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stringlengths 14
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|---|---|---|
Military and Security Developments
Middle East & World
Muhib Salha
Fuad Hamira
SHORESH DARWISH
Lama al-Atasi
Munther Khaddam
Muhammad Seyid Rasas
Meghan Bodette
Dr. Sarbast Nabi
Khorshid Delli
Marah al-Bukai
Hussein Jemmo
Nabil Al-Melhem
VIDEO GRAPHIC
60 SECOND SYRIA
TO BE AN IDP
WASHINGTON ONLINE
north press - North press agency
Poor harvest leaves farmers struggling in Syria's northeast
Wheat harvest in Hasakah countryside- North Press
HASAKAH, Syria (North Press) – This year's rain shortage has dashed the hopes of farmer Farhad Mousa for an acceptable season after the lack of wheat production of his land, despite his efforts to irrigate it with wells and sprinkle fertilizers.
As far as the area of Tel Tamr in the countryside of Hasakah, northeastern Syria, the rain-fed lands appear devoid of crops due to the lack of rain last winter, while the damage to the irrigated areas varied according to the number of watering times and the nature and fertility of the land.
Farhad Mousa, who rents agricultural land west of Tel Tamr after the Turkish army seized his village of Qasimia in the north, said he had hoped for better production despite the lack of rain.
He added that he expected the land to produce three times the amount of seed he planted, but it did not produce even twice the amount of seed.
The farmer estimated that he lost about 50% of the costs he incurred for sowing, plowing, watering, fertilizing, and harvesting fees.
"I planted 40 kilos of wheat as seed per acre. Now it only produces one or one and half bags of wheat. Ten hectares produces 15 bags, maybe less. We are greatly affected; the loss rate exceeds 50%."
Damage to rainfed crops
On the level of rainfed production, like most of the northern regions of Syria, the climate changes, accompanied by high temperatures, the lack of rain, especially during April and the interruption of water streams by Turkey caused total damage to the rainfed crops in the Tel Tamr region.
Sa'id Nayef, co-chair of the Agriculture Committee in Tel Tamr, said that the production of irrigated lands, despite the damage, is acceptable when compared to rainfed crops.
"The production of the irrigated areas considered acceptable. The agricultural areas in Tel Tamr are estimated at about 900,000 hectares.We licensed about 55,000 acres of irrigated lands and 70,000 acres of rainfed lands," he said.
Nayef pointed out that thousands of hectares in the countryside of Tel Tamr were not planted for the second year in a row due to the Turkish occupation or their presence near the front lines.
"There are thousands of hectares located within the areas occupied by Turkey. This was not cultivated by the people of the area."
Affected workers and families
Amid the deteriorating living conditions in the country, crop damage caused the absence of work opportunities for hundreds of laborers awaiting the annual harvest season.
The number of operating harvesters decreased, as many of them preferred not to work in light of the low production and high maintenance costs.
Fawaz Amin, a harvester owner from the city of Hasakah, does not seem satisfied with the results of his work after ten days of harvesting.
"In the past seasons we did well. The harvester's work the past two years was good. The workers who worked with us during the last season benefited, but this season is very bad," Amin said.
"But we regret harvesting this season. First of all, because of the insufficient fuel allocations we received, and also due to low production. The costs of maintaining the harvesters are high. This is reason behind the small number of harvesters," he explained.
He points out that the lack of production and the presence of moisture and green weeds in the agricultural lands damage the harvester, which means additional costs.
"There are very few harvests this year because of the low production. Many refrained from operating their harvesters," he added.
In mid-May, the AANES set the purchase price of wheat at 1,150 SYP, while the purchase price of barley was 850 SYP per kilogram.
Naming the price took into account the general conditions of the current season, such as drought, lack of rain and the decrease in irrigated areas of wheat, according to statements by AANES officials.
The region faces challenges regarding food security and the living situation due to the low production of wheat, despite north and east Syria being widely known as the country's "bread basket."
The challenges include providing sufficient stocks of flour needed to produce bread, and sufficient quantities of planting seeds for the next season, in addition to the loss of many of the region's residents as their source of income and job opportunities in dependent on agriculture.
The administration is ready to import wheat to meet the needs of the region in case the local production is not enough, said Salman Barudo, co-chair of the Economy and Agriculture Board of the Autonomous Administration of North and East Syria (AANES), in early June.
In a previous statement to North Press, Barudo expected that the production of wheat would not exceed 400,000 tons, which is about two-thirds of the needs of northeast Syria that range between 500,000 and 600,000 tons annually.
Mousa believes that it is the duty of the Autonomous Administration to support farmers this year so that they can continue planting next season.
"We ask the concerned authorities to help the farmers to be able to continue next year.
We've had a huge loss this year."
Reporting by Dilsoz Youssef
One person killed by gunmen in Syria's Daraa
Turkish opposition's memo pledges to deport Syrian refugees
An Autonomous Administration for South Syria?
Syrian expatriate Opens Iraq's First Coffee Academy
"Finnish and Swedish security go together" – Finland's FM
North Press Agency is a Syrian news agency that works on covering events in Syria as a whole through a team of correspondents and editors. North Press Agency deals with latest developments on all sides as well as preparing visual and written reports. North Press Agency is for all Syrians.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,915
|
Q: How to perform ng-include with button? I am having trouble trying to perform an ng-include when I click a button. Here is my code:
Main.html:
<div class="main">
<div ng-include="(manage.show && 'dsManage.html') || dsSelect.html"></div>
<h1>Data Sources</h1>
<div ng-controller="MainCtrl" >
<div ng-show="reportManage && (reportManage.canManage || reportManage.canManageAll)">
<div class="row searchRow">
<span class="col-sm-6" style="padding-left: 0px;">
<span class="input-group">
<input type="text" class="form-control searchInput" placeholder="Search" ng-model="search">
<span class="input-group-addon"><img ng-if="search" src="../images/Search_clear.png" ng-click="clearSearch()"/></span>
</span>
</span>
<div class="col-sm-2 col-sm-offset-4 text-right">
<a ng-show="tileLayout" class="btn-list-view" ng-click="tileLayout=false">List View</a>
<a ng-show="!tileLayout" class="btn-tile-view" ng-click="tileLayout=true">Tile View</a>
</div>
</div>
<div class="text-left">
<button ng-click="showDatasources()">All/Default</button>
<button ng-click="showManagePage()">Manage</button>
<!--<a href="{{dsSelectAllDefault}}" id="btn-default-all"></a>-->
</div>
</div>
</div>
</div>
Main.js:
$scope.showManagePage = function() {
console.log("True");
$scope.manage.show = true;
}
Basically what I want is that when I click the Manage button, showManagePage() will trigger and cause the ng-include at the very top of the code to trigger and show dsManage.html. This will replace the existing view, dsSelect.html, with dsManage.html. However, whenever I press this button, nothing occurs. The thing is these pages are in my root directory and when i add the ./ prefix to my files, my page seems to crash. Can anyone help me out? Thanks!
EDIT: Updated Main.html file:
<div class="main">
<div ng-controller="MainCtrl" >
<div ng-show="manage.show" ng-include="'dsManage.html'"></div>
<div ng-show="!manage.show">
<h1>Data Sources</h1>
<div ng-show="reportManage && (reportManage.canManage || reportManage.canManageAll)">
<div class="row searchRow">
<span class="col-sm-6" style="padding-left: 0px;">
<span class="input-group">
<input type="text" class="form-control searchInput" placeholder="Search" ng-model="search">
<span class="input-group-addon"><img ng-if="search" src="../images/Search_clear.png" ng-click="clearSearch()"/></span>
</span>
</span>
<div class="col-sm-2 col-sm-offset-4 text-right">
<a ng-show="tileLayout" class="btn-list-view" ng-click="tileLayout=false">List View</a>
<a ng-show="!tileLayout" class="btn-tile-view" ng-click="tileLayout=true">Tile View</a>
</div>
</div>
<div class="text-left">
<button ng-click="showDatasources()">All/Default</button>
<button ng-click="showManagePage()">Manage</button>
<!--<a href="{{dsSelectAllDefault}}" id="btn-default-all"></a>-->
</div>
</div>
</div>
</div>
</div>
</div>
I am now getting an issue saying WARNING: Tried to load angular more than once. and I notice I only get this error when I perform the ng-include. Any ideas on how to fix this?
A: I believe your manage.show is outside the scope of controller MainCtrl.
Try moving the DIV with ng-include inside the controller DIV. or move the controller declaration to the div with class main.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,745
|
{"url":"https:\/\/itprospt.com\/num\/1581491\/her-concepts-di-x-coordinate-the-vertex-the-parabola-ea","text":"5\n\n# Her Concepts di- x-coordinate the vertex the parabola ea? Find the 4x 12 by averaging the x-coordinates f(x) the y-intercept and its symmetric point x-coordinate th...\n\n## Question\n\n###### Her Concepts di- x-coordinate the vertex the parabola ea? Find the 4x 12 by averaging the x-coordinates f(x) the y-intercept and its symmetric point x-coordinate the vertex the parabola Find the f(x) =x + 4x 12 by averaging the x-coordinates of the x-mntercepts: Compare the methods you used parts (a) and (b)_ Are your results the same\" Which method from parts (a) and (b) is easier t0 use find the X-coordinale of the vertex the parabola t0 g(+) 54x2 19Sx 216? Explain. Which method(s) can b\n\nher Concepts di- x-coordinate the vertex the parabola ea? Find the 4x 12 by averaging the x-coordinates f(x) the y-intercept and its symmetric point x-coordinate the vertex the parabola Find the f(x) =x + 4x 12 by averaging the x-coordinates of the x-mntercepts: Compare the methods you used parts (a) and (b)_ Are your results the same\" Which method from parts (a) and (b) is easier t0 use find the X-coordinale of the vertex the parabola t0 g(+) 54x2 19Sx 216? Explain. Which method(s) can be used t0 find the x-coordinate of the vertex of the parabola h(x) X2 + 4x 62 Explain. E Summarize your findings in this exercise 70. In this exercise; You will discover how to convert the standard form of a quadratic function to its vertex form;: Find the vertex of the parabola f(x) 3x2 6x + 7. b Recall that if (h, k)is the vertex of the graph of a quadratic function,then f(x) a(x h)? k.Use your result from part (a) to determine h and k for f(x) 3x? 6x + 7. Substitute the values of h and k you found from part (b) in f(x) a(x h)? + k. Compare f(x) 3x2 6x + with your result in part (c): Determine the value of a. [Hint: You may see this immediately If not; write your result in part (c) in stan- dard form and compare again with f(x) 312 6x 7] Substitute your value of a into your result from part (c) 6 Use a graphing calculator to verify your result by comparing the graph of f(x) 3x2 6x with the graph of f(x) 0(35 6)2 + k for the values ofa,h,and k you found\n\n#### Similar Solved Questions\n\n##### ~\/5.26 points BBBasicStatBAcc 3.3.005.MI:Consider the following ordered data; 2 5 5 6 7 7 8 9 10(a) Flnd the low; Q1n median_ Q3r and high_ (For each answer; enter number: )mcdiannigh(b} Find the interquartile range (Enter number: }Make box-and-whisker plot: (Select the correct graph.)Need Help?Raad ItalktoTulor\n~\/5.26 points BBBasicStatBAcc 3.3.005.MI: Consider the following ordered data; 2 5 5 6 7 7 8 9 10 (a) Flnd the low; Q1n median_ Q3r and high_ (For each answer; enter number: ) mcdian nigh (b} Find the interquartile range (Enter number: } Make box-and-whisker plot: (Select the correct graph.) Need He...\n##### Ordered _ pair: Write as an (I,y) find the vertex coordinates. For problems & - f(r) - 21? 5.t3 = 0\nordered _ pair: Write as an (I,y) find the vertex coordinates. For problems & - f(r) - 21? 5.t3 = 0...\n##### When planning party you want to know whom to invite. Among the people you would like to invite are three touchy friends_ You know that if Jasmine attends; she will become unhappy if Samir is there Samir will attend only if Kanti will be there; and Kanti will not attend unless Jasmine also doesLet j s and k denote the propositions that Jasmine, Samir and Kanti attend, respectively: Express the given conditions using logical connectives(You must provide an answer before moving to the next part )\nWhen planning party you want to know whom to invite. Among the people you would like to invite are three touchy friends_ You know that if Jasmine attends; she will become unhappy if Samir is there Samir will attend only if Kanti will be there; and Kanti will not attend unless Jasmine also does Let j...\n##### Consider the following function:f() t} 2x} 311 = T2 13 with the following two constraints;lambd:A :1 Itz 7 12Ml;1UT1 + T2 + x3 = 15\nConsider the following function: f() t} 2x} 311 = T2 13 with the following two constraints; lambd: A : 1 Itz 7 12 Ml; 1U T1 + T2 + x3 = 15...\n##### Ifp > 0 and #-6 0 as k _ show that 6#fon E (and thus that there IS subsequcnce I tae iD\nIfp > 0 and #-6 0 as k _ show that 6#fon E (and thus that there IS subsequcnce I tae iD...\n##### 1) Let's find a root of the function f (x) =x? + 4x 3 iteratively starting with Xi =-landxz = 1. If applicable; what is the next point _ 13 usiug the Bisection method (3pts)Newton method (3pts)Secant method (4pts)\n1) Let's find a root of the function f (x) =x? + 4x 3 iteratively starting with Xi =-landxz = 1. If applicable; what is the next point _ 13 usiug the Bisection method (3pts) Newton method (3pts) Secant method (4pts)...\n##### Use the coordinates of the five quarter points of y = cos x to determine the corresponding quarter points on the graph ofy = 2 cos (6x Quarter points of 37 (0,1) 2,0 (1, - 1) Y = cosX (21,1) Quarter points of Y=2 cos (6x + x) + 2 (Simplify your answers. Type ordered pairs. Type exact answers, using 1 as needed. Use integers or fractions for any numbers in the\nUse the coordinates of the five quarter points of y = cos x to determine the corresponding quarter points on the graph ofy = 2 cos (6x Quarter points of 37 (0,1) 2,0 (1, - 1) Y = cosX (21,1) Quarter points of Y=2 cos (6x + x) + 2 (Simplify your answers. Type ordered pairs. Type exact answers, using ...\n##### Given g(z) =evaluate and simplify: 59(3 + h) - 9(3) hPrevicw Enter your answer a5 an expression Example: 3x*2+1 , x\/5, (atb)lc Be_sure YOur variables match those in the question Points possibe: This is attempt of 3_ Message instructor about this questionSubmit\nGiven g(z) = evaluate and simplify: 5 9(3 + h) - 9(3) h Previcw Enter your answer a5 an expression Example: 3x*2+1 , x\/5, (atb)lc Be_sure YOur variables match those in the question Points possibe: This is attempt of 3_ Message instructor about this question Submit...\n##### The truss shown is one of several supporting an advertising panel. Determine the force in each member of the truss for a wind load equivalent to the two forces shown. State whether each member is in tension $(T)$ or compression $(C).$\nThe truss shown is one of several supporting an advertising panel. Determine the force in each member of the truss for a wind load equivalent to the two forces shown. State whether each member is in tension $(T)$ or compression $(C).$...\n##### 6.4b Amount HCI UnansweredIfthere was enough Mg to displace the Cu, how much 6 M HCl (in mL) would be required to decompose the excess Mg present? Use the correct number of significant figures in your answer.\n6.4b Amount HCI Unanswered Ifthere was enough Mg to displace the Cu, how much 6 M HCl (in mL) would be required to decompose the excess Mg present? Use the correct number of significant figures in your answer....\n##### Calculate the derivative of the following functions. $$y=\\tan \\left(x e^{x}\\right)$$\nCalculate the derivative of the following functions. $$y=\\tan \\left(x e^{x}\\right)$$...\n##### A particle at rest decays into tWO with masses 3 kg and 5.33 kg and respective speeds 0.8c and 0.6c. What was the mass of the particle? Show your work parent\nA particle at rest decays into tWO with masses 3 kg and 5.33 kg and respective speeds 0.8c and 0.6c. What was the mass of the particle? Show your work parent...\n##### Find the exact area of the surlace obtained by rotating the curve y = V2r + 1, 0 <* <1about T-axis\nFind the exact area of the surlace obtained by rotating the curve y = V2r + 1, 0 <* <1 about T-axis...\n##### What Quastlon is Ihe rate at , 8ol z0 the rate Bro, of disappearance which Br ~(aq) . 5 Br + 6 H\" Bro; disappears 3 Brz '~(aq) 3H,0 is 0.028 Ihe reaction Ms? below if\nWhat Quastlon is Ihe rate at , 8ol z0 the rate Bro, of disappearance which Br ~(aq) . 5 Br + 6 H\" Bro; disappears 3 Brz '~(aq) 3H,0 is 0.028 Ihe reaction Ms? below if...\n##### CJTJCILCndAA solmor1 Subm WL H @FTAM \u00e2\u201a\u00ac\nCJTJCILCndAA solmor 1 Subm WL H @FTAM \u00e2\u201a\u00ac...\n##### In the sketch below, what is the potential difference between points B and F?\nIn the sketch below, what is the potential difference between points B and F?...","date":"2022-09-29 01:37:32","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.520930826663971, \"perplexity\": 3226.5830392627927}, \"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-2022-40\/segments\/1664030335303.67\/warc\/CC-MAIN-20220929003121-20220929033121-00480.warc.gz\"}"}
| null | null |
using UnityEngine;
using UnityEditor;
using System.Collections;
using System.Collections.Generic;
using System.Linq;
using tk2dEditor.SpriteCollectionEditor;
namespace tk2dEditor.SpriteCollectionEditor
{
public interface IEditorHost
{
void OnSpriteCollectionChanged(bool retainSelection);
void OnSpriteCollectionSortChanged();
Texture2D GetTextureForSprite(int spriteId);
SpriteCollectionProxy SpriteCollection { get; }
int InspectorWidth { get; }
SpriteView SpriteView { get; }
void SelectSpritesFromList(int[] indices);
void SelectSpritesInSpriteSheet(int spriteSheetId, int[] spriteIds);
void Commit();
}
public class SpriteCollectionEditorEntry
{
public enum Type
{
None,
Sprite,
SpriteSheet,
Font,
MaxValue
}
public string name;
public int index;
public Type type;
public bool selected = false;
// list management
public int listIndex; // index into the currently active list
public int selectionKey; // a timestamp of when the entry was selected, to decide the last selected one
}
}
public class tk2dSpriteCollectionEditorPopup : EditorWindow, IEditorHost
{
tk2dSpriteCollection _spriteCollection; // internal tmp var
SpriteView spriteView;
SettingsView settingsView;
FontView fontView;
SpriteSheetView spriteSheetView;
// sprite collection we're editing
SpriteCollectionProxy spriteCollectionProxy = null;
public SpriteCollectionProxy SpriteCollection { get { return spriteCollectionProxy; } }
public SpriteView SpriteView { get { return spriteView; } }
// This lists all entries
List<SpriteCollectionEditorEntry> entries = new List<SpriteCollectionEditorEntry>();
// This lists all selected entries
List<SpriteCollectionEditorEntry> selectedEntries = new List<SpriteCollectionEditorEntry>();
// Callback when a sprite collection is changed and the selection needs to be refreshed
public void OnSpriteCollectionChanged(bool retainSelection)
{
var oldSelection = selectedEntries.ToArray();
PopulateEntries();
if (retainSelection)
{
searchFilter = ""; // name may have changed
foreach (var selection in oldSelection)
{
foreach (var entry in entries)
{
if (entry.type == selection.type && entry.index == selection.index)
{
entry.selected = true;
break;
}
}
}
UpdateSelection();
}
}
public void SelectSpritesFromList(int[] indices)
{
OnSpriteCollectionChanged(true); // clear filter
selectedEntries = new List<SpriteCollectionEditorEntry>();
// Clear selection
foreach (var entry in entries)
entry.selected = false;
// Create new selection
foreach (var index in indices)
{
foreach (var entry in entries)
{
if (entry.type == SpriteCollectionEditorEntry.Type.Sprite && entry.index == index)
{
entry.selected = true;
selectedEntries.Add(entry);
break;
}
}
}
}
public void SelectSpritesInSpriteSheet(int spriteSheetId, int[] spriteIds)
{
OnSpriteCollectionChanged(true); // clear filter
selectedEntries = new List<SpriteCollectionEditorEntry>();
foreach (var entry in entries)
{
entry.selected = (entry.type == SpriteCollectionEditorEntry.Type.SpriteSheet && entry.index == spriteSheetId);
if (entry.selected)
{
spriteSheetView.Select(spriteCollectionProxy.spriteSheets[spriteSheetId], spriteIds);
}
}
UpdateSelection();
}
void UpdateSelection()
{
// clear settings view if its selected
settingsView.show = false;
selectedEntries = (from entry in entries where entry.selected == true orderby entry.selectionKey select entry).ToList();
}
void ClearSelection()
{
entries.ForEach((a) => a.selected = false);
UpdateSelection();
}
// Callback when a sprite collection needs resorting
public static bool Contains(string s, string text)
{
return s.ToLower().IndexOf(text.ToLower()) != -1;
}
// Callback when a sort criteria is changed
public void OnSpriteCollectionSortChanged()
{
if (searchFilter.Length > 0)
{
// re-sort list
entries = (from entry in entries where Contains(entry.name, searchFilter) select entry)
.OrderBy( e => e.type )
.ThenBy( e => e.name, new tk2dEditor.Shared.NaturalComparer() )
.ToList();
}
else
{
// re-sort list
entries = (from entry in entries select entry)
.OrderBy( e => e.type )
.ThenBy( e => e.name, new tk2dEditor.Shared.NaturalComparer() )
.ToList();
}
for (int i = 0; i < entries.Count; ++i)
entries[i].listIndex = i;
}
public int InspectorWidth { get { return tk2dPreferences.inst.spriteCollectionInspectorWidth; } }
// populate the entries struct for display in the listbox
void PopulateEntries()
{
entries = new List<SpriteCollectionEditorEntry>();
selectedEntries = new List<SpriteCollectionEditorEntry>();
if (spriteCollectionProxy == null)
return;
for (int spriteIndex = 0; spriteIndex < spriteCollectionProxy.textureParams.Count; ++spriteIndex)
{
var sprite = spriteCollectionProxy.textureParams[spriteIndex];
var spriteSourceTexture = sprite.texture;
if (spriteSourceTexture == null && sprite.name.Length == 0) continue;
var newEntry = new SpriteCollectionEditorEntry();
newEntry.name = sprite.name;
if (sprite.texture == null) {
newEntry.name += " (missing)";
}
newEntry.index = spriteIndex;
newEntry.type = SpriteCollectionEditorEntry.Type.Sprite;
entries.Add(newEntry);
}
for (int i = 0; i < spriteCollectionProxy.spriteSheets.Count; ++i)
{
var spriteSheet = spriteCollectionProxy.spriteSheets[i];
if (!spriteSheet.active) continue;
var newEntry = new SpriteCollectionEditorEntry();
newEntry.name = spriteSheet.Name;
newEntry.index = i;
newEntry.type = SpriteCollectionEditorEntry.Type.SpriteSheet;
entries.Add(newEntry);
}
for (int i = 0; i < spriteCollectionProxy.fonts.Count; ++i)
{
var font = spriteCollectionProxy.fonts[i];
if (!font.active) continue;
var newEntry = new SpriteCollectionEditorEntry();
newEntry.name = font.Name;
newEntry.index = i;
newEntry.type = SpriteCollectionEditorEntry.Type.Font;
entries.Add(newEntry);
}
OnSpriteCollectionSortChanged();
selectedEntries = new List<SpriteCollectionEditorEntry>();
}
public void SetGenerator(tk2dSpriteCollection spriteCollection)
{
this._spriteCollection = spriteCollection;
this.firstRun = true;
spriteCollectionProxy = new SpriteCollectionProxy(spriteCollection);
PopulateEntries();
}
public void SetGeneratorAndSelectedSprite(tk2dSpriteCollection spriteCollection, int selectedSprite)
{
searchFilter = "";
SetGenerator(spriteCollection);
foreach (var entry in entries)
{
if (entry.type == SpriteCollectionEditorEntry.Type.Sprite && entry.index == selectedSprite)
{
entry.selected = true;
break;
}
}
UpdateSelection();
}
int cachedSpriteId = -1;
Texture2D cachedSpriteTexture = null;
// Returns a texture for a given sprite, if the sprite is a region sprite, a new texture is returned
public Texture2D GetTextureForSprite(int spriteId)
{
var param = spriteCollectionProxy.textureParams[spriteId];
if (spriteId != cachedSpriteId)
{
ClearTextureCache();
cachedSpriteId = spriteId;
}
if (param.extractRegion)
{
if (cachedSpriteTexture == null)
{
var tex = param.texture;
cachedSpriteTexture = new Texture2D(param.regionW, param.regionH);
for (int y = 0; y < param.regionH; ++y)
{
for (int x = 0; x < param.regionW; ++x)
{
cachedSpriteTexture.SetPixel(x, y, tex.GetPixel(param.regionX + x, param.regionY + y));
}
}
cachedSpriteTexture.Apply();
}
return cachedSpriteTexture;
}
else
{
return param.texture;
}
}
void ClearTextureCache()
{
if (cachedSpriteId != -1)
cachedSpriteId = -1;
if (cachedSpriteTexture != null)
{
DestroyImmediate(cachedSpriteTexture);
cachedSpriteTexture = null;
}
}
void OnEnable()
{
if (_spriteCollection != null)
{
SetGenerator(_spriteCollection);
}
spriteView = new SpriteView(this);
settingsView = new SettingsView(this);
fontView = new FontView(this);
spriteSheetView = new SpriteSheetView(this);
}
void OnDisable()
{
ClearTextureCache();
_spriteCollection = null;
tk2dEditorUtility.CollectAndUnloadUnusedAssets();
}
void OnDestroy() {
tk2dEditorSkin.Done();
}
string searchFilter = "";
void DrawToolbar()
{
GUILayout.BeginHorizontal(EditorStyles.toolbar, GUILayout.ExpandWidth(true));
// LHS
GUILayout.BeginHorizontal(GUILayout.Width(leftBarWidth - 6));
// Create Button
GUIContent createButton = new GUIContent("Create");
Rect createButtonRect = GUILayoutUtility.GetRect(createButton, EditorStyles.toolbarDropDown, GUILayout.ExpandWidth(false));
if (GUI.Button(createButtonRect, createButton, EditorStyles.toolbarDropDown))
{
GUIUtility.hotControl = 0;
GUIContent[] menuItems = new GUIContent[] {
new GUIContent("Sprite Sheet"),
new GUIContent("Font")
};
EditorUtility.DisplayCustomMenu(createButtonRect, menuItems, -1,
delegate(object userData, string[] options, int selected) {
switch (selected)
{
case 0:
int addedSpriteSheetIndex = spriteCollectionProxy.FindOrCreateEmptySpriteSheetSlot();
searchFilter = "";
PopulateEntries();
foreach (var entry in entries)
{
if (entry.type == SpriteCollectionEditorEntry.Type.SpriteSheet && entry.index == addedSpriteSheetIndex)
entry.selected = true;
}
UpdateSelection();
break;
case 1:
if (SpriteCollection.allowMultipleAtlases)
{
EditorUtility.DisplayDialog("Create Font",
"Adding fonts to sprite collections isn't allowed when multi atlas spanning is enabled. " +
"Please disable it and try again.", "Ok");
}
else
{
int addedFontIndex = spriteCollectionProxy.FindOrCreateEmptyFontSlot();
searchFilter = "";
PopulateEntries();
foreach (var entry in entries)
{
if (entry.type == SpriteCollectionEditorEntry.Type.Font && entry.index == addedFontIndex)
entry.selected = true;
}
UpdateSelection();
}
break;
}
}
, null);
}
// Filter box
GUILayout.Space(8);
string newSearchFilter = GUILayout.TextField(searchFilter, tk2dEditorSkin.ToolbarSearch, GUILayout.ExpandWidth(true));
if (newSearchFilter != searchFilter)
{
searchFilter = newSearchFilter;
PopulateEntries();
}
if (searchFilter.Length > 0)
{
if (GUILayout.Button("", tk2dEditorSkin.ToolbarSearchClear, GUILayout.ExpandWidth(false)))
{
searchFilter = "";
PopulateEntries();
}
}
else
{
GUILayout.Label("", tk2dEditorSkin.ToolbarSearchRightCap);
}
GUILayout.EndHorizontal();
// Label
if (_spriteCollection != null)
GUILayout.Label(_spriteCollection.name);
// RHS
GUILayout.FlexibleSpace();
// Always in settings view when empty
if (spriteCollectionProxy != null && spriteCollectionProxy.Empty)
{
GUILayout.Toggle(true, "Settings", EditorStyles.toolbarButton);
}
else
{
bool newSettingsView = GUILayout.Toggle(settingsView.show, "Settings", EditorStyles.toolbarButton);
if (newSettingsView != settingsView.show)
{
ClearSelection();
settingsView.show = newSettingsView;
}
}
if (GUILayout.Button("Revert", EditorStyles.toolbarButton) && spriteCollectionProxy != null)
{
spriteCollectionProxy.CopyFromSource();
OnSpriteCollectionChanged(false);
}
if (GUILayout.Button("Commit", EditorStyles.toolbarButton) && spriteCollectionProxy != null)
Commit();
GUILayout.EndHorizontal();
}
public void Commit()
{
spriteCollectionProxy.DeleteUnusedData();
spriteCollectionProxy.CopyToTarget();
tk2dSpriteCollectionBuilder.ResetCurrentBuild();
if (!tk2dSpriteCollectionBuilder.Rebuild(_spriteCollection)) {
EditorUtility.DisplayDialog("Failed to commit sprite collection",
"Please check the console for more details.", "Ok");
}
spriteCollectionProxy.CopyFromSource();
}
void HandleListKeyboardShortcuts(int controlId)
{
Event ev = Event.current;
if (ev.type == EventType.KeyDown
&& (GUIUtility.keyboardControl == controlId || GUIUtility.keyboardControl == 0)
&& entries != null && entries.Count > 0)
{
int selectedIndex = 0;
foreach (var e in entries)
{
if (e.selected) break;
selectedIndex++;
}
int newSelectedIndex = selectedIndex;
switch (ev.keyCode)
{
case KeyCode.Home: newSelectedIndex = 0; break;
case KeyCode.End: newSelectedIndex = entries.Count - 1; break;
case KeyCode.UpArrow: newSelectedIndex = Mathf.Max(selectedIndex - 1, 0); break;
case KeyCode.DownArrow: newSelectedIndex = Mathf.Min(selectedIndex + 1, entries.Count - 1); break;
case KeyCode.PageUp: newSelectedIndex = Mathf.Max(selectedIndex - 10, 0); break;
case KeyCode.PageDown: newSelectedIndex = Mathf.Min(selectedIndex + 10, entries.Count - 1); break;
}
if (newSelectedIndex != selectedIndex)
{
for (int i = 0; i < entries.Count; ++i)
entries[i].selected = (i == newSelectedIndex);
UpdateSelection();
Repaint();
ev.Use();
}
}
}
Vector2 spriteListScroll = Vector2.zero;
int spriteListSelectionKey = 0;
void DrawSpriteList()
{
if (spriteCollectionProxy != null && spriteCollectionProxy.Empty)
{
DrawDropZone();
return;
}
int spriteListControlId = GUIUtility.GetControlID("tk2d.SpriteList".GetHashCode(), FocusType.Keyboard);
HandleListKeyboardShortcuts(spriteListControlId);
spriteListScroll = GUILayout.BeginScrollView(spriteListScroll, GUILayout.Width(leftBarWidth));
GUILayout.BeginVertical(tk2dEditorSkin.SC_ListBoxBG, GUILayout.ExpandWidth(true), GUILayout.ExpandHeight(true));
bool multiSelectKey = (Application.platform == RuntimePlatform.OSXEditor)?Event.current.command:Event.current.control;
bool shiftSelectKey = Event.current.shift;
bool selectionChanged = false;
SpriteCollectionEditorEntry.Type lastType = SpriteCollectionEditorEntry.Type.None;
foreach (var entry in entries)
{
if (lastType != entry.type)
{
if (lastType != SpriteCollectionEditorEntry.Type.None)
GUILayout.Space(8);
else
GUI.SetNextControlName("firstLabel");
GUILayout.Label(GetEntryTypeString(entry.type), tk2dEditorSkin.SC_ListBoxSectionHeader, GUILayout.ExpandWidth(true));
lastType = entry.type;
}
bool newSelected = GUILayout.Toggle(entry.selected, entry.name, tk2dEditorSkin.SC_ListBoxItem, GUILayout.ExpandWidth(true));
if (newSelected != entry.selected)
{
GUI.FocusControl("firstLabel");
entry.selectionKey = spriteListSelectionKey++;
if (multiSelectKey)
{
// Only allow multiselection with sprites
bool selectionAllowed = entry.type == SpriteCollectionEditorEntry.Type.Sprite;
foreach (var e in entries)
{
if (e != entry && e.selected && e.type != entry.type)
{
selectionAllowed = false;
break;
}
}
if (selectionAllowed)
{
entry.selected = newSelected;
selectionChanged = true;
}
else
{
foreach (var e in entries)
{
e.selected = false;
}
entry.selected = true;
selectionChanged = true;
}
}
else if (shiftSelectKey)
{
// find first selected entry in list
int firstSelection = int.MaxValue;
foreach (var e in entries)
{
if (e.selected && e.listIndex < firstSelection)
{
firstSelection = e.listIndex;
}
}
int lastSelection = entry.listIndex;
if (lastSelection < firstSelection)
{
lastSelection = firstSelection;
firstSelection = entry.listIndex;
}
// Filter for multiselection
if (entry.type == SpriteCollectionEditorEntry.Type.Sprite)
{
for (int i = firstSelection; i <= lastSelection; ++i)
{
if (entries[i].type != entry.type)
{
firstSelection = entry.listIndex;
lastSelection = entry.listIndex;
}
}
}
else
{
firstSelection = lastSelection = entry.listIndex;
}
foreach (var e in entries)
{
e.selected = (e.listIndex >= firstSelection && e.listIndex <= lastSelection);
}
selectionChanged = true;
}
else
{
foreach (var e in entries)
{
e.selected = false;
}
entry.selected = true;
selectionChanged = true;
}
}
}
if (selectionChanged)
{
GUIUtility.keyboardControl = spriteListControlId;
UpdateSelection();
Repaint();
}
GUILayout.EndVertical();
GUILayout.EndScrollView();
Rect viewRect = GUILayoutUtility.GetLastRect();
tk2dPreferences.inst.spriteCollectionListWidth = (int)tk2dGuiUtility.DragableHandle(4819283,
viewRect, tk2dPreferences.inst.spriteCollectionListWidth,
tk2dGuiUtility.DragDirection.Horizontal);
}
bool IsValidDragPayload()
{
int idx = 0;
foreach (var v in DragAndDrop.objectReferences)
{
var type = v.GetType();
if (type == typeof(Texture2D))
return true;
else if (type == typeof(Object) && System.IO.Directory.Exists(DragAndDrop.paths[idx]))
return true;
++idx;
}
return false;
}
string GetEntryTypeString(SpriteCollectionEditorEntry.Type kind)
{
switch (kind)
{
case SpriteCollectionEditorEntry.Type.Sprite: return "Sprites";
case SpriteCollectionEditorEntry.Type.SpriteSheet: return "Sprite Sheets";
case SpriteCollectionEditorEntry.Type.Font: return "Fonts";
}
Debug.LogError("Unhandled type");
return "";
}
void HandleDroppedPayload(Object[] objects)
{
List<int> addedIndices = new List<int>();
foreach (var obj in objects)
{
Texture2D tex = obj as Texture2D;
string name = spriteCollectionProxy.FindUniqueTextureName(tex.name);
int slot = spriteCollectionProxy.FindOrCreateEmptySpriteSlot();
spriteCollectionProxy.textureParams[slot].name = name;
spriteCollectionProxy.textureParams[slot].colliderType = tk2dSpriteCollectionDefinition.ColliderType.UserDefined;
spriteCollectionProxy.textureParams[slot].texture = (Texture2D)obj;
addedIndices.Add(slot);
}
// And now select them
searchFilter = "";
PopulateEntries();
foreach (var entry in entries)
{
if (entry.type == SpriteCollectionEditorEntry.Type.Sprite &&
addedIndices.IndexOf(entry.index) != -1)
entry.selected = true;
}
UpdateSelection();
}
// recursively find textures in path
List<Object> AddTexturesInPath(string path)
{
List<Object> localObjects = new List<Object>();
foreach (var q in System.IO.Directory.GetFiles(path))
{
string f = q.Replace('\\', '/');
System.IO.FileInfo fi = new System.IO.FileInfo(f);
if (fi.Extension.ToLower() == ".meta")
continue;
Object obj = AssetDatabase.LoadAssetAtPath(f, typeof(Texture2D));
if (obj != null) localObjects.Add(obj);
}
foreach (var q in System.IO.Directory.GetDirectories(path))
{
string d = q.Replace('\\', '/');
localObjects.AddRange(AddTexturesInPath(d));
}
return localObjects;
}
int leftBarWidth { get { return tk2dPreferences.inst.spriteCollectionListWidth; } }
Object[] deferredDroppedObjects;
void DrawDropZone()
{
GUILayout.BeginVertical(tk2dEditorSkin.SC_ListBoxBG, GUILayout.Width(leftBarWidth), GUILayout.ExpandHeight(true));
GUILayout.FlexibleSpace();
GUILayout.BeginHorizontal();
GUILayout.FlexibleSpace();
if (DragAndDrop.objectReferences.Length == 0 && !SpriteCollection.Empty)
GUILayout.Label("Drop sprite here", tk2dEditorSkin.SC_DropBox);
else
GUILayout.Label("Drop sprites here", tk2dEditorSkin.SC_DropBox);
GUILayout.FlexibleSpace();
GUILayout.EndHorizontal();
GUILayout.FlexibleSpace();
GUILayout.EndVertical();
Rect rect = new Rect(0, 0, leftBarWidth, Screen.height);
if (rect.Contains(Event.current.mousePosition))
{
switch (Event.current.type)
{
case EventType.DragUpdated:
if (IsValidDragPayload())
DragAndDrop.visualMode = DragAndDropVisualMode.Copy;
else
DragAndDrop.visualMode = DragAndDropVisualMode.None;
break;
case EventType.DragPerform:
var droppedObjectsList = new List<Object>();
for (int i = 0; i < DragAndDrop.objectReferences.Length; ++i)
{
var type = DragAndDrop.objectReferences[i].GetType();
if (type == typeof(Texture2D))
droppedObjectsList.Add(DragAndDrop.objectReferences[i]);
else if (type == typeof(Object) && System.IO.Directory.Exists(DragAndDrop.paths[i]))
droppedObjectsList.AddRange(AddTexturesInPath(DragAndDrop.paths[i]));
}
deferredDroppedObjects = droppedObjectsList.ToArray();
Repaint();
break;
}
}
}
bool dragging = false;
bool currentDraggingValue = false;
bool firstRun = true;
List<UnityEngine.Object> assetsInResources = new List<UnityEngine.Object>();
bool InResources(UnityEngine.Object obj)
{
return AssetDatabase.GetAssetPath(obj).ToLower().IndexOf("/resources/") != -1;
}
void CheckForAssetsInResources()
{
assetsInResources.Clear();
foreach (tk2dSpriteCollectionDefinition tex in SpriteCollection.textureParams)
{
if (tex.texture == null) continue;
if (InResources(tex.texture) && assetsInResources.IndexOf(tex.texture) == -1) assetsInResources.Add(tex.texture);
}
foreach (tk2dSpriteCollectionFont font in SpriteCollection.fonts)
{
if (font.texture != null && InResources(font.texture) && assetsInResources.IndexOf(font.texture) == -1) assetsInResources.Add(font.texture);
if (font.bmFont != null && InResources(font.bmFont) && assetsInResources.IndexOf(font.bmFont) == -1) assetsInResources.Add(font.bmFont);
}
}
Vector2 assetWarningScroll = Vector2.zero;
bool HandleAssetsInResources()
{
if (firstRun && SpriteCollection != null)
{
CheckForAssetsInResources();
firstRun = false;
}
if (assetsInResources.Count > 0)
{
tk2dGuiUtility.InfoBox("Warning: The following assets are in one or more resources directories.\n" +
"These files will be included in the build.",
tk2dGuiUtility.WarningLevel.Warning);
assetWarningScroll = GUILayout.BeginScrollView(assetWarningScroll, GUILayout.ExpandWidth(true));
foreach (UnityEngine.Object obj in assetsInResources)
{
EditorGUILayout.ObjectField(obj, typeof(UnityEngine.Object), false);
}
GUILayout.EndScrollView();
GUILayout.Space(8);
GUILayout.BeginHorizontal();
GUILayout.FlexibleSpace();
if (GUILayout.Button("Ok", GUILayout.MinWidth(100)))
{
assetsInResources.Clear();
Repaint();
}
GUILayout.EndHorizontal();
return true;
}
return false;
}
void OnGUI()
{
if (Event.current.type == EventType.DragUpdated)
{
if (IsValidDragPayload())
dragging = true;
}
else if (Event.current.type == EventType.DragExited)
{
dragging = false;
Repaint();
}
else
{
if (currentDraggingValue != dragging)
{
currentDraggingValue = dragging;
}
}
if (Event.current.type == EventType.Layout && deferredDroppedObjects != null)
{
HandleDroppedPayload(deferredDroppedObjects);
deferredDroppedObjects = null;
}
if (HandleAssetsInResources()) return;
GUILayout.BeginVertical();
DrawToolbar();
GUILayout.BeginHorizontal();
if (currentDraggingValue)
DrawDropZone();
else
DrawSpriteList();
if (settingsView.show || (spriteCollectionProxy != null && spriteCollectionProxy.Empty)) settingsView.Draw();
else if (fontView.Draw(selectedEntries)) { }
else if (spriteSheetView.Draw(selectedEntries)) { }
else spriteView.Draw(selectedEntries);
GUILayout.EndHorizontal();
GUILayout.EndVertical();
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 952
|
Q: appropriate minimum support for itemset? Please suggest me for any kind material about appropriate minimum support and confidence for itemset!
::i use apriori algorithm to search frequent itemset. i still don't know appropriate support and confidence for itemset. i wish to know what kinds of considerations to decide how big is the support.
A: The answer is that the appropriate values depends on the data.
For some datasets, the best value may be 0.5. But for some other datasets it may be 0.05. It depends on the data.
But if you set minsup =0 and minconf = 0, some algorithms will run out of memory before terminating, or you may run out of disk space because there is too many patterns.
From my experience, the best way to choose minsup and minconf is to start with a high value and then to lower them down gradually until you find enough patterns.
Alternatively, if you don't want to have to set minsup, you can use a top-k algorithms where instead of specifying minsup, you specify for example that you want the k most frequent rules. For example, k = 1000 rules.
If you are interested by top-k association rule mining, you can check my Java code here:
http://www.philippe-fournier-viger.com/spmf/
The algorithm is called TopKRules and the article describing it will be published next month.
Besides that, you need to know that there is many other interestingness measures beside the support and confidence: lift, all-confidence, ... To know more about this, you can read this article: "On selecting interestingness measures for association rules" and "A Survey of Interestingness Measures for Association Rules" Basically, all measures have some problems in some cases... no measure is perfect.
Hope this helps!
A: In any association rule mining algorithm, including Apriori, it is up to the user to decide what support and confidence values they want to provide. Depending on your dataset and your objectives you decide the minSup and minConf.
Obviously, if you set these values lower, then your algorithm will take longer to execute and you will get a lot of results.
A: The minimum support and minimum confidence parameters are a user preference. If you want a larger quantity of results (with lower statistical confidence), choose the parameters appropriately. In theory you can set them to 0. The algorithm will run, but it will take a long time, and the result will not be particularly useful, as it contains just about anything.
So choose them so that the result suit your needs. Mathematically, any value is "correct".
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,847
|
Q: A type of local minimum (2) Data: $\Omega \subset \mathbb{R}^{n}$ is an open connected (may be unbounded) set, and locally $\partial \Omega$ is a Lipschitz graph. $S \subset \partial \Omega$ is measurable and $H^{n-1}(S)>0$. The Dirichlet data on $S$ are given by non-negative function $u^0 \in ^{1}_{Loc}(\Omega)$ with $\nabla u^0 \in L^{2}(\Omega)$. The given force function $Q$ is non-negative and measurable.
Consider the convex set
\begin{equation}
K:=\{ v \in L^{1}_{Loc}(\Omega): \nabla v \in L^{2}(\Omega) \quad \mbox{and} \quad v=u^0 \quad \mbox{on } S\}.
\end{equation}
We are looking for an absolute minimum of the functional
\begin{equation}
J(v):= \int_{\Omega}(|\nabla v|^{2} + \chi(\{v>0\})Q^2)
\end{equation}
in the class $K$.
Definition: We call $u \in K$ a local minimum if for some small $\varepsilon>0$ we have $J(u)\le J(v)$ for every $v \in K$ with
\begin{equation}
\|\nabla (u-v)\|_{L^{2}(\Omega)} + \| \chi(\{v>0\}) -\chi(\{u>0\})\|_{L^{1}(\Omega)} \le \varepsilon.
\end{equation}
Lemma1: If $u$ is a minimum local, then $u$ is subharmonic, hence we can assume that
\begin{equation}
u(x) = \lim_{r\downarrow 0} \oint_{B_r(x)}u \quad \mbox{for} \quad x \in \Omega,
\end{equation}
where $\oint $ denotes the mean value.
Proof: For non-negative functions $\xi \in C^{\infty}_{0}(\Omega)$ we have
\begin{equation}
0 \le \limsup_{\varepsilon\downarrow 0} \dfrac{1}{2\varepsilon} (J(u- \varepsilon \xi) - J(u)) \le - \int_{\Omega} \nabla \xi \nabla u,
\end{equation}
that is, $u$ is subharmonic. Then the limit in the assertion exists for every $x \in \Omega$, and coincides with $u(x)$ for almost all $x$.
1.How can I prove the part "hence we can assume that
\begin{equation}
u(x) = \lim_{r\downarrow 0} \oint_{B_r(x)}u \quad \mbox{for} \quad x \in \Omega,''
\end{equation}
2.How can I do the details in the lemma
Lemma 2: If $u$ is local minimum, then $u$ is harmonic in the open set $\{u>0\}$.
proof: Use $u + \varepsilon \xi$ as first variation.
If you want the details can be found in the article Alt, H. M. and Caffarelli, L. A. Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math., 325, (1981), 105–144. and related question A type of local minimum. I thank any hint.
My thoughts for lemma 2 is that I need to prove that for all $\xi \in C^{\infty}_{0}(\{ u>0\})$ we have
$\int_{\{u>0\}} \langle \nabla u, \nabla \xi\rangle dx = 0 $. This should follow by
\begin{eqnarray}
0 &=& \lim_{\varepsilon\rightarrow 0} \dfrac{J(u + \varepsilon \xi) - J(u)}{2 \varepsilon} \\
&=&\int_{\{u>0\}} \langle \nabla u \nabla \xi \rangle dx + \lim_{\varepsilon\rightarrow 0^+ } \dfrac{1}{2\varepsilon}\int_{\Omega} Q^2(\chi_{\{u + \xi>0\}} - \chi_{\{u>0\}})
\end{eqnarray}
Then we must have
$$ \lim_{\varepsilon\rightarrow 0^+ } \dfrac{1}{2\varepsilon}\int_{\Omega} Q^2(\chi_{\{u + \xi>0\}} - \chi_{\{u>0\}}). $$
Am I rigth here?
A: First, note that in stating your Lemma 2, Caffarelli uses Corollary 3.3 which says that $u\in C^{0,1}(\Omega)$. So we can assume this fact. Moreover, on the proof of this corollary, he uses your Lemma 1, and by using it he says that $$\tag{1}u(x)\leq \lim_{r\downarrow 0} \oint_{B_r(x)}u$$
My conclusion is that in the statement of your Lemma 1 (Lemma 2.2 in the paper), there is a typo, so what needs to be proved is just $(1)$, but this is equivalently to a function being subharmonic.
To prove Lemma 2, we just need to prove that $$\tag{2} \lim_{\epsilon\rightarrow 0^+ } \dfrac{1}{\epsilon}\int_{\Omega} Q^2(\chi_{\{u + \epsilon\xi>0\}} - \chi_{\{u>0\}})=0 $$
Let $K=\operatorname{support}\xi$. Note that $K$ is a compact set contained in $\{u>0\}$, hence, we can find $\delta>0$ such that $u>\delta $ in $K$. Choose $\epsilon_0$ in such a way that $|\epsilon_0\xi|<\delta$ and consider $\epsilon<\epsilon_0$.
If $x\in K$, then $u(x)+\epsilon\xi(x)>\delta-\delta>0$ and $u(x)>0$,
If $x\in \{u>0\}\setminus K$, then $u(x)+\epsilon\xi(x)>0$ and $u(x)>0$,
If $x\in \Omega\setminus\{u>0\}$, then $u(x)+\epsilon\xi(x)\leq 0$ and $u(x)\leq 0$.
By combining the three cases above, we conclude that the limit in $(2)$ is in fact $0$, then Lemma 2 is proved.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,720
|
package org.demis.codegen.core.db.reader;
public class DatabaseReadingException extends Exception {
public DatabaseReadingException() {
super();
}
public DatabaseReadingException(String message) {
super(message);
}
public DatabaseReadingException(String message, Throwable cause) {
super(message, cause);
}
public DatabaseReadingException(Throwable cause) {
super(cause);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,402
|
Q: In MVC Who is Responsible for Animations? This question is about a Cocoa app I'm working on but it could apply to anything using an MVC or related architecture.
Who is responsible for animations?
I can see two arguments:
1) Animation code should exist in a view (part of a view's presentation, how it draws itself) but be controlled by a controller (interpreting user input, etc).
2) Animations and their lifecycle should be managed completely by a controller and act on the views belonging to that controller.
A: I think first option is better because it would be fast if we can handle the events at view itself. Animation is most of the cases do not need additional data...so there is no need to reach to controller.
A: I think an animation is a modification of the model which is shown by the view. That's why I see the animation handling in the controller.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,640
|
# NEW Brewing Lager Beer
Other books by Brewers Publications include:
How to Brew
By John J. Palmer
Radical Brewing
By Randy Mosher
Designing Great Beers
By Ray Daniels
The Compleat Meadmaker
By Ken Schramm
The Brewers Association's Guide to Starting Your Own Brewery
Edited by Ray Daniels
North American Brewers' Resource Directory
A bi-annual directory of breweries and suppliers.
**For further information, see** www.BrewersPublications.com
# NEW Brewing Lager Beer
# The Most Comprehensive Book for Home- and Microbrewers
Completely Revised and Expanded
GREGORY J. NOONAN
An Imprint of Brewers Publications
A Division of the Brewers Association
Boulder, Colorado
_New Brewing Lager Beer_
Copyright 1986, 1996 by Gregory J. Noonan
Copy Editor: Merilee Eggleston
Technical Editor: Darryl Richman
Book Project Editor: Theresa Duggan
The illustration "Hardness of Ground Waters" on pages 38 and 39 is from _Water Atlas of the United States,_ by James J. Geraghty, and was used with permission of the publisher.
Published by Brewers Publications, a division of the Brewers
Association: PO Box 1679, Boulder, Colorado 80306-1679;
(303) 447-0816 www.BrewersAssociation.org
Direct all inquiries/orders to the above address.
All rights reserved. Except for use in a review no portion of this book may be reproduced in any form without written permission of the publisher.
Neither the author, editor, nor the publisher assumes any responsibility for the use or misuse of information contained in this book.
Cover design by Marylin Cohen and Stephanie Johnson
Cover illustration by Molly Gough
All illustrations by Gregory J. Noonan unless otherwise noted.
**Library of Congress Cataloging-in-Publication Data**
Noonan, Gregory J.
New brewing lager beer / Gregory Noonan : foreword by Charlie Papazian.
p. cm.
Includes bibliographical references (p. - ) and index.
ISBN 0-937381-82-9
EISBN 978-1-938469-23-7
1. Brewing—Amateursʹ manuals. 2. Beer. I. Title.
TP570.N583 1996 95-25594
541.8ʹ73—dc20 CIP
# DEDICATION
To the brewing of better beer, and to those who brew it.
# TABLE OF CONTENTS
List of Tables
Foreword
Preface
Acknowledgments
Introduction
Outline of Brewing
Reference Units
Classic Lager Types
PART 1: BREWING CONSTITUENTS
1. Barley
2. Malted Barley
Evaluation
Carbohydrates
Protein
Other Malt Fractions
3. Water
Using the Water Analysis
Turbidity, Sediment, Color, and Odor
pH
Ions
Measuring pH
pH Adjustment
Total Dissolved Solids/Specific Conductivity
Hardness/Alkalinity
Molarity, Equivalence, and Normality
Mineral Ions Common in Water
Water Treatment
Mineral Salt Treatment
4. Hops
On the Vine
In the Brewery
5. Yeast
Culturing Pure Yeast Strains
Storing Yeast
Washing Yeast
6. Bacteria
Gram-Positive Bacteria
Gram-Negative Bacteria
7. Enzymes
PART 2: THE BREWING PROCESS
8. Malting
Steeping
Germination
Kilning
9. Crushing the Malt
10. Mashing
Doughing-In
Mash pH
Acid Rest
Why Decoction Mash?
Three-Decoction Mash
Wort Separation
11. Boiling the Wort
Kettle Hops
Hop Rates
Establishing the Evaporation Rate
Hot Break (Kettle Break)
Cold Break
Finishing Hops
Straining the Wort
Cooling the Wort
12. Fermentation
Preparing for Pitching
Kraeusening
Pitching the Yeast
The Fermentation Lock
Primary Fermentation
Temperature
Temperature Control
Density and pH Monitoring
Primary Fermentation: Lag Phase
Low Kraeusen
High Kraeusen
Post Kraeusen
Real and Apparent Attenuation
Racking
Gauging Yeast Performance
Yeast Collection
13. Secondary Fermentation
Lagering
Fining
Clarifying with Beech Chips
Real Terminal Extract
Bottling
Imbibing
14. Planning the Brew
A Note on Mixing
Planning
Recipes
15. Brewing Procedure
Malt Examination
Three-Decoction Mash
Single-Decoction Mash
Sparging/Filtering
Boiling the Wort
Hot Break
Cold Break
Finishing Hops
Filtering and Cooling the Wort
Pitching the Yeast
Kraeusening
Yeast Starter
Primary Fermentation
Secondary Fermentation
Bottling
Troubleshooting
16. Cleaning and Sterilizing
Construction
Equipment
Alkaline Cleansers
Acid Cleansers
Alkaline Sterilants
Acid Sterilants
17. Equipment
Appendix A
Basic Homebrewing from Malt-Extract Syrup
Appendix B
The Infusion Mash
Infusion Mash Procedure
Appendix C
The Step Mash (Modified Infusion Mash)
Appendix D
Weights and Measures
Brewer's Glossary
Bibliography
About the Author
Index
# Tables
1. Malt Analysis, Dry Basis, Comparative Analysis of Acceptable Ranges
2. Grading by Size and Character
3. Degree of Modification and Malt Colors
4. Water: Typical Analysis U.S. Public Health Service Units
5. pH
6. Water Composition Indicators
7. Estimated Characters of the Classic Brewing Waters
8. Mineral Salt Treatment
9. Hops
10. Part A: Hop Varieties, and Part B: Hop Producing Districts
11. Brewing Contaminants
12. Rootlet Growth as an Acrospire-Length Indicator
13. Lauter-Tun Grist Profile
14. Planning
15. Grain Bill
16. Brew-house Yield from Various Sources
17. Brewing Liquor, Volume, and Treatment
18. Hop Rates
19. Brew Log
20. Brew-house Efficiency
21. Calculating Pitching Volume
22. Primary Fermentation Temperature/Time Guideline
23. Bottle Priming
# FOREWORD
It is difficult to improve a classic. Yet as all master brewers know, the process of learning about beer and brewing is never-ending. Ten years ago, Greg Noonan captured the essence of fine brewing and presented his knowledge of and appreciation for brewing great beers in his original _Brewing Lager Beer_ (Brewers Publications, 1986). Now Greg brings us all up to date with the best cutting-edge brewing information and techniques in this new and expanded revision, _New Brewing Lager Beer._
I had the great privilege of being the technical editor for the first edition of _Brewing Lager Beer_. After initially reviewing the original manuscript, I was awed and spellbound. It was during those moments that I really began to appreciate how much I didn't know about brewing's finer points. Indeed, what I was reading was about to help improve the quality of homebrewed beer as no other advanced book on applied brewing had.
With its release in 1986, _Brewing Lager Beer_ became Brewers Publications' cornerstone and the standard by which the Association of Brewers and all of its other divisions have measured their work. Greg's original book was a first of its kind, and the research and experience that went into it has benefited brewers around the globe.
American homebrewers were the first to embrace Greg's practical and applied approach to brewing lager beers. Then as the homebrewing hobby gained the respect it still enjoys today, professionals came to cite this classic work as a reference, one that is still unequaled.
In fact, _Brewing Lager Beer_ has become a standard for aspiring and veteran small brewers everywhere. Greg's devotees sometimes refer to themselves as "Noonanites." I have encountered their enthusiasm, appreciation, and most excellent beers during my travels through South America, Asia, Africa, Australia, and Europe. Just as I know when I've tasted a great beer, I know that when I encounter a Noonanite, I have come upon the "best of the best" among homebrewers — someone who has been infected with Greg's contagious enthusiasm and respect for the art of brewing.
As Greg states in his introduction to this new edition, ten additional years of brewing experience is an invaluable resource that can't be bought. His success in the microbrewing industry and his gracious efforts to share his increased knowledge provide us all with continued inspiration and even better beers.
_New Brewing Lager Beer_ is the new touchstone for those of us who choose to advance our knowledge of beer and brewing. Like me, you probably thought it couldn't get any better. Thanks, Greg, for showing us otherwise.
Charlie Papazian, Founding President
Association of Brewers
and author of _The New Complete Joy of Home Brewing_
and _The Home Brewer's Companion_
January 3, 1996
# PREFACE
A lot has changed in the world of craft brewing in ten years, and the updating of _Brewing Lager Beer_ is long overdue. It is surprising to me that the book has endured, and that so may brewers still refer to their dog-eared copies.
_Brewing Lager Beer_ never really was just about lager brewing. The lager tradition is really the culmination of the brewing theories of all the other European brewing traditions. By examining lager techniques and technology, any student of brewing is being exposed to brewing dissected to its vital principles. Before 1985, Dave Line's still-classic _Big Book of Brewing_ (Amateur Winemaker Publications, 1985) was the only readily available reference for homebrewers and the pioneers of microbrewing; in deference to him, I chose the title so as not to compete with his groundbreaking work.
This edition retains the title, but broadens its scope to include more of the spectrum of craft-brewing techniques and more information specific to ale brewing. Besides updating the work and including more useful information for serious brewers, a lot of factual and editing errors that appeared in the original have been corrected.
The trickle of knowledge available to craft and homebrewers when this book was first published has become a flood; I am proud to have _Brewing Lager Beer_ continue to be part of most serious brewers' libraries.
Greg Noonan
Burlington, Vermont
July 1, 1995
# ACKNOWLEDGMENTS
Thanks to my wife, Nancy, Martha and John Murtaugh, Darryl Richman, George Fix, Charlie Papazian, Charles Kochenour, and everyone else who contributed to the publication of this book.
# INTRODUCTION
## Outline of Brewing
Brewing begins with malted barley, which is the sprouted, then dried and/or roasted, barley seed. The malt is coarsely crushed and mixed with hot water to form a mash. During the mash, compositional changes occur, brought about by enzymes in the malt. The hot aqueous solution leaches the contents out of the crushed malt kernels and gelatinizes the starch; the enzymes reduce them to soluble fractions. This extract is rinsed from the insoluble malt particles during sparging. The sweet solution is called wort.
The wort is boiled with the flowerlike cone of the hop vine for bitterness and flavor, and to clarify the wort. The wort is cooled, and active brewers' yeast is added to ferment it. The yeast forms carbon dioxide and alcohol from the sugars extracted from the malt; carbon dioxide carbonates the beer. This fermentation may be carried out in several stages, and the beer moved from one vessel to another to separate it from flavor-impairing sedimented yeast, malt, and hop residues.
The beer is bottled, kegged, or casked so that it may be served carbonated.
## Reference Units
The volume, temperature, and density (weight) of a solution all have various terms by which they may be expressed. Throughout this book, U.S. gallons, degrees Fahrenheit, and density as degrees Plato have been given, with liters, degrees centigrade, and specific gravity parenthetically appended. The term original gravity (OG) refers to the density of the boiled wort (original extract, OE), and final gravity (FG) to the density of the fermented beer (apparent extract, AE). Specific gravity (sp gr) refers to the usual metric system (sp gr 1.046), gravity (SG) to a simpler form of expression preferred by brewers (1046), and excess gravity (G) to an abbreviated form (46) commonly used in equations.
Other abbreviations used in this book include:
ASBC | American Society of Brewing Chemists
---|---
EBC | European Brewing Convention
IOB | Institute of Brewing (British)
°F | Temperature, degrees Fahrenheit
°C | Temperature, degrees Centigrade
°Plato | Density, degrees Plato
°SRM | Wort color, standard reference method
°L | Malt color, degrees Lovibond (often used interchangeably with SRM)
°HCU | Homebrew color units (sum of °L of goods)
°EBC | Color, degrees EBC
°IOB | Color, degrees IOB
DBFG | Laboratory extract potential, dry basis fine grind
DBCG | Brewhouse extract potential, dry basis coarse grind
FG-CG | Malt modification, fine grind minus coarse grind
HWE | Hot water extract, extract potential, IOB method
CWE | Cold water extract, modification, IOB method
TN | Total nitrogen
% P | Percent protein (TN/6.25)
TSN | Total soluble nitrogen
SNR | Soluble nitrogen ratio (TSN/TN)
DP | Diastatic power
IBU | International bittering units, iso-alpha acid in solution
BU | Bittering unit (IBU x 1.125)
HBU | Homebrew bitterness units, identical to AAU (alpha acid units, ounces of hops x % alpha acid)
DU | Dextrinizing units
MC | Moisture Content
Ppm | Parts per million
mg/L | Milligrams per liter
For those more familiar with standards other than those used herein, tables are provided in appendix D for conversion of U.S. standards to metric and British units. Other useful tables of measure and conversion formulas for brewing are also found in the appendixes.
## Classic Lager Types
The range of lager beers is represented by seven recognized types, which are identified by the brewing centers where they evolved. They are:
_Pilsener_ (Pilsner, Pils), Bohemia, Czech Republic. 11 to 14 °Plato (OG 1044 to 1057). Golden color, distinctive hop palate, medium body, well carbonated, dry. 4 to 5.2 percent alcohol by volume (3.2 to 4.1 w/w). After the style set by the classic Pilsner Urquell (Plzensky Prazdroj). Served at 48 to 50 degrees F (9 to 10 degrees C).
_Dortmunder_ (Dort, Export), Westfalen, Germany. 12 to 14 °Plato (OG 1049 to 1057). Light, golden-blond color, moderate hop palate, medium body; rich, mellow, and yet sharp. 5 to 5.7 percent alcohol by volume (3.9 to 4.5 w/w). Served at 48 to 50 degrees F (9 to 10 degrees C).
_Vienna_ (Wiener/Spezial, Märzenbier/Oktoberfest), Lower Austria and Bavaria. 12.5 to 15 °Plato (OG 1050 to 1061). Reddish-amber or copper color, moderately hopped, medium body; malt to toasted-malt flavor. Viennas tend to be somewhat drier and more bitter than Oktoberfests. 4.9 to 5.7 percent alcohol by volume (3.8 to 4.5 w/w). Served at 50 to 55 degrees F (10 to 13 degrees C).
_Dark Munich_ (Münchener dunkel), Bavaria, Germany. 12 to 14 °Plato (OG 1048 to 1057). Dark-brown color, lightly hopped, medium to full body; slightly sweet, malty flavor. 4.5 to 5.5 percent alcohol by volume (3.5 to 4.3 w/w). Served at 35 to 45 degrees F (2 to 7 degrees C).
_Light Munich_ (Münchener helles), Bavaria, Germany. 11 to 12 °Plato (OG 1044 to 1049). Amber color, lightly hopped, medium bodied; slightly sweet, malty character. 4 to 5 percent alcohol by volume (3.2 to 3.9 w/w). Served at 45 to 50 degrees F (7 to 10 degrees C).
_Bock,_ Bavaria, Germany. 16 to 18 °Plato (OG 1065 to 1074). Light- to dark-brown color, rich and malty flavor, medium to full body. Darker bocks usually have a roasted-malt and hop flavor. 6 to 7.5 percent alcohol by volume (4.7 to 5.9 w/w). Served at 50 to 65 degrees F (10 to 18 degrees C).
_Dopplebock,_ Bavaria, Germany. Most often the brand name ends in the suffix -ator. 18 to 28 °Plato (OG 1074 to 1112). Light- to dark-brown color, often having a roasted-malt flavor. Medium to very full bodied. 7.5 to 13 percent alcohol by volume (5.9 to 10.2 w/w). Served at 50 to 65 degrees F (10 to 18 degrees C).
Myriad variations of the classic lager types exist: eisbocks that are partially frozen to increase the concentration of alcohol; schwarzpils and dark Pilseners, some with and some without roast-malt character; smoked lagers; spiced lagers; and American Pilseners, the much-maligned fizz-water that many Americans still think is the only kind of beer.
Lagers are defined by the process used to brew them. They are essentially distinguished from ales, the other beer family, by relatively slower fermentation at cool (40 to 55 degrees F [4 to 13 degrees C]) temperatures, followed by a relatively longer period of cold conditioning. By definition, lagers have low levels of esters and vicinal diketones; devoid of significant "yeast character," they rely solely upon the interplay of the malt, hops, and water for their aromas and flavors.
# PART ONE
# BREWING CONSTITUENTS
# CHAPTER 1
# Barley
Barley has been the brewer's preferred source of fermentable extract since prehistory. The barley kernel is readily malted, contains adequate enzymes to convert its starch to sugars, and upon mashing, yields a very complete, highly soluble extract that is relatively free of unwieldy constituents. Barley gives beer its characteristic sweet-to-malty taste, as well as its body, head, and color. Other grains such as wheat, oats, and rye do not lend themselves so readily to brewing.
The barley kernel is the seed of a plant of the grass family (Gramineae). It is one of the hardiest of all the cereal grains and is able to grow under widely varying conditions from Alaska to the equator; it grows best, however, in cool, dry climates.
The seeds grow in two, four, or six rows, called _heads,_ along a central stem. The number of seed rows is determined by the number of fertile flowers; in two-row types only two of the six flower clusters are fertile, whereas all the flowers of six-row barley are fertile. Long, thin bristles protrude from each seed, producing a "bearded" grain that closely resembles wheat.
Barley
Two-Row
Six-Row
Harvested when fully ripe, malting barley is dried from a moisture content of 15 to 25 percent down to below 14 percent and is binned for six to eight weeks before being malted. It is then graded; only the largest (greater than three-thirty-seconds of an inch in width) of the four standard grades is considered suitable for malting.
One bushel of brewing-grade barley weighs forty-eight pounds, and it should be at least 95 percent germinant. The kernels should be of uniform size, glossy, uniformly straw colored (light to dark yellow), plump, and have a clean grain aroma. There should be no rancid, moldy, or musty smell, nor any slender, immature grains with greenish-white husks. The husks should be thin, finely wrinkled, and tightly jacket the kernels. The endosperm (starch) must be opaque, white, and mealy, not translucent, grayish, steely, or glassy. Steely grains are excessively high in nitrogen, contain a greater percentage of poorly soluble protein, and do not take up moisture as readily. Consequently, nitrogen content should not exceed 1.6 percent where the barley will not be well modified or where the malt will be infusion mashed. Except where the wort will contain a high percentage of adjuncts, the nitrogen content should never exceed 2 percent. The protein-to-carbohydrate ratio should suit the type of beer being brewed and the processes used to brew it.
European brewers prefer superior two-row, thin-husked, large-berried barley varieties, which give the best brew-house yield, clarity, and flavor. These have a more favorable starch-to-protein/husk ratio than other barleys and yield a mellow flavor and good clarity.
Six-row barley is the most economical to grow because the greater number of rows per head increases the per-acre yield. It is a warm-climate barley and is the type most widely grown in the United States. It is favored by domestic commercial breweries because it is rich in the enzymes needed to convert the adjuncts they employ (adjuncts may account for up to 60 percent of the extract in U.S. brews). Because six-row barley is high in protein, in all-malt brews it presents problems with clarity and stability. Its higher husk content improves mash filterability but can contribute harsh flavors, especially when brew-house practices cause hot-side aeration.
Dehusked Barley Kernel
Barley Corn
Barley Head, top view
Four-row barley is still grown and malted in Northern Europe, where it has long been prized for its hardiness in the cold climate. It is not widely used elsewhere, however, because of its steeliness and low yield. It is actually a six-row barley that appears to form four rows, rather than six, because of its thin, elongated head.
Modern barley varieties have been selected for improved field and maltings yields. Improved varieties have replaced older cultivars several times during the past twenty years, and will continue to do so. Thus far, the only appreciable improvement in brewing quality that has been realized is the lower nitrogen of six-row varieties, and in many cases "improved" varieties have been rejected by brewers.
# CHAPTER 2
# Malted Barley
Because malt is organic, and subject to the meteorological and soil conditions under which the barley type is grown, malts of the same type can vary substantially in appearance and composition, depending on the season and where and how they are grown. For instance, moist, cool seasons produce more starch and less protein and fat, whereas nitrogen-based fertilizers applied too heavily or too late in the season increase the protein content. A lack of nutrients and minerals will also affect growth and composition. Malt quality depends as well on the circumstances of its malting and subsequent handling. Furthermore, the different barley types are each composed of several varieties.
For these reasons, it is advisable to ask for a lot analysis of any malt to guide adjustments in the mash program so that a wort consistent with previous brews can be made. Commercial breweries evaluate malt at the time of purchase. Craft brewers and homebrewers should too, using the lot analysis provided by the maltster. At the minimum, the lot analysis should give information about the moisture content, color, dry basis fine grind extract, fine-coarse difference, diastatic power, total nitrogen (protein), soluble nitrogen ratio, mealiness, and size distribution of the particular batch of malt.
The lot analysis is generally given on a "dry basis"; that is, the parameters given in the analysis are adjusted to give values as if the malt were oven-dried to 0 percent moisture content. This makes comparisons between lots of malt easier, because no adjustments need to be made for varying percentages of moisture in the malt.
Maltsters generally publish "typical" analyses so that brewers can evaluate their products before purchase. A typical analysis should not be accepted as a substitute for a lot analysis; unless the brewer knows the laboratory parameters of the malt actually on hand, adjustments to brewing procedures cannot be made, and brew-to-brew consistency is not possible. Lot analyses that vary considerably from the parameters stated in the typical analysis are reason to find a new source of malt.
Typical and lot analyses provide the brewer with essential data regarding the general quality of a malt and the barley it is made from, and the suitability of a particular malt for a given mashing method. Without referring to a lot analysis, the brewer cannot know whether a given malt is suitable for infusion mashing or if it requires a protein rest, and if it does, what the duration of the rest should be. Lot-to-lot adjustments to the brew house program usually need to be made in response to changes in the parameters of the malt on hand.
"Base" malts are those malts that form the basis of the brew, usually making up 60 to 100 percent of the grain bill. They usually provide the bulk of the extract and the enzymes to produce it. Lager, Pilsen, Brewers', Pale, Vienna, light Munich, and British mild ale malts are base malts. The last three resemble British pale malt, except that their colors are higher, and during malting, Vienna and Munich malts are converted somewhat less (fine grind– coarse grind difference 1.5 to 2.0 percent, lower soluble nitrogen ratio) than the British ale malts. Vienna malts have a diastatic power similar to European lager malts, while the range of diastatic power of Munich malts is similar to British malt. Consequently, Munich malt presents the potential for conversion problems if it is at all undermodified by the maltster, and cannot be used as a base malt in high-adjunct mashes.
Pilsen and lager malts originating in Britain and America usually resemble the brewers' malts of those countries in everything but color; they need to be handled in the brew house accordingly. Generally, malts purchased from the country of origin of the beer style give better results than those originating elsewhere.
Any analysis is only of use to brewers if they understand the significance and acceptable ranges of the quoted values. For lot analyses, the common parameters are:
Moisture content, percent: The upper limit for acceptable moisture content in any malt is 6 percent. The closer a malt is to 1.5 percent moisture content, the less risk of mold growth and the less flavor and aroma loss there will be in storage. For the last two reasons especially, colored malts should never be higher in moisture content than 4 percent. Of the base malts, British ale malts have the lowest moisture content.
The moisture content reflects upon the quality of the malting; high moisture-content malts are very often lacking in other areas as well. Furthermore, each 1 percent increase in the moisture content adds 1 percent to the real cost of the malt.
Table 1
Malt Analysis, Dry Basis, Comparative Analysis of Acceptable Ranges
---
Malt Analysis
Dry Basis
Acceptable Ranges | Traditional
2-Row
Lager | European
2-Row
Lager | American
2-Row
Brewers | American
6-Row
Lager | British
2-Row
Best Pale
Moisture content, % | 3.5–4.5 | 3.5–4.5 | 3.5–4.5 | 3.5–4.5 | 1.5–3.5
Color, SRM (°L, ASBC) | 1.2–1.6 | 1.4–1.8 | 1.4–2.0 | 1.4–2.0 | 2.0–2.7
Color, °EBC | 2.0–3.0 | 2.5–3.5 | 2.5–4.0 | 2.5–4.0 | 4.0–6.0
Extract, DBFG | 80.0–82.0 | 80.0–82.0 | 79.0–81.5 | 78.0–79.5 | 80.5–82.5
Extract, DBCG | 78.0–80.5 | 78.0–80.5 | 77.0–80.5 | 76.0–78.1 | 79.0–81.5
FG-CG, difference | 1.5–2.2 | 1.5–2.0 | 1.0–2.0 | 1.4–2.0 | 1.0–1.5
HWE (L°/kg at 20° C), .2M | 309–316 | 309–316 | 305–315 | 301–307 | 311–318
CWE | 16–18 | 18–20 | 18–21 | 18–21 | 18–22
Hartong 45°, % | 30–34 | 32–38 | 36–42 | 38–42 | 36–42
Alpha-amylase, DU | 44–48 | 40–48 | 40–45 | 35–45 | 25–35
DP, °Lintner | 70–100 | 75–110 | 100–150 | 125–160 | 50–70
DP, °WK | 200–325 | 220–350 | 375–450 | 400–500 | 150–220
Conversion, min | 10–15 | 10–15 | 5–10 | 5–10 | 10
Protein, % | 9.0–11.0 | 9.0–11.0 | 11.5–12.5 | 12.0–13.5 | 9.0–10.0
Soluble protein, % | 3.0–4.2 | 3.5–4.6 | 4.2–5.3 | 4.8–5.8 | 3.5–3.8
TN, % | 1.4–1.75 | 1.4–1.75 | 1.8–2.0 | 1.9–2.2 | 1.4–1.55
TSN, % | .48–.67 | .50–.75 | .65–.85 | .7–.93 | .5–.6
SNR, % (S/T, SN/TN, %) | 33–38 | 36–42 | 36–42 | 40–45 | 38–42
Mealy, % | 92–97 | 95–97 | 95–97 | 92–97 | 97–99
Half-glassy, % | 2–8 | 2–4 | 2–4 | 2–8 | 1–3
Glassy, % | 0–1 | 0–1 | 0–1 | 0–1 | 0
Friability, % | 80–85 | 80–90 | 80–90 | 75–85 | 85–95
On 7/64" scrn.-plump % | 75–85 | 75–85 | 60–70 | 45–55 | 80–85
On 6/64" screen % | 10–20 | 10–20 | 20–30 | 35–45 | 10–20
On 5/64" scrn. %<2.2 mm | 0–3 | 0–3 | 7–13 | 10–20 | 0–2
Thru 5/64" scrn.-thin % | 0–2 | 0–2 | 0–2 | 0–2.5 | 0
Wort pH | 5.8–5.9 | 5.7–5.9 | 5.7–5.9 | 5.7–5.9 | 5.5–5.8
Viscosity, cP | 165–1.75 | 1.55–1.65 | 1.52–1.62 | 1.60–1.70 | 1.45–1.55
Odor of mash | Aromatic | Aromatic | Aromatic | Aromatic | Aromatic
Speed of filtration | Norm.–Slow | Norm.–Rapid | Normal | Norm.–Slow | Norm.–Rapid
Degree of clarity | Clr.–Sl. hazy | Clear | Clear | Sl. Hazy | Clear
The brewer always needs to take into account the moisture content of each lot of malt to calculate its real extract potential. Given two malts with 81 percent DBFG extract (see below), malt at 6 percent moisture content will give only 76.1 percent potential extract as-is, while a malt at 3 percent moisture content will give 78.6 percent.
Color, SRM (°Lovibond, ASBC color): The color given for the lot of malt being evaluated should be within, or close to, the color parameters given in tables 1 and . Moreover, for batch-to-batch beer consistency, the grain bill needs to be adjusted to reflect malt color change, especially regarding highly colored malts.
Color may not be stated as SRM/ASBC/°L units. Where EBC units are given, the formula ASBC = (°EBC + 1.2)/2.65 gives reasonable but not entirely accurate transposition. The color for English malts is often given as Color IOB, EBC method. IOB color is only approximately 80 percent of EBC color. Where IOB units are given, the formula ASBC = [(IOB/.80) + 1.2]/2.65 can be used.
HCU color units are a summary measurement of wort color that is equal to the sum of the SRM/°Lovibond of all the malts used to make a wort. HCU units and SRM wort color are comparable up to about 10 °SRM; after that, corrections must be made. See the conversion chart in table 15 to convert from HCU color to approximate °SRM.
The color ranges for most malt types vary widely, depending on the country of origin and the maltster. From the maltster's point of view, it is of some advantage to have a "unique" color range, since it requires any brewer using the malt to make serious adjustments in the brew house program in order to switch suppliers. To some extent, this diversity also benefits brewers, but on the whole, color variation in the same nominal malt types is a problem for the brewer seeking predictable results.
Extract, DBFG: Extract yield, dry basis (0 percent moisture content) fine grind, from an ASBC laboratory mash. The "fine grind" is standardized as being crushed on a Buhler-Miag disc mill at a .2 millimeter setting (often given as 2 Miag), giving 10 percent (±1 percent) of the yield held on a .0232-inch screen, or 40 percent flour. The fine-grind extract gives the maximum potential yield of any malt. It indicates the quality of the barley and the proficiency of its malting. The higher the DBFG extract, the better the malt. Any base malt that does not give at least 78 percent DBFG extract is substandard.
Extract, DBCG: Extract yield, dry basis coarse grind, from an ASBC standard laboratory mash, to 113 degrees F (45 degrees C), malt ground at .7 millimeters on a Buhler-Miag disc mill (7 Miag), giving 75 percent (±1 percent) of the yield as grits held on a .0232-inch screen, or 20 percent flour. Again, extract is given on the dry basis of the malt, for ease of comparison. DBCG indicates the yield that the malt is capable of giving, because it approximates the crush experienced in the brew house. Brew house yield, however, is always lower than the coarse-grind figure predicts, because the value is given on a dry basis (see above), and because the laboratory crush and mash are always more efficient than that actually experienced in the brewery.
In reality, brew-house efficiency is only 85 to 95 percent of what coarse grind predicts. The brew-house yield can be calculated from the dry basis coarse grind by adjusting it to account for the moisture content given in the lot analysis (as-is coarse grind), then factoring in the anticipated brew-house efficiency:
DBCG/(1 + moisture content) - .002 = AICG (as-is coarse grind)
For example, where DBCG is 80 percent and moisture content is 5 percent, the result would be
(.80/1.05)-.002 = .7599
Seventy-six percent is the yield a brewer could expect from the malt if mashing and lautering were 100 percent efficient. For a brew house averaging 92 percent efficiency, anticipated brew-house yield would be
[(.80/1.05)-.002].92 = .699, or 70% brew-house yield
The adjusted brew-house extract yield can be used to predict the degrees Plato or specific gravity that each pound of malt will give in a gallon of wort by the formulas
Adjusted yield x 11.486 = °Plato of 1 lb. of malt in 1 gal. of wort
Adjusted yield x 46.214 = excess gravity, 1 lb. in 1 gal. of wort
.699 x 11.486 = 8.02 °Plato
.699 x 46.214 = G 32.3, or SG 1032.3
Conversely, the actual density of the wort yielded can be used to calculate the brew-house efficiency:
Brew-house efficiency = [(°Plato x gal. of wort)/11.486]/[DBCG/(1 + moisture content) - .002)] x lb. of malt
FG-CG, difference: The fine grind–coarse grind difference indicates the modification of the malt. When a "steely" (vitreous) malt is crushed at 7 Miag, the large grits are not as accessible to diastatic enzymes as grits from a mealy malt. The difference between the extract yield of the fine-grind and coarse-grind mashes will be greater with a steely malt. An FG-CG difference of up to 2.2 percent is acceptable if a decoction or step mash is being employed, but brewers are well advised to purchase malts with a difference less than 2.0 percent, and below 1.8 percent when infusion mashing.
HWE (hot-water extract, liter degrees/kilogram at 68 degrees F [20 degrees C], dry basis): The IOB method of measuring the dissolved extract from malt. A loose laboratory EBC Congress mash is made from fine-ground malt. L°/kg expresses the excess gravity of one liter of wort that a kilogram of the given malt could produce. The HWE can be compared to ASBC DBFG by dividing it by the factor 3.86 (300/3.86 = 77.7 percent). A reasonably good malt gives at least 300 L°/kg, dry basis, and a good malt 305 to 315 L°/kg.
An IOB lot analysis may give the HWE at 7 Miag, which is equivalent to the DBCG rather than the DBFG; the yield from such a lot will be correspondingly lower. It can be compared to the DBCG by dividing it by 3.86.
HWE (pounds per quarter): Expressing the hot water extract in pounds per quarter, or pounds per barrel, is an outmoded British means of measuring extract potential. It was based on dividing the weight of a barrel of extract derived from a quarter of malt (336 pounds) by the weight of one UK barrel of plain water (360 pounds). The weight of the wort in excess of the weight of the water gives the pounds of extract per barrel:
Brewer's pounds per barrel = weight of wort - 360, or G (excess gravity) x .36
Brew-house extract was expressed as pounds per quarter of malt:
Pounds per quarter = [(G - 360) x barrels of wort)/(pounds of malt/336)]
Specific gravity was obtained by the formula
sg gr = weight of 1 UK barrel of the wort/360
CWE (cold-water extract): The amount of soluble material that is present in any malt before mashing, obtained from a three-hour stirred mash of coarse-ground malt at 68 degrees F (20 degrees C), containing ammonia to inhibit enzyme activity. It is an indicator of malt modification; the higher the CWE value, the more complete the beta-glucanase dissolution of endosperm cell walls has been, and the more soluble starch and protein the malt contains. A CWE of 18 percent indicates well-modified malt, and higher than 20 percent "forced" or overmodified malt. The range of 18 to 22 percent is acceptable for infusion mashing, and 15 to 18 percent for decoction and step mashing; for Vienna and Munich malts, 17 to 20 percent is more usual. The CWE as an indicator of modification should be judged in relationship to the soluble nitrogen ratio (see below) for a more complete picture of the malt character.
Hartong 45 degrees: An expression of malt modification, based upon mashing a lot sample by the Congress mash method and at 113 degrees F (45 degrees C). The yield obtained from the 45-degree mash as a percentage of the yield obtained from the Congress mash is given as the Hartong 45-degree percentage, or VZ45 value. Like CWE and SNR, it is an indicator of malt modification.
Another value, known as the Hartong extract, compiled from mashes at four different temperatures, was used to indicate malt quality, with well-modified malt giving a value greater than 5. The Hartong extract is no longer commonly used.
Alpha-amylase, DU: ASBC measurement of dextrinizing units/100 grams, dry weight. Each unit represents the quantity of alpha-amylase that dextrinizes one gram of soluble starch in one hour at 68 degrees F (20 degrees C). One hundred grams of six-row malt should give thirty-five to forty-five units, and American two-row malts forty to fifty. Pilsener malts, because they are kilned at lower temperatures, should range between forty-four and forty-eight, and higher-kilned lager and Vienna malts in the range of forty to forty-five. Munich and ale malts give the least alpha-amylase, because extensive conversion and higher kilning temperatures have caused greater enzyme depletion.
Overly modified malts generally have low DU values. On the other hand, the less well modified a malt is, the greater the DU value needs to be for it to be infusion mashed. When a lot analysis gives an alpha-amylase value below the quoted ranges and modification indicators are relatively usual, the mash may need to be thicker and longer for it to yield the usual brew-house efficiency.
DP, °Lintner (diastatic power ASBC, .25 maltose equivalent): Diastatic power expresses the combined alpha- and beta-amylase strength of the malt. DP measures digestion of a standard starch solution by titration with ferricyanide. Diastatic power is compared to dextrinizing units as a measure of beta-amylase strength, and considered together with mealiness/vitreosity as a measure of how well a malt will respond to mashing. The DP may be as low as 35 to 40 for a well-converted, low-protein ale malt, 100 for a lager malt, 125 for an American high-protein two-row lager malt, and as high as 160 for six-row brewers' malt. Diastatic power decreases as malt color increases.
IOB DP units give values similar to °Lintner.
DP, °WK (diastatic power, degrees Windisch-Kolbach, WK units): The EBC unit of measurement for diastatic strength. °WK may be as low as 100 for ale malts or as high as 600 for green malt. The °WK value can be converted to degrees Lintner by the formula °Lin = (°WK + 16)/3.5. The reciprocal equation is °WK = (°Lin x 3.5) - 16.
Conversion time, min: Indicates modification and diastatic power under real, albeit ideal, mash conditions, by the time it takes a laboratory mash to reach iodine end point. "Base" malts such as pale and Pilsener should convert within fifteen minutes, British ale malts within ten, and better converted, high-enzyme base malts, such as brewer's two- and six-row, in less than ten minutes.
Protein, percent: Equal to the total nitrogen times a factor of 6.25. For all-malt beers, protein should not exceed 12 percent; 9 to 10 percent is usual for European lager and British ale malts.
Soluble protein, percent: See TSN, percent, below.
TN, percent: Expresses the total nitrogen as a percentage of the weight of the malt. Since 30 to 40 percent of malt nitrogenous matter ends up as "permanently soluble nitrogen" and composes 5 to 6 percent of the dissolved solids in the boiled wort, the nitrogen content of any malt is of concern to the brewer. Total nitrogen includes poorly soluble proteins and polypeptides, and affects extract yield, clarity, head formation, and fermentation. There needs to be enough nitrogen, but not too much; malt of 2 percent or more nitrogen creates the risk of haze problems in an all-malt beer.
TSN, percent (total soluble nitrogen, percent of total nitrogen by weight): The percentage of nitrogen in solution in a wort. Generally, about half the soluble nitrogen in wort is formed by proteolysis during mashing; the other half exists in the malt. TSN is used in the soluble nitrogen ratio, below.
SNR (Kolbach Index, S/T, SN/TN, soluble nitrogen/total nitrogen ratio): The soluble nitrogen (or protein) figure, divided by the percent total nitrogen (or protein) is another indicator of malt modification. The higher the percentage, the more highly modified the malt sample is. For traditional lager malts, 30 to 33 percent indicates undermodification, and 37 to 40 percent overmodification. Malts to be infusion mashed should have an SNR of 38 to 42 percent. Malt is seldom rejected on the basis of the SNR unless it approaches 50 percent for infusion mashing, and 40 percent for decoction mashing. At 50 percent SNR, the beer brewed will inevitably lack body. Generally, when the percentage given exceeds that recommended for the type of malt, the brewer increases the temperature or shortens the duration of protein and saccharification rests; when it is lower, the saccharification rest is lengthened or the temperature is lowered, or a protein rest is added or extended.
Mealy, percent: By convention, malt is classified as being mealy, half-glassy/glassy-ends, and glassy (steely or vitreous). Mealy kernels are those in which the endosperm is not more than 25 percent glassy. Half-glassy endosperm is 25 to 75 percent glassy, and the endosperm of glassy kernels is more than 75 percent vitreous. The better and more extensive the malting, the higher the percentage of mealy kernels. Glassy malt does not crush well and is not readily hydrolyzed because it contains a higher percentage of beta-glucans and complexed protein, so it gives less extract than mealy malt. If a malt is to be infusion-mashed, it must be at least 95 percent mealy, and for any mash program should be at least 90 percent mealy. Where mealiness is expressed as mealy/half-glassy/glassy, for base malts the ratio should be 92/7/1 percent or better for decoction and step mashing, and 95/4/1 percent or better for infusion mashing.
Friability, percent (vitreosity): Another method of analyzing malt conversion. The ASBC/EBC method measures the percentage friable and the percentage unmodified by standardized crushing, sieving, and weighing. Any malt should be at least 80 percent friable, and for infusion mashing 85 percent friable.
By another method sometimes used, a sample of thirty or so kernels are split lengthwise in a farinator box, then examined and classified. The factor 1 expresses vitreous, glassy endosperm, .5 is half-glassy, .25 indicates glassy ends, and 0 is the value given to completely modified (mealy) kernels. The average result is given. The range is 0 to 1.0, with a vitreosity of 0 to .25 preferred. By itself, the vitreosity value is of questionable use, since it doesn't clearly pinpoint either the exact problem or the extent to which it occurs. Moreover, it is very subjective.
Size (plump/thin, screen separation less than 2.2 millimeters): Size is most clearly expressed as screen separations, but is just as commonly given as plump/thin. European malts often list only the percentage that can be sieved through 2.2-millimeter openings (thins, 5/64 inch diameter and smaller). Generally, the plumper the kernels, the better the yield; however, the uniformity of malt size is just as important. Because of this, some analyses list homogeneity instead of sizes. Any lot of malt that will crush reasonably well must have kernels that are at least 90 percent adjacent sizes, regardless of the plumpness. Brewers generally demand malt that is of 95 percent or greater homogeneity.
Malt is also rejected for containing more than 1 percent thin kernels or more than 2 percent of kernels smaller than 2.2 millimeters, because their presence indicates unmodified kernels.
Wort pH: Base malts should give a pH of 5.5 to 5.8, with ale malts giving the lower value and lager malts the higher. The pH value is used by the brewer to adjust liquor acidity for the particular lot of malt.
Viscosity, cP (centipoise units): The viscosity is a measure of the breakdown of beta-glucans (endosperm cell walls) during malting. A malt that gives a laboratory wort of high viscosity (more than 1.75) will not run off well during sparging. The higher the viscosity, the slower lautering and filtering will be, and the greater the need for a decoction program, or less effectively, a step mash. Boiling and a protein rest help break down beta-glucans to less viscous gums and polysaccharides.
Where given, the IOB 70-degree mash viscosity should be 6.3 to 6.8.
Odor of mash: This measure is primarily used to indicate off-odors, such as moldy, green-malt, or burnt. With colored malts, very aromatic should be expected.
Speed of filtration: The lower the modification or the higher the protein and beta-glucan content, the slower any malt will filter in the lab. Slow filtration with presumably well-converted malts indicates conversion problems.
Degree of clarity: Brewers generally accept slightly hazy as an acceptable degree of clarity, but it does indicate protein or starch conversion shortcomings. A review of the other parameters of the lot analysis should pinpoint changes in the brew-house program that can be made to improve the runoff from the mash.
Several other parameters may be given in a lot analysis. Some of these, and their usual values, are the following: 1,000-kernel weight (36 to 45) or bushel weight (42 to 44), DMS-P (DMS precursor, 5 to 15 parts per million for lager malts), and growth (acrospire 0 to ¼, ¼ to ½, ½ to ¾, ¾ to full, overgrown; 80 percent minimum of ¾ to full growth for American and British/fully modified malts, and ½ to ¾ for multirest lager malts). Other evaluation techniques may be performed by the brewer to judge the quality of any malt.
Ultimately, only experience with the particular barley type, its origin, season of growth, age, and malting permits finite procedural adjustments.
## Evaluation
Good malt should be plump and firm, even in size and shape, and of a light, straw-colored hue. A sample handful should contain almost no straw, rootlets, dust, or debris. The malt kernel should be easily crushed between the fingers, and uniformly soft from end to end. When broken, the kernel should write like chalk and show no air pockets within the husk.
If no growth specifications are given in the lot analysis, it is advisable to check growth when the malt is received. The acrospire can be exposed by cutting or rubbing away the husk over the acrospire of fifty kernels, or by boiling the kernels for an hour to turn the husks translucent. At least 80 percent of the kernels of any malt delivery must fall within a single size group for the malt to be uniformly crushed and mashed. Moreover, malt that is 5 percent or more ungerminated (zero to one-fourth growth) is unacceptable for decoction mashing; 2 percent or more is unacceptable for infusion mashing. Malt to be mashed by only one temperature rest must have been uniformly sprouted, and the acrospire growth should uniformly be from three-fourths to the full length of the kernel. Malt that has been kilned when the acrospire growth is only from one-half to three-fourths the length of the grain can yield a greater amount of extract than fully modified malt, but it should be step or decoction mashed. Malt of widely varying acrospire growth also should be step or decoction mashed. Brewing-quality malt should yield no appreciable number of kernels with acrospire growth less than one-half the length of the grain.
Malt
Acrospire Growth
Acrospire growth should be considered in conjunction with examination of the endosperm. Thirty or so kernels are split lengthwise, and the starch is examined for steeliness/vitreosity; short acrospires and steely ends indicate insufficient sprouting, whereas steeliness irrespective of acrospire length indicates poor drying/kilning.
Another test of germination/modification is made by the "sinker" test. Fifty kernels are shaken into water, and then let rest for ten minutes. After ten minutes, the floaters and sinkers are counted. Ungerminated and undermodified kernels will sink. At the very least, thirty-five of the kernels (70 percent) should float; with good malt, 95 percent will float parallel to the surface of the water. Vertical floaters indicate poor modification.
The malt should yield 65 to 80 percent sugar by weight after mashing and test between 1.4 and 2.0 percent nitrogen. Malt of greater than 2 percent nitrogen should only be used when brewing with low-protein adjuncts.
Malt should be selected by its color and modification to suit the type of beer being brewed and the brewing method.
Avoid malt of very high protein content because it has correspondingly fewer carbohydrates. The protein can retard starch conversion during mashing and increases the likelihood of hazing in the beer. Finally, there is no substitute for tasting the malt. The better the flavor and aromatics of the malt, the better the malt complement of the finished beer will be.
Table 2
Grading by Size
---
Grade | Characteristic
1 | Remains on 7/64" screen, plump
2 | Remains on 3/32" screen
3 | Remains on 5/64" screen, "thin" European
4 | Falls through 5/64" screen, "thin" American
The two larger sizes will mash well and produce high yields. The small kernels have greater husk content and will give lower extracts.
Grading Pale Malt by Character
---
Mealy | (mellow, soft) | Will mash well
Half-glassy | (semihard) | Must employ a longer, more thorough mashing
Glassy | (hard, translucent, grayish white) | Unsuitable for brewing
## Carbohydrates
Carbohydrates are compounds formed by molecules of carbon, hydrogen, and oxygen. They include simple sugars, chains of four or more sugars called oligosaccharides, and polysaccharides, formed by the union of ten or more monosaccharides. Barley and malt are largely made up of carbohydrates, as are 90 to 95 percent of the wort solids. The barley carbohydrate group is composed of insoluble cellulose (the membranes and casing of the barley corn) and soluble hemicellulose, starch, dextrins (alpha-glucans), and the simpler mono-, di- and trisaccharides that we call sugars.
Cellulose is an insoluble compound-sugar (50 to 5,000 molecules of C6H10O5) that makes up the barley husk; it is irreducible by malt enzymes. It constitutes roughly 5 to 10 percent of the malt by weight. Although it is structurally similar to starch, cellulose does not contribute fermentable extract or desirable flavors to the malt extract. But as malt husks, cellulose is employed in the lauter-tun to form a filter bed through which the mash extract is strained. Extraction by high temperature and alkaline sparging leaches harsh tannins from the husk into the extract to the detriment of the beer.
Hemicellulose and soluble gums bind the aleurone layer of the barley kernel together, and are constituents of endosperm cell walls that encapsulate starch granules. They account for about 10 percent of the weight of the barley kernel. They are both polysaccharide mixtures complexed with protein. The malt gums are largely pentosans of xylose, arabinose, galactose, and complexed glucose, which may be dissolved at temperatures of 160 to 176 degrees F (71 to 80 degrees C). They compose about 20 percent of the endosperm cell walls. They can be fully reduced to simple sugars in a thorough mash cycle.
Table 3
Degree of Modification
---
Acrospire Growth: | Characteristic of:
0"–¼" | Cereal malt, unmodified
½"–⅔" | Steely lager malt
½"–¾" | Lager malt
¾"–Full | British malt, American brewers' malt
Malt Colors
---
Malt | Type | SRM | EBC | IOB
Diastatic | Base malt | 1.0–1.4 | 1.5–2.5 | 1.2–2.0
Pilsener | Base malt | 1.2–1.6 | 2.0–3.0 | 1.6–2.4
Lager | Base malt | 1.4–1.8 | 2.5–3.5 | 2.0–2.8
Brewers | Base malt | 1.4–2.0 | 2.5–4.0 | 2.0–3.2
CaraPils | Vitreous | 1.4–4.0 | 2.5–9.0 | 2.0–7.2
British pale | Base malt | 2.0–2.7 | 4.0–6.0 | 3.2–4.8
Vienna | Base malt | 3–5 | 7–12 | 5.6–9.6
Munich | Toasted | 6–12 | 15–30 | 12–24
Light caramel/Carastan | Vitreous | 5–45 | 12–120 | 10–96
Amber/Biscuit/Dunkles | Toasted | 20–30 | 50–80 | 40–64
Crystal/Caramel 50 | Vitreous | 45–60 | 120–160 | 96–130
Brown | Roasted | 45–60 | 120–160 | 96–130
Caramel 80 | Vitreous | 75–85 | 200–225 | 160–180
Caramel 120 | Vitreous | 100–122 | 265–325 | 210–260
Chocolate | Roasted | 340–500 | 900–1325 | 720–1060
Roast barley | Roasted | 270–650 | 725–1700 | 580–1400
Black/Roast | Roasted | 550–650 | 1450–1700 | 1220–1400
Hemicelluloses are more complex and stubbornly resist hydrolysis. They give a purple-to-black color with iodine. Very viscous beta-glucan is the major constituent of barley hemicellulose, accounting for about 75 percent of the cell walls. In intermediate stages of cell-wall solubilization, the release of beta-glucan inhibits lautering and filtering and cause hazes. One consequence of forced malting is insufficient hydrolysis of beta-glucans to glucose by beta-glucanase enzymes, which are denatured above 140 degrees F (60 degrees C) and do not usually survive kilning. High wort viscosity indicates excessive beta-glucans in the malt. With reasonably well converted malt, manageable amounts of beta-glucans may be liberated from hemicellulose by proteolytic enzymes during a 95 to 113 degree F (35 to 45 degree C) mash rest and contribute to a beer's fermentability, body, and foam head.
### Starch
Native starch (in malt, 20 to 400 molecules of C6H10O5) occurs as granules that are insoluble in cold water, accounting for about 60 percent of the malt's weight. If the native starch granule is crushed to flour, it readily combines with moisture to form a pastelike gel. In warm water, amylose, the small fraction of a granule that is water soluble, can be diffused out of native starch. In hot water, the starch granule swells until it finally bursts, exposing insoluble amylopectin as well.
Amylose occurs as straight helical chains of glucose molecules joined by 1-4 links. Exposed to amylose, iodine becomes entrapped within the coils, giving an intense blue-black color. Amylose is reduced to maltose and maltotriose from the nonreducing end of the chain by the beta-amylase enzyme. The alpha-amylase enzyme facilitates this process by randomly severing amylose chains, yielding glucose, maltose, and maltotriose, and exposing more chain ends to beta-amylase activity.
Starch
Cellulose
Amylopectin accounts for about 75 percent of the malt starch; consequently, its decomposition is central to mashing. It occurs as highly branched glucose chains that give a red color with iodine. The chain and branches are formed by 1-4-linked glucose molecules, but the glucose molecules at the branching points are joined in a 1-6 configuration.
### Dextrins
Dextrins, or alpha-glucans (C6H10O5) ⋅ H2O, are residual, unfermentable fractions of amylopectin of from four to thirty glucose molecules. Dextrins occur in brewing because neither alpha- nor beta-amylase can sever the 1-6 links of amylopectin or the 1-4 links near those branching points. In malting, limit dextrinase debranches amylopectin by separation of the 1-6 links, but this enzyme group does not survive malting and mash temperatures. In the mash, beta-amylase is only able to effect a slight reduction of amylopectin, liberating maltose only at the nonreducing ends of the many branches. The large dextrins it leaves behind are called ß-limit dextrins. ß-limit dextrins give a mahogany (reddish-brown) color to iodine.
Amylose
Amylopectin Fragments
Starch
Maltose
Isomaltose
Sucrose
ß-limit dextrins are dismantled in the mash by alpha-amylase. It randomly breaks 1-4 glucose links in the amylopectin complex, exposing new nonreducing ends for beta-amylase to convert to maltose. Acting together, the amylases reduce amylopectin to much smaller a-limit dextrins. Small a-limit dextrins give no color with iodine, while the largest give a faint red-to-violet color.
Melibiose
Fructose
Mannose
Galactose
Glucose
### Sugars
Monosaccharides (C6H12O6) are single-molecule carbohydrates. Glucose, photosynthesized in the leaves of green plants from carbon dioxide and water, is the primary sugar associated with grains. However, it accounts for only 1 to 2 percent of the malt's weight. After mashing, it accounts for 7 to 10 percent of the wort extract. Fructose (an isomer, or molecular rearrangement, of glucose), galactose and mannose are other monosaccharides found in malt. They are all readily and wholly fermentable. Galactose is a constituent of many oligosaccharides and polysaccharides, occurring especially in gums and pectins.
The disaccharides are double sugars (C12H22O11). Maltose, the sugar most closely associated with brewing, is the simplest compound sugar. It is formed by two molecules of glucose, joined together at their 1 and 4 carbon atoms by the removal of a water molecule (C6H12O6 \+ C6H13O6 \- H2O = C12H22O11 x H2O). Although it makes up only 1 percent of malt's weight, it is the sugar that is primarily derived from the hydrolysis of starch, and accounts for 40 to 55 percent of the wort solubles. It is slowly but very surely fermentable.
Other disaccharides present in malt are sucrose (one molecule each of glucose and fructose), lactose (an isomer of sucrose), and melibiose (one molecule each of galactose and glucose). Sucrose is readily fermentable, lactose is unfermentable, and melibiose is fermentable only by _Saccharomayces uvarum._ Of these disaccharides, only sucrose is present in malt in any appreciable amount, composing 3 to 5 percent of the wort extract.
Trisaccharides are three-molecule sugars. In malt, they are maltotriose, glucodifructose, and fructosant; raffinose, which is present in barley, disappears during malting. Maltotriose (three molecules of glucose) is slowly fermentable by most strains of brewing yeast. It is the only significant trisaccharide in brewing, accounting for up to 15 percent of the wort solids.
## Protein
Carbohydrates alone do not form a satisfactory brewing extract; malt protein is essential as a yeast nutrient and to give the beer body and head. In the barley kernel, protein serves as plant food for the embryo. During malting and mashing, it is reduced by protein-metabolizing enzymes to less-complex _albumins_ (an outdated but useful term that groups together all coagulable and water soluble protein fractions) or even simpler amino acids. In solution with mineral salts, some of the protein forms acids, causing the pH of the mash to drop.
The term _protein_ strictly defines very complex polymeric coils of amino acid chained together by peptide links (molecular weight 17,000 to 150,000), but in a wider sense, it also includes the products of their decomposition. This reference to protein on the one hand as an unwieldy malt constituent (e.g., high-molecular-weight albumin and less soluble globulin, glutelin, and hordein), and on the other as a source of a beer's body and head can be confusing. In fact, the brewer does not strive to eliminate the proteins but to simplify them to a range of colloidable and soluble fractions. Only when an excessive number of complex proteins (molecular weight 10,000 to 100,000, average 30,000) are carried into the beer will they result in chill haze, an irreversible cloudiness, or off-flavors caused by oxidation.
The large protein complexes of the malt must be largely reduced to intermediate albumins (molecular weight 5,000 to 12,000), such as proteoses and peptones, and peptides (molecular weight 400 to 1,500) during mashing, so that they are simple enough to dissolve, or colloid with hop resins, and be carried over into the finished beer. Considerable nitrogenous matter must be present: amino acids and peptides to sustain culture-yeast activity, and simple albumin to form the body of the beer. Three to 6 percent of the wort extract is usually nitrogen-based. A beer brewed so that it had little or no low-molecular-weight protein fractions (molecular weight 12,000 to 20,000) left in it would be unable to form or support a foam head. Generally, 25 to 50 percent of the malt protein is carried into the ferment as simple nitrogen complexes (amino acids, peptides, albumin).
Because enzymes are proteins, the protein content of any malt is an indication of its enzymatic strength. In general, protein-rich, six-row malts are more apt to cloud beer than are other malts, but they do produce stronger enzymatic activity. Low-protein malts are less apt to cause hazing, but as a consequence have less enzymatic power and must be carefully mashed.
## Other Malt Fractions
Nonprotein nitrogen accounts for up to 10 percent of the total malt nitrogen. Peptides, although not true proteins, are intermediate forms between protein and amino acids. They are nitrogen based, as are vitamins and nucleic acids. Peptides, which enhance the beer's viscosity (palate fullness), may also oxidize to high-molecular-weight polypeptides. Polypeptides are unstable and contribute to nonbiological haze formation.
The vitamins in malt are principally of the B complex; they are necessary for yeast growth. Products of the decomposition of nucleic acids account for only .1 percent of the wort extract, but contribute to both yeast nutrition and flavor enhancement.
Polyphenols, phosphates, lipids, and fatty acids are other significant malt constituents. Polyphenols from the husk, pericarp, and aleurone layers of the malt kernel are acidic precursors of tannins and give beer an unpleasantly bitter, astringent taste and reddish hue. With highly kilned malt, oxidation of phenolic melanoids (pigments) to aldehydes, such as by hot-side aeration, contribute to stale flavors in beer. Complex polyphenol polymers are true tannins and counteract the solubility of otherwise stable proteins. They should be eliminated from the extract by a well-roused kettle boil; in the cooled wort and the ferment, the oxidative polymerization of polyphenols to tannins causes medicinal off-flavors, astringency, and haze formation.
Phosphates in the malt, principally organic phytin, are major factors in the acidulation of the mash. They give up phytic acid (phytate) at high kiln temperatures during malting, and by enzymatic reduction during an acidifying mash rest at below 128 degrees F (53 degrees C).
Lipids are fatlike substances composing roughly 3 percent of the malt; they range from straight-chain to complex branched-ring hydrocarbons, including neutral fats, fatty acids, alcohols, aldehydes, and waxes. About one-third of malt lipids occur as reserves in the embryo, and most of the rest are concentrated in the aleurone layer. Triacylglycerol and other triglycerides (triesters of glycerol and long-chain fatty acids) are the predominant neutral lipids in barley. They support respiration of the embryo during malting.
Fatty acids (CnH2nO2), or vegetable oils, are fat-derived _(aliphatic)_ hydrocarbon chains. In barley malt, these are the relatively long-chain linoleic (C18H32O2), palmitic (C6H32O2), and oleic (C18H34O2) acids. Proportionally more of these than other malt lipids are carried into the wort. Although they are essential in the yeast cell as reserves and they comprise only .05 percent of the malt extract, if excessive amounts of lipids are washed into the runoff by overzealous sparging they will reduce foam stability and give rise to "cardboardy," "goaty," and "soapy" stale flavors in beer.
Finally, considerable silica and inorganic phosphate are leached out of the malt, along with a wide range of trace minerals necessary for yeast metabolism.
# CHAPTER 3
# Water
Water constitutes 85 to 90 percent of the volume of any beer, and therefore the mineral content of the brewing water has a marked effect on the flavor and appearance of the finished beer — and on the brewing process. Certain beer styles are suited to waters of very specific mineral composition, and an otherwise well-brewed example will always be diminished by the use of totally inappropriate brewing water.
By looking closely at the geology of any given area, it is often possible to find wells or springs that will perfectly suit a given brew. Local, regional, and state water departments and services can be very helpful in locating such sources. For most brewers, however, mineral treatment of the local supply is the more reasonable alternative.
Only brackish, polluted water and sea water are entirely unsuitable for brewing. Most potable fresh waters, whether too "hard," too carbonate, too "soft," or iron contaminated, may be boiled, aerated, sedimented, filtered, or treated with an appropriate mineral salt or acid to be made suitable for brewing almost any type of beer. Practically speaking, however, brewing water should be clear, bright, unpolluted, and have agreeable taste and reasonably uniform composition from day to day. It should not be corrosive, have a detectable odor, or throw an appreciable amount of sediment upon resting or boiling.
All naturally occurring waters are dilute solutions of minerals in which small quantities of gases and organic matter may be dissolved. Rainwater should be the purest natural source of water, but because it assimilates atmospheric gases and organic mineral particles wherever the air is the least bit polluted, most rainwater is absolutely unsuitable for use in brewing. Precipitation in areas far removed from large fossil-fuel burning plants more often than not is still polluted by highly corrosive sulfuric acid (H2SO4). Free hydrogen carbonates (HCO3, usually referred to as _bicarbonates_ ) are also common in rainwater. They rob the calcium from the mash, wort, and ferment by forming bicarbonate salts that are precipitated from solution during boiling.
Surface waters, besides having the dissolved materials and gases found in rainwater, usually contain large amounts of organic matter, vegetable coloring, soil, silica, clay, and microflora. Especially in marshy terrains and industrial areas, surface water is likely to be heavily contaminated with organic acids and nitrates and is completely unsuitable for use in brewing.
Surface waters suitable for brewing are generally limited to clear-running, spring-fed brooks and streams that flow over gravel, sand, or rocky beds, and deep reservoirs with carefully protected watersheds. Water from old soft-bottomed streams, which flow sluggishly or carry topsoil and vegetation in suspension, and rivers, ponds, and lakes needs to be filtered, at the very least, before use, and is seldom a good choice for brewing. If the water tastes like it comes from a pond, so will the beer.
Municipal water supplies are usually gathered from several deep wells and reservoirs, and sometimes from rivers. They are invariably filtered and treated (most commonly with up to 0.5 parts per million chlorine) to inhibit microbial contamination. Such tap water is often perfectly suitable for use in brewing, after being filtered, rested, or boiled and aerated to drive off free chlorine and induce sedimentation of carbonates, silicates, and incrustants. Chlorine and organics may be removed by activated carbon filtration. Water departments often vary the inflows from several different sources, and water composition varies accordingly.
The quantity and composition of underground water at any given location and depth are contingent upon subsurface geological formation. Although the elementary minerals and metals dispersed in the earth's crust are relatively few (rock is largely silica with aluminum, iron, calcium, potassium, magnesium, manganese, zinc, and copper), the soluble components (mineral salts) they form yield water of varying composition from place to place. The mosaic geological structure of the earth's crust causes water to rise to different levels at different locations, and its level may be fairly constant or may fluctuate with changing patterns of precipitation, freezing, and thawing.
The value of a spring or well as a brewing source should first be judged by the seasonal consistency of its flow. There is less fluctuation in the composition of any ground water that has a reasonably constant flow year-round. Shallow wells and seasonal springs do not usually yield acceptable brewing water. Their composition varies widely, and they often carry soil and other surface contamination in suspension. They commonly yield unacceptable levels of bacterial contamination.
Deep wells originating in large subterranean aquifers, and mountain springs percolating up through fissures in bedrock formations without leaching through soil or disintegrated rock are usually of very stable composition and free from surface contamination. Subterranean springs (with the notable exception of mineral springs) emerging from inorganic rock complexes commonly yield water with less dissolved materials than does deep-well water. Deep wells usually tap water sources that have traveled farther than the water of springs, and having contacted more mineral-bearing substrata consequently have more minerals in solution than does spring water. Which minerals occur in ground water is dependent on the nature of the strata the water has contacted. Igneous and metamorphic rocks (granite, basalt, marble, gneiss, and quartz) are both very dense and compositionally very stable; they yield the fewest mineral ions to hydrolysis of any rock formations. Because they are hard, they do not filter the water passing over them as do most sedimentary formations. They are not likely to bear much water unless they are extensively fissured or enclose voids left by the dissolution of softer strata. Granite is the most common of the igneous rocks, composing the greatest part of the continental plates. It commonly yields very "soft" water of less than 100 ppm hardness (as CaCO3), and water of less than 50 ppm hardness is not at all uncommon.
These rocks are eroded by freezing water, scaling, hydrolysis, or friction, and the particles are carried away by wind and water and ultimately deposited in topographical depressions. This alluvia forms unconsolidated sediments, often far from the parent rock.
Sand, which is largely silica dioxide split off from quartz, passes water freely while filtering out most suspended solids. Where it lies above impervious bedrock, it yields excellent water. Clay, on the other hand, is impervious and yields very little water, but water pools above it wherever it makes an unbroken sediment. Clay underlies many excellent aquifers. Clay is mostly hydrous silicates of aluminum, often colored red by iron oxide or dark by carbon-based impurities. Gravel (pebbles of ⅛ to 2 ½ inches in diameter) is usually found with sand or clay, either on river terraces or as the residue of glacial retreat. It often serves to make clay subsoil pervious to penetration by some surface runoff.
Consolidated sediments are deposits of sand, clay, or gravel that have hardened under pressure or have been cemented by lime. They are typically deposits left on the floors of prehistoric seas and lakes. Sandstone is very common; it is porous and generally rather coarse. It filters water very effectively but also gives up its mineral ions to hydrolysis very readily. Sandstone formations yield predominantly "permanent" hardness, but the mineral composition and "total" hardness vary widely from place to place. (For an explanation of water hardness, see pages 61 through 65.) Water drawn from older sandstone may be soft but is usually moderately hard, averaging 50 to 300 ppm hardness. New red sandstone usually bears very hard water (150 to 400 ppm). Although the hardness is largely sulfate, water drawn from new red sandstone is often objectionably inundated by iron.
Formed from loose sediment, limestone (calcium carbonate) is not an extremely dense rock, but it is impervious and yields water only if it has been extensively fissured by seismic activity or eons of hydrolysis. The passage of water along joints and bedding plates has carved out huge subterranean caverns and labyrinths in some limestone formations, where abundant water may be tapped. Carbonic acid from the atmosphere readily dissolves calcium carbonate, forming very soluble calcium hydrogen-carbonate (bicarbonate) salts. Consequently aquifers and springs from limestone tend to yield water high in "temporary" hardness, with other minerals in solution. Some limestones are largely made up of magnesia and may yield considerable sulfate as well. Hardness of 150 to 350 ppm is common.
Underground Water
Chalk formations are very soft, fine-grained limestone from mudlike sea-bottom sediment. They yield from 150 to 375 ppm hardness, largely as calcium carbonate. Soapstone (talc) is insoluble hydrous magnesium silicate, often found with magnetite (iron oxide) and chlorite. Formed from consolidated clay, shale is impervious but may pass water along its bedding plates. Like other sedimentary rock, it commonly contains many minerals.
Marl is consolidated clay with sand, and most often calcium carbonate, potash, or phosphorus. It is commonly found stratified with sandstone or limestone. Marl generally yields little water, and of an undesirable mineral composition. Conglomerates are impervious, hard rocks from lime-cemented gravel and sand; they vary widely in composition but usually have little effect on the water coming in contact with them.
## Using the Water Analysis
Any potential source of brewing water should be analyzed for organic and inorganic composition and biological purity. Because the composition of any water supply is likely to vary seasonally, and even within each season, it is advisable to make or obtain an analysis of it before brewing, or when changes in the brew may be due to changes in the water. Where periodic testing is not possible or practical, a simple pH test demonstrates changes in the mineral content and can indicate what the changes in the mineral distribution may be.
The standard water analysis identifies the amounts of the mineral ions present in the water and indicates the presence of organic pollution as well. A bacterial analysis may be included or made separately. Analyses for community water supplies are available upon request from local water departments; other sources are analyzed by private labs for a fee. Do-it-yourself kits are also available for identifying the pH, hardness, and alkalinity of water, and the presence of various mineral ions (calcium, magnesium, iron, chlorine, sulfate, nitrate, nitrite); they are easy to use, handy, and inexpensive over the long run if analyses would otherwise have to be made by a private lab.
Table 4
Water: Typical Analysis U.S. Public Health Service (U.S.P.H.S.) Units
---
Turbidity | Color | pH | Sediment | Odor
Specific Conductivity (micromhos/cm)
Hardness – Total (CaCO3, in ppm [mg/L])
Alkalinity – Total (CaCO3, in ppm)
Major Constituents 1–1000 ppm
---
Calcium (Ca)
Magnesium (Mg)
Sodium (Na)
Silica (SiO2)
Sulfate (SO4)
Chloride (Cl)
Secondary Constituents .01–10 ppm
---
Potassium (K)
Nitrogen (Ammonia, NH4)
Nitrogen (Nitrite, NO2)
Iron (Fe)
Nitrogen (Nitrate, NO3)
Minor Constituents .001–.1 ppm
---
Manganese (Mn)
Copper (Cu)
Trace Constituents less than .001 ppm
---
Zinc (Zn)
Coliform bacteria (colonies per 100 mL)
If testing by a lab is necessary, certain procedures for obtaining the specimen are advisable. Rinse a clean quart jar several times with the water to be tested; fill the jar and immerse it nearly to its neck in a kettle of water. Heat and boil for twenty minutes. Decant the water from the jar and invert the jar on a clean paper towel to drain. When cool, fill it with the water to be tested. If it is tap water, allow the cold water to run for a minute before taking the sample to flush clear any mineral deposits jarred loose by the initial release of water from the tap. Cap the jar tightly and rush it to the lab where you have made arrangements to take it. The longer the sample sits, the less accurate the analysis will be. Where pollution is suspected, the water must be tested within twelve hours of collection, and in all other cases, within seventy-two hours.
When the brewing water is to be treated with mineral salts or boiled to precipitate carbonates, an analysis after treatment will pinpoint the resultant mineral distribution, but it is hardly necessary. The carefully weighed addition of salts and a simple pH test after treatment usually adequately indicates the subsequent mineral distribution.
Brewing with an untreated natural water supply is possible in almost all cases by manipulating the brewing procedure. A water analysis should be made before brewing to determine the formulation and procedure that best suits the particular water composition.
## Turbidity, Sediment, Color, and Odor
Turbidity and sediment may be caused by suspended clay and other inorganic soil, organic topsoil or waste, colloidal ferrous and aluminum oxides, or manganese and silicon dioxide.
Color is usually due to colloidal vegetable pigments, although a yellow-to-brown hue may be from suspended clay or silt. This sediments out upon resting; particles in colloidal suspension may be eliminated only by filtering. Odor may be from dissolved gases or organic decay.
Overall character may be improved by activated-carbon filtration. Where this is not satisfactory, a clearer water should be found to brew with.
U.S. Public Health Service Drinking Water Standards suggest as limits that should not be exceeded: turbidity — five units; color — fifteen units; odor — threshold number three. For brewing, it is recommended that these all be less than one.
## pH
The pH indicates acid to alkalinity ratios and the mineral composition of water as well, both of utmost importance to the brewer. Appropriate acidity is a prerequisite of a successful brewing cycle. Enzyme activity, kettle break, and yeast performance rely upon conducive acidity in the mash, wort, and beer. The acidity of the brewing water source is therefore of concern to the brewer.
pH is the measure of the acidity or alkalinity of a solution. Acid solutions taste sour; alkaline solutions taste bitter and flat. In other terms, acidity is expressed as a greater number of positively charged particles in solution than negatively charged ones; conversely, alkalinity marks an excess of dissociated negative particles.
## Ions
All elements are reducible to single atoms. Atoms are made up of an equal number of positive and negative charges, respectively termed protons and electrons. The atoms of each element are distinguished from those of every other element by the number of protons in their nuclei and the number and arrangement of electrons they have in orbit. Atoms are chemically inactive because the charges of their protons and electrons neutralize each other. They may, however, be unstable. Only atoms having two electrons in their first orbit and eight electrons in their outermost orbit are stable; only these elements occur in their true atomic form.
Atoms
An unstable atom either gives electrons to or receives electrons from another unstable atom, each thereby forming an ion of the element that has a stable electron configuration.
Ions
Ionic Compounds
Unstable Elements Sodium and Chloride
Electron Transfer
Ionic Compound, Sodium Chloride
Because this results in an imbalance in the number of positively charged protons and negatively charged electrons, all ions have an electromagnetic charge. Ions that have a positive attraction are cations; negatively charged ions are anions.
Cations and anions combine to form ionic compounds; the strength of their bond and the charge of the compound itself are dictated by the relative electromagnetic attraction of the ions involved. The greater the difference in their charges, the stronger the ionic bond, and the more acid or alkaline their compound.
Ionic compounds are formed by the exchange of electrons. Other compounds are formed by sharing electrons; these are termed covalent. Most organic compounds (compounds containing carbon) are covalent.
The water molecule, H2O, is covalent. Only a single electron orbits the nucleus of a hydrogen atom; the outer orbit of an oxygen atom contains only six electrons. Both atoms are unstable, hydrogen needing to give up an electron and oxygen needing to receive two. In the covalent compound of the water molecule, the hydrogen atoms achieve stable single orbits by sharing their electrons with the oxygen atom, which then has a stable outer orbit of eight. Since no electrons are exchanged, the bond is nonionic, and the molecule is electromagnetically neutral. The negative valence (-2) of the oxygen atom (0\--, or lacking two electrons) is precisely neutralized by the combined positive valence (+2) of the two hydrogen atoms (H2++).
Although neutral, the water molecule retains a strongly polar character, because the eight protons of its oxygen molecule have slightly more attraction for the negatively charged electrons of the shared orbits than do the single protons of the two hydrogen atoms. The oxygen side of the water molecule is thus slightly negatively charged and the hydrogen side equally positively charged; it is even polar enough to disrupt the ionic bonds of many inorganic compounds, causing them to dissolve into their component ions. When all of the water present is absorbed in the reaction with an ionic compound, that compound is said to be hydrated by water of crystallization. For example, gypsum (CaSO4⋅2H2O) is a hydrated salt. Like other compounds, it can be dissolved into its component ions by the introduction of more water. In fact, water is the most universal solvent known, and it is able to react chemically with most inorganic acids and bases, and to dissolve many salts.
Although pure water is characterized as being covalent, a certain number of its molecules react in pairs to form hydroxide ions (OH-) and hydronium ions (H3O+, most often expressed as simply H+). This is an equilibrium reaction, and the combined number of hydroxide and hydronium ions always remains constant; in pure water at 77 degrees F (25 degrees C), the concentration of electropositive hydronium ions and electronegative hydroxide ions are each .000,000,1 moles per liter. When mineral compounds dissolve in water, their positive mineral ions bond to the oxygen side of the water molecule, freeing hydronium (H+) ions, and their negative ions react with the electropositive hydrogen side, releasing hydroxide (OH-) ions.
Water Molecule
When an excess of either hydroxide or hydronium ions is released, the ionic equilibrium of pure water is disturbed. Because the combined value of hydroxide and hydronium ions always remains constant, an increase in either always results in a proportional decrease in the other.
## Measuring pH
The pH scale represents the relative molar concentration of hydrogen and hydroxide ions present in any solution by measuring the activity of the free hydronium ions in solution. If the hydronium ion concentration exceeds .000,000,1 moles per liter, the solution is acidic; if it is less than .000,000,1 moles per liter, the solution is alkaline and will neutralize acids and liberate CO2.
The reaction that the hydrogen ion concentration represents causes a color change in litmus paper and other pH indicators. The degree of color change is gauged against a standard scale to identify the pH of any solution. For a more detailed analysis, electromagnetic equipment is employed.
The pH, however, is not expressed as the molar concentration of hydrogen and hydroxide ions. The pH scale uses the exponent of 10 in the logarithm (see table 5) to identify the acidity of a solution. Because it is a logarithmic scale, a solution at pH 4 is 10 times more acidic than a solution at pH 5, 100 times more acidic than a solution at pH 6, and 1,000 times more acidic than a solution at pH 7.
Table 5: pH
Part A.
---
Moles Per Liter, H+ | pH | Moles Per Liter, OH-
.1 | 1⋅10-1 | 1 | 1⋅10-13 | .000 000 000 000 1
.000 1 | 1⋅10-4 | 4 | 1⋅10-10 | .000 000 000 1
.000 01 | 1⋅10-5 | 5 | 1⋅10-9 | .000 000 001
.000 000 1 | 1⋅10-7 | 7 | 1⋅10-7 | .000 000 1
.000 000 001 | 1⋅10-9 | 9 | 1.10-5 | .000 01
Part B. Hydrogen Ion Concentration in the pH Scale
---
pH | 4 | 5 | 6 | 7
H+, moles/L | .000 1 | .000 01 | .000 001 | .000 000 1
## pH Adjustment
All changes in the pH of the brewing liquor, the mash, the wort, or the beer, whether induced or consequential, are due to the formation, addition, or precipitation of mineral ions or organic acids. The pH of the brewing water may be adjusted by precipitating out alkaline carbonate salts or by adding organic acids or mineral salts. Salts are added when additional mineral character is also desired. As an alternative to water treatment, the mash may be made more acidic by the metabolism of certain bacteria, which causes the formation of organic acids.
The pH of the mash affects the level of enzyme activity within it, and the acidity of the wort and beer. The pH significantly affects hop extraction and protein precipitation in the kettle, and yeast performance and clarification in the ferment. Because ideal pH levels often cannot be attained, or because the pH at one stage conflicts with the pH optimum of a more critical reaction, concessions are sometimes made; shortcomings can usually be overcome by time and temperature manipulation. For example, in the mash the enzymatic reduction of proteins to soluble nitrogen is most efficient at pH 5 or below, a level of acidity that conflicts with starch reduction optimums during mashing. Therefore, it is necessary to make a longer rest at temperatures conducive to protein degradation, especially when the mash begins at above pH 5.5.
At above pH 6, any mash suffers from sluggish enzyme activity, and in the lauter-tun, troublesome tannins and silicates are leached into the extract. With "soft" water and pale malt, the few acid ions cannot overcome ("invert") buffers leached from the malt, which results in a strong resistance to further acidulation of the mash.
pH 5.2 to 5.5 should be the target acidity of the saccharification rest for all mashes. This is the range at which enzyme activity, filtering, color, and clarity are best. A mash at pH 5.2 to 5.5 can be expected to yield a sweet wort of 5.5 or slightly above, which best serves hop extraction and flocculation of protein in the kettle.
## Total Dissolved Solids/Specific Conductivity
The _total dissolved solids_ are the mineral ions in solution. Total dissolved solids are measured by passing water through a filter fine enough to screen out all sediments and colloids, then weighing the residue left after the filtered water has been evaporated.
The specific conductivity of water measures the mineral ions in solution by gauging the solution's ability to conduct an electric current at 77 degrees F (25 degrees C). Most inorganic acids, bases, and salts are good conductors of an electric current because ions are by definition electrovalent, or charged, particles, whereas organic compounds conduct current very poorly, if at all. Specific conductance, therefore, is a measure of the total dissolved solids in any water.
The specific conductance in micromhos per centimeter ( _mho_ is the opposite of the basic unit of electrical resistance, the _ohm_ ) is roughly equatable to the total dissolved solids in solution in parts per million (ppm is the same measurement as milligrams per liter [mg/L]) by multiplying the specific conductance by a factor that ranges from .55 (for water of pH less than 7.2 or more than 8) up to .90 (for saline water). For most natural waters, multiplying the specific conductance by a factor between .55 and .70 gives a reasonable assessment of the total dissolved solids.
## Hardness/Alkalinity
The _hardness_ of water is gauged by measuring the dissolved cations of the alkaline-earth elements, most significantly calcium (Ca++) and magnesium (Mg++). These common minerals inhibit the sudsing of sodium-based soap in "hard" water and are precipitated as an insoluble, furry residue.
Alkaline-earth ions are weakly electropositive and give water slight acidity. The polar bonds they form with water molecules are weak and readily broken by strongly basic anions, causing them to precipitate as insoluble salts. Calcium exhibits a fragile solubility. Magnesium is more electropositive than calcium but is not correspondingly less soluble; it is actually more stable in solution. The cations of the larger, less electropositive alkali metals sodium (Na+) and potassium (K+) are even more weakly acidic. They are much more stable in water, give an essentially neutral pH reaction, and do not contribute to water hardness.
Calcium is the most widely occurring "metal" found in water, followed by magnesium, sodium, and then iron. Potassium and manganese are much less common; when considering their effects, potassium is often grouped with sodium, and manganese with iron.
The metal cations all occur out of solution bonded to acid anions in crystalline-structured ionic compounds called _mineral salts._ The bicarbonate, sulfate, chloride, nitrate, borate, and phosphate ions are all classified as acids because they are derived from carbonic, sulfuric, hydrochloric, nitric, boric, and phosphoric acids; their effects, however, are decidedly alkaline.
The several salts that may be formed by the acid anions with any given metal vary in solubility and acid or alkaline reaction according to the electronegative valence of the particular anion involved. A weak metal and a weakly alkaline anion in solution have very little attraction for each other and stay in solution. The same metal with a moderately alkaline anion may be precipitated out of solution under certain conditions. With a strong base, the metal may even be insoluble. Thus, calcium chloride is freely soluble, calcium sulfate is of limited solubility, and calcium carbonate is nearly insoluble. Sodium and potassium, because they are only slightly electropositive at best, are freely soluble not only with both chloride and sulfate ions but also with the carbonate ion.
Similarly, the pH of a solution is determined by the relative electrovalence of the several cations and anions dissolved together. Since soluble metals are all only weakly acidic and acid anions range from very weakly alkaline to extremely alkaline, their solution in water may be slightly acidic (calcium and the sulfate ion), neutral (sodium and chloride ions), or very alkaline (calcium and carbonate ions). Mild alkalinity is usually indicative of solutions containing more than one anion (calcium with the sulfate and carbonate ions, for instance).
The anions found in water are almost exclusively sulfates (SO4\--), chlorides (Cl-), and bicarbonates (HCO3-). The sulfate ion is weakly basic, and the chloride ion only slightly more so, whereas the unstable bicarbonate ion is a very strong buffer, because in its formation from the carbonate ion (CO3\--), it pulls a hydronium ion off a water molecule, freeing hydroxide ions into solution.
Far less common than the sulfates, chlorides, and bicarbonates are the weakly basic nitrate, borate, phosphate, and silicate ions; few of their salts are even soluble. In fact, only six salts commonly dissolve in ground water: calcium bicarbonate, magnesium bicarbonate, calcium sulfate, magnesium sulfate, sodium sulfate, and sodium chloride. Only three other salts are occasionally present in significant amounts: calcium chloride, magnesium chloride, and sodium bicarbonate. Potassium compounds are rarely present in any quantity.
Calcium and magnesium significantly affect the brewing process. When calcium and magnesium occur primarily with the bicarbonate ion, calcium precipitates out of solution during boiling, potentially robbing the yeast of a necessary element of its composition and causing high mash pHs that may react sluggishly to acidification. On the other hand, calcium in solution with the sulfate ion provides a very stable vehicle for the transmission of calcium into the ferment, and aids rather than retards mash acidulation.
The hardness of water expresses the calcium and magnesium in solution. The hardness of any water is determined by titration, as with EDTA, after addition of a dye, to an end point. It is expressed as "hardness as CaCO3," although it represents all of the calcium and magnesium ions in solution, arbitrarily combined with carbonate ions.
Bicarbonates, and to a lesser extent, carbonates, constitute most of the alkalinity of natural waters. Combined with calcium, they are expressed as the temporary or carbonate hardness of water, or that part of the hardness that will precipitate out of solution by boiling or with the addition of lime.
After boiling, calcium and magnesium ions remain in solution with noncarbonate ions. These are expressed as the permanent or noncarbonate hardness. In fact, some carbonate ions remain, bonded to water molecules; calcium carbonate is soluble to 20 ppm, and with magnesium in solution, magnesium carbonate may remain at up to 300 ppm.
The alkalinity of water measures the buffering capacity of dissolved anions, especially the bicarbonate (HCO3; the carbonate ion CO3 is only a significant factor in waters of pH 8.3 and above). By titrating the alkalinity, the temporary hardness can be assessed more readily than by boiling the water and can be used to precisely indicate treatment. An indicator dye, such as Bromcresol Green-Methyl Red, whose color in neutral and alkaline solutions is known, is added to a measured volume of water. The number of drops of a strong mineral-acid solution, such as .035 _N_ sulfuric acid, it takes to neutralize the water (overcome its alkalinity) is indicated by the color change of the dye, from blue-green to methyl-orange end point. When multiplied by a factor based upon the sample size, the number of drops it takes to reach end point expresses the bicarbonate alkalinity as parts per million of calcium carbonate. For more alkaline waters with a pH above 8.3, phenolphthalein is used as an indicator dye, and the carbonate alkalinity is titrated before the bicarbonate is measured.
Hardness actually measures calcium and magnesium in solution, and alkalinity measures all the alkaline ions. By expressing hardness and alkalinity in the same terms, as CaCO3 (calcium carbonate), the two values are readily compared; "as CaCO3" is the accepted standard because its cation is the primary mineral of hardness and its anion is the principal cause of alkalinity in most waters. This convention also simplifies water treatment.
Hardness and alkalinity nicely define the permanent and temporary hardness of water. _When the alkalinity_ _as CaCO_ 3 _exceeds the hardness, then the hardness is largely temporary. When the hardness value exceeds the alkalinity, the difference is indicative of permanent sulfate hardness._ Especially where hardness greatly exceeds alkalinity, the water is eminently suitable for brewing and responds well to acidulation during mashing.
Where alkalinity as CaCO3 is unknown, the hardness before and after boiling must be measured to define permanent and temporary hardness. This method is at least as satisfactory an indicator as is the alkalinity reading of a water analysis, but a great deal more difficult to make.
Temporary hardness is always strongly alkaline; permanent hardness is usually only slightly acidic, so only soft waters or waters where hardness well exceeds alkalinity yield a proper mash pH with pale malt.
Most water supplies are slightly alkaline, due to the buffering of any calcium and magnesium in solution by the strongly basic reaction of even a small amount of bicarbonate. At over 50 ppm alkalinity as CaCO3, water reacts sluggishly to acidulation in the mash and kettle. This water becomes weakly acidic upon precipitation of its carbonate salts. The carbonates can be sedimented out by boiling or the addition of slaked lime, or overcome by the addition or formation of organic acids, and to some extent by adding calcium or magnesium as sulfate or chloride salts to the mash or the mash liquor.
The larger of the two readings, hardness or alkalinity, can also indicate how much of the total dissolved solids are sodium, potassium, and chloride, which contribute to neither the hardness nor the alkalinity, by subtracting it from total dissolved solids.
Where calcium and magnesium measurements are not given in an analysis, dividing the hardness reading by 1.25 and by 5.0 roughly indicates the calcium and magnesium in solution (assuming a four-to-one Ca/Mg ratio).
## Molarity, Equivalence, and Normality
The chemist's tools molarity, normality, and equivalence, as described by Dr. George Fix in _Principles of Brewing Science_ (Brewers Publications, 1989), are of significance to brewers, especially for understanding liquor acidification.
Molarity is a method for quantitative analysis of a substance in solution. Molarity employs the mole (mol, gram-molecular weight, gmw, gram mole, combining weight) as its unit of measure. Molecular weight is the sum of the atomic weights of the elements that compose any given substance; a mole is the molecular weight in grams. A molar solution ( **M** ) of any given substance is equal to one mole in a liter of solution.
The atomic weights of the elements sulfur and oxygen are 32.066 and 15.9994. The sulphate molecule, SO4, is composed of one atom of sulfur and four atoms of oxygen. The molecular weight of the sulphate ion is 96.0636, because 32.066 + (4 x 15.9994) = 96.0636. A mole of the sulphate ion, then, is 96.0636 grams, and a molar solution of sulphate is equal to 96.0636 grams in one liter of solution.
A molar solution of sulfuric acid (H2SO4) is 98.08 grams, of phosphoric acid (H3PO4) 97.995 grams, citric acid (C6H8O7) 192.13 grams, and lactic acid (C3H6O3) 90.08 grams in one liter of solution.
Molarity is generally given as a decimal percentage; .01 **M** sulfuric acid means that the solution contains .01 moles of lactic acid per liter, or .9808 grams per liter.
The concentration in parts per million (milligrams/liter) of a molar solution can be derived by the formula ppm = molarity x mole x 1000. So for sulfuric acid, .01 **M** : .01 x 98.08 x 1000 = 9.808 ppm.
Equivalence measures the number of moles of hydrogen or hydroxyl ion that a substance can liberate. The moles and equivalency of some common ions are:
Ion | Mole weight | Equivalent Weight
---|---|---
H+ | 1.00794 | 1.00794
Ca++ | 40.078 | 20.039
Mg++ | 24.3050 | 12.1525
Na+ | 22.989768 | 22.989768
K+ | 39.0983 | 39.0983
SO4- | 96.0636 | 96.0636
CO3\-- | 60.0092 | 30.0046
HCO3- | 61.01714 | 61.01714
Cl- | 35.4527 | 35.4527
The strength of an acid is measured by its ability to release H+ ions, lowering pH. Sulfuric acid (H2SO4) disassociates when it is added to water, releasing two hydrogen ions into solution: 2H++ \+ SO4\--. It has an equivalency, then, of 2. In a solution containing the carbonate ion (CO3\--), the two hydrogen ions released by sulfuric acid exert a strong enough attraction on the unstable atoms of the carbonate ion to pull it apart:
H2SO4 \+ CaCO3 → H2CO3 \+ Ca++ \+ SO4\-- → H2O + CO2 \+ Ca++ \+ SO4\--
The alkaline carbonate ion is thus eliminated, because one mole of sulfuric acid neutralizes one mole of carbonate.
Phosphoric acid (H3PO4) is another mineral acid. Each molecule of phosphoric acid contains three hydrogen ions, but still only has an equivalence of 2, because it only partially disassociates in water and releases only two of its three hydrogen ions:
H3PO4 \+ CaCO3 → H2CO3 \+ CaHPO4, precipitated
Two organic acids commonly used by brewers are citric and DL-lactic acid. Citric acid (2C6H8O7) partially disassociates and releases three hydrogen ions in solution, giving an equivalence of 3. Lactic acid (2C3H6O3) has an equivalence of 1, because it releases only one hydrogen ion. Only two-thirds of a mole of citric acid is needed to neutralize a mole of carbonate, but two moles of lactic acid are needed to neutralize a mole of carbonate:
2C6H8O7 \+ 3CaCO3 → 3H2CO3 \+ 3Ca++ \+ 2C6H5O7\---
2C3H6O3 \+ CaCO3 → H2CO3 \+ Ca++ \+ 2C3H5O3-
Normality _(N)_ is how the strength of an acid is expressed. It may be given as a decimal or fraction: .02 _N_ or _N_ /50. One mole of a 1 _N_ compound releases one mole of hydrogen (H+) or hydroxyl (OH-) ion. A .02 _N_ acid solution releases .02 or one-fiftieth of a mole of hydrogen, and a 5 _N_ acid solution releases five moles of hydrogen. Normality is equal to the molarity of the acid times the equivalence of the acid. So the normality of one mole of the four acids are:
Lactic acid, with an equivalence of 1: 1 x 1 **M** = 1 _N_
Sulfuric acid, with an equivalence of 2: 2 x 1 **M** = 2 _N_
Phosphoric acid, with an equivalence of 2: 2 x 1 **M** =2 _N_
Citric acid, with an equivalence of 3: 3 x 1 **M** = 3 _N_
Reversing the equation, 1 _N_ solutions of these acids give:
Lactic acid, 1 _N_ : gram mole 90.08/equivalence 1= 90.08 grams C3H6O3/liter of solution
Phosphoric acid, 1 _N_ : gram mole 97.995/equivalence 2 = 48.998 grams H3PO4/liter of solution
Sulfuric acid, 1 _N_ : gram mole 98.08/equivalence 2 = 49.04 grams H2SO4/liter of solution
Citric acid, 1 _N_ : gram mole 192.13/equivalence 3 = 64.043 grams C6H8O7/liter of solution
These values allow for ease of calculations for alkalinity adjustments, since they represent milligrams per milliliter as well as grams per liter, and multiplied by 1,000 they give milligrams per liter, or ppm, the measurement by which alkalinity as CaCO3 is expressed. Alkalinity reductions require acid additions that are equivalent to the molarity of the alkaline cations, and the equivalence of acids is measured by normality.
