteyn Astronomical Institute
is not fundamentally different from the laboratory of Kapteyn's days.
Astronomers obtain their data from elsewhere, often without the need to travel
and observe personally. We sometimes indeed do
travel to observe at observatories the Netherlands
participates in such as on La Palma, La Silla or at Paranal,
but in many cases we can use service observing and this
is increasingly an option. We use other facilities through collaborations
with astronomers that have access to facilities, in which we are
not a stakeholder, such as in my case Palomar or Siding Spring, sometimes
traveling there, sometimes not.
In all cases material comes to Groningen for analysis
much like the plates that reached Kapteyn and van
Rhijn. Or we use space facilities, where data arrive at
your doorstep. As a PhD student in Leiden in
the 1960s I did not have to go to the
Dwingeloo for radio observations (although I did for the larger part of my
observations) and only had to fill in a form to 'order' observations.
The same has
been the normal modus operandi in Westerbork from the start in 1970.
Obviously, satellites and observatories in space by definition are operated
this way. Dedicated staff ensures efficient use of the telescope or satellite.
The problem van Rhijn faced was the reliance on other observatories to
provide date without having the authority Kapteyn had. The concept failed if
it {\it relied entirely} on the generosity of directors of observatories with
telescopes, but with funding to arrange access to facilities it is a model
working well in many places. In fact, going to the telescope oneself is seen
more and more as a poor and inefficient use of manpower and budgets. Kapteyn
could concentrate on research without having the burden of running an
observatory. We still conduct our research that way.
\bigskip
\noindent
{\it What precisely caused the Kapteyn Laboratorium to loose its prominent
place in the international context?}
There is no single cause. One simple answer is that van Rhijn was not in the
class of Kapteyn. Indeed, Kapteyn really was a though act to follow.
It would be difficult to rival the discovery of the Star
Streams, the vision of the {\it Plan of Selected Areas}, the two concepts of
statistical astronomy and galactic dynamics that Kapteyn had developed. But
there must have been more to it. Part of that was van Rhijn's attitude to give
data collection in the context of the {\it Plan of Selected Areas} priority
over analysis to develop an overall picture of Galactic structure. And there
of course was Leiden, which with no less than three world class leaders in
Willem de Sitter, Ejnar Hertzsprung and Jan Oort was impossible to better by
any single person. And then there is the lack of funding, resulting among
others from the remoteness to the seat of
government. This affected all of the University of Groningen but more so
astronomy, where in addition
funding the Sterrewacht in Leiden was more opportune
than the small scale Laboratorium in distant Groningen.
On the other hand, one should not forget that the Kapteyn Laboratorium and
Pieter van Rhijn did not become insignificant in the global ranking of
astronomy.
\bigskip
\noindent
{\it How important was the Plan of Selected Areas for the progress of astronomy
and did it ever reach the goals Kapteyn had in mind for it?}
Astronomy profited from the coordinated approach in various ways. First of
course the availability of a database in general. For photometry the existence
of photometric sequences in the Selected Areas, sometimes to very
faint magnitudes has been important, as discussed by Kinman in the Legacy
Symposium \cite{TK2000}. The existence of reasonably consistent and uniform
catalogues of spectral types, color indices, proper motions and radial
velocities must have been of value for many ongoing investigations.
Was this what Kapteyn had in mind? Not in the first place. His focus was an
analysis to determine what we now call the structure and dynamics of the
Galaxy, which he pioneered in his seminal `first attempt' paper of 1922
\cite{JCK1922}, in which he was the first to apply dynamics to observational
information. The dynamics was studied extensively by,
in particular, James Jeans,
Arthur Eddington, Bertil Lindblad and Jan Oort, so not in van Rhijn's
research establishment. But it seems faur to say that the ultimate goal of
understanding Galactic structure was brought within reach by the
data collecting of the {\it Plan of Selected Areas}. On the other hand, the
Plan did not fundamentally contributed to the discovery
of extinction.
Kapteyn's original goal had been to use the data collected in the Plan
to determine the distribution of the stars in space as a road-map for the
future of his Laboratorium.
In his presentation of the Plan \cite{SA} he wrote in the Preface:
\par \begingroup \leftskip2.5em \rightskip2em \large
As will be explained more fully in the introduction, the present plan was
originally meant only as a working plan for the astronomical laboratory of
Groningen. Having first determined on its main lines it was my task to try to
convince astronomers, who were in a situation of executing the necessary
plates and observations, of its urgency and thus win their cooperation.
\par \endgroup
\noindent
And in the Introduction:
\par \begingroup \leftskip2.5em \rightskip2em \large
In the following pages a plan is outlined, wich may be realised by the
cooperation of few astronomers in a, relatively speaking, moderate time.
The aim of it is to bring together, as far as is possible with such an effort,
all the elements which at the present time must seem most necessary for a
successful attack on the sidereal problem, that is: the problem of the
structure of the sidereal world.
\par \endgroup
What he had in mind was a final attack on this problem when all the data would
be in along the lines of his 'first attempt'. I have argued that Oort's paper
in 1938 \cite{JHO1938} was just that, except that much of the data used where
not yet in the final form.
This concerned the {\it Systematical Plan}. The {\it Special Plan}, I stress,
adopted under pressure from Pickering, had a few goals which can be appreciated
from his justification of the choice off the {\it Special Areas} in his
presentation of the Plan. As far as I am aware only the study of those few
Areas that were targeted to dark clouds was actually realized in the
thesis by Broer Hiemstra.
\bigskip
\noindent
{\it Did van Rhijn get the support he deserved from his university and the
government?}
The correspondence between van Rhijn and the Curators of Groningen University
show a healthy level of support for the Kapteyn Laboratorium. The problem is
the bias, at least in the 1930s,
of the Ministry for observational facilities in favor of Leiden and
Utrecht and against Groningen. The statement that computers in Groningen
should not expect to be paid at the same level as in Leiden is curious, to say
the least. After all, the work is similar and the salary should not depend on
the success or prominence of an institution on an international level
or the fame of the
professor in charge. It is much to van Rhijn's credit that he persevered in
his attempts to obtain his own telescope.
A comment that should be added here is that van Rhijn's
idea in the 1930s to address
the important and urgent issue of the wavelength dependence of interstellar
extinction by comparing similar stars with strong effects of reddening and
insignificant amounts, was timely. The difficulties to obtain the funding and
the crippling effects of the economic depression and the War delayed the
project by about two decades, so that by that time photoelectric techniques
had made photographic approaches obsolete, while the results obtained by van
Rhijn and Borgman have shown the program to be viable and capable of providing
definitive answers. The lack of support from the Ministry in funding the
telescope and refusal to
fund at least the dome, has been a major factor in the delay and has therefore
been a major cause of van Rhijn missing this important opportunity.
\bigskip
\noindent
{\it How influential was van Rhijn and what was his role in the prominence of
Dutch astronomy that followed during the twentieth century?}
Obviously, in the field of stellar and Galactic
astronomy he was overshadowed by de Sitter,
Hertzsprung and Oort in Leiden, but probably not by Pannekoek in
Amsterdam. The latter's legacy after all is in the area of astrophysics and
stellar atmospheres. Minnaert in Utrecht was prominent in solar research.
Van Rhijn kept Kapteyn's legacy alive, but the dominance
of Dutch galactic and stellar astronomy in the twentieth century resulted
from the work of the trio from Leiden and constitutes a major part of
Kapteyn's legacy.
Van Rhijn's influence on the
development of Dutch astronomy was limited, but his coordination of the work
on the Selected Areas remained important for astronomy as a whole.
That is not to say, van Rhijn did not contribute.
He did produce — or at least
supervised in the final stages — a number of excellent PhD theses and
students, that later
would occupy very prominent positions in the Netherlands and abroad,
such as Jan Schilt, Jan Oort, Peter van de Kamp, Bart Bok and Adriaan Blaauw.
In the same period there had been in Leiden
Dirk Brouwer (with de Sitter), and Pieter Oosterhoff,
Gerrit (Gerard) Kuiper and Adriaan Wesselink (with Hertzsprung).
In this respect van Rhijn did extremely well, also
compared to Leiden.
Of course there have been more outstanding astronomers in the Dutch school,
but these had their PhDs only after
van Rhijn's active period: Oort's most
prominent students Maarten Schmidt and Lodewijk Woltjer and van de Hulst's
Gart Westerhout followed in 1956 and 1958. Utrecht with Marcel Minnaert
contributed with Henk van de Hulst and Cees de Jager, and Pannekoek in
Amsterdam with Bruno van Albada, but only after WWII. This selection is of
course a subjective inventory
and somewhat arbitrary, but does show van Rhijn did
contribute very significantly to the reputation of Dutch astronomy in the first
half and a bit of the twentieth century.
\bigskip
\noindent
{\it What made Adriaan Blaauw accept his appointment as van Rhijn's successor?}
I have argued that a significant factor was Blaauw's appetite for adventure and
challenges. This made him accept Oort's request to join the Kenya expedition
and give up leave his secure, tenured position in Leiden for an
uncertain, temporary one at Yerkes. But adventure or challenge is not enough.
The advent of radio astronomy and the allocation upon his request for a new
staff position and of funds for travel abroad to collect
new observational
material together made an effort to effectuate a revival of Groningen
astronomy one with a reasonable chance of success. I suppose that had it
failed, Blaauw would easily have found
some position elsewhere. What helped him to pursue the
kind of career he desired, was the attitude of his wife to agree to
let him go away from his family for often long periods and her consent to move
the family across the Atlantic twice in only a few years. Adriaan Blaauw was
exactly what Groningen needed after van Rhijn.
Of course, the University of Groningen had been keen as well on attracting
Blaauw and even provide opportunities for growth of the astronomy
laboratory. I have not been able to find any written statement on this in the
Archives, but to back this up
I would think the university leaders and administrators were
still sufficiently aware of Kapteyn's significance and his stature as one of
the greatest scientists in its history to keep his legacy intact, even though
the focus of Dutch astronomy had moved with Kapteyn's prot\'eg\'es de Sitter,
Hertzsprung and Oort to Leiden.
\bigskip
In conclusion, Pieter van Rhijn may not have been
of the same caliber as Kapteyn (who could possibly blame him for
that?), and his work and leadership may have been not highly imaginative,
but still an enormous amount of
important and useful work has been done at the
Laboratorium under his directorship to the great benefit of astronomy.
And an admirable number of extraordinary students wrote their theses under his
supervision. He deserves much credit for keeping the
{\it Plan of Selected Areas} alive and
coordinating the enormous effort that was not likely to lead to much fame
or praise for himself. Van
Rhijn disliked travel and attending large meetings, and it was his misfortune
to have much of his directorship coincide with the Great
Depression, World War II
and his own protracted suffering of tuberculosis, while
the Minister denied him funding at the level he was
prepared to supply Leiden, yet under these unfavorable circumstances,
in the end in his modest and unassuming manner
his efforts left the spirit and heritage of his inspiring predecessor intact
until new leadership took over.
\bigskip
\noindent
{\bf Acknowledgments}
I thank my colleague astronomer Jan Willem Pel and historian Klaas van Berkel,
for a critical reading of a draft of this paper and making many detailed and
very useful comments and suggestions. David Baneke and Ton Schoot Uiterkamp
also read the manuscript and made a number of important remarks.
I am grateful to Kalevi Mattila of Helsinki Observatory for providing the
information on the correspondence of Kapteyn with Donner on the issue of
Kapteyn's succession.
I am especially grateful to Gert Jan van Rhijn,
grandson of van Rhijn's brother Maarten,
who maintains the van Rhijn Website \cite{PJgenea}, for good quality
photographs and much extensive biographical information on
Pieter van Rhijn.
I thank the staff of the Kapteyn Astronomical Institute, in particular the
secretariat and computer group, for support and help
and the director, Prof. L\'eon Koopmans, for hospitality extended to an
emeritus professor as guest scientist.
\bigskip
}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,015
|
package com.mygdx.pmd.model.components;
import com.badlogic.ashley.core.Component;
import com.badlogic.gdx.math.Vector2;
/**
* Created by Cameron on 4/17/2017.
*/
public class PositionComponent implements Component {
private Vector2 fPos;
public PositionComponent(Vector2 pos) {
fPos = pos;
}
public PositionComponent(float x, float y) {
this(new Vector2(x, y));
}
public Vector2 getPos() {
return fPos;
}
}
/*
public class PositionComponent {
public int x;
public int y;
private Tile currentTile;
public PositionComponent(TestEntity entity){
this.entity = entity;
}
public PositionComponent(TestEntity entity, int x, int y){
this.entity = entity;
this.currentTile = entity.tileBoard[y/ Constants.TILE_SIZE][x/ Constants.TILE_SIZE];
this.x = x;
this.y = y;
}
public void setCurrentTile(Tile nextTile) {
this.currentTile = nextTile;
this.x = currentTile.x;
this.y = currentTile.y;
}
public Tile getCurrentTile() {
return currentTile;
}
public void removeFromCurrentTile() {
currentTile.removeEntity(entity);
}
}
*/
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,662
|
package com.naughtyzombie.missingelement;// you can also use imports, for example:
// import java.util.*;
import java.util.Arrays;
// you can use System.out.println for debugging purposes, e.g.
// System.out.println("this is a debug message");
class Solution {
public int solution(int[] A) {
int n = A.length;
int maxTotal = ((n + 1) * (n + 2)) / 2;
int totalA = Arrays.stream(A).reduce(0, Integer::sum);
return maxTotal - totalA;
}
public static void main(String[] args) {
Solution sol = new Solution();
System.out.println(sol.solution(new int[] {2,3,1,5}));
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,822
|
package com.squarespace.cldr;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.List;
import java.util.concurrent.TimeUnit;
import org.openjdk.jmh.annotations.Benchmark;
import org.openjdk.jmh.annotations.BenchmarkMode;
import org.openjdk.jmh.annotations.Fork;
import org.openjdk.jmh.annotations.Measurement;
import org.openjdk.jmh.annotations.Mode;
import org.openjdk.jmh.annotations.OutputTimeUnit;
import org.openjdk.jmh.annotations.Scope;
import org.openjdk.jmh.annotations.State;
import org.openjdk.jmh.annotations.Warmup;
import org.openjdk.jmh.infra.Blackhole;
@Fork(1)
@Measurement(iterations = 5, time = 5)
@Warmup(iterations = 3, time = 2)
@BenchmarkMode(Mode.Throughput)
@OutputTimeUnit(TimeUnit.SECONDS)
public class LanguageMatcherBenchmark {
private static final List<String> SUPPORTED = Arrays.asList(
"es-MX", "es-419", "es-ES", "zh-HK", "zh-TW", "zh", "en-GB", "en-AU", "en-CA"
);
public static final List<String> SUPPORTED3 = select(3);
public static final List<String> SUPPORTED6 = select(6);
public static final List<String> SUPPORTED9 = select(9);
@Benchmark
public void build3(BenchmarkState state, Blackhole blackhole) {
blackhole.consume(new LanguageMatcher(SUPPORTED3));
}
@Benchmark
public void build6(BenchmarkState state, Blackhole blackhole) {
blackhole.consume(new LanguageMatcher(SUPPORTED6));
}
@Benchmark
public void build9(BenchmarkState state, Blackhole blackhole) {
blackhole.consume(new LanguageMatcher(SUPPORTED9));
}
@Benchmark
public void match3(BenchmarkState state, Blackhole blackhole) {
blackhole.consume(state.matcher3.match("en-AR"));
}
@Benchmark
public void match3Miss(BenchmarkState state, Blackhole blackhole) {
blackhole.consume(state.matcher3.match("de-DE"));
}
@Benchmark
public void match6(BenchmarkState state, Blackhole blackhole) {
blackhole.consume(state.matcher6.match("zh-MO"));
}
@Benchmark
public void match6Miss(BenchmarkState state, Blackhole blackhole) {
blackhole.consume(state.matcher6.match("de-DE"));
}
@Benchmark
public void match9(BenchmarkState state, Blackhole blackhole) {
blackhole.consume(state.matcher9.match("en-ZA"));
}
@Benchmark
public void match9Miss(BenchmarkState state, Blackhole blackhole) {
blackhole.consume(state.matcher9.match("de-DE"));
}
@State(Scope.Benchmark)
public static class BenchmarkState {
private final LanguageMatcher matcher3 = new LanguageMatcher(SUPPORTED3);
private final LanguageMatcher matcher6 = new LanguageMatcher(SUPPORTED6);
private final LanguageMatcher matcher9 = new LanguageMatcher(SUPPORTED9);
}
private static List<String> select(int count) {
List<String> result = new ArrayList<>();
for (int i = 0; i < count; i++) {
result.add(SUPPORTED.get(i));
}
return result;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,884
|
Norske Løve (dt. norwegischer Löwe) steht für:
das norwegische Staatswappen, siehe Staatswappen Norwegens
Orden des norwegischen Löwen, ein von König Oscar II. von Norwegen gestifteter Orden
Norske Løve (Horten), eine Festung bei Horten
Norske Løve ist der Name folgender Schiffe:
Norske Løve (Schiff, 1634), bis 1653 in der dänisch-norwegischen Marine
Norske Løve (Schiff, 1654), bis 1666 in der dänisch-norwegischen Marine
Norske Løve (Schiff, 1665), Schiff der dänischen Flotte in der Seeschlacht in der Køgebucht, bis 1679 in der dänisch-norwegischen Marine
Norske Løve (Schiff, 1680), bis 1715 in der dänisch-norwegischen Marine
Norske Løve (Schiff, 1704), ein bewaffnetes Kauffahrteischiff der Dänisch-Ostindischen Kompanie
Norske Løve (Schiff, 1735), ein Linienschiff 3. Ranges in der dänisch-norwegischen Marine
Norske Løve (Schiff, 1765), ein Linienschiff 3. Ranges in der dänisch-norwegischen Marine
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,246
|
Q: Interpeting ACF PACF plots time series forecasting After deseasonalizing using ec_rawhstt=ec_rawhst-ec_rawhst.shift(24)
are those plots valid for determining Ac(p) for autoregression models
sorry, I am a beginner
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,347
|
The media committee of the Freedom and Dignity Strike said on Monday that a serious deterioration has occurred to the health condition of MP Marwan al-Barghouthi who has been on hunger strike along with 1,500 prisoners for the 8th day in a row.