## Mineral Ions Common in Water
### Cations — Earths
_Calcium_ (Ca++, atomic weight 40.08). Calcium is the principal mineral of hardness. It comes from the water's passage over limestone, dolomite, gypsum, or calcified gypsiferous shale. Calcium increases mash acidity and inverts malt phosphate to precipitated alkaline phosphate by the following reaction:
In appropriate amounts, calcium is advantageous to the brew. Calcium stimulates enzyme activity and improves protein digestion, stabilizes the alpha-amylase, helps gelatinize starch, and improves lauter runoff. It also extracts fine bittering principles of the hop and reduces wort color. A calcium precipitate formed with potassium phosphate improves hot-break flocculation. It is also an essential part of yeast-cell composition. Small amounts of calcium neutralize substances toxic to yeast, such as peptone and lecithin. It improves clarification during aging, as well as the stability and flavor of the finished beer.
Precipitation of Calcium Carbonate
Boiling
Adding Slaked Lime
In excess, however, calcium precipitation with organic phosphates interferes with runoff filtering and robs the wort of phosphate, a necessary yeast nutrient. Calcium levels are usually 5 to 200 ppm; its solubility is greatly affected by anions in solution with it.
_Magnesium_ (Mg++, atomic weight 24.32). Magnesium is the secondary mineral of hardness. It is essential as a cofactor for some enzymes, and as a yeast nutrient. In concentrations of 10 to 30 ppm, magnesium accentuates the beer's flavor, but it imparts an astringent bitterness when it is present in excess. At levels higher than 125 ppm it is cathartic and diuretic. Usually found at levels of 2 to 50 ppm, its solubility is less affected by carbonate anions in solution than is calcium.
### Cations — Metals
_Sodium_ (Na+, atomic weight 22.991). The sour, salty taste of sodium can accentuate beer's flavor when it is found in reasonable concentrations, but it is harsh and unpleasant in excess. It is poisonous to yeast, and brewers generally avoid water that contains sodium in excess of 50 ppm, especially where softness is characteristic of the beer flavor. Usually found at levels of 2 to 100 ppm, it is very soluble.
_Potassium_ (K+, atomic weight 39.1). Potassium imparts a salty taste. In excess of 10 ppm, it inhibits enzyme activity and acts as a laxative. It is difficult to measure and is usually grouped with sodium. Levels seldom exceed 20 ppm, although potassium is very soluble.
_Iron_ (Fe++, atomic weight 55.85, Fe+++). Common in ground water, iron gives an unpleasant, inky taste detectable at levels as low as .05 ppm. Above 1 ppm, iron weakens yeast and increases haze and oxidation of tannins. It blackens porcelain and spots fabrics at .02 ppm, causes white turbidity in water, and corrodes metal. Levels should be less than .3 ppm. Reduce iron content to .1 ppm by aerating and filtering the water through sand.
_Manganese_ (Mn++, atomic weight 54.94). Trace amounts of manganese are found in most ground and surface waters. It imparts an unpleasant taste and streaks porcelain at .05 ppm. The manganese level should be less than 2 ppm and optimally below .05 ppm. It can be reduced to .02 ppm by aeration.
_Ammonia_ (NH4+, atomic weight 18.04). Ammonia is a corrosive ion from microbial organic decomposition. Most volatile of the nitrates, it is reduced by oxidation to the corrosive alkaline gases NH3 and NH2. Always indicative of pollution, ammonia is never present in unpolluted water. Levels of ammonia are normally .00 to .03 ppm and should never exceed .05 ppm.
_Copper_ (Cu++, atomic weight 63.54). Elevated levels of copper cause yeast mutation and haze formation. Copper in a water supply is evidenced by blue-green stains on porcelain. Levels of copper should be less than 1 ppm.
_Zinc_ (Zn++, atomic weight 65.38). Zinc is a yeast nutrient when it is found at .1 to .2 ppm, but toxic to yeast and inhibiting to enzymes above 1 ppm.
### Anions
_Carbonate_ (CO3\--, atomic weight 60.0092). Carbonate is a strongly alkaline buffer formed by the reaction of atmospheric carbon dioxide with hydroxides of alkaline-earth and alkali metals. Carbonates go into solution as hydrogen carbonates (HCO3-, "bicarbonates"), which are strong buffers. Bicarbonates form by the reaction of a carbonate ion with a molecule each of carbon dioxide and water.
Bicarbonate resists increases in the mash acidity by neutralizing acids as they are formed. It also hinders gelatinization of starch by alpha-amylase, impedes trub flocculation during the cold break, and increases risk of contamination in the ferment. It contributes a harsh, bitter flavor that is overwhelming in delicate lagers. Carbonate in excess of 200 ppm is tolerable only when dark-roasted malts are employed to buffer its excessive alkalinity. Carbonates in the brewing liquor should be less than 50 ppm if the mash is from only pale malts and no liquor acidulation is employed. Where carbonates exceed 50 ppm, water treatment is generally in order.
_Sulfate_ (SO4\--, atomic weight 96.0576). Sulfate is weakly basic, and its alkalinity is overcome by most acids. It is fairly soluble. It gives beer a dry, fuller flavor, although the taste can be objectionably sharp. With sodium and magnesium it is cathartic. Above 500 ppm it is strongly bitter, and levels are generally kept at less than 150 ppm unless the beer is very highly hopped. With intensely bitter beers, sulfate at 150 to 350 ppm gives a cleaner, more piquant bitterness.
_Chloride_ (Cl-, atomic weight 35.453). Chloride is very weakly basic, and readily neutralized. It accentuates bitterness, but also increases mellowness; it increases the stability of any solution and improves clarity. The "salt" taste of chloride generally enhances beer flavor and palate fullness, but the salt flavor is reduced by the presence of calcium and magnesium. Usually found at levels of 1 to 100 ppm, chloride levels in the brewing liquor may be as high as 250 ppm for British mild ales.
_Silica_ (SiO2, atomic weight 60.0843), silicon dioxide. Originating from sand or quartz, silica is insoluble. As a colloid, it interferes with the filtering of the mash. Under certain conditions, silica forms silicate (HSiO3-), which causes hazes, precipitating out of the boil as scale with calcium and magnesium. Silica levels are usually less than 10 ppm but may be as high as 60 ppm.
_Nitrate_ (NO3-, atomic weight 62.0049). Nitrate is the most highly oxidized naturally occurring form of nitrogen. It may be from geological strata or originate from contact with sewage or oxidized organic matter. Above 10 ppm, it is indicative of pollution by sewage. It is alkaline, and during fermentation in the presence of chlorides, it forms nitrites, which are more strongly alkaline yet.
_Nitrite_ (NO2-, atomic weight 46.0055). Strongly basic, nitrite originates from nitrates during decomposition of organic matter by coliform bacteria. It rarely exceeds .1 ppm, and is always indicative of pollution. Nitrite is toxic to yeast in minute concentrations; as little as .1 ppm may retard or terminate yeast growth.
Table 6: Water Composition Indicators
Hardness and Mineral Content of Water
---
Nature | Hardness as CaCO3, ppm
Very soft | Less than 50
Soft | 50–100
Slightly hard | 100–150
Moderately hard | 150–250
Hard | 250–350
Very hard | 350 and above
Multiplying the ions below by the corresponding factors yields hardness as CaCO3.
Ca–2.497 Mg–4.116 Fe–1.792 Mn–1.822 Zn–1.531
Total Dissolved Solids/Specific Conductivity
---
|
Total Dissolved Solids | Specific Conductivity
Water low in ionized matter | Below 50 ppm | Below 90 micromhos/cm
Range of average water supplies | 30–275 ppm | 50–500 micromhos/cm
Very highly mineralized water | Above 275 ppm | Above 500 micromhos/cm
pH and Hardness as a Treatment Indicator
---
pH | Hardness, ppm | Character
Below 7 | 0–100 | Soft and acidic, little bicarbonate alkalinity to overcome. May need to treat with acid to correct mash pH for light-colored worts. Add calcium salts to correct deficiency, and as appropriate to style.
Above 7 | 0–100 | Soft, but largely bicarbonate alkaline hardness, treat with acid to correct mash pH; add calcium salts as appropriate to style.
Up to 7.6 | 100–150 | Slightly hard, bicarbonate alkalinity is easily overcome by acid liquor treatment.
7.0–7.2 | 150 and up | Moderately hard, predominately sulphate hardness. Excellent water for all but Bohemian Pilseners.
7.2 and up | 150–250 | Moderately hard, largely bicarbonate hardness. For Dortmund lagers and Burton ales, treat with gypsum; for most other beer styles, precipitate carbonates or adjust pH with acid.
7.2 and up | 250–1000 | Hard to very hard water, predominately bicarbonates. Appropriate for Dortmund/Vienna lagers and British ales. For other styles, precipitate carbonates and/or adjust pH with acids.
pH as an Alkalinity and Treatment Indicator
---
pH | Alkalinity, as CaCO3
|
% HCO3 | % CO3 | % H2CO3
10 | 68 | 32 | 0
9 | 95 | 5 | 0
8 | 97 | 0 | 3
7 | 81 | 0 | 19
6 | 30 | 0 | 70
5 | 4 | 0 | 96
From table A–1, _Principles of Brewing Science,_ by George Fix.
### Other
_Coliform bacteria._ Measures the amount of any fecal bacteria, such as _Escherichia coli, Streptococcus faecalis,_ pathogenic _Salmonella_ strains, _Shigella dysenteriae,_ and _Vibrio cholerae._ The U.S.P.H.S. standard for drinking water is that there should be less than 2.2 colonies per 100 milliliters. For brewing water, it is recommended that this be zero.
Where bacterial population is not given in an analysis, nitrate, nitrite, and ammonia values suffice to indicate water pollution.
## Water Treatment
Brewing water sources should be chosen first for their purity and second for their mineral composition. In fact, treatment is only necessary when the mineral distribution of any water is unsatisfactory, or when accentuation of bitterness or saltiness is desired.
The most common correction of brewing water is the reduction of bicarbonate to yield a satisfactory mash acidity/pH. The bicarbonate is alkaline, and if it is reduced, so is the alkalinity.
Where liquor of less than 50 ppm of alkalinity is mashed with pale malt, proper mash acidity is usually realized without liquor treatment, because phosphates dissolved from the malt react with calcium bicarbonate, precipitating calcium phosphate and releasing CO2. With more alkaline water, an excess of carbonate remains in solution, and the mash pH will be too high. Decomposition of the carbonates may be accomplished in several ways.
The most basic is by bringing water to a boil and aerating it thoroughly to decompose bicarbonates to carbonates, which are precipitated as calcium or magnesium carbonate salts, and to decompose carbonic acid to CO2, which is driven off. After a reasonable rest to allow the carbonates to sediment the water should be decanted off the sediment, so that gradual dissolution of atmospheric CO2 back into the water does not result in bicarbonates re-forming from the precipitate.
The use of naturally acidic toasted malt, or a portion of dark-roasted malt, is a time-honored manner of water treatment. The acidity released by intensive kilning can overcome the alkalinity of even moderately alkaline waters.
Where only pale malts are being mashed with soft to moderately alkaline waters (less than 250 ppm alkalinity or 150 ppm HCO3), proper mash acidity is most often achieved by the formation or addition of mild acid. Mixing a portion of sourmalt or lactic-acid mash into the main mash reduces alkalinity and contributes flavor nuances that help round out a beer's flavor. Sourmalt is made by allowing limited lactic-acid bacterial activity prior to the malt's being kilned. In the brew house, formation of lactic acid may be accomplished by _Lactobacillus delbruckii_ activity in a partial mash, held closely covered at 95 to 120 degrees F (35 to 50 degrees C) for forty-eight to seventy-two hours, until its pH drops below 4.
Where acids (most commonly lactic, phosphoric, sulphuric, or citric) are used to reduce alkalinity, carbonates are decomposed with the formation of carbonic acid and lactate, phosphate, sulphate, or citrate anions, but the reactions are to some extent reversible. Moreover, if the water is more than moderately alkaline, the excessive amounts of acid required give the beer a noticeable sourness and characteristic taste. Generally, DL-lactic acid is preferred by brewers, but orthophosphoric, monohydrate citric, and sulfuric acids are also commonly used.
Approximate carbonate reduction can be made by gradual acid addition, checking the pH of the liquor after each dose, until it drops below 7. For moderately modified pale malt that won't undergo acid or protein rests, the pH reduction of the liquor may need to be as low as pH 6.
Carbonate and alkalinity reduction of a water can be more accurately made, where the parts per million of alkalinity as CaCO3 is known, by calculating the treatment beforehand. Some difficulty arises because various dilutions of the acids are offered, and may be expressed as percentage solutions or percentage-of-normality solutions _(N)._ Normality is discussed below. The following quantities of the commonly used acids are equivalent to 1 _N_ :
Lactic acid, 1 _N_ = 90.08 milligrams C3H6O3 per milliliter of solution
Phosphoric acid, 1 _N_ = 48.998 milligrams H3PO4 per milliliter of solution
Sulfuric acid, 1 _N_ = 49.04 milligrams H2SO4 per milliliter of solution
Citric acid, 1 _N_ = 64.043 milligrams C6H8O7 per milliliter of solution
And as percentage solutions, common dilutions give:
Lactic acid 85 to 90% w/w: 1,020 milligrams per milliliter of solution
Phosphoric acid 85 to 88% w/w: 1,445 milligrams per milliliter of solution
Sulfuric acid 95 to 98% w/w: 1,766 milligrams per milliliter of solution
Citric acid is generally available as the monohydrate, in granular or powder form, and so can be weighed out, each milligram of the monohydrate giving .9143 milligrams of citric acid.
These values allow ease of calculations for alkalinity adjustments. Given a water with alkalinity as CaCO3 of 220 ppm and pH 7, the brewer wants to reduce the alkalinity to below 50 ppm; to, say, 35 ppm: 220 - 35 = 185 ppm. 185 ppm of alkalinity should be removed. Reference to table 6 shows that at pH 7, 81 percent of that alkalinity is bicarbonates, and the remainder is harmless carbonic acid. 185 x .81 = 150 ppm of alkalinity as CaCO3 (at this pH, actually bicarbonate, HCO3) needs to be disassociated. The acid treatment required to accomplish this is predicted by calculating the total alkalinity to be removed; that is, the ppm of alkalinity as CaCO3 times the total volume of liquor.
Where the brewer is using liters as a measure, this is simply done by multiplying the alkalinity as CaCO3 times the number of liters of liquor needed for the brew. Where the brewer is working with gallons, the conversion factor of liters in a gallon (3.7854) needs to be included in the formula:
Alkalinity as CaCO3 x 3.7854 x number of gallons of liquor
Divided by the milligrams per milliliter that the particular acid solution on hand bears, the formula gives the milliliters of acid needed to disassociate the carbonate/bicarbonate alkalinity. For example, for 185 ppm of alkalinity to be disassociated, and 7.5 gallons of water to be treated:
185 x 3.7854 x 7.5 = 5,250 ppm of alkalinity as CaCO3 to be disassociated.
Using 85 percent lactic acid, which bears 1,020 milligrams of lactic acid per milliliter:
5,250/1,020 = 5.2 milliliters of 85 to 90 percent lactic acid will reduce the alkalinity as CaCO3 of 7.5 gallons of water by approximately 185 ppm.
## Mineral Salt Treatment
Ion-exchange water softeners should never be used to reduce hardness. They do not remove the carbonate ion from solution, but precipitate calcium and magnesium by exchanging them for more soluble sodium ions, correspondingly increasing the sodium concentration.
Table 7
Estimated Characters of the Classic Brewing Waters
---
|
Ca | Mg | K | Na | SO4 | HCO3 | Cl | Hardness | Total Dissolved Solids
Pilsen | 7 | 2
| |
2 | 5 | 15 | 5 | 30 | 35
Munich | 75 | 18
| |
2 | 10 | 150 | 2 | 250 | 275
Vienna | 200 | 60
| |
8 | 125 | 120 | 12 | 750 | 850
Dortmund | 225 | 40
| |
60 | 120 | 180 | 60 | 750 | 1000
London | 90 | 5
| |
15 | 40 | 125 | 20 | 235 | 300
Dublin | 120 | 5
| |
12 | 55 | 125 | 20 | 300 | 350
Yorkshire | 100 | 15
| |
25 | 65 | 150 | 30 | 275 | 400
Edinburgh | 120 | 25
| |
55 | 140 | 225 | 65 | 350 | 650
Burton | 275 | 40
| |
25 | 450 | 260 | 35 | 875 | 1100
Iron, manganese, and colloids that cause hazes are best removed by aeration, followed by filtration or sedimentation.
Mineral salts may be added to the brewing water when additional hardness or other mineral character is desired, or to precipitate carbonates. All salts should be carefully weighed (on a gram scale for small batches) before they are added to the brewing water. Mineral salts cannot be accurately dispensed by volume. One level teaspoon of finely powdered gypsum might weigh 3.65 grams. Tightly packed, it weighs 5 grams. A teaspoon of more crystalline magnesium sulfate weighs 4.55 grams, finely granular potassium chloride 5.05 grams, and sodium chloride 6.45 grams. For accuracy, salts need to be measured by weight.
It is advisable to first mix salts into a small quantity of boiling water before introducing them to the brewing water. Salts should never be added directly to the mash because uniform dispersal is unlikely.
Table 8
Mineral Salt Treatment
One gram of a freely soluble mineral salt in one U.S. gallon of water at 68 degrees F (20 degrees C) can be expected to increase the total dissolved solids by 264.2 ppm. The amount of any ion in the salt being added may be estimated from the salt composition percentages given; for instance, one gram of gypsum (calcium sulfate dihydrate) yields 264.2 x .2328 = 61.5 ppm of calcium and 264.2 x .5579 = 147.4 ppm of the sulfate ion. Brewing water profiles in table 7 and the analysis of your water supply may be used to guide salt additions.
Calcium Sulfate (Gypsum), CaSO4⋅2H2O
Ca 23.28%, SO4 55.79%, H2O 20.93%. Mol. wt. 172.172 (CaSO4 136.142). Increases calcium content and lowers pH. Improves the quality of hop bitterness, gives drier and fuller flavor. Soluble to 2,650 ppm in cold water, to 2,000 ppm upon heating. Apparently most beneficial at 150 to 350 ppm. In excess, precipitates with calcium phosphate in the kettle. One gram in one gallon yields 61.5 ppm Ca, 147.4 ppm SO4. The anhydrous salt CaSO4 (plaster of Paris), mol. wt. 136.14, gives 77.8 ppm Ca, 186.5 ppm SO4.
Magnesium Sulfate (Epsom Salts), MgSO4⋅7H2O
Mg 14%, SO4 55%, H2O 31%. Mol. wt. 246.475 (MgSO4 120.369). Increases magnesium content. Freely soluble. Most satisfactory at 150 to 300 ppm; reduce when adding with calcium sulfate. Very bitter in excess. Generally avoided in pale lagers. One gram in one gallon gives 37 ppm Mg, 145.3 ppm SO4.
Calcium Hydroxide (Slaked Lime, Hydrated Lime), Ca(OH)2
Mol. wt. 74.093. Reacts with calcium bicarbonate, causing both to precipitate as carbonates [Ca(OH)2+Ca(HCO3)2 → 2CaCO3+2H2O]. Also precipitates magnesium bicarbonate. Its calcium ion may replace any magnesium that is in solution with the chloride ion. Addition of slaked lime should not exceed the ppm of alkalinity as CaCO3 given in the water analysis. One gram in one gallon yields 264.2 ppm of the salt.
Calcium Carbonate (Precipitated Chalk), CaCO3
Ca 40.04%, CO3 59.96%. Mol. wt. 100.087. Strongly buffers mash acidity. Partially precipitates in the wort boil. One gram in one gallon gives 106 ppm Ca.
Sodium Chloride (Common Table Salt), NaCl
Na 39.34%, Cl 60.66%. Mol. wt. 58.443. Accentuates bitterness and enhances flavor and fullness of beer. Also promotes diastatic enzyme activity and the release of acid malt phosphates. Usually less than .75 grams per U.S. gallon (198 ppm) as treatment, or so that neither sodium nor chloride contents exceed 100 ppm (250 ppm of each for very dark and full beers). Objectionable in excess. Inhibits or even kills yeast over 850 ppm. One gram in one gallon equals 104 ppm Na, 160.25 Cl.
Potassium Chloride, KCl
K 52.44%, Cl 47.56%. Mol. wt. 74.551. A substitute for part of sodium chloride treatment. In excess of 150 ppm inhibits enzyme activity. One gram in one gallon yields 138.6 ppm K, 125.6 ppm Cl.
Calcium Chloride, CaCl2
CaCl2⋅2H2O⋅Ca 27.26%, Cl 48.23%. Mol. wt. 147.014 (CaCl2 110.983). Adjusts calcium and adds saltiness. Commonly used. One gram in one gallon gives 72 ppm calcium, 127.4 ppm chloride. The anhydrous salt CaCl2, mol. wt. 110.99, gives 161.4 ppm Ca, 168.8 ppm Cl2.
Sodium Sulfate, NaSO4
NaSO4⋅Na 14.27%, SO4 29.89%. Mol. wt. 119.053. Rarely used. One gram in one gallon gives 37.7 ppm Na, 79 ppm SO4.
Potassium Metabisulfite, K2S2O5
K 35.2%, S 28.8%, O 36%. Mol. wt. 222.33. Removes chloride from solution; in excess antifermentative. Not commonly used, and not usually more than 1 to 2 ppm added.
# CHAPTER 4
# Hops
Hops are the conelike female "flowers," _strobiles,_ of the vining _Humulus lupulus._ The strobiles are formed by a cluster of petallike, yellowish-green bracts and bracteoles emerging from a central stem. Each bract bears many tiny glandular sacs (trichomes) of _lupulin_ at its base. Lupulin accounts for as much as 15 percent of the weight of the hop. The yellow lupulin is composed of essential oils, resins/bittering principles, polyphenols, nitrogen, sugars, pectin, lipids, and wax.
The resins may be classed alpha acids, beta acids, and gamma resins. The alpha and beta acids are "soft," whereas the gamma resin is hard, and contributes nothing to the brewing.
The alpha resin group is the most important hop fraction, and the most stable. Alpha acids have no aroma but are intensely bitter. They are responsible for the hops' bacteriostatic contribution to the brew. Alpha acids (humulone, cohumulone, and adhumulone) may be are _isomerized_ during boiling (their atoms are rearranged) to somewhat soluble and even more bitter iso-alpha acids.
In contrast, the beta acids (lupulone, colupulone, and adlupulone) are far less stable than alpha acids, are extremely subject to oxidation, and are only slightly soluble. They become even less soluble as the hot wort cools, and are deposited in the trub as an amorphous yellow precipitate. Very little of their antibiotic, aromatic, and bittering properties are carried into finished beer unless they are oxidized, in which case they are extremely bitter, but the hops may taint the beer with an unpleasant spoiled-vegetable taste.
Varieties of hops that are high in alpha acids are generally preferred for bittering beer, because less hops are thereby required to achieve target bitterness levels. Because the amount of alpha acid that is isomerized by boiling is in large part time-dependent, hops for bittering are generally boiled in wort for thirty to ninety minutes. Boiling, however, drives off hop flavor and aromatics. Where hop flavor is desired, some isomerization efficiency is given up, and "flavor" hops are only boiled for five to thirty minutes.
Hop aromatics are even more fugitive; although some aromatics survive boiling for as long as twenty to thirty minutes, the full, fresh aroma of hops is only captured by "dry-hopping," or adding whole hops or their extract to the conditioning beer after fermentation. Hop-oil esters and floral ketones are generally evaporated more quickly than the terpene and sequiterpene oxides and alcohols that give beer a spicy flavor/aroma and greater mouthfeel.
The other major contribution of hops to the finished beer is the _tannins_ they dissolve into the boiling wort. Tannins are complex, generally oxidized polyphenol polymers. When proteins and proteoses come into contact with the astringent-tasting amorphous flakes, they adhere to them and, by virtue of their increased mass, are precipitated out of solution. Because there are generally fewer hop tannins than proteins, they do not usually carry over into the finished beer and significantly affect flavor.
Although the bitter flavor and tangy aroma of the hop is now considered an essential complement to the malt sweetness of beer, hops were first employed in brewing as a preservative. Beer made with hops stored better than beer brewed without them, although the reason why was not understood. Only toward the end of the nineteenth century did brewers discover that hops prevented the growth of many waterborne and airborne bacteria. The role that hop polyphenols play in precipitating unstable proteins in the kettle, thereby reducing the potential for chill haze in the beer, also began to be understood.
## On the Vine
Hops are grown on perennial vines that trail along wires strung on trellises fifteen to twenty-five feet above ground level. Each year new stems twine clockwise around the wire strands supporting the strobiles. When the strobiles are mature, the plant is cut loose from the trellis, and the clusters of hop cones are stripped from the stems of the plant.
Hops that are of brewing quality must be harvested during the five to ten days of their prime. Immature hops are very green and have a haylike aroma; overripe cones have rusty-colored petals, tend to shatter easily, and have a harsh smell.
The cones, which contain 70 to 80 percent moisture at harvesting, are dried to 8 to 10 percent moisture (usually at 140 to 150 degrees F [60 to 65 degrees C] but sometimes below 130 degrees F [55 degrees C] when a very strong aroma is characteristic) over a period of eight to twelve hours. They are sometimes fumed with sulfur to lighten their color and give them a softer, silkier, more appealing feel. Dried hops are cured in cooling bins for five to ten days to equalize their moisture content, improve their aroma and appearance, and make the cones more resilient against shattering. They are compressed and baled, each bale measuring approximately twenty by thirty by fifty-four inches and weighing between 185 and 205 pounds. They are traditionally stitched in burlap hopsack, but modern foil-mylar laminate vacuum-packaging under nitrogen or CO2 atmosphere is gradually replacing burlap.
Hops on the Vine
The harvest is purchased by hop merchants, who hold the hops in cold storage until they are sold to the brewer. If the hops have been properly dried and baled, protected from direct sunlight, and stored at low humidity and low temperatures (33 degrees F [1 degree C] is ideal), little destructive oxidation occurs; hops that have a beta-acid content equal to their alpha acids store better than hops with a low percentage of beta acids. Under ideal conditions hops will keep for up to two years. Under adverse storage conditions, however, the essential oils are driven off, and many of the alpha-acid resins are oxidized to uncharacterized bitter substances or to useless hard resins. Such hops may have an "off" cheesy, soapy, or other disagreeable aroma and may be yellow or brown. At this point, their bittering strength has been greatly diminished and their flavor contribution to beer is abnormal.
## In the Brewery
Brewing hops should be whole cones of a light yellowish-green color, not mottled or spotted, roundish or slightly elongated in shape, and of less than 6 percent stem and leaf content. They should have a pleasant aroma; it is this bouquet that indicates the condition of the essential oils. The hop cones should be silky, glossy, and springy to the touch. Small hops tend to be of finer quality than large hops.
Two or three small cones should be rubbed in the palm of the hand and sniffed to assess the aroma. The lupulin quality and content can be judged by the stickiness left on the hand. Several cones should be broken lengthwise and the quantity and color of the lupulin assessed. It should be lemon colored and plentiful. Old, deteriorated hops have powdery, light-brown lupulin and range in color from green or greenish-yellow to yellow or brownish-green. Usually the discoloration is obvious. Old, dry, and powdery hop cones should be avoided, as the alpha-acid content will be considerably reduced. The deterioration of old or mishandled bales may account for a 50 percent or greater loss of alpha acids. This is also true in the case of lots containing an excessive number of broken cones. Improved oxygen-barrier packaging is becoming an important factor to brewers, since it can cut alpha-acid losses to a fraction of what can be expected with burlap-baled hops.
An alpha-acid analysis is usually made of samples taken from each hop bale. Alpha acidity is given as a percentage of the sample, by weight. This is stated by the dealer and is used by the brewer to adjust bittering-hop rates.
Alpha acids go into solution only after boiling has isomerized them to iso-alpha acids. Their bittering contribution is dependent upon isomerization efficiency in the kettle and the quantity and alpha-acid content of the hops used. Several methods for the quantitative analysis of iso-alpha acids make it possible to estimate the hop bitterness in beer; the internationally agreed upon standard is bitterness units (IBU). One IBU equals .0001335 of an ounce (avoirdupois) of iso-alpha acid per gallon of solution, or one milligram per liter.
Where stated in a beer profile, bitterness units are used to target bittering-hop rates, adjusting the amount to reflect the widely varying alpha-acid content of hops from lot to lot and season to season.
Dave Line ( _The Big Book of Brewing_ ) devised the alpha acid unit (AAU) to simplify these adjustments; his method was adopted by the American Homebrewers Association as the homebrew bitterness unit, or HBU. One AAU/HBU equals one ounce of a l percent alpha-acid hop. Using this system, two ounces of a 5 percent alpa-acid hop gives 10 HBU, and so on.
Assuming 30 percent isomerization/utilization of the alpha resins by a 90- to 120-minute rolling boil, each HBU will contribute 22.472 IBUs to a gallon of wort, or 85 IBUs per liter. Where utilization is high, dividing the IBU given for any beer by 22.472 can approximate the HBUs required per gallon of wort. Likewise, where bitterness units aren't given in a recipe, they can be roughly figured by multiplying the AAUs given by 22.472 (per gallon of wort). More accurate predictions of bitterness can be made using table 18. In any case, bitterness units can only be accurately matched if fresh, properly stored hops are used; with oxidized hops, alpha acidity is diminished and the proper hop rate becomes guesswork.
Hop Cross-section
Table 9: Hops
Color Range
---
Low-quality hops | High-quality hops | Deteriorated hops
Dark green | Yellowish-green | Yellow
Olive green | Greenish-yellow | Brownish-yellow
Mottled brown-green
| |
Brown
| |
Brownish-green
Cone Size
---
Large | Medium | Small
2 ¼–3" long | 1 ¼–2" long | ¾–1" long
Analysis
---
MC | 8–13%
Resins | 7–20%
Alpha acid | 4–15%
Cohumulone % | 20–40%
Alpha:beta ratio | .8–3.5
Essential oils | .2–3%
Tannins | 2–5%
Nitrogen | 2–4%
Fats and waxes | 2–5%
Hop storage index | 50–85%
Varietal and lot analyses of hops include several commonly quoted parameters; only the percentage of alpha acid, the alpha:beta ratio, the percentage of cohumulone, and the percentage of total oils seem to be of definite significance. Other indicators are much more subjective; the hop storage index (percent alpha acid after six months storage at 68 degrees F [20 degrees C]), for instance, is a guideline only, since merchants and brewers store hops under varying temperature and packaging conditions. Percentages of the significant hydrocarbon fractions, taken into consideration with the percentage of total oils, may to some extent indicate the aroma that a hop will give, but the compounds actually responsible for the flavors and aromatics that brewers prize have not been defined. Furthermore, although it is known that oxygenated hydrocarbons, especially terpenes and sesquiterpenes, are less volatile and contribute a major part of hop flavor and aroma to wort and beer, synergic relationships between them and their esters, alcohols, and acids are not understood. Consequently, aroma and flavor characteristics cannot be defined from laboratory data.
The ratio of alpha to beta acid and the percentage of cohumulone do seem to have some significance regarding the fineness of wort bitterness, but there are exceptions to the brewer's rule that the best hops have a high beta-acid ratio and a low percentage of cohumulone.
Hop bitterness is accentuated by magnesium, carbonate, and chloride ions, and hop rates must generally be reduced as these increase; historically, brewers have employed carbonate and chloride waters only for malty, low-hopped beers. Increasing amounts of sulfate, on the other hand, give a cleaner hop flavor. Well-hopped beers brewed with gypsiferous liquor commonly exhibit a finer, less coarse bitterness than is obtained with other liquor profiles.
Hop pellets have gained in popularity with brewers in recent years because they are less susceptible to oxidation during storage, especially under adverse conditions. Pellets stored under the same conditions as whole hops will lose only about one-third the alpha acids that whole hops will lose. They are compressed from fresh hops and foil-packaged in an oxygen-free environment. Because pellets are almost invariably less deteriorated than whole hops, utilization is usually 3 to 10 percent better (5 to 35 percent; as opposed to 3 to 30 percent for whole hops).
Table 10
Part A: Hop Varieties
---
Hop Varieties | % Alpha Acid | % Beta Acid | Cohumulone, % of AA | Total Oils mL/100g | M | H | C | F | %AA after 6 mo. @ 20°C | Origin
% of each:
Czech Saaz | 2.5–4.5 | 2.5–4.0 | 22–28 | .4–.7 | 23 | 43 | 11 | 13 | 50
|
Polish Lublin | 3.0–4.5 | 2.5–3.5 | 25–30 | .7–1.2 | 30 | 38 | 10 | 11 | 50
|
Spalt | 4.0–5.5 | 4.0–5.5 | 23–28 | .5–1.1 | 20 | 22 | 13 | 13 | 55
|
U.S. Spalt | 3.0–6.0 | 3.0–5.0 | 20–25 | .5–1.0 | 45 | 15 | 5 | 13 | 50
|
Tettnang | 3.5–5.5 | 3.5–5.0 | 23–29 | .6–1.1 | 23 | 23 | 8 | 14 | 60
|
U.S. Tettnang | 3.0–5.0 | 2.5–4.5 | 20–25 | .4–.8 | 41 | 21 | 7 | 7 | 60
|
H Hersbruck | 2.0–5.5 | 3.0–5.5 | 19–25 | .7–1.3 | 20 | 20 | 10 | <1 | 60
|
U.S. Hersbruck | 3.5–5.5 | 5.5–7.0 | 20–30 | .6–1.2 | 45 | 25 | 8 | <1 | 50
|
H Hallertau | 2.5–5.5 | 2.5–5.5 | 18–24 | .6–1.2 | 20 | 33 | 12 | <1 | 55
|
U.S. Hallertau | 3.0–5.5 | 3.0–5.5 | 18–24 | .6–1.0 | 40 | 34 | 11 | <1 | 55
|
U.S. Perle | 5.5–9.5 | 3.5–5.0 | 27–32 | .7–.9 | 50 | 31 | 11 | <1 | 85 | NBrwr+
Mt Hood | 3.0–6.5 | 3.0–5.5 | 24–30 | .6–1.2 | 38 | 38 | 11 | <1 | 50 | Hal+
Liberty | 3.0–5.5 | 3.0–4.5 | 24–30 | .6–1.2 | 38 | 38 | 11 | <1 | 45 | Mtlfh+
Crystal | 2.0–4.5 | 4.5–6.5 | 20–26 | 1.0–1.5 | 52 | 21 | 6 | <1 | 50 | Hal+
Cascade | 4.5–8.0 | 4.0–8.0 | 30–40 | .8–1.5 | 53 | 13 | 4 | 6 | 50 | Fug+
Bav N Brewer | 6.0–10. | 3.0–5.0 | 28–33 | 1.6–2.1 | 33 | 28 | 9 | <1 | 75 | BGold+
U.S. N Brewer | 6.5–10. | 1.5–5.0 | 20–30 | 1.5–2.0 | 55 | 25 | 8 | <1 | 80
|
Nugget | 12.0–16. | 4.0–8.0 | 24–30 | 1.7–2.3 | 55 | 18 | 9 | <1 | 75 | BGold+
Eroica | 9.5–14. | 2.5–5.0 | 36–42 | .8–1.3 | 60 | 1 | 10 | <1 | 60 | BGold+
Centennial | 8.5–12. | 2.5–5.0 | 29–30 | 1.5–2.3 | 50 | 14 | 7 | <1 | 65 | BGold+
Galena | 12.0–15. | 7.0–9.5 | 38–42 | .9–1.2 | 58 | 13 | 4 | <1 | 65 | BGold+
Brewers Gold | 7.0–10. | 3.0–4.5 | 40–45 | 1.8–2.5 | 63 | 15 | 8 | <1 | 55
|
Chinook | 10.0–14 | 2.5–4.0 | 29–34 | 1.5–2.5 | 38 | 23 | 10 | <1 | 70 | Fugl+
Styrian Gldng | 4.0–6.0 | 2.0–3.0 | 26–30 | .5–1.3 | 30 | 36 | 10 | 3 | 65 | Fugl+
Willamette | 4.0–7.0 | 3.0–4.5 | 26–35 | 1.0–1.5 | 50 | 25 | 8 | 6 | 65 | Fugl+
U.S. Fuggle | 4.0–6.0 | 1.5–3.0 | 25–32 | .7–1.2 | 45 | 23 | 8 | 5 | 65
|
UK Fuggle | 4.0–5.5 | 2.0–3.0 | 25–30 | .7–1.1 | 26 | 37 | 12 | 9 | 60
|
BC Goldings | 4.0–6.5 | 1.5–2.5 | 22–28 | .8–1.1 | 26 | 41 | 13 | <1 | 60
|
UK Goldings | 4.5–6.5 | 2.5–3.5 | 22–32 | .8–1.0 | 24 | 45 | 14 | <1 | 55 | Kent+
Pride Rngwd | 6.0–10. | 3.0–6.0 | 33–39 | 1.0–2.0 | 38 | 6 | 8 | <1 | 55
|
Cluster | 5.5–10. | 4.5–8.0 | 36–42 | .4–.8 | 50 | 17 | 7 | <1 | 85 | BGold+
UK Challenger | 6.5–8.5 | 3.0–4.5 | 22–28 | 1.0–1.5 | 31 | 29 | 9 | 2 | 55 | Chal+
UK Northdown | 7.0–9.0 | 4.5–7.0 | 28–32 | 1.2–2.2 | 25 | 43 | 15 | <1 | 60
|
Chart compiled from references provided by Morris Hanbury and HopUnion.
Notes: % alpha acid and % total oils vary widely year to year.
M=Myrcene, H=Humulene, C=Caryophyllene, F=Farnesene. These are significant hop oil hydrocarbons; their respective amounts help define aroma characteristics.
Part B: Hop Producing Districts
---
Country of Origin | Hop Type | Grown
Czech Republic | Saaz/Zatec Red | Zatec, Auscha, Raudnitz, Dauba
Germany | Hallertauer/Mittlefrueh | Hallertau, Baden
Tettnanger | Wurtenburg
Spalt | Spalt
Hersbrucker/Gebirg | Hersbruck
Yugoslavia | Savinja/Styrian | Wojwodina, Slovenia
Goldings
|
Belgium | Hallertau/Saaz | Alost, Poperinghe
United States | Various | Yakima Valley (Washington)
Willamette Valley (Oregon)
Boise and Snake River
Valleys (Idaho)
Australia | Pride of Ringwood | Victoria, Tasmania
The most highly prized hops in the world are the mild southern-English and central-European varieties. Although these and similar types are being more widely cultivated in the western United States, they are not frequently used by large domestic commercial breweries. New disease-resistant, high-alpha-acid varieties are emerging that also have desirable aromatic qualities, giving better kettle utilization and economy without forsaking fine hop character, as has been the case with high-alpha-acid-percentage strains previously developed for economical use.
# CHAPTER 5
# Yeast
Yeast are nonphotosynthetic, relatively sophisticated, living, unicellular fungi, considerably larger than bacteria. Brewers' yeast are of the genus _Saccharomyces._ In an aciduric aqueous solution, they absorb dissolved vitamins, minerals, and simple nitrogenous matter (amino acids and very simple peptides) through their hemicellulose cell membranes. Then they employ a structured series of reactions known as _metabolic pathways_ to break down these substances into nutrients, mainly amino acids to nitrogen and sugars to carbon. They obtain oxygen for metabolism from what is dissolved in the solution, or they split it off of molecular compounds.
Yeast, although living organisms, are actually highly organized enzyme collectives, each and every reaction of the yeast cell being controlled by a separate enzyme. In one reaction, simple sugars are reduced to alcohol and carbonic gas in the presence of a constitutive intracellular enzyme group called _zymase_ and a phosphoric coenzyme. This process is known as _fermentation._ Yeast metabolism directly determines the degree of attenuation of any wort, and its character greatly affects the flavor of the finished beer. In fact, just as every living organism varies from every other, every yeast strain, and even every fermentation, has qualities distinctly its own. These depend on a number of factors. Particular strains of yeast produce different flavor characteristics; variations in the pH, temperature, or composition of each ferment result in slight to significant changes in the metabolic products.
Although brewing dates back to prehistory, it was not until 1841 that Mitcherlich discovered that yeast were essential to fermentation. Further research by Pasteur and Buchner revealed that yeast produced alcohol only as a by-product of carbon metabolism, and that it was in fact the nonliving zymase enzyme that was responsible for the fermentation of sugar.
Bottom fermenting began with Gabriel Sedlmayer in Munich and Anton Dreher in Vienna in 1841, using mixed strains of yeast that were not purely bottom fermenting. Emil Hanson, working at Jacob Christian's Carlsberg brewery in Copenhagen, set the stage for modern lager brewing by isolating two distinctly different _pure-culture_ yeasts, that is, strains propagated from a single cell and therefore all exhibiting the same characteristics. These were a top-fermenting _Saccharomyces cerevisiae_ and the bottom-fermenting _Saccharomyces carlsbergensis (S. uvarum)._ Pure-culture, bottom-fermenting yeast were first employed at Carlsberg in 1883; within the decade, lager culture yeast were being employed in refrigerated fermentation throughout Europe and America.
Besides their visually different flocculating characteristics, the yeast operate at different temperatures and ferment different sugars. The top-fermenting yeast strains are generally only effective at 55 to 75 degrees F (13 to 24 degrees C). They form colonies that are supported by the surface tension of the beer and create a very thick, rich yeast head; in general, they ferment glucose, fructose, mannose, galactose, maltose, sucrose, xylulose, and maltotriose, and partially ferment the trisaccharide raffinose. ( _S. cerevisiae_ splits off and ferments the fructose molecule from raffinose, leaving the disaccharide melibiose.) "Bottom-fermenting" lager yeasts, which don't have as great an ability to chain and cling together, form smaller colonies that make a thinner, less tenuous head and that sediment out on the bottom of the fermenter more rapidly. They operate best at temperatures below 50 to 55 degrees F (10 to 13 degrees C). They ferment glucose, fructose, mannose, galactose, maltose, sucrose, melibiose, xylulose, and maltotriose, and fully ferment raffinose. Neither yeast ferments lactose, and all but the monosaccharide sugars need to be reduced by specific yeast enzymes before they can be fermented; sucrose must be split into glucose and fructose by invertase (sucrase), and maltose and maltotriose must be reduced to glucose by maltase (a-glucosidase). Maltose is able to be absorbed into the yeast cell before being hydrolyzed, but all the other disaccharides need to be reduced to monosaccharides by excreted enzymes before they can be transported into the yeast cell; this is the basis of maltose's ready fermentability.
Yeasts
Top, left to right: _Saccharomyces carlsbergensis, Lactobacillus, Pediococcus, Exiguus_
Middle: _Acetobacter, Acetomonas, Hafnia_
Center: _Pichia membranaefaciens_ (wild yeast)
Bottom: _Torulopsis_ (wild yeast), _Klebsiella, Zymomonas, Mycoderma_
There are two distinctive subdivisions of the bottom-fermenting yeast _S. carlsbergensis._ The Frohberg type (F.U., dusty or "powdery" yeasts) ferment very strongly, and attenuation is very rapid. Because they do not clump well, they remain in suspension longer and consequently have a greater effect upon wort attenuation. They ferment isomaltose as well as maltose. The Saaz type (S.U., or "break" yeasts) settle out of the ferment more satisfactorily than do the powdery yeast strains. Consequently, they are very weak fermenters and reduce the extract very slowly. They do not ferment isomaltose.
Different yeast strains span the spectrum between these two major classifications, producing very different aspects of taste, mouthfeel, alcohol, and clarity in the finished beer; the yeasts that ferment the quickest and most completely are not often the yeasts that produce the best beer. Yeast strains are selected for the character of their fermentation, their ability to form colonies, their ability to ferment with or without forming esters, and their viability rather than their ability to attenuate the wort rapidly.
Chemically, yeasts are constituted of proteins (especially _volutin,_ a nucleoprotein visible as small, shiny bodies in the vacuoles and cell plasma), glycogen (a starchlike reserve not usually found in older or stressed cells), minerals, enzymes, and vitamins (especially those of the beta complex).
Yeasts require various nutrients to renew these elements of their cellular structure. They absorb simple protein from hydrolytic solution, which they refine to a very high quality amino-acid group that composes roughly half of the yeast cell. Another 10 percent of the cell is calcium based and requires renewal, as do the minerals and trace elements that account for up to 5 percent of its structure. The minerals, besides calcium, are mostly the inorganic salts of phosphorus and potassium, with some magnesium, sodium, and sulfur. Yeast obtain these from mineral compounds in the ferment. The trace elements, especially zinc, boron, and manganese, are almost always available in small amounts from the malt, hops, or water. Yeast cells also require readily available oxygen for membrane synthesis; this is particularly important during the reproductive phase.
Yeasts reproduce by cell division, known as _binary fission_ or budding. They reproduce only in a nutrient-rich environment; one daughter cell emerges and grows to the size of the mother cell in two to six hours in a suitable solution.
There are numerous strains of yeast, and each operates successfully within a very narrow pH and temperature range. It is necessary to carefully control these factors during brewing because the metabolic reactions and the reproduction rate of the yeast greatly influence the nature and flavor of the beer being brewed.
Yeast operate in suspension in a sugar solution, until they clump together and are brought to the surface by attached CO2 or are sedimented by virtue of their increased mass. They cease to have a considerable effect on attenuation once they have clumped.
As yeast cells age, their previously colorless, homogeneous plasma (protoplasm) becomes bubbly, then separates into solids and liquid substances by forming vacuoles that envelop the liquid plasma secretion; later they become granulated, and gradually the plasma turns to fat (visible as round bodies of varying sizes within the cell walls). Although they are incapable of sporulation, yeast can be sustained in an unsuitable environment for long periods by these fatty bodies.
In solutions lacking obtainable nutrients, the culture yeast will cease reproducing. When they can no longer sustain their own metabolic functions, albumin-, hemicellulose-, and vitamin-dissolving enzymes are activated, which reduce the yeast cell to amino acids and other simple substances. This autolization releases typical organic decomposition flavors into beer that is not racked off its sediment.
Because a ferment lacks nutrients needed by the culture yeast, or because the temperature or the pH of the ferment does not suit the particular yeast strain does not mean that wild yeast strains, mutations, or other microbes will not find the conditions ideal. Under normal conditions, one in a million yeast cells spontaneously mutates; under hostile conditions mutations increase dramatically. Either a wild yeast strain or one of these genetically altered mutations may become the dominant fermentation organism, to the detriment or ruin of the finished beer.
Wild yeast cause spoilage, including clove, sour, vinegar, sulphuric, phenolic/medicinal, fusely, and diacetyl flavors, and create film formation on the beer surface. Because wild yeast do not tend to cling together as well as culture yeast, and consequently remain in suspension longer, they almost invariably cloud the beer. The offending yeast may even be a wild strain of _S. uvarum (S. carlsbergensis)_ or _S. cerevisiae,_ but this does not make their presence any more desirable. Other common spoilage yeasts are _Torulopsis, Candida, Dekkera,_ and _Pichia_ species. It is essential to ferment with solely the culture yeast alone, maintaining its purity, ensuring its adequate nutrition, and carefully controlling its metabolic functions through manipulation of the nutrient spectrum, pH, and temperature of the ferment.
## Culturing Pure Yeast Strains
Pure-culture yeasts are strains propagated from a single cell. Yeast from a successful primary fermentation that has exhibited good brewing characteristics are collected and mixed into a small amount of distilled water until the solution just becomes cloudy (approximately 100,000 cells/milliliter). One drop of the yeast solution is then mixed into one fluid ounce of diluted wort gelatin (beer wort diluted with sterile water at 4 to 8 °Plato [SG 1016 to 1032] mixed with 5 to 10 percent pure vegetable gelatin). The yeast is distributed by thorough agitation before the mixture is thinly spread over a clean cover glass, allowed to congeal, and placed in a sterile, moist container. The glass is then fixed to a graduated stage and microscopically examined at powers of 400 to 1,000 magnification, and the location of isolated, healthy-looking (white, hemispherical, nonreflective, uniformly sized) cells marked.
Pure Yeast Culture Growth
After twenty-four hours at 68 degrees F (20 degrees C), the glass is reexamined. Colonies should have formed. If they appear healthy, sample yeast cells from several isolated chains that are known to have grown from a single cell are removed with a flame-sterilized platinum or stainless-steel wire loop. Where a microscope is unavailable, the loop can be used to take a sample directly from a cloudy yeast solution, and an isolated colony can be chosen from the petri dish in the next step.
Each sample is streaked onto the surface of a sterile, staining nutrient agar (WL nutrient agar) in a petri dish. The inoculating streak is cross-hatched to isolate individual cells from it. When visible colonies have formed, an isolated clump ("rosette") is microscopically examined, and if it is uncontaminated, it is used to inoculate an agar wort slant (eight fluid ounces of wort diluted to 4 to 8 °Plato [SG 1016 to 1032] with sterile water, mixed into five grams of prepared agar or vegetable gelatin, at room temperature, heated to boiling after fifteen minutes, and boiled [or autoclaved at 15 psi] for fifteen minutes). This is poured into sterile twenty-milliliter test tubes tilted fifteen degrees from the horizontal, that are then capped or plugged with cotton and allowed to cool.
The temperature is maintained at 50 to 68 degrees F (10 to 20 degrees C), until fermentation is apparent (usually two to four days); then the culture may be refrigerated for three months at 39 degrees F (4 degrees C) for lager yeast, or at above 50 degrees F (10 degrees C) for ale yeast. The medium can be covered with a layer of sterile mineral oil to maintain an anaerobic environment.
The slants may be recultured by adding one-half inch of wort to each of the older tubes, and after fermentation begins, using that mixture to inoculate four freshly prepared slants. All culturing must be done under strictly sanitary conditions using sterile labware.
For the brewer who does not have the laboratory equipment necessary to isolate and incubate pure cultures (basically, a microscope, wire loop, and the several items of glassware mentioned), purchase of commercially prepared vials or slants is the best source of a yeast culture. Frozen yeast is a reasonable alternative, as are properly handled liquid cultures. Granulated dry yeasts are the least-desirable alternative, as they are likely to contain many dead cells and be contaminated by bacteria during the drying process.
Slants are activated by covering the culture with one-half inch of wort. After forty-eight hours, that pure liquid-culture is used to inoculate a sterile, narrow-necked eight- or twelve-ounce vessel (or Erlenmeyer flask) filled with four fluid ounces of (sterile) aerated wort; this volume can be successfully inoculated directly from the cover glass if slant culturing must be omitted. The bottle must be capped or covered with a fermentation lock.
After twenty-four hours at 82 degrees F (28 degrees C), each four fluid ounces of wort should yield two to four grams of pure culture yeast. Cooler temperatures, however, are generally employed to retard the yeast's reproduction rate; for lager yeasts, 50 to 68 degrees F (10 to 20 degrees C) for two to three days is usual. If capped, the lid must periodically be loosened to release pressure. When strongly fermenting, the culture may be roused into one quart of wort at a slightly warmer temperature than is usual for the brewery fermentation. It may be cooled to as low as 39 degrees F (4 degrees C) if the culture is not needed immediately. It is ready to pitch when it comes into active kraeusen. Each quart of starter should yield sixteen grams or more of pure culture yeast.
Aerate starter cultures often and well, so that the increase in available oxygen will stimulate greater yeast growth. In professional practice, a swab of yeast from the starter is cultured on a slide and microscopically examined for contamination before the parent culture is pitched.
## Storing Yeast
Starters may be held at 39 degrees F (4 degrees C) for up to three weeks, or until fermentation subsides. The beer above the yeast can then be decanted, and the yeast covered with cold wort before it is refrigerated again.
For longer storage, after one week at 50 degrees F (10 degrees C), the yeast may be forced to sediment by lowering the temperature. The liquid above the yeast is decanted, and the yeast sediment pressed to remove at least all of the free liquid. The yeast mass is formed into a ball, tightly covered with plastic wrap, placed in chipped ice, and frozen.
Yeast prepared in this manner may be stored for several months. When reactivation is desired, it is crumbled into a quart of well-aerated wort.
Yeast may be collected from each brewing and successively subcultured until undesirable changes occur in the beer flavor or the strain's fermentation profile. Most breweries repitch yeast only three to fifteen times before going back to the pure culture.
## Washing Yeast
Usually yeast requires only rinsing before reuse, but periodically cultures should be washed to destroy bacterial contaminants. (This will not, however, destroy wild yeast; they can only be eliminated by reculturing.)
To wash, chill the vessel to sediment the yeast, then decant off the liquid above the yeast cake. Rinse the yeast by covering with, and then decanting off, distilled or biologically sterile water. Cover again with a solution of sodium metabisulfite, phosphoric or tartaric (winemakers) acid at pH 2.8, or a .75 percent solution of acidified ammonium persulfate (one teaspoon of tartaric acid with two teaspoons of ammonium persulfate in one quart of water, at pH 2.8) in water or sterile beer, equal in volume to the amount of yeast being washed. Agitate the yeast into temporary suspension. When the yeast have completely settled, or within two hours, decant off the liquid above the yeast, rinse several times, and cover with sterile wort. Some yeast may display abnormal characteristics in the first fermentation cycle following an acid wash; they should be cultured through at least one fermentation cycle before being pitched. The above precautions notwithstanding, many breweries wash their yeast regularly, often pitching the yeast, still in the acid solution at pH 2 to 2.5, after two hours.
# CHAPTER 6
# Bacteria
Some bacteria are almost invariably present during brewing, having been transported into the brew by air, water, or the yeast culture. Certain bacteria may be present in small quantities without noticeably affecting the finished beer, but small concentrations of other bacteria can quickly ruin it. In some beer styles, bacteria in the ferment give a beer its particular character, but on the whole, bacterial contamination and growth need to be discouraged by strict sanitation. Bacteria are only tolerated in a lager brewery where they are cultured to reduce the mash acidity.
The countless types of bacteria oxidize or ferment a wide variety of organic substances. Fortunately, only a relatively few types of bacteria are encountered during brewing, and no pathogenic bacteria can survive in beer. Bacteria grow on sugar, wort, beer, protein, and hop residues, and even on the yeast. By careful control and strict sanitation, flavor and stability problems caused by bacterial contamination of beer can be kept in check. The brewer must work to eliminate any contaminant, or reduce it to a level where its growth will not appreciably affect the finished beer.
Because bacteria adapt and mutate so readily (far more readily than yeast), they can emerge as the dominant fermentation microbe from a relatively small number of cells. It is also because they mutate so readily that bacteria are so difficult to classify. Although they have a host of characteristics, they are initially categorized by whether or not they are stained by gentian violet (the Gram stain).
## Gram-Positive Bacteria
The gram-positive bacteria encountered during brewing are the Peptococcaceae (family) _Pediococcus_ (genus) and the Lactobacillaceae _Lactobacillus._ These are grouped together as _lactic acid_ bacteria, and were formerly referred to as "beer sarcina" (a term more specifically applied to _Pediococcus_ ). Both operate anaerobically and ferment simple sugars to lactic acid; they have little effect upon protein.
The several species of _Pediococci_ are strictly anaerobic, globular ("cocci") bacteria occurring singly, paired, or in cubicle groups called _tetrads._ They form diacetyl and inactive lactic acid from dextrins and glucose. Heterofermentive strains also ferment maltose, fructose, and sucrose, producing acetic acid as well. In general, the _Pediococci_ produce a disagreeable taste, odor, and turbidity (cloudiness). Contamination is most often from calcified trub deposits on poorly cleaned equipment; it is _Pediococcus's_ ability to survive even rigorous cleaning, sheltered by "beerstone" deposits that make them the most pernicious brewery contaminants.
The _Lactobacillus_ are the single genus of their family. These long, thin, curved rods occur singly or paired at obtuse angles. Microaerophiles, they form lactic acid by the fermentation of carbohydrates in even oxygen-poor solutions. Although they do not cause odor, they may produce a sour taste and turbidity. Several species of _Lactobacillus_ and _Pediococcus_ also cause a _ropy_ or gelatinous fermentation, or a silky turbulence, formed by the excretion of an extracellular slime. It disappears briefly during stirring, but reforms as chains upon the beer surface.
_Lactobacillus delbruckii_ (pH 5.6 to 5.8, active below 131 degrees F [55 degrees C]) is a heat-tolerant or _thermophilic_ homofermentive species that grows on malt. It is especially well suited to the acidulation of the mash, without producing undesirable flavors or turbidity. It metabolizes glucose and yields only lactic acid. Because it is favored by anaerobic conditions, its growth is encouraged by holding a tight, saccharified mash, closely covered, at above 95 to 120 degrees F (35 to 49 degrees C). This inhibits both aerobic and nonthermophilic bacteria.
When lactic acid bacteria, and especially thermophilic heterofermentive strains, are active in the mash or its extract, they may spoil it by acidification and souring, turbidity, or the formation of off-flavors, most notably a rancid-butter taste from the diacetyl diketone. The same symptoms in cooled wort, in young beer, and sometimes in aged beer are more likely due to contamination by _Pediococcus cerevisiae_ (pH 5.5, active over a wide temperature range, most active at 70 to 77 degrees F [21 to 25 degrees C]) and other nonthermophilic strains.
The thermophilic gram-positive bacteria are inhibited by isohumulones from the hops and usually will not survive in bitter wort or in beer. Lactic-acid bacteria are often contaminants from pitching yeast or from air. They may be the most significant infectious organism in the fermentation.
The gram-positive bacteria are rarely a problem in the aged beer because they have highly complex nutritional requirements and are inhibited by hops. During fermentation, the yeast will have absorbed many of the essential amino acids, making them unavailable to the lactic acid bacteria. Only when the proteolysis and precipitation of protein has been poor, when aging beer is not separated from deteriorating yeast sediment, or when temperature shock to the yeast causes it to autolyze will the bacteria be able to obtain enough amino acids to support reproduction.
## Gram-Negative Bacteria
The most significant gram-negative bacteria commonly affecting the lactic-acid mash is the coliform _Clostridium butyricum_ (butyric acid bacteria). These are thick, principally anaerobic rods that putrefy the mash by forming rancid-smelling ethylacetic acid (butyric or butatonic acid). They are inactive above 112 degrees F (45 degrees C) and very active below 104 degrees F (40 degrees C). This spore-forming bacteria may also occur in beer.
Acetic-acid bacteria sometimes taint mashes that come in contact with the air below 122 degrees F (50 degrees C) (they are quite active below 95 degrees F [35 degrees C]), but they present a far greater danger to fermenting beer. The acetic-acid bacteria are active over the entire pH range of the brewing cycle and are not inhibited by isohumulone from the hops. They are strong oxidizers and are usually responsible for any overwhelming sour-fruit or vinegary taste and odor, and oftentimes turbidity. The surface contamination they cause is often apparent as an oily or moldy (pellicle) film. They are usually introduced to the ferment during racking; aeration of the beer by rousing or splashing provides them with sufficient oxygen for respiration. Active yeast in kraeusen beer, when added during racking, can consume the dissolved oxygen quickly enough to prevent their growth, but in racking "quiet" beer, it is imperative that all aeration be avoided. Dispensing equipment should be given frequent and complete sterilization, as it is also a point of contamination. Sour-tasting or -smelling draft beer has almost certainly been contaminated by these bacteria. The lactophilic Achromobacteraceae _Acetobacter_ (significantly _A. aceti_ and _A. suboxydans_ ) oxidize ethanol to acetic acid. They are short, chain-forming, ellipsoidal-to-rod-shaped aerobic bacteria. The glucophilic Pseudomonodaceae _Acetomonas (Gluconobacter)_ excrete vinegar and gluconic acid. They are short, rod-shaped-to-ovoid, polarly flaggelated aerobic bacteria occurring singly, paired, or in chains. _Achromobacter_ and _Pseudomonas_ are infrequently encountered, and then only in sweet wort, because they are acid intolerant and inhibited by alcohol.
The coliform bacteria (termobacteria) commonly taint the wort by adding a dimethyl-suphide-related cooked- or spoiled-vegetable odor, caused by very rapid metabolism of wort sugars. They are waterborne, nonsporulating aerobes and faculative anaerobes most active at 98.6 degrees F (37 degrees C). Enterobacteriaceae _Escherechia_ are straight rods occurring singly or in pairs. Enterobacteriaceae _Klebsiella (Aerobacter)_ are nonmotile, encapsulated rods occurring singly, paired, or in chains. The cooled wort is an ideal medium for their growth. _Aerobacter aerogenes_ is a commonly encountered source of pungent, vegetable, or sulfuric taste and ropy fermentation in both wort and green beer. They adapt and reproduce far more quickly than culture yeast. It is essential that the starter or kraeusen beer used to inoculate the wort be strongly fermenting and of similar temperature and composition to the wort into which it will be pitched. Otherwise coliform bacteria may become strongly established in the lag phase.
The source of these coliform bacteria is most commonly rinsing water. Although they are active over a wide temperature range and are unaffected by hop resins, most are inhibited below pH 4.4 and are not commonly encountered during the later stages of brewing.
Table 11
Brewing Contaminants
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Brewing Stage | Symptoms | Bacteria Responsible | Solution
Mash, at below 140°F (60°C) | Acidity, sourness, turbidity | Thermophilic lactic acid bacteria | Raise temp. to above 131°F (55°C)
Rancid odor | Butyric acid bacteria | Raise temp. to above 112°F (45°C)
Sour, vinegar taste and odor | Acetic acid bacteria | Raise temp. to above 122°F (50°C)
Cooled Wort | Fruity or vegetable odor | Coliform bacteria | Pitch quickly
Primary Fermentation | Celery odor | _Hafnia protea_ | Go to new yeast culture
Secondary Fermentation | Sour taste, silky turbidity | Lactic acid bacteria | Lower temp.
Sour taste, odor and turbidity | Lactic acid bacteria | Lower temp.
Aging/conditioning | As above | Acetic acid bacteria | None
As above | Lactic acid bacteria | None
Stench | _Zymomonas_ | None
Enterobacteriaceae _Hafnia (Obesumbacterium)_ are short, nonmotile, straight rods of variable shape that commonly taint the early stages of fermentation. _Hafnia protea_ _(O. proteus)_ are fat rods (pH optimum 6.0) that, when present, are almost always in the yeast culture and only rarely in the wort (other _Hafnia_ strains, as well as acetic and lactic-acid bacteria, may also contaminate pitching yeast). They produce sourness, diacetyl, and a dimethyl-sulphide smell like parsnips or celery. Like others of their family, they are intolerant of very acidic solutions and do not usually affect the aging beer.
Acetic and especially lactic-acid bacteria are the most prevalent contaminants of aging and bottled or kegged beer, but Pseudomonaceae _Aeromonas_ is also encountered. The plump rods of Pseudomonaceae _Zymomonas (Achromobacter anaerobium,_ pH 3.5 to 7.5, temperature optimum 86 degrees F [30 degrees C], active as low as 40 degrees F [5 degrees C]) are relatively uncommon, but when they are present, they produce an objectionable acetaldehyde and rotten-egg stench in a very short time. _Zymomonas anaerobia_ or _Z. mobilis_ are then the usual contaminants, fermenting fructose and glucose to ethanol, hydrogen sulfide, and acetaldehyde. Ground water or soiled equipment are the usual inoculants.
Bacterial contamination can be assessed both by perceptory analysis of the wort or beer (accentuated by "forcing" closed samples at 85 degrees F [30 degrees C]) and by culturing the wort or beer on a staining or yeast-inhibiting nutrient agar in a petri dish and estimating the microbial population after several days.
# CHAPTER 7
# Enzymes
Enzymes are complex, protein-based biological catalysts that induce reactions between substances without being changed by the reaction or appearing in its end product. Enzymes may be _constitutive,_ that is, normally present within the cell, or _inducible,_ formed only in the presence of a particular substrate. They may be _intracellular,_ operating only within the cell, or _extracellular,_ excreted by the cell into solution.
During the malting and brewing cycle, the barley kernel is decomposed to soluble simple sugars and albuminoids by diastatic and proteolytic enzymes. These sugars are in turn fermented to carbon dioxide and ethyl alcohol by the zymase enzyme group, while other enzymes form organic acids, aldehydes, fusel alcohols, and esters.
The traditional decoction mash is constructed largely upon a series of conditions that reactivate enzyme activity that was prematurely checked by kilning the green malt. It completes the reduction of the native barley proteins and carbohydrates to a soluble extract. In the decoction mash, proteolytic enzymes associated with malting are employed to overcome flaws in the malt.
The proteolytic (peptonizing) group reduces proteins of high molecular complexity to simpler peptides and amino-acid constituents through a structured series of interdependent reactions that sever the peptide links (CO–NH) between protein coils and replace them with a water molecule. This restores the amine (NH2) and carboxyl (COOH) groups of the amino acid
Protease and proteinase (optimum range 122 to 140 degrees F [50 to 60 degrees C] pH 4.6 to 5.0), and then peptase and peptidase (optimum range 113 to 122 degrees F [45 to 50 degrees C] pH below 5.3) solubilize protein and sequentially reduce it to proteose, peptones, polypetides, peptides, and amino acids.
Phytase and phosphatase acidify the malt by forming phytic acid, and they are primarily responsible for the acidulation of the mash at 95 to 122 degrees F (35 to 50 degrees C). They also increase the soluble mineral content of the wort. Cellulase, hemicellulase, collagenase, and pectinase are active within the same temperature range, dissolving the cell walls, endosperm case, gelatin, and pectins.
The diastatic enzymes reduce starch to fractions. Primarily, these are the amylolytic enzymes — alpha- and beta-amylase. The alpha-amylase liquefies native starch and reduces amylose and amylopectin to a stew of carbohydrate fractions. By randomly separating 1-4 linked glucose molecules within the length of polysaccharide chains, it liberates glucose, maltose, maltotriose, and dextrins, leaving "a-limit" dextrins wherever it is stopped by 1-6 link branching points in amylopectin. It reduces complex starch to a-limit dextrins very rapidly and completely, so that its solution gives only a faint-red reaction with iodine. Yet it further generates a predominance of maltose only very slowly and ineffectively. It is present in the unmalted barley.
Debranching Enzymes
Beta Amylase
Alpha Amylase
Beta-amylase, on the other hand, does not appear until malting. It has no effect on the native starch. In solution it detaches glucose molecules from the nonreducing ends of amylose and amylopectin chains, rejoining them with a water molecule to produce maltose. Alone, it breaks down amylose very slowly and amylopectin very incompletely, because it proceeds in a linear fashion and only from one chain end. It is ineffective within two or three glucose molecules of amylopectin's outermost branching points, leaving a very large "ß-limit" dextrin that gives a deep mahogany color reaction with iodine. Where alpha-amylase activity splits soluble starch into smaller fractions, beta-amylase operates more efficiently, capitalizing upon the increased number of exposed chain ends.
Both amylases are made more effective by the activity of debranching enzymes. A-glucosidase (maltase), limit dextrinase, and pullulanase reduce amylopectin and limit dextrins to amylose by cleaving the linkages at their branching points. The debranching enzymes are most active during malting, and very few survive kilning, even with low-color Pilsen malt. At low mash temperatures they may dismantle some amylopectin, but not at hotter saccharification temperatures in the mash.
During fermentation, the zymase enzymes and a phosphoric coenzyme convert glucose to alcohol and carbonic gas; other enzymes are formed during fermentation that split and invert the more complex sugars present in the ferment. Intracellular maltase and glucase reduce maltose to two molecules of glucose; extracellular invertase splits sucrose into glucose and fructose. Finally, proteolytic enzymes within the yeast cell, triggered by a decline in the yeast's metabolism, autolyze the cell contents to other enzymes, minerals, and vitamins that are slowly released into solution.
It is the enzymatic composition of the yeast cell that determines the nature and vigor of fermentation. Various yeast strains have widely varying enzymatic capability. When the yeast cells do not contain the specific enzymes to reduce the sugars in a wort, they synthesize them. Fermentation lag times, however, are dangerously extended.
# PART TWO
# THE BREWING PROCESS
# CHAPTER 8
# Malting
Barley must be malted before it is mashed. The starch of unmalted barley is too complex to be readily converted to sugars, so the grain must pass through a series of steps to activate its constitute enzymes. The first is steeping.
During _steeping,_ or soaking, many enzymes in the grains are either formed or activated, and the starchy endosperm mass is solubilized to gummy _polysaccharides._ During _sprouting,_ the hydrolytic enzymes inside the developing plant embryo increase and penetrate the endosperm, reducing proteins and hemicellulose to soluble fractions. Polysaccharides and protein are reduced nearly proportional to the degree of the acrospire growth until both are arrested by kilning.
The _acrospire,_ or germinal stem of the barley plant, is grown to the full length of the kernel in British malts, almost fully modifying the endosperm to readily saccharified polysaccharides. American "brewers'" malts are less completely modified, and traditional continental malts modified the least of all. The acrospire growth of continental malt may be stopped when it is only one-half the length of the grain, in order to minimize the loss of starch by the digestion of the endosperm by the germinating embryo. The lesser degree of conversion is responsible for the lower enzyme strength and greater nitrogen complexity of continental malt. Haze-forming proteins that remain in continental malt must be decomposed during a low-temperature mash that is unnecessary when using British malt and most modern malts.
American malts are sprouted more fully than traditional continental malts. Acrospire growth is usually two-thirds to three-fourths the length of the kernel, but because American barley, especially six-row, is of much higher nitrogen content than traditional lager malt, its enzyme strength is correspondingly greater. High kiln temperatures can be held longer than with continental malt without risking a serious depletion of the malt's enzyme strength. Traditional continental malts require a protein rest to degrade large proteins, and albumin in excess of the amount required for body, head, and yeast nutrients. Although American malt can usually be infusion mashed, it is more often given a protein rest to reduce the potential for chill haze.
## Steeping
The barley to be malted is examined, and if it is judged suitable, steeped to thoroughly wet the endosperm mass and float off dust, debris, and lightweight, unmaltable grains. The kernels are stirred in an aerating fashion into water at 50 to 65 degrees F (10 to 18 degrees C), which is allowed to overflow the steeper to carry off the debris, then soaked for two to four days (preferably in alkaline water) to 35 percent moisture content. The water is drained, and the moist grains are turned several times during steeping to increase oxygen uptake by the respiring barley. The barley may be aerated for up to twelve hours before it is re-covered with water. Kernels are periodically removed and the extent of the moisture penetration is determined (wet endosperm is off-white). The grains will have swollen 1 ⅓ times their original size. Before sprouting, the malt should be of 45 percent moisture content.
Steely or nitrogenous barley must be wetted even more completely. The moisture content is verified by weighing a sample of the moist grain, drying it completely at a low oven temperature, and reweighing the dried grain; the weight loss should generally be 40 to 45 percent.
The white tips of the rootlets may be just emerging (chitting) when steeping is complete.
## Germination
Barley may be sprouted in many different ways, but the traditional floor malting produces the most uniform growth as well as the mellowest possible malt. The grain is laid eight to twelve inches thick (thirty-inch maximum) on waterproof concrete, at an ambient temperature of 45 to 60 degrees F (7 to 15 degrees C), for six to ten days. It is wetted and turned periodically to aerate it and to keep the temperature at an even 50 to 70 degrees F (10 to 21 degrees C).
Cooler temperatures encourage greater enzyme production and soluble-carbohydrate yield by impeding acrospire and rootlet growth. Reducing the initial sprouting temperature to below 55 degrees F (13 degrees C) produces the mellowest and most enzyme rich malt. Temperature control is achieved by lowering room temperature and reducing the depth of the sprouting grain to allow the heat being produced by the respiring grain to dissipate.
Growth should start during the first day, as the embryo internally forms immature rootlets; during the second day, the grains are wetted, then aerated by lifting and turning. This is done regularly thereafter. Growth speeds up, and by the fourth day rootlets usually have appeared.
Generally, the acrospire will have grown to one-half the length of the kernel by the sixth day of germination.
The degree of modification can be judged with some degree of accuracy by comparing the length to which the rootlets have grown against the length of the kernel, and by cutting through the hull and examining the endosperm and the length of the acrospire.
Table 12
Rootlet Growth as an Acrospire-Length Indicator
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Rootlet Length | Acrospire Length
½–¾ the length of the kernel | ½
1–1½ times the length of the kernel | ¾
1½–2 times the length of the kernel | Full
The modification of the endosperm proceeds in the same direction and at approximately the same rate as the growth of the acrospire, although modification tends to exceed acrospire growth in grain malted at lower temperatures.
The green malt should have a clean, wholesome smell and appear plump, with healthy, unwithered rootlets. The endosperm mass should feel chalky when it is rubbed between two fingers. Hard, watery, or gummy malt endosperm is poorly modified.
The objectives of malting for lager malt are even modification, reduction of beta-glucan cell-walls, gums, and protein matrixes, reduction of native starch to "mashable" fractions, and the development of proteolytic and diastatic enzymes. For malts for infusion mashing, the reduction of the sprouted kernel needs to be carried even further. British pale malt must be partially "mashed" as part of the malting program by a longer sprouting cycle and by saccharification at the start of the kilning cycle.
## Kilning
Kilning dries the malt, facilitates removal of the rootlets, and gives malt its character. It also reduces the pH in the mash.
The temperature of green lager malt is generally raised to 90 degrees F (32 degrees C) over twenty-four hours to allow the enzymes to continue starch modification and proteolysis. The lumps of tangled grain are gently broken up after drying has begun but while the green malt is still moist. The temperature is slowly raised to 120 degrees F (49 degrees C) and held for twelve hours to dry the malt, then raised to roasting temperature. It is essential that the malt be bone dry before it is heated above 120 degrees F (49 degrees C) so that enzyme destruction is minimized.
Domestic lager malt may be kilned-off at 130 to 180 degrees F (55 to 82 degrees C), while British pale-ale malt is dried to 2 to 3 percent moisture content and kilned, usually at 200 to 220 degrees F (94 to 105 degrees C). Temperatures for Czechoslovakian and Bohemian malts are raised very slowly from 120 to 153 degrees F (49 to 67 degrees C) to completely dry the malt before it is roasted at 178 degrees F (81 degrees C). Dortmund is roasted at 195 to 205 degrees F (90 to 95 degrees C). Enzymatic malts are slowly germinated at cooler than usual temperatures for six-row, high-nitrogen malts. They are slowly dried to 6 to 8 percent moisture content and cured at below 145 degrees F (63 degrees C). Vienna malt is dried to near 5 percent moisture content before kilning for about an hour at 210 to 230 degrees F (99 to 110 degrees C), while Munich malts are brought up to 210 to 220 degrees F (99 to 105 degrees C) for light, and up to 244 degrees F (118 degrees C) for darker Munich (dunkles).
Vienna and Munich malts give richer, maltier flavor and slightly fuller color to beer. Melanoids developed during kilning, especially those that are amino-acid derived, give these malts their characteristic flavor. Melanoids act as antioxidants (reductones), improving a beer's storage stability. Historically, Vienna and Munich malts were kilned from well-modified malt to ensure good color and flavor development. In modern practice they are likely to be kilned from overmodified barley so that they will be less problematic in an infusion mash. Just as often, however, these malts are made from inferior or high-protein barley, because maltsters expect them to be used only in small (5 to 15 percent) portions of the malt bill. Where they will compose a larger percentage of the grist, the brewer needs to review typical analyses provided by the maltsters and choose high-quality malt suited to the mash program being employed.
Amber malt is made from very well modified grain (slightly overmodified by British standards; similar to mild ale malt) that is dried to 2 to 3 percent moisture content and heated from 122 degrees F (50 degrees C) to 340 degrees F (171 degrees C) over the span of one hour. Some maltsters use lower temperatures, in the range of 280 to 300 degrees F (138 to 149 degrees C), for a longer time.
"Aromatic" and biscuit malts are very similar but employ different kiln temperatures; 240 degrees F (115 degrees C) for a longer time for aromatic, and 430 to 460 degrees F (221 to 238 degrees C) for biscuit. Amber, aromatic, and biscuit malts give intensifying degrees of a coppery color, dry, "biscuity" flavors, and toasty aromatics to beer.
Brown malt is dried before kilning at up to 355 degrees F (180 degrees C) for twenty minutes. It gives a deeper color and a more bitter/burnt flavor to beer.
Amber and brown malts contribute considerable fermentable extract to a brew, but the darker-roasted and crystal/caramel malts contribute very little. Caramel and crystal malts give a red to red-brown color to beer. They also increase the sweetness, fullness, foam retention and storage stability of beers; pound for pound, the fully crystallized versions impart these characteristics more than the caramel ones do.
Crystal and caramel (caramalt, carastan) malts are similar products, but they should not be considered interchangeable. Caramel malts have a higher moisture content, are not completely saccharified, and are not kilned to the point that the endosperm is entirely vitrified/glassy. Both are commonly kilned in drums. Both are surface dried at temperatures rising to 150 degrees F (65 degrees C) over an hour's time, during which period a significant amount of proteolysis occurs. "Caramel" versions of CaraPils (CaramelPils, CaraPilsen) malt are generally raised to 212 degrees F (100 degrees C) within five to ten minutes, then held there for thirty to forty-five minutes without ventilation to "caramelize" the endosperm. After kilning, the temperature is reduced to 175 degrees F (80 degrees C) to cure for forty-five minutes; "crystal" versions of the malt are completely saccharified during kilning. They both increase the sweetness, fullness, foam retention, and storage stability of beers without appreciably increasing the color.
Other caramel malts are treated similarly but are colored at higher kiln temperatures — 240 to 275 degrees F (116 to 135 degrees C). Caramel malts as 5 to 15 percent of the grain bill give a caramel, often raisinlike flavor and "chewy" character to beers. As caramel malt color increases, bitterness and roastiness increase accordingly.
Caramel malts were traditionally used by continental lager brewers, whereas crystal malts were favored by British ale brewers. The distinctive, complex flavors of caramel malts have their place in brewing, but unfortunately, modern maltsters are eschewing the production of crisper-flavored crystal malts in favor of the easier-to-process caramel malts. In fact, most modern maltings no longer make a distinction between caramel and crystal malts.
True crystal malts are completely saccharified in a moisture-saturated environment at 158 degrees F (70 degrees C) for 11/2 to 2 hours. This stewing of the malt liquefies and completely saccharifies it before it is brought up to kilning-off temperatures. Crystal malts are drum kilned at 250 to 275 degrees F (120 to 135 degrees C) for 1 ½ to 2 hours, depending on the color desired. They give flavors that are crisper and cleaner than caramel malts; the lighter-colored crystal malts especially give less bitterness and pungency than caramel malts.
Undermodified ("lager") malts are used to produce chocolate and black malts. They are generally dried to 5 percent moisture content, loaded in the kiln at 158 degrees F (70 degrees C), and roasted for up to two hours, to 420 degrees F (215 degrees C) for chocolate malt and to 435 to 480 degrees F (224 to 249 degrees C) for black malt, before being quenched by spraying water over the roasted grain.
When using specialty malts, the brewer needs to adjust the amounts lot to lot to keep beer color consistent. Moreover, a chocolate malt, for instance, at 350 °Lovibond, gives vastly different character to a beer than does a "chocolate" malt at 600 °Lovibond. At 2 to 5 percent of the grist bill, roasted malts richen color and contribute a burnt or nutty flavor to beer. They are used at 10 to 15 percent of the grist bill for porters and stouts.
Malts that are kilned over open hardwood (beech wood, fruit woods) or peat fires have special "smoked" flavors (from phenols released from the wood), which are characteristic of certain beers. These malts may be sold as mild, medium, or heavily smoked as measured by the phenol content. For peat malts, lightly peated malt gives 3 to 6 ppm phenols, moderately peated 7 to 11 ppm, and heavily peated 12 to 15 ppm.
In kilning, the maximum temperature is usually held only until the grains are evenly roasted; then the malt is cooled to below 100 degrees F (38 degrees C) and cleaned to remove rootlets and debris. Rootlets are easily screened from the dried grain, but care must be taken that the malt is not injured and the husk not broken during cleaning.
Weight loss during malting and kilning should be roughly 7 to 15 percent. Losses are more extreme where the rootlet growth is excessive or the malt has been caramelized during kilning.
Before being mashed, the malt must be binned for twenty to thirty days in a cool, dry place to mellow it and improve wort clarity.
# CHAPTER 9
# Crushing the Malt
Barley malt should be milled so that the husk is not shredded; ideally it should be split along its length. In this manner, the contents of the crushed kernel is released, and the maximum surface area of starch grits is exposed to enzyme activity without tearing the hulls. Only reasonably intact husks will form a suitably porous filter bed in the lauter-tun. Shredded hulls also contribute to a rough, harsh palate in the finished beer.
The best grist is obtained from six-roll malt mills, which crush the malt by running it between three successive pairs of rolls ten to twelve inches in diameter, each pair being set closer together and turning faster than the previous pair. Screens are placed between each set of rolls to allow the fine grits to fall through, so they are not pulverized into flour by further crushing. More than 75 percent of the malt may be reduced to grits in this manner — the remaining part being hulls and flour. More than 10 percent flour is undesirable because it balls or cakes readily. Balled flour is inaccessible to enzymes and results in unconverted starch, some of which washes into the wort during sparging, causing an irreversible haze in the beer.
Mills
Six-Row Malt Mill
Hammer Mill
Home Grain Mill
Crushed Malt
Well-Crushed
Poorly-Crushed
Milling equipment other than a six-roll mill is employed, but none yields as good a grind. Only mills equipped with blades that cut the grain are entirely unsuitable, however. Hammer mills and the more commonplace grain mills that employ radially grooved, opposing-face grinding wheels are used, although they tend either to grind grain too finely and shred the husks, or to leave large chunks of the kernel intact.
Coarsely ground malt also does not yield the extract that it should. Heavy, gummy, insufficiently modified starch particles interfere with mash filtering. The hard ends of poorly malted grain are particularly subject to being left uncrushed. In general, it is better to crush the malt too finely (at the risk of a set mash), taking extra care doughing-in, and sparge very slowly than not to mill it finely enough. Only well-modified malts give up their extract when very coarsely ground; poorly modified malts especially require adequate milling. If the iodine test after a sufficient mash saccharification rest shows significant blue-black starch particles and grain ends (not husk fragments, which always deeply discolor iodine), and the spent malt gives any sweet taste, then crushing was probably insufficient.
Milling for lauter-tun brewing should yield predominantly fine grits.
Table 13
Lauter-Tun Grist Profile
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U.S. Standard Mesh | Mesh Width | Characteristics | Lauter-Tun Grist Composition
10 | .0787 inches | husks held | 15%
14 | .05512 | husks held | 15%
18 | .03937 | husks held | 15%
30 | .02323 | coarse grits held | 25%
60 | .00984 | fine grits held | 30%
100 | .00587 | flour held | 20%
100 | .00587 | fine flour falls through | 10%
# CHAPTER 10
# Mashing
During mashing, the reduction of complex sugars and of insoluble proteins to simpler amino acid chains is entirely an enzymatic process. Before mashing, the malt is only 15 to 25 percent soluble (CWE). Mashing should yield an extract equal to 65 to 80 percent of the weight of the dry malt. Not all of this extract is fermentable; in fact, the varying percentages of unfermented "rest extract," i.e., dextrins, proteins, and peptides, give each beer its malt character. The part of any extract that is unfermentable dextrins (sweetness and flavor) and proteins and peptides (body) is controlled by manipulation of the times and temperatures of the mash.
## Doughing-In
Enzymes act on the malt only in an aqueous solution; water induces the enzymes, encased in the aleurone layer of the malt kernel, to go into extracellular solution with soluble starch and hemicellulose. Because crushed malt, and especially floury malt, tends to "ball" into a dry mass that isolates it from enzyme activity, it is essential to mix the grains with liquor in a way that does not saturate any part of the mash while another part is still dry. The intention is to create conditions conducive to dissolving all of the endosperm, including the enzyme-rich particles of the aleurone layer, and not to induce enzymatic activity just yet. Even moistening prevents the starch from balling and entirely solubilizes the enzymes.
Time/Temperature Graph
Traditional Three-Decoction Mash
A successful mixture is most readily accomplished by the gradual addition of liquid to the grain. Small amounts of liquor are sprinkled onto and then kneaded into the whole of the grain mass until the crushed malt can absorb no more. When a mash is doughed-in cold, only a small amount of liquor should be standing free at the bottom of the well-kneaded mash.
Time/Temperature Graph
Two-Decoction Mash
Using the least possible amount of liquor to form a very thick mash improves enzyme effectiveness early in the mash and simplifies rest-temperature maintenance later on. Liberal infusions of boiling-hot liquor may then be made to hold the saccharification temperature without overly thinning the mash. In decoction mashing, this is best accomplished by doughing-in the malt with twenty-four to forty fluid ounces of cold liquor at a deep-well water temperature of about 58 degrees F (14 degrees C) per pound of crushed grain, the higher end of the range being for larger brew lengths and mechanized systems.
The doughed-in mash is allowed to stand for fifteen to thirty minutes, the longer time being for malts that are dark, weakly enzymatic, hard tipped, or poorly malted. At the same time, roughly half the volume of liquor used to dough-in the malt is brought to boiling. It is infused into the mash to raise the temperature of the whole from 60 degrees F (16 degrees C) or so up to 95 to 105 degrees F (35 to 41 degrees C).
Where the grist is not doughed-in cold, even more attention needs to be paid to evenly hydrating it. Even with well-modified British malts, balling is a problem as the temperature at doughing-in increases.
## Mash pH
The correct initial pH of any mash depends on the type and color of the malts employed and the planned mashing technique. It must never be begun at above pH 6.2, or below 4.7. When the mash is from enzyme-poor malt that will be fully decoction mashed, the mash cycle may begin at a pH as high as 5.5 to 5.8 (as measured at the reference temperature, usually 68 degrees F (20 degrees C).
"Acid" sourmalt is used to lower the alkalinity of the mash. Sourmalt has been treated with lactic acid or lactic-acid bacteria to a percentage-lactic-acid of 3.5 to 4.0 percent, giving a wort pH of 4.0 to 5.0. It is employed as 5 to 10 percent of the grist to give an initial mash pH of 5.2 to 5.8. Colored malts also have a significant effect on wort pH; the darker-roasted the malt, the greater its effect on wort acidity. While lager malts generally give a wort pH of 5.7 to 6.0 in a laboratory mash from distilled water, more highly kilned Vienna and Munich give pH 5.5 to 5.7, better-modified pale pH 5.3 to 5.7, the darker brown and caramel/crystal-50 pH 4.5 to 4.8, chocolate 4.3 to 4.5, and black malt 4.0 to 4.2.
Depending on their proportion in the grist and the alkalinity of the liquor, colored malts may be enough to give proper mash acidity. Until the proper initial mash acidity is approximated (within pH 0.2), however, the mash cycle should not be begun.
## Acid Rest
The pH of calcium-bearing liquor always drops as it is mashed with malt because phosphates in the malt react with calcium and carbonate, precipitating alkaline calcium phosphate and lowering the mash pH. Where malt color and calcium content of the liquor are low, however, this reaction is insufficient to give an acceptable mash acidity.
The _acid rest_ is made solely to correct the initial mash pH. Significant phytase activity in conjunction with very limited bacterial fermentation of glucose to lactic acid acidifies the mash without imparting a harsh flavor to its extract. The rest is most successfully employed when mashing with reasonably soft or sulfate water; it cannot overcome the alkalinity of strongly carbonate waters. Excessive carbonate salts present in the brewing water must first be precipitated by boiling or adding slaked lime, or overcome by the inherent acidity of dark-roasted malt, or by adding lactic, phosphoric, sulfuric, or citric acid to the brewing liquor or sourmalt or lactic-acid mash to the mash itself.
Acidulation of the mash is primarily by the enzyme phytase, active at 86 to 128 degrees F (30 to 53 degrees C), which dismantles insoluble _phytin,_ a salt in which most of the malt phosphate is bound up, to significantly acidic phytic acid. Generally referred to as an enzyme of malting, reactivation of phytase by the acidic hydrolysis at 95 degrees F (35 degrees C) accounts for a twofold or threefold increase in the phytic acid in a decoction mash from lager malt. It benefits the mash not only by lowering its pH but by increasing the mineral content of its liquid extract and producing a rich and accessible source of yeast nutrients, especially myoinositol, a B-vitamin necessary for yeast growth.
Phytase activity is most dramatic when mashing undermodified malt, since less of the malt phosphate has been inverted during malting. Mashes made from highly kilned malts show little pH reduction during an acid rest because phytase is destroyed by the high kiln temperatures. The natural acidity of these malts, however, can be sufficient for establishing a proper mash pH.
During the rest, the pH of a mash from low-modified lager malt generally drops from 5.5 to 5.8 to pH 5.2 to 5.3. A mash sequence employing an acid rest thus needs to rely less on the brewing water being naturally acidic or calcium rich, or on using colored malts, or acid malt, mash, or treatment, and begins at a higher pH than an infusion mash.
The rest is not usually held for longer than it takes to boil the first decoction. When the acidulation occurring within this period is inadequate, some manner of acidifying treatment must be made.
Traditionally, where at mashing-in the pH would be above 5.8, a separate "lactic-acid mash" is made prior to the main mashing. A 5 to 15 percent fraction of the mash volume is doughed-in and saccharified, cooled to below 130 degrees F (55 degrees C), inoculated with crushed malt, and then rested, closely covered, at 95 to 122 degrees F (35 to 50 degrees C) for up to several days. The saccharified mash at 95 to 122 degrees F (35 to 50 degrees C) creates an ideal environment for the fermentation of glucose to lactic acid by _Lactobacillus delbruckii_ (temperature range 86 to 131 degrees F [30 to 55 degrees C], most active at 107 to 111 degrees F [41 to 44 degrees C]; pH optimum 5.6 to 5.8). The closely covered lactic acid mash is held until its pH drops below 4.0. It is then intermixed into the main mash to correct its pH.
If the lactic-acid mash begins to smell the least bit solventy or rancid, if turbulence or ropiness develops on its surface, or if it is "off" in any way, it should be skimmed and its temperature raised to above 122 degrees F (50 degrees C) to destroy the spoiling mold, aerobic butyric or acetic-acid bacteria. Where thermophilic anaerobes ( _L. bulgaris, L. brevis,_ or any of the thermophilic strains of _Bacillus)_ are the source of the spoilage, temperatures may have to go above 140 degrees F (60 degrees C) to terminate the activity. The most common contaminant is the putrefying _Clostridium butyricum,_ which turns the mash rancid and renders it unusable. Anaerobic conditions prevent the growth of most organisms that might otherwise spoil a lactic-acid mash, so elimination of air space between the mash and its cover is of primary importance.
## Why Decoction Mash?
Although decoction mashing serves to raise the temperature of the mash to the protein, saccharification, and lauter-rest temperatures, its more important function is to cause several significant changes in the boiled portions.
Not even the most thorough infusion mash can eke out the quantity or quality of extract that is obtained by decoction mashing. There are several reasons for this. During decoction mashing, the thick part of the mash passes through the diastatic-enzyme temperature range two to three times. Boiling also reduces the size and complexity of malt starch and protein — a process that is absolutely essential when mashing-in difficult malts. Malts such as dark Munich, having only one-third the enzyme strength of pale malt, cannot otherwise be satisfactorily mashed.
Since boiling destroys enzymes, the enzymes in the unboiled mash portion must be preserved. In a mash that is satisfactorily solubilized during doughing-in, the enzymes are washed into the free liquid when the mash is flooded to raise its temperature to 95 degrees F (35 degrees C). The thickest part of the mash — containing the heaviest and least accessible concentration of native starch and protein — can then be boiled without decimating the enzyme population.
The heavy decoction is quickly heated to 150 to 158 degrees F (65 to 70 degrees C), without resting at 122 degrees F (50 degrees C), so that no further enzymatic acidulation of the mash occurs.
There are two good reasons for passing by this rest, and both are based on the pH sensitivity of the diastatic enzymes. First, the dextrinization of native starch by alpha-amylase (pH optimum 5.7) is far more effective at the higher pH of this first decoction than later when the whole mash comes into the saccharification/dextrinization range at a far lower pH. Second, the same high pH that stimulates alpha-amylase activity retards beta-amylase activity (pH optimum 4.7). Because native starch is far too complex to be successfully reduced by beta-amylase until alpha-amylase has reduced it to shorter amylose chains and smaller amylopectin fragments, resting the decoction at saccharifying temperatures is not productive. It is quite enough that the manageable small dextrins replace the native starch, even when mashing for a high-maltose extract.
Regardless of the diastatic power of the malt, unconverted starch is invariably entrapped within poorly solubilized malt particles. As the decoction is heated above 167 degrees F (75 degrees C), the particles burst, and their contents are absorbed into the liquid extract. This makes them accessible to alpha-amylase activity during the diastatic-enzyme rest of the main mash. This otherwise lost extract increases both the quality and the quantity of the extract yield.
The acidity of worts generally decrease after mashing, as they are heated to boiling (the more so with rising liquor alkalinity), because heat disassociates carbonic acid in solution to H2O and CO2, causing the pH to rise. This phenomenon is of less concern to the decoction-mash brewer because the mash is stirred and portions of the mash are boiled, both of which actions decompose HCO3, and because the acid and protein rests lower the mash acidity. Boiling the decotions also precipitates more inorganic calcium phosphate than is otherwise achieved.
Boiling also dissolves protein gum. At lower temperatures, protein gum is unaffected by enzyme activity and passes through mashing largely unconverted. Only when thick mash is boiled can the proteolytic enzymes successfully degrade dissolved gum to albuminous fractions; instead of clouding the beer, the smaller proteins enhance its body and head. Protein trub precipitated during wort cooling is also dramatically decreased.
Boiling also deoxygenates the mash, reducing hot-side aeration and allowing it to settle in well-defined layers in the lauter-tun. Only the absence of residual protein gum makes this effective filter bed possible; when an infusion mash is employed, such a dense filter bed likely results in a set mash.
In mashing techniques that do not use a decoction sequence, the proteolytic and diastatic enzymes are destroyed before the mash achieves optimum temperatures for the dissolution of starch particles and protein gum. The extract content, clarity, character, fullness, maltiness, and body of the finished beer are negatively affected. Where the malt is reasonably well modified and evenly crushed, however, the traditional three decoctions may not be necessary; even a single-decoction (or step) mash is never advisable with British ale or brewers' malts that have been thoroughly modified during malting and would be "overmodified" by exposure to low temperature rests.
Doughing-in with boiling water to 95 degrees F (35 degrees C), followed by a second infusion to 122 or 131 degrees F (50 or 55 degrees C), only a limited protein rest, and a single, thick decoction before saccharification is sufficient for all but the most undermodified malts, and doughing-in at 122 or 131 degrees F (50 or 55 degrees C) is sufficient for most modern lager malts.
During decoction mashing, the brewer is at his busiest, because two mashes must be handled at the same time, with great care. The brewer must be thoroughly organized before plunging into the sometimes hectic decoction mash cycle.
Decoction mashing is often met with open skepticism by brewers who have no experience with it. The fact remains that the beguiling maltiness of European lagers is only achieved by boiling undermodified malt. Extract yield is increased. Moreover, hot-side aeration is reduced, because the boiling and mixing of the mash deaerates it. Boiling of the mash does not lead to astringent harshness in the brew, probably because the density and pH of the decoction prevents phenols from being leached out of the husks.
There are two widely accepted programs for decoction mashing. The first, described here, employs a protein rest at 122 degrees F (50 degrees C) and saccharification/dextrinizing at 149 to 158 degrees F (65 to 70 degrees C). It is best suited to malts of below 37 percent soluble nitrogen.
The second program is better suited to higher-protein modern malts with a soluble nitrogen ratio of 37 to 40 percent. A combined proteolysis/saccharification rest is made at 131 degrees F (55 degrees C), allowing proteases to reduce large proteins to body- and head-building polypeptides and beta-amylase (temperature optimum 126 to 149 degrees F [52 to 65 degrees C], pH 5.4) to reduce amylose to maltose and glucose, and amylopectin to ß-limit dextrins. Dextrinization is then accomplished separately, usually at 158 to 162 degrees F (70 to 72 degrees C).
## Three-Decoction Mash
### First Decoction
The volume of thick mash to be boiled, relative to the volume of the whole mash, is dependent upon mash thickness. A very thick mash requires that only its heaviest one-third part (mostly grain mass, with only enough liquid to fill the spaces between the grain particles) be boiled, along with very little of the mash liquid. Thinner mashes require that proportionally more of the mash be boiled, along with more liquid, because even if most of the malt is removed and boiled, in a thin mash it would not contribute enough heat to the resting mash to sufficiently raise its temperature to the next rest. One pound of crushed malt contributes about the same amount of heat to the mash as does one pint of water, yet displaces only as much volume as six fluid ounces of water. In a thick mash, the favorable heat-to-volume ratio of the malt is such that the heaviest one-third part can raise the temperature of the whole mash to the next rest. In a thin mash, however, the heat value of the malt is not enough to overcome the far greater amount of water. A greater percentage of the mash must therefore be boiled — but usually not more than 40 percent.
After the decoction has been pulled, the rest mash (cold settlement) is closely covered and held undisturbed, except for occasional mixing to disperse temperature and enzyme activity. At the end of each decoction cycle, the mashes are remixed to raise the temperature of the whole to the next rest.
### Protein Rest (First Thick Mash)
With traditional lager malts, the character of the finished beer — its body, clarity, lack of chill haze, stability, and resistance to spoilage — is largely established during the protein or "albumin" rest. This "softens" poorly modified malt and improves mash runoff by decomposing heavy, gummy, insufficiently modified malt particles. During the rest, complex protein globules are decomposed by proteolytic enzymes to less troublesome fractions. With relatively unmodified lager malts, proteases, peptases, and proteinases progressively dissolve the peptide links within the protein coils to liberate coagulable albuminous fractions, peptides, and amino acids. It is albumin (proteoses, peptones, and polypeptides), not protein, that gives beer its body and enables it to raise and support a frothy foam head.
The protein-rest temperature should be 122 to 131 degrees F (50 to 55 degrees C), although temperatures from 113 to 140 degrees F (45 to 60 degrees C) support proteolytic enzyme activity. It should be kept in mind, however, that at the lower end of the temperature range, head-and-body polypeptides may be denatured to peptides and amino acids, reducing the body of the beer. Proteinase (temperature range 104 to 140 degrees F (40 to 60 degrees C), optimum 122 to 140 degrees F (50 to 60 degrees C), pH 4.6 to 5.0, solubilizes and breaks down simple proteins to peptones, polypeptides, and peptides. Peptidase (optimum range 113 to 122 degrees F [45 to 50 degrees C] pH below 5.3) dissolves polypeptides and peptides to individual amino acids, which fuel yeast growth in the early stages of fermentation. In any all-malt beer, however, there are generally sufficient amino acids to support fermentation.
Extract efficiency is also enhanced by the protein rest. Extract is exposed by the dissolution of membranous proteins, and complex amylopectin may to some extent be dismantled by debranching enzymes (maltase, dextrinase).
Phytase continues its activity during the rest, reducing phytin from the aleurone layer and embryo of the malt to phytic acid. Other acids also rapidly form during the rest, further lowering the pH toward the optimum values for saccharification, the clarification of the wort during boiling, and subsequent yeast fermentation. During this rest, the pH should drop again to below 5.4.
Other nonproteolytic enzymes (most notably cytase, temperature range 113 to 131 degrees F [45 to 55 degrees C], pH 5.0, and beta-glucanase, range 95 to 131 degrees F [35 to 55 degrees C]) actively dissolve pectins and other constituents of the malt hemicellulose during the protein rest.
A thick mash improves enzyme performance. In a thin mash, proteolytic and other heat-labile enzymes are destroyed in the course of the rest; in a thick mash, they may survive into the saccharification range.
The protein digestion can be overdone, however. Devoid of proteoses and peptides, the beer would lack body and a froth head. It would be very stable, but very empty-tasting. Without any coagulable proteins to adhere to, hop tannin would not precipitate from the boil, and the beer would taste "rough." Reducing nitrogen complexes too far would result in the presence of an excessive amount of simple nutrients in early fermentation, which would encourage bacterial contamination.
The degree of protein degradation achieved during this rest may be fairly judged later on by the thickness and slickness of the protein sludge covering the settled grist in the lauter-tun. It should be moderately thick and powdery rather than gummy.
The rest temperature must be reached by effective mixing, accomplished by lifting the mash from the bottom of the tun. It is essential that the return of the decoction be competently handled so that temperature dispersal is absolutely even. Attempting to correct wide temperature fluctuations within the remixed mash is never easy.
In the event that the strike temperature is reached before all the decoction has been returned, the remainder of the boiled mash is force-cooled to 122 degrees F (50 degrees C) before it is returned to the main mash.
The objectives of the albumin rest should be accomplished in less than two hours, or the malt is entirely unsuitable for use in brewing. Usually after five to twenty minutes at 122 to 131 degrees F (50 to 55 degrees C), the heaviest part of the mash is again drawn off, to begin the second decoction. Up to 30 percent of the malt nitrogen can be expected to have gone into solution at its conclusion.
### Saccharification Rest
Malt starch occurs as long straight or complex-branched chains of linked glucose, C6H10O5. During the saccharification rest, alpha- and beta-amylase reduce that starch to simpler fractions. This yields flavorful dextrins and fermentable sugars.
Alpha-amylase very rapidly reduces insoluble native starch to smaller polysaccharide fractions (a mix of some glucose, maltose, maltotriose, and straight-chain dextrins with a predominance of branched "a-limit" dextrins). Given long enough, the alpha-amylase continues to sever 1-4 glucose links, producing more glucose, maltose, and maltotriose, but starch-chain fragments are more effectively saccharified to fermentable sugars by the faster-acting beta-amylase. Beta-amylase has no effect on native starch, but in hydrolytic solution, it reduces soluble starch by cleaving glucose molecules from one end of starch chain fragments and rejoining them in pairs with a water molecule to create maltose, C12H22O11.
It is inadvisable to reduce all starch to fully fermentable maltose. Significant quantities of more complex polysaccharides must be carried over into the ferment for the beer to have a sweet flavor and sense of fullness. Partially fermentable dextrins, oligosaccharides, and especially maltotriose support the yeast during the long, cold aging period.
The temperature of the rest may be from 149 to 160 degrees F (65 to 71 degrees C), depending on the nature of the beer being brewed. Precisely hitting the appropriate rest temperature is essential, as a variation of two or three degrees for even five minutes will dramatically alter the maltose/dextrin ratio of the extract.
Mash thickness also affects the fermentability of the wort. A thick mash (less than three-tenths of a gallon of water per pound of malt) induces the greatest overall extraction. A much thinner mash increases the proportion of maltose, and thus wort attenuation.
The reduction of the large starch chains in a thick mash at 155 to 158 degrees F (68 to 70 degrees C), almost excludes any maltose formation whatsoever. The richly dextrinous wort produces a fullness and sweetness complementary to Munich-style lagers and darker beer with a contrasting burnt-malt bitterness. It is seldom suitable for light-colored beers.
Above 160 degrees F (71 degrees C), strong enzyme action ceases; temperatures below 149 degrees F (65 degrees C), on the other hand, seriously limit dextrin formation (by alpha-amylase, temperature optimum 149 to 158 degrees F [65 to 70 degrees C], pH 5.1 to 5.9) while favoring the formation of maltose by beta-amylase. Because starch granules are not gelatinized or dispersed below 149 degrees F (65 degrees C), beta-amylase activity at lower temperatures serves only to eliminate the straight-chain dextrins formed in the decoction, without further significant starch reduction.
For very light beers, the release of ungelatinized starch into solution at 149 degrees F (65 degrees C) is capitalized upon by raising the temperature of the mash from 131 to 149 degrees F (55 to 65 degrees C) over fifteen to thirty minutes; this largely eliminates the amylose liberated during the decoction. The mash is brought to rest at 149 to 151 degrees F (65 to 66 degrees C) to gelatinize and further dextrinize the starch, and to produce a maltose/dextrin ratio that favors lightness on the palate and rapid maturation.
For rather more usual palate fullness and fermentability, the recombination of the mashes requires even more careful handling. The decoction must be returned to the rest mash as quickly as possible, but without creating wide temperature variations within it. The rest temperature should be evenly attained within less than ten minutes.
Most of the amber and gold lagers, and even the pale Pilsener/Dortmunder types, rely on the heavier, richer dextrin complement formed at 152 to 155 degrees F (67 to 68 degrees C). This is the strike temperature of most "character" beers brewed with undermodified malts and a protein rest at 122 degrees F (50 degrees C). Saccharification in this temperature range encourages an alpha-/beta-amylase activity ratio greater than five to one; as a result, the dextrin content of the wort is 25 percent or greater, and the alcohol content by weight of the finished beer is roughly one-third the value of the wort density (°Plato). Where a combined protein/saccharification rest at 131 degrees F (55 degrees C) has been made, similar results are acheived by a dextrinizing rest at 158 to 160 degrees F (70 to 71 degrees C).
As the mash saccharifies, it becomes thicker, brighter, and browner. The brewer may decide to add brewing liquor to thin an overly thick mash and to speed up saccharification (beta-amylase is more effective in the looser mash). Caution must be used, however; as mash temperatures rise above 149 degrees F (65 degrees C), enzymes are rapidly destroyed in a thin mash. Even in a thick mash, beta-amylase is destroyed in less than an hour at above 149 degrees F (65 degrees C), and alpha-amylase is destroyed within two hours at above 154 degrees F (68 degrees C) in a mash below pH 5.5. This fact must be remembered when mashing to yield a dextrinous wort; a satisfactory dextrinous wort cannot be formed at below 153 degrees F (67 degrees C).
As opposed to British infusion mashes, which are entrained with air and "float," a decoction mash is stirred regularly, in a nonaerating fashion, to break up any pockets of unmodified starch and ensure uniform conversion. After fifteen minutes at the rest temperature, testing for saccharification with iodine should begin.
### Iodine Starch-Conversion Test
Place a small sample of the extract in a porcelain dish. Float common iodine (.02 _N_ solution; 1.27 grams iodine and 2.5 grams potassium iodide in 500 milliliters water) onto the extract, drop by drop, until a distinct layer of iodine is formed. Note any color change in the iodine at its interface with the mash liquid. Also observe the intensity of the color: is it trace, faint, or strong? Blue-black indicates the presence of native starch (amylose); deep mahogany/red-brown evidences gelatinized starch (amylose fragments and large a-limit dextrins), faint red simple a-limit dextrins. A faint mahogany to violet-reddish reaction denotes a mix of small dextrins. Total mash saccharification (a solution of some small a-limit dextrins with maltotriose, maltose, and simple sugars) causes no change in the yellow color of iodine.
Iodine is a poison. DISCARD ALL TESTS. Ensure that there is no iodine contamination by washing any article that comes into contact with it. Conduct iodine tests some distance from the mash so that no iodine will inadvertently contaminate the mash.
Most of the starch should be reduced to at least small a-limit dextrins by alpha-amylase. Even for sweet, less-fermented beers, the reaction with iodine should be no more than faintly mahogany-to-red. There should never be a strong color reaction with the iodine; neither, however, should rich beers be saccharified to the point that a negative iodine reaction occurs. A faint mahogany-to-reddish reaction indicates an acceptable extract composition for these beers.
The mash should be held at the strike temperature until saccharification is complete. Infusions of boiling water may be made with less regard to enzyme viability as conversion nears completion. The looser mash improves filter bed formation.
(Note: Koji or other diastatic enzyme preparations should not be used to increase enzyme activity. Although they convert hundreds of times their weight in soluble starch to simple sugar, they do not form dextrins.)
If the mash does not saccharify within one hour, it should be stirred, restored to temperature, verified for proper pH (5.2 to 5.5), and held for thirty minutes more. If the iodine color is not further reduced, addition of diastatic malt or extract may be required.
The efficiency of the malt crushing can be gauged by pressing a sample of the goods until all of the kernel ends and malt particles have been crushed, then separating the liquid from the particulate matter. Repeat the iodine test on the liquid, as above. If the color at the iodine-mash interface is intensely black or blue, then crushing was insufficient or the malt was poorly doughed-in. (Some isolated color from insignificant amounts of exposed starch is to be expected; husk particles themselves will always turn intensely black). Suspicions regarding the efficiency of milling may be verified either by a wort density that is less than predicted or by tasting the dried spent mash. If extract efficiency is below 65 percent or the spent grains taste sweet, the malt has probably been insufficiently crushed.
### Final Decoction
When the starch end point has been verified, the very thinnest part of the mash is removed to be boiled. The decoction is usually 40 percent of the volume, although a very thin mash may require boiling half of the mash. Because there are fewer starch and albuminous particles in the thinner portion, there is less risk of these being decomposed during the boiling of the runoff. These remain with the rest mash. On the other hand, enzymatic reduction of the dextrins in the thin part of the mash is more quickly terminated, preventing oversimplification of the extract.
The lauter decoction is brought to boiling, while being stirred, in ten to fifteen minutes and may be held at a strong boil for a further fifteen to forty-five minutes, although in modern practice it is uncommon to boil the mash for more than five or ten minutes. The temperature of the rest mash is held at or slightly above the strike temperature during boiler-mash processing.
The boiled extract must be well mixed with the rest mash. Care should be taken that the strike temperature of the final rest, 167 to 170 degrees F (75 to 77 degrees C), is not exceeded. Temperature adjustments may be made by the infusion of either cold or boiling brewing liquor, as required.
Exact temperature maintenance is, as before, critically important. Lower temperatures do not terminate enzyme activity or expand particles of intermediate starch degradation enough to keep them in temporary suspension, up and away from the bottom of the mash filter bed. At higher temperatures, the starch granules burst, and insufficiently modified carbohydrate and albuminous matter becomes dissolved and unfilterable. Because the diastatic and proteolytic enzymes have been destroyed by the high temperatures, the starch and protein gum have no opportunity to be reduced to manageable fractions. High temperatures also induce the extraction of tannins from the husk.
The mash-out temperature may be maintained for up to one-half hour while the mash is roused up. Thorough mixing allows the mash to settle very slowly and form a well-delineated filter bed.
The lauter mash should be very thin and thoroughly intermixed to encourage the absorption of the malt extract into solution and to temporarily force small starches and proteins into suspension, allowing the husks to freely settle.
## Wort Separation
The purpose of sparging/filtering is to rinse the soluble extract free from the malt husks and to trap insoluble, poorly modified starch, protein, lipids, and silicates within the husks. Without adequate filtering, extract is lost, while the mash runoff is clouded by starch, proteins, tannins, and husk particles. This produces beer likely to be cloudy, astringent, and unstable.
The mash may be transferred to a lauter-tun for filtering. The diameter of the vessel should allow the filter bed to form to a depth of twelve to eighteen inches. A filter bed of six-row barley, however, may need to be only six inches thick; the greater percentage of husks in six-row barley increases its filtering efficiency. The mash filter bed when brewing with infusion-mashed British ale malts is more commonly twenty-four inches deep at the start of the runoff.
The husks accumulate on a false bottom, or filter plate, that fits one-eighth to two inches above the real bottom of the lauter-tun. Slots or perforations in the plate allow the sparge water to slowly and evenly filter through the husks. The lauter-tun itself is equipped with a spigot located below the level of the false bottom to draw the extract-rich sweet wort off from below the grain mass.
### Setting the Lauter Bed
The lauter-tun should be filled to one-half inch above the false bottom with water of 175 to 212 degrees F (80 to 100 degrees C). This preheats the lauter-tun and reduces the amount of debris that is otherwise carried into the space below the false bottom. This practice largely eliminates the need for flushing the space prior to filtering and improves the clarity of the runoff. The thin lauter mash is quickly transferred to the tun, given a last thorough stirring, and allowed to settle.
Mash Filter Bed
An infusion mash, on the other hand, is handled gently. Stirring is avoided. An unstirred infusion mash does not settle in as well-defined stratification as does a decoction mash, and tends to "float." The suspended particulate matter somewhat offsets the lack of a clearly defined filter bed, as it entraps less extract. However, it never yields so clear a runoff as does a decoction-mash filter bed. Commercially, false bottoms for infusion mashes generally have larger slots than those used for decoction mashes.
Within ten to twenty minutes, the liquid displaced by the settling mash should show clear and "black" above a nebulous cloud of trub. If it doesn't clear, then filtering efficiency can be expected to be poor. The temperature of the mash is likely to drop during the setting of the filter bed; every effort, however, should be made to limit its heat loss.
In an ideal stirred-mash filter bed, the heavy hulls that settle onto the false bottom are covered by a deeper layer of lighter hull fragments and endosperm particles. Until this porous filter-mass has formed, tiny, gelatinized particles of starch and protein remain suspended in the liquid; after it has formed, they should settle out, creating the pasty "protein-sludge" or upper dough.
If this trub precipitates too early and settles within the hulls in any appreciable quantity, some will wash into the sweet wort runoff. It may also cake within the filter bed and cause a set mash that blocks the flow of liquid down through the filter bed; either a set mash or a runoff that doesn't clear may be due to ineffective crushing and mashing.
### Sparge Liquor
A volume of liquor roughly 25 percent greater than the liquor used for mashing-in is heated to 170 to 176 degrees F (77 to 80 degrees C) in preparation for sparging. The temperature of the sparge liquor is critical because sugars flow more freely in hot solution than in cold. Its temperature must be maintained throughout the sparging to dissolve and rinse free the extract cupped in the hulls or adhered to the malt particles.
Excessive temperatures in the mash itself, however (above 170 degrees F [77 degrees C]), rupture balled native-starch particles and decompose the protein sludge, causing them to be carried away in the runoff. Because only very simple protein and carbohydrate fractions can be managed by culture yeast, none of these more complex fragments should be allowed in the wort. Runoff temperatures above 170 degrees F (77 degrees C) also cause extraction of husk polyphenols and marked astringency in the beer.
If necessary, the salt content or acidity of the sparge liquor should be adjusted, preferably with calcium salts or lactic or phosphoric acid, to limit extraction of harsh-tasting malt fractions and improve clarity. The pH of the runoff should never rise above pH 6; better results are achieved when the runoff pH does not rise above 5.8.
Carbonate waters are not useful for sparging, because they induce haze fractions and silicates into solution and may induce renewed enzyme activity. These waters become even more alkaline upon heating. Carbonate ions must be precipitated or disassociated before the liquor is used for sparging.
### Preparing to Sparge
The space below the false bottom can be purged of particulate matter by flushing it with clear, 170 degree F (77 degrees C) liquor, either through an inlet below the false bottom, opposite the spigot, or through a tube thrust down through the mash. The inlet and runoff rates must be carefully matched to avoid disturbing the filter bed above. Flushing can be eliminated if the cloudy runoff is refiltered through the mash until it runs clear (vorlauf vehrfahren), or the runoff is starch free.
The degree of clarity that should be obtained in the runoff is a matter of debate. A lot of draff carried into the kettle is a recipe for astringent beer, but a small amount may improve trub coagulation. The majority of brewers recycle until the runoff is no longer heavily clouded; this is generally accomplished in less than ten minutes. Excessive recycling may lead to greater lipid levels in the wort and ought to be avoided.
To set the filter bed and settle the protein sludge, the lauter-tun spigot is opened once the liquid above the grains clears. The liquid is run off very slowly until it stands one-half to one inch deep above the surface of the mash. This liquid level must be maintained throughout sparging. Draining below the mash surface level causes the mash to settle too tightly and the protein sludge to cake. Extract efficiency is reduced and the potential for developing a set mash increases. Too great a liquid depth, on the other hand, acts as weight on the grains and leads to stuck mashes. The mash surface should be periodically leveled and smoothed to fill in all the depressions and vertical channels.
### Sparging
The liquid level above the filter bed is maintained by the introduction of sparge liquor. Sparge liquor should be gently and evenly dispersed over the top of the mash so it will evenly percolate through the mash and diffuse all the extract from it. The sparging rate should be free from surges and matched to the runoff rate so that the liquid level in the lauter-tun is not changed.
The sparging/runoff rate may be gradually increased, but not so much so that turbidity is caused in the runoff. Set mashes also result from too rapid a flow rate.
High-husk, six-row barley may be run off in less than an hour (a six-inch-deep bed may be filtered in as little as one-half hour). Maximum extraction, however, is achieved with a very slow runoff rate, a deeper filter bed, and raking the mash to within six inches of the false bottom. Raking restructures the filter bed, ensuring even percolation of the sparge liquor through the grain and complete extraction of the sugars. A mash that is raked, or that is from finely ground malt, or shows a tendency to set, must be run off slowly. Set mashes that don't respond to being stirred must be cut repeatedly during sparging in order to reopen channels of extract flow.
Within 1 ½ hours, the greater part of the extract will have been leached from the malt. Although the maximum yield is obtained by restricting the runoff rate so that it takes two to four hours to collect the sweet wort, the small percentage of extract gained is not worth the time and effort.
The temperature within the mash should be carefully maintained during sparging and filtering, although the early runoff will usually be well below 168 degrees F (75 degrees C). In the interest of preventing further enzyme activity, the wort collecting in the copper should be heated to above 170 degrees F (77 degrees C) as it accumulates. Where a large amount of evaporation is required, the sweet wort is brought to boiling and partially hopped as soon as it has covered the bottom of the kettle.
As the color of the runoff pales, its extract content is periodically checked with a hydrometer; when the reading drops to below 3 °Plato (SG 1012, corrected to 68/60 degrees F [20/15.56 degrees C]), the runoff is diverted from the wort kettle.
Below this density, the runoff pH is likely to rise above pH 6, increasing the likelihood that malt tannins, lipids, and silicates will be leached into it. Malt tannins give an astringent taste and are harsher flavored than hop tannins. They are more soluble and are not as readily precipitated in the kettle. Lipids interfere with foam stability, increase ester formation, and are precursors to cardboardy, stale flavors in beer.
# CHAPTER 11
# Boiling the Wort
Vigorously boiling the mash runoff produces several desirable effects: it destroys mash enzymes, sterilizes the wort, and stabilizes salts in solutions. It extracts hop resins, drives off kettle-harsh hop oils, and coagulates and precipitates unstable protein. Boiling also evaporates excess water, lowers the wort pH, and creates a stable medium for controlled fermentation by the culture yeast. Boiling may begin when enough wort has been collected to cover the bottom of the kettle.
## Kettle Hops
Hops should be added to the kettle by being scattered over the surface of violently boiling wort. They may be added all at once, but more commonly they are meted out in portions throughout the boil. The actual sequence is determined by the hop character that is meant to be carried over into the finished beer.
Adding hops early on in the boil ensures greater utilization of bittering principles and a more complete precipitation of proteins, hop tannins, and hop particles. A sixty-to-ninety-minute boil succeeds in isomerizing 25 to 30 percent of the alpha resins and in bonding them to the wort as iso-alpha-acids. With pelletized hops, ruptured and better-exposed lupulin glands give greater utilization, even as high as 35 percent. This is the greatest percentage of hop bittering and preservative principles that is normally ever carried over into the finished beer. On the other hand, the bitterness derived from long boiling is coarser than that from a more moderate period; for this reason, it is usual to add only a fraction of the hops at the start of the boil.
Some of the hop polyphenols are transported into the ferment in combination with simple albumins, forming tiny substances-in-solution known as _colloids._ This colloidal matter is not significantly precipitated and is involved in forming the body and head of the finished beer. Because their surface area is disproportionately greater than their volume, colloids do not readily settle out of solution. Consequently, their contribution to the beer's body is not offset by inherent instability, as is the case with noncolloidal protein.
It is common to add 5 to 15 percent of the hops at or before the onset of boiling to break the surface tension of the wort so that it does not throw up as voluminous a protein head and boil over. When the wort is the product of an infusion mash, it should be boiled vigorously for fifteen to thirty minutes before more of the hops are added to allow the boiling action to decompose and precipitate some of the proteins. If this is not accomplished before the hops are added, then hop polyphenols will combine with the coarse protein flocks and be precipitated out of solution, carrying hop resins with them.
Even an intense initial boil, however, does not eliminate the large proteins as effectively as do the processes of decoction mashing. Although the proteins can be precipitated, they cannot be dissolved into albumin, peptides, and amino acids, because all enzyme activity has been terminated by the boil.
When the wort is the product of a decoction mash, excessive complex proteins aren't usually a problem. The several boilings and rests largely reduce or eliminate them. Decoction-mashed wort can therefore be hopped somewhat more conservatively than infusion-mashed wort, simply because the hops need not overcome a great amount of protein.
All of the aromatic hop character of the beer is lost during a long boil. The hops' volatile essential oils and esters can be preserved by adding hops later in the boil. It is usual, in fact, to add the hops in two, three, or even four portions. Only lightly hopped beers that employ hops for their preservative contribution rather than for their flavor and aroma fully extract the entire quantity of hops during a sixty-minute or longer boil.
Beers that are heavily hopped in the beginning of the boil exhibit a cleaner kraeusen fermentation head and are more stable than beers hopped later, but the hop bitterness will be coarser and less pleasant. It is essential, however, that most of the hops should be vigorously boiled in the uncovered wort for forty-five minutes or more to efficiently isomerize alpha acid and precipitate tannin and proteins. Generally, a small portion of the bittering hops is added to the kettle with the first mash runoff. The largest part is added to boil for forty-five to sixty minutes. A smaller portion may be cast onto the wort fifteen to thirty minutes before the boil ends. Finishing hops, which give the beer a spicy hop flavor and bouquet, may be added within the last minutes of the boil, as the wort is struck from the kettle, or as an extract during fermentation.
## Hop Rates
The quantity of hops is determined by several factors: the desired bitterness level, the hop flavor and aromatic character in the finished beer, the alpha-acid content and condition of the hops, and the efficiency of the hop extraction.
Hop acids have limited solubility and ability to isomerize, which lessen with increasing wort gravity. Usual lager hop rates are approximately .2 to .4 ounces of hops per gallon of cooled wort, but may be as low as .15 ounces or as high as .75 ounces per gallon, depending on hop quality, the alpha acidity of the hops, the beer type and its density. Contrary to what might be expected, hop acids become less soluble as wort density increases. Kettle-hop rates may or may not be increased to balance the terminal density of some beers; rather, finishing-hop rates may be increased so that hop flavor, not bitterness, balances the sweetness.
## Establishing the Evaporation Rate
As soon as all of the sparging runoff has been brought to a full boil, the wort's extract content and volume can be measured. With the hops fully submerged, correct the volume to 60 or 68 degrees F (15.56 or 20 degrees C). Correct the volume to 60 degrees F by multiplying volume at full boil by .959, to 68 degrees F by .960 (the displacement of wort by the hops is insignificant). These figures can be used to establish the evaporation rate necessary to evaporate the wort to the desired volume and concentration within the prescribed parameters of the boil. The usual evaporation rate in an uncovered boil is 10 percent per hour, although the actual rate depends upon the wort's surface area, surface tension, kettle geometry, the amount of energy applied, ventilation, and the ambient atmospheric pressure.
It may happen that the brewer will need to proceed with a boil that will yield less wort than is needed to satisfy fermenting, priming, topping-up, or kraeusen and yeast-culturing requirements. If a correction must be made, then the volume is generally allowed to vary from what was expected. Too great or too small a volume of sweet wort is of less concern than the correct density. When extract-poor malt, inefficient mashing or sparging, or miscalculation results in a wide disparity between the density that was expected and what occurs, a lighter-density beer must be accepted or the extract content increased with malt-extract or wort, should any be available.
Never boil for less than the prescribed time. The kettle may be partly covered for part of the boil to control evaporation, but the wort must be vigorously boiled, uncovered, for at least the final thirty minutes to drive off harsh, volatile kettle-hop and malt oils, sulphur compounds, ketones, and esters.
Never simmer the wort in lieu of a vigorous boil. Efficient hop-resin isomerization, albumin/resin bonding, and protein/tannin precipitation is achieved only through the agitation of the boil. In fact, a violent boil has the greatest influence on the stabilization of the wort. If movement cannot be induced by the circulation of thermal currents in the wort (heating the kettle asymmetrically improves circulation), then agitation becomes increasingly important. Oxygenation improves flocculation, but at the unacceptable cost of oxidizing and discoloring the wort. Aeration of the mash, wort, or beer at any time except after wort cooling should be avoided.
Once adjustments to the volume of the wort have been made, the pH of the boil should be checked in a sample cooled to 68 degrees F (20 degrees C). Optimum protein flocculation occurs at above pH 5.5, but an initial pH of 5.2 to 5.5 is more appropriate to satisfy the other pH requirements of wort boiling and fermentation. Corrections to the wort acidity should be made with acid or calcium carbonate. Lower pH values produce fewer, smaller flocks; below pH 5.0, the protein does not coagulate. Whenever the pH is less than optimal, agitation and movement within the kettle become increasingly important to flock size.
The pH of the wort drops during boiling as calcium phosphate is precipitated out of solution (sodium and potassium phosphate are unaffected by boiling); usual pH reduction is approximately 0.2 for a sweet wort of 5.5, and .3 for a pH of 5.8.
## Hot Break (Kettle Break)
Samples periodically taken from the wort and viewed in a glass container should reveal the progressive flocculation of albuminous protein with hop tannin (polyphenols). Invisible in suspension, they first appear as a mist of tiny flakes that cloud the wort soon after boiling commences. The rolling motion of the boil causes the malt proteins to collide with and adhere to the sticky hop polyphenols. The particles rapidly coagulate into a much smaller number of larger flocks one-eighth inch across, roughly composed of 50 to 60 percent protein, 20 to 30 percent polyphenols, 15 to 20 percent resins, and 2 to 3 percent ash. Upon resting, these large flocks should readily precipitate, leaving the sample brilliantly clear.
## Cold Break
As the end of the prescribed boil approaches, samples taken and force-cooled to below 50 degrees F (10 degrees C) are examined. The wort that showed clear when it was hot should cloud slowly as it cools, as previously invisible coagulum loses its solubility in the cooler solution. This cold break should settle, again leaving the wort clear, bright, and sparkling.
The wort must be boiled past a positive cold break in the sample, and flavoring hops should not be added until after the break has been achieved. It is important that the break samples be evaluated; however, boiling should not be extended beyond the recommended time even when the break is poor. A scarcity of flocculum in a well-agitated, strong boil at the proper pH may be caused by malt of poor quality or by either an excessively long or insufficient albumin rest. In the first case, almost all the albumin has been reduced to amino acids or retained in the spent grain, and the beer can be expected to be thin. In the latter case, the protein is too complex to coagulate, and the beer will lack stability and be prone to serious oxidation and taste impairment.
In any case, no correction in the kettle is possible if temperature, pH, and movement of the wort are all satisfactory. The boiling should not be extended unless it is subsequent to a pH or temperature adjustment to the wort.
If a satisfactory break cannot be established because proteolysis has been insufficient, the only recourse is to rack the beer off its sediment several times during fermentation and lagering to separate it from proteins in the trub, and to chill it or tightly filter it before packaging. Even so, the beer may form a chill haze.
## Finishing Hops
Finishing hops are usually the very finest hops, chosen for their flavor and aromatics. Generally they are only a fraction of the quantity of kettle hops employed. Fragrant hops are broken up and added to the kettle or the hop back, or an extraction of their hop oils is infused into the cooled wort or fermented beer.
The later in the brewing cycle that finishing hops are added, the greater their bouquet will be. Flavoring hops are commonly added ten or fifteen minutes before the end of the boil for lager beer, so that humulene, carophyllene, and their oxidation products are effectively extracted by exposure to the boiling-hot wort. Late hops contribute little bitterness to the beer and only subdued aroma, but they give the beer a crisp hop flavor. Some aroma hops may be added as the wort is filtered through the hop bed.
Later addition of hops is made only when a distinctive hop aroma is desired. Traditionally, whole hops are added to British ales, even up to the point of packaging, but a hop extract is more appropriate for lager styles. A hop extract can be made by steeping aroma hops for ten minutes or more at pH 5.5 or above in four fluid ounces of wort per each half ounce of hops. An extract is usually added to the wort post-primary, so that none of it is lost in the hop and trub residue and the aroma is not scrubbed out during primary fermentation. Extracts give a cleaner kraeusen head than adding loose or bagged hops post-kettle and present less risk of contamination. The aromatic character of an extract varies substantially from that achieved by dry-hopping; boiling drives off some volatile essential oils (myrecene, thioesters) while extracting others (humulene, carophyllene). Overall, aroma from an extract is milder, spicier, and less grassy/weedy than that derived by dry-hopping.
The hop nose and flavor characteristic of most lagers is obtained by adding loose hops to the wort at or shortly before the conclusion of the boil. Even the very hoppy character of some lagers is attributable to liberal kettle finishing-hop rates rather than to dry-hopping.
Hop nose and flavor are matters of personal preference; finishing-hop rates may be adjusted to suit the brewer's preference, as well as to reflect the aromatic quality of the hops being used.
## Straining the Wort
At the end of the recommended boiling period, the wort should be at its desired volume and concentration (both corrected to the reference temperature, 60 or 68 degrees F [15.56 or 20 degrees C]). The hot wort may simply be siphoned off its hop and trub residue, but this causes an unreasonable amount of extract to be lost. Where pellets are used, the wort is generally whirlpooled for several minutes, settled until it is clear (generally ten to fifteen minutes), and then run off from a side outlet. Otherwise, it is more efficient to strain the wort through a loose bed of hops, two inches thick, in a large strainer (hop back) or on a false bottom (for example, perforated with sixteenth-inch holes on eighth-inch centers, or slots .062 inches wide covering 30 percent of the surface). The wort may be recycled, very slowly at first, to settle the hops, and returned to the liquid above the filter bed until it runs clear.
In all cases, the wort should be run off or filtered through the hops before it cools below 170 degrees F (77 degrees C). The first clear runoff may immediately be force-cooled and mixed with the yeast starter to facilitate adaptation of the yeast upon pitching. When all of the clear wort has been run off, the hops can be slightly sparged with up to eight fluid ounces of boiling water per ounce of hops, or until the density of the runoff drops below 5 °Plato (SG 1020). The extract still retained by the hops is insignificant — never attempt to press or wring out the last of it. Great care should be taken to see that only clear runoff is taken for cooling and fermentation.
## Cooling the Wort
The clear runoff must be quickly cooled to separate the cold break trub from the wort. Fast cooling is essential; the more slowly the wort cools, the more protein and tannin is trapped in suspension, giving rise to chill haze and harsh aftertastes in the beer. The cold break is generally 10 to 20 percent of the volume of the hot-break sediment, and much less coarse.
Cooling in lager breweries traditionally took place in shallow, open coolships to present maximum surface for air cooling. Better flocculation is achieved, however, by force-cooling the wort and employing a deeper settling tank, closely covered against contamination. Below 145 degrees F (63 degrees C), great care must be taken to prevent contamination of the wort by airborne wild yeast and bacteria or unsterilized equipment. The wort should be force-cooled to below 50 degrees F (10 degrees C) to secure the maximum break. Complete precipitation of tannin/proteins — and thus brilliantly clear beer — is achieved by cooling the wort until it becomes slushy, but cooling to 39 to 43 degrees F (4 to 6 degrees C) before racking the beer off of its settlement is generally sufficient.
Since boiling the wort drives its oxygen out of solution, it must be aerated to force oxygen back in. Yeast require considerable (4 to 14 ppm) molecular oxygen during respiration; without it, they cannot reproduce. Cells that survive an oxygen-starved respiratory phase taint the ferment with abnormal, estery flavors. Their lag phase is characteristically shortened, reproduction is limited, and their fermentation is sluggish. Oxygen starvation produces "petite mutants," which ferment weakly and often incompletely, giving a peculiar and cloying diacetyl taste and other off-flavors.
In an oxygenated wort, the yeast splits the sugar molecule in such a fashion that it produces more CO2 than alcohol. The carbonic gas rising to the surface quickly forms a blanket above the ferment, which insulates it from airborne infection. It may also carry with it a film of debris that can be readily skimmed from the head during the kraeusen stage of fermentation.
Aeration by rousing the wort when it is hot saturates the wort more completely than does aeration of the cooled wort. The risk of airborne contamination is less while the wort is above 145 degrees F (63 degrees C), and aeration of the hot wort causes some of the oxygen to combine with protein fractions, improving the cold break. It would seem that the wort should be aerated when hot, but oxidative polymerization of polyphenols to tannins and oxidation of wort constituents create very objectionable flavors. The color darkens when the hot wort is aerated and flavor suffers irreversible oxidation damage. Aerating the cooled wort (at 60 degrees F [16 degrees C] or below) is always preferred to aerating hot wort, and it yields satisfactory dissolved oxygen (up to 8 ppm). It is essential that the air or oxygen be sterile to preclude contamination of the extract.
If a settling tank is being employed, the cold break should be well settled before the wort is racked into the fermenter. The pH of the wort should be 5.0 to 5.5. With infusion-mashed and ale worts, a pH of 5.0 to 5.2 is usual, but for lager beers a cooled-wort pH of 5.3 to 5.5 is still considered normal.
# CHAPTER 12
# Fermentation
Any successful fermentation proceeds along a predictable course dictated by the composition of the wort and the characteristics of the yeast strain. The amount of extract, its dextrin/maltose ratio, the amount and complexity of the nitrogenous matter, the availability of yeast nutrients, the pH, and the oxygen saturation and biological purity of the wort are values that have been fixed by mashing particular malts and boiling, cooling, and aerating the wort.
The only significant influences upon the fermentation that can be manipulated by the brewer are those of the yeast — its character, purity, vitality, quantity, and its rousing — and the temperature and duration of the ferment. Changes in any one of these can affect the residual sugar, mouthfeel, clarity, aroma, and flavor of the beer.
## Preparing for Pitching
The fermentation cycle should never be started with a weak yeast; such yeast will only be made weaker upon being diluted into the wort. The yeast to be pitched should have been cultured in a wort similar to that being brewed. Yeast that must undergo significant adaptation suffers a high rate of mortality, causing fermentation to start slowly and be relatively weak. The likelihood of contamination increases and decomposition of dead yeast cells mars the beer flavor.
pH in Typical Lager Brewing
Sugar solutions should not be used to culture yeast. When yeast is cultured in solutions lacking maltose, it loses its ability to absorb maltose, and suffers from unreasonably long periods of adaptation upon being diffused into the wort.
Approximately .5 to .6 fluid ounces (10 to 14 grams) of pasty, thick yeast is needed for each gallon of wort to be pitched to give 12 to 15 x 106 cells per milliliter of wort. Up to one fluid ounce (21 grams) of yeast is necessary for each gallon of wort when a very strong start is needed, or when the yeast is a weak fermenter. That much is also needed for worts of very low or high extract content. In the first case, more yeast ensure an adequate start in the nutrient-poor environment, and in the latter case are needed to ferment the greater amount of extract.
Pitching too much yeast, however, overtaxes the supplies of dissolved oxygen, simple sugars, and yeast nutrients and can result in yeast autolysis (self-digestion). The rapid fermentation and premature autolysis can result in fusely, estery and rubbery, yeasty and sulfurous flavors.
Less yeast than the amount recommended above should be pitched when the yeast strain has proven to be a very strong fermenter. A good culture should require only .4 fluid ounces (8.5 grams) of yeast slurry per gallon of wort. Conservative pitching rates (.8 to 1 x 106 cells/milliliters per each °Plato of the wort) of healthy yeast are the rule unless experience dictates otherwise.
Conservative pitching rates should not be confused with inadequate pitching rates. Pitching too few yeast results in long lag and reproductive phases, estery aromas, and increased risk of contamination. English ales are commonly pitched at far lower rates (.6 to .8 x 106 cells/milliliter per each °Plato of the wort) where esters are meant to be prominent in a beer's aroma. In lager brewing, ester formation is repressed in part by limiting reproduction through higher pitching rates. Ale yeasts may grow to five to eight times the amount pitched, while lager yeasts generally increase three to five times.
The pitching yeast should be clear, white-to-tan, its sediment should be thick and rich, and its aroma pure and pleasant. If the culture is collected from the parent ferment at the height of kraeusen and repitched reasonably quickly, the sediment will be composed mainly of healthy yeast. Regular rousing of the culture produces more yeast and a richer sediment. Dusty yeast that remains suspended and does not form a rich sediment by the time it has thrown up a rocky foam cover is unsuitable for pitching.
The yeast culture can be the most significant source of microbial infection to the ferment. A culture that smells or tastes off will produce disastrous results if pitched. Yeast must be handled carefully, so that the brewing strain is cultured, not wild yeast or bacteria.
If a dry yeast absolutely must be pitched, then slurry two to four grams of granulated yeast per gallon of cooled wort into twice its volume of sterile 100 degrees F (38 degrees C) water to minimize shock excretion while the desiccated yeast resuscitates. Cover it and rest it for thirty minutes. It should be raising a frothy head before it is pitched.
(A note of caution: granulated dry yeast may be contaminated by significant quantities of bacteria. It is probably the least viable and most often contaminated source of brewing yeast. Subculturing from a slant, frozen, or liquid culture, or kraeusening from a healthy ferment is more likely to produce a satisfactory fermentation cycle.)
Before pitching, the yeast can be forcefully roused into the first clear runoff from the kettle (force-cooled to the pitching temperature and racked off of its sediment). The purpose is to aerate and evenly distribute the yeast, and allow it to adapt to the extract while the rest of the wort is being force-cooled and sedimented.
Cooled wort in excess of that to be fermented can be run off into the containers in which it will be stored until it is used for topping-up, priming, kraeusening, or for yeast culturing. The amount of wort removed should at least be equal to the requirements listed in table l4. Refrigerate the tightly capped containers until needed; they will keep for at least six months at 33 degrees F (1 degree C).
## Kraeusening
_Kraeusen_ is the German word used to describe the infusion of a strongly fermenting young beer into a larger volume of wort or beer that is past the stage of strong fermentation. Kraeusen beer introduces vigorous yeast in its own sugar-rich substratum. It is characterized by the active raising of a tightly knit or rocky foam head. Yeast colonies should visibly cloud the liquid below.
Kraeusen beer should be taken only from ferments that exhibit textbook characteristics. Although successive kraeusening may encourage the culturing of wild or dusty yeast, it still remains the best method by which fermentation may be induced in the cooled wort.
The culture must be strong, so that it is neither overwhelmed by the larger volume of wort nor unable to renew active fermentation in a well-aged, extract-depleted beer. Kraeusen is traditionally obtained from a strong primary fermentation, but it may be made from wort and a yeast culture _(yeast starter)_. Sterile wort is pitched with a culture, and its volume is increased to at least 5 percent of the wort volume by doubling. (The starter volume is built by adding wort in up to a ten-to-one ratio each time vigorous fermentation becomes apparent.) It should be in full kraeusen when it is pitched.
When 10 percent new beer (just coming into high kraeusen) is used to induce fermentation in the cooled wort, the yeast lag-phase is virtually eliminated. The yeast, having adapted to the solution, require no respiratory phase to develop a cell membrane and enzymes appropriate to the given wort. Initial fermentation is stronger. Employing kraeusen beer to top up the secondary (lager) fermenter induces a strong temporary fermentation and reduces diacetyl and the risk of oxidation and contamination at racking.
When wood chips are used to clear the lager beer, _aufkraeusening_ is absolutely necessary. There must be movement of the aged beer so that every part comes into contact with the latticework of chips. The introduction of kraeusen beer at a temperature 5 degrees F (3 degrees C) warmer than the aging beer and the subsequent fermentation create the necessary movement of the whole volume of beer. The chips alone would not otherwise clarify an aged beer.
Kraeusening also reduces lagering time by introducing vigorous fermentation, capable of more rapid metabolism of the small amount of fermentable sugar in the aging beer than the few yeast cells already in the solution.
Good sanitary procedures are an absolute necessity, and whether it is being mixed with wort or with aged beer, the new beer should be well roused in.
## Pitching the Yeast
For lager beers, yeast is generally pitched into wort that is at or near the lower end of the intended fermentation temperature range. Although yeast will generally reproduce more quickly if pitched into relatively warmer wort, it will also produce more diacetyl, fusel alcohols, and esters.
Where practical, it is advisable to separate the chilled wort from the cold break in a settling tank. The yeast may be pitched in the settling tank up to twelve hours before the wort is transferred to a fermenter, or during the transfer. The wort temperature is generally allowed to rise to 39 to 43 degrees F (4 to 6 degrees C) before it is racked off the cold break into the fermenter along with the yeast starter, slurry, or kraeusen beer. Where a settling tank is not employed, the wort is generally pitched at 42 to 47 degrees F (6 to 8 degrees C).
The pitching yeast is commonly added at a temperature up to 5 degrees F (3 degrees C) warmer than the wort, and well roused into it in an aerating fashion. Because brewers' yeast requires considerable dissolved oxygen (eight to twelve milligrams/liter) to synthesize fats to cell-wall consituents, a stream of air is often used to effect this mixture of yeast and wort. An aeration stone or other device that increases air-to-wort surface contact improves oxygenation. Gentle rousing should be continued throughout the transfer to achieve an intimate admixture and to dissolve oxygen into the wort.
Care should be taken that the wort's trub sediment is not disturbed, especially as its draining nears completion. Good hot and cold breaks are meaningless if a significant amount of trub is carried into the ferment. Racking should cease as soon as the runoff shows the least bit cloudy; trub carried into the ferment taints the beer with objectionable flavors and aromas.
Proteinaceous precipitate from the hot and cold breaks forms the greatest part of the trub. Although amino acids are absolutely necessary for yeast metabolic functions, yeast react to an excess of simple protein by generating aromatic fusel alcohols. Even more of these volatile carbonyl compounds are excreted when the wort has been underoxygenated. Fusel alcohols are subject to esterization, which produces fruity and solventlike odors that are inappropriate in a lager beer, and to oxidation, forming "stale"-tasting aldehydes.