The committee, which was created by the Palestinian Committee of Prisoners and Ex-Prisoners Affairs and the Palestinian Prisoner Society, said in a statement that Barghouthi refused to take the medicine offered to him by the al-Jalama prison warden.
The al-Jalama warden asked the hunger-striking prisoner Nasser Abu Hmeid to convince Barghouthi to take the medicine, but Abu Hmeid refused saying, "If Marwan al-Barghouthi dies, he will die a martyr," according to the committee. The Israel Prison Service (IPS) punished Abu Hmeid by transferring him to Eshel prison.
The IPS continues to prevent lawyers from visiting the striking prisoners since the start of the hunger strike, making it very difficult to obtain information about the conditions in the prisons.
Israeli Prisons Service (IPS) closed Hadarim Israeli jail and transferred Palestinian captives held there to Negev desert prison, the media committee of the hunger strike follow up announced on Monday.
The committee said, in a statement, that 100 out of 120 detainees in Hadarim jail joined the open hunger strike since the first day.
They were transferred to isolation sections in other prisons. The 20 remaining captives who did not join the hunger strike because of illness were moved on Monday to Negev prison.
The transfer of the sick detainees aimed at penalizing the striking prisoners as well as the ill captives by repeated transfers, the statement added.
Some 1,500 Palestinian captives held in Israeli jails have been on an open hunger strike for the eighth consecutive day in order to demand their human rights.
Deputy chairman of Hamas political bureau, Ismail Haneyya, called Monday for mobilizing support for the Palestinian detainee Fouad al-Shoubaki, held in Israeli jails.
Haneyya's solidarity was expressed a few days after the national campaign to push for the release of al-Shoubaki appealed to the International Red Cross, among other concerned bodies, to urgently work on releasing the elderly detainee.
The campaign's coordinator, Abed al-Mutalib al-Nakhal, said prisoner al-Shoubaki, aged nearly 80 years old, has been diagnosed with life-threatening diseases, most notably prostate cancer, renal insufficiency, abdominal muscle weakness, and sight disorders.
Haneyya called on all concerned institutions to immediately step in and push for the release of al-Shoubaki before it is too late.
Haneyya expressed his support for the protest moves initiated by the Palestinians in Israeli lock-ups, vowing that the prisoners' cause will forever top Hamas's resistance agenda.
In 2002, prisoner al-Shoubaki was sentenced to four years in the Palestinian Authority (PA) Jericho jail, under British and American supervision.
Al-Shoubaki, who served as former chief of financial affairs at the General Security Service and President Yasser Arafat's financial adviser, was sent to Israeli prisons after the Israeli occupation forces kidnapped him from Jericho jail on March 14, 2006, in collaboration with the PA security apparatuses.
The Israeli Occupation Authorities (IOA) renewed the administrative detention of Sheikh Bajes Nakhla, who is currently held in solitary confinement, for three months.
Sheikh Nakhla, from Jalazoun refugee camp to the north of Ramallah, spent more than 18 years in Israeli jails, mostly in administrative detention.
In 1992, he was deported to Marj Al-Zuhour in Southern Lebanon for a whole year.
His house was recently notified with demolition, his wife affirmed.
A mass hunger strike for Palestinian prisoners in Israeli jails has entered its eighth consecutive day on Monday.
Some 1,580 Palestinian detainees, up from 1,500 on Saturday, have joined the mass hunger strike staged in Israeli jails to protest their dire conditions of captivity and to press for basic rights, including family visits, health care, lawyer visits, better quality of food, and humanitarian treatment in addition to an end to the policy of administrative detention and solitary confinement.
A series of rallies, marches, and protest moves has been ongoing across the occupied Palestinian territories and overseas in solidarity with the Palestinian hunger strikers.
Dozens of Israeli soldiers invaded, on Monday at dawn, the towns of Beit Ummar and Yatta, in the southern West Bank governorate of Hebron, searched and ransacked many homes, and abducted two young men.
Mohammad Awad, a media activist with the Popular Committee in Hebron, said the soldiers invaded Beit Ummar town, northwest of Hebron, before breaking into homes and violently searching them.
He added that the soldiers abducted a high-school student, identified as Ahmad Adel Za'aqeeq, 18, and took him to Etzion military base and security center, north of Hebron.
Clashes took place between the soldiers and many local youngsters, especially near the Old Mosque, and the soldiers fired many live rounds and concussion grenades.
In addition, Rateb Jabour, the coordinator of the National and Popular Committee in southern Hebron, said the soldiers invaded Yatta town, south of Hebron, searched and ransacked homes belonging to Mohammad Ahmad Abu Arram, Mahmoud Ayed Abu Arram and Sami Issa al-Adra.
Israeli soldiers invaded, on Monday at dawn, several villages and towns, in the northern West bank governorate of Jenin, searched many homes, and abducted two young men.
The Jenin office of the Palestinian Prisoners' Society (PPS) said the soldiers invaded and ransacked many homes in Maraka village, and abducted a teacher, identified as Talal Mustafa Abu Jalboush, 30, in addition to Mo'taz Yousef Abu Jalboush, 24.
Furthermore, many army vehicles invaded Sanour, Zababda and Msalya towns, conducted extensive and violent searches of homes, and interrogated many Palestinians while inspecting their ID cards.
Israeli soldiers abducted, on Monday at dawn, two young Palestinian men, from their homes, in Sawahra and Abu Dis towns, in Jerusalem governorate, in the occupied West Bank.
Several military jeeps invaded the two towns from different directions, stormed and violently searched homes, and abducted Morad Ayyad, from Abu Dis, and Hakem al-A'raj, from Sawahra, southeast of Jerusalem.
The two Palestinians were cuffed and blindfolded, before the soldiers took them to an unknown destination.
Palestinian medical sources have reported that two children were injured, on Sunday evening, after being struck by a speeding Israeli settler's car, while walking with their parents in Sur Baher town, south of occupied East Jerusalem.
The sources stated that Mohammad Mousa Attoun suffered a head wound, while his sister, Nada, suffered several cuts and bruises.
An Israeli ambulance was called to the scene, and moved the two children to a hospital for treatment; their condition has been described as stable.
Eyewitnesses said a settler, who was accompanied by a passenger, entered the center of town from its main road, and was speeding through its streets, apparently while being chased by an Israeli police car, before ramming the two children in addition to striking several parked cars.
The settlers fled the scene, and hid in a medical center, before dozens of soldiers invaded Sur Baher, clashed with many local youths, and evacuated them.
In Ramallah, several military jeeps and trucks invaded Bani Saleh village, northwest of the city, and searched many homes before abducting two children, identified as Ahmad Tamimi, 15, and Mohammad Tamimi, 14, and took them to an unknown destination.
The joint media committee of the Palestinian Prisoners' Society and the Palestinian Detainees' Committee, has reported in a press release, Sunday, that the Israeli Prison Authority is still denying all lawyers' visitations with the hunger striking detainees, except for those held in Ofer prison.
The committee stated that this illegal ban is now facing ongoing legal efforts, including various appeals, while human rights groups, including organizations defending the detainees' rights, are preparing to file an appeal with the Israeli High Court.
It also stated that the lawyers, who requested to meet with hunger striking detainees in different prisons, have faced repeated denials, under various allegations, including, the claims that certain detainees have been transferred to other prisons, and in some cases, the lawyers were allowed to visit with detainees who have not joined the hunger strike yet.
The committee added that the administrations in Asqalan, Majeddo and Ramla prisons, have denied all visitations with the hunger striking detainees, while Eshil and Nafha claimed they are still deciding on the issue.
Former Palestinian detainee Mahmoud Balboul spoke out, on Sunday, about his experience undertaking a weeks-long hunger strike, last year, and urged Palestinians everywhere to stand in solidarity with a mass hunger strike currently underway in Israeli prisons.
Mahmoud Balboul refused food for 79 days, after launching a hunger strike last July, and was joined by his brother Muhammad two days later, who refused food for a total of 77 days.
They both suspended their strikes after reaching a deal with Israel, stipulating their release from administrative detention — internment without charges or trial, a practice that has become a main target of the ongoing mass hunger strike.
According to Ma;an News Agency, the Balboul brothers attended a sit-in tent in their hometown of Bethlehem, in the southern occupied West Bank, on Sunday, among a number of similar tents set up across the occupied Palestinian territory in solidarity with approximately 1,500 imprisoned hunger strikers.
"It is of the utmost importance for the people to stand with the families of Palestinian prisoners first, and the hunger strikers themselves second, because a hunger-striking prisoner has prepared himself politically and psychologically for the experience — he has already gone through so many difficulties that have enabled him to adapt to any situation, despite all obstacles," Balboul continued.
Speaking of his own experience, Balboul stressed that his brother Muhammad's participation emboldened him to continue in the hunger strike against all odds. "A hunger striker needs support and solidarity to be able to continue," he said.
He said that there were only two occasions upon which he felt defeated by the weight of his emotions. First, when an International Committee of the Red Cross (ICRC) employee delivered him a letter from his mother that told him to stay determined; and second, when he was put in a solitary confinement cell under exceptionally miserable conditions.
Balboul directed a message to all Palestinian prisoners on hunger strike, telling them to stay strong and not listen to Israeli authorities' lies and rumors, which he said were only circulated to break the will of the hunger strikers. "Just focus on achieving your goal and stay calm. Think of freedom, and you will stay determined. When you are determined, you will reach victory," he said.
At least 40 Palestinians being held in the Israeli prison of Megiddo have joined the open-ended mass hunger strike, said the strikers' Media Committee of "Freedom and Dignity".
The committee said, according to WAFA, that the prison administration prevented the lawyer from visiting the hunger strikers.
The mass strike entered its seventh day on Sunday.
The detainees aim to restore many of the rights that were taken from the, by the prison administration, which they had achieved through many past strikes.
They are demanding to be moved to prisons in the occupied territories as per the Fourth Geneva Convention, which would make it easier for their families to visit them, as well as lifting restrictions on family visits and better treatment at military checkpoints, along with other demands.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,727
|
{"url":"https:\/\/stats.hohoweiya.xyz\/2021\/11\/18\/IRM\/","text":"# Infinite Relational Model\n\n##### Posted on Nov 18, 2021 (Update: Dec 07, 2021) 0 Comments\n\nSuppose we are given one or more relations involving one or more types.\n\nGoal: partition each type into clusters, where a good set of partitions allows relationships between entities to be predicted by their cluster assignments.\n\n\u2022 organize the entities into clusters that relate to each other in predictable ways\n\u2022 predicate types: multiple relations defined over the same domain, group them into a type and refer to them as predicates\n\ne.g. several social predicates defined over the domain people x people: likes(,), admires(,), \u2026, then introduce a type for these social predicates, and define a ternary relation applies(i, j, p) which is true if predicate $p$ applies to the pair (i, j).\n\nSuppose that the observed data are $m$ relations involving $n$ types. Let $R^i$ be the $i$-th relation, $T^j$ be the $j$-th type, and $z^j$ be a vector of cluster assignments for $T^j$. The task is to infer the cluster assignments, and ultimately interested in the posterior distribution\n\n$P(z^1,\\ldots, z^n\\mid R^1,\\ldots, R^m)$\n\n## HIRM\n\nGiven a set of relations defined over a collection of domains, the model first infers multiple non-overlapping clusters of relations using a top-level Chinese restaurant process.\n\nWithin each cluster of relations, a Dirichlet process mixture is then used to partition the domain entities and model the probability distribution of relation values.\n\nThe HIRM generalizes the standard infinite relational model and can be used for a variety of data analysis tasks including dependence detection, clustering, and density estimation.\n\nPresent new algorithms for fully Bayesian posterior inference via Gibbs sampling.\n\nFor relational data, we observe attributes and interactions among a set of entities and the goal is to learn models that are useful for explaining or making predictions about the entities, their attributes, and\/or their interactions.\n\nA relational system $S$ consists of $n$ domains $D_1,\\ldots,D_n$ and $m$ relations $R_1,\\ldots,R_m$. Each domain $D_i$ ($1\\le i\\le n$) is a countably infinite set of distinct entities ${e_1^i,e_2^i\\ldots,}$. Each relation $R_k$ ($1\\le k\\le m$) is a map from the Cartesian product of $t_k$ domains to an arbitrary codomain $C_k$. The symbol $d_{ki}$ ($1\\le k\\le m, 1\\le i\\le t_k$) denotes the domain index of the $i$-th argument of $R_k$.\n\nConsider a system $S$ with $n$ domains and $m$ relations. For each $i=1,\\ldots,n$, the IRM assumes that entities ${e_1^i,e_2^i,\\ldots}$ in domain $D_i$ are associated with integer cluster assignments ${z_1^i,z_2^i,\\ldots}=:z^i$.\n\nThe IRM defines a joint probability distribution over cluster assignments and relation values with the following factorization structure:\n\n$P(z^1,\\ldots,z^n, R_1,\\ldots,R_m) = \\prod_{i=1}^nP(z^i)\\prod_{k=1}^mP(R_k\\mid z^1,\\ldots,z^n)$\n\nTo allow IRM to discover an arbitrary number of clusters for each domain $D_i$, the cluster assignments $z^i$ for the entities are given a nonparametric prior that assign a positive probability to all possible partition using the Chinese restaurant process (CRP).\n\nFor each $i=1,\\ldots,n$, the cluster assignment probabilities $P(z^i) = P(z_1^i,z_2^i,\\ldots,)$ are defined inductively with $z_1^i:=1$, and for $l\\ge 2$,\n\n$P(z_l^i=j\\mid z_1^i,\\ldots,z_{l-1}^i) \\propto \\begin{cases} n_j & \\text{if } 1\\le j\\le M\\\\ \\gamma & \\text{if }j=M+1 \\end{cases}$\n\nwhere $n_j$ is the number of previous entities at cluster $j$, and $M$ is the number of clusters among the first $l-1$ entities, and $\\gamma > 0$ is a concentration parameter.\n\nNext, for each relation $R_k(1\\le k\\le m)$, a set of parameter $\\theta_k(j_1,\\ldots,j_{t_k})$ is used to dictate the distribution of $R_k(i_1,\\ldots,i_{t_k})$.\n\nThe generative model of the IRM is\n\n\\begin{align} \\{z_1^i,z_2^i,\\ldots\\} & \\sim CRP(\\gamma_i)\\\\ \\theta_k(j_1,\\ldots,j_{t_k}) & \\sim \\pi_k(\\lambda_k)\\\\ R_k(i_1,\\ldots,i_{t_k}) &\\sim L_k(\\theta_k(z_{i_1}^{d_{k1}}, \\ldots,z_{i_{t_k}}^{d_{kt_k}})) \\end{align}\n\n## Limitations of The IRM\n\n\u2022 enforcing shared domain clusterings leads to overfitting\n\u2022 restrictions when clustering multiple relations\n\nPublished in categories Note","date":"2022-01-20 14:16:09","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\": 2, \"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\": 1, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.915641188621521, \"perplexity\": 744.4426371927793}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"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-2022-05\/segments\/1642320301863.7\/warc\/CC-MAIN-20220120130236-20220120160236-00216.warc.gz\"}"}
| null | null |
Michael Bloomfield may refer to:
Michael J. Bloomfield (born 1959), American astronaut
Mike Bloomfield (1943–1981), American guitarist
|
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|
Exactly how McLaren's outdated F1 facilities hold it back – The Race
McLaren's not made any secret of the fact it's currently at a facilities disadvantage to many of its Formula 1 rivals.
Work is underway on an all-new simulator and a windtunnel at the team's Woking base. It currently still uses its very first simulator and is relying on Toyota's windtunnel in Cologne.
F1's measures to improve teams' financial situations plus the £185million deal with a consortium of US investors at the end of 2020 have allowed McLaren to commit to major facilities upgrades.
But the timescale for the sim and tunnel projects is such that they won't have any impact on McLaren's F1 designs until the 2024 car.
And when you hear the team's technical director James Key talking about the precise ways in which the windtunnel and simulator deficits restrict McLaren's design work, it really is no surprise that the team has made a relatively underwhelming start to F1's new rules era.