Trub also contains polyphenols, ketones, and sulfur compounds that may be absorbed into the ferment. Polyphenols give astringent-tasting, mouth-puckering flavors. Volatile sulfur compounds (H2S, DMS, thiois, and mercaptans) produce rotten-egg, skunky, onionlike, rubbery, and burnt-match flavors and odors.
## The Fermentation Lock
Although the release of carbonic gas from the fermentation gives it some measure of protection against oxidation and contamination, covering the fermenter immediately after pitching and fitting it with a fermentation lock is advisable. The airlock allows the pressure created by the carbonic gas to push past the liquid in the lock without allowing air in. This prevents the reverse passage of airborne wild yeast and bacteria into the culture-yeast fermentation. Oxygen trapped within the fermenter by the lock is readily driven off by the rising blanket of heavier carbon dioxide produced by the ferment.
The liquid in the lock should be maintained at a constant level, but not so deep that it puts the fermenting beer under any appreciable pressure. During the primary fermentation, it is essential that virtually no carbon dioxide remain in solution, since it carries malt and hop debris into the head and sulphur compounds and esters out of the ferment.
The purpose of the fermentation lock is to prevent infection. It must be kept perfectly clean. The trap can be filled with an antiseptic solution in which microbes cannot exist, a practice that is advisable during later fermentation. Of course, this solution must not contact the ferment either by splashing caused by excess pressure or by careless handling.
## Primary Fermentation
Five to twelve days may elapse from the time the yeast is pitched until vigorous fermentation abates; six or seven days is usual. Ales are fermented at higher temperatures over a relatively shorter period of time to develop characteristic esters and other fermentation flavors. Lagers are suited by lower temperatures, which retard fermentation times and ester development.
The duration of the primary fermentation is also subject to the strength and reducing characteristics of the pitched yeast strain. Nonselective strains that completely ferment the extract work very quickly but produce a thin, inferior-tasting beer. Temperamental strains such as Saaz yeasts incompletely convert the extract during a relatively long, weak ferment but produce a richer-tasting and fuller beer.
Normal primary fermentation is verified by its characteristic low, high, and post-kraeusen stages. Where deviations are encountered, the source of the irregularity should be investigated, identified, and corrected as soon as possible.
## Temperature
The temperatures quoted here apply to dextrin-rich lager worts of 10 °Plato (SG 1040) or greater, producing full-bodied beer. If a high-maltose wort relatively free of haze-forming protein fractions is being fermented, or if a top-fermenting yeast strain is used, temperatures should be higher by 6 to 20 degrees F (3 to 10 degrees C). Fermentation times will be correspondingly foreshortened.
Do not exceed the recommended temperatures when fermenting a dextrinous wort with lager yeast. The yeast requires the longer fermentation time at the lower temperatures to break down and convert the less readily fermentable dextrinous sugars. Moreover, higher temperatures invariably cause an increase in esters, fusel alcohols, and solventlike flavors that are inappropriate in lagers.
## Temperature Control
Fermentations generate heat. The temperature of any ferment must be monitored and the excessive heat drawn off by lowering the ambient temperature. In no case should the internal temperature of a lager ferment exceed 60 degrees F (15 degrees C), and it should be limited to a cumulative increase of 7 to 14 degrees F (4 to 7 degrees C) relative to the starting temperature.
Ideally, the maximum temperature should not rise above 47 to 52 degrees F (8 to 11 degrees C) when employing traditional lager strains. The maximum temperature may be maintained through high kraeusen until yeast nutrients are depleted, yeast activity slows, and heat generation ends, or even raised for "diacetyl" rest, but it is usually lowered soon after the maximum temperature is reached.
Controlling the temperature at the beginning of fermentation is more important than controlling the temperature near the end of fermentation, because esters and fusel alcohols are largely produced when the yeast is respiring, during the lag and reproductive phases of fermentation.
Temperature changes at any stage of the fermentation should not exceed 5 degrees F (3 degrees C) daily. Abrupt reduction in temperature will shock the yeast and may arrest fermentation completely. The sudden death of many yeast cells deleteriously affects flavor; moreover, yeast mutations tend to adapt to a sudden temperature change more readily than culture yeast. Temperature maintenance and modification must be handled carefully.
Convection currents within the ferment (formed by asymmetrical cooling of the fermentation vessel) improve temperature distribution and yeast performance, producing a more even fermentation.
## Density and pH Monitoring
The acidity of the ferment increases as the yeast adapt and respire glucose to succinate and other organic acids. With top-fermenting yeast, the pH drop during respiration is dramatic; it falls .4 to .6 within twelve hours of pitching and to pH 4.0 within twenty-four hours, before it levels off as fermentation begins in earnest. With lager yeast, the pH drop is much less precipitous; for a wort of pH 5.3, a .5 drop requires forty-eight hours, and the pH only falls to 4.5 or so by the end of primary fermentation. One measure of consistent yeast performance is its effect on the pH of the extract solution. To this end, monitoring the pH is critical during the lag phase of a top-fermenting culture, and at the low, high, and post-kraeusen stages of a lager ferment.
The liquid pressure of the fermenting beer also makes its most dramatic drop during primary fermentation and should be regularly checked and logged to define yeast activity and pinpoint racking time.
## Primary Fermentation: Lag Phase
After pitching, yeast take some time adapting to the conditions of their new environment. During this "lag" phase, there is little visual evidence of their activity. How successfully the yeast culture adapts to the wort depends upon the number and condition of the yeast cells and the nature of the wort itself. Temperature, density, glucose/maltose content, amino acid availability, and the level of dissolved oxygen all influence yeast behavior.
At pitching, yeast rely on free oxygen, wort fatty acids, and glycogen, an intracellular carbohydrate reserve, to provide energy for the synthesis of wort-specific enzymes and a permeable cell membrane. Glycogen is structurally similar to the amylopectin of malt starch, but with a greater number of shorter branches. Without adequate glycogen reserves, the pitched yeast cannot survive until they can develop the ability to absorb and metabolize wort sugars and nutrients.
During the lag phase, yeast employ a complex enzyme system to hydrolyze the polymeric glycogen to glucose. The glucose molecule is phosphorylated and its carbon links broken (glycosis) to yield two triose phosphates. These simpler three-carbon compounds are oxidized to pyruvic acid (pyruvate, an important yeast oxo-acid) with the release of energy by formation of energy-rich ATP, adenosine triphosphate, from energy-depleted ADP.
An inadequate starch reserve may be characteristic of a particular yeast strain, but more often this is due to depletion of glycogen from storing a culture for too long or at too warm a temperature. A culture that survives glycogen deprivation produces abnormal levels of vicinal diketones (especially diacetyl), marring the beer flavor. Fermentation takes longer and is less vigorous, because there are fewer cells, and consequently, slower yeast growth.
Yeast cells also store simple acids, alcohols, nitrogen, and phosphates catabolized from compounds assimilated during culturing. At pitching, yeast employ other enzyme groups to combine these simple chains, synthesizing many of their complex structural and metabolic requirements. Pyruvic acid is reduced to oxaloacetate, from which amino acids and proteins may be synthesized, or to acetyl Co A, an acyl Co A, an acetic-acid-related sulfur compound that can be oxidized to a host of fatty acids, triglycerides, and lipids required for cell-membrane synthesis. Molecular oxygen is required for these reactions. The synthesis of a ten-fold increase of sterols in the yeast to make cell walls permeable requires oxygen. Unless there is sufficient dissolved oxygen in the wort, the formation of a cell wall able to react to and regulate uptake of the particular sugars and nutrients in that wort will cease. Unable to selectively absorb nutrients from solution, many yeast cells will autolyze, and surviving cells will not develop normally. Both scenarios produce off-flavors in beer.
Lacking dissolved oxygen, acetyl Co A esterifies alcohols, including fusel alcohols. These solventlike and harsh-tasting "higher" alcohols are intermediate products of amino-acid metabolism and are normally oxidized back to organic oxo-acids. When respiring yeast lack oxygen, fusel alcohols may be excreted or dehydrated by acetyl Co A to esters. The principle ester formed is ethyl acetate, which irreversibly flavors the beer with fruity/solventy aromatics.
Inadequate oxygenation also causes pyruvic acid, fatty acids, and amino acids to be decarboxylated to aldehydes. These too are normally metabolic intermediates, but without enough oxygen, brewers' yeast must absorb trub to fuel sterol synthesis, and "staling" compounds are excreted. They may be reabsorbed by the yeast during fermentation, but they are just as likely to be further decarboxylated to fusel alcohols or remain after fermentation ceases. Acetaldehyde, the aldehyde of pyruvic acid, usually predominates, giving an odor like green apples.
As the yeast depletes its glycogen reserves, it starts to absorb glucose and fructose from solution and begins to manufacture the enzymes and permeases necessary to reduce other wort sugars. Only the monosaccharides and sucrose in wort can be absorbed by yeast that have not adapted to the wort into which they have been pitched.
_Permeases_ are enzymelike transports that carry specific compounds through the plasma membrane and into the yeast cell. The yeast must synthesize permeases to absorb maltose and maltotriose, and the enzyme a-glucosidase to hydrolyze them to glucose. Lager yeast ( _S. uvarum_ ) synthesize and excrete melibiase to split and absorb the disaccharide melibiose. Other inducible enzymes are formed and secreted to the yeast's outer cell-membrane surfaces after maltose and maltotriose are depleted. These sever glucose molecules from dextrins to fuel subdued metabolism.
The lag and respiratory phases are generally longer when yeast have not been cultured in a solution similar to the wort into which they will be pitched. Prolonged adaptation, common with granulated dry yeast, can lead to the increased formation of fusel alcohol and esters. Culturing in solutions that contain a high percentage of corn sugar or glucose inhibits the formation of maltose permeases. Worts high in corn sugar suffer prolonged lag phases; some yeast strains even lose the ability to ferment maltose in high glucose worts (catabolite repression, or the glucose effect).
The first evidence of yeast activity is usually the formation of wisps of lacy white foam on the surface of the beer eight to twenty-four hours after pitching. Gradually this foam forms a wreath at the rim of the fermenter, and the beer below becomes milky-white from the haze of suspended yeast colonies. Carbon dioxide production is prodigious, although attenuation of the wort is slight. Most of the CO2 is being released as a byproduct of pyruvic acid decarboxylation to acetyl Co A and oxo-acids; the carbon source for this reaction is glycogen, and only very little of the wort sugar is being metabolized.
If the yeast lag-phase extends beyond twenty-four hours, and the wort and yeast starter were originally well roused and oxygenated, then more yeast should be pitched. Rousing the quiet beer may cause the yeast to start fermenting, but as a rule, more yeast should be pitched. If the extended lag phase appears to be characteristic of the yeast strain, it should not be recultured; if it is due to prolonged storage of the culture before pitching, or to wort composition, the problem should be remedied.
As the yeast build cell walls and reserves up during the lag phase, they begin reproducing. As long as dissolved oxygen remains available in the wort, the yeast will utilize it to fuel anabolic reproduction. This respiratory period marks the period of greatest culture growth. Yeast growth, then, is most dramatic in a well-oxygenated wort.
## Low Kraeusen
Although free molecular oxygen is not necessary for yeast growth, it does facilitate it. Yeast can utilize carbon as an energy source fourteen times more efficiently by respiring molecular oxygen than it can by fermenting it anaerobically. Yeast cells reproduce more rapidly in an oxygenated solution, but normally continue reproduction at least until maltose is depleted.
The yeast begin budding as the lag phase ends, and rapidly scavenge the free oxygen from the wort in the early part of the low-kraeusen stage. As long as readily assimilable nutrients remain, the yeast continues reproducing, albeit at a gradually slowing rate.
As they deplete the molecular oxygen, the yeast begin anaerobic wort metabolism. Within six to thirty-six hours of pitching, the foam wreath should begin to migrate toward the surface center, marking the commencement of the low kraeusen stage of primary fermentation. It characterizes the start of intense catabolism of maltose, the uptake of a wide range of wort amino acids, the gradual transition from respirations to anaerobic fermentation, and a period of exponential yeast growth.
Low Kraeusen
As the head rises to form low, rich mounds and curls of foam, it carries with it protein, hop residues, and degenerated yeast cells, which are visible as a brown scum that collects on the head and at the surface edge. In closed fermentations, the scum may be eliminated by being "blown off" along with some of the liquid supporting it, or allowed to drop. In open systems, it is eliminated from the fermenter by skimming. If oxidized scum is allowed to fall back into the ferment, it will impart harsh, bitter tastes to the beer and provide a nutrient source for bacterial contaminants. Care must be taken that the fermentation is not contaminated if it is exposed for skimming. A low humidity improves atmospheric purity, reducing the likelihood of contamination.
Anaerobic Glucose Fermentation
The ferment may be topped-up with sterile wort or liquor to compensate for evaporation and blowoff or skimming losses, although it is more practical to compensate for these losses by increasing initial wort volume.
At this point, the yeast have completely adapted to the conditions of the ferment and are rapidly multiplying. Extract reduction should be about .5 °Plato (SG 1002) during this brief low-kraeusen phase. The pH also declines as organic acids are released as by-products of the yeast metabolism of amino acids.
The major fermentation products are ethyl alcohol (ethanol) and carbon dioxide, but even during a normal fermentation cycle, other compounds are excreted by the yeast. The metabolism of the yeast is fueled primarily by the hydrolysis of carbohydrates, but amino acids and fatty acids from the wort also serve as energy sources.
Normal carbohydrate metabolism follows the EMP pathway to pyruvic acid so that ATP may be regenerated to fuel, continuing biosynthesis of the yeasts' metabolic requirements. The ADP-ATP cycle, however, cannot continue if pyruvic acid buildup is left to block it. Pyruvic acid in excess of that required for acetyl Co A and oxaloacetate synthesis is metabolized to CO2 and acetaldehyde by the yeast enzyme pyruvate decarboxylase. The CO2 is excreted. Acetaldehyde is hydrated to ethyl alcohol by the enzyme alcohol dehydrogenase so that hydrogen buildup blocking the ADP-ATP synthesis can also be eliminated.
Nitrogen metabolism is closely related to glucose hydrolysis. The yeast enzymatically split amino acids in the wort and separately absorb the nitrogenous amino groups (NH2) and oxo-acid skeletons. These can be reassembled as amino acids, or as proteins, appropriate to the yeasts' requirements. Oxo-acids necessary for amino acid synthesis may also come from carbohydrate metabolism, and similarly there are several other ways in which oxo-acids are used with consequences affecting beer flavor. They may be decarboxylated to aldehydes, and the aldehydes hydrated by the enzyme alcohol dehydrogenase to form fusel alcohols. Lack of dextrinous sugars, trub in the yeast cake, and elevated temperatures all contribute to the formation of piquant, solventlike, and highly aromatic harsh fusel alcohols.
Oxo-acids may be metabolized to acetohydroxy acids, which are not metabolized by the yeast and are therefore expelled by them. During low kraeusen, this provides for the elimination of oxo-acids (primarily from pyruvic acid), which carbohydrate metabolism is producing in excess.
Excreted into solution, acetohydroxy acids can be oxidized to vicinal diketones, principally diacetyl (dimethyl diketone, C4H6O2). Diacetyl has a perceptible buttery flavor, which is objectionable in amounts above .15 ppm; during low kraeusen it may be as high as .35 ppm. That there is some oxygen in the ferment during low kraeusen and that the temperature is not unreasonably depressed in later fermentation are both important to diacetyl control, because after yeast growth has slowed, healthy cells reabsorb vicinal diketones and metabolize them to harmless diols. When the acetohydroxy acid is not oxidized to vicinal diketones during vigorous early fermentation, later diacetyl formation may irreversibly mar beer flavor. The presence of dissolved oxygen later in fermentation also increases the likelihood of oxidation of acetohydroxy acids to diacetyl; vigorous anaerobic fermentation after the low kraeusen stage is essential to diacetyl control.
## High Kraeusen
With top-fermenting yeast, a thick head of clumped yeast covers the beer soon after intense fermentation becomes apparent. At their normal operating temperatures, top-fermenting yeast have largely metabolized the sugars in solution at a time when lager yeast are still undergoing growth. Not until eighteen to seventy-two hours after the mounds of foam begin to form (two to four days after pitching) do the lager yeast weave a tightly knit cover over the surface of the beer. This cover rises further and finally breaks into cream-colored, less dense, "rocky heads."
In a lager fermentation, extract reduction approximates l °Plato (SG 1004) daily over the two to five days of high kraeusen, while the yeast may still be reproducing logarithmically. The temperature must be exactly controlled.
High Kraeusen
Although 60 degrees F (16 degrees C) is given as the maximum allowable temperature, with most lager yeast strains every effort should be made to hold it to 47 to 52 degrees F (8 to 11 degrees C). When this temperature is reached within the ferment, the ambient temperature can begin to be lowered if diacetyl levels in the beer are not a problem. The temperature should not be lowered more than 5 degrees F (3 degrees C) daily and should be reduced to 38 to 40 degrees F (3 to 4 degrees C) over several days. The temperature at the conclusion of a traditional kraeusen fermentation is usually about 45 degrees F (7 degrees C).
The continued release of organic acids during high kraeusen reduces the pH, depending on the yeast strain and wort characteristics, to 4.0 to 4.5. If the pH drops too rapidly, the yeast will settle out of suspension prematurely; if the pH drops too slowly, it may prevent the beer from clearing properly.
## Post Kraeusen
The extract is largely metabolized by the yeast during high kraeusen. As the yeast activity slows, carbonic gas production slows, and consequently the agglutinated yeast colonies sink out of suspension and the foam head is no longer formed.
Late Kraeusen
At this point, all of the head can be floated, siphoned, or skimmed off, even as more is forming, so that it does not fall back through the beer. An exception to this procedure is made when a low-extract, quickly maturing beer is being brewed. Such a beer requires the peptides and amino acids from clean foam for yeast nutrients and for body. The increase in the alcohol content of the beer induces the reabsorption of the albuminous matter into solution. Only a clean head should be allowed to fall back through the ferment. Residual scum gives the beer a harshly bitter background flavor. The stability of the beer is invariably less than if the head is removed or the beer is quickly separated from its trub after the head falls.
Generally, four to six days after high kraeusen begins (six to ten days after pitching), the formation of the foam cover ceases. As the availability of fermentable extract drops during the post-kraeusen stage, the yeast adapt to changing conditions by accelerating their secretion of extracellular enzymes capable of splitting off glucose from dextrins in solution. Reasonable levels of diacetyl and the related diketone 2,3-pentane dione are also absorbed and metabolized by the yeast. It is important that the fermentation temperature not be prematurely lowered and that the beer not be racked off its yeast sediment until the diacetyl has been reabsorbed. With a clean fermentation, it is usual for the beer to be held in the primary fermenter for two or three days after the kraeusen head has fallen, with the temperature being lowered from 45 degrees F (7 degrees C) to 38 to 40 degrees F (3 to 4 degrees C). The extract drop over the final twenty-four-hour period of primary fermentation should be about .5 °Plato (SG 1002), and the density should be about one-third what the wort density (OG) was.
An entirely different approach is relatively common in modern fermentation cycles. When the density drops to about 1.5 °Plato (SG 1006) above the target terminal gravity, the brewer raises the temperature of the post-kraeusen beer to 52 degrees F (11 degrees C) or higher, and holds that temperature for two to seven days for a _diacetyl rest_ to reinvigorate the yeast culture so that it will metabolize diacetyl, removing it from solution.
Because fermentable extract is rapidly consumed at the higher temperatures of the diacetyl rest, subsequent conditioning can be foreshortened. Secondary fermentation will be both subdued and brief, and lagering may require only fourteen to twenty-one days to achieve the same clarity and flavor stability (but not the same flavor development) that would be expected with the usual five-to-seven-week secondary fermentation and lagering.
## Real and Apparent Attenuation
Normal primary fermentation ends when head formation ceases; this may take as few as five days when the wort is below 10 °Plato (SG 1040), or eight to ten days for a very rich and dextrinous wort. Roughly 50 to 65 percent of the extract will have been converted to alcohol and carbon dioxide, although the hydrometer may show a 65 to 80 percent reduction in density ("one-third gravity"). The difference between the real attenuation of the beer and the apparent attenuation as gauged by the hydrometer is usually about 15 percent. This phenomenon occurs because the hydrometer measures liquid pressure, and does not reflect the fact that this pressure has been reduced by the formation of alcohol as well as by the reduction of the fermentable extract. Because alcohol is far lighter than water (the liquid pressure of pure water is SG 1000; of alcohol, 798), the hydrometer sinks further into a solution in which alcohol is present, and the hydrometer reading is lower than the extract loss alone can account for.
The real attenuation can be determined. First, a volume of beer is measured at the temperature the brewer's hydrometer is calibrated to, usually 60 or 68 degrees F (15.56 or 20 degrees C), and is decarbonated, usually by membrane filtration. This volume of beer is raised to a temperature of 173 degrees F (78 degrees C) or slightly higher and roused for thirty to sixty minutes to drive off the alcohol. The sample is cooled to 60 degrees F and topped-up to its original volume with distilled water. The volume of water required to replace the lost volume of beer, divided by original volume, is equal to the percent alcohol by volume of the beer. The hydrometer reading of the dealcoholized sample, after topping-up, accurately reflects the real extract content of the beer. The real attenuation is measured by subtracting this reading from the original °Plato (OG) of the wort.
## Racking
The beer is carefully racked off its settlement when its density is one-third or less of the wort density (OG) and its drop over the preceding twenty-four hours is .5 °Plato (SG 1002) or less. A reducing-sugar analysis usually shows less than 5 percent. The beer should be free of any foam cover. The transfer to a closed secondary fermenter should be made under antiseptic conditions, and all equipment should be sanitized before use.
The purpose of racking is to separate the beer from decaying yeast cells and flavor-impairing precipitates. Care should be taken that no yeast sediment or trub is carried along into the secondary fermentation. Siphoning or decanting must be terminated just as soon as the runoff becomes the least bit cloudy.
Employing up to 5 percent strongly fermenting kraeusen beer at racking produces a stronger start of secondary fermentation and a better overall fermentation. This absolutely must be done when yeast performance during primary fermentation has been poor, as it replenishes the degraded culture.
Racking must be done without rousing or splashing to prevent oxygen from entering the solution. Oxygen in beer past early kraeusen poses serious consequences to the beer flavor: oxidation of acetohydroxy acids in the secondary fermenter produces diacetyl that the yeast may not reabsorb; alcohols may be oxidized to aldehydes; amino acids may be oxidized to fusel alcohols; acids may oxidize alcohols to esters; and phenolic material may polymerize and become haze fractions. It is also advisable that the secondary fermenter be topped-up with kraeusen beer or wort so that only enough airspace remains to allow for very mild foaming. Cone-bottomed "unitank" fermenters preclude the necessity for racking to a secondary fermenter/lagering tank. Trub is discharged from the bottom outlet throughout the course of the primary fermentation, and yeast is collected as it sediments.
## Gauging Yeast Performance
A sample of the beer at racking should show very clear and bright. It should demonstrate a good break. When held up to the light, it should show clear. When agitated, distinctly visible suspended yeast colonies may float about, but upon resting, should settle out rapidly and firmly. Such a yeast is satisfactory for collecting to be repitched and for employment in a long secondary and lager fermentation.
Only "break," or _Bruchhefen,_ sedimentary yeast form colonies as the yeast nutrients in the beer diminish. Powdery or dusty _Staubhefen_ yeast do not sediment in so clean a break and are likely to remain as a foam cover on the beer surface even after measurable attenuation falls off.
Dusty yeast ferment more of the extract than do break yeast, and do so more quickly. Beer brewed with dusty yeast is unsuitable for long secondary fermentation and lagering, because the yeast have largely eliminated the less readily fermentable extract necessary to support aging. Lagering must be conducted at lower temperatures (as low as 30 degrees F [-l degree C]) to increase sedimentation and retard fermentation. The beer should be racked into the secondary/lagering fermenter before the density has dropped much below one-third that of the wort (OG).
The appearance of dusty yeast in a ferment is usually due to the propagation of an inferior strain and/or harvesting late in the fermentation cycle. It should not be used for repitching.
## Yeast Collection
Fermentations displaying normal characteristics and desirable flavors are the best sources of yeast suitable for culturing. If the yeast have deteriorated (the beer is tainted with burnt-rubber or sulfury flavors, or fermentation is sluggish), are unable to reabsorb diacetyl (buttery taste and aroma), or are contaminated by wild yeast (cloudy beer after kraeusen, medicinal flavors) or bacteria (abnormal, sour, vegetal, or rancid-butter taste and aroma), the culture is not suited for repitching or culturing.
Seed yeast for subsequent brewings, culturing, and bottle priming should be collected only from the middle layer of the primary-fermentation sediment. The sediment should be relatively clean; an undisturbed sediment is composed of three distinct layers. The very thin, dark upper layer and the bottom layer of dead, inferior cells and trub sandwich between them the active, healthy, white yeast, or _barm._ Barm has the best fermenting qualities — strong cells that agglutinate well and settle out properly. Where an open fermenter is used, after the beer is transferred from the primary fermenter, all of the top layer is scraped aside before the middle layer is gathered up into a sterile container. When a closed fermenter that has no yeast-collection system at its base is used, the entire sediment is washed out and the barm separated from the trub by several rinsings, which float off the dead cells and organic residues. With cone-bottomed fermenters, collection begins when the sediment begins to run clean, and ceases when it becomes discolored again by settling trub.
Yeast collected for repitching can be covered with very cold, biologically clean water and agitated into suspension. When most of the yeast has settled, the water is decanted off, taking with it dead cells and trub. The rinsing is repeated. A subsequent acid wash with phosphoric acid or ammonium persulfate solution destroys bacteria, but the yeast culture may need to be recultured to restore its normal fermentation characteristics.
Depending upon the strain, yeast covered with sterile wort in a container fitted with a fermentation lock can be stored at 32 to 40 degrees F (0 to 4 degrees C) for from seven to twenty-one days without significant deterioration. If the seed yeast will not be pitched within that time, it must be fed again, or drained, pressed, and frozen.
Again depending upon the strain and the sterility of the conditions, lager yeast may be subcultured through as many as twenty successive brewings if they are repitched within twenty-four hours of collection. If the period between repitching is longer, or the strain is prone to mutation, it may be usable for only four or five intermittent brewings. As a general rule, the greater the frequency of use, the more times a strain may be directly subcultured.
# CHAPTER 13
# Secondary Fermentation
Whether or not a beer will be extensively lagered, a secondary fermentation in a closed fermenter allows for the slow reduction _(conditioning)_ of the remaining fermentable extract. In the secondary fermentation stage of lagering, the beer is free from the flavor impairment of sedimented trub and degenerating yeast cells. Seven to twenty-one days may be required for the yeast to deplete the fermentable sugar left after the kraeusen period has ended. Traditional lagering entails holding the beer for a further two to seven weeks for clarification and stabilization.
During the secondary fermentation phase, the beer is slowly attenuated. It is slowly cooled to 33 to 37 degrees F (1 to 3 degrees C) — or to as low as 30 degrees F (-1 degree C) to settle dusty yeast — to allow the yeast to settle thoroughly and to inhibit the activity of any microorganisms possibly contaminating the ferment.
Because the potential risk of airborne contamination is great during the slow, cold ferment, the beer absolutely must be protected from contact with the atmosphere by being fermented in a closed vessel fitted with a fermentation lock. Contamination is otherwise a major risk at this point; yeast activity is slow, because the beer no longer contains abundant extract and nutrients, yet bacteria may be capable of significant dextrin, protein, or yeast-waste fermentation. Even though the pH is below their optimum, given a warm enough temperature even a few wild yeast or lactic-acid bacteria might rapidly propagate and ruin the beer.
Reproduction by the culture yeast will have entirely ceased during this stage of fermentation; further attenuation relies solely on the metabolic activity of the relatively few remaining yeast cells. It is imperative that conditions be conducive to the continued metabolism of the fermentable extract by these yeast cells and that they not be subjected to temperature shock. If the yeast culture needs regeneration, then an active starter culture or 5 percent kraeusen beer is added.
The duration of the secondary fermentation and the temperature at which it should be conducted are determined by the maltotriose and dextrin content of the post-kraeusen beer. If its reduction in density has naturally slowed and the hydrometer reading is still about one-third the value of the original wort reading, then the beer is rich in dextrins. It should be fermented out in the secondary at 33 to 41 degrees F (l to 5 degrees C), depending on the temperature preferences of the yeast strain, for at least fourteen days. When the post-kraeusen density is much less than one-third the value of the wort density, it indicates that the beer is lacking in dextrins and should undergo a secondary fermentation at 34 to 37 degreesF (1 to 3 degrees C) for not more than ten days before the temperature is reduced for lagering. Lagering times will also be shortened.
Long fermentations often darken the color of the beer. Consequently, when lagers are brewed for paleness, secondary fermentation may be carried out at higher temperatures (36 to 39 degrees F [2 to 4 degrees C], but not above 40 degrees F [5 degrees C]) over the shorter time period.
Since beer for draft need not be brewed for a long shelf life, it may also be fermented at higher temperatures (34 to 41 degrees F [1 to 5 degrees C]) and for a shorter period of time. When the hydrometer reading drops less than .2 °Plato/1 degree of gravity over a twenty-four-hour period and is within .4 °Plato/2 degrees of gravity of its anticipated terminal gravity, it can be assumed that there is just enough yeast and fermentable extract left in solution to support cask carbonation. The beer is racked into a keg or cask.
## Lagering
A long, cold, post-fermentive rest is usually employed when the wort is from a decoction mash. It yields a more stable beer with a smoother flavor.
_Lagering_ mellows harsh flavors by the combined effects of the falling rate of yeast metabolism, increased acidity, and low temperatures. Astringent tannins coagulate with haze-forming proteins, precipitating these and other, sulfurous compounds out of solution.
Yeast cells are not usually decomposed during lagering, but the culture becomes progressively dormant as fermentable extract (and to varying extents, glycogen reserves) are depleted. With the decline in available carbohydrates, the yeast reabsorb some of the esters and sulfur compounds from the beer.
Successful lagering requires that the beer not be subjected to temperature fluctuations or oxygenation. Oxygen in nearly fermented beer causes the irreversible formation of diacetyl and the oxidation of fusel alcohols and lipids. Where lagering temperatures are too warm, aldehyde formation is accelerated.
"Staling" aldehydes give beer stale, papery, cardboardy or sherrylike flavors. When higher temperatures decompose yeast cells, sulfury, stale, and soapy flavors arise.
When the kraeusen tradition is being followed, a lattice of beech chips is laid on the bottom of the secondary fermenter and covered with the nearly fermented, or _ruh,_ beer. From 5 to 15 percent new beer at up to 39 degrees F (4 degrees C) is roused into it. Where tank construction permits pressurization, and the tank is fitted with some manner of pressure relief, it is common to lager the beer under .2 to 2 atmospheres (3 to 28 psi) of pressure, after the vessel is purged of the atmosphere in the headspace. This can be accomplished by various pressure-regulating arrangements.
The lagering period is determined by referring back to the mash program, the hydrometer reading of the cooled wort, and the primary fermentation time and temperature. Dextrinous beer from a decoction mash should undergo a secondary fermentation and lagering period of seven to twelve days at 33 to 34 degrees F (1 to 2 degrees C) for each 2 °Plato (SG 1008) of cooled-wort hydrometer reading (OG). Lighter beer, lacking dextrins, is usually held for only three to seven days for each 2 °Plato of the wort density. Very strong, kraeusened beer, on the other hand, may be lagered for six to eight months before it is bottled.
Reducing the temperature to near freezing several days after secondary fermentation falls off reduces lagering time; in fact, the decrease in the solubility of body-forming colloids at 30 to 33 degrees F (-1 to +l degree C) necessitates a briefer lager period.
## Fining
Whether or not the beer is being lagered, fining improves its head retention, lacing, and clarity, and reduces aging times. It precipitates degenerated yeast cells, haze proteins, and tannins out of the beer. Gelatin and isinglass act as fining substances by enveloping suspended particles in their matrix, and gelatin further combines with tannic acid to form an insoluble precipitate. Either brewers' gelatin, unflavored 95-percent-pure gelatin, or isinglass may be used, so long as it is dry, smooth, pale colored, and odor free. Finings spoil if they absorb moisture during storage.
Isinglass finings are made from the shedded air bladders of certain fish. They are even more subject to spoiling than gelatin finings, and have a more limited shelf life. Isinglass works quickly and is generally added to the beer only two or three days before the beer will be racked. Isinglass precipitates lipids and can dramatically improve head retention.
When the beer has fermented out, it is ready for fining. A reducing-sugar analysis should show less than 2 percent. The beer must be colder than 50 degrees F (10 degrees C) for gelatin to react with the ferment; the closer to freezing temperature the beer is, the more efficient the action of gelatin finings will be.
Isinglass finings act more rapidly than gelatin finings do under the same conditions, and far better at warm temperatures. Isinglass can be used at up to 60 degrees F (16 degrees C), although it performs better in colder beer.
Dissolving finings into beer, wort, or water must be intelligently handled so that the finings are completely liquefied. They need to be evenly dispersed into the aged beer in a sanitary fashion. Finings cannot combine with yeast, polyphenols, and albumins unless they come into intimate contact with them. To accomplish this, the finings must be diffused throughout the entire volume of beer.
Where the beer is to be filtered, insoluble polyvinylpyrrolidones, such as Polyclar, and polyamides, such as nylon, can be used to remove polyphenols, especially chill-haze anthocyanogens, by bonding to them. Other media used are diatomaceous earth (kieselguhr) and silica gel. Diatomaceous earth selectively removes particles of a certain and greater size and is the filtering agent most commonly used by small breweries. Silica gel absorbs large polypeptides and proteins, removing them from solution so they cannot combine with polyphenols.
If a strong hop aroma is desired in a lager, liquid hop extract may be added with the gelatin finings, or before other treatment. Even with British ales, hops for strong aroma are never usually added after fining. Hops, and especially extracts, when added with finings, can improve the clarification of beer from a well-mashed and boiled wort by increasing the polyphenols available for coprecipitation with albumin.
If the bottled beer lacks a good foam head or is thin, then the clarifying treatment should be reduced in future batches. Use of particular malts and brewing techniques precludes the need for clarifying.
## Clarifying with Beech Chips
Beech or hazelnut chips one-eighth inch thick by one-half inch wide are laid on the bottom of the lagering vessel to form a loosely woven lattice. As the wood absorbs moisture, extract coats the many surfaces of the chips, bonding weak yeast cells to them and thus clearing the beer.
The chips are first soaked and then boiled for twelve to twenty-four hours in a sodium bicarbonate solution (11/2 pounds/gallon) before being rinsed. The process is repeated, sometimes substituting bisulfate of lime. Finally, the shavings are rinsed in a cold-hot-cold water cycle. The pH of the last rinse should be neutral, indicating that all of the bicarbonate has been washed from the chips. They may be washed and reused until they crack from age.
## Real Terminal Extract
When the aged beer is ready to be bottled, its real-extract content can be determined by boiling off the alcohol from a measured volume of the beer, topping it up to its original volume with distilled water, and gauging its density with a hydrometer. The liquid pressure exerted by the fermented beer (sugar analysis less than 2 percent) is due to unfermentable dextrins, maltotetraose, and soluble nitrogen. Taking a hydrometer reading of a dealcoholized sample is the only means by which the "real extract" may be quantified with absolute accuracy, by removing the "apparent density" effect of alcohol from it.
The amount of unfermentable extract remaining in the finished beer is the direct result of the duration and effectiveness of the proteolytic, alpha-amylase and beta-amylase mash rests, the amount and nature of malt used, the effectiveness of protein/polyphenol bonding in the boil, and the ability of the yeast to ferment maltotriose and isomaltose. Typical bottled lagers have a real density of 3.0 to 5.0 °Plato (SG 1012 to 1020); richer types average 5.5 °Plato (SG 1022) to 6.5 °Plato (SG 1026). Bocks may range even higher. The real density of a fully fermented beer is generally 40 percent greater than its apparent density.
Full-bodied beers should show 45 to 60 percent real attenuation, light beers up to 70 percent. Apparent attenuation is usually 60 to 75 percent in the former case and up to 85 percent in the latter.
## Bottling
Naturally carbonated beer must be racked off its sediment into a sterile, closed container and mixed with a quantity of actively fermenting kraeusen beer or priming solution sufficient to produce the desired bottle pressure (see table 23). When corn sugar (dextrose) is used to carbonate the bottled beer, it should be made up into a solution; dry primings are not recommended. High-quality dextrose should be the only sugar employed for bottling.
Whether kraeusen beer or sugar fuels bottle carbonation, measurements need to be exact, and the solution mixed with the aged beer very thoroughly, without splashing, or inconsistent carbonation results. The character of the finished beer is greatly influenced by the degree of carbonation.
Mixing the kraeusen beer or priming solution into the aged beer should not be done in an aerating fashion but should raise a foam head, indicating CO2\-- release. This aids in the prevention of oxidation and contamination of the beer at bottling by forming a protective blanket of carbonic gas above the beer and driving some atmosphere from the head space above.
The bottles into which the beer is siphoned must be biologically as well as physically clean. Commercially, bottles are washed with 3 percent caustic soda, rinsed, then sterilized at above 170 degrees F (77 degrees C), drained, and rinsed. Clean bottles may be "pasteurized" by soaking them for at least thirty minutes in clean water at above 140 degrees F (60 degrees C) or placing them wet in an oven at 200 degrees F (93 degrees C) for twenty minutes before they are inverted to dry and be inspected.
Any type of bottle may be used, so long as it can be sealed and can withstand the pressure of bottle fermentation. Carbonation in excess of three atmospheres requires the use of a heavy-gauge bottle. Under carefully controlled conditions, thin-walled, "nonreturnable" bottles are sufficient when brewing lager beer of normal carbonation.
Each bottle should be filled to within at least three-quarters of an inch of the top and be left to rest, loosely capped, for several minutes before the caps are secured. This allows air trapped in the neck space to be driven off by the release of carbonic gas. Oxygen that is not displaced may be scavenged by the yeast, but it is possible that where there is a significant amount of air it may be absorbed into the beer and diminish its stability and mar its flavor. Oxygenation at bottling from splashing the beer or from trapping air in the neck space can be a problem as serious as contamination.
The bottled beer should be held at 50 degrees F (10 degrees C) or above for several days to allow fermentation to be established within the bottle before lowering the temperature. Temperature reduction should be gradual, not exceeding 5 degrees F (3 degrees C) daily. Bottle conditioning should take place away from direct sunlight, and the bottles should not be subjected to major temperature fluctuations.
The beer should be aged in the bottle for an absolute minimum of ten to fourteen days, and preferably thirty days, before serving. Lagered, bottle-conditioned beers usually keep for at least several months. There is an optimum storage period for every beer, when chemical changes within the bottle produce the best taste and aroma, and this should dictate the length of the bottle-aging period.
## Imbibing
Handcrafted beers are not usually consumed with the same carelessness as a can of beer at a ball game. They are brewed to be drunk at the right temperature, with discernment and appreciation for the subtler aspects of their character. When bottle-conditioned beer is ready for drinking, it should be carefully poured so that the yeast sediment is not disturbed; a good yeast strain cakes solidly on the bottom of the bottle.
A beer is judged by its flavor, aroma, body, head, color, and clarity. All of these values are subjective, and various standards depend on the type of beer being brewed, the balance of one characteristic against another, and the predisposition and preference of the consumer.
Areas for taste
Color, clarity, head formation and retention, and aroma can be assessed before the beer is even tasted. The aroma is usually sampled by swirling the glass and holding it under the nose. The cleanliness and sharpness of the hop bouquet, the balance of roastiness or maltiness, and the intensity and character of fermentation products are the major criteria by which aroma is judged. Aroma may be characterized as being ethereal, aromatic, floral, spicy, malty, roasty, nutty, yeasty, fruity, vegetative, sulfury, buttery, phenolic, solventy, or musty.
Flavor is evaluated by the first sensation, the taste, then by the aftertaste once the beer is swallowed. The body of beer is judged by its fullness and texture, and is closely related to "mouthfeel." The palate fullness of beer is mostly due to the low molecular weight albumins and dextrins carried over to the finished beer, and its viscosity.
Taste perceptions are very generally classified as sweet, sour, bitter, or salty. _Sweetness_ is produced by unfermentable sugars and the breakdown of simple dextrins by saliva. _Sourness_ (acidity) is directly affected by the pH of the beer, and below pH 4.0 it becomes both noticeable and in most cases, objectionable. _Bitterness_ is due to the iso-alpha acids from the hops, and from roasted malt. _Saltiness_ is formed by the mineral content of the wort and generally complements bitterness and accentuates dryness.
Because our taste perceptions are highly sophisticated, we can differentiate between far more flavor characteristics than the four basic tastes can account for. Olfactory perceptions greatly affect flavor assessment. Beer flavor is likely to be expressed using olfactory terms such as malty, roasty, caramel, grainy, hoppy, vegetative, medicinal, metallic, sulphury, skunky, burnt, nutty, yeasty, buttery, and fruity.
In the final analysis, the character of the beer should cater to the preferences of the ultimate consumer.
# CHAPTER 14
# Planning the Brew
Before each brewing cycle is begun, prepare everything that will be needed so the operation may proceed smoothly and without interruption. Make sure that an adequate volume of healthy yeast are available.
Be aware that density and pH readings vary with temperature, so the hotter the mash or wort sample is, the lower the density and pH appear to be. The pH and density should both be gauged at the reference temperature they are calibrated at, whether that is 60 or 68 degrees F (15.56 or 20 degrees C).
Where periods of time are quoted, the longer times generally suit full-bodied, fully lagered beer, or the mashing, sparging, and boiling of difficult malts. The shorter times favor light, quickly maturing beer and beer made from well-modified malt.
If you have not read the whole text, do so before beginning to brew. The procedural outline immediately following is inadequate by itself; the brewer should be familiar with the wider scope of information found elsewhere in the text.
## A Note on Mixing
Whenever two substances are to be mixed, mixing should be accomplished gradually to ensure even distribution. When one of the substances is liquid and the other is dry, the dry substance is gradually and evenly moistened before it is flooded with the liquid. Unless this precaution is taken, dry "balls" may become isolated by being encased in paste.
Always mix by the gradual dilution of the smaller amount of any two substances by the larger. Always mix liquids into dry substances, and mix them gradually.
To oxygenate any solution, rouse it splashingly and force air into it, as by pushing downward with an inverted spoon. To rouse a solution without aerating it, very gently rock the vessel or stir slowly, pivoting the spoon at the surface level of the solution to prevent vortexing and splashing.
## Planning
Paperwork is probably not your favorite pastime, but predicting a beer's density, color, and flavor and achieving reasonable batch-to-batch consistency are only possible by planning. Use the worksheets in the following tables to plan any recipe and brew, using the text and tables to calculate requirements. Worksheets are provided rather than formulas, because many brewers are not comfortable with algebra.
Due to the cumbersome nature of parenthetically including all the varying units of measurement commonly used by brewers (barrels, hectoliters, liters, kilograms), weight is only given in pounds or avoirdupois ounces, and volume in U.S. gallons. For brewers who use other units of measurement other than pounds, ounces and gallons as given, use the weights and measures in appendix D to convert results to other units of measure. Where EBC color is given in a malt lot analysis, convert it to °Lovibond/SRM by the formula EBC(.375) + .46 = SRM. Where extract potential is given as liter°/kilogram 7 Miag, convert it to percent extract/gallon by the formula L°/kg/385.65 = % extract/gal.
Table 14
Planning
Batch Parameters
1. Beer style: ________________________________________
2. Liquor mineral character: ____________________________
3. Wort density (OG), °Plato: ____________________________
4. Beer density (FG), °Plato: _____________________________
5. % ABV (OG-FGx .52): ________________________________
7. Wort color, SRM: ___________________________________
6. Hop bitterness, IBU: _________________________________
8. Carbonation, volumes/CO2: ___________________________
Wort Volume
9. Beer, bottling volume: + ________
10. Kraeusen beer, volume (optional, usually 10%): - ________
11. Blowoff/racking losses, volume (3–5%): + ________
12. Wort removed for kraeusen, starters (optional): + ________
13. Cooled wort, volume required: = ________
14. Wort boil, duration: ______ hours
15. Evaporation rate per hour (.10 av.): _____x 14:_____= _________
Evaporation loss: + ________
16. Wort retained in the hops; oz./hops x .045 gal. (pellets .02): +
17. SWEET WORT, VOLUME REQUIRED: = ________
Planning Data
18. Mash rests, temperature and duration: | 1 | ________ | ________
---|---|---|---
2 | ________ | ________
3 | ________ | ________
4 | ________ | ________
5 | ________ | ________
19. Mash-liquor temperature: For infusion mashes and mashes doughed-in at 95° F (35°C) or warmer, heat liquor to 10 to 15°F (6 to 8°C) warmer than the rest temperature. For decoction mashes doughed-in cold, dough-in with liquor at 57 to 70°F (14 to 2°C) and then heat liquor to boiling. Use boiling liquor to raise the temperature of the cold mash to the next rest. Volumes of mash to be pulled for each decoction can be predicted by completing this worksheet:
A. Total lb. of malt in mash: _________ x 3.33 = _______
B. Total volume of liquor in mash: _________ x 8.32 = _______
C. Add lines A and B: _________ = _______
D. Temperature of next mash rest: _______ °F
E. Temperature of mash before decoction: _______ °F
F. Subtract line E from line D. Temperature rise: = _______ °F
G. Multiply line C by line F: _______
H. Subtract line E from 212°F: _______ °F
I. Divide line G by line H: _______ °F
J. Divide line I by line C: _______
K. For a thick mash/thick decoction, multiply line J by 1.25; for a thin mash/thick decoction, multiply line J by 1.4: ___
Line K is the percentage of the mash that should be withdrawn for the decoction.
20. Decoction 1, volume: _____ Rests at: ______ °F ______ °F ______ °F.
Boil for: ______minutes. Next rest temperature: ____ °F.
Decoction 2, volume: _____ Rests at: ______ °F ______ °F ______ °F.
Boil for: ______minutes. Next rest temperature: ____ °F.
Decoction 3, volume: _____ Rests at: ______ °F ______ °F ______ °F.
Boil for: ______minutes. Next rest temperature: ____ °F.
21. Sparge liquor temperature (c.170–175°F): ___________ °F.
22. Cooled wort/pitching temperature: _______
23. Wort aeration method, amount:: ________
24. Yeast strain: ________________ Yeast/kraeusen, amount: ________
25. Primary fermentation, maximum temperature: ________
26. Rack to secondary at: _______ °Plato (see table 22)
27. Secondary fermentation, temperature: ______ duration: ________
29. Lagering/aging, temperature: ______________ duration: ________
30. Finings, type: ________________ amount: _______
31. Primings, type: ________________ amount: _______ (see table 23)
Table 15
Grain Bill
1. Wort volume required (table 14, line 17): __________
2. Wort color desired, homebrew color units: HCU/lb. __________
Convert SRM color to linear HCU color scale. At 1–10°, SRM equals HCU. At more than 10 °SRM, convert to HCU:
SRM: | 10.5 | 11 | 11.5 | 12 | 12.5 | 13 | 13.5 | 14 | 14.5 | 15
---|---|---|---|---|---|---|---|---|---|---
HCU: | 10.8 | 11.6 | 12.4 | 13.3 | 14.1 | 14.9 | 17.7 | 18.6 | 20.5 | 22.4
SRM: | 15.5 | 16 | 16.5 | 17 | 17.5 | 18 | 18.5 | 19 | 19.5 | 20
HCU: | 24.3 | 26.2 | 28.1 | 30 | 32.9 | 35.8 | 38.8 | 41.9 | 45 | 47.8
3. Multiply line 1 by line 2: TOTAL HCU REQUIRED = ____________
4. Wort density (table 14, line 3) ________ divide by 11.47.
(For specific gravity, divide by 46.16) LB. EXTRACT/GAL.: __________
5. Multiply line 1 by line 4: TOTAL LB. OF EXTRACT REQUIRED = ______
Begin formulating any recipe with the specialty malt or adjunct that has the most critical impact on flavor. Follow with specialty malts and adjuncts in diminishing order of effect on flavor, and then color. List each one, and subtract its color and extract contribution from lines 3 and 5. Where no lot analysis is available, skip lines 10, 11, 18, and 19, and use values from table 16, columns 3 and 5 for lines 7, 12, and 20.
SPECIALTY MALT or ADJUNCT _________________ Lot number: ____
6. Pounds: __________
7. Color, °L: x _________
8. Multiply line 6 by line 7: HCU COLOR CONTRIBUTION = _________
9. List color contributions, line 8, of all specialty malts and adjuncts as they are accumulated:
__________ + __________ + __________ + __________ = _________
TOTAL HCU COLOR CONTRIBUTION
10. Specialty malt/adjunct, extract potential, DBCG: ._________
11. Moisture content: /1._________
12. Divide line 10 by line 11: as-is coarse grind extract = ._________
13. Brew-house efficiency (usually .85–.95): x .________
14. Multiply line 12 by line 13: Extract contribution/lb. = ._________
15. Multiply line 14 by line 6: EXTRACT YIELD = ___.______
16. List extract yields, line 15, of all specialty malts and adjuncts as they are accumulated:
__________ + __________ + __________ + __________ = _________
TOTAL EXTRACT YIELD
Repeat lines 6 through 16 for each specialty malt or adjunct, then go on to calculating the amount of base malt you will need:
BASE MALT: _____________________________ Lot number: _______
17. Subtract line 16 from line 5: REMAINING EXTRACT REQUIRED = _____
18. Base malt, extract DBCG: __________
19. Base malt, moisture content: /1.________
20. Divide line 18 by line 19: as-is coarse grind extract = _________
>21. Divide line 17 by line 20: LB. OF BASE MALT = _________
22. Base malt color, °L: x __________
23. Multiply line 21 by line 22:
BASE-MALT COLOR CONTRIBUTION =
24. Add line 23 to line 9: TOTAL HCU COLOR CONTRIBUTION =_________
Line 24 should match table 14, line 7.
Table 16
Brew-house Yield from Various Sources
Brew-house yield is the actual extract that can be expected from a given source in a given brewery. It reflects the extract potential of the malt or adjunct and its moisture content, as well as the degree of crushing, the skill of the brewer, and the efficiency of the brew-house equipment.
This table can be used to predict brew-house yield when no grain analysis is available.
Columns 3 and 4 give typical color and extract-potential values for the grains and adjuncts shown in column 1. Column 4, dry basis coarse grind extract (DBCG) reflects the extract potential of grains given a usual brew-house crush, but does not account for the grain's moisture content or the brew-house efficiency; it is the yield at 100 percent brew-house efficiency if the malt were oven-dry.
Grains usually have a moisture content of 2 to 5 percent, and usual brew-house efficiency is 85 to 95 percent. As-is coarse grind (AICG) is the value derived by adjusting the DBCG of grains to reflect the moisture content but not the brew-house efficiency. Column 5 gives the AICG for grains, assuming a moisture content of 4 percent. Columns 7 and 8 are provided for novice brewers to use in simplified recipe formulation. They give the brew-house yield one could expect from average grains processed with reasonably good efficiency (92 percent brew-house efficiency for grains to be mashed, except for infusion-mashed lager and wheat malts, which are given at 90 percent); for kettle adjuncts, extract efficiency is assumed to be 100 percent.
Brew-house Yield from Various Sources
---
Source | Modification | Color, °L/ASBC (Avg.) | Extract Potential | Mash Type | Brewhouse Yield 1 Lb./1 Gal., 92%
DBCG | AICG | °P | SG
Lager malt, 2-row tradit'l | Low | 1.4 | 80.5% | 77.4% | Multi-rest | 8.2 | 1033
Infusion | 8.0 | 1032
Lager malt, 2-row European | Moderate | 1.6 | 80% | 76.9% | Multi-rest | 8.1 | 1033
Infusion | 8.0 | 1032
Lager malt, 2-row American | Full | 1.7 | 79.5% | 76.4% | Multi-rest | 8.1 | 1032.5
Infusion | 7.9 | 1032
Lager malt, 6-row American | Full | 1.7 | 77.5% | 74.5% | Multi-rest | 7.9 | 1031.5
Infusion | 7.7 | 1031
Pale malt, 2-row British | Full | 2.4 | 80% | 76.9% | Infusion | 8.1 | 1033
Diastatic, 6-row | Moderate | 1.3 | 80.5% | 77.4% | Infusion | 8.2 | 1033
Wheat malt, Red winter | Full | 2.0 | 82.5% | 79.3% | Multi-rest | 8.4 | 1034
Infusion | 8.2 | 1033
CaraPils | Sacch'r | 3.0 | 78% | 75.0%
| |
7.9 | 1032
Vienna | Moderate | 5.0 | 80% | 76.9%
| |
8.1 | 1033
Munich | Moderate | 9.0 | 80% | 76.9%
| |
8.1 | 1033
C-30 | Sacch'r | 30 | 77% | 74.0%
| |
7.8 | 1031.5
Crystal | Sacch'r | 55 | 77% | 74.0%
| |
7.8 | 1031.5
C-80 | Sacch'r | 80 | 76.5% | 74.0%
| |
7.8 | 1031
C-120 | Sacch'r | 120 | 76% | 73.1%
| |
7.7 | 1031
Amber | Over | 25 | 77% | 74.0%
| |
7.8 | 1031.5
Brown | Full | 55 | 75% | 72.2%
| |
7.6 | 1030
Chocolate | Low | 400 | 70% | 67.3%
| |
7.1 | 1028.5
Black malt | Low | 600 | 70% | 67.3%
| |
7.1 | 1028.5
Roast barley
| |
450 | 70.5% | 68.8%
| |
7.3 | 1029
Flaked barley
| |
1.5 | 60% | 57.7%
| |
6.1 | 1024.5
Flaked wheat
| |
2.0 | 87% | 83.7%
| |
8.8 | 1035.5
Flaked corn
| |
1.0 | 88% | 84.6%
| |
8.9 | 1036
Oatmeal
| |
1.0 | 62% | 59.6%
| |
6.3 | 1025
Rice
| |
0.5 | 88% | 84.6%
| |
8.9 | 1036
Kettle Adjuncts
---
Source | Color, °L/ASBC (Avg.) | Extract Potential | Brewhouse Yield 1 Lb./1 Gal., 92%
DBCG | AICG | °P | SG
Malt syrup, diastatic | 3.0 | 78%
| |
9.0 | 1036
Malt syrup, light | 5.0 | 78%
| |
9.0 | 1036
Malt syrup, amber | 10.0 | 78%
| |
9.0 | 1036
Malt syrup, dark | 15.0 | 78%
| |
9.0 | 1036
Malt syrup, wheat | 3.0 | 78%
| |
9.0 | 1036
DME, light | 7.5 | 97%
| |
11.1 | 1045
DME, amber | 12.5 | 97%
| |
11.1 | 1045
DME, dark | 18.0 | 97%
| |
11.21 | 1045
Malto-dextrin | 6.0 | 92%
| |
10.6 | 1042.5
Glucose
| |
90%
| |
10.4 | 1041.5
Sucrose
| |
100%
| |
11.5 | 1046.2
Honey
| |
76%
| |
8.8 | 1035
CaraPils | 3.0 | 78% | 75.0% | 8.6 | 1034.5
C-30 | 30 | 77% | 74.0% | 8.5 | 1034
Crystal | 55 | 77% | 74.0% | 8.5 | 1034
C-80 | 80 | 76.5% | 73.6% | 8.5 | 1034
C-120 | 120 | 76% | 73.1% | 8.4 | 1034
Chocolate | 400 | 70% | 67.3% | 7.7 | 1031
Black malt | 600 | 70% | 67.3% | 7.7 | 1031
Roast barley | 450 | 70.5% | 68.8% | 7.9 | 1032
Table 17
Brewing Liquor, Volume, and Treatment
1. Sweet wort, volume required (table 14, line 17): __________
2. Grain bill, total weight (table 15, lines 6 and 21): __________
3. Multiply line 2 by .165 gal.: Wort retained, spent grains + __________
4. Add line 3 to line 1: LIQUOR REQUIRED, TOTAL VOLUME = __________
5. Mash liquor, per lb./grain (.28–.45 gal./lb.): x __________
6. Multiply line 2 by line 5: MASH LIQUOR, VOLUME = __________
If line 6 is greater than 45% of line 4, recalculate the amount of mash liquor using a lower volume per pound, but not less than .28 gal. per pound of grain.
7. Subtract line 6 from line 4: SPARGE LIQUOR, VOLUME = __________
If line 7 is greater than 65% of line 4, reduce it to 65%, and add the difference to the wort after it is in the kettle. Regardless of the volume given on line 7, stop sparging when the mash runoff reaches pH 5.8 or drops to 3.0 °Plato (SG 1012) and add the excess liquor directly to the kettle.
To treat any volume of brewing water:
1. List the ions present in your water source as ppm (mg/L) in column 3.
2. Subtract column 3 from the ppm of ions for the style of beer that you are brewing (column 2).
3. Use the figures in column 5 (ppm of ions per gram of salt) to determine which mineral salts to use and how much to add.
4. Multiply column 5 by column 6; list in column 7. Column 7 should be approximately similar to column 4.
5. Multiply column 7 by the gallons of water you will be treating; list in column 8. Column 8 gives the grams of salts to add.
6. Add columns 3 and 7 to display the final treated-liquor profile in column 9 It is not possible, or necessary, to absolutely duplicate the quoted mineral distribution; approximation is all that is needed, so that column 9 more or less equals the chosen water type from column 2.
Brewing Liquor, Volume, and Treatment
---
1 | 2
| | | | | | | | |
-3 | 4
| |
5
| | | | |
6 | 7 (5x6) | 8 (COL 6 | 9 (3+7)
IONS | PILSEN | DORTMUND | MUNICH | VIENNA | BURTON | YRKSHR | DUBLIN | LONDON | EDINBRGH | Your Water | Difference
| |
IONS: 1 GRAM ADDED TO 1 GAL. | X GRAMS PER GAL. | =PPM ADDED | X GAL.) = TREATMENT | FINAL LIQUOR PROFILE, PPM
CaSO4 | MgSO4 | NaCl* | CaCl2 | CaCO3
Ca | 7 | 225 | 75 | 200 | 275 | 100 | 120 | 90 | 120
| | | |
61.5
| | | | | | | |
| | | | | | | | | | | | | | | |
72.
| | | | |
| | | | | | | | | | | | | | | | |
106.
| | | |
| | | | | | | | | | | | | | | | | | | | |
Mg | 2 | 40 | 18 | 60 | 40 | 15 | 5 | 5 | 25
| | | | |
37.
| | | | | | |
| | | | | | | | | | | | | | | | | | | | |
Na | 2 | 60 | 2 | 8 | 25 | 25 | 12 | 15 | 55
| | | | | |
104.
| | | | | |
| | | | | | | | | | | | | | | | | | | | |
SO4 | 5 | 120 | 10 | 125 | 450 | 65 | 55 | 40 | 140
| | | |
147.4
| | | | | | | |
| | | | | | | | | | | | | |
145.3
| | | | | | |
HCO3 | 15 | 180 | 150 | 120 | 260 | 150 | 125 | 125 | 225
| | | | | | | |
158.4
| | | |
| | | | | | | | | | | | | | | | | | | | |
Cl | 5 | 60 | 2 | 12 | 35 | 30 | 20 | 20 | 65
| | | | | |
160.3
| | | | | |
| | | | | | | | | | | | | | | |
127.4
| | | | |
HDNS | 30 | 750 | 250 | 750 | 875 | 275 | 300 | 235 | 350
| | | | | | | | | | | |
| | | | | | | | | | | |
**HARDNESS ADDED, ONE GRAM/1 GAL.**
| | | |
| | | | | | | | | | | |
CaSO4 | 153.8
| | | | | | | |
| | | | | | | | | | | |
MgSO4
| |
92.5
| | | | | | |
| | | | | | | | | | | |
NaCl
| | |
0.
| | | |
0.
| |
0.
| | | | | | | | | | | |
CaCl2
| | | |
180.
| | | | |
| | | | | | | | | | | |
CaCO3
| | | | |
106.
| | | |
| | | | | | | | | | | | | | | | | | | | |
TDS | 35 | 1000 | 275 | 850 | 1100 | 400 | 350 | 300 | 650
| | | |
**TDS ADDED, ONE GRAM IN 1 GAL.**
| | | |
| | | | | | | | | | | |
CaSO4 | 208.9
| | | | | | | |
| | | | | | | | | | | |
MgSO4
| |
182.3
| | | | | | |
| | | | | | | | | | | |
NaCl
| | |
264.2
| | | | | |
| | | | | | | | | | | |
CaCl2
| | | |
199.4
| | | | |
| | | | | | | | | | | |
CaCO3
| | | | |
264.2
| | | |
* Calcium chloride, dihydrate (CaCl2⋅H2O)
Table 18
Hop Rates
To calculate hop additions, refer to the utilization chart following the worksheet. Always calculate the bitterness contribution of anti-boilover, aroma, and flavoring hop additions first, so that the remainder of bitterness required can be adjusted by the kettle hop addition(s):
1. Hop bitterness, IBUs, from table 14, line 6: >_________
2. Wort volume in U.S. gallons, from table 14, line 17: _________
3. Multiply line 2 by 3.785 to convert gallons to liters: _________
4. Multiply line 1 by line 3: TOTAL IBUs REQUIRED = _________
Hops added at start of boil to prevent boil over, usually ¼ oz. / 5 gal., ¾ oz./bbl.
**Variety:** ____________________ **Boil time, minutes:** ________
5. % alpha acid, as a whole number (5%=5): ________
6. Amount by weight, as avoirdupois ounces: OZ. OF HOPS x
7. Multiply line 5 by line 6: AAUs = ________
8. Expected utilization, as IBUs per gallon (chart, below): ________
9. Multiply line 7 by line 8: ________
10. Divide line 9 by line 2: IBUs = ________
11. Subtract line 10 from line 4: REMAINING IBUs REQUIRED = ________
**Aroma hops, variety:** ________________ **Boil, minutes:** ________
Aromatic oils, % by weight: ________
12. % alpha acid as a whole number (5%=5): ________
13. Amount by weight, as avoirdupois ounces: OZ. OF HOPS x ________
14. Multiply line 12 by line 13: AAUs = ________
15. Expected utilization, as IBUs per gallon (chart, below): ________
16. Multiply line 14 by line 15: ________
17. Divide line 16 by line 2: IBUs = ________
18. Subtract line 17 from line 11: REMAINING IBUs REQUIRED =
**Flavor hops, variety:** ________________ **Boil, minutes:** ________
Aromatic oils, % by weight: ________
19. % alpha acid as a whole number (5%=5): ________
20. Amount by weight, as avoirdupois ounces: OZ. OF HOPS x ________
21. Multiply line 19 by line 20: AAUs = ________
22. Expected utilization, as IBUs per gallon (chart, below): ________
23. Multiply line 21 by line 22: ________
24. Divide line 23 by line 2: IBUs = ________
25. Subtract line 24 from line 18: REMAINING IBUs REQUIRED = ________
**Kettle hops, variety:** ________________ **Boil, minutes:** ________
26. % alpha acid as a whole number (5%=5): ________
27. Amount by weight, as avoirdupois ounces: OZ. OF HOPS x ________
28. Multiply line 26 by line 27: AAUs = ________
29. Expected utilization, as IBUs per gallon (chart, below): ________
30. Multiply line 28 by line 29: ________
31. Divide line 30 by line 2: IBUs = ________
32. Subtract line 31 from line 25: REMAINING IBUs REQUIRED = ________
**Kettle hops, variety:** ________________ **Boil, minutes:** ________
33. Expected utilization, as IBUs per gallon (chart, below): X ________
34. Multiply line 33 by line 2: = ________
35. Divide line 32 by line 34: AAUs REQUIRED = _______
36. Hop variety, % alpha acid as a whole number (5%=5): / ________
37. Divide line 35 by line 36: OZ. OF HOPS = ________
Brew-house Utilization and IBUs of 1 HBU/Gal. for Various Densities/Gravities
---
Wort Density: | 8–12.5 °P
SG 1032–50 | 12.5–16 °P
1050–1065 | 16–18 °P
1065–1075 | 18–20.5 °P
1075–1085 | 20.5–23 °P
1085–1095
Boil Time, Minutes: | UTL' | IBU | UTL' | IBU | UTL' | IBU | UTL' | IBU | UTL' | IBU
Hops | 90 | 31% | 22.5 | 28% | 21 | 27% | 20 | 26% | 19.5 | 24% | 18
Pellets | 90 | 33% | 24.5 | 30% | 22.5 | 28% | 21 | 27% | 20 | 54% | 18.5
Hops | 60 | 28% | 21 | 26% | 19.5 | 24% | 18 | 23% | 17 | 21% | 15.5
Pellets | 60 | 31% | 23 | 28% | 21 | 26% | 19.5 | 25% | 18.5 | 23% | 17
Hops | 30 | 15% | 11 | 14% | 10.5 | 13% | 10 | 13% | 10 | 12% | 9
Pellets | 30 | 18% | 14 | 17% | 13 | 16% | 11.5 | 16% | 11.5 | 15% | 11
Hops | 15 | 8% | 6 | 8% | 6 | 7% | 5 | 7% | 5 | 7% | 5
Pellets | 15 | 12% | 9 | 12% | 9 | 10% | 7.5 | 9% | 7 | 8% | 6
Hops | 5 | 5% | 3.5 | 5% | 3.5 | 5% | 3.5 | 4% | 3 | 4% | 3
Pellets | 5 | 6% | 4.5 | 6% | 4.5 | 5% | 3.5 | 4% | 3 | 4% | 3
Hops | 0 | 5% | 3.5 | 4% | 3 | 4% | 3 | 4% | 3 | 3% | 2
Pellets | 0 | 5% | 3.5 | 5% | 3.5 | 4% | 3 | 4% | 3 | 3% | 2
Table 19
Brew Log
Date: ____/_____/____
Brew number: ____ Beer style: ___________ Brand: _______________ | Lot # | Weight | HCU | Extract Contribution
---|---|---|---|---
Base malt: _____________ | ________ | ________ | ______________
Colored ______________ | ________ | ________ | ______________
malts ________________ | ________ | ________ | ______________
and __________________ | ________ | ________ | ______________
adjuncts: ______________ | ________ | ________ | ______________
Treated liquor: Volume ________ pH _____ Hardness _____ TDS _____
Yeast strain: ________ Generation #:________ | Time | Temp | pH | Density | Volume | Iodine | Amount
---|---|---|---|---|---|---|---
Mash liquor:
| |
____ | ___
| |
______
| |
Dough-in: | ____ | ____ | ___
| |
______
| |
_______
Mash-in: | ____ | ____ | ___
| |
______
| |
Decoction: | ____ | ____
| | |
______
| |
Protein rest: | ____ | ____ | ___
| | | |
Decoction: | ____ | ____ | ___
| | | |
Sacch' rest: | ____ | ____ | ___
| | |
_____
|
|
____ | ____
| | | |
_____
|
Decoction: | ____ | ____
| | |
______
| |
Lauter mash: | ____ | ____ | ___
| | | |
Recycle: | ____ | ____
| | | | |
Runoff start: | ____ | ____
| | | | |
Sparge liquor: | ____ | ____ | ___
| | | |
Sparge: | ____ | ____
| | | | |
Runoff end: | ____ | ____ | ___ | _____
| |
_____
|
Kettle top-up:
| | | | |
______
| |
Wort boil, start: | ____ | ____ | ___ | _____ | ______ | _____
|
Hopping: | ____
| | | | | |
_______
|
____
| | | | | |
_______
|
____
| | | | | |
_______
|
____
| | | | | |
_______
Copper finings: | ____
| | | | | |
Cast-out: | ____ | ____ | ___ | _____ | ______
| |
Volume adjustm':
| |
____ | ___ | _____ | ______
| |
Chill start: | ____
| | | | | |
Chill finish: | ____ | ____ | ___ | _____ | ______
| |
Pitching: | ____
| | | | | |
_______
Oxygenation: | ____
| | | | | |
_______
Lag end: | ____ | ____ | ___ | _____
| | |
Yeast growth: | ____
| | | | | |
_______
|
____ | ____
| |
_____
| | |
|
____ | ____
| |
_____
| | |
|
____ | ____
| |
_____
| | |
Kraeusen end: | ____ | ____ | ___ | _____
| | |
_______
|
____ | ____ | ___ | _____
| | |
_______
Racking: | ____ | ____ | ___ | _____
| | |
|
____ | ____
| |
_____
| | |
Dry hops: | ____ | ____
| |
_____
| | |
_______
Finings: | ____ | ____
| | |
______
| |
_______
Primings: | ____ | ____
| |
_____
| | |
_______
Bottled: | ____ | ____ | ___ | _____ | ______
| |
Evaluation: Carbonation ____ Color ____ Clarity ____ Head ____ Lace____ Aroma____ Flavor____ Sweet____ Malty____ Roasty____ Bitter____ Hoppy____ Sulphur____ Esters____ Phenols____ Fusels____ DMS____ Diacetyl____ Smooth____ Fullness____ Other____
## Recipes
Recipes are for five-gallon "closed" fermentation based upon 90 percent brew-house yield, 15 percent evaporation in the kettle, a two-quart yeast starter, 5 percent kraeusen wort added to the secondary, 5 percent racking losses, and corn sugar for bottle conditioning. There should be 5 ¼ gallons of cooled wort present after boiling, of which three quarts should be bottled; one quart for kraeusening at racking to secondary and two quarts for future yeast culturing. Pitch with a two-quart yeast starter. If you are repitching an active strain, remove less wort, so that five gallons are present after pitching. If you are not going to kraeusen the beer, 5 ¼ gallons should be fermented. Where type 90 hop pellets will be used instead of whole hops, reduce kettle-hop rates by 10 percent.
### Pilsener
OG 12 °Plato (SG 1048)
FG 3.5 °Plato (SG 1012)
4.5% alcohol by volume
Color 4.5 °standard reference method
40 international bitterness units
Soft water; treat 7.5 gallons with lactic acid to below pH 7.0
6.5 lb. two-row lager malt
1 lb. Munich malt
.5 lb. CaraPils malt
Doughing-in: 7 quarts of liquor
Mashing-in: 3.5 quarts
Saccharification strike temperature: 153°F (67°C)
Rest temperature maintenance: 1.5 quarts
Sparging: 4.5 gallons
1.25 homebrew bitterness units Saaz hops at start of wort boil
7 HBU Saaz for 45 minutes
2.5 HBU Saaz for 30 minutes
.5 oz. Saaz finishing hops 15 minutes before strike
.5 oz. Saaz aroma hops at strike
Neutral or slightly estery lager yeast, 12⋅106 cells/mL
### Dortmunder
OG 13.5 °Plato (SG 1054)
FG 3.2 °Plato (SG 1013)
5.4% ABV
Color 5 °SRM
30 IBU
Very hard, high-sodium water (see table 18), treat 7.75 gallons
6.5 lb. two-row lager malt
1 lb. Munich malt
1.25 lb. CaraPils malt
Doughing-in: 7.5 quarts
Mashing-in: 3.75 quarts
Strike temperature: 150°F (65°C)
Temperature maintenance: 1.5 quarts
Sparging: 4.5 gallons
1.5 HBU Hallertau/Perle hops at start of boil
4.5 HBU Hallertau/Perle hops for 60 minutes
1.5 HBU Hallertau/Perle hops for 30 minutes
.25 oz. Hallertau/Perle hops for 15 minutes
Neutral or slightly estery lager yeast, 12–14⋅106 cells/mL
### Vienna
OG 13 °Plato (SG 1052)
FG 3.5 °Plato (SG 1013)
4.8% ABV
Color 11 °SRM
26 IBU
Hard water (see table 19), treat 7.75 gallons
4.5 lb. two-row lager malt
3 lb. Munich malt
.25 lb. CaraPils malt
.5 lb. caramel-30 °L malt
Doughing-in: 8 quarts
.25 lb. crystal-50 malt
Mashing-in: 4 quarts
Strike temperature: 150°F (65°C)
Temperature maintenance: 1.5 quarts
Sparging: 4.25 gallons
1.25 HBU Hallertau/Perle at start of boil
6.0 HBU Hallertau/Perle for 45 minutes
3.0 HBU Hallertau/Perle for 15 minutes
Aromatic lager yeast, 10–12⋅106 cells/mL, 50–55°F (10–13°C)
This recipe can be brewed from 100% Vienna malt (8.25 lb.) for a drier flavor.
Add ½ oz. of Hallertau/Perle at strike for a hoppier aroma.
### Munich Dark
OG 13.5 °Plato (SG 1054)
FG 4.0 °Plato (SG 1016)
5.0% ABV
Color 16.5 °SRM
25 IBU
Carbonate water (see table 18), treat 8 gallons
4 lb. two-row lager malt
3.5 lb. Munich malt
1 lb. crystal-50 malt
.25 lb. caramel-80 malt
.25 lb. chocolate malt
Doughing-in: 8 quarts
Mashing-in: 5 quarts
Strike temperature: 155°F (68°C)
Temperature maintenance: 1.5 quarts
Sparging: 4.25 gallons
1.5 HBU Hallertau/Perle at start of boil
3.0 HBU Hallertau/Perle for 60 minutes
3.0 HBU Hallertau/Perle for 30 minutes
.25 oz. Hallertau at strike
Neutral lager yeast, 14⋅106 cells/mL
This recipe can be brewed from 100% Munich malt (8.75 lb.).
### Munich Light
OG 11.5 °Plato (SG 1046)
FG 3.0 °Plato (SG 1012)
4.4% ABV
Color 5 °SRM
23 IBU
Mildly carbonate water (see table 18), treat 7.5 gal. with lactic acid to below pH 7.0
5 lb. two-row lager malt
1 lb. Munich malt
1.25 lb. CaraPils malt
Doughing-in: 6.5 quarts
Mashing-in: 4.5 quarts
Strike temperature: 153°F (62°C)
Temperature maintenance: 1.75 quarts
Sparging: 4.25 gallons
1.5 HBU Hallertau Kettle at start of boil
3.0 HBU Hallertau for 45 minutes
3.0 HBU Hallertau for 30 minutes
.25 oz. Hallertau at strike
Neutral lager yeast, 12⋅106 cells/mL
### Hellesbock
OG 16.5 °Plato (SG 1067)
FG 4.5 °Plato (SG 1018)
6.6% ABV
Color 5 °SRM
25 IBU
Soft or hard water, 8 gallons
9 lb. two-row lager malt
1.5 lb. CaraPils malt
.25 lb. caramel-30 malt
Doughing-in: 9 quarts
Mashing-in: 5 quarts
Strike temperature: 152°F (66°C)
Temperature maintenance: 1.5 quarts
Sparging: 4.5 gallons
1.5 HBU Perle at start of boil
1.5 HBU Perle for 60 minutes
3.0 HBU Perle for 45 minutes
.5 oz. Tettnang for 15 minutes
.25 oz. Tettnang at strike
Neutral lager yeast, 16–18⋅106 cells/mL
### Bock
OG 16.5 °Plato (SG 1067)
FG 4.5 °Plato (SG 1018)
6.6% ABV
Color 20 °SRM
25 IBU
Soft or hard water, 8 gallons
7.75 lb. two-row lager malt
1.5 lb. crystal-50 malt
.5 lb. caramel-120 malt
.75 lb. brown malt
.5 lb. chocolate malt
Doughing-in: 9 quarts
Mashing-in: 5 quarts
Strike temperature: 150°F (65°C)
Temperature maintenance: 1.5 quarts
Sparging: 4.25 gallons
1.5 HBU Hallertau at start of boil
3.0 HBU Perle for 60 minutes
4.5 HBU Perle for 15 minutes
.5 oz. Hallertau at strike
Neutral lager yeast, 16–18⋅106 cells/mL
### Dopplebock
OG 18.5 °Plato (SG 1075)
FG 4.5 °Plato (SG 1018)
7.3% ABV
Color 16 °SRM
22 IBU
Soft or hard water, 8.25 gallons
7.25 lb. two-row lager malt
3.5 lb. Munich malt
1.5 lb. crystal-50 malt
Doughing-in: 10 quarts
Mashing-in: 5.5 quarts
Strike temperature: 150°F (65°C)
Temperature maintenance: 1.5 quarts
Sparging: 4 gallons
1.5 HBU Perle at start of boil
3.0 HBU Perle for 60 minutes
3.0 HBU Perle for 45 minutes
.5 oz. Perle at strike
Neutral lager yeast, 16–18⋅106 cells/mL
# CHAPTER 15
# Brewing Procedure
The following is a step-by-step guide to the classic lager brewing process, using a three-vessel brew house (mash-tun, lauter-tun, combination decoction/wort kettle). The same program can be followed using either a two-vessel (combined mash-/lauter-tun) or four-vessel (separate decoction kettle) brew house.
Mash temperatures given are for 122 degree F (50 degrees C) and 153 to 155 degree F (67 to 68 degrees C) rests; 131 degree F (55 degrees C) and 158 to 160 degree F (70 to 71 degrees C) rests can be substituted. With well-modified modern malts (over 36 percent SNR, and especially over 40 percent SNR), peptidase activity during a 122 degree F (50 degrees C) rest risks depleting the wort of body- and head-forming polypeptides. A brief, combined proteolysis and saccharification rest at 131 degrees F (55 degrees C) better suits malts with an SNR of 37 to 40 percent than a 122 degree F (50 degrees C) rest, and the higher temperatures should be substituted in the decoction program.
For infusion mashing, see appendix B. For step mashing, refer to appendix C.
## Malt Examination
Peel the husk away from the dorsal side of twenty or so kernels of malt to expose the acrospire. Chew another dozen or so kernels. Unless the malt is easily chewed and the acrospire has been uniformly grown to from three-fourths to the full length of the kernel, the malt should be step or decoction mashed. This is also true for base malts that are of questionable enzymatic strength, are high in protein, or that are unevenly malted.
## Three-Decoction Mash
Expect a three-decoction mash to take from 3 ½ to 9 hours. Begin by crushing the amount of malt called for in the recipe. The malt should be weighed, but it may be measured assuming that one pound equals 4.25 cups, U.S. liquid measure.
### Doughing-In
Dough-in the crushed malt by sprinkling it with small amounts of brewing liquor at roughly 58 to 70 degrees F (14 to 21 degrees C) until twenty-four to twenty-eight fluid ounces of liquor per pound of malt has been kneaded in. For a thin mash, when brewing light, quickly fermenting beer, thirty-two to forty fluid ounces of water per pound are required.
Hold for fifteen minutes (or for up to thirty minutes for steely or enzyme-poor malt). Mix regularly and thoroughly to distribute moisture evenly throughout the mash. The malt should be uniformly and universally solubilized. Check for successful moisture penetration of the coarser grits and hard ends of the kernels by pulverizing several between the fingers. Make sure there are no dry pockets or balled flour within the mash.
### Acid Rest
For each pound of malt doughed-in, bring twelve to fourteen fluid ounces of liquor to a boil. For a thin mash, sixteen to twenty fluid ounces should be boiled. Knead it into the grain to raise the mash temperature evenly to 105 degrees F (40 degrees C). After twenty minutes, check the mash pH; if it is at or below 5.8 (preferably 5.2 to 5.5), proceed with the first decoction.
Otherwise, acidifying measures must be taken before mashing should proceed. The mash acidity can be reduced by adding acidified mash from a lactic-acid mash. To make such a mash, one or two days before brewing, mash-in 5 percent of the malt to 155 degrees F (68 degrees C). Rest for one hour. Cover and cool, undisturbed, down to 125 degrees F (52 degrees C). Knead in a small amount of dry crushed malt (to introduce _Lactobacillus delbrueckii_ ) and cover. No airspace should be left above the mash; cover the mash by pressing food-wrap down over it to seal it from contact with air.
After twenty-four hours at 95 to 120 degrees F (35 to 49 degrees C), the pH will drop below 5. After two days, it will be below 4.5. A lactic-acid mash is most effective when the brewing water is relatively soft. Five percent acidified mash at pH 4.8 can be expected to reduce a mash at pH 6 to below 5.8. At pH 4.5, it will lower the mash pH to about 5.6.
More commonly, the pH of an overly alkaline mash is corrected by employing a portion of sourmalt or cautiously mixing minute amounts of lactic or phosphoric acid into it until the mash pH drops to 5.5. In future brews, adjust the pH of the liquor, rather than the mash, using proportionally similar treatment.
### First Decoction
Pull the heaviest one-third part of the mash to the side of the tun and withdraw it to the decoction kettle. The "heaviest" part of the mash means mash with only enough liquid to fill the spaces between the grains, but not so much as to cover them. The decoction should always be thick, with just enough free liquid to prevent scorching. Return any free-standing liquid that settles above the boiler mash back to the main mash. If the mash itself is thin, however, 40 percent or more of the mash volume may need to be pulled for the decoction, and boiling some free-standing liquid with the decoction is inevitable. Use the "Planning Data" worksheet in table 14 to calculate decoction volumes.
Closely cover the cold settlement in the mash-tun and maintain its temperature as nearly as possible at 95 to 105 degrees F (35 to 40 degrees C).
Heat the decoction to 150 to 158 degrees F (66 to 70 degrees C) in ten minutes (or as rapidly as possible), and hold it there for ten minutes to dextrinize it. Heat it to 167 degrees F (75 degrees C) over five minutes, then bring it to a boil in five to ten minutes while lifting the grain up and away from the bottom of the kettle to prevent scorching. Cover and boil it vigorously for five to ten minutes, the longer time being for steely or enzyme-poor malt.
### Protein Rest
Return the darkened decoction to the starchy cold settlement by degrees, while lifting and breaking up the mash, over a period of five or so minutes, to evenly raise the temperature of the whole to 122 degrees F (50 degrees C) or within the range of 118 to 128 degrees F (48 to 53 degrees C). Check the temperature throughout the mash, making sure it is even.
Monitor the mash pH; it should drop to pH 5.2 to 5.3. The usual rest period is only five minutes before the heaviest part of the mash is drawn off for the second decoction.
### Second Decoction
Withdraw the heaviest 33 to 45 percent of the mash to the decoction kettle (as before, adjust the decoction volume to reflect consistency). Cover both mashes. Heat the decoction to 150 degrees F (66 degrees C) within ten minutes, then through the alpha-amylase range and to 167 degrees F (75 degrees C) over ten to fifteen minutes, and then to boiling, while lifting and stirring. Boil vigorously for five minutes, or for up to twenty minutes for very steely malts, stirring frequently.
### Saccharification/Dextrinization Rest
Return the decoction to the rest mash evenly, so as not to scald any portion of the rest mash. Temperature dispersal should be absolutely uniform. The saccharification temperature, usually 153 to 155 degrees F (67 to 68 degrees C), should be reached within five minutes when the beer is to be fully lagered. Returning the decoction gradually, over a period of fifteen to thirty minutes, to a rest temperature of 149 to 151 degrees F (65 to 66 degrees C) favors maltose production when brewing lighter, drier beers.
Hold the saccharification temperature for fifteen minutes; the mash will darken in color.
Test for successful starch conversion. Float tincture of iodine (.02 _N_ solution; usual medicine-cabinet variety) drop by drop above a small sample of the mash in a porcelain dish. Check the color at the interface of the iodine and the mash. Continue mashing until there is no color change, or for a sweet, full beer until the reaction is only very faintly mahogany reddish. Disregard discoloration caused by husk particles; it in no way indicates lack of conversion.
**Use caution. Iodine is a poison. Do not let it inadvertently taint the mash. Discard all samples and rinse the dish and any equipment that has been contacted by the iodine.**
The precise saccharification rest temperature should be maintained. This may be simply accomplished by infusing small amounts of boiling liquor into the mash. Because the mash liquid absorbs a great deal of extract during saccharification, it may become too thick to satisfactorily settle into a well-stratified filter bed. These temperature-maintenance infusions serve to improve filtering of a thick mash and are generally necessary when less than 1 ½ quarts of liquor have been used to mash-in each pound of malt. Thinning a thick mash assures that its density does not interfere with filter bed settlement in the lauter-tun.
### Lauter Decoction
When starch end point has been verified, rack off the very thinnest 40 to 50 percent of the mash. Bring it to a boil. Boil vigorously while stirring for five minutes.
### Final Rest
Remix the mashes thoroughly, evenly raising the temperature to near 170 degrees F (77 degrees C). Rouse and mix the mash for five minutes while maintaining the lauter rest temperature to force insoluble mash particles into a temporary suspension. This causes a clearly stratified settlement of first the hulls, then endosperm particles, and finally the protein gums to be formed in the lauter-tun.
## Single-Decoction Mash
_Part 1._ Using malt of 33 to 36 percent SNR, mash-in the malt with thirty-six to forty-eight fluid ounces of liquor at 130 to 135 degrees F (55 to 58 degrees C) per pound of malt (the colder the malt is, and the less the volume of liquor being used, the hotter the liquor temperature will need to be).
After five minutes, pull the heaviest 33 to 45 percent of the mash volume and heat it to 150 degrees F (66 degrees C) in ten minutes, to 167 degrees F (75 degrees C) in ten to fifteen minutes, and then to boiling. Boil for five minutes.
Return the decoction to the rest mash for a 153 to 155 degree F (67 to 68 degrees C) saccharification rest. After fifteen minutes, begin testing for starch conversion. Maintain the temperature by infusing with boiling water as necessary. Depending on the fullness desired for the beer, conversion at the conclusion of the rest should give a negative to red-mahogany iodine reaction.
As soon as the desired iodine reaction is achieved, add boiling liquor to bring the dilution up to forty-eight fluid ounces per pound of malt and the mash temperature to near 170 degrees F (77 degrees C), then transfer the goods to the lauter-tun as quickly but as gently as possible.
_Part 2._ Using malt of 37 to 40 percent SNR, mash-in the malt with thirty-six to forty-eight fluid ounces of liquor at 140 to 145 degrees F (60 to 63 degrees C) per pound of malt for a 131 degree F (55 degrees C) protein/saccharification rest temperature.
After five minutes, pull the heaviest 33 to 45 percent of the mash volume and heat it to 158 degrees F (70 degrees C), rest it for ten minutes, and then heat it to boiling. Boil the decoction for five minutes.
Return the decoction to the rest mash for a 158 to 160 degree F (70 to 71 degrees C) dextrin rest. After fifteen minutes, begin testing for starch conversion. Maintain the temperature by infusing with boiling water as necessary.
As soon as the desired iodine reaction is achieved, add boiling liquor to bring the dilution up to forty-eight fluid ounces per pound of malt and the mash temperature to near 170 degrees F (77 degrees C), then transfer the goods to the lauter-tun as quickly but as gently as possible.
## Sparging/Filtering
Fill the lauter-tun to one-half inch above the false bottom with sparge liquor. Give the mash one final stirring and transfer it to the lauter-tun. Maintain the temperature at as close to 170 degrees F (77 degrees C) as possible for ten to fifteen minutes while the filter bed forms undisturbed.
In the meantime, bring the appropriate volume of sparging water to 170 to 175 degrees F (77 to 80 degrees C).
After the malt particles (husks, acrospires, and any starch granules) have settled and the liquid above the protein coagulum has cleared, set the filter bed and flush debris from the space below the false bottom by opening the draincock until a steady trickle of runoff forms. Drain the mash until the protein has settled and the clear liquid above it lies only about one-half inch deep. Smooth and level the mash surface. Maintain the liquid depth above the mash by returning the runoff to the lauter-tun until the wort runs clear. When all the cloudy runoff has been recycled, begin sparging. Open the sparging-water tap, matching the trickle of 170 to 175 degree F (77 to 80 degrees C) liquor to the runoff rate. Carefully balance the inlet and outlet flow-rates so that the liquid level above the mash is not disturbed.
Manipulate the flow, balancing the sparging rate to the runoff, so that the filtering takes 1 ½ hours to complete. Divide the amount of sweet wort to be collected by ninety (minutes) to define the required runoff flow per minute. (Six-row barley with a high husk content may be run off in as little as thirty minutes.) During a slow filtering, carefully rake the lauter mash to within six inches of the false bottom to close vertical channels and improve extract yield. Smooth the mash surface as any cracks appear, and keep the liquid level above the surface of the mash, but not more than two inches deep. Immediately begin heating the wort collecting in the copper to above 170 degrees F (77 degrees C). Sparge until the density of the runoff drops below 3.0 °Plato (SG 1012). Discontinue sparging and allow the mash to drain.
## Boiling the Wort
Add a small portion of loosely broken-up boiling hops, to reduce surface tension and the likelihood of boil over, to the sweet wort as soon as all of the extract has been collected.
Check the wort acidity. It should be pH 5.2 to 5.5. If it is below pH 5.0, protein precipitation will be retarded; adjust with a carbonate salt if necessary. Measure and record the density, corrected to 68 degrees F (20 degrees C), or 60 degrees F (15.56 degrees C) if that is the temperature that the hydrometer is calibrated at.
Lauter-tun
## Hot Break
Periodically examine samples of the boiling wort to assess protein/polyphenol flocculation. Remove a glassful of wort. The tannin/proteins that mist the wort early in the boil should coagulate into a much smaller number of larger flakes, one-eighth inch long or longer, as the boil progresses. Check the pH. It should drop during the boil to 5.0 to 5.3.
## Cold Break
As the end of the boiling period draws near, the flocks in the hot sample should readily precipitate and leave the wort clear. Force-cool the sample; it should slowly cloud again as it cools. This cold break should then settle out from the wort, leaving it clear and bright. If a clean break cannot be established during the designated boil, and the intensity of the boil and wort pH are acceptable, then either the malt or the brew-house program is unacceptable. The finished beer will likely be cloudy and astringent. Do not, however, exceed quoted boiling times, especially when brewing light beers. Longer boiling is not likely to improve clarification unless it is subsequent to a significant correction of the boiling intensity or pH, and will increase the bitterness and discoloration of the wort.
## Finishing Hops
Add finishing hops ten to fifteen minutes before the end of the boil, as the heat is shut off, or when the wort is being run from the kettle. The later the addition, the less the bitterness contribution, but the greater the flavor and aroma.
At kettle knock-out, turn off the heat. Check and record the wort acidity. The pH should be 5.0 to 5.3. Measure the wort volume and density, corrected to 68 degrees F (20 degrees C), or 60 degrees F (15.56 degrees C) if that is the temperature that the hydrometer is calibrated at. Correct the volume to the 68 degree F (20 degrees C) reference temperature by multiplying by .96, or by .958 for 60 degrees F (15.56 degrees C). If the density is high, and the volume is less than required, restore it with cold water. Determine extract efficiency using the table below.
Table 20
Brew-house Efficiency
---
1. Volume of wort, in gallons, at 68°F (20°C): | 1.__________
2. °Plato of wort, at 68°F: | 2.__________
3. Multiply line 1 by line 2: | 3.__________
4. Divide line 3 by table 15, line 1 (anticipated wort volume): | 4.__________
5. Planned °Plato of wort, from table 14, line 3: | 5.__________
6. Divide line 4 by line 5: | 6.__________
7. Multiply line 6 by line 13, table 15: | 7.__________
Line 7 is the actual brew-house efficiency. Use this efficiency in future brews to more accurately predict original gravity.
## Filtering and Cooling the Wort
If the wort will be run through a hop back, lay a bed of fresh hops over the false bottom to form a filter bed. If the wort will either be siphoned from the kettle or run off from a side tap, or if the wort was hopped with pellets, whirlpool the wort with a paddle for two to three minutes so that the hops and trub form a cone at the center of the kettle's bottom. Allow the hops and trub to settle for ten to fifteen minutes.
Force-cooling the wort quickly (fifteen to forty-five minutes) gives a more complete break than slow or passive attempering, and reduces DMS development. The colder the temperature the wort is chilled to, the better the cold break will be. A wort chiller should always be used. Where an immersible chilling coil is employed, the wort is cooled before it is run off from the kettle. For lager beers, cooling to at least 39 degrees F (4 degrees C) will reduce the chances of chill haze in the finished beer.
Run off the wort, slowly at first, returning it to the kettle without splashing, until the runoff clears. If the wort is being siphoned from the kettle, keep the pick-up close to the side of the kettle and away from the bottom of the kettle until near the end of the run. Run the wort into the fermenter or settling tank.
If the wort is being run into a closed settling tank, allow it to rest undisturbed for several (two to sixteen) hours before racking it off its trub to the fermenter. Expect the wort to precipitate a cold-break sediment equal to 10 to 20 percent of the hot-break trub. If the wort is being run directly to the fermenter, let it splash in freely to aerate it. Keep the cooled wort covered, and work using as sanitary a method and in as clean, dry, and draft-free an environment as possible.
Decant any wort in excess of the amount needed for primary fermentation into sterile containers. Reheat to boiling, cap and refrigerate until needed.
## Pitching the Yeast
Most lagers require the pitching of .40 to .66 fluid ounces (8.5 to 14 grams) of thick, pasty yeast per gallon of wort to be pitched, for a rate of 10 to 15 million cells per milliliter. Up to 1 fluid ounce (21 grams) of yeast may be required to ferment a gallon of wort at greater than 15 °Plato (SG 1061).
Table 21
Calculating Pitching Volume
If the culture vessel is not graduated, the volume of the yeast culture, in fluid ounces, can be measured by the formula:
(πr"2)h"
————
1.8046
1. Diameter of yeast-culture container:
| |
d" = _____________
---|---|---
2. Divide line 1 by 2 (r=d/2):
| |
r" = _____________
3. Multiply line 2 by itself:
| |
r"2 = _____________
4. Multiply line 3 by 3.1416:
| |
πr"2 = _____________
5. Multiply line 4 by the thickness (depth, in inches) of the yeast sediment:
| |
(πr"2)h" =
6. Divide line 5 by 1.8046: | (πr"2)h"
————
1.8046 | = _____________
Line 6 gives the fluid ounces of yeast sediment in the starter vessel.
Shake the starter vessel to mix the yeast sediment into a milky solution. Pitch only the amount of yeast necessary to ensure rapid initial fermentation. Any remainder may be frozen, or covered with fresh wort, then capped with an airlock and refrigerated, for use later in the brewing cycle or for subsequent brewings.
## Kraeusening
When kraeusening, "new" beer equal to 10 percent of the primary fermenting volume at up to 5 degrees F (3 degrees C) above pitching temperature is used to introduce active yeast for fermentation. Kraeusening produces strong initial fermentation.
## Yeast Starter
A yeast starter is the equivalent of kraeusen beer, but made up from a yeast culture roused into ten times its volume of wort. The ten-to-one dilution is repeated each time strong fermentation becomes evident until the starter is at pitching strength (5 to 10 percent of the volume it will be pitched into). Yeast starters, like kraeusening, promote stronger and faster fermentation starts and blanket the ferment with CO2 much sooner than does pitching yeast sediment.
Yeast starters should be made up one to two days before brewing, from sterile wort and 10 to 20 percent of the pitching volume of yeast. The wort may be from bottled wort saved from a previous brewing, or may be made up in a small brewing to resemble the wort that will be pitched. If the yeast can be aerated, oxygenate it for several hours before pitching to strengthen the culture and reduce the lag phase.
Pitch the yeast as the wort runs into the fermenter. Allow the wort to splash into the fermenter and rouse it splashingly and thoroughly. Fill the fermenter and fit the airlock (blowoff) to it. The temperature at pitching should usually be 42 to 50 degrees F (6 to 10 degrees C). A low initial temperature will produce a beer with less esters and fusel alcohols. To encourage growth, the yeast can be oxygenated by trickling sterile air or oxygen up through the wort for up to an hour after pitching.
## Primary Fermentation
Primary fermentation takes five to fourteen days. Temperatures quoted are for full-bodied, dextrin-rich lagers above 10 °Plato (SG 1040) that are to be fully lagered. For maltose-rich wort to be fermented in less time, add 6 to 8 degrees F (3 to 4 degrees C) to the temperature. Do not, however, exceed 60 degrees F (16 degrees C) during primary fermentation and 50 degrees F (10 degrees C) during secondary fermentation if at all possible. Keep the beer out of direct sunlight and avoid drafts and temperature fluctuations.
Check and record the temperature, density, pH, and yeast-cell count of the beer while it is fermenting and lagering only if it can be done conveniently and without any risk of contaminating the beer.
Table 22
Primary Fermentation Temperature/Time Guideline
---
Temperature | Duration
50–55°F | (10–13°C) | 5–8 days
48–50°F | (9–10°C) | 6–10 days
41–48°F | (5–9°C) | 7–14 days
Every yeast strain has a temperature optimum, where it ferments well and gives the desired fermentation character. Quoted primary and secondary fermentation temperatures should be adjusted to suit the specific requirements of any particular yeast strain.
Hold the pitched wort, covered and fitted with a fermentation lock, at an ambient temperature of 45 to 50 degrees F (7 to 10 degrees C).
A foam wreath forming at the sides of the fermenter indicates that the lag phase is ending. A light shone on the surface reveals active carbonic gas release.
### Low Kraeusen
Six to thirty-six hours after pitching, the foam migrates to the center of the beer surface. The temperature should begin rising. Extract should drop approximately .5 °Plato (SG 1002), and the pH should drop noticeably. The foam cover becomes tightly knit. A light shone on the beer reveals even more CO2 release and a milky turbidity, indicative of the amount of yeast in suspension.
Carboy with Blowoff and Airlock
### High Kraeusen
Two to four days after pitching, the foam rises up to form looser-knit, cream-colored "rocky heads." When the temperature of the ferment reaches the maximum recommended for the yeast strain, or 55 degrees F (12 degrees C), attemper the fermenter so that the temperature of the fermenting beer does not rise further. The liquid pressure should fall by .75 to 2 °Plato (SG 1003 to 1008) daily. The pH should drop to about 4.5.
### Post Kraeusen
Six to fourteen days (six to ten days is usual) after pitching, the foam head begins to diminish as CO2 production falls off. Extract reduction should slow dramatically, and with a reasonably flocculant yeast strain the cell count will drop below ten million cells per milliliter.
If a diacetyl rest is being employed, force the temperature of the post-kraeusen ferment to rise to 55 to 60 degrees F (13 to 16 degrees C). After two days, lower the ambient temperature again, bringing the beer down to 38 to 40 degrees F (3 to 4 degrees C) at 3 to 5 degrees F (1 to 3 degrees C) per day.
When the extract drop slows to 0.5 °Plato (SG 1002) over twenty-four hours, the head will have completely fallen. Rack the beer into a secondary fermenter/lagering vessel. The beer should be clear and bright. If the yeast does not seem to have largely settled out and still clouds the beer, then the yeast strain is dusty or mutated, the beer may ferment past end point, and it will probably need to be fined.
Make a hydrometer reading of the sample and record it. For beers from worts of 10 to 15 °Plato (SG 1040 to 1060), it should be roughly 30 to 40 percent of what the wort density (OG) was, or 1 to 2 °Plato (SG 1004 to 1008) above the expected final density (FG). Testing with Dextrocheck (reducing-sugar indicator) should show less than 5 percent reducing sugar. Check and record the pH; it should not have dropped much below pH 4.5.
## Secondary Fermentation
Clean and sterilize the lagering vessel, preferably rinsing it with sterile, boiled water. Carefully rack the beer in the primary fermenter off of its settlement into the closed lagering vessel. Take care not to carry along any of the sedimented yeast or trub; cease racking when the runoff becomes the least bit cloudy.
Run the beer into the lagering vessel as quickly as possible, absolutely avoiding splashing. Only enough head space to allow for mild foaming should be left above the nearly fermented beer; if necessary restore the volume by topping-up the fermenter with sterile water. Kraeusen beer should be added to produce the smoothest possible beer. Kraeusening generally reduces diacetyl, corrects poor primary-fermentation yeast performance, entrains CO2 in the beer, and gives fuller, mellower flavors.
Seed yeast for starters and culturing should be collected from the primary-fermenter yeast cake.
Fit the fermenter with a fermentation lock. If the hydrometer reading at racking is one-third the density of the original gravity or greater, ferment for seven to twenty-one days, reducing the ambient temperature from 38 to 40 degrees F (3 to 4 degrees C) down to 33 to 37 degrees F (1 to 3 degrees C) when carbon dioxide production falls off. If the reading is much less than one-third of the original hydrometer reading, the beer lacks slowly fermenting dextrins and should nearly ferment out in seven to ten days before the temperature is reduced for lagering.
Beer that will not be lagered is usually held in a secondary fermenter for only one to two weeks, and is fermented down to the terminal extract value before being chilled to clarify and sediment it in preparation for packaging.
### Lagering
Lager tradition calls for seven to twelve days secondary fermentation and lagering per each 2 °Plato (SG 1008) of the original wort hydrometer reading, with the beer temperature falling to as close to 33 to 36 degrees F (1 to 2 degrees C) as possible. The above notwithstanding, if the hydrometer reading at racking was much less than one-third the value of the wort reading (OG), the beer should not be secondary fermented/lagered for more than one week for each 2 °Plato of the wort reading; four or five days per each °Plato is usual. Lowering the temperature to 30 to 33 degrees F (-1 to 1 degree C) immediately after secondary fermentation reduces lagering times.
Beer that will be bottle-conditioned is lagered at atmospheric pressure. Lagering may be carried out under pressure, but only if the lagering vessel is able to be safely pressurized, is fitted with a pressure-relief valve, and the beer will not be bottle-conditioned. _Do not attempt to lager under pressure in a glass carboy._ Use only a Cornelius keg or the like fitted with a pressure-relief valve. The vessel should be closed when the beer is .2 to .5 °Plato (SG 1001 to 1002) above the anticipated terminal gravity.
Fermentation is complete when the hydrometer reading is at or near the terminal extract value and no drop or visible activity has been experienced in the last five days. A reducing-sugar analysis should show less than 2 percent, indicating that the beer has fermented out.
If the beer is to be fined, prepare the finings. For fining with gelatin, measure out one gram (one-eighth teaspoon) of 95 percent pure gelatin and two fluid ounces (sixty milliliters) of cold water or beer per gallon of beer to be fined. Cover the solution and let the gelatin hydrate for one hour.
Gently heat the solution to 150 to 160 degrees F (65 to 70 degrees C) to dissolve the gelatin; do not let it come to a boil. Do not let it cool below 120 degrees F (50 degrees C) before thoroughly mixing the hot solution into the aged beer as a stream, by stirring, or by rocking the lager vessel for two to three minutes, without aerating the beer. Allow the beer to rest undisturbed for seven to fourteen days at below 50 degrees F (10 degrees C) before racking for bottling.
Prepare isinglass finings by measuring out one-half gram of the finings and four fluid ounces (120 milliliters) of sterile beer or water for each gallon of beer to be fined (for batches of more than one barrel, use five grams of isinglass in .15 gallons, per barrel of beer). Check the pH; reduce the acidity to pH 2.5 to 3.0 with phosphoric acid if necessary. Closely cover the finings while they undergo acid hydrolysis at 55 to 65 degrees F (12 to 18 degrees C) for twenty-four to thirty-six hours. Mix thoroughly into the aged beer.
Isinglass will clear the beer within twenty-four to seventy-two hours. The beer temperature should be held stable or be let rise slightly until it is racked.
Dry hops or hop extract can added either anytime before the finings or with them, but they are not usually added afterward.
## Bottling
The ruh beer is usually quietly racked off of its sediment to a third vessel for mixing with the priming solution, whether that be priming sugars, wort, or kraeusen beer. Refer to table 23 for guidelines for the amounts of each to add for a desired level of bottle carbonation.
Table 23
Bottle Priming
---
For Bottle Pressure* At: | Approximate: Oz. Dextrose/Gal. | Fl. Oz. Kraeusen/Wort/Gal
50°F | 32°F | 40°F | 60°F | 10°P | 12°P | 14°P
atmospheres
2 | 1.2 | 1.5 | 2.5 | 1.2 | 13.5 | 11 | 9.5
2.5 | 1.5 | 1.9 | 3.2 | 1.5 | 17 | 14 | 12
3 | 1.9 | 2.35 | 3.8 | 1.8 | 20.5 | 17 | 14.5
3.5 | 2.3 | 2.8 | 4.4 | 2.1 | 24 | 20 | 17
* At sea level. For elevations other than sea level, reduce bottle priming by approximately 3.3% for each 1,000' of elevation. 50°F is the standard temperature at which bottle pressure is gauged.
Bottling should take place in a clean, dry area that is free from drafts. Mix the carefully measured priming solution into the ruh beer without splashing, rousing it for several minutes to ensure uniform dispersal and to induce carbon dioxide to release from and blanket the beer.
Inspect the bottles. Reject any with chipped rims or residue deposits. Wash in warm water, using cleansing solution only if necessary. Thoroughly brush the inside surfaces. Sterilize bottles in a dishwasher on the sanitizer cycle, with live steam by stacking wet bottles in an oven and heating them to 200 degrees F (93 degrees C), in a .2 percent chlorine bath, or in a water bath at above 170 degrees F (77 degrees C). Invert the bottles on clean paper towels to drain.
Carefully dispense the primed beer into the sanitized bottles, filling them at least to within three-fourths of an inch of the rim. Loosely cover the bottles for several minutes to allow carbon-dioxide release to drive off oxygen in the head space before securing the caps and inverting the bottles to disclose leakage.
Hold at 50 to 65 degrees F (10 to 18 degrees C) for fourteen to twenty-one days before beginning to reduce the temperature to 32 to 45 degrees F (0 to 7 degrees C) for conditioning and storage. Do not subject the bottled beer to drafts or temperature fluctuations.
Soda Keg and Filtering System
Soda Keg
Soda Keg and Filter
### Draft Beer
Clean the keg in the same way you would clean bottles; if the keg is wooden, check its pitch or paraffin surfacing for cracks or deterioration, and melt out the old wax and reline as necessary. Usually beer for draft is drawn into the keg with just less than 1 percent fermentable extract left in it to provide carbonation for cask conditioning. Otherwise, add kraeusen beer or priming solution in sufficient quantity to produce 1.5 atmospheres (20 psi) pressure at 50 degrees F (10 degrees C). Fill the keg to within two inches of the bung hole and rouse thoroughly at priming to induce a mad condition. After several minutes, drive in the bung with a rubber or wooden mallet. Condition for ten to fourteen days at 50 to 65 degrees F (10 to 18 degrees C) before tapping.
A CO2 source may be used to carbonate the beer. At its simplest, this may be accomplished by applying CO2 pressure to the keg, with the regulator set for the pressure (psi) called for in the "Volumes of CO2" table found in appendix D, and shaking the keg several times over a fifteen-minute period, until the flow of CO2 upon shaking falls off.
## Troubleshooting
### Mash
_Balled starch in mash_ : Poorly handled doughing-in of the crushed malt; malt too finely crushed.
_Mash pH too high_ : Brewing water overly alkaline; liquor for pale beer not acidified, poor quality malt. Correct by adding acid or mineral salt.
_Mash pH too low_ : Brewing water overly acidic; lactic-acid- or acetic-acid-bacteria-spoiled malt. Correct pH with alkaline mineral salts.
_Mash doesn't saccharify_ : Lack of diastatic enzymes in mash, either destroyed at malting or by mash temperatures above 160 degrees F (71 degrees C); inappropriate pH. Add crushed diasatic malt and continue mashing; check pH. As a last resort, add diastatic enzyme preparation.
_Set mash_ : Poorly converted mash; malt poorly doughed-in; malt too finely crushed; mash poorly stirred up before filtering; too fast a runoff rate. In most cases, correct by thoroughly stirring up the mash and allowing it to resettle; cut the mash to within six inches of the bottom to reopen channels of extract flow in the mash. Press hot water up from under the false bottom. Reduce runoff/sparging rate.
### Wort
_Low extract_ : Not enough malt used; poor sparging efficiency; insufficient crushing; balled starch from poorly handled doughing-in; wrong mash pH; insufficient mash enzyme activity. Check for unconverted starch by making an iodine test of a wort sample. Continue brewing the same volume at the lower density.
_Poor kettle break_ : Flocks do not sediment; excessive protein decomposition during malting or mashing; wort agitation insufficient; temperature too low; improper pH; too few hop tannins in the boil; poor quality malt. Increase heat and wort movement. Check pH and hop rate; correct if necessary. If the wort clears but throws very little sediment, then malting or mashing protein digestion may have been overdone.
_Poor cold break_ : Wort cooled too slowly; improper pH; wort lacks tannins; excessive protein digestion. Cool to below 32 degrees F (0 degrees C) to encourage sedimentation.
_Wort tastes sour_ : Wort pH too low; coliform, acetic, or lactic-acid bacteria contamination. Check pH; heat wort to above 140 degrees F (60 degrees C) and re-cool; pitch quickly. Sourness may diminish with aging or be masked by adding burnt malt, finishing hops, or calcium carbonate.
_Sour or vinegar taste/smell_ : Acetic acid bacteria. As above.
### Fermentation
_Insufficient lag phase_ : Wort insufficiently aerated; too much trub carried into ferment; temperature too high. Ferment has strong solvent and fruity aroma. Irreversible.
_Excessive lag phase_ : Too little yeast pitched; yeast weak, degenerated, poisoned, or shocked by temperature change; wort extract too low; inappropriate pH; wort too cold; inadequate oxygenation of wort. Rouse in new yeast. Correct temperature to suit yeast strain.
_Sluggish fermentation_ : Weak or inappropriate yeast strain; yeast degenerated, poisoned, or shocked by temperature change; fermentation temperature too cold; iron, chlorine, or nitrates in the water supply; low wort extract; wort lacking in readily fermentable sugars or soluble nitrogen. Establish correct temperature and pH; repitch.
_Spotty low-kraeusen head, unable to support trub_ : Weak yeast; temperature too low; wort inadequately hopped; deteriorated hops. Raise fermentation temperature; rouse in new yeast.
_Fermentation ceases before high kraeusen_ : Temperature too cold; extract too low or from wort lacking readily fermentable sugars or yeast nutrients. Yeast strain mutated by exposure to too much glucose (catabolic repression) or lack of oxygen (petite mutants). Establish correct temperature; repitch.
_Yeast break, head falls prematurely_ : Temperature lowered prematurely; mutated yeast strain cannot ferment maltotriose; too fast a pH drop in the ferment (bacterial contamination); inadequate extract. Establish correct temperature, pH; repitch.
_Poor yeast flocculation; yeast do not settle; porous sediment_ : Dusty yeast strain; wild yeast contamination; inappropriate pH; temperature too high. Reduce temperatures. Fine the aged beer.
_Haze in beer after high kraeusen_ : Insufficient protein reduction during mashing; inadequate boil; wild yeast contamination; dusty yeast pitched. Reduce temperature before racking and during lagering; fine; as last resort treat with papain or similar proteolytic-enzyme extract.
_Yeast floats to surface after high kraeusen_ : Fermentation contaminated by wild yeast; culture yeast degenerated; brewing water too soft; sudden temperature rise; pH too high. Skim. Degenerated yeast have an unpleasant smell; when yeast performance is poor or abnormal, sample the bouquet.
### Bottled or Kegged Beer
_Roughness:_ Abnormal water composition; insufficient boil; excess tannin; excessive/alkaline sparging; insufficient kettle evaporation; hot-side oxygenation.
_Fruity aroma/flavor_ : From esters, higher alcohols, acetates of higher alcohols. The consequence of underoxygenation of the pitching yeast, too high a fermentation temperature, too low a pitching rate, or a deteriorated yeast strain. When unpleasant and combined with vegetal aroma and flavor, from coliform contamination of the wort or yeast culture.
_Celery odor: H. protea_ contamination, probably from tainted yeast culture. Irreversible.
_Bitter-vegetable taste:_ From deteriorated hops (oxidized beta-acids).
_Buttery (diacetyl) flavor_ : Where strong and like rancid butter, from lactic-acid bacteria, significantly _Pediococcus._ Slight or pleasant diacetyl flavor more often from low pitching rate, underoxygenation, petite mutants in culture yeast, or characteristic of the yeast strain; beer racked off its primary sediment too early or oxidized in the secondary.
_Cardboardy taste_ : Oxidation; from too much air in bottle headspace; trub carried into/lipids oxidized in the ferment; warm storage, mishandling; insufficient boil; when with poor head retention, from lipids in the beer.
_Sulfury aroma/flavor_ : Too low a fermentation temperature; poor rinsing of sulfur-based sterilant; from wild yeast, _Zymomonas_ or coliform bacteria. May be characteristic of yeast strain, or from autolization of sedimented yeast. Except when from bacterial contamination, may be reduced by aging or by scrubbing with carbon dioxide. Corny aroma and flavor is characteristic of dimethyl sulphide (DMS), from poorly malted barley, especially six-row; high-moisture malt; hot wort not chilled quickly enough; coliform bacteria contamination.
_Sour taste_ : From too low a pH; from acetic- or lactic-acid bacterial contamination.
_Medicinal aroma/flavor_ : From wild yeast or bacteria; chlorine in the ferment; plastic contamination; excess of phenolic material from oversparging or weak wort boil. Accentuated by high fermentation temperatures.
_Astringency_ : Excessive sparging, hot-side aeration, yeast autolization, excessive or oxidized trub.
_Skunky odor_ : Beer light-struck. Avoid direct sunlight during brewing and in package; reduce headspace.
_Rotten egg odor_ : Hydrogen sulphide; yeast-strain characteristic; fermentation by wild yeast; weak fermentation; in bottled or kegged beer, may be from contamination by _Zymomonas_ bacteria.
_Disagreeable smell/taste; turbidity, acidity_ : _Pediococcus_ or _Bacillus_ contamination of the primary ferment. Irreversible.
_Green apple flavor_ : Acetaldehyde, the principle volatile acid in beer. From too high a fermentation temperature; yeast-strain characteristic; bacterial contamination.
_Banana aroma/flavor_ : Acetates. Yeast-strain characteristic; wild-yeast contamination; too high a fermentation temperature.
_Thinness:_ Wort extract too low; excessive mash protein digestion; dextrin-poor extract.
_Haze:_ Poor mash protein digestion; insufficient boil; wild yeast; bacteria; oxidation of beer; poor starch conversion in mash.
_Gelatinous precipitate_ : Excessive sparging; poorly degraded hemicellulose.
_Lack of head_ : Excessive protein rest; overmodified malt; too high an adjunct ratio; lipids in ferment (excessive sparging; autolized yeast); overfoaming in fermentation; overboiling; insufficient or deteriorated hops; contact with oil.
_Gushing:_ Excess of priming sugars; beer not fermented out before packaging; temperature fluctuation; mishandling; old malt; iron; wild-yeast contamination.
Problems encountered with bottled beer can be accentuated by holding a sample at 85 degrees F (30 degrees C) and visually monitoring it for several days, then evaluating the forcing sample. Precipitates or surface formations generally indicate microbial contamination. Culturing or flavor evaluation may pinpoint organism responsible.
# CHAPTER 16
# Cleaning and Sterilizing
All brewing equipment must be kept scrupulously clean. Wort-cooling, yeast-culturing, and fermenting equipment must also be made sterile. Equipment should be cleaned immediately after each use and all residues and particles washed off before it is sterilized. No abrasives should ever be used to scour brewing equipment, with the exception of copper or stainless steel kettles.
Cleaners should be used sparingly, and then only on equipment that has been badly neglected or is too soiled to be cleaned by being soaked in water and then rigorously scrubbed. Where deposits have formed, only the deposits themselves should be treated for removal, and then preferably by nothing more than a thick paste of fresh yeast.
Sterilizing with boiling water or steam is preferable to using chemical sterilants. Heating equipment to above 140 degrees F (60 degrees C) for twenty minutes eliminates most microbes, while heating to above 170 degrees F (77 degrees C), boiling, or steaming sterilizes any equipment that does not have impenetrable deposits. For homebrewers, sterilizing wet bottles and small pieces of equipment in an oven at 200 degrees F (93 degrees C) or in a dishwasher with a heat-drying cycle is oftentimes the most practical method.
Sterilizing solutions should only be applied where equipment size, shape, or manufacture — or severe contamination — makes heat sterilization impractical. The cleansers and sterilants specified below should not be misconstrued as being substitutes for prompt and thorough cleaning with hot water or sterilizing with pure, clear 180 degree F (82 degrees C) water. All cleaning and antiseptic solutions should be freshly made up as needed, as most lose their potency during storage.
A thorough rinsing after using a cleanser or sterilant is very important to protect the taste of the beer. Spray or swab all surfaces with rinse water. Grossly inefficient rinsing can be identified by monitoring the pH of the rinse water for change.
The usual cleaning cycle includes a cold-water soaking, either a hot or cold cleaning as appropriate, and a cold rinse. For fermenting equipment, heat sterilization or the application of a sterilant (usually in cold water, followed by a hot rinse) is also necessary. See "Percentage Solutions" in the "Useful Information" section of appendix D for the preparation of percentage solutions. Following is a detailed look at how to clean various materials with different solutions.
## Construction
_Stainless steel_ : Clean with 2 to 3 percent hot caustic-soda solution and mild detergent at 170 degrees F (77 degrees C), followed with one or more rinses with water hotter than 140 degrees F (60 degrees C). Use 2 percent phosphoric, sulfuric, or sulfurous acid or 3 percent iodophor as a sterilant. Acid sterilants remove _beerstone,_ or deposits of oxalic lime and organic matter and passivate stainless steel.
_Copper:_ Only noncorrosive cleansers should be used. Caustic soda especially must be avoided. As copper is commonly used only for kettle fabrication, it should only need to be scoured with sand or other silica abrasive.
_Glass, glazed porcelain_ : Most cleansers and sterilants are suitable for use on glass or glazed porcelain; only phosphoric acid should not be used, as it can etch glass. A 2 to 3 percent caustic-soda solution is the preferred cleanser. Trisodium phosphate or mild detergents may be substituted. Microorganic contamination may be countered by application of chlorinated trisodium phosphate, or more commonly by a subsequent sterilant application. Household bleach in .2 percent aqueous solution is the most commonly used sterilant, but after a clear rinse, any of the acid sterilants can be used.
_Plastics, enamelware_ : A 2 to 3 percent caustic-soda solution will not harm these surfaces, but mild detergents are more commonly used. Either material requires application of a cleanser more frequently than glass or metal surfaces. Sterilants may be .2 percent chlorine bleach, 5 percent chlorinated trisodium phosphate solution (alkaline), sodium or potassium metabisulfite, 2 to 3 percent phosphoric acid, or 3 percent iodophor (acid).
_Aluminum_ : Only mild detergent with an acid cleanser (2 to 5 percent nitric or phosphoric acid) should be used. Compatible sterilizing agents include nitric and sulfurous acid.
_Wood:_ Rinse and brush wooden articles and then immerse them in boiling water immediately after use. Because of wood's natural porosity, use of cleansers should be avoided. Where absolutely necessary, a hot 2 to 3 percent solution of caustic soda or 5 percent sodium carbonate or bicarbonate can be applied, but it must be rinsed away by several long baths, from hot to cold. Five percent metabisulfite sterilizing followed by several clear-water rinses may be employed, but boiling or steaming is preferred. Any solution applied to wood is likely to leach into it, and is very difficult to coax back out of it.
Do not let wood implements sit damp between brewings. They must be dried thoroughly after cleansing.
## Equipment
_Malt mills and screen_ : These need only be cleaned with a stiff brush after every use. The cleaning, however, should be thorough.
_Mash and lauter-tuns_ : These should be kept wet after use and then scrubbed clean, rinsed, and dried. _A false bottom, strainmaster, or grain bag_ in the tuns must at least occasionally be boiled in a 2 to 3 percent caustic-soda solution or be scrubbed with an appropriate cleanser.
_Brew kettles of copper, stainless steel, or aluminum_ : Articles of this construction are the only ones used in brewing upon which an abrasive should be used. In fact, they benefit by its employment. The abrasives not only scour away calcified deposits, but pitting and etching of the kettle surface improves heat transfer and protein coagulation during wort boiling. A thick paste of fresh yeast should be brushed onto any stubborn deposits and kept moist before scrubbing. The yeast is finally rinsed away with clean, cold water.
_Immersible heating elements_ : These must periodically be soaked in 10 percent trisodium phosphate solution and then scrubbed to remove scale.
_Fermentation locks_ : The locks should always be soaked in a detergent or caustic-soda solution and then flushed clean after every use. Use of an antiseptic solution in the fermentation lock is not necessary, but it is advisable. A fermentation lock should be changed as soon as it becomes dirty.
_Fermenters:_ Insofar as possible, fermenters should only be cleaned by a thorough brushing in cold water after use and then be heat sterilized before their next use. Any beerstone deposits that do not come free should be covered with yeast paste. A mild detergent or cleanser (3 percent caustic soda, 5 percent trisodium phosphate, or 2 percent nitric or phosphoric acid) may be added. The paste should be kept moist for a day and then the deposit scrubbed away. Where a cleanser is necessary, use caustic soda. Where a sterilant is necessary, use household bleach, 2 to 3 percent phosphoric acid, or 1 percent iodophor.
_Plastic tubing_ : Used for wort cooling or siphoning, plastic tubes should be boiled in caustic soda and thoroughly flushed. Hydrogen peroxide can also be used to clean and sterilize tubing very effectively. Storing tubing in a sterilant solution largely eliminates the risk of contamination, but bleach turns tubing opaque, and iodophor discolors it, although this is not a serious problem to most brewers.
_Bottles:_ Clean-looking bottles can be washed in 170 degree F (77 degrees C) water, boiled, or steamed. Dirty bottles should be soaked and then scoured with 2 to 3 percent caustic soda, 5 percent trisodium phosphate, or a mild detergent solution, followed by several rinses with water hotter than 140 degrees F (60 degrees C).
## Alkaline Cleansers
_Sodium hydroxide_ : Caustic soda, NaOH. It is effective only when it is used hot. A most effective solvent, it hydrolyzes most malt and wort residues. It may, however, leave calcium salt residues, except when it is applied with a detergent or the sequestering agent EDTA (C10H12N2Na4O8). It is highly corrosive and should not be used on copper or aluminum surfaces. It dissolves skin and burns mucous membranes; use only with rubber gloves and adequate eye protection. Potassium hydroxide gives the same results.
_Sodium carbonate_ : Washing soda, Na2CO3⋅10H2O. It is applied hot. It is far less effective than other cleansers mentioned.
_Sodium bicarbonate:_ Commonly called baking soda, NaHCO3, similar to sodium carbonate in effect.
_Trisodium phosphate:_ A common household cleanser, TSP, Na3PO4⋅12H2O, is a very effective solvent, especially for calcified deposits such as boiler scale. Chlorinated trisodium phosphate adds some bactericidal capability to its cleansing.
## Acid Cleansers
_Nitric acid_ : Called aqua fortis, HNO3, this substance is caustic but does not corrode aluminum. It is corrosive to other metals. It can also be used as a sterilant, or to passivate steel.
_Phosphoric acid_ : Phosphoric acid, H3PO4, is a very effective bactericide at below pH 3.0 (2 to 3 percent solution). It passivates steel, dissolving scale deposits and protecting it from oxidation. It is a substitute for nitric acid when cleaning aluminum. Do not use on porcelain.
## Alkaline Sterilants
_Sodium hypochlorite_ : Common household chlorine bleach, Chlorox, or chlorinated soda, NAClO⋅5H2O. It is a very effective sterilant. It is usually used in strengths less than 1 percent in aqueous solution, but it is effective in distilled water at less than .05 percent (.3 fluid ounce in five gallons) with reasonable contact time. Usual dilution is .2 percent/one fluid ounce in four gallons; at this concentration, the sterilant does not need to be rinsed, so long as objects are well drained.
Caution: This cleanser produces poisonous chlorine gas in contact with acid compounds. This reaction also strongly corrodes stainless steel.
## Acid Sterilants
_Iodophor:_ Iodine combined with phosphoric acid. One-half fluid ounce of iodophor per gallon of solution (four milliliters/liter) is an effective sterilant with thirty minutes' contact time at below 120 degrees F (49 degrees C). At this concentration it does not require rinsing.
_Phosphoric acid:_ A commonly used acid sterilant. It is very effective at 2 to 3 percent concentration (pH 2.0 to 3.0), and residues at this strength may not mar beer flavor. Phosphoric acid passivates stainless steel against corrosion.
_Hydrogen peroxide:_ H2O2 is an oxidizing sterilant that is also a very effective solvent for cleaning flexible tubing in 2 to 3 percent solution.
_Sodium metabisulfite_ : Na2S2O5 is antifermentive, not bactericidal. In the presence of acids, it produces noxious sulfur dioxide (burnt-match) odor. It is not widely favored by brewers as a sterilant.
_Potassium metabisulfite:_ K2S2O5 is similar to the sodium salt.
_Calcium sulfite_ : CaSO3⋅2H2O was once commonly used by commercial breweries. In contact with the atmosphere, it oxidizes to CaSO4.
_Sulfuric acid_ : Sulfuric acid, H2SO4, is corrosive to most metals. As sulfurous acid, SO2, it is commonly used in commercial breweries in 6 percent aqueous solution.
_Alcohol, ethyl or isopropyl_ : This is useful only for sterilizing small articles. It is effective only with reasonable contact time.
_Quaternary ammonium sterilants_ : Should be avoided, as the residues from these compounds negatively affect foam retention.
# CHAPTER 17
# Equipment
The following is a list of equipment used for homebrewing. It is by no means complete, comprehensive.
_pH papers_ : These indicate acidity/alkalinity by gauging the color change of the paper wetted with a solution against a scale. Homebrewers commonly use wide range, pH 2 to 10, and narrow range, pH 4.2 to 6.2. Narrow range 5.2 to 6.8 and 5.2 to 7.4 are also available. The best papers are only accurate to pH .2 at best, and most are less accurate. Inexpensive digital pH meters are generally only accurate to ±pH .2, and so are only an improvement over papers because they are easier to read. More accurate (± pH .1 or better), calibratable pH meters are used by commercial brewers but exceed the budgets of most homebrewers. Temperature compensation accounts for the drift in readings made at other than the reference temperature, but readings made at other than 68 degrees F (20 degrees C) still need to be factored back to that temperature to be compared with standard brewing pH values. The decision to purchase any meter should be based upon the unit's accuracy.
_Water test kits_ : Kits are available to test for hardness, alkalinity, calcium, sulphate, chloride, chlorine, nitrite, nitrate, ammonia, iron, and pH. Many are inexpensively available at aquarium-supply shops, and others through laboratory-supply houses. Test kits, especially for water hardness, are more valuable to any brewer — and more versatile — than municipal water-supply analyses, the moreso when the two are used in conjunction.
pH Papers
pH and Hardness Test Kit
_Kettle for water treatment_ : For boiling or mineral-salt treatment of brewing water. The brewing kettle is generally used.
_Gram scale_ : A gram scale is necessary for accurately dispensing mineral salts. These are usually quite inexpensive, but because the accuracy of a cheap scale is likely suspect, it should be calibrated using a substance of known weight to correct inaccuracies.
Constructing a Balance Scale
Construction of a simple balance scale can eliminate the need to purchase this or any scale. To use the balance scale, suspend a substance known to be of the weight desired from one side, and add the substance being measured to the other side until the loads level. Coins in good condition make excellent balance weights; a U.S. dime weighs 2.27 grams, a penny weighs 3.1 grams, a nickel 5 grams, and a quarter 5.6 grams.
_Covered liquor-storage vessel_ : For treated water. Any idle fermenter or food-safe bucket can serve the purpose.
_Graduated one-quart measuring cup_ : A Pyrex cup is more serviceable than either glass or plastic. Use it to measure and dispense water, malt, and mash. It is indispensable as a dipper and for mixing during mashing and wort boiling.
Mill
_Malt scale_ : Some brewers use malt scales in the one-to-ten-pound range; however, malt may be measured by volume with reasonable accuracy to eliminate the need for this piece of equipment.
_Malt mill_ : Unless crushed malt is purchased, some sort of a mill is necessary. Countertop grain/maize mills such as that pictured yield a satisfactory grist, but roller mills give the best grist of any of the home mills. Coffee mills and others equipped with cutting blades are wholly unsatisfactory. Roller mills specifically constructed for crushing malt give the best results, but at a dear price.
_Screens:_ Screens are used to gauge the degree of crushing. A kitchen colander, sieve, or flour sifter, or various gauges of screening can be used to separate several proportions of malt particle sizes. A rough comparison can then be made against separation by standard screens to guide mill adjustments.
_Sourmash-tun_ : Used when a lactic-acid mash is being made to reduce pH prior to brewing. A well-insulated thermos, jug, or picnic cooler maintains temperature well during the long rest; it should be sized to the volume of the sourmash.
_Mash-tun_ : This may be a food-safe plastic bucket, kettle, or insulated plastic picnic or beverage cooler. Capacity should be roughly equal to fermenter capacity.
If it is to double as a lauter-tun, it needs to be equipped with a spigot and a false bottom or filter bag. Some brewers equip the lauter-tun with the means to flush the space below the false bottom. On the whole, it is far easier to manage mashing and filtering in separate containers; this leaves the mash-tun to be fitted with a spigot from which to dispense the sparge water.
The most important criteria for any mash-tun are that it hold heat well and be readily and thoroughly cleanable. Food-safe, insulated, hard-plastic picnic chests hold mash temperature exceedingly well. High temperatures may distort the surfaces, however (high-density polyethylene at 170 degrees F [77 degrees C]); check the manufacturer's specifications.
Large stainless steel pails or kettles are easy to clean, rugged, and lightweight, but they cannot maintain rest temperatures unless they are insulated. A jacket cut from Styrofoam or a wrapping for the tun from any of the myriad insulating materials will improve its performance. It should be used to stabilize the temperature of a stovetop mashing.
A heating element, whether it is integral or a hand-held immersible unit, is somewhat effective, but even temperature dispersal can be achieved only by demonic stirring. Some overheating and caramelization of the mash is inevitable and cleanup may be tedious.
An alternative to the immersible heating element is the RIMS (Recirculating Infusion Mash System), utilizing a picnic cooler, pump, heating element in copper tubing, and electronic parts to fabricate a temperature-controlled mash-tun capable of upward-infusion mashing. Its fabrication is described in Zymurgy, Special Issue 1992, Vol. 15, No. 4; it is also available as an off-the-shelf unit by a homebrew manufacturer.
Rigid plastic pails and buckets hold heat somewhat better than do those of stainless steel, but not as well as is required. They are very inexpensive and can be fitted with an insulating jacket and a hardware-store spigot with relative ease. Use only food-safe, heat-resistant plastic.
Enameled steel kettles should be avoided as they chip easily and expose the mash to certain iron contamination.
_Lauter-tun_ : This is the most sophisticated piece of equipment required by the homebrewer. It should be either insulated or constructed of material having insulating properties. It should be of the proper diameter for optimal mash thickness, and at its base it must have a spigot and some manner of false bottom or filter bag set above it. The means to flush the space between them may improve the clarity of the runoff collected from it.
Picnic Cooler as a Mash-tun
A fine-mesh bag, or one fashioned of canvas sides and a mesh bottom, is commonly used for holding the mash, but presents several disadvantages. These literally "bag" at their bottoms, even when resting on a slotted or perforated base such as is used for steaming vegetables, encouraging uneven percolation of the sparge water through the mash. With canvas-sided bags, the loss is compounded by sparge water flowing down outside of the bag. False bottoms fitted tightly to the inside of the lauter-tun, and one-eighth to two inches above its base, are employed by commercial brewers. Similar designs available to homebrewers give excellent results.
Lauter-tun
A mash strainer can be used instead of a false bottom to separate the wort from the spent grains. It can be constructed from one-half-inch PVC or copper pipe, using hardware-store fittings. Fitted to an insulated picnic chest, the mash strainer allows mashing and sparging to be accomplished in the single vessel.
False Bottom Cross Sections
The lauter-tun should be of roughly the same capacity as the mash-tun and must be selected with consideration for the effect of its diameter upon mash depth. Mashing ten pounds of malt at grain depth of six inches requires a diameter of fourteen inches; for a twelve-inch-deep filter bed, a diameter of ten inches is needed, and for eighteen inches, a diameter of eight inches. For twenty pounds of malt, these figures become twenty inches, fourteen inches, and twelve inches.
Lauter-tun
Sparging System
Some manner of spigot is also a prerequisite of any lauter-tun.
A dedicated means of flushing the space under the false bottom and above the lauter-tun floor is not generally necessary; either an inlet set opposite the spigot or a tube thrust down through the mash to the false bottom suffices.
_Sparger_ : Any manner of introducing a regulated flow of liquor at 170 to 176 degrees F (77 to 80 degrees C) evenly over the top of the mash, without unduly disturbing it, serves the purpose of a sparger. A perforated stainless steel, copper, or plastic tube or sprinkling head attached by a length of flexible tubing to the mash-tun, a pail, a kettle, or a jug with a spigot serves well as a sparger. Any method of introducing the sparge liquor that does not "drill" the liquor into the mash is entirely suitable.
An insulated mash-tun, filled with the hot sparge liquor while the mash is setting in the lauter-tun, makes the most economical and easily managed arrangement. This fact should greatly influence the selection of the mash-tun.
_Large stainless steel spoon_ : Preferably of one-piece construction. It is used throughout the brewing. Use the spoon to calibrate your kettle by standing the long-handled spoon in the kettle, adding water one quart at a time, and scratching calibration lines into the spoon's handle after each addition.
_Mesh strainer_ : Useful for pulling the goods for a decotion, because the free liquid drains off through the mesh.
Making a Paddle
Floating Thermometer
_Paddle_ : Can be used to stir the mash and boiling wort; it is easier to use and more effective than a paddle. It may be cut from a hardwood board.
A good combination is using the one-quart measuring cup to mix the mash and the long-handled paddle for keeping extract from caramelizing on the kettle's bottom.
_Immersible thermometer:_ In the 32 to 212 degree F (0 to 100 degrees C) range, this is indispensable. The floating type that is sealed in a concentric Pyrex bulb offers the best alternative, as it may be readily cleaned, fits through the neck of a carboy, and floats in the mash-tun. Digital electronic thermometers are preferred, if they are accurate to .25 percent of the scale.
Caution: Mercury is highly toxic; never use a cracked or damaged thermometer, and always handle it carefully. Should mercury taint a brew, the brew must be discarded! Alcohol thermometers are vastly preferred.
_Decoction kettle:_ The kettle should be roughly 50 percent of the capacity of the mash-tun. Stainless steel or copper is best for boiling decoctions and, in British mashing, for boiling water for infusions.
_Porcelain plate_ : This is essential for starch-conversion testing with iodine. It should be kept solely for this purpose and washed and rinsed separately.
_Hydrometer/saccharometer_ : Measures sugar in solution by displacement, sinking less deeply into solutions of increasing density. A hydrometer measures specific gravity. A saccharometer is marked with a scale in degrees Balling, Plato, or Brix; otherwise, saccharometers and hydrometers are functionally identical. One equipped with an integral thermometer is handy, although better accuracy is obtained by heating or cooling the solution to the reference temperature of the instrument.
Hydrometer
A hydrometer may be retained within a trial tube and immersed directly into the wort or beer to make a reading, or set into a beaker filled with a sample of the wort or beer.
_Scrub brush, sponges_ : For cleaning brewing equipment. Use only a soft-bristled brush on plastic; a stiffer brush should be used on porcelain, metal, and glass.
_Wort kettle_ : This is usually of 40 to 50 percent greater capacity than the closed fermenter. The kettle may be equipped with a spigot — and a false bottom, stainless steel or copper "scrubby," or cheesecloth pressed down over the kettle's outlet — to filter hop and break residues from the wort. If the kettle has no spigot, filtering may be accomplished by siphoning the wort through flexible tubing and a racking cane fitted with a stainless steel or copper scrubby. For grain brewers, pouring the wort into the lauter-tun and using it as a hop back gives excellent results. Where hop pellets are used, the wort can be whirlpooled and then siphoned off its trub.
Copper Jam Boiler
A kettle is best fabricated from copper for heat-transfer efficiency, but stainless steel is far easier to obtain, clean, and maintain. The exception here is the not-uncommon copper jam boiler, which makes an excellent asymmetrically heated kettle for five-gallon brewing. At this volume any metal kettle may be set directly on a range top to heat (although electric heating elements may not be able to heat five or six gallons of wort to boiling). Nontoxic plastic boilers (polypropylene or Teflon) equipped with integral heating elements also perform satisfactorily, although they are more difficult to maintain.
Calibrate your kettle by making one-quart additions of water to it and etching calibration lines with each new addition into the handle of a long-handled spoon.
The brew kettle is commonly used to boil the brewing water where carbonate salts need to be precipitated.
_Wine thief_ : For taking samples from the kettle or the ferment. It should be of Pyrex or glass manufacture.
_Hop scale_ : Used to dispense hops by weight. An inexpensive plastic "calorie-counter" scale suffices. Calibrate it with an object or substance of known weight.
_Wort chiller_ : The usual practice of cooling the wort by setting the kettle into a tub of ice or cold running water is woefully ineffective. Slow cooling doesn't precipitate the cold break well, and prolongs the time the wort must spend at temperatures conducive to bacterial growth.
Running the wort through twenty-five feet of three-eighths-inch inside diameter copper tubing coiled into chipped ice will reduce wort temperature from boiling down to at least 90 degrees F (32 degrees C). A concentric-tubing cooler can be made by jacketing such tubing with five-eighths-inch inside diameter tubing or garden hose, using a "running tee" for the connection where the inner tubing emerges, and filling the space between with pressurized tap water running in counterflow to the bitter wort. Wort run through a concentric tubing cooler can be cooled to within 10 degrees F (5 degrees C) of the tap-water temperature.
Hop Scale
The arrangement is more simply constructed than might be expected. However, it is difficult to rinse the cooler free of extract deposits and impossible to know whether or not one has. The copper tubing must be thoroughly flushed with water immediately after use, followed by rinsing and storage with iodophor or an acid sterilant solution.
Counterflow Wort Chiller
An immersion wort chiller can be made up of fifteen to thirty feet of copper tubing coiled so that it can be set directly into the wort kettle. Because tap water is run through the tubing, not wort, it is easier to sanitize. It allows the cold break to be precipitated in the kettle, eliminating the need for running the wort into a separate cooling and settling tank. Wort cooling is, however, less efficient than running the wort itself through copper.
Immersion Chiller
_Wort-storage jars_ : For a five-gallon brewing, vacuum seat canning jars of half-pint, pint, or quart capacities should be used. Several quart jars are needed to hold wort for kraeusening or topping-up. Otherwise, ordinary beer bottles may be used.
_Yeast-culturing equipment_ : This equipment is needed to culture yeast to pitching strength by repeatedly doubling volumes with sterile wort. Unless the yeast strain is very strong, for a five-gallon batch, half-pint, pint, quart, and half-gallon glass containers or Erlenmeyer flasks are needed. Preferably, containers should have constricted necks and should be fitted with a stopper and fermentation lock. Culturing from a single cell requires petri dishes and test tubes, a stainless steel loop, and culture medium (agar-agar or pure vegetable gelatin). A 600X to 1200X microscope fitted with a precision graduated stage, as well as slides and cover glasses, complete a professional lab. A good-quality used lab microscope with glass lenses is superior to any department-store model and can be bought for about the same price. The expense, however, can equal the cost of all other equipment combined. If you want to avoid staining cultures to differentiate lactic-acid bacteria, you will need a phase-contrast or dark-field microscope, and such specialty scopes are expensive.
Culturing from a single cell is usually approximated by thinning the parent culture with wort and streaking a nutrient, bacteria-inhibiting culture medium with the parent culture. Any isolated colonies that develop can be assumed to be from a single cell. Streaking the culture onto a yeast-suppressant medium to disclose bacterial contamination largely eliminates the necessity for a microscope. Prepared staining, yeast-or-bacteria-suppressant nutrient agar in sterile petri dishes is available.
Glass bottles or jugs for culturing yeast to pitching strength should be fitted with fermentation locks; for twelve- and sixteen-ounce bottles, you will need No. 1 and No. 2 drilled-and-tapered rubber stoppers; for quart bottles, use a No. 3 stopper.
_Pump-spray or squeeze bottles_ : These are for dispensing freshly made-up cleaning and antiseptic solutions.
_Primary fermenter_ : The primary fermenter should be constructed of easily cleaned material.
For closed-fermentation systems, a six-gallon carboy with a fermentation lock is preferred. For "blowoff" systems, a five-gallon carboy fitted with three feet of one-inch-interior-diameter flexible tubing is used. This Burton-Union inspired arrangement was developed to separate top-fermenting yeast and trub from the ferment.
Inverted-carboy single-stage fermentations reduce the risks of contamination and oxidation and allow for easy trub separation and yeast collection. These carboy kits allow the homebrewer to mimic commercial unitank fermentations, with similar benefits.
An "open" fermenter should not have a constricted neck, but must be of up to 25 percent greater volume than the beer and be covered by a close-fitting lid equipped with a fermentation lock. Food-safe plastic pails are inexpensive, but are a nuisance to clean and sterilize after a bit of use. They must be replaced or relegated to use as mash-tuns or the like as their surfaces become badly abraded.
A fermentation lock must be fitted through the cover of the primary fermenter so that all exchange between the fermenter and the atmosphere takes place through the airlock. It should be of adequate size to allow the release of copious amounts of carbon dioxide produced during the primary fermentation without splashing the liquid within the lock. The concentric type serves best. The traditional chemist's airlock is more appropriate for use during the less volatile secondary fermentation and for yeast culturing.
Five-Gallon Glass Carboy
An inexpensive adhesive-backed liquid-crystal thermometer strip should be attached to the fermenter so that the temperature of the fermentation can be easily monitored.