"In terms of the facilities, the difference they will make, it'll make quite a big difference to us because we're working in probably the oldest simulator in the pitlane," Key told The Race.
"It's the original, albeit updated, and it's fantastic considering how well it's served the team for 20 years or whatever it is, but it's definitely generation zero.
"And the windtunnel, although it's been brilliant support from Toyota, it's well out of date compared to the state of the art as well.
"So when both of those come online, and they're almost at a testing phase now so we're really beginning to see these projects coming together, the windtunnel will allow us to do things we can't do at all at the moment and that will give us a huge amount more information."
Slow-speed corner performance was a particular weakness for McLaren across 2020 and '21, and was an issue that the team's current facilities were really inadequate for tackling.
"Some of the issues last year for example, or the year before that which are very similar, whilst you can replicate them in CFD, CFD has its pros and cons with looking at transient things," Key explained.
"Windtunnel is actually slightly better than CFD for that.
"We couldn't replicate it in the current tunnel we're using. So we knew that we were totally hopelessly out of bed with replicating a low-speed corner.
"We compensated as best we could. But we knew that from a tunnel perspective, it was pretty inaccurate compared to what all other tunnels can do.
"This new tunnel will allow us to go to a much more representative condition."
Key feels it's hard for both fans and media to truly appreciate how much difference disparities around design technologies such as simulators and windtunnels can make because there are so much secrecy around teams' facilities.
"In all honesty, I find it tremendously frustrating and a great shame but entirely understandable that the engineering side of Formula 1 can't be more public," he said.
"Because it's just as interesting as what we're seeing here [at the track].
"You never see a real driver-in-loop simulator on TV, all you see is a bloke with three screens and a static chassis, it's a kid's toy.
"If you see a real driver-in-loop sim, it'll blow your mind. They are absolutely incredible – the amount of movement, and it's a massive thing as well.
"And we never get to show them publicly because they're all IP, very confidential, of course major tools.
"And it's the same as windtunnels, people see these as old dinosaurs, the word windtunnel is a little bit kind of antiquated and simple.
"But it's a laboratory which has strobing lasers in it, which has robotic arms putting probes in all different sorts of places, which has a very, very dynamic car, which has tyre pressures which change and squash, and has active suspension, can go around a corner to a certain level to give you effectively live continuous motion data based on a car going around a corner, and are immensely complicated things.
"And again, you never really get to see that. But it's not a tunnel with a bloke blowing a bit of smoke. It's really complex stuff, which is a high-tech laboratory environment. I can't explain the details of that.
"But if you ever got to see one, you'd be like 'Now I understand why!'.
"The accuracy, the data collections, I mean, you could go on forever with the array of ways you can collect data.
"And if your windtunnel runs come down [unless the new aero testing restrictions], the accuracy and the amount of information you can gather for a single run becomes increasingly important.
"And that's what we can't do."
Key pinpointed the 2022 McLaren MCL36's main weaknesses as being excessive drag and sensitivity to high track temperatures.
Despite his certainty that McLaren's facilities disadvantages are hurting it, he thinks those two problems were actually just a consequence of misjudgements around the 2022 rules.
"I think that's more just experiencing reference points. Thinking, 'OK, I wasn't expecting that to be a problem. So let's go and work out what we missed with that'," he said.
"Everything will have that – porpoising was an awkward thing for everyone. So every team has it and it's not unique to us.
"But in our case, the high track temperatures thing we could have just done a better job and thought more about it and set more ambitious targets. So that's not, with hindsight, difficult to understand, it's just 'fair point, let's go and address that'.
"The straightline speed thing was a surprise because there are methods of predicting where you should be but of course, it's always relative. And with such fresh regulations and the biofuel content in the engine, all sorts of things, to actually predict that is incredibly difficult.
"So that's turned out to be a weakness, other things have turned out to be a strength."
Formula 1 aspirant Andretti Global has announced plans for a $200million new headquarters in Indiana for its worldwide motorsport operations
Former Alfa Romeo Formula 1 driver Antonio Giovinazzi will drive the 2022 Haas car in two practice sessions this season
Lando Norris's suggestion that George Russell wasn't as much fun since joining Mercedes wasn't the blunt dig it was widely interpreted as. But did Norris have a point?
Daniel Ricciardo's current struggle at McLaren isn't the first occasion in his Formula 1 career he's been floundering at a critical time. But when it happened before, he turned things around spectacularly
Walrus noses, twin tusks, X-wings, zero sidepods, double wings and a seagull landing on the nose tip – there have been some truly bizarre Formula 1 designs over the years. And the worse the car has looked, the slower it's usually gone
Max Verstappen may be dominating the 2022 Formula 1 season, but he's doing it in a Red Bull that doesn't really suit his driving style and can't be made to
The Race started in February 2020 as a digital-only motorsport channel. Our aim is to create the best motorsport coverage that appeals to die-hard fans as well as those who are new to the sport.
Ex-Schumacher Ferrari F1 car fetches huge sum at auction - Racingnews365.com
ANALYSIS: Why Alonso accepted Aston Martin's advances and signed a shock deal – and why it could be a perfect match - Formula 1
F1 revenue doubles to US$360m in Q1 2022 – SportsPro – SportsPro Media
Hamilton: F1 'might as well not have a cost cap' if breaches get slap on the wrist – Motorsport.com
F1 is currently waiting to hear from the FIA about any potential action against Red Bull after the team was deemed to have breached...
F1 News: Daniel Ricciardo – Alonso and Hamilton prove I can make a comeback – Sports Illustrated
Ricciardo is confident that he can make a comeback.The next twelve months will be crucial for Daniel Ricciardo as he looks to re-establish himself...
Formula 1's Toto Wolff: 'You need to push people out of their comfort zone' – Financial Times
Then $69 per monthNew customers onlyCancel anytime during your trial OR BEST VALUE – SAVE 20% Then $74.75 every 3 months Sign in Check...
|
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\section{Introduction}
It has long been acknowledged that neutron stars (NSs) have strong
magnetic fields, which may play a variety of roles in the physics of
these objects. Magnetic fields can change the stability of a
NS, affect its evolution and cooling properties, induce precession and
provide the restoring force for a class of oscillation modes
\citep{harding_lai,mereghetti}. They may also produce crust-core
coupling and affect the post-glitch spin behaviour of NSs
\citep{easson,dib_kg}. All of these effects are likely to be sensitive
to the details of the
magnetic-field configuration in the stellar interior, whereas we are
only able to observe the exterior, dominantly dipolar, field. For this
reason considerable effort has gone into modelling the interior
magnetic fields of neutron stars; we give an overview of this in
section 2.
Despite considerable progress, the vast majority of studies
on magnetised neutron stars treat them as barotropic single-fluid
bodies. Whilst this is a sensible way to begin
producing simplified models, more realistic studies need to confront
some of the detailed interior physics of NSs. For example, over forty
years since the first predictions that neutron
star matter would contain superfluid neutrons and superconducting
protons (see, e.g., \citet{baym_pp}), there have been no attempts to
construct multifluid
magnetic equilibria along these lines. The issue is highly topical,
too --- recent observations of the cooling of the young NS in Cassiopeia A
are well explained by a model with superfluid neutrons and
superconducting protons \citep{casA_sht,casA_page}.
In the case of no magnetic field there has been more progress,
including studies of the equilibria and oscillation modes of
superfluid NSs (e.g. \citet{prix_r,prix_nc,yosh_eri,passa_sf}). The
closest related work on magnetised stars is probably the study of
\emph{single-fluid} barotropic NSs with superconductivity, by
\citet{akgun_wass}.
A simpler problem than a NS with superfluidity/superconductivity is a
stratified (i.e. non-barotropic), magnetised star composed of normal
fluid. Stratification is expected to exist in NSs \citep{reis_gold},
where it can
allow for a wider range of equilibrium configurations as well as
providing a stabilising effect on the magnetic field
\citep{reis_strat}. Again, the limitations of barotropic stellar
models have been known for decades \citep{mestel_nonbar}, but there
have been few attempts to construct stratified magnetised
equilibria. Most of the work in this direction has been that of
Braithwaite and collaborators, who perform nonlinear
magnetohydrodynamics (MHD) evolutions of
a stratified star and show that the system appears to relax to an
equilibrium over time \citep{braith_nord,braithtorpol}. \skl{In addition,
\citet{mastrano} very recently studied single-fluid non-barotropic
models of NSs, albeit using pre-specified magnetic field configurations.}
In this paper we attempt to plug some of the gaps in the
literature, presenting the first results for magnetised NS
equilibria modelled as a two-fluid system. We treat the neutrons as
being a superfluid, which mainly interacts with the proton fluid through the
star's gravitational potential. By choosing different degrees of
compressibility for the two fluid species, we can introduce
stratification. We begin by considering the case where protons are a
magnetised normal fluid and find
equilibria with purely poloidal, purely toroidal and mixed-field
configurations. After this, we treat the `full' problem where the
neutron star is composed of superfluid neutrons and type-II
superconducting protons for the first time, specialising to the case
of a purely toroidal magnetic field. This paper takes a nonlinear
approach to the problem, but we have also worked on a separate study
using perturbation theory; the results of this are presented in a
parallel paper, \citet{GAL}.
This paper is laid out as follows. In section 2 we present the
equilibrium equations for our stellar model, independent of the
details of the magnetic field. We then specialise to the case where
the protons are a normal fluid governed by standard MHD. In section 3
we consider the case where the protons are instead a type-II
superconductor, presenting simplified equations which nonetheless are
likely to be accurate enough for an initial study of equilibria. In the case of
purely toroidal fields we derive equations in a form suitable for
numerical integration. In section 4 we give details of our numerical
procedure, whilst in section 5 we show how to convert code output (in
terms of dimensionless code variables) into useful physical
quantities. Section 6 contains our results for two-fluid
equilibria, exploring the effect of stratification and type-II
superconductivity and section 7 is a discussion.
\section{Two-fluid equilibrium equations and normal MHD}
\subsection{Equilibrium equations for two-fluid magnetic stars}
We begin by modelling a neutron star as axisymmetric, with the
magnetic symmetry axis being the same as the rotation axis. For this
system it is natural to use cylindrical polar coordinates
$(\varpi,\phi,z)$, where the $z$-axis is aligned with the symmetry axis
of the star. Although relativistic effects will certainly play a role
in neutron star physics, we will neglect these for this study, and
work in Newtonian gravity. We also ignore the elastic crust
and any non-hadronic matter that may form an inner core, and assume
that the neutron star is a multifluid system, with a neutral fluid of
neutrons and a charged fluid of protons and electrons.
\skl{Our multifluid model should be applicable for the bulk of a
neutron star's interior during much, but not all, of its life. A
neutron star is born hot, with its charged and neutral components
strongly coupled through various dissipative mechanisms; at this
stage it behaves as a \emph{single},
non-barotropic, fluid. When the star cools below a critical
temperature $T_c$ however, the core neutrons begin to condense into
a superfluid
state\footnote{\skl{Neutron superfluidity in the inner crust is likely to
occur at a higher temperature --- above $10^9$ K --- due to its
singlet pairing type (as opposed to the triplet type in the core),
making this part of the star a multifluid system too.}}. The
recent observations of the cooling of the Cassiopeia A NS indicate
that $T_c\sim (5-9)\times 10^8$ K; a NS core should start to drop
below this temperature $\sim 100$ years after birth
\citep{casA_page,casA_sht}. For this
reason, even young pulsars are likely to have substantial multifluid
regions --- i.e., regions where the neutrons are superfluid. Models
of magnetar temperature profiles \citep{kaminker_magprof} indicate
they are cool enough to have
multifluid regions too. For this preliminary study into stratified
NSs we model the \emph{entire} star as multifluid; this is obviously
a simplification, and more detailed
modelling should account for the layered structure (including
single-fluid regions) expected in real NSs.}
Now, since electrons have a much smaller mass than either of the other
components, we take the standard approach of neglecting their
inertia. Our final model is
then a two-fluid system, with a proton fluid (whose variables are
denoted with a subscript roman index p) and a neutron fluid (with
index n). Neutrons and protons are assumed to be of equal mass
$m_\rmn=m_\mathrm{p}\equiv m$. Next, we denote the particle chemical
potentials by $\mu_\rmn$ and $\mu_\mathrm{p}$, then define $\tilde{\mu}_\rmn\equiv\mu_\rmn/m$ and
$\tilde{\mu}_\rmp\equiv(\mu_\mathrm{p}+\mu_\mathrm{e})/m$; note that we use $\tilde{\mu}_\rmp$ to
denote the chemical potential of the whole charged component, as it
also contains a contribution from the electrons $\mu_\mathrm{e}$.
In normal MHD, only the protons are coupled to the magnetic field, but
for a superfluid/superconducting system there is
generally a magnetic force acting on the neutrons as well
\citep{mendell,GAS2011}. In this work however, we only consider the case
without entrainment, for which the neutron superfluid is decoupled
from the magnetic field in the superconducting case too. This issue is
discussed in more detail in \citet{GAL}.
Given these assumptions, we may now state the separate
Euler equations which govern the two fluids:
\begin{equation} \label{n_Euler}
\nabla\brac{\tilde{\mu}_\rmn+\Phi-\frac{\varpi^2\Omega_\rmn^2}{2}}=0,
\end{equation}
\begin{equation} \label{p_Euler}
\nabla\brac{\tilde{\mu}_\rmp+\Phi-\frac{\varpi^2\Omega_\mathrm{p}^2}{2}}=\frac{{\bf F}_{\mathrm{mag}}}{\rho_\mathrm{p}},
\end{equation}
where ${\bf F}_{\mathrm{mag}}$ denotes the magnetic force; the form of
this force will be the only difference between the normal-MHD and
superconducting cases to our order of working. As usual, $\Phi$ denotes the
gravitational potential, $\rho$ is density and $\Omega$ denotes
rotation rate (with indices for individual fluid species). We will
only consider the case where there is no
entrainment and the two fluids corotate,
i.e. $\Omega_\rmn=\Omega_\mathrm{p}\equiv\Omega$. Since we are only studying
stationary configurations,
the equations do not contain mutual friction terms.
As in the single-fluid case we have Poisson's equation for the
gravitational potential, from which we see that the behaviour of the
two fluids is linked despite the neutrons being a superfluid:
\begin{equation}
\nabla^2\Phi = 4\pi G\rho = 4\pi G(\rho_\rmn+\rho_\mathrm{p}).
\end{equation}
Let us multiply equation \eqref{n_Euler} by $\rho_\rmn$, equation
\eqref{p_Euler} by $\rho_\mathrm{p}$, and add them:
\begin{equation}
\rho_n\nabla\tilde{\mu}_\rmn+\rho_\mathrm{p}\nabla\tilde{\mu}_\rmp
+\rho\nabla\Phi-\rho\nabla\brac{\frac{\varpi^2\Omega^2}{2}}={\bf F}_{\mathrm{mag}}.
\end{equation}
Now since
$\rho_n\nabla\tilde{\mu}_\rmn+\rho_p\nabla\tilde{\mu}_\rmp = \nabla P$ (the gradient of the
total fluid pressure), this allows us to recover the familiar
`single-fluid' Euler equation:
\begin{equation}
\frac{\nabla P}{\rho}+\nabla\Phi-\nabla\brac{\frac{\varpi^2\Omega^2}{2}}
= \frac{{\bf F}_{\mathrm{mag}}}{\rho},
\end{equation}
except that in this two-fluid case one will typically have
stratification, and hence can no longer replace the pressure term with an
enthalpy gradient. Instead, it is more useful to work with the
proton-fluid Euler and a `difference-Euler', given
by \eqref{p_Euler} minus \eqref{n_Euler}:
\begin{equation} \label{d_Euler}
\nabla\brac{\tilde{\mu}_\rmp-\tilde{\mu}_\rmn} = \frac{{\bf F}_{\mathrm{mag}}}{\rho_\mathrm{p}}.
\end{equation}
Taking the curl of this equation, we see that (as in the single-fluid
case) there exists a scalar $M$ such that
\begin{equation} \label{M_defn}
\frac{{\bf F}_{\mathrm{mag}}}{\rho_\mathrm{p}}=\nabla M.
\end{equation}
The magnetic force --- whether the normal MHD `Lorentz force' or the
flux-tube tension force of a type-II superconductor (see section 3)
--- therefore depends on a scalar function $M$ and the
proton-fluid density $\rho_\mathrm{p}$, as opposed to the single-fluid case
where the dependence is on the total mass density: ${\bf F}_{\mathrm{mag}}/\rho=\nabla
M$. In deriving the equations that govern the behaviour of the
magnetic field in the two-fluid case, the only differences from the
single-fluid case will be in this density factor; the final magnetic
equations will be the same but with $\rho$ replaced by $\rho_\mathrm{p}$.
For our numerical scheme we need to work with the equilibrium
equations in integral form; in this section we discuss those equations
which hold for all magnetic-field configurations. Firstly, we always
have the same form of Poisson's equation for the gravitational potential:
\begin{equation} \label{int_Poisson}
\Phi({\bf r})
= -G \int \frac{\rho_\rmn({\bf r}')+\rho_\mathrm{p}({\bf r}')}{|{\bf r}-{\bf r}'|}\ \mathrm{d}{\bf r}'.
\end{equation}
We also have first-integral forms of the proton-fluid Euler
equation \eqref{p_Euler}:
\begin{equation} \label{pEuler}
\tilde{\mu}_\rmp+\Phi-\frac{\varpi^2\Omega^2}{2} = M+C_\mathrm{p}
\end{equation}
and the difference-Euler \eqref{d_Euler}:
\begin{equation} \label{dEuler}
\tilde{\mu}_\rmp-\tilde{\mu}_\rmn = M + C_d,
\end{equation}
where $C_\mathrm{p}$ and $C_d$ are the integration constants for the proton
and difference-Euler equations, respectively.
Finally, for the equation of state we choose an energy
functional $\mathcal{E}$ given by
\begin{equation} \label{eos_functional}
\mathcal{E} = k_\rmn\rho_\rmn^{\gamma_\rmn}+k_\mathrm{p}\rho_\mathrm{p}^{\gamma_\mathrm{p}},
\end{equation}
a two-fluid analogue of a polytropic model
\citep{prix_ca,passa_sf}. This could also be written
in terms of the polytropic indices $N_\mathrm{x}=1/(\gamma_\mathrm{x}-1)$,
$\mathrm{x}=\{\rmn,\mathrm{p}\}$. Note that \eqref{eos_functional} can be
generalised to include cross-terms, corresponding to symmetry energy
and entrainment. The chemical potentials are then defined from the
energy functional by
\begin{equation} \label{mux_defn}
\tilde\mu_\mathrm{x}\equiv\left.\pd{\mathcal{E}}{\rho_\mathrm{x}}\right|_{\rho_{\mathrm{y}}},
\end{equation}
where the index x represents one particle species (neutrons or
protons) and the y index represents the other. The other equations of
the system depend on the details of the magnetic-field configuration;
we discuss these next.
\subsection{Normal MHD}
The simplest case is when the charged component of the stellar matter
is governed by normal MHD. The magnetic force ${\bf F}_{\mathrm{mag}}$ is then the
familiar Lorentz force, given by
\begin{equation} \label{lor_force}
\lor={\bf j}\times{\bf B}=\frac{1}{4\pi}(\nabla\times{\bf B})\times{\bf B},
\end{equation}
where ${\bf B}$ is the magnetic field and ${\bf j}$ the current.
Most studies of NS equilibria have used the normal-MHD equations and
have additionally adopted a single-fluid model, where the charged
component is the entire fluid --- this ansatz has been used by
innumerable authors. Recent examples include
\citet{tomi_eri,kiuchi,haskell,1f_eqm} and \citet{ciolfi}, but there
are decades of earlier work on single-fluid models --- see, e.g.,
\citet{ferraro} and \citet{roxburgh}.
An obvious way to progress is to work with a two-fluid model, but
still in normal MHD: the protons are a normal fluid and the neutrons
superfluid. This is not just a stepping stone towards the
superfluid/superconducting model of \citet{baym_pp} --- it already allows
us to study the role of stratification in magnetised
stars. Furthermore, the interior magnetic field strength in magnetars
may well exceed the second critical field $H_{c2}\sim 10^{16}$ G at
which superconductivity is destroyed; in this case magnetar matter
could indeed be predominantly normal protons and superfluid neutrons
\citep{GAL}.
Since only the proton fluid is magnetised, the derivations for all
equations with magnetic terms proceed in the same manner as the
single-fluid case, with factors of $\rho$ replaced by $\rho_\mathrm{p}$;
otherwise they are identical. For this reason we do not give the
details here, but the essentially identical single-fluid derivation
may be found in various papers (see, e.g., \citet{1f_eqm}). The
non-magnetic equations of the system are those discussed in the
previous subsection.
In the mixed-field (or purely poloidal field) case we have a Poisson
equation for the $\phi$-component of the magnetic vector potential
too, \skl{allowing for the magnetic field to extend outside the star}:
\begin{equation} \label{int_Aph}
A_\phi({\bf r})\sin\phi
= \frac{1}{4\pi}
\int\ \frac{ \frac{1}{\tilde\varpi}f(\tilde u)f'(\tilde u)
+ 4\pi\tilde\varpi M'(\tilde u)\rho_\mathrm{p}(\tilde{\bf r})
}{|{\bf r}-\tilde{\bf r}|}
\ \sin\tilde\phi\ \mathrm{d}\tilde{\bf r},
\end{equation}
where $u$ is a streamfunction; see the next section. In this
equation $f$ and $M$ are functions of $u$, tildes denote dummy
variables and primes denote
derivatives. In addition, $u$ and $A_\phi$ are related through
$u=\varpiA_\phi$, so the integral in equation \eqref{int_Aph} may be
rewritten in terms of $A_\phi$. As mentioned earlier, this integral
equation for $A_\phi$ is identical to the single-fluid version
except with a $\rho$ term in the integrand replaced by $\rho_\mathrm{p}$.
\skl{The magnetic functions $M(u)$ and $f(u)$ appear to be arbitrary, but
in practice we have found only one acceptable functional form for
each; this was also the case in our earlier study of single-fluid
models \citep{1f_eqm}. In the mixed-field case, we use the following
form of $M$:}
\begin{equation}
M_{mix} = \kappa u = \kappa\varpiA_\phi
\end{equation}
where $\kappa$ is a constant governing the strength of the
Lorentz force. \skl{We find that choosing $M(u)$ as a lower or higher power
of $u$ results in the code, through equation \eqref{int_Aph}, iterating
to an unmagnetised equilibrium solution --- and hence only the above
choice of $M_{mix}$ is acceptable in our work.}
The other magnetic function $f$ in the integral equation for
$A_\phi$ relates to the toroidal component of the field and is defined
to be non-zero only within the largest interior closed field line, so there is
no exterior toroidal field \skl{(and hence no exterior current)}:
\begin{equation} \label{f_defn}
f(u)=a(u-u_{\max})^{1.1} H(u-u_{\max}),
\end{equation}
where $u_{max}$ is the maximum value attained by $u$ within the star
and $H$ is the Heaviside step function. The strength of the toroidal
component is adjusted by varying the constant $a$. Other choices of
the exponent than $1.1$ are possible; we choose this value as it gives
a relatively strong toroidal-field component. Choosing a value of
unity or less for the exponent gives undesirable behaviour in $f'(u)$,
however \citep{1f_eqm}. \skl{Finally, whilst adjusting the exponent in
\eqref{f_defn} allows for configurations with very modest differences,
we do not know of any genuinely different functional form of $f$
which is still dependent on $u$ and restricts the toroidal component to the
stellar interior.}
In the toroidal-field case there is no separate integral equation for the
magnetic field. $M$ has a different functional form in the
pure-toroidal field case $M_{tor}$ from that in the
mixed/pure-poloidal field case $M_{mix}$. As with other quantities,
the two-fluid form of $M_{tor}$ may be found from the single fluid
version by replacing $\rho$ with $\rho_\mathrm{p}$:
\begin{equation}
M_{tor}
= -\frac{1}{4\pi}\int_0^{\rho_\mathrm{p}\varpi^2}
\frac{h(\zeta)}{\zeta}\td{h}{\zeta}\ \mathrm{d}\zeta,
\end{equation}
where $h$ is a function of $\rho_\mathrm{p}\varpi^2$, related to the magnetic
field by
\begin{equation}
B_\phi=\frac{h(\rho_\mathrm{p}\varpi^2)}{\varpi}.
\end{equation}
We choose $h(\rho_\mathrm{p}\varpi^2)=\lambda\rho_\mathrm{p}\varpi^2$, where $\lambda$
is a constant governing the strength of the toroidal field. \skl{Once
again, all other functional forms appear to result in either
divergent magnetic quantities or our numerical scheme iterating to
an unmagnetised solution \citep{1f_eqm}.}
Note that the difference-Euler \eqref{d_Euler} for an unmagnetised
star shows that the two fluids are in chemical equilibrium. In the
magnetic case there is an extra ${\bf F}_{\mathrm{mag}}$ term --- this could be
interpreted as a statement that the star is out of chemical
equilibrium \citep{GAL}. \skl{Alternatively, one could adjust the notion of
what is meant by `chemical equilibrium'. Since the energy functional
$\mathcal{E}$ gains an extra contribution from the presence of a magnetic field
\citep{GAS2011}, the chemical potentials --- defined through
\eqref{mux_defn} --- also change accordingly. One could then
see equation \eqref{d_Euler} as quantifying the difference
between magnetised and unmagnetised chemical equilibria.}
\section{Magnetic forces in type-II superconductivity}
In a typical NS the protons are likely to form a type-II superconductor,
where the field penetrates the star through thin fluxtubes
\citep{baym_pp}. This results in a magnetic force dependent on the averaged
magnetic field ${\bf B}$ but also on the first critical field
$H_{c1}$. The result is a fluxtube tension force
\citep{easson_peth,mendell,GAS2011}, quite
different from the Lorentz force \eqref{lor_force}.
The aim of this section is not to give a complete description of the
equations governing a star with type-II superconducting protons, but
instead to explore how many of the steps from the normal-fluid
Grad-Shafranov derivation (see, e.g., \citet{1f_eqm}) also hold in this
case. We derive a general `interim' expression
for the magnetic field, analogous to the normal-MHD case, at which
point one has to specialise to either purely toroidal fields or mixed
fields (with purely poloidal fields as a particular limiting
case). A key result for mixed fields from the normal-MHD case no
longer holds in the superconducting case, and we defer a detailed
study of this to a later paper. Instead we concentrate on purely
toroidal magnetic fields, where the derivation is more straightforward.
\subsection{General form}
In \citet{GAS2011}, a full derivation of the equations governing
a type-II superconducting neutron star is presented. For
our purposes, however, only a reduced and simplified set are
needed. We consider
the case where there is no magnetic force on the neutrons, and neglect
small terms like the `London field'. Given this, our equations reduce
to those of section 2.1,
by setting ${\bf F}_{\mathrm{mag}}$ to be the superconducting magnetic force
$\boldsymbol{\mathfrak{F}}_{\mathrm{mag}}$. Before discussing this force, we have one useful result from
normal MHD which still holds in this case. Using the fact that
$\div{\bf B}=0$ together with the assumption
of axisymmetry allows us to write ${\bf B}$ in terms of a streamfunction
$u$ \citep{1f_eqm}:
\begin{equation}
{\bf B} \equiv {\bf B}_{pol}+{\bf B}_{tor} = \frac{1}{\varpi}\nabla u\times{\bf e}_\phi +
B_\phi{\bf e}_\phi.
\end{equation}
As in normal MHD then, we have ${\bf B}\cdot\nabla u=0$.
At this point we turn to equation (153) from \citet{GAS2011} for the
explicit form of $\boldsymbol{\mathfrak{F}}_{\mathrm{mag}}$:
\begin{equation} \label{orig_GAS_force}
\boldsymbol{\mathfrak{F}}_{\mathrm{mag}} = \frac{1}{4\pi}
\Big[
({\bf B}\cdot\nabla)(H_{c1}\hat{\bf B})-B\nabla H_{c1}
\Big]
-\frac{\rho_\mathrm{p}}{4\pi}\nabla\brac{B\pd{H_{c1}}{\rho_\mathrm{p}}},
\end{equation}
\skl{where $\hat{\bf B}$ is a magnetic unit vector and $B$ is the
(position-dependent) magnitude of the field, so that ${\bf B}=B\hat{\bf B}$.
From the calculation in appendix A2 of \citet{GAS2011}, the first critical
field $H_{c1}=h_c\rho_\mathrm{p}/\varepsilon_\star$ to a good approximation, where
$\varepsilon_\star$ is entrainment and $h_c$ a constant parameter. Since
we are assuming that there is no neutron-fluid magnetic force, and
hence no entrainment, we have $\varepsilon_\star=1$.} The magnetic force then becomes
\begin{equation} \label{orig_fmag}
\boldsymbol{\mathfrak{F}}_{\mathrm{mag}} = \frac{h_c}{4\pi}
\left[
({\bf B}\cdot\nabla)(\rho_\mathrm{p}\hat{{\bf B}})
-\nabla(B\rho_\mathrm{p})
\right].
\end{equation}
Now by using a vector identity to rewrite the first term on the RHS and
rearranging the result we arrive at:
\begin{equation} \label{brack_term}
-\frac{4\pi}{h_c}\boldsymbol{\mathfrak{F}}_{\mathrm{mag}}
= \rho_\mathrm{p}\nabla B
+ {\bf B}\times\brac{ \rho_\mathrm{p}\nabla\times\hat{\bf B}
+ \nabla\rho_\mathrm{p}\times\hat{\bf B} }.
\end{equation}
Next we want to rewrite the bracketed term on the RHS of equation
\eqref{brack_term}. Let us start by
defining a `unit current' $\boldsymbol{\hat{\j}}\equiv\nabla\times\hat{\bf B}$; as in the
single-fluid case it may be shown that
\begin{equation}
\boldsymbol{\hat{\j}} = \frac{1}{\varpi}\nabla(\varpi\hat{B}_\phi)\times{\bf e}_\phi
+ \hat{\j}_\phi{\bf e}_\phi.
\end{equation}
We then decompose both of the bracketed terms from \eqref{brack_term}
into poloidal and toroidal components. The resultant expression may
then be rearranged to show that
\begin{equation}
\rho_\mathrm{p}\nabla\times\hat{\bf B} + \nabla\rho_\mathrm{p}\times\hat{\bf B}
= \frac{1}{\varpi}\nabla(\rho_\mathrm{p}\varpi\hat{B}_\phi)\times{\bf e}_\phi
+ \brac{\rho_\mathrm{p}\hat\j_\phi
- \frac{\nabla\rho_\mathrm{p}\cdot\nabla u}{\varpi B} }{\bf e}_\phi
\end{equation}
The magnetic force becomes
\begin{equation} \label{2and3RHS}
-\frac{4\pi}{h_c}\boldsymbol{\mathfrak{F}}_{\mathrm{mag}}
= \rho_\mathrm{p}\nabla B
+\frac{1}{\varpi}{\bf B}
\times\bigg(\nabla(\rho_\mathrm{p}\varpi\hat{B}_\phi)\times{\bf e}_\phi\bigg)
+\brac{\rho_\mathrm{p}\hat\j_\phi
- \frac{\nabla\rho_\mathrm{p}\cdot\nabla u}{\varpi B} }{\bf B}\times{\bf e}_\phi.
\end{equation}
Again, it is useful to rearrange some terms in this expression. The
second term on the RHS of \eqref{2and3RHS} may be rewritten as:
\begin{equation}
\frac{1}{\varpi}{\bf B}\times
\brac{\nabla(\rho_\mathrm{p}\varpi\hat{B}_\phi)\times{\bf e}_\phi}
= \frac{1}{\varpi^2}\nabla u\times
\nabla(\rho_\mathrm{p}\varpi\hat{B}_\phi)
+ \frac{B_\phi}{\varpi}\nabla(\rho_\mathrm{p}\varpi\hat{B}_\phi),
\end{equation}
using standard vector
identities. To simplify the third term on the RHS
of \eqref{2and3RHS} we use
\begin{equation}
{\bf B}\times{\bf e}_\phi={\bf B}_{pol}\times{\bf e}_\phi=-\frac{\nabla u}{\varpi}.
\end{equation}
Now, putting these two results into the expression for the magnetic
force \eqref{2and3RHS} we find that:
\begin{equation}
-\frac{4\pi}{h_c}\boldsymbol{\mathfrak{F}}_{\mathrm{mag}}
= \rho_\mathrm{p}\nabla B
+\frac{1}{\varpi^2}\nabla u\times
\nabla(\rho_\mathrm{p}\varpi\hat{B}_\phi)
+\frac{B_\phi}{\varpi}\nabla(\rho_\mathrm{p}\varpi\hat{B}_\phi)
+\brac{\frac{\nabla\rho_\mathrm{p}\cdot\nabla u}{\varpi B}
-\rho_\mathrm{p}\hat\j_\phi}\frac{\nabla u}{\varpi}.
\end{equation}
By axisymmetry, the toroidal component of $\boldsymbol{\mathfrak{F}}_{\mathrm{mag}}$ must be zero. Now,
all of the terms in the above expression are gradients of scalars (and
hence poloidal), except the cross product. This term
is purely toroidal and hence must be zero:
\begin{equation} \label{parallelnablas}
\frac{1}{\varpi^2}\nabla u\times\nabla(\rho_\mathrm{p}\varpi\hat{B}_\phi)=0.