_Refrigeration_ : Unless the heat produced during primary fermentation is attempered by an appropriate ambient temperature, a refrigerator is necessary for controlling lager fermentation temperatures. Serviceable but cosmetically second-class refrigerators may be found at reasonable prices. Tighter temperature control can be had by adding an after-market controller with a wider and higher range of set temperatures and tighter differential than a refrigerator's internal thermostat is capable of.
Carboy Cap and Chemist's Lock
Carboy Brush
_Siphon tubing:_ To rack wort or beer off its sediment, a four-foot length of flexible tubing is affixed to Pyrex or rigid-plastic tubing fitted with an end cap. Plastic tubing should be replaced as it becomes discolored, cracked, or stiff with use.
_Secondary/lager fermenter:_ This should have a constricted neck to reduce airspace above the ferment. Glass presents the most readily cleaned surface. For five-gallon brewings, the traditional glass carboy is the uncontested choice (and presents a compelling reason for brewing in five-gallon batches). Use a carboy cap or No. 6 ½ or No. 7 tapered neoprene or rubber stopper to seal the carboy and hold the fermentation lock.
For lagering under pressure, use a Cornelius keg that is equipped with a pressure-relief valve.
When selecting any closed fermenter, bear in mind that its capacity determines the relative sizes of all other brewing vessels.
Cornelius Keg System
_Carboy brush_ : Used to scour the inside of carboys or other closed-neck containers. It should be roughly two feet long.
_Reducing-sugar analysis kit:_ This positively identifies when aged beer has fermented out. Dextrocheck and other urine-sugar reagents are inexpensive and available at any pharmacy. Glucose-specific analysis kits can be useful indicators of the glucose content of worts and green beer.
_Priming bucket_ : For rousing priming solution into aged beer in preparation to bottle. A fermenter can be employed, but a bucket equipped with a spigot is vastly preferred.
_Keg_ : For draft beer, a five-gallon pre-mix syrup tank (Cornelius keg) used by the soft drink industry is the first choice of homebrewers.
_Bottle filler_ : When the priming tub has no spigot, the beer will need to be siphoned into bottles. A bottle filler with a shut-off valve opens only when it is pressed against the bottom of a bottle. This reduces foaming and oxygen uptake by the beer, and makes bottling less messy.
_Bottles_ : Bottles should be clean, with unchipped rims. Heavy "bar bottles" are preferred over the lighter gauge retail bottles, but in a carefully controlled fermentation, the latter are sufficient. Bottles with porcelain/plastic swing-tops and PET plastic soft-drink bottles with screw-tops are also commonly used; a cap designed to fit PET bottles allows the homebrewer to artificially carbonate beer in the bottle.
Bottle Filler
Bottle Brush and Bottle Rinser
_Bottle brush_ : A brush that reaches the bottom of the bottle is used to scour them during cleaning. A bottle-washer faucet attachment or a multiple-bottle washer made from copper tubing may also be handy.
_Bottle rack_ : A bottle rack holds inverted empties after the work of emptying them is done. A Christmas-tree type rack conserves counter space.
_Caps (or other suitable stoppers)_ : Crown-type caps are commonly used for sealing bottled beer.
_Capper_ : To secure crown-type caps. It should be chosen for its ability to evenly seal bottles without chipping their rims or cracking the bottle necks. The best arrangement is the bench-type; other cappers may not be so expensive, but neither do they work so well.
# APPENDIX A
## Basic Homebrewing from Malt-Extract Syrup
Brewing from grain requires a fair amount of time, expertise, and specialized equipment. It is not really practical or possible for everybody to brew from grains. For beginners, it is not even advisable. Learning about brewing is facilitated by simplifying the process and mastering one step before progressing to another.
Brewing can be reduced to its simplest elements with malt extract syrup, which is simply condensed wort. The homebrewer reconstitutes it with water so that it can be boiled, cooled, pitched with yeast, and fermented. This allows the novice homebrewer to master the mechanics of fermentation and hopping, uncomplicated by crushing, mashing, and sparging grain.
For most hobby brewers, malt extracts present the best choice for producing their beer without giving up the better part of a day each time they brew. An extract brew can be started at noon and the cleanup finished by three. By using unhopped extract, hop pellets, and small amounts of specialty malts in the wort kettle, the homebrewer can create any one of a wide variety of excellent beers in under four hours.
In brewing good beer from extracts it is important to substitute Dry Malt Extract (DME) for any cane or corn sugar called for in the recipe. Sugar increases the alcohol content of a beer without a balancing increase in flavor and fullness. Moreover, sugar gives beer a peculiar flavor, often characterized as "cideriness," that is unpleasant. For every pound of sugar called for in the recipe, substitute 1 ¼ pounds of dry malt extract or 1 ½ pounds of liquid malt extract.
The best beers are generally produced by boiling the full amount of wort for 1 ½ hours, but most households do not have a twenty-four- to forty-quart/liter stainless kettle and a wort chiller among the pots, pans, and kitchen widgets. Many homebrewers and virtually all novices boil a more concentrated wort in a twelve- to twenty-quart/liter stockpot for forty-five to sixty minutes and dilute that high-gravity wort down with sterile, chilled water.
For a five-gallon (nineteen liters) batch, the malt extract is boiled with two gallons (eight liters) of liquor, and that high-gravity wort is cooled down with three gallons (eleven liters) of sterilized, chilled liquor as the wort is transferred to the fermenter.
Brewing really needs to begin with planning and preparation. You must choose a recipe and purchase suitable ingredients. Unless you will be pitching granulated dry yeast, you will need to activate liquid or harvested yeast at least twenty-four hours before you pitch it. And you need to sterilize three gallons of water and give them enough time to chill down to 35 degrees F (2 degrees C) or so (the colder the better).
The beginning brewer also needs to accumulate three plastic one gallon spring-water jugs or six two-liter PET soda bottles to hold the chilled liquor. Rinse out soda bottles immediately after they are emptied, put a little detergent and hot water in each, screw the rinsed lids on, and shake them well. Uncap each bottle and pour out the detergent solution, then rinse them several times with very hot water.
Measure out three-quarters of a teaspoon of bleach into each two-liter bottle, or one teaspoon into each gallon jug. Fill the bottles with tap water to the top, cap them, and let them sit and sterilize until needed.
In addition to containers for the liquor, you will need to collect bottles to package the beer into. Heavy glass bottles, such as are used by most import beers, are preferable to light weight bottles. For five gallons, you will need fifty 12-ounce bottles, or forty 16-ounce, or twenty-eight 22-ounce bottles. Rinse each bottle several times just as it is emptied.
If you are pitching liquid yeast, prepare it the day before brewing, or as appropriate for the amount of yeast on hand. At least a quart of starter, or its equivalent volume of yeast, should be actively fermenting on brew day.
On the day before brewing, heat 3 ½ gallons of water in a clean pot(s) to a boil. Empty the sterilant solution out of your sterilized bottles/jugs. With a very clean, just-rinsed funnel and a Pyrex measuring cup for a ladle, pour a cup or so of boiling water into each jug, swirl it around to rinse out chlorine residues, and dump the rinse. Leave your jugs draining on a paper towel.
Put a clean lid on the kettle, stopper up your clean sink, and fill it with cold water. Immerse the kettle into the water bath. After five minutes or so, drain and refill the sink. Once the brewing liquor has chilled to near body temperature, use the measuring cup and funnel to fill the bottles. Cap and refrigerate them.
The beer will be fermented in either a six- or seven-gallon (twenty-four- to thirty-liter) lidded, food-grade plastic pail with no visible scratches, or a five- or six-gallon (nineteen- to twenty-five-liter) glass carboy. These containers are relatively inexpensive and easily sanitized. Overall, beginning brewers are best advised to begin with the cheaper, and more rugged, food-grade plastic bucket and move on to two-stage or inverted-carboy fermentations after they have gained some experience.
The fermenter, and any and all parts and utensils that contact the wort and beer after the kettle boil, need to be sanitized. Put 1 to 1 ½ ounces of bleach into the bottom of the fermenter and fill it with water. Fill the fermenter one quart at a time and calibrate its volume as you do so by marking the level after each addition on the outside of the bucket with an indelible marker.
It's a good idea to calibrate your kettle, too, by immersing a long-handled spoon into the water and scratching calibration lines into its handle after each one-quart addition.
Fill the fermenter to the brim. Put every part and utensil that won't rust right into the sterilant solution. Cover it and leave it overnight.
On brew day, fill a kettle with two gallons of water and heat it to boiling. Add any mineral salts called for in the recipe. If you are using malt-extract syrup, open the can and set it in a pot of hot water to warm the syrup so that it will run out of the can more readily.
If you are using specialty grains in the recipe, crush them coarsely and place them in a nylon or muslin bag. Suspend the bag in the brew kettle; the contents of the grains will be leached out into solution as the liquor heats up. Only add caramelized or fully roasted grains to the kettle; starchy malts and grains need to be mashed first. Lager, pale, Vienna, Munich, and amber malts should not be used as kettle adjuncts.
Remove the grains when the liquor reaches 165 to 175 degrees F (74 to 79 degrees C), so that harsh-tasting tannins aren't leached from the husks by hotter temperatures. Squeeze all the free liquid from the bagged grains as they are removed.
As the water comes to a boil, use a rubber spatula to scrape the extract out of the can and into the brew pot. Stir and lift the extract up away from the bottom of the kettle. Scrape the can clean, and mix the wort again until the extract has evenly dispersed.
Add any DME called for in the recipe and part of the kettle hops to your extract, and boil the wort for at least forty-five minutes. The wort boil should be rolling and intense, but not so wild that it boils over.
While the wort boils, heat two quarts or so of water to boiling. Empty the sanitizing solution out of the fermenting bucket, rinse it with a quart or so of the boiling water, and invert the fermenter onto paper towels on a clean drainboard to drain. Keep an eye on the boiling wort and a clock, and make hop additions as the recipe calls for them.
If you are using granulated dry yeast to pitch the brew, wash and rinse out a pint (500-milliliter) glass container or Pyrex measuring cup with hot water, and then pour about four fluid ounces (120 milliliters) of boiling water into it. Cover it tightly with plastic wrap and force-cool it in the freezer. Cool it to 95 to 110 degrees F (35 to 43 degrees C, about body temperature, so that it feels warm, not hot), and uncover it just long enough to pour the granulated dry yeast into it. The yeast will rehydrate in the warm water and should show a frothy head within a half hour or so.
Shut off the heat to the kettle at the conclusion of the boil. Check the volume level of the wort. If needed, make up the volume to 5 ¼ gallons (twenty-one quarts, twenty liters) with cold water. Cover the kettle and remove it from the burner. Stopper up the sink again and fill it with cold water. Place the kettle into it for a few minutes to reduce the heat of the wort from boiling to near 160 degrees F (70 degrees C).
Remove the kettle to a clean counter and block it up on one side so that the bottom slopes. Wait for the trub to settle to the low side of the kettle bottom. Wash your arms and hands and sanitize the area around the kettle. Set the sanitized and drained fermenter below in the kettle, away from any drafts. Splash the three gallons of chilled liquor into it.
Siphon or ladle the clear wort into the fermenter, letting it splash into the chilled liquor to aerate it. Leave the last trub-laden wort behind in the kettle, collecting five gallons, or a little more, in the fermenter.
Fit the fermenter with its sanitized lid and an airlock. Cool the wort to below 80 degrees F (27 degrees C), preferably by force-cooling the fermenter, as in a sink full of cold water and ice cubes.
Pitch the yeast. Uncover the fermenter, splash the yeast into the wort and re-cover the fermenter. Give the fermenter a few circular twists to disperse the yeast, and set the fermenter out of harm's way at an ambient temperature, or at slightly below, the fermentation temperature called for by the beer style and the yeast strain's preferred operating range. For ales, the general range is 60 to 72 degrees F (16 to 22 degrees C), and 45 to 55 degrees F (7 to 13 degrees C) for lagers. Fermentation temperatures can be inexpensively monitored by applying an adhesive-backed liquid-crystal temperature indicator to the outside of the fermenter.
Fermentation temperatures can be controlled to some extent by evaporative cooling or by insulation. For evaporative cooling, set the fermenter in a pan of cold water, drape a dampened towel over it, and let the towel trail into the pan of water. Evaporative cooling will draw heat out of the fermenter, reducing its temperature by 5 to 10 degrees F (3 to 5 degrees C). Where a higher temperature is needed, wrapping the fermenter with a dry towel or blanket will let the fermentation temperature rise as much as 10 degrees F (5 degrees C) above room temperature.
Once fermentation has subsided, move the fermenter to a cooler location (ideally 50 to 60 degrees F [10 to 16 degrees C] for ales, 35 to 50 degrees F [2 to 10 degrees C] for lagers). The temperature is reduced for conditioning because it produces mellower flavors. One of the first equipment upgrades that most homebrewers make is either a two-stage or inverted-carboy fermentation (using the BrewCap, for instance). The purpose of racking to a secondary fermenter, after intense primary fermentation subsides, is to separate the beer from its trub. Inverted-carboy systems accomplish the same thing by draining trub and yeast as it settles.
Novice brewers are advised to begin using a single-stage fermentation solely because racking to secondary introduces significant risk of contamination; until the beginning brewer learns the importance of sanitation and how to ensure it, a beer with an astringent bite or phenolic character is at least preferable to a contaminated one. Most beer styles benefit by two to ten weeks of conditioning, but a beer that has not been separated from its trub should not remain on it for more than two weeks, even if it can be aged very cold. So for single-stage fermentations, the beer should be bottled a week to two weeks after the fermentation head has dropped and CO2 generation has ceased. This is not usually a major concern for beginning brewers, who are impatient to try their handcrafted beer anyway.
The first step of bottling is to sterilize all the bottles you have collected. If your dishwasher has a sanitizing cycle, you can load it up with your bottles and let it sanitize them. Or you can soak them overnight in a clean ten- to fifteen-gallon (forty- to sixty-liter) bucket of water treated with the standard sanitizing solution (approximately one fluid ounce/thirty milliliters per each four gallons/sixteen liters). Another method of sanitizing bottles is to put a half ounce/fifteen milliliters of water into each rinsed bottle and stack them in your oven, set at 200 degrees F (94 degrees C), for an hour to heat-sterilize them. Shut the heat off about an hour or so before you will begin bottling.
A couple of hours before you will begin bottling, carry your fermenter over and place it gently on a countertop with a solid chair below it, in an area that is free from drafts and that will be easy to clean up. Block the fermenter up at a slightly tilted angle with a book to let the yeast sediment settle to the low side. Fill your priming bucket with sterilizing solution, and cover it.
When you are ready to begin your bottling session, measure out your priming sugar and a pint or so of water into a pan or small pot. Set up your sanitized, drained, and lidded priming bucket on the chair below the fermenter. You will need to start a siphon through the racking cane/tubing to get clear beer from the fermenter into the priming bucket.
Homebrewers have devised dozens of methods for starting a sanitary siphon. One of them is to boil a quart or so of water in a pot, and then pour it hot into one of your chilled-water bottles. Cut a two- to three-inch (fifty- to seventy-five-millimeter) piece off one end of your tubing and put both pieces into sterilant solution in a pot or bucket on the counter with the fermenter. Wash your hands very thoroughly (you should do this again before you prime your beer, and again before you bottle), and assemble your racking cane to one end of the tubing. Pull the plastic cap off the working end of the racking cane and slide the short piece of tubing over the end of the cane.
Insert the tubing into the bottle of hot water and suck on the end of the short piece of tubing to fill the tubing. Use your thumb to cap the end of the short tubing, and lower the tubing into your sanitizing solution. Pull the plastic cap off the working end of the racking cane and replace it with the trub cap.
Hot water should be siphoning from the jug to the sanitizing solution. Slide the fermenter and priming bucket lids over slightly to one side, lift the racking cane out of the sanitizer and up into the fermenter, then quickly drop the free end of the tubing into the priming bucket. Water, and then beer should start flowing into the priming bucket. Keep the free end of the tubing submerged in the priming bucket, and keep a watchful eye on the end of the racking cane. Unless it is kept below the beer level, you will lose the siphon; on the other hand, you want to avoid sucking up trub from the bottom. Pull the beer from the uptilted side of the fermenter until the end of the transfer.
As you get near the bottom, watch the end of the racking cane. Leave the sediment, and enough beer to take a hydrometer reading, behind. The hydrometer reading gives you the final gravity (FG).
Cover the priming bucket. Rinse and then disassemble the racking cane from the tubing. Put both back into the sanitizing solution. If your bottling bucket is not fitted with a spigot, you will need to start a siphon again to fill the bottles. Attach the tubing to the bottle filler.
Heat the pint or so of water to boiling, adding the priming sugar to it. Stir until the sugar dissolves, then lift the bottling-bucket lid. With your sterilized spoon, gently stir the beer to mix the priming solution into the beer, without splashing it around. Cover the bucket and give it a series of sharp twists to complete the mixing.
Set up your bottles ready to fill. Drain any water or sanitizing solution out of them. Set up your bag of crown caps and the capper within easy reach.
You are ready to bottle. If you are siphoning, run the sterilant through the tubing and bottle filler to a bottle until beer emerges, then move to another bottle and begin filling. If you are bottling from a spigot, try to let the beer run down the inside wall of each bottle without a lot of splashing, or attach a bottle filler. Fill each bottle until the beer comes up nearly level with the rim of each bottle. Set a cap loosely on top of each bottle after it is filled. Set the first bottles that you fill an arm's length away so that you won't knock the caps off as you accumulate more and more full bottles.
Tilt your priming bucket as you get near the bottom so that you get all the beer you can. When the racking cane sucks air, you will lose the siphon.
Go back and crimp the caps onto the bottles with the capper. Rinse off the bottles, set them in cases, and put them out of the way at room temperature. Thoroughly rinse and clean your fermenter, priming bucket, and equipment, being careful not to scratch the plastic.
Let your beer carbonate in the bottles at room temperature for a week or two before storing them in a cooler place while the bottles condition. Be patient; most homebrew will not come into its prime state until at least three to four weeks after bottling.
# APPENDIX B
## The Infusion Mash
Infusion mashing follows different principles for extracting malt than decoction or step mashing. Infusion mashes have only one temperature rest; they do not include a protein rest. Only well-modified malts can be infusion mashed. Mashing and sparging generally take place in a dual-purpose tun over a two- to three-hour period.
The classic infusion mash is not stirred. It entrains and retains a great deal of air; consequently, it "floats."
The greatest challenge in an infusion mash is to achieve and maintain a reasonably even saccharification temperature. Unless the mash can be raked during sparging, an infusion mash can't be stirred to disperse temperature evenly, because stirring deaerates the mash, and the lauter mash will set. Entrained air keeps an infusion mash from settling until late in the sparging cycle.
Temperature variation within the mash, then, is almost inevitable, and within limits is considered acceptable. So long as variations do not exceed ±2 degrees F (1 degree C) of the target saccharification temperature, attenuation consistency will be acceptable.
A mash program should not be chosen to suit the style of beer being brewed, with the single exception that a decoction mash is specific to developing the maltiness that is characteristic of some lager beer styles (Munichs, fests, alts). All ales and most lagers can be made from infusion, decoction, or step mashes.
The choice of a mash program is determined by the character of the malt being used. The infusion mash is designed for use with well-modified malt that can be extracted by a single rest in the 149 to 158 degree F (65 to 70 degrees C) range. English pale and mild malts are not the only malts that are suitable for infusion mashing. Most modern malts, including those of continental origin, can be infusion mashed. The criteria for whether or not a malt can be infusion mashed are: a fine/coarse extract difference of less than 1.8 percent, a soluble nitrogen ratio (S/T) of at least 38 percent, and malt that is at least 95 percent mealy. Where no lot analysis is available, acrospire growth examination should give at least 90 percent grown to two-thirds the length of the kernel, and the majority at three-quarters-to-full-kernel length. Beers that will be served very cold may show a protein haze if malt of more than 1.6 percent nitrogen (10 percent protein) with a soluble nitrogen ratio less than 40 percent is mashed without a protein rest.
Traditional Infusion Mash
If the malt is well converted and perfectly crushed, if the saccharification temperature is reasonably uniform, if the mash floats well, if sparging is evenly dispersed, if the lauter mash is raked and the sweet-wort runoff is restricted so that it takes ninety minutes or so to collect, an infusion mash will give nearly the same extract as a step or decoction mash would. In practice, however, infusion mashes give 3 to 10 percent lower extract, depending upon the particular brew house's efficiency.
The saccharification temperature for an infusion mash almost always falls between 149 degrees F (65 degrees C) and 158 degrees F (70 degrees C). And 149 degrees F is the temperature at which malt starch gelatinizes, so the saccharification temperature should be at least 149 degrees F for malt starch to be made easily accessible to diastatic enzymes. Beta-amylase is still very active at 149 degrees F, while alpha-amylase is somewhat subdued, so the maltose-to-dextrins ratio of the wort will be very high, and the wort from a 149 degree F mash will be composed primarily of fermentable sugars. A final gravity (apparent) of 20 to 25 percent of the OG usually results. At 158 degrees F, beta-amylase is almost entirely deactivated, while alpha-amylase is performing at its peak. The greater percentage of maltotriose, maltotetraose, and dextrins in a wort from a 158 degree F mash will give a beer with a high end-gravity (33 to 35 percent of the OG). Mashes within the range bracketed by these temperatures give intermediate results; at 153 to 155 degrees F (67 to 68 degrees C), the FG might be expected to be 28 to 32 percent of the OG.
The duration of the saccharification rest also affects fermentability and flavor. A 120-minute mash is going to eke out every bit of diastatic power that the malt has to offer, while a 45-minute mash at the same temperature is going to leave more large polysaccharides. Consequently, a two-hour mash at 149 degrees F is going to give a beer with a lower final gravity than a 45-minute mash at the same temperature.
Finally, mash thickness will affect fermentability. The thicker the mash, the more effective the enzymes will be, and the longer their power will last. Alpha-amylase is especially sensitive to mash thickness. When brewing for a dextrinous wort, it is important that the mash be kept thick, so that alpha-amylase will not be degraded before all the malt starch is reduced to at least dextrins. The greater the degree of attenuation desired, the thinner the mash should be. It is common to gradually thin an infusion mash with boiling liquor when it is for a well-attenuated beer.
The quality and uniformity of crushing is more important for an infusion mash than for a multitemperature program. The crush needs to be relatively coarse, so the mash will float, but not so coarse that all the kernels aren't at least fully cracked, or the starch won't all hydrolyze. On the other hand, predominantly shredded husks and a high percentage of flour almost ensure a stuck mash.
Historically, the extract of infusion mashes was run off completely after conversion, and the mash was reflooded with hotter liquor and run off a second time in lieu of sparging (double mash). In modern practice, sparging is carried out as for step and decoction mashing, except that the liquid level above the settling grain is not usually lowered as rapidly. Sparging may begin almost immediately after the runoff is begun, and the liquid level may be lowered only gradually.
Since there is little remedy for temperature, thickness, or pH shortcomings in a mash that can only be stirred moderately, the mashing-in for an infusion mash needs to be competently handled. The malt is usually mashed-in with liquor at a strike temperature that is 10 to 20 degrees F (6 to 11 degrees C) hotter than the target mash temperature, depending on the temperature of the malt itself, the ambient temperature, the temperature and insulating properties of the mash-tun, and the mash thickness desired. With a poorly insulated tun, the mash needs to be made up thick, so that subsequent temperature-maintaining liquor infusions won't dilute it so much that the liquid weight forces the grains to sink. In general, where the crushed malt, whether dry or hydrated, drops into liquor/mash in the tun, the mash floats better than if the liquor is added to dry malt in the tun.
The alkalinity of the liquor is critical to the mash pH. When mashing pale malts, the liquor pH needs to be adjusted to below 6.8 for soft waters and as low as 5.8 for hard waters to realize a pH of 5.2 to 5.3 in the mash. As the percentage of dark malts increases in a mash, it becomes less sensitive to the pH of the liquor.
Where very dark or crystallized malts are called for in a recipe, many brewers wait until the end of the mash to add these malts so that exposure to the hot liquid does not unnecessarily extract tannins from the husks. Since these malts do not contain raw starch, they do not need to be saccharified.
## Infusion Mash Procedure
Crush the malts. Infuse with 1 ¼ to 1 ½ quarts of liquor at 160 to 180 degrees F (71 to 82 degrees C) per pound of malt for a saccharification rest at 149 to 158 degrees F (65 to 70 degrees C) and mix gently. Check and record the pH and temperature of the mash. Maintain the rest temperature for forty-five to sixty minutes, or for up to a maximum of two hours for greater attenuation. Make boiling-liquor infusions as necessary to maintain the mash temperature. Crush and hydrate roasted and crystal malts as the end of the mash nears. Confirm the degree of saccharification by iodine testing; for a dextrinous wort, the mash can be concluded once the iodine reaction falls off to a mahogany color.
Begin to run off the sweet wort slowly. Begin sparging with liquor at 170 to 175 degrees F (77 to 80 degrees C) when the grist begins to sink. As the mash level drops, let the liquid level drop proportionally; not more than two inches of liquid should stand above the grains during sparging.
Terminate the runoff when the pH of the sweet wort rises above 5.8 or the density drops to 3 °Plato (SG 1012).
# APPENDIX C
## The Step Mash (Modified Infusion Mash)
The modified infusion method (step mash, temperature-programmed mash) mimics the traditional decoction-mash sequence, but with less satisfactory results (primarily because no part of the mash is ever boiled). It is, however, far more effective in dealing with undermodified malt than an infusion mash, and a great deal simpler and less time-consuming than a decoction mash, taking only 2 ½ to 5 hours. It has displaced both continental and British tradition in most commercial breweries.
Generally, heat is applied directly to the mash-tun to raise the temperature and to restore temperatures as they fall off during the mash rests. Consequently, a very thick mash can be formed. Caution should be used when applying heat, as rest temperatures can be unwittingly exceeded if not carefully monitored, and if the mash is not constantly and thoroughly mixed.
Crush the malt, dough-in, and bring it to rest at 95 degrees F (35 degrees C), as for decoction mashing. The pH should be near 5.5. After thirty minutes or so, apply heat to the mash-tun to raise the temperature to an even 122 degrees F (50 degrees C) in fifteen minutes; hold it for ten to fifteen minutes. The pH should have dropped to 5.3. Heat to saccharification temperature, quickly for dextrinous worts, or over a span of fifteen to thirty minutes for highly attenuable worts. After fifteen minutes begin testing for starch conversion. When a faint to negative iodine reaction is observed, infuse the mash with boiling water to raise its temperature to 167 to 170 degrees F (75 to 77 degrees C). Hold the temperature and thoroughly mix the mash for five minutes before transferring it to the lauter-tun for sparging.
Step Mash
# APPENDIX D
## Weights and Measures
The standard density/specific gravity reference temperature was changed in the United Kingdom from 60 degrees F (15.56 degrees C) to 68 degrees F (20 degrees C) in 1990. The same standard temperature is now used in the British, U.S., and metric systems.
In the same year the Congress of the International System of Units (SI) made corrections to the Celsius temperature scale; what was 20.005 degrees C is now 20 degrees C (68 degrees F), and what was 15.564 degrees C is now 15.56 degrees C (60 degrees F).
The SI size of the liter was changed slightly in 1964. One liter now equals one cubic decimeter exactly, or 99.9972 percent of the pre-1964 liter (the "old" liter was defined as the volume occupied by one kilogram of water at 4 degrees C). Since 1990, the SI standard weight for one liter of pure water in vacuo at 20 degrees C is .998202 kilogram, and in dry air .997151 kilogram. The National Institute of Standards and Technology (NIST) in the U.S. gives the weight of one liter of water at 20 degrees C at one atmosphere, measured in vacuo, as .998232 kilogram, and in dry air as .998229 kilogram. The SI standards are used here.
Abbreviations
---
U.S. Liquid Measure | Liquid Measure
dram | dr. | liter | L
fluid ounce | fl. oz. | milliliter | mL
pint | pt. | deciliter | dL
quart | qt. | dekaliter | dkL
gallon | gal.
| |
barrel | bbl.
| |
Metric System | Metric System Units
micro | μ | .0001 | gram | g
milli | m | .001 | meter | m
centi | c | .01 | square meter | m2
deci | d | .1 | cubic meter | m3
deka | dk | 10
| |
hecto | h | 100
| |
kilo | k | 1000
| |
## Capacity
To express any volume in another unit of measure, convert the measure in the left-hand column to any unit in the right-hand column by multiplying by the factor shown.
microliter | = .000 000 1 L
---|---
milliliter | = 1 cm3 = .2705 fl. dr. = .033814 fl. oz. = .001 L = 0.06102 cu. in.= .997151 g of water at 20°C
fluid dram | = 3.697 mL = .125 fl. oz.
centiliter | = 10 mL = .3381 fl. oz. =.01 L
tablespoon | = 14.7868 mL = .5 fl. oz.
Imperial fluid ounce | = 28.4131 mL = .9608 fl. oz. (U.S.)
deciliter | = 100 mL = 3.3814 fl. oz. = .1 L
fluid ounce | = 29.5735 mL = 2 tblsp. = 1.8046 cu. in.
cup | = 236.6 mL = 8 fl. oz. = .2366 L
pint | = 473.176 mL = 16 fl. oz. = 2 cups = .8327 Imp. pt. = .47321= 28.875 cu. in.
Imperial pint | = 568.29 mL = 20 Imp. fl. oz. = 19.2152 fl. oz. (U.S.) = 1.2009 pt. (U.S.) = .56825 L
quart | = 946.3530 mL - 32 fl. oz. = 4 cups = .8327 Imp. qt. = .94633 L = 57.75 cu. in.
liter | = 1000 mL = 33.8140 fl. oz. = 2.1134 pt. = 1.0567 qt. = .2642 gal.= 997.151g of water at 20°C = 61.0234 cu. in.
Imperial quart | = 1136.6 mL = 40 Imp. fl. oz. = 38.4304 fl. oz. (U.S.) = 1.2009 qt. (U.S.) = 1.1366 L
gallon | = 3785.4118 mL = 128 fl. oz. = 16 cups = 8 pt. = 4 qt. = 3.7854 L = .8327 Imp. gal. = 231 cu. in. = 8.3216 lb. of water at 20°C
Imperial gallon | = 4546 mL = 160 Imp. fl. oz. = 153.7234 fl. oz. (U.S.) = 4.5459 L = 1.20095 gal.
dekaliter | = 10 L = 2.6417 gal.
⅛ barrel | = 3.875 gal.
¼ barrel | = 7.75 gal.
½ barrel | = 15.5 gal.
hectoliter | = 100 L = 26.4172 gal. = 21.9969 Imp. gal. = .8522 bbl. = .6110 Imp. bbl.
barrel | = 31 gal. = 1.1 735 hL = .717 Imp. bbl. = 7056 cu. in.
Imperial barrel | = 43.2342 gal. (U.S.) = 36 Imp. gal. = 1.63659 hL = 1.3946 bbl. (U.S.)
Submultiples of Capacity (U.S. Liquid Measure)
---
Gallons | Fl. Oz. | Fl. Pt. | Fl. Qt.
.0078 | 1
| |
.0156 | 2
| |
.0313 | 4 | ¼
|
.0469 | 6
| |
.0625 | 8 | ½ | ¼
.0938 | 12 | ¾
|
.125 | 16 | 1 | ½
.1563 | 20 | 1¼
|
.1875 | 24 | 1½ | ¾
.2188 | 28 | 1¾
|
.25 | 32 | 2 | 1
.375 | 48 | 3 | 1½
.50 | 64 | 4 | 2
.75 | 96 | 6 | 3
1.00 | 128 | 8 | 4
## U.S. and Metric Systems Cubic Measure (Volume)
cubic centimeter | = 1 mL at 20°C = .06102374 cu. in.
---|---
cubic inch | = 16.387064 cm3
cubic decimeter | = 61.0237cu. in. = 1 L at 20°C
cubic foot | = 1,728 cu. in. = 28.317 dm3 = 28.316846592 cm3 L = .0283 m3
cubic meter | = 35.31467 cu. ft.
## Dry Measure (Volume)
dry pint | = 33.6003 cu. in. = .5506 L
---|---
dry quart | = 67.2006 cu. in. = 2 pt. = 1.1012 L
liter | = 61.0255 cu. in. = .9081 qt.
peck | = 537.605 cu. in. = 8 qt. = 8.8096 L
bushel | = 2,150.42 cu. in. = 32 qt. = 4 pecks = 35.2383 L = .3524 hL
barrel | = 7,056 cu. in. = 105 qt.
hectoliter | = 6,102.5461 cu. in. = 90.81 qt. = 100 L = 2.8375 bu.
## Weight
microgram | = .000 000 1 g
---|---
milligram | = .015432 grains = .001 g
centigram | = .1543 grains = .01 g
grain | = 64.7989 mg = .0648 g
gram | = 1,000 mg = 15.432358 grains = .035274 oz. (avoir.) = .00220462 lb.
ounce | = 437.5 grains = 28.349523125 g (avoirdupois)
pound | = 7,000 grains = 453.59237 g = 16 oz. = .4536 kg
kilogram | = 1,000 g = 35.27396 oz. (avoir.) = 2.2046226 lb.
short ton | = 2,000 lb. = 20 hundred weight = 907.18474 kg = .9072 metric ton
metric ton | = 1,000 kg = 2204.6226 lb. = 1.1023 short tons
## Linear Measure
millimeter | = .03937 in. = .001 m
---|---
centimeter | = .3937 in. = .01 m
inch | = 25.4 mm = 2.54 cm
decimeter | = 3.937 in. = .1 m
foot | = 30.48 cm = 12 in. = .3048 m
yard | = 91.44 cm = 36 in. = 3 ft. = .9144 m
meter | = 39.370079 in. = 3.28084 ft. = 1.09361 yd.
## Square Measure
square centimeter | = .1550 sq. in.
---|---
square inch | = 6.4516 cm2 = .006944 sq. ft.
square decimeter | = .1076 sq. ft.
square foot | = 929.0304 cm2 = 144 sq. in. = .0929 m2
Temperature
---
°Fahrenheit | °Centigrade | °Réaumur
30 | -1.11
|
32 | 0 | 0
35 | 1.67
|
39.2 | 4.0
|
40 | 4.44
|
41 | 5 | 4
45 | 7.22
|
50 | 10 | 8
55 | 12.78
|
59 | 15 | 12
60 | 15.56
|
65 | 18.33
|
68 | 20 | 16
70 | 21.11
|
77 | 25 | 20
80 | 26.67
|
86 | 30 | 24
90 | 32.22
|
95 | 35 | 28
100 | 37.78
|
104 | 40 | 32
110 | 43.33
|
113 | 45 | 36
122 | 50 | 40
131 | 55 | 44
140 | 60 | 48
145 | 62.78
|
149 | 65 | 52
155 | 68.33
|
158 | 70 | 56
160 | 71.11
|
167 | 75 | 60
170 | 76.67
|
176 | 80 | 64
180 | 82.22
|
185 | 85 | 68
194 | 90 | 72
200 | 93.33
|
203 | 95 | 76
212 | 100 | 80
221 | 105 | 84
230 | 110 | 88
## Conversion
## Density
°Plato (°P; Balling, °B; or Brix) expresses a solution's density as grams of sucrose per 100 grams of solution, measured at 68 degrees F (20 degrees C). 10 °Plato, then, is a 10 percent weight/weight solution. Plato, Balling, Brix and specific gravity are all now commonly calibrated at the international standard of 20 degrees C (68 degrees F). Although 60 degrees F (15.56 degrees C) is being abandoned as a brewing reference temperature, some hydrometers are still calibrated at a reference temperature of 60 degrees F.
Specific gravity (sp gr) measures the density of a solution as compared to the density of pure water (sp gr 1.000). Brewers rarely use specific gravity notation in its usual form. For instance, 1.040 is more often given as "1040" and called "gravity" (SG). British brewers further abbreviate SG 1040 to "40," call it "excess gravity," and write it as "G." Both are useful in simplifying formulas.
Specific gravity is a weight/volume measurement. A 10 percent weight/weight solution gives SG 1040.03 (sp gr 1.04003) at 68 degrees F, rather than 1040 as might be expected, because specific gravity-to-°Plato conversions are not linear.
To convert from specific gravity to °Plato with absolute accuracy requires the use of a regression equation. At 68 degrees F (20 degrees C):
°P = 135.997 x sp gr3 -630.272 x sp gr2 \+ 1111.14 x sp gr - 616.868
A simpler formula gives reasonably accurate results:
°P = 260-(260/sp gr) and inversely, sp gr = 260/(260-°P).
Multiplying °Plato by four roughly gives excess gravity (G):
10 °P x 4 = 40 G
Conversely, dividing G by four gives the approximate °Plato:
40/4 = 10 °P
Also:
Extract = G/46.21. 40/46.21 = .866 lb. of extract/gallon required.
Extract is the amount of any substance in solution. In brewing, extract is based on sucrose, dry basis, giving 100 percent yield. What density 100 percent yield gives depends on the ratio of the extract to the volume (or weight) of the solution.
The ASBC laboratory mash assesses extract on a weight/weight basis. In an ASBC mash at 68 degrees F, 100 percent extract gives 11.11 °P, SG 1044.65, and 372.94 L°/kg.
In practice, brewers need to use a weight/volume basis. In the U.S., this is usually pounds per gallon, or pounds per barrel. As one pound in one gallon, 100 percent extract at 68 degrees F is 11.486065 °P, SG 1046.21415, and 385.6458 L°/kg. At 60 degrees F, 100 percent extract/one pound in one gallon gives 11.47 °P, SG 1046.15, and 385.11 L°/kg.
Liter degrees per kilogram (L°/kg) is the IOB (British) standard for extract. It is a metric weight/volume measurement. The most accurate method of converting L°/kg to percent extract is by the formula:
% extract = 10.13 x (.2601 x L° per kg / 10.13 - .03025)
A less accurate but simpler formula is:
.2601 x L°/kg -.3064
Example:
.2601 x 308 L°/kg - .3064 = 79.8% extract
The reciprocal formula is:
L°/kg = 3.845 x % extract + 1.178
For all intents and purposes, °Plato, Balling, and Brix are interchangeable, although the Balling scale was calibrated at 17.5 degrees C. Where a hydrometer calibrated at 60 degrees F (15.56 degrees C) is being used, the reference temperature can be corrected to 68 degrees F (20 degrees C) by the formula:
SG at 60°F x .99548 + 4.53 = SG at 68°F
The reciprocal formula is:
SG at 68°F x 1.00454 - 4.55 = SG at 60°F
For all practical purposes, a hydrometer calibrated at 60 degrees F gives acceptably accurate results, as is seen in the tables below.
The density tables are based upon these values:
1. °Plato = 135.997 x SG3 \- 630.272 x SG2 \+ 1111.14 x SG - 616.868
2. Extract required for one gallon of solution at a given specific gravity or density: % extract required, at 68°F (20°C) = sp gr x °P x 8.321628; given that one gallon equals 3785.411784 milliliters, and that one milliliter of pure water at 68°F (20°C) in dry air weighs .997151 grams.
A. Gravity and °Plato at 68°F (20°C):
---
°P | SG
0.000 | 1000
.255 | 1001
.513 | 1002
.771 | 1003
1.028 | 1004
1.284 | 1005
1.541 | 1006
1.796 | 1007
2.052 | 1008
2.306 | 1009
2.561 | 1010
2.815 | 1011
3.068 | 1012
3.321 | 1013
3.574 | 1014
3.826 | 1015
4.078 | 1016
4.329 | 1017
4.580 | 1018
4.831 | 1019
5.081 | 1020
5.331 | 1021
5.580 | 1022
5.829 | 1023
6.077 | 1024
6.325 | 1025
6.572 | 1026
6.819 | 1027
7.066 | 1028
7.312 | 1029
7.558 | 1030
B. Gravity, corresponding °Plato, and pounds of extract required for one gallon of solution at the given gravity, at 68°F (20°C°:
---
For Density of: | Lb. Extract for 1 Gal.:
SG | °P
1031 | 7.804 | .6695
1032 | 8.049 | .6912
1033 | 8.293 | .7129
1034 | 8.537 | .7346
1035 | 8.781 | .7563
1036 | 9.024 | .7780
1037 | 9.267 | .7997
1038 | 9.510 | .8214
1039 | 9.752 | .8432
1040 | 9.994 | .8649
1041 | 10.234 | .8866
1042 | 10.475 | .9084
1043 | 10.716 | .9301
1044 | 10.956 | .9518
1045 | 11.196 | .9736
1046 | 11.435 | .9953
1046.214 | 11.486 | 1.0000
1047 | 11.674 | 1.0171
1048 | 11.912 | 1.0389
1049 | 12.150 | 1.0606
1050 | 12.388 | 1.0824
1051 | 12.625 | 1.1042
1052 | 12.862 | 1.1260
1053 | 13.098 | 1.1477
1054 | 13.334 | 1.1695
1055 | 13.570 | 1.1913
1056 | 13.805 | 1.2131
1057 | 14.039 | 1.2349
1058 | 14.274 | 1.2567
1059 | 14.508 | 1.2785
1060 | 14.741 | 1.3003
1061 | 14.975 | 1.3221
1062 | 15.207 | 1.3440
1063 | 15.440 | 1.3658
1064 | 15.672 | 1.3876
1065 | 15.903 | 1.4094
1066 | 16.135 | 1.4313
1067 | 16.365 | 1.4531
1068 | 16.596 | 1.4750
1069 | 16.826 | 1.4968
1070 | 17.056 | 1.5187
1075 | 18.198 | 1.6280
1080 | 19.331 | 1.7374
1085 | 20.454 | 1.8468
1090 | 21.568 | 1.9564
1095 | 22.673 | 2.0660
1100 | 23.769 | 2.1758
1105 | 24.855 | 2.2856
1110 | 25.933 | 2.3954
1115 | 27.002 | 2.5054
1120 | 28.062 | 2.6154
1125 | 29.113 | 2.7255
C. °Plato, corresponding gravity, and pounds of extract required for one gallon of solution at the given °Plato, at 68°F (20°C)
---
For Density of: | Lb. Extract for 1 Gal.:
°P | SG
8.0 | 1031.8 | .6869
8.5 | 1033.9 | .7324
9.0 | 1035.9 | .7758
9.5 | 1037.96 | .8206
10.0 | 1040.03 | .8655
10.5 | 1042.11 | .9108
11.0 | 1044.19 | .9560
11.486 | 1046.214 | 1.0000
11.5 | 1046.28 | 1.0014
12.0 | 1048.37 | 1.0469
12.5 | 1050.48 | 1.0929
13.0 | 1052.59 | 1.1388
13.5 | 1054.71 | 1.1850
14.0 | 1056.83 | 1.2312
14.5 | 1058.97 | 1.2779
15.0 | 1061.11 | 1.3245
15.5 | 1063.26 | 1.3715
16.0 | 1065.42 | 1.4186
16.5 | 1067.58 | 1.4658
17.0 | 1069.76 | 1.5134
17.5 | 1071.94 | 1.5611
18.0 | 1074.14 | 1.6091
18.5 | 1076.33 | 1.6570
19.0 | 1078.54 | 1.7054
19.5 | 1080.75 | 1.7538
20.0 | 1082.97 | 1.8024
21.0 | 1087.44 | 1.9004
22.0 | 1091.95 | 1.9991
23.0 | 1096.49 | 2.0987
24.0 | 1101.06 | 2.1991
25.0 | 1105.67 | 2.3003
D. Gravity and pounds of extract required for one gallon of solution at the given SG, at 60°F (15.56°C)
---
Density SG | Lb. Extract for 1 Gal.:
1031 | .6706
1032 | .6923
1033 | .7141
1034 | .7358
1035 | .7575
1036 | .7793
1037 | .8010
1038 | .8228
1039 | .8445
1040 | .8663
1041 | .8881
1042 | .9098
1043 | .9316
1044 | .9534
1045 | .9752
1046 | .9970
1046.15 | 1.0000
1047 | .0188
1048 | .0406
1049 | .0624
1050 | .0842
1051 | .1060
1052 | .1278
1053 | .1496
1054 | .1714
1055 | .1933
1056 | 1.2151
1057 | 1.2369
1058 | 1.2588
1059 | 1.2806
1060 | 1.3024
1061 | 1.3243
1062 | 1.3462
1063 | 1.3680
1064 | 1.3899
1065 | 1.4117
1066 | 1.4336
1067 | 1.4555
1068 | 1.4774
1069 | 1.4992
1070 | 1.5211
1075 | 1.6306
1080 | 1.7402
1085 | 1.8498
1090 | 1.9596
1095 | 2.0694
1100 | 2.1793
1105 | 2.2893
1110 | 2.3993
1115 | 2.5095
1120 | 2.6197
1125 | 2.7299
Temperature Corrections for Hydrometers Calibrated at 68°F (20°C).
---
If Temperature Is: | SG: 1010- | 1030- | 1040- | 1050- | 1060- | 1070- | 1080- | 1090-
°F | °C | Adjust Hydrometer Reading By:
35 | 2 | -2 | -2 | -3 | -3 | -3 | -3 | -4 | -4
40 | 4 | -2 | -2 | -3 | -3 | -3 | -3 | -3 | -3
50 | 10 | -2 | -2 | -2 | -2 | -2 | -2 | -2 | -2
60 | 15.5 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1
80 | 27 | +2 | +2 | +2 | +2 | +2 | +2 | +2 | +2
90 | 32 | +3 | +3 | +3 | +3 | +4 | +4 | +4 | +4
95 | 35 | +4 | +4 | +4 | +4 | +4 | +5 | +5 | +5
105 | 40 | +5 | +5 | +5 | +6 | +6 | +6 | +6 | +6
115 | 45 | +6 | +7 | +7 | +7 | +7 | +8 | +8 | +8
125 | 50 | +7 | +8 | +8 | +8 | +9 | +9 | +9 | +10
Adjust reading; for example, reading at 80°F is 1055: add 2 = 1057.
Temperature Corrections for Hydrometers Calibrated at 60°F (15.56°C).
---
If Temperature Is: | SG: 1010- | 1030- | 1040- | 1050- | 1060- | 1070- | 1080- | 1090-
°F | °C | Adjust Hydrometer Reading By:
35 | 2 | -1 | -2 | -2 | -2 | -2 | -2 | -2 | -3
40 | 4 | -1 | -2 | -2 | -2 | -2 | -2 | -2 | -2
50 | 10 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1
70 | 20 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +2
80 | 27 | +2 | +3 | +3 | +3 | +3 | +3 | +3 | +3
90 | 32 | +4 | +4 | +4 | +4 | +5 | +5 | +5 | +5
95 | 35 | +4 | +5 | +5 | +5 | +5 | +5 | +6 | +6
105 | 40 | +6 | +6 | +6 | +6 | +7 | +7 | +7 | +7
115 | 45 | +7 | +7 | +8 | +8 | +8 | +8 | +9 | +9
125 | 50 | +8 | +9 | +9 | +9 | +10 | +10 | +10 | +11
Adjust reading; for example, reading at 80°F is 1055: add 3 = 1058.
Temperature Corrections for Saccharometers Calibrated at 68°F (20°C).
---
If Temperature Is: | °P: 2.5- | 7.5- | 10.0- | 12.5- | 15.0- | 17.0- | 19.0- | 21.5-
°F | °C | Adjust Saccharometer Reading By:
35 | 2 | -.5 | -.5 | -.8 | -.8 | -.8 | -.8 | -1.0 | -1.0
40 | 4 | -.5 | -.5 | -.8 | -.8 | -.8 | -.8 | -.8 | -.8
50 | 10 | -.5 | -.5 | -.5 | -.5 | -.5 | -.5 | -.5 | -.5
60 | 15.5 | -.3 | -.3 | -.3 | -.3 | -.3 | -.3 | -.3 | -.3
80 | 27 | +.5 | +.5 | +.5 | +.5 | +.5 | +.5 | +.5 | +.5
90 | 32 | +.8 | +.8 | +.8 | +.8 | +1.0 | +1.0 | +1.0 | +1.0
95 | 35 | +1.0 | +1.0 | +1.0 | +1.0 | +1.0 | +1.3 | +1.3 | +1.3
105 | 40 | +1.3 | +1.3 | +1.3 | +1.5 | +1.5 | +1.5 | +1.5 | +1.5
115 | 45 | +1.5 | +1.8 | +1.8 | +1.8 | +1.8 | +2.1 | +2.1 | +2.1
125 | 50 | +1.8 | +2.1 | +2.1 | +2.1 | +2.3 | +2.3 | +2.3 | +2.6
Adjust reading; for example, °Plato at 85°F is 13.5: add .5 = 14.0.
## Alcohol
Expected Alcohol by Volume and by Weight
---
Apparent Attenuation: | Alcohol by Volume/Weight
°P | G | ABV | ABW
3 | 12 | 1.55 | 1.22
4 | 16 | 2.06 | 1.62
5 | 20 | 2.58 | 2.03
6 | 24 | 3.10 | 2.43
7 | 28 | 3.61 | 2.84
7.5 | 30 | 3.87 | 3.04
8 | 32 | 4.13 | 3.24
8.5 | 34 | 4.39 | 3.44
9 | 36 | 4.64 | 3.65
9.5 | 38 | 4.90 | 3.85
10 | 40 | 5.16 | 4.05
10.5 | 42 | 5.42 | 4.25
11 | 44 | 5.68 | 4.46
11.5 | 46 | 5.93 | 4.66
12 | 48 | 6.19 | 4.86
12.5 | 50 | 6.45 | 5.06
13 | 53 | 6.71 | 5.27
14 | 57 | 7.22 | 5.67
15 | 61 | 7.74 | 6.08
16 | 65 | 8.26 | 6.48
17 | 70 | 8.77 | 6.89
18 | 74 | 9.29 | 7.29
Formulas for deriving approximate alcohol by weight (ABW) or alcohol by volume (ABV) from apparent attenuation (OG-TG):
ABW, °P | = [.8192 (OG-TG)]/[2.0665 - (.010665 x OG)]
---|---
or:
ABV (v/v), °P | = OE-TE x .516
ABW (w/w), °P | = OG-FG x .405
ABV (v/v), G | = OG-FG x .129
ABW (w/w), G | = OE-TE x .102
ABV | = 1.26 x ABW
ABW | = .785 x ABV
True attenuation/fermentability = apparent attenuation x .814
## Pressure
Absolute pressure, lb./sq. in. = psi on a gauge + 14.7 psi
Hydrostatic pressure, psi = head, in feet x .434
Head, in feet = psi/.434
Pressure at 50°F at sea level:
1 atmosphere | = water at 212°F, sea level = 14.6959 psi = 1.013250 bar = 101.325 kilopascals = 760.002 mmHg = 1.03323 kg/cm2 = 406.8 inches of H2O
---|---
1 psi (1 lb./sq. in.) | = .068046 atmospheres = 6.89476 kPa = 70.308 mm/Hg = .0703086 kg/cm2 = 27.68"/H2O
1 bar | = .9869 atmospheres = 14.5 psi = 100 kPa = 750.062 mm/Hg = 1.01972 kg/cm2 = 401.47"/H2O
Pressure at Various Altitudes
---
Altitude | psi
0' | 14.70
500' | 14.43
1,000' | 14.18
1,500' | 13.90
2,000' | 13.67
2,500' | 13.41
3,000' | 13.19
3,500' | 12.92
4,000' | 12.70
4,500' | 12.45
5,000' | 12.23
10,000' | 10.10
15,000' | 8.28
Volumes of CO2: Pressure at Various Temperatures
---
°F | °C | psi
6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 | 26 | 28
Volumes of CO2 at given temperature and pressure:
30 | -1 | 2.36 | 2.60 | 2.82 | 3.02
| | | | | | | |
32 | 0 | 2.27 | 2.48 | 2.70 | 2.90
| | | | | | | |
35 | 1.7 | 2.14 | 2.34 | 2.52 | 2.73 | 2.93
| | | | | | |
40 | 4
| |
2.10 | 2.30 | 2.47 | 2.65 | 2.84 | 3.01
| | | | |
45 | 7.2
| | |
2.08 | 2.26 | 2.42 | 2.60 | 2.77 | 2.94
| | | |
50 | 10
| | | | |
2.21 | 2.38 | 2.54 | 2.70 | 2.86 | 3.02
| |
55 | 12.7
| | | | |
2.04 | 2.19 | 2.33 | 2.47 | 2.63 | 2.78 | 2.93 | 3.01
60 | 15.6
| | | | | |
2.01 | 2.14 | 2.28 | 2.42 | 2.56 | 2.70 | 2.84
At 50°F (10°C):
To carbonate 1 gallon of beer to 1 volume requires 7.5 grams of CO2.
To carbonate 1 hL of beer to 1 volume requires .2 kg of CO2.
1 gram CO2 per 100 mL = 5.06 volumes per volume of beer
1 volume of CO2 per volume of beer = .198 grams of CO2 per 100 mL
1 Atmosphere = 14.7 psi = .968 kg/cm2.
Although the amount of CO2 in a packaged beer doesn't change as temperature increases, the pressure does. If bottled beer is let warm up, it may gush upon opening, or even explode if it is overcarbonated:
Pressure Increase at Rising Temperatures
---
Volumes of CO2: | Temperature:
40°F | 50°F | 60°F | 70°F | 80°F
2.0 | 7 psi | 11 psi | 16 psi | 20 psi | 25 psi
2.5 | 12 psi | 18 psi | 23 psi | 28 psi | 34 psi
3.0 | 18 psi | 24 psi | 30 psi | 36 psi | 42 psi
## Water
At sea level, pure water (H2O; 11.188% hydrogen, 88.812% oxygen by weight) freezes at 32 degrees F (O degrees C), boils at 212 degrees F (100 degrees C). It is most dense at 39.2 degrees F (3.98 degrees C). One gallon at 60 degrees F (15.56 degrees C) weighs 58,310 grains. At 68 degrees F (20 degrees C), one cubic foot weighs 64.3 lb., one ounce (avoir.) equals .96 fluid ounces, one pound equals .12 gallons, and one cubic inch equals 16.36 grams. At 32 degrees F (O degrees C), water weighs 1.000429 grams/milliliter, but as ice weighs .919 grams/milliliter.
1 mL = 1 cc of water at 68°F (20°C)
1 L = 1 kg of water at 68°F (20°C)
Density of Water, in Air
---
°F | °C | g/cm3 (g/mL) | oz./fl. oz. | lb./gal. | lb./UK gal.
32 | 0 | 1.000429 | 1.043622 | 8.348982 | 10.027
39.2 | 4 | 1.000693 | 1.043898 | 8.351188 | 10.029
60 | 15.56 | .998772 | 1.041895 | 8.335156 | 10.010
68 | 20 | .997151 | 1.040204 | 8.321628 | 9.994
77 | 25 | .994836 | 1.037789 | 8.302309 | 9.971
212 | 100 | .957344 | 0.998678 | 7.989423 | 9.601
Pressure at Temperatures Other Than 50 Degrees F
---
Atmosphere | Temperature At: | 50°F | 40°F | 59°F | 68°F | 77°F
|
lb. per sq. in.
1.0
| |
14.7 | 10 | 20 | 22 | 24
2.0
| |
29.4 | 20 | 34 | 40 | 43
2.5
| |
36.8 | 25 | 43 | 50 | 53
3.0
| |
44.1 | 30 | 52 | 63 | 65
4.0
| |
58.8 | 40 | 70 | 79 | 88
## Water Hardness
Parts per million (in metric usage, mg/L) | =1 part CaCO3 per 1,000,000 parts (water = 1 mg/L = .1 parts per 100,000 = .07 °Clark = .0583 gpg (U.S.) = .056 °German
---|---
Grain per gallon (U.S.) | =1 part CaCO3 per 58,310 parts water = 17.1497 ppm = 1.2 °Clark
Clark degree (grain per gallon, U.K.) | =1 part CaCO3 per 70,000 parts water = 14.25 ppm = .833 gpg (U.S.) = .8°German = .7 millival
German degree | = 1 mg Ca per 1000 L = 1 part CaO per 100,000 parts water = 17.9 ppm as CaCO3 =1.4285 °Clark
French degree | = 1 mg CaCO3 per 1000 L
Millival | = .001- ion concentration = 20.357 ppm
Expansion/Contraction of Water:
---
°F | °C | Relative Volume
32 | 0 | .998362
39.2 | 4 | .998230
60 | 15.56 | .999190
68 | 20 | 1.000000
77 | 25 | 1.001162
212 | 100 | 1.041581
Boiling Point at Various Elevations
---
Elevation | Boiling Point
-1000 | 213.8°F (101°C)
sea level | 212°F (100°C)
1000' | 210.2°F (99°C)
2000' | 208.4°F (98°C)
3000' | 206.5°F (97°C)
4000' | 204.7°F (96°C)
5000' | 202.9°F (95°C)
6000' | 201.1°F (94°C)
7000' | 199.2°F (93°C)
8000' | 197.4°F (92°C)
9000' | 195.6°F (91°C)
10,000' | 183.7°F (90°C)
Boiling point drops approximately 1°F (.56°C) for each 550 feet (170 meters) of increase in altitude.
Boiling Temperature at Various Pressures
---
Atmospheres of Pressure | Boiling Point
°F | °C
1 | 212 | 100
1.5 | 234 | 112.2
2 | 249 | 120.6
3 | 273 | 134
4 | 291 | 144
10 | 356 | 180
17 | 401 | 205
## Useful Information
One bushel of barley weighs 48 lb.
One bushel of malt weighs 34 lb.
One cup of whole malt weighs approximately 5 oz. (142 g).
After malting, barley yields 85 to 93 percent malt by weight.
Ethyl alcohol boils at 173°F (78.5°C).
No amount of heat applied to a liquid will increase its temperature once it has reached its boiling point. Only pressure can increase a boiling liquid's temperature, by raising its boiling point.
Specific heat at 20°C: Beer = .92, Water = 1.00, Malt = .4.
1 BTU = 1°F temperature change per gallon of water = 2.928 x 10-4 KWH
1 Calorie = 1°C rise per kilogram of water
1 BTU/hr. = 2.931 W
1 Boiler HP = 33.479 BTU/hr.
One gallon per minute equals 225 liters per hour.
For refrigeration load at 68°F: 8.3217 x SG = lb./gal.
Freezing point: -(.42 x ABW + .04 x OG + .2) °C
Approximate Calories in Beer, per 12 Fluid Ounces
---
OG | TG
1010 | 1012 | 1014 | 1016 | 1018 | 1020
1040 | 132 | 132 | 133 | 134 | 135 | 136
1050 | 164 | 165 | 166 | 167 | 168 | 169
1060 | 196 | 197 | 198 | 199 | 200 | 202
To test or calibrate a thermometer, insert it in chipped ice, and into the steam just above the surface of boiling water; adjust the second reading to account for elevation above sea level.
1 Pasteur unit = 1 minute at 140°F (60°C). Pasteur units increase 10 times for every 12.6°F (7°C) increase in temperature.
Sterilizing at 170°F (77°C) is equal to 268 Pasteur units, at 160°F (72°C) is equal to 45 PU.
Homebrew Sterilant Solutions
---
Volume of Sterilant Solution: | Add Bleach:
1 gal./4 L | 1 ½ teaspoons or 7 mL
4 gal./15 L | 1 fl. oz. or 30 mL
5 gal./19 L | 1 ¼ fl. oz. or 40 mL
6 ½ gal./25 L | 1 ½ fl. oz. or 50 mL
## Approximate Displacement Values of Sucrose, Malt, and Hops
1 lb. of sucrose displaces .074 gal. of water
1 g of sucrose displaces .6165 mL of water
1 kg of sucrose displaces .6165 L of water
1 lb. of malt displaces .13 gal. in the mash
1 kg of malt displaces 1.07 L in the mash
1 lb. of spent malt retains .108 gal. of wort
1 kg of spent malt retains .90 L of wort
1 lb. of hops displaces .12 and absorbs 1.8 gal. of wort in the copper
1 kg of hops displaces 1 L and absorbs 15 L of wort in the copper
1 lb. of hops contributes .07 lb. of extract to 1 gal.
1 kg of hops contributes .265 kg of extract to 1 L of wort
1 lb. of spent hops retains .72 gal. of wort
1 kg of spent hops retains 6 L of wort
Percentage Solutions
---
% | Oz. (Avoir.) | Grams | W/V | W/W Dissolved In (to make up 1 qt.):
1 | .33 | 9.4 | in enough water to make up 1 quart of solution | 31.66 fl. oz.
2 | .67 | 18.9 | 31.36
3 | 1 | 28.4 | 31.04
4 | 1.33 | 37.8 | 30.72
5 | 1.67 | 47.3 | 30.4
10 | 3.33 | 94.5 | 28.8
For water, weight/volume (w/v) and weight/weight (w/w) solutions are identical. Otherwise, a w/w% solution at 10% will be 1% stronger than a w/v% solution at 10%.
% solutions: | in 10 mL | 100 mL | 1 L water
---|---|---|---
5% | .5g | 5 g | 50 g
10% | 1.0g | 10 g | 100 g
25% | 2.5g | 25 g | 250 g
Decimal Equivalents
---
1/64 | .015
1/32 | .031
3/64 | .046
1/16 | .062
5/64 | .078
3/32 | .093
7/64 | .109
⅛ | .125
9/64 | .140
5/32 | .156
11/64 | .171
3/16 | .187
13/64 | .203
7/32 | .218
15/64 | .234
¼ | .250
9/32 | .281
5/16 | .312
11/32 | .343
⅜ | .375
13/32 | .406
7/16 | .437
15/32 | .468
½ | .500
9/16 | .562
⅝ | .625
11/16 | .687
¾ | .750
13/16 | .812
⅞ | .875
15/16 | .937
1 | 1.000
Percentage/Decimal Equivalents
---
% | Decimal
.01 | .0001
.5 | .005
1.0 | .01
5 | .05
10 | .1
12.5 | .125
## Useful Formulas
Diameter of a circle | = circumference x .31831
---|---
Circumference of a circle | = diameter x 3.1416
Area of a circle | = diameter2 x .7854
= circumference2 x .0796
= radius2 x 3.1416
Volume of a cylinder | = (πr2)h = cu. in., ft., m, etc. =
Doubling the diameter of a cylinder increases its volume 4 times.
Volume of a dome | =
Volume of a cone | = = cu. in., ft., m, etc.
## Conversions
Gallons (U.S.) to pounds (avoir.) = (8.33 x sp gr) gal.
Pounds (avoir.) to gallons (U.S.) = lb./8.33 (sp gr)
Milliliter to grams = mL (sp gr)
Grams to milliliters = g/sp gr
Milliliters to ounces (avoir.) =
Ounces (avoir.) to milliliters =
# BREWERS GLOSSARY
achroodextrins. Simple "border" dextrins, from the reduction of starch (amylopectin) by alpha-amylase; simple a-limit dextrins; negative reaction with iodine.
acrospire. The germinal plant-growth of the barley kernel.
adjuncts. Fermentable extract other than malted barley. Principally corn, rice, wheat, unmalted barley, and glucose (dextrose).
aerate. To saturate with atmospheric air; to force oxygen into solution. Introducing air to the wort at various stages of the brewing process.
aerobic. An organism requiring oxygen for metabolism.
agar. Agar-agar. A nonnitrogenous, gelatinous solidifying agent, more heat-stable than gelatin. A culture medium for microbial analysis.
agglutination. The grouping of cells by adhesion.
airlock. See _fermentation lock._
airspace. See _ullage._
albumin. Intermediate soluble protein subject to coagulation upon heating. Hydrolyzed to peptides and amino acids by proteolytic enzymes.
alcohol by volume (v/v). The percentage of volume of alcohol per volume of beer. To calculate the approximate volumetric alcohol content, subtract the final gravity from the original gravity and divide the result by 75. For example: 1.050 – 1.012 = .038/0.0075 = 5% v/v.
alcohol by weight (w/v). The percentage weight of alcohol per volume of beer. For example: 3.2% alcohol by weight = 3.2 grams of alcohol per 100 centiliters of beer. Alcohol by weight can be converted to alcohol by volume by multiplying by 0.795.
aldehyde. An organic compound that is a precursor to ethanol in a normal beer fermentation via the EMP pathway. In the presence of excess air, this reaction can be reversed, with alcohols being oxidized to very complex, unpleasant-tasting aldehydes, typically papery/cardboardy/sherry notes.
ale. 1. Historically, an unhopped malt beverage; 2. Now a generic term for hopped beers produced by top fermentation, as opposed to lagers, which are produced by bottom fermentation.
aleurone layer. The enzyme- and pentosan-bearing layer enveloping, and inseparable from, the malt endosperm.
all-extract beer. A beer made with only malt extract as opposed to one made from barley, or a combination of malt extract and barley.
all-grain beer. A beer made with only malted barley as opposed to one made from malt extract, or from malt extract and malted barley.
all-malt beer. A beer made with only barley malt with no adjuncts or refined sugars.
alpha acid. a-acid. The principle bittering agent of the hop, more soluble when isomerized by boiling. From the alpha resin of the hop.
alpha-acid unit. A measurement of the potential bitterness of hops, expressed by their percentage of alpha acid. Low = 2 to 4%, medium = 5 to 7%, high = 8 to 12%. Abbrev: AAU.
alt. The german word for old. This is an old-fashioned, top-fermenting style of beer that undergoes a cold lagering for maturation.
ambient temperature. The surrounding temperature.
amino acids. The smallest product of protein cleavage; simple nitrogenous matter.
amylodextrin. From the diastatic reduction of starch; ß-limit dextrin; the most complex dextrin from hydrolysis of starch with diastase. Mahogany (red-brown) color reaction with iodine.
amylopectin. Branched starch chain; shell and paste-forming starch. Unable to be entirely saccharified by amylolytic enzymes; a-limit dextrins, or amylodextrins, remain.
amylolysis. The enzymatic reduction of starch to soluble fractions.
amylose. Straight chain of native starch; a-D-glucose (glucose dehydrate) molecules joined by a-(1-4) links. Gives deep blue-black color with iodine.
anaerobic. Conditions under which there is not enough oxygen for respiratory metabolic function. Anaerobic microorganisms are those that can function without the presence of free molecular oxygen.
anion. An electronegative ion.
aqueous. Of water.
attempter. To regulate or moderate process temperature, as by maintaining ambient temperature cooler than the fermentation temperature.
attenuate. Fermentation, reduction of the extract/density by yeast metabolism.
attenuation. The reduction in the wort's specific gravity caused by the transformation of sugars into alcohol and carbon-dioxide gas.
autolysis. Yeast death due to shock or nutrient-depleted solutions.
bacteriostatic. Bacteria inhibiting.
Balling, degrees. A standard for the measurement of the density of solutions, calibrated on the weight of cane sugar in solution, expressed as a percentage of the weight of the solution (grams per 100 grams of solution).
beerstone. Brownish-gray, minerallike deposits left on fermentation equipment. Composed of calcium oxalate and organic residues.
blow-off. A single-stage homebrewing fermentation method in which a plastic tube is fitted into the mouth of a carboy, with the other end submerged in a pail of sterile water. Unwanted residues and carbon dioxide are expelled through the tube, while air is prevented from coming into contact with the fermenting beer, thus avoiding contamination.
brewers gravity. SG. See _gravity._
buffer. A substance capable of resisting changes in the pH of a solution.
carbonates. Alkaline salts whose anions are derived from carbonic acid.
carbonation. The process of introducing carbon-dioxide gas into a liquid by: injecting the finished beer with carbon dioxide; adding young fermenting beer to finished beer for a renewed fermentation (kraeusening); priming (adding sugar) to fermented wort prior to bottling, creating a secondary fermentation in the bottle; finishing fermentation under pressure.
carboy. A large glass, plastic, or earthenware bottle.
cation. Electropositive ion.
chill haze. Haziness caused by protein and tannin during the secondary fermentation.
chill-proof. Cold conditioning to precipitate chill haze.
closed fermentation. Fermentation under closed, anaerobic conditions, to minimize risk of contamination and oxidation.
coliform. Waterborne bacteria, often associated with pollution.
colloid. A gelatinous substance-in-solution.
decoction. Boiling, the part of the mash that is boiled.
density. The measurement of the weight of a solution, as compared with the weight of an equal volume of pure water.
dextrin. Soluble polysaccharide fraction, from hydrolysis of starch by heat, acid, or enzyme.
diastase. Starch-reducing enzymes; usually alpha- and beta-amylase, but also limit dextrinase and a-glucosidase (maltase).
diketone. Aromatic, volatile compound perceivable in minute concentration, from yeast or _Pediococcus_ metabolism. Most significantly the butter flavor of diacetyl, a vicinal diketone (VDK). The other significant compound of relevance to brewing is 2,3-pentanedione.
dimethyl sulfide (DMS). An important sulfur-carrying compound originating in malt. Adds a crisp, "lager-like" character at low levels and corn or cabbage flavors at high levels.
disaccharides. Sugar group; two monosaccharide molecules joined by the removal of a water molecule.
dry hopping. The addition of hops to the primary fermenter, the secondary fermenter, or to casked beer to add aroma and hop character to the finished beer without adding significant bitterness.
dry malt. Malt extract in powdered form.
EBC (European Brewery Convention). See _SRM._
enzymes. Protein-based organic catalysts that effect changes in the compositions of the substances they act upon.
erythrodextrin. Tasteless intermediate dextrin. Large a-limit dextrins. Faint red reaction with iodine.
essential oil. The aromatic volatile liquid from the hop.
esters. "Ethereal salts" such as ethyl acetate; aromatic compounds from fermentation composed of an acid and an alcohol, such as the "banana" ester. Formed by yeast enzymes from an alcohol and an acid. Associated with ale and high-temperature fermentations, although esters also arise to some extent with pure lager yeast cultures, though more so with low wort oxygenation, high initial fermentation temperatures, and high-gravity wort. Top-fermenting yeast strains are prized for their ability to produce particular mixes of esters.
excess gravity. G. A form of expressing specific gravity, for convenience and in formulas, as a whole number: sp gr 1.046 is given as G 46, etc.
extract. Soluble constituents from the malt.
extraction. Drawing out the soluble essence of the malt or hops.
fecal bacteria. Coliform bacteria associated with sewage.
fermentation lock. A one-way valve, that allows carbon-dioxide gas to escape from the fermenter while excluding contaminants.
final specific gravity. The specific gravity of a beer when fermentation is complete.
fining. Clarifying beer, with isinglass, gelatin, bentonite, silica gel, polyvinyl pyrrolidone.
flocculation. The coagulation of phenols and proteins by boiling; the hot break during the boil, and the cold break upon cooling.
germination. Sprouting of the barley kernel, to initiate enzyme development and conversion of the malt.
glucophilic. An organism that thrives on glucose.
gravity. SG. Specific gravity as expressed by brewers; sp gr 1.046 is expressed as 1046. Density of a solution as compared to water; expressed in grams per milliliter (1 mL water weighs 1 g, hence sp gr 1.000 = SG 1000; sp gr 1.046. = SG 1046).
hexose. Sugar molecules of six carbon atoms. Glucose, fructose, lactose, mannose, galactose.
homebrewers bittering units. HBU. A formula adopted by the American Homebrewers Association to measure bitterness of beer. Example: 1.5 ounces of hops at 10 percent alpha acid for five gallons: 1.5 x 10 = 15 HBU per five gallons.
homofermentive. Organisms that metabolize only one specific carbon source.
hop pellets. Finely powdered hop cones compressed into pellets. Hop pellets are less subject to alpha-acid losses than whole hops.
hydrolysis. Decomposition of matter into soluble fractions by either acids or enzymes, in water.
hydrometer. A glass instrument used to measure the specific gravity of liquids as compared to water, consisting of a graduated stem resting on a weighed float.
hydroxide. A compound, usually alkaline, containing the OH (hydroxyl) group.
inoculate. The introduction of a microbe into surroundings capable of supporting its growth.
international bitterness unit. IBU. This is a standard unit that measures the concentration of iso-alpha-acids in milligrams per liter (parts per million). Most procedures will also measure a small amount of uncharacterized soft resins so IBUs are generally 5 to 15 percent higher than iso-alpha acid concentrations.
isinglass. A gelatinous substance made from the swim bladder of certain fish and added to beer as a fining agent.
isomer. Iso-. Organic compounds of identical composition and molecular weight, but having a different molecular structure.
kraeusen. The period of fermentation characterized by a rich foam head. Kraeusening describes the use of actively fermenting beer to induce fermentation in a larger volume of wort or extract-depleted beer.
lactophilic. An organism that metabolizes lactate more readily than glucose.
lager. "To store." A long, cold period of subdued fermentation and sedimentation subsequent to active (primary) fermentation.
lauter. The thin mash after saccharification; its clear liquid. From the German word, to purify/strain.
lauter-tun. A vessel in which the mash settles and the grains are removed from the sweet wort through a straining process. It has a false, slotted bottom and spigot.
lipids. Fatlike substances, especially triacylglycerols and fatty acids. Negatively affect ability of beer to form a foam head. Cause soapy flavors; when oxidized contribute stale flavors.
liquefaction. The process by which alpha-amylase enzymes degrade soluble starch into dextrin.
malt. Barley that has been steeped in water, germinated, then dried in kilns. This process converts insoluble starchs to soluble substances and sugars.
malt extract. A thick syrup or dry powder prepared from malt.
maltodextrin. Isomaltose; also amylodextrin, or an impure mixture of glucose with compounds formed of it.
maltose. A disaccharide of two glucose molecules, and the primary sugar obtained by diastatic hydrolysis of starch. One-third the sweetness of sucrose.
mash, mashing. The process of enzymatically extracting and converting malt solubles to wort, in an aciduric aqueous solution.
microaerophile. An organism that is inhibited in an well-oxygenated environment, and yet requires some oxygen for its metabolic functions.
modification. The degree to which the malt endosperm is converted, manifested by the solubilization of malt protein.
mole. Gram-molecular weight. The sum of the atomic weights of all the atoms of any molecule, in grams.
monosaccharides. Single-molecule sugars.
oligosaccharides. Sugars of more than three molecules, less complex than dextrins.
original gravity. The specific gravity of wort previous to fermentation. A measure of the total amount of dissolved solids in wort.
oxidation. The combination of oxygen with other molecules, oftentimes causing off-flavors, as with aldehydes from alcohols.
pH. A measure of acidity or alkalinity of a solution, usually on a scale of one to fourteen, where seven is neutral.
ppm. Parts per million. Equal to milligrams per liter (mg/L). The measurement of particles of matter in solution.
pectin. Vegetable substance, a chain of galacturonic acid that becomes gelatinous in the presence of sugars and acids.
pentosan. Pentose-based complex carbohydrates, especially gums.
pentose. Sugar molecules of five carbon atoms. Monosaccharides from the decomposition of pentosans, unfermentable by yeast. Xylose, arabinose.
peptonizing. The action of proteolytic enzymes upon protein, successively yielding albumin/proteoses, peptides, and amino acids.
phenols. Aromatic hydroxyl precursors of tannins/polyphenols. Phenolic describes medicinal flavors in beer, from tannins, bacterial growth, cleaning compounds, or plastics.
phosphate. A salt or ester of phosphoric acid.
pitching. Inoculating sterile wort with a vigorous yeast culture.
plasma. Protoplasm. The substance of cell bodies, excluding the nucleus (cytoplasm), in which most cell metabolism occurs.
Plato, degrees. Commercial brewers' standard for the measurement of the density of solutions, expressed as the equivalent weight of cane sugar in solution (calibrated on grams of sucrose per 100 grams of solution). Like degrees Balling, but Plato's computations are more exact.
polymer. A substance having identical elements in the same proportion as another substance, but of higher molecular weight. For example, polyphenols from phenols, polypeptides from peptides.
polyphenol. Complexes of phenolic compounds involved in chill haze formation and oxidative staling.
polysaccharides. Carbohydrate complexes, able to be reduced to monosaccharides by hydrolysis.
precipitation. Separation of suspended matter by sedimentation.
precursor. Matter subject to polymerization.
primary fermentation. The first stage of fermentation, during which most fermentable sugars are converted to ethyl alcohol and carbon dioxide.
priming solution. A solution of sugar in water added to aged beer at bottling to induce fermentation (bottle conditioning).
priming sugar. A small amount of corn, malt, or cane sugar added to bulk beer prior to racking or at bottling, to induce a new fermentation and create carbonation.
protein. Generally amorphous and colloidal complexed amino acid, containing about 16 percent nitrogen with carbon, hydrogen, oxygen, and possibly sulfur, phosphorous, and iron. True protein has a molecular weight of 17,000 to 150,000; in beer, protein will have been largely decomposed to a molecular weight of 5,000 to 12,000 (albumin or proteoses), 400 to 1,500 (peptides), or amino acids. Protein as a haze fraction ranges from molecular weight 10,000 to 100,000 (average 30,000), and as the stabilizing component of foam, 12,000 to 20,000.
proteolysis. The reduction of protein by proteolytic enzymes to fractions.
racking. The transfer of wort or beer from one vessel to another.
reagent. A substance involved in a reaction, that identifies the strength of the substance being measured.
resin. Noncrystalline (amorphous) plant excretions.
rest. Mash rest. Holding the mash at a specific temperature to induce certain enzymatic changes.
ropy fermentation. Viscous, gelatinous blobs, or "rope," from bacterial contamination.
rousing. Creating turbulence by agitation; mixing.
ruh beer. The nearly fermented beer, ready for lagering. Cold secondary fermentation.
saccharification. The naturally occurring process in which malt starch is converted into fermentable sugars, primarily maltose.
saccharometer. An instrument that determines the sugar concentration of a solution by measuring the specific gravity.
sparging. Spraying the spent grains in the mash with hot water to retrieve the remaining malt sugar.
solubilization. Dissolution of matter into solution.
sparge. The even distribution or spray of water over the saccharified mash, to rinse free the extract from the grist.
specific gravity. sp gr. Density of a solution, in grams per milliliter.
SRM (Standard Reference Method) and EBC (European Brewery Convention). Two different analytical methods of describing color developed by comparing color samples. Degrees SRM, approximately equivalent to degrees Lovibond, are used by the ASBC (American Society of Brewing Chemists) while degrees EBC are European units. The following equations show approximate conversions:
(°EBC) = 2.65 x (°Lovibond) – 1.2
(°Lovibond) = 0.377 x (°EBC) + 0.45
starter. A batch of fermenting yeast, added to the wort to initiate fermentation.
strike temperature. The target temperature of a mash rest, the temperature at which a desired reaction occurs.
substratum. The substance in or on which an organism grows.
tannin. Astringent polyphenolic compounds, capable of colloiding with proteins and either precipitating or forming haze fractions.
terminal extract. The density of the fully fermented beer.
thermophilic. "Heat loving"; bacteria operating at unusually high temperatures.
titration. Measurement of a substance in solution by addition of a standard disclosing solution to initiate an indicative color change.
trisaccharide. A sugar composed of three monosaccharides joined by the removal of water molecules.
trub. Precipitated flocks of haze-forming protein and polyphenols.
turbidity. Sediment in suspension; hazy, murky.
ullage. The empty space between a liquid and the top of its container. Also called airspace or headspace.
v/v. See _alcohol by volume._
valence. The degree to which an ion or radical is able to combine directly with others.
viscosity. Of glutinous consistency; the resistance of a fluid to flow. The degree of "mouthfeel" of a beer.
volatile. Readily vaporized, especially esters, essential oils, and higher alcohols.
w/v. See _alcohol by weight._
water hardness. The degree of dissolved minerals in water.
wort. Mash extract (sweet wort); the hopped sugar solution before pitching (bitter wort).
wort gelatin. Culture medium made up from wort as a nutrient source and gelatin to solidify it, for surface-culturing yeast.
# BIBLIOGRAPHY
Abel, Bob. _The Book of Beer._ Chicago: 1976.
American Public Health Association. _Standard Methods for the Examination of Water and Wastewater._ Washington: American Public Health Association, 1971.
American Society of Brewing Chemists. _Methods of Analysis of the American Society of Brewing Chemists._ St. Paul, Minn.: American Brewing Chemists, 1976.
Anderson, S. F. _The Art of Making Beer,_ New York, 1971.
Bailar, J. C., Kleinberg, J., Moeller, T. _University Chemistry._ Boston, 1965.
Baker, Pat. _The New Brewers Handbook._
Baron, Stanley Wade. _Brewed in America._ Boston: Little, 1962.
Berry, C. J. J. _Home Brewed Beers and Stouts._ Andover, England, 1963.
Briggs, D. E., Hough, J. S., Stevens, R., and Young, T. W. _Malting and Brewing Science._ 2 vols. London: Chapman and Hall, 1971.
Burch, Byron. _Brewing Quality Beer._ San Rafael, Calif.: 1974.
Chapman, A. C. _Brewing._ Cambridge, 1912.
Comer, Jay and Tobey, Alan. "Hop Pellets vs. Whole Hops." _A Transcription of the Proceedings of the 5th Annual National Homebrew and Microbrewery Conference._ Boulder, Colo.: American Homebrewers Association, 1983.
Despain, R. O. _The Malt-Ease Flagon._ Berkeley, 1978.
Eckhardt, Fred. _The Essentials of Beer Style,_ Portland, Ore.: Fred Eckhardt Communications, 1989.
———. _Mashing for the North American Home Brewer._ Portland, Ore.: Fred Eckhardt Communications, 1974.
———. _A Treatise on Lager Beers._ Portland, Ore.: Fred Eckhardt Communications, 1970.
———. "The Use of Hops in Your Beer." _Amateur Brewer #4._ Portland, Ore.: Fred Eckhardt Communications, 1977.
Ferguson, W. B. "The Chemistry of a Brewer's Vat," _Science for All._ London, 1978.
Findlay, W. P. K., ed. _Modern Brewing Technology._ Cleveland, 1971.
Fix, George. _Principles of Brewing Science._ Boulder, Colo.: Brewers Publications, 1989.
Forget, Carl. _Dictionary of Beer and Brewing._ Boulder, Colo.: Brewers Publications, 1988.
Foster, Terence. "Yeast Culture and Propagation." _Best of Beer and Brewing._ Boulder, Colo.: Brewers Publications, 1987.
———. "Yeast, An Essential Ingredient." _Zymurgy,_ Spring 1984 (Vol. 7, No. 1).
———. "In the Beginning — There Was Malt." _Zymurgy,_ Winter 1984 (Vol. 7, No. 4).
Garret, A. B., Richardson, J. S., Montague, _E. J. Chemistry._ Boston, 1976.
Geraghty, James J. _Water Atlas of the United States._ Port Washington, New York, 1973.
Gold, Elizabeth, ed. _Evaluating Beer._ Boulder, Colo.: Brewers Publications, 1993.
Hahn, Peter C. _Chemicals from Fermentation._ New York, 1968.
Henius, Max. _Danish Beer and Continental Beer Gardens._ New York, 1914.
Hopkins, R. H. and Krause, B. _Biochemistry Applied to Malting and Brewing._ New York, 1937.
Hopkins, R. H. "Brewing; Alcohol; Wine and Spirits." _What Industry Owes to Chemical Science._ Brooklyn, 1946.
Hunt, Brian. "Spoilage and Sanitation in Homebrewing." _Zymurgy,_ Summer 1984 (Vol. 7, No. 2).
Jackson, Michael. _The World Guide to Beer._ London: Courage Books, 1977.
Kenny, Stephen T. "Hop Varieties." _Zymurgy,_ Spring 1985 (Vol. 8, No. 1).
Kieninger, H. "The Influence of Raw Materials and Yeast on the Various Beer Types." _Best of Beer and Brewing._ Boulder, Colo.: Brewers Publications, 1987.
Lewis, Michael. "Microbiological Controls in Your Brewery." _Best of Beer and Brewing._ Boulder, Colo.: Brewers Publications, 1987.
Line, Dave. _The Big Book of Brewing._ Andover, England: Amateur Winemaker Publications, 1974.
———. _Brewing Beers Like Those You Buy._ Andover, England: Amateur Winemaker Publications, 1981.
Matson, Tim and Lee Ann. _Mountain Brew._ Thetford Center, Vermont, 1975.
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# ABOUT THE AUTHOR
Greg Noonan is one of the best-known craft brewers in America as well as an internationally recognized expert on brewing. Since 1988, Greg has been the owner and brewmaster at the renown Vermont Pub and Brewery, and in 1994 he founded the Seven Barrel Brewery of West Lebanon, New Hampshire.
Since Greg became brewmaster, the Vermont Pub and Brewery has won a gold medal in 1993 for its Auld Tartan Wee Heavy in the ale strong ale category, and a bronze medal in both 1991 and 1992 for its Vermont Smoked Porter in the smoked flavored beers category at the Great American Beer Festival®.
Greg is author of _Scotch Ale_ (Brewers Publications, 1993) and the _Seven Barrel Brewery Brewers' Handbook_ (G. W. Kent, 1996), as well as numerous beer style and technical articles for brewing periodicals, including _Zymurgy_ and _The New Brewer._ Greg really enjoys good beer.
# INDEX
AAU. _See_ Alpha acid unit
Abbreviations, listed, xviii-xix
ABV. _See_ Alcohol by volume
ABW. _See_ Alcohol by weight
Acetaldehyde, ; odor of,
Acetic-acid bacteria,
_Acetobacter,_ 91 (illustration),
Acetohydroxyl acids,
_Acetomonas,_ 91 (illustration),
Acetyl Co A, , ,
_Achromobacter,_ ,
Acid cleansers, cleaning with,
Acidity, ; measuring, ; problems with,
Acid rest, 130-33,
Acid sterilants, cleaning with,
Acidulation, 130-31, ; enzymatic,
Acrospire, ; growth of, 21 (illustration), , ,
Acyl Co A,
Aeration, , ; avoiding, ; hazing and, ; hot-side, , ,
_Aerobacter aerogenes,_
Airlock, 241 (illustration)
Albumin rest, ,
Albumins, , , ; finings and,
Alcohol, cleaning with,
Alcohol by volume (ABV), deriving, ,
Alcohol by weight (ABW), deriving, ,
Aldehydes, , ; flavor and,
Ales, fermentation of,
Alkaline cleansers, cleaning with, 258-59
Alkalinity, , 55-59, ; adjusting, , , 72-73; measuring,
Alpha acids, , , , ; analysis of, ; beta acids and, ; isomerized, ; loss of,
Alpha acid unit (AAU), ,
Alpha-amylase, , , , 109 (diagram), , , , ; mash rests,
Alpha resin group,
Altitude, pressure and,
Aluminum, cleaning, ,
Amber malt, ; fermentation and, 118-19
American Homebrewers Association,
Amino acids, , , ; as energy sources,
Ammonia,
Amylases, 109-10
Amylopectin, , 27 (diagram), , ,
Amylose, 26-27, 27 (diagram)
Anions, , , 66-68
Anthocyanogens, removing,
Apparent attenuation, , ; real attenuation and, 185-86
Apparent density,
Aroma, , ; problems with, ,
ASBC laboratory mash,
Astringency, problems with,
Atomic weights,
Attenuation, , ; apparent, 185-86, , ; real, 185-86,
_Aufkraeusening,_
_Bacillus,_ ,
Bacteria, , 100-6; fermenting and, ; gram-negative, 103-6; gram-positive, 101-3
Balance scale, ; constructing, 263 (illustration)
Balled starch, ; problems with, , ,
Balling, ,
Barley, ; aerating, 114-15; four-row, 5 (illustration), ; harvesting, ; improved varieties of, ; six-row, 2 (illustration), , 5 (illustration), ; two-row, 2 (illustration), , 5 (illustration); ungerminated/undermodified, ; weight/yield of, . _See also_ Malted barley
Barley corn, 4 (illustration)
Barley kernels, ; decomposition of, ; dehusked, 4 (illustration)
Barm, fermenting qualities of,
Beech chips, clarifying with, 196-97
Beta acids, , ; alpha acids and,
Beta-amylase, , , , 109 (diagram), , , , , ; mash rests,
Beta-glucan, ,
Bicarbonate ions, 56-57
Bicarbonates, , , , 66-67, ; reduction of,
_Big Book of Brewing_ (Line), xiv,
Biscuit malt,
Bitterness, ,
Bitterness units (IBU), ,
Black malt, ; pH of,
Blowoff, , , , 241 (illustration)
Bock, xx; real density of, ; recipe for,
Bohemian malt, kilning,
Boiling, , 134-35, , , ; clarification and, ; elevation and, ; pressure and, ; sterilizing by,
Bottle-aging, ,
Bottle brush, , 287 (illustration)
Bottle filler, , 286 (illustration)
Bottle priming, 246 (table)
Bottle rack,
Bottle rinser, 287 (illustration)
Bottles, 286-87, ; cleaning, 198-99, , ; inspecting,
Bottling, 198-99, 245-46, , 294-95, ; oxygenation at, ; problems with, 251-53; racking and,
Bottom fermenting, ,
BrewCap,
Brew-house efficiency, , , 236 (table)
Brew-house extract, ,
Brew-house yield, various sources for, 208-9 (table)
Brewing, 226-53; outline of, xvii; preparing for, 202-25,
Brewing liquor, 211-13 (table)
Brew log, 216-17 (table)
British ale malts, , ; balling and, ; kilning, ; mash filter bed for, ; partial mashing of, ; single-decoction mash and,
British ales: fining and, ; whole hops in,
Brix, ,
Bromcresol Green-Methyl Red,
Brown malt: drying, ; fermentation and, 118-19; pH of,
_Bruchhefen,_
Calcium, , , , ; measuring, ; precipitation of,
Calcium carbonate, ; precipitation of, 64 (diagram)
Calcium sulfite, cleaning with,
_Candida_ sp.,
Capacity: expressing, 307-8; submultiples of,
Capper,
Caps,
Caramel malt, ; crystal malt and, ; pH of,
CaraPils malt, caramel versions of,
Carbohydrates, 24-26,
Carbonate ions, , ; sparging and,
Carbonates, , , ; free hydrogen, ; reduction of, 71-72; sparging and,
Carbonation, , ; degree of, ; sugar and,
Carbon dioxide, , ; pressure and, 322-23; production of, 177-78; volumes of,
Carbonic acid, , , ,
Carboy brush, 284 (illustration),
Carboy cap, 283 (illustration)
Carboys, 241 (illustration), 282 (illustration), ; fermentation with, ; lagering with,
Carlsberg, bottom-fermenting yeast at,
Casks, conditioning for,
Cations, , , , 65-66; metal,
Cellulose, , 26 (diagram)
Celsius scale, changing,
Chemical sterilants, using,
Chemist's lock, 283 (illustration)
Chillers, 276-78; counterflow, 278 (illustration); immersion, 279 (illustration)
Chloride, 49 (diagram),
Chloride ions, 49 (diagram),
Chlorides, ,
Chocolate malt, ; pH of,
Christian, Jacob: lager brewing by,
Citric acid, 61-62, ,
Clarification, ; boiling and, ; reducing treatment for,
Clarity, , ,
Clay, 40-41,
Cleaning, 254-60
_Clostridium butyricum,_ ,
Cloudiness. _See_ Haze
Cold break, 158-59, ; problems with, ; settling tank and,
Cold ferment, contamination during,
Coliform bacteria, , ,
Colloids, ,
Color, , ; water, 46-47
Conditioning, ,
Cones, ,
Congress of the International System of Units,
Construction, cleaning, 255-57
Contamination, 105 (table), , ; preventing, 100-1, , , , ; risk of, ,
Convection currents,
Conversions, various,
Copper, cleaning, , ,
Copper jam boiler, 275 (illustration)
Cornelius keg, , 285 (illustration)
Corn sugar, carbonating with,
Counterflow wort chiller, 278 (illustration)
Crystal malts, , ; caramel malt and, ; pH of, ; saccharified,
Cubic measures, U.S./metric,
CWE (cold-water extract), ,
Czechoslovakian malt, kilning,
Dark Munich, xx; enzyme strength of, ; recipe for,
DBCG extract, 12-13
DBFG extract, ,
Debranching enzymes, 109 (diagram)
Decimal equivalents,
Decoction, , ; first, ; second,
Decoction mashing, , 107-8, , , , , , , ; described, 133-36; filter bed for, ; hopping, ; skepticism about, ; wort and,
_Dekkera_ sp.,
Density, , 215 (table); apparent, ; checking, ; measuring, 312-15; reference units for, xviii; temperature and, ; wort, , 186-87, , ,
Dextrinization rest, 230-31,
Dextrinous beer, secondary fermentation for,
Dextrins, 27-28, , , , , ; a-limit, , , , ; ß-limit, , , ; maltose and,
Dextrocheck, testing with,
Diacetyl, , , , , ; flavor and, ; formation of, ; kraeusening and, ; reducing,
Diacetyl diketone,
Diacetyl rest, ; fermentable extract and,
Diastatic enzymes, 108-9, , ,
Diatomaceous earth, using,
Disaccharides,
DME. _See_ Dry malt extract
DMS, ; reducing development of,
Dopplebock, xx; recipe for,
Dortmund, kilning,
Dortmunder, xix; mash thickness and, ; recipe for,
Doughing-in, 126-29, , , 227-28,
Dreher, Anton: bottom fermenting and,
Dry Malt Extract (DME), 288-89,
Dry measures, 309-10
EBC method, ,
Eisbocks, xx
Electron transfer, 49 (diagram)
EMP,
Enamelware, cleaning,
Enzymes, 107-10; constitutive, ; debranching, 109 (diagram); diastatic, 108-9, , , ; extracellular, ; improving effectiveness of, ; inducible, ; intracellular, ; nonproteolytic, ; pH and, ; proteolytic, ,
Equipment, 261-87; cleaning, 257-58,
Equivalence, 60-63
_Escherichia coli,_ ,
Esters, , ,
Ethyl, cleaning with,
Ethyl acetate,
Ethyl alcohol, ; boiling,
Evaporation rates, establishing, 156-58
_Exiguus,_ 91 (illustration)
Extract efficiency, ,
Extracts: diacetyl rest and, ; expressing, ; post-kraeusen, ; problems with, ; terminal, 197-98, ; unfermentable, ; unhopped,
False bottom cross sections, 269 (illustration)
Fatty acids, 33-34, ; as energy sources,
Fermentation, , , 164-90, , , 293-94; acidity of, ; clean, ; culture-yeast, , ; flavor and, ; gelitanous, ; heat from, ; inducing, ; kraeusening and, , ; lag times for, ; mash thickness and, 140-41, ; problems with, 250-51; products of, ; ropy, , ; single-stage, ; temperature and, 172-73, , . _See also_ Primary fermentation; Secondary fermentation
Fermentation lock, , , , ; cleaning,
Fermenters, ; blocking, ; cleaning, , ; cone-bottomed, ,
Filtering, , , , 233-34, 247 (illustration); improving, ; purpose of, ; speed of,
Final rest,
Fine grind-coarse grind difference, 13-14
Finings, ; isinglass, 195-96,
Finishing hops, , 159-60; adding, 235-36
Fix, George,
Flavor, , ; fermentation and, ; kraeusening and, ; problems with, , , ; temperature and, 172-73,
Flocculation: improving, ; oxygenization and, ; pH and, ; polyphenol, ; problems with, ; protein, ,
Force cooling, , , ,
Formulas, various,
Freezing point,
Friability, percent, 18-19
Fructose, 29 (diagram)
Fusel alcohol, , , , ; esterization of, 170-71; oxidation of,
Galactose, , 29 (diagram),
Gallons, pounds and,
Gelatin, , 244-45
Gelatinous precipitate, problems with,
Germination, , , 115-16
Glass, cleaning,
Glazed porcelain, cleaning,
Glucose, 29 (diagram), ; anaerobic fermentation of, 180-81, 180 (diagram)
Glycogen, ,
Grain bill, 206-7 (table)
Grams, milliliters and,
Gram scale,
Gravity, 215 (table), , , , ; excess, xviii; final,
Grist, ; lauter-tun, 125 (table)
_Hafnia,_ 91 (illustration),
Hammer mills, 123 (illustration),
Hanson, Emil: lager brewing by,
Hardness, 38-39 (map), 55-59, , , 68 (table), ; reducing, ; temporary, ; as treatment indicator, 69 (table)
Hardness test kit, 262 (illustration)
Hartong extract,
Hartong 45 degrees,
Haze, , , ; malts and, ; problems with, ,
Hazelnut chips, clarifying with, 196-97
HBU, , 82-83
Heads, ; formation/retention of, ; problems with,
Heating elements, cleaning,
Hellesbock, recipe for,
Hemicellulose, 24-25, , ,
Home grain mill, 123 (illustration)
Hop acids, solubility of,
Hop aromatics, ,
Hop extraction, ; efficiency of, ; pH and, ,
Hop nose,
Hop oils, ,
Hop pellets, , ,
Hop rates, , 214-15 (table)
Hop resins, ; isomerization of,
Hops, xvii, 77-88, 80 (illustration), ; analysis of, 84 (table); aroma, ; bitterness of, ; color range of, 84 (table); cone size of, 84 (table); cross-section of, 83 (illustration); displacement values of, ; dry, , , ; finishing, , 159-60, 235-36; flavoring, ; growing, 79-80, 88 (table); kettle, 153-55, , ; as preservative, , ; varieties of, , 86-87 (table),
Hop scale, , 277 (illustration)
Hot break, ,
_Humulus lupulus,_
HWE, 14-15
Hydrochloric acid,
Hydrogen, 48 (diagram), 50-51
Hydrogen ions, 48 (diagram),
Hydrogen peroxide, cleaning with,
Hydrometer, 244 (illustration); calibrating, ; measuring with, , , ; temperature corrections for,
Hydronium ions, , ,
Hydroxide ions, ,
Hydroxyl ions,
IBU. _See_ Bitterness units
Immersion chiller, 279 (illustration)
Indicator dyes,
Infusion mash, , , , 298-303; hopping, ; malts for, , ; modified, 304-5; traditional, 299 (graph),
IOB lot analysis,
Iodine starch conversion test, 143-44, , , ,
Iodophor, cleaning with,
Ionic compounds, 48 (diagram),
Ions, 47-48, 48 (diagram); equivalent/molecular weight of,
Iron, 65-66; removing,
Isinglass finings, 195-96,
Iso-alpha-acids, ,
Isomaltose, 28 (diagram), ,
Isopropyl, cleaning with,
Kegging, problems with, 251-53
Kegs, ; cleaning,
Kettle adjuncts, 210 (table)
Kettle break, ; pH and, ; problems with,
Kettle hops, ; adding, 153-55,
Kettles, , 292-93; calibrating, ; cleaning, , ; decoction, ; wort, , 275-76,
Kilning, 117-21
_Klebsiella,_ 91 (illustration),
Kraeusen, , , , , , 168-69, ; aroma hops and, ; fermentation and, , , , ; hazing in, ; high, 182-83, 183 (illustration), 184-85, , , ; late, ; low, 178-79, 179 (illustration), , , ; pH of, ; post, 183-84, 242-43; problems with,
Lactic acid, 61-62, , , , , , ,
Lactic-acid mash, 132-33,
_Lactobacillus,_ 91 (illustration), ,
_Lactobacillus brevis,_
_Lactobacillus bulgaris,_
_Lactobacillus delbruckii,_ , , ,
Lagering, , , , 193-95; bottle-conditioning and, ; fermenting and, , , 284-85; length of, , 244-45; pH and, 165 (graph); yeast and,
Lager malts, , , ; wort pH of,
Lagers, xix-xxi, ; real density of, ; temperatures for,
Lag phase, ; primary fermentation and, 174-78; problems with, ; yeast in, ,
Lautering, , , , ,
Lauter-tuns, , , , , , , 266-71, 268 (illustration), 270 (illustration), ; cleaning, ; filtering with, ; milling for, ; sparging and,
Light beers, mash thickness and,
Light Munich, xx, ; recipe for,
Line, Dave, xiv,
Linear measures, expressing,
Lipids, ; oxidation of, ; sparging and,
Liquid measures, abbreviations for,
Liquor-storage vessels,
Lot analysis, ,
Magnesium, , , ; measuring,
Malt, , 21 (illustration), ; aromatic, ; base, 8-9, ; binning, ; coarsely ground, ; colored, , 25 (table), ; crushed, 122-25, 124 (illustration), , , ; dark, ; diastic power of, ; displacement values of, ; dry analysis of, 10 (table); enzymatic, ; evaluation of, 21-23; fractions, 32-33; glassy, ; gums, ; half-glassy, ; infusion mash, ; kilning, , ; lager, , , , ; mealy, ; modification of, 25 (table); moisture content of, , ; roasted, ; size grading for, 23 (table); specialty, 118-20; weight of, ; well-modified/poorly modified, ,
Maltase, ,
Malted barley, xvii, 7-34, ; milling, 122-25
Malt-extract syrup, homebrewing with, 288-97
Malting, 113-21
Malt mills/screens, ; cleaning,
Maltose, 28 (diagram), , , , ; dextrin and,
Maltotriose, , , ,
Malt scale,
Manganese, , ; removing,
Mannose, 29 (diagram),
Mash: acidity of, , ; compositional changes during, xvii; fermentability and, ; lactic-acid, 132-33, ; odor of, ; pH of, 129-30, , , ; problems with, 248-49; rest mash, , , , , , ; saccharified, ; set, ; single-decoction, , 231-32; step, , , , , , , 304-5, 304 (graph); temperature of, , , , , 304-5; thick, , 138-41, , , ; thin, 136-37, ; three-decoction, 127 (graph), 136-52, 227-31; two-decoction, 128 (graph). _See also_ Decoction mashing; Infusion mash
Mash filter bed, 147 (illustration)
Mashing, , , 126-52, , , 298-99; extract from,
Mash-tuns, , , 265-66, ; cleaning, ; picnic cooler as, 267 (illustration)
Measures, 306-29
Measuring cups, 264-65,
Melanoids, 117-18; phenolic,
Melibiose, 29 (diagram),
Mesh strainers,
Metabolic pathways,
Metric units, abbreviations of,
Mho,
Microaerophiles,
Mill, 264 (illustration)
Milliliters, grams/ounces and,
Milling: efficiency of, ; equipment for, 122-25
Mineral ions, 63-64
Mineral salt treatment, , , 75-76 (table)
Mitcherlich, yeast and,
Mixing, described,
Molarity, 60-63
Monohydrate acid,
Monosaccharides, , , ,
Munich lagers, , ; mash thickness and,
Munich malt, ; kilning, , ; pH of,
_Mycoderma,_ 91 (illustration)
National Institute of Standards and Technology (NIST),
Nearly fermented beer, 245-46, ; contamination of,
Neon, 48 (diagram)
NIST. _See_ National Institute of Standards and Technology
Nitric acid, ; cleaning with,
Nitrogen, metabolism of,
Normality, 60-63
Odor, 46-47; problems with, ,
Ohm,
Oligosaccharides, , ,
Orthophosphoric acid,
Ounces, milliliters and,
Oxaloacetate, ,
Oxidation, ; prevention of,
Oxo-acids, , , ,
Oxygen, 48 (diagram); contamination by, ; saturation, ; starvation,
Oxygenation, , , ; bottling and, ; flocculation and, ; yeast and,
Oxygen ion, 48 (diagram)
Paddle, 272 (illustration),
Pale malts, , ; character grading for, 23 (table); mashing, ; pH of,
Pasteur, Louis: yeast and,
Pasteur units,
Pectins, dissolving,
_Pediococcus,_ 91 (illustration), , , ,
Peptides, , , ,
Percentage equivalents/solutions,
Petite mutants,
PET soda bottles,
pH, 53 (table), ; as alkalinity indicator, 69 (table); checking, , 53-54, , , , ; mash, , ; papers, 261-62, 262 (illustration); problems with, , ; temperature and, ; testing, , , , ; test kit, 262 (illustration); as treatment indicator, 69 (table)
Phosphates, , , ; leaching,
Phosphoric acid, , , , , , ; cleaning with, ,
Phosphoric coenzyme, ; fermentation and,
Phytase, , 130-31,
Phytic acid, ,
_Pichia membranaefaciens,_ 91 (illustration)
_Pichia_ sp.,
Pilseners, xix; American, xx; dark, xx; mash thickness and, ; recipe for,
Pilsen malts, , , ,
Pilsner, xix
Pitching, 169-71, , , , 237-38, , , ; preparing for, , 166-68; volume, 238 (table)
Planning, 202-25,
Plastics, cleaning, ,
Plato, , , , ,
Polypeptides, , , ,
Polyphenols, , , , ; finings and, ; flavors and, ; polymerization of, ; removing,
Polysaccharides, , , , ; saccharified,
Polyvinylpyrrolidones,
Porcelain plate,
Potassium, , , ,
Potassium metabisulfite, cleaning with,
Pounds, gallons and,
Pressure: absolute, ; altitude and, ; boiling and, ; carbon dioxide and, 322-23; regulating, ; temperature and, , , ,
Primary fermentation, , , , , ; described, 239-40; lag phase of, 174-78; temperature and, , , 240 (table); time guideline for, 240 (table)
Primary fermenter, ,
Priming bucket, , , ; cleaning,
_Principles of Brewing Science_ (Fix),
Protein, 31-32; percent,
Proteinase, ,
Protein precipitation, ; pH and,
Protein rest, 137-39,
Protein sludge, ,
Proteolysis, , ,
Proteoses, ,
_Pseudomonas,_
Pump-spray bottles,
Pyruvic acid, , ,
Quaternary ammonium sterilants, cleaning with,
Racking, 186-87, , , ; bottling and, , ; cloudiness and, ; sampling at,
Racking cane, ,
Raking, ,
Real attenuation, ; apparent attenuation and, 185-86
Recipes, 218-25
Recirculating Infusion Mash System (RIMS),
Reducing-sugar, analysis of,
Reducing-sugar analysis kit,
Reference units, xviii
Refrigeration, 283-84,
Repitching, , ,
Resins, ,
Respiratory phase, yeast and,
Rest mash, , , , , ; alpha-amylase/beta-amylase,
RIMS. _See_ Recirculating Infusion Mash System
Rootlet growth, as acrospirelength indicator, 116 (table)
Roughness, problems with,
Rousing, , ; aeration by, 162-63; racking and,
Ruh beer. _See_ Nearly fermented beer
Runoff, ; boiling, ; cloudy, ; restricting, 150-51; temperature of,
Saaz yeasts,
Saccharification, , , , , , , , ; mash, ; problems with, ; temperature for, , ,
Saccharification rest, 140-43, 230-31; duration of, ; temperature for, ,
Saccharometers, 274-75; temperature corrections for,
_Saccharomyces,_
_Saccharomyces carlbergensis,_ , 91 (illustration), ,
_Saccharomyces cerevisiae,_ , ,
_Saccharomyces uvarum,_ , ,
_Salmonella_ strains,
Sanitization, 292-93,
Screens,
Scrub brush,
Scum, ,
Secondary fermentation, , , 191-201, 243-44; duration of, 192-93; lagering and, ; length of, 244-45; temperature for,
Secondary fermenter, 284-85
Sediments, ; cold-break, ; filtering, ; problems with, ; water, 46-47
Sedlmayer, Gabriel: bottom fermenting and,
SG 1040,
_Shigella dysenteriae,_
Shredded hulls,
Silica, ; leaching,
Silica gel, using,
"Sinker" test,
Siphoning, , 296-97
Siphon tubing,
Six-row malt mill, 123 (illustration)
Size,
Skimming, ,
Slants,
SNR, 17-18
Soda keg, 247 (illustration)
Sodium, 49 (diagram), , ,
Sodium bicarbonate, cleaning with,
Sodium carbonate, cleaning with,
Sodium chloride, 49 (diagram)
Sodium hydroxide, cleaning with,
Sodium hypochlorite, cleaning with,
Sodium ion, 49 (diagram)
Sodium metabisulfite, cleaning with,
Softeners, ionic-exchange,
Soluble gums,
Soluble protein, percent,
Sourmalt,
Sourmash-tun,
Sparge liquor, 148-49
Sparger,
Sparging, , 150-52, , , , , , , ; lipids and, ; preparing for, 149-50; purpose of, ; rate for, 233-34; system for, 271 (illustration)
Specific conductivity, 54-55, 68 (table)
Specific gravity (sp gr), xviii,
Sp gr. _See_ Specific gravity
Splashing, , , , ; avoiding, , ; contamination during, ; racking and,
Sponges,
Spoon,
Springs, as brewing source, ,
Sprouting, , ,
Square measures, expressing,
Squeeze bottles,
SRM,
Stabilization,
Stainless steel, cleaning, , 255-56,
Starch, 26-27, 26 (diagram), 28 (diagram); conversion of,
_Staubhefen_ yeast,
Steeping, , 114-15
Step mash, , , , , , , 304-5, 304 (graph)
Sterilant solutions, making,
Sterilization, , 254-60, , ,
_Streptococcus faecalis,_
Subculturing, ; fermentation and,
Sucrose, 28 (diagram), ; displacement values of, ; yeast and,
Sugars, 30-31; carbonation and,
Sulfates, , ,
Sulfuric acid, , , , , , ; cleaning with, ; pollution by,
Tanks, pressurization in,
Tannins, , ,
Taste: areas for, 200 (illustration); perceptions, ; problems with, 249-50, ,
Temperature: density and, ; fermentation and, 172-73, , ; flavor and, 172-73, ; maintaining, , , , , , ; pH and, ; pressure and, , , , ; primary fermentation, , , 240 (table); problems with, , ; reference, xviii, ; rest, , , ; saccharification, , ; standard density/specific gravity reference, ; strike, ; various measures of,
Thermometers: calibrating, 326-27; floating, 273 (illustration); immersible,
Thermophilic homofermentive species, ,
Thick mash, , , , ; enzyme performance of, 138-39; fermentability and, 140-41
Thin mash, 136-37, ,
Three-decoction mash, 136-52, 227-31; final decoction of, 144-45; first decoction of, 136-37; time/temperature graph for,
TN, percent,
Tongue, taste areas on, 200 (illustration)
Top fermenting, , ,
_Torulopsis_ sp., 91 (illustration),
Total dissolved solids, 54-55, 68 (table)
Treatment, 211-13 (table)
Trisodium phosphate, cleaning with,
Troubleshooting, 248-53
Trub, , ; contents of, 170-71; discharging, , ; settling,
TSN, percent,
Tubing, , , ; cleaning,
Turbidity, 46-47; problems with,
Underground water, , 42-43 (diagram)
U.S. Public Health Service Drinking Water Standards, ,
Valence, negative/positive,
Varietal analysis,
_Vibrio cholerae,_
Vicinal diketones, ,
Vienna, xix-xx, ; recipe for,
Vienna malt, , , ; kilning, , ; pH of,
Viscosity, cP,
Volume, 211-13 (table); reference units for, xviii
_Volutin,_
Water, 35-76; alkalinity of, ; analysis of, 44-46, 45 (table); characteristics of, 68-69 (table), 74 (table); density of, ; expansion/contraction of, ; hardness of, 38-39 (map), 40-41, , , 55-59, 68 (table), 69 (table), , ; molecules, , 51 (diagram), ; pH of, , ; softness of, 40-41, , ; treatment of, , 70-73; underground, 42-43 (diagram); volume of,
Water test kits,
Weights, 306-29
Weight/volume measurement, converting, 313-14
Wells, as brewing source, ,
Wine thief,
Wood, cleaning, 256-57
Wort: acidity of, , ; aerating, ; anaerobic metabolism of, 178-79; bitterness of, ; boiling, xvii, 153-63, , , 234-35, , ; cooling, 161-63, 236-37; decoction mash and, ; density of, , 186-87, , , ; filtering, 236-37; heating, ; high-gravity, ; oxygenated, ; pH of, , , ; problems with, 249-50; running off, , ; straining, 160-61; volume of,
Wort chiller, 276-78
Wort-storage jars,
Yeast, xxi, 89-99, 91 (illustration); activity of, 177-78; adaptation and, ; aerating, ; aging by, ; bottom-fermenting, , ; break, , ; characteristics by, ; collecting, 188-90; dry, ; dusty, , , , ; fermentation and, ; finings and, ; flavor and, ; Frohberg type, ; growth of, 96 (illustration), 96-98; head, ; lager, , ; lag phase and, , , , ; metabolism, 89-90; nutrients for, ; operation of, ; oxygenation and, ; performance of, , , , ; pitching, 166-67, 169-71, , , , 237-38, , ; problems with, ; pure-culture, 95-98; reproduction by, 93-94, ; respiratory phase and, ; Saaz type, ; starter, , 238-39; storing, ; strains of, 92-93, ; subculturing, ; temperature for, ; top-fermenting, , , ; washing, ; wild, 94-95,
Yeast-culturing equipment, 280-81
_Zymase,_ ; fermentation and,
_Zymomonas,_ 91 (illustration), ,
|
{
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| 4,934
|
Cette page recense des événements qui se sont produits durant l'année 1840 en Belgique.
Chronologie
18 avril : formation du gouvernement Lebeau II
18 mai : mise en service de la ligne de chemin de fer entre Bruxelles-Midi et Tubize.
30 mai : des pétitions demandant le rétablissement de la langue flamande dans certaines provinces pour les affaires communales et provinciales (13000 signatures) sont prises en considération à la Chambre.
Naissances
3 janvier : Père Damien, missionnaire, saint catholique.
7 janvier : Eugène Dumont de Chassart, homme politique.
23 janvier : Xavier Neujean, homme politique († ).
2 mars : François Vaxelaire, entrepreneur.
15 mars : Charles de Hemricourt de Grunne, homme politique, diplomate.
11 avril : Paul Janson, homme politique.
26 avril : Alexandre Cousebandt d'Alkemade, militaire et homme politique belge († ).
19 mai : Adhémar de Steenhault de Waerbeeck, homme politique.
7 juin : Charlotte de Belgique, impératrice consort du Mexique.
11 juin : Henri de Braekeleer, artiste peintre.
25 novembre : Hugo Verriest, prêtre, écrivain de langue néerlandaise.
15 décembre : Émile de Neve de Roden, homme politique.
Décès
17 mai : Charles Surmont de Volsberghe, juriste, homme politique.
Bibliographie
1840
00
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 3,379
|
{"url":"https:\/\/gateoverflow.in\/763\/gate2001-22","text":"488 views\n\nWe wish to construct a $B^+$ tree with fan-out (the number of pointers per node) equal to 3 for the following set of key values:\n\n80, 50, 10, 70, 30, 100, 90\n\nAssume that the tree is initially empty and the values are added in the order given.\n\n1. Show the tree after insertion of 10, after insertion of 30, and after insertion of 90. Intermediate trees need not be shown.\n2. The key values 30 and 10 are now deleted from the tree in that order show the tree after each deletion.\nedited | 488 views\nfanout means no of records per node but b+ trees contain an additional pointer pointing to next node so in this case would splitting will take place after insertion of 2 key values or 3 key values? Plz explain...\nAfter 2 key values. Also, that additional pointer appears only in leaves.\nwhy split after 2 key value ??\n\nsince order = 3, so we can have 2 key elements in a node with no problem but when we have 3 key then it should be a overflow right ??\nplz explain this one....\nSame like btree","date":"2018-01-17 20:28:38","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\": 0.270546019077301, \"perplexity\": 1151.5557462900792}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-05\/segments\/1516084886964.22\/warc\/CC-MAIN-20180117193009-20180117213009-00303.warc.gz\"}"}
| null | null |
Momsregistreringsnummer är ett unikt serienummer som i EU tilldelas alla som registreras för redovisning av moms av medlemsstatens skattemyndighet. I internationella sammanhang kallas det för VAT-nummer, där VAT är en förkortning av det engelska value-added tax.
Numrets konstruktion
Det finns en internationell standard för momsregistreringsnummer. De består av en två-bokstavskod och innehåller ett antal, max 12 siffror (ibland bokstäver). Två-bokstavskoden kommer från standarden ISO 3166-1 alpha-2, utom för Grekland som har EL.
Ett svenskt momsregistreringsnummer är baserat på ett 10-siffrigt personnummer eller organisationsnummer. Det inleds med den svenska landskoden SE och avslutas alltid med 01. Eftersom varken bindestreck eller mellanrum skall förekomma är svenska momsregistreringsnummer alltid 14 tecken: SE999999999901, där de tio niorna ersätts med personnummer eller organisationsnummer.
Momsregistreringsnummer i andra EU-länder kan ha varierande format, men brukar baseras på de organisationsnummer som finns i landet. Dock börjar de alltid med det aktuella landets landskod.
Även de flesta europeiska länder utanför EU och många länder i Sydamerika och några andra länder, har infört internationella momsregistreringsnummer.
Referenser
Externa länkar
Validering av VAT-nummer på EU:s webbplats
Företagsekonomi
|
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| 96
|
using System;
using System.Collections.Generic;
using System.Linq;
using System.Web.Mvc;
using Telefonbuch.Models;
namespace Telefonbuch.Controllers
{
[Authorize]
public class HomeController : Controller
{
public ActionResult Index()
{
return View();
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,626
|
idrinth.tier = {
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typeof key !== "function" &&
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};
if (!isValidParameter(name)) {
return;
}
let boss = this.list[name];
/**
*
* @param {int} x
* @param {string} name
* @returns {undefined}
*/
let make = function(x, name) {
let makeElement = function(label, number, description) {
return {
content: label + " " + idrinth.ui.formatNumber(number),
attributes: [
{
name: "title",
value: description
}
]
};
};
let info = [
makeElement("FS", boss.fs.nm, idrinth.text.get("tier.FS")),
makeElement("AP", boss.ap, idrinth.text.get("tier.AP"))
];
if (boss.os && boss.os.nm) {
info.push(makeElement("OS", boss.os.nm, idrinth.text.get("tier.OS")));
info.unshift(
makeElement(
"MA",
boss.nm[boss.nm.length - 1],
idrinth.text.get("tier.MA")
)
);
info.unshift(
makeElement("MI", boss.nm[0], idrinth.text.get("tier.MI"))
);
}
info.unshift({
type: "strong",
content: boss.name.replace(/\(.*$/, "")
});
idrinth.tier.taggedSlots[x] = idrinth.ui.buildElement({
id: "idrinth-tier-box-" + name,
css: "idrinth-hovering-box idrinth-tier-box",
children: [
{
children: info
}
],
attributes: [
{
name: "title",
value: idrinth.text.get("tier.clickClose")
},
{
name: "onclick",
value: "idrinth.ui.removeElement(this.id);idrinth.tier.taggedSlots['" +
x +
"']=null;"
},
{
name: "style",
value: "left:" +
x +
"px;background-image: url(https://dotd.idrinth.de/static/raid-image-service/" +
boss.url +
"/);"
}
]
});
idrinth.ui.base.appendChild(idrinth.tier.taggedSlots[x]);
};
for (var key in this.taggedSlots) {
if (isFreeSlot(key)) {
return make(key, name);
}
}
idrinth.core.alert(idrinth.text.get("tier.maxBoxes"));
},
/**
* initializes this module
* @returns {undefined}
*/
start: function() {
let pos = 1;
/**
* parsed a json-response and fills tier list and exclusion list
* @param {string} data
* @returns {undefined}
*/
let importData = function(data) {
data = JSON.parse(data);
if (data) {
idrinth.tier.list = data;
/**
*
* @param {string} name
* @param {string} url
* @returns {undefined}
*/
let create = function(name, url) {
if (!idrinth.settings.data.bannedRaids[name]) {
idrinth.settings.data.bannedRaids[name] = false;
window.localStorage.setItem(
"idotd",
JSON.stringify(idrinth.settings.data)
);
}
document.getElementById("idrinth-raid-may-join-list").appendChild(
idrinth.ui.buildElement({
name: "bannedRaids#" + name,
rType: "#input",
type: "checkbox",
id: "idrinth-raid-may-join-list-" + name,
label: idrinth.text.get("raids.disableJoining") + name
})
);
document
.getElementById("idrinth-raid-may-join-list")
.lastChild.setAttribute(
"style",
"background-image:url(https://dotd.idrinth.de/static/raid-image-service/" +
url +
"/);"
);
};
for (var key in data) {
if (data[key].name) {
create(data[key].name, data[key].url);
}
}
} else {
idrinth.core.timeouts.add("tier", idrinth.tier.start, 1000);
}
};
while (0 < window.innerWidth - 140 * (pos + 1)) {
this.taggedSlots[(pos * 140).toString()] = null;
pos++;
}
idrinth.core.ajax.runHome(
"tier-service/",
importData,
function() {
idrinth.core.timeouts.add("tier", idrinth.tier.start, 10000);
},
function() {
idrinth.core.timeouts.add("tier", idrinth.tier.start, 10000);
}
);
},
/**
* displays bosses that match both name and type
* @returns {undefined}
*/
getMatchingTiers: function() {
idrinth.workers.run('tiers', {
name: document.getElementById("idrinth-tierlist-namesearch").value,
type: document.getElementById("idrinth-tierlist-typesearch").value,
list: idrinth.tier.list
});
},
/**
*
* @param {Boolean} yes
* @returns {undefined}
*/
allCheck: function(yes) {
let boxes = document
.getElementById("idrinth-raid-may-join-list")
.getElementsByTagName("input");
for (var counter = boxes.length - 1; counter >= 0; counter--) {
if (
boxes[counter].getAttribute("type") === "checkbox" &&
boxes[counter].checked !== yes
) {
boxes[counter].checked = yes;
idrinth.settings.change(
boxes[counter].getAttribute("id").replace(/idrinth-/, ""),
yes
);
}
}
}
};
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,914
|
Ribhan (रिभान, transcribed also as "'Reevan'" and "'Rivan'") is a village and Village Development Committee in Kaski District in the Gandaki Zone of northern-central Nepal. At the time of the 2001 Nepal census it had a population of 1,617 persons living in 372 individual households. Brahmin and Gurung are main ethnic groups in this village. Ribhan is mainly in the bank of Mardi river which is main source of irrigation for people of this agriculture-based village. The forest in this village has many endangered species such as Himali red panda and leopard. Much of its forest is still unexplored as it is very remote with high hills and sometimes snow. It is a part of Annapurna Conservation Area project .
References
External links
UN map of the municipalities of Kaski District
Populated places in Kaski District
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,453
|
Kyrie Irving said Friday at his introductory press conference in Boston that he did not speak to LeBron James in the month before his eventual trade from the Cleveland Cavaliers to the Boston Celtics. In fact, he didn't talk to the media either, he says, because of his strategic plan to not have his quotes misconstrued.
"I haven't spoken to him and my intent, like I said, was for my best intentions. To look back at the amount of ground we covered in the last three-year span, or even before that because ... to really realize how special that was and how much stuff happened in that amount of time, I'd be sitting up here and telling you guys a lie if I didn't tell you how much I really learned from that guy.
"The perfection of the craft comes in a variety of forms and you watch, you ask a lot of the great players, what does it take to be great? I've had the unique opportunity to play with one of the greats and it was awesome."
Irving spent six years in Cleveland after the team selected him No. 1 overall in the 2011 NBA Draft. He won a championship with the team -- the first in the franchise's history -- and he joined forces with James to form a dynamic duo the league could rarely stop. Ultimately, though, Irving made a decision to leave Cleveland. He says it was a difficult one but he's happy to have spent time alongside The King.
"This was a very, very challenging decision at first, but after a while you understand and you have that confidence in yourself to understand the magnitude of what you actually can accomplish and potentially can do with other great people," Irving said. "And now that I'm sitting here, it just echoes in terms of me just being very appreciative of not only the Cleveland fans, but all of Ohio, as well as Bron in incorporating me into that special team we had in Cleveland."
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,993
|
Jacob "Jack" Pressman (October 26, 1919 – October 1, 2015) was an American Conservative rabbi. He served as the rabbi of Temple Beth Am in Los Angeles, California, from 1950 to 1985. He was a co-founder of the American Jewish University in Bel Air. He penned a weekly column in The Beverly Hills Courier, from 2004 to 2015.
Early life
Jacob Pressman was born on October 26, 1919 in Philadelphia, Pennsylvania. His father was Solomon Pressman and his mother, Dora (Levin) Pressman. Raised in the Jewish faith, he attended Temple Beth Am in Philadelphia.
Pressman graduated from the University of Pennsylvania in 1940, where he received a Bachelor of Arts degree. He attended the Jewish Theological Seminary of America and was ordained as a conservative rabbi in 1945.
Career
Pressman served as rabbi at Forest Hills Jewish Center in Queens, New York City, from 1944 to 1946. In 1946, he moved to Los Angeles, California, where he first served as associate rabbi at Sinai Temple until 1950. At the time, the senior rabbi was Jacob Kohn (or Cohen).
In 1950, he became rabbi of the Olympic Jewish Center. It was renamed Temple Beth Am in 1957. Pressman established Beth Am Manor, a low-rent residence for senior citizens as well as the Rabbi Jacob Pressman Academy, a combination of nursery, elementary and secondary schools connected to the synagogue.
Pressman spearheaded the 'Save Soviet Jewry' campaign in 1964, leading to greater awareness of the plight of the Jews in the Soviet Union. This in turn led to more immigration of Soviet Jews to Israel. Moreover, he joined Martin Luther King Jr. in the Selma to Montgomery marches in 1965.
Pressman served as President of the Western Region of the Rabbanical Assembly and Chairman of its convention in 1979. Additionally, he served as President of the Board of Rabbis of Los Angeles. He served on the Executive Council of the Jewish Federation of Greater Los Angeles. He was also Chair of the Los Angeles campaign for Israel Bonds. Moreover, he was involved with the United Jewish Appeal.
Pressman spearheaded the establishment of Camp Ramah in California, a summer camp in Ojai. He was a co-founder of the Brandeis-Bardin Camp Institute. He served as Chairman of the Los Angeles Zionist Youth Commission. He played a critical role in the founding of the Los Angeles Hebrew High School. He was a co-founder of the University of Judaism, later known as the American Jewish University (AJU), located in Bel Air. He was a recipient of the 2004 Rabbi Simon Greenberg Award from AJU.
Pressman was the founding President of the Maple Counseling Centre, a non-profit organization which offers free counseling sessions, based in Beverly Hills, California. He wrote a weekly column in The Beverly Hills Courier, as well as two books.
Personal life
Pressman married Marjorie Steinberg in 1942. They resided in Beverly Hills, California. One of their sons, Joel, was a Performing Arts teacher and director of the Madrigal Singers at Beverly Hills High School from 1975 to 2013, and died from cancer in 2013. Their second son, Daniel, is a rabbi. They also had a daughter, Judith, who made Aliya to Israel and has been a pioneer of Ma'ale Tzvia community village in the Galilee.
Death
Pressman died in Los Angeles on October 1, 2015. He was ninety-five years old. His funeral was held at Temple Beth Am, and he was buried at the Eden Memorial Park Cemetery, a Jewish cemetery in Mission Hills, Los Angeles.
Bibliography
This Wild and Crazy World as Seen From Beverly Hills by Rabbi Jack (1999).
Dear Friends: A Prophetic Journey Through Great Events of the 20th Century (Hoboken, New Jersey: KTAV Publishing House, 2002).
References
1919 births
2015 deaths
Clergy from Philadelphia
People from Beverly Hills, California
University of Pennsylvania alumni
Jewish Theological Seminary of America alumni
American Conservative rabbis
American Jewish University
American columnists
Journalists from California
Journalists from Pennsylvania
21st-century American Jews
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3
|
Alchemy::Picture.class_eval do
image_accessor :image_file do
after_assign { |a| a.process!(:optimize) }
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 733
|
Ultra Eczema is een Belgisch onafhankelijk platenlabel uit Antwerpen dat werd opgericht door Dennis Tyfus.
Naast muziek op vinyl en cd brengt het label ook posters, boeken en magazines uit. Het label legt zich voornamelijk toe op obscure en experimentele muziek. Het drukwerk is vaak van de hand van Dennis Tyfus en sluit aan bij het avant-gardistische karakter van de muziek.
Belgisch platenlabel
Bedrijf met zetel in Vlaanderen
Economie in Antwerpen (stad)
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,195
|
Q: auth0 auth0-spa-js must run on a secure origin angular I hope someone can help me out here, I have found no help anywhere else.
I have created a site using auth0 and angular universal.
I followed their documentation and it all seemed to be working ok; but now I have deployed it to azure, the site never loads. I have done some digging and found that it's saying this:
auth0-spa-js must run on a secure origin. See https://github.com/auth0/auth0-spa-js/blob/master/FAQ.md#why-do-i-get-auth0-spa-js-must-run-on-a-secure-origin for more information.
I have no idea why this is happening as I am running it on a secure host (https://86sparks-develop-website.azurewebsites.net/) and I have tried running it on a server with a valid SSL certificate, as well as locally using a self generated certificate, but they all throw this error.
I believe it has something to do with using Angular Universal, but I can't see what is causing the issue.
Has anyone else had issue with auth0 and angular universal?
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,767
|
__all__ = ['Trigger','TGOP_INSERT','TGOP_UPDATE','TGOP_DELETE','TGEV_AFTER','TGEV_BEFORE']
def allbases_object(obj):
bases = []
if obj.__name__ not in ['FormModel','dict','Rules','object','DB']:
bases.append(obj.__name__)
for o in obj.__bases__:
if o.__name__ not in ['FormModel','dict','Rules','object','DB']:
bases.extend(allbases_object(o))
return bases
TGOP_INSERT = 'INSERT'
TGOP_UPDATE = 'UPDATE'
TGOP_DELETE = 'DELETE'
TGEV_AFTER = 'AFTER'
TGEV_BEFORE = 'BEFORE'
triggers = {
TGOP_INSERT: {TGEV_AFTER:{},TGEV_BEFORE:{}},
TGOP_UPDATE: {TGEV_AFTER:{},TGEV_BEFORE:{}},
TGOP_DELETE: {TGEV_AFTER:{},TGEV_BEFORE:{}}
}
class Trigger():
@staticmethod
def connect(sender, functor, tgop, tgev):
can_attach = True
for klass in allbases_object(sender):
if triggers[tgop][tgev].has_key(klass):
if functor in triggers[tgop][tgev][klass]:
can_attach = False
if can_attach:
if not triggers[tgop][tgev].has_key(sender.__name__):
triggers[tgop][tgev][sender.__name__] = []
triggers[tgop][tgev][sender.__name__].append(functor)
@staticmethod
def on_trigger(ar, tgop, tgev):
for klass in allbases_object(ar.__class__):
if triggers[tgop][tgev].has_key(klass):
for t in triggers[tgop][tgev][klass]:
t(ar, tgev == TGEV_BEFORE)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,754
|
If marijuana becomes legal for recreational use in California, who do we want growing and selling it? My research and writing for the Drug Law and Policy Blog will explore two of the barriers to entry into the recreational marijuana industry: capital requirements and criminal records. By looking at the economic barriers to entry and legislation that prevents the justice involved from receiving licenses in states who have legalized recreational marijuana, I want to explore who is being excluded from the "green rush." The ultimate question is whether we are excluding communities of color and the indigent, communities who have been the most adversely affected by the war on drugs.
The arena of drug policy interests me because it is closely tied to the field of criminal law and disproportionately impacts indigent communities. I came to law school interested in pursuing a career in social justice and public service and I have since narrowed my interest to indigent criminal defense. I interned at the Santa Clara Office of the Public Defender for six months in 2014 and found the work to be both challenging and rewarding. Currently, I am working at the Northern California Innocence Project, a clinic providing representation for the wrongfully convicted, run through Santa Clara University School of Law.
This entry was posted in Uncategorized and tagged cannabis, disparate impacts, drug reform, marijuana. Bookmark the permalink.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,104
|
Pyroppia tridentifera är en kvalsterart som beskrevs av H. och Zhan Wang 1994. Pyroppia tridentifera ingår i släktet Pyroppia och familjen Ceratoppiidae. Inga underarter finns listade i Catalogue of Life.
Källor
Spindeldjur
tridentifera
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,689
|
Investing & Analysis
Money Wire
AT&T's DirecTV Cancels NEWSMAX in Censorship Move, See More Here
Home | Markets
Tags: Japan | Prime Minister | Inflation | Economy
Inflation Must Be Accompanied by Strong Economy, Japan Prime Minister Says
Tuesday, 12 February 2013 07:15 AM EST
Japanese Prime Minister Shinzo Abe said on Tuesday the central bank is responsible for achieving its new 2 percent inflation target, but that a rise in prices must be accompanied by strong economic growth driven by fiscal stimulus.
"It is possible to achieve the 2 percent inflation target with monetary policy, but fiscal stimulus is needed to support the economy as well because it would take time for wages to start rising," Abe told parliament.
Abe also said a "correction" of past sharp yen rises is driving up Japanese corporate profits, which should eventually help boost wages.
© 2023 Thomson/Reuters. All rights reserved.
Japanese Prime Minister Shinzo Abe said the central bank is responsible for achieving its new 2 percent inflation target, but that a rise in prices must be accompanied by strong economic growth driven by fiscal stimulus. It is possible to achieve the 2 percent...
Japan,Prime Minister,Inflation,Economy
Tuesday, 12 February 2013 07:15 AM
MONEYNEWS.COM
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,516
|
Figuring out how to present your media with the greatest impact can be a daunting and expensive task. We focus on creating optimal solutions that result in the production of both small and large projects across multiple platforms. Our innovative team prioritizes the seamless integration and delivery of your work, respecting budget parameters, marketing objectives and schedule. Below are just a few of the services we provide that will have your media exceeding all expectations.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,574
|
Q: C# checking if expression is brackets valid The expression:
"( a[i]+{-1}*(8-9) )"
should return true since it is valid to write syntax like this. Every left bracket has a right closer in the correct place and all brackets are at legal positions.
I tried to do this via one stack and I know where I'm wrong but I want to know a relevant way to solve this.
thx!
My poor poor wrong code:
string expression = "( a[i]+{-1}*(8-9) ) ";
Stack<char> expStack = new Stack<char>();
List<char> rightBracketsHolder = new List<char>();
for (int i = 0; i < expression.Length; i++)
{
if (expression[i] == '{')
{
expStack.Push('}');
Console.Write("}" + " ");
}
else if (expression[i] == '(')
{
expStack.Push(')');
Console.Write(")" + " ");
}
else if (expression[i] == '[')
{
expStack.Push(']');
Console.Write("]" + " ");
}
}
Console.WriteLine();
for (int i = 0; i < expression.Length; i++)
{
if (expression[i] == '}')
{
rightBracketsHolder.Add('}');
Console.Write(expression[i] + " ");
}
else if (expression[i] == ')')
{
rightBracketsHolder.Add(')');
Console.Write(expression[i] + " ");
}
else if (expression[i] == ']')
{
rightBracketsHolder.Add(']');
Console.Write(expression[i] + " ");
}
}
Console.WriteLine();
bool stackResult = checkValidity(expStack, rightBracketsHolder);
if (stackResult)
Console.WriteLine("Expression is Valid.");
else
Console.WriteLine("\nExpression is not valid.");
Console.ReadKey();
}
private static bool checkValidity(Stack<char> expStack, List<char> leftBracketsHolder)
{
Console.WriteLine();
int length = leftBracketsHolder.Count;
for (int i = 0; i < length; i++)
{
if (expStack.Peek().ToString().Contains(leftBracketsHolder.ToString()))
{
leftBracketsHolder.Remove(expStack.Peek());
expStack.Pop();
}
}
if (expStack.Count == 0 && leftBracketsHolder.Count ==0)
{
return true;
}
return false;
}
}
A: This code will solve your purpose -
static void Main(string[] args)
{
bool error = false;
var str = "( a[i]+{-1}*(8-9) )";
Stack<char> stack = new Stack<char>();
foreach (var item in str.ToCharArray())
{
if (item == '(' || item == '{' || item == '[')
{
stack.Push(item);
}
else if(item == ')' || item == '}' || item == ']')
{
if (stack.Peek() != GetComplementBracket(item))
{
error = true;
break;
}
}
}
if (error)
Console.WriteLine("Incorrect brackets");
else
Console.WriteLine("Brackets are fine");
Console.ReadLine();
}
private static char GetComplementBracket(char item)
{
switch (item)
{
case ')':
return '(';
case '}':
return '{';
case ']':
return '[';
default:
return ' ';
}
}
A: You need to pop things off the stack as the closing occurs. Try the following code. It will push an open brace/bracket/parenthesis on the stack and the first thing then it will be popped from the stack by a corresponding close. Otherwise it is invalid. If you have no opens on the stack when a close is encountered, it is invalid. If you have any extra opens when you are complete it is invalid.
I also used a switch statement instead of an if statement just because I thought it was easier to read.
using System;
using System.Collections.Generic;
public class Program
{
public static void Main()
{
string expression = "( a[i]+{-1}*(8-9) ) ";
bool stackResult = checkValidity(expression);
if (stackResult)
Console.WriteLine("Expression is Valid.");
else
Console.WriteLine("\nExpression is not valid.");
}
private static bool checkValidity(string expression)
{
Stack<char> openStack = new Stack<char>();
foreach (char c in expression)
{
switch (c)
{
case '{':
case '(':
case '[':
openStack.Push(c);
break;
case '}':
if (openStack.Count == 0 || openStack.Peek() != '{')
{
return false;
}
openStack.Pop();
break;
case ')':
if (openStack.Count == 0 || openStack.Peek() != '(')
{
return false;
}
openStack.Pop();
break;
case ']':
if (openStack.Count == 0 || openStack.Peek() != '[')
{
return false;
}
openStack.Pop();
break;
default:
break;
}
}
return openStack.Count == 0;
}
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,194
|
Norops sagrei är en ödleart som beskrevs av Duméril och BIBRON 1837. Norops sagrei ingår i släktet Norops och familjen Polychrotidae.
Underarter
Arten delas in i följande underarter:
N. s. ordinatus
N. s. sagrei
N. s. mayensis
N. s. greyi
N. s. luteosignifer
N. s. nelsoni
Källor
Externa länkar
Fjällbärande kräldjur
sagrei
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,332
|
\section{Introduction}\label{sec:introduction}}
\input{src/sec1-introduction}
\input{src/sec2-related_work}
\input{src/sec3-overview}
\input{src/sec4-data}
\input{src/sec4-dminer}
\input{src/sec4-dminer2}
\input{src/sec-evaluation}
\input{src/sec7-discussion}
\input{src/conclusion}
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
\section{Mining Design Patterns}
\subsection{\yanna{Design Rule Mining}}
\label{sec:design_patterns}
The goal of DMiner{} is to find effective arrangements and coordination for the multiple views of a dashboard.
Thus, we aim to model the mapping from data and encoding features to arrangement and coordination as well as the interrelations between arrangement and coordination within multiple dashboard views, as shown in \autoref{fig:relationship}.
Here, we adopt the decision rule approach \cite{greco2016decision} to achieve this goal.
The reasons for choosing the decision rule approach are as follows:
1) it allows us to investigate the mutual influences among those features in a data-driven manner;
2) it embraces algorithmic explainability and is easy for humans to interpret.
Specifically, given a set of dashboard features $\mathcal{F}$ as detailed in~\autoref{fig:features},
we aim to mine rules that imply $ X \Rightarrow Y$, where $X, Y \subseteq \mathcal{F}$.
All these rules are automatically learned and extracted from the collected dashboard dataset.
\textbf{Model Selection}. As mentioned above, we have formulated the problem as mining decision rules.
To solve the problem,
we chose one of the decision rule algorithms, RuleFit Binary Classifier~\cite{imodels2021}.
It is efficient since it produces a set of unordered independent rules, which can be checked in parallel rather than in series~\cite{friedman2008predictive}.
The extracted rules of RuleFit follow the structure:
$condition \rightarrow target$ with each rule's coefficient and importance.
The coefficient and importance of a rule describe how and how much the \textit{condition} contributes to the \textit{target}.
\yanna{Specifically, if the \textit{condition} is obeyed, then the corresponding \textit{target} tends to be true for positive coefficients and false for negative coefficients.}
\textbf{Feature Processing}.
Since the \textit{condition} and the \textit{target} of rules extracted by RuleFit have to be Boolean variables, we need to process the extracted features before feeding them into the model.
Specifically, we convert numerical and categorical features into Boolean variables by setting thresholds or asserting equivalence.
To handle numerical values, we set manual thresholds by splitting the interval by the average value of two-edged cut-offs ({i.e.,}\xspace~larger or smaller than the mean).
The purpose of two-edged cut-offs is to improve interpretability.
It can be challenging for humans to understand more intervals,
{e.g.,}\xspace~\textit{the distance of two views smaller than 0.5} is easier for sense-making than \textit{the distance either within $(0.0, 0.15]$ or $(0.3, 0.45]$}.
\yanna{
The final derived Boolean features are represented using binary vectors \cite{binaryvector}. }
\textbf{Training and Post-processing}.
We train a RuleFit binary classifier for each \textit{target} feature of each mapping identified in \autoref{fig:relationship}.
In total, we train 208 models.
\yanna{
Specifically, to increase the interpretability of the rules, we reduce the condition complexity of each rule by limiting each \textit{condition} with at most 2 features~\cite{ming2018rulematrix}.}
We randomly partition dashboards into a 75\% \yanna{(640)} training set and a 25\% \yanna{(214)} test set.
The trained models achieve an average of \yanna{73\% accuracy on the training set and} 71\% accuracy on the test set.
Considering the huge amount of models,
we select the top 3 rules with positive coefficients and the highest importance for each model to avoid overfitting issues, resulting in 624 decision rules.
\begin{figure*}[!t]
\centering
\includegraphics[width=1\linewidth]{figures/recommendation.pdf}
\caption{The recommender framework.
Given several views and pre-computed possible arrangements, we first generate the candidate dashboards.
Then we apply S2S rules to prune unreasonable candidates to improve efficiency.
The remaining candidates are further assigned scores and ranked using other rules.
Finally, we recommend the candidate with the highest score.
}
\label{fig: recommendation}
\end{figure*}
\subsection{\yanna{Recommender}}
\label{sec: recommender}
Given the design rules extracted in \autoref{sec:design_patterns},
the recommender of DMiner{} (\autoref{fig:teaser} \cc{C}) can further automatically recommend appropriate view arrangement coordination for the views of a dashboard.
\autoref{fig: recommendation} provides an overview of the recommender pipeline.
Similar to prior optimization-based visualization recommendation approaches~\cite{wu2021ai4vis}, our recommender also
enumerates all the possible designs and recommends the designs with the highest scores.
Specifically, it first searches all possible arrangements ({i.e.,}\xspace~positions and sizes for each view) to generate candidates.
Then, it computes the score for each candidate by checking whether it obeys or violates the extracted decision rules.
\yanna{Inspired by~\cite{moritz2018formalizing}, the cost score is a weighted sum of violated decision rules, since different rules make different contributions to the final designs}.
We use the importance of each decision rule as the weight, which measures how the rule is important to the prediction introduced in \autoref{sec:design_patterns}.
However, it is time-consuming and infeasible to calculate scores for all candidates.
The number of possible candidates is huge, since it grows exponentially with the number of views.
To reduce the computational complexity, we adopt the common strategy in previous studies on visualization recommendations ({e.g.,}\xspace SeeDB~\cite{vartak2015seedb} and QuickInsights~\cite{ding2019quickinsights}), {i.e.,}\xspace~pruning, to balance efficiency and performance.
We first consider only the rules that map \textbf{S}ingle-view data and encoding features to \textbf{S}ingle-view layout arrangement features (denoted as \textbf{S2S} rules),
since it only considers single view and has smaller complexity than pairwise-view computations.
The top 1\% recommendations with the highest average score are subsequently fed into the next step, {i.e.,}\xspace~to compute scores based on all other rules.
The percentage 1\% is decided on an empirical basis to reduce the running time to seconds, which is desirable in real-world scenarios.
Finally, we recommend the arrangement and coordination with the highest score.
\section{Discussion}
\label{sec: discussion}
In this section, we further discuss a few issues that need further clarification regarding DMiner{}.
\begin{reviseyanna}
\textbf{High-quality multiple-view dashboard dataset construction.}
Recent years have witnessed an increasing interest in applying AI and machine learning (ML) for visualization research~\cite{wu2021ai4vis,wang2021survey}, where high-quality datasets are fundamental to making the AI/ML-based approaches work well in practice.
However,
little research has been
dedicated to constructing large-scale, high-quality multiple-view dashboard datasets.
In this paper, we construct a dashboard dataset by crawling GitHub repositories.
However, the designers of the dashboards published on GitHub may have different levels of visualization expertise.
Our results in \autoref{subsec: quantitative_analysis} have shown that our collected dashboard designs (i.e., \textit{Human} in \autoref{fig:result}) are not significantly different to those designed by \textit{Designer} in \autoref{fig:result}.
We believe that the quality of the dashboard designs can be further improved.
For example, appropriate dashboard designs also depend on other important factors like target users, user intention, and analysis tasks, which, however, are not included in our current dashboard dataset.
Thus, we would encourage the whole visualization community to contribute more high-quality and informative dashboard datasets.
\textbf{Human agency vs. machine automation.}
DMiner{} proposes a data-driven framework to mine design rules and automatically recommend dashboard designs.
Our evaluation in \autoref{subsec: quantitative_analysis} has shown that the dashboard designs recommended by DMiner{} (i.e., DMiner{}-Full) are not significantly different from those created by experienced designers.
Recent research on visualization recommendations has tried to balance human agency and machine automation by keeping humans in the loop \cite{cao2022visguide}.
For dashboard design recommendations, it is worth further exploring how the human agency can be involved in dashboard design mining.
For instance, it will be interesting to investigate how expert knowledge can be incorporated into refining the extracted design rules for dashboards and
further improve the quality of recommended dashboards.
\textbf{DMiner{} for dashboards created by other software or packages.}
In this paper, DMiner{} extracts dashboard design rules from our collected dashboard datasets created by Tableau, one of the most popular software for creating multiple-view dashboards, and is further evaluated on only Tableau dashboards.
However, DMiner{}
can be extended to dashboards created using other software or packages.
For example, the source files of dashboards created by Power BI~\cite{powerbi} also specify visual coding, data operation, data types, and arrangement and coordination between views in a dashboard, which is quite similar to those of Tableau.
With a proper data parsing module, DMiner{} can also work for dashboards by Power BI.
However, this requires considerable research and engineering efforts to improve the interoperability,
since different visualization software and packages have different methods for specifying and rendering visualizations~\cite{satyanarayan2019critical}.
With more multiple-view dashboard datasets available in the future, we would like to extend DMiner{} further to various dashboard datasets.
\end{reviseyanna}
\textbf{More factors for dashboard design.}
DMiner{} has considered a series of features, including data and visual encoding characteristics, for designing dashboards in terms of the arrangement and coordination of its views.
However, our evaluation result in \autoref{evaluation} has demonstrated its effectiveness.
However, there exist other factors that warrant future research.
\yanna{
As pointed out in \autoref{subsec: quantitative_analysis}, the semantic meaning of the visualized data
can also influence the arrangement and coordination of dashboard views.
For example, it is better to arrange the views showing the data of different seasons in the order of spring to winter (\autoref{subsec: quantitative_analysis}).
Also, the decision rules of DMiner{} focus on the effects of single features, and it will be interesting to further explore the combined effects of multiple feature types
in a single decision rule for dashboard design.}
\section{Conclusion}
\label{conclusion}
This paper proposes DMiner, a data-driven framework for mining dashboard design and recommending appropriate arrangement and coordination for multiple-view dashboards.
Building upon our Tableau dashboard dataset collected from GitHub,
DMiner{} extracts a series of relevant features, {i.e.,}\xspace~single-view and pair-view features in data and encoding, arrangement and coordination.
With these features, DMiner~employs a decision rule approach to mine the design rules from the collected Tableau dashboard dataset.
Further, a recommender is proposed to recommend appropriate arrangements and coordination of dashboard views.
\begin{reviseyanna}
We conduct an expert study and a user study to evaluate the effectiveness of DMiner{}.
The expert study demonstrates that the extracted design rules are reasonable and align well with the design practices of experts.
The user study further confirms that the dashboard designs recommended by DMiner{}, considering all design rules, are not significantly different from those of experienced designers.
They are also significantly better than those using the default settings of \textit{Tableau} in both arrangement and coordination.
In summary, as the first work for automating dashboard arrangement and coordination, DMiner~is appreciated by visualization experts and study participants who need to design and use dashboards for data analysis.
\end{reviseyanna}
In the future, we hope to further improve the efficiency of our recommender for dashboards with more views.
\yanna{
It is also promising to regard DMiner{} as a feature and integrate it into existing visualization authoring tools, such as Voyager~\cite{wongsuphasawat2015voyager}, to help designers in their daily work.}
\section{Evaluation}
\label{evaluation}
\begin{reviseyanna}
To evaluate DMiner{}, we conducted an expert study to evaluate the appropriateness of our extracted design rules (\autoref{sec: expert_study}), as well as a user study to demonstrate the effectiveness of the recommender (\autoref{sec: user study}).
\subsection{Expert Study}
\label{sec: expert_study}
We conducted a study with four experts to gather quantitative and qualitative feedback about the appropriateness of extracted design rules.
Our goal is to investigate the extent to which the extracted rules accord with the expert knowledge of dashboard designs.
\subsubsection{Study Setup}
\textbf{Design Rules.}
It is time-consuming to ask our experts to evaluate all the extracted rules.
Thus, we selected Top 10\% (62) rules with the highest accuracy on the test dataset.
To increase the readability, we translated rules into natural language sentences.
For example, the rule in \autoref{fig:case example} \cc{1} was translated into \textit{``\textbf{If} View A is Text, \textbf{and} it does not have fields on Y-axis, \textbf{then} View A should be of the height of 1''}.
\yanna{Moreover}, considering that some rules may present similar conditions or semantic meanings, we further organized them into 22 \textit{processed} rules.
For example, the second and the third rule in~\autoref{tab:problematic_rules} were merged from several rules with the same \textit{if} conditions.
These 22 processed rules with the associated original rules were presented for the experts to evaluate and justify their validity.
\textbf{Participants}. We invited 4 experienced visualization or dashboard design experts (3 males, with age $31.25 \pm 2.87$).
\textbf{Expert 1} (\textbf{E1} for short), a full professor at a local university, has engaged in visualization analytics for ten years, and has been working on MV dashboard design for the past few years.
\textbf{E2} has worked as a research scientist in a company for nine years, specializing in using dashboards for text analytics and business analysis.
\textbf{E3}, a research assistant professor, has been designing dashboards for spatial-temporal data analytics for the past seven years.
\textbf{E4}, a strategy analyst in a finance company for 3.5 years, needs to analyze trading data daily using dashboards.
None of them is color-blind.
\textbf{Procedure.}
Each expert study lasted about 1.5 hours, and we sought the experts' consent to record the entire process.
The studies were preceded by a 10-minute introduction to our work, including the goals and the corresponding features in \autoref{sec: feature_enginnering}.
In the expert studies, each expert was presented with 22 processed rules with 62 original rules.
Also, we provided a user interface for experts to explore some supporting example dashboards for each original rule.
Experts were asked to score the appropriateness of 22 processed rules on a 7-point Likert scale from 1 (the least reasonable) to 7 (the most reasonable) and give qualitative feedback about their scores.
All the experts were encouraged to think aloud throughout the process.
\begin{figure*}[!ht]
\centering
\includegraphics[width=1\linewidth]{figures/tvcg-caseexample.pdf}
\caption{An example dashboard to illustrate identified design rules. The dashboard consists of four views, \cc{A} - \cc{D}. Part of the data and the information of each view in the dashboard are shown in two tables, \cc{E} and \cc{F}, respectively. In the dashboard, the application of four design rules (\cc{1} - \cc{4}) has been observed. \yanna{For example, \cc{1} represents that the height of \textit{Text} view in \cc{A} tends to be 1, and \cc{3} describes that this \textit{Text} view in \cc{A} tends to be placed on top of other views.}}
\label{fig:case example}
\end{figure*}
\subsubsection{Result Analysis}
Our extracted design rules were appreciated by the experts.
Considering that each processed rule represented a different number of original rules, we got a weighted average score of $4.44 \pm 1.32$, with the number of original rules as the weight.
According to the qualitative feedback from experts, our extracted design rules were reasonable and aligned well with their design knowledge, thus reducing the burden of dashboard design.
\textbf{E4} commented that our rules used \textit{low-level and more specific features}, and were therefore easier to understand and follow.
In this subsection,
we present the top five rules with the highest average scores (\autoref{tab:problematic_rules}), and summarize the feedback from the experts.
An example case is used to illustrate four of these rules, as shown in \autoref{fig:case example}.
\input{tables/problematic_rule3}
\textbf{Adjust the size of the view according to the importance of the view.}
The first rule receiving a score of 6.5 from experts is that the \textit{Text} view
(not the \textit{Text} table)
tends to have the smallest width or height, such as \autoref{fig:case example} \cc{A} and \cc{1}.
\textbf{E2} and \textbf{E3} confirmed that ``\textit{Text view is usually used as an assistive view to describe the data analysis purpose
and does not have too many meanings.
Thus, it is often equipped with either the smallest height or the smallest width}".
\textbf{E1} mentioned that ``\textit{This reminds me that the size of the view should be related to the importance of the view, especially given the limited space available in the dashboard}".
\textbf{Configure the two views used for comparison in a similar and close manner.}
The second rule is that, if two views possess the same mark type and use the same fields on the Y-axis, they are preferred to be arranged side by side ($score = 6.5$).
All the experts inferred that these two views were for comparison, and recognized that they preferred to arrange views horizontally for comparison.
\textbf{E4} gave this rule a score of 7 and commented, ``\textit{Two views with the same chart type and Y-axis must be in the same information hierarchy in a dashboard. I would not hesitate to arrange them side by side}".
Both \textbf{E2} and \textbf{E4} said that the views for comparison should be configured ``\textit{as similar as possible, like the same width, the same height, and the same scale}".
In this way, users can compare them effectively.
While \textbf{E3} mentioned, ``\textit{In some cases, I might also arrange two comparison views vertically, if there are also other views that need to be arranged horizontally.
However, most of the time, I prefer the horizontal orientation}".
\textbf{E1} pointed out, ``\textit{This condition reminds me of the concept of `small multiples' (proposed by Tufte~\cite{tufte1985visual}), which describes the practice of arranging two views of the same visual type together to facilitate comparison}".
\textbf{Add some necessary coordination to facilitate data exploration.}
The third rule is about coordination, {i.e.,}\xspace~if two views encode the same fields in color and use more than 50\% of the same fields, they will be coordinated by the \textit{brushing} ($score = 6.25$).
For example, as described in \autoref{fig:case example} \cc{2}, Views \cc{C} and \cc{D} have a total of 5 fields, with 4 being identical, so 80\% of the fields are shared. Moreover, both of them present the field \textit{Profitable?} using colors, so that they brush each other.
\textbf{E3} mentioned, ``\textit{This coordination is necessary, since it can help users identify the related points in another view and thus explore the data efficiently}".
\textbf{E4} pointed out, ``\textit{This is really useful if there are too many colors or objects in one view, like a scatterplot}".
\textbf{Overview first, then details-on-demand.}
The last two rules are about the relative position of views.
The first one shows that \textit{Text} view (not \textit{Text} table) tends to be on the top right, top left, or top of the other view types ($score = 5.5$).
For example, \autoref{fig:case example} \cc{A} was arranged on the top left or top right of the other three views, as described in \autoref{fig:case example} \cc{3}.
\textbf{E1} said, ``\textit{Text view providing the statistical values works as an overview and attracts the audiences' attention. Hence, the overview should be at the top of a dashboard}".
While \textbf{E2} and \textbf{E4} argued, ``\textit{it is correct to put the statistical Text view at the top, but the word cloud should not always be put at the top}".
The last rule is that if View A encodes more fields than View B, then View A prefers to be on the bottom left, bottom right, or bottom of View B ($score = 5.5$).
\textbf{E3} mentioned, ``\textit{the view with more fields tends to be more detailed than the view with fewer.
According to Schneiderman's visualization mantra `overview first, then details on demand'~\cite{shneiderman2003eyes}, the view with more fields should be at the bottom or on the right}".
\textbf{E4} agreed, ``\textit{the view with fewer fields may have more coarse information, thus having a higher hierarchy at the top}".
\textbf{E2} considered that ``\textit{the overview should be a simple view with fewer fields}".
For example, \autoref{fig:case example} \cc{4} described that View \cc{A} had the least fields, thus being arranged on top of other views.
While View \cc{D} encoded more fields than View \cc{B}, {i.e.,}\xspace~4 vs. 3, then View \cc{D} was put at the bottom right of View \cc{B}.
\textbf{Limitations and lessons.} We also received some feedback on those processed rules with low scores, which can be summarized into two main reasons, {i.e.,}\xspace~different personal experiences and shortcut learning of the algorithm.
From personal experience, experts may have different opinions on a few rules.
For example, the rule, \textit{Scatterplot should be put on the right-most}, received a score of $3.5\pm1.73$, denoted as \textbf{the 6th rule}.
\textbf{E1} mentioned, ``\textit{I have previously used scatterplots to show specific details, such as the distribution relationship of certain variables in two selected cities. I tend to put the detailed view on the right. Thus, I agree with this extracted rule}".
In contrast, \textbf{E3} often used T-SNE to reduce high-dimensional data to a two-dimensional scatterplot to overview the system.
Therefore, he thought ``\textit{the scatter plot should be on the left or top rather than right-most}".
Another rule with a higher score variance of 1.91 is that \textit{the view raising brushing should have a width or height of 3 or 4}, denoted as \textbf{the 7th rule}.
\textbf{E4} has developed systems that use slender panels on the left or wide panels on the top.
Thus, she thought it was reasonable for views raising \textit{brushing} to have a larger width and height.
While \textbf{E3} mentioned that ``\textit{the view proposing brushing should be a high-level view and therefore have a smaller size. 3 or 4 is too large for such a view}".
Another problem is that the decision rule approach learns some highly-correlated relationships between attributes but lack semantics for humans to interpret.
For example,
for the rule (\textit{If View A and View B encode the same field in color, then the height of View A or View B is 3}),
\textbf{E2} commented that ``\textit{DMiner{} may
derive this rule from the
rules mentioned before (the 3rd rule and the 7th rule).
However, it is challenging for humans
to understand such a derivation or inference and make sense of this rule
}".
\end{reviseyanna}
\begin{figure*}
\centering
\includegraphics[width=1\linewidth]{figures/result-tvcg.png}
\caption{An overview of the study result.
Specifically, each column describes the score of a metric in three clusters, {i.e.,}\xspace~the cluster considering all dashboards, the cluster with 3- and 4-view dashboards, and the cluster with 5- to 8-view dashboards.
Six metrics cover 3 perspectives, {i.e.,}\xspace~arrangement, coordination, and overall ({i.e.,}\xspace~both).
Each sub-plot presents the score of 6 methods regarding the corresponding metric and cluster.
The error bars are 95\% confidence intervals.
\yanna{
\textbf{$[$ } indicates the significant difference between a pair of methods ($p < 0.05$).}
}
\label{fig:result}
\vspace{-1em}
\end{figure*}
\subsection{User Study}
\label{sec: user study}
We conducted a user study with 12 participants to \yanna{compare 6 methods} of designing MV dashboards and collect their feedback about the effectiveness of our approach for recommending appropriate MV dashboards.
\subsubsection{Study Setup}
\textbf{Participants.}
We recruited 12 participants (9 females and 3 males, ages 22 to 27, with an average of $24.8$) \yanna{by posting online advertisements on social media platforms ({e.g.,}\xspace~email and microblogging websites)}.
They are all well-educated (7 PhD students, 3 MSc students, 1 research assistant, and 1 data analyst) and from diverse backgrounds, including \yanna{ visualization (4), data science (3), finance (2), human-computer interaction (2) and recommendation system (1).
None of them has color blindness.
All the participants are
experienced in the data analysis according to their self-reports on a 7-point Likert scale ($\mu = 5.25, \sigma = 0.45$), where 1 indicates ``no experience'' and 7 represents ``highly experienced''.}
Each participant was compensated with \$10 after finishing the user study.
\textbf{Dashboards and Baseline Methods.}
\begin{reviseyanna}
We selected 15 dashboards from the test dataset
covering almost all the mark types (10/13) and various topics (e.g., business, COVID-19, and patient analysis).
The number of views ranges from 3 to 8, obeying the distribution in \autoref{fig:basic_info} \cc{A}.
To evaluate the effectiveness, we built a comparative group designed by five experienced designers
(3 females and 2 males, aged $27 \pm 2.74$).
Two are postdocs at the university, focusing on the visual analytics of computational social science and narrative storytelling, respectively.
The remaining three designers are data analysts in automobile, outlet malls, and retail companies.
They are all experienced in data analysis (with an average of 4.1 years) and dashboard design (with an average of 3 years).
Each designer was randomly assigned three dashboards.
They were given the views of each dashboard and started their design after becoming well familiar with the view content.
Specifically, the design space of the dashboard was fixed at the common display size of 1080p ({i.e.,}\xspace~1920 x 1080) to enable comparisons.
All designers finished the design in around 1 hour with \$15 compensation.
In this study, we leveraged 6 methods to recommend appropriate designs for 15 dashboards. The 6 methods are as follows:
\begin{compactitem}
\item \textit{Default}: the \textit{Default} dashboard designs recommended by Tableau~\cite{tableau}. Note that Tableau does not support the recommendation of coordination. Thus, the corresponding dashboards do not have coordination;
\item \textit{DMiner-Basic}: the dashboard designs recommended by \textit{DMiner-Basic} using design rules concerning only the single-view data and encoding features;
\item \textit{DMiner-Partial}: the dashboard designs recommended by \textit{DMiner-Partial} considering part of design rules, {i.e.,}\xspace~all black relationships in \autoref{fig:relationship}, and ignoring those rules on interrelations between arrangement and coordination;
\item \textit{DMiner-Full}: the dashboard designs recommended by \textit{DMiner-Full} considering all extracted design rules;
\item \textit{Human}: the original dashboard designs
we crawled online, which are designed by general \textit{Human} users; and
\item \yanna{\textit{Designer}}: the dashboard designs created by the recruited experienced \textit{Designers}.
\end{compactitem}
\end{reviseyanna}
The above 6 methods were used to recommend dashboard designs for a given dashboard.
All the generated dashboard designs were of consistent size (i.e., 1920 x 1080 pixels),
except for \textit{Human}, whose corresponding dashboard design was in the original size.
\yanna{For the selected 15 dashboards, it took designers around eight minutes to design a dashboard. Our three methods speeded up this process by a factor of eight. In addition, our three methods generated designs within five seconds for dashboards with fewer than six views.}
For simplicity, the resulting six dashboard designs are called \textit{a dashboard group}.
Therefore, we presented 15 dashboard groups
for participants to evaluate their design effectiveness.
\textbf{Procedure.}~The user study lasted about 1.5 hours, and we gained participants' consent for video recording the whole user study.
During the user study, we
first briefly introduced our work.
Then,
due to the long time cost of evaluating dashboard designs, we randomly split the 15 dashboard groups into 3 clusters (each with 5 dashboard groups).
Each participant was asked to score the arrangement and coordination of 1 cluster ({i.e.,}\xspace~5 dashboard groups).
\yanna{
For each dashboard group,
the 6 dashboard designs were ordered randomly and anonymously assigned to participants.}
Thus, each dashboard group was evaluated by 4 participants.
Before participants started to evaluate each dashboard group,
we introduced every dashboard view that needed to be arranged.
\begin{reviseyanna}
Once participants were familiar with these views, they freely explored the dashboards for around ten minutes.
After the free exploration, they further scored the dashboard designs using a 7-point Likert scale.
\end{reviseyanna}
Participants were asked to evaluate their arrangement (including logic, aesthetics, and helpfulness), coordination (including logic and helpfulness), and overall, which refers to the previous studies~\cite{smith2013data, wang2019datashot}.
Specifically, the \textit{logic} measures how well the arrangement and coordination follow human users' \textit{logical} analysis workflow~\cite{dowding2018development}.
The \textit{helpfulness} describes how well the arrangement and coordination help users analyze and explore data.
The \textit{aesthetics} evaluates how users perceive the visual appearances of the dashboard arrangement, while the \textit{overall} score measures how well the dashboard has been designed concerning both the arrangement and coordination.
All the participants were encouraged to
report why they gave the corresponding scores in a think-aloud manner.
\subsubsection{Result Analysis}
\label{subsec: quantitative_analysis}
This section introduces and discusses results from the user study, \yanna{including the ratings and the corresponding qualitative feedback.}
We received positive feedback on the recommender of DMiner{}.
Participants appreciated the automated dashboard designs in terms of both arrangement and coordination by our approach.
Participant 11 (P11 for short) commented, ``\textit{I hope it can be integrated into my workflow, [...], I believe it will greatly facilitate my analysis process}".
To analyze the results, we further split all dashboards into two \textit{dashboard clusters} according to the number of views in dashboards, {i.e.,}\xspace~the dashboards with 3-4 views and the dashboards with 5-8 views.
The rationale for the split stemmed from the feedback from our participants.
They indicated that they cared more about the arrangement and coordination when the number of views is more than 4, since a well-organized dashboard can reduce their cognitive load of viewing many visualizations.
We performed the one-way ANOVA to compare the six methods across six metrics
and three dashboard clusters with different view numbers ({i.e.,}\xspace~\textit{3 and 4 views}, \textit{5 to 8 views}, and \textit{all}).
LSD post-hoc tests were used if the scores obeyed the homogeneity of variance test; otherwise, Tamhane's T2 post-hoc tests were used~\cite{lee2018proper}.
Detailed information, including the average ratings, standard deviation, and post-hoc results, are presented in \autoref{fig:result}.
\autoref{fig:result} shows that all the other methods achieved better results than \textit{Default} on all metrics, except for the 3- and 4-view dashboard arrangement.
Specifically, \textit{Default} mainly scored below 5, while the other five methods scored mostly above 5.
\begin{reviseyanna}
\end{reviseyanna}
In the following, we first introduce some interesting results regarding arrangement, coordination, and overall performance, then summarise these results, and finally discuss some recommendations that can be further improved.
\begin{figure*}[!ht]
\centering
\includegraphics[width=1\linewidth]{figures/tvcg-positiveexample.pdf}
\caption{\cc{A} is a case identified in our user study, \yanna{and \cc{B} is the coordination type distribution.} In \cc{A}, \cc{a} and \cc{b} were the dashboards created by \textit{Human} and our method \textit{DMiner-Full}, respectively. Our recommended dashboard was highly rated due to outstanding arrangement logic and coordination type (see \cc{a5} and \cc{b5}).
Participants recognized that \cc{b1} was more suitable than \cc{a1} to act as a starting point to explore the dashboard.
For the same view pair, our recommended coordination type was \textit{brushing}, which kept the context for comparison.
However, \textit{Human} chose \textit{filtering}, which led to a change in the display and confused the users.
In \cc{B}, our three recommenders ({i.e.,}\xspace~\textit{DMiner-Basic}, \textit{DMiner-Partial} and \textit{DMiner-Full}) suggested more coordination than \textit{Human}.
}
\label{fig: positive_example}
\end{figure*}
\textbf{Arrangement}.
For \textbf{logic}, \autoref{fig:result} shows our method \textit{DMiner-Full} was not significantly different to \yanna{\textit{Designer}} and \textit{Human} regardless of the number of views.
Specifically, it was slightly better than \textit{Human} for 3- and 4-view dashboards.
\yanna{Moreover, our method \textit{DMiner-Full} performed significantly better than \textit{Default} for 5- to 8-view dashboards ($p < 0.01$).}
\autoref{fig: positive_example} \cc{A} provides one example, where \cc{a} was created by \textit{Human} and \cc{b} was the recommended dashboard design by \textit{DMiner-Full} (with an average score of logic as 5 and 5.25, respectively).
Participants recognized that the arrangement logic of \cc{b} was better than \cc{a}, since \cc{b1} was more suitable than \cc{a1} to act as a starting point to guide the data analysis.
Furthermore, \textit{DMiner-Full} achieved better arrangement logic than \textit{Default} for dashboards with 5 to 8 views.
Participants acknowledged that the arrangement logic was essential for dashboards with many views, while \textit{Default} arranged the views alphabetically by view name, leading to worse logic that confused participants.
For \textbf{aesthetics}, our method \textit{DMiner-Full} offered a slightly better visual aesthetic design than \textit{Default}.
Participants appreciated that \textit{DMiner-Full} set different sizes for different views, which was better than \textit{Default} which set all views with equal width and height.
P8 mentioned that ``\textit{when there are too many views, there should be a dominant one telling me what it is attempting to express.
The exactly-equal size confuses me, and it is challenging for me to identify what I should pay attention to first}".
Moreover, participants suggested that arrangement aesthetics was less important compared to arrangement logic for facilitating data analysis.
Thus, for \textbf{helpfulness}, \autoref{fig:result} shows
observations similar to the logic.
\iffalse
For dashboards with 3-4 views, we cannot observe any significant differences among the \yanna{6 methods}.
And we can notice that the \textit{DMiner-Full} has a slightly higher score than \textit{Human}.
\autoref{fig: positive_example} provides one example, where \cc{A} is created by \textit{Human} and \cc{B} is our \textit{DMiner-Full} recommendation.
Participants recognized that the position logic of \cc{B} is better than \cc{A}, since \cc{B1} is more suitable than \cc{A1} to act as a starting point to guide the data analysis.
And the symmetrical position of \cc{B2} and \cc{B4} are also plausible.
However, they thought our \cc{B1} (map) should be with a higher height, but still better than \cc{A1}, where the bar with a too larger height.
What's more, we can notice that the \textit{Default} has a higher score in terms of all three arrangement metrics.
This is probably because the \textit{Default} divides the largest view into two even views each time, resulting in a balanced aspect ratio and a neat arrangement.
The participants prefer such a balanced arrangement, especially when there are relatively less views and their interrelations are simple.
While for dashboards with 5-8 views,
\begin{reviseyanna}
\textit{Default} is slightly worse than \textit{DMiner-Basic} but significantly worse than the other methods in the logic and the ability to help analyze the data.
This may be attributed to that \textit{Default} arranges the views according to the alphabetical order by view name, leading to the worse logic confusing the participants.
And in these two perspectives, our \textit{DMiner-Full} recommendation performs similarly to \yanna{\textit{Designer}} and \textit{Human}, as participants recognized that our \textit{DMiner-Full} mode recommended a logically sound position.
However, it sometimes recommended problematic size ({i.e.,}\xspace~too small or too large in height or width) to cause deformation and extrusion, such as \autoref{fig: negative_example} \cc{B2}.
This leads to significantly poorer performance in aesthetics.
However, there is still a small difference in the ability to help analyze since participants felt that position logic is more important than size to help analyze.
\end{reviseyanna}
Furthermore, though \textit{Default} still recommends the arrangement with the balanced aspect ratio, it suffers the same deformation and extrusion problems as our three recommendations.
P8 has mentioned that ``\textit{when there are too many views, there should be a dominant one telling me what it wants to express.
The exactly equal size confuses me.
And it is challenging for me to identify what I should pay attention to first}."
P9 had similar comments.
This may be the main reason why our \textit{DMiner-Full} and \textit{DMiner-Partial} are slightly better in aesthetics, and significantly better in the logic and ability to help analyze than the \textit{Default}.
\begin{reviseyanna}
Thus, for group with all dashboards, it presents the similar phenomena to the dashboard group with 5-8 views.
The exception is that \textit{DMiner-Full} has slightly but not significantly better results than \textit{Default} since \textit{Default} has really high score in 3- and 4-view dashboards.
\end{reviseyanna}
What's more, we further notice that, among our three recommendations, \textit{DMiner-Full} achieves the best arrangement performance, followed by \textit{DMiner-Partial} and \textit{DMiner-Basic}.
\textit{DMiner-Basic} is the worst among our three recommendations and only slightly better than \textit{Default} for the dashboards with 5-8 views.
While \textit{DMiner-Partial} is better than \textit{DMiner-Basic}, it has a great difference from \textit{Default} in terms of the logic and the powers to help analyze \yanna{in 5- to 8-view dashboards, and it even achieves the same logic with \textit{Human}}.
While \textit{DMiner-Full} is the best, its arrangement achieves not only human-like logic but also human-like powers to help analyze. \fi
\textbf{Coordination.}
\yanna{
\autoref{fig:result} shows that all our methods ({i.e.,}\xspace~\textit{DMiner-Full}, \textit{DMiner-Partial}, and \textit{DMiner-Basic}) offered significantly better coordination than \textit{Default} in terms of logic and helpfulness, regardless of the number of views (with all $p<0.001$).
Furthermore, \textit{DMiner-Partial} and \textit{DMiner-Full} performed similarly to \textit{Human} and \yanna{\textit{Designer}} in 5- to 8-view dashboards,
and were significantly better than \textit{Human} in 3- and 4-view dashboards (with all $p < 0.05$).}
Participants all confirmed the indispensability of the coordination among views in a dashboard.
P1 mentioned, ``\textit{Without coordination, it is challenging for me to identify the related data across views and focus the data of interest}".
As shown in \autoref{fig: positive_example} \cc{B}, compared to \textit{Human}, participants appreciated that our methods were equipped with more brushing or filtering between views for dashboards, as it provided a stronger power of exploration.
\yanna{Moreover, P5 and P11 appreciated the coordination type recommended by our approach in some cases.
They mentioned, ``\textit{\textit{Filtering} can help identify relevant information efficiently in those views with many visual elements, such as text tables with thousands of lines of text.
\textit{Brushing} works better for those views without visual clutter, since it maintains context and enables comparisons}"}.
For example, as shown in \autoref{fig: positive_example} \cc{A}, \cc{a5} was the filtered result of \cc{a2}, while \cc{b5} was the brushed result of \cc{b4}.
It is clear to see that \cc{b5} well preserved the context before and after the interaction.
That is why our three methods gained the highest coordination score for 3- and 4-view dashboards.
\textbf{Overall.}
From \autoref{fig:result}, our method \textit{DMiner-Full} achieved significantly better results than \textit{Default} regardless of the number of views (with all $p<0.001$).
\yanna{Moreover, compared to \textit{Human} and \yanna{\textit{Designer}}, the average scores obtained by \textit{DMiner-Full} were similar without significant differences, except that \textit{DMiner-Full} performed significantly better than \textit{Human} for 3- and 4-view dashboards ($p < 0.01$).}
Participants mentioned that the arrangement of views in dashboards was slightly more important than the coordination for data analysis, especially for dashboards with more views.
For 3- and 4-view dashboards, \textit{DMiner-Full} was significantly better than \textit{Human},
as \textit{DMiner-Full} can achieve similar arrangements as \textit{Human}, but with much better and more coordination.
Regardless of the number of dashboards, our method \textit{DMiner-Full} achieved similar performance to \yanna{\textit{Designer}}.
\iffalse
Though \textit{Default} achieves similar arrangement performance to other recommendations, it gets a significantly worse result due to the lack of coordination across 3 groups.
Participants all recognized the indispensability of the coordination.
For the all-view group, \textit{DMiner-Full} has a slightly but not significantly better overall score than the \textit{Human},
though \textit{DMiner-Full} has a significantly great coordination score to the \textit{Human}.
This is because the participants thought that the arrangement was slightly more important than the coordination for data analysis.
And for 3- and 4-view dashboards, \textit{DMiner-Full} is significantly better than the \textit{Human}, due to a similar result in arrangement but better and more comprehensive coordination provided \yanna{by} \textit{DMiner-Full}.
While for 5- to 8-view group, \textit{DMiner-Full} still achieves human-like performance.
\yanna{What's more, regardless of the number of dashboards, \textit{DMiner-Full} achieves the same performance as \yanna{\textit{Designer}}.}
And our 3 recommendations have no significant difference from each other.
\fi
\begin{figure*}[h]
\centering
\includegraphics[width=1\linewidth]{figures/TVCG-negativeexample.pdf}
\caption{A case identified in our user study.
\yanna{\cc{A} is created by \yanna{\textit{Designer}}, \cc{B} is created by \textit{Human}, while \cc{C} is created by our method \textit{DMiner-Full}.
\cc{A} and \cc{B} received similar scores in all metrics (with a difference of less than 0.5), though they have different designs with different analysis focus.
\textit{DMiner-Full} in \cc{C} received a lower score, and our participants pointed out that the arrangements should be improved from two perspectives, {i.e.,}\xspace~the size of \cc{C2} should be equipped with more space like \cc{A2} and \cc{B1}, and \cc{C1} about ``winter" should be above \cc{C3} about ``summer".}}
\label{fig: negative_example}
\end{figure*}
\textbf{Summary.}
In summary,
\yanna{\textit{DMiner-Full} performed significantly better than \textit{Default} for dashboards with more views, and similarly to \textit{Human} and \yanna{\textit{Designer}} without significant differences.}
It was the best among our three methods, followed by \textit{DMiner-Partial}, and then \textit{DMiner-Basic}.
With an increased number of views, \textit{DMiner-Full} and \textit{DMiner-Partial} always performed better than \textit{DMiner-Basic}, demonstrating the importance of the pairwise-view relationship in modeling complex dashboards.
\yanna{
Moreover, no significant difference between \textit{Human} and \yanna{\textit{Designer}} in 6 metrics was observed, indicating the quality of our collected dashboard dataset.
\autoref{fig: negative_example} shows an example where \textit{Human} and \yanna{\textit{Designer}} had different designs given the same views, where \cc{A} was created by \yanna{\textit{Designer}} and \cc{B} was created by \textit{Human}.
\yanna{\textit{Designer}} regarded \cc{A1} as an overview to analyze the bike trips every day in a week,
while \textit{Human} adopted \cc{B1} as the overview to analyze the most common trips that bike riders like.
Participants recognized that they had different analysis focus, and both were designed with plausible arrangement and coordination (both with average scores larger than 5.75)}.
\textbf{Limitations.} DMiner{} occasionally provides some recommendations that can be further improved.
According to participants' feedback, the underlying reasons for it stems from two perspectives, {i.e.,}\xspace~without considering the number of visual elements within each view and the semantic meaning of data.
\yanna{Take \autoref{fig: negative_example} as an example, where the MV dashboard designed by \textit{Designer} \cc{A}, \textit{Human} \cc{B} and our method \textit{DMiner-Full} \cc{C} are shown.}
Compared with \autoref{fig: negative_example} \cc{A2} and \cc{B1},
the same view in the dashboard design recommended by our approach (\autoref{fig: negative_example} \cc{C3}) does not have enough space to show the heatmap.
The underlying reason for such a result is that DMiner{} does not explicitly consider the number of visual elements ({i.e.,}\xspace~many rectangles in the heatmap) due to the lack of underlying data tables.
Also, a few participants pointed out that \autoref{fig: negative_example} \cc{C} placed the line chart showing the ``Peak hour in Winter Month'' above the line chart displaying the ``Peak hour in Summer Month'', which was contradictory to their expectation, {i.e.,}\xspace~the line chart for ``Summer Month" should be positioned above that for ``Winter Month". Unfortunately, such semantic meanings of the data are not considered in DMiner{}.
\iffalse
\subsubsection{Design Suggestions}
\label{subsec: design_suggestions}
Combining the feedback from the participants and the findings from the unreasonable recommendations, we summarized the following design suggestions, which supplement our proposed method.
\textbf{Arranging views by proximity.}
All participants commented that views encoding similar information or the same chart types should be positioned closely with equal sizes, such as \cc{B1} and \cc{B2} in \autoref{fig: negative_example}.
This perspective confirms the proximity principle in visual designs that elements near each other are perceived as related.
We encourage future research to investigate the effects of view proximity on their layout arrangement through empirical experiments.
\textbf{Reducing view ``islands''.}
Most participants mentioned that it was less describable to have a view that was an ``island'' and disconnected with other views in terms of encoded attributes or coordination.
Instead, the dashboard should compose views as a cohesive whole.
Furthermore, P10 expected the dominant view to be coordinated with the remaining views.
This comment echoed that of P12, ``\textit{the top-left most view is an overview, but it is strange that there is no coordination. It should guide me to explore other views.}"
\textbf{Adding cues or hints for coordination.}
Participants commented that it was difficult to find coordination among views.
They need to carefully explore each view and observe other views to identify whether there is coordination between them.
This process is tedious and poses high cognitive loads.
Thus, it is important to provide user-friendly hints on coordination among views.
\textbf{Considering the semantics.}
Participants deemed semantics as an important concern in arranging views.
For instance, P1 said ``\textit{the view about peak hour in summer month should on the top of that in winter}", (a counter example shown in \autoref{fig: negative_example} \cc{B1} and \cc{B3}).
This requires understanding the semantic meaning of each view and their ``orders''.
However, visualization sequencing remains a challenging problem that warrants future research~\cite{lin2020dziban}.
\textbf{Paying attention to cross-view filtering.}
Participants noted that cross-view filters could result in a huge difference in the filtered view ({e.g.,}\xspace~ \cc{A1} leading to a large change from \cc{A2} to \cc{A5} in \autoref{fig: positive_example}).
Such changes catch attention but also bring about a cognitive load.
For example, P1 said, ``\textit{I got lost for seconds, wondering what makes the sudden change. It is a waste of time since I soon realize the reason, and it does not bring new data insights.}''
This problem suggests the need for careful consideration of cross-view filtering.
\fi
\section{Related Work}
\label{related_work}
Our research is related to prior studies on multiple-view visualizations,
visualization recommendations, and design knowledge mining.
\subsection{Multiple-view Visualization}
Multiple-view visualizations (MVs) have gained extensive research interest in the visualization community.
A large body of research aims to advance the theoretical underpinnings of MVs through empirical methods.
Baldonado~{et~al\xperiod}\xspace~\cite{wang2000guidelines} drew from a workshop to present eight guidelines for using MVs in information visualization along three dimensions:
the selection, presentation, and interaction of views.
Roberts~{et~al\xperiod}\xspace\cite{roberts2007state} contributed a state-of-the-art report highlighting seven fundamental research areas of MVs:
data processing, view generation, exploration techniques, coordination, tool infrastructure, human interface, usability and perception.
Sarikaya~{et~al\xperiod}\xspace~\cite{sarikaya2018we} summarized the design space of dashboards by analyzing the intentions of a chosen corpus of 83 dashboards.
\begin{reviseyanna}
Followed by this work, Bach~{et~al\xperiod}\xspace~\cite{bach2022dashboard} gathered 36 more dashboards and proposed a more fine-grained design space focusing on the dashboard structure, visual design, and interactivity.
\end{reviseyanna}
Researchers have also contributed several experiments to understand the design of MVs.
For instance, Qu and Hullman~\cite{qu2017keeping} studied how visualization authors consider the importance of encoding consistency between views when designing MVs.
Langner~{et~al\xperiod}\xspace~\cite{langner2018multiple} investigated how users interacted with MVs on large screens and concluded with design suggestions.
While those studies contributed valuable design knowledge,
they rely on large-scale experiments and require considerable human effort.
We take a different perspective by automatically extracting MV design knowledge from a larger scale corpus.
One of the key challenges in designing MVs is to relate data relationships among views~\cite{sun2021towards}.
Coordination techniques such as brushing and linking can reveal cross-view data relationships and have been widely used in visualization tools ({e.g.,}\xspace~Tableau~\cite{tableau}, Power BI~\cite{powerbi}, Jigsaw~\cite{stasko2008jigsaw}).
Despite being useful,
they require users to trigger interaction events ({e.g.,}\xspace~brushing) and pay adequate attention to subsequent changes on other views.
Thus,
researchers have advocated the use of proximity-based methods that spatially organize relationship components~\cite{sun2021sightbi}.
Inspired by those studies,
we aim to automatically recommend coordination and spatial arrangement of views to facilitate the design of MVs.
We focus on MV dashboards, which are one of the most popular genres of multiple-view visualizations~\cite{sarikaya2018we}.
Through a review of existing work, we develop a set of features that influence the arrangement and coordination of MV dashboards, extract design rules from a corpus of Tableau dashboards and further recommend appropriate arrangement and coordination of multiple views.
\subsection{\yanna{Visualization Recommendation}}
Authoring effective and elegant visualizations is a challenging task even for professionals,
since it requires the consideration of many aspects such as data insights, perceptual effectiveness, and aesthetics.
Researchers have proposed many visualization recommendation systems to assist in data analysis.
Several recommenders ({e.g.,}\xspace~\yanna{Data2Vis~\cite{dibia2019data2vis}, VizML}~\cite{hu2019vizml}, Draco~\cite{moritz2018formalizing}, and KG4Vis~\cite{li2021kg4vis}) focus on recommending visual encodings of a single visualization.
However, a single visualization is often insufficient in supporting in-depth data analysis, as it often requires progressive and iterative exploration of different data subsets~\cite{wongsuphasawat2015voyager}.
Thus, researchers have started to investigate the problem of recommending multiple visualizations.
For instance,
VizDeck~\cite{key2012vizdeck} utilizes users' preferences to select data for presentation and organize dashboards.
Voder~\cite{srinivasan2018augmenting} supports the interactive exploration of data facts associated with charts and natural language descriptions.
To further reduce manual effort,
\yanna{
Tundo {et~al\xperiod}\xspace~\cite{tundo2020declarative} allowed users to select dashboard templates to transform the declarative definition they create into dashboards.}
MultiVision~\cite{wu2021multivision} and Dashbot~\cite{deng2022dashbot} recommend analytical dashboards given an input dataset in an end-to-end manner.
\begin{reviseyanna}
However, those systems are either template-based and do not consider the underlying data, or do not recommend layout arrangement and view coordination, which requires considerable human effort.
Our work fills this gap by developing a recommender for arranging views in an MV dashboard.
\end{reviseyanna}
Research in the field of data storytelling has studied approaches for composing multiple visualizations in a logical manner.
For instance,
researchers have proposed methods to arrange visualizations in a logical sequence to enhance storytelling~\cite{hullman2013deeper,kim2017graphscape}.
DataShot~\cite{wang2019datashot} and Calliope~\cite{shi2020calliope} group multiple visualizations into a coherent topic according to insights derived from the visualizations.
\yanna{
Different to them, we study the layout arrangement and view coordination in MV dashboards for data analysis and comprehension.}
\subsection{Design Mining Visualizations}
\begin{reviseyanna}
There are an increasing number of multiple-view visualizations created and shared by people from different domains, which has inspired many researchers to mine and extract multiple-view visualization design knowledge from them.
\end{reviseyanna}
Some researchers have invested effort into adopting the statistical method to \yanna{mine} the visualization usage.
Al-maneea and Roberts \cite{al2019towards} collected MVs from the published papers, and answered what the most common number of views is and what the popular tiles are by statistically counting their manual-label chart type and layouts.
Beyond counting, Chen {et~al\xperiod}\xspace\cite{chen2020composition} further leveraged some statistical methods like condition probability to perform the configuration and composition analysis, and integrated the findings into a system for exploration and recommendation.
Following this work, Shao {et~al\xperiod}\xspace \cite{shao2021modeling} employed Bayesian probabilistic inference to analyze the effects of design factors on layouts of MVs, and discovered some insightful layout design patterns, {e.g.,}\xspace~views for exploration with a more scattered area ratio.
Though inspiring, these analyses and findings were limited to arrangements and view types, since the MVs collected in image format prevented them from accessing the underlying data and the coordination among views.
Recently, Lu {et~al\xperiod}\xspace \cite{lu_exploring_2020} detected the position and underlying semantics (such as the numbers 1-9) of components in infographics and explored how different components are linked.
\yanna{Inspired by this work, we aim to mine the mappings from the characteristic of individual views and view pairs to the arrangement and coordination in an MV dashboard.}
\section{DMiner}
\label{dminer}
\begin{reviseyanna}
DMiner{} aims to automatically extract design rules for MV dashboards from existing dashboard datasets and guide the subsequent design of MV dashboards, which is common in visualization recommenders ({e.g.,}\xspace~Draco~\cite{moritz2018formalizing}).
\autoref{fig:teaser} provides an overview of~DMiner{}.
We first survey previous relevant studies to identify a set of key features that indicate important considerations for MV dashboard design.
We then extract those features from our dataset and mine design rules.
Finally, we develop a recommender for MV dashboard design based on the extracted rules.
The detailed procedures of feature engineering, rule mining, and recommender will be introduced in the following subsections.
\end{reviseyanna}
\subsection{Feature Engineering}
\label{sec: feature_enginnering}
\autoref{fig:relationship} provides an overview of the extracted features and their mappings.
Unlike existing works that propose single-view features ({e.g.,}\xspace~the absolute position of views),
we further introduce pair-wise features ({e.g.,}\xspace~the relative position of views),
which allows us to delineate the arrangement and coordination between views in a more fine-grained manner.
\begin{reviseyanna}
Our design choices for feature selection and the mappings among them are built upon prior theoretical research about dashboards and multiple visualizations~\cite{tufte1985visual, Roberts1988MV, north2000snap, chen2020composition,wu2022computableviz}.
Specifically, they have
suggested that the data and encoding features of views are crucial for arranging and coordinating the view into a cohesive dashboard.
Therefore, we choose those mappings, which are denoted as black arrows in \autoref{fig:relationship}.
\end{reviseyanna}
The bottom part of \autoref{fig:teaser}(B) shows an example mapping from \textit{data and encoding features} of a view to its \textit{arrangement features}, which describes that
a text view (i.e., \textit{A.Mark $=$ Text}) without data encoding on Y-axis (i.e., \textit{A.\#Y $<$ 1}) tends to be placed at the top (i.e., \textit{A.y $=$ 0}).
\begin{reviseyanna}
Also, prior studies have shown that the arrangement and coordination of a dashboard are interrelated~\cite{al2019towards, shao2021modeling}.
Therefore, we have further added such mappings in DMiner, as indicated by the red arrows in \autoref{fig:relationship}.
\end{reviseyanna}
For example, the upper part of \autoref{fig:teaser}(B) shows that two views, with one brushed by the other (\textit{coordination features}), should be put closely (\textit{arrangement relationship features}).
In summary, we have defined ten mappings among the features describing dashboard designs.
\autoref{fig:features} shows the formal definitions of all the extracted features.
They can be divided into two major groups, {i.e.,}\xspace~single-view features and pairwise-view features.
Single-view features include features about the data, encoding, and arrangement of each view in a dashboard.
Pairwise-view features are extracted to represent the relationship between two views.
Specifically, we extract the data relationship, encoding relationship, arrangement relationship, and coordination relationship between two views.
In summary, there are 33 single-view features and 41 pairwise-view features.
A detailed feature list is available in the supplementary material.
\begin{figure}[t!]
\centering
\includegraphics[width=1\linewidth]{figures/relationship1.png}
\caption{The targeted mappings in our design rule mining. We identified \yanna{ten mappings in black and red between features in designing dashboards, while the gray dotted lines show the derivation relationship. These black mappings focus on how to utilize data and encoding information of single views for arranging these views in a dashboard and adding coordination among them. The red mappings between arrangement features and coordination features are used to further comprehensively describe the dashboard design.}}
\label{fig:relationship}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=1\linewidth]{figures/features.pdf}
\caption{The formal definition and representation of extracted features. We classify features into single-view and pairwise-view features: (A) The former describes data, encoding, and layout of a single view. (B) The latter describes relationships of data, encoding, and layout between two views as well as the coordination.}
\label{fig:features}
\end{figure*}
\subsubsection{Single-view Features}
Single-view features are extracted to describe each individual view in the dashboard.
These features can be roughly categorized into two types.
\textbf{Data and encoding features} can reflect what data is used in the view and how the data is visually encoded, which has been considered essential for view arrangements in a previous study~\cite{chen2020composition}.
\yanna{
Inspired by previous studies that utilize features to characterize single-view visualizations~\cite{hu2019vizml, li2021kg4vis}, we first extract data features regarding data types ({i.e.,}\xspace~numerical, nominal, and ordinal) and data operations ({e.g.,}\xspace count and sum).}
Since the numbers of data \yanna{fields} in different views are inconsistent, it is infeasible to use each field's data type and operation as features.
\yanna{Instead, as shown in \autoref{fig:features}, we define an operator, i.e., \textit{CountFunc}, to compute the frequency of different data types and data operations as features of views.}
For example, the bar chart in \autoref{fig:basic_info} \cc{A} uses one ordinal field on X-axis and one numerical field on Y-axis.
Regarding encoding information, we extract features about the mark type of the view and the usage of different encoding channels ({i.e.,}\xspace~position, size, color, shape).
Similar to data features, we also summarize the usage of each channel, for example, how many data fields are encoded using the position channel on X-axis.
\textbf{Layout arrangement features} describe how the view is placed in the dashboard.
We propose a series of arrangement features to delineate
the
position and size information of each view.
The display area of a dashboard is evenly divided into $n \times n$ grids ($ n > 0$), and we assume that each view takes a rectangle area and does not overlap with each other.
In this paper,
$n$ is set to 4 to strike a balance between performance and efficiency in our subsequent evaluations.
However, our mining algorithm can also work well when $n$ is other values.
As a result, besides the features delineating a view's arrangement with its original position and size, the arrangement features using the grids are also extracted.
Given that each dashboard has a different size,
the size of all dashboards is normalized to 4*4 to enable comparisons.
That is, the width and height of each dashboard is 4, and the width and height of each view is one of the values $[1,2,3,4]$.
\subsubsection{Pairwise-view Features}
Previously we introduced how we extract the single-view features to characterize each individual view in the dashboard.
To further reveal the relationships across different views in the dashboard, we introduce a group of pairwise-view features.
In our study, we propose three types of pairwise-view features:
\textbf{Data and encoding relationship features} are computed via aggregation functions on two views.
Similarly, due to the inconsistent number of fields among views, it is infeasible to calculate the statistical values of two views like correlation~\cite{hu2019vizml}.
We propose two main types of aggregation operators shown in \autoref{fig:features}.
The first type is binary operators to compare the number of data fields, {e.g.,}\xspace~\textit{IsEqual} determines whether two views encode the same number of fields for a given encoding channel.
\begin{reviseyanna}
The second type is set operators concerning the overlapping between data encodings, {e.g.,}\xspace~\textit{IsOverlapping} and \textit{CountOverlapping} decide whether and how many data fields encoded by two views are overlapping.
\textbf{Layout arrangement relationship features} concern the relationships of both the sizes and relative layouts of two views. For the former, we similarly apply aggregation to compare the sizes between two views, {e.g.,}\xspace~\textit{IsLarger} compares whether the size of one view is larger than the other. For the latter, we compute the relative angle and distances between two views.
We further introduce \textit{IsNeighbour} to decide whether two views are adjacent.
\end{reviseyanna}
\textbf{Coordination features} describe the interactions between two views.
As mentioned in \autoref{data}, there are two types of coordination in Tableau, namely brushing and filtering.
Specifically, \textit{filtering} removes irrelevant data objects while \textit{brushing} highlights the related visual elements and keeps the context~\cite{bartram2002filtering}.
\iffalse
\begin{reviseyanna}
Creating dashboards involves multiple stages: view selection, arrangement, and coordination~\cite{wang2000guidelines}.
Most of previous studies ({e.g.,}\xspace MultiVision~\cite{wu2021multivision}) aim to assist the selection of views and provide little help to the other two steps.
In our paper, we present DMiner~to fill the research gap.
DMiner~mines the design patterns and automate the view arrangement and coordination in dashboard creation.
First, we surveyed previous research and identified the relationship between data and encoding features and view arrangement and coordination (see \autoref{sec: feature_extraction} and \autoref{fig:teaser} \cc{A1}).
Then the corresponding features are extracted (see \autoref{sec: feature_extraction} and \autoref{fig:teaser} \cc{A2}).
Based on the features, we mine the design patterns using machine learning models (see \autoref{sec:design_patterns} and \autoref{fig:teaser} \cc{B}).
Finally, a recommender is proposed to suggest view arrangement and coordination based on the extracted design patterns (see \autoref{sec: recommender} and \autoref{fig:teaser} \cc{C}).
\end{reviseyanna}
\subsection{Feature Engineering}
\label{sec: feature_extraction}
\begin{reviseyanna}
\textbf{Features definition (No correct enough).}
Existing research suggests that the data and encoding features of views, and their relationship between views are crucial to arranging and coordinating the view into a cohesive whole ~\cite{tufte1985visual, Roberts1988MV, north2000snap, chen2020composition}.
Based on these works, we conclude that the features that influence the arrangement and coordination include both the single-view data and encoding features and the data and encoding relationship (\textbf{black} lines in \autoref{fig:relationship}).
What's more, some researchers proposed that arrangement and coordination of a dashboard are interrelated~\cite{al2019towards, shao2021modeling}.
Therefore, we have added their relationship as \textcolor{red}{\textbf{red}} lines in \autoref{fig:relationship}.
\textbf{Simplification.}
Before extracting features, some simplification are first introduced.
One simplifying assumption is that the display area of a dashboard is evenly divided into $n \times n$ grids ($ n > 0$).
\end{reviseyanna}
Furthermore, in a dashboard, each view must be a rectangle and fully occupy some grids without overlapping.
In our paper, to weigh performance and efficiency, $n$ is set to 4.
However, our mining algorithm can be applied when $n$ is other values.
While for the coordination between views, we focus on two primary types, {i.e.,}\xspace~\textit{Filter} and \textit{Brush}\yanna{~\cite{bartram2002filtering}}.
Specifically, \textit{Filter} only presents the data of interests separately and can help users identify the related ones, but it may lead to the content change in views, and disable the comparison sometimes, {e.g.,}\xspace~the change from \cc{A2} to \cc{A5} in \autoref{fig: positive_example}.
While \textit{Brush} highlights the data of interest and does not change the context, which is suitable for comparison but not suitable for those views with overloaded details.
For example, \textit{Filter} is better for a text table with thousands of lines of text, while \textit{Brush} works better when it comes to identifying the position of a circle in the scatter plot.
\begin{reviseyanna}
\textbf{Feature Extraction.}
Based on the definition and simplification before, we extract features to delineate characteristics of dashboard comprehensively.
\end{reviseyanna}
Our extracted features can be divided into two major groups, single-view features, and pairwise-view features.
Single-view features include the features about data, encoding, arrangement of each view in a dashboard.
Pairwise-view features are extracted to represent the relationship between two views.
Specifically, we extract the data relationship, encoding relationship, arrangement relationship, and coordination relationship between two views.
\autoref{fig:features} provides an overview of our extracted features, which are described in the following text.
\textbf{Single-view Features}. Single-view features are extracted to describe each individual view in the dashboard.
These features can be roughly categorized into two classes:
\begin{compactitem}
\item \textbf{Data and encoding features} can reflect what data is used in the view and how the data is visually encoded, which has been considered essential for view arrangement in a previous study~\cite{chen2020composition}.
Inspired by previous studies which utilize features to characterize single-view visualizations~\cite{hu2019vizml, li2021kg4vis}, we first extract data features regarding data types ({i.e.,}\xspace~\yanna{numerical}, nominal, and ordinal) and data operations ({e.g.,}\xspace count and sum).
Since the numbers of data \yanna{fields} in different views are inconsistent, it is infeasible to use each column's data type and operations as features.
Instead, we compute the frequency of different data types and data operations as features of views, \yanna{({i.e.,}\xspace~CountFunc)}.
Regarding encoding information, we extract features about the mark type of the entire view and the usage of different encoding channels ({i.e.,}\xspace~position, size, color, shape).
Similar to data features, we also summarize the usage of each channel, for example, how many data fields are encoded using the position channel on X-axis.
\item \textbf{Arrangement features} describe how the view is placed in the dashboard.
We propose a series of arrangement features to delineate views' position and size information.
As we mentioned \yanna{before}, to simplify the view arrangement of dashboards, each dashboard is divided into grids.
As a result, besides the features that describe a view's arrangement using its original position and size, the features using the grids are extracted.
Given that each dashboard has a different size,
\begin{reviseyanna}
the size of all dashboards is normalized to 4*4 to enable comparisons.
\end{reviseyanna}
\end{compactitem}
\textbf{Pairwise-view Features}.
Previously we introduce how we extract the single-view features to characterize each individual view in the dashboard.
To further reveal the relationship across different views in the dashboard, we introduce a group of pairwise-view features.
\begin{reviseaoyu}
In our study, we propose three types of pairwise-view features:
\begin{compactitem}
\item \textbf{Data and encoding relationship features} are computed via aggregation functions on two views.
\yanna{Similarly, due to the inconsistent number of fields among views, it is infeasible to calculate the statistical values of two views like correlation in~\cite{hu2019vizml}. And we propose two main types of aggregation functions.}
The first type is a set of binary operators to compare the number of data fields, {e.g.,}\xspace~IsEqual determines whether two views encode the same \yanna{number of fields} for a given encoding channel. The second type is set operators concerning the overlapping between data encodings, {e.g.,}\xspace~IsOverlapping and CountOverlapping decide whether and how many data fields encoded by two views are overlapping, respectively.
\item \textbf{Arrangement relationship features} concern the size relationships and relative layout. For the former, we similarly apply aggregation to compare the sizes between two views ({e.g.,}\xspace~IsLarger). For the latter, we compute the relative angle and distances between two views. We further introduce IsNeighbour to decide whether two views border each other.
\item \textbf{Coordination features} describe the interactions between two views.
\yanna{As we have mentioned before, there are two types of coordination, namely brushing and filtering.}
\end{compactitem}
\end{reviseaoyu}
In summary, there are 33 single-view features and 41 pairwise features.
A detailed feature list is available in Supplementary {S2}.
\fi
\section{Recommender}
\label{sec: recommender}
In this section, we apply the extracted design patterns to the arrangement and coordination recommender, and conduct a comparative user study to evaluate the effectiveness of the extracted design patterns.
\subsection{Recommendation}
To demonstrate the usefulness of extracted decision rules, we develop a recommender that automates view arrangement and coordination.
Specifically, given $n$ views, the recommender aims to provide arrangement and coordination that best aligns with the design patterns.
\autoref{fig: recommendation} provides an overview of the recommender pipeline.
It takes the common method in optimization-based visualization recommenders, that is, to enumerate possible designs and recommend designs with the highest scores~\cite{wu2021ai4vis}.
Specifically, it firstly enumerates all possible arrangements ({i.e.,}\xspace~positions and sizes for each view) to generate candidates.
Then it computes the score for each candidate by checking whether it obeys or violates the extracted decision rules.
Inspired by~\cite{moritz2018formalizing}, the cost score is a weighted sum of violated decision rules.
We use the importance of each decision rule as the weight, which measures how the rule is important to the prediction.
However, it is time-consuming and infeasible to calculate scores for all candidates.
The number of possible candidates is huge since it grows exponentially with the number of views.
To reduce the computational complexity, we adopt the common strategy in previous studies on visualization recommendation ({e.g.,}\xspace SeeDB~\cite{vartak2015seedb} and QuickInsights~\cite{ding2019quickinsights}), {i.e.,}\xspace~pruning, to balance the efficiency and performance.
We first consider the \textbf{S2S} patterns ({i.e.,}\xspace~single-view data encoding features to single-view layout arrangement features), since it only considers single view and therefore has smaller complexity than pairwise-view computations.
We use cost scores computed from \textbf{S2S} patterns to get the top 1\% results, which are subsequently fed into the next step, {i.e.,}\xspace~to compute scores based on all other patterns.
The number 1\% is decided on an empirical basis to reduce the running time to seconds, which is desirable in real-world scenarios.
Finally, we recommend the arrangement and coordination with the highest score.
\iffalse
Given such set of views, we firstly regard it as $n$ independent views and $n * (n-1)$ view pairs.
Then we extract the corresponding $n$ sets of $SF_{D\&E,i}$ features and $n * (n-1)$ sets of $SF_{D\&ER,(i, j)}$ features by conducting the feature extraction and feature processing described in \autoref{sec: feature_extraction} and \autoref{sec:design_patterns}.
For each individual view and view pairs, we can infer the \textit{mapped target} is true or false of each pattern based on their data and encoding features.
For example, for a view $i$, we can infer its $SF_{Arr, i}$ by observing whether the $SF_{D\&E,i}$ features obey the \textit{conditions} of the \textbf{S$\rightarrow$S} pattern or not.
If $view \ i$ obeys the condition of a pattern, the corresponding \textit{mapped target} tends to be true, since the coefficients of all the patterns are positive.
We denote the inferred \textit{mapped target} for $view \ i$ as $\tilde{Y}_{view \ i} = \{y_1, y_2, ..., y_k\}$, where $y_j \in \{0, 1\}$.
However, this forwarding inference has some limitation since there may be some conflicts between views.
For example, view $A$ and view $B$ are all preferred to be on the top left position.
Thus, the final arrangement and coordination must break some design patterns, and our goal is to recommend those with best overall score.
To achieve this goal, we adopt the brute-force and greedy algorithm.
We firstly brute-force all potential arrangements for the given number of views $n$.
We formulate this problem as dividing the display space into $n$ rectangles, then using memory searching to recursively find every possible rectangles given the previous selected rectangles.
The details can refer to the Supplementary xxx.
And to remove some inappropriate division, we calculate the distribution of \textbf{aspect ratio with the number of views} \yanna{how to describe aspect ratio $\times$ the number of views}, and remove those aspect ratio with the number of views above the 95th percentile or below the 5th percentile (should we need to clarify this one?).
In this case, we get 200 kinds of divisions for $n=5$, while 1392 for $n=6$.
The number of divisions increase exponentially with the number of views.
And it will increase by $n!$ times due to the permutation and combination of view for each division, {i.e.,}\xspace~ there will up to $200 \times 5! = 24000$ arrangement candidates for $n=5$.
And each candidates need to consider $n$ single-view relations ({i.e.,}\xspace~S2S) and $5*n*(n-1)$ other pair-related relations.
Thus, it is time-consuming to calculate scores considering all relations for all arrangement candidates.
To make the trade-offs between
the performance and efficiency, we adopts greedy algorithm by firstly filtering some arrangement candidates with higher score considering only the \textbf{S2S} relation, and then applying all other relations.
We only consider the \textbf{S2S} relation, since it consider each view as individuals and ignored the interrelations.
Specifically, for each arrangement candidates, we can conduct the same process to extract the $SF_{Arr, i}$.
We then convert the $SF_{Arr, i}$ features to vectors aligning the prediction target order of $S2S$ relations.
Then, we can get the arrangement features for $c$th arrangement candidate as $Y_{candidate \ c} = \{y_1, y_2, ..., y_k\}$.
Thus, for $view \ i$ arranging in $c$th arrangement candidate, we can get the score as:
\begin{equation}
\begin{aligned}
\centering
S_{view \ i, candidate \ c}= w^T (\tilde{Y}_{view \ i} \odot Y_{candidate \ c}),
\end{aligned}
\end{equation}
where the $\odot$ is inclusive or, {i.e.,}\xspace~1 for two same elements and 0 for two different ones.
Same elements means that the $c$th arrangement candidate is aligned well with what the design patterns predicted.
Thus, the $c$th arrangement candidate gets the corresponding scores for obeying the proposed patterns.
Considering the number of views is $n$, the whole score for $c$th arrangement candidate is the sum of the score of all individual view:
\begin{equation}
\begin{aligned}
\centering
S_c = \Sigma_{i=1}^{n} S_{view \ i, candidate \ c},
\end{aligned}
\end{equation}
In this case, we get the score of each arrangement candidate considering \textbf{S2S} relations.
To increase the arrangement diversity, we group the candidates with same rectangle division but different view combinations as one and consider their average score as the final score.
Then, we select the $maximum(1\% \times \# \ divisions, 10)$ divisions with largest average score.
This means that we will take at least $10\times n!$ candidates.
Following the similar process, we further calculate the score for each filtered candidates considering all other relations.
Specifically, for the coordination, we also adopts the brute-force methods, {i.e.,}\xspace~trying every coordination type for each pair and finally selecting the one with highest score.
Finally, we recommend arrangement and coordination with the highest score for the given $n$ views.
We have tried another greedy algorithm, that is, every time we find a position for a view with the highest score.
Though efficiently, xxxx.
\yanna{need to re-consider the reason}
Thus, we chose the first choice.
\fi
\begin{figure*}
\centering
\includegraphics[width=1\linewidth]{figures/Result.pdf}
\caption{An overview of the study result.
Specifically, each column describes the score of a metric in three groups, {i.e.,}\xspace~the group considering all dashboards, the group with 3- and 4-view dashboards, and the group with 5- to 8-view dashboards.
And six metrics cover 3 perspectives, {i.e.,}\xspace~arrangement, coordination, and overall ({i.e.,}\xspace~both).
Each sub-plot presents the score of 5 recommendations regarding the corresponding metric and group.
Here, we show some results from 3 metric perspectives:
(1) \textbf{Arrangement}: For dashboards with 5-8 views, \textit{Human} performs significantly better than all others in three arrangement metrics, except that the \textit{Full} recommendation achieves similar results in terms of the arrangement logic and helpful to analyze.
(2) \textbf{Coordination}: \textit{Default} without coordination receives a significantly worse result than others. And our \textit{Full} and \textit{Partial} recommendations achieve significantly better performance than \textit{Human}, considering all dashboards and the dashboards with 3-4 views.
(3) \textbf{Overall}: \textit{Full} recommendation suppresses \textit{Human} regarding dashboards with 3-4 views, and achieves \textit{Human}-like performance considering all dashboards and dashboards with 5-8 views.}
\label{fig:result}
\vspace{-1em}
\end{figure*}
\subsection{User Study}
\label{sec: user study}
To verify the effectiveness of our recommender which is based on learned design patterns, we conducted a user study with novice dashboard users.
\paragraph{Participants.}
In this study, we recruited 12 participants (9 females and 3 males, ages 22 to 27, with an average of $24.8$).
They are all well educated (7 Ph.D. students, 3 MSc students, 1 research assistant, and 1 data analyst) and from diverse backgrounds, including visualization (4), finance (2), recommendation system (1), human-computer interaction (2) and data science (3).
All the participants are reported as experienced in the data analysis by giving a 7-point Likert scale ($\mu = 5.25, \sigma = 0.45$).
Each participant was compensated with \$10 after finishing the user study.
\paragraph{Dataset.}
In the user study, 15 representative dashboards with four recommended variants were presented to the participants for scoring.
Despite the dashboard created by \textit{Human}, the four recommended variants are:
\begin{itemize}
\item \textit{Default}: the default recommended dashboard by Tableau~\cite{tableau}. One point to mention is that the default dashboards do not have coordination since Tableau does not support the recommendation of coordination;
\item \textit{Basic}: the recommended dashboard using only \textbf{S2S} patterns;
\item \textit{Partial}: the recommended dashboard considering all patterns except those about interrelations between arrangement and coordination; and
\item \textit{Full}: the recommended dashboard leveraging all design patterns.
\end{itemize}
For simplicity, each original dashboard with four variants is called \textit{a group of dashboards}.
Therefore, we totally have 15 groups of dashboards for participants' scoring.
The dashboards cover almost all of the mark types (10/13), and are about various topics, like business and patient analysis.
What's more, the number of views ranges from 3 to 8, obeying the distribution in \autoref{fig:basic_info} \cc{A}.
\paragraph{Procedure.}~The user study lasted about 1 hour, and we asked for participants' consent for recording the whole user study.
Before the user study, we conducted a 5-minute introduction of our work.
In the user study, each participant was asked to score the arrangement and coordination of 5 groups of dashboards sampled from the 15 groups. As a result, each group of dashboards is evaluated by 4 participants.
Before evaluating each group of dashboards, we introduced each view that needs to be arranged.
When the participants felt familiar with the views, they freely explored the dashboards.
After the free exploration, they were asked to score the dashboards using the 7-point Likert scale.
They were asked to score arrangement (including logic, aesthetics, and the ability to help analyze data), coordination (including logic and the ability to help facilitate exploring data), and the overall score, which refers to the previous studies~\cite{smith2013data, wang2019datashot}. Specifically, the ``logic" measures how well the arrangement and coordination follow a human's ``logical" analysis~\cite{dowding2018development}.
All the participants were encouraged to think aloud about why they gave the corresponding score.
\subsubsection{Quantitative Analysis}
This section introduces and discusses the quantitative results of our user study.
To analyze the results, we further split all dashboards into two groups according to the number of views in dashboards, {i.e.,}\xspace~the dashboards containing 3-4 views and the dashboards containing 5-8 views.
The rationale for the split stems from the feedback from our participants.
They indicated that they care more about the arrangement and coordination when the number of views is more than 4 since a well-organized dashboard reduces their cognitive load of viewing many visualizations.
Figure~\ref{fig:result} presents the average ratings of recommended dashboards using six metrics.
To further analyze the results, we performed the one-way ANOVA (5 recommendations) on 6 metrics between the dashboards with different numbers of views. In the following, we will introduce the result regarding arrangement, coordination, and overall.
\begin{figure*}[!ht]
\centering
\includegraphics[width=1\linewidth]{figures/positive_example.pdf}
\caption{A case identified in our user study. \cc{A} and \cc{B} are the dashboards created by \textit{Human} and our \textit{Full} recommendation, respectively. Our recommended dashboard is highly rated due to outstanding logic and coordination (see \cc{A5} and \cc{B5}).
Participants recognized that the \cc{B1} is more suitable than \cc{A1} to act as a starting point to explore the dashboard.
And for the same view pair, our recommended coordination type is \textit{Brush}, which keeps the context for comparison.
However, \textit{Human} chooses \textit{Filter}, which leads to a change in the display and can confuse the users.}
\label{fig: positive_example}
\end{figure*}
\textbf{Arrangement}.
For all dashboards regardless of view numbers, no significant difference is observed among 5 recommendations for all three perspectives, {i.e.,}\xspace~logic, aesthetics, and ability to help analyze data.
While for dashboards with 3-4 views, we still cannot observe any significant differences among the 5 recommendations.
And we can notice that the \textit{Full} has a slightly higher score than \textit{Human}.
\autoref{fig: positive_example} provides one example, where \cc{A} is created by \textit{Human} and \cc{B} is our \textit{Full} recommendation.
Participants recognized that the position logic of \cc{B} is better than \cc{A}, since \cc{B1} is more suitable than \cc{A1} to act as a starting point to guide the data analysis.
And the symmetrical position of \cc{B2} and \cc{B4} are also plausible.
However, they thought our \cc{B1} (map) should be with a higher height, but still better than \cc{A1}, where the bar with a too larger height.
What's more, we can notice that the \textit{Default} has a higher score in terms of all three arrangement metrics.
This is probably because the \textit{Default} divides the largest view into two even views each time, resulting in a balanced aspect ratio and a neat arrangement.
The participants prefer such a balanced arrangement, especially when there are relatively less views and their interrelations are simple.
While for dashboards with 5-8 views, \textit{Human} recommendation significantly differs from all others.
The only exception is the \textit{Full} recommendation in terms of the logic and the ability to help analyze data.
Participants recognized that the position recommended by our \textit{Full} mode is logically plausible.
However, it sometimes has problematic sizes, {i.e.,}\xspace~too small or too large height or width to cause deformation and extrusion, such as \autoref{fig: negative_example} \cc{B2}.
Thus, it led to a slightly lower score on the logic but a great performance drop on the aesthetics.
Moreover, the problematic size does not greatly influence the ability to help understand since the position is more important.
What's more, both \textit{Full} and \textit{Partial} recommendations perform significantly better than \textit{Default} in terms of the logic and the ability to help analyze.
This may be attributed to that \textit{Default} arranges the views according to the alphabetical order by view name, leading to the worse logic confusing the participants.
Furthermore, though \textit{Default} still recommends the arrangement with the balanced aspect ratio, it suffers the same deformation and extrusion problems as our three recommendations.
P8 has mentioned that ``\textit{when there are too many views, there should be a dominant one telling me what it wants to express.
The exactly equal size confuses me.
And it is challenging for me to identify what I should pay attention to first}."
P9 had similar comments.
This may be the main reason why ours are slightly better in aesthetics, and significantly better in the logic and ability to help analyze than the \textit{Default}.
What's more, we further notice that, among our three recommendations, \textit{Full} achieves the best arrangement performance, followed by \textit{Partial} and \textit{Basic}.
\textit{Basic} is the worst among our three recommendations and only slightly better than the \textit{Default} for the dashboards with 5-8 views.
While \textit{Partial} is better than \textit{Basic}, it has a great difference from \textit{Default} in terms of the logic and the powers to help analyze, and it even achieves human-like logic.
And \textit{Full} is better than \textit{Partial}, and its arrangement achieves not only human-like logic but also human-like powers to help analyze.
\textbf{Coordination.}
Since \textit{Default} has no coordination, it has significantly worse results than all other recommendations in terms of both logic and the ability to facilitate exploration.
P1 mentioned that ``\textit{Without coordination, it was challenging for me to identify the related data cross views and focus the data of interest}".
P2 commented that ``\textit{The views without any interaction are really like a decoration, and it is strange to composite such views with others}."
What's more, \textit{Full} and \textit{Partial} recommendations possess significant differences to \textit{Human} among all dashboards and dashboards with 3-4 views.
This may be because we recommend either \textit{Brush} or \textit{Filter} for almost all view pairs.
As shown in \autoref{fig:basic_info} \cc{D}, our three recommendations recommend more coordination than \textit{Human}.
And participants prefer comprehensive coordination, since it gives a stronger power of exploration.
What's more, P5 and P11 appreciate our recommendation of the interaction type.
As shown in \autoref{fig:basic_info} \cc{D},
\textit{Human} recommendation prefers the \textit{Filter} rather than \textit{Brush}.
And participants thought our \textit{Brush} recommendation was more appropriate in some cases, since it does not change the context and enables comparisons.
For example, in \autoref{fig: positive_example}, \cc{A5} is the filtered result of \cc{A2}, while \cc{B5} is the brushed result of \cc{B4}, and \cc{B5} keeps the context well to compare.
When it comes to dashboards with 5-8 views, \textit{Full} and \textit{Partial} have no significant difference from \textit{Human}, while \textit{Human} gets a very high score.
This is because when more views are involved, participants are not just concerned with the completeness of coordination; they recognised that a clearer logic of coordination is more important.
P7 said, ``\textit{Too many coordination relationships will confuse me.
One view will lead to the change of lots of views, and I do not know where to look.}"
In this perspective, most of the \textit{Human} dashboards are more appropriate, while our \textit{Full} recommendation provides a more suitable coordination type.
For our three recommendations, \textit{Full} and \textit{Partial} achieve almost the same performance and are better than \textit{Basic}.
When it comes to the dashboards with 3-4 views, \textit{Basic} performs significantly better than \textit{Human} in terms of both the coordination logic and the power to facilitate exploration.
However, the coordination logic of \textit{Human} suppresses \textit{Basic} when there are at least 5 views, and they are almost the same for all dashboards regardless of the number of views.
\textbf{Overall.}
Though \textit{Default} achieves similar arrangement performance to other recommendations, it gets a significantly worse result due to the lack of coordination across 3 groups.
Participants all recognized the indispensability of the coordination.
For the all-view group, \textit{Full} has a slightly but not significantly better overall score than the \textit{Human},
though \textit{Full} has a significantly great coordination score to the \textit{Human}.
This is because the participants thought that the arrangement was slightly more important than the coordination for data analysis.
And for 3- and 4-view dashboards, \textit{Full} is significantly better than the \textit{Human}, due to a similar result in arrangement but better and more comprehensive coordination provided \textit{Full}.
While for 5- to 8-view group, the \textit{Full} still achieves human-like performance.
And our 3 recommendations have no significant difference from each other.
In summary, we earn nice feedback about our recommendations from participants.
Participants appreciated the automating results for both the arrangement and coordination.
And P11 expected that our recommendations can combine to her work, ``\textit{I really love this work, and I hope it can be integrated into my workflow, ..., I believe it will greatly facilitate my analysis process.}"
However, our approach occasionally provides some unreasonable recommendation.
\autoref{fig: negative_example} illustrates a case where \cc{B} recommended by our \textit{Full} mode got a low score in the user study.
And \cc{A} is created by human.
Specifically, the participants' confusion mainly came from three perspectives: 1) the \cc{B2} was inserted between two similar views; 2) the size of \cc{B2} is too small to show the details; 3) \cc{B1} about the ``winter" is on the top of \cc{B2} about the ``summer".
\begin{figure}[h]
\centering
\includegraphics[width=1\linewidth]{figures/negative_example.pdf}
\caption{A case identified in our user study.
\cc{A} is created by \textit{Human}, while \cc{B} is created by our \textit{Full} recommendation.
Compared to the original dashboard, the recommended one received a lower score and our participants pointed out that the problematic arrangement from 3 perspectives, {i.e.,}\xspace two similar views \cc{B1} and \cc{B3} are not positioned together, too small in size of \cc{B2}, and \cc{B1} about ``winter" is on top of the \cc{B3} about ``summer".}
\label{fig: negative_example}
\end{figure}
\subsubsection{Design Suggestions}
Combining the feedback from the participants and the findings from the unreasonable recommendations, we summarized the following design suggestions, which supplement our proposed method.
\textbf{Arranging views by proximity.}
All participants commented that views encoding similar information or the same chart types should be positioned closely with equal sizes, such as \cc{A1} and \cc{A2} in \autoref{fig: negative_example}.
This perspective confirms the proximity principle in visual designs that elements near each other are perceived as related.
We encourage future research to investigate the effects of view proximity on their layout arrangement through empirical experiments.
\textbf{Reducing view ``islands''.}
Most participants mentioned that it was less describable to have a view that was an ``island'' and disconnected with other views in terms of encoded attributes or coordination.
Instead, the dashboard should compose views as a cohesive whole.
Furthermore, P10 expected the dominant view to be coordinated with the remaining views.
This comment echoed that of P12, ``\textit{the top-left most view is an overview, but it is strange that there is no coordination. It should guide me to explore other views.}"
This comment aligns well with one of the patterns mentioned in \autoref{sec:case_study}.
\textbf{Adding cues or hints for coordination.}
Participants commented that it was difficult to find coordination among views.
They need to carefully explore each view and observe other views to identify whether there is coordination between them.
This process is tedious and poses high cognitive loads.
Thus, it is important to provide user-friendly hints on coordination among views.
\textbf{Considering the semantics.}
Participants deemed semantics as an important concern in arranging views.
For instance, P1 said ``\textit{the view about peak hour in summer month should on the top of that in winter}", (a counter example shown in \autoref{fig: negative_example} \cc{B1} and \cc{B3})
This requires understanding the semantic meaning of each view and their ``orders''.
However, visualization sequencing remains a challenging problem that warrants future research~\cite{lin2020dziban}.
\textbf{Paying attention to cross-view filtering.}
Participants noted that cross-view filters could result in a huge difference in the filtered view ({e.g.,}\xspace~ \cc{A1} leading to a large change from \cc{A2} to \cc{A5} in \autoref{fig: positive_example}).
Such changes catch attention but also bring about a cognitive load.
For example, P1 said, ``\textit{I got lost for seconds, wondering what makes the sudden change. It is a waste of time since I soon realize the reason, and it does not bring new data insights.}''
This problem suggests the need for careful consideration of cross-view filtering.
\section{Introduction}\label{sec:introduction}}
\maketitle
\begin{figure*}
\centering
\includegraphics[width=1\linewidth]{figures/teaser-tvcg.png}
\caption{The workflow of DMiner. This paper proposes DMiner~as a framework for dashboard design mining \begin{reviseyanna}
and automatic recommendation. With the dashboard dataset as the input, DMiner: \cc{A} first surveys a set of features important for dashboard design, and then extracts those features to delineate dashboard designs comprehensively. These are categorized into two types, {i.e.,}\xspace~single-view features such as data and encoding and pairwise-view features such as coordination and relative position; \cc{B} then mines design rules using decision rule approach, and further filters them; and \cc{C} finally leverages these rules for recommending dashboard arrangement and coordination.
\end{reviseyanna}
}
\label{fig:teaser}
\end{figure*}
\IEEEPARstart{M}{ultiple-view} (MV) visualizations composite multiple visualizations into a single cohesive representation.
Due to its power to support users in exploring several perspectives of data simultaneously, a large number of MV visualizations have been created and shared on the web by various domains, from biomolecular to multimedia to business.
Creating an MV visualization typically starts with selecting views of interest, followed by presenting the selected views and adding interactions between views \cite{wang2000guidelines}.
This paper refers to the presentation and interactions as layout arrangements and coordination, respectively.
Arranging and coordinating views are vital when explaining the widest range of usability problems in visualizations~\cite{camilla2010Evaluation}.
Specifically,
a proper arrangement can maximize the utility of the limited display space and improve the effectiveness and expressiveness of the information exchange, improving the usability of the system~\cite{sadana2016designing}.
Besides,
coordination among visualizations can make cross-view data relationships more apparent and reduce users' cognitive burden~\cite{chen2021nebula}.
However,
it remains challenging to create effective MV visualizations with a proper view arrangement and coordination.
From a theoretical perspective,
existing guidelines on MV visualization designs focus on high-level recommendations,
{e.g.,}\xspace~drawing users' attention to the right \yanna{view}
and making cross-view data relationships more obvious~\cite{wang2000guidelines}.
They are insufficient in providing lay users with actionable suggestions to adjust their designs.
From a practical perspective,
while researchers have recently developed some recommenders or authoring tools to assist lay users in creating visualizations and MV visualizations,
few of them have focused on the arrangement and coordination among views.
Existing tools ({e.g.,}\xspace~Tableau~\cite{tableau}, \yanna{Power BI}~\cite{powerbi}, and MultiVision~\cite{wu2021multivision}) provide default layout templates that require manual adjustment to achieve satisfactory MV visualization designs.
This process is tedious and time-consuming,
given that the potential layouts increase exponentially with the increasing number of views.
\yanna{
We present DMiner, a data-driven framework for mining dashboard design and automating the layout arrangement and view coordination for MV dashboards (\autoref{fig:teaser}), thus reducing the design burden of designers.
Specifically, MV dashboards are one of the most common genres of MV visualizations \cite{sarikaya2018we}.
In this paper, we use the term MV dashboards and dashboards interchangeably to represent multiple-view dashboards.
Given that end-to-end ML-based visualization recommenders suffer from poor explainability and can confuse
end-users~\cite{saket2018beyond}, we aim to develop an explainable approach by first mining design rules from an MV dashboard dataset and further recommend appropriate layout arrangement and view coordination in MV dashboards}.
Due to the lack of MV dashboard datasets, we first crawled a large number of dashboards created by Tableau, a common dashboard authoring tool, from GitHub~\cite{github}.
We deduce a set of features that influence the arrangement and coordination of these views through reviewing prior studies, and further identify the mappings among them (\autoref{fig:teaser}~\cc{A}).
With the collected dataset, we then extract features from two perspectives: 1) the single-view features
describing each view in terms of its visual encodings ({e.g.,}\xspace color), encoded data ({e.g.,}\xspace data types), and its layout ({e.g.,}\xspace position); and 2) the pairwise-view features describing pairwise relationships between views, including the data relationship ({e.g.,}\xspace~overlapping data columns), encoding relationships ({e.g.,}\xspace~same encoding), spatial relationships ({e.g.,}\xspace~relative angle), and coordination ({e.g.,}\xspace~filter).
Building upon those features, we mine the mappings using the decision rule and filter them to distill the design rules of dashboards (\autoref{fig:teaser} \cc{B}).
Specifically, based on the data and encodings of every single view and the data relationship and encoding relationship between views, the rules infer the layout of each view and the spatial and coordination relationship between view pairs.
For instance,
one design rule in \autoref{fig:teaser}~\cc{B} shows that a text visualization (``A.Mark = Text'') without data encoding on the Y-axis (``A.\#Y $<$ 1'') tends to place at the top (``A.Y = 1'').
Using the extracted rules, we develop a recommender that recommends the optimal arrangement and coordination by ranking the obedience of design rules (\autoref{fig:teaser}~\cc{C}).
\yanna{We demonstrate the usefulness and effectiveness of DMiner~through an expert study and a comparative user study using the collected Tableau dashboard dataset.
We invite four experts to rate the appropriateness of the 10\% of extracted rules, and showcase those rules with higher expert scores.
Also, we compare our recommender with the existing tool, \yanna{the crawled dashboard}, and experienced designers.}
We believe that our work can be a starting point for automating the layout arrangement and view coordination in MV dashboards.
Our main contributions are as follows:
\begin{compactitem}
\item DMiner, a novel data-driven framework for dashboard design mining and recommendation.
\item \yanna{An expert study to demonstrate the appropriateness of extracted rules and a comparative study to show the effectiveness of the recommender.}
\item A dataset of 854 real-world dashboards with detailed information such as metadata, encodings, and coordination, which can benefit future research on dashboards.
\end{compactitem}
\section{Dashboard Dataset}
\label{data}
Our study aims to mine design rules from existing dashboards
and further automatically recommend optimal MV dashboards.
However, there is a lack of off-the-shelf dashboard datasets that can enable automated MV dashboard design.
Specifically, existing datasets of MV dashboards~\cite{chen2020composition, shao2021modeling, al2019towards} contain only the information of individual views, such as position, size, and chart type.
They do not provide the metadata ({e.g.,}\xspace~the data operations) and the interactions between views, which are necessary information for developing visualization and dashboard recommenders~\cite{hu2019vizml,wu2021multivision}.
To address this problem, we build a new dataset from the Internet.
The detailed procedures for constructing the dashboard datasets will be introduced as follows:
\textbf{Data Crawling.}
We first crawl Tableau dashboards from GitHub,
{i.e.,}\xspace~searching and downloading Tableau workbooks with \textit{.twbx} extensions.
There are multiple advantages to using Tableau dashboards.
First, Tableau workbooks contain all the necessary information to render the corresponding dashboards ({e.g.,}\xspace~the underlying metadata, visual encodings, and coordination among views).
Second, the information is stored in a structured XML format, which can be processed automatically at scale and can avoid the heavy manual labelling procedures required in previous studies~\cite{chen2020composition, shao2021modeling, al2019towards}.
Finally, Tableau is one of the most popular tools for creating dashboards, and its workbooks are widely shared on GitHub.
Thus, the collected dataset is diverse regarding its creators, design themes and styles, offering opportunities to mine common design rules.
\textbf{Automatic Data Processing and Cleaning.}
The datasets crawled from GitHub
suffer from noise, affecting our analysis results.
To improve the dataset quality, we first perform data cleaning by parsing and filtering the raw workbooks (.twbx).
As shown in \autoref{fig:data_info}, we parse the workbooks to derive information including the metadata ({e.g.,}\xspace~data types and data operations), encodings for each view, the coordination among views, and the arrangement of each view in a dashboard.
Subsequently, we remove workbooks without dashboards ({e.g.,}\xspace~the workbooks containing only individual views).
Dashboards using multiple data sources are also abandoned since the lack of relational schema between data sources hinders data processing.
Finally, we discard dashboards without any view coordination ({e.g.,}\xspace~cross-view interactions), since we aim to mine and recommend view coordination.
According to Tableau, there are two coordination types, namely filtering and brushing.
Specifically, \textit{filtering} removes irrelevant data objects while \textit{brushing} highlights the related visual elements and keeps the context~\cite{bartram2002filtering}.
\textbf{Manual Data Processing and Cleaning.}
Tableau workbooks offer a feature that automatically decides the mark type.
Therefore, about half of the visualizations have an ``automatic'' mark type.
\yanna{
To address this problem, we manually label the mark types according to both the definition of marks in Tableau and Vega-Lite~\cite{satyanarayan2016vega}, a widely-used grammar of visualizations.
}
A detailed list of the mark types and corresponding examples is available in {the supplementary material}\footnote{The supplementary and dataset are available at \url{https://github.com/Yannahhh/DMiner}}.
To ensure the quality of labels, each view is labeled by two co-authors.
\yanna{The co-authors first finish the labelling individually, resulting in 99\% agreement.
Conflicts are then discussed until reaching a consensus.}
\yanna{We remove dashboards containing views with multiple mark types, {e.g.,}\xspace~the network that includes the \textit{circle} and \textit{line}.}
\textbf{Results.}
We finally collected 854 dashboards with 2990 views in total for further exploration, \yanna{and more details can be found in the supplementary material}.
Specifically, \autoref{fig:basic_info} shows the basic information of the dashboards, where \cc{A}, \cc{B}, and \cc{C} represent the distribution of the number of views of each dashboard, the mark type of each view, and the coordination among view pairs, respectively.
\autoref{fig:basic_info} \cc{A} and \cc{B} show that the distribution of the number of views aligns well with the observations from previous studies~\cite{al2019towards, chen2020composition}.
For exmaple, 2- and 3-view dashboards are the most common, and \textit{bar} is the most popular mark type.
\yanna{\autoref{fig:basic_info} \cc{C} shows that half of the view pairs possess coordination, which is almost \textit{filtering}.}
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Diede importanti contributi alla cronologia.
Biografia
Anniano fu il primo computista a riconoscere il ciclo di 532 anni, con cui ricorre la Pasqua nel calendario giuliano. Questo ciclo è spesso attribuito a Vittorio d'Aquitania, il primo in Occidente a riconoscere tale ciclo nel 457. Il ciclo nasce dalla necessità di combinare il ciclo solare di 28 giorni, che determina il giorno della settimana in cui cade il Capodanno dell'anno solare, con il ciclo metonico di 19 anni, che determina la data del Capodanno lunare. Prima di Anniano in oriente e di Vittorio in occidente gli astronomi ecclesiastici avevano proposto cicli più brevi, ma imprecisi, che avevano determinato controversie sulla data della pasqua fra la chiesa di Alessandria e quella di Roma.
Anniano criticò la storia universale del suo confratello contemporaneo Panodoro di Alessandria per aver fatto troppo affidamento su fonti profane piuttosto che sulle fonti bibliche e, quindi, sviluppò la propria cronologia, secondo la quale la data della creazione è il 25 marzo 5492 a.C. Dato che il calendario alessandrino pone il Capodanno il 29 agosto, lepoca da lui proposta, cioè il primo giorno del calendario prolettico civile alessandrino, venne a cadere sette mesi prima, cioè il 29 agosto 5493 a.C. Quest'anno era esattamente undici cicli pasquali, di 532 anni ciascuno, prima dell'anno alessandrino che iniziava il 29 agosto 360, che a sua volta era di quattro cicli di 19 anni successivo al 29 agosto 284, l'epoca dellera di Diocleziano (detta anche "era dei martiri"). Il 29 agosto 360, invece è l'epoca dell'era nota come era di grazia nella chiesa copta.
Nessuna delle opere di Anniano è sopravvissuta. È principalmente noto dalla discussione delle sue opere da parte di Giorgio Sincello durante il IX secolo, sebbene frammenti minori appaiano altrove. Elia di Nisibi lo cita nella sua cronografia dell'XI secolo.
Tuttavia, la tabella pasquale di Anniano di 532 anni, contenente un ciclo pasquale di 532 anni basato su un ciclo lunare metonico di 19 anni, è sopravvissuta. Il suo ciclo lunare metonico di 19 anni fu adottato dal vescovo Cirillo di Alessandria, che utilizzó nella sua tabella pasquale di 114 anni (in lingua greca e con date del calendario alessandrino). Poco prima della morte di Cirillo (AD 444) fu dato principio a una tabella pasquale nella lingua latina e con date del calendario giuliano, probabilmente per uso nella parte latina dell'Europa; ne risultò la tabella pasquale di 95 anni, nota come 'la tabella pasquale attribuita a Cirillo', che copriva l'intervallo di tempo AD 437-531. Un secolo più tardi (AD 525) questa tabella pasquale fu estesa da Dionigi il Piccolo coprendo l'intervallo di tempo AD 532-626. Solo due secoli dopo Beda il Venerabile la ampliò redigendo una tabella per un intero ciclo di 532 anni dal 532 al 1063. Concludiamo che è precisamente la variante del ciclo lunare metonico di 19 anni (inventato da Anatolio) scelta da Anniano che è stata il nocciolo del Calcolo della Pasqua nel mondo cristiano latino dal sesto al sedicesimo secolo.
Quanto a Vittorio d'Aquitania, Jan Zuidhoek, fingendo di menzionare esplicitamente tutti i cicli lunari metonici di 19 anni rilevanti, ha perso un'opportunità per menzionare quello di Vittorio Tuttavia, Alden Mosshammer lo ha menzionato esplicitamente. In linea di principio, ogni data del ciclo pasquale di 532 anni della tabella pasquale di Vittorio può essere calcolato applicando la vecchia regola romana "Domenica pasquale è la prima domenica dopo il primo giorno dopo la luna piena pasquale" alla data della luna piena pasquale corrispondente del suo ciclo lunare (su richiesta con l'aiuto del concorrente essendo il numero che indica il giorno feriale di 1 gennaio).
Note
Bibliografia
William Adler. Time immemorial: archaic history and its sources in Christian chronography from Julius Africanus to George Syncellus . Washington, DC: Dumbarton Oaks Research Library and Collection, 1989. (Tempo immemorabile: la storia arcaica e le sue fonti nella cronografia cristiana da Giulio Africano a Giorgio Sincello)
William Adler, Paul Tuffin, The chronography of George Synkellos: a Byzantine chronicle of universal history from the creation . Oxford: Oxford University Press, 2002. (L'opera di Giorgio Sincello, tradotta in questa opera, contiene grandi blocchi di testo scritti da Anniano).
Georges Declercq (2000) Anno Domini (The Origins of the Christian Era): Turnhout (ISBN 9782503510507)
Alden A. Mosshammer (2008) The Easter Computus and the Origins of the Christian Era: Oxford (ISBN 9780199543120)
Otto Neugebauer (1979) Ethiopic Astronomy and Computus: Wien (ISBN 9783700102892)
Jan Zuidhoek (2019) Reconstructing Metonic 19-year Lunar Cycles (on the basis of NASA's Six Millennium Catalog of Phases of the Moon): Zwolle (ISBN 9789090324678)
Voci correlate
Calendario alessandrino
Collegamenti esterni
Cinque cicli lunari metonici di 19 anni
Monaci cristiani egiziani
Storici bizantini
Teologi bizantini
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\section{Introduction} \label{sec:intro}
Low-mass pre-main sequence stars, or T Tauri stars (TTS), are formed surrounded by disks and evolve accreting mass from these disks.
The accretion of the material from the inner region of protoplanetary disk onto the star follows the magnetospheric accretion paradigm \citep{hartmann2016}.
Under this framework, gaseous material, heated by the stellar radiation field and other processes, flows along the magnetic field lines onto the star, creating an accretion shock at the base of the flow.
The emission from the accretion shock is observed as an excess over the stellar photosphere \citep{calvet1998}.
Emission lines form in the magnetospheric accretion flows, so that the kinematics of the flow can be inferred from the line profiles.
Numerical simulations of magnetized stars \citep[e.g.][]{romanova2003} as well as magnetospheric accretion models \citep{hartmann1994,muzerolle1998a,muzerolle2001,kurosawa2011} have confirmed this picture for accreting T Tauri stars (Classical T Tauri star; CTTS) and have provided insight into the physical properties of accretion, including the geometry of the flows.
One of the most important properties of accretion is the mass accretion rate {$\dot{\text{M}}$}. This can be estimated by either directly measuring the excess over the photosphere and inferring the accretion luminosity,
L$_{\rm acc} = {\rm GM}_{\star}\dot{\rm M}/{\rm R}_{\star}$,
or by using emission lines.
Specifically, the excess flux over the photosphere can be extracted from the optical flux by measuring the veiling of photospheric absorption lines and measured directly in the UV \citep[e.g.][]{ingleby2013}.
The accretion luminosity can then be measured using accretion shock models \citep{calvet1998,robinson2019},
or slab models \citep[e.g.][]{gullbring1998,herczeg2008,manara2016,alcala2017},
to fit the excess and account for the flux outside the wavelength regions where the excess is observed.
Emission lines have been used to infer the mass accretion rate via empirical relationships between the line luminosity or line width
and the accretion luminosity \citep{muzerolle1998b,natta2004,calvet2004,ingleby2013,alcala2014}.
These relationships have been calibrated with direct measurements of accretion luminosity from the excess over the photosphere, obtained simultaneously in many cases.
A more direct method to measure accretion rates from emission lines is by modeling the line profiles with magnetospheric accretion models.
This has been done for hydrogen Balmer lines \citep{muzerolle2001,natta2004,espaillat2008}, Na~D line \citep{muzerolle2001}, and {{\ion{He}{1}}$~\lambda$10830} line \citep{fischer2008,kurosawa2011}. Modeling emission lines also provides information about accretion geometry.
Finding the method that best estimates the mass accretion rates is especially relevant for the stars with the lowest accretion rates, the {\it low accretors}. Population studies show that
the
mass accretion rate scales with stellar mass \citep[e.g.][]{muzerolle2003a,herczeg2008,alcala2014,manara2015}, suggesting that a mass range is implied when a star is designated as a low accretor. For example, the current detection limit of mass accretion for solar-mass star is $\sim10^{-10}\,\rm{M_{\sun} \, yr^{-1}}$ \citep{ingleby2013,manara2013}, whereas this accretion rate is not unusual for mid-M type stars \citep[e.g.][]{alcala2017}.
As expected from viscous evolution of protoplanetary disks, the mass accretion rate onto the star decreases with time \citep{hartmann1998}.
Studies of many star-forming regions also show that the frequency of accretors, as well as the frequency of disk-bearing T Tauri stars, in a given population decreases as the age of the population increases \citep{fedele2010,hernandez2008,briceno2019}.
However, it is unclear how accretion proceeds at very low accretion rates and how it finally stops.
To understand processes occurring at the last stages of accretion it is necessary to carry out systematic studies of T Tauri stars accreting at very low accretion rates.
This is the main motivation for our ongoing observational and modeling program to search and characterize low accretors.
As the initial result of our study of low accretors, we presented the characterization of the inner disk of three 5\,Myr T Tauri stars in \citet{thanathibodee2018}.
One target, CVSO~1335, was particularly interesting. The star is a pre-main sequence solar analog with M$_{\star}=0.87\,\rm M_{\sun}$, R$_{\star}=1.58\,\rm R_{\sun}$, and spectral type of K5.
The star, located in the 5 Myr old Ori~OB1b subassociation \citep{briceno2019}, is a CTTS, based on the presence of redshifted absorption in the {{\ion{He}{1}}$~\lambda$10830} line.
\citet{thanathibodee2018} showed that the protoplanetary disk surrounding the star is gas-rich, as indicated by fuorescent FUV H$_2$ emission, while the spectral energy distribution indicates that it is a transitional disk with a gap depleted of small dust.
That work also showed that the star had complex {H$\alpha$} line profiles in several epochs, which complicated the measurement of its mass accretion rate.
The accretion rates determined with different indicators differed by more than 3 orders of magnitude.
The lower range of the measured mass accretion rate at $\sim10^{-10}\,\rm{M_{\sun} \, yr^{-1}}$ would suggest that the star is a low accretor for its mass.
The disagreement between the accretion diagnostics and the complex features in {H$\alpha$}, especially the persistent low-velocity redshifted absorption which is found in AA~Tau-like stars \citep[e.g.][]{bouvier2007,fonseca2014} and other dippers \citep[e.g.][]{alencar2018}, makes this star an ideal target for a detailed study of accretion properties in low accretors.
Here we report the observations and the characterization of the accretion rate and the accretion geometry in the low accretor CVSO~1335. Optical spectra of the star are presented in \S\ref{sec:observation}, with the analysis and modeling in \S\ref{sec:result}. The implication of the model results are presented in \S\ref{sec:discussion} with a summary in \S\ref{sec:summary}.
\section{Observations} \label{sec:observation}
The analysis of the {H$\alpha$} line in \citet{thanathibodee2018}
was based on observations with the moderate-resolution (R$\sim$4100, 73\,$\rm \, km \, s^{-1}$) MagE Spectrograph on the 6.4m Magellan Baade telescope at the Las Campanas Observatory in Chile, as well as the high-resolution (R$\sim$14000, 22\,$\rm \, km \, s^{-1}$) Goodman Spectrograph on the CTIO SOAR Telescope.
The MagE spectrograph covers the full optical range from 3200-8200{\AA}, giving access to the Balmer jump.
We report here additional spectra observed on 2017 November 27 and 28, which we reduced similarly to the MagE spectra reported in \citet{thanathibodee2018}.
In total, we have four MagE spectra observed in consecutive nights, and one Goodman spectra observed about two months earlier.
Table \ref{tab:obs} shows the details of all observations of CVSO~1335.
\begin{deluxetable}{llccc}[t]
\tablecaption{Summary of Observations \label{tab:obs}}
\tablehead{
\colhead{Instrument} &
\colhead{Start Date} &
\colhead{Exp. time} &
\colhead{Airmass} &
\colhead{SNR\tablenotemark{a}} \\
\colhead{} &
\colhead{(UT)} &
\colhead{(sec)} &
\colhead{} &
\colhead{}
}
\startdata
Goodman & 2017 Sep 18\tablenotemark{b} & $3 \times 600$ & 1.22 & 35 \\
MagE & 2017 Nov 27 & $3 \times 600$ & 1.75 & 290 \\
& 2017 Nov 28 & $3 \times 600$ & 1.14 & 260 \\
& 2017 Nov 29\tablenotemark{b} & $2 \times 900$ & 1.14 & 280 \\
& 2017 Nov 30\tablenotemark{b}& 600+900 & 1.14 & 260 \\
\enddata
\tablenotetext{a}{Signal-to-noise at 6560\,{\AA}.}
\tablenotetext{b}{Reported in \citet{thanathibodee2018}.}
\end{deluxetable}
The top panel of Figure~\ref{fig:observation} shows the optical spectra of CVSO~1335 observed with the MagE spectrograph.
Several emission lines are presented in the spectra, including the hydrogen Balmer lines and the Ca H \& K doublet.
We do not detect the \ion{O}{1}\,$\lambda6300$ in the MagE spectra, suggesting that the stellar or disk wind is very weak in the system, and the contrast between the line and the continuum could be low for a K5 star.
The lower panel in Figure~\ref{fig:observation} shows the corresponding {H$\alpha$} profiles from the MagE and Goodman spectra,
as well as other emission lines. To remove the photospheric contribution from the \ion{O}{1} lines and the chromospheric and photospheric contributions from the {\ion{H}{1}} lines, which are generally found in active young stars \citep{manara2013,manara2017}, we subtracted the line profile of RECX~1, a standard non-accretor (Weak T Tauri star; WTTS), from the CVSO~1335 profiles. The standard star has similar stellar parameters as those of CVSO~1335 (SpT=K5-6, M$_{\star}=0.9\,\rm M_{\sun}$, R$_{\star}=1.8\,\rm R_{\sun}$, age$\sim$5-9\,Myr), so its spectrum can be used to represent the underlining photospheric and chromospheric emission of the target \citep{ingleby2011b}.
The spectrum of RECX~1 was obtained by the UVES spectrograph and was retrieved from the ESO archive. We convolved the UVES spectrum down to the resolution of the MagE and Goodman spectrographs before subtraction.
\begin{figure*}[t]
\epsscale{1.18}
\plotone{f1.pdf}
\caption{\emph{Top:}
Optical spectra of CVSO~1335 observed with the MagE spectrograph (R$\sim$4100).
The spectra are normalized to unity at $\sim$6000\AA.
\emph{Bottom:} Profiles of the {H$\alpha$}, {H$\beta$}, \ion{O}{1} lines, and the \ion{Ca}{2}~HK doublet. We also include a high resolution {H$\alpha$} profile observed by the Goodman spectrograph (R$\sim$14000). For the hydrogen lines, the chromospheric contribution have been subtracted by the convolved spectrum of RECX~1, a non-accretor observed by UVES. The hydrogen Balmer lines shows complex and variable features, in particular
a low-velocity redshifted absorption. This absorption is especially strong for the {H$\beta$} line observed on the last 3 epochs. The \ion{O}{1}\,$\lambda6300$ lines show no detection. The \ion{Ca}{2} lines are symmetric, showing minimal variation
in epochs separated by 1 day, and increases in strength on 20171130. The emission to the right of the \ion{Ca}{2}~H line is from H$\epsilon$. The \ion{Ca}{2} lines of RECX~1 are comparable to those of CVSO~1335 observed on 20171130.}
\label{fig:observation}
\end{figure*}
\section{Analysis and Results} \label{sec:result}
Using the data in \S \ref{sec:observation}, we carry out detailed modeling to determine mass accretion rates in CVSO~1335 at the different epochs of observations, and estimate its accretion geometry.
\subsection{Accretion Shock Model} \label{ssec:shock}
We use accretion shock models from \citet{calvet1998} to measure the mass accretion rate. This model is based on an assumption that the accretion flow is a cylindrical column in which the material flows vertically onto the stellar surface. At the height where the ram pressure of the flow and the thermodynamic pressure of the stellar photosphere are equal, an accretion shock occurs in which the kinetic energy of the flow is released. Approximately half of the X-rays emitted from the shock are absorbed by the material in the incoming flow, the \emph{pre-shock}, and the rest by the \emph{post-shock} region and the stellar photosphere below the shock. Energy reprocessed by these regions emerges as the shock emission. The input parameters of the model are the stellar mass, radius, and effective temperature, the energy flux of the accretion column, and the filling factor \citep{calvet1998}. In general, the energy flux $F$ determines the spectral slope around the Balmer jump, while the filling factor $f$, the fraction of the stellar surface area covered in the accretion flow, influences the strength of the Balmer jump.
To compare the predictions of the model with the observations, we need to add the shock emission to a photospheric+chromospheric template for the star's spectral type. Since TTS are magnetically active, their chromospheric emission may be significant in the UV region for the low accretors \citep{ingleby2013}. We use the spectrum of the non-accreting T Tauri star RECX~1 \citep{ingleby2013} as the template for the stellar photosphere+chromosphere. We used the low-resolution X-shooter spectra of the star, taken from the ESO Archive, for this analysis. Finally, the model spectra are generated by adding the spectral template, the pre-shock, the post-shock, and the heated photosphere.
We created a grid of models varying the energy flux $F$ between $1.0\times10^{10} - 9.0\times10^{11}$\,ergs\,cm$^{-2}$\,s$^{-1}$ and the filling factor $f$ between 0.05 and 2.05\% to find the best fit to the observed spectra. Since the seeing during the observation was higher than the size of the slit, we expected some uncertainty in the absolute flux level. Without simultaneous photometric measurement during the MagE observation, we assumed that the optical spectra were approximately constant as we did not observe an evidence of optical veiling nor did we expect any such veiling in such a low accretor. Therefore, we adjusted the flux level of all spectra to correspond to the star's SDSS~r' magnitude. With this uncertainty, we assumed a conservative estimate of the flux calibration in the UV part at the 10\% level.
To determine if there is a detection of the Balmer jump due to the accretion shock, in addition to that from the chromosphere, we calculated the integrated flux in the spectral range $3300-5000$\,{\AA} of the observed spectra and compared the measurements with the integrated fluxes of spectra in our grid of shock models.
We found that the predicted fluxes were higher than the observed fluxes in all models,
but the excess was within the 10\% uncertainty of the flux measurement for a set of models with low accretion rates. Nevertheless, these results suggest that there is no detection of the accretion shock contribution to the Balmer jump in all of our observations and that the jump is mainly chromospheric as is the case in the template WTTS. Therefore, we estimate an upper limit of the mass accretion rate by adopting the highest accretion rate among the models with predicted flux within 10\% of the observed one. Figure~\ref{fig:shock} shows the models with the highest accretion rates of which the fluxes are still consistent with the measurement.
Table \ref{tab:model_shock} shows the upper limits to the mass accretion rates for each epoch. These limits allow for the possibility of accretion variability on a daily timescale.
\begin{figure*}[ht!]
\epsscale{1.18}
\plotone{f2.pdf}
\caption{The results from the accretion shock model of four MagE spectra of CVSO~1335.
The observed spectra are normalized to the star's SDSS i' magnitude.
No significant excess over the non-accretor standard is detected in all of the spectra. Upper limits are calculated assuming 10\% uncertainty in the flux and photometric calibration. For comparison, the orange line shows the model of the star at the mass accretion rate of $1.0\times10^{-8}\,\rm{M_{\sun} \, yr^{-1}}$, a typical accretion rate for T Tauri stars.}
\label{fig:shock}
\end{figure*}
\begin{deluxetable}{cccc}[t]
\tablecaption{Results of Accretion Shock Models \label{tab:model_shock}}
\tablehead{
\colhead{Obs. Date} & \colhead{$\dot{\text{M}}$$_{\rm shock}$} & \colhead{log(L$_{\rm acc}$/L$_{\odot}$)} & \colhead{log(L$_{\rm acc}$/L$_{\star}$)} \\
\colhead{UT} & \colhead{($10^{-10}\,\rm{M_{\sun} \, yr^{-1}}$)} & \colhead{} & \colhead{}
}
\startdata
20171127 & $\lesssim9.39$ & $\lesssim-1.37$ & $\lesssim-1.19$ \\
20171128 & $\lesssim4.93$ & $\lesssim-1.65$ & $\lesssim-1.47$ \\
20171129 & $\lesssim7.04$ & $\lesssim-1.49$ & $\lesssim-1.31$ \\
20171130 & $\lesssim7.27$ & $\lesssim-1.48$ & $\lesssim-1.30$ \\
\enddata
\end{deluxetable}
\subsection{Magnetospheric Accretion Model} \label{ssec:acc_model}
To estimate the mass accretion rates and the geometry and properties of the accretion flows we modeled the profiles of {H$\alpha$} with magnetospheric accretion models from \citet{muzerolle2001}. The model is described in \cite{hartmann1994,muzerolle1998a,muzerolle2001}, and here we summarize the main assumptions. The model assumes that the geometry of the magnetospheric accretion flows is axisymmetric, following the dipole magnetic field, with the magnetic pole aligned to the stellar and Keplerian disk rotation poles, assumed to be the same. Material flows toward the star inside the boundary controlled by the magnetic dipole geometry and specified by the innermost radius of the disk $R_{i}$ and the width at the base of the flow $W$. The mass flow is steady and set by the total mass accretion rate {$\dot{\text{M}}$}. The temperature at each point in the flow is a free parameter, and each model is specified by the maximum temperature in the flow $T_{\rm max}$. The models use the extended Sobolev approximation to calculate mean intensities, which in turn are used to calculate radiative rates in the statistical equilibrium for the level populations of a 16-level hydrogen atom \citep{muzerolle2001}. The line flux is determined by using a ray-by-ray method, in which the specific intensity and the total optical depth at each ray are calculated at a given inclination $i$. The final {H$\alpha$} line profile is calculated from the spatially integrated specific intensity.
We calculated a large grid of models with parameters covering ranges shown in Table~\ref{tab:model_param}. The ranges of parameters are chosen to cover all possible values based on previous parametric exploration of \citet{muzerolle2001}. In particular, they found that as {$\dot{\text{M}}$} decreases, {$T_{\rm max}$} needed to be increased in order to reproduce the observation; {$T_{\rm max}$} between 10k and 12k is required for {$\dot{\text{M}}$ = 10$^{-9}\,\rm{M_{\sun} \, yr^{-1}}$}. We adopted this range for our modeling since the expected mass accretion rate is lower than this value, based on the accretion shock model.
To compare the line profile results with the observations, we convolved the model profiles with the instrumental profile of the spectrograph and use the $\chi^2$ statistics to determine the best fit for each observation.
We show examples of fitting line profiles using the standard model in Figure~\ref{fig:one_layer}. First, we attempted to fit the {H$\alpha$} profile for night 20171130 using the entire velocity range of the line. The best fit, shown on the left panel, could not reproduce the the multiple emission and absorption components in the line profiles. Therefore, we attempted to fit each component in the line profile separately.
To fit the low-velocity redshifted absorption component of the line, we selected the model that gave the best fit to observations in the velocity range of 0-150 $\rm \, km \, s^{-1}$. Qualitatively, we found that models with low mass accretion rates, large magnetospheres, high inclinations, and high temperatures could reproduce the narrow absorption component at $\sim 75 \,\rm \, km \, s^{-1}$. We then created a grid of models with parameters suitable for exploring these ranges of parameter space. The ranges and the value of the parameters are shown in Table~\ref{tab:model_param}, and an example of the best fit model for these parameters are shown in the center panel of Figure~\ref{fig:one_layer}.
For the rest of the line profile with the standard model, excluding the 0-150 $\rm \, km \, s^{-1}$ region, we found that the best fit models tended to favor higher accretion rates, small magnetospheres, high inclinations, and high temperatures. We therefore created another set of models to explore these parameters, shown in Table~\ref{tab:model_param}. By excluding the low-velocity redshifted absorption from the fit, the model could fit the wings of the line, including the redshifted absorption component near the free-fall velocity ($\sim$ 200\,$\rm \, km \, s^{-1}$). Shown in the right panel of Figure~\ref{fig:one_layer} is an example of the best fit for the small magnetosphere.
The general trend from these fits is that the components of the {H$\alpha$} line profile could be reproduced using low mass accretion rate, high temperature, and high inclination.
\begin{deluxetable}{lccc}[h!]
\tablecaption{Range of Model Parameters \label{tab:model_param}}
\tablehead{
\colhead{Parameters} & \colhead{Min.} & \colhead{Max.} & \colhead{Step}
}
\startdata
\multicolumn{4}{l}{\emph{Standard Model}} \\
{$\dot{\text{M}}$} ($10^{-9}\,\rm{M_{\sun} \, yr^{-1}}$) & 0.1 & 9.0 & 0.1, 1.0 \\
$R_{\rm i}$ (R$_{\star}$) & 2.2 & 5.4 & 0.4 \\
$W$ (R$_{\star}$) & 0.8 & 2.0 & 0.4 \\
$T_{\rm max}$ (K) & 10000 & 12000 & 250 \\
$i$ (deg) & 10 & 85 & 5 \\ \hline
\multicolumn{4}{l}{\emph{Small Magnetosphere}} \\
{$\dot{\text{M}}$$_{\rm in}$} ($10^{-10}\,\rm{M_{\sun} \, yr^{-1}}$) & 1.0 & 9.5 & 0.5 \\
$R_{\rm i, in}$ (R$_{\star}$) & 1.4 & 3.8 & 0.4 \\
$W_{\rm in}$ (R$_{\star}$) & 0.2 & 0.6 & 0.2 \\
$T_{\rm max, in}$ (K) & 11000 & 12000 & 200 \\
$i_{\rm in}$ (deg) & 55 & 85 & 5 \\ \hline
\multicolumn{4}{l}{\emph{Large Magnetosphere}} \\
{$\dot{\text{M}}$$_{\rm out}$} ($10^{-10}\,\rm{M_{\sun} \, yr^{-1}}$) & 1.0 & 9.5 & 0.5 \\
$R_{\rm i, out}$ (R$_{\star}$) & 5.5 & 8.0 & 0.5 \\
$W_{\rm out}$ (R$_{\star}$) & \multicolumn{3}{c}{0.3, 0.5, 1.0, 1.5} \\
$T_{\rm max, out}$ (K) & 11000 & 12000 & 200 \\
$i_{\rm out}$ (deg) & 65 & 85 & 5 \\
\enddata
\end{deluxetable}
\begin{figure*}[t]
\epsscale{1.1}
\plotone{f3.pdf}
\caption{Best fits for the {H$\alpha$} line using the standard magnetospheric accretion model. The convolved model line profiles are shown in black, while the observations, taken from 20171130, are shown in red.
\emph{Left:} The best fit for the entire velocity range range. The model could not reproduce the observation.
\emph{Center:} The best fit model for the low-velocity redshifted absorption using a large magnetosphere and a low accretion rate.
\emph{Right:} The best fit model for the entire line profile, excluding the low-velocity redshifted absorption. The profile can be reproduced with a small magnetosphere and a slight increase in the mass accretion rate.}
\label{fig:one_layer}
\end{figure*}
\subsection{Modified Magnetospheric Accretion Model} \label{ssec:mod_acc_model}
\subsubsection{Computations with the new geometry} \label{sssec:new_geometry}
To fit the entire line profile, we modified the magnetospheric accretion model to include two magnetospheric flows in concentric shells; the \emph{inner flow} resembles a small magnetosphere, and the \emph{outer flow} corresponds to a larger magnetosphere covering the entire accretion structure. A schematic drawing of the new geometry is shown in Figure~\ref{fig:schematic}. We also slightly modify the calculation of the line profile. For a given inclination, the standard model calculates the emerging specific intensity $I_{\nu, p, q}$ and the total optical depth $\tau_{\nu, p, q}$ at each location in the projected coordinate system $(p, q)$ on the sky. To calculate the composite profile with both magnetospheric flows, we assume that the flows are geometrically separated and the source function and level populations of each flows are independent. For the inner flow, the specific intensity includes the emission from the stellar photosphere and the accretion shock, which is absorbed by the accreting material, and the emission from the flow itself. The emission from the outer flow includes only that from the accretion flow. The total emission map of the entire geometry is then given by
\begin{equation}
\mathcal{I}_{\nu, total} = \exp(-\mathcal{T}_{\nu, outer}) \cdot \mathcal{I}_{\nu, inner} + \mathcal{I}_{\nu, outer},
\end{equation}
where $\mathcal{I}$, $\mathcal{T}$ are 2D maps of the specific intensity and optical depth, respectively.
Finally, the model line flux is calculated as
\begin{equation}
F_{\nu} = \iint \mathcal{I}_{\nu, total} \,dp\,dq.
\end{equation}
\begin{figure}[]
\epsscale{1.15}
\plotone{f4.pdf}
\caption{The side view schematic of the geometry of the modified magnetospheric accretion model. The material flows from the gas disk onto the star along two geometrically separated, axisymmetric flows. The inner flows originates from $\sim 2.5$\, R$_{\star}$ and the outer flow, covering the entire geometry, originates at $\sim 7$\, R$_{\star}$. The color at each point in the flow represents the relative velocity of the emission from that point as view from an observer at the inclination of 70\degree.}
\label{fig:schematic}
\end{figure}
It is computationally impractical to compute the profile from all possible combinations of all small flows and large flows. For each observed profile, we selected 100 best fits for the small and large magnetosphere models, as outlined above. Even though the large magnetosphere could physically be similar in all epochs, the 100 best-fit model profiles of the magnetosphere could be different from epoch to epoch. This is because the depth and shape of the low-velocity redshifted absorption, to which the large magnetosphere models are fitted, still depend on the strength of the emission from the small magnetosphere, especially when that emission is strong. Assuming that the large magnetospheres are similar, which is supported by the persistent nature of the low-velocity component, we produced a global best fit of the large magnetosphere by combining the 100 best fits from each observed model. The final model line profiles for each observed profiles are then calculated by the combination of the 100 best fits of the small magnetosphere and the global best fits of the large magnetosphere.
We selected the best fits in each step using the minimization of $\chi^2$, given by
\begin{equation}
\chi^2 = \sum_i\, \frac{\left(F_{{\rm obs}, i} - F_{{\rm model}, i}\right)^2}{F_{{\rm obs}, i}},
\end{equation}
where $i$ indexes over the pixels in the observed spectra in relevant velocity ranges. In addition, we have experimented with other fitting methods including Root Mean Square Error (RMSE), the Mean Absolute Deviation (MAD), and the Mean Absolute Percentage Error (MAPE) of each models. The mean and the standard deviation of the first 100 best fits of these statistics are very similar to those using the $\chi^2$ fit, suggesting that the choice of statistical tools does not affect the general results.
\subsubsection{General Results for {H$\alpha$}} \label{sssec:halpha}
Figure~\ref{fig:two_geometry} shows the $\chi^2$ best fit model for each of the five {H$\alpha$} observations.
Our two-shell models are able to qualitatively reproduce the observed profiles in all epochs.
Table \ref{tab:model_results} shows the mean and standard deviation of model parameters from
the first 1000 $\chi^2$ best fits for each observed profile. These model profiles are qualitatively similar.
For both inner and outer flows the models requires high temperatures, with T$_{\rm max}\sim11000$\,K. The inclination for the outer flow is fairly constant at $\sim70\degree$, while that of the inner flows vary slightly. Similarly, the mass accretion rates for the outer flow are quite steady at $\sim2\times10^{-10}\,\rm{M_{\sun} \, yr^{-1}}$, while the accretion rates for the inner flow are somewhat more variable. These results suggest that the inner flow and the outer flow are slightly misaligned, and the outer flows are more stable than the inner flows.
The corotation radius R$_c$, outside which mass cannot accrete onto the star, is an absolute upper limit of the size of the magnetosphere. As a consistency check, we calculate R$_c$ of the star, assuming that the disk plane is aligned with the equatorial plane of the stellar rotation. Since the rotation period of the star is still undetermined, we use the measured projected rotational velocity $v\sin(i)$ of the star as a proxy. In this case, the corotation radius is given by
\begin{equation}
R_c = \left(GM\right)^{1/3}\left(\frac{R_{\star}\sin(i)}{v\sin(i)}\right)^{2/3},
\end{equation}
where $i$ is the inclination of the system.
With $v\sin(i)=11.5\,\rm \, km \, s^{-1}$ based on APOGEE results (J. Hernandez, private communication), we found that R$_c\sim8.9$\,R$_{\star}$ for $i=70\degree$. This is consistent with the model results for the outer flows (Table~\ref{tab:model_results}) and suggests that the infall originates close to the stellar corotation radius.
We calculated the residual of the model from the observation (Figure~\ref{fig:two_geometry}, lower panel), and found that the model systematically over-predicts some emission on the red side of the profile at $\sim 100\,\rm \, km \, s^{-1}$. This may indicate that the absorption component of the model is not extended enough, suggesting that there is additional absorbing material that is not accounted for.
\begin{figure*}[t]
\epsscale{1.18}
\plotone{f5.pdf}
\caption{The best fits from the modified magnetospheric accretion model for {H$\alpha$}. The top row shows the comparisons between the observed profiles (red) and the convolved model profiles (black). The green dotted lines and the blue dashed lines are the flux from the outer and inner flows, respectively. The bottom row shows the residual between the observed and the model profile (F$_{\rm model}$ $-$ F$_{\rm obs}$). Our model with two accretion flows can qualitatively reproduce the observed line profiles in all epochs, spanning several months. The model has some difficulty fitting the Goodman spectra, possibly due to its higher resolution that the model could not account for. However, the general results are clear that multiple flows are required to fit the profile. Evidently, the model is missing some absorption component at $\sim100\,\rm \, km \, s^{-1}$, which may indicate even more complex geometry.}
\label{fig:two_geometry}
\end{figure*}
\subsubsection{Testing the Model with {H$\beta$}} \label{ssec:hbeta}
To test the consistency of the two-shell model, we applied the modeling set up to the {H$\beta$} lines observed simultaneously with {H$\alpha$}. We found that, similarly to the case of {H$\alpha$}, the one-flow geometry could not reproduce the {H$\beta$} observations. We therefore followed the procedure outlined in \S\ref{sssec:new_geometry} to model the {H$\beta$} lines.
Figure~\ref{fig:hbeta} shows the best fits of the four {H$\beta$} profiles. The models are able to qualitatively reproduce the observation in all epochs. However, the models
cannot entirely fit the strong low-velocity redshifted absorption in three epochs, suggesting that extra absorbing material is needed. As shown in Table~\ref{tab:model_results}, the geometries, temperatures, and inclinations of both the inner flows and the outer flows for {H$\beta$} are consistent with those of {H$\alpha$}.
In fact, {H$\beta$} profiles calculated using the parameters that produce the best fits for {H$\alpha$} show two redshifted absorption components as seen in the observed {H$\beta$} profiles and the best fit profiles. This seems to indicate that the global properties of the flows derived from both lines are similar.
However, in the best-fit models for H$\beta$ the mass accretion rates in the outer flows for {H$\beta$} are slightly higher than those for {H$\alpha$}, and the opposite is found for the inner flow.
In addition, the discrepancy between the accretion rates in the inner flows and outer flows become smaller for {H$\beta$}. This could indicate that {H$\alpha$} and {H$\beta$} are formed in a slightly different region in the accretion flows with different filling factors, or it could indicate deviations in the temperature distribution assumed in our model.
\subsubsection{Model Limitation}
Stellar winds and disk winds may be present in accreting stars surrounded by protoplanetary disks. In fact, high-sensitivity and high spatial resolution observation have shown the presence of neutral hydrogen emission component from an extended region close to the central accreting stars, which is likely to be from winds \citep{gravity-collaboration2017,koutoulaki2018}. Our magnetospheric accretion model does not include a wind component, but we do not expect a significant contribution of the line emission from winds, since the mass accretion rate is low \citep{muzerolle2001}.
Another limitation of the model is based on the axisymmetric assumption. Simulations \citep{romanova2003,romanova2004} have shown that the accretion flow is generally not axisymmetric and the magnetic pole is likely not aligned to the rotation axis. However, without a measurement of the magnetic properties of CVSO~1335, a parametric study using MHD simulations would be prohibitively expensive. Our model allows a parametric study with a significantly smaller resources requirement.
Lastly, the model assumes only a strictly dipolar geometry, which is likely not the case for more evolved T Tauri stars such as CVSO~1335. However, adding a complex flow prescription would add more parameters to the study and the model would lose its generality. By keeping the geometry simple, we can make simple inferences about the relationship between the parameters in the model.
\begin{figure*}[t]
\epsscale{1.18}
\plotone{f6.pdf}
\caption{The best fits from the modified magnetospheric accretion model for {H$\beta$}. The legends are the same as in Figure~\ref{fig:two_geometry}. The modified geometry qualitatively reproduced the observation in all epoch, especially the emission and the high-velocity redshifted absorption. The deviation between the model and the observation at the low-velocity redshifted absorption is more prominent for this line compared to the {H$\alpha$} line models.
}
\label{fig:hbeta}
\end{figure*}
\begin{deluxetable*}{lcccccccccc}
\tablecaption{Results of the Modified Magnetospheric Accretion Model \label{tab:model_results}}
\tabletypesize{\scriptsize}
\tablehead{
\colhead{Spectrum} &
\colhead{$\dot{\text{M}}$$_{\rm in}$} &
\colhead{$R_{\rm i, in}$} &
\colhead{$W_{\rm in}$} &
\colhead{$T_{\rm max, in}$} &
\colhead{$i_{\rm in}$} &
\colhead{$\dot{\text{M}}$$_{\rm out}$} &
\colhead{$R_{\rm i, out}$} &
\colhead{$W_{\rm out}$} &
\colhead{$T_{\rm max, out}$} &
\colhead{$i_{\rm out}$} \\
\colhead{} &
\colhead{$10^{-10}\,\rm{M_{\sun} \, yr^{-1}}$} &
\colhead{R$_{\star}$} &
\colhead{R$_{\star}$} &
\colhead{$10^4\,$K} &
\colhead{deg} &
\colhead{$10^{-10}\,\rm{M_{\sun} \, yr^{-1}}$} &
\colhead{R$_{\star}$} &
\colhead{R$_{\star}$} &
\colhead{$10^4\,$K} &
\colhead{deg}
}
\startdata
\multicolumn{11}{l}{H$\alpha$} \\
Goodman-20170918 & 8.2$\pm$1.4 & 2.2$\pm$0.0 & 0.4$\pm$0.1 & 1.15$\pm$0.03 & 72$\pm$3 & 2.4$\pm$0.4 & 5.8$\pm$0.5 & 1.5$\pm$0.1 & 1.14$\pm$0.03 & 70$\pm$0 \\
MagE-20171127 & 7.9$\pm$1.7 & 2.2$\pm$0.1 & 0.2$\pm$0.0 & 1.15$\pm$0.04 & 75$\pm$8 & 2.4$\pm$0.4 & 5.9$\pm$0.5 & 1.5$\pm$0.1 & 1.14$\pm$0.03 & 70$\pm$0 \\
MagE-20171128 & 8.8$\pm$0.6 & 2.2$\pm$0.0 & 0.6$\pm$0.1 & 1.15$\pm$0.03 & 85$\pm$1 & 2.3$\pm$0.6 & 6.4$\pm$0.9 & 1.5$\pm$0.1 & 1.14$\pm$0.03 & 71$\pm$2 \\
MagE-20171129 & 5.3$\pm$1.0 & 2.2$\pm$0.0 & 0.6$\pm$0.1 & 1.14$\pm$0.03 & 77$\pm$3 & 2.4$\pm$0.5 & 6.6$\pm$0.9 & 1.5$\pm$0.1 & 1.14$\pm$0.03 & 70$\pm$1 \\
MagE-20171130 & 5.2$\pm$1.9 & 2.1$\pm$0.2 & 0.6$\pm$0.0 & 1.14$\pm$0.03 & 69$\pm$4 & 2.3$\pm$0.4 & 6.3$\pm$0.8 & 1.5$\pm$0.1 & 1.15$\pm$0.03 & 70$\pm$0 \\ \hline
\multicolumn{11}{l}{H$\beta$} \\
MagE-20171127 & 3.1$\pm$1.5 & 2.2$\pm$0.0 & 0.2$\pm$0.0 & 1.13$\pm$0.03 & 72$\pm$14 & 3.5$\pm$0.7 & 6.8$\pm$0.8 & 1.5$\pm$0.1 & 1.15$\pm$0.03 & 70$\pm$0 \\
MagE-20171128 & 3.6$\pm$1.2 & 2.2$\pm$0.0 & 0.5$\pm$0.1 & 1.14$\pm$0.03 & 85$\pm$0 & 3.5$\pm$0.7 & 7.1$\pm$0.8 & 1.5$\pm$0.1 & 1.15$\pm$0.03 & 71$\pm$2 \\
MagE-20171129 & 3.9$\pm$1.1 & 2.2$\pm$0.0 & 0.5$\pm$0.1 & 1.14$\pm$0.03 & 85$\pm$0 & 3.5$\pm$0.8 & 7.0$\pm$0.9 & 1.5$\pm$0.1 & 1.15$\pm$0.03 & 71$\pm$2 \\
MagE-20171130 & 3.9$\pm$1.0 & 2.0$\pm$0.2 & 0.6$\pm$0.0 & 1.15$\pm$0.04 & 82$\pm$2 & 3.5$\pm$0.8 & 7.1$\pm$0.9 & 1.5$\pm$0.1 & 1.15$\pm$0.03 & 71$\pm$2 \\
\enddata
\end{deluxetable*}
\subsection{Effects of Inclination and Mass Accretion Rate} \label {ssec:inclination_mdot}
Given that there are some similarities of the {H$\alpha$} and {H$\beta$} line profiles of CVSO~1335 to those of dipper stars such as AA~Tau \citep[][see \S\ref{ssec:redshift}]{bouvier2007}, it is insightful to explore the effects that could change the observed line profiles while keeping the two-flow geometry, especially in assessing the frequency of a type of profile. In Figure~\ref{fig:inclination}, we show {H$\alpha$} profiles using the two-flow geometry with parameters similar to the average results in Table~\ref{tab:model_results} but changing the inclination and the total mass accretion rate. Specifically, we selected R$_{\rm i, in}=2.2$\,R$_{\star}$, W$_{\rm in}=0.6$\,R$_{\star}$, with constant ratio of mass accretion rate between the inner and outer flow $\dot{\text{M}}$$_{\rm in}$/$\dot{\text{M}}$$_{\rm out}$=3. For the model with high {$\dot{\text{M}}$}, we also calculated the profiles with a slightly lower temperature ($T_{\rm max, in}$=10000\,K and $T_{\rm max, out}$=11000\,K, compared to 11400\,K and 11600\,K for the fiducial model). We found that the low-velocity redshifted absorption (v$\,\lesssim100\rm \, km \, s^{-1}$) only appears in models with low total accretion rates in moderate to high inclination, $i\gtrsim65\degree$, in agreement with our findings in \S\ref{ssec:acc_model}. The velocity of the deepest absorption moves closer to the line center as the inclination increases. On the other hand, the high velocity redshifted absorption (v$\gtrsim200\,\rm \, km \, s^{-1}$) appears in almost all inclinations at low accretion rates, and the velocities of such absorptions are fairly constant. There is some degree of degeneracy between the mass accretion rate and temperature, as models with lower temperature and higher accretion rate are similar to those with higher temperature and lower accretion rate (e.g. green dashed line and black solid line in Figure~\ref{fig:inclination}).
\begin{figure*}[h!]
\epsscale{1.1}
\plotone{f7.pdf}
\caption{
{H$\alpha$} line profiles for the two-shell accretion geometry with varying inclinations, mass accretion rates, and temperatures. Note that the y-axis is in the log scale. The black lines show the {H$\alpha$} profiles with parameters similar to those in Table~\ref{tab:model_results}, with $\dot{\text{M}}$$_{\rm in}=6\times10^{-10}\,\rm{M_{\sun} \, yr^{-1}}$ and $\dot{\text{M}}$$_{\rm out}=2\times10^{-10}\,\rm{M_{\sun} \, yr^{-1}}$. The orange, green, and red lines show the profiles at 0.5, 2.0, and 5.0 times the total mass accretion rate of the black line, respectively. For each {$\dot{\text{M}}$} the dashed lines show the profiles with a slightly lower T$_{\rm max}$ (see text.)
The low-velocity redshifted absorption starts appearing at $i\geq65\degree$, shifting closer to the line center as the inclination increases. The line profiles at low inclinations are similar to those typically observed.
\label{fig:inclination}}
\end{figure*}
Figure~\ref{fig:fixout} shows the {H$\alpha$} line profiles with the same parameters as those in Figure~\ref{fig:inclination}, but with a constant $\dot{\text{M}}$$_{\rm out}=2\times10^{-10}\,\rm{M_{\sun} \, yr^{-1}}$ while varying the $\dot{\text{M}}$$_{\rm in}$ from $3\times10^{-10}$ to $3\times10^{-9}\,\rm{M_{\sun} \, yr^{-1}}$.
Interestingly, the low-velocity redshifted absorption appears conspicuously regardless of the mass accretion rate of the inner flow. This suggests that the outer flow could be responsible for a significant amount of emission if the accretion rate is high enough, as in Figure~\ref{fig:inclination}. The outer flow acts as an absorber only in a very low accretion regime ($\sim2-4\times10^{-10}\,\rm{M_{\sun} \, yr^{-1}}$).
\begin{figure}[h!]
\epsscale{1.1}
\plotone{f8.pdf}
\caption{Model of {H$\alpha$} line in the same two-shell accretion geometry at different mass accretion rates and temperatures. The legends are the same as in Figure~\ref{fig:inclination}, but the accretion rate of the outer flow {$\dot{\text{M}}$$_{\rm out}$} is fixed at $2\times10^{-10}\,\rm{M_{\sun} \, yr^{-1}}$.
\label{fig:fixout}}
\end{figure}
\section{Discussion} \label{sec:discussion}
\subsection{Measuring Mass Accretion Rates in Low Accretors}
As we have shown in \S \ref{ssec:shock}, the mass accretion rate of CVSO~1335 could be variable, with an upper limit of the order of $\sim 4-9\times10^{-10}\,\rm{M_{\sun} \, yr^{-1}}$. These estimates disagree with the mass accretion rate measurements using traditional methods such as the full width at the 10\% height \citep[{W$_{10}$};][]{natta2004}. In a previous study \citep{thanathibodee2018}, we calculated the accretion rate of CVSO~1335 using the {W$_{10}$} method and found that the accretion rates should be of the order of $10^{-8}-10^{-7}\,\rm{M_{\sun} \, yr^{-1}}$ to account for the width of the {H$\alpha$} line. To account for the presence of the redshifted absorption, we estimated {W$_{10}$} by measuring the half-width on the blue side, in which no absorption is present. The resulting full width of $\sim600\,\rm \, km \, s^{-1}$ still gives $\dot{\text{M}}$$\sim8\times10^{-8}\,\rm{M_{\sun} \, yr^{-1}}$. As shown in Figure~\ref{fig:shock}, such an accretion rate would produce a significant excess in the blue part of the optical spectrum of the star, in disagreement with the observation. Our results show that in low accretors such as CVSO~1335, the width at the 10\% height of the {H$\alpha$} line is not a reliable method for measuring the mass accretion rate. The absorption near the line center complicates the measurement of the W$_{10}$ as the actual height cannot be reliably measured.
On the other hand, the mass accretion rates of CVSO~1335 determined in this study by using the magnetospheric accretion model and accretion shock model to fit Balmer line profiles and the Balmer jump are consistent with the accretion rates determined from the {H$\alpha$} luminosity in \citet{thanathibodee2018}, suggesting that {H$\alpha$} line luminosity is a reliable mass accretion rate estimator. Nevertheless, there is a limit to the estimates of mass accretion rates using line luminosities imposed by the chromospheric contribution to the emission lines \citep{manara2013}. The chromospheric contribution is significant in low accretors, since the accretion-originated line emission is weak, and the resulting mass accretion rate would have a high relative uncertainty. On the other hand, line profile modeling can disentangle between the narrow chromospheric core emission and the broad line wings from the accretion flow \citep{espaillat2008}. Therefore, direct modeling of the {H$\alpha$} lines, with chromospheric emission taken into account, is required for an accurate measurement of accretion rates in the low accretion regime and/or with a presence of low-velocity redshifted absorption.
As shown by \citet{ingleby2011b}
emission at the Balmer jump is difficult to detect in low accretors, including the case of CVSO~1335 as shown here. This is because the accretion shock emission is weak compared to the photospheric and chromospheric emission in the UV. Nevertheless, the non-detection of the Balmer jump sets a useful upper limit on the mass accretion rate for objects in which other accretion indicators, such as redshifted absorption, are present.
\subsection{The Origin of the Low-Velocity Redshifted Absorption} \label{ssec:redshift}
The {H$\alpha$} profiles of CVSO~1335 show two systems of redshifted absorption. One is a persistent low-velocity ($v\sim75\,\rm \, km \, s^{-1}$) redshifted absorption seen in all epochs; another is very variable complex system, located at velocities consistent with free-fall velocities (c.f. Fig.~\ref{fig:observation}.)
Stars classified as dippers, such as AA~Tau \citep{bouvier1999,bouvier2003,bouvier2007} and LkCa~15 \citep{alencar2018}, show comparable redshifted absorptions. Using line profile decomposition, these studies indicate the presence of low-velocity blueshifted and redshifted absorption components, attributed to a hot wind and to the magnetospheric accretion flow, respectively, in a system viewed at high inclination. The velocities of the blue and redshifted absorption components are correlated and vary in absolute value between $\sim$ 10 and 60 $\rm \, km \, s^{-1}$, such that the highest redshifted velocity corresponds to the lowest blueshifted velocity. The low velocity absorption components do not seem to be correlated with the stellar rotation period, unlike the system at nearly free-fall velocities, which is highly variable and tends to appear near the photometric minimum in the dippers; these absorption are also attributed to magnetospheric infall \citep{bouvier1999,bouvier2003,bouvier2007,alencar2018}.
The observed low- and high-velocity redshifted absorptions in the {H$\alpha$} profiles of CVSO~1335 could in principle correspond to those observed in dippers. The star does not seem to exhibit any blueshifted absorption in any of its emission lines, possibly due to its low mass accretion rate, and consequently low mass loss rate. The spectral resolution of our MagE data could not significantly detect temporal variations in the location of the low velocity absorption of the order of those seen in dippers, although the comparison between the MagE and Goodman spectra, spanning over two months, seems to suggest that the absorption are fairly stable within the low velocity range.