\end{equation}
Hence we can remove this term from the magnetic force to arrive at a
somewhat simplified result, in terms of the
gradients of various scalar functions:
\begin{equation} \label{fmag_general}
-\frac{4\pi}{h_c}\boldsymbol{\mathfrak{F}}_{\mathrm{mag}}
= -\frac{4\pi}{h_c}\rho_\mathrm{p}\nabla M
= \rho_\mathrm{p}\nabla B
+\frac{B_\phi}{\varpi}\nabla(\rho_\mathrm{p}\varpi\hat{B}_\phi)
+\brac{\frac{\nabla\rho_\mathrm{p}\cdot\nabla u}{\varpi B}
-\rho_\mathrm{p}\hat\j_\phi}\frac{\nabla u}{\varpi},
\end{equation}
where $M$ is defined by equation \eqref{M_defn}, as before. For
comparison, the equivalent form for the magnetic force in the
normal-MHD derivation (see, e.g., the appendix of \citet{1f_eqm}) is:
\begin{equation} \label{lor_vs_fmag}
-4\pi\rho\nabla M = \frac{B_\phi}{\varpi}\nabla(\varpi B_\phi)
- 4\pi j_\phi \frac{\nabla u}{\varpi}
\end{equation}
where $j_\phi=\frac{1}{4\pi}[\nabla\times{\bf B}]_\phi$. We see that in the
superconducting case there are extra terms with $\nabla B$ and
$\nabla\rho_\mathrm{p}$ factors.
The above derivation is similar in approach to that for the
Grad-Shafranov equation of barotropic normal MHD \citep{1f_eqm}, up
until the result \eqref{fmag_general}. One remaining step from the normal-MHD
derivation cannot, however, be generalised straightforwardly for
superconducting matter: we no longer have ${\bf B}\cdot\nabla M=0$ and so
in general $M\neq M(u)$. This prevents us from continuing the derivation for
poloidal/mixed fields in the normal-MHD manner. Rather than discussing
this issue in more detail now, we
stop at the interim general result
\eqref{fmag_general} and only consider the simplest specific case,
where the magnetic field is purely toroidal. We intend to return to
the cases of purely poloidal and mixed poloidal-toroidal fields in
future work.
\subsection{Purely toroidal fields}
For purely toroidal fields, let us return to equation
\eqref{fmag_general}. In this case we have $\nabla u=0$ and the
expression for the magnetic force reduces to
\begin{equation} \label{fmag_tor}
\boldsymbol{\mathfrak{F}}_{\mathrm{mag}} = \rho_\mathrm{p}\nabla M
= -\frac{h_c}{4\pi}\frac{1}{\varpi} \nabla(\varpi\rho_\mathrm{p} B_\phi),
\end{equation}
where we have also used $\hat{B}_\phi=1$. Again, we recall the
normal-MHD result \citep{1f_eqm} for comparison:
\begin{equation}
\lor = \rho\nabla M
= -\frac{B_\phi}{4\pi\varpi}\nabla(\varpi B_\phi).
\end{equation}
Now, dividing equation \eqref{fmag_tor} by $\rho_\mathrm{p}$ and taking the
curl of it yields
\begin{equation}
0 = \nabla\brac{\frac{1}{\varpi\rho_\mathrm{p}}}
\times\nabla(\varpi\rho_\mathrm{p} B_\phi)
= -\frac{1}{\varpi^2\rho_\mathrm{p}^2}
\nabla(\varpi\rho_\mathrm{p})\times\nabla(\varpi\rho_\mathrm{p} B_\phi).
\end{equation}
Wherever the magnetic field is non-zero $\varpi\rho_\mathrm{p}\neq 0$ too,
since this is the geometry of a toroidal field, so we have
\begin{equation}
0 = \nabla(\varpi\rho_\mathrm{p})\times\nabla(\varpi\rho_\mathrm{p} B_\phi)
\end{equation}
and hence the two arguments of the gradient operators are related by some
function $\eta$:
\begin{equation}
\eta(\varpi\rho_\mathrm{p}) = \varpi\rho_\mathrm{p} B_\phi.
\end{equation}
Putting this into \eqref{fmag_tor} and defining
$\zeta\equiv\varpi\rho_\mathrm{p}$ we get
\begin{equation} \label{nabM_tor}
\nabla M = -\frac{h_c}{4\pi}\frac{1}{\zeta}\nabla\eta(\zeta)
= -\frac{h_c}{4\pi}\frac{1}{\zeta}\td{\eta}{\zeta}\nabla\zeta
\end{equation}
where the corresponding magnetic field is
\begin{equation}
{\bf B}=B_\phi{\bf e}_\phi=\frac{\eta(\zeta)}{\zeta}{\bf e}_\phi.
\end{equation}
The first integral of \eqref{nabM_tor} gives our final result, which
may be used in the code:
\begin{equation}
M = -\frac{h_c}{4\pi}
\int_0^{\varpi\rho_\mathrm{p}} \frac{1}{\zeta}\td{\eta}{\zeta}\ \mathrm{d}\zeta.
\end{equation}
Not all functional forms of $\eta(\zeta)$ are acceptable choices ---
for example, taking $\eta$ to be a linear function of $\zeta$ leads to a
magnetic force which diverges on the polar axis. In this paper we will
work with two other choices of $\eta$. The first (which we will
refer to as `$\zeta^2$-superconductivity' for brevity) is
\begin{equation}
\eta=\eta_0\zeta^2
\end{equation}
with $\eta_0$ a constant, which may be varied to adjust the magnetic
field strength. With this we have
\begin{equation}
M = -\frac{h_c\eta_0}{2\pi}\varpi\rho_\mathrm{p}
\ \ \ \textrm{and}\ \ \
B_\phi = \eta_0\varpi\rho_\mathrm{p}.
\end{equation}
This form of $B_\phi$ is the same as that in our normal-MHD
toroidal-field case. \skl{Note that
$\zeta^2$-superconductivity is the two-fluid equivalent of the case
considered by \citet{akgun_wass}.}
The second functional form we
will work with (hereafter `$\zeta^3$-superconductivity') is
\begin{equation}
\eta=\eta_0\zeta^3.
\end{equation}
The magnetic force scalar and magnetic field in this case are given by
\begin{equation}
M = -\frac{3h_c\eta_0}{8\pi}\varpi^2\rho_\mathrm{p}^2
\ \ \ \textrm{and}\ \ \
B_\phi = \eta_0\varpi^2\rho_\mathrm{p}^2.
\end{equation}
\skl{At this stage, our motivation for choosing these two functional forms
is their simplicity. Later, we will find that the resulting equilibria
in the two cases are very similar, suggesting that our results may be
quite generic.}
\section{Numerics}
\subsection{Overview of code}
The code we use iteratively solves the equilibrium equations for our neutron
star model, and being non-linear is not restricted to the perturbative
regime of slow rotation and weak magnetic fields. Our numerical scheme
is based on the Hachisu
self-consistent field (SCF) method \citep{hachisu}, a more robust
extension of an earlier SCF method by \citet{ostr_mark}. A version of
the Hachisu SCF method for magnetised stars was presented by
\citet{tomi_eri}. Since our scheme is a fairly straightforward extension
of these previous ones, we content ourselves with a summary here and
focus on the differences between them; we refer the reader to
\citet{hachisu} and \citet{tomi_eri} for more details.
To find equilibrium models, the user must specify a number of
stellar parameters at the outset. The major ones are related to:\\
1. the EOS --- through the polytropic indices $N_\rmn$ and $N_\mathrm{p}$ (related to the
compressibility of each fluid);\\
2. the shape of the star (related to the rotation rate) --- specified
through the ratio of polar to equatorial radii $r_{pole}/r_{eq}$, and the
ratio of the neutron-fluid equatorial surface to the proton-fluid
equatorial surface $r_{eq}^\rmn/r_{eq}^\mathrm{p}$;\\
3. the magnetic field --- through $\kappa$ and $a$ for normal-MHD
mixed fields (or poloidal fields), $\lambda$ for normal-MHD toroidal
fields, and $\eta_0$ for toroidal fields in a superconductor. See
sections 2.2 and 3.2 for more details.
We nondimensionalise all quantities within the code using the
requisite combination of $r_{eq}$, $G$ and $\rho_{\mathrm{max}}$ and use a hat
to denote these dimensionless quantities. As a consequence we have
$\hat{r}_{eq}=\hat\rho_{\max}=1$. When calculating integrals we first
decompose the integrands into radial pieces and Legendre polynomials
(note that there is no azimuthal piece, since we work in
axisymmetry). The contributions over all relevant grid points are then
summed up appropriately. We include angular contributions up to degree
$l=32$ in the Legendre polynomials, thus allowing for magnetic
configurations of high multipolar structure, in
contrast with the dipolar models of \citet{GAL}.
\subsection{Finding integration constants and the rotation rate}
To find the stellar rotation rate and integration constants of the
Euler equations we have to evaluate these equations at suitable
boundaries. Typically the two fluid surfaces do not coincide, and
there are different procedures to calculate these constants depending
on whether the proton or neutron fluid is outermost. However, we know
\emph{a posteriori} that all our configurations have the proton fluid
either outermost or coincident with the neutron surface, so we only
consider that case here.
For this, we work with the proton-fluid Euler and the difference Euler
and aim to find $C_\mathrm{p},C_d$ and $\Omega$; recall that we
assume that both fluids have the same rotation rate,
$\Omega_\rmn=\Omega_\mathrm{p}=\Omega$. In this section we define each fluid
surface by the vanishing of its corresponding chemical potential
$\tilde\mu_\mathrm{x}=0$; this approach was also used by \citet{yosh_eri}
and avoids numerical difficulties associated with dealing with the
true fluid surfaces --- those given by $\rho_\mathrm{x}=0$. In general the
two definitions do not agree, due to coupling between the different
fluids \citep{prix_nc}, and in practice one would expect an interior phase
transition rather than a sharp fluid surface. Despite these issues,
working with the $\tilde\mu_\mathrm{x}=0$ surfaces is sufficient for our
simplified model of a two-fluid NS. In section 5.2 we describe how to
check the surfaces of the redimensionalised, physical, star.
This subsection requires the user-specified stellar axis ratio
$r_{pole}/r_{eq}$ (which is the same as the proton-fluid axis ratio
$r_{pole}^\mathrm{p}/r_{eq}^\mathrm{p}$). In our dimensionless units the axis ratio is
equal to the dimensionless polar radius of the proton fluid
$\hat{r}_{pole}^\mathrm{p}$. The neutron-surface equatorial radius
$\hat{r}_{eq}^\rmn=r_{eq}^\rmn/r_{eq}^\mathrm{p}$ is also used here.
We begin by evaluating \eqref{pEuler} at the equatorial stellar
surface $\hat{r}_{eq}=1$:
\begin{equation}
\hat{\mu}_\mathrm{p}(1)=0
=\hat{C}_\mathrm{p}+\hat{M}(1)+\frac{\hat\Omega^2}{2}-\hat\Phi(1).
\end{equation}
and at the polar stellar surface $\hat{r}_{pole}(=\hat{r}_{pole}^\mathrm{p})$:
\begin{equation}
\hat{\mu}_\mathrm{p}(\hat{r}_{pole})=0
=\hat{C}_\mathrm{p}+\hat{M}(\hat{r}_\mathrm{p})-\hat\Phi(\hat{r}_\mathrm{p}).
\end{equation}
We may combine the above expressions to find $\Omega$ and $C_\mathrm{p}$:
\begin{equation}
\hat\Omega^2
=2\Big( \hat{M}(\hat{r}_{pole})-\hat{M}(1)
-\hat\Phi(\hat{r}_{pole})+\hat\Phi(1) \Big)
\end{equation}
\begin{equation}
\hat{C}_\mathrm{p}=\hat\Phi(1)-\frac{\hat\Omega^2}{2}-\hat{M}(1).
\end{equation}
Note that \skl{we have} $M\propto\rho_\mathrm{p}$ for a toroidal field (in
both normal and
superconducting matter) and so is zero at
the stellar surface; in this case the expressions for $\Omega$ and
$C_\mathrm{p}$ have no explicit dependence on the magnetic force.
Now evaluating the difference Euler at the \emph{neutron-fluid surface},
rather than the stellar surface, we determine $C_d$:
\begin{equation}
\hat{C}_d = \hat\mu_\mathrm{p}\brac{\hat{r}_{eq}^\rmn}
- \hat{M}\brac{\hat{r}_{eq}^\rmn}.
\end{equation}
\subsection{Iterative step}
We recall that the chemical potentials are defined from the energy
functional $\mathcal{E}$ by
\begin{equation}
\tilde\mu_\mathrm{x}\equiv\left.\pd{\mathcal{E}}{\rho_\mathrm{x}}\right|_{\rho_{\mathrm{y}}}.
\end{equation}
For our particular functional \eqref{eos_functional}, the resulting
expressions for the chemical potentials may be rearranged to give
\begin{equation} \label{otherinvert}
\rho_\rmn = \brac{\frac{\tilde{\mu}_\rmn}{k_\rmn\gamma_\rmn}}^{N_\rmn}\ ,\
\rho_\mathrm{p} = \brac{\frac{\tilde{\mu}_\rmp}{k_\mathrm{p}\gamma_\mathrm{p}}}^{N_\mathrm{p}}.
\end{equation}
Evaluating these at the centre of
the star we obtain relations between the maximum values of the
chemical potential and density for each fluid. In dimensionless form
these may be rearranged to show that
\begin{equation}
\hat{k}_\mathrm{p}\gamma_\mathrm{p}=\hat\mu_\mathrm{p}^{\max} x_\mathrm{p}(0)^{-1/N_\mathrm{p}}\ ,\
\hat{k}_\rmn\gamma_\rmn=\hat\mu_\rmn^{\max}(1-x_\mathrm{p}(0))^{-1/N_\rmn},
\end{equation}
where $x_\mathrm{p}(0)$ is the central proton fraction, using the fact that
$\hat\rho_\mathrm{p}=\rho_\mathrm{p}/\rho=x_\mathrm{p}$ and similarly
$\hat\rho_\rmn=(1-x_\mathrm{p})$.
\skl{Comparing these relations with those in \eqref{otherinvert} we obtain
expressions which will allow us to iterate for new density
distributions within our numerical scheme:}
\begin{equation} \label{it_rhopn}
\hat{\rho}_\mathrm{p}
= x_\mathrm{p}(0)
\brac{\frac{\hat{\mu}_\mathrm{p}}{\hat{\mu}_\mathrm{p}^{\max}}}^{N_\mathrm{p}}
\ \ ,\ \
\hat{\rho}_\rmn
= \big(1-x_\mathrm{p}(0)\big)
\brac{\frac{\hat{\mu}_\rmn}{\hat{\mu}_\rmn^{\max}}}^{N_\rmn}.
\end{equation}
\subsection{Iterative procedure}
We now have all equations needed for the iterative procedure that
generates our equilibria. As discussed at the start of this section,
the procedure is an extension of the Hachisu SCF method
\citep{hachisu,tomi_eri}. \skl{To begin the iterative process (step 0),
we set the two fluid densities and the magnetic potential to be
constant within the star; however, we have found that the final
equilibria are independent of the values chosen at this step.} Note
that for a toroidal field --- in
normal or superconducting matter --- there is no separate iteration for
the magnetic field and hence no step 2; instead the behaviour of the
field is directly linked to $\rho_\mathrm{p}$. The iterative steps are as
follows:\\
\\
0. For the initial iteration, start with a guess that
$\rho_\rmn,\rho_\mathrm{p}$ and $A_\phi$ are all constant;\\
1. Calculate the gravitational potential $\Phi$ from the $\rho_\rmn$ and
$\rho_\mathrm{p}$ distributions and Poisson's equation \eqref{int_Poisson};\\
2. Calculate the new magnetic potential component $A_\phi$ from its
value at the previous iteration $A_\phi^{old}$, using the magnetic
Poisson equation \eqref{int_Aph} with $A_\phi^{old}$ and $\rho_\mathrm{p}$ in the
integrand;\\
3. Evaluate the proton-fluid Euler \eqref{pEuler} at the equatorial
and polar surface, using boundary conditions on $\tilde{\mu}_\rmp$; this gives two
equations which fix $\Omega^2$ and then $C_\mathrm{p}$;\\
4. We are now able to use the proton-fluid Euler to find $\tilde{\mu}_\rmp$
throughout the star;\\
5. Evaluate the difference-Euler \eqref{dEuler} at the equatorial
neutron-fluid surface to find $C_d$;\\
6. Now use the difference-Euler to find $\tilde{\mu}_\rmn$ throughout the star;\\
7. Calculate the new density distributions from the chemical
potentials, using the expressions in \eqref{it_rhopn};\\
8. Return to step 1 using the new $\rho_\rmn,\rho_\mathrm{p}$ and $A_\phi$;
repeat procedure until satisfactory convergence is achieved
(i.e. until the difference between quantities at consecutive iterative
steps is less than some small tolerance).
\section{Stellar parameters and sequences}
\subsection{Redimensionalising}
When looking at sequences of configurations, to study the effect of
magnetic fields or rotation, we need to ensure we are always working
with the same physical star --- the dimensionless quantities produced
by the code are not sufficient. Two stars are physically the same if
they have the same mass and equation of state:
$\mathcal{M},\gamma_\rmn,\gamma_\mathrm{p},k_\rmn$ and $k_\mathrm{p}$ must all be equal for
both. Since we specify the (dimensionless) indices $\gamma_\rmn$ and
$\gamma_\mathrm{p}$ at the outset, we only need to fix the physical values
of $\mathcal{M},k_\rmn$ and $k_\mathrm{p}$.