Another characteristic of dippers are episodes of dust obscuration observed via reddening of the stellar spectra. In CVSO~1335, the spectrum observed on 2017-11-27 is redder than in other epochs (cf. Figure \ref{fig:observation}), which could be due to a similar reddening process. In support of this possibility, we found that we could reproduce the mean of the spectra in other epochs, adopted as template, by correcting the 2017-11-27 spectrum for extinction using the opacity of silicate dust grains with a size distribution $n(a) \propto a^{-3.5}$ between $a_{min} = 0.005 \mu$m and $a_{max} = 0.25 \mu$m, and a dust-to-gas mass ratio of 0.004, comparable to ISM grains \citep{dalessio2001}. We estimate that a low column density of N$_{\rm gas+dust}\sim9\times10^{-3}$\,g\,cm$^{-2}$ is required to produce the observed obscuration. A more detailed analysis using time-series optical spectra is required to confirm the existence of dust obscuration events in this star.
The high velocity redshifted absorption in CVSO~1335 is highly variable, as it is in dippers, but our limited number of observations does not allow us to determine if it correlates with the stellar period, which we expect to be $\sim 6.5$ days based on a $v\sin(i)$ measurement (\S \ref{sssec:halpha}). We require multi-band photometric monitoring of the star to test the hypothesis that the star is a dipper and observations are under way. Contemporaneous optical-NIR spectroscopic observations are also needed to link the light-curve variation in the photometry to the structure of accretion. Finally, HST observation using high-resolution FUV spectrographs would give insights into the connection between all components in the inner disk and the accretion in this low accretor. These properties will be followed up in future studies.
\subsection{Magnetospheric Accretion in Two-Shell Geometry} \label{ssec:acc_in_two_shell}
The presence of accretion-originated low-velocity redshifted absorption \emph{in addition} to redshifted absorption near free-fall velocity and broad wings suggests that there are two distinct accreting components, which we called two accretion flows in a two-shell geometry (c.f. Fig.~\ref{fig:two_geometry}, Table~\ref{tab:model_results}). This structure is an idealization of the real geometry that must be much more complicated. Depending on the actual structure of the accretion geometry, the difference in mass accretion rate between the inner and outer flows could be explained in different ways. If the inner and outer flow are physically disconnected, the mass accretion rate of the flows would naturally be different. On the other hand, if the two flows represent one complex, but physically connected, accretion flow, the actual mass accretion rate should be the same. The difference in {$\dot{\text{M}}$} we find could be due to different azimuthal coverage of the inner and outer flows. We note that since the low-velocity redshifted absorption component is stable, the azimuthal coverage of the outer flow must be fairly uniform. In this case, if the inner flow is concentrated in a smaller azimuthal region, the density in the flow would be higher than that in the outer flow which covers a larger volume. Since the model assumes axisymmetric flows, the mass accretion rate inferred from the concentrated rate in the inner flow would be higher than the actual rate.
Although our model could not determine the actual geometry in the accretion flow, the agreement between our simple models and the observations suggests that the two-flow structure must be representing conspicuous features in the real situation that give rise to the observed features. Here we explore the implication of such structure.
\subsubsection{Complex Stellar Magnetic Field Structure} \label{sssec:complex-mag}
The two-shell geometry could be reminiscent of a complex structure of the stellar magnetic field that gives rise to complex accretion flows. For example, it could be that the accreting material is the combination of magnetic dipolar and multipolar fields \citep{long2007,long2012}. Spectropolarimetric observations of accreting T Tauri stars have shown evidences of higher order magnetic fields in stellar magnetosphere; for example \citet{donati2007,donati2011a} showed that both the dipole and octupole components are present in the (K5) T Tauri star V2129~Oph. Based on these results, \citet{romanova2011} used 3D MHD simulations to show that some material could be channelled onto the star via octupole fields at lower latitude in addition to the dipole accretion at higher latitude. CVSO~1335 could exhibit a similar magnetic field structure, and due to its high inclination the dipole flow could occult the higher-order flows at lower latitude. \citet{alencar2012} calculated {H$\alpha$} line profiles based on the results of the simulations, and the low-velocity redshifted absorption was not present in the profiles, while it is conspicous in CVSO~1335. However, this could be due to the combined effect of a lower inclination and higher mass accretion rate in V2129~Oph compared to CVSO~1335 (see \S \ref{ssec:inclination_mdot}).
Another explanation for the two flow geometry is that the accretion in CVSO~1335 is at the interface between stable and unstable accretion in a strictly dipolar magnetic field. Results from 3D MHD simulation show that in this transition region, matter flows along two large funnels and through several small accretion tongues that deposite matter near the stellar surface \citep{romanova2008,kulkarni2009}. If this is the case for CVSO~1335, the outer flows could be the larger funnels that occult the several small tongues, which are the inner flows. To test either of these hypotheses, spectropolarimetric observations and MHD simulations with the exact stellar parameters of CVSO~1335 are required.
Lastly, the process responsible for the low-velocity redshifted absorption in this star could be similar to that proposed by \citet{bouvier2003}, based on results of numerical simulations \citep{romanova2012,miller1997}, namely an ``inflated'' magnetosphere, resulting from differential rotation between the star and the disk.
No radiative transfer model of such geometry has been applied to the Balmer lines in AA~Tau yet, although some progress has been made for modeling the low-velocity blueshifted absorption \citep{esau2014}.
\subsubsection{Complex Inner Disk Structure due to a Planetary Companion} \label{sssec:planet}
Another possibility that gives rise to geometrically separated accretion flows is that the inner disk that provides the material is not radially uniform. One example of such condition is that the inner disk, inside the corotation radius, may exhibit a gap structure at $\sim5\,$R$_{\star}$. In this case, very little material at that radius is flowing into the star along the magnetic field lines, thus leaving a gap between two shells of material flowing from the inner ring and from the ring at corotation. Rings and gaps in protoplanetary disks have been observed in larger scales in sub-mm \citep[e.g.][]{alma2015,long2018b} and infrared scattered light \citep[e.g.][]{avenhaus2018}. Hydrodynamic simulations of protoplanetary disks have shown that these rings and gaps could be the consequences of a planet forming in the disk \citep{bae2017}. However, it is unclear that similar process could happen in the inner disk. Nevertheless, if the gap is produced by a planet, it is unlikely that the planet's orbit is stable inside an actively accreting magnetosphere. \citet{bae2017} have shown that a planet could open a secondary gap at $\sim0.5r_p$, where $r_p$ is the orbital radius of the planet, which implies that the planet could be at $\sim10$\,R$_{\star}$ (a$\sim$0.087\,au, P$\sim$10\,d). Since the Shakura-Sunyaev $\alpha$ parameter could be high in the MRI-driven innermost region of the disk \citep{mohanty2018}, and the mass of the gap-opening planet increases with $\alpha$ \citep{bae2017}, the planet, if it exists, should be massive (several M$_{\rm Jup}$).
Planets have been found to orbit the central stars with orbits of a few days, i.e. hot Jupiters. Many studies have indicated the presence of close-in planets around (non-accreting) weak TTS \citep[e.g.][]{van-eyken2012,mann2016,david2017}, but evidence of such planets around an actively accreting pre-main sequence stars are still sparse. So far, the only proposed planets in such category are a 11.3\,M$_{\rm Jup}$ planet with $\sim$\,9 days period orbiting the 2\,Myr star CI Tau \citep{johns-krull2016}, and a M$_p\sin i\sim19.3$\,M$_{\rm Jup}$ planet with a 24.8 day period around the 0.5\,Myr star AS~205A \citep{almeida2017}, based on the radial velocity method. With the parameters of CVSO~1335, a comparable planet would cause an RV signature of $\sim1\,\rm \, km \, s^{-1}$; spectroscopic monitoring of the star is required to test this hypothesis.
\subsubsection{Observability of Low-Velocity Redshifted Component}
We have shown that the modified magnetospheric accretion model with a
two-flow geometry is able to reproduce the observed {H$\alpha$} and {H$\beta$} profiles of CVSO~1335. Therefore, it is insightful to consider this model for other T Tauri stars. As shown in Figure~\ref{fig:inclination}, producing two redshifted absorption components requires low accretion rates and high inclinations;
for high accretion rates, the two-flow model produces similar profiles to the standard model since {H$\alpha$} has become optically thick. Nevertheless, the model could be used for other lines that are more optically thin even with high accretion rate, but this remains to be calculated.
For low mass accretion rates the presence of the low velocity component still depends on the inclination of the system (c.f. Figure~\ref{fig:inclination}). Assuming that all stars orient randomly, one would expect that a significant portion of the low accretors exhibit a similar type of profiles. Several studies have shown that complex {H$\alpha$} line profiles are not uncommon \citep[e.g.][]{reipurth1996,antoniucci2017}, but {H$\alpha$} profiles with two redshifted absorption components, as seen in CVSO~1335, are rare even in CTTS with low accretion rates. This suggests that other factors contribute to the formation of such geometry.
The two scenarios discussed in this section could explain the rarity of multiple redshifted absorption profile.
In order to have complex magnetic fields, required for scenario discussed in \S\ref{sssec:complex-mag}, pre-main sequence stars need to have solar or higher mass and/or old age \citep[e.g.][]{villebrun2019}. In this regard, solar mass low accretors are rare since stars in this mass range account for only $\lesssim10\%$ of a young population \citep[e.g.][]{briceno2019}, and the number of accretors decreases sharply with age. The frequency of low accretors are not yet available, but the number should be less than the frequency of all CTTS at 5\,Myr, $\sim$ 15\% \citep{briceno2019,fedele2010}. Therefore, the upper limit of the frequency of solar-mass low accretor is $\sim$1.5\% of all T Tauri stars in a given population, and $\sim$0.75\% would have a high inclination. This fraction means that only a few stars in a given population could show line profiles similar to those of CVSO~1335, since the number of T Tauri stars in a given population is on the order of a few hundreds \citep[e.g.][]{luhman2018a,luhman2018b} to thousands \citep[e.g.][]{sung2009,briceno2019}. It is therefore conceivable that many solar mass stars could have a magnetosphere in the two-flow geometry at some point in their life, but the frequency of observing them is low and only CVSO~1335 has been identified as such so far.
In the gap opened by a planet scenario, the required mass of the planet is several Jupiter mass. If the planet exist, it would be classified as a hot Jupiter due to its close orbit. Since, only $\sim$ 1\% of solar mass stars, and even less in M-type stars, host such planets \citep{dawson2018}, the expected number of CTTS with CVSO~1335-type profiles is as equally small as in the case of complex magnetic fields.
\section{Summary} \label{sec:summary}
We applied accretion shock models and magnetospheric accretion models to MagE and Goodman optical spectra of the 5\,Myr old, $\sim$ solar mass star CVSO~1335 to characterize the accretion properties of this low accretor. Here we summarize our findings:
\begin{enumerate}
\item The Balmer jump of CVSO~1335 does not show any significant excess over the WTTS used as template, confirming that CVSO~1335 is a low accretor. Using accretion shock models, we find variable upper limits of the mass accretion rates $\sim4-9\times10^{-10}\,\rm{M_{\sun} \, yr^{-1}}$. These limits are in agreement with estimates based on {H$\alpha$} line luminosity. However, they contradict the measurements based on the line width, which would indicate rates as high as $10^{-8}\,\rm{M_{\sun} \, yr^{-1}}$.
\item The excess at the Balmer jump does not provide an
estimate of the mass accretion rate in CVSO~1335, or in general in low accretors. On the other hand, line profile fitting provides a
measurement of the accretion rate, as well as the geometry of
the accretion flows. Therefore, modeling line profiles is the only reliable method to accurately measure mass accretion rate in low accretors.
\item Redshifted absorption components superimposed on
the {H$\alpha$} and {H$\beta$} emission lines are conspicuous in CVSO~1335
and are found in all epochs. The line profiles are variable with clear multiple components, which cannot be explained using the standard magnetospheric accretion model. A modified magnetospheric accretion model, with two separated accretion flows, can explain the low-velocity redshifted absorption simultaneously with broad wings and high-velocity redshifted absorption in the {H$\alpha$} and {H$\beta$} lines. The inner flows have higher and more variable {$\dot{\text{M}}$} compared to the outer flows. High inclination ($i\sim70\degree$) and high accretion flow temperature (T$_{\rm max} \gtrsim 11000$) are required to reproduce the profiles.
\item The required high inclination, in addition to the presence of a persistent low velocity component and a highly variable high velocity component, with velocities
comparable to free-fall velocities, may indicate that CVSO~1335 is a dipper. Multi-band photometric monitoring is required to test this hypothesis.
\item Our simple two-flow geometry represents a more complex accretion geometry. This could suggests
a complex magnetic field structure including higher-order fields, unstable accretion, or an inflated magnetosphere. Alternatively, a ringed structure in the inner disk resulting from the presence of a companion, could be responsible for this geometry. The rarity of {H$\alpha$} profiles with multiple redshifted absorption components is compatible with both scenarios.
\end{enumerate}
\acknowledgments
\noindent
This project is supported in part by NASA grant NNX17AE57G. This research is based on data obtained from the ESO Science Archive Facility under request number thanathi/412978. We thank Lee Hartmann, Jaehan Bae, Zhaohuan Zhu, Susan Edwards, and Jerome Bouvier for insightful suggestions.
We thank the telescope operators and the staff at Las Campanas Observatory for the help during the MagE observations.
\facilities{Magellan:Baade (MagE), SOAR (Goodman Spectrograph)}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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Evans Blue är ett kanadensiskt hårdrocksband från Toronto som bildades 2005. Sammanlagt har de sålt en miljon album världen över.
Album
The Melody and the Energetic Nature of Volume (2006)
The Pursuit Begins When This Portrayal of Life Ends (2007)
Evans Blue (2009)
Graveyard of Empires (2012)
Letters from the Dead (2016)
Nuvarande medlemmar
Dan Chandler - sångare (2009–nu)
Parker Lauzon - gitarr (2005–nu)
Vlad Tanaskovic - gitarr (2005–nu)
Joe Pitter - gitarr (2005–nu)
Kanadensiska rockgrupper
Kanadensiska metalgrupper
Kanadensiska indierockgrupper
Musikgrupper bildade 2005
Musikgrupper från Toronto
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
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package ca.uwaterloo.joos.ast;
import static ch.lambdaj.Lambda.*;
import static org.hamcrest.Matchers.*;
import java.util.ArrayList;
import java.util.List;
import java.util.logging.Logger;
import ca.uwaterloo.joos.Main;
import ca.uwaterloo.joos.ast.AST.ASTConstructException;
import ca.uwaterloo.joos.ast.descriptor.ChildDescriptor;
import ca.uwaterloo.joos.ast.descriptor.ChildListDescriptor;
import ca.uwaterloo.joos.ast.descriptor.Descriptor;
import ca.uwaterloo.joos.ast.descriptor.SimpleDescriptor;
import ca.uwaterloo.joos.ast.descriptor.SimpleListDescriptor;
import ca.uwaterloo.joos.ast.visitor.ASTVisitor;
import ca.uwaterloo.joos.parser.ParseTree.LeafNode;
import ca.uwaterloo.joos.parser.ParseTree.Node;
import ca.uwaterloo.joos.parser.ParseTree.TreeNode;
import ca.uwaterloo.joos.parser.ParseTreeTraverse;
import ca.uwaterloo.joos.parser.ParseTreeTraverse.Traverser;
public abstract class ASTNode implements Traverser {
protected static final Logger logger = Main.getLogger(ASTNode.class);
@SuppressWarnings("serial")
public static class ChildTypeUnmatchException extends ASTConstructException {
public ChildTypeUnmatchException(String msg) {
super(msg);
}
}
private static class ASTChild {
public Descriptor descriptor;
public Object value;
public Descriptor getDescriptor() {
return descriptor;
}
public Object getValue() {
return value;
}
public ASTChild(Descriptor descriptor, Object value) {
this.descriptor = descriptor;
this.value = value;
}
}
private ASTNode parent = null;
private String identifier = new String();
private List<ASTChild> childrenList = new ArrayList<ASTChild>();
public ASTNode(Node node, ASTNode parent) throws Exception {
if(node == null) return;
this.parent = parent;
ParseTreeTraverse traverse = new ParseTreeTraverse(this);
traverse.traverse(node);
}
/**
* @return the parent
*/
public ASTNode getParent() {
return parent;
}
/**
* @return the identifier
*/
public String getIdentifier() {
return identifier;
}
protected void setIdentifier(String identifier) {
this.identifier = identifier;
}
protected void addChild(Descriptor descriptor, Object child) {
this.childrenList.add(new ASTChild(descriptor, child));
}
private List<?> getChildrenByRawDescriptor(Descriptor descriptor) throws ChildTypeUnmatchException {
List<Object> children = extract(select(this.childrenList,
having(on(ASTChild.class).getDescriptor(), is(descriptor))),
on(ASTChild.class).getValue());
return children;
}
public List<?> getChildByDescriptor(ChildListDescriptor listDescriptor) throws ChildTypeUnmatchException {
return this.getChildrenByRawDescriptor(listDescriptor);
}
public ASTNode getChildByDescriptor(ChildDescriptor childDescriptor) throws ChildTypeUnmatchException {
List<?> children = this.getChildrenByRawDescriptor(childDescriptor);
if(children.size() == 0) {
return null;
} else if(children.size() > 1) {
throw new ChildTypeUnmatchException(childDescriptor + " is mapping to more than one item");
}
ASTNode child = (ASTNode) children.get(0);
if (childDescriptor.getElementClass().isAssignableFrom(child.getClass())) {
return (ASTNode) child;
}
throw new ChildTypeUnmatchException(childDescriptor + " is not mapping to a " + childDescriptor.getElementClass().getSimpleName());
}
public List<?> getChildByDescriptor(SimpleListDescriptor listDescriptor) throws ChildTypeUnmatchException {
return this.getChildrenByRawDescriptor(listDescriptor);
}
public Object getChildByDescriptor(SimpleDescriptor childDescriptor) throws ChildTypeUnmatchException {
List<?> children = this.getChildrenByRawDescriptor(childDescriptor);
if(children.size() == 0) {
return null;
} else if(children.size() > 1) {
throw new ChildTypeUnmatchException(childDescriptor + " is mapping to more than one item");
}
Object child = children.get(0);
if (childDescriptor.getElementClass().isAssignableFrom(child.getClass())) {
return child;
}
throw new ChildTypeUnmatchException(childDescriptor + " is not mapping to a " + childDescriptor.getElementClass().getSimpleName());
}
public final void accept(ASTVisitor visitor) throws Exception {
visitor.willVisit(this);
logger.finest("Visiting <" + this.getClass().getSimpleName() + ">");
if (visitor.visit(this)) {
for (ASTChild child : this.childrenList) {
Descriptor descriptor = child.getDescriptor();
if (descriptor instanceof ChildListDescriptor ||
descriptor instanceof ChildDescriptor) {
ASTNode node = (ASTNode) child.getValue();
node.accept(visitor);
}
}
}
visitor.didVisit(this);
}
@Override
public final String toString() {
String str = "";
str += "<" + this.getClass().getSimpleName() + ">";
if(this.identifier.length() > 0) str += " " + this.identifier + " |";
for (ASTChild child : this.childrenList) {
Descriptor descriptor = child.getDescriptor();
if( descriptor instanceof SimpleDescriptor ||
descriptor instanceof SimpleListDescriptor ) {
str += " " + descriptor.getElementClass().getSimpleName() + ": " + child.getValue() + " |";
}
}
if (this.parent != null)
str += " parent: " + this.parent.getClass().getSimpleName();
return str;
}
@Override
public List<Node> processTreeNode(TreeNode treeNode) throws Exception {
List<Node> offers = new ArrayList<Node>();
offers.addAll(treeNode.children);
return offers;
}
@Override
public void processLeafNode(LeafNode leafNode) throws Exception {
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 318
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\section{Introduction}
In this note, we discuss several different definitions of an extended $1$-perfect bitrade in Hamming graphs $H(n,q)$, $q>2$,
prove their equivalence, and prove the nonexistence of extended $1$-perfect bitrades for odd $n$, generalizing
some of the results of~\cite{Bespalov:ext} on the nonexistence of extended $1$-perfect codes.
As usual, for different combinatorial objects like some classes of codes, designs, orthogonal arrays, etc.,
bitrades are defined as a generalization of the difference pair
$(C_0\backslash C_1, C_1\backslash C_0)$ of two objects $C_0$ and $C_1$ from the initial class.
The main question is the following:
which property of the initial class has to be generalized to form the definition of a bitrade.
The five definitions of extended $1$-perfect bitrades considered below reflect diferent definitions of an extended $1$-perfect
code as
\begin{itemize}
\item a code whose projection in any direction
(puncturing in any coordinate)
gives a $1$-perfect code
(this definition reflects
a traditional meaning of the concept
``extended code'');
\item a uniformly packed
(in the wide sense \cite{BZZ:1974:UPC}) code,
with certain parameters;
\item a completely regular code \cite{Neumaier92},
with certain parameters;
\item a diameter-$3$ perfect code \cite{AhlAydKha};
\item a distance-$4$ code with certain spectral constraints,
namely, the characteristic function of code
belongs to the sum of three eigenspaces of $H(n,q)$,
with the eigenvalues $(q-1)n$
(the maximum eigenvalue of $H(n,q)$),
$q-2$, and $-n$ (the minimum eigenvalue of $H(n,q)$);
these spectral restrictions can be also expressed
in terms of the components of the dual distance distribution.
\end{itemize}
We note that our definitions are not in agree with known definition of
extended $1$-perfect bitrades in the binary Hamming graph $H(n,2)$.
The reason is that $H(n,2)$, or $n$-cube, is a bipartite graph,
and it is natural to require that the components of the bitrade
lie in one of its parts, either even or odd (which also allows
to consider bitrades as objects in the halved $n$-cube).
Relaxing this restriction leads to some degenerate cases,
which are not interesting from theoretical point of view.
Of course, $H(n,q)$ can also be considered as a $q$-partite graph,
but the parts are not uniquely defined, and we cannot guarantee
for an extended $1$-perfect code that it lies in one of them;
neither it is natural to require this from a bitrade.
We finish the introduction by the observation that
extended $1$-perfect codes and extended $1$-perfect bitrades
do exist in $H(2^k+2,2^k)$, so the equivalences proved are about
the objects from non-empty class.
\section{Preliminaries}
Given a graph $G$ denote by ${\scriptscriptstyle\mathrm{V}}{G}$ its vertex set.
A code in $G$ is an arbitrary nonempty subset of ${\scriptscriptstyle\mathrm{V}}{G}$.
Denote $e^a_i=(0,\ldots,0,a,0,\ldots,0)$, where $a \in \mathbb Z_q$ is placed in the $i$-th position..
Denote by $B(x)=\{z:d(x,z) \le 1\}$ the \emph{radius-$1$ ball} with center $x$.
Denote by $S_i(x)$ the \emph{radius-$i$ sphere} with center $x$, i.e. $S_i(x)=\{y:d(x,y)=i\}$.
The set $C_{x,i}=\cup_{a \in \mathbb Z_q} B(x+e^a_i)$
is called \emph{cylinder}.
A pair $(T_{+},T_{-})$ of disjoint subsets of ${\scriptscriptstyle\mathrm{V}}{H(n,q)}$ is called a \emph{perfect bitrade} in $H(n,q)$ if
$|B(x) \cap T_{+}|=|B(x) \cap T_{-}| \le 1$ for any $x \in {\scriptscriptstyle\mathrm{V}}{H(n,q)}$.
\begin{lemman}\label{l:dist3}
If $(T_{+},T_{-})$ is a perfect bitrade in $H(n,q)$, then
$T_{+}$ and $T_{-}$ are codes with code distance $3$.
\end{lemman}
For a vertex $x=(x_1,\ldots,x_n) \in {\scriptscriptstyle\mathrm{V}}{H(n,q)}$
denote by $x^a_i$ the vertex $(x_1,\ldots,x_{i-1},a,x_{i},\ldots,x_n)$ of $H(n+1,q)$.
Let $C$ be a code in $H(n+1,q)$.
The set $\{x \in {\scriptscriptstyle\mathrm{V}}{H(n,q)}: x^a_i \in C \text{ for some } a \in \mathbb Z_q\}$
is called the \emph{$i$-projection} of $C$ (or simply, projection).
Let $(T_{+},T_{-})$ be a pair of disjoint sets of vertices of $H(n+1,q)$,
and let $A_{+}$ and $A_{-}$ be $i$-projections of $T_{+}$ and $T_{-}$ respectively.
The pair $(T'_{+},T'_{-})$, where $T'_{+}=A_{+} \backslash A_{-}$ and $T'_{-}=A_{-} \backslash A_{+}$,
is called the \emph{$i$-projection} (or simply, projection) of $(T_{+},T_{-})$.
A function $f:{\scriptscriptstyle\mathrm{V}}{H(n,q)} \to \mathbb C$ is called an \emph{eigenfunction} of $H(n,q)$ with eigenvalue $\lambda$ if
$Af=\lambda f$, $f \not\equiv 0$, where $A$ is the adjacency matrix of $H(n,q)$.
The Hamming graph $H(n,q)$ has the following eigenvalues: $\lambda_i=n(q-1)-iq$, $i=0,1,\ldots,n$.
Denote by $U_i(n,q)=\{f: Af=\lambda_i f\}$
the eigenspace of eigenfunctions of $H(n,q)$ with eigenvalue $\lambda_i$.
For functions $f,g: {\scriptscriptstyle\mathrm{V}}{H(n,q)} \to \mathbb C$ define
$(f,g)=\sum_{x \in {\scriptscriptstyle\mathrm{V}}{H(n,q)}} f(x) \overline{g(x)}$.
\section{Definitions of bitrade}
\begin{definitionn}\label{d:matrixeq}
A pair $(T_{+},T_{-})$ of disjoint subsets of ${\scriptscriptstyle\mathrm{V}}{H(n,q)}$ is called an extended perfect bitrade in $H(n,q)$, if there is an $(q^n,3)$-matrix $F$,
whose columns are indexed by the vertices of $H(n,q)$, such that:
\begin{itemize}
\item the first column is the characteristic $(0,\pm 1)$-function of $(T_{+},T_{-})$
\item in each row the sum of elements equals $0$ and there are at most $2$ nonzero elements
\item there holds the matrix equation $AF=FS$, where $A$ is the adjacency matrix of $H(n,q)$ and
\[\displaystyle{S=\begin{pmatrix}
0 \ \ \ & n(q-1) & 0 \\
1 \ \ \ & q-2 & (n-1)(q-1) \\
0 \ \ \ & n & n(q-2)
\end{pmatrix}}.\]
\end{itemize}
\end{definitionn}
\begin{definitionn}\label{d:proection}
A pair $(T_{+},T_{-})$ of disjoint subsets of ${\scriptscriptstyle\mathrm{V}}{H(n,q)}$
is called an extended perfect bitrade in $H(n+1,q)$,
if $T_{+}$ and $T_{-}$ are codes with code distance $4$ and
for every $i \in \{1,\ldots,n\}$ the $i$-projection of $(T_{+},T_{-})$
is a perfect bitrade in $H(n,q)$.
\end{definitionn}
Let $A$ be an arbitrary subset of ${\scriptscriptstyle\mathrm{V}}{H(n,q)}$.
For a vertex $x \in {\scriptscriptstyle\mathrm{V}}{H(n,q)}$,
define its weight $w_A(x)$ with respect to $A$ in the following way:
$w_A(x)=|S_0(x) \cap A|+|S_1(x) \cap A|+\frac{2}{n}|S_2(x) \cap A|$.
\begin{definitionn}\label{d:weighball}
A pair $(T_{+},T_{-})$ of disjoint subsets of ${\scriptscriptstyle\mathrm{V}}{H(n,q)}$ is called an extended perfect code in $H(n,q)$ if for any vertex $x$ we have $w_{T_{+}}(x)=w_{T_{-}}(x) \le 1$.
\end{definitionn}
\begin{lemman}\label{l:dist4}
Let $(T_{+},T_{-})$ be an extended perfect bitrade according to Definition~\ref{d:weighball}.
Then $T_{+}$ and $T_{-}$ are codes with code distance $4$.
\end{lemman}
\begin{proof}
Let there be vertices $x$ и $y$ in $T_{+}$
such that $d(x,y) \le 3$.
Therefore, there is a path $x=z_0,z_1,\ldots, z_l=y$, where $l \le 3$.
So, $w_{+}(z_1) \ge 1 + 2/n$ and we have contradiction Definition~\ref{d:weighball} of bitrade.
For $T_{-}$ the proof is similar.
\end{proof}
For a pair $(T_{+},T_{-})$ of subsets of ${\scriptscriptstyle\mathrm{V}}{H(n,q)}$ and $x \in {\scriptscriptstyle\mathrm{V}}{H(n,q)}$,
denote $w_{+}(x)=w_{T_{+}}(x)$, $w_{-}(x)=w_{T_{-}}(x)$.
\begin{definitionn}\label{d:eigensubspace}
A pair $(T_{+},T_{-})$ of disjoint subsets of ${\scriptscriptstyle\mathrm{V}}{H(n,q)}$ is called an extended perfect bitrade in $H(n,q)$,
if $T_{+}$ and $T_{-}$ are codes with code distance $4$ and
the characteristic $(0,\pm 1)$-function of pair belongs to the direct sum
of eigenspaces, corresponding to eigenvalues $-n$ and $q-2$.
\end{definitionn}
\begin{definitionn}\label{d:bochka}
A pair $(T_{+},T_{-})$ of disjoint subsets of ${\scriptscriptstyle\mathrm{V}}{H(n,q)}$ is called an extended perfect bitrade in $H(n,q)$,
if for any $x \in {\scriptscriptstyle\mathrm{V}}{H(n,q)}$ and $i \in \{1,\ldots,n\}$ it holds
$|C_{x,i} \cap T_{+}|=|C_{x,i} \cap T_{-}| \le 1$
\end{definitionn}
\begin{lemman}\label{l:weight1}
Let $A$ be a code in $H(n,q)$ with code distance $4$. Then for any vertex $x$ of $H(n,q)$
\begin{enumerate}
\item $w_A(x) \le 1$.
\item If $w_A(x)=1$, then either $|S_0(x) \cap A|=1$, $|S_1(x) \cap A|=0$, $|S_2(x) \cap A|=0$,
or $|S_0(x) \cap A|=0$, $|S_1(x) \cap A|=1$, $|S_2(x) \cap A|=0$,
or $|S_0(x) \cap A|=0$, $|S_1(x) \cap A|=0$, $|S_2(x) \cap A|=n/2$.
\end{enumerate}
\end{lemman}
\begin{proof}
If $x \in A$, there are no vertices from $A$ in $(S_1(x) \cup S_2(x))$.
If $x \not\in A$ and $x$ is adjacent with some vertex from $A$, then
$|S_0(x) \cap A|=0$, $|S_1(x) \cap A|=1$, $|S_2(x) \cap A|=0$.
In the other case, the statement follows from the fact that
there are at most $n/2$ vertices in $S_2(x)$ with pairwise distances not less than $4$.
\end{proof}
\section{Equivalence of the definitions}
Since all definitions of bitrade are symmetric with respect to $T_{+}$ and $T_{-}$,
we will omit one of symmetric cases in the proofs of this section.
\begin{predln}\label{p:23}
The definitions~\ref{d:proection} and \ref{d:weighball} are equivalent.
\end{predln}
\begin{proof}
$~\ref{d:weighball} \Longrightarrow \ref{d:proection}$.
Let $(T_{+},T_{-})$ be an extended perfect bitrade in $H(n,q)$ according to Definition~\ref{d:weighball}.
Consider its $i$-projection for arbitrary $i$, a pair $(T'_{+},T'_{-})$.
Let us prove that this pair is a perfect bitrade in $H(n-1,q)$.
$T'_{+}$ and $T'_{-}$ are codes with code distance not less than $3$
(since the code distance of the projection of some code $A$ decreases at most by $1$
with respect to the code distance of $A$).
For a vertex $x \in H(n-1,q)$, denote by $c_x$ the clique $\{x^a_i:a \in \mathbb Z_q\}$ in $H(n,q)$
(pre-image of $x$ in the projection) and denote by $c_x(y)$ the vertex in the clique
that has $y \in \mathbb Z_q$ in position $i$.
Let $x \in T'_{+}$.
So, $c_x(y) \in T_{+}$ for some $y \in \mathbb Z_q$.
From the code distance we have $|S_1(x) \cap T'_{+}|=0$.
Suppose that $|S_1(x) \cap T'_{-}|=0$.
Then the vertex $c_x(y)$ has no neighbours in $T_{-}$.
Therefore, there are $n/2$ vertices from $T_{-}$ in $S_2(c_x(y))$
(by Lemma~\ref{l:weight1}).
Without loss of generality, assume that $c_x(y)$ is the zero vertex.
Since $T_{-}$ has code distance $4$,
the supports of nonzero positions do not intersect.
Hence, the cardinality of the union of the supports equals $n$.
But $i$ does not belong to this union
(as in this case, one of the vertices belongs to a clique $c_z$ such that $d(z,x)=1$).
Therefore, the union of the supports has cardinality at most $n-1$, and we have contradiction.
Let $x \not\in (T'_{+} \cup T'_{-})$.
If there is a vertex from $T_{+}$ and a vertex from $T_{-}$ in the clique $c_x$,
then from the code distance we have $|S_1(x) \cap (T'_{-} \cup T'_{+})|=0$.
Assume that there are no vertices from $(T_{+} \cup T_{-})$ in $c_x$.
Suppose that there is a vertex $z$ from $T'_{+}$ and there are no vertices from $T'_{-}$ in the neighbourhood of $x$.
Then $c_z(y) \in T_{+}$ for some $y \in \mathbb Z_q$; hence, $w_{+}(c_x(y))=1$.
Without loss of generality, we assume that $c_x(y)$ is the zero vertex.
So, there are $n/2$ vertices from $T_{-}$ in $S_2(c_x(y))$ and the supports of nonzero positions do not intersects.
As before, we have that $i$ does not belong to the union of these supports.
Since $n-1$ positions remain, we have the contradiction.
$\ref{d:proection} \Longrightarrow \ref{d:weighball}$.
Let $(T_{+},T_{-})$ be an extended perfect bitrade in $H(n,q)$ according to Definition~\ref{d:proection}.
Consider an arbitrary vertex $x$.
Without loss of generality, we assume that it is the zero vertex.
Let us prove that $w_{+}(x)=w_{-}(x)$.
Consider cases.
Suppose $x \in T_{+}$.
In this case, $w_{+}(x)=1$.
If there is a vertex from $T_{-}$ in the neighbourhood of $x$, then $w_{-}(x)=1$.
Otherwise, let $y_1,\ldots,y_k$ be vertices from $T_{-}$ at the distance $2$ from $x$ and
let $I_j$ be the support of nonzero positions of $y_j$.
The supports $I_1,\ldots,I_j$ do not intersect.
Let $i$ does not belong to the union of their supports.
Consider the $i$-projection.
The projection of $x$ has no neighbours from one part of perfect bitrade that contradicts the definition of perfect bitrade.
Therefore, $k=n/2$ and $w_{-}(x)=1$.
Let $x \not\in (T_{+} \cup T_{-})$.
If there is a vertex from $T_{+}$ and a vertex from $T_{-}$ in the neighbourhood of $x$, then $w_{+}(x)=w_{-}(y)=1.$
Suppose that $|S_1(x) \cap T_{+}|=1$ and $|S_1(x) \cap T_{-}|=0$.
As before, there are $n/2$ vertices from $T_{-}$ at the distance $2$ from $x$, hence, $w_{-}(x)=1$.
If there are no vertices from $(T_{+} \cup T_{-})$ in the neighbourhood of $x$, then analogously
there is the same number of vertices from $T_{+}$ and $T_{-}$ at the distance $2$ from $x$
(as the union of the supports of nonzero positions is the same in both cases; otherwise,
there is some position $j$ such that $x$ has a neighbour only in one part of the perfect bitrade in the $j$-projection).
Therefore, in all cases we have $w_{+}(x)=w_{-}(x)$, and $(T_{+},T_{-})$
is an extended perfect bitrade according to Definition ~\ref{d:weighball}.
\end{proof}
\begin{predln}\label{13}
The Definitions~\ref{d:matrixeq} and \ref{d:weighball} are equivalent.
\end{predln}
\begin{proof}
$\ref{d:weighball} \Longrightarrow \ref{d:matrixeq}$
Let $(T_{+},T_{-})$ be an extended perfect bitrade in $H(n,q)$ according to Definition~\ref{d:weighball}.
Let us construct a matrix $F$ that satisfy Definition~\ref{d:matrixeq}.
The first column of $F$ is the characteristic $(0,\pm 1)$ function of $(T_{+},T_{-})$.
For a vertex $v$ denote by $f(v)$ the $v$-row of $F$.
Let $x \in T_{+}$. If there is a vertex from $T_{-}$ in the neighbourhood of $x$ then define $f(x)=(+1,-1,0)$,
otherwise, $f(x)=(+1,0,-1)$.
Let $x \in T_{-}$.
Analogously, if there is a vertex from $T_{+}$ in the neighbourhood of $x$ then define $f(x)=(-1,+1,0)$,
otherwise, $f(x)=(-1,0,+1)$.
Let $x \not\in (T_{+} \cup T_{-})$. If $|S_1(x) \cap T_{+}| = |S_1(x) \cap T_{-}|$, then $f(x)=(0,0,0)$.
Otherwise, $f(x)=(0,+1,-1)$ if $|S_1(x) \cap T_{+}|=1$ and $f(x)=(0,-1,+1)$ if $|S_1(x) \cap T_{-}|=1$.
Thus, we defined the matrix $F$.
It remains to prove that $AF=FS$.
Let $x$ be an arbitrary vertex of $H(n,q)$.
Without loss of generality, we assume that $x$ is the zero vertex.
The neighbourhood of $x$ is the union of $n$ cliques of cardinality $q-1$; denote them by $N_1,\ldots,N_n$.
Consider some cases depending on $f(x)$.
Let $f(x)=(+1,-1,0)$.
By definition, $x \in T_{+}$,
there is a vertex $y \in T_{-}$ in $S_1(x)$, and
there are no vertices from $(T_{-} \cup T_{+})$ in $S_2(x)$.
The vertex $y$ belongs to $N_i$ for some $i$ and $f(y)=(-1,+1,0)$.
The remaining vertices of $N_i$ have value $(0,0,0)$.
The remaining $(n-1)(q-1)$ vertices from the neighbourhood have value $(0,+1,-1)$.
Hence, $AF_x=(-1,+1,0)+(n-1)(q-1)(0,+1,-1)=(-1,(n-1)(q-1)+1,-(n-1)(q-1))$.
On the other hand, $(+1,-1,0)S=(-1,(n-1)(q-1)+1,-(n-1)(q-1))$.
Let $f(x)=(+1,0,-1)$.
By definition, $x \in T_{+}$,
there are no vertices from $T_{-}$ in $S_1(x)$, and
there are $n/2$ vertices from $T_{-}$ in $S_2(x)$.
So, in each clique $N_i$ there is exactly one vertex adjacent to some vertex from $T_{-}$ .
Therefore, there are $n$ vertices with value $(0,0,0)$ and $n(q-2)$ vertices with value $(0,+1,-1)$ in the neighbourhood of $x$.
Hence, $AF_x=n(q-2)(0,+1,-1)$. On the other hand, $(+1,0,-1)S$ equals $(0,n(q-2),-n(q-2))$.
Let $f(x)=(0,+1,-1)$.
By definition, $x \not\in (T_{+} \cup T_{-})$,
there is a vertex $y \in T_{+}$ in $S_1(x)$ and
there are $n/2$ vertices from $T_{-}$ in $S_2(x)$.
The vertex $y$ belongs to $N_i$ for some $i$.
If $y$ is adjacent to some vertex from $T_{-}$,
then there is a vertex $y$ with value $(+1,-1,0)$,
$q-2$ vertices in $N_i$ with value $(0,+1,-1)$ (not adjacent with vertices from $T_{-}$),
$n-1$ vertices with value $(0,-1,+1)$ in the neighbourhood of $x$.
All remaining vertices from the neighbourhood have value $(0,0,0)$ .
Hence, $AF_x=(+1,-1,0)+(q-2)(0,+1,-1)+(n-1)(0,-1,+1)=(+1,-n+q-2, n-q+1)$.
If $y$ is not adjacent with any vertex from $T_{-}$,
then the neighbourhood of $x$ containing a vertex $y$ with value $(+1,0,-1)$,
$q-3$ vertices in $N_i$ with value $(0,+1,-1)$, and $n-1$ vertices with value $(0,-1,+1)$.
All remaining vertices from the neighbourhood have value $(0,0,0)$.
Hence, $AF_x=(+1,0,-1)+(q-3)(0,+1,-1)+(n-1)(0,-1,+1)=(+1, q-n-2, n-q+1)$.
This value is also equal to $(0,+1,-1)S$.
Let $f(x)=(0,0,0)$. In this case, by definition, $x \not\in (T_{+} \cup T_{-})$.
Suppose that there are $y \in T_{+}$ and $z \in T_{-}$ in the neighbourhood of $x$.
Let $y$ and $z$ belong to the same clique $N_i$ for some $i$.
So, $f(y)=(+1,-1,0)$, $f(z)=(-1,+1,0)$ and all remaining vertices from the neighbourhood have value $(0,0,0)$.
Hence, $AF_x=(0,0,0)$.
Let $y$ and $z$ belong to different cliques.
So, the neighbourhood of $x$ contains the vertex $y$ with value $(+1,0,-1)$, the vertex $z$ with value $(-1,0,+1)$,
$q-2$ vertices with value $(0,+1,-1)$, $(q-2)$ vertices with value $(0,-1,+1)$.
All remaining vertices from the neighbourhood have value $(0,0,0)$.
Hence, $AF_x=(0,0,0)$.
It remains to consider the case when there are no vertices from $(T_{+} \cup T_{-})$ in the neighbourhood of $x$.
In this case, the number of vertices in $S_1(x)$
that adjacent to some vertex from $T_{+}$ and not adjacent with any vertex from $T_{-}$ is equal
to the number of vertices in $S_1(x)$
that adjacent to some vertex from $T_{-}$ and not adjacent with any vertex from $T_{+}$.
All remaining vertices from the neighbourhood have value $(0,0,0)$.
Hence, $AF_x=(0,0,0)$.
This is also equal to $(0,0,0)S=(0,0,0)$.
$\ref{d:matrixeq} \Longrightarrow \ref{d:weighball}$
Let $(T_{+},T_{-})$ be an extended perfect bitrade in $H(n,q)$ according to Definition~\ref{d:matrixeq}.
Consider an arbitrary vertex $x$.
Without loss of generality, we assume that $x$ is the zero vertex.
Let us prove that $w_{+}(x)=w_{-}(x)$.
Remind that by definition $T_{+}$ and $T_{-}$ are codes with code distance $4$, so,
we can use Lemma~\ref{l:weight1}.
Consider some cases depending on the value of $f(x)$.
Let $f(x)=(+1,-1,0)$.
So, $x \in T_{+}$ and $w_{+}(x)=1$.
From the matrix equation, in the vector $AF_x$ the value in the first position equals $-1$.
Therefore, there is a vertex from $T_{-}$ in the neighbourhood of $x$, and hence $w_{-}(x)=1$.
Let $f(x)=(0,+1,-1)$ ($f(x)=(0,-1,+1)$).
So, there is a vertex $y$ from $T_{+}$ ($T_{-}$) and there are no vertices from $T_{-}$ ($T_{+}$) in the neighbourhood of $x$.
So, $w_{+}(x)=1$ ($w_{-}(x)=1$).
Let $y \in N_i$, and let $z_1,\ldots,z_k$ be vertices from $T_{-}$ in $S_2(x)$.
Let $f(y)=(+1,-1,0)$.
In this case, excluding $y$, the neighbourhood of $x$ contains $(q-2)$ vertices with value $(0,+1,-1)$,
$2k-1$ vertices with value $(0,-1,+1)$, and the remaining vertices have value $(0,0,0)$.
Let $f(y)=(+1,0,-1)$.
In this case, excluding $y$, the neighbourhood of $x$ contains
$q-3$ vertices with value $(0,+1,-1)$,
$2k-1$ vertices with value $(0,-1,+1)$, and the remaining vertices have value $(0,0,0)$.
In both cases, from the matrix equation we have $2k=n$ and $w_{-}(x)=1$.
Let $f(x)=(+1,0,-1)$.
So, $w_{+}(x)=1$ and there are no vertices from $T_{-}$ in the neighbourhood of $x$.
Hence, the vertices from the neighbourhood have values $(0,0,0)$ or $(0,+1,-1)$.
In the other hand, the sum of these values is equal to $(0,n(q-2),-n(q-2))$.
Hence, the neighbourhood of $x$ contains $n$ vertices that has neighbour in $T_{-}$.
Since each vertex from $S_2(x)$ is adjacent with $2$ vertices from $S_1(x)$,
there are $n/2$ vertices from $T_{-}$ in $S_1(x)$; hence, $w_{-}(x)=1$.
Let $f(x)=(0,0,0)$.
If there is a vertex from $T_{+}$ and a vertex from $T_{-}$ in the neighbourhood of $x$, then $w_{-}(x)=w_{+}(x)=1$.
Otherwise, the number of vertices with value $(0,+1,-1)$ is equal to
the number of vertices with value $(0,-1,+1)$.
All remaining vertices have value $(0,0,0)$.
Therefore, $S_2(x)$ has the same number of vertices from $T_{+}$ and $T_{-}$; i.e. $w_{+}(x)=w_{-}(x)$.
\end{proof}
Consider two functions $f: {\scriptscriptstyle\mathrm{V}}{H(n,q)} \to \mathbb C$ and $\varphi: {\scriptscriptstyle\mathrm{V}}{H(1,q)} \to \mathbb C$.
Define the function $f *_i \varphi: {\scriptscriptstyle\mathrm{V}}{H(n-1,q)} \to \mathbb C$ in the following way:
$f *_i \varphi (x)=\sum_{y=x^a_i: a \in \mathbb Z_q}f(y)\overline{\varphi(a)}$.
\begin{lemman}\label{l:fiproekt}
Let $f \in U_j(n,q)$, and let $\psi \equiv 1$ be a constant function on $H(1,q)$. Then
\begin{enumerate}
\item \cite{MV:eq2eigen} If $j \ne n$, then $f *_i \psi \in U_{j}(n-1,q)$ for any $i \in \{1,\ldots,n\}$.
\item \cite{MV:eq2eigen} If $j=n$, then $f *_i \equiv 0$ for any $i \in \{1,\ldots,n\}$.
\item If $j \ne n$, then there is $i$ such that $f*_i \psi \not\equiv 0$
\end{enumerate}
\end{lemman}
\begin{proof}
Let $j \ne n$.
Denote by $c_{x,i}$ the following set of vertices $\{x+e^a_i:a \in \mathbb Z_q\}$,
which form a clique in $H(n,q)$.
Denote by $f_{x,i}$ the restriction of $f$ on the set $c_{x,i}$.
If $f*_i \psi \equiv 0$ for any $i$,
then $f_{x,i}$ is an eigenfunction of $H(1,q)$ with eigenvalue $-1$ for any $x$ and $i$
(indeed, $(g,\psi)=0$ for any $g \in U_1(1,q)$ and $(h,\psi) \ne 0$ for any $h \in U_0(1,q)$ such that $h \not\equiv 0$ ).
In this case, $f$ is eigenfunction of $H(n,q)$ with eigenvalue $\lambda_n=-n$
and we have a contradiction.
\end{proof}
The following fact is well known and straightforward.
\begin{lemman}\label{l:perfbitreigen}
A pair $(T_{+},T_{-})$ of disjoint subsets of ${\scriptscriptstyle\mathrm{V}}{H(n,q)}$ is a perfect bitrade in $H(n,q)$ if and only if
$T_{+}$ and $T_{-}$ are codes with code distance $3$ and
the characteristic
$(0,\pm 1)$-function $f$ belongs to $U_l(n,q)$, where
$\lambda_l=-1$ and, respectively, $l=\frac{(n-1)(q-1)+1}{q}$.
\end{lemman}
\begin{predln}\label{p:24}
The definitions~\ref{d:proection} and \ref{d:eigensubspace} are equivalent.
\end{predln}
\begin{proof}
$\ref{d:eigensubspace} \Longrightarrow \ref{d:proection}$
Let $(T_{+},T_{-})$ be an extended perfect bitrade in $H(n,q)$ according to Definition~\ref{d:eigensubspace}.
Let $f$ be the characteristic $(0,\pm 1)$-function of the bitrade.
By definition, $f \in U_n(n,q) \oplus U_l(n,q)$, where $\lambda_l=q-2$, i.e. $l=\frac{(n-1)(q-1)+1}{q}$.
Therefore, $f=g+h$, where $g \in U_n(n,q)$, $h \in U_l(n,q)$.
Let $i \in \{1,\ldots,n\}$ be an arbitrary position, and
let $\psi \equiv 1$ be a constant function on $H(1,q)$.
We have, $f *_i \psi = g *_i \psi + h *_i \psi = h *_i \psi \in U_{l}(n-1,q)$.
By Lemma~\ref{l:perfbitreigen}, $f *_i \psi$ is the characteristic function of some perfect bitrade in $H(n-1,q)$.
On the other hand, the pair $(T'_{+},T'_{-})$ corresponding to $f *_i \psi$
is a projection of $(T_{+}, T_{-})$.
Therefore, $(T_{+}, T_{-})$ is an extended perfect bitrade by Definition~\ref{d:proection}.
$\ref{d:proection} \Longrightarrow \ref{d:eigensubspace}$
Let $(T_{+},T_{-})$ be an extended perfect bitrade in $H(n,q)$ according to Definition~\ref{d:proection}, and
let $f$ be its characteristic $(0,\pm 1)$-function.
Let $f \in U_{j_1}(n,q) \oplus \ldots \oplus U_{j_k}(n,q)$.
So, $f=g_1+\ldots+g_k$, where $g_t \in U_{j_t}$.
Denote $l=\frac{(n-1)(q-1)+1}{q}$ ($\lambda_l=-1$ in $H(n-1,q)$ for this $l$).
Suppose that there is $m \in \{j_1,\ldots,j_k\} \backslash \{l,n\}$.
By Lemma~\ref{l:fiproekt}, there is such $r$ that $g_m *_r \psi \not\equiv 0$.
Hence, $f *_r \psi = g_1 *_r \psi + \ldots + g_k *_r \psi=h_1+\ldots+h_k$,
where $h_t \in U_{j_t}(n-1,q)$ and $h_m \not\equiv 0$. Therefore, $f*_r \psi \not\in U_{l}(n-1,q)$,
and we have a contradiction with the fact that $f *_r \psi$ is a characteristic function of some bitrade.
\end{proof}
\begin{predln}\label{p:235}
The Definitions~\ref{d:proection},~\ref{d:weighball}, and \ref{d:bochka} are equivalent.
\end{predln}
\begin{proof}
$\ref{d:bochka} \Longrightarrow \ref{d:proection}$.
Let $(T_{+},T_{-})$ be an extended perfect bitrade in $H(n,q)$ according to Definition~\ref{d:bochka}, and
let $i \in \{1,\ldots,n\}$ be an arbitrary position.
Consider the $i$-projection of the bitrade; denote it by $(T'_{+},T'_{-})$.
Let $x$ be an arbitrary vertex of $H(n-1,q)$ and $B(x)$ be radius-$1$ ball in $H(n-1,q)$.
The pre-image of $B(x)$ is a cylinder $C_{y,i}$, where $y=x^0_i$.
If there are no vertices from $(T_{+} \cup T_{-})$ in $C_{y,i}$,
then there are no vertices from $(T'_{+} \cup T'_{-})$ in $B(x)$.
Let $C_{y,i} \cap T_{+} =\{v\}$, $C_{y,i} \cap T_{-}=\{u\}$.
If the vertices $u$ and $v$ differ only in position $i$,
then there are no vertices from $(T'_{+} \cup T'_{-})$ in $B(x)$; otherwise,
we have $|B(x) \cap T'_{+}|=|B(x) \cap T'_{-}|=1$.
$\ref{d:weighball} \Longrightarrow \ref{d:bochka}$.
Let $(T_{+},T_{-})$ be an extended perfect bitrade in $H(n,q)$ according to Definition~\ref{d:weighball}.
Consider $C_{x,i}$ for arbitrary $x \in {\scriptscriptstyle\mathrm{V}}{H(n,q)}$ and $i \in \{1,\ldots,n\}$.
Since $T_{+}$ and $T_{-}$ are codes with code distance $4$ by Lemma~\ref{l:dist4}, there
are at most one vertex from $T_{+}$ and at most one vertex from $T_{-}$ in $C_{x,i}$.
It is sufficient to prove that if there is a vertex $y$ from $T_{+}$ in $C_{x,i}$,
then there is a vertex from $T_{-}$ in $C_{x,i}$.
Since $y \in T_{+}$, there is a vertex $z$ from $C_{x,i}$ such that $w_{+}(z)=1$ and
$z-x=e^a_i$ for some $a \in \mathbb Z_q$.
So, we have that either $z$ is adjacent to some vertex $u$ from $T_{-}$,
or there are $n/2$ vertices from $T_{-}$ at the distance $2$ from $z$.
In the first case, $u$ also belongs to $C_{x,i}$.
In the second case, there is a vertex $v \in T_{-}$ at the distance $2$ from $z$ such that
$z$ and $v$ differ in position $i$ and some other position
(otherwise, there are vertices $v_1$ и $v_2$ that differ from $z$ in the same position,
but in this case $d(v_1,v_2) \le 3$).
So, $v$ also belongs to $C_{x,i}$.
$\ref{d:proection} \Longrightarrow \ref{d:weighball}$.
By Proposition~\ref{p:23}.
\end{proof}
From Propositions~\ref{p:23} --~\ref{p:235} we have the following.
\begin{theoreman}
The Definitions~\ref{d:matrixeq} --~\ref{d:bochka} are equivalent.
\end{theoreman}
A pair $(T_{+},T_{-})$ of disjoint subsets of ${\scriptscriptstyle\mathrm{V}}{H(n,q)}$ is called an \emph{extended perfect bitrade} in $H(n,q)$
if it satisfies one of definitions~\ref{d:matrixeq} - \ref{d:bochka}.
\begin{theoreman}
If there is an extended perfect bitrade $(T_{+},T_{-})$ in $H(n,q)$, then
\begin{enumerate}
\item $n$ is even and
\item $n=lq+2$ for some $l \in \{0,1,\ldots\}$.
\end{enumerate}
\end{theoreman}
\begin{proof}
1) Let $x \in T_{+}$.
Then there is a vertex $y \in S_1(x)$ such that $y$ is not adjacent to any vertex from $T_{-}$.
Since, $w_{+}(y)=1$, there are $n/2$ vertices from $T_{-}$ in $S_2(y)$.
Hence, $n$ is even.
2) By definition, $n(q-1)-iq$ equals $q-2$ for some $i$.
Hence, $n\equiv2 \mod q$.
\end{proof}
\providecommand\href[2]{#2} \providecommand\url[1]{\href{#1}{#1}}
\def\DOI#1{{\small {DOI}:
\href{http://dx.doi.org/#1}{#1}}}\def\DOIURL#1#2{{\small{DOI}:
\href{http://dx.doi.org/#2}{#1}}}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,501
|
Movie Review: The Resurrection of Jake "The Snake" Roberts
Jake the Snake Roberts is a legend in the wrestling industry. For 2 things. His in ring performance and his known substance abuse. This film was not what I expected but it turned into an uplifting story of battling ones demons, overcoming addiction, and making things right with everyone he wronged.
Jake has gotten a bad rap over the years. From mid carding WrestleMania's with 30k people in attendance to his current state of wrestling in front of 300 people in a high school gymnasium, this proves he is the poster child of throwing one's life/talents away. But, he has also been exploited more than once by the media and what little credibility he had left was destroyed by the movie, "Beyond the Mat." Understanding that is a matter of debate depending on who you ask, at the end of the day the facts were correct.
A) Jake Roberts was an addict to alcohol and cocaine.
B) He ruined with marriage and wrestling career because of it.
C) He was an end product of being raised poorly as a child
I saw Jake years ago at a small wrestling event in the Niagara Region and he looked horrible then. But when I saw him in this film, he looked worse. I didn't even think it was possible. He returned to WWE programming for a small stint about 10 years ago and at that time the WWE released a documentary called "Pick Your Poison," I honestly believed that Jake might finally have woken up to sobriety. No Such luck. This documentary starts in 2011/2012, years well after the fact. The film follows around DDP(Diamond Dallas Page) as he picks up Jake Roberts and helps him to the road of recovery. He facilitates Jake's living arrangements, diet and even introduces him to DDP Yoga. This was not an easy road as he stumbles off the path of sobriety multiple times. He sets the goal to wrestle in the royal rumble after he goes 100% clean living and gets back into shape.
This film was a real tearjerker. If you have watched any of the other the documentaries, you did see the human side of Jake before, but never the 10% sober version of him. Small spoiler though, along Jake's path and unexpected "Bad Guy" passenger joins the journey. I would strongly suggest giving this film a watch and I pray that Jake kept his sobriety.
Jake if you stayed on the path, we are very proud of you!
#wwe #jakeroberts #thesnake #DDT #documentary
The Most Explicit Episode Ever *** Uncut Version ***
Drinking In The Park: Let It Be The Firemen
Drinking In The Park: The Long and Windup Road
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,807
|
Now for our most demanding test settings, using the ultra quality preset at 2560x1600. Here the GeForce GTX 260 averaged 39fps with a minimum of 31fps. There were 13 graphics cards that were slower than the GeForce GTX 260, all of which failed to provide a minimum frame rate equal to or greater than 30fps. That said, the $200 GeForce GTX 460 is all gamers will need to fully appreciate StarCraft II, as it was just as fast as the GeForce GTX 480 and Radeon HD 5870 graphics cards.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,983
|
JOIN, RENEW your memberships, CONTRIBUTE here!
See what is happening in "real time" with up-to-the-minute Twitter and Facebook posts. No accounts required!
Super easy online voter registration. Instructions on how and where to register if you prefer to go in person.
Take Action - Join In!
Volunteer to help your community. Come to our meetings and social gatherings.
Come Join In! Find out what is happening with TheNewBlue of Okanogan County!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 169
|
Q: Unique "key" prop warning in render method - ReactJS My app have a warning say that: Each child in an array or iterator should have a unique "key" prop. Check the render method of 'SortableTable' Hear is my SortableTable file:
import React from 'react';
import {Link} from 'react-router';
import {v4 as uuid} from 'node-uuid';
export default class SortableTable extends React.Component {
render() {
return (
<table style={{width: '100%'}} className={this.props.className} id={this.props.id}>
<thead>
<tr>
{this.props.headers.map(this.generateHeaders)}
</tr>
</thead>
<tbody>
{this.props.children}
</tbody>
</table>
);
}
generateHeaders = (value) => {
if (Object.keys(value).length === 0) return
let sort, colspan
if(value.sort) {
let {query} = this.props;
let nQuery, title, icon, colspan = 1;
if(query.sort === value.sort && query.sortDirection === 'desc') {
nQuery = Object.assign({}, query, {sort: value.sort, sortDirection: 'asc', page: 1})
title = 'asc';
icon = String.fromCharCode(0xe630)
} else {
nQuery = Object.assign({}, query, {sort: value.sort, sortDirection: 'desc', page: 1})
title = 'desc';
icon = String.fromCharCode(0xe62d)
}
sort = <Link to={this.props.link} query={nQuery} className="icon order active" title={title} data-icon={icon} />
}
let className = value.className ? value.className : ''
if(value.colspan) {
colspan = value.colspan
}
return <th className={className} colSpan={colspan}><span>{value.name}</span>{sort}</th>
}
Can someone show me how to set key prop to resolve this warning?
A: The idea is to have a key attribute with a unique value. You can use index, the second argument of Array#map callback:
generateHeaders = (value, index) => {
// ...
return <th key={index} className={className} colSpan={colspan}><span>{value.name}</span>{sort}</th>
}
Alternatively, if value object has a unique property like id, you may use it instead.
A: The <th> element returned by generateHeaders needs to have a key attribute set on it, since you're returning an array of them from your render method.
The simplest way to achieve this is to use the index:
generateHeaders = (value, index) => {
// ...
return <th key={index} //...
This will make the warning go away but it's better to use a unique property of each value, if one exists. This is because React uses the key to determine whether two items are the same. If you're using the array index and you insert a new value somewhere into the array, the indexes will change, causing the items to be re-rendered unnecessarily.
A: You should insert key into th tag:
generateHeaders = (value, index) => {
...
return <th key={index} className={className} colSpan={colspan}><span>{value.name}</span>{sort}</th>
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,698
|
{"url":"https:\/\/brainly.com\/question\/163239","text":"2014-10-25T23:53:06-04:00\n3z-4=6z-17\n-3z-4=-17 (subtracted the 6z from both sides)\n-3z=-13 (added the 4 to both sides)\nz= (divided both sides by -3z)\nz=\n2014-10-26T08:19:04-04:00\n3z - 4 = 6z - 17. \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 First try to take all the zs to one side of the equation\npreferrably\u00a0to the left. And the sign of any expression\nchanges on crossing the equality sign.\n\n3z-6z = -17+4\n-3z \u00a0 \u00a0 = \u00a0+4-17\n-3z \u00a0 \u00a0\u00a0= \u00a0 - 13. \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 Divide both sides by -3.\n-3z\/-3 \u00a0 = \u00a0-13\/-3\n\nz = 13\/3.","date":"2017-01-19 02:50:36","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\": 0.8022774457931519, \"perplexity\": 13256.510973526025}, \"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-2017-04\/segments\/1484560280425.43\/warc\/CC-MAIN-20170116095120-00226-ip-10-171-10-70.ec2.internal.warc.gz\"}"}
| null | null |
Q: Cannot retrieve initial cluster partitions from initial URIs [RedisURI [host='127.0.0.1', port=7001]] I am working on Lettuce cluster Java client. It is set up inside a bolt topology (Apache Strom). spout is reading data from kafka and passing it to bolt. However, when I am starting my topology, I am getting below error message and program terminated. Am I missing something? what is causing this?
Stack trace
29502 [Thread-17-RecommendationLettuceBolt-executor[2 2]] ERROR o.a.s.util - Async loop died!
io.lettuce.core.RedisException: Cannot retrieve initial cluster partitions from initial URIs [RedisURI [host='127.0.0.1', port=7001]]
at io.lettuce.core.cluster.RedisClusterClient.loadPartitions(RedisClusterClient.java:865) ~[lettuce-core-5.1.7.RELEASE.jar:?]
at io.lettuce.core.cluster.RedisClusterClient.initializePartitions(RedisClusterClient.java:819) ~[lettuce-core-5.1.7.RELEASE.jar:?]
at io.lettuce.core.cluster.RedisClusterClient.connect(RedisClusterClient.java:345) ~[lettuce-core-5.1.7.RELEASE.jar:?]
at com.projectName.indexer.lettuce.LettuceClusterClientProvider.getConnection(LettuceClusterClientProvider.java:72) ~[classes/:?]
at com.projectName.indexer.lettuce.LettuceCacheRepopulationHandler.openLettuceConnection(LettuceCacheRepopulationHandler.java:42) ~[classes/:?]
at com.projectName.indexer.bolts.RecommendationLettuceBolt.prepare(RecommendationLettuceBolt.java:35) ~[classes/:?]
at org.apache.storm.daemon.executor$fn__8058$fn__8071.invoke(executor.clj:795) ~[storm-core-1.0.2.jar:1.0.2]
at org.apache.storm.util$async_loop$fn__624.invoke(util.clj:482) [storm-core-1.0.2.jar:1.0.2]
at clojure.lang.AFn.run(AFn.java:22) [clojure-1.7.0.jar:?]
at java.base/java.lang.Thread.run(Thread.java:844) [?:?]
Caused by: io.lettuce.core.RedisConnectionException: Unable to establish a connection to Redis Cluster at [RedisURI [host='127.0.0.1', port=7001]]
at io.lettuce.core.cluster.topology.AsyncConnections.get(AsyncConnections.java:89) ~[lettuce-core-5.1.7.RELEASE.jar:?]
at io.lettuce.core.cluster.topology.ClusterTopologyRefresh.loadViews(ClusterTopologyRefresh.java:73) ~[lettuce-core-5.1.7.RELEASE.jar:?]
at io.lettuce.core.cluster.RedisClusterClient.doLoadPartitions(RedisClusterClient.java:871) ~[lettuce-core-5.1.7.RELEASE.jar:?]
at io.lettuce.core.cluster.RedisClusterClient.loadPartitions(RedisClusterClient.java:844) ~[lettuce-core-5.1.7.RELEASE.jar:?]
... 9 more
Input Code
private void init() {
redisUri = RedisURI.Builder
.redis(lettuceConfig.getLettuceClusterHost())
.withPort(lettuceConfig.getLettuceClusterPort())
.withTimeout(Duration.ofMillis(lettuceConfig.getLettuceClusterTimeout()))
.build();
}
public StatefulRedisClusterConnection getConnection() {
if (connection == null || !connection.isOpen()) {
redisClusterClient = RedisClusterClient.create(redisUri);
final ClusterTopologyRefreshOptions topologyRefreshOptions = ClusterTopologyRefreshOptions.builder()
.enablePeriodicRefresh(Duration.ofMinutes(BoltConstants.Lettuce.PERIODIC_REFRESH_TIME_IN_MIN))
.enableAdaptiveRefreshTrigger()
.build();
final ClusterClientOptions clusterClientOptions = ClusterClientOptions.builder()
.autoReconnect(true)
.topologyRefreshOptions(topologyRefreshOptions)
.build();
redisClusterClient.setOptions(clusterClientOptions);
connection = redisClusterClient.connect(SnappyCompressor.wrap(new StringCodec()));
log.info("Connected to Redis client lettuce. lettuce connection is up and running.");
}
return connection;
}
Environment
compile 'io.lettuce:lettuce-core:5.1.7.RELEASE'
A: The error was coming because I have not installed redis cluster locally on my machine.
I solved using these steps
https://redis.io/download and run below commands Installation
$ wget http://download.redis.io/releases/redis-5.0.5.tar.gz
$ tar xzf redis-5.0.5.tar.gz
$ cd redis-5.0.5
$ make //The binaries that are now compiled are available in the src directory
$ src/redis-server
Creating a Redis Cluster using the create-cluster script
Now check redis-5.0.5/utils/create-cluster directory in the Redis distribution. There is a script called create-cluster inside, it's a simple bash script. In order to start a 6 nodes cluster with 3 masters and 3 slaves just type the following commands:
$ create-cluster start // start
$ create-cluster create // create cluster
$ create-cluster stop // for stoping
$ create-cluster clean // clean all the cluster
You will see the following output in the terminal
$ ps aux |grep redis
300067846 28222 0.6 0.0 4379932 2612 ?? Ss 7:41PM 0:00.73 ../../src/redis-server *:30004 [cluster]
300067846 28227 0.5 0.0 4380956 2604 ?? Ss 7:41PM 0:00.71 ../../src/redis-server *:30006 [cluster]
300067846 28218 0.4 0.0 4381980 2676 ?? Ss 7:41PM 0:00.73 ../../src/redis-server *:30002 [cluster]
300067846 28216 0.4 0.0 4374812 2576 ?? Ss 7:41PM 0:00.50 ../../src/redis-server *:30001 [cluster]
300067846 28225 0.4 0.0 4380956 2632 ?? Ss 7:41PM 0:00.68 ../../src/redis-server *:30005 [cluster]
300067846 28220 0.3 0.0 4379932 2596 ?? Ss 7:41PM 0:00.52 ../../src/redis-server *:30003 [cluster]
300067846 85550 0.0 0.0 4309420 1128 ?? S 3:08PM 0:07.76 redis-server *:6379
see all master slaves nodes
../redis-5.0.5/utils/create-cluster$ cat nodes-3000*.conf
You can update the Redis cluster config. open vi create-cluster and update the following config as you needed.
// default config are these
CLUSTER_HOST=127.0.0.1
PORT=30000
TIMEOUT=2000
NODES=6
REPLICAS=1
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,452
|
Aaron Kaufman (born January 26, 1982) is an American television personality, racing driver, and owner of Arclight Fabrication, a Dallas enterprise that supplies aftermarket components for the Ford F-100 pickup.
He is the former lead mechanic for Gas Monkey Garage. He formerly starred in a one-season television show entitled Shifting Gears with Aaron Kaufman which premiered on March 5, 2018, featuring his new business with Arclight.
In September 2015, Kaufman competed in the Speed Energy Formula Off-Road (Stadium Super Trucks) series, driving a Toyo Tires-sponsored truck at the Sand Sports Super Show in Costa Mesa, California. He ran two rounds during the weekend, finishing fifth in both his heat races, followed by tenth- and eighth-place runs in the features. Kaufman struggled during the races, including one incident in which his truck partially rode on a K-rail; after the race, fellow driver Robby Gordon joked it was "some of the best, worst driving I have ever seen from the Toyo Tires driver."
Motorsports career results
Stadium Super Trucks
(key) (Bold – Pole position. Italics – Fastest qualifier. * – Most laps led.)
References
American company founders
Participants in American reality television series
Living people
1982 births
Stadium Super Trucks drivers
People from Crowley, Texas
21st-century American businesspeople
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,265
|
package org.onosproject.bmv2.api.runtime;
import java.util.Objects;
import static com.google.common.base.Preconditions.checkArgument;
import static com.google.common.base.Preconditions.checkNotNull;
/**
* An entry of a match-action table in a BMv2 device.
*/
public final class Bmv2TableEntry {
private static final int NO_PRIORITY_VALUE = -1;
private static final int NO_TIMEOUT_VALUE = -1;
private final String tableName;
private final Bmv2MatchKey matchKey;
private final Bmv2Action action;
private final int priority;
private final double timeout;
private Bmv2TableEntry(String tableName, Bmv2MatchKey matchKey,
Bmv2Action action, int priority, double timeout) {
this.tableName = tableName;
this.matchKey = matchKey;
this.action = action;
this.priority = priority;
this.timeout = timeout;
}
/**
* Returns a new BMv2 table entry builder.
*
* @return a new builder.
*/
public static Builder builder() {
return new Builder();
}
/**
* Returns the name of the table where this entry is installed.
*
* @return table name
*/
public final String tableName() {
return this.tableName;
}
/**
* Returns the match key of this table entry.
*
* @return match key
*/
public final Bmv2MatchKey matchKey() {
return matchKey;
}
/**
* Returns the action of this table entry.
*
* @return action
*/
public final Bmv2Action action() {
return action;
}
/**
* Returns true is the entry has a valid priority.
*
* @return true if priority is set, false elsewhere
*/
public final boolean hasPriority() {
return this.priority != NO_PRIORITY_VALUE;
}
/**
* Return the priority of this table entry.
*
* @return priority
*/
public final int priority() {
return priority;
}
/**
* Returns true is this table entry has a valid timeout.
*
* @return true if timeout is set, false elsewhere
*/
public final boolean hasTimeout() {
return this.timeout != NO_PRIORITY_VALUE;
}
/**
* Returns the timeout (in fractional seconds) of this table entry.
*
* @return a timeout vale (in fractional seconds)
*/
public final double timeout() {
return timeout;
}
@Override
public final int hashCode() {
return Objects.hash(matchKey, action, priority, timeout);
}
@Override
public final boolean equals(Object obj) {
if (this == obj) {
return true;
}
if (obj == null || getClass() != obj.getClass()) {
return false;
}
final Bmv2TableEntry other = (Bmv2TableEntry) obj;
return Objects.equals(this.matchKey, other.matchKey)
&& Objects.equals(this.action, other.action)
&& Objects.equals(this.priority, other.priority)
&& Objects.equals(this.timeout, other.timeout);
}
@Override
public final String toString() {
return com.google.common.base.MoreObjects.toStringHelper(this)
.addValue(matchKey)
.addValue(action)
.add("priority", priority)
.add("timeout", timeout)
.toString();
}
public static final class Builder {
private String tableName;
private Bmv2MatchKey matchKey;
private Bmv2Action action;
private int priority = NO_PRIORITY_VALUE;
private double timeout = NO_TIMEOUT_VALUE;
private Builder() {
// hide constructor
}
/**
* Sets the table name.
*
* @param tableName a string value
* @return this
*/
public Builder withTableName(String tableName) {
this.tableName = checkNotNull(tableName, "table name cannot be null");
return this;
}
/**
* Sets the match key.
*
* @param matchKey a match key value
* @return this
*/
public Builder withMatchKey(Bmv2MatchKey matchKey) {
this.matchKey = checkNotNull(matchKey, "match key cannot be null");
return this;
}
/**
* Sets the action.
*
* @param action an action value
* @return this
*/
public Builder withAction(Bmv2Action action) {
this.action = checkNotNull(action, "action cannot be null");
return this;
}
public Builder withPriority(int priority) {
checkArgument(priority >= 0, "priority cannot be negative");
this.priority = priority;
return this;
}
/**
* Sets the timeout.
*
* @param timeout a timeout value in fractional seconds
* @return this
*/
public Builder withTimeout(double timeout) {
checkArgument(timeout > 0, "timeout must be a positive non-zero value");
this.timeout = timeout;
return this;
}
/**
* Build the table entry.
*
* @return a new table entry object
*/
public Bmv2TableEntry build() {
return new Bmv2TableEntry(tableName, matchKey, action, priority,
timeout);
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,377
|
# Forgotten Footprints
Lost Stories in the Discovery of Antarctica
John Harrison
For Celia, who found me there
# Contents
Title Page
Dedication
PROLOGUE:
1: The Master Craftsman:
2: William Who?
3: The Admiral and the Boy Captain:
4: Charted in Blood:
5: The Circumnavigators:
6: Deception Island
7: The First Antarctic Night 1897– 99:
8: A Permanent Foothold:
9: The Great Escape
10: Antarctica with a Butler:
11: Red Water: Whaling
12: The Boss: Ernest Shackleton (1877–1922)
13: Living in Antarctica: The Bases
14: Who Owns Antarctica?
15: The Future of Antarctica
A Glossary of Specialist Polar and Maritime Words
Acknowledgements
About the Author
Copyright
### **Who is this book for?**
The history of Antarctica is largely written in magisterial biographies and coffee-table tomes that require three polar heroes to lift them. My own collection can swiftly be piled up into a fortress against the harshest blizzard. Some of the explorers are household names, bywords for bravery and suffering. But none of those heroes discovered Antarctica: neither Cook, Scott, Shackleton nor Amundsen. You would think it was a conspiracy. The Russians will tell you Admiral Bellingshausen discovered it, Americans make a claim for a sealing captain called Nathaniel Palmer. No.
It was a Briton no one has heard of. This book tells his tale and those of the other jobbing sailors who were the first to find Antarctica, and to start to chart its shape. There's a few heroes too, where they went to places we can visit. That is the second aim of the book: to describe the discovery and exploration of Antarctica through the places that feature in the visits of the cruise ships which now land over thirty thousand passengers a year on the last continent. It focuses on the Antarctic Peninsula down to the Antarctic Circle, the South Shetland Islands, and the fringe of the Weddell Sea; around 98% of tourists come here. There are some fine general histories but they give equal weight to expeditions and activities which took place on remote shores and in harsh interiors visited only by a handful of professional scientists and daring adventurers. This book gives the background to the things and places visitors actually see. Whether you are planning a trip, or you are just an armchair adventurer it will explain how mankind found the last continent and how we have treated it.
# PROLOGUE:
The Ghosts of Detaille Island
I did not want to go ashore. We had never been asked to start an Antarctic landing at midnight. I'd get no sleep before and very little after. The Expedition Leader was showboating for the guests, perhaps because they included a Belgian Prince and Baron von Gerlache, the grandson of the Belgian explorer Adrien. Another guest owned tankers, six hundred of them. At Ushuaia, the Argentine gateway to Antarctica, their entourage brought the airport to a halt, as the authorities tried to find parking places for their fifteen private jets.
I climbed into my Zodiac, a five-metre-long open inflatable powerboat, and was lowered into the bay to ferry ashore the descendants of Adrien de Gerlache and their friends. The guests had spent the evening partying and celebrating their farthest south, just below the Antarctic Circle. We landed on a tiny block of rock on which stood the old British Detaille Base, and explored the huts by torchlight. The season was advanced and night had returned. The days had become mortal again; they had a beginning and an end.
Detaille Base seemed more authentic in the torchlight, and I enjoyed picking my way round. In its few years of active life from 1956 to 1959, it was lit by fitful generators and hot-handled hurricane lamps. It was a time capsule from my childhood. On a table lay the August 1953 edition of _World Sports_ magazine. The middle-distance runner Christopher Chataway was on the cover, breasting the tape in a number 1 shirt. He was one of the athletes who would, the following spring, help Roger Bannister to run the first four-minute mile. Cupboard doors opened on tins whose primary-coloured fifties' cheerfulness was corroded by rust along the joints: the war was over, custard and porridge for all. The party enjoyed their visit; their boisterousness subsided to pleasant thoughtfulness. The Prince took off his mink-lined hat and absent-mindedly stroked the fur with his thumb. They left in trickles; I ran shuttles ferrying them back across the bay.
Each time I returned to the ship the bar sounds from the lounge high above the stern were more raucous. Each shore visit was more tranquil. The Expedition Leader asked if any of the Zodiac drivers wanted to be lifted. I said I was fine. I drove the last passengers back at 03:30, leaving the empty base to its ghosts, dropping the other staff at the gangway, before pulling away to prepare the Zodiac to be lifted by crane onto the top of the rear deck. I unfolded the web of three pairs of straps, coming to a central ring, then clipped the hooks onto the shackles at the front and centre of the boat. The back ones are fixed. I stowed my backpack where it would not tangle with them when I hitched the ring to the hook. Snug in my one-piece survival suit, I balanced, on my back, on the side pontoon of the boat. Above me, in the black blanket of sky, rents worked open, and the cover began to break up. Haphazard stars appeared. The radio crackled: 'John, we have slight problem with anchor.' When Polish officers speak English the articles disappear. 'Stand by fifteen minutes.'
She was an old Swedish ice-breaker built for coastal patrol, launched as the _Njord_ , the god of the sea, but renamed the _Polar Star_. She does not go to Antarctica now. Under the delightful Captain Jacek Mayer, who taught me to use a sextant, and whose first command was a square rigger ('She is in a museum now, I am still sailing.'), she hit an uncharted rock off Detaille Base in 2011 and ripped her belly open. The company went bust as she sat in a Canaries dockyard and the repair bill mounted.
Now I had turned off the 60 HP Yamaha outboard engine, I could hear the fans and motors which keep the ship alive, the tinkling party, and the Filipino sailors I had worked with for six years, Alan, Lito and Dallas, calling to each other tiredly in Tagalog. The water cooling my engine piddled into the sea. I felt the wavelets ping the rubber hull. I saw the stars brighten and then fog in the confusion of the cloudless sweep of the Milky Way: the edge of the galaxy opening up with light from times when men walked in animal skins. The ship was a solitary pool of warmth and light. I let my Zodiac drift away from her into the darkness at the edge of the bay. Her deck lights were a cluster in the sky, a constellation without a name: a new, unpredictable star sign. The bar music faded and all I could hear was the creases of ocean slipping under the boat, the cold air hosing through my throat, and the thump of my heart. The Zodiac turned slowly as she drifted, and I was the only soul alone in an empty world. The only man abroad in a continent. The last continent. But who discovered it? I didn't care. I was unspeakably happy.
# The Master Craftsman:
James Cook (1728–1779)
## VOYAGE I (1768–1771)
VOYAGE II (1772–1775)
VOYAGE III (1776–1779)
_A ditch-digger's son becomes one of the greatest navigators of his own age, or any other age, and is sent out three times to look for the one land he does not believe exists._
In a locked, unlit village shop, the gangly youth eased himself out from under the counter where he had slept the night. He was the son of a farm labourer who had left Scotland when it was reduced to desolation after the crushing of the Jacobite rebellion of 1715–16, and moved to the straggling hamlet of Marton in Yorkshire, in north-east England. Two years later, the labourer's wife gave birth to a son, James, who at eight years old was already helping his father clean ditches and cut hedges. A reputation for honest muscle earned the father a job as bailiff to the local Lord of the Manor, Thomas Skottowe, at a larger village: Great Ayton. Skottowe saw something in the boy and paid for him to attend school. To his humble parents, apprenticing James to a shopkeeper in the nearby fishing village of Staithes was a way to rise above a life of physical drudgery. But to Cook an apprenticeship to a draper-cum-grocer was just a genteel jail. One day a lady paid for goods with a brilliant silver shilling. When the shop closed, the youth read on it the letters SSC: South Sea Company. He put it in his pocket and replaced it with one of his own. The shopkeeper William Sanderson noticed it missing and accused Cook of theft. Cook's fervent denials were accepted, but Sanderson's lack of trust rankled, and Cook's pocket already held the shiny promise of another world. He asked to be released from the apprenticeship to go to sea. Sanderson agreed and Cook was bound as a seaman apprentice for three years with the Quaker family of John Walker of Whitby, a town famous for its ruined monastery, its whaling, and, for me, as the port where Dracula enters England.
The Walkers gave James a room of his own and paid his fees to attend sea school in the town. At this time London's hearths and furnaces devoured over a million tons of coal a year, much of it supplied by north-east England in a thousand sturdy, blunt-bowed colliers known as cats, carrying six hundred tons of coal. He first sailed in February 1747, aged eighteen, becoming a Master Mate in 1755, when he was offered his own command: the _Friendship_. But Cook's ambition had been shaped by reading the accounts of the great explorers of the Pacific. War with France was looming and he might be pressed into the Navy. With great courage he decided to meet fate on his own terms. He turned down promotion to a merchant captain and signed on as an Able Seaman in the Royal Navy. The Navy was so short of experienced men that Cook made First Mate in less than a month. He learned surveying during the successful war to break French rule in Quebec, and after the war he was tasked with surveying the islands off Newfoundland which the peace treaty had ceded to France. A century later Admiral Wharton said of Cook's charts 'their accuracy is truly astonishing.'
In Cook's day, the supposed southern continent _Terra Australis Incognita_ , lying in temperate latitudes and ripe for exploitation, was an idea that would not die. Cook doubted its existence long before he ever sailed south. The man described as 'one of the greatest navigators our nation or any other nation ever had' spent much of the rest of his life looking for something he doubted existed, but proving that conviction would re-write the map of the world and make him immortal. The Antarctic passages of his voyages would be forceful expeditions pressing as far south as was humanly possible, but he did so in a period of history when the earth was suffering a miniature Ice Age and it was the worst time in two centuries to be doing it.
The first problem he faced was the envy of a rival: Alexander Dalrymple. He was the creature of the Astronomer Royal, Nevil Maskelyne, the man who had so frustrated clockmaker John Harrison, the genius who solved the longitude problem. Maskelyne recommended that the expedition should be led by Dalrymple, a scientist and East Indiaman captain who was later the first Chief Hydrographer; he was also vain, cussed, and a man with a genius for backing the wrong horse. He was convinced his career would be crowned by charting the _Terra Australis Incognita_ , which his readings of old and often dubious charts and memoirs convinced him was a known fact.
Dalrymple shot himself in the foot by demanding absolute overall control of the expedition. The Navy had made one famous exception to the rule that the senior Naval officer was the overall commander. The first Astronomer Royal, Sir Edmund Halley, had been given absolute control to observe a transit of Venus in the Pacific. Chaos ensued and mutiny threatened. Halley was Maskelyne's immediate predecessor as Astronomer Royal, so it was idiotic to imagine the lesson had been forgotten. The First Lord of the Admiralty, Sir Edward Hawke, made his oppositions plain enough even for Dalrymple, declaring 'I would rather cut off my right hand than sign such a commission.'
Cook shared his cabin with the naturalist, the twenty-five-year-old son of a rich, land-owning agricultural improver who had made a fortune draining the marshy fenlands of east England: a man who had employed numberless ditch-diggers and labourers like Cook and his father without ever knowing their names. Young Joseph Banks owned an estate near Lincoln worth £6000 per annum. He could have bought both the expedition ships out of a year's income and still have had cash spare to live like a gentleman. At Eton College, bored with Greek and Latin, he walked home from a swim in the river through a flower meadow and decided to study the beautiful flora. Oxford had no botanist, so he paid for one to move there from Cambridge. The whim became a stellar career. Once on board he impressed Cook with his diligence. Nothing was beneath his attention. He even noted that the weevils in the ship's biscuits were three species of the genus _Tenebrios_ and one of _Ptinus_ , but the white Deal biscuit was favoured by _Phalangium cancroides_.
The orders governing the expedition read comically today, governing as they do transactions with people and places which often did not exist; it is the bureaucracy of Never Never Land. If Cook found Antarctica, he had to take possession 'with the consent of the natives.'
For a ship, he chose a Whitby cat, renamed the _Endeavour_. The cruise took them to the main island of Tierra del Fuego where Banks showed his disdain for superstition by shooting a wandering albatross. Despite this, the Horn was a millpond. They sailed for the heat of Tahiti where the astronomers would observe the transit of Venus across the face of the sun, a rare and irregular event that allows accurate calculations to be made of the distance from the earth to the sun. Their next orders were to go to 40° south, heading into the south-west sector of the Pacific. To appreciate how good Cook's gut instincts were about _Terra Australis Incognita_ , you have to look at the diaries of the other officers on his two ships to see how long, and how longingly, they clung to the romance of a temperate continent. Every time they go ashore in New Zealand, Banks refers to it as a visit to the continent. He calls his party of believers 'the Continents' but refrains from calling the opposition the Incontinents. They all but circumnavigate the North Island, but the officers refuse to accept it is an island until Cook goes back north to Cape Turnagain where he had begun. Although the weather was poor for much of the time, making surveying difficult, when the explorer Julien Marie de Crozet, a fine navigator himself, sailed these waters with a Cook chart he 'found it of an exactitude beyond all powers of expression.'
When Cook reached home he had been 1074 days away from his wife Elizabeth, whom he had left pregnant and with young children. But he first visited the Admiralty, before going home to find their baby had been born and died, and another child was also dead. Within one month he was preparing to leave on a circumpolar trip. There were still southern latitudes where _Terra Australis Incognita_ might have been skulking since the Flood. Cook didn't believe it was there but he reasonably conceded that there were latitudes to the west of Cape Horn so little visited that sizeable land masses could have been missed.
Cook made two more Whitby cats famous: the _Resolution_ , 462 tons, with 118 men, accompanied by the _Adventure_ , 336 tons, with 82 men under Captain Tobias Furneaux. There were names already famous on board, and famous names to be. The Able Seamen included George Vancouver, aged fifteen, later to explore the Pacific NW Coast. AS Alex Hood, aged fourteen, was cousin to two admirals: Hood and Lord Bridport. Dalrymple didn't even make the long-list, but he had been busy redrawing the world from his desk. While Cook was away he had published a new chart of the southern oceans showing the extent of _Terra Australis Incognita_ , incorporating the few sightings of land which could be given any credence, such as the French navigator Lozier Bouvet's claim to have seen a snowy headland on the first day of 1739 and named it Cape Circumcision. Dalrymple's chart was shoddy work, based on a map by Abraham Ortelius: a masterpiece in its day, but now nearly two hundred years old. In his early thirties, Dalrymple was already a young fogey.
Their best chronometer was justifiably famous: K2, the commercial rebuild of the John Harrison masterpiece H4 that won him the £20,000 prize for solving the longitude problem. Its error was less than one second a day. One of the scientists was Johann Reinhold Forster, a fan of Dalrymple and an overpaid complainer with the sharp hunger for recognition and respect which only the truly insecure display. But his son Georg was altogether better company. The handsome Banks was missing; grown vain while he was the toast of London society, especially its women. He sulked about the choice of small ships again, and flounced off to botanise in Iceland. Cook's expedition would make four passes south and come within half a days' sailing of seeing Antarctica, never, of course, knowing it.
In Cape Town, they heard the sea gossip. The Breton noble Yves-Joseph de Kerguelen-Trémarec with two French ships had recently put into harbour, with orders similar to Cook's to hunt for the southern continent. They had returned from the seas where Cook was heading, and claimed to have discovered land at 48°S, and coasted along it for 180 miles. They had seen a bay in which they hoped to anchor but when their boats were sent in to sound for an anchorage, a gale blew up and the ships stood off, abandoning the men ashore. Cook's lieutenant Charles Clerke fumed that there existed no circumstance so bad that Kerguelen had to abandon his men. Clerke was a man whose candid diaries make you long to have worked with him. He comes across as witty, earthy, pragmatic and humane. One officer summed him up like this: 'Clerke is a right good officer, at drinking and whoring he is as good as the best of them.' Cook was too preoccupied wondering whether the French had stolen a march on him to criticise their personnel policies. Kerguelen, despite the distance between him and the shore, was sure it was a land of plenty. The motto of believers in the southern continent was _Often Wrong: Never in Doubt_.
On 22 November 1772, Cook headed for Bouvet's Cape Circumcision. Bouvet claimed to have sailed for twelve hundred miles along a shore covered in snow and ice but with land beneath. His pilot thought it was nothing of the sort, but fools make work for wise men. By late November Cook began to see icebergs. Nothing prepares you for your first sight of large icebergs. In dull light the turquoise blue shines out brilliantly. They can look as if violet lamps have been lit in the recesses. The largest I have ever seen took three hours to steam past; it was twenty-nine miles long (forty-six kilometres) but was only a broken fragment of a much larger one. In mid-December Charles Clerke reported horrible fogs 'so thick we can scarcely see the length of the quarterdeck' and out of the murk came bergs which towered over the ship. He reported one as big as St Paul's Cathedral. Even the taciturn Cook took a moment in his diary to appreciate the grandeur of seas so fierce they broke over the tallest icebergs, but his responsibilities soon resumed precedence over aesthetics: the scene 'for a few moments is pleasing to the eye, but when one reflects on the danger this occasions, the mind is filled with horror, for was a ship to get against the weather side of one of these islands when the sea runs high she would be dashed to pieces in a moment.'
It would soon be midsummer, but the weather was getting worse. Ice smothered the rigging; a rope as thick as two fingers became as broad as your wrist and would not run through blocks and pulleys, so sails and yards could not be moved. Men were continuously employed hacking off ice, and clearing it from deck. If they did not keep pace, the weight aloft would capsize the ship. At least most of the livestock died and everyone enjoyed fresh meat.
By 13 December they were at 54°S, the latitude where Bouvet saw his coast, but the weather had driven them 354 miles east of his location for Cape Circumcision, so Cook tried to force his way south, into the pack. He tried for four weeks before the weather eased on 15 January and he enjoyed gentle breezes and serene weather for five consecutive days. Fresh water was a problem; in Antarctica most of it is frozen. He expected most of the sea ice would be saltwater, but tested it by bringing brash ice on board in rope slings and baskets and melting it. All the ice was fresh water. In two days they made more water than they had taken on at Cape Town. The wittiest account comes from an Irish gunner's mate called John Marra who concealed his journal from the enforced collection at the end of the voyage. 'I believe [this] is the first instance of drawing fresh water out of the ocean in hand baskets.' The reason is that when seawater freezes, the salts do not freeze with it, so freezing purifies water just as boiling does. By killing seals and birds, they could be self-sufficient in the south: vital if extensive voyages were to be conducted so far from known land. Cook was the first to prove this could be done.
His perseverance was rewarded when they pressed south once more and punched their way through the pack. At noon on 17 January, at around 40°E he wrote 'we were by observation four and a half miles south of [the Antarctic Circle] and are undoubtedly the first and only ship that ever crossed that line' – which must have raised eyebrows over on the _Adventure_. The next day thirty-eight icebergs could be seen from the masthead and later in the day at 67°15´S a wall of ice barred their path. He was blocked seventy-five miles from the continent.
There was still a Breton bubble to puncture. In good seas they cruised for a week with remarkable ease right across the land seen by Kerguelen the previous year. Charles Clerke relished it. 'If my friend Monsieur found any land, he's been confoundedly out in the latitudes and longitudes of it for we've searched the spot he represented it in and its environs too pretty narrowly and the devil of an inch of land is there.'
They headed east but on 8 February the two ships were separated in fog and forced to abandon searching for each other, and head for their agreed rendezvous in Queen Charlotte Sound, New Zealand. Captain Furneaux in the _Adventure_ used the separation to head rapidly north and hightail it directly to New Zealand. Cook used it to push south once again, this time lacking a support vessel to come to his aid, but there are hints that Cook feels safer without Furneaux's assistance. On 10 February he was passing south through 50° of latitude when they saw many penguins in the water, raising hopes of land. A week later they were treated to shows of the _aurora australis_. Later in the month they suffered a gale which brought sleet and snow. It's the most miserable mix of all; few garments are warm when wet. The night had enough light for them to see huge icebergs all around and pray for daylight, but when it came they were even more scared as it spelled out just how dangerous their position was. One four times the size of the ship exploded into four pieces just as they were passing.
Forster Senior whined about the 'whole voyage being a series of hardships such as had never before been experienced by mortal man.' The rest reacted in the ambivalent way that typifies most first impressions of Antarctica: awe, fear and beauty.
Cook surveyed his men, and found their bodies and clothing were damaged and worn. Among the meagre livestock left on board the poor sow chose this time to farrow nine piglets. Despite efforts to help her care for them, all were dead by the end of the same afternoon. They sailed north to New Zealand, and found the _Adventure_ had been and gone, but not before a shore party seeking water and greens had been killed by Maoris. Their cooked remains were later found in parts, in a basket. Not even this could prevent Cook being even-handed about the Maoris: 'Notwithstanding they are cannibals, they are naturally of a good disposition.' They stayed in warmer climes all winter but in the spring, on 25 November 1773, Cook turned the bow south again. On 12 December the first iceberg welcomed them back to the ice-realm. Soon ice jammed the blocks again and icicles an inch long were on each deckhand's nose. They crossed the Antarctic Circle a second time but were again soon blocked. On Christmas Day the crew stupefied themselves on hoarded brandy. Cook headed north east a while then turned back south on 11 January 1774, to make his third crossing of Antarctic Circle. They were desperate for land, willing it into view. Cook recorded that he 'saw an appearance of land to the east and south-east' and Wales: 'a remarkably strong appearance of land.' Next day Clerke soberly reassessed these hopes: 'We were convinced to our sorrow that our land was nothing more than a deception.' When the atmosphere lies in layers of differing densities it bends light back to earth showing images of land hundreds of miles away. At the end of the month they drove farther south than ever, reaching 71°10' S, 235 nautical miles beyond their previous best, though in this longitude, they were actually farther from the coast.
'We perceived the clouds over the horizon to the south to be of an unusual snow white brightness which we knew announced our approach to field ice.... It was indeed my opinion as well as the opinion of most on board, that this ice extended quite to the Pole or perhaps joins to some land to which it has been fixed from the creation. As we drew near this ice some penguins were heard but none seen and but few other birds or any other living thing to induce us to think any land was near.' This may have been the poignant sound of penguins in water, where some species vocalise in a way they never do ashore, crying like a man lost.
He was 'now well satisfied no Continent was to be found in this ocean but what must lie so far to the south as to be wholly inaccessible for ice.' They headed for the warmth of Easter Island. Cook went down with a stomach illness. Perhaps it never left him, because his personality underwent a sea change, witnessed but not understood by all around him. The diary of one young officer, Trevenen, calls him a tyrant. They next headed west for Tierra del Fuego, reaching there just before Christmas, naming a dramatic double-turreted rock York Minster, after the great cathedral of Cook's native Yorkshire. The island they discovered on the 25th became Christmas Island. On Christmas Eve they shot six geese for Christmas dinner and Marine Bill Wedgeborough celebrated the birth of Christ with such fervour that he went forward at midnight to relieve himself and was never seen again.
They did some charting but it was half-hearted by Cook's standards, not even discerning that Cape Horn, whose whereabouts it was always nice to be sure of, was on an island. The men yearned to set course for home, but Cook turned south-east to explore the missing piece in the jigsaw of the Southern Ocean: the seas below and between Cape Horn and the Cape of Good Hope. This was the sector where Dalrymple, unknown to Cook at this point, had been publishing charts of the lands to be seen there. By 6 January they were sailing through the land Dalrymple had drawn around the mythical Gulf of St Sebastian. Cook sailed north to the place Dalrymple had argued was the peninsula marking the north-east limit of the gulf. This assignation had better provenance: a merchant, English despite his name, Antoine de la Roche, had seen land here in a storm exactly one hundred years before in 1675. Arriving at la Roche's co-ordinates he had taken, Cook found only fog. It cleared next day to reveal an iceberg, which in the mild latitude of 54°S suggested high and cold land was close by. It was first seen by Midshipman Willis, who was known for his heavy drinking, and possibly only on deck to relieve his bladder. The off-lying island he saw is named for him and soon there was more, a long run of coast, with high icy mountains towering into the clouds, separated by steep glaciers. Cook wrote: 'Not a tree or shrub was to be seen, no not even enough to make a toothpick. I landed in three different places and took possession of the country in his Majesty's name under a discharge of small arms.' They sailed down the dramatic north-east coast, Clerke discerning subtle variations in misery and desolation when comparing it to other Godforsaken places they had seen and concluding 'I think it exceeds in wretchedness both Tierra del Fuego and Staten Land (Island), which places till I saw this I thought might vie with any of the works of providence.' He didn't appreciate the 'sublime' nature of wild landscapes; that would become popular taste in the next century.
But as they went south-east and the shore continued, mile after mile, for a hundred miles, some of the men began to believe this was at last the ancient dream: _Terra Australis Incognita_. Then the coast began to turn; there were off-lying islands separated by reefs and rocky shoals which catch tabular icebergs blown north. The fjord now named Drygalski opened up, then a string of islands and rocks forced them further out and when they could come close again they saw the land running back north-west, the way they had come; it was an island. Cook named the southern tip Cape Disappointment, adding. 'I must confess the disappointment I now met with did not affect me much, for to judge of the bulk by the sample it would not be worth the discovery.' His sense of tact seemed to be wearing a little thin because he then named it after the King: South Georgia. It is still governed by Britain; it must be, there is a Post Office. Cook turned south-east and spent the first three days of February in conditions that appalled even his most experienced officers. The land they suddenly saw, when they could see at all, threw peaks straight from the sea some four thousand feet into the clouds. Cook was happy to declare it 'the most horrible coast in the world' then name it after his friend, the Lord of the Admiralty, the Earl of Sandwich. Unsure of whether they were looking at islands or a peninsula they called it Sandwich Land.
He declared:
I firmly believe that there is a tract of land near the Pole, which is the source of most of the ice which is spread over this vast Southern Ocean. The greater part of this Southern Continent (supposing there is one) must lie within the Polar Circle where the sea is so pestered with ice that the land there is inaccessible. The risk one runs in exploring a coast in these unknown and icy seas, is so very great, that I can be bold to say that no man will ever venture farther than I have done, and that the lands which may lie to the south will never be explored. Thick fogs, snowstorms, intense cold and every other thing that can render navigation dangerous one has to encounter, and these difficulties are greatly heightened by the inexpressibly horrid aspect of the country, a country doomed by nature never once to feel the warmth of the sun's rays, but to lie forever buried under everlasting snow and ice.
The natural harbours which may be on the coast are in a manner wholly filled up with frozen snow of a vast thickness, but if any should so far be open to admit a ship... she runs a risk of being there forever.
Future travellers would read these words and feel an icicle enter their hearts.
Elizabeth Cook must have been delighted to have one of the most talked about men in the Navy back in her house, when he eventually got there after first visiting the Admiralty. She was also glad to be pregnant within two months, while he made plans to settle in London. He had nothing more to prove. After six months he was invited to a dinner with three of the most powerful men in the Admiralty to advise on the manning of an expedition to find the North West Passage by sailing across the Canadian Arctic to the spices and silks of the east. The prize equalled that for solving longitude, £20,000, much of which would go to the captain. By the end of the evening Cook had risen to his feet with his glass in hand and volunteered to lead it. It was probably just what their Lordships had intended would happen. What Elizabeth thought is another matter. In another six months he was gone. She would live to be ninety-one, but she would never see her James again. She would outlive all her children.
His orders were to take the latest cats, the _Resolution_ and _Discovery_ , to look for the land Kerguelen-Trémarec had claimed in the sub-Antarctic waters, then probe the west coast of North America, loosely claimed as New Albion by the British without actually having visited it. He was to press north into the Arctic through the Bering Straits, then retire to overwinter in Kamchatka, Siberia. In the spring he was to return to the high latitudes and look for the most promising passage to the east. One of his young sailing masters was a highly determined and skilful man called William Bligh.
The Antarctic element of his final expedition was modest. He checked the positions of Crozet and Marion Islands and found the French navigators had located them accurately. At the Kerguelen Islands, despite finding a bottle claiming them for France, they left a Union Jack and a counterclaim. The rest of the trip was spent carrying out orders in a slow and aimless way. It was not the Cook any of them knew. His temperament, once so sober and serious, began to be subject to violent fits of anger, bouts of indecision, and irrational attacks of urgency.
Ironically Cook's final discovery was Hawaii, where he was revered, arriving exactly how and when ancient myths predicted a God would, but when he was forced back to repair his ship he stepped outside the myth: the god did not return. He was clubbed to death in the surf, cut up and most of him eaten. Just a few remains were returned to a heartbroken Charles Clerke, already dying of tuberculosis at thirty-eight, for burial at sea.
In almost any other decade Cook would have met less ice and discovered Antarctica. He did so by inference; penguins, seals and icebergs all required land. Although he never saw Antarctica, he added much information about what it was not, and made it impossible for anyone of reason and judgement to believe that any large new land mass lay in the southern temperate zone. The area we know as Antarctica contains huge swathes of ocean he first sailed through, peppered with small islands as wild as the day he saw them. He concluded that there was nothing more to be gained by further exploration only because his vision of exploration was bound up with early ideas of empire: the utility of a place was all. He also failed to appreciate that another age would bring other young bloods eager to exceed their predecessors.
The boy who saw a silver world shining in a shilling piece set the gold standard. Yet he didn't discover the last continent. And if Cook didn't, who did? Navigators from all nations would follow like sharks behind a great ship, seeking glory. But their discoveries did not lead to conquest and empires, or to riches. The greater share of polar fame has gone to the heroic failures: Franklin, Scott and Shackleton.
Who discovered it in the first place? Who indeed.
#### THE ANCIENT MARINER
Samuel Taylor Coleridge is often remembered for a cliché of poetic inspiration: he wrote _Kubla Khan_ in an opium-inspired trance. It is a fine tale, but false. The inspiration for his epic poem _The Rime of the Ancient Mariner_ is much stranger, and is completely true. The albatross of his tale was a real bird, shot in 1726 near Staten Island north-east of Cape Horn. However Coleridge did not begin intending the albatross would be a central motif and it was not even his own idea.
Poems are written for many reasons and his reason was money. He and his friends William and Dorothy Wordsworth were on holiday in the English West Country and finances were failing. On a walk along the coast near the small port of Watchet they hatched a project to publish a book of ballads, a form then bringing popular success to Robert Burns and Thomas Chatterton. Coleridge was turning over his mind a nightmare, dreamed by his friend and local bailiff John Cruickshank, of a spectre-ship that still sailed although all its planks were gone: a skeleton of a ship.
Wordsworth warmed to the idea. 'I suggested; for example, some crime was to be committed which should bring upon the Old Navigator, as Coleridge after delighted to call him, the spectral persecution. I had been reading Shelvocke's _Voyages_ , a day or two before, that while doubling Cape Horn, they frequently saw albatrosses in that latitude. Suppose you represent him as having killed one of these birds on entering the South Sea, and that the tutelary spirits of these regions take it upon them to avenge the crime.' Coleridge seized the idea, and studied the original account. Shelvocke wrote 'we had not had the sight of one fish of any kind, since we were come to the Southward of the straits of _le Mair_ , nor one sea-bird, except a disconsolate black _Albitross_ , who accompanied us for several days, hovering about us as if he had lost himself, till Hatley, observing, in one of his melancholy fits, that this bird was always hovering near us, imagin'd from his colour, that it might be some evil omen... after some fruitless attempts, at length, shot the _Albitross_ , not doubting (perhaps) that we should have a fair wind after it.' These words were reborn as:
'God save thee, ancient Mariner!
From the fiends, that plague thee thus!-
Why lookst thou so?' – 'with my cross-bow
I shot the ALBATROSS.'
Behind this coming together of ideas another scarcely believable coincidence is hidden. Hatley's 'melancholy fits' had a cause. Some years before he had been sailing on a privateer called the _Cinque Ports_ along the west coast of South America when they put in at the uninhabited offshore island group of Juan Fernández. The ship was in poor condition and the first mate, Alexander Selkirk, remonstrated with his young captain when he gave Selkirk orders to sail before repairs could be completed. Selkirk said the ship was unsafe and he would rather be left there, and he was. He survived four years and four months before being picked up by another privateer, Captain Woodes Rodgers. The _Cinque Ports_ had sunk, but one of the survivors had been captured by the Spanish and worked for years as a slave in a silver mine. That man was Simon Hatley. Selkirk was briefly a celebrity, but he was immortalised under another name by a writer who turned his adventure into a novel. The author was Daniel Defoe; the book was _Robinson Crusoe._ The otherwise unknown Hatley helped inspire two of the classics of his age.
The narrative of the _Rime_ takes place in the cold south (Antarctica's discovery was still twenty-one years off) then the tropics, but nowhere is clearly recognisable. Coleridge didn't want it to be. The ship has no name nor does any character in it. There are no officers, no cargo and no ports. At the time he wrote the _Rime_ the longest sea voyage Coleridge had made was to cross the River Severn near Bristol. The _Rime_ is a spiritual journey; every section except the last ends with a direct or indirect reference to the Cross of the Sacrament. He employed two animal images, the killing of the albatross, for sin, and the blessing of the water snakes, for redemption. He used them to affirm that all life is part of one moral system.
# William Who?
A Merchant Goes Astray
## WILLIAM SMITH (1790–1847?)
_The captain of a small English merchantman sails far south of Cape Horn to avoid a storm and finds an unknown island on which a ghost ship has been wrecked, providing him with his coffin._
You'd think discovering one of the seven continents would make you famous, although America preserves the name of the navigator Amerigo Vespucci, who definitely didn't discover it. Columbus long got the credit for discovering the Americas, despite two objections. Firstly Leif Erikson had already been there in the tenth century, and secondly Columbus spent the rest of his life denying it was a new continent; you can hardly discover what you deny.
The last continent to be discovered was Antarctica. Two hundred years ago – two human lives – no one knew it existed, although in summer it is the size of the USA and Mexico combined, and doubles in size in winter when the surrounding sea freezes. Surely its finder would become a historic figure on a par with Columbus. He didn't.
Odder still, the ancient Greeks named the continent long before it was discovered. They believed that the world must be in balance, so to counterpoise the vast earth, there must be an anti-earth, or _Antichthon_ , moving unseen round the other side of the sun. As there were large temperate land masses in the north of the earth, there must be similar temperate continents in the south, beyond the broiling tropics. They called it the Unknown Southern Land, translated into Latin as _Terra Australis Incognita_. The southern land would be different, and must be separated from the world the Greeks knew by a strait of water, for these two worlds could not touch: like matter and anti-matter. Centuries rolled by; Europeans voyaged towards all points of the compass, but no one found a land that matched the ancients' ideas of a balmy southern continent. Yet belief never wavered; _Terra Australis Incognita_ became a thirst for a southern world which the Americas, for all their riches and romance, did not slake. Every scrap of rock glimpsed through a gale became the tip of a peninsula reaching out from the southern paradise. In 1598 five Dutch ships sailed through the Strait of Magellan and one, under Dirck Gherritz, was first separated from the others by fog, and then driven south in a storm. Accounts said that at 64°S he had seen mountains covered in snow, looking very like Norway, which might have located him at the south end of the South Shetland Islands off the west coast of the Antarctic Peninsula. Until the nineteenth century he was given credit for sighting Antarctica. But his own writings made no such claim; only when they were translated in 1622 was that passage added.
In 1811, forty years after Captain James Cook concluded there was a frozen southern continent, a consortium of five businessmen was gathering funds to build a 216-ton brig. One of them was a young man called William Smith, born on 11 October 1790 in the coastal village of Seaton Sluice, Northumberland. In the baptismal register his name is sandwiched between those of babies born to pitmen, glass-workers, farmers and sailors. He grew up three miles farther north in the coal port of Blythe, where, judging from the standard of his letters, he received a competent education. Aged twenty-one he was the master and owner of the _Three Friends_. Williams had also worked in the Greenland whale fishery and was a confident ice-pilot: a specialist job. The brig was finished the next year, and since two of the other principal three shareholders were also called Williams, they imaginatively christened it the _Williams_. Within three years they were early players in the new trade with South America. Another four years on, she was in Buenos Aires with her twenty-two-man crew preparing to make a voyage which usually took around six weeks, sailing west round Cape Horn to Valparaiso in Chile. Chile was one of the new republics rising from the ruins of the Spanish Empire, declaring independence in 1818. Britain actively assisted the fledgling republics. It wasn't high-principled support for personal freedom which spurred Britain on; at this time no British woman had the vote, and only those men who owned property. It was eagerness to open up new markets for its trade goods, displacing Spain, its trading and colonial rival. Britain provided vital naval support under the brilliant maverick, Lord Cochrane. The busy man in Valparaiso charged with protecting British interests in Chile during this turmoil was Captain Shirreff of HMS _Andromache_.
However, Spain's New World empire had propped up her antique economy and European ambitions for three hundred years; she would not surrender South America without a fight. One of King Ferdinand VII's responses was to send out more troops from Spain in a squadron of four ships including the powerful seventy-four-gun warship _San Telmo_ under Captain Joaquín Toledo. The squadron encountered terrible storms off Cape Horn and was driven far south of their usual sailing routes. Dismasted and rudderless, the _San Telmo_ was towed until she began to imperil the towing vessel, and she was cut loose on 4 September 1819 at 62°S. The _San Telmo's_ 644 men, coffined in the helpless hulk, drifted south.
Into the hold of the _Williams_ went the usual miscellany of trade goods, including nine cases of pianos, five cases of hats, and one of eau-de-cologne, tobacco, medicine and cloth. There was also a consignment that typified how British manufacturing created dependent markets round the world. Samples of hand-woven ponchos had been taken back to England and replicated on northern steam looms for a fraction of the cost. Gauchos would now buy their ponchos from Yorkshire. Although it was still the southern summer, Captain William Smith met strong contrary winds south of Cape Horn and decided to head further south and take a calculated risk of encountering ice, whose dangers he was familiar with, rather than be at the mercy of a storm over which he had no control. He was not an explorer, adventurer or a geographer, he was a businessman reducing his risks.
What happened next is poorly documented. There is no surviving original document, just partial copies and contemporary reports referring to papers now lost. On 19 February, he found himself in a south west gale stiffened with sleet and snow. Coming up to daybreak at 06:00, where the charts showed open ocean, he saw land. He recorded in his log, 'Land or Ice was discovered ahead bearing SE by S distant about two leagues, blowing hard gales with flying showers of snow.' He logged his position as 62°15´S and 60°01´W. In the afternoon the weather improved: '4pm tacked ship, the land or islands bearing SEbE to ESE about 10 miles, the weather fine and pleasant, when discovered to be land a little covered with snow.' The skies soon told Smith the foul weather was ready to return; at 18:00 with whales and seals all around, he resumed his course to Valparaiso.
Smith anchored beneath the city's steep hills, only to find the British representative Captain Shirreff was in Santiago, so Smith spoke to the most senior officer, Captain Thomas Searle, of the forty-two-gun warship _Hyperion_. Searle was so startled at the news he forbade them to go ashore where grog would loosen sailors' tongues. When Shirreff returned, he quizzed Smith. Had he landed, explored any length of shore? – or at least taken soundings to prove it was real land, and not ice, or one of the many islands seen once at a distance and never again? (A century before, Jonathan Swift had lampooned such fugitive sightings when he invented the flying island of Laputa in _Gulliver's Travels_.) Williams had done none of these things. It wasn't that Shirreff didn't believe him, but in Chile he had many tasks of clear and pressing importance to British interests. He required solid evidence before pursuing other ends. Smith said he would sail a similar course on the way back, but when he got to the same area on 16 May, the weather was poor; winter had begun. The encircling ice began to rip off the expensive copper plating which protected the vessel against shipworms. He left without re-sighting land.
But Smith knew what he had seen. With his next cargo, in October, he swept south again and at 18:00 on the 15th, he reached the position where he had logged land, and saw it again. This time he was determined to prove his case. He was probably looking at King George Island, and he followed the land eastward and named features. On 17 October, at a place now called Venus Bay on Ridley Island, also known as Ridley's Island, 'Finding the weather favourable, we lowered down the boat, and succeeded in landing; found it barren and covered in snow; seals in abundance. The boat having returned, and been secured, we made sail.' That laconic, business-like record was the last time anyone could record finding a continent. He claimed it for Great Britain, and named it New South Britain. They turned and went 150 miles WSW along the coast of an island chain, and when they finally decided to turn north, still more land was in view. He was later persuaded that since the Scottish Shetland Islands occupied an equivalent latitude in the north, he should re-christen the group the South Shetlands. Smith invested a lot of time in this diversion. This trip took an extra six weeks, so he planned to come back and take seals, to recoup his investment. He also recorded that sperm whales, the most valuable whales of all, were abundant. There were no great words or thoughts on landing, as Antarctica was at last discovered; he had no way of realising the significance. On 18 October he wrote, 'I thought it prudent, having a merchant's cargo on board, and perhaps deviating from insurance, to haul off to the westward on my intended voyage.' He wasn't going to be an uninsured driver.
Alexander von Humboldt sarcastically observed that there were three phases in the popular attitude to a man who makes a great discovery. Firstly men doubt its existence, next they deny its importance, finally they give the credit to someone else. Smith suffered all these. Some British journalists were not convinced Smith's reports were genuine. _The Edinburgh Magazine_ mocked: 'At first we treated it as an Irish or American report, both of which are generally famous for not being true.' Then, as soon as word of Smith's discovery spread, American newspapers proclaimed that these islands had already been discovered by US sealers. Baltimore's _Niles Weekly Register_ for 23 November 1822 read: 'It is now well-known that some of these hardy people had visited what is regarded by the English as newly discovered land as early as 1800.' There was no written evidence but that, they said, was not surprising since sealers were secretive about their movements: knowledge was money. In which case why were the beaches still swarming with seals after twenty years' sealing? Rookeries were wiped out in two seasons. Tellingly, records of sealers in the period 1812 –19 show that no vessel came home with catches exceeding expectations from known rookeries, but when the sealers followed Smith to the Peninsula, catches soared, and the Antarctic fur seal was all but exterminated by 1822.
The partially informed sharpened their quills and recycled old lies. The prestigious _Edinburgh Philosophical Journal_ carried a report of Williams's discovery illustrated with a chart showing a southern continental mass labelled _Terra Australis Incognita_ lying between 54-58°S and 40-53°W. They claimed this was the previously reported land mass which William Smith had merely re-discovered. This despite the fact that Captain Cook had sailed through that area and seen only ocean and ice. The chart was an old fraud, based on a Mercator map of 1569, copied in turn from a small world map of Oronce Finé drawn in 1531. But if there was money to be made, any pretext might do to claim prior discovery. What was really needed was an official voyage to formally take possession and claim it for their sponsor's country. Britain, handily established in Chile, was first off the mark. One factor which helped Shirreff make a case for diverting effort south was that Britain had no convenient base from which to help exploit the emerging trade markets in South America. The Falklands had been abandoned in 1774 and would not be effectively occupied again until 1833. Their nearest colonies were South Africa and Australia, too remote to help.
Smith's landing secured Shirreff's ear. Shirreff was only thirty-four but he had been in the Navy since he was eleven years old. He wanted to help and he could spare some men from the _Andromache_ , but the vessels themselves were committed. The _Williams_ had been chartered to an entrepreneur called John Miers to ship heavy technical equipment from Valparaiso to Concón. But Miers was also a man with strong scientific interests; he would later become a Fellow of both the Linnaean Society and the Royal Society, and bequeath over twenty thousand specimens to the British Museum. He kindly agreed to shelve his plans and let Shirreff charter the _Williams_ to investigate the South Shetlands.
Command of the vessel was given to Captain Edward Bransfield, born in Cork, Ireland in 1785. He had been pressed into service when only eight years old. Now the word was out, it would be a race to go south and take the initiative in exploring the new territories. That summer, fifty British and American vessels followed the _Williams's_ wake, over the toughest seas in the world, to see what money a man might make there. Shirreff's orders to Bransfield looked for a longer commitment. They read:
You will ascertain the natural resources for supporting a Colony and Maintaining a population, or if it be already inhabited, will minutely observe the Character, habits, dresses, customs and state of civilisation of the Inhabitants, to whom you will display every friendly disposition.
It seems quaint to imagine native Antarcticans waiting to exchange diplomatic niceties, but at this time William Parry and John Ross were voyaging high above the Arctic Circle and meeting native peoples surviving in far harsher climates than Smith had reported. Might they not meet the southern equivalent? What's more, _The Literary Gazette_ had reported William Smith as saying of the South Shetlands that 'firs and pines were observable in many places.' It continued: 'The climate of New Shetland would seem to be very temperate, considering its latitude; and should the expedition now sent out bring assurances that the land is capable of supporting population, – an assumption which the presence of trees renders very probable, the place may become a colony of considerable importance.' _The Literary Gazette_ had exaggerated a line in William's log of an observation he made in thickening weather as he prepared to leave: 'could perceive some trees on the land the southward of the cape.' He was deceived. Antarctica's largest plant is a few inches high.
Captain Edward Bransfield, with William Smith and forty-three others on board, sailed from Valparaiso in late December and sighted Livingston Island in the South Shetlands on 16 January 1820. They made their way north-east along the northerly coasts of the large islands of the group, then round to the south side of King George Island, where Bransfield landed in what is now King George's Bay, after George III of England. It is one of the nice perks of being a monarch that you sit snugly on your throne while others suffer the discomforts of finding places to name after you.
These islands are a relatively warm part of Antarctica, their climate softened by the surrounding ocean; during their voyage, the lowest temperature recorded was –1°C. Because of this, and the ease and economy of servicing a base there, it is now the home of many Antarctic stations including those of nations with subterranean profiles in the annals of Polar history, such as Papua New Guinea. However it isn't hot, and many of the seamen transferred from the _Andromache_ in Valparaiso had settled down to life in a Mediterranean climate and sold off all their warm clothes. They now suffered miserably from exposure.
Although they had taken ninety days' water, leaking barrels made them anxious to replenish stocks. From 22 to 27 January they anchored off a small island dominated by an extinct volcano and named it after the main inhabitants, Penguin Island, before beating them aside to go ashore. It is a short climb up from the beach, through soft ash and cinders, to the main crater, within which lies a smaller cone. There was still breath to spare for more words of possession and a volley of small arms. From the 170-metre-high summit, there is a breathtaking view across the narrow channel guarded on the far side by Turret Point on King George Island, a cluster of three stacks linked to the land by a wishbone of shingle beaches. They crossed over and soon found a more convenient spring, and encountered elephant seals on the beach. They decided to kill some to make oil. At that time of year elephant seals, which weigh up to four tons, are moulting, which is uncomfortable and puts them in a bad mood. Although they are not agile on land, they are powerful and deceptively flexible. One sailor approached a seal from behind, but was caught flat-footed when it swivelled round and savaged his hand in its immensely strong jaws. The officers contented themselves with safer work: collecting mosses and stones for study.
They landed again the next day and killed twenty-one elephant seals in thirty minutes, startled at the quantity of blood which came from the carcases. Other wildlife didn't wait to be approached. Skuas are brown birds built like a heavy gull. Although they have webbed feet they act more like birds of prey, and are aggressive and skilled hunters. I have watched two knock down a large man by co-ordinated swoops on his head. When sailors came ashore with a dog, the skuas of King George Island were faced with two new species at once, and, true to form, decided to attack both. The dog was soon covered in blood, and the sailors had to fetch staves to defend themselves.
Unknown to them, on 27 January, as they left Penguin Island and cruised down the South Shetlands, King George III was released from his final madness by death. The news did not reach Valparaiso until May that year. So they continued south, and named and claimed new bleaknesses on his behalf. They passed Livingston Island again, a fiercely beautiful island of jagged peaks and ridges skirted with swooping snow slopes: 'every atom was covered with snow.' The talk on deck 'was the idea of having, by the direction the land took, found what might possibly lead to the discovery of the long-contested existence of a Southern Continent.'
We know about such chatter because of the survival of one original document from this trip: the journal of Midshipman Charles Wittit Poynter. He was baptised in 1798, the son of an East Indiaman Captain, and joined the Royal Navy in 1811 serving on HMS _Inconstant_. It was more common to name vessels after virtues rather than vices. It conjures up images of a whole fleet of disreputable ships and men: all the drunken sailors gathered on HMS _Inebriate_ , enjoying adultery on HMS _Inconstant_ , before retiring to HMS _Incontinent_. Poynter's journal comprises seventy pages of handwritten ink entries bound in red calf leather with marbled boards. We have few descriptions from the first two summers around the peninsula; making money from taking animals was uppermost in most minds. But there was a strong emotional reaction to these landscapes, and it is consistent in its nature: awe at beauty forged in desolation. Robert Fildes, shipwrecked in the sealer _Cora_ in 1821, was forced to live ashore for a while, he and his men sleeping in barrels inside a tent. The ship's cat moved into an empty barrel and was promptly joined by two penguins, living together amicably. Fildes initially tried to be amusing about the landscapes, writing, 'Madam Nature had been drinking too much gin when she made this place.' But he soon became serious: it possessed 'an awfully grand though terrific and desolate appearance.' By the time he reached the shores of volcanic ash at gloomy cloud-capped Deception Island, he was seduced: 'the mind is forced into pious contemplation of the grandeur of the scene.' Just over sixty years before, the philosopher Edmund Burke had written _A Philosophical Enquiry into the Origin of Our Ideas of the Sublime and Beautiful_. Burke suggested, 'The sublime is the supreme, strongest emotion, [and] can be excited by terrible, painful and dangerous things.... When danger or pain press too nearly, they are incapable of giving any delight, and are simply terrible; but at certain distances, and with certain modifications, they may be so capable.' This perfectly fitted these new Antarctic landscapes which, in the same moment, could be both breathtaking in their beauty and terrifying in their hostility.
They met no Antarcticans. It has never been inhabited by humans. On the stony shores of Desolation Island, where once I landed in bone-biting cold at dusk, with snow darkening the sky, a human skull was found next to a primitive stone hut. Simple stone and bone tools were found. Initial excitement cooled when they were studied carefully. The remains date to the early nineteenth century, and the artefacts are all from known cultures in Tierra del Fuego. The hut is a sealer's hut, and the skull was that of a woman around twenty-one years old. She was probably one of the Patagonian natives whom sealers were known to take on board for company and, doubtless, sex.
On 30 January, Bransfield and Williams saw what we now know is the continental land of the Antarctic Peninsula, though they had no way of knowing it. It looked no different from the islands. They named the strait between the South Shetlands and the new land after Bransfield. They thought it might be a gulf; it is now the Bransfield Strait. Penetrating to more than 63°S, they named some of this Trinity Land, after the maritime organisation Trinity House, which manages navigation in British waters. Smith would soon need them. They went on, writing names on the whiteness of the land and the maps, sketching the coast. A map annotated with remarks from the log summarises the experience: 'supposed land', then 'lost in fog' and 'here the coast was lined with icebergs.' After reaching the limit of the land and naming it Hope Island (for the hope that the eastern shore they could not see extended away into continental land) the farewell was once again: 'lost in fog.' They reached Elephant Island where Shackleton would stagger ashore after his escape from the crushing of the _Endurance_ in the Weddell Sea ice. They found a driftwood log and speculated where this old tree had once taken root and sent tender buds into the spring air. They took fur seals and found them tough to kill. As soon as the seabirds smelled blood, they swooped down and attacked both men and seals. Union Flags were scattered over the rocks and squirrelled away in protecting caves, then the first prospective colonisers returned to Chile. Winter returned to Antarctica, freezing all ambition. William Smith made a hazily documented fifth voyage in 1821. It was probably on this voyage that Smith suffered an eerie _memento mori_. Standing off in Blythe Bay, Desolation Island he saw ships' timbers on Half Moon Beach. Landing, he found spars, and sails tangled like winding sheets, all from the galleon _San Telmo_ , the Spanish vessel abandoned in the Drake Passage. It had made a stubborn path south through the tempests, before the Furies riding the Antarctic winds broke it apart, and the sea and rocks had prised open its carcase like monstrous skuas. There were no bodies, so it is likely that it was broken up at sea, and the debris dispersed. Smith brooded over the ship's remains in melancholy mood. He found the anchor, a symbol of hope, its wooden stock hooped with iron and bolted with copper. Some inner demon made him order its removal, to be made into a coffin. He said he would be buried in the anchor that had waited for him in the cold rocks at the end of the world. Financially he was badly stretched, having lent his brother money for an unsuccessful project to build a steam vessel in Bristol, ironically called the _Hope_. But there was profit to be taken;
Smith said the seals were so thick on the ground they looked as if they had been stowed in bulk. He returned to Britain on 17 September 1821 with thirty thousand fur seal skins. On the market they were worth £7,500, a rich return for him and his three partners. But while he was away, they had gone broke. The skins were seized by debtors and sold off cheaply, leaving him bankrupt. Smith wrote a letter to the Admiralty, received on New Year's Eve 1821, begging a reward for the discovery, and claiming for Britain, of new territories. They offered him nothing. He continued to press the Admiralty for reward for sixteen more years. They continued to refuse. He was forced to sell the _Williams_ on 13 June 1822, to a consortium in London's East End. The proceeds bought him release from bankruptcy two weeks later. For the next twenty years he would live in the same area, first at 3 Commercial Place in Whitechapel. His neighbours were tradesmen and the lower middle classes. None of them had discovered a continent. He worked in humble maritime jobs: from 8 June 1824 he was licensed as a Trinity House pilot on the Thames. The licence describes him as living in 54 Lucas Street off Commercial Road, and being five feet eight inches tall, with black hair and a dark complexion. He subsequently met James Weddell who was a partner in the Leith whaling firm of Messrs Wood, and secured him work as a master on the _William and Ann_ whaling in the Davis Strait off west Greenland. In his first year he caught a respectable twenty-five whales. But every year his catches were lower, each one less than the season's average. In 1830 he returned 'clean': no whales caught; the following season the _William and Ann_ had a new skipper. In 1837 he resided at 44 Jamaica Street, in the heart of Whitechapel. Next year he applied for one of the two Trinity House almshouses, pitiably not realising he was a year too young to qualify. A small courtyard of those almshouses still stands behind locked wrought gates on the Whitechapel Road, whispering of old dignities, beneath a weathercock in the form of a small brig. Next year his licence was withdrawn because he was 'a pilot greatly reduced by age and infirmity' though he was only forty-eight years old. He was pensioned off by Trinity House in 1839, but could not fully support himself financially. He petitioned the Admiralty again for a reward for his discovery of Antarctica. There is an extant letter asking for a statement from Captain Shirreff describing Smith's services in the discovery, but no record of further correspondence survives, and the following year he was admitted to the charity of the almshouses. His date of death is not known, but his will was proved in 1847, to the value of £100, about £6,000 in modern money. There wasn't much for the British Columbus to dispose of. His real bequest was Antarctica. In the _Dictionary of National Biography_ there are twenty-three Williams Smith but not the man who discovered Antarctica.
#### THE NAMING OF NAMES
Naming is powerful; it asserts control. Stories like Rumpelstiltskin and Turandot have at their core the power of knowing a name. Explorers often ask the locals the names of places. If they share no common language, confusion results, sometimes comic, occasionally poignant, memorials to the failure of two peoples to understand one another. Near Cape Horn is Tekenika Bay, _Teke unika_ being the Indians' response when asked what it was called. The words mean 'I don't understand.'
Uniquely, Antarctica was huge and uninhabited when discovered. It was a blank to be written on: no locals to help. Antarctic place names reflect western values. If naming is power, the powerful want a share. Kings, princes, politicians, naval officials and sponsors of all kinds go nowhere and watch the world being inscribed with their names. South Georgia and King George Island still commemorate King George III of England. The Chilean name for the Peninsula, Tierra San Martín, commemorates the founder of their state, and their Antarctic bases González Videla and Frei record the names of the two presidents to have visited the continent. The Peninsula is also Grahamland, named in 1832 for Sir James Graham, First Lord of the Admiralty. Americans name part of the Peninsula Palmer Land after Nathaniel Palmer who did at least explore for himself. Money can buy immortality on a map. The world's largest glacier, Beardmore, is named for one of Shackleton's backers. The first man to overwinter below the Antarctic Circle, Adrien de Gerlache, remembered the generosity of sponsors in the beautiful (Paul) Errera Channel which threads its way between the Peninsula and (Madame) Rongé Island. Luck also matters. Two sponsors of Shackleton, Messrs Stancomb-Wills and Dudley Docker, had the ship's boats named after them converted into huts and are remembered only by polar historians. The third boat, named after jute millionaire Sir James Caird, was sailed to South Georgia and into history; his name endures. The greatest price paid for a name is death. De Gerlache's seaman Carl Wiencke was washed overboard in the Gerlache Strait, and is now recalled in picturesque Wiencke Island. From the same expedition we have Danco Island, named after the geophysicist who tasted seal meat just once before announcing he would rather die than eat more. Soon the late Mr Danco had his memorial. And, of course, people name things after family, wives, sweethearts and each other. Adélie Land, like the Adélie penguin, is named for the wife of Dumont D'Urville, who, as well as charting parts of Antarctica, purchased for France a statue of Venus, patron of sweethearts every-where.
# The Admiral and the Boy Captain:
Thaddeus Bellingshausen (1778–1852) and Nathaniel Palmer (1799–1877)
_A teenage captain creeping across uncharted water in dense fog hears his small bell answered by two greater ones. He meets the leader of a Russian expedition. One of them has just seen the continental mainland of Antarctica for the first time, but which one?_
Poor Bellingshausen. He achieved success without fame, suffered toil without recognition. Even his name is a mouthful that needs rehearsing before you can remember it. He was one of two captains involved in the most famous sea encounter in Antarctic history: the polar equivalent of Livingston and Stanley. But in later life, the other man, embellishing yarns in his well-heeled retirement, got his name entirely wrong, in print.
Russia's defeat of Napoleon before the gates of Moscow in 1812 inspired her to view the world with the confidence of a great power. Her empire stretched from the borders of Europe, across all Asia, continuing so far east it crossed the International Date Line into the Americas and became the west again; modern Alaska was then Russian. But the east was so remote it might as well have been on the moon. Land transport was little farther advanced than at the dawn of history. The only effective way to move bulk goods was by water. Russia's coastline was huge, but practically every mile of it was ice-locked from autumn to spring or even summer. This changed just when Bellingshausen was born, on a small island off modern Estonia, in 1799. Russia had just won control of the mouths of three mighty rivers, the Bug, Dnieper and Don, which flowed into the Black Sea and gave Russia a gateway to the world. From there, through the Hellespont, lay the Mediterranean, then the Atlantic. But it was a world in which Russia had no chain of possessions 'establishing future permanent sea communications or places for the repair of ships', as the brief for a new circumnavigation of the world put it. They would look for such havens in the Southern Ocean. Their first choice for commanding officer didn't work out. Second choice was Fabian Gottlieb von Bellingshausen, usually known by his Latinised name: Thaddeus Bellingshausen. Despite receiving just six weeks' notice of departure, he sailed on time.
Bellingshausen had gone to sea aged ten as a cadet, and from 1803–06 had been a humble Fifth Officer under Admiral Krusenstern on Russia's first circumnavigation of the world. Krusenstern had served in the British Navy from 1793–97; this wasn't unusual. Russia knew it was not rich in maritime experience and had a policy of placing its officers in other navies and appointing foreign officers to its own ships. The second practice, putting foreigners in charge of the military, was not as reckless as it sounds, as a Russian captain was not the absolute ruler he was in other navies. Major decisions were referred to a cabinet of the senior officers on board in which native Russians were always in the majority.
At this time the government was still forcing the eastern governors to build wooden ships from local timber and fit them with anchors made in pieces in the west, and rope cut into fifty-foot lengths, then carried four thousand miles overland on carts, barges and sledges. The work employed four thousand horses. When Bellingshausen returned to Russia, he lobbied for vessels to be sailed round to the east and stationed in the north-west Pacific, independent of the overland transport. The problem with going away and broadening your mind is that your masters stay at home. His petitions fell on deaf ears until a new Czar, Alexander I, came to the throne, and one of Alexander's projects was opening up the east.
Bellingshausen's mentor Krusenstern idolised James Cook. Krusenstern copied the Englishman's obsessions with the sciences of navigation and charting, and his manias for accuracy and cleanliness. In his two ships, the _Nadezhda_ and _Neva_ , not one man died: an astonishing record for a vessel passing through every kind of climate, sea and tempest. Bellingshausen learned the same disciplines from Krusenstern. His devotion did not induce any special warmth from his mentor at the end of three years' work. In fact Krusenstern's account of the trip seldom raises itself above the narrative thrill of a laundry list. But he did manage to stir his pen for a sentence on Bellingshausen: 'His reputation as a skilful and well-informed officer in the different branches of navigation... I found to be perfectly just.'
While Bellingshausen was being damned with faint praise, a seven-year-old boy called Nathaniel Palmer was playing in his father's shipyard at Stonington, Connecticut, 110 miles up the coast from New York City. When he was thirteen, learning to sail ships, the US was at war with Britain. To make life difficult for foreign mariners unfamiliar with the waters, all lighthouses were decommissioned and the buoys marking the channels were lifted. For six years he traded in unmarked waters, running the British blockade of the ports as a smuggler or a free-trader, depending on your passport. By nineteen, he was a captain. Like Drake and Cook, he had learned the skills of coastal sailing in diverse waters: invaluable when exploring.
Stonington was in the heart of whaling country, but Palmer's town prospered on sealing, killing fur seals for their skins and elephant seals for oil, rendered down from the fat layer beneath the skin to a product that was as fine as good whale oil. These ships typically paid no wages; crew members received a share of profits according to their rank on board. Sealing was as good an example of a ruinous free-market as you could devise. The seals came ashore each year to pup and mate on the rocky shores of islands so remote that no government actively governed them: it was a free-for-all. Any animals you leave to breed will not be there next year, they will be in someone else's warehouse. They killed the females then the males, often blinding the beachmasters in one eye so they would continue to force half of their harem of females to stay onshore, while the men on its blind side killed the rest with a blow to the head from a hickory club. They didn't bother killing the pups; the skins were too small. They just starved to death without mother's milk.
Sealskin is very warm, with an extremely dense layer of short, fine fur. This is protected by longer, tougher guard hairs that take the rough and tumble of rocks, stone beaches. Every visible hair covers up to forty under-hairs; there are 330,000 hairs per square inch. Americans and Europeans didn't know how to take off the guard hairs without damaging the fine fur. The Chinese did. In the 1780s the ship the _States_ had gone sealing in the Falklands and brought back 13,000 skins which a New York dealer bought for 50¢ each. He took them to China where they were a high fashion object, especially in the north, with its bitter winters. They sold in Canton for $5 each, making a profit of $58,500 for a single voyage. In 1792 the _Eliza_ of New York took 38,000 skins to Canton. The voyage was long enough for the least numerate crew member to multiply $4.50 by 38,000 and get the answer: riches. When they got there they found the market was flooded; a skin sold for 50¢. It must have been a long trip home. In 1796 the London furrier Thomas Chapman invented a process for removing the guard hairs and the lure of sealing turned to lust. The trick is to shave the back of the skin close. The coarse guard hairs have deeper roots which are severed by this process, and can then be combed out.
The _Eliza's_ failure didn't stop the New Englanders trying. In a voyage from 1797 to 1799, the _Betsey_ went first to the Falklands, and then on to Más-a-Fuera in the Juan Fernández Islands, five hundred miles west of Valparaiso, Chile. On this rocky island, just a few miles across, they killed 100,000 seals. The _Betsey_ sailed for Canton leaving men ashore to continue killing until they returned. The sealer Captain Amaso Delano guessed that three million seals were taken from this tiny island group in seven years. With the profits from the skins and the Chinese goods later sold in New York, the _Betsey_ , a one hundred-ton ship which had cost only $3000 to build, made $52,300 profit. One trip could make a man. An ordinary deckhand's share bought him a 40% stake in a similar vessel. A bad trip would disgorge him onto a wooden pier in Connecticut, chill with the morning dew, after two years' hard, dangerous work, without a cent in his pocket, watching a wife's joyful smile collapse into a a bent pin of desperation. But he was alive to try again.
An early portrait of Nathaniel Palmer shows a confident-looking man who you would be pleased to share a beer with in a quayside bar. He was a man who would try again. Returning to old sealing grounds didn't pay much. Sealers had to be prepared to take ever greater risks and sail further from safely charted waters. One neighbour of Palmer was Edmund Fanning, a sealer, and author of _Voyages Round the World_. When Palmer was a baby, Fanning took 57,000 fur seal skins to Canton. He was also a man who liked to spend time with dog-eared charts and ancient deep-water tales. One was the account of Dirck Gherritz in 1599. Driven far south of Cape Horn, he saw islands which he christened the Auroras. Or at least the part of the account added without his knowledge by a translator twenty-three years later said he did. Fanning didn't know about the fraud, but bad information can lead you to good things, like Columbus looking for China and discovering the Caribbean. Fanning eliminated from his search the sea areas reliable navigators had sailed through, and looked at the gaps. He concluded the Auroras were real, and lay in a window bounded by 60°-65°S and 50°-60°W, which, by chance, boxes off the tip of the Antarctic Peninsula and the more northerly of the South Shetland Islands. When sealing failed in 1817, for the unsurprising reason that they had killed all the ones they knew of, eyes turned south, and Palmer went with them.
Bellingshausen adopted his mentor's spare journal style; there is little fun in reading him, though he lacks the literary anaesthesia that makes Krusenstern so numbingly dull. He was only interested in learning as much formal science as could help navigation, but considering this, his speculations during the voyage on scientific matters were rich in sound insights. The log shows he was a disciplinarian, and lash-happy. Portraits show a man you wouldn't joke with unless you were extremely sure how the remark would be received. His fellow commander Lt. Mikhail Lazarev was much more social and a more popular officer.
The ships were not ideal; they seldom are in Polar work. They were what was available and fit to sail. The _Vostok_ , meaning East, was a frigate 129 feet long crewed by 117 officers and men. The _Mirnyi_ (Peace) was a sloop under Lt. Lazarev crewed by seventy-three officers and men. The smaller, _Mirnyi_ , was ice reinforced unlike her companion ship, but was so slow that _Vostok_ had to sail the whole voyage under reduced sail.
Expeditions under military command usually receive standard pay for unusually demanding and dangerous exceptional work. Some private expeditions offered no pay at all: just experience and adventure. This one was different. All officers and crew received eight times their regular pay plus a year's extra pay. Their orders included studying the colour, stature and constitution of any inhabitants of new regions, plus details of their 'inner anatomy should it be possible to obtain bodies for dissection.' Natives resisting this unwanted contact often became museum specimens.
Pickled cabbage, a traditional food in Russia and Germany, would be good protection against scurvy. Other health measures went down even better; spruce essence and molasses were taken to make anti-scorbutic beer. 'As beer is the healthiest drink at sea, it would seem beneficial to give it frequently to the men.' If only all employers were as enlightened.
Bellingshausen's brief of 'establishing future permanent sea communications or places for the repair of ships' seems impossibly loose to build a voyage plan around it. In fact, the tropical and temperate latitudes and most navigable Arctic seas were well explored and useful discoveries were unlikely. The real opportunities lay in the open spaces of the southern oceans. He would plan a course to complement Cook's, focusing on those coasts and seas which were not seen by him because of bad ice or weather, or simply because even Cook could not sail every mile of this vast space.
They sailed from Kronstad, near St Petersburg, on 16 July 1819. Only one civilian scientist was taken: the astronomer Simanov. They anchored off Portsmouth while officers went to London to buy charts, instruments and favourite and luxury foods. Bellingshausen wrote, as if from experience, 'there are more prostitutes in England than anywhere else, particularly in the chief ports.' As they entered the Atlantic, the surgeon examined the crew and to the officers' surprise found the crew free of sexual disease. At the equator, Bellingshausen was reminded just how parochial the Russian navy was; he was the only person on either ship who had previously crossed the Line. The ceremonies were civilised by the standards of the English navy which left men half-drowned, covered in tar and filth, and pole-axed by drink. The Russian officers were merely sprinkled with water; it was more like a decorous Mass in honour of Neptune. More seriously, only he, Zavodovski and Lazarev knew how to take astronomical observations. The rest now trained under severe eyes.
Bellingshausen's real work began on 27 December 1819 when they sighted Willis Island, off the north-west tip of South Georgia. Cook had charted the north-east shore but left for home after seeing just enough of the south-west coast to prove it was an island. Bellingshausen kept to port the stone pyramids of Willis Island, Bird Island and their guard islets, skirted with spume thrown up by the grey South Atlantic. They met elephant sealers, dropped in a cove to live under their boat from spring to autumn, killing and boiling down fat and flesh in crude iron cauldrons. One was a Russian Navy sailor who had jumped ship in England. If he had known Bellingshausen's reputation for flogging he might have remained ashore, but Bellingshausen let sleeping dogs lie. The men spent a short time on board remembering there was a world outside South Georgia, where cold had respite, and killing was not the only pastime. With gifts of grog, sugar and butter, they climbed down the rope ladder and cut away into the spray and back towards the loveless shore.