Fixing the stellar mass is straightforward --- we choose
$\mathcal{M}=1.4\mathcal{M}_\odot$ (where $\mathcal{M}_\odot$ is one solar mass) --- but
more care is needed with the polytropic constants. We choose to fix
their physical values in the following way: take a spherical star in
hydrostatic equilibrium (nonrotating and with no magnetic field), with some
fixed $\gamma_\rmn$ and $\gamma_\mathrm{p}$ and coincident neutron and
proton surfaces. Now find the polytropic constants
$k_\rmn,k_\mathrm{p}$ that give it a radius of 10 km. More specifically,
the dimensionless mass and polytropic constants are related to their
physical counterparts through:
\begin{equation}
\hat{\mathcal{M}} = \frac{\mathcal{M}}{\rho_{\mathrm{max}} r_{eq}^3}\ \ ,\ \
\hat{k}_\mathrm{x}=\frac{k_\mathrm{x}}{Gr_{eq}^2\rho_{\mathrm{max}}^{\gamma_\mathrm{x}-2}}.
\end{equation}
Having specifying a spherical configuration with some particular
$\gamma_\mathrm{x}$, the code may be used to calculate $\hat{\mathcal{M}}$ and
$\hat{k}_\mathrm{x}$. \skl{The requirement that $\mathcal{M}=1.4\mathcal{M}_\odot$ and
$r_{eq}=10$ km for the spherical configuration then fixes the physical
value of $k_\mathrm{x}$ for the stellar sequence; from this we may find
$\rho_{\mathrm{max}}$ and $r_{eq}$ for any star in the sequence, and hence
redimensionalise any other quantities (e.g. rotation rates and field
strengths). Any rotating, magnetised configuration redimensionalised in this
manner will be the same physical star: if you stopped it from rotating and
removed its magnetic field, it would return to the same 10 km
spherical body.}
\subsection{Adjusting the position of the fluid surfaces}
Our iterative procedure does not automatically adjust the
location of the neutron surface, even though it will change with
respect to the proton surface when the star is rotating or
magnetised. A rotating or magnetised star in which the two fluids
coincide at the equator would \emph{not} return to our canonical
unmagnetised nonrotating spherical star with coincident fluid
surfaces. This means that the physical values of the polytropic
constants $k_\mathrm{x}$ would be different in the two cases. Instead
we have to manually adjust the location of the neutron-fluid
equatorial surface so that the values of $k_\mathrm{x}$ \emph{do} agree with those
of our canonical physical star. In practice the difference between the
fluid surfaces is only significant at high rotation rates or magnetic
field strengths, and for the purposes of this study a simple
root-finder algorithm can be employed to run the code a few times,
adjusting the neutron-surface until satisfactory results are achieved.
\subsection{Virial test}
The scalar virial theorem, which is derived from the (sum) Euler
equation, states that a certain combination of energies balances the
acceleration of the system. For a single-fluid polytrope:
\begin{equation}
2T+\clE_{\mathrm{mag}}+W+3(\gamma-1)U=\frac{1}{2}\td{^2 I}{t^2}
\end{equation}
where $T,\clE_{\mathrm{mag}},W$ and $U$ are, respectively, the kinetic, magnetic,
gravitational and internal energies of the system, and $I$ is the
moment of inertia. For our `separable' equation of state, where each
fluid obeys its own polytropic relation, this relation may be
generalised simply:
\begin{equation}
2T+\clE_{\mathrm{mag}}+W+3(\gamma_\rmn-1)U_\rmn+3(\gamma_\mathrm{p}-1)U_\rmn
= \frac{1}{2}\td{^2 I}{t^2}.
\end{equation}
Since we seek equilibrium configurations, we can test the accuracy of
our code by how close the residual acceleration of the system
(i.e. the right-hand side of the previous equation) is to zero. We
then form a dimensionless
quantity by normalising this acceleration through division by $W$:
\begin{equation} \label{VT}
\textrm{`virial\ test'}\equiv \textrm{error} = \frac{|2T+\clE_{\mathrm{mag}}+W+3(\gamma_\rmn-1)U_\rmn+3(\gamma_\mathrm{p}-1)U_\mathrm{p}|}{|W|}.
\end{equation}
\subsection{Ellipticity}
Our numerical scheme involves specifying some axis ratio
$r_{pole}/r_{eq}$, corresponding to the surface shape of the
star. A more informative quantity however is the ellipticity, which measures
the deviation of the whole mass distribution from sphericity. Starting
from the mass quadrupole moment tensor
\begin{equation}
Q_{jk} = \int\rho x_j x_k \ \mathrm{d} V
\end{equation}
where $x_j,x_k$ are Cartesian coordinates, we define the ellipticity
of the star through the components of the quadrupole moment
at the equator $Q_{xx}$ and pole $Q_{zz}$:
\begin{equation}
\epsilon=\frac{Q_{xx}-Q_{zz}}{Q_{xx}}.
\end{equation}
\subsection{Magnetic-field quantities}
The conventional definition of the (normal-MHD) magnetic energy is as an
integral out to infinite radius:
\begin{equation}
\clE_{\mathrm{mag}}=\frac{1}{8\pi}\!\!\int\limits_{\mathrm{all\ space}}\!\! B^2\ \mathrm{d} V,
\end{equation}
which would have to be truncated on our finite numerical grid. One
alternative but equivalent form of $\clE_{\mathrm{mag}}$ is as the integral of
${\bf r}\cdot\lor$; this may be seen from the relevant part of the virial
theorem derivation \citep{chand_hydro}. Since $\lor$ is only non-zero
over the star (by virtue of its $\rho_\mathrm{p}$ factor), we may write the
magnetic energy as
\begin{equation}
\clE_{\mathrm{mag}}=\int\limits_{\mathrm{star}}{\bf r}\cdot\lor\ \mathrm{d} V.
\end{equation}
The magnetic energy in the toroidal component of the field is also an
integral over the star:
\begin{equation}
\clE_{\mathrm{tor}}=\frac{1}{8\pi}\int\limits_{\mathrm{star}}B_\phi^2\ \mathrm{d} V,
\end{equation}
since $B_\phi=0$ outside the star. Finally, it is clear that the
poloidal-field contribution to the energy is simply $\clE_{\mathrm{mag}}-\clE_{\mathrm{tor}}$,
so there is no need to use an integral out to infinity for this case
either.
When looking at mixed-field configurations, we want some measure of the
relative strength of the toroidal and poloidal field components. We
do this in two ways: one related to the global contributions of
each component and one which looks at their maximum values. For the
former, we take the percentage of the magnetic energy stored in the
toroidal component $\clE_{\mathrm{tor}}/\clE_{\mathrm{mag}}$; this is relevant for understanding
quantities like the ellipticity, which scale with $B^2$. For the
latter we compare the maximum value attained by each field component,
using the ratio
\begin{equation} \label{poltor_max}
\frac{B_{\mathrm{tor}}^{\max}}{B_{\mathrm{tor}}^{\max}+B_{\mathrm{pol}}^{\max}}.
\end{equation}
Unlike the more obvious choice of $B_{\mathrm{tor}}^{\max}/B_{\mathrm{pol}}^{\max}$, the
ratio \eqref{poltor_max} varies from zero (no toroidal field) to unity
(purely toroidal field), which is perhaps clearer.
Although a number of studies have used the energy ratio $\clE_{\mathrm{tor}}/\clE_{\mathrm{mag}}$
to assess the stability of a magnetic-field configuration (see, e.g.,
\citet{braithtorpol}), this is a
global quantity, whereas the relevant instability is highly localised ---
around the neutral point for a purely poloidal field
\citep{markey}. To remove this instability, what is more important is
whether the toroidal component is \emph{locally} comparable in
strength with the poloidal one \citep{wright}. Since the strength of the poloidal
component in the closed-field line region is directly related to its
maximum value, we argue that the ratio \eqref{poltor_max}
may be a better indication of the stability of a configuration.
Since we are interested in the effect of a magnetic field on the
global properties of neutron stars, we work with a
volume-averaged form of the magnetic field $\bar{B}$ rather than a surface
value:
\begin{equation}
\bar{B}^2\equiv\frac{1}{V}
\!\!\int\limits_{\mathrm{all\ space}}\!\!
B^2\ \mathrm{d} V=\frac{8\pi\clE_{\mathrm{mag}}}{V},
\end{equation}
where $V$ is the star's volume. This volume-averaged value is
approximately double the polar surface value $B_{pole}$ for a poloidal
field (for a toroidal field the surface value is zero).
For superconducting configurations we have to fix the value of the
first critical field strength $H_{c1}$. This is not a constant, but is
related to $\rho_\mathrm{p}$ and hence position-dependent. Unless stated
otherwise, we fix the central (i.e. maximum) value of the field to be
$H_{c1}(0)=10^{16}$ G, to fit with the estimate from
\citet{GAS2011}. This corresponds to a volume-averaged value of
$\bar{H}_{c1}\approx 3\times 10^{15}$ G.
\section{Results}
Our results are divided into four subsections. We begin by testing the
convergence properties of the code and comparing our results with
those of previous studies. We then present results for two-fluid stars
in normal MHD,
focussing on the effect of stratification. In the third subsection we
study configurations where the protons form a type-II superconductor,
and in the fourth subsection we give approximate relations for the magnetic
ellipticity of stars as a function of field strength.
\subsection{Tests of the code}
\begin{figure}
\begin{center}
\begin{minipage}[c]{0.5\linewidth}
\psfrag{virial}{virial test}
\psfrag{resolution}{grid points}
\includegraphics[width=\linewidth]{cvg_test.eps}
\end{minipage}
\caption{\label{cvg_test}
Testing the order of convergence for our code, by plotting the
virial test result against the number of grid points. All
results are for a rotating magnetised NS model, stratified
with $N_\rmn=1$ and $N_\mathrm{p}=2$, with $\bar{B}=1.2\times 10^{17}$
G and $\Omega=650$ Hz. The line drawn shows the expected
result for second-order convergence; it agrees well with the
actual points from different resolutions.}
\end{center}
\end{figure}
Before we present results for our NS models, we test how accurate our
code is by using the virial test \eqref{VT} to determine the relative
error in calculating an equilibrium configuration for different
numbers of grid points; see figure \ref{cvg_test}. We see that the
error decreases quadratically with resolution, and hence is
second-order convergent, as intended.
Next, we compare our results with the three nonrotating and
unmagnetised two-fluid
models presented in \citet{prix_r}. To do this we need to convert from
their dimensionless quantities to ours, using the speed of light
$c=3.00\times 10^{10}$ cm s${}^{-1}$ and
nuclear density $\rho_{nuc}=1.66\times 10^{14}$ g cm${}^{-3}$. Some
algebra shows that our dimensionless polytropic constants (denoted by
the index $LAG$) and those of \citet{prix_r} (with index $PR$) are related by
\begin{equation}
\hat{k}_\mathrm{x}^{LAG}
= \frac{c^2}{Gr_{eq}^2\rho_{nuc}}
\brac{\frac{\rho_{\max}}{\rho_{nuc}}}^{\gamma_\mathrm{x}-2}
\ \hat{k}_\mathrm{x}^{PR}
= 8.11\times 10^3 \brac{\frac{r_{eq}}{\mathrm{km}}}^{-2}
\brac{\frac{\rho_{\max}}{\rho_{nuc}}}^{\gamma_\mathrm{x}-2}
\ \hat{k}_\mathrm{x}^{PR}.
\end{equation}
Next we consider the conversion of dimensionless masses:
\begin{equation}
\hat{M}^{LAG} = 12000 \brac{\frac{\rho_{\max}}{\rho_{nuc}}}^{-1}
\brac{\frac{r_{eq}}{\mathrm{km}}}^{-3}
\ \hat{M}^{PR}.
\end{equation}
Now using these, we can make a direct comparison between our results
and those of \citet{prix_r} --- see table \ref{PR_compare}.
\begin{table}
\begin{center}
\caption{\label{PR_compare}
Comparing with the unmagnetised two-fluid models presented in
Prix and Rieutord. The values shown for the polytropic constants
and masses are our
results, in our dimensionless units, with the discrepancy
from those of Prix and Rieutord indicated in brackets.}
\begin{tabular}{cccccc}
\hline
& $\gamma_\rmn$ & $\gamma_\mathrm{p}$ & $\hat{k}_\rmn$ & $\hat{k}_\mathrm{p}$ & $\hat{M}$\\
\hline
Model I & 2.0 & 2.0 & 0.7074 (0.1\%) & 6.366 (0.1\%) & 1.273 (0.5\%) \\
Model II & 2.5 & 2.1 & 0.7060 (0.5\%) & 9.034 (1.4\%) & 1.834 (0.7\%) \\
Model III & 1.9 & 1.7 & 0.6802 (0.7\%) & 3.466 (0.9\%) & 1.083 (1.0\%) \\
\hline
\end{tabular}\\
\end{center}
\end{table}
\subsection{Two-fluid equilibrium configurations in normal MHD}
Our two-fluid formalism gives us a great deal of flexibility, but for
clarity we will present results using just a few canonical
values. For unstratified models, we set $N_\rmn=N_\mathrm{p}=1$; whilst when
we refer to stratified models we have taken $N_\rmn=1$ and $N_\mathrm{p}=2$
unless otherwise stated \skl{(see the following figure for
justification of this choice)}. In addition, we have fixed the
central value of the proton fraction $x_\mathrm{p}(0)$ to be equal to $0.15$
for all results, \skl{although we have found that our results are
virtually independent of the value chosen}. In many figures, we show
stars whose magnetic fields are
(probably) unphysically high --- of the order of $10^{17}$ G. This is
to emphasise effects which would be less obvious at lower values. We
do, however, discuss the scaling of our results to more typical NS
field strengths too.
In this section we present results for neutron star models composed of
superfluid neutrons and normal-fluid protons, subject to a magnetic
field. As discussed above, this situation may apply to an interior
region of magnetars, where it is conceivable that the second
critical field $H_{c2}$ will be exceeded. Below this critical field
the protons will be superconducting, and we cover this case in the
following subsection.
\begin{figure}
\begin{center}
\begin{minipage}[c]{0.9\linewidth}
\psfrag{rho_tot}{$\displaystyle{\frac{\rho_{tot}}{\rho_{\max}}}$}
\psfrag{r_eq}{$\ r$}
\psfrag{x_p}{$x_\mathrm{p}$}
\psfrag{N_p=1}{$N_\mathrm{p}=1$}
\psfrag{N_p=2}{$N_\mathrm{p}=2$}
\psfrag{N_p=3}{$N_\mathrm{p}=3$}
\includegraphics[width=\linewidth]{EOS_fits.eps}
\end{minipage}
\caption{\label{EOS_fits}
Demonstrating the additional flexibility possible with a
two-fluid stellar model. Left: an approximation to the
Douchin-Haensel EOS for a fluid core and a solid crust,
\skl{plotting the total fluid density
$\rho_{tot}=\rho_\rmn+\rho_\mathrm{p}$ against radius $r$}. In
Douchin and Haensel, the core is fairly close to a polytrope with index
$N_{\rm core}\approx 0.7$ and the crust is similar to a different
polytrope, having $N_{\rm crust}\approx 1.5$, with a
crust-core boundary at $\approx 0.03\rho_{\mathrm{max}}$ and central
proton fraction $x_\mathrm{p}(0)=0.15$. In our model we have
$x_\mathrm{p}(0)=0.15$ and $N_{\rm core}\approx N_\rmn=0.7$, while
$N_{\rm crust}\approx N_\mathrm{p}=1.3$. This gives us a density of
$\sim 0.02\rho_{\mathrm{max}}$ at the crust-core boundary, with a 2
km-thick `crust' for a 10 km radius star. Right: the change in
proton fraction $x_\mathrm{p}$ against radius. We fix
$N_\rmn=1$ and look at the variation of proton fraction with
$N_\mathrm{p}$. In the unstratified case, $N_\mathrm{p}=N_\rmn=1$, the
proton fraction is constant within the star.}
\end{center}
\end{figure}
In figure \ref{EOS_fits} we show the additional flexibility possible
in two-fluid models. We can model the effect of having a fluid
core surrounded by a crust, by allowing the two fluids to terminate at
different points, so that there is an outer region with only one of
the fluid species --- in this case the protons. This is shown in the
left-hand plot, where we attempt to fit the equation of state
described in \citet{douchin} by adjusting parameters in our model
accordingly. Two-fluid models also allow us to have stratification and
consequently a non-uniform proton fraction, as one would expect in
real NSs \citep{kaminker_hy}; see the right-hand plot. Comparing
this plot with figure 1 from \citet{GAL} showing proton
fractions in more sophisticated models, we see that choosing $N_\mathrm{p}=2$
produces a reasonable-looking proton-fraction profile.
\begin{figure}
\begin{center}
\begin{minipage}[c]{0.8\linewidth}
\includegraphics[width=\linewidth]{breakup.eps}
\end{minipage}
\caption{\label{breakup}
Density contours of two unmagnetised stellar configurations
rotating at breakup velocity, left: our canonical
unstratified model ($N_\rmn=N_\mathrm{p}=1$), right: our canonical
stratified model ($N_\rmn=1$ and $N_\mathrm{p}=2$). The bold lines
show the fluid surfaces --- which are
coincident for the unstratified star. For the stratified star the
neutron fluid is not rotating at breakup velocity, but the
proton fluid is.}
\end{center}
\end{figure}
Before turning to the effect of a magnetic field on a two-fluid star, we
consider the role of rotation on it. This allows us to make contact
with models in previous work, as well as giving us some
intuition about what to expect from magnetic fields. In figure
\ref{breakup} we show two configurations rotating at their Keplerian
frequency. The left-hand star is an unstratified model
($N_\rmn=N_\mathrm{p}=1$) and both fluid
surfaces coincide. The interior density contours are different,
however, because the central proton fraction $x_\mathrm{p}(0)=0.15$; were it
$x_\mathrm{p}(0)=0.5$ then each species would have the same density contours
too. The right-hand star is a stratified model and in this case it is
the outer fluid, the protons, that determine the star's breakup
velocity --- the neutron surface is more rounded, without the
characteristic cusped-shape of a fluid at breakup \citep{prix_nc}. If
we had $N_\rmn>N_\mathrm{p}$, the neutron fluid would be outermost; however,
we do not consider this case. Instead the protons in our models are always
outermost, for rotating configurations but also magnetised ones, as we
will see later in this section.
\begin{figure}
\begin{center}
\begin{minipage}[c]{0.8\linewidth}
\includegraphics[width=\linewidth]{Bpol_vs_Np.eps}
\end{minipage}
\caption{\label{Bpol_vs_Np}
Contours of the streamfunction $u$ for a star with a purely
poloidal field, corresponding to magnetic field lines. The
thick arc indicates the stellar surface, at a dimensionless
radius of unity; the shaded area indicates the closed-field
line region, and the small circle on the $x$-axis shows the
location of the neutral line, where the field strength
vanishes. Left: no stratification ($N_\rmn=N_\mathrm{p}=1$),
right: a stratified model, with $N_\rmn=1$ and
$N_\mathrm{p}=3$. The closed field line region is bigger in the
stratified case.}
\end{center}
\end{figure}
Whilst our formalism is able to deal with stars that are both highly
magnetised and rapidly rotating, we will concentrate on non-rotating
models from now on. This is because we are chiefly concerned with the
effect of stratification on magnetic fields in stars; allowing for the
extra effect of rotation obfuscates the picture. This makes our
results most directly applicable to magnetars (which rotate very
slowly), but the ellipticity formulae presented at the end of this
results section are equally applicable to the magnetic distortions in
less-magnetised NSs (like pulsars).