They went south to examine Cook's South Sandwich Land, where atrocious seas and weather had worn down even that Yorkshireman's resolve before he could establish with certainty what he was looking at. Naming it Sandwich Land, not Islands, was an act of faith and flattery to his sponsor, Lord Sandwich. Bellingshausen found more land than Cook had described, and made the first landings, including, on 5 January 1820, one on Zavodovski Island, named for Lt. Commander Ivan Zavodovski. If you remember David Attenborough, in one of the opening scenes of _Life in the Freezer_ , standing amid volcanic fumes and one million chinstrap penguins braying like asses, that's Zavodovski Island. The Russians climbed halfway up the volcano until fumes of sulphur and hot guano drove them down. The place names say it all: Acrid Point, Stench Point, Fume Point, Reek Point, Pungent Point and Noxious Bluff. If the murk clears, you may glimpse its highest point, at 1805 feet: Mt Asphyxia. They stayed until 5 January labouring under weather much the same as Cook had suffered. By then he had proved South Sandwich Land was the South Sandwich Islands. They took live penguins back on board, where, despite being surrounded by an ocean full of fish, they fed the birds on pork. They all died. They pressed on south, passing the Antarctic Circle for the first time on 15 January.
On 17 January 1820 Lazarev on the _Mirnyi_ wrote of an 'ice shore of extreme height'. Bellingshausen recorded:
At 6:00am the broken ice had become so dense and thick that the attempt to continue farther to the southward at this point was impossible. A mile and a half from there we could see blocks of ice piled on top of the other. In the farther distance we saw ice-covered mountains similar to those mentioned above and probably forming a continuation of them. We were then in Latitude 69°06´24"S Longitude 15°51´45"E. The farther we proceeded, the more dense became the ice until about 3:15pm, when we observed a great many large high flat-topped icebergs surrounded by small broken ice, in places piled up high. The ice towards the south-south-west adjoined the high icebergs which were stationary. Its edge was perpendicular and formed into little coves, whilst the surface sloped upwards towards the south to a distance so far that its end was out of sight even from the masthead.
There has been endless debate over what they saw that day. With the benefit of modern charts, we know that if you stood on their ships' mastheads, and looked in the direction they did, continental ice-fields are visible, though perhaps not bare rock to confirm land lay beneath. Moreover neither Bellingshausen nor Lazarev used words loosely. If they write 'mountains' and 'shore' they mean it. They do not mean 'like a mountain or shore'. The letters Bellingshausen wrote to his masters make clear his own convictions about what he was seeing then and in the following days. To the Marquis of Traversey, 8 April 1820: 'After midnight between February 5 and 6 we reached latitude 68°07´S and longitude 16°15´E. There, behind small ice floes and islands, the continent of ice was seen. It had edges broken perpendicularly and stretched beyond the limit of our vision, sloping up towards the south like a shore.' In the official journal for the Minister of the Navy for 24 January: 'I designate as continental the huge ice which, on approaching the South Pole, rises into sloping mountains, as I believe that when it is four degrees of frost on the best summer day, then further to the south the frost does not, of course, become any milder, and therefore I can conclude that the ice spreads over the Pole and must be immovable, touching the ground or islands in some places.' On 17 February he observed 'ice islands near that continent show clearly that pieces are broken off this continent, because they have edges and surfaces similar to the continent.' They then probed where Cook had achieved his farthest south, not to compete with the master, but to compare ice conditions in different years. They were halted sixteen miles short of his position.
A healthy diet kept both crews fit, but it was now thirteen weeks since they had left the warmth of Rio de Janeiro. That time had been spent almost entirely in cold and exposed waters. Bellingshausen had fostered good spirits by observing all the many Russian feast days, although in the cold, the officers might not always have appreciated being required to observe the holidays in formal dress uniform. They now sailed east, keeping south of 60° until the first week in March when, with the temperature at –13°C, the commander decided he had endured enough of 'dark, harsh climes: it seems as if men's hearts grow cold in sympathy with the surrounding objects, men become gloomy, depressed, harsh and to a certain extent indifferent to everything.' The two vessels separated, _Mirnyi_ sailing a more northerly course, though it was still 150 miles south of Furneaux's timid route in the _Adventure_ after he and Cook separated. The Russians tied up in Sydney at the end of March after 131 days at sea. All the livestock and two sailors were suffering from scurvy. He received a dispatch from the Russian ambassador in Rio describing William Smith's discovery of the South Shetlands. The news flew round the port. When they sailed, on 31 October, local sealers were just a few days behind their wake. The Russians made a bad start. The _Vostok_ sprang a bad bow leak but they laboured on to Macquarie Island where one of the forty sealers they found had been living for six years. They were killing elephant seals; the fur seals had already been exterminated. Leaving there on 2 December, they sighted no land for the next two months. On 22 January 1821 they began to see terns and skuas when 250 miles further south than any previously sighted land. When land appeared, the first ever seen south of the Antarctic Circle, even Bellingshausen managed to let himself go a little: 'Words cannot describe the delight.' They named it after the Czar: Peter I Island. Their feet were itching to go ashore but the ice had other ideas, keeping them fourteen miles off. It was not landed on until whalers from the _Norvegia_ made it ashore in 1929. Now the Russians were eager to follow up the news of Smith's South Shetlands. As soon as they arrived and anchored there, they had the strangest encounter. But first, we need to back-track a little.
The _Hersilia_ was a new brig under Captain James A. Sheffield; Nat Palmer was second mate. In warmer climes, seal skins were cured for storage simply by drying them out. Where the _Hersilia_ was heading, they doubted there would be enough fine weather for that, so they sailed via the Cape Verde islands off the tip of West Africa and took on six hundred pounds of salt: enough for ten thousand skins. They next called at the Falklands, then lawless and ungoverned, to obtain water, greens, and fresh meat from wildfowl and the abandoned cattle which thrived in huge numbers. Palmer and one sailor were dropped ashore to do this, while the _Hersilia_ cruised the off-lying islands looking for seals. Another brig appeared on the horizon, and it headed for shore. It was the _Espirito Santo_ out of Buenos Aires, with British owners and crew. The lack of clear written records of these early sealing voyages is often put down to sealers being secretive in the extreme, to protect their biggest trade secret: where the seals were. In fact knowledge was shared by word of mouth, but they did not need to justify their every action to the owners in writing so little down; the test of a successful voyage was skins, not good paperwork. They also felt a comradeship with all seafarers, and confided good anchorages, safe harbours and supplies of fresh water and food. Palmer and his shipmate helped the new crew reprovision from the land and probed for intelligence on the _Espirito Santo's_ voyage. All they would confide was that they were provisioned for only a short voyage, and were following up reports of new islands sighted by William Smith. The English sailed first, Palmer watched them to the horizon, noting their course. All his life he was credited with excellent long-distance vision. But were they sailing a blind, knowing he would watch? He had three days to remember that course before the _Hersilia_ returned. Captain Sheffield decided they would follow.
Four days later they saw land: the South Shetland Islands. The received a typical reception: two days in the lee of an island, cowering from vile weather. When it calmed enough to show their heads above deck they made for a harbour fleetingly glimpsed before the storm. Palmer claimed, and it seems rather too pat, that they found the _Espirito Santo_ at anchor there. When the captain finished salting his own skins he guided Palmer to the rookeries, and the butchery began again. Antarctica has virtually no harbours where a ship can moor safely against the land. So the men were dropped ashore from whaleboats, slim wooden craft about twenty-five feet long and five feet wide, and lived on the exposed, narrow beaches, made huts from timber and canvas, and ate seal meat and more seal meat. As long as it is not overcooked, it has enough vitamin C to stave off scurvy. The heart and liver cooked up as haslet was said to be like pork haslet. Particularly if you hadn't had pork in a while. One man could kill and skin about fifty animals a day, rounding them up the beach and striking them on the head with the tip of the hickory clubs, sheathed in iron to stop them shattering on the rocks when a blow missed. It was then heavy work to scrape the blubber off the skin with a heavy beaming knife. Despite having six hundred tons of salt, they could see they would run out of it before they ran out of seals. They first tried to kill the non-breeding males, whose large fine skins fetched the highest prices. That season they sold in Stonington for $2 each, an 800% return on investment. Young Nat probably got the share of 1 skin in 35: a bonus of about $560.
Next year Stonington sent more ships. The roustabouts of New York, New Haven, Boston, Nantucket and Salem geared up for the plunder. The Stonington men even built a new ship specially. They agreed it would be useful to have a smaller, manoeuvrable vessel to fetch and carry once they were back in the South Shetlands, and to go off, lightly crewed, looking for new rookeries. For that work, they took a sloop, the _Hero_ , fifty feet long and forty tons, with a shallow draught for the unchartered seas. First she had to sail across the worst stretch of water in the world: the Drake Passage. The _Hero's_ log is a budget book bought from Samuel A. Burtus at his Book Store and Lottery Office on the corner of Water Street, in the south-east corner of Manhattan. A cover of old sail canvas and sail twine binds a few hundred sheets of soft writing paper. In it Palmer, skipper of the little sloop, would write history. The log begins: 'Commences with fair weather with breeze from WSW.' His writing is small, ornamented and clear; his spelling is usually sound. The two vessels left for the South Shetlands on 5 November. By the 9th they sighted Smith Island, at the south-west end of the Archipelago, and anchored in President's Harbour. The rest of the Stonington fleet was there, under the leadership of Captain Benjamin Pendleton. The area had rocks and the rocks had seals, but the 'harbour' was nothing more than space between two islands. Above them, usually hidden by cloud, Smith Island rose to 6,900 feet, the highest point in all the South Shetlands. This is an exposed area; I have landed fewer than half the times I have tried, rebuffed by surf crashing on steep shingle beaches. Their light wooden whale-boats whipped in, men jumped over the bows into the numbing waters, straining to hold the boat against the outward drag of the receding waves, then turning to stop the next roller throwing her up the beach and stranding or smashing her. The other men hurled gear beyond the waves' reach, scrambling to land everything quickly before the boat was stove. The second it was done, the boat was pushed out, the last man jumping up, weighed down with freezing water, throwing himself over the gunwale, his mates grabbing his shoulders and arms, hauling him unceremoniously into the bilges, laughing with relief when it was safely done. The shore party dragged everything to the top of the beach and turned a few pieces of lumber and old sail canvas into a hut: their home for months to come. Each evening was the same: arms, shoulders and back exhausted from the labour, the mind a little coarser from the killing, despite the gallows humour they used to escape the moment. Three oars would be crossed in a tripod, and a cast-iron pot hung over a fire. They savoured hot tea and coffee sweetened with molasses, full bellies, stinging mugs of grog. Never enough sleep.
Killing was done by rounding up a segment of the herd, and driving them up beach to stop the main herd becoming alarmed. They did this slowly, not out of kindness, but so as not to damage the fur. Then they clubbed them; seal skulls are not especially strong. The danger to the men came from charging males. I have been charged on a rough beach by a 250-pound male fuelled by a riot of hormones, moving faster than I could run. Knowing that they usually stop if I stand my ground lends me courage. But I would prefer something more certain than _usually_. At the last minute I take a step forward, raise my arms to look tall, and make some kind of loud grunt. By this time, grunting comes easily. If it doesn't stop, the seal will mow me down, and when I roll face down to protect my face and other favourite organs, pick me up by the buttocks with its long yellow teeth, and throw me in the air. The bite is filthy, and, before antibiotics, always became infected. Even if your immune system saw you through, you would eat standing up for weeks.
The work went on, boats returned through the surf to take the skins on board for salting. The fleet's commander Captain Pendleton thought the seals at President Harbour would not provide sufficient for the whole fleet. Nat Palmer was sent to look for new rookeries. The _Hero_ now earned its keep. He departed at 14:00 on 15 November 1820. The story is famous. He headed south and west, and soon saw an island that, oddly, had less snow on it than its neighbours. He made his way along its eastern shore which was straight as a ruler but too rugged to land. At the southern end was the massive bluff of Bailey Head and beneath it a beach which, even today, in tough inflatable powerboats, is a trophy landing. I've managed it once, and even then, the surf was wild enough to make an able seaman miss his leap into the last departing boat and go swimming. The afternoon of 15 November 1820 was not a fine day. Snow was blowing up into a storm. Palmer felt his way down the coast and, at 20:00, finding no shelter, he stood off to ride out the night. At five next morning he continued to where the cliffs had a bite out of them, before they reared up again in gigantic cliffs teeming with pintado (painted) petrels with irregular white flecks over the wings looking like paint spills. In mid-morning, a slot-like gap opened in the coast and he turned to starboard, rust-red rocks to his left. Instead of ending in a gully where one or perhaps two small ships might take precarious shelter, he found he had sailed into one of the world's great natural harbours, up to five miles across. It is shaped like a horseshoe that has been bent until almost wholly closed, giving protection from bad weather, whatever point of the compass it comes from. He named the entrance Neptune's Bellows, and the island whose shape had fooled him Deception Island. In thick weather he went ashore and climbed the south-facing crags to the bite in the cliffs, and named it Neptune's Window. Mobbed by petrels, he collected eggs and trained his superb eyes on the south and the east. On the horizon he saw land. Soon he would sail there and find land which proved to be the Antarctic mainland. He christened the anchorage Port William. It was soon well-used by American sealers and a nickname, Yankee Harbour, stuck. To avoid confusion with another anchorage of the same name, it is now Whalers Bay.
It's a story I have told many times. My research for this book showed me it isn't true. His own log confirms it. Firstly he wrote that he was leaving President Harbour for Deception Island, which you can't write unless Deception had already been discovered and named. Palmer's biographer John Spears says the island had been named on 'the previous voyage' without explaining clearly what he means. The answer lies in the fourth voyage of William Smith. Under Captain Bransfield they landed at King George Island, claiming it for that monarch, and headed further south on 27 January 1820. Two days later worsening weather forced them to stand off south from the land they had in view, and they passed a group of small islands then a larger one. In that location it could only be Deception. Smith and Bransfield were there ten months before.
Secondly, Palmer's log suggests that in contrary winds he gave up trying to reach Deception. The log consists of spare notes with few place-names, many nautical terms and abbreviations and little useful punctuation; quoting it would not help clarify the point. But Edouard Stackpole in a paper entitled _The American Sealers and the Discovery of the Continent of Antarctica_ , reads the log as taking him east to Livingston, to the bay that Weddell would later name Palmer's Bay (not its modern name, but lying immediately west of Livingston's most southerly limb: Barnard Point) and claims that subsequent log entries make much better sense if you read him as starting from there.
Wherever he was, Palmer's landing is also a suspect legend. Edmund Fanning's book says: 'From Captain Pendleton's report, as rendered on their return, it appeared that while the fleet lay at anchor in Yankee Harbor, Deception Island, during the season of 1820 and 21, being on the lookout from an elevated station, on the mountain of the island during a very clear day, he [that is Pendleton] had discovered mountains (one a volcano in operation) in the south; this was what is now known by the name of Palmer's Land.'
What did he see? I have climbed those cliffs two dozen times. If he climbed right to the top of Cathedral Crags, above the Bellows, he could see farther than me, because access to the very top is now prohibited to protect nesting birds. On two occasions I have seen a low snow hill far out to the south-east, scarcely distinguishable from an iceberg. Checking my compass bearing against the ship's charts, it was the nearest peninsula on the mainland, some thirty miles off: Cape Roquemaurel. It is possible he saw the mainland, though if he was on the south shore of Livingston, the distance was still farther. Palmer's own papers were later lost in a house fire, so unless luck throws more information our way, the evidence will remain ambiguous.
Nat reported back to the fleet in President Harbour. His news was timely; he found that the _Frederick_ had dragged her anchors in the inadequate shelter of that anchorage and nearly been wrecked. Five vessels now relocated to the haven of Whalers Bay, Deception. They had leisure to realise that the reason there was less snow, and that the springs along the beach were hot, was that they were anchored inside a volcano. They dug pits in the black volcanic sand and luxuriated in the hot water, washing clothes and bodies caked with blood and fat. It is a caldera, a volcano that had collapsed after a fierce eruption, allowing the sea to come in, as he had, through the Bellows, and fill the crater. For the rest of the season, the larger vessels acted like factory ships, staying in shelter and salting the skins brought back by the _Hero_ , which acted as a tender between the shore parties and Whalers Bay. Palmer found time to scout around Livingston Island, a short distance to the north-east. It is the most beautiful of the South Shetland Islands. Its high mountains rise in swooping ridges that are both wild and elegant. It is as though tents had been overtaken by a blizzard and fossilised in rock and ice. It was after sailing past Livingston that sealer Robert Fildes wrote, 'the mind is forced into pious contemplation of the grandeur of the scene.' Sailing alone into a narrow strait, Palmer grounded on a hidden rock, but managed to ease her off without damage. He then saw a whale swimming slowly ahead, declared 'Where a whale can go, I can go!' and followed it safely through to discover more rookeries. They now knew the whereabouts of sufficient seals to fill all the ships. It was late November, and the seals were arriving in ever greater numbers. On 26 November he picked up 465 skins from the shore-men. The log then falls strangely silent until 3 December when he records 905 skins ferried to Deception. On 7 December the log records ten thousand skins picked up, so many that he and his crew did not finish unloading at Deception until midnight. Then he weighed anchor and sailed back to President Harbour. Two vessels which had been working Livingston Island sailed into Deception with skins and grim news: their companion ship the _Clothier_ had been driven ashore and although the men were saved, the ship was wrecked. The busiest period of all now followed, but Palmer still has time for one-line log entries. Then there are more silent days. What could have been happening, what kept him too busy to write a single line?
The sealers wrapped up the season taking elephant seals for oil, which they rendered in Whalers Bay. The _Hero_ took it home, departing 22 February 1821, having ferried 50,598 skins to Whalers Bay. A quarter of a million skins were taken by all sealers in the 1820 –21 season. The average return of all sealers was 8,333 skins per vessel but the Stonington fleet did best, averaging 11,000 per vessel. They also harvested fifteen hundred barrels of elephant seal oil at $10 a barrel. When they left Antarctica, they probably thought that the loss of one vessel was a cost they could afford to carry, but the full price had not yet been paid. Captain Sheffield in the _Hersilia_ left their company to go fur-sealing at St Mary's Island off Chile, where he took fifteen thousand skins. He and his vessel were then seized by Spanish. All the hard-won skins were confiscated.
There is still the question of Palmer's missing days. The answer lies in the journal of another Stonington sealer, Edward Fanning. He wasn't on the voyage, but in later life he recorded Palmer's memories of those missing days. He might have forgotten a Russian name, but facts slip when feelings and impressions remain green, even in old age. Palmer had sailed towards whatever he had seen to the south and east from the heights of Deception. He recollected, 'I pointed the bow of the little craft to south'ard and with her wings spread, the mainsail abeam, and the jib abreast on the opposite bow. She speeded on her way like a thing of life and light. With her flowing sheet she seemed to enter into the spirit which possessed my ambition, and flew along, until she brought into the sight of land not laid down on my chart.' Fanning wrote: 'To examine this newly discovered land, Captain N. B. Palmer, in the sloop _Hero_ , a vessel but little rising forty tons, was despatched; he found it to be an extensive mountainous country, more sterile and dismal if possible, and more heavily loaded with ice and snow, than the South Shetlands; there were sea leopards on its shore, but no fur seals; the main part of its coast was ice-bound, although it was in the midsummer of this hemisphere, and a landing consequently difficult.' He reached land around what is now Mount Hope, and turned south. The leopard seals were of no interest; they are mostly solitary animals fortunate to have neither desirable fur nor thick blubber.
The _Hero_ bumped her way to 68°S before the pack ice became impenetrable, and he turned the helm back north. Few modern cruise ships make it that far south in this area. As he passed Mount Hope once more, fog came. He lowered a line; it was too deep to anchor. He kept the minimum sail needed to manoeuvre and crept forward, fearful of sailing onto a reef or into ice. The wind softened away. They drifted. Such times have a special solace. When I work in Antarctica there are times when I stand by in a small boat until needed. It might be five minutes or an hour. I turn off the engine. If it's calm, there is a peaceful solitude I feel at such times which I ever never known anywhere else. In moments, the ship has ceased to be there. No time passes, there is just now. Just me and the universe, above, around and below me. Antarctica blows its crystals into your mind.
As night fell, a light breeze sprang up. Bow ripples and the gentle creak of cord and canvas replaced the faint dripping from ropes and spars. At midnight Nat relieved the Mate, Phineas Wilcox. At 00:30 he gave bell a single tap. He heard a second bell answer from off the bow. While he was still telling himself he was alone in the world and there could be no answering bell, he heard a third. Palmer said: 'The response startled me, but I soon resumed my pace, turned my thoughts inward, and applied myself to making castles in the air.' At one he struck two bells and heard two more answers. 'I could not believe my ears. Save for the screeching of the penguins, the albatrosses, the pigeons and Mother Cary chickens, I was sure no living object was within leagues of the sloop.'
Many of the crew thought the bell supernatural. At 03:30 Wilcox came on deck saying he had heard human speech. The voices were unfamiliar, speaking a strange language. Soon the fog lifted a little. It was only twenty-three years since the publication of _The Rime of the Ancient Mariner_. What spectre-barks had come to them in the night? Parts of the mist became darker and took shape. A frigate was seen off the starboard bow, and a war sloop off the port quarter, and ship's cutter coming towards them bearing a man in full uniform. Palmer hoisted the Stars and Stripes. The ships raised the Russian flag. The officer spoke good English. 'Capt Bellingshausen requests your company on the _Vostok_.' It was a strange embassy. The Russians wore naval uniform, Palmer a home-made sealskin coat and boots with a sou'wester. Bellingshausen's English was minimal. Palmer offered to guide him round the South Shetlands. He freely told them where they were sealing, and about his foray south past over 150 miles of coast, and sent the cutter back to fetch the _Hero's_ chart and log. It wouldn't have helped the Russians much. Palmer was a businessman not an explorer, and because he had found no fur seals, he had not charted his route or the coast in any detail. A chart on which he soon after collaborated with the English sealer George Powell showed the known segments of what we now know is the Antarctic Peninsula. Powell's memoir on the chart shows he thought the Peninsula was like the South Shetlands: an archipelago but with the channels between blocked by ice.
American accounts stress Bellingshausen's admiration for the young adventurer whose little ship lay in the shadow of his frigate. In them the Russian mocks his own hopes of discovering and naming the neighbouring islands of the South Shetlands, as Palmer, this Yankee boy wonder, has come down in his cockleshell and done it all. Fanning wrote: 'His astonishment was yet more increased, when Captain Palmer informed him of the existence of an immense extent of land to the south, whose mountains might be seen from the masthead when the fog should clear away entirely.' Fanning claims Bellingshausen immediately named it Palmer Land. This is fiction. Firstly, flowery words were not Bellingshausen's style, especially on a cold foggy morning when discussing navigation in a language he knew poorly. Fanning's eulogy also forgets that Bellingshausen had come to the spot because he had already heard at Sydney that Smith had discovered islands in the area. He had no expectation of being the first.
Bellingshausen's account of the meeting is matter-of-fact. Well, dull would be more honest. Its focus is not a magical encounter in the fog (he doesn't mention there being any fog) but in the feverish activity of the sealers who had rushed south like gold prospectors. He was prescient about the short-sightedness of the massacre.
At 10 o'clock we entered the strait and encountered a small American sealing boat. I lay to, despatched a boat, and waited for the Captain of the American boat. Soon after Mr Palmer arrived in our boat and informed us that he had been here for four months sealing in partnership with three American ships. They were engaged in killing and skinning seals whose numbers were perceptibly diminishing. There were as many as eighteen vessels about at various points, and not infrequently differences arose among the sealers, but so far it had not yet come to a fight. Mr Palmer told me that the above mentioned Captain Smith, the discoverer of New Shetland, was on the brig _Williams_ , that he had succeeded in killing as many as 60,000, whilst the whole fleet had killed 80,000. As other sealers were competing in the destruction of the seals there could be no doubt that round the South Shetland Islands just as at South Georgia and Macquarie Islands the number of these sea animals will rapidly decrease. Sea elephants, of which there had also been many, had already moved from these shores further out to sea.
More practically, it wasn't a place to linger. 'The anchorage was thin sand, and two British ships and one American one had dragged their anchors and been wrecked.' Bellingshausen then rouses himself for some striking lines to encapsulate the momentous meeting of these two genuinely great men and put his finger on the emotional impulse of the moment. He wrote: 'Mr Palmer soon returned to his ship, and we proceeded along the shore.'
They sailed along the islands, charting. On 10 February they left the north shore of Elephant Island, with neighbouring Clarence, the most northerly islands in the chain, lashed by a blizzard. It is a wild shore facing thousands of miles of empty ocean, and a marvellous last sight of Antarctica. One dusk I jammed myself into corners of the deck trying to get enough shelter from the Force 12 katabatic winds screaming off the island's ice-cap to photograph the savage beauty of the scene, while the wind kept tearing the camera from my face. They headed for balmy Rio de Janeiro, reaching it on 11 March. The home port of Kronstadt was finally regained on 5 August 1821 after 751 days and 57,073 miles. Just three men had been lost, a record Cook couldn't have bettered. In his absence, Russian politics had moved on; land ambitions had overtaken marine. His brilliant voyage was forgotten; recognition has been little and late. It took ten years fighting bureaucratic obstruction to publish, and there was no English version of his voyage until 1945. There was no official visit to Antarctica by Russia or the Soviet Union until 1946. Bellingshausen remained in the Navy, was appointed Admiral in 1831, and became Governor of Kronstadt, where he died in 1852.
The year after the meeting in the fog, despite the warning signs that all the easy sealing had been done, Stonington geared up a sizeable fleet, giving Palmer a larger sloop than before; the _James Monroe_. Once in Antarctica, Palmer was again despatched south to scout. Once more, he reached 68°S before the pack ice barred his way. Next, in the company of the English sealer the _Dove_ , Capt George Powell, he went farther north looking for new rookeries. North and east from the tip of the Antarctic Peninsula, they found barren islands lying around 60°S and between 044° and 047° W, and named them after Powell. Soon they were re-named the South Orkneys, because they lay in the equivalent latitude to the northern Orkneys.
Already, the big seal catches were just a memory. All told, the entire fleet took just fifteen hundred skins. The _Hero_ was sold at Coquimbo, Chile, and its cargo was brought back by the _Alabama Packet_. Palmer never claimed he had seen continental land, but as the map of the Peninsula was filled in, it looked less and less like an archipelago.
The newly trained navigating officers of the _Vostok_ and _Mirnyi_ were zealous in the huge quantity and fine quality of their observations. The charts made of the South Shetlands set new standards for Antarctic surveying in this area. Right until the 1930s, the coastal sketches of South Georgia made by Paul Mikhailov were still used in the Admiralty's _The Antarctic Pilot_ , the international handbook for sailing those waters.
A portrait of Nathaniel Palmer as an old man shows another face: a patriarch whose opinions are law. Even allowing for the immobile expression demanded by long photographic exposures, the jaw juts sullenly, and the mouth bends down in a grim bow. He seems to have repeated too often the dictum of the skippers who went south down and past the roaring forties, and begun to believe it: 'Up there you obey your captain and God Almighty. Down here I am God Almighty!' He had helped Simon Bolívar in his wars of liberation against the Spanish, and escaped a firing squad by giving the officer in charge a Masonic handshake, and finding a fellow member. He led the packet trade to Europe and the Clipper trade to China, designing ships with slimmer hulls which flew through the sea. He loaded the _Samuel Russell_ until the deck was only a foot above water; she then sailed from New York to San Francisco and beat the record. He rebuilt the biggest all-wooden ship ever made: the _Great Republic_ of 4555 tons after she burnt out in Liverpool docks in 1853, but he never believed in iron ships. He stood up a hull plate on the dockside, fired a musket ball through it, and said, 'They'll never work.' He died on Midsummer's Day 1877. Iron ships took over the world.
# Charted in Blood:
James Weddell (1787–1834) and Other Sealers
## JAMES WEDDELL'S VOYAGE OF 1822–24
_Seduced by Antarctica, Weddell puts business aside, passes James Cook's farthest south, and finds an open sea. Ancients were right: you can sail to the South Pole._
James Weddell's no-nonsense approach is epitomised by his writing. After the customary meandering title, _A Voyage to the_ _South Pole Performed in the Years 1822 – 24_ , he describes everything about the aims and resources of the voyage in two sentences. 'Our adventure was for procuring Fur-seal skins, and our vessels were the brig _Jane_ , of Leith, of 160 tons, and the cutter _Beaufoy_ , of London, of 65 tons, both fitted out in the ordinary way and provisioned for two years. The former, with a crew of twenty-two officers and men, was under my command; the latter, with a crew of thirteen, was commanded by Mr Matthew Brisbane.'
His plan was to head for the South Orkneys, and if the sealing was not good 'to prosecute a search beyond the tracks of former navigators.' He could now sail through the area between South Georgia and the Falklands without fear of wrecking himself on the Aurora Islands. They had been recorded by the Spanish since 1762 when their ship _Aurora_ not only sighted them but sailed between them. For sixty years everyone sailed carefully around them without seeing them. In 1820, on his first voyage in the area, Weddell had surveyed their supposed position in fine weather and proved they did not exist. He then spent two months navigating in the channels around Cape Horn. Everyone who rounded the Horn did so as far out at sea as possible, and next to nothing was known about the geography, or the safe harbours and anchorages. He provided his own rough surveys and urged further work. It became the project that brought Robert Fitzroy and the _Beagle_ here with a young graduate called Charles Darwin.
At daylight on 12 January 1823 they were in sight of the South Orkneys, and hoping to gain harbour. But the wind dropped and left them wallowing in a heavy swell surrounded by icebergs 'which made our navigation truly hazardous.' Weddell had briefly been to the islands the year before, heading home with a hold full of seal skins. They kept the ships out in the bay of Saddle Island, which lies off the centre of the north coast of Laurie Island, and sent in a boat from each vessel. He surveyed the sweep of the mountains that rise straight from the stark shoreline. 'This coast is,' he thought, 'if possible, more terrific' [terrifying] 'in appearance than South Shetland. The tops of the mountains, for the most part, terminate in craggy towering peaks, and look not unlike the mountains of a sunken land.'
After going ashore and having a fright when fog hid the ships from view, Weddell ordered his second mate ashore to take leopard seals. He 'captured' six, or as we usually say, shot them. Back in Edinburgh, they were the first leopard seals to be scientifically described. Weddell's own painting of them, inscribed 'drawn from nature' is carefully done, and does have the trademark ferocious teeth and unholy reptilian smile, but otherwise it is quite inaccurate. The tiny head at the end of a stretched neck makes it look more like a fossil plesiosaur than a seal.
He did not find fur seals in any numbers, so he continued as planned, 'beyond the tracks of former navigators' going south-south-east. At noon on 27 January they reached nearly 65°S. He was already south of the latitude of the South Shetland Islands, but on the other side of the Peninsula. The weather and ice conditions were good, but he saw the nights beginning to lengthen, and their darkness deepen: not the time for ice navigation in unknown waters. He decided to retreat north and explore east 'considering it probable that land might be found between the South Orkneys and Sandwich Land.' They immediately had a scare. At eleven o'clock that night a dark object appeared in the tricksy light of dusk, two ship-lengths away. It looked like rock. They stopped the ships, threw over the lead sounding line and found no bottom. As they cautiously edged closer, 'the chief mate ascertained it to be a dead whale very much swollen'; by such incidents are lands invented.
Alone in a sea of ice, winter darkness creeping north each day, they looked for Sandwich Land, not knowing Bellingshausen had already proved it to be islands, because the Russian had not published his findings. Weddell knew what motivated ordinary sailors: 'I had offered a gratuity of £10 to the man who should first discover land. This proved the cause of many a sore disappointment; for many of the seamen, of lively and sanguine imaginations, were never at a loss for an island.' All of them evaporated on approach. Showing his courage at sailing into the unknown, and generous to his precursors, Weddell 'carefully avoided the tracks of Captains Cook and Furneaux; and I may remark how narrowly Captain Furneaux in the _Adventure_ , in December 1773 and January 1774, escaped seeing South Shetland and the South Orkneys. He passed within 45 miles of the east end of Shetland and 75 miles of South Orkneys.' But by 4 February they were back so close to Cook's course that Weddell was convinced no land connected Sandwich 'Land' to any southern continent: they were islands. But he 'conceived it probable that a large tract might be found a little farther south than we had been.'
Fogs and gales kept the decks soaked and the men suffered from colds and rheumatism. Weddell moved the ship's cooking stove near to their quarters and had fires lit to dry clothes. Added to the three glasses of rum per day, this removed, or at least took their mind off, the discomforts. At daylight on 10 February land was called, not by bonus-hungry seamen, but the first officer. Weddell came on deck and agreed. Here, at last, was the goal of mariners through the ages: the southern continent. By afternoon they could see it was just ice embedded with dirt. Weddell speculated whether many of the isolated rocks which were scattered over the charts of the Atlantic were similar casual observations of dirty bergs which might delay mariners for years to come if they were not investigated and deleted from the charts. However it was still 'a new disappointment, and seriously felt by several of our crew, whose hopes of having an immediate reward for their patience and perseverance were again frustrated.' It's easy to forget they were not explorers or scientists but common seamen being paid a share of profit: no seals meant no pay, no land meant no bonus.
Exceptionally for a sealer, Weddell had spent the huge sum of £240 on three chronometers for the trip. One by Murry of London proved exceptionally sound; even its slight rate of gain was extremely regular and could be allowed for precisely. He was therefore puzzled to find that his calculation of longitude by chronometer was very different from his dead reckoning using the ship's log. He reluctantly concluded that in difficult conditions, among the ice, his observations were not as good as he wished. In time he realised that observations made from the same spot, an hour apart came out differently, because near the magnetic poles, the earth's magnetic field dips downwards. The horizontal element of the force, which aligns the compass needle, becomes weaker, and the ship's iron equipment, such as anchors, began to influence the compass much more. Small movements around the deck could change the angle of north on the compass between three and five degrees. These confusing magnetic measurements were taken very seriously by the British Admiralty, and clarifying their variations would be the prime task of James Clark Ross's voyage of 1839 – 43. Meanwhile Weddell's calculations (estimates would be a better word) were being affected by the gyre or clockwise circular current that would later drag Shackleton's _Endurance_ hundreds of miles while it was gripped in the ice. He correctly concurred with Cook that the great ice formations originated from land.
18 February became a historic day. They were more than 72° south, and by evening, with whales all around the ship, it was noted with capital letters on the log: NOT A PARTICLE OF ICE OF ANY DESCRIPTION WAS TO BE SEEN. They sailed on in hope. 'The evening was mild and serene, and had it not been for the reflection that probably we should have obstacles to contend with in our passage northwards through the ice, our situation might have been envied.' On 20 February at 74° 15´S they saw three bergs so big they called them ice islands; one was covered in penguins. Open water still beckoned them on, but the wind turned to the south, and February was almost out: 'taking into account the lateness of the season, and that we had to pass homewards through 100 miles of sea strewed with ice islands, I could not determine otherwise than to take advantage of this favourable wind for returning.'
Weddell's open sea was to mislead generations of future navigators. He had been spectacularly lucky in arriving late in an unusually ice-free season. It breathed new life into the old story of open polar waters drawn by Gerhardt Mercator back in 1606. Many geographers had argued that in high latitudes, the long summer days would heat up the oceans and melt the ice. He wrote: 'the antarctic polar sea may be found less icy than is imagined, and a clear field of discovery, even to the South Pole, may therefore be anticipated.' However often his successors hit dense pack ice farther north, they pushed on, spurred by Weddell's experience.
He paid tribute to his crew. 'I expressed my approbation of their patient and orderly behaviour, and informed them that they were now to the southward of the latitude to which any former navigator had penetrated. Our colours were hoisted, and a gun was fired, and both crews gave three cheers. These indulgences, with an allowance of grog, dispelled their gloom, and infused a hope that fortune might yet be favourable.' Yet the crew still faced returning home with empty pockets to anxious wives, their credit run to the limit at every local store. They would be honoured and broke.
He was deep in the south-east corner of the sea he would name King George IV Sea, but would later be renamed for him: the Weddell Sea. As he sailed north he passed through 63°21´S and 45°22´W and looked at his South Atlantic chart, one in common use. It showed they were sailing through the mythical kingdom of South Iceland. Weddell had earned the right to be contemptuous: 'It is much to be regretted that any men should be so ill-advised as to propagate any hydrographical falsehoods; and I pity those who, when they meet with an appearance that is likely to throw some light on the state of the globe, are led through pusillanimity to forego the examination of it.'
They headed for South Georgia which they reached on 12 March 1823. He climbed a peak to measure the altitude of the sun; the weather was perfect: still, without a breath of wind. He set up the mercury dish which acted as an artificial horizon from which to measure the angle. He saw it trembling. This was the first record of seismic activity on South Georgia. He looked at the toll his own industry had taken on the wildlife of the island. The elephant seals were nearly extinct, twenty thousand tons tons of oil having been shipped to London alone; 1.2 million fur seal skins had been taken. Unlike the American sealers, the British sold their skins cheaply, despite Chapman having discovered how to dress them in 1799. Sealskin soon replaced the much more expensive beaver pelts for hats and other articles.
On 11 May he arrived at New Island in the Falklands, then frequented by whalers and sealers. Weddell moved to Quaker Harbour on Swan Island for the winter. While here, he was told that four years before quantities of ice seen had been seen at 50°S, but he admits he never saw it north of South Georgia. He also observed why, some time before, Commodore John Byron, grandfather of the future poet, had thought that fur seals could give birth to eighteen or more pups: he had mistaken the male's harem of much smaller females for a mother and her young. During his earlier stay in 1820 he was also caught up in a little piece of history that was to have a long fuse. He was visiting Port San Salvador on the north shore of East Falkland when on 2 November he received a letter telling him of the claim made by the thirty-gun frigate _Heoind_ of Buenos Aires. Colonel Jewett's commission, in the name of the United Provinces of South America, was to take possession of the islands. In deference to the colonel's commission and thirty-eight guns Weddell paid a social call without contesting possession. He left the Falklands on 7 October, bound once more for the South Shetlands.
On 16 October they saw their first tabular iceberg; the previous season's conditions had been so freakish they hadn't seen one. Later that day, still ninety-five miles from land, they met ice too dense to sail in. It wasn't until the next day that they found a channel leading into it. They probed the pack around a wild, high island for a week. On the 28th, a hurricane hit them. Everything movable on deck was washed away and replaced by so much ice that Weddell credited it with reinforcing the rigging and fixtures and reducing damage. There was a downside. The rudders became locked solid. Unable to steer, the vessels lost sight of each other.
Next day they regained contact but 'many of our crew were hurt in the early part of the gale, by being thrown down, and nearly all of them were frost bitten. They had been above a twelvemonth from home, and consequently their clothes were nearly worn out.'
They fought the ice with little success, Weddell and Brisbane's perseverance being all the braver since the bottoms of the ships had only 2½" thick planks. Their bow was damaged and temporary repairs made, nailing lead sheets and canvas over to slow the leaks. Weddell was familiar with navigation around Greenland, yet wrote 'sailing among ice in these southern latitudes is attended with a much greater risk.'
In 1820 James Weddell's had been the first vessel fitted out from England to anchor in the South Shetlands; William Smith had been trading out of South America. Weddell now singled out the island he thought the 'highest and most forbidding in aspect', the one they had fought round in the second half of October, and made the first landing on it, naming it for himself: James Island.
He began sealing. In the South Shetlands they killed two thousand elephant seals for oil, and noted the vast profusion of fur seals. His observations on elephant seals are highly professional. When the males first arrived, the fat from three or four of them would yield one ton of oil. If both males and females were taken, it required an average of seven animals to make a ton. The males might try to prevent a boat landing by swimming out and attacking it; the size and outline of a whaleboat's submerged hull would be comparable to an adult male, and they have mistaken boats for rival males. Descriptions of their death throes are tinged with whimsy: 'the sea-elephant, when lying on the shore, and threatened with death, will often make no effort to escape into the water, but will lie still and shed tears, merely raising the head to look at the assailant; and though very timid, will wait with composure the club or the lance which takes its life.' Their power was not doubted: 'in the agony of death, stones are ground to powder between its teeth.'
He realised the males were polygamous, and that Ross was wrong about multiple births, but he thought the harem was where males protected females throughout their gestation; he did not know how seasonal their visits to land were. He observed moulting, and the differing timings of seals' arrivals, dependent on age. He made fair guesses at the ages of maturity, and their longevity. He thought that after they had experienced hunting they began hauling out on rocks convenient to launch straight into sea. He mocked the myth that they would throw stones at pursuers with their tails when fleeing; it was just part of their clumsy flight. He tamed pups, hand-rearing them from a few weeks to two months until 'by some accident they were allowed to fall or walk overboard.' Until they escaped, we would say. In 1821–22, over 320,000 fur seals were taken. A sustainable yield was around a 100,000 a year. Before he left, on the western isles of the South Shetlands Weddell saw pieces of wreck and the scantling, or small timbers, of a seventy-four-gun ship; he thought it must be the _San Telmo_ , the first ghostly discoverer.
Fifty years before, the Master, James Cook, had achieved a farthest south and written 'I can be bold to say that no man will ever venture farther than I have done.' He continued that if anyone did so, 'I shall not envy him the honour of the discovery.' Weddell had gone 214 miles farther south. He had done it as a sideline while doing his other job, sealing, and lost no one. He was a man who read all he could about where he went, but wore his learning lightly: with humour, without pretension.
After the voyage was over, the admirable Captain Brisbane, who had skippered a cutter the size of a Victorian racing yacht to the farthest south ever achieved, would go to the Falklands to manage a new livestock venture and be murdered by convicts left during the brief Argentine presence there. Weddell offered himself to lead or organise another expedition south but was rebuffed by John Barrow, effectively the Permanent Secretary at the Admiralty. Barrow, like Clements Markham at the RGS during the Heroic Era, could not believe that anyone but a naval officer could manage an exploring adventure. This was an interesting notion, for if excitement is the product of poor preparation, as William Speirs Bruce's men would later write, Naval expeditions were certainly exciting. Five years later, Weddell sailed from Buenos Aires in the ageing _Jane_ , Gibraltar-bound. His part-share in her represented much of his wealth. Weakened by her battles in the Antarctic, she leaked so badly that he barely nursed her into the Azores, docking in the pretty harbour of Horta. A survey found her unseaworthy and beyond repair. He was forced to board another vessel to come home, but she in turn was wrecked off the neighbouring island of Pico. He had to take waged work as a captain on others' ships. In 1831 he was on the Derwent River, Hobart, Tasmania, as skipper of the sealer _Eliza_ , helping the scurvy-rotted crew of the _Tula_ ashore. Their captain was John Biscoe, making a circumnavigation on Antarctica described later in this chapter. It was Weddell's last voyage; he died in poverty in lodgings in London on 9 September 1834 and was buried at St Clement Danes on the Strand, Samuel Johnson's old parish church. Time and London air have eroded his headstone. Today, it cannot be found.
## THE SEALING FLEETS ARRIVE
_War in the South, and the first footprint is left on the continental mainland, but whose?_
The sealers were hot on the heels of the Royal Navy. Two months after Smith's first landing, the _Espirito Santo_ under Captain Joseph Herring and financed by British businessmen in Buenos Aires landed at an unknown spot in the South Shetlands, looking for seals. Soon, a mob of sealers and whalers would follow. In 1821, Sherriff Cove in the north west of Livingston Island was the anchorage for the _Williams, John, Lady Frances_ and _Mercury_ , all of Liverpool, the _Dragon_ of Valparaiso and two American ships. The four English crews alone took 95,000 seal skins from there.
In his memoir _Voyages_ , the highly successful sealer Edmund Fanning from Stonington, Connecticut describes sealing around the globe. He had interesting comments about the Antarctic hunt. Elephant seals should be caught when they first came ashore and yielded three barrels of oil each; after a few weeks ashore, fighting and fasting, the yield halved. Young elephant seals could be clubbed, female and younger male adults could be lanced, but the biggest males had to be shot with muskets. They were provoked into bellowing open-mouthed then shot through the open mouth into the brain. Two musket balls were fired. They might not kill outright but would stun them so heavily they could be safely lanced. If they came across elephant seals sleeping they would hold the gun to the head and fire. The noise seldom woke the others.
The skin, with the fat, but not the meat beneath it, was cut into five- to ten-inch wide strips, washed, diced into two-inch cubes, and put in a kettle to render out the oil. The fire was fed by scraps of rendered-out skin. The tongue was a delicacy; Fanning thought it more delicious than ox tongue.
They also took 'sea leopards'. Sometimes it is hard to know whether they meant leopard seals or the other seal with a mottled coat, the Weddell seal, but Fanning specifies it was a species which would follow the boats and attack them. Weddell seals do not do that; leopard seals, as I know from experience, love biting boats. Leopards are now mostly solitary creatures, but the early Antarctic sealers saw sizeable rookeries, now unknown: 'on Palmer's Land, and the south part of Sandwich Land, they are found herded together in rookeries of many hundreds and furnish oil, as the elephant, in proportion to their size.'
Fur seals were killed like elephant seals; the females, known as clapmatches (the males were wigs), and the pups would let them approach within clubbing range. Only when they had become shy from what Fanning euphemistically calls disturbance, did the sealers have to shoot them, adding a cost. The killing was described by Captain Amaso Delano, a man immortalised by Herman Melville in his masterful but little-known story 'Benito Cereno', in which Delano appears in his own name as the weak-moralled master of a vessel which meets a slaver in which discipline has broken down. Delano had a descendant who shared his name: US President Franklin Delano Roosevelt. 'The method practised to take them was to get between them and the water, and make a lane of men, two abreast, forming three or four couples, and then drive the seal through this lane; each man furnished with a club, between five and six feet long and as they passed, he knocked down such of them as he chose, which are commonly the half-grown. When stunned, knives are taken to cut and rip them down on the breast from the under jaw, to the tail, giving a stab in the breast that will kill them. After this the hands got to skinning. I have seen men, one of whom would skin sixty in an hour.'
The skin was best cured by leaving half an inch of fat on it. They then washed it and, while it was still wet, rubbed salt well into the fat especially around the neck and the edges. It took about two kilos of salt per skin, usually picked up in the Cape Verde Islands on the way south. They then packed the skins flat, in tiers, or folded in books known as kenches. For the China market, all the fat was cut away, and the skins stretched on wooden pins to dry. An experienced man could dress thirty a day.
With few exceptions, the sealers did not explore inland Antarctica. Their prey stayed on the shore and so did they. But they competed to find new rookeries, going as far as the pack would allow. Despite the brutality of their business, running a mobile abattoir in a pristine wilderness, their ranks included men of science and curiosity. They made important discoveries and took risks with no promise of profit, to know more about this newfound world.
Two American fleets had historic impacts. One was Pendelton's Stonington fleet, the other was a group including the two hundred and fifty-ton ship _Huron_ from New Haven, Connecticut and the eighty-ton schooner _Huntress_ , a Nantucket schooner. But it was the _Cecilia_ , a little schooner acting as tender to the larger vessels, which would make history. She was brought south 'knocked down', that is taken apart and stowed, on the _Huron_. There were two experienced seamen of forty-one and thirty, the average age of the rest was twenty-one.
When they arrived, the larger ships hove to, and the _Cecilia_ went off to look for a safe harbour, quickly finding Yankee Harbour at the west end of Livingston Island, which was to be the main base for American sealing. You can still see their rusting iron try-pots on its shingle beach. The sheltered inner harbour is shallow. One captain I worked with, new to Antarctica, attempted to enter Yankee Harbour in the relatively deep-draughted ice-breaker _Polar Star_. The novices are always more cocksure than the veterans. I had been in on my Zodiac earlier in the season and dared to point to shallows I had seen on the inside of the bend. I still recall the patronising smile he gave me, a failed camouflage for anger. I retired to the top deck, from where, four minutes later, I photographed the yellow mud thrown up when we grounded. Cue astern. The fur seals had come back. When I returned to my Zodiac as the landing was wound up, a pup was leaning against it.
The _Cecilia_ spent the following days ferrying men to the seal beaches where they cached their stores and pitched crude tents. It would return in a few days to pick up the men and the skins. On 19 December, Captain Burdick on the _Cecilia_ made the first of three landings on the south shore of Livingston. At the next bay, 'it blowing a gale on Shore we Could neither Land nor they git off.' Burdick's spelling enlivens all the log entries. When he uses the expression 'wore off' (turning with the wind to stern); he adds an 'h' in the unluckiest place. The next day they made it, retrieving the men and 500 skins, and, at the third site, another 480.
Soon after the Stonington fleet arrived they used knowledge gained the previous season to head straight to the richest beaches and they creamed off the known rookeries. The other fleets had to discover new ones, or go without. Captain John Davis made a final pick up from the shore camps then left on 28 December to reconnoitre the other islands of the South Shetlands lying to the north and east. Despite going to the far end of King George Island and circumnavigating the principal islands over twelve days he 'found no Seal to speak of.' His only cargo had been the crew of an English sealer, possibly the _Lady Trowbridge_ , sunk on 20 December, which he returned to their own fleet, before calling at their old shore-sites for 2470 skins which were in poor condition because they had waited two weeks without salt.
Captains Davis and Burdick sometimes sent a whaleboat to the west end of Yankee Sound to hunt for seals. On 24 January the _Huron's_ bosun Charles Laing, twenty-one years old and five feet three, returned from a four-day cruise along the north shore of Livingston Island, to Shirreff's Cove. He had only fifty-two skins, and they had clashed with the English sealers. Another American whaleboat had taken a drubbing. Captain Burdick's log gives details in a brand of English which suggests his real schooling was the sea: 'a boat came in belonging to Captain Barnard brig _Charity_ having ben robed of Eighty Skins by the English at Sheriff's Cape and Drove off the Beach 4 p.m. our Boat came in from a Cruce with 52 having Likewis ben Drove from the beach at Sheriff's Cape by the English where he said there was plenty of Seal.' A contemporary map of Livingston Island names one cove Robbery Beach.
The Yankees had not come so far to watch the British take the spoils. They would fight. They mustered 120 men: 'by the best information we can git the English have but about 80 men there.' One of them, Captain Barnard, would have been especially keen to get in a good fight. He had rescued a shipwrecked English party in the Falklands, who then seized his vessel and marooned him. A battle seemed inevitable: 'then we got under way and stood to the South & Westward in company for Sheriff's Cape at 11 p.m. Capt. Bruno came alongside in his boat and reported that he had examined the Beaches round Sheriffs Cape and Saw but a very few Seal nothing to make an object to stop for.' A few days later, John Davis sent his first mate Goddard to Shirreff's Cape and 'at 2 P.M. the Boat returned from Shore not being allowed to Land as the English had collected in numbers say from 65 to 70 men, all armed with Guns, Pistoles, & Swords and appeared in a hostile manner, Hoisted in the Boat and Proceeded on to the westward.'
The undeclared war ended as suddenly as it began because there was little left to fight over. The day before John Davis had admitted as much in his laconic log entry: 'Concluded to make the best of our way out for our people that is stationed on the South beach, and then to go on a cruise to find new Lands, as the Seal is done here.' I have never seen a log entry that wonders what they might do when they have killed their own livelihood. There would be other trades.
The American sealers returned to their vessels at Yankee Harbour. Captain Robert Johnson took the _Cecilia_ and pushed as far south and west as he could, returning on 27 January after three weeks' absence. The log read 'he had ben to the Lat. 66° South and the Long. of 70° West and still found what appeared to be nothing but Sollid Islands of Ice and Snow.' Captain Davis decided to look for himself. On the morning of Tuesday 30 January 1821, he slipped the _Cecilia_ south down the Morton Strait with Snow Island to starboard and Livingston to port. He visited his three shore camps, picking up a total of 902 skins, then steered south, into unknown waters beyond the South Shetland chain. Tracking their route now is hampered by the fact that they did not know their compass deviation was around 25°E. On occasion they locate their position relative to well-known landmarks, and we can pin down where they are. But when they give latitude and longitude, what you would really see from those co-ordinates is seldom what they describe. In fine weather, the clarity of the atmosphere deceived them, and here they often thought land was close when it might be forty miles off. On 1 February they reached the south shore of Smith Island, the westerly outpost of the South Shetlands. They closed on the shore and lowered a boat, hoping for seals. They were there, but so were seventeen British sealers on the beach killing them. John Davis pointed the _Cecilia's_ bows south east to Low Island, a low shield of an isle compared with the pyramidal profile of Smith. This was new land to Davis when he landed on 2 February. The seals were there, the British were not. That day they took 200, and the next day 422. They stayed until 6 February, taking 109 skins on the final day. Twenty-two miles, and a little east of due south, lay Hoseason Island, named after James Hoseason, first mate on the _Sprightly_ , an Enderby Brothers sealing ship which would operate in these waters in 1824–25. History was also in sight.
'Wednesday 7th February 1821 Commences with open Cloudy Weather and Light winds a standing for a Large Body of Land in that direction SE at 10 A.M. close in with it, out Boat and Sent her on Shore to look for Seal at 11 A.M.' The place is now called Graham Land on the Antarctic Peninsula near Cape Sterneck, which forms the eastern arm of Hughes Bay. On 7 February 1821, at ten in the morning, the _Cecilia_ dropped anchor. A boat was immediately lowered and made for the shore. The mate of the _Huron_ Samuel Goddard was on board. At Low Island we know the second mates Smith and Philips went ashore, so they may have joined him. John Davis seems to have stayed on board. One or both of their black seamen might have been among them. Someone is in the bow, holding the painter (bow rope) watching for rocks. Like all the others, he is wearing seal-skin boots, warm and waterproof. He jumps into the shallows just before the keel scrapes the pebble beach, and steadies her in the surf. The other men step nimbly to the bow and jump for the dry shingle. Someone has left the first human footprints on continental Antarctica. We do not know the name of the man. In less than an hour they get back in the boat and return to ship. The land seems more extensive than other islands they had visited. It doesn't matter much. What does matter is there are no seals; they move on. The incoming tide washes away the lower tracks. The boot marks above high tide linger longer.
Unless a new document falls from unsorted archives or is found among forgotten papers in a New England attic, we will never know the name of the owner of those boots. I am fascinated by the possibility he might have been black. It's possible something will come to light; Davis's own papers were only discovered in 1952. The crew had no way of knowing if they had been on an island or on the mainland, but Davis had a gut feeling and got it right. His last log entry for the day reads: 'I think this Southern Land to be a Continent.'
At the end of the century, Dr Frederick Cook on the _Belgica_ would describe the spot more lyrically: 'every projection seemed a continuous mass of impenetrable crystal solitude.' But the anonymous men of the _Cecilia_ , who stayed long enough to confirm it was another disappointing landing in their bloodthirsty trade, were the first. There were now no other continents to discover.
## JOHN BISCOE'S (1794–1843) VOYAGE OF 1830–33
_Biscoe, working for a company so famous they are described in_ Moby Dick _as whaling royalty, gains exploring fame through commercial madness._
If John Biscoe's name is known today, it is by people who have visited Deception Island and asked why one of the British wooden huts is called Biscoe House. John Biscoe was a sealer for the great Enderby Brothers, so famous they were name-checked in _Moby Dick_ when Captain Ahab meets the whaler _Samuel Enderby_ , whose Captain Boomer has lost an arm to Moby Dick. Two of Samuel Enderby's grandsons were among the founding members of the RGS, and an Enderby sister was the mother of Gordon of Khartoum. Melville thought their family as noble as the Tudors and Bourbons. His life is illuminated only for the three years of one voyage. This is why.
Biscoe left Berkeley Sound in the Falklands on 27 November 1830 and sailed a little south of due east, passing to the north of South Georgia. Aged thirty-six, he was a former Navy officer recruited for his education and willingness to explore and take samples of plant and animals as he went. His ships were both slight; the brigantine _Tula_ was just 157 tons, and the _Lively_ much smaller, maybe less than 50 tons, a former Cowes pilot-boat.
Ice conditions were good and they turned south-east until they were pressing close to the Antarctic Circle. On 25 February 1831, Biscoe entered in the log, 'At 8pm saw an appearance of land to the southward.' In log entries, 'appearance of land' is the stock phrase for 'I think it's land but I'm not sure.' As they came closer it seemed Biscoe's caution was justified. 'At noon our latitude was 66°29´S, longitude 45°17´E; that which lately had the appearance of land' proved to be ice, and 'I could trace it in extent for at least 30 to 40 miles from the foretop with a good telescope; it was then lost in the general glow of the atmosphere.' It was ice, but although he did not know it, he was looking at cliffs of continental ice: the edge of the continent.
On 28 February 1831 Biscoe's luck turned: 'In the morning more regular sea. Tacked to the southward. Wind SE. Noon more clear. Latitude 65°57´S 047°20´30" E. pm, passed to the southward through much broken field-ice. 4pm saw several hummocks to the southward, which much resembled tops of mountains, and at 6pm clearly distinguished it to be land, and to considerable extent; to my great satisfaction what we had seen being the black tops of mountains showing themselves through the snow on the lower land, which, however, appeared to be a great distance off, and completely beset with close field-ice and icebergs. The body of the land bearing away SE.' He sensibly named it after his boss: Enderby Land. He ventured a boat into the dense ice but it was soon driven back, and he failed to land.
The weather soon turned on them, viciously, but first they were treated to a magnificent display of the _aurora australis_ , playing the trick of seeming to be right above their heads, just out of arm's reach. It was 'without exception the grandest phenomenon of nature of its kind I ever witnessed.' On 5 March, a regulation gale force blizzard reduced visibility until the ships could not see each other or advancing icebergs. By midnight it was hurricane strength and the larger _Tula_ , which even in normal conditions was cussed to handle, was almost beyond control. Ice and snow built up on the decks and in the rigging and spars, making her dangerously top-heavy. Seas broke over the deck and smashed one ship's boat and carried another away. The winds drove them 120 miles to the north. When the storm cleared, they could see the horizon, but no sign of the _Lively_. Biscoe doggedly turned back south with every hope that the southerly winds had dispersed the pack that had prevented him getting to shore. He fought his way to Cape Ann where they had been when the storm started. The pack was intact. Some men had been injured in the storm, three others were bedridden with scurvy, including the carpenter, who was vital in performing the running repairs needed to keep the ship working. By 16 March Biscoe concluded 'I feel myself absolutely obliged to give up all further pursuit.' Yet he could not quite let go. He persevered in probing the ice until 6 April but never achieved a landing. They headed for New Zealand but on 23 April the carpenter died, and soon after, another seaman. Only two mates, one seaman, one boy were fit for work. When night fell they hove to, drifting until dawn, under the pale green light of the dwindling _aurora_ before dragging their exhausted bodies from their bunks to painfully haul on ropes, hoist sail, begin again. He encouraged his men but his diary reveals that I 'often had a smile on my face, with very different feelings within.'
In Hobart he began repairing the ship, waiting for the _Lively_ to appear. Months passed; no incoming ship had seen her. He decided he had to continue. You can imagine his feelings sailing alone down the Derwent River watching the banks fall away to show him the sea that had claimed his companions. At the river mouth a familiar set of sails was coming towards them: the _Lively_. They turned and sailed back to port where Captain Avery described how disease and the hurricane had scythed through the crew until there was just one seaman and a boy left alive. The man was exhausted and the boy's hand was broken. They had arrived near Melbourne a month before the _Tula_ had docked, and barely had the strength to go ashore for help. They returned to find the ship gone. It was two weeks before they found her grounded in an obscure inlet, her mooring lines having been broken or cut.
On 8 October 1831 the two ships joined company under sail again, making New Zealand, then the Chatham Islands, 450 nautical miles east of Christchurch. It was the second leg of Biscoe's circumnavigation. The seal colonies were there, but most had already been butchered. This didn't stop them killing the survivors. Twenty-three skins sailed with them south to the Bounty Islands, where there were no seals at all. They continued east, setting course for the archipelago marked as the Nimrod Islands on the chart, and finding the islands were as fugitive as the seals: another map-maker's myth. The last hope was the South Shetland; although all known seal rookeries had been wiped out by 1823, there were large areas of uncharted whiteness, on the map and on the ground. A few gravel beaches full of the stink and bark of fur seals would remedy all. He decided to head south of Cook's course in this same sea, hoping to find new land, not knowing Bellingshausen had sailed here eleven years before. They met fields of icebergs, and though short of water, Biscoe did not melt ice because he blamed it for past dysentery outbreaks.
On 15 February 1832, approaching the base of the Antarctic Peninsula from the west, they suffered 'strong gales from the southward. Water smooth. Latitude at noon 67°01´S, longitude 71°48´W. At 5pm saw land bearing ESE which appeared at a great distance – run for it all night with a light breeze from the SW. It has a most imposing and beautiful appearance, having on very high peak running up into the clouds, and occasionally appears both above and below them; about one-third of the mountains, which are about four miles in extent from north to south, have only a thin scattering of snow over their summits. Towards the base the other two-thirds are buried in a field of snow and ice of the most dazzling brightness. This bed of snow and ice is about 4 miles in extent, sloping gradually down to its termination; a cliff ten or twelve feet high, which is split in every direction for at least two or three hundred yards from its edge inwards, and which appears to form icebergs, only waiting for some severe gales or other cause to break them adrift.' Biscoe decided: 'This island being the farthest known land to the southward, I have honoured it with the name of Her Majesty Queen Adelaide.'
'February 19 – At 4pm I sent the boat to an island, which appeared to join the mainland, and some naked rocks lying off the mouth of a considerable entrance. I had great hopes of finding seal in them. At 10am the boat returned, not having found anything alive on the island, put having pulled quite round what Mr White informed me was an excellent harbour for shelter, although a rocky bottom. I have named this place Pitt's Island, from the likeness of an iceberg to that statesman in a sitting posture, and which for some time I took to be a rock.' Pitt Island is now the most northerly of the Biscoe Islands, which Charcot would later survey properly.
From naming an island after the Queen, there was only one more step to go. 'February 21 – On the 21st I again stood towards the mainland [actually islands] and at 8am went into the boat myself and pulled into a large inlet; the bottom appeared to be rocky in places where it could be seen, but I found no bottom with 20 fathoms, but as we found no seal nor indeed anything but penguins and a few birds, I did not sound with a deeper line. This being the mainland I took possession of it in the name of His Majesty King William the Fourth, the highest mountain I named Mount William on the same occasion.'
The ship, still all but empty, reached a harbour on the west end of Livingston Island called New Plymouth. Here Biscoe was looking for elephant seals: oil not skins. He met an English sealer, Adam Kellock, who enthused him with reports of rich pickings before sailing home. But after two weeks, Biscoe had found just a few pups. They prepared to sail north: the _Lively_ returned to England, the _Tula_ went whaling in a desperate attempt to salvage some profit. Before they could sail, a storm pinned them in the harbour, seventeen-foot waves driving the _Tula_ with its deeper draught into shallows where its bottom scraped on the rocks and its safety hung on a single anchor chain. They evacuated to the _Lively_ and watched the storm blow out. The anchor held; they re-boarded and sailed to the Falklands. Having built a temporary furnace in Berkeley Sound to repair the _Tula'_ _s_ rudder, damaged in the storm, they sailed west to New Island, a pretty island off West Falkland favoured by sealers and whalers. On the way the _Lively_ went missing, but they had learned to be optimistic, she was a tough ship. There were no seals at New Island, and a month after leaving Berkeley Sound Biscoe began looking for his companion. Soon he heard that the _Lively_ was sunk, but the men had got ashore. Slowly they made their way back to Biscoe. His orders were to wait where he was and pick up supplies shipped out from England, and complete another season's work beginning in the next southern summer. But the men began complaining about rations, and wouldn't be satisfied. Their real complaint was they had suffered enough, and their most effective argument was to desert, which they did, in numbers. Biscoe was forced to sail home.
Biscoe completed a circumnavigation of the continent, the third in all, that was at once fearless and unimaginative. He was a working skipper not much interested in geography, except as a business opening, but he discovered two areas of the continent that still bear the names he gave them. He did correctly deduce that all the land and ice he had seen was part of a whole. How much was land he could not say, but land and ice together made up a continent: 'I have not the least doubt that the whole spaces, from the latitudes I have visited to the Pole, are one solid mass'. It was a magnificent achievement for a small vessel at work in a now forgotten trade. The Enderby Brothers, one of whose ships had, in 1773, had its tea dumped in Boston Harbour by Revolutionaries dressed as Indians, would continue to encourage exploration alongside commerce. It was admirable but it would cost them their business. Funding Biscoe's voyage cost the Enderbys £6,147, but he limped into London on 8 February 1833, with just thirty seal skins in his hold. Charles Enderby swallowed his disappointment, and invited Biscoe to speak at the new Royal Geographical Society, presenting the paper _Recent Discoveries in the Antarctic Ocean_. The Society gave Biscoe their second ever gold medal.
In 1838 he was in Tasmania, and destitute enough for a public subscription to be held to pay for his passage home. £109 12s 0d bought tickets for him and his wife and children. A contribution of ten guineas came from the Lieutenant Governor John Franklin. Yet poor Biscoe died on the voyage. His memory is preserved in the RRS _John Biscoe_ , two BAS supply and research ships which carried that name between 1947 and 1991, when the RRS _James Clark Ross_ replaced her. His name also survives on Biscoe House, Deception Island.
## JOHN BALLENY'S (?–1857) VOYAGE OF 1838–39
_The first footprints south of the Antarctic Circle._
One final sealing voyage deserves mention, also sponsored by the Enderbys. They continued to direct voyages south and they continued to lose money. Two of the senior partners died in 1829 and young Charles Enderby was more interested in adventure than commerce. When they bought the _Eliza Scott_ and the _Sabrina_ in 1838–39 the bulk of the shares in it were immediately sold on. They had both been built as pleasure yachts and were poorly ballasted. They moved violently in bad weather, and the _Sabrina_ was lost in a storm on the return journey. The Enderbys were evangelical in religion, as well as geography, and appointed masters with more regard for their piety than their competence. They had problems with officers and crew, and seem to have dragged John Balleny out of retirement at the last minute to skipper the _Eliza Scott_. He may have been in his late sixties, though details of his birth and upbringing are unknown. He was a very knowledgeable and capable navigator, but with no whaling or sealing experience. He was at least pious.
Most of his voyage is not relevant to the Peninsula area, but he did discover, 1250 miles south of New Zealand at nearly 67°S, a new group of islands which he named after himself. There they made the first landing south of the Antarctic Circle on 12 February 1839. They also sighted new continental coast which they named Sabrina Land, two remarkable firsts for a working sealer, but they returned with just 178 seal skins. Balleny slipped back into obscure retirement, enriched only by his name on an obscure corner of the map. There is no record of his death, no will. Charles Enderby blithely continued to suggest further money-losing ventures, and was for a while saved from himself as the RGS had already committed to supporting James Clark Ross on his voyage of 1839–43. In fact when Balleny sailed up the Thames he had passed Chatham Dockyard where the _Erebus_ and _Terror_ were being fitted out. When he departed twelve days later, Ross had copies of Balleny's chart and log, including a suggestion that from Balleny's observations of the ice, a drive farther south might be fruitful if made between 170° and 180°E. Those waters would soon be christened the Ross Sea.
#### THE DRAKE PASSAGE
When he returned to England in September 1580, Drake was adamant. He had seen open ocean south of Cape Horn. The land they had reached on 24 October 1578 held the key. They had been blown south and fought their way back north to land on an island. They refreshed their water, collected wood and berries, there were 'herbs and trees flourishing.' He named it Elizabeth Island, and the harbour, Port Francis Drake. Fletcher drew the island as square, with a central lake, about thirty miles north to south. Later in life Drake yarned with Richard Hawkins, telling how he had gone 'seeking the part of the island, cast himself down on the uttermost point, grovelling, and so reached out his body over it. Presently he embarked, and then told his people that he had been on the southernmost known land in the world.'
They were there three or four days, and their descriptions should be accurate. The problem is, Cape Horn is not like that. The island is not square and has no central lake. There are no trees to offer wood, no sweet herbs or berries. Drake is known to have cooked the books and spread misinformation about his course to confuse the Spanish.
In 1908, in Mexico City an American named Zelia Nuttall was searching through the archives of the Inquisition into native witchcraft. On the floor, in a dark, dusty corner, lay a disposition to that Inquisition by Nuño da Silva. It included descriptions of his time with Drake.
In the General Archive of the Indias in Seville, a storehouse of documents on the exploration of the Americas. She arrived, chatting to friends, in a room banked high with documents which were only partly sorted. Still talking, she put out a hand at random to the label of a sheaf of documents. It was a bundle of papers about Francis Drake in the South Sea.
Her charmed search led to sixteen pages stitched roughly together. Da Silva's log is brief and factual: a navigator's record, not a narrative. 'October 24 Came to an anchor off an island in 57°S. October 25 This day we went ashore. October 26 We procured wood.' Not wanting to overdo the local detail, he continues 'October 27 Went ashore. October 28 We sailed.'
The log has no helpful description of Elizabeth Island but it did enable historians to track the wanderings of the _Golden Hinde_ during the seven or more weeks of the storm, and solve the mystery of the supposed landing at Cape Horn. But when they plotted the course described by da Silva, the result was a location for Elizabeth Island which was well out to the south west of the Horn, in open ocean where there is no land.
Over the years, the original chart entries for Elizabeth Island began to be marked PD, position doubtful, then ED, existence doubtful.
In 1938, Felix Reisenberg calculated their movements and put Elizabeth Island back in the empty ocean in fifteen thousand feet feet of water. He added the W-E current to the equation and brought Elizabeth Island's supposed position back nearly a hundred miles east. But there was no island there either. He wrote to the US Navy Hydrographic Office who replied:
Captain W D Burnham reports that while running before a gale off Cape Horn in command of the American ship _Pactolus_ , at 4 o'clock in the morning of 6 November 1885, the wind lulled and the sea fell, and noticing very highly discoloured water, he hove the ship to and sounded three times, obtaining each time from 67 to 70 fathoms, black sand and small rocks. Then ran South for thirty miles before the water, which all the time was very thick and yellow, resumed its natural colour.
It is named after him, Burnham Bank. It was in the right place, thirty miles north to south, like Fletcher's Island, and for the _Golden Hinde_ , with a northerly wind, two days sailing from Tierra del Fuego, as Nuño da Silva had said. A rock soaring from depths of fifteen thousand feet, to two hundred feet below the waves. Reisenberg had it. Small islands with a central lake are not common. They tend to be volcanoes with flooded craters.
# The Circumnavigators:
Jules Dumont D'Urville, Charles Wilkes and James Clark Ross
_Adventurers from three of the great marine powers made circumnavigations of the Southern Ocean between 1837 and 1843. They were, in order of departure, the Frenchman Jules-Sébastien-César Dumont d'Urville, the American Charles Wilkes, and the Englishman James Clark Ross. Long stretches of their cruises were spent far from the Antarctic Peninsula but each contributed to the understanding of the continent, and to their own nation's claims to a stake in its future: one a handsome, sociable dandy; the other two solitary egoists. _
## THE FRENCH ARRIVE 1837– 40
JULES DUMONT D'URVILLE (1790–1842)
Jules-Sébastien-César Dumont d'Urville was born on 23 May 1790 at Condé-sur-Noireau near Caen, Normandy. Aged ten, he was told that no famous men had come from his town, and vowed 'I promised myself to work twice as hard to place my name on the wings of fame.' He joined the French Navy in 1807, when it was licking its wounds after Trafalgar. He was twenty-five years old before he left harbour, as a botanist on board the _Chevrette_ to the Mediterranean. On this voyage his career took off.
In April 1820, they lay off Milo in the Greek Islands. Dumont d'Urville went ashore to see the French representative on the island, a Monsieur Brest, who told them of some new statues discovered three weeks before by a local farmer. From the pose, d'Urville recognised Venus at the Judgement of Paris. It was extremely beautiful and finely made. They thought about buying it but the captain said he did not have room for it on board. In Constantinople d'Urville obtained permission to purchase the statue for France. But when he returned, it had been sold and loaded onto a Greek ship. Accounts of what happened next vary. It was either recovered by bribing the Greeks or they fought over it. Whichever it was, the arms were lost in the melee. It was taken to Court, then to the Louvre, and named after the place it was found: the Venus de Milo. A misunderstanding arose that d'Urville had discovered the statue personally; he never corrected it.
He added successful circumnavigations to his experience. With his friend Louis-Isidore Duperrey, he successfully promoted a trip to assess colonising parts of New Zealand and Australia, not yet secure as English colonies. It was led by Duperrey, who was more senior, with Dumont d'Urville as second-in-command. Relations between the two were soon strained. Dumont d'Urville was solitary and patronising; his coolness irked Duperrey. He often dressed scruffily, a colleague said he had 'the indifference of a tramp to his appearance.' He relished the misunderstandings when officers from other ships blinked at being addressed on familiar terms by someone without a uniform. An anonymous commentator observed that while he had 'the heart of an aristocrat, his outer husk was plebeian.'
On their return they anchored in Marseilles on 9 March 1825 after thirty-one months, 75,000 miles, and not a single fatality. Their work was praised by Cuvier and von Humboldt, but Dumont d'Urville thought Duperrey had been too timid and hampered the science. His remarks betray a paranoid strain that would be echoed by his American contemporary, Charles Wilkes: 'I was disgusted to see that officer get all the credit for an expedition that was almost wholly due to me.'
Comments from fellow officers are consistent, revealing both his competence, which was widely respected, and his character, which inspired no affection. In January 1837 he wrote to the Minister of Marine proposing a visit to the South Seas; D'Urville had become obsessed with equalling Cook's three Pacific cruises. The French King himself added a requirement to include Antarctica. It led to the first close circumnavigation of the Antarctic, nailing it as a sizeable landmass, and throwing a lasso around mapmaker's imaginations. From now on, they had to draw it in the latitudes south of his course. D'Urville took the _Astrolabe_ which he had used on his previous voyage, and his old friend Charles Jacquinot captained the corvette _Zelée_. The crew were offered a one hundred Francs bonus for achieving 75°S and an additional twenty Francs for each extra degree attained. Dumont d'Urville was forty-seven, looked older, and his wife did not want him to go. When he boarded someone remarked, presumably very quietly, 'That old fellow isn't going to take us very far.' One additional scientist was recruited at the last minute when d'Urville met a Monsieur Dumoutier, who analysed character from the shape of the head using the fraudulent science of phrenology, and invited him along. He would, in the oddest way, be the only one to know his commander's body.
They left Staten Island, the last outpost of South America, on 10 January 1838, and within three days they saw their first ice. He quoted the inscription Dante gave to the gates of Hell: _Lasciate ogni speranza voi ch'entrate_ : Abandon all hope you who enter. He wanted to chart the edge of the ice field, so they sailed east and south from Clarence Island, off the tip of the Peninsula, encountering the pack on 21 January and cruising along its edge for three days. Land glimpsed on 23 January proved an illusion. Illusions are big in Antarctic history. On 24 January they made a farthest south at 63°23´S and 045°17´W, and began to talk of reaching 80°S; after all, they were in the area where Weddell had reported open water. Like everyone who followed Weddell, they found ice. Fogs and gales drove them north to the South Orkneys hoping to kill birds and seals to augment their diet and strength.
When the winds eased on 2 February they decided to foray south once more. In two days they were again surrounded by massive bergs, and the other officers were alarmed when he decided to enter a narrow lead since the bergs were moving and soon menaced his exit. They were the first ship to sail into the pack and try to force a passage, just beating James Clark Ross, often wrongly credited with being the first. Despite taking such risks, d'Urville only just passed 62°S. He warped his way along on ice anchors: carrying anchors out in boats to distant bergs, securing them, then using the capstans to winch the ship forward. In ten hours they made one mile, and it was five days before they gained open water. At the time d'Urville could not decide whether to doubt Weddell's navigation or his honesty: 'Either Weddell struck an exceptionally favourable season, or he played on the credibility of his readers.' But when writing his formal account he later admitted, 'the recent success of Captain James Ross has altered my opinion.'
On Valentine's Day they headed west for the South Shetlands. Hopes for more open ice conditions were dashed; one berg measured eleven miles long and 120 feet high. Gales began to blow them back towards the South Orkneys, and they went with the weather. After failing to land at Saddle Island they made a landing at Weddell Island, where they took samples of rocks and penguins. On 22 February they made a second attempt to reach the South Shetlands. Bad weather and poor sketch-charts made d'Urville decide to name all the lands he saw around tip of peninsula and withdraw them later if they proved to have been previously named by Weddell. The first rocks seen were named Zelée, and the main island at the tip of the peninsula still bears name they gave them it: Joinville, named for King Louis-Phillipe's son, the Prince de Joinville. D'Urville Island next door was named for the leader by a later explorer.
This time they went down the Bransfield Strait and on 5 March reached 63° 27´S at a position west of Trinity Island, where Lt. Dubouzet on the _Zelée_ wrote: 'The heavy north west swell forced us out to sea, the whole night we had freezing, penetrating rain which exhausted our sailors. Several of them were starting to get sore mouths, pain and fever, the early symptoms of scurvy, and this made a longer stay in these regions dangerous.'
That same day Dr Hombron informed d'Urville that scurvy risked becoming rampant on the _Zelée_ ; fourteen were bedridden, fourteen more showed early symptoms. D'Urville was shocked; despite two long voyages in his youth, this was the first time scurvy had broken out on any ship he was on. He banned even mentioning the word scurvy or referring to his plans to come south again the next Austral summer; such was the disease's power to undermine morale. They sailed for Chile on 20 March with a total of twenty-seven cases on the two ships. They were so short-handed that officers had to go aloft. When they disembarked at Talcahuano, some crewmen were so disfigured that their officers had to ask their names.
In port Dumont d'Urville received awful news. A letter from his wife was waiting. Their young son had died of cholera. He recorded, 'I no longer pray to God, I am cursed by him.' He could not replace his sick crew because other French seamen in port thought the expedition was disaster waiting to happen. During a spell in the Far East, the expedition picked up such virulent fevers that Dumont d'Urville made his will. They headed for Hobart then, once more, the purification of the ice.
D'Urville's task was to see where the edge of the pack lay in those longitudes south of Tasmania where no one had ever probed. At midnight on 18 January at 64°S they saw five huge tabular bergs, making them think land was close. Next morning there were more, even bigger. During one spell of fog, they suspected there was land to the west-south-west under the dark cloud oppressing that quarter, but as the sun rose higher the island vanished. At 18:00 Dumoulin went up the mast, and called land – new land! Others swarmed up and confirmed it, but Dumoulin had second thoughts and began doubting what he had seen. He ended up being the first to see it and the last to be convinced.
They crossed the Antarctic Circle on 21 January and the crew improvised a Crossing the Line ceremony, featuring Father Antarctic. An engraving of this is the most vivid image of the voyage. Many of the crew are in fancy dress, one is a stock carnival character slung about with animals skeletons, another is a giant penguin. The officers huddle above on the quarterdeck, hands jammed in pockets or armpits, looking glumly tolerant. Around them towered the icebergs: 'the spectacle they presented was both magnificent and terrifying. One could have believed oneself in the narrow streets of a city of giants.'
Nearing midnight the sun went behind a cloud and the glare dropped. Both ships were now becalmed, but the _Zelée_ launched a boat which hailed the _Astrolabe_ , saying they were sure it was land, but they could not see bare rock. The wind rose and huge bergs gathered about them, higher than their masts. At 22:30 d'Urville launched two boats which rowed towards the alleged shore until they were so far off only the masts of the ships were visible, knowing all the time that if the wind shifted, the ships would have to desert them and sail out to sea for their own safety. The boats reached small islands five hundred metres from the coast, and went ashore to raise the _tricolore_ 'on this land that no human being before us had either seen or set foot on.'
They were struck by the extreme barrenness of their new realm; there were no shells or lichen, and just one piece of seaweed, dropped by a passing cormorant. The land lacked something else: a name. It is touching that d'Urville's wife, Adèle, was so much in his mind that he not only called this new coast after her, but he used his pet name for her: Adélie. With picks, the landing party prised out samples of a hard reddish granite or gneiss, and took penguins. They were a new species, and they were named Adélie penguins, a fine name for these chic, stylishly marked birds: the most Parisienne of penguins.
They made their way along 150 miles of coast, naming another section the Clarie Coast, for Captain Jacquinot's wife; Cape Pépin was named after his wife's family. Once the Pépins had not been good enough for him to marry, according to d'Urville's snobbish mother; now their name was written on the earth. D'Urville formally claimed Terre Adélie for France, and it became important as the foundation for France's future Antarctic claims. Ice conditions were difficult, and they were near enough to the South Magnetic Pole for the compass to be unreliable. Gales stacked so many icicles in the rigging that it was not safe to go aloft, and falling lumps and daggers of ice made simply being on deck dangerous. At one point the ships became separated and they were forced to fire cannons to locate each other.
On 29 January they saw the one thing that seemed impossible: another ship. More on that in the next chapter. On 1 February at 65°20´S 113°E they left the coast behind, the men exhausted. Their first port was Hobart, Tasmania where d'Urville compared notes with sealer John Biscoe. The French squadron did not anchor back home in Toulon until 6 November 1840, by which time twenty-five men had died, and all the living were worn down. He was visited by Captain Matterer, an old accomplice in the purchase of the Venus de Milo, who found 'he was no more than a ghost, a worn body that dragged itself painfully around, but his calm and austere features were still highly expressive, and his eagle eyes shone when he told me' of his voyage. Chronic gout had caused renal colic, and he complained: 'My friend I am finished, a being who is now worn out.'
Recognition for his suffering was everything to the small-town boy. When d'Urville was promoted to rear-admiral, Matterer found him with letter 'against his heart and his eyes full of tears.' A Société de Géographie gold medal followed. The ten-year-old who had lamented the lack of heroes in Condé-sur-Noireau's cemetery had his fame at last. Dumont d'Urville published his account of the voyage in 1841 to great acclaim, but he was no longer fit for sea. He retired in 1842, aged only fifty-two. The official history of the voyage came out in 1846 in ten volumes, and the scientific results were published in twenty-three volumes and seven atlases completed in 1854. But d'Urville was already dead.
On 8 May 1842 d'Urville, his wife Adélie, and their son took the new railway out to Versailles on the Sunday of the King's official birthday. They returned on the 17:30 train, and the stationmaster locked the doors as the signal was given to depart. The train was crammed with 768 people in its seventeen carriages, and was too full to stop at any of the stations between Versailles and Paris. It departed three minutes late; the engineers were urged to go full speed to make up time. Just after Bellevue Station the axle of the lead engine shattered, the second engine and carriages piled up to the height of a house. Part caught fire. Dozens died, and the d'Urvilles were missing. Rear Admiral Jules-Sébastien-César Dumont d'Urville's skull was identified by the phrenologist Dumoutier. His wife was identified by the partly melted gold chain about her neck, their son only by his proximity to them.
#### SCURVY
The first book you used to see as you walked into the Greenwich Observatory in London was a large ledger whose pages rippled with lines of browned hand-written ink. It was the log of HMS _Wager_ under Commander George Anson on her voyage to attack Spanish treasure ships sailing from the west coast of South and Central America to the Far East. Financially it was Eldorado; they brought home treasure which filled thirty-two carts and paraded it round London. In human terms it was a massacre. They set out with 1972 men to actively engage with the enemy. They lost four men in combat and thirteen hundred to disease, many to scurvy. Such loss of life was intolerable even in a society where life was cheap. It triggered the first serious study by Edinburgh naval surgeon James Lind of a disease that killed more sailors than any human enemy.
We now know that the body needs vitamin C to make collagen, a protein which helps hold the body together. The vitamin is vital for the maintenance and repair of soft tissue, bones and teeth. It cannot be stored by the body; what isn't used leaves in urine. Humans are unusual among animals in not being able to make vitamin C; they must consume it in their food. If healthy people are put on a diet without vitamin C, their bodies start to run out of it after three months, so the disease was little known until men began to make long oceanic voyages, unable to take on fresh food. Without vitamin C, skin falls off, fingernails and toenails bleed, gums shrink, teeth loosen and drop out, long-healed bone fractures come apart, and the mind is afflicted with depression. Victims bruise easily and succumb more readily to infections.
The first detailed account of the symptoms comes from Sir Richard Hawkins on his voyage to the Pacific in 1593. He blamed it on the long time they had spent close to the equator and thought it could be remedied by eating the produce of the tropic zone. He recommended citrus fruit. Although his cure was exactly the right solution, subsequent voyages tried different fruits without success, and fresh food became just another recommendation, along with cleanliness.
James Lind's work concluded with a trial where he separated scurvy victims into groups and treated each one with a different contemporary cure. One was a cordial of lime and lemon juice. At the end of a week, all the men following that treatment were fit for work. All the other patients were dead. It still took the British Navy another fifty years to act. Finally, in 1795, lemon juice was issued on all naval vessels and by 1805 scurvy was virtually eliminated, although it later resurged because captains bought limes, which were cheaper than lemons. They did not know that levels of vitamin C in limes are half those of lemons. Lind saved the nation. Had health not improved in the Napoleonic Wars, Britain would have run out of sailors and lost.
## IT'S A CONTINENT!
CHARLES WILKES AND THE EXPLORING EXPEDITION (1838–42)
CHARLES WILKES (1798–1877)
In 1825 US President John Quincy Adams's inaugural address advocated exploring the Pacific North West; expansionism would make the United States. It was a sound idea; but the idea which launched Wilkes's Exploring Expedition was utterly mad. In 1818 John Cleves Symes, a thirty-eight-year-old retired army captain, had noted that some animals migrated north in summer and returned fat. He argued that the animals were going down through a hole at the pole to feed on richer pastures. The world was hollow with other worlds within, like Russian dolls. The curvature was gentle enough to allow you to sail over the rim and head into the interior without being aware of it. In March 1822, he petitioned Congress for money for a voyage north to demonstrate its truth. Jeremiah Reynolds, a talented public speaker, took it up, gathered support, but split with Symes when the author of the theory changed his plans to advocate a voyage south seeking the open water found by James Weddell. Symes died in 1829, before any voyage could be launched, and was buried with a hollow globe on his tombstone. Edgar Allan Poe's short story 'The Narrative of Arthur Gordon Pym of Nantucket' was inspired by Symes's theory. But the Exploring Expedition would take place, and a commander was needed.
That dryly witty stylist, Mark Twain, wrote that Charles Wilkes 'had gone wandering about the globe in his ships and had looked with his own eyes upon its furthest corners, its dreamlands – names and places which existed rather as shadows and rumours than as realities.' Wilkes led the last naval squadron to circumnavigate the world by sail without auxiliary steam. But when he died, his achievement as leader of the Exploring Expedition was already forgotten.
His officers on that trip began by loving and respecting him. The twenty-two-year-old Midshipman William Reynolds opened his journal declaring 'I like Captain Wilkes very much. He is a most wonderful man, possesses a vast deal of knowledge and has a talent for everything.' He was 'the most proper man who could have been found in the Navy to conduct this expedition.' In Polynesia Reynolds would revisit this entry and write over it: 'great mistake, did not at this time know him.'
Wilkes was born in New York City in 1798; his mother died when he was two. He was put in the care of his aunt Elizabeth Seton, who became an abbess and the USA's first native-born saint. His Welsh nanny had no objection to her own reputation as a witch, and predicted he would become an admiral although there was no such rank in the US Navy. After the end of the Napoleonic Wars in 1815, all navies cut back hard. You didn't need to lay off the men, they gladly walked, but the officers stayed on the books, inactive, on half pay, manoeuvring for positions. Wilkes's family contacts were good, but not good enough. Landlubbered, he practised the scientific aspects of seamanship such as sextant work, and convinced himself that homework mattered more than experience. This caused problems when he went to sea with men who thought sailing provided the best experience. In desperation he eventually signed on as a midshipman. He did not like the fo'c'sle, it was chaotic, communal and familiar; he was aloof, scholarly and condescending.
After Wilkes returned from a voyage to the Pacific in 1823 he made a one-year trip to the Mediterranean. After that, he spent no significant time at sea for fifteen years until, through political manoeuvring and luck, he obtained command of the Exploring Expedition, though he had less sea time than many of the midshipmen. Private notes reveal his ambition outreached his confidence: 'It required all the hope I could muster to outweigh the intense feeling of responsibility that hung over me. I may compare it to that of one doomed to destruction.'
They sailed on 18 August 1838 with a varied fleet. The flagship was the 127-foot sloop _Vincennes_. The next largest was the _Peacock_ , 118 feet, She had spent sixty hours on a reef earlier that year, and her captain, Hudson, was worried about her condition. The _Porpoise_ , eighty-eight feet, was under Cadwallader Ringgold, two schooners, the _Flying Fish_ and the _Sea Gull_ , commanded by midshipmen were seventy-foot New York pilot cutters; with a crew of fifteen they were like yachts, they even had tillers instead of wheels.
Their orders, which Wilkes shared with no one, were to investigate reported mid-Atlantic shoals, go to Rio de Janeiro for provisions, survey the estuary of the Rio Negro, and establish a base in the islands north of Cape Horn. One vessel, the _Relief_ , proved so slow she was ordered to sail straight to Rio.
Wilkes was scared of officers with real experience. He promoted two junior lieutenants from his coterie of favourites to skipper the cutters _Sea Gull_ and _Flying Fish_ , upsetting all his senior lieutenants. The resentful senior lieutenants were then despatched to other ships, to serve under officers who were also lieutenants. On 16 February they attained Cape Horn in sunny, warm weather and calm seas, and anchored in Orange Bay. The _Porpoise_ and _Peacock_ would go south with the schooners. Hudson in the _Peacock_ , together with the _Flying Fish_ , was ordered to head for the region around 106°W where Cook had made his farthest south, and try to beat it. This contrasted with the approach of previous leaders who shunned waters previously sailed in preference for new seas offering the prospect of discoveries.
Wilkes on the _Porpoise_ would head, with the _Sea Gull_ , for the South Shetlands, then to the Weddell Sea and the area where Weddell had found open sea. Possibly the lateness of the season could be converted into an advantage. In the ice, Wilkes found courage, or bravado. He wrote to his wife, Jane, of ice 'islands fifty times as large as the Capitol and much whiter.' When they arrived off the Peninsula they found the Government-issue cold-weather clothing was useless, which was especially hard on the crew of the little cutter _Sea Gull_ , with waves breaking over her deck. Wilkes cut short her suffering, ordering her back to South America. He himself did not press farther south but went up the east side of the South Shetlands.
In the other party, a gale soon separated Hudson from the cutter _Flying Fish_ and when she did not show up at the pre-planned rendezvous, he went on. If Wilkes got a buzz out of ice navigation, Hudson disliked it: 'This fancy kind of sailing is not all that it is cracked up to be.' On 23 March, Hudson stumbled across the _Flying Fish_. In storms she had lost her binnacle, and several seamen had broken ribs; the ship had leaked and all their clothes and bedding were wet. Five men were so hypothermic they could barely stand. Despite this they took time to be astonished at the rich wildlife. At one point a sleeping right whale had to be pushed out of the way with poles. In a comparison that reminds us that the world powers were now industrial countries, the spouts were likened to city chimneys. Size and distance befuddled them; they said that seeking ice for water they hove to against a berg 830 feet high. Including the invisible ice below the water, that would mean a piece of floating ice eight thousand feet high. There is none so large. But their sense of wonder was genuine, they felt like 'a mere skiff in the moat of a giant's castle'. Fog fell and a man called Palmer wrote that when they spoke: 'The voice had no resonance, words fell from the lip and seemed to freeze before they reached the ear.' The fog dispersed on 20 March, and they followed a lead through the ice between bergs like 'pale masses, like tombs in a vast cemetery.' On the 22 March they had reached 70°S 101°W, the farthest south for a US ship. Even James Cook had gone only ten miles farther. Two days later the sea began to freeze with an audible clicking like a death watch beetle.
Now came two incidents where officers coped well and Wilkes pilloried them. An officer called Dale was absurdly blamed for being trapped on a beach by a gale, and Midshipman Reynolds called it a 'turning point' in the feelings of the officers. In the Court Martial that followed back in the USA, one testimony proved most damning because it was temperate but acute. Asked if Wilkes's outbursts were due to provocation he responded 'No sir, in most cases directly the contrary; I have noticed that those most attentive to their duty would fall under his displeasure the soonest.'
Reynolds fumed: 'Here forward, there was no affection for his person, and consideration, humanity or justice was no longer hoped for at his hand.' That is chilling, though at this stage he was probably speaking more of himself and his intimates than the whole rank of officers. Reynolds developed a pathological hatred of Wilkes, and ultimately it blinded him to Wilkes's strengths, but his journal gives insight into the electric tensions between Wilkes's circle of favourites and the general run of his officers. Reynolds himself was a competent sailor and a popular man. He had little to learn from Wilkes, except, perhaps, how to hate.
When they sailed to South America, there was some drunkenness and desertion. The maximum number of lashes a captain could order without convening a Court Martial was twelve, a regulation often flouted. Wilkes gave punishments of between twenty-four and forty-one.
Rear Admiral Charles Wilkes
On 15 July 1839, one day out of Valparaiso, Wilkes appeared on deck in an 'immense pair of Epaulettes'. He had promoted himself to captain, commander, and commodore, and without authority flew a commodore's pennant from his masthead. 'It will give [Hudson and me] much more respect,' he wrote, as if respect were earned by tokens not action. His timing of this, as they entered the Pacific, meant that news of his presumption would be slow to reach home. It is fascinating to contrast two outward forms of the same arrogance: d'Urville, with a duke-like sense of superbia, dressing down; Wilkes collecting scrambled egg like a Latin dictator. You can imagine, if they had worn sunglasses, what size Wilkes would have bought.
On 13 August they reached the Tuamotus, north-east of Tahiti. The scientists chafed to explore. But science was permitted only as an afterthought, or not at all. Wilkes spitefully ordered that fraternising between ranks should be reduced, although he had begun by socialising with junior officers, before quarrelling with many of them. From this time forward there was war to the knife between Captain Wilkes and most of his officers. By mutual agreement, he and other officers now spoke only on business, and he was nicknamed Stormy Petrel because when he appeared there was bad weather ahead. Maybe compensating for his lack of general approval he drove himself very hard, seldom taking more than five hours sleep a night, some days none. Loss of sleep reduces emotional control, and makes it difficult to make complex social judgements. He recognised 'constant anxiety' in himself which he blamed on having to work with incompetents.
They went on to Sydney where Wilkes found it convenient to keep the US consul waiting two hours for an audience: 'I am now a great man, and others will wait patiently.' The scientists decided to stay in Australia. There, they would have leisure and freedom to conduct their investigations unhindered by Wilkes. They would also miss the cruise that proved Antarctica was a continent.
When the fleet next sailed towards the ice, south of New Zealand, Hudson's _Peacock_ became detached from the other two large ships: the _Vincennes_ and the _Porpoise_. The remaining two then lost contact with each other. Their continental sighting was handled shambolically. On 15 January the _Peacock_ reached 65°25´ and no ice was visible ahead. Reynolds swaggered: 'Soon we would pass 70 degrees – eclipse Cook & distance the pretender Weddell.' The ice soon taught them respect. They noted the compass dip was so steep that an iron button could deflect it 20°. The South Magnetic Pole was assuredly on this side of the continent. At 16:00 the _Peacock_ simultaneously saw the _Porpoise_ and the pack ice: welcome and unwelcome news in the same vista. The _Porpoise_ had been cruising along its edge for several days. There would be no 70° here.
On 16 January, Reynolds and a colleague called Eld climbed to the masthead to enjoy the scene. He remarked on the entrancement of perpetual daylight: 'what a rooster would do here I cannot imagine.' Soon they saw three peaks at bearing 160°. They observed them for half an hour and fetched a telescope: 'We actually beheld the long sought for _Terra Firma_ of the Antarctic continent.' They informed the deck officer Budd and Captain Hudson, but unbelievably neither would go aloft or send someone up to confirm. In fact Hudson would not leave his stove, and gave orders to tack away, adding contrarily that he had no doubt it was land but he would not enter it in the log. It would not be the only example of Hudson seeming to forget he was on an exploring expedition.
Fog descended to end any further arguments about what they were seeing. When vision is reduced, noises seem louder, the creaking of the rigging would have filled the misty hollow of their world. The sound of the small ice tinkling, the floes grating, and bergs grinding along the hull accompanied them every waking and sleeping hour. When the ship moved into deep clear water in the lee of a big tabular iceberg, the noise died, and many of the off-duty men woke up, spooked by the sudden hush. Wilkes now announced that there was no need for the ships to stay together, which promoted rivalry between crews to make new discoveries, but undermined safety. With a companion vessel, men can be removed from damaged ships to other vessels, but for a ship on its own, your only refuges are your open boats or possibly an iceberg.
On Sunday 19 January at 65°20´S and 154°30´E, Lt. James Alden on the deck of the _Vincennes_ saw land, just as Wilkes was going below. Wilkes heard the call but went below anyway. He came out on deck later and, ignoring the officers, discussed with a gunner whether he thought it had been land. The gunner said if that wasn't land he'd never seen it. Still Wilkes did not enter it in the log.
On 23 January they sounded 350 fathoms, and killed an emperor penguin. The bird had stones in its belly. Having failed to prosecute enquiries when land was in sight, they now, on the basis of nothing more than shallow water and penguin ballast, celebrated with double grog rations the proximity of land.
On 28 January, in weather as 'clear as a bell', Lt. Alden made the first sighting of land that everyone could agree on. It was exciting, but islands had already been discovered in this sector by Balleny. On 30 January, still in beautiful weather, more land was seen, and a channel led through the ice towards it. This wasn't a group of small islands like the Ballenys; a chain of mountains ran east and west as far as the eye could see, sixty miles each way. It was a sizeable landmass, the length of Long Island or Wales. Wilkes named it the Antarctic Coast, and was desperate to land on it. In Piner Bay, named for their quartermaster, they lowered boats in a rising wind and got to half a mile from shore before the ice milling around made their position too precarious. Wilkes decided that any landing would have to be farther west. Before they could do so, Wilkes received a communication from one of the few groups of men he had yet to fall out with. The surgeons signed a letter on 31 January advising that the men were so weakened by cold and lack of sleep that any further reduction in the numbers able to report for work would place the ships in hazard. Wilkes's response was to warn they would sail west along the ice and shore 'until the ship should be totally disabled, or it should be evident that it was impossible to persist any longer.' Think about what he is saying: as many men as necessary will be sacrificed for his ambition.
On 2 February in fine weather there were twenty on the sick list. The next day blew a gale and thirty-six reported sick, many with ulcers and boils, a typical consequence of prolonged exposure to seawater and cold coupled with a poor diet. On 9 February darkness fell only to be relieved by a stunning display of the _aurora australis_. By day they cruised along the coast, Wilkes naming it after the officers he had not fallen out with: a dwindling choice. But he was still very fond of his young nephew, Henry Wilkes, who is always mentioned kindly in letters home to Jane. The boy must have found it trying to live in an atmosphere where his uncle was loathed by more and more men. He must also have dealt with it extraordinarily well; no one's private diary has anything but praise for him.
They landed on two large icebergs for magnetics and frolics. Magnetic measurements were more reliable taken safely away from the iron on the ship. The most memorable image of the trip is an informal painting of this day. It's a charming piece, showing the men at leisure. In the background, three men have planted a flag on hill. One man is tobogganing down on his belly, another tumbles over at its foot. Others swing pickaxes through a foot of ice to reach a pond of fresh water which their mates scoop into large leather flagons to take to ship. Young Midshipman Charles Erskine, good-looking with striking blue eyes, said: 'We had a jolly time, sliding and snowballing one another, and playing with penguins and seals.' In the foreground Wilkes's large Newfoundland dog, Sydney, lies watching his master slide decorously down a small bank in a sitting position, his large seal-skin boots braking his progress. Wilkes is taking his pleasure slowly, cautiously, alone. The painting is by Wilkes.
On 17 February the coast forced them north. Wilkes named it Termination Land. He had overridden the unanimous advice of his surgeons and forced his way along fifteen hundred miles of coast, while Dumont d'Urville had packed it in after a tenth as much, confessing: 'I very much doubt whether I could have stood it much longer.' Fifteen hundred miles of coast does not belong to an island. Wilkes's cussedness had identified it as a continent. Meanwhile Ringgold, commanding the _Porpoise_ , had sailed a conservative course north-west, largely remaining too far north to see land. What he did see, by a remarkable coincidence, were the ships of Dumont d'Urville, and they had seen him. The French vessels were slow, so d'Urville put up more sail to close the gap. Ringgold thought they were sailing away to avoid him, and veered off in a huff. The French saw this and blamed Ringgold; each felt rebuffed, so the Americans did not learn for some time that the French had sighted land on the 19 January: the precise day when a listless Wilkes refused to stay on deck and check Alden's sighting of land. Nor did he learn that on 21 January, the French had come to Piner Bay and done what they had failed to do and land, doubly pre-empting the Americans.
The Yankees headed for warmer water. Soon after landing at Sydney on 11 March, Wilkes heard the news of d'Urville's achievements. He now wanted to acknowledge Alden's sighting on 19 January and may have altered his own diary to say so; the entry looks suspicious. On the _Peacock_ , Reynolds and Eld had seen land on 16 January, three days before the French, when the torpid Hudson had refused to abandon stove duty. Wilkes now conferred with Hudson who was in a dilemma. He could help Wilkes pip the French by endorsing Reynolds's and Eld's claims for prior discovery, but their claims also pre-dated Wilkes's discoveries, which would wound Wilkes's pride. It would also highlight Hudson's negligence in not checking for himself. In character, Hudson opted for the easy life, said nothing about Reynolds and Eld, and declared that Mr Wilkes had all the luck, and had beaten him once more. Wilkes now wrote in the _Sydney Herald_ that he too had seen land on 19 January. When Wilkes later found out about Hudson's negligence on the 16 January, he wrote to Jane: 'for all we gained by the others they might as well been elsewhere employed'; one of his more justified gripes.
Wilkes wrote to James Clark Ross, who was soon expected in Tasmania, to tell him of the US discoveries. Wilkes included a chart of their discoveries, although divulging such details was completely against orders. His motive was vanity; the last time they had met Ross had been the senior navigator. Although Ross would not accept Wilkes's observations as conclusive, he took Wilkes's chart and later sailed though areas marked as land, generously observing that it was hard to distinguish ice from land without being much closer to shore than ships normally achieved.
Wilkes continued to Fiji then Sydney. Then tragedy struck close. Landings were resisted by the Aborigines; a party under the intelligent and well-educated Lt. Underwood was attacked on a beach and overwhelmed. Nephew Henry Wilkes was clubbed in the head and lay in the shallows while the fight continued. When it was over someone rushed to him, but he had drowned. By chance Wilkes arrived back at the ship at the same time as the boat carrying the bodies. When they pulled back the canvas he fainted. They returned to the island and in revenge for two deaths killed eighty natives.
Late in the cruise they received newspapers; Reynolds had been promoted, Wilkes had not. Hurt and frustration filled his letters home. Jane, whom he called his moderator, was supposed to placate him, but even she questioned the wisdom of his flying, without authority, a commodore's pennant. The homecoming was not to be an opportunity to recognise a team performance, it would be stage-managed to feed Wilkes's needy ego. The _Oregon_ and _Porpoise_ were diverted to Rio de Janeiro ostensibly for minor observations, but in reality so the _Vincennes_ would enter New York harbour first and alone. Only a little over half the original crews remained, and two ships. The _Flying Fish_ had been found to be unsound and was sold off in Singapore.
What he could not control was the politics at home.
They had departed under the Democrat President Jackson, but the Oval Office was now occupied by the Whig John Tyler, who was embroiled in delicate negotiations with Britain over the border between Canada and Maine. Tyler did not want attention drawn to the Pacific North West where Britain was fiercely jealous of the charter of her Hudson Bay Company. He imposed a news blackout. Wilkes fumed.
Several officers and scientists dismissed by Wilkes had arrived home long before and a tawdry exchange of accusations began. Wilkes was largely exonerated by a Court Martial, being found guilty only of unlawful flogging. But the whiff of bad eggs hung around his management. Having awarded himself captaincy years before and paraded in the costume, he was awarded only the lesser rank of commander, and not until the following year. He did not make captain until thirteen years later.
His character should not overshadow his achievement. His scientists had out-collected Cook's: 4000 ethnographic items, 10,000 species of plants, 2150 birds, 588 species of fish, 300 species of fossils and 400 species of corals. They had drawn 241 new charts, and surveyed fifteen hundred miles of Antarctic coast, confirming it as a continent. Dana's volume _Crustacea_ identified over five hundred new species of crabs, lobsters, shrimps and barnacles. Charles Darwin read it with admiration, and wrote to him to say so: if Dana had 'done nothing else whatever, it would have been a _magnum opus_ for life. I am really lost in astonishment at that which you have done in mental labour. And then, beside the labour, so much originality in all your works.' Being called original by Darwin is praise for life. Botanist Asa Gray worked on the botany specimens at Harvard. After corresponding with Darwin on similarities between geographically distant plants he became the USA's leading promoter of the theory of evolution. Humboldt called the expedition's results 'the most splendid contribution to science of the present day.'
They brought back so many objects and records that the US did not have the resources in its institutions to handle them. However four years before, in 1838, James Smithson had left the Government half a million dollars in gold for the 'increase and diffusion of knowledge.' Joel Poinsett created the National Institution for the Promotion of Science, as a vehicle to handle Wilkes's results. It became Washington's Smithsonian Museum.
Wilkes petitioned to have the copyright of both the official journal and his personal narrative, despite having heavily used his officers' accounts. Astonishingly this was granted. The volumes of the official account matched the leader's ego: vast and hide-bound. A limited edition of one hundred Morocco-covered sets of five principal volumes was produced: some of the most expensive books in the history of American publishing. Wilkes included much secondary material, the sign of an insecure writer. Charles Davies, writing for the _North American Review_ , said it was 'crushed under the weight of irrelevant matter.'
The expedition had an impact on wider American policy and the creation of nationhood and policy. From 1840 – 60, Government-driven surveying and exploring expeditions, domestic and international, took between 25% and 33% of the federal budget. But Wilkes was not given another command until the Civil War, when he seized Confederate commissioners from a British vessel. He met President Lincoln, and was promoted to one of the new posts of rear admiral, just as his Welsh nanny had predicted. Persistent flouting of orders led to him being brought to Court Martial, and suspended for three years; he never again saw active service.
Charles Wilkes died at home on 8 February 1877, aged seventy-eight. Many obituaries made no reference to the Wilkes Expedition. The Civil War had inflicted such a bloody trauma to the American psyche that a blind had been drawn across US history. Behind it, obscured, lay another country, and Wilkes's fame.
## DARING TO PIERCE THE PACK 1839–43
JAMES CLARK ROSS (1800–1862)
Born in London in 1800, James Clark Ross was the nephew of Captain John Ross. James joined the Navy in 1812, and between 1819 and 1827 made four Arctic expeditions under William Parry, his uncle's Arctic rival. In 1825 James witnessed the loss of the _Fury_ which made Parry conclude that 'a vessel of whatever magnitude, or whatever strength, is little better than a nutshell, when obliged to withstand the pressure of the unyielding ground on one side, and a moving body of ice on the other.' Young Ross would make his name hazarding his vessels to such risks.
He was back with his uncle from 1829–33, when he led the journey which claimed one of the great geographical prizes: the determination of the North Magnetic Pole, located on the Boothia Peninsula, Canada, in May 1831. In a delightful touch he planted the flagstaff of his boat to mark the position, so there was an actual pole at the Magnetic Pole. As overall commander, his uncle took the credit, the prize money and a knighthood; however, James was promoted. The actual magnetic pole is constantly shifting, so the flagstaff has been retrieved and is in the National Maritime Museum in Greenwich, London, a more convenient place to explore. They also have one of the numerous portraits of him; he was reputed to be the handsomest man in the Navy at a time when playing the peacock was one of the perks of working in uniform. He carried it off in grand style.
The eighth meeting of the British Association for the Advancement of Science at Newcastle, England, in 1838, noted of the earth's magnetism: 'That great and notorious deficiencies exist in our knowledge of the course of the variation lines generally, but especially in the antarctic seas, and that the true position of the southern magnetic pole or poles can scarcely even be conjectured with any probability from the data already known.' They resolved 'that the deficiency... should be supplied by observations of the magnetic direction and intensity, especially in the high southern latitudes between the meridians of New Holland (Australia) and Cape Horn.' They recommended a naval expedition, and command was awarded to James Clark Ross. He must have thought this was his chance to bag a double: the North and South Magnetic Poles. The scheme was endorsed by the Royal Society, then the Government, and the Admiralty allocated the 370-ton _Erebus_ and 340-ton _Terror_ , both bomb ships: platforms to fire heavy mortars. Built to extremely strong specifications, they were considered suitable for demanding work in ice. Because of past problems with mixing military and civilian personnel, the civilian scientists necessary for the voyage were taken as 'surgeons'; the botanist J. D. Hooker was one, while on the _Erebus_ was Robert McCormick, who had been with Darwin on HMS _Beagle_ and with Parry on the _Hekla_ in 1827. Each ship had a crew of sixty-four. Although much of Ross's trip took place far from the Peninsula, I describe it here because Ross's discoveries influenced all later explorers.
They left Cornwall, England on 5 October 1839, and sailed via St Helena, which they reached on 31 January 1840, pausing to establish the first observatory there. Their next passage, to the Kerguelen Islands via the Cape of Good Hope, took sixty-eight days, for forty-five of which they suffered gale force winds or worse. Two days after they left Kerguelen they were separated in a storm, and did not see each other until they sailed into their default rendezvous in Hobart on the 15 and 16 August 1840. Governor John Franklin was waiting with two hundred convicts who built a new observatory in nine days. In mordant irony, Franklin would later be given Ross's two ships in another effort to seek the North West Passage, and die a death that made him more famous than any deed in his life.
They left Hobart on 12 November and drove south via the Auckland Islands, then Campbell Island. On 27 December they saw whales and the first icebergs; fifteen were visible at one time during the evening. On 29 December 'a great many whales were seen, chiefly of the common black kind' (perhaps southern right whales, very profitable for oil and baleen); 'we might have killed any number we pleased: they appeared chiefly to be of unusually large size, and would doubtless yield a great quantity of oil, and were so tame that our ships sailing close past did not seem to disturb them.' This report became common knowledge at a time when Arctic and Pacific catches were declining, and it aroused fresh excitement over the possibility of a southern bonanza for whalers. On 30 December 'a beautiful white petrel was seen in the evening [a snow petrel] giving notice of our approach to a large body of ice.' These beautiful creatures feed on the zooplankton which graze beneath the ice, and are true ice-birds.
They crossed the Antarctic Circle on New Year's Day 1841, but 5 January was to prove a more historic date. Firstly, in the early hours, the number of large icebergs dwindled. At 08:00 they met the pack and sailed along its edge. 'From the mast-head it seemed sufficiently open to admit of our penetrating as far as we could see to the southward; and although other circumstances were not so favourable for taking the pack as I could have wished, owing to the unsettled state of the weather and the wind blowing so directly upon the ice as to preclude our regaining the open water if thought desirable, I nevertheless determined to make the attempt, and push the ships as far into it as we could get them.'
Now that we have maps of the world which show the knowledge gained by Ross and his successors, it is easy to underestimate how courageous his move was. Amundsen, not a man to scatter compliments, rated it the most audacious decision in all polar history. One reason Ross dared do it was the brute strength of his ships, but just as important were his ambition and his observant mind. He had noticed that the main body of the pack was often lighter than the ice at the edge. His audacity soon paid off. 'After an hour's hard thumping, we forced our way into some small holes of water, connected by narrow lanes' and 'found the ice much lighter and more scattered than it appeared to be when viewed from the distance.' By midnight, they were seventy miles inside the pack. On 8 January they took a new species of seal, one which lives deep in the pack, now known as the Ross seal.
Ross is remembered as the man who dared to force the pack because, unlike d'Urville who entered it first, on 9 January Ross forced through it and out of the other side. He discovered open water behind the ice, and called it the Ross Sea. For future explorers it was, like Weddell's farthest south in an open sea, an encouragement to push south regardless. It would often prove a siren call. On 10 January Ross wrote, 'We now shaped our course directly for the Magnetic Pole,' following the compass's arrow. Fog cleared during the morning and noon revealed that 'not a particle of ice could be seen in any direction from the mast-head.' Excited anticipation ran through the ship. They were sailing straight to the Pole! The very next day, he saw land blocking his path. His readings suggested that the Magnetic Pole was still two hundred miles away. He rightly guessed the South Magnetic Pole would not be found at sea, and it would be a land expedition, not a naval voyage that would claim it.
With Ross himself boarding a ship's boat, they threaded their way through hazardous ice travelling in a swift current, and leaped ashore on an island. On Possession Island the toasts and ceremonies of colonisation were conducted with relish. The new territory became Victoria Land on 12 January 1841, the first land to be claimed for the young queen, and the most southerly ever seen by man, surpassing Bellingshausen's discoveries. They tracked along the coast, still increasing their latitude, and on 22 January they passed Weddell's farthest south, a record which had stood for eighteen years.
Whales were everywhere, blue whales, humpbacks and sperm whales: 'here they have enjoyed a life of tranquillity and security; but will now, no doubt, be made to contribute to the wealth of our country, in exact proportion to the energy and perseverance of our merchants.' He could not have guessed such energy, allied with new technology, would almost wipe them from the seas.
On 28 January they saw a lofty peak which appeared to have snowdrift blowing far from its summit, 'but as we drew nearer, its true character became manifest.' It streamed flame and smoke, and was an active volcano, which they named Erebus. The eastern promontory at the foot of Mount Erebus was named after the commander of the _Terror_ : Cape Crozier. It would feature as the base for Scott's attempts to reach the South Pole. Their sister ship _Terror_ was remembered by a slightly smaller, inactive peak.
Their magnetic readings now suggested the Magnetic Pole was around 75°S and 174 miles away. On the same day as they discovered Mount Erebus they made another startling discovery, and one of the greatest sights in Antarctica: 'we perceived a low white line extending... as far as the eye could see to the eastward. It presented an extraordinary appearance, gradually increasing in height, as we got nearer to it, and proving at length to be a perpendicular cliff of ice, between one hundred and fifty and two hundred feet above the level of the sea, perfectly flat and level at the top, and without any fissures or promontories on its even seaward face.' It was much higher than the tips of their masts, and formed a tantalising curtain closing off all views of the interior. He named it the Victoria Barrier, but it was later rechristened in his honour: the Ross Ice Shelf. It was often known simply as The Barrier, for as Ross wrote, damping down his disappointment, 'we might with equal chance of success try to sail through the Cliffs of Dover, as penetrate such a mass.' They tracked it for 250 miles, the edge of the continent, a vast bight filled with the outpourings of the world's largest glaciers. It was to be the base for all the early attempts on the pole.
On 15 February the sea began to freeze. The new ice was already four to five inches thick: 'had we not been favoured with a strong breeze of very precarious duration, which enabled me to force our ships through it, we should certainly have been frozen in.' In contrast to Wilkes's willingness to destroy men and bring ships to the brink of destruction, Ross thought their work did not justify such a terrible risk. As for work left undone, Ross devoutly wrote: 'we bowed in humble acquiescence to the will of Him who had so defined the boundaries of our researches.' They set a course northwards, but leaving the pack was much tougher work than entering it. They sailed through the line of Wilkes's mountains, drawn in the wrong place. When air sits in layers with denser air below, light is refracted downwards making visible objects normally below the horizon, and making them appear much closer to the American than he thought. Ross reached Hobart on 6 April 1841, with both crews healthy.
On 7 July 1841, with three years' more provisions stowed away, he set out once more to pick up his work where he had left it, sailing east towards the Antarctic Peninsula. By April 1842 he was back in the Falklands, with healthy men but tired and quarrelsome officers. There was one more sally south; on 17 December 1842 they sailed for the Antarctic Peninsula, and in the area now known as the Erebus and Terror Gulf, they picked their way around the north-west corner of the Weddell Sea, discovering, naming and taking possession. The island named after him dates from this venture. On 4 February 1843 they entered the pack at 64° south then tracked eastwards along its edge crossing the Antarctic Circle on 28 February 1843. On 5 March at 71°30´S Ross decided the season was over, and he headed home. His men were paid off in Woolwich, London on 23 September 1843, after an absence of four and a half years.
He was awarded the Founder's Medal of the RGS, the Gold Medal of the RGS of Paris; in 1844 he was knighted. When his account of the voyage was published he was elected to the Royal Society, but afterwards wrote very little. He married Ann Coulman and became a country gent near Aylesbury in Buckinghamshire. Ross refused command of a new expedition to find the North West Passage, having promised his bride's father he would make no more polar expeditions. Leadership passed to the governor who had shown them hospitality and friendship in Hobart: John Franklin. In 1848 Ross broke his promise by accepting command of the first expedition to discover Franklin's fate. He suffered terrible losses in the effort, but never found his friend.
# Deception Island
_There is one island I say goodbye to with poignancy at the end of each season in Antarctica. I know that one winter, vulcanologists' sensors planted in its dirt will tremble and satellites will spy plumes rising from its peaks. The first ships to break the spring ice will find the island changed, its harbour may be entirely closed, its anchorages sealed off from the sea by fresh lava, or uplifted land, the island almost unvisitable._
The stone horseshoe of Deception Island is one of the most visited places in Antarctica, but don't hope for a sunny day or rich wildlife. Because its volcanic rocks are warm, it makes its own weather and prefers a lid of glowering cloud. Despite a harbour, Port Foster, that seems to shelter you for 350°, the winds can whistle over the surrounding mountains and come right down to sea level again, and rip through your clothing. Lieutenant Edward Kendall of the _Chanticleer_ was on Sir John Franklin's second voyage and went through a particularly tough Arctic voyage on HMS _Griper_ , but he complained of Deception: 'it was withal so raw and cold, that I do not recollect having suffered more at any time.' In rapidly deteriorating weather I once shared the last Zodiac to ship with Peter Hillary, son of Sir Edmund Hillary of Everest, who has climbed many of the world's major peaks and skied to the South Pole. He pulled up his hood as we crossed Port Foster in a wind rising to full gale and remarked 'Feisty, isn't it?', which is as near as he gets to complaining about the weather. Although its waters are only slightly warmer than elsewhere, it's enough to make life tough for species of krill and fish that are extreme Antarctic specialists, so wildlife inside the harbour is scarce, and staff biologists on cruise ships usually loathe this visit. I love it.
First you squeeze through the narrow gap of Neptune's Bellows between the towering cliffs of Cathedral Crags and the barely submerged hazard of Raven Rock. To the left, red iron-rich rocks grip the last remains of the _Southern Huntress_ , wrecked in 1957. Each spring the wreck is a little smaller, its carcase picked over by the winter gales. You may, if air pressure is low, smell the sulphur from the smoking fumaroles along that shore, for you are sailing into the flooded crater of an active volcano.
The key features of the shoreline are visible from on board as you turn starboard and see the arc of Whalers Bay before you. At the right hand end of the panorama, the bite out of the cliffs is Neptune's Window, famed from the Palmer story. Beneath it, and scattered all over the shore as you scan left, are the remains of whaling days. Many materials were abandoned in 1931 by men who left thinking business would soon recover, and never came back. The relics are all the more poignant for their smallness in the great and harsh landscape: staves still waiting to be made up into barrels, boat huts, and small, decked boats probably used to ferry fresh water to ships, a few whale-bones: a clavicle stabbing at the sky or a vertebra the size of a cable spool, but most bones now carpet the floor of the bay. Landings begin at a peculiar landmark: a huge U-shaped iron box. When flooded it could be slid under the hulls of catchers whose plates had been damaged by ice or rocks, then pumped with compressed air to raise the hull for repairs. In the centre of the bay is a mix of old huts from the days of Operation Tabarin, Falkland Islands Dependencies Survey (FIDS) and BAS, dominated by the rusting machinery and drunken storage tanks of the Hektor whaling station. In the middle of them is a gap where the _lahar_ triggered by the 1969 eruption cut a swath through the heart of the old settlement and flattened the cemetery where sealers, whalers and one BAS scientist were buried. The few crosses that now stand are 1970 re-interments of graves left partially exposed by the _lahar_. If you visit please ignore the fad of laying cairns around them. Such things were not part of the original cemetery. Away to the left the single large shed is a hangar, not from the pioneering days but from the post-war use by BAS.
Every human activity that has ever gone on in Antarctica has happened here: exploration, sealing, science, whaling, flying and spying. It's the continent in miniature. Initial exploration and sealing were dealt with in the chapter on Nathaniel Palmer, so we'll start with the first scientific visitors to Deception.
## THE PIT AND THE PENDULUM
In 1829 a small British Naval barque of 234 tons called the _Chanticleer_ , commanded by Captain Henry Foster, sailed through Neptune's Bellows to the head of the harbour, and slipped into a creek lined with coarse black sand. He anchored securely in the manner of someone intending to stay. They began building strange structures, and erecting huts and tents. When these were completed to Captain Foster's satisfaction, unopened boxes were brought ashore under his strict supervision. They were handled like newborn babes. Under pain of severe punishment, no one, of whatever rank, was allowed to open or even touch one without Foster's express orders.
Foster was a rising star of the new breed of well-educated scientific Royal Naval officers, elected as a Fellow of the Royal Society aged twenty-seven. In 1827 he had won the Society's highest award, the Copley Medal, for magnetic and other observations in the Arctic on Sir William Parry's third voyage in search of the North West Passage. His current orders defined two main tasks: firstly, to make measurements of the power of the Earth's gravity here, in order to calculate the exact shape of the Earth in southern regions. Secondly he was to record the force and direction of its magnetic field. These might not seem priorities for military men, but the Navy had a life and death interest in both, because they affected navigation.
Magnetic research would make compass readings more reliable. A compass does not show true north and south, the geographic poles around which the earth rotates. Its needle points to the magnetic poles, and observations showed that they moved over time. Without an up-to-date map of the earth's magnetic field, you don't know where your compass is pointing.
The shape of the earth is important because distance travelled north or south over a curved surface like the earth is most conveniently measured using degrees, measured from the equator. If the earth were a perfect sphere, all these degrees would represent the same number of miles at the surface of the earth, but it isn't. Ninety years before, Newton's theory that the earth's rotation made it belly out at the equator had been proved by painstaking measurements on the ground. The circumference around the equator was shown to be eighty-five miles greater than that around the poles, so a degree of latitude at the equator is slightly larger than one at the poles. If you didn't correct for this, you would, as you approached the poles, always be nearer than you thought: a good recipe for sailing aground. Newton had also shown that the gravity of a large object like the earth behaves as if all its mass were located in a single point at its centre. Since gravity increases as objects get closer, gravity near the poles would be greater. As gravity influences the speed at which a pendulum swings, if you can measure the speed of a pendulum with fiendish accuracy, you can detect tiny differences in gravity and plot the earth's shape. Foster's forbidden boxes contained instruments that would allow him to do this. They were the best in the world: rare, expensive and delicate. He was one of the few men who knew how to use them with the necessary accuracy. The precision required to extract accurate results was exacting in a controlled laboratory. To do it in field conditions in Antarctica was a trial of science, skill, assiduousness, determination and perhaps sanity.
The spot where he worked is still named Pendulum Cove, but the creek has gone, and the beach is chiefly visited to find a pool of warm volcanic water for an Antarctic swim. The _Chanticleer_ stayed two months and the men were heartily sick of it before they left. Midshipman Joseph Kay, only fourteen years old when the voyage began, heard the men muttering that Deception was 'the last place that Nature made.' When they were at last ready to leave, the shape-shifting island was not ready to release them. Contrary winds and icebergs blocking the Bellows kept them there another week.
They must have looked forward to reaching the turquoise anchorages of the Caribbean. In Panama, Foster made a canoe trip up the Chagres River to measure the longitude of the east and west coasts of Panama. That section of river is now the north end of the Panama Canal. Returning downstream Captain Foster was pitched overboard and drowned and the body not recovered for some time. He was thirty-four years old. Natives went through his pockets, and turkey vultures through his flesh. 'What a loss to England and to Science,' wrote Captain Fitzroy of the _Beagle_.
It was later found that the volcanic rocks of Deception Island were so full of iron that it had affected the pendulum's swing. All his painstaking measurements were worthless.
#### PENDULUM SCIENTISTS: THE SULTANS OF SWING
The time it takes a pendulum to complete a swing depends on two things: the length of the pendulum (longer = slower) and the strength of gravity (stronger = faster). This was discovered by the French astronomer Jean Richer when in 1672, he left Paris, close to 49° north, and made measurements at Cayenne, French Guiana, at 5° north. To keep good time, he had to shorten his pendulum by 0.0226 cms. This means that you can measure gravity by timing the swing of a pendulum of known length. For example, a 40-inch pendulum (101.6 cms) has a one-second beat in Britain.
Foster had two models of pendulum to use in tandem. They worked best in buildings, so he first had to erect one: a wooden hut, double-skinned, and lined with blankets. The test pendulum was set up in front of a long-case clock with a marker on the pendulum of each. They swung at different rates and the simplest method of timing the swings was to time the interval between successive events where the swings coincided.
To correct for the expansion and contraction of the metals, parallel observations of air temperature were made from three thermometers at different heights in the room. He also measured the moisture of the air, the barometric pressure, the height above sea level, the state of the tide, and the line of swing of the pendulum. A telescope was used to make astronomical observations to determine the precise time of day. An average observation took 2 hours 40 minutes and he made 1017 of them. It was so time-consuming that he did not have time to calculate results from his own measurements, and the results were returned to Greenwich.
The magnetic results were still more complex. The lines of magnetic force were found to dip down into the earth as ships neared the poles. The horizontal part of the force became weak, and when a ship changed course the compass was sluggish in responding. The magnetic results were mostly passed to Woolwich Academy who made superficial remarks, published nothing, and filed them somewhere never since found.
## WHALING
In Antarctica's brief history, the only land-based activity to make much money was the brief slaughter carried out by the sealers. The oceans yielded more: the riches of whale oil. Until the middle of the nineteenth century, when coal gas and then mineral oil began to be heavily exploited, whales were the major source of high quality oils used in lighting, lubrication and various chemical industries. The last place on earth to be exploited was the Southern ocean round Antarctica, because it was far from the world's markets, dangerous to sail, and many of the whales were too powerful and fast to catch. By the late nineteenth century, technology had changed that.
Although whaling was a massive industry in Antarctica it was almost entirely ship-based. When the industry died and the ships sailed away they left few signs of their presence. In places like Neko Harbour where many people, including me, have taken their first steps on the continent, only the name remains of the whaler _Neko_ which anchored here a few seasons. In other places, like Mikkelsen Harbour, there are bones and boat timbers, greying and crumbling, slowly becoming more like each other. Only at Deception are there significant remains.
In 1820, the year after the discovery of the South Shetlands, when Captain Edward Bransfield RN came down, there was a flotilla of sealers in his wake. Many whales were seen, but most were the massive rorqual species like blue, fin and sei whales. James Clark Ross found his wake was incessantly attended by whales. But although the whalers who followed him could and did take southern right whales, the great whales were too powerful for their harpoons and boats. Planting a hand-thrown harpoon in a rorqual was like throwing darning needles into a bull; it was barely enough to irritate it, never mind catch it. The industry wisdom was that even if you could catch whales, you were so far from the markets that most of your cruise would be inactive, sailing to and from the whaling grounds; the sums would never add up.
In 1870 a Norwegian named Svend Foyn perfected a small cannon which fired a explosive harpoon head that penetrated the whale's body deeply and detonated a few seconds later. Four fins would open up like an umbrella, and lock securely in the body. For the first time in whaling, securing a line and killing the whale had become a single operation. But they still did not have the boats to get close to the athletic whales. Steam-powered steel-hulled catcher ships of about 160 tons were being developed that could endure ice floes and keep pace with a sprinting whale. They brought a whole new way for whalers to operate, delivering the carcases to factory ships, then going straight out to hunt for more. In 1892 the first Norwegian whalers came down to Graham Land and the Weddell Sea. The steam factory ship _Admiralen_ was in Antarctica in 1905, but lost money that season because of the coaling costs. Slowly they learned how to make it pay. While the catchers were purpose-built, the factory ships were mostly converted old freighters and passenger vessels of some three thousand to five thousand tons, bought cheaply for £6000 to £9000 in the years before the First World War, and lodged in safe harbours to await the catch. They soon realised that ageing sailing vessels were even cheaper to buy and didn't burn coal. The beautiful old Cape Horn clipper _Pisagua_ was one such ship. In early photographs of Whalers Bay its graceful lines stand above the ugly shoeboxes of the steamers. The men usually slept on board ship though some bunkhouses were built ashore.
The British recognised the terrible wastefulness of the industry. Even though the southern whales were much bigger than those now caught in northern waters, the oil yield per whale was only three-quarters. In part, this was deliberate. Whales were so abundant it was more profitable to take only the oil-rich blubber, then push the rest of the carcase out into the bay, with the meat and bones untouched. There were plenty more whales in the sea. Company balance sheets tell the story. 1911 was the last year the Hektor Company relied solely on ship-based processing with their single factory ship, the _Ronald_ , and three catchers. They paid a dividend of 32%.
A flensed carcase (one stripped of blubber) might sink, drift away, or be blown back on shore. When there was an onshore wind at Whalers Bay, it was sometimes so deep in rotting carcases that men attempting to land waded through a swamp of rotting flesh being pillaged by clouds of scolding gulls and terns, blizzards of pintado petrels, and bloated southern giant petrels, their lumpen heads scarlet with gore. The shore station was encouraged by the British, who wanted an industry that had a chance of surviving the Klondike days of wanton killing, and they granted The Hektor Company a twenty-one-year lease. Their steam pressure cookers used far more of each whale, including the sawn bones which held up to 30% of the oil. They were also granted the rights to process any carcases discarded by other stations. The seas administered by FIDS, including Antarctica and South Georgia, contributed 65% of world production in 1913–14. The following year Hektor's shore-based operation, the rusting remains you see today in the centre of the bay, became operational. The improved efficiency showed in profits; by 1915 Hektor had taken over two other companies, and paid a dividend of 50%.
The Great War provided respite for the whales as ships were requisitioned for other uses and the seas became too dangerous to sail. But two inventions changed the industry. One was the stern slipway, rising through the stern of the factory ship, to haul whales smoothly and quickly onto deck. The other was the perfection of reverse osmosis to make freshwater from seawater. By the late 1920s the industry was independent of the shore and free of regulation. The industry overfished so savagely in 1930 – 31 that the price collapsed. Next year Hektor's twenty-one-year lease expired, and soon the Great Depression was on everyone. No land station ever operated again in Antarctica.
## AVIATION AT DECEPTION ISLAND
The first ever flight in Antarctica was made from just above the beach on a bumpy 600 yard (550 metre) ash track enlivened by two dog-legs, three ditches and two mounds. The pilot was Alaskan Carl Eielson but the imaginative force and directing genius was Hubert Wilkins. He was born on 31 October 1888 into a family forging a living on marginal land 100 miles (160 kilometres) north of Adelaide, South Australia, where unpredictable droughts wiped out the profit and stock painstakingly accumulated in the good times. In one three-year drought his father lost ninety thousand sheep. Cattle died, families starved. Wilkins wondered: 'Why should it not be possible to learn of the laws which govern the movements of the atmosphere, the conditions that bring about the seasonal rains and droughts? I determined to devote my life to that work.' He vowed to spend twenty years familiarising himself with polar regions, then twenty more promoting international co-operation in weather and climate studies. He did just that, and in doing so Hubert Wilkins advanced polar aviation beyond recognition and became one of the most famous people in the world.
He first came to Antarctica as one of the four members of the British Imperial Antarctic Expedition of 1920 under Dr John Lackland Cope. He returned in 1928 after making the first aircraft flight over the North Pole, pioneering the circumpolar routes flown by modern airliners, despite Roald Amundsen warning him: 'What you are trying to do is beyond the possibility of human endeavour.' He had done it in a new plane, the Vega, manufactured by a then unknown company called Lockheed. The company was so impressed with Wilkins they sold him the third one ever made at cost price. It was a wooden monoplane with a monocoque fuselage, that is, it needed no internal frame. Wilkins began making various adaptations for cold-weather flying. While he did so, Vega number one vanished in a race to Hawaii, and Vega number two crash-landed while he watched.
Lockheed were keen to help him because he was already one of the groundbreaking aviators of the day. Less successfully Wilkins would nearly drown in a low-budget failure to take an ageing submarine to the North Pole.
With Wilkins's public profile, funding his Antarctic trip was easy. He intended to make the first Antarctic flight, to see if he could confirm the existence, reported by several land and sea expeditions, of channels crossing the Peninsula from the Bransfield Strait to the Weddell Sea. The climax was a possible continental crossing of some three thousand kilometres to the Ross Sea. His principal backer was the radio and newspaper magnate William Randolph Hearst, the man Orson Welles parodied in _Citizen Kane_. Hearst offered $40,000 for exclusive press rights plus a $10,000 bonus should Wilkins land at the South Pole, something Wilkins said he didn't want to do. But Wilkins was allowed to create just the kind of no-fuss small team he craved: two planes, five men, and a lift south on the Norwegian whaler _Hektoria_. He stuck with the plane that had proved itself, but in honour of new sponsors, he renamed his existing Vega _Los Angeles_ and a new one _San Francisco_.
When Wilkins called at the Falkland Islands on his passage south, he met the British Governor who had secret instructions to ask him to drop the Union Jack on any newly discovered territory. Britain was concerned that a parallel attempt to fly over the continent by American Naval Officer Richard Byrd was a covert means of mounting a bid for US sovereignty, and wanted to stay one step ahead, piggy-backing financially on Wilkins. Despite having American backers, he quietly agreed, because of his loyalty as an Australian to the Mother Country.
He had done all the homework he could about local conditions but Antarctica makes its own rules. At Deception Island, after fourteen consecutive years of solid spring ice holding fast in the harbour until Christmas, Wilkins found that this year the ice was too thin to land on and too thick to allow float landings on the water. In the end, the _Los Angeles_ was fitted with a pontoon undercarriage and lowered directly from the ship onto a patch of open water. They were reluctant to use the sea as their main method of take-off and landing as pontoons are liable to damage by small, hard-to-see pieces of ice, and in flight, their drag reduces a plane's range. On 16 November 1928 Carl Eielson and Wilkins lifted off from the water of Port Foster in the first ever Antarctic flight. Their reserve pilot, Joe Crosson, Eielson's best friend from Alaska, took up the _San Francisco_ soon after. The first task was to look for a better site for a landing strip, but they found nowhere more promising than where they'd come from: the small hummocky land at Whalers Bay, a mix of volcanic material ranging from coarse sand to fine gravel.
They now prepared to reconnoitre the Peninsula. A picture taken on 20 December 1928 shows Wilkins and Eielson boarding the _Los Angeles_ carrying two large, angular backpacks and long snow-shoes. They had two months' food and some survival gear, but no one really expected to survive a crash. If they did not return, their support pilot, Crosson, had orders not to search for them.
Wilkins had found no new lands on his North Pole flight, but now, for the first time in history, he could be sure that new lands would be seen by man for the first time from the air. He had fuel for more than 1250 miles (2000 kilometres) of flying, and set course down the Bransfield Strait, past Trinity Island and the Danco Coast, then over Waterboat Point where Wilkins' two young companions Bagshawe and Lester had overwintered during the John Cope Expedition. They crossed the west side of the Peninsula at Forbidden Plateau, then turned south to follow the east coast of Graham Land. Reading unfamiliar landscapes from the air is tricky, and they now made a mistake which gave ammunition to Wilkins's detractors. They included rival aviators like the American egotist Byrd and, sadly, fellow Aussie hero Douglas Mawson, as well as conservatives of all nations who thought ship and sledge journeys were the only manly road to achievement. Wilkins and Eielson saw valleys cutting across the Peninsula and recorded that some of them were chasmic channels, big enough to swallow skyscrapers. These channels would, they reported to the world, block any expeditions attempting to exploring the south overland. There were no such features. They were an illusion.
Near the end of their range they saw the weather behind was deteriorating. Before turning around at 71°20´S 64°15´W they saw a new plateau and named it Hearst, while quietly dropping a British flag with a note claiming sovereignty. Needless to say, these have not been seen since. A full storm was brewing and they could not take a second look at the channels. Wilkins instead plotted a course taking the shortest route north, into darkening skies. The wind shifted to blow on the nose and reduce their range further. It was a race against time. After a flight lasting 9 hours 25 minutes, covering 1200 miles (1920 kilometres) of which 625 miles (1000 kilometres) were over new territory, their Vega bumped down into the soft dirt of Deception Island, and the history of exploration had taken a new direction.
Today you cannot see much evidence of the two ash runways which once crossed like open scissors outside the later hangar of the British Antarctic Survey. Until a few years ago, one of BAS's tiny orange De Havilland Otter planes still stood outside the rear of the hangar, looking halfway in size between a big toy and a real plane. Its wings stood against the wall inside, steeped in the snow which drifted through the partly open doors each winter. The plane has now been brought back to England to be restored and exhibited at an aviation museum. Outside, heavy girders lie in the dirt. They were brought to enlarge the hangar, but when the volcano erupted again in 1970, expansion plans were put on a high shelf and never taken down. It's hard to justify investment when a blow from a side vent of Deception could melt five years' work in an afternoon.
Wilkins contracted a long series of articles about the expedition and meditated on the nature of exploring. He was candid about its unfulfilling nature; a feeling no other explorer I know has voiced so plainly. The next time he was down here it was without his old comrade Carl Eielson who had chosen to pioneer commercial flying in Alaska, and been immediately been killed by a storm coming in from Russia. The Russians had known about it but were not sharing information with America. Wilkins's dream of international co-operation improving understanding of weather and climate seemed more urgent and personal than ever.
Wilkins was the organiser of one more historic polar flight. It was funded by Lincoln Ellsworth, a playboy heir who contributed nothing to the planning, except letting his obsession with the Wild West frontier life dictate the name of the plane: the _Wyatt Earp_. Wilkins organised the first trans-polar flight, taking off at 10:00 on 23 November 1935. It was planned to last fourteen hours and took twenty-two days because they came down twenty kilometres short of the Bay of Whales, at the gates of the Ross Sea, and had to walk the rest.
After a lifetime of adventure, Wilkins died in November 1958, suffering a heart attack alone in the cheap hotel room where he preferred to work when in America. He still had one last dramatic journey to make. His ashes were scattered at the North Pole by Captain James Calvert when the nuclear-powered submarine USS _Skate_ surfaced there on 17 March 1959. A new generation had not forgotten his worth.
## VOLCANOES
Deception is a relatively new island, and is a form of collapsed volcano called a _caldera_ , a Spanish word for a pot or cauldron. When liquid rock, or magma, forces its way up from around twenty-five miles (forty kilometres) below the surface of the earth, and erupts onto the surface as a volcano, it creates huge stresses which are often relieved by ring-faults: cracks in the rocks which are circular when seen from above. The central area often has the most pressure under it and is driven up by the terrific pressure of the rising lava. When a major eruption dies down, it has, through lava flows and explosions, relieved the pressure, causing the centre to collapse downwards. When this happens on continental land, rain may fill it, creating a lake. In a few rare instances, where the volcano is a small island, the sea may flood in. At Deception this happened through a small gap in a nearly complete crater rim: Neptune's Bellows. Visitors often look around the island and ask where the volcano was. The answer is: You are in it.
The first sailors knew it was active, and Wilkes, who saw it in 1842, saw 'the entire south rim of the crater on fire.' In 1820 sealer Robert Fildes surveyed the harbour and found a depth in the centre of the harbour of 1080 feet (329 metres). Lieutenant Edward Kendall of the _Chanticleer_ resurveyed it in 1829 but was unable to sound more than 582 feet (177 metres). Both were trustworthy surveyors, it was the harbour bottom which had moved in an intervening eruption. It currently has a maximum depth of 630 feet (190 metres).
Foster's Pendulum Cove later became the site of a Chilean research base which was damaged by the eruption of 1967 and then destroyed by the 1969 eruption. Only the stumps of concrete bases and tangles of steel reinforcing rods remain. Each time all the personnel in the island's three bases (the other two were Argentinian and British) escaped, though it was a close thing, as molten lava bombs fell side by side with ice blown out of the ice-cap.
Since 1948 the Argentines have maintained their small Base Deception at the head of Port Foster, and the Spanish Gabriel de Castilla Base is nearby. Both monitor the levels of vulcanism. If you see remote sensors around the island it's rude to jump up and down near them, as it panics the staff. No one now overwinters after a third eruption in 1970 highlighted the risks that rescuers would run attempting to pick up researchers during winter.