We begin by looking
at stars with a purely poloidal magnetic field - figure
\ref{Bpol_vs_Np}. We show contours of the streamfunction $u$, which
are parallel to magnetic field lines, shading the closed-field line
region and marking the `neutral line' (where the field vanishes) with
a circle. The field configuration of the unstratified model is very
similar to that of a single-fluid polytrope with index $N=1$ (see,
e.g., figure 3 from \cite{1f_eqm}). Stratification has the effect of
moving the neutral line inwards and increasing the volume of the
closed-field line region. Fixing $N_\rmn=1$, we find that the greater
the value of $N_\mathrm{p}$, the larger the closed-field line region; we
have emphasised the effect by taking $N_\mathrm{p}=3$ rather than our
canonical choice of $N_\mathrm{p}=2$.
\begin{figure}
\begin{center}
\begin{minipage}[c]{\linewidth}
\includegraphics[width=\linewidth]{increase_a.eps}
\end{minipage}
\caption{\label{increase_a}
Effect of increasing $a$, the coefficient of the magnetic
function $f(u)$, within a stratified star. Contours of
the toroidal field component are
shown. As $a$ is increased, from left to right, the
percentage of toroidal field increases (1\%,3\% and 4.5\%),
with a corresponding increase in the ratio
$B_{\mathrm{tor}}^{\max}/(B_{\mathrm{pol}}^{\max}+B_{\mathrm{tor}}^{\max})$ --- from 0.10 to
0.18, and finally 0.34. At the same time, however, the size
of the toroidal-field region is decreased.}
\end{center}
\end{figure}
The toroidal-field component of a mixed-field star is defined by a
function $f(u)$, which can only be non-zero within the closed-field
line region, and by the coefficient $a$ (see equation
\eqref{f_defn}). Increasing $a$ increases the
percentage of magnetic energy stored in the toroidal-field component,
and also the maximum value it attains relative to the poloidal
component, but it \emph{decreases} the volume of the torus containing the
toroidal-field component; see figure \ref{increase_a}. This seems to
limit the size of the toroidal component, as in the barotropic case
\citep{1f_eqm,ciolfi}. The volume of
the torus is increased somewhat for higher values of $N_\mathrm{p}$, however.
\begin{figure}
\begin{center}
\begin{minipage}[c]{0.5\linewidth}
\includegraphics[width=\linewidth]{P2_Btot.eps}
\end{minipage}
\caption{\label{P2_Btot}
Magnetic field strength within a stratified nonrotating NS
with 4.7\% toroidal energy. The toroidal field is contained
within the small region near the equatorial surface, reaching
its maximum at $x\approx 0.8$. The poloidal field pervades the
rest of the star, reaching a maximum at the centre. The
surface field strength is about 20\% of the central
value, 50\% of the maximum toroidal-component
strength and 50\% of the average value $\bar{B}$.}
\end{center}
\end{figure}
In figure \ref{P2_Btot} we show the variation of magnetic field
strength within a typical mixed-field stratified configuration. The
field reaches a local maximum in two places --- the centre of the star
(corresponding to the maximum value of the poloidal component) and
near the equatorial surface (the toroidal-component maximum). The
toroidal component vanishes at the stellar surface, but the poloidal
component extends outside the star (not shown). The configuration
shown has 4.7\% toroidal energy, around the maximum value possible
within our approach. The surface field strength is about 20\% of
the central value, about 50\% of the maximum toroidal field
strength and also about 50\% of the volume-averaged field
strength $\bar{B}$.
\begin{figure}
\begin{center}
\begin{minipage}[c]{0.8\linewidth}
\includegraphics[width=\linewidth]{pol_densconts.eps}
\end{minipage}
\caption{\label{pol_densconts}
Density contours for proton fluid (solid line) and neutron
fluid (dashed) in non-rotating stars distorted by purely
poloidal magnetic fields. The bold lines represent the fluid
surfaces, which coincide at the pole but not the equator. The
proton fluid is seen to be more distorted, due to the
magnetic field. The left and right plots show our canonical
unstratified and stratified models, respectively.}
\end{center}
\end{figure}
Next we turn to the effect of the magnetic field on the density
distribution of the star. We begin with purely poloidal fields, in
figure \ref{pol_densconts}. As in
the single-fluid barotropic case, these make the star oblate, but
virtually all of the distortion is in the proton fluid, with the
neutron fluid remaining nearly spherical. The two fluid surfaces
coincide at the pole, but at the equator the Lorentz force distorts
the proton fluid, resulting in a single-fluid region composed only of
protons. To make the effect noticeable, we have shown a highly
magnetised model, with $\bar{B}\sim 10^{17}$ G. The single-fluid region
is analogous to the case of a rotating stratified star with no
magnetic field; see figure \ref{breakup}. As mentioned in figure
\ref{EOS_fits} and also by \citet{prix_nc}, this single-fluid region
can be thought of as an approximation to a neutron-star crust.
The unstratified (left) and stratified (right) plots of figure
\ref{pol_densconts} are very similar; the only obvious difference is that when
$N_\mathrm{p}=2$ the protons
have a larger low density region than the $N_\rmn=1$ case (this is
true in unmagnetised stars too). We have not included density
contours for mixed-field stars, as these are very similar to those in
figure \ref{pol_densconts}, with the contour lines pushed outwards
slightly in the region where the toroidal component is located.
\begin{figure}
\begin{center}
\begin{minipage}[c]{0.4\linewidth}
\includegraphics[width=\linewidth]{tor_densconts.eps}
\end{minipage}
\caption{\label{tor_densconts}
Density contours for proton fluid (solid line) and neutron
fluid (dashed) in a non-rotating stratified star with a purely
toroidal magnetic field. The bold line represents the two
(virtually coincident) fluid surfaces, and hence the stellar
surface. Again, the proton fluid is seen to be more
distorted, due to the magnetic field.}
\end{center}
\end{figure}
In figure \ref{tor_densconts} we show density contours of a stratified
star with a purely toroidal magnetic field. Again, the neutron fluid is
virtually spherical and unaffected by the magnetic field, whilst the
proton fluid is highly distorted --- into a prolate configuration this
time. Note that whilst the innermost proton-density contours are
noticeably prolate, the outer ones become virtually spherical; the same effect
was seen in the (single-fluid) models of \citet{ostr_hart}, who
considered mixed fields but with a dominant toroidal
component. Related to this effect, the surfaces of the two fluids are
seen to be coincident (or very close to being so). The
unstratified case is very similar, just with more evenly-spaced
proton-density contours (as for the previous figure) --- it will be
shown in the following subsection, when we compare toroidal fields
in normal MHD and superconductivity.
\begin{figure}
\begin{center}
\begin{minipage}[c]{0.6\linewidth}
\psfrag{Np}{$N_\mathrm{p}$}
\psfrag{etor_pc}{$\displaystyle\frac{\clE_{\mathrm{tor}}}{\clE_{\mathrm{mag}}}$ (\%)}
\psfrag{Btor_max}{$\displaystyle\frac{B_{\mathrm{tor}}^{\max}}{B_{\mathrm{tor}}^{\max}+B_{\mathrm{pol}}^{\max}}$}
\includegraphics[width=\linewidth]{maxtor_vs_Np.eps}
\end{minipage}
\caption{\label{maxtor_vs_Np}
Mixed-field configurations with the maximum possible toroidal
component, as a function of $N_\mathrm{p}$. All configurations have
$N_\rmn=1$, are nonrotating and have axis ratios close to
unity. The central proton fraction $x_\mathrm{p}(0)=0.15$ in all
cases, but the results are virtually invariant of the value
chosen. We see that in more stratified stars the percentage
of toroidal energy can increase, but the maximum toroidal
field strength decreases with respect to the maximum poloidal
field strength.}
\end{center}
\end{figure}
In figure \ref{maxtor_vs_Np} we explore mixed-field configurations
with the maximum possible toroidal component (within our models, at
least). We use two different measures of the
relative strength of the toroidal component: comparing the maximum
values of the two field components using the ratio
$B_{\mathrm{tor}}^{\max}/(B_{\mathrm{tor}}^{\max}+B_{\mathrm{pol}}^{\max})$ and looking at the the
contribution of the toroidal component to the total magnetic energy,
$\clE_{\mathrm{tor}}/\clE_{\mathrm{mag}}$. Fixing $N_\rmn=1$ as usual, we
find that for increasing $N_\mathrm{p}$ the energy percentage increases, but
the relative maximum value of the toroidal component decreases. In all
cases the toroidal component is smaller --- especially in terms of
$\clE_{\mathrm{tor}}/\clE_{\mathrm{mag}}$. Note that all configurations plotted are close to
spherical (with an axis ratio of $r_{pole}/r_{eq}=0.996$). Very highly
distorted stars (with consequently strong magnetic fields) can have
slightly larger values of $\clE_{\mathrm{tor}}/\clE_{\mathrm{mag}}$.
\begin{figure}
\begin{center}
\begin{minipage}[c]{0.5\linewidth}
\psfrag{Bav}{$\displaystyle\brac{\frac{\bar{B}}{10^{17}\ \mathrm{G}}}^2$}
\psfrag{ellip}{$\epsilon$}
\psfrag{a}{(a)}
\psfrag{b}{(b)}
\includegraphics[width=\linewidth]{normal_polmix_ellips.eps}
\end{minipage}
\caption{\label{polmix_ellips}
The ellipticity of nonrotating unstratified stars as a function of
magnetic field strength. Line (a) shows the dependence for a
poloidal field, whilst (b) shows a mixed-field configuration
with 3\% toroidal energy. The toroidal component decreases
the oblateness of the star, as expected. There is no
discernable difference in the corresponding figure for a
stratified star; the ellipticity appears to be virtually
independent of stratification.}
\end{center}
\end{figure}
Finally in this section, we plot the dependence of ellipticity on
magnetic field strength, figure \ref{polmix_ellips}. The relation is
quadratic, as expected from the single-fluid case, and does not appear
to depend on the stratification of the star. We show distortions of a
purely poloidal field star, and one with a relatively strong mixed
field, 3\% toroidal energy. The toroidal component decreases the
oblateness of the star, as expected, but we are not able to generate
mixed-field configurations where the toroidal component is strong
enough to induce a \emph{prolate} distortion. We are only able to
produce prolate distortions when the magnetic field is purely
toroidal; this case is covered in the next section.
\subsection{Superconducting two-fluid equilibria with toroidal fields}
In the previous section we investigated magnetic NS models with superfluid
neutrons and normal protons. This situation
could apply in (some) magnetars, if their interior magnetic fields
exceed $H_{c2}$. Most NSs, however, are likely to be
composed predominantly of superfluid neutrons and type-II superconducting
protons. In this section we present the first NS models that account
for this, specialising to configurations with purely toroidal magnetic
fields.
\begin{figure}
\begin{center}
\begin{minipage}[c]{0.8\linewidth}
\includegraphics[width=\linewidth]{Btor_magnitude.eps}
\end{minipage}
\caption{\label{Btor_magnitude}
The variation in field strength within stars with purely
toroidal magnetic fields. The left-hand plot shows a
unstratified model in normal MHD; the
right-hand plot shows $\zeta^3$-superconductivity
in a stratified star. Both plots have the same maximum field
strength, for easier comparison. Stratification and superconductivity
both have the effect of concentrating the field in a smaller,
more central region of the star.}
\end{center}
\end{figure}
In figure \ref{Btor_magnitude} we consider the variation of
magnetic-field magnitude within the interior of two
different NS models. The left-hand plot shows a star with normal
protons and hence subject to the usual Lorentz force of MHD. It is
also unstratified. The field strength is seen to drop off fairly
slowly before vanishing at the stellar surface. In the right-hand plot
we show a toroidal magnetic field in a superconducting star, with a
particular choice of toroidal-field function which we refer to as
`$\zeta^3$-superconductivity' --- see section $3.2$. This star is also
stratified, and both this and the choice of toroidal-field function
act to `bury' the field deeper into the star. Our other choice for the
toroidal-field function, `$\zeta^2$-superconductivity', produces
configurations which are (in the unstratified case) indistinguishable
from the left-hand
plot. This is not surprising, since the expression for the toroidal
field in this case is of the same form as for normal MHD (again, see
section $3.2$).
\begin{figure}
\begin{center}
\begin{minipage}[c]{\linewidth}
\includegraphics[width=\linewidth]{norm_vs_sc_dens.eps}
\end{minipage}
\caption{\label{norm_vs_sc_dens}
The distorting effect of a toroidal field. From left to right:
normal MHD, $\zeta^2$-superconductivity,
$\zeta^3$-superconductivity. The plots are all
for very high field strengths (of the order of
$10^{17}$ G), since the aim of the figure is to illustrate
the different ways the density distribution is distorted in
each case. We discuss more realistic values later.}
\end{center}
\end{figure}
In figure \ref{norm_vs_sc_dens} we show density distributions for
three NS models with toroidal magnetic fields. We consider very highly
magnetised stars, which serve to emphasise the magnetic
distortions. All three models are broadly
similar: each has a spherical neutron-fluid distribution, coincident
neutron and proton-fluid surfaces, and prolate distortions of the
proton fluid. Comparing
the density contours for each model, we see that the superconducting
models have less distortion to the proton fluid in the outer region,
and more in the centre. The central panel ($\zeta^2$-superconductivity)
has the unusual feature of cusps in its proton-fluid distribution
around the pole. We show unstratified models, but the only obvious
effect of stratification is that the spacing of the proton-fluid density
contours is altered, as seen earlier in figure \ref{pol_densconts}.
We mentioned that figure \ref{norm_vs_sc_dens} shows models with
extremely strong magnetic fields, and we will show models with similar field
strengths in the next figure too. Whilst these high fields are
useful for emphasising certain features, the main reason for using
them comes from our numerical method, which calculates configurations
based on a given distortion (axis ratio). Since magnetic distortions
are typically very small, this means that even the axis ratio for the
least non-spherical star we can specify (constrained by the grid resolution)
corresponds to a very strong magnetic field. This is not necessarily a
problem for normal MHD, but superconductivity will be broken at these
field strengths, which exceed the second critical field of
$H_{c2}\approx 10^{16}$ G. We are able to produce these models because the
destruction of superconductivity is not built into them; the equations
may be solved for any field strength. Having done so, however, we need
to check that these models are consistent with our expectations for
NSs with superconducting protons. If they are, we may extrapolate our
results back to more realistic models, where $\bar{B}<\bar{H}_{c2}$.
\begin{figure}
\begin{center}
\begin{minipage}[c]{0.45\linewidth}
\psfrag{Bav}{$\displaystyle\brac{\frac{\bar{B}}{10^{17}\ \mathrm{G}}}^2$}
\psfrag{ellip}{$-\epsilon$}
\includegraphics[width=\linewidth]{normal_tor_ellips.eps}
\end{minipage}
\hspace{0.05\linewidth}
\begin{minipage}[c]{0.45\linewidth}
\psfrag{a}{(a)}
\psfrag{b}{(b)}
\psfrag{c}{(c)}
\psfrag{Bav}{$\displaystyle\frac{\bar{B}}{10^{17}\ \mathrm{G}}$}
\psfrag{ellip}{$-\epsilon$}
\includegraphics[width=\linewidth]{sc_ellips.eps}
\end{minipage}
\caption{\label{toroidal_ellips}
Left: in normal MHD the ellipticity scales with
$B^2$; right: in superconductivity it scales with $H_{c1}B$ - this is
illustrated by showing the results for a central
critical-field value $H_{c1}(0)=10^{16}$ G (a), and also
$H_{c1}(0)=2\times 10^{16}$ G (c); the latter line has double
the gradient. (a) and (c) are for $\zeta^2$-superconductivity; (b)
is for $\zeta^3$-superconductivity with $H_{c1}(0)=10^{16}$ G.}
\end{center}
\end{figure}
We perform this sanity check next, in figure
\ref{toroidal_ellips}. We plot the scaling of ellipticity
with average field strength $\bar{B}$ for NS models with toroidal
magnetic fields, for normal (left) and type-II superconducting protons
(right). Note that all ellipticities are negative, since these
configurations are prolate. In the left-hand plot we find that the
measured numerical values agree very well with the expected scaling
$\epsilon\propto B^2$, with small deviations when $\bar{B}\gtrsim
10^{17}$ G.
We now turn to the right-hand plot of figure
\ref{toroidal_ellips}. This time we plot the ellipticity against
$\bar{B}$, not $\bar{B}^2$. First we look at configurations with
$\zeta^2$-superconductivity, for a central critical field value
$H_{c1}(0)=10^{16}$ G (the points on the line marked (a)) and also for
$H_{c1}(0)=2\times 10^{16}$ G (the points on line (c)). In both cases we
see that the points lie virtually on straight lines, showing that
$\epsilon\propto\bar{B}$. In addition, line (c) has twice the gradient of
line (a). This confirms that despite the high field strengths we are
obliged to use (see the discussion above), we find the correct scaling
of the ellipticity: $\epsilon\propto H_{c1}\bar{B}$. This gives us more
confidence about our results and means we can safely extrapolate to
more typical NS field strengths. We also present ellipticities for
$\zeta^3$-superconductivity (points
along line (b)). The level of distortion in this case is very similar
to that in the $\zeta^2$ case.
Note that for both plots in figure \ref{toroidal_ellips} we consider
unstratified models, but we find that stratification makes no
discernable difference to the ellipticity results. This is the same as
for the previous plot showing ellipticities, figure
\ref{polmix_ellips}.
\subsection{Ellipticity formulae}
In figures \ref{polmix_ellips} and \ref{toroidal_ellips} we plotted
ellipticities of two-fluid stars as a function of field strength for
poloidal, toroidal and mixed fields in normal MHD, and toroidal fields
in superconductivity. In all cases we found the scaling of points on the
plots was in convincing agreement with the expected results:
$\epsilon\propto\bar{B}^2$ in the normal case and $\epsilon\propto H_{c1}\bar{B}$ in
superconductivity. In addition, for the very highest field strengths
we were able to see slight deviations from this linear result. From
the lines fitted to our data points we get quantitative relations
between ellipticity and magnetic field strength, which we present here
scaled to somewhat lower field strengths. In all cases the formulae
are for two-fluid models without stratification, but we found that
stratified stars obeyed the same relations, to a good approximation.
We begin with results for stars with superfluid neutrons and normal
protons. In the case of purely toroidal fields we have
\begin{equation} \label{norm_tor_ellip}
\epsilon=-3.0\times 10^{-6}\brac{\frac{\bar{B}}{10^{15}\ \mathrm{G}}}^2,
\end{equation}
whilst for purely poloidal fields the relation is
\begin{equation}
\epsilon=4.3\times 10^{-6}\brac{\frac{\bar{B}}{10^{15}\ \mathrm{G}}}^2.
\end{equation}
We also consider mixed fields with $3\%$ toroidal energy. This is
relatively strong for our unstratified models, but still only produces a
14\% reduction in oblateness from the purely poloidal case.
Next we consider models comprising superfluid neutrons and type-II
superconducting protons with purely toroidal magnetic fields. As
discussed in section 3.1, we have some flexibility in choosing the
function $\eta$ which governs the magnetic field. For our first choice of
function, $\zeta^2$-superconductivity, we have
\begin{equation} \label{sc_tor_ellip1}
\epsilon=-2.3\times 10^{-5}
\brac{\frac{H_{c1}(0)}{10^{16}\ \mathrm{G}}}
\brac{\frac{\bar{B}}{10^{15}\ \mathrm{G}}},
\end{equation}
whilst for $\zeta^3$-superconductivity the relation is
\begin{equation} \label{sc_tor_ellip2}
\epsilon=-2.6\times 10^{-5}
\brac{\frac{H_{c1}(0)}{10^{16}\ \mathrm{G}}}
\brac{\frac{\bar{B}}{10^{15}\ \mathrm{G}}}.
\end{equation}
\skl{We recall that these results were obtained using a density-dependent
first critical field $H_{c1}=h_c\rho_\mathrm{p}$, where $h_c$ is a constant};
in the above expressions we use the central critical field value
$H_{c1}(0)$, normalised to $10^{16}$ G. The
equivalent volume-averaged value is
$\bar{H}_{c1}\approx 3\times 10^{15}$ G. We note that the functional
form of $\eta$ has little effect on the ellipticity relation. The
results in the superconducting case are in very good agreement with
the barotropic study of \citet{akgun_wass}; using their ellipticity
formula (71) gives a result just 5\% different from our
$\zeta^2$-superconductivity relation \eqref{sc_tor_ellip1} and 15\%
different from \eqref{sc_tor_ellip2}.
\begin{figure}
\begin{center}
\begin{minipage}[c]{0.5\linewidth}
\psfrag{Bav1}{$\displaystyle\frac{\bar{B}}{10^{16}\ \mathrm{G}}$}
\psfrag{ellip}{$-\epsilon$}
\psfrag{superconductivity}{superconductivity}
\psfrag{normal MHD}{normal MHD}
\includegraphics[width=\linewidth]{sc_to_normal.eps}
\end{minipage}
\caption{\label{sc_to_normal}
Below the (volume-averaged) second critical field
$\bar{H}_{c2}$ a NS has type-II
superconducting protons and its ellipticity scales with
$H_{c1}\bar{B}$ --- considerably higher than the normal-MHD
curve plotted below it. Above the critical field
superconductivity is destroyed and
the ellipticity has the normal-MHD scaling, quadratic in
$\bar{B}$.}
\end{center}
\end{figure}
We summarise these last two extrapolated formulae in figure \ref{sc_to_normal},
where we show the expected variation of ellipticity with field
strength in a two-fluid NS model with a toroidal magnetic field. We
assume that below the volume-averaged value of the second critical
field $\bar{H}_{c2}\approx 10^{16}$ G the
protons are superconducting, whilst above it they are normal. The
neutrons are a superfluid in both cases. In the superconducting
regime, $\epsilon\propto\bar{B}$ and the star is more distorted than one
would expect from normal-MHD models. As the field strength is
increased superconductivity is broken and the protons obey normal MHD, with
$\epsilon\propto\bar{B}^2$.
Note that superconductivity can make a huge difference to the
ellipticities predicted for most pulsars \citep{wasserman}; the
discrepancy from normal MHD scales with $\bar{H}_{c1}/\bar{B}$. Let us
take a pulsar
with a volume-averaged field $\bar{B}=10^{13}$ G, and assume this field is purely
toroidal (since our only superconducting models are for this case). If the
protons form a normal fluid we may apply equation
\eqref{norm_tor_ellip} to predict
that its ellipticity will be $-3.0\times 10^{-10}$; if they are a
type-II superconductor then equation \eqref{sc_tor_ellip1} predicts that the
ellipticity of the pulsar will be $-2.3\times 10^{-7}$ --- a factor of
around 800 larger.
\section{Discussion}
This paper, together with a companion paper \cite{GAL}, presents the
first results on the equilibria of multifluid neutron stars with
magnetic fields. Although it has been thought for decades that the
fluid interiors of \skl{all but the youngest} neutron stars are likely
to comprise
(predominantly) a neutron superfluid and a magnetised proton fluid,
no previous studies have constructed equilibria of this kind. A
related problem is the role of stratification in NSs. This renders the
stellar matter non-barotropic and prevents the construction of
single-fluid equilibria using the usual Grad-Shafranov equation
\citep{reis_strat}. Virtually no studies have attempted to confront
this extra difficulty; to our knowledge the only previous studies of
magnetic equilibria in stratified stars are by Braithwaite --- although
these adopt an ideal-gas EOS more suited to main-sequence stars
\citep{braith_nord,braithtorpol} --- and the recent paper by
\citet{mastrano}. Finally, in most neutron stars the
protons are likely to form a type-II superconductor, which
significantly changes the form of the magnetic force from the familiar
Lorentz force. Once again, this effect has been neglected in most
studies; the only related work on superconducting NS equilibria is
that of \citet{akgun_wass}, although these are single-fluid models
which do not have a separate neutron superfluid.
Within this paper we explore some of these issues. In
the normal-MHD case, we show how the equations for a magnetised
barotropic star may be generalised quite straightforwardly to a
two-fluid model in which each fluid species obeys its own polytropic
relation. By choosing different proton and neutron polytropic indices,
we have a natural way of introducing stratification into the
models. \skl{For simplicity we assume that all the stellar matter is
multifluid, whereas NSs will also have regions which behave as a
single fluid; when purely multifluid stars are better understood, it
will be a logical future step to account for the effect of having
these different fluid regions.}
\skl{The equilibrium equations governing our stellar equilibria
feature a number of apparently arbitrary magnetic functions, related
to the strength of the magnetic field and its poloidal and toroidal
field components. We argue that there are actually a number of
restrictions on these functions, with little freedom in choosing
them --- and hence limited scope for producing qualitatively
different equilibria from those we present here. Given this, we
believe our results are quite generic to multifluid stars.}
We also discuss the equations governing equilibria with type-II
superconducting protons, starting from the relations derived in detail
by \citet{GAS2011}. The main difference from the normal-MHD two-fluid
case is in the form of the magnetic force, which depends on the usual
magnetic field ${\bf B}$ but also on the first critical field $H_{c1}$,
related to the presence of quantised fluxtubes. We use an approach inspired by
the derivation of the Grad-Shafranov equation (see, e.g.,
\citet{1f_eqm}) to try to produce equilibrium equations that may be
readily solved. As in the normal-MHD case, we arrive at a general
equation governing the magnetic field, at which point one must
specialise to different field configurations to get a solution. We
remark that one important step from the normal-MHD derivation does not
carry through to the superconducting case (and so may lead to
significant differences between the two states), but we postpone a detailed
discussion of this point to a later paper. For a purely toroidal
magnetic field however, a solution may be obtained quite easily, so we
specialise to this case in this paper.
In all cases, the equilibrium equations are solved numerically using
an iterative procedure, which is a variation of the Hachisu SCF method
\citep{hachisu}. This method is quite versatile and allows us to
consider extremely strong magnetic fields, in contrast with the study
reported in \citet{GAL}, where the magnetic field is regarded as a
perturbation on a spherical background star. The good agreement
between the results of this paper and \citet{GAL}, however, vindicate
the latter work's perturbative ansatz.
One main aim of this paper is to explore the role of
stratification in a magnetised neutron star. We increase the
stratification of the star by increasing the value of the proton-fluid
polytropic index $N_\mathrm{p}$, whilst fixing $N_\rmn=1$. For a purely
poloidal field, this results in a larger closed-field line region,
with the `neutral line' (where the field strength vanishes) moving
inwards; see figure \ref{Bpol_vs_Np}. In mixed-field configurations
the toroidal field component is contained in this region, and is
consequently larger in stratified stars. Even in the stratified case,
however, the energy in the poloidal component is always considerably
greater than
that in the toroidal component. This is because the parameter which
increases the toroidal-component strength also \emph{decreases} the
size of the region occupied by this component (figure
\ref{increase_a}). These results are in good agreement with those of
\citet{GAL}. \skl{Finally, for magnetic ellipticity relations our results
suggest that stratification is not very important; there is little
difference from the single-fluid barotropic relations.}
We have not investigated the stability of any configurations presented
here, \skl{although purely toroidal fields are unstable in both stratified
and unstratified stars}
\citep{goos_tayler}, and it seems likely that the same will be true for
purely poloidal fields, as in the unstratified case
\citep{markey}. This leaves mixed poloidal-toroidal fields as the only
candidates for stable configurations, and hence the best models for NS
magnetic fields. Although our configurations have quite small
percentages of magnetic energy in the toroidal component, due to the
small volume of the star it occupies, the magnitude of the two
field components can be comparable. This suggests that these models
could indeed be
stable \citep{wright}. Note that \citet{braithtorpol} suggested that a
considerably higher percentage of toroidal-field energy was needed for
stability. That work considered very different stellar models, however,
more applicable to main-sequence stars than neutron stars. \skl{There
is no consensus, as yet, about what percentages of each field
component are likely to exist in real NSs.}
Our other results concern equilibrium configurations of neutron stars
with superfluid neutrons and type-II superconducting protons. In this
initial study we specialise to purely toroidal magnetic fields, even
though the equilibria we generate are likely to be unstable
\citep{akgun_wass}. From our results we are able to produce the first
formulae for magnetic ellipticities of superconducting NSs based on a
two-fluid model. Our results suggest that the distortion is
around 50\% greater than the estimate used by \citet{cutler_prec}
(having accounted for the fact that we also use a larger value for the
critical field $H_{c1}$). There is very good agreement, however, with the
ellipticity formula (equation 71) from \citet{akgun_wass}. If a
substantial proportion of protons in a typical NS interior form a
type-II superconductor, then ellipticity formulae based on models
in normal MHD (e.g. \citet{haskell} and \citet{1f_eqm}) greatly
underestimate the potential magnetic distortion.
With the two-fluid model discussed in this paper, we are able to
produce a wide range of different NS equilibria, accounting for
stratification, superconductivity (with toroidal fields) and
rotation. As well as being interesting in their own right, these
equilibria could be used as background models on which to study
perturbations. This would help us understand the stability of these
various models, and their oscillation modes. The next logical step,
however, is to consider superconducting stars with purely poloidal and
mixed poloidal-toroidal magnetic fields. We hope to tackle this
problem in a future study.
\section*{Acknowledgments}
SKL acknowledges funding from the European Science Foundation (ESF)
for the activity entitled `The New Physics of Compact Stars', NA
acknowledges support from STFC in the UK through grant number
ST/H002359/1 and KG is supported by an Alexander von Humboldt fellowship
and by the German Science Foundation (DFG) via SFB/TR7. We thank the
referee Andreas Reisenegger for his detailed reading of this paper
and useful criticism.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,902
|
Q: Why my xcode 6 break on crash? I have some error in my app,
It report in console:
exc:*** -[__NSArrayI objectAtIndex:]: index 0 beyond bounds for empty array
but it just crashed and terminated, just didn't break on the crash line of code.
Is it a bug or something wrong on my settings ?
I know what is the problem of my code. I just want to know why xcode not break at the line of the code, not just terminated the app running.
Xcode 6.1 Mac OSX 10.10
A: somewhere in your code, you are trying to access objects within an array but the array is empty. Before accessing elements of array, just check if array is empty or not by using [myArray count].
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 586
|
Q: How to store info about a user in a file and then read and modify them I'm trying to build a channel where users can update a bot message via command. The channel would work like this: if X sends + Item the bot adds "Item" to an array linked to the user, that later is written on a file. To build the message, the bot reads the file and writes the elements of the array related to the user. How can I make that the array is someway linked to the user, like through the id, so that I can then read it?
Thanks for the help
A: In case you did not knew, you can get the sender's ID via message.author.id, where message is Message object.
You could find some node API around to help you with that.
Otherwise, you simply create a dictionary, with the user ID as the key value. You can check it out here or here.
(Either one of these would work. This can also help: clickme!, or you can google around for more.)
Storing into a textfile and maintaining the data is much trickier. If you can find some API that simplifies the process for you, then thats good.
But if alternative, you can store your data to a textfile somehow like strings are stored in a program. (Check out data-structures on how strings are stored and read.)
Basically, lets say you have 2 users with 2 element each. Your data in textfile is going to look like this:
user1-ID
element1
element2
NULL-VALUE
user2-ID
element1
element2
NULL-VALUE
And basically whenever you want to read from the textfile, just tell your program to search for the respective UserID, and read down the elements until it reaches a NULL-VALUE. And once it reaches there, your program stops reading it.
(Of course you can use something else other than NULL-VALUE, that was an example. I suggest making your elements wrapped around brackets. As some user might enter NULL-VALUE in the command arguement, which would cause some trouble. )
A: I finally ended up using a JSON object file, and reading the properties using list[id], since that id is a string. I set up the file in this way:
{"id1" : ["element1", "element2"], "id2": ["element3"]}
In this way, I can read the data stored in the file and write the modified values by simply writing the object with the fs module.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,350
|
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