## OPERATION TABARIN
A snazzy Second World War poster hangs above the partition between the bunks and the kitchen in the living quarters at the restored historic base of Port Lockroy. It shows a cabaret girl at the Tabarin Club, Paris. In its smoky atmosphere British servicemen hatched a smoke and mirrors scheme. By 1944, under the guidance of a man called Moxon, it emerged from the Admiralty as a plan for a top secret operation in Antarctica. Winston Churchill's officials waited for one of the handful of occasions during the war when the Prime Minister left the country, to rubber-stamp the initiative in his absence. In 1943, fourteen men were taken from different parts of the British Armed Forces to join Operation Tabarin. They were sworn to secrecy. Their leader, 'Jimmy' J. W. S. Marr, then in Ceylon, was a geologist who had been one of the two boy scouts chosen in a competition run by the _Daily Mail_ to accompany Shackleton on what turned out to be his final expedition in the _Quest_ in 1922. They were to form bases in Antarctica. To convey them there, they were allocated a Norwegian sealer renamed HMS _Ramsey_. They were given little information but one piece of advice proved very true: 'She may leak.' She did little else. At frequent intervals in the painfully sluggish Atlantic passage Marr would call the men together and lecture them on the need for total secrecy: 'Don't let anyone, family friends or sweethearts, know what you are doing.' After a while one man protested: 'But sir, we don't _know_ what we're doing!'
They first made for Deception where, since the war had begun, some Argentines had painted their flag. The British promptly painted out the blue and white and painted in the red, white and blue. They set up at a comfortable wooden hut, code-named Base B, and another at Port Lockroy, Base A. A projected third at Hope Bay, on the tip of the Peninsula, proved impractical due to ice conditions. Further bases were added later.
At last they learned their top secret task: monitoring German naval communications. In 1916 during the First World War, a British squadron had won a major naval battle in Falkland Sound, between East and West Falkland. Perhaps the theatre of war might move even farther south; German warships might hide in Antarctica and take coal from British bases and whaling stations. The first thing that strikes you looking at the location of these bases is how poorly located they are to monitor anything, being tucked away in nice sheltered bays. Deception's Base B, for instance, was built inside the island at sea level not outside on the mountain, commanding the Bransfield Strait. A few years ago the records of Operation Tabarin were made public. The daily report sheets were all bare. Not one enemy communication had ever been intercepted. The specialist teams, pulled from all over the world and sworn to secrecy, had returned blank forms day after a day. There was no radio activity of any sort. Was it a foolish mistake or something more subtle?
A coding machine known as Chiffriermaschine E was invented by the Dutch in 1919 and was on general sale until 1929. It had, for the first time, solved a fundamental problem of all coding: that in any language some letters occur more often than others, in a known pattern. Any code that substitutes one symbol for a letter of the alphabet repeats that pattern and is easily cracked. Chiffriermaschine E had mechanical rotors and electrical switches that made seven changes to every typed letter. Each one was routed in a continually changing way to the next rotor, hiding the frequency at which individual letters appeared. The rotors of the machine were changed each day and codes let the receiver know how they had changed. The final letter was given to a Morse code man and transmitted separately from the main coding transmission. It seemed foolproof.
In 1932, Polish mathematical cipher-crackers, fearing it might make an extremely tough code to crack in any future war, set a team of mathematicians to break the code. They found they could crack it, but too slowly to defeat the daily change of rotors. In 1938 the German state added two more rotors to increase to the complexity. The German Navy added three, and was convinced it was now far too complex to break. They reserved it for top secret strategic communications. When war got under way, the British knew such a code was being used, but could not devise a way of decoding the complex transformations before they changed again. They christened it the Enigma Code. In February 1944, Alan Turing, working at Britain's Bletchley Park intelligence centre, built a machine called Colossus I to extract and analyse patterns for further analysis. Ten were made. They were capable of sifting through unheard-of amounts of data, and extracting patterns before the rotors were re-positioned the next day. The enigma was solved. Today we have these machines in our homes and workplaces; we call them computers.
Cracking the enemy code required agonising decisions from those conducting the war. If they acted on all the new knowledge it would quickly become obvious the code had been broken. It would be changed and the advantage lost. Some information must be ignored, and decisions taken that appear to be made in ignorance of the actual plans. Sometimes, lives would be sacrificed to maintain the illusion. More kindly, men will be sent to the other end of the world to carry out surveillance on operations that the enemy never intended to perform: like occupying Antarctica. There is clear evidence that British officers in Antarctica occasionally broke normal security by sending parcels through Uruguay, which was sympathetic to Germany, knowing they were likely to be intercepted and betray a British presence in Antarctica.
It seems that Operation Tabarin was a smoke screen designed to cover up the fact that Britain had cracked the Enigma Code. In French dialect _tabarin_ means a cloak, and one of the islands chosen for a base was called Deception. Just because there is a war, there's no reason to lose your British sense of humour.
#### THE CHEEK OF MR BORCHGREVINK
In 1894, schoolmaster Carsten Borchgrevink, living in Melbourne, was desperate for polar experience. He persuaded expedition leader Henrik Johan Bull and Captain Leonard Christiansen to take him on board the whaler _Antarctic_ , bound for Ross Sea, as an unpaid Ordinary Seaman. Although he had an English mother and wife, he was born in Norway and, as a boy, played with Roald Amundsen.
It was a primarily a whaling expedition, sponsored by Svend Foyn, the inventor of the exploding harpoon. On 24 January 1895 at one in the morning, they lowered a boat off Cape Adare in the Ross Sea. They knew this was the mainland and thought landing would make them the first men to set foot on the continent; knowledge of the sealers' landing on the peninsula had been lost. Its crew included Capt Kristensen, Bull, Borchgrevink and four others. Conditions were calm, with little swell. For an hour they rowed through loose ice, the water swarmed with vast red jellyfish trailing yard-long tendrils. As leader, Bull expected to be given the honour of landing first, but the captain prepared to go first. As the boat ran into the pebbly beach, Borchgrevink slipped over the bows before either of them, and turned to hold the boat. Bull and Christiansen were furious at his insolence. Bull would later claim he was the first to land, and Borchgrevink had only stood in the shallows: playground boasts. It is a blessing that the sealers had long pre-empted them, and the quarrel does not need to be resolved. They were there less than ninety minutes, but as Bull recognised 'Mr Foyn would no doubt have preferred to exchange this pleasing sensation on our part for a Right whale even of small dimensions.' It was a commercial failure.
In 1898 Borchgrevink secured £40,000 from British publisher George Newnes and led his own expedition back to Cape Adare in the _Southern Cross_. Most of the men were Norwegian or Finnish, but Newnes's money bought the right to fly the British flag. Historians have sometimes damned their efforts with faint praise, but they successfully executed their plan to set up Camp Ridley and be the first people to overwinter on the continent. They also pioneered the used of sled dogs, climbed onto the Ross Ice Shelf, and saw it extended as far as the eye could see. This would become the route by which Amundsen and Scott attained the Pole. Borchgrevink's farthest south here: 78°50´ was a new record. Their magnetic measurements showed the South Magnetic Pole must be much farther north and west than had been assumed. It would be discovered on Shackleton's 1907–09 _Nimrod_ expedition by the sled party of David, Mawson and Mackay.
# The First Antarctic Night 1897– 99:
Adrien de Gerlache
_A surgeon who will soon go mad takes a book of Edgar Allan Poe's stories to the Antarctic. Their expedition is locked into the ice and their own story begins to outdo in horror anything in the author's dark imagination. Black doubts walk the ship like spectres: were they trapped by accident, or design?_
The 250-ton barquentine _Belgica_ sailed out of Ushuaia, then just twenty-five sheet metal houses at the foot of Argentina. Before they reached open sea they ran aground and had to be rescued by a farmer. 'Early in the morning of New Year's Day, 1898,' wrote Lucas Bridges, in his father's remote mission farm on the north shore of the Beagle Channel, 'I looked out of a window of the Cambaceres house. Across the outer harbour to the southward, about half a mile away, lay a little vessel on a shoal.' They were hooked on a rock which was sawing through their hull as they swung about, helpless and out of control. The captain was in his cabin and crying his eyes out with frustration. They had run aground trying to enter an uncharted, unfam iliar harbour in near-darkness during a storm. Mariners have a technical term for this manoeuvre. It's called being stupid.
The expedition commander, Lt. Adrien Victor Joseph de Gerlache y Gomery, had taken special leave from the Belgian Navy to gain sailing experience off East Greenland in the sealer _Castor_. The physician, Dr Frederick Cook, although a last minute replacement, was a veteran who had travelled with Peary in the Arctic. The only other man with polar experience was the second officer, who had sailed on the famous sealer _Jason_ , a veteran of both Antarctic and Arctic seas: Roald Amundsen. When they returned in triumph, Amundsen declared: 'Modern scientific exploration has now been initiated and de Gerlache has won his place.' The voyage was de Gerlache's idea. He secured private, public, then government support in Belgium against the tenor of the times, as King Leopold was trying to focus Belgium's colonial ambition on his personal scheme: the murderous exploitation of the Congo that inspired Joseph Conrad's _Heart of Darkness_ , and later the film _Apocalypse Now_.
Novices sometimes innovate where experts stick to a formula. Harland and Wolff, who made the _Titanic_ , were professionals; Noah was an amateur. With Otto Nordenskjöld, William Speirs Bruce and Jean-Baptiste Charcot, de Gerlache was one of four men who, at the turn of the century, mounted compact expeditions that put science to the fore and jingoism to the rear. The money de Gerlache raised was barely adequate. His crew was the minimum needed to sail the vessel, just fourteen plus scientists who would have to double as officers. Only two seamen, the bosun Adam Tollefsen and Johansen, were able to lead a watch. It was only possible to sail at all because they had Cunningham's patent furling topsails which could be handled from deck without sending men aloft. He was not short of supplies. In fact they seemed to have more than they might ever use. When they left, the deck was just fifty centimetres above water. The few seamen they had were so ill-disciplined that when they bunkered coal in Punta Arenas on the Straits of Magellan, he had to call the local police to regain control of them. They sailed without signing the formal articles of discipline. De Gerlache remarked that the only disciplinary power he had was to put men ashore, but that's quite persuasive when you are going to Antarctica. The scientists were to be unpaid simply because there was no money to pay them, but he recruited talented men because he was offering a rare chance to carry out extended scientific research in a virtually unknown environment. Despite the Belgian flag, they were a multinational crew. Many of the other seamen were Norwegian and the _lingua franca_ of the trip was going to be French until the arrival of Cook, who had no French. English and Norwegian were spoken as necessary to communicate orders. The international mix was in large part a gesture to defy the nationalistic expeditions of the great powers.
The oldest man on board, at thirty-four, was the Chief Engineer; de Gerlache was thirty-one. The zoologist cum botanist was Emile-Georges Racovitza, a twenty-nine-year-old Romanian. The Pole Henryk Arctowski, after whom the modern Polish Base is named, was geologist, meteorologist and oceanographer. The assistant meteorologist was the Pole Antoine Dobrowolski, twenty-five and the other scientist was a man known to de Gerlache's family: the Belgian magnetist, Emile Danco. De Gerlache harboured reservations about accepting Danco's application. Young Emile had a weak chest and a strict father who frowned on frivolity. In protective isolation Emile grew into a man full of juvenile opinions and naïve enthusiasms. When his father died, he was left with an independent income but no close family. He had had enough of being cosseted; he wanted adventure, and believed fresh cold air would restore his chest. De Gerlache relented only when he heard that if he was not accepted Danco's back-up plan was to go to the Congo which was so unhealthy that most men arriving were expected to die of disease somewhere between passport control and customs. De Gerlache sent him to Vienna, Paris, Jena, and Berlin to buy instruments, and be trained directly by the manufacturers or expert local academics.
De Gerlache had prepared thoroughly for many aspects of the trip, including consulting the Norwegian explorer Fridtjof Nansen and Nansen's captain Otto Sverdrup, a giant of polar navigation who shunned the limelight. The plan, which would alter frequently, was to reconnoitre the Weddell Sea then sail the long route east to Victoria Land and overwinter with three other men while the ship retreated to Melbourne. Their last call before leaving South America was at Staten Island, the desolate slab of rock east off the tip of Tierra del Fuego. For the Argentine official administering the island, it was the posting from hell. Most of its occupants were convicts, the sweepings of Argentina's jails. Its Sub-Prefect was the cultivated Nicanor Fernández, who served dinner to the adventurers on fine china marked _Esmeralda_ , the name of a ship wrecked there a month before. The visitors brought ashore a machine called a _Coelophone_ that played music from a punched paper roll. Deprived of music since his arrival, Fernández listened, remembering what polite society had been like, and burst into helpless tears. They sailed on 14 January 1897, already so late that their original plan was reined in. They would remain in the Peninsula area for the rest of the southern summer, study the area around Hughes Bay in Graham Land, and leave a trip to Victoria Land for the next summer. Come winter, they would return to Ushuaia in the Beagle Channel to study zoology and the ethnography of the Ona Indians. The latter was deeply ironic as de Gerlache was spiteful to the Bridges family in his memoir, implying they were more interested in trade than saving souls: 'He will nevertheless be leaving his sons in an enviable situation when eventually the last of the Yaghans dies of consumption on the threshold of the mission.' In fact Bridges senior, with a wife and six children and no income, had resigned from his missionary society because he insisted evangelism on its own was useless to natives without skills for the modern world. Meanwhile de Gerlache attended a most agreeable picnic with a millionaire whose 100,000 sheep were being harassed and killed by mad dogs. He always called them mad dogs, and he paid a £1 a head for each one shot. The 'mad dogs' were the Ona Indians.
Their work began, lowering the oceanographic gear donated by the Danish Navy, netting samples of sea-life and measuring depth. Off Staten Island they found 296 metres, but late the same afternoon the bottom plunged to 1564 metres. The next day they saw three ships, all Europe-bound, and thought how those sails were spread for home, whereas theirs would take them to two months of solitude. They could not know it was their last sight of the outside world for well over a year. Or perhaps one person could. That day they sounded at 4040 metres. Though mariners had sailed these water regularly for nearly three hundred years, no one had hung around to take measurements. Before they even saw Antarctica, they had made an important discovery: the Drake Passage was a basin, deepest in the middle.
On 20 January 1898 at 16:00 they saw land at 63°11´S 061°37´W. No one forgets their first sight of Antarctica. They were awe-struck. The next day, before they could even plan a landing they lost steerage among icebergs and rocks with steam pressure too low to pull them clear. Through luck, they gained steerage before they hit anything. De Gerlache claimed he told the engineer that the boiler pressure was low fifteen minutes before this happened. Even if you credit this, you wonder why he continued into a hazard before confirming the boiler pressure was restored. The next day, the cost of carelessness was tragedy.
They had sailed down the Bransfield Strait and were south of Deception Island. The weather turned ugly. The underlying cause of the accident was overloading: the fifty centimetres of freeboard. In stormy weather, some coal stored on deck, because there was nowhere else for it, burst loose. It crashed down the deck and blocked the scuppers, the gaps that drain deck water under the ship's rails. On the undermanned ship, Auguste Wiencke, twenty years old, with the invincibility of youth, worked alone. Against standing orders, he tied himself to a line and let himself over the side to help free the coals. A cry was heard. The details are lost in the confusion of desperate action. The watch officer was studying an iceberg lying in their course, and it took him some time to see Wiencke's danger. Dr Cook saw Wiencke had come free of his line and, in water only a degree above freezing, the seaman's hands would only work for four or five minutes. Cook threw the ship's log line to him: a slender rope with a flat piece of wood at its end, ideal to grasp. Wiencke caught and held it. Cook began to haul him in, desperate to bring Wiencke close before his grip failed, but fearful of ripping the wood from his grasp. Feeling drained from Wiencke's numbed hands. The wood slid from his grasp. He began to drift away. One man showed true heroism. First Officer Lecointe tied a line about his waist, and leapt overboard. He swam to Wiencke and was within arm's length of grasping him when a vicious gust within the storm tore them apart. Wiencke was borne into the night, and suffered for brief minutes before solitary resignation overcame him, and unconsciousness relieved him from experiencing the iciness of mortality. The watch officer felt he should have been more aware of the problem with the deck cargo; he had focused on the iceberg to the exclusion of other matters. No one else blamed him, but the officer's standards were high. He was Roald Amundsen.
De Gerlache's account of the incident may reflect his anxiety at losing a member of a slender crew so soon after reaching Antarctica. He was young and it was his first command. 'On the afternoon of the 22nd... our young sailor August Karl Wiencke paid with his life for a minor act of disobedience... heedless of the danger and believing he was doing his duty Wiencke went over the side on a rope.' This puts pride before compassion. Maybe he felt responsible for the fifty centimetres freeboard that made any blockage potentially fatal. De Gerlache added: 'Wiencke was popular with his shipmates, and his superior officers recognised an unusual intelligence, an excellent character, and a devotion to duty. His loss was keenly felt by all.' Wiencke Island on which I have twice camped, enjoying spectacular views of the highest peak in the region, Mount Français, is named for him.
Their first landing, on 23 January 1898 at 19:00, was on an island named Auguste Island after de Gerlache's father. Cook's reactions are fascinating because he had previously seen the Canadian Arctic and Greenland. He said there was nothing in the landscape that reminded him of anywhere. Antarctica, above all else, is itself, like nowhere else.
Henryk Arctowski observed the sense of disconnection, both with past experience and with the artificiality of his expectations, with the sketchy records that were all that was known of a whole continent. 'At 7pm we passed close to a headland, which very probably was Cape Cockburn, but as we went on the charts became valueless; what we saw corresponded to nothing that they represented...'
Adrien de Gerlache in furs
He was first ashore, followed by Racowitza; de Gerlache and Cook had to scramble back on board to stop the surf throwing the boat onto the rocks. Cook, itching to spend a few moments ashore, was forced into observing the lucky pair, walking where no one had ever walked. 'We tried to follow them with our glasses as we rocked about in our boat, but soon lost sight of their movements in the dark ness. We were able to locate Arctowski by the dull echo of his hammer, and we were able to trace Racowitza by the chorus of penguins which greeted him from rock to rock. The alternate interchange of the music of the hammer and the war song of the penguins was an entertainment which to Gerlache and myself, will be a long and weird remem brance.'
The landing lasted only thirty minutes. Danco brought back souvenirs: two live penguin chicks which he looked after, said Lecointe, with maternal care. Many photos of other early expeditions show them playing with penguins as if they were toys. Racowitza was a rare, kind observer, withdrawing when the birds seemed stressed. He was especially intrigued by the sociable and noisy chinstraps. 'From a distance they resembled nothing as much as two fishwives discussing the freshness of their respective wares.' He characterised them well, and though he anthropo morphises them shamelessly, he has clearly observed them with patience, empathy and humour:
Imagine a small, upright gentleman, with two paddle-like flippers in place of his arms imagine him dressed in a dark coloured coat with blue patches, ending in a pointed tail dragging along the ground... now put him in motion on his two feet and give him an odd waddling gait and a constant movement of his head, and you have before your eyes something both irresistibly attractive and comic... We began to think in our surprise that we had come across a political meeting in the middle of an election, and I was taken ashore to find out what was happening.... The moment I set foot ashore I was received with a violent outcry and shrieks of vehement indignation which left me in doubt of the opinion these birds had of me. I thought I would in time become acceptable to them, and sat down on a rock a little distance away. But my patience and friendly demeanour were a waste of time. All the birds turned to face me, drawn up to their full height, with the feather on their heads raised in anger and beaks agape, and assaulted me in language which I judged from the tone to be decidedly insulting.
When you first see Antarctica you cannot tear yourself away from it. Seamless light does not encourage sleep. Cook is always better than de Gerlache at the detail that stirs the heart. 'Though the sun was sliding eastward just under the high mountains to the south-west it seemed perfectly dark. Nevertheless, on the water, as we paddled over it, there was a curious luminous gray light, by which it was possible to read coarse print, even at midnight. The sky, however, continued black, made so by the sooty clouds which cease lessly rose out of the Pacific to drop their white cargoes of snow on the neighbouring lands.'
In February 2008, I worked on an expedition taking a party headed by the grandson of de Gerlache along his ancestor's route. The present baron is a man who enjoys his family's obsession with the Antarctic (each generation has followed Adrien down there). We flew the same Belgian flag that Adrien de Gerlache had flown. The problem with following de Gerlache's route was that he landed where he could, and that is not the most interesting itinerary. After a few days of squeezing onto ribbons of shingle and paving stones of rock, some of the less obsessed travellers wanted to know why there was nothing to see. We began to slip in landings unrelated to Gerlache's route, but with lots of penguins. Morale soared. That trip cured me of another problem. I am sometimes asked if I have ever trodden on land which no one had ever walked before. The honest answer is I really don't care, but saying so sounds churlish. On 29 January 2007 we made the first recorded landing on Ghent Island. The organisers had asked that the baron be the first ashore. We asked if they realised this would mean him leaping out of the bow of the boat and holding it in the surf. When the day came, the staff landed first to check out the site, walking in different directions. I walked on new land.
The films and documentaries that form our images of Antarctica only paint a one-dimensional view of the weather: killing cold. In fact, Scottish visitors to the Peninsula often receive emails telling them it's colder at home. Visitors from the Mid West and north-east of the USA, never mind Canada, have certainly left colder weather behind. A sunny calm day can take the thermometer well into double figures, centigrade. On 25 January Arctowski noted it was 'fine and calm, the air perfectly transparent, the sky cloudless, and the heat of the sun intense.' When they landed at Harry Island, de Gerlache observed: 'the radiant sun showed nature "grandiose" and savage. It was so hot one of us caught sun stroke.' The day before on Moreno Island, Arctowski had recorded: 'I had the pleasure of discovering the first Antarctic insect, almost microscopic in its dimensions.' This was christened _Belgica antarctica_. It's a nice bar-room question on an Antarctic cruise to ask what is the largest land-predator on the continent. Leopard seals are usually mentioned, but they and all the other big beasts hunt in the sea; on land, the top predator is a wingless midge. On Louise Island Dr Cook remarked on something else that startles many Antarctic visitors. 'Our greatest surprise here was the discovery of large quantities of moss and lichens, which gave the spot an appearance of life that to us, after having seen nothing but ice and rocks for so many days, made it a true oasis.' The mosses can be as green as in any oak wood and the lichens the same brilliant orange as those on the rocks of temperate seashores.
At the end of the month, they landed with ease on snow reaching down to the shore at d'Ursel Point on Brabant Island, and made the first ever Antarctic sled journey. They aimed to survey the island and obtain a view far into the west from the Solvay Mountains above them. The team comprised de Gerlache, Cook, Arctowski, Danco and the first man to master Antarctic sled travel: Amundsen. They had two Nansen sleds, the lightweight wooden models developed by Fridtjof Nansen after he crossed Greenland successfully in 1888. The Inuit sled was lashed together rather than jointed or screwed, and designed to flex with the stresses; all rigid structures broke eventually. The last dog sleds used in Antarctica were the direct heirs of his design. Arctowski was not impressed: 'The sledges were horribly heavy' and 'there was far too much luggage.' He helped haul them up a snow slope of 40° for four hours until they found level ground to camp at 1100 feet. Cook described a night of stormy discomfort in their silk tent: 'A wind came out of the bed of a glacier above us, against which we could hardly stand.' It is strange to read in Amundsen's journal: 'The doctor, an experienced polar explorer, went first and I followed.' Little more than a decade after this trip ended, he would never need to defer to any living explorer, except his mentor: Nansen. That night they were very warm in their reindeer bags. Next day they began walking but soon camped again, because of fog. At 13:00 they tried again, aiming for high land, but after four hours of heavy labour were frustrated by a huge crevasse, and had to retreat to the plateau. On 1 February it was snowing. De Gerlache and Danco were roped together hauling the same sled. 'Suddenly,' wrote de Gerlache, 'my friend vanished as if a trapdoor had opened under his feet.' De Gerlache was pulled towards the yawning crevasse, trying to dig his feet into the snow, unable to hold the weight. He was approaching the lip of the chasm when Danco's skis wedged tight in the crevasse and stopped him falling further. He emerged unhurt. There were many narrow crevasses, and they were hard to see. The following day they got to the summit of the nunatak at 315 metres but frustratingly the visibility closed in before the theodolite could be set up. Next morning revealed vast panoramas of Danco Land, a clear light illuminating the snow plains of the interior. Cook wrote, 'The view before us was even more beautiful, if possible, than anything we had seen since our first entrance into this new white world. To the south-west there was an opening through a new land and into a new sea, which remained for us to explore later.' Below them icebergs split with 'mysterious explosions' like 'a thousand cannons.' On 4 February Cook and Amundsen cut their way across a very steep ice face which fifty feet below ended in crevasse they estimated as over a hundred feet deep. Above, overhanging ice-masses prevented them climbing away from danger. They gingerly went forward, Amundsen trusting to Cook's judgement and experience. If Cook slipped, Amundsen remembered thinking, it was 'doubtful if I could have hung on to him.' Then they crossed the crevasse by crawling over a bridge of thin ice. Crevasses cut off access to all the higher peaks; a glimpse of unknown territory was quickly obscured by fog and snow. Descending what has seemed from above to be a more appealing return route, they found it much more heavily fissured. Without trees or buildings or any object to give scale, the Antarctic landscape fools you into seeing what you want. Amundsen: 'Until you are standing in the middle of it, so you can touch and feel everything, you know nothing with certainty.'
At the time Amundsen was thrilled with Antarctic life: 'I hope that there are more of these wonderful trips to come.' But when he looked back at his first experience of man-hauling sledges, lasting just one week, he damned it. It wasn't heroic but 'unpleasant, sweaty, toilsome and stupid.' Fifteen years later Scott was still measuring manliness by the yardstick of misery endured, and extolling its virtues. Amundsen's men would return from the Pole heavier than when they began while Scott's men suffered death by scurvy and starvation.
From the top of Brabant Island, Danco and de Gerlache thought they recognised Two Hummock Island which would join their present survey areas to their previous ones, uniting the known geography. Sailing to Hughes Bay, on the mainland opposite Two Hummock Island, confirmed their guess. On 8 February de Gerlache navigated 'a series of the most picturesque locations we were ever privileged to see.' This is now the Neumayer Channel. Many seasoned Antarctic travellers prefer it to the much-hyped Lemaire Channel, which has become such a must-see item for some cruise passengers that they feel swindled if they don't make a passage of it. Frederick Cook wrote of the Neumayer: 'It was a photographic day. As the ship steamed rapidly along, spreading out one panorama after another of this new world, the noise of the camera was as regular as the tap of a stock ticker.' Three hundred photographs were taken that day. He continued: 'Everybody was on deck with pencil and paper, some making nautical and geographical notes, other geological and topographical notes, and all recording the strange, other-world scenic effects. Even the sailors, the cabin-boy, and the cooks were out with paper and notebooks, taking long looks and then bending over their paper.'
Despite this Arctowski was frustrated. He recorded privately on 10 February:
Since we were always in sight of the coast, I never ceased to ask for more landings; I urged Lecointe, de Gerlache, and the others again and again, but not with so much effect as I could wish. For this eighteenth landing he conducted me himself, but for ten minutes only. A few strokes of his oars brought us to the beach amid cries of 'Hurry up, Arctowski.' I gave a hammer to Tollefsen, with orders to chip here and there down by the shore, while I hurriedly climbed the moraine, picking up specimens as I ran, took the direction with my compass, glances to the left and right, and hurried down again full speed to get a look at the rock in situ; meanwhile Cook had taken a photograph of the place from the ship – and this is the way geological surveys had to be conducted out in the antarctic.
On the evening of 11 February, a message to the engineer was misunderstood and they crashed into an iceberg losing their figurehead: a loss often regarded as bringing bad luck. On 12 February they made their twentieth and last landing, at Cape Renard, then sailed south through the Lemaire Channel without commenting on the scenery: just another channel. The next day the expedition landed at Wiencke Island and later on the Wauwermans group, Arctowski described them as 'like great whale-backs appearing above the sea.' The water was as calm as glass.
Around 1873 the German Edouard Dallman had been sealing in the _Grönland_ in this area and attained 65°S, going beyond Anvers Island, and discovering Kaiser Wilhelm I Islands. To the south he could see only sea leading east from between himself and Booth Island (the island that forms the seaward side of the Lemaire Channel). He called it the Bismarck Strait, and believed it to be the western end of a strait Carl Larsen thought he had seen looking westward from the other side of the peninsula. From 8 February to 11 February the _Belgica's_ probings proved Dallman had seen nothing more than a large bay, now called Flandres Bay. De Gerlache sailed out to the open ocean via the Argentine Islands area, where the Ukrainian Vernadsky Station now stands, and saw another large opening which he thought might have been the Bismarck Strait. Phantasmal sightings of this non-existent strait would continue until the British Graham Land Expedition finally banished it in 1937.
On 15 February they guessed, without being able to confirm by observation, that they were crossing the Antarctic Circle. In other sectors of Antarctica it is possible to sail much further south. In recent years it has become possible for vessels with only moderate ice-strengthening to cross it in February and early March before winter returns, but the line still has a magic to it. De Gerlache moved on, threading his way among heavy ice on 17 February to 67˚58´S 069˚53´W, into a sea where no vessel had sailed for seventy-six years since Bellingshausen had named the shore Alexander Land, and the wild neighbouring islands, Alexander Islands. Chances to take navigational sightings were rare, now they had one at 69˚06´S 078˚28´W and everyone reminisced of home, where it was breakfast time, 9989 miles away. Cook, always subtly watchful of the psychological state of his charges, likened it to a starving man finding comfort that, somewhere, food exists. He reflected on the reality: 'we are as hopelessly isolated as if we were on the surface of Mars, and we are plunging deeper into the white antarctic silence.' A decision would soon be taken in secret that would plunge him into the mad heart of that silence, and turn him, for a time, into the head of a small and very exclusive lunatic asylum.
As de Gerlache approaches this decision, his account, the _Voyage of the Belgica_ , is revealing in a way he does not intend. They probe the ice edge trying to edge further south. He recalls Ross forcing his way through the pack and finding open water beyond, discovering the ice shelf, and achieving a new furthest south. He notes that Ross and later Larsen in the sealer _Antarctic_ both escaped the pack 'in a few days'. But they had done it earlier in the season. A week before, the _Belgica_ had three times become surrounded by ice and only escaped, at de Gerlache's own admission, 'with some difficulty'.
There is no doubt he was tempted by the prospect of entering unchartered waters, philosophising that there were two types of sailor, one for whom uncharted means 'Avoid!' and another for which it means 'Come!'. He had three scientists on hand to consult. They all opposed delving into the pack because they had valuable specimens which they wanted to get safely home. That didn't carry any weight with de Gerlache who concluded that 'it was our duty, it seemed to me, at least to try' to force a route deeper into the pack. He chatted to Lecointe on the bridge; they agreed and shook hands. Amundsen was informed but not consulted. Everyone else would find out the hard way. It was 09:00 on 28 February, 'a date that will forever be remembered in the history of the expedition.' They proceeded south under sail with occasional assistance from the engine. The wind, 'already strong' at noon, reached 60mph by 18:00: an inauspicious start. De Gerlache's account doesn't mention one fact that probably trumped any scientific argument: money. They only had £640 remaining, quite inadequate for their plans; returning to a port, and creditors, would end the expedition.
Entering the ice greatly reduces your room to navigate and manoeuvre. When it is tight, you are a prisoner in its grip. They soon saw the perils of trusting to luck. On 3 March a massive tabular iceberg passed one hundred metres away when they were helplessly trapped in dense ice. The sea was also beginning to freeze. The next day the pack was too dense to allow them further south and when they turned north, the leads they had used already had fresh ice closing over them.
Even the familiar became strange. On 4 March they saw an odd light moving low over the horizon and changing shape. Whispers spread through the ship and all hands gathered at the rail. The light grew higher and clearer and began changing shape. When it settled on one shape they saw it was the moon. That day Cook admits they are stuck fast, at 71˚22´S: 'To be caught in the ice is... a life of certain dangers and uncertain rewards.'
The following day the ice thickened; they made no progress. The day after they made some headway until the lead became blocked. The promise of hope was withdrawn almost before they could start to believe. Morale was subtly dented on 7 March when Bébé, the last survivor of three gentoo penguins taken on 9 February, died of convulsions. To lighten the mood, ski practice was organised for 10 March, and a one-mile run was marked out. One diary suggests this was still a novelty for many, the writer feeling the need to explain that skis were 'long flat pieces of wood, tapered and turned up at each end.' It suggests Belgian's Alpine industry was in its infancy.
By 12 March winter had arrived, the thermometer dipping to –18.6°C. The men had been set to cut and break a channel but it was now done with apathy as they felt de Gerlache had intended they should be trapped and did not want to succeed in freeing them from the ice's terrible grasp. The first aurorae on 14 March did little to lift the crew, though Dr Cook was so entranced he decided, without telling anyone, to sleep out on the ice to enjoy them. The temperature fell to –20˚C. He found out his hair had frozen to his hood when he turned round to face away from wind, and tore out a clump. In the grey morning light he was woken by a tapping: penguins were pecking at him. He returned to ship to find Lecointe putting away a rifle. He had been waiting for better light to shoot a 'seal' when it got up and turned into Dr Cook. The physician wrote that Antarctica reminded him of 'a period of the earth in its infancy, long before the advent of man. Everything about it is new yet old; every sight is simple yet clothed in mystery.' He mused that one of the lures of the polar areas was seeing the earth in its youth: a grand extension of an individual's nostalgia for his youth.
De Gerlache was ready with a quote from Nansen: 'The winter is approaching in great strides.' It is, in turn, an echo from Coleridge's _The Rime of the Ancient Mariner_ : 'At one stride comes the night.' A show of trying to escape is maintained. On 16 March 'we again entertained the hope for a few hours that it would be possible to escape the clutches of the ice' when a gale from the ENE sent swells that cracked the ice. A great fissure snaked towards them, they fired the boilers, but the wind soon veered to east and fell. The ice closed. 'Without doubt a winter station had become inevitable.' Lecointe wrote, 'It is certain that we _honestly_ tried to return northwards, but it is also certain that de Gerlache and myself were happy at the failure of our attempt.' His use of _italics_ for honestly undermines every syllable of the sentence.
They were drifting three miles a day to the south-west, and calculated they could easily survive a five-month passage over the pole and out into the Ross Sea. Edgar Allan Poe had anticipated them in 'Manuscript Found in a Bottle': 'a curiosity to penetrate the mystery of these awful regions predominates even over my despair, and will reconcile me to the most hideous aspect of death. It is evident we are hurrying onward to some exciting knowledge – some never-to-be-imparted secret, whose attainment is destruction. Perhaps the current leads us to the Southern Pole itself. It must be confessed that a supposition apparently so wild has every probability in its favour.'
In fact, de Gerlache admitted he had two years' provisions on board, an extraordinary excess for an expedition short on cash. His father Colonel de Gerlache was interviewed by the _Geographical Journal_ when the _Belgica_ did not show up as first planned in Melbourne, Australia. He said his son had three years' provisions on board. It looked like they were trapped accidentally on purpose.
On long expeditions, men always stayed healthier when rations were augmented by fresh local game. They soon realised there were fewer seals and penguins around. Early explorers generally assumed that Antarctica's animals stayed there all year. In fact nearly all of them come only in summer to breed and feed. When they leave, they go quickly. Men like Shackleton, Larsen and Nordenskjöld who were forced by circumstance to overwinter were caught out by this and did not lay in sufficient food to augment their stores.
For the time being, de Gerlache was still caught up in the romance of their predicament. Even his prose bucks up. In a single paragraph of _Voyage of the Belgica_ one can read: 'The future seemed full of menace and mystery. We were about to spend a winter in the pack, and this fact alone promised plenty of data to collect and phenomena to study. Was that not what we had asked for and for which we had been looking?' It wasn't what one group of men had been looking for. None of them wonders how the sailors might take it: the wage-men who had not been told that a three-month voyage would become the first experience of a full Antarctic night, if they ever got out at all. Most of their usual duties would cease. They had no reason to be there, no science to pursue, no motive to get up at all in the months of darkness that would be relieved only by moonlight and the supernatural glow of the aurorae.
Perhaps the gentlemen now took out the expensive, inscribed editions of books gifted by the sponsors for Christmas. Each was chosen specially for that officer or scientist. The bear-like Arctowski received Balzac, de Gerlache had chosen for himself the _Uilenspiegel_ of Charles de Coster, a romance about Ancient Flanders, and the delicate Danco received Flaubert's slender masterpiece _Trois Contes_. With what feelings did Dr Cook take out that masterpiece of American Gothic: Edgar Allan Poe's _Tales of Mystery and Imagination_? When would he feel ready to read 'Manuscript Found in a Bottle', a story of a man rescued from the ocean by a ghost ship, propelled south to disaster with men who cannot see him?
Overwintering initially creates a lot of work, and keeps hands and minds busy. They banked snow against the sides to reduce the heat lost from radiation, and put an awning over part of the deck, using materials intended for accommodation ashore when overwintering. On deck they built a shed finished with waterproof canvas and tar paper. A forge, water distillation plant and changing area were set up. Provisions had to be sorted to leave only frost-proof materials in the hold. The scientists erected observation huts on the pack and maintained a hole in the ice on the starboard side for soundings, fishing and a ready supply of water in case of fire. On 26 March an act of lasting symbolism was carried out; the boiler fires were left to die out. When you live on a powered vessel the background noise of the engine is always with you. When it is turned off, it feels as if the ship's heart has been stilled. They had left seventy tons of coal and forty of anthracite. Most sails were stowed, but the topsails, one staysail and the spinnaker were kept to hand in case a sudden break up of ice left them without power or steerage. To save coal, only blubber was used from now on to fuel their water distiller.
De Gerlache recognised the importance of discipline and a structured day to keep men healthy and sane. He abandoned the ship's watches and decreed everyone's day should consist of eight hours' work, eight hours' sleep and eight hours of leisure. For safety there was a ban on going out of sight of masts. They had noticed that seemingly fixed landmarks like distinctively shaped icebergs could move about, making navigation across the ice treacherous.
Food was vital not just for nutrition but for morale. When there is not much to do, meals give shape to the day and are something to look forward to. Ideally, de Gerlache only manages memorable humour when writing of the cook: 'If the provisions were not wanting, the same could not be said for the cook.' Michotte had been an Algerian missionary 'unwilling to tell me he was not possessed of a genius for cooking... imagination was not lacking, in so far as he was capable of putting together the most disparate of ingredients.' The only thing consistently good was the tomato soup because he could think of nothing to add to it. De Gerlache took charge and drew up a cycle of twenty-eight daily menus but the meals didn't quite arrive as planned. Dinner was seldom the item on the menu, but a mélange of last two days' leftovers. However Michotte's enthusiasm bought forgiveness, especially since no one else thought they could do better.
The menus looked varied but most tinned meats and foods tasted the same, whatever it said on the label. The men first fantasised on what animals might be in the tin and then what they would do to the manufacturers. Tinned meat and fish with cream were the most despised and, top of the hate-list was Fiskabolla: Norwegian tinned fish balls. They were used in bets; the loser had to eat them. There was little fibre in their diet, and de Gerlache obliquely referred to the consequences: 'Our organs of secretion functioned with difficulty.' Sugar and milk were running out, and sorely missed. Living off the land did not enthuse anyone. Dr Cook diagnosed the problem: 'We have tried penguin and cormorants, but the majority have voted them unpalatable.' In fact de Gerlache had another, rather peculiar motive for not serving fresh meat. On 26 March they had shot many seals and caught penguins for food, but they did not appear on de Gerlache's menus. Five weeks later Lecointe asked where they were and de Gerlache replied he was worried what the press back home would say about killing and eating seals.
They concluded the sea was 'too deep' to fish, and penguins and seals were 'more plentiful than tasty.' They learned with relief that numbers of emperor and Adélie penguins would stay in the area all winter, the poor Adélies often tobogganing towards the hunter out of curiosity. Seal and penguin meats were both 'black and hard, very fat and oily but without the least taste of fish, as is commonly supposed.' It was just bad cooking. I have eaten seal which was as tender as fine beef, and richer. Grog was served only on Sunday nights and celebrations; otherwise spirits were banned. Every Sunday a 15cl glass of wine was drawn for each man.
Scientific work could continue. They trawled with filter nets for fine organisms; their catches were the first at this latitude. After each trawl, Racowitza would spend two to three days in the laboratory examining, drawing and describing those specimens which were hard to conserve, and making notes for future watercolours. They found beautifully coloured corals and sponges. Arctowski and Dobrowolski led the meteorological observations, assisted by Lecointe, Amundsen and de Gerlache. Danco, whose health during the coming winter was of special concern to Dr Cook, kept busy making three sets of magnetic observations per day.
Cook began to monitor the men's health. They were fortunate in being able to keep the living quarters around 10°C which meant there was little condensation, the bane of polar accommodation. All the men were encouraged to bathe often, and at least once a week. In May, Cook gave everyone a check including temperature, pulse and weight. That month saw a thaw and their observation hut first settled drunkenly in the ice, then broke off on a floe and had to be recaptured. When it grew cold again head-sized lumps of ice began crashing to deck. By mid-May the sun showed for only a few moments each day. Then 'the Polar night descended on us.' They would live 1600 hours before seeing it again.
In the dim light new distortions taunted them. Scale and distance became confused. De Gerlache himself set off to see why packing cases had been left on the ice some hundred metres away and after a few strides reached what he had seen: a newspaper blown onto the ice. He lamented: 'How much more, therefore did we value the fine clear nights too rare alas! – in spite of their atmosphere of intense melancholy, above the few daily hours of nebulous light.' He considered the ship their only hope of salvation. It was 'the dead star lighting up a dead world.' Being suspended above the ocean made them muse on the vast unknown of the waters beneath and how all creatures perishing at the surface 'fall down into the abyssal depths to feed other mysterious and shadowy beings.' Once the ice silenced the water they learned the subtle new sounds 'indecipherable and almost imperceptible, like the gentle and regular breathing of a world asleep.' Animal compan ions were now few. The seals went to sea, among the birds, only a few snow petrels remained. Of all the flighted birds, it is the true ice bird.
Exercise days were organised. The dress for recreation was Icelandic wool jerseys, a balaclava, a leather cap with ear-flaps like the Norwegian sealers wore, a canvas anorak, and Inuit-style finnesko boots lined with sennegras insulation. In cold weather they added canvas pants or Siberian wolfskin outfits.
Once they were reconciled to overwintering, morale began at a good level. Each night Racowitza posted a cartoon of something topical that day. One revealed that after Arctowski had carefully planted a measuring stick to measure new snowfall, Danco had sneaked out and hammered it deeper into the ground. Danco was still in good spirits and told everyone that all this fresh air would help his chest.
But as the darkness continued, in varying degrees for different people, optimism was steadily eroded. Cook caught the disintegration of daily routine: 'In the first days of our life in the pack, we ate when we were hungry, slept when we were tired, and worked when the spirit moved us. But later we were never hungry, always tired, and the spirit never moved us.' Even de Gerlache, who felt bound to both justify his decision and lead by positive example, reflected that 'for the time being we were no longer navigators, but rather a little colony of convicts serving out time.'
Lecointe thought he was going mad on the morning of 17 May when at 07:00 he report seeing a light, like a lantern being waved as a signal. Everyone congregated half-dressed on deck, peering into the gloom, getting colder. After ten minutes they all saw it, a pale glow slowly waving to and fro. Amundsen was one of the best skiers, and more fully dressed than most. He skied alone across the ice and found a piece of ice had popped out of the pack and was bobbing up and down. Its underside was coated with luminous algae.
One man felt the winter first, the man they all feared for: tall, kindly Emile Danco. As soon as sun went he became short of breath very easily and withdrew from exercise. Normally extremely conscientious, on 20 May he gave up all work. His spirits shone as his health declined. He alone believed he'd survive or he affected to believe; it was nothing, he insisted, and would pass when sun returned. He smiled always and the men used their leisure time to play whist with him.
Up until May, Dr Cook had not been anxious about the general health of his charges, but his fears began to grow.
The truth is, that we are at this moment as tired of each other's company as we are of the cold monotony of the black night and of the unpalatable sameness of the food. Now and then we experience affectionate moody spells and then we try to inspire each other with a sort of superficial effervescence of good cheer, but such moods are short lived. Physically, mentally and perhaps morally, then, we are depressed, and from my past experience in the arctic, I know that this depression will increase with the advance of the night, and far into the increasing dawn of next summer. Physically we are losing strength, though our weight remains nearly the same, with a slight increase in some. All seem puffy about the eyes and ankles, and the muscles, which were hard earlier, are now soft, though not reduced in size. We are pale and the skin is unusually oily. The hair grows rapidly and the skin about the nails has a tendency to creep over them seemingly to protect them from the cold.
There was one widespread symptom which threatened to reduce them to helplessness.
The heart action is failing in force and is decidedly irregular. Indeed, this organ responds to the slightest stimulation in an alarming manner. The observers, going only one hundred yards to the observatories come in almost breathless after their short run. The usual pulse, too, is extremely changeable from day to day, in one case it was yesterday 43, today it is 98, but the man complains of nothing and does his regular work. The sun seems to supply an indescribable something which controls and steadies the heart. In its absence it goes like an engine without a governor.
About half the men complain of headaches and insomnia; many are dizzy and uncomfortable about the head, and others are sleepy at all times, though they sleep nine hours. Acid dyspepsia and frequent gastric discomforts are often mentioned, there are also rheumatic and neuralgic pains, muscular twitchings and an indefinite number of small complaints, but there is but one serious case on hand.
This is Danco:
He has an old heart lesion, a leak of one of the valves, which has been followed by an enlargement of the heart and a thickening of its walls. In ordinary conditions, when there was no need for unusual physical or mental strain, and when liberal fresh food and bright sunshine were at hand, he felt no defect. The hypertrophied muscular tissue is beginning to weaken. If it continues at the present rate it will prove fatal within a month.
At intervals, the ice attacked. From 28 to 31 May they were squeezed tightly in the pack. In a premature rehearsal of Shackleton's later dilemma, when the _Endurance_ sank in the Weddell Sea, they looked at various options should they lose their ship. They proposed to haul the ship's boats over the ice, and sail to the South Shetlands, then on to South America. De Gerlache candidly assessed their chances of success at 1 in 100.
On 5 June Dr Cook came to de Gerlache, who recalled: 'His expression was disordered and his voice trembled as he said, "Commander, today is the day." He meant the day Danco would die. Danco was installed on stateroom sofa and the officers ate in captain's cabin, then later gathered round his bed. He said, "I can breathe lighter and will soon get strength."' They were his last words.
They improvised a memorial. Rysselberghe produced flowers, given to him fresh by his mother as he left Belgium, now withered and dried. They were sewn into Danco's canvas burial bag. He was buried on 7 June on a day so bitter the ice was hard to cut. At 11:00 they followed the sled bearing Danco to the ice hole, and bared their heads. When the service for the burial of the dead was concluded, they tipped the bag into the ring of dark water. The bag did not slide below, but stood upright before sinking, making the sailors recoil in horror. Dr Cook was rightly worried about the impact of kindly Danco's death on a group he thought was already approaching dissolution. He had urged de Gerlache to order the playing of the _Coelophone_ music machine at the funeral, but de Gerlache thought happy music might trivialise the ceremony, and instead sent grog to the men. Cook was right. Death changed the atmosphere; it became profoundly demoralised. It was hard to tell whether men's constitutions were slipping into ruin, or they were hiding in psychosomatic illnesses.
De Gerlache himself was sicker than he would admit. Dr Cook confided to his diary: 'At present I have the captain in the baking treatment.' [Giving patients a dry sauna.] 'He is pale and yellowish, with a feeble almost imperceptible pulse of 100 –140 – his recovery, while hopeful, is uncertain.' Lecointe was in such a poor state he sorted out his affairs and made a will. Cook suspected anaemia was exhibiting itself in confusing symptoms and prescribed a diet to remedy this, including eating lightly cooked seal and penguin meat. It is fortunate that this would also cure scurvy, which is what they were actually suffering from. Danco had been asked to follow this regime and had replied he would rather die. Lecointe's initial response to the doctor's request to serve more fresh meat was to be offended, taking it as criticism of his issuing of the stores. As he got sicker he changed his mind, saying he would sit on the stove for two hours a day and eat penguin for the rest of his life if that would help. He followed the prescription and in two weeks he was back at work.
In these months Cook earned the admiration of all. Amundsen, not hasty to scatter praise, described 'Cook, the calm and imperturbable never losing his temper. He gives advice on everything. He gives it in a likeable and tactful manner; not with fuss and noise.' In his later career, Cook made absurd and soon disproven claims about reaching the North Pole and climbing the USA's highest peak, Mount McKinley, then spent seven years in jail for fraudulent promotion of oil shares. But Amundsen could not be persuaded to change his opinion of the character of the man he considered had, more than anyone else, held the _Belgica_ expedition together. It was 'during these thirteen long months in which almost the certainty of death stared us in the face, that I came to know Dr Cook intimately and to form the affection for him and the gratitude for him which nothing in his later career could ever cause me to alter. He, of all the ship's company, was the one man of unfaltering courage, unfailing hope, endless cheerfulness, and unwearied kindness.'
Van Rysselberghe complained of heart pain, Arctowski and Racowitza suffered persistent stomach pain without diagnosed physical cause. For a while Lecointe was described as 'in danger' and, wrote de Gerlache, 'for some days I too was severely ill.' Cook told de Gerlache it was scurvy, and the symptoms were seldom mistaken. De Gerlache, who had ordered the stores and devised the menus would not have it. He complained that 'seal and penguin literally had to be burned to a cinder to get rid of the excess fat.' Fresh meat would cure scurvy if undercooked, but nearly raw meat was nauseous to most of the men. Cook also prescribed exercise. In poor weather they tramped round the ship on skis, muffled to the eyeballs like the damned.
Compare the image with Poe's 'Manuscript Found in a Bottle': 'It is a long time since I first trod the deck of this terrible ship, and the rays of my destiny are, I think, gathering to a focus. Incomprehensible men! Wrapped up in meditations of a kind which I cannot define, they pass me by unnoticed.'
In good weather they escaped to an iceberg two miles off. Nevertheless Cook was alarmed to observe a high pulse of 130/140 in many men after just thirty minutes walking. More had vertigo while still on the ground. Prolonged intellectual work became impossible, complaints multiplied to embrace insomnia, nightmares and constipation. Cook soon found symptoms of cardiac and cerebral malfunction.
One man on the _Belgica_ went deaf and dumb for days, apparently without physical cause. The Norwegian bosun Adam Tollefsen had been trained up to help prepare animals for the microscope and the specimen jar, but he suddenly went on strike saying it was unworthy of him, and that colleagues were out to kill him. He abandoned his bunk and moved to a corner of between-decks, living like an animal in a den, where he was clandestinely watched for fear of aggression to himself or others.
There were moments when nature provided the relief of beauty. On 21 June their midwinter night was filled with aurorae so bright one could read a book. Finally, on 21 July, a Sunday, they climbed to the top of a berg and saw a long-missed friend: the sun. The few seconds were treasured but fleeting. They all looked bloated and yellow, older and tired, with sad grey hair which they grew long for warmth. De Gerlache thought Racowitza looked like a Turk's Head brush for cleaning ceilings. On 2 August de Gerlache celebrated his thirty-second birthday.
Before any spring ice melts the animals return: the whales and seals. Birds lifted their heavy souls, but the return of the light brought no other relief, and the warmth brought an infestation of rats, which must have boarded when they were bunkering at Punta Arenas. Winters which come early often linger late, and sometimes they do not let go. A second winter would destroy them; that was the unspoken wisdom of the senior officers and Dr Cook. Did Dr Cook re-read his Poe?: 'But I could not help feeling the utter hopelessness of hope itself' and 'my very soul has become a ruin.'
On 8 September they experienced the lowest temperature of the whole trip: –43.1°C. They experimented with the Tonite explosive brought to break ice, and found it useless. The penguins heard the explosions without alarm, but when some imp prompted Van Mirlo to play his cornet to them, they fled in terror. On 20 September they enjoyed a thaw and ice began to fall off the rigging. The leaders' thoughts were not about getting out and going home. The expedition would continue. On 21 September, officers met to discuss the coming summer programme. They planned to break out of the ice and conduct oceanographic studies off Graham Land, then explore the Dirck Gherritz Archipelago (a mythical group). Their key objectives: firstly, not to get iced in again, secondly to connect their surveys to d'Urville's. In anticipation of spring they dismantled their deck awning, but October came and went without the sea ice weakening its grip. November should have brought early summer weather, but blizzards dropped so much snow it was hard to open the doors to go out on deck. By 21 November, almost the whole stern was buried.
They counted that in the year from March 1898 to March 1899 they had 257 days of snow. The midnight sun began on 27 November and thawing progressed. Before the month ended, Amundsen made a discovery about de Gerlache's instructions for the transfer of command in the event of his death. He found the spirit of international co-operation did not extend to the transfer of power, which would pass not to the next most senior officer, but to the next most senior Belgian. Amundsen found both the instruction and its concealment from him unacceptable, and resigned on the spot, though continuing to perform his duties as before.
By mid-December, Cook, Amundsen and Tollefsen, whose mind had rallied, got to a distant berg thirty-five metres high and saw small movements in another berg nearer the open sea. On 20 December, a day short of midsummer, the engineers began preparing the engines, and on 23 December they were able to free the rudder with boiling water. But the sea ice was still an unbreakable eight metres thick. On 31 December, the year's end found them at 70°03´S and 82°20´W. They had drifted thirteen hundred miles in the ice and come back to where they had first been frozen in. They had not drifted deep into the pack where they might never escape. This gave hope that if this area had been ice-free last year, it would thaw again. In the next days a tantalising crack appeared before and behind the _Belgica_. On 7 January, at Dr Cook's urging, the men set to work to saw ice and set explosives. A penguin stood watching the explosions. It had little effect. They tried reversing along the fault risking damage to their rudder and propeller. A returning storm petrel flew over. Did that mean there was open water nearby? They worked in three eight-hour shifts, twenty-four hours a day. A bottle-nosed whale surfaced in their new channel, offering encouragement. Tollefsen suffered a relapse but denied anything was wrong.
On 31 January, after sitting in their lead for three weeks, a crack opened up ahead. They now lay at the centre of a crescent-shaped lead with a lagoon between the two tips. But the ice was teasing them, and the channel astern, which they had laboriously sawn and blasted, began to fill with ice. The officers were nearly resigned to spending another winter in the pack. Cook assessed their prospects, and concluded at least four would not be strong enough to survive it. De Gerlache reduced their meagre butter and sugar rations immediately, and reviewed food stocks to devise how they could be made to last until April 1900. On 1 February the temperature fell and so did the snow. The pack tightened and de Gerlache became fully resigned to another Antarctic night. They also prepared to be able to evacuate if the ship became damaged, going as far as bagging up a spare set of clothes ready for emergency evacuation.
Then, on 4 February, the ship began to gently rock. A sliver of hope crystallised. In the morning all was once again still and there was new ice in the channel thick enough to be walked on. On 11 February it loosened again, and at 17:30 the crack and their channel re-opened. In the early hours of 15 February they slipped the _Belgica_ into the small lagoon. The sea between them and the long-unseen open ocean loosened then closed, but on 20 February a dark line was seen from crow's nest. It was open sea at the edge of pack, and appeared about ten miles off. By early March it was less than a mile away, and surf breaking at the limit of the pack was visible from the deck. It had to break soon, as they were reminded every minute, by the flocks of moulting penguins surrounding them on the ice; summer was nearly over.
On 14 March a new threat appeared. Two large bergs began to move towards the ship in a pincer movement. The sickening prospect loomed of being crushed to pieces when escape was in the fingertips of their grasp. The swell rose and helped break up the ice around ship. At 02:00 de Gerlache ordered the engines to be fired up, and at 04:00 they began to delicately work a passage through the channels, evading the bergs. By noon, they were clear.
They broke out of the ice 335 miles from their position when they had entered it on 2 March 1898. The ice had taken them on a journey of 1700 miles. In their absence from the world, they had missed Marconi's invention of radio. It would, in time, transform exploration of all types, by removing the dread of isolation. They sailed home via Punta Arenas, from where Racowitza, Cook and Arctowski went ahead on steamers to prepare their return. Amundsen stuck by his resignation and refused to accompany them. He booked himself, Tollefsen and two other men who had gone mad onto a regular passenger vessel and escorted them safely home.
As well as being the first people to experience an Antarctic winter, the expedition made the first Antarctic sled journey and added hugely to the systematic knowledge of the Peninsula and its islands. Before de Gerlache, the cartography of the Peninsula had been constructed partly working by speculative theories and partly through reconciling observations which either did not join up or did not tally with each other when they did. De Gerlache redrew the chart of the area. Arctowski added greatly to geological knowledge, zoological specimens were taken from new parts of the ocean, and animal behaviour had been observed over time. They more than doubled the number of plant species recorded from the last continent. Under the direction of the _Belgica_ Commission, the records and samples they collected kept seventy-eight international scientists busy for many years. Frederick Cook's highly readable _Through the First Antarctic Night_ became a best-seller.
# A Permanent Foothold:
William Speirs Bruce (1867–1921)
_A tale of self-effacing scientific achievement, quiet and noble self-sacrifice, and the first confirmed sighting of bagpipes in the south._
In the photo-portrait he gives nothing away. He does not look at the camera, and the drooping lids he inherited from his father half cover his eyes. A heavy beard hides his mouth, though you can see the folds of his cheeks above the corners of the mouth are raised slightly as though he has been asked to look cheery but the effort is all too much. His editor wrote: 'He was seldom seen to smile; life for him was earnest and demanding.' The man is William Speirs Bruce, leader of the Scottish National Antarctic Expedition of 1902 – 04.
Bruce was not part of what would be styled the Heroic Age of polar exploration. Peary, Amundsen, Scott and Shackleton would have chosen whatever form of public endeavour was most likely to bring them advancement and acclaim; in their era, it was being a polar hero. Bruce was a scientist with work to do; he did not care if he came back to crowds and medals so long as he came back with specimens and measurements. He was as self-effacing as that portrait. Only the sleepy eyes deceive. He was a man in whom sense and judgement were energised by great drive and courage.
He read medicine at Edinburgh, where a friendship with William Burn Murdoch persuaded him that science was part of the romantic Scots-Celtic revival being promoted by Professor Patrick Geddes's circle. With Otto Nordenskjöld, Adrien de Gerlache, and Jean-Baptiste Charcot, Bruce went on to pioneer small, agile expeditions that were conceived and led by private individuals, even if they received public funding and government endorsement. Aged thirty-five when he headed his own expedition, he had already been to the Arctic four times and Antarctica once. His first Antarctic voyage had been an imaginative innovation. In 1874 the brothers David and John Gray, whalers from Peterhead, Scotland, had published a pamphlet entitled _A Report on New Whaling Grounds in the Southern Seas_ , including Ross's reports of bountiful whales during his voyages from 1839 – 43. The Grays urged whalers to look south, stressing: 'We think it is established beyond doubt that whales of a species similar to the right or Greenland whale found in northern high latitudes exist in great numbers in the Antarctic seas, and that the establishment of a whale fishery within that area would be attended with successful and profitable results.' In 1892, faced with declining Arctic catches, Robert Kinnes of Dundee sent four steam whalers south: the _Balaena_ under Captain Alexander Fairweather, the _Diana_ , the _Active_ , and the _Polar Star_.
The whalers would transport scientists south to conduct research; in return, the ships would receive doctors on each vessel and superior instruments, including chronometers, compasses and meteorological kit. The captains would assist by allowing their routes to be closely charted, and by stopping as often as was feasible for observations of the weather and magnetics. Bruce was appointed surgeon for _Balaena_ despite having no training in surgery. It's hard to know whether to feel reassured or scared by reports that he embraced his new role with enthusiasm and confidence. The captains were equally nonchalant about the spirit and word of the agreement. On the _Balaena,_ Bruce soon found the old Arctic hand Captain Fairweather interpreted 'stopping whenever feasible' as 'never', but Bruce's choice had been an unsatisfactory voyage or no voyage at all, so he persevered and did what he could. In the Erebus and Terror Gulf, they saw the whaler _Jason,_ and went on board on 24 January to meet the veteran skipper Captain Larsen who would later establish the Grytviken whaling station on South Georgia. 'He is a frank and very pleasant man, and one who seemed to take an enthusiastic interest in everything connected with the expedition.' He would have been a perfect captain for Bruce. From the last week in December to the end of January the Dundee men hunted whales without success.
Publicly Bruce complained diplomatically: 'commerce was the dominating note. A great deal more might have been done for the geology and the biology of these Antarctic regions if some opportunity for landing had been afforded me.' His assistant surgeon Murdoch expressed how they really felt: 'Perhaps it seems incredible that gentle, loving, law-abiding Bruce and the writer, if possible more so, could have lain swinging in our hammocks abiding under an inverted whale-boat watching the blue lace-lined seas slipping past and planning how to tip the old man overboard or slit his throat.'
On 26 December Bruce made himself more miserable by dropping and losing his only bucket for sampling surface water. Unable to land, they sampled rocks recovered from ice or retrieved from penguin stomachs. Even this modest haul was sabotaged by a vindictive sailor heaving some of the biggest specimen boxes overboard. The captains gave up on whales and began sealing for skins and oil; the scientists were made to help with skinning. Donald, on the _Active_ , put his finger on the flaw in the scheme: 'It is useless to trust sealers for exploring purposes, for as long as they can fill their ships with blubber in latitude 64° they will never penetrate to 65°.'
But despite the aggravations the boffins made the first full description of salps, which are tunicates, or sea-squirts; they look and feel like a pulpy yellow sock. The whalers took seals and even penguins for blubber. Penguins proved deceptively difficult to kill. One walked off after being pronounced dead on the reasonable grounds that it had an ice pick through its head. The whalers turned home on 18 February, without whales, but with enough seal oil and skins to cover their costs.
Bruce settled down for a while. He was director of the Ben Nevis meteorological station from 1894 – 96 and, unusually for a British polar explorer, became a competent skier, and the first president of the Scottish Ski Club. He worked as a naturalist on the Jackson-Harmsworth expedition of 1896 – 97 to Franz-Josef Land in the Arctic. In 1900 he was eager to go south once more. Bruce had unwittingly offended Sir Clements Markham, easily done, the influential President of the RGS, when, after the Jackson-Harmsworth Arctic Expedition, Bruce had presented his paper first to the Royal Scottish Geographical Society, not the London RGS. Markham was now putting together the team for what became the _Discovery_ Expedition of 1901– 04. With a predilection for handsome young naval officers, he wanted his protégé Scott to lead the team, and Bruce was dangerous in having the qualifications and experience Scott lacked. Markham first excluded Bruce from the _Discovery_ team then, when he proposed a Scottish venture, challenged him in writing to dissociate himself from any rival venture, an impertinence bordering on the delusional. Bruce made peace by undertaking to stay clear of the _Discovery_ _'s_ route and not to fund-raise outside Scotland, which hampered him severely.
While Bruce was driven by science, there was a broad streak of jingoism running through his project, and it was not British, but Scottish. So although Bruce contributed fewer than twenty lines as a preface to his colleagues' popular account of the voyage, he found space to say: 'While "Science" was the talisman of the Expedition, "Scotland" was emblazoned on its flag.' First they needed a ship. Late in 1901 Bruce went to Norway to purchase the four hundred-ton whaler _Hekla_ and bring her to Troon, West Scotland for a refit under yacht and naval architect G. L. Watson who worked without fee. Designed for polar waters, she was barque-rigged with an auxiliary steam-powered screw; 140 feet long, she drew 15 feet. The refit was so thorough that 'very little of the old _Hekla_ was left.' She now had a nine-foot depth of timber in the bow, and her main hull was twenty-five inches thick and sheathed with greenheart. She was renamed _Scotia_ , the Roman name for Scotland.
The refurbished quarters provided a well-lit deck-house for scientific work, comfortably warm next to the galley, and a second lab beneath it. Her stores and equipment included a thousand gallons of methylated spirits for preserving samples, and 36,000 feet of cable for trawling. They were the first expedition to record sound, on a phonograph, and to use a cine camera, although it jammed most times it was used.
Their captain was Thomas Robertson from the _Active_ , who had been one of the more helpful skippers on the 1892 – 93 Antarctic trip. The staff included Robert Rudmose Brown, a botanist who had worked at Kew, J. H. Harvey Pirie who came as geologist, bacteriologist and Medical Officer, while R. C. Mossman was both meteorologist and magnetist. Those three would write the popular account of the trip _The Voyage of the Scotia_. Assisting Bruce was Wilton, a second zoologist, who had lived in north Russia and was an experienced skier and sledder. Their major sponsors were textile tycoons, the Coats brothers, James, Andrew and Thomas, who donated £30,000 of the trip's slim budget of £35,000. The next largest donation was £250.
Like most expeditions, the last days were chaotic. When they sailed on 2 November 1902 'the ship was like a midshipman's bag, everything on top and nothing handy.' Pirie wrote: 'we got away fairly quietly, – a proceeding much to our taste.' One reason was that it was Sunday, prompting the local newspaper to muse: 'Stands Scotland where she did, when a ship can sail on the Sabbath with pipes playing and people singing, not psalms, but profane songs, such as "Auld Lang's Syne".'
Pirie is one of the livelier chroniclers and is soon putting on his medical officer's hat and defining the second stage of seasickness: it was characterised by deep fear, the fear that ship is never going to sink and put you out of your misery. En route they measured the sea temperature and salinity, vital to track currents. The first sight of southern land was the bare slopes of the Falklands. They leaned on the rails, warmed by their Fair Isle woollen knitwear. Some of their socks were a macabre but very warm mix of goat and human hair. Workmen were still finishing the squat tower of Stanley cathedral, but the city immediately endeared itself to them; the smell of burning peat from the chimneys was the scent of rural Scotland.
Continuing south, on 28 January they saw their first tabular icebergs: the spectacular calvings from the great ice-shelves. On 2 February 1903, at 01:00 they approached the pack; at 60°28´S 043°40´W they had met it much further north than d'Urville (63°30´S) or Ross (65°S). The wind and swell were driving directly onto the ice edge throwing up surf and agitating the floes deep into the pack. Men who had not sailed in ice before, including the ship's carpenter, James Rice, who had worked on the refit, could not believe a ship could survive being driven into that fury. He went to the bow to watch. Twelve miles in they laid up snug in the ice on a fine calm night, and the contented carpenter went below.
The next day the South Orkneys were visible about twenty-five miles off to the WSW, but they had to run north to escape dense pack ice closing in on them. On 4 February they tried again and made their first landing, getting ashore at Saddle Island, only the third ever visit there after its discovery by the sealer John Powell in the _Dove_ in 1821, and then d'Urville in 1838. They landed on the north side, and took penguins, including two live ones, and eggs, as both specimens and food. The penguins were loosed on deck to be tormented by men and dogs playing with them, but soon proved smarter than they looked by escaping overboard.
The season was growing late, and they were anxious to push south, leaving Laurie Island at noon to steam south west; by 21:00 they were laid to, in heavy ice. Next day it was so dense at 07:00 that further progress would only be possible by burning precious coal. Captain Robertson turned north-west, the only direction in which there was still loose ice; even so, some of it was fifteen to twenty feet thick. They followed the pack roughly along the line of latitude 60°S but by mid-month they had inched their way back into the pack as far as 62°S and now sailed between the wakes of Ross and Weddell.
On 18 February they crossed the Antarctic Circle in open sea; at times no ice was visible. It was eighty years ago to the day that a little west of there James Weddell had made his famous farthest south in an ice-less sea, at 74°15´S. They hoped to go beyond Weddell's farthest south by Sunday, but progress immediately slowed: 'We made little southing next day, owing to the increasing tightness of the pack, which Captain Robertson did not consider expedient to negotiate.' It was five years since de Gerlache had been trapped, and it was at the forefront of their minds: 'Any mistake now would finish our work for the season, and we had no wish to repeat the experiences of the _Belgica_ and drift helplessly about all winter, frozen up in a sea of shifting ice.'
Reading early accounts it is interesting what attracts their interest. Those who journey to the interior soon quit the ocean and wildlife, so accounts of attempts on the pole are all about snow and ice conditions, weather, navigation, logistics and the health of men and animals. For them, sea voyages are just the parentheses around the real business of the trip. But _The Voyage of the Scotia_ is written by three ship-based scientists and they are very good at describing the sea ice that controls their movement and safety. Because they observe closely, the writing is vivid. The following passage tells you sea-water freezes in a way you would never imagine:
The formation of 'young ice,' as met with in the polar seas, is an interesting process. The first stage, which takes place when the cold is very intense, is the ascent of large clouds of vapour from the sea, so dense it looks as if a veil had been spread over the surface of the ocean. If there are spaces of water and portions covered with ice, the water spaces look like boiling cauldrons, so dense are the masses of vapour that ascend.... Soon a thickening is observed on the surface of the water, and threads like the web of a spider radiate in all directions.... The salt-water ice is now a pasty mass, so elastic that every movement of the water on which it floats is clearly discernible. The ice, however, is quite different from that formed on fresh water; for although it becomes tougher with very intense cold, it will not support a man with safety until after thirty or thirty-six hours. Even after twenty-four hours, with a temperature of 40° below zero, the new ice is still so soft that, in spite of its thickness, a stick can still be easily pushed through it. After three days it is still in no way brittle, and bends under the weight of a man without breaking. The impression it gives is as if one were walking on well-stretched leather; and this characteristic it retains for a considerable time. After a fortnight, when over a foot and a half thick, the young ice looks as if the water had been surprised by the cold and every wave turned suddenly into ice.
Antarctica's rare fine days are sweet. The afternoon of 21 February brought the cheeriest conditions since Stanley: bright, clear and calm. They passed 70°S, a feat achieved in this sector only by Weddell and Ross, and by no one for sixty years. The sea was a blue sheet mirroring the sky until their wake creased the reflections and rocked the ice; the ripples softly shushed along the ice edge, and the plates of sea ice grated against each other. Sometimes a crack would snake across a large floe watched by the permanently alarmed white-ringed eyes of the Adélies. Today there were many emperors, stately and slow. Mossman tore himself away to go back to work; good weather or bad, it was time to take another barometer reading. Bracing his eyes for the sudden blackness inside, he entered the deck laboratory and found he could see perfectly well because the stove was one sheet of flame, and in a second remembered he was surrounded by bottles of alcohol and benzene, and in the deck immediately below, were a thousand gallons of methylated spirits. Someone, the account is generous in leaving them unnamed, had put a pan of sealing wax to melt on the stove, left the lab, and forgotten all about it. As Mossman entered, the pan had just burnt through and caught fire. He put it out; two minutes more, and it might have been uncontrollable. They narrowly avoided climbing into open lifeboats, warmed only by the blaze of their sinking ship.
Normal weather returned, bringing with it tricks and mirages. The next day they met ice, newly formed after the coldest February frost then recorded in the Weddell Sea, and only breakable under steam. At 03:00 there was a bank of fog in front of which was a thin bluish vapour. Here and there through the fog could be seen castellated towers and battlements rising above it. At 08:00 on 22 February they reached their farthest south at 70°25´S and 017°12´W surrounded by new ice cementing together the old. The captain decided to retreat north, then go east and try to probe south again where Ross and Weddell had reported meeting freer seas. The ice had other ideas. They made only half a degree north, where they found the line of least resistance lay to the west. Deferring to Nature, they followed.
Sometimes bad weather elsewhere helped them: a storm to the north brought in strong swells which loosened the pack on 27 February, but the next day the ice repeatedly threatened to grip them. On 1 March they crossed Weddell's path again, but he had been in open sea and they were clamped in tight pack. Once you enter March on the Peninsula you are on borrowed time. Bruce had to decide where to winter. Seas were being kept open only by ocean swell, and would soon freeze over; the nights were growing long and dark. The Chief decided to winter the ship at the South Orkneys, where it was expected that the spring ice melt would arrive in October. There would be no de Gerlache moment, succumbing to the temptation of a final push, no impulsive gamble. On 8 March the _Scotia_ was shaken to her keel as a following gale drove her hard onto ice, but the old whaler's hull showed its mettle, emerging undamaged. In three more days they crossed the Antarctic Circle going north. The nights were dark once again, so unless there was moonlight, dense ice forced them to find a pool of open water each night to spend the night. On 20 March they reached open sea but any celebrations were brief since, in the shelter of the ice, most men had lost their sea-legs. By 20:00 all the officers and scientists except the captain and Bruce were seasick. They got little rest. By 23:00 a gale had blown up, and all hands were called to deck when icebergs were seen. Early next day they spied the South Orkneys, near Cape Dundas at the east end of the group. The islands strike a wild profile with sharp peaks rising straight from the sea. The wind carves the snow into grand sweeps, and the surrounding reefs and rocks snare massive oceanic bergs. What followed shows how Antarctica can make you suffer very hard for small gains.
The southern shores were ice-choked, so in a 'tremendous swell' they looked for a place known to them only from Powell's chart and a passing glimpse on their way south: Spence Harbour. On 22 March they steamed past Saddle Island and at 09:00 suffered heavy squalls from the land which blinded everyone on deck with sharp snow. Round noon, the wind and sea fell, but visibility was bad, the lookouts suddenly shouting 'Full astern!' to avoid bergs looming out of the murk. Preoccupied by bergs, they narrowly missed rocks that appeared out of the snow-choked air just yards from the ship. At dusk, frustrated at being so close to shore but not being able to land and feel earth under their feet, they were forced to move out to sea.
Next day they saw Spence Harbour in good weather for the first time and felt misled enough by previous accounts to call it a 'fraud.' It was very deep, which is bad because it requires a lot of anchor chain to be let out, allowing the ship to make large swings when wind and tide change. It was also exposed. A second choice, Ellison Harbour, proved too small, and Bruce and his captain began to feel their charts were not to be trusted. Not wanting to denigrate the two fine navigators who had preceded them, they speculated whether, in the eighty years since Powell and Weddell had mapped the Orkneys, receding glaciers might have altered the shape and size of harbours, making them less suitable. Still unable to land, but still uncomplaining, they put out to sea at dusk, where snow squalls and poor visibility made life 'unpleasant': a very Scottish understatement. On 24 March around 18:00, the wind moved to the west and built to a full gale, but they enjoyed good protection in the lee of the island, and 'but for the proximity of bergs [we] would have felt in clover.' Then, at 04:15, they were woken by a horrible noise: they'd drifted out beyond the lee and into ice which had torn the rudder until it hung by one pintle. They fixed it up as best they could, which wasn't much. The jury rig reduced their manoeuvrability, and hence their ability to miss big bergs and meet the wind and waves on their best quarter. They suffered a beating until they made open water at 10:00.
On 25 March they thought it had improved enough to close in on the land again which they did by 13:00, finding, south of Laurie Island, a good anchorage in sixty feet. Four days had passed between arriving and landing, but Bruce's account has no word of complaint or self-pity. It still bears their name: Scotia Bay. At a little over 60°S, it seemed a kind latitude to overwinter, equivalent to Bergen in Norway or Anchorage, Alaska. Their shortest winter day would still have five hours of light. Bruce's would be the first study of the islands. Because the group was on the border of the Antarctic and sub-Antarctic regions they anticipated interesting and new results in biology and meteorology. The beach on which they started to make camp is the only extensive area of flat land. Although extensive, it is exposed: a storm beach 350 metres wide connecting two islands, and making a bay each side. In a flash of inspiration they named it The Beach. What the sea builds, it can also destroy.
Brown noted that Weddell had referred to the island's 'terrific appearance', meaning terrifying, but as Brown sensibly observed, Weddell might have been influenced by the poor weather he endured. In sun, Brown wrote, 'I can imagine nothing more beautiful', but gloom made it 'a weary and depressing landscape', 'one that made me often long for some colour – any colour – to rest my eyes on; then the black hull of the ship was a delightful relief, or better still, the bright red of Copeland Observatory.' Brown looked in the dusk light at the deep violets of the ice-floe: 'There is, I feel sure, no region of the world more grand in its scenery than the Antarctic, and no place more transcendent in its beauty. Its very vastness, no less than its beauty, while it quickens the traveller's daily wonder and deepens his reverence, forces him to feel that it is a world that he can never conquer, a world in which the forces of nature are too tremendous to overcome, and which must resignedly be bowed before in the hope that they will suffer him to come and pass again unscathed.' It is a return to an earlier world where nature was feared and had to be propitiated. He declines to use the military language of conquest, and talks of reverence and respect: a modern sensibility.
Reflecting on their winter realm, they were pleased it belonged to no country, and they were its only human inhabitants. Less idealistically, Second Officer Fitchie thought it would make a fine penal settlement with convicts house building in summer and shovelling snow off glaciers in winter. Brown agreed: 'It is an excellent plan, with much to recommend it.'
The _Scotia_ was anchored a quarter mile off, her stern to shore. The bay was relatively shallow: good, because large bergs could not be driven in. They had expected to take the ship out of the harbour, now called Scotia Bay, to trawl for specimens, but three days after their arrival, the ice blown into the bay froze together. Soon snow papered over the cracks between the ice, and for a week they fell through the gaps when relaying stores and equipment ashore. The first week in April saw them securely frozen in, and the ship was prepared for overwintering. Down came the topgallant yards, the furnaces were allowed to go cold, and the boiler emptied. The sailors banked snow to the _Scotia's_ sides and on the decks as insulation to limit the damage severe frosts inflict on the surface of the wood. They cut a fire-hole in ice to maintain a supply of water in emergencies.
As they tramped from ship to shore it was hard to see where the land actually began. The tidal range is about six feet and they took the last of several fissures parallel to the shore to be the high tide mark. They chose a site for their hut a little above it, and fifty yards from the cliffs running inland from the left hand side of the beach, as you come ashore. Although the ship was overwintering, they immediately began constructing shore accommodation for when the _Scotia_ returned to the Falklands for repairs in the spring. It would also provide a base to continue work in future years after the expedition, an aim already central to Bruce's plans for continuing scientific study. Britain had declined to take it over (Markham's malevolent finger might have been in that pie) but Argentina had expressed interest. They had a prefabricated wooden hut, which they could surround with stone walls. Ironically, they had assumed they would winter in Antarctica proper where the weather would not allow them to build in stone, so they had no masonry tools or cement. As a result, they could only build clumsy dry stone walls five feet thick, buttressed at the corners.
A yard-arm sawn in half formed the gable pole, from which they suspended a roof of doubled canvas with insulating felt between, and the outside coated in oil and grease. The interior was canvas-lined, and the captain sacrificed his tween-deck hatches for flooring. Inside it was fourteen feet square, with six to eight feet headroom. When the ship's spare cooking range was installed, to the left of the door, it was snug and warm in all but the fiercest blizzards. To the west, they built a coal store and store-room whose walls were made of boxes of ship's biscuits, many sailors over the centuries having noted their masonry-like qualities. It was roofed with their thirty-year-old Arctic whaleboat. On the oak door lintel salvaged from the whaleboat, Brown carved Professor Geddes's educational motto _vivendo discimus_ (by living we learn), and they christened it Omond House after R. T. Omond, the meteorologist who promoted the Ben Nevis observatory at which Bruce had learned his cold weather meteorology, and who had been a major supporter of the expedition.
They soon found their choice of site had been both unlucky and careless. The unlucky part was that snow drifted deeply in this corner of The Beach. A hard winter's shovelling lay ahead. Spring would show how they had been careless. Building continued with a cairn for surveying and a magnetic hut beyond that, located clear of distorting iron and jostling humans. This Copeland Observatory was named for Professor Ralph Copeland, an Edinburgh astronomer and a supporter. Next to the ship they preserved two holes in the ice and maintained a six hundred-foot whale line in the water between them to pull a trawl through. This raised _isopods_ , the group of _crustacea_ which includes woodlice, and they found a new ten-legged kind of _pycnogon_ (sea-spider). They employed patent traps like lobster pots to tempt bottom-living creatures to donate themselves to science. When the skeletons of larger specimens needed a final clean, they lowered the bones in cages to the sea floor where the _amphipods_ (shrimp-like crustacea) nibbled them spotlessly clean in a week. They would discover that the seas were startlingly rich and diverse. A BAS survey of the South Orkneys carried out in 2008 recorded 1224 species, more than in the Galapagos islands.
Thousands of penguins walked through The Beach, mostly gentoos and Adélies, which they took for food and specimens. Although they professed themselves uncomfortable at killing animals with no fear of man, they often saved ammunition by knifing seals to death, a brutal business which Brown observed with awed insensitivity. 'I have seen a Weddell seal, with a six-inch blade and the hilt of the knife as well, buried right in its side, live for forty minutes.'
Although the islands were largely volcanic, Pirie found fossils in a small deposit of sedimentary rocks near Cape Dundas. There was sufficient work to keep everyone busy; sometimes they were even competing for labour. The solitude of the night watch, when it came round on the rota, could be a welcome refuge from the daily bustle. In their leisure time they had use of a good library including the account of the _Belgica's_ forced overwintering, which it was wise to save for an optimistic mood. There were few men with musical talent, their harmonium had not sounded the same since it had been deluged by a wave, and only Chief Engineer Ramsay played it properly. There were fifty records, and a major attraction was simply hearing a voice that was from the outside world. Talk often gravitated to fantasies of their favourite foods, feasts they would eat when they got home. The cook Florence made a hit with seal heart stuffed with sage and onion, and mastered penguin cooking, browning then stewing the breasts to stop it being too fatty. Adélies were judged the best; two birds' breasts would feed twelve to fifteen men.
In September, still winter, they grew impatient, and tried to saw and blast their way free, but the weather turned colder and they abandoned their efforts. On 1 November, with the impending departure of the _Scotia_ in mind, they transferred the meteorological observations from ship to shore. Their results quantified the gloom: the mean cloud cover was 82% and there were 180 sunless days a year. They were growing tired; on a grey day, thought Brown, snow was a blessing 'since it obscures the monotony of a wider view.'
Spring spirits were lifted by preparations for sled journeys to survey farther afield. The first six penguins to arrive made a mistake as the men were pining for fresh meat. When the penguins began to lay eggs each man ate an average of fifteen a day. This was not science with a light ecological touch. When the weather was slow to improve, the whalers blamed it on a 'jinker', an unlucky presence. A mannequin of the person believed responsible was made and revealed in a dramatic flourish: Brown was obliged to burn an effigy of himself. The weather remained cold. Winds are often more important than temperature in loosening the pack, and over the days of 22–23 November, north and north east winds broke the pack in twenty-four hours. They were itching to get away; the _Scotia_ sailed on 27 November.
The South Orkney party's agenda was prompted by recalling their dangerous search for a harbour, which had shown the inaccuracy of the charts. Sled journeys would now allow accurate surveying from a stable platform. They didn't have many dogs; there were two Siberian Samoyeds and six collies. In the winter, one Samoyed and three collies died from accidents, and one collie of old age. Pirie, Wilton and able seaman William Martin made an eight-day trial trip to Delta Island in the mouth of the bay. They ate a light lunch on the ice, but waited to get back to the tents for the highlight of the day: a mug of tea. 'How the thought of that hot tea kept us going all day! The recollection of it is the strongest I have of our camping-out experiences, – how both hands having clasped the cup so as not to lose any heat, the warm glow gradually spread and spread, till at last even the toes felt warm ere the cup was drained. Truly it was a cup that cheered.'
On Midwinter's Day the sun could no longer be seen from the ship, and just forty-five minutes was enjoyed on the highest part of the beach. They broke open one of the barrels of porter donated by Guinness. The first sailors to sample it quickly became drunk, then it ran out far too soon for the size of the barrel. They found all the water in the drink had frozen and only the alcohol remained liquid.
The three authors of the official account are silent on one man's suffering, until the consequences force them to mention him. Alan Ramsay, the Chief Engineer, had been ill with heart disease since March; on 27 June Bruce records that 'he is not at all well.' A month later he suffered an evening where he could scarcely breathe. Bruce charts his decline with tender sensitivity. _Mitral stenosis_ was diagnosed: a malfunction of a heart valve, reducing blood supply to the body. He had felt twinges in his chest before they reached Stanley but said nothing because he realised they would find it very hard to find an engineer to replace him. He did not want to let anyone down. On 6 August Bruce recorded Ramsay had a bad night.
He thanked me for my kindness to him, and asked me to stay with him to the last. I held his head for several hours, as he wanted me to – for which he was most grateful. He had many terrible paroxysms of pain, in each of which he prayed and hoped to die. Ramsay held out his hand feebly, and said, 'Good-bye, I'm dying,' and in another two minutes or less all was over. Pirie had his hand on his heart, and 'Hands off' were the last words he said. He never said much but he always did his duty. This morning he opened up his heart to me more than he ever did before: 'I came here with the intention of doing my best, and making a name out of this job.' He said to me 'I've not been very good, but I haven't led a bad life.'
Two days later he was buried away from Omond House in Uruguay Bay. 'We placed Ramsay's head to the southward, his face looking to the sun, northward and homeward.' Three Adélie penguins came to them and stood with them by the grave. There was little they could do to raise a monument, but they collected boat-loads of green moss to lay on Ramsay's grave, signifying the new life of the Resurrection. His grave is still on the beach, looking north with other servants of the base. The doctor, Pirie, wrote, 'That scene we can never forget: one of those perfect days we sometimes had – crisp, clear, and cold, but absolutely calm; the melancholy little procession to the shore, headed by the piper playing _The Flowers of the Forest_ and _The Old Hundredth_. He gave his life for others, and gave it uncomplainingly. There are those, I know, who envy him his last resting place beneath the shadow of the ice-capped hill that is named after him, where throughout the ages the seabirds wheel in their restless flight, and the waves crash upon the shore save when frost holds them in its mighty grip, and there is stillness deep as death.' They named the most imposing peak Mount Ramsay.
They mounted a second expedition to the west. It was hard progress, covering six miles in three hours. Wilton said: 'If this is Science, she's a hard mistress; give me Art.' They were rewarded with fossils: Silurian _graptolites_ (an extinct colonial form of plankton) and a crustacean. The geology prompted them to guess, quite correctly, that the South Orkneys were part of the Andean chain, and they even asked why the Drake Passage had opened up across it: a question which would, in time, be answered by the theory of plate tectonics. Wilton and Martin became snow-blind after two weeks of bad weather. Cocaine eye drops eased the pain. Pirie, still unaffected, climbed a nunatak from which he was able to survey a wide area. Then he decided to walk Wilton back and on the way he lost sight in one of his own eyes for several days. It didn't blind him to the beauty: 'But what a delightful sensation it was to breast such a hill with the feeling that you were the first human being whose foot had trodden its summit, and whose vision had drunk in the beauty of the scene that lay spread out below you!'
In August ice began to break up around the ship, and Bruce wrote: 'I have been discussing next year's programme with the captain. I have been emphasising the importance of the oceanographical side of the work. He is keen for a high latitude; so am I – _provided always it is compatible with good oceanographic results_. I will not satisfy oceanography or other scientific research for the sake of getting one degree – or mile – further than somebody else.'
On the last day of the month they saw Weddell seals which had come ashore to pup. They killed two pups, one in its second day of life, the other five days old. 'They looked very beautiful on the sledge with their great dark eyes and their soft, long, furry coats. They were bleating like lambs. The smaller one had almost a human infant's cry.'
On 1 September, they fired up the engine. But the cold returned, and the sea re-froze. One pup was kept alive. It would suck everything except the milk bottle. They only succeeded when they sucked the milk into a football bladder and used a tube to expel the milk into its stomach. They used the phonograph to record the seal's cries, and Kerr playing the bagpipes. The seal died of convulsions later that day.
It wasn't until 27 November that the ice allowed the _Scotia_ to return north. Four days later Bruce took a break from his usual post monitoring the fathom-counter on the derrick. One minute later the derrick broke and smashed through the ship's rail right where he had been standing. The _Scotia_ safely reached Stanley in the Falklands where Lt. Marriott from HMS _Beagle_ came over to tell them that Otto Nordenskjöld (see next chapter) had not been seen for thirteen months, but Argentine and Swedish ships were looking for them. It hammered home how lucky anyone was to return home from Antarctica. The _Beagle's_ officers asked what culinary treat they would like after their dietary deprivations; they chose beer and any lunch with fresh potatoes. Bruce hurried ahead to Buenos Aires on the SS _Orissa_. They saw the ship _Français_ heading south, but did not know it was Charcot (chapter ten) or his purpose.
Ashore they met the head of the Argentine Meteorological Office, Walter Davies, who agreed to take over Omond House if the _Scotia_ would take down their men and stores. They met the _Uruguay_ which had just rescued Nordenskjöld and the _Frithjof_ , the Swedish official rescue ship begrudgingly sent down too late to help. On 21 January they left Buenos Aires with new passengers on board: naturalist Luciano Valette, and Hugo Acuña and the German-Argentine Edgar Szmula as meteorological assistants. Calling at the Falkland Islands, the Scots felt more homesick than last time; with each visit it looked more like the west of Scotland. At the South Orkneys, they quickly dropped off the land party at Laurie and sailed on 21 February. The waiting shore party said it was the first day in five weeks when it would have been safe to land a boat.
The Scots party that had remained on South Orkney for the rest of the summer was commanded by Mossman, supported by Pirie, Cuthbertson, Ross, Martin, an Able Seaman trained up as a meteorologist, and Smith, the cook. Their orders were to make hourly meteorological observations and conduct as much other science as they could. It was a comfortable base. Initially there was enough water to be able to take regular baths. They had a year's food supplies, not counting penguins. This was not as long as the supply of yarns held by Bill Smith, the second steward, who filled the evenings with so many reminiscences the others calculated he was at least 143 years old.
When the bay ice broke up it revealed an unpleasant surprise. The crack they had assumed was the shore line was in fact some way out to sea; they had built the hut just yards above high tide. The shore ice now ended in a fifteen-foot ice wall into which they had to cut a ramp for the boat. They took the first Antarctic macaroni penguin specimens to make it home, and the first ever eggs of pintado petrels, a single pure white egg, rather large for the bird. The naturalists found that the flying sea birds produced more heat and maintained a body temperature similar to temperate birds: 108-110°F, while penguins, despite having up to one inch of blubber, maintained 103-105°F. Other observations reveal dubious attitudes to wildlife: the leopard seal 'cannot be played with in the same way as Weddell seals, on whose backs we used to ride... but is best approached behind a loaded rifle.'
By January the men were eagerly looking forward to the return of the _Scotia_. They ran a sweepstake on the likely date; the loss of Nordenskjöld's _Antarctic_ was fresh in memory. As January drew to a close they began, like de Gerlache in eerie anticipation of Shackleton, to consider their options: 'our chances of making Cape Horn or the Falklands, 800 miles away and against contrary winds, were small enough.' February came, the ship didn't. They began surveying their food stores, and calculated they would have to go onto reduced rations by 15 February. They practised repairing their boots with sealskin. Pirie, mindful of the long slow nights, started to make a chess set. On 14 February, just before the last full rations lunch was served, the ship came into the north bay. They waved and returned to eat lunch, suddenly self-conscious about their anxiety, wishing to keep it to themselves.
Ship-based work resumed. One day, sounding with a 60lb weight when surrounded by sperm whales, the lead paused at 6000 feet then descended to 15780 feet. They could think of no other reason except it had landed on a whale's back, over a mile below, but they could scarcely believe it. It was the first indication of how deep that species dives for its prey. The summer of 1903–04 was one of low ice. Under canvas, the _Scotia_ sailed in open water past the latitude where the pack had stopped them the previous year. They regretted that the lateness of the season stopped them exploring as far as they might. They met dense pack at 72°18´S and 017°59´W and sounded 6786 feet instead of the usual 15,000 feet. The masts were surrounded by large numbers of birds, as Weddell had also recorded in this area. The skipper went straight to the crow's nest and saw the edge of a large ice-shelf and, he believed, land. It was 3 March, and they named it Coats Land after their principal sponsors. Land had not been expected here. James Clark Ross had sounded 24,000 feet and found no bottom, causing Sir John Murray, back home, to calculate that the coast was four hundred miles farther south. Bruce proved the Weddell Sea was smaller and shallower than previously thought. It was their first sight of the ice barrier, the near-permanent cliffs of multi-year glacier ice forming shelves fast to the shore. The _Scotia_ battled to within two miles of it then followed the face for 150 miles. Many explorers might have logged a sense of excitement and wonder, laced with speculation. Bruce's reaction is to make a sounding of the bottom and record: 'I had two water Buchanan-Richard bottles and thermometers, and attached one for a bottom sample. A screw of one bottle broke and I had to get another.' Bruce can be safely read without your cardiologist standing by in case of intolerable excitement.
Their dredges lifted glacial boulders of granite, gneiss, schists, quartzite, sandstone, limestone and slate: all typical of continental land. Bruce correctly guessed it would join up with Enderby Land seen farther east by the sealer John Biscoe in 1831. This deduction was one of the most important geographical discoveries of the expedition, since it was evidence of continuous and extensive continental land. It also explained why Cook and others were consistently obstructed in this area by heavy ice; the continental land mass was drained by huge glaciers despatching vast fleets of bergs northwards.
As Shackleton would discover to his cost, this is a sea where currents press the ice against the shore. It was soon piling up as high as their deck. The ship was pinched by the floes and lifted four feet: disconcerting, but it relieved the pressure. Two days of gales drove them into a bight in the ice barrier. When it eased they were at 74°01´S and 022°00´W. It would be their farthest south, four degrees beyond the previous season, and fourteen miles past Weddell.
As they listened to the grinding and groaning of the pack, the spectre of overwintering haunted them. Three days later it looked worse. The _Scotia_ was lifted up by the lateral pressure of the ice, there was no open water to the north, and the pack was freezing hard. Captain Robertson brought down the topgallant spars to reduce the wind stress on her top gear. Bruce ordered an inventory of their stores, and reduced the use of lamps to a minimum. To maintain spirits they played football on the ice. After the gale, the weather turned clear, crisp and calm, and they experienced the 'awful silence' of the barrier.
The zoologists captured emperor penguins weighing up to 80lbs. They had now concluded strangling was the best way to kill penguins, as it minimised the risk of getting blood on the feathers, and was preferable to hydrocyanic acid, which stopped the blood coagulating and made skinning messy. One emperor featured in the cruellest picture in Antarctic history. OS Gilbert Kerr dressed himself in full highland rig and played the bagpipes to it. The bird displayed 'no excitement, no sign of appreciation or disapproval, only sleepy indifference.' In the photo (there's a copy near the dartboard in the bar of Vernadsky Base) the bird stands six feet away looking happy enough, until you look carefully and see there is a line tethering it to Kerr's foot. Some of the emperors were intended for Cape Town zoo. Like many birds, they need a very precise stimulus to feed, and would not eat fish from a bucket, though they would swallow them if hand-fed. For unstated but bizarre reasons the crew did not persevere with this although fresh fish were brought up with each trawl. They tried giving them preserved fish, but the emperors grew thin and died. The sheathbills they tamed should have been easier to look after, being scavengers, but they were killed by their own curiosity; one drowned in a barrel of tar and another suffered death by inebriation in methylated spirits. One live sheathbill was accidentally washed clean in a solution of 50% alcohol, an error highlighted after it preened its feathers and fell over drunk. It survived both the alcohol and the hangover.
On 12 March a south-west wind eased the pressure of the ice, and at 20:30 the ship slipped back down into the water. The spectre of overwintering receded. Said Bruce, 'never during the whole voyage had I seen everybody in such exuberant spirits.' At 16:00 leads opened to the north-east, and they tried to free the vessel with Tonite explosive, by hacking with poles, and jumping up and down. On 13 March they took risks to drive the ship through the pack, and the ice took a bite out of the propeller. Next day they had escaped into brash ice with fresh pancake ice forming on the water. The wind changed to the north. In such conditions the engines were vital in escaping up wind. They decided it was too dangerous to re-enter the pack that season and headed for the 'Ross Deep'. On 18 March they made their richest trawl, lifting many deep-sea fish. They noted that many of them were coloured with reds and purples, and speculated on why creatures living beyond sunlight still made use of colour.
On 23 March they arrived at 68°32´S and 012°49´W over the Ross Deep where Ross had found '4000 fathoms no bottom.' The winch began lowering the 60lb weight and they were surprised when it hit mud at 2660 fathoms (15,960 feet). The current had carried out Ross's line sideways, misleading him. Soundings and deep water sampling, carried out by lowering several miles of wire, were strenuous and sometimes dangerous affairs, with four to six' tons of tension on the cable. The wire could kink, break, and jam, or be taken away by currents. Sometimes seven hours work revealed that the sampling bottles had not worked, and nothing had been learned. But Bruce was doggedness itself, and redrew both the boundaries and bottom contours of the Weddell Sea.
The summer over, they laid course to Cape Town via the sub-Antarctic island of Gough. The _Scotia_ rolled heavily, making sleep impossible; the worst roll took them 56° to starboard then 43° to port. Harvey Pirie wrote that although the Roaring Forties had a reputation for wild weather, comparing the Forties with the Furious Fifties was like comparing a lamb to a hyena. Their soundings would reveal a sea floor ridge, emerging above water in various islands: South Georgia, the South Sandwich Islands and South Orkneys, curving from Cape Horn east then back west to the Antarctic Peninsula. This is now known as the Scotia Arc, after the ship, and the tectonic plate to the south is the Scotia Plate.
Returning to Scotland they made such good progress from the Azores they had to sit our a few days in Dun Laoghaire, Ireland, on 15 July as their reception on the Clyde was not ready until 21 July. They had sailed thirty thousand miles in twenty-two months.
A few of the men had remained ashore in the South Orkneys. The second winter, under the command of Mossman, was a tougher affair. The comforting ship was not lying a short walk away, and the island was not brisk with the duties of a full ship's crew. The Argentine party included a 'youth named Acuña, whose profession was hard to define.' He was from the Ministry of Agriculture, not directly relevant to the Antarctic, but he brought a letter franker and some books in a mail bag so the Scots called him the Post Master General. Before the _Scotia_ continued south beyond the South Orkneys, Omond House was improved and extended. A shed and coal store was erected alongside and the passage between them was roofed.
Since November they had realised that Omond House was precariously close to the sea, but the prevailing winds gave them shelter. When early March brought onshore gales they had undermined the seaward walls. From 8 to 29 March they toiled to erect a breakwater in front of their only shelter. The biggest blocks weighed over one ton, and took five men seven hours to drag into position with a block and tackle. On 30 March there was a spring tide which they hoped would strengthen their breakwater by washing shingle into the voids. The night of 3 April mocked those hopes. A high spring tide coincided with low pressure, which draws the sea up higher, and hurricane force winds driving right into Scotia Bay. It sounded as if the waves were beating at their door. They looked outside. They were. Waves were sweeping past them and up the beach, threatening to pass right over the storm beach to the other shore. They evacuated Omond House. Masses of ice had been hurled a hundred yards above the high water mark. Cases of stores had been thrown all over the beach. They made what shelter they could. Next morning at 11:30 when the tide had fallen, they re-entered their house and found the sea had broken in but, by some magic, had not put out the fire. However their only shelter for the coming Antarctic winter was precarious: the porch undermined and collapsed, the aisle roof torn off. The corner of the house had gone and the reinforcing buttresses were only standing because frozen spray had cemented them together. Bags of stones and boxes were shoved underneath. The next tide was lower and the weather calmer; the sea merely lapped at the walls. Fetching new rocks for repairs was made difficult when snow froze up the scree-slope where they quarried their stone. They put a sixty-ton dyke around the buttress, finishing it on 22 April. Three days later came Argentine Independence Day, and whatever their nationality they celebrated it with feeling. On 29 April they suffered a hard gale from the south-east. They braced themselves for another beating but big seas did not form, suggesting the sea offshore was beginning to freeze. At 02:00 next morning the night watch observed pack ice advancing, and by 11:00 Scotia Bay was full. They harvested late penguins to store and caught shoals of small fish.
Smith and Mossman suffered from poor health in the mid-winter days. Compared with the previous winter's bustle, it was harder to maintain morale with just a handful of men, three of whom spoke limited English. 'Life had become very dull. The social atmosphere was by no means brilliant,' Mossman wrote. 'There were no quarrels, but during the winter, conversation flagged because there was nothing to talk about.' Even Smith, the garrulous cook, remarked 'Life's too slow for a funeral.' They noted that in great cold, minus 18°C, pockets of air escaping from crevasses looked like smoke. It had been observed at Queensferry on the Firth of Forth in the great frost of 1814.
On 3 August they recorded minus 40°C, 7°C colder than the previous year's minimum. The morning of 8 August brought a thunderstorm. They noted with irony that the ice was hard, dense and rigid, ideal for the sledging they'd had to perform in terrible conditions the previous year. But spirits lifted on 17 August when the first snow petrels returned to their nests. The health of Smith and Mossman improved as the days lengthened. No one suffered the severe nutritional problems experienced by the crews of de Gerlache and Charcot. On 28 August a Föhn wind raised the temperature by 22°C, which was a mixed blessing as they suffered 'silver thaws' when rain froze as it landed. They froze shut the door of the meteorological hut; the met men had to cut the ice so as not to tear out lumps of the wood. To make magnetic observations, Mossman developed a routine: first wiping frost from the instruments. Some days the observer's breath misted the mirrors, and he had to leave the door open. The wind could be too loud for him to hear chronometer's beats from two feet, but if one of the timed observations was missed, the whole series had to begin again. As winter went on, the snow built up. The morning dig-out grew from a fifteen-minute job to two hours.
Further signs of spring began on 2 September when the first Weddell seals came ashore to pup. 11 October brought a cold snap, the last of the winter, and repeating a pattern they had seen the previous year, the first penguins arrived immediately after. On 27 October the ice in Uruguay Cove broke up and went out to sea. They were glad of fresh eggs when the penguins began to lay and were ruthless in their take: eighteen hundred eggs in all. Penguins can relay if eggs are lost immediately, but success is lower for the second batch. The skuas, curious and intelligent birds, were getting used to their new neighbours, and enjoyed cleaning up after the seal-kills. Mossman came across two tern eggs which he put in a bag. He put it down to search a penguin rookery for new chicks, and returned to find a skua had taken the eggs out and eaten them. Trying to take the skuas' own eggs launched open warfare. They bombed him so vigorously he was knocked off balance and fell eighty feet down a steep ice-shelf, escaping injury only by great luck.
Though in 1903 the _Scotia_ had sailed out on 23 November, in 1904 Scotia Bay had not thawed by 31 December. The ship was due 'about the end of the year.' When Mossman climbed the hill above Omond House on New Year's Eve and looked south-west it was 'very wintry: nothing but heavy pack to be seen, with very few lanes of water among the floes. As usual, I crossed over to the east side of the glacier to get a view of the conditions to the north and east.' He saw a ship. It was not the _Scotia_ , it was the _Uruguay_ , the Argentine vessel deputed the work of tidying up.
The _Scotia's_ scientific results were published in six volumes from 1907–20. Then the money ran out and the log of the _Scotia_ , the popular narrative usually used as a vehicle to promote the men as heroes, was not published until SPRI did so in 1992. Bruce failed to raise money for a second Antarctic trip to achieve the Pole, and instead went north to Spitsbergen with Prince Albert. In 1907 he used much of his own money to set up the Scottish Oceanographic Laboratory with the aim of it being co-opted by Edinburgh University, but the outbreak of the First World War stopped that and it closed for lack of funds in 1920. From 1915–16 he managed a whaling station in the Seychelles, but he didn't enjoy the tropics. He died in 1921 after a long illness followed by depression. The whaler _Smyrna_ scattered his ashes in the waters off South Georgia. In 1957, the Commonwealth Trans-Antarctic Expedition closely followed Bruce's planned route and took his Saltire (the Scottish flag) from the 1902–04 voyage and flew it at the South Pole.
In his career he made eleven trips to the Arctic and two to the Antarctic. His Antarctic discoveries were economical, original, and dove-tailed effectively with other contemporary expeditions. His expeditions were conducted with much bravery and no recklessness. When Bruce studied the fossils, he correctly concluded that the South Orkney _graptolites_ they had collected fitted biological evidence, gathered by many authors, of historical land connections between Antarctica and New Zealand, Australia and South America, South Africa and South America, and the Auckland Islands near New Zealand. Between Bruce's gathering of the fossils and his paper, a man called Alfred Wegener had from 1906–09 conducted research from a small, sturdy hut I have visited in Danmarkshavn, East Greenland. He postulated a theory which earned him controversy and ridicule but which is now the organising thesis of how we understand the face of the planet on which we live: continental drift. Antarctic fossils would provide dramatic evidence that plants and animals had thrived in environments quite different from their present locations. His oceanographic results added greatly to the geography of the sea floor.
Let's return to where we began: the portrait of Bruce. Genuinely self-effacing, and valuing science more than adventure, he received little recognition outside the circles where the scientific disciplines he helped to advance are valued for their own sake, not as vehicles to fabricate heroes. Robert Rudmose Brown wrote of his old chief, 'Not only did Bruce hold a remarkable record in the number of polar expeditions he had led or accompanied during the last 30 years, but he was also acknowledged to be the highest authority on polar exploration of his time. It was only his intense dislike of publicity and his natural shyness that prevented him becoming one of the outstanding personalities of his generation.' Although nationalistic for Scotland, he was happy for Argentina to continue his science, and Omond House, now Orcadas Base, has the longest history of continuous use of any Antarctic Base. He achieved new standards for science and his zoological collections set the benchmark for decades. Yet he was never a public figure, never remotely famous, which was exactly what he would have wanted.
#### HOW MANY ANTARCTICAS?
If you ask scientists what defines a continent, the answers will reflect their discipline. Some geographers define a continent strictly, as the main landmass but not the islands or seas around. Geologists may see them as part of the same thing, just separated by a little local flooding. Antarctica is the most awkward of the seven.
It is the only polar continent: the Arctic is sea surrounded by land. Antarctica is also covered by up to two miles of ice. Remote sensing has revealed the details of the continent's two surfaces: the ice which forms 98% of the visible surface, and the rocks lying beneath. The familiar shape in an atlas, a disc with a tail coiled up towards South America, is the ice Antarctica. If you could lift off this icy lid to reveal the rocky landscape beneath, it would look very different. West Antarctica, that is the Antarctic Peninsula side, would dissolve before your eyes. The Peninsula might be attached by a slender strip of land to the main disc, but the coast between around 80° and 180° west would retreat so far towards the pole that Byrd Land would be left as a large oceanic island.
If we carefully replace the lid, the familiar shape still does not include the sea ice which rings the coast all winter, doubling its size in a cold year. In the twentieth century the pack ice twice reached as far north as South Georgia, equivalent to polar ice appearing off the English Lake District.
In ecological terms, the most important boundary is not on the land but in the sea, which is the home of nearly all Antarctic life. The Polar Front, previously known as the Antarctic Convergence, is a well-marked change from cool waters of the main oceans to the icy. Its line and location vary with the seasons and currents, but it lies mainly between 50°S and 60°S. Each side of this temperature boundary has distinct sea-life which can seldom survive on the other side. The Polar Front therefore defines Antarctica physically and ecologically.
These moving boundaries cannot be used to define the legal limits of Antarctica for the purposes of international treaties. It has been agreed that for such purposes only, Antarctica is everything below 60°S.
# The Great Escape
1901 – 03: Otto Nordenskjöld
_Can the Antarctic night wear you down so much that when you encounter men on the ice by chance, you do not recognise them as members of your own expedition?_
Otto Nordenskjöld (the _skjöld_ approximates to _shelt_ as in shelter) was born on 6 December 1869 in Småland, in the centre of southern Sweden. His uncle, Baron Adolf Nordenskiöld (that side of the family spelled the name differently) had completed a voyage which had defeated the most determined navigators for over three hundred years, completing the North East Passage over the north of Russia. In the steam whaler the _Vega_ , he nearly made it in one season, but was forced to overwinter from 1878 – 79, before completing the first circumnavigation of the Eurasian-African landmass. Young Otto gained a doctorate on the geology of Småland and became a lecturer at the University of Uppsala, but he had inherited his uncle's hunger for adventure, and led a Swedish expedition to the Magellan Straits in 1895–97, studying Ice Age deposits in Tierra del Fuego and estimating how much larger the ice caps had once been. In 1898 he took part in an expedition to Alaska, and in 1900 was geologist under Admiral Amdrup on the Danish East Greenland Expedition.
Aged thirty-two he was ready to conduct a polar expedition of his own. In contrast to other jingoistic expeditions, his was to be science-led, privately funded, and low-budget. No one had penetrated the ice to the south shore of the Weddell Sea so it wasn't known if the Antarctic Peninsula and Wilkes-Victoria lands further east were connected, though he candidly admitted it was probable that the problem of how much land was connected might never be solved. Nordenskjöld's ambitious plan was to explore new land east and west of the Antarctic Peninsula: to go as far south as possible in the Weddell Sea, stay with five other staff to overwinter, continue work the next summer, and be picked up before the second winter. In the meantime, the ship would ferry various staff to Tierra del Fuego, Staten Island and South Georgia to pursue other projects.
Their ship, _Antarctic,_ was formerly the sealer of Henrik Bull (see _The Cheek of Mr Borchgrevink_ ). It left Gothenburg on 16 October 1901, and paused at Falmouth, in the south west of England, while Otto dashed to London. He met Scotsman William Speirs Bruce who was preparing his own compact, low-budget Scottish National Antarctic Expedition in the _Scotia._ They agreed to offer mutual support if either got into trouble. Crossing the tropics, some of Nordenskjöld's sled dogs died of heat. He had an offer of support from the Argentine Government, which was beginning a major political and diplomatic drive, that still continues, to establish claims on sub-Antarctic and Antarctic territories. In return for donating supplies of food and expensive coal, Argentina wanted one of their own nationals to be a full member of the expedition. Nordenskjöld was sympathetic to their ambitions, but he was wrong-footed on his departure to Buenos Aires when he received messages making it clear that Argentina expected their man to be one of the small overwintering party, jeopardising Nordenskjöld's careful plans to take only those few judged to have the temperament, as well as the need in scientific terms, to endure two months of continuous night. Tactfully, he delayed responding until he met the man: Lt. José Maria Sobral. Nordenskjöld was relieved to find him 'so unaffected and affable, so interested in the [project] and so intrepid [that] the matter was definitely decided the very same day.' Sobral was qualified to assist with meteorology, magnetism, astronomy and hydrography. Nordenskjöld's five other selections for overwintering anxiously wondered who would be ordered to give up their priceless bunk in the tiny wooden hut to make way for this late interloper. It was a diplomatic time bomb ticking in the young leader's hands. In all, crew and staff, twenty-nine men began the voyage out of Buenos Aires in 1901.
Nordenskjöld secured as captain the remarkable Norwegian, Carl Anton Larsen. The son of a sea captain, Larsen had gone to sea as soon as he could walk. Enterprising, brave, and a good skier, he was as passionate and informed about science as he was about business. In 1901 he was manager of the great Finnmarken whaling station in Norway, but Nordenskjöld tempted him to return to the seas he had explored in 1892– 93 on the sealer _Jason_ , when he discovered Oscar II Land on the west shore of the Weddell Sea and got as far as 64°40´S, picking up the first Antarctic fossils from Seymour Island. The staff also included the American artist F. W. Stokes, a landscape painter welcomed for his past experience with Robert Peary in Greenland 1893–94. His goal was to record the _aurora_ _australis,_ the southern lights, and compare them with the northern lights. They proceeded to Staten Island then east to the Falklands, where they replaced the dogs which had perished in the tropics. These poor beasts are easy to recognise in the photographs; the substitutes are much slighter than the others, and must have suffered dreadfully from the cold. The _Antarctic_ arrived in the South Shetlands on 11 January 1902. They began the first part of the mission: exploring islands west of the Peninsula, probing the Bransfield Strait, and going ashore at Nelson Island to sample plants and rocks. They attempted to explore the Orleans Channel trying to determine whether it connected to the Gerlache Strait further south, but they were defeated by dense ice.
They worked their way back north through the pack to the entrance of the sound between the Antarctic Peninsula and Joinville Island. It had been seen when ice-filled by Dumont d'Urville in 1833, but not sailed. The _Antarctic_ made the first passage of the sound, naming it after their ship. On the way they observed Hope Bay, an inviting snow-free valley on the mainland shore. They continued and landed on Paulet Island before crossing the Erebus and Terror Gulf to leave a depot of fuel and food on Seymour Island. They continued south down the east coast of the Peninsula, following the Oscar II Coast a decade after Larsen had named it, exploring and mapping until they met sea ice at 66°15´S. They had hoped to push much farther south, but this season conditions would not allow exploration of the south shore of the Weddell Sea. Nordenskjöld's assessment that it would be difficult to link up previously explored sections of coast proved correct: most of the easy discoveries had been made. In February, with the brief southern summer ending, and thin ice coating the surface of the sea each morning like a film of grease, Nordenskjöld was set down on Snow Hill Island, with five companions, sledges and supplies. Nordenskjöld's problem of whose bunk Sobral would take had been solved by Antarctica. Since they arrived they had not seen the _aurora australis_ once. The artist Stokes decided an Antarctic winter with nothing to paint was a trial he didn't need. He would return north with the ship. We now know that the Southern Magnetic Pole, which draws in the solar particles that create the aurorae, is on the opposite side of the continent, and the Peninsula seldom enjoys displays.
#### PEMMICAN
Pemmican sounds like, and is, a North American Native word. Made by the Cree nation, it was a paste of meat and fat which they took on hunting trips as high-energy rations. Men like John Rae from the Hudson Bay Company found it convenient and sustaining as field food. It was adapted for explorers and frontiersmen and manufactured commercially; both Bovril and Batchelors made varieties. It is not a source of carbohydrates and it was often mixed with broken hard biscuits like those served on ships; the British called this mix hoosh. Some later mixes included carbohydrates. Getting this balance right was important to keep healthy and feel satisfied and full. The science of nutrition was poorly understood; even the word _vitamines_ was not coined until 1912. Looking back we can see that men endured extreme hardships on an inadequate diet. When Nansen crossed Greenland he did not check the recipe for the pemmican; the variety he bought had little fat, but Nansen had assumed it comprised 50% fat. Although he had other sources of fats, this unbalanced his diet.
Ideal formula | | Nansen's rations
---|---|---
| |
10% | protein | 40%
30% | fat | 10%
60% | carbohydrates | 50%
A similar food was devised for dog rations though they did not prosper unless it was mixed with fresh meat, usually penguins and seal. The dog food had low carbohydrates and when men were forced to eat it, they suffered diarrhoea if this could not be made good. Shackleton's men, on top of all their other woes, had to switch to dog pemmican before beginning their boat journey from the peninsula to Elephant Island making use of the 'head' or toilet – meaning over the side of the boat – even more miserable.
The Snow Hill hut
From the ship's hold flowed boxes of supplies which took three days to land, using planks placed across two ship's boats to ferry them ashore. Among them was a partially pre-fabricated hut measuring 13´ 6" x 21´ 0" (4.1 metres by 6.4 metres). It had two wooden skins with insulation between. Short journeys were made to set up supply dumps, in readiness for journeys in the spring, and they prepared to overwinter. Nordenskjöld wrote: 'I stood there amid the grandeur of the scenery, while the sun sank slowly behind the haughty, ice-covered crown of Mount Haddington [on James Ross Island] and gilded the ice-field far away on the eastern horizon. Shall we succeed? At the moment I felt a strong faith in the future, everything lay bright and promising before me.'
Nordenskjöld's party comprised Ole Jonassen, the sledder; Erik Ekelöf, a bacteriologist doubling as doctor; a sailor to handle their boats, Gustaf Åkerlundh; Gösta Bodman (hydrology, meteorology and magnetism), and José Sobral. Heating huts in Antarctica is a problem. They had an iron stove but the temperature was 20°C colder on the linoleum floor which was lethally slippery whether wet or iced. They could live without suffering and they planned to be there only for a year, and while a tent is shelter, a hut is home. In March 1902 they went south with winter on their heels, to lay a last depot, and in late April they returned north to Seymour Island to check their depot and cairn, and collect fossils. Science seemed to have been on their minds more than logistics. On many journeys small oversights eroded their safety margins, and Antarctica doesn't tolerate mistakes. On this trip they forgot their lamp and then let the dogs ransack all the butter from their sledging supplies. Fats are very important both in using the other food groups efficiently and making the stomach feel full.
Back at their Snow Hill some dogs died of exposure, and base storms broke their magnetic observations hut to pieces. The wreckage was salvaged to build kennels. One perk of not having penetrated as far south as they had hoped was that it is never wholly dark at 64°30´S, even in midwinter. However it is savagely cold for its latitude, with strong raw winds. The coldest open boat work I have ever done in Antarctica was at Snow Hill in March. A rattling noise on my survival suit told me the spray from the bow was freezing in flight before falling on me as hail. That was late summer, not winter.
In midwinter, June winds wrenched one boat free from its lashings and threw it seventy metres. Come October 1902, and early spring, Nordenskjöld, Sobral and Jonassen, went back to explore the Oscar II Coast. Their preparation had not been encouraging; on a trial run in July to test the gear, the tent poles collapsed. Impatience gnawed at them and they set out again on 30 August to lay depots: too early for serious sledging. Two of the three dogs broke loose and ran home, and the journey was rescheduled for late September. Few dogs were now in good condition, and they could only muster five working animals for two sleds. Men made up the shortfall.
Modern travellers to Antarctica, having bought the latest clothing and read up about layering and wicking, assume the earlier explorers had vastly inferior clothing. In fact, once the Orkneyman John Rae, in the 1830s, and other workers of the Hudson's Bay Company had learned how to dress and travel like the Inuit, there was no need for anyone to be cold. Nansen reinforced the message after overwintering among the West Greenland Inuit in 1886–87. Nordenskjöld had an outer wind-breaking layer made from canvas, and thermal layers of guanaco and reindeer. They also had a reasonably varied diet. As well as the polar staple of pemmican, they had meat biscuits, bacon, lentils and peas, bread, butter, sugar and coffee.
By 8 October they had covered eighty-four miles. They scaled Robertson Island, a lone vantage point off the east coast, and found it was volcanic. The visibility was good and they could clearly see that the Trinity Coast, a relatively well-surveyed coast facing the South Shetlands, and Larsen's Oscar II coast were two sides of the same peninsula. They had accomplished one of their prime objectives and joined up two previous discoveries on the map. They made the first ever landing on the east side of what we now call the Peninsula at 65°57´S next to a towering nunatak which they named Borchgrevink after the first man to overwinter on the continent. It was during this journey that Nordenskjöld's sled dogs disappeared down a crevasse. He was dragged in after them. The other men ran to the lips and saw him tangled up below; his skis had stopped him plunging out of reach. He was badly hurt, and their most expert sledder, Ole Jonassen, had injured his arm. In the confusion, the dogs got at their own rations and gorged all the pemmican that was supposed to sustain them for the rest of the journey.
The party rested up in a tent while a storm raged for two days. They spent their unwelcome leisure considering their chances of getting back to base. 'One feels like a fever patient lying there without occupation staring at the roof and making spots assume different patterns.' It was no fun 'to listen to the howl of the storm and to know nothing but that our provisions are coming to an end and that our poor dogs are becoming weaker.'
It was now that their previous preparations paid off. They made their way to the depot near Robertson Island and, warmed by food and more fuel, they made it back to the Snow Hill base hut five days after leaving their storm refuge. Despite the fall, they reviewed the trip with some satisfaction: they had travelled 380 miles in thirty-three days and pieced together two pieces of Antarctic geography. However, the rations they had carried were not sufficient to support a much longer journey; on average, each man lost fifteen pounds. In December, they sledged to Seymour Island, and collected important fossils, including plants, ammonites, ferns and firs. Many plainly demonstrated that Antarctica's rocks had once enjoyed a much warmer climate. They also discovered that the emperor penguin, weighing up to forty-five kilos in prime condition, was not the largest that had ever lived. They found the fifty million-year-old bones of a giant penguin: _megadyptes_ , as tall as a man. I had a nightmare about these birds; they had ganged up with blue tits to steal milk. The penguins kicked my door down, then they stood in the kitchen helping themselves from the fridge.
But when Nordenskjöld's men returned to base only two things were on their minds; why wasn't the ice melting, and where was the _Antarctic_? On 9 January Jonassen climbed a hill and looked around. It was a melancholy panorama of densely packed ice with little water. On 18 February 1903, the sea froze, and they knew they were trapped for another year. Reluctantly they looked at the penguins that had been their companions and started to kill them. They took four hundred. They even killed skuas. If you are what you eat, skuas are foul. Nordenskjöld glumly recorded: 'It was only bitter need which could compel us to this horrible slaughter. It is especially repulsive here, where the creatures have not yet learned to fear man.' Someone thought it would be a good idea to skin them and wash the meat in seawater so the salt could help preserve them.
Their mood would have deepened farther if they had known why the _Antarctic_ had not appeared. She had sunk in Antarctica, on her way to them, six days before. No one in the world knew.
The six at Snow Hill settled down for a second winter. Science kept them busy but the isolation of their position began to settle on their souls. Nordenskjöld wrote: 'With what delight should we not have greeted one blade of grass: red, green and yellow, the colours which have a stimulating influence on the senses. We saw only pale tints. They attract the beholder with wondrous power, although they seem to radiate a something which resembles the chill of death.'
The greatest part of Nordenskjöld's party was on the ship. Sub-groups had variously overwintered in Tierra del Fuego, South Georgia and the Falklands, while Larsen carried out hydrographic work round South Georgia and a shore party of three men studied its biology and geology. They found one of the few fossils ever discovered there: a mollusc. On 1 May, Andersson walked over the top of the mountains enclosing Cumberland Bay and found himself overlooking a bay within a bay. It was a perfect anchorage framed by high peaks, austerely beautiful. When he clambered down to the shore he found he was not the first person to have visited: on the beach was an abandoned ten-metre sealer's boat, and seven try-pots. He named it Pot Cove which in Norwegian is Grytviken. Two of those pots stand in front of the museum today.
On 5 November 1902, the _Antarctic_ had left the Falklands, but she met ice four days later at just 59°30´S. After a spell trapped by sea-ice beneath a huge iceberg in a snow-storm, they freed themselves and made Deception Island in the South Shetlands. They continued south and surveyed the west coast of the Peninsula as far as Brabant Island. In doing so they achieved what they had failed to do the previous season, and sailed from the Orleans Channel into the Gerlache Strait. This linked two important navigable channels for the first time. They began to make their way north, intending to pass through the Antarctic Sound once again. Ice conditions were difficult and the weather poor. They forced their way through sea ice dodging the huge icebergs that drift in here from the ice-shelf on the east side of the Peninsula which is still named Larsen after their captain. On 5 December they began to make their way to the entrance of the sound. In dense ice, with little room for manoeuvre, they were struck by a savage storm.
When it eased they anxiously reviewed their position. The ice might not allow the ship to repeat the passage of the sound, and even if it did, there was no guarantee that conditions on the other side, in the Erebus and Terror Gulf, would be good enough to drive on south to Snow Hill. However if the area were unseasonably iced up, an overland party might get there by skiing over the land and the frozen sea. They remembered the beautiful snow-free bay on the tip of the Peninsula. It might make a suitable base. On 29 December, deputy expedition leader Gunnar Andersson, Samuel Duse, and Toralf Grunden were put ashore at Hope Bay (now Base Esperanza). First, Andersson supervised the laying of a big depot in case the _Antarctic_ could not reach Snow Hill Camp, and everyone had to overwinter in Hope Bay waiting to try next again spring.
More immediately, Captain Larsen and Andersson determined that each would head independently for Snow Hill. In the days before radio, you had to think through, and plan for, all the likely outcomes. Once they separated, each party would be ignorant of the other's circumstances. If the ship successfully rounded Joinville and Dundee Islands to the north and arrived first at Nordenskjöld's base, they would wait there for the Hope Bay three until 25 January. If they did not arrive, the ship would sail to Hope Bay and seek them there. If the Hope Bay party got to Nordenskjöld but found no ship they would wait at Snow Hill until 10 March. Then both Nordenskjöld's and Andersson's parties would return to Hope Bay to rendezvous with the ship.
The ship sailed out of Hope Bay. As soon as their tents were pitched and stores organised, Andersson realised that in the haste of unloading, he had landed only half of their bread rations. Duse and Grunden did not blame him; such generosity of spirit would soon be tested to the full. The three began to ski the two hundred miles to Snow Hill.
The _Antarctic_ retreated from the sound and began to sail round to the north towards Snow Hill. It rounded the tip of the islands lying off the peninsula but when they turned south they met dense ice and were soon trapped. She was a strong ship designed for polar work so they did not worry, but a storm blew up and pressed the pack ice tight against her hill. Carl Skottsberg, writing on 10 January 1903, said, 'the ship began to tremble like an aspen leaf, and a violent crash sent us all up on deck to see what the matter was. The pressure was tremendous; the vessel rose higher and higher, while the ice was crushed to powder along her sides.' Descending to the ice they found the rudder had been damaged. They were held tight for three weeks. When the pressure eased enough to float free into a narrow lead, they found that they were leaking badly. The nearest land was a small island they had visited last season: Paulet. They planned to beach the _Antarctic_ there but the ship was so grievously damaged that on 12 February all twenty men plus the ship's neurotic cat were forced to camp on the ice, where they built a crude tent from sails. The whaleboat was lowered and a ton of provisions loaded into her. They began an exhausting journey, sailing and rowing, then hauling across the sea ice when there was no open water. It took fourteen days to ice-hop the twenty-five miles to Paulet Island, reaching there on 28 February 1903.
Meanwhile Andersson's group worked their way south using a copy of a chart drawn by James Clark Ross. Through an error in the surveying or the copying, it was not correctly orientated, and they were led south-west instead of south, taking them into the fractured ice of Crown Prince Gustav Channel. Instead of land they found a frozen strait full of bergs. Their confusion did not overshadow their sense of the marvellous: 'One can actually imagine that a gigantic snow-clad city lies before us, with houses, palaces, and all the wonders of the world.' They headed for an island they could see to the south-east, thinking it was James Clark Ross Island, but it was Vega Island, named after the ship Adolf Nordenskjöld had navigated through the North East Passage, and close to the north shore of James Ross Island. They toiled up the heights of Vega Island, and saw open sea running away south towards Snow Hill. They could not ski to Snow Hill, and had no boat, but it seemed certain that the _Antarctic_ would have found it easy to cross such open water and pick up Nordenskjöld and his companions. They returned north to the agreed rendezvous at Hope Bay, arriving on 13 January, and expecting their comrades to sail into view any day, blissfully unaware that the _Antarctic_ was then fighting for her life in the pack. Meanwhile they ascended the hill behind the camp, which is one of the best places in Antarctica to collect fossils, and is now named Mount Flora for the richness of its plant fossils. Clambering around the rocks, they found their boots were not up to the job. The seams opened and holes were worn in their socks. Andersson wouldn't let the prospect of frostbite and lameness get him down, there were compensations: 'we could cut our toenails without taking off our boots.' But when there was no sign of ship or companions by 11 February, they realised something had gone terribly wrong. It seemed likely that the ship had been trapped in ice, perhaps destroyed, and that the crew and returning staff who comprised the majority of their party were dead: drowned or frozen to milken statues. It was certain that no one outside knew anything of their troubles, or could learn of them until the only service that could be conducted for them was a belated funeral.
They began to construct a hut, one man digging up rocks, and the other two carrying the blocks on two poles, like a stretcher. They planned a crude cubic structure within which their larger storm tent could be pitched. On 17 February they laid the foundations, over a yard thick, and began building upwards, filling the voids with fine gravel. They hadn't built the roof when heavy snow fell on 6 March. Desperate to have better shelter before winter closed in hard, they continued working while the snow fell steadily, until on 10 March the weather was too harsh to work and they laid up in the tent for a day. The snow could at least make their walls windproof, and they packed it against the outside. The following day they lifted the sled onto the walls to support their tarpaulin roof. They made a tunnel like an igloo's, with an entrance, between boxes of fossils, just seventy centimetres square. The tunnel also doubled unappetisingly as a food store and foul weather winter toilet area. Duse finished a penguin-skin carpet from the skins of the birds killed for the larder. Finally a house-proud Andersson swept out the interior with the wing of a southern giant petrel. When the three lay down side by side to sleep, they entirely filled it; home was a stone triple coffin. On a cold morning the temperature in their inner tent was –14°C near the roof, and –20°C near the floor. They hated thaws which produced mud and cold showers of melting rime. In early March they reviewed their stores and switched sharply from eating depot stores to living off the land. They were fortunate that in this area some penguins linger late after their end of season moulting. Their frying pan was a flat tin with a wooden handle nailed on to it. At first they struggled to raise enough heat to cook; it took them six hours to make penguin stew. But with practice they could get a fire roused in two to three minutes. Grunden sang when he cooked, liking folk songs, shanties, and American Negro songs learned on the Florida coast. One began: 'And a seaman's life it took my fancy most, since I turned o' fifteen year.' It sounded more and more plaintive as the long winter wore them down. As mid-winter approached, they calculated their fuel would not last until spring, and after 1 June, they cut their meals from three a day to two. Canned herrings, tinned meats and dried vegetables provided a welcome break from seal and penguin. There were a few bottles of spirits to enable them to take a dram of Dutch gin once a month, and each had a birthday during the winter. At Christmas they went on a spree, lighting three candles at the same time.
Hope Bay
In the evenings they took turns giving mini-lectures. Duse and Andersson tried to remember the plot of The _Three Musketeers_ and _The Count of Monte Cristo_ to tell to Grunden, who didn't know them. They made sealskin overshoes to protect their boots; it was immensely time-consuming, which was a blessing. The official account says that time slipped by quickly, with no sense of the oppressive slowness of time, before admitting that they suffered from awful 'intellectual nothingness.' Later they were reduced to passing round food tins to read the labels.
Meanwhile Larsen's party found Paulet: a small, volcanic island with a long, steep storm beach on one shore, behind it a stone ridge rising to a lake whose waters are a strong green colour, enriched by the guano washed down from thousands of Adélie penguins nesting on the slope above. They collapsed ashore, acutely conscious that no one in the world knew of their fate, and that the two other groups whose lives depended on Larsen were beginning a makeshift overwintering, with nothing but fear for their companions' fate to feed their thoughts.
Paulet's rocks are basalt, and naturally fracture into blocks a convenient size for building, if not always a handy shape. For their new accommodation Larsen's men chose a site on the ridge behind the beach, commanding a view of the sea, and built a two-room hut, roofed with sails lashed over oars and salvaged wood. One room was a kitchen; they sensibly isolated the cooking area, because Antarctica is dry and windy, and fires easily flare out of control. The other room had two broad, low stone benches running the interior length of the two longer walls. This was to drain the cold dense air into the trench between. They sat and slept on the benches, a few precious degrees warmer than the floor. The floor was lined with penguin skins; they killed 1100 to see them through the winter. They wanted more, but the Adélies moulted and left before they were finished. The biologist Carl Skottsberg described how any disturbance to the rotting penguin skins sent up a sickening odour: 'When one came out of the fresh air into the hut it was at first almost impossible to breathe, so bad was the atmosphere.'
One of the seamen was already ill when they arrived. Ole Wennersgaard 'had been a long time poorly, had violent attacks of coughing as soon as he went into the open air, and, as we thought at the time, showed other signs of consumption. [It was later diagnosed as heart disease]. He would sit night after night, moaning softly and slowly, for he seldom had any rest, and if one happened to look up during the course of the night, one met the terrified gaze of his large sorrowful eyes. We seldom heard him complain; he only moaned softly.
'On the morning of the 7th of June he had said goodnight with a "Now I shall sleep well." And he fell asleep in a sitting posture, the only one possible for him. Then his neighbour suddenly felt how Wennersgaard sank softly down upon his shoulder – a few rattling breaths, and life had fled.' He was nineteen years old. 'It was dim and silent in the hut; cold, clear and silent in the open air. Death, the one guest who could reach us, had laid his hand heavily upon the circle of comrades who had so long striven together for life. Sewn up in his sleeping bag, the only coffin we could give him, he was carried out to one of the boats. A couple of days later we buried his body in an immense snow-drift. Not until the spring came could we build him a lasting resting place. Slowly we wander home and assemble in the hut, where everything speaks of death and corruption; we assemble there – _nineteen_ of us.'
His grave is still there, a simple wooden cross on a stone cairn. Adélie penguins nest all around and on top of it. It is some distance from their camp, around the steep flank of a hill that comes nearly to the water. They did not want a daily reminder of their fragile mortality. The days shortened, the last of the wildlife left.
For the three men at Hope Bay, wrote Andersson, 'The winter passed very quickly.' It makes you wonder what he normally did for entertainment in the long nights. He cherished 'the strong power that warmth and honest friendship has to proudly subdue the dark night of isolation and extreme distress.'
On 29 September 1903 they set off sledging towards Vega Island to reconnoitre ice conditions. It was early in the season for a long journey, and cold; Duse and Grunden got frostbite. Grunden's left foot froze until he could not feel pin pricks, and ugly black blisters covered the toes of his left foot. Duse presented some signs on one little toe. The blisters were lanced with a sail needle, before applying disinfectant and wrapping them in dirty bandages, the only ones they had.
At Snow Hill, Nordenskjöld's group had no reason to doubt that all their friends were in a single group returning to pick them up. They maintained meteorological observations all winter and were surprised to find that they differed from the first year's. Until then, it had been assumed that Antarctica's climate was simple: it was cold in summer and colder in winter, unchanging from year to year. They found the weather was much more complex. It would prompt men like Hubert Wilkins (see chapter six on Deception Island) to lobby for a network of Antarctic meteorological stations to better understand how polar weather systems affect the inhabited continents.
Their dogs were further reduced when the sled expert Jonassen left the two Falkland Islands sheepdogs untethered. They got into one-sided fights with the big Greenlandic dogs and were killed. On the same day that the Hope Bay party set out, Nordenskjöld and Ole Jonassen began a surveying trip towards the Peninsula. They set out in poor weather and forgot half their bread. Their tent was blown down and torn, forcing them to return to the hut. They tried again on 4 October, leaving messages at key locations, including last year's cairn on Seymour Island. By 12 October they had explored the whole west coast of James Ross Island thereby discovering Prince Gustav Channel, which separates it from the continental land. On 12 October at 13:00 they halted beneath a high square cliff called Cape Dreyfus. It appeared there were seals scattered on ice. But two seemed to rear up to an impossibly high posture. Nordenskjöld's tepid writing doesn't capture the drama of the moment but it does bring out the absurd humour.
'What the deuce can those seals be, standing up there bolt upright?' says one of us, pointing to some small, dark objects far away on the ice, in towards the channel. 'They are moving,' cries another.
A delirious eagerness seizes us. A field glass is pulled out. 'It's men! It's men!' we shout.
There was a moment when they wondered whether they had met the first natives, coming out of primeval isolation to meet them. Perhaps, at last, the ancient protocols on meeting Native Antarctic people would be used.
'Who are they? Is it a sledge-party from the winter ing station or can it be people from the _Antarctic_?'
Then there was a pistol shot, the other party had seen them.
Two men, [Grunden was sitting down cooking] black as soot from top to toe; men with black clothes, black faces and high black caps, and with their eyes hidden by peculiar wooden frames, which are so attached to the face that they remind one of black silk masks with pierced pieces of wood for the eyes. Never before have I seen such a mixture of civilisation and the extremest degree of barbarousness; my powers of guessing fail me when I endeavour to imagine to what race of men these creatures belong. They hold out their hands with a hearty 'How do you do?' in the purest English.
Their manners become more English than a Duke's.
'Thanks, how are you?' was my answer.
'Have you heard anything of the boat?' they continue.
'No!'
'Neither have we! How do you like the station?'
'Oh, very well in every respect.'
There comes a moment's pause, and I puzzle my brains without result.
Duse worked out the puzzle quicker than Nordenskjöld:
'Don't you know who I am?'
'No, it's not very easy to recognise you!'
'Oh, I am Duse and this is Gunnar Andersson.'
Joyfully they set off for the Snow Hill base hut. They sighted it two years, to the day and the hour, after they had left Sweden. A celebration dinner featured roast emperor penguin. The name of the meeting place was changed from Cape Dreyfus to Cape Well-met, one of the few name changes you cannot argue with.
Back in Larsen's group at Paulet they were eating well; some of the Norwegians and Swedes were experienced hunters who regularly shot seals on the ice. They were also skilled at ice-fishing, providing some welcome variation in their diet. On 31 October 1903, nineteen days after the sled parties met on the ice, Larsen re-launched the whale boat with five companions and left Paulet for Hope Bay. At Hope Bay he found an empty hut and a note saying the three had left for Snow Hill to see what had gone wrong. The ice had broken up enough for Larsen to try to sail after them, using one of the tent poles for a mast and the canvas roof from the hut for a sail. Despite their isolation and danger, Larsen paused on his journey to collect fossils. William Speirs Bruce said that Larsen, even when sailing on a sealing voyage, 'showed a zeal for extending our knowledge of these regions would that not have been unworthy of the leader of a purely scientific expedition.'
During this time, Nordenskjöld led his reunited group to Hope Bay, hoping against hope to see the ship. Finding nothing, they left a message and returned to Snow Hill. Perhaps William Speirs Bruce would learn of their troubles and search for them. Perhaps. On 8 November, around the first date it was possible the ship might come, Bodman and Akerlündh were collecting penguin eggs. They stopped when they saw four men coming from the north, and a ship. They thought it was the _Antarctic_ but as they hurried close they saw an unfamiliar vessel. It was an Argentine naval sloop, the _Uruguay_ , under Lt. Julián Irízar, Argentina's Naval Attaché to Britain. When the expedition had not returned at the end of the summer, those who had promoted Sobral's attachment to the expedition had organised a search expedition, relieving Bruce of his promise to come to aid. The _Uruguay_ had reached Seymour Island on 7 November 1903, and seen a boathook Nordenskjöld had set in a cairn to attract attention to his message left two weeks earlier. The shore party were greeted by two figures on the ice; one was Irízar. They led the visitors off to Snow Hill for another reunion celebration. Irízar told them that Sweden had launched a rescue mission using the _Fridtjof_. During the evening, choices began to weigh on Nordenskjöld's mind and his party mood chilled. He was a thoroughly decent man who had to decide quickly whether to look for Larsen now, or wait for the _Fridtjof_ , so as not to snub his country's efforts to rescue him. He reflected, 'the thought of the enormous responsibility resting in the decision that would have to be made within the next few hours, seemed, for the moment, to deprive me of all power of motion. Sorrow was depicted on every countenance for everyone saw how small was the hope of ever again seeing the comrades we had left on board the _Antarctic.'_ At 22:30 the whole party atmosphere had dampened, with everyone thinking of Larsen and their other comrades. Annoyingly the dogs disturbed their sober reflections with barking and howling. Bodman went outside and was stunned to see six bedraggled men on the ice, he yelled: 'Larsen! Larsen is here!' Nordenskjöld wrote: 'No pen can describe the joy of this first moment, when I saw among us these men, on whom I had been thinking a few minutes before, with feelings of the greatest despondency. We conducted the newcomers in triumph to the building.' Larsen had sailed to within sixteen miles of the base before dense ice blocked him. They had walked the rest of the way.
Nordenskjöld is able to indulge in wistful nostalgia as he takes leave of their little hut. 'I go for a last turn through the rooms where we have experienced so much during the past years, and which now look so lonely and disordered. The door is barred as carefully as possible... We cast one look back at the house to which so many memories are attached, and then start off along the ice.' They made their way to the shore but the wind had blown up. It wasn't safe to take the boat out to the ship one final time. They sat until on the shore until dusk, watching the surf, looking at the warm lights on the ship. Then they returned to sleep one final night in the hut.
I visited the hut in February 2008 when the Argentine conservation team who had been renovating it over several seasons were preparing to leave. It is a snug, homely place. The technician had just finished restoring the hurricane lamp and fired it up. We sat hundreds of miles from any other human beings. I thought of how it might have looked to someone able, over a century before, to look down on the Weddell Sea and in the warm slate of winter dusks see this yellow light spring up in the darkness, fierce and bright. Moving round it, six wraiths, parcels of human warmth in the ocean of blue cold all around them.
Nordenskjöld sailed the next day and reached Paulet the following morning at 04:00, just as the sun was rising, sparkling each snow crystal, glittering the scene. No one ashore had stirred. Anticipating another winter vigil, botanist Carl Skottsberg was supervising the collection of six thousand Adélie eggs. Nordenskjöld reflected: 'Everything is so different from those remarkable moments when I myself have been the one affected by the event. Here I am nothing but a spectator, but my feelings are perhaps the deeper, that we come as the unexpected rescuers to men, in a position the gloom of which could hardly be surpassed.'
He ordered the ship's whistle to be sounded and went ashore. The whistle scared the cat witless. It tore round the hut then out among the penguins creating pandemonium. Men followed, staggering, gesticulating: 'sooty, dirty, emaciated, in tattered clothes, but with their countenances, on which suffering has impressed its melancholy seal, beaming with joy.'
Anticipating future expeditions might suffer difficulties in this area, the rescuers now cached a large depot. The Paulet men gazed enviously at the boxes of jams, sugar and bread being deposited only when they were leaving. The new arrivals paid their respects to the teenager's grave. 'We stood in silence around the stone heap... far from our native land and all those dear to us, here upon this desolate shore, where the crowd of penguins would soon be the only watchers around our comrade's grave.' The final task was to sail to Hope Bay and pick up the botanical fossils Andersson had collected from Mount Flora. They included _Glossopteris_ , a plant which enjoys temperate climates and beech species, which indicated mild weather in Antarctica as recently as fifteen million years ago. _Glossopteris_ is also found in rocks of the same geological period in Australia and South Africa. It would later be used as evidence in favour of the theory we now call plate tectonics, to help break up the continents and piece them back together as they were hundreds of millions of years ago when they formed part of the ancient continent of Gondwanaland. In the hold, he also had fossils of _Zeuglodon_ , an extinct early whale. On Cockburn Island they discovered huge numbers of scallops on a wave-cut platform at 750´(228m) asl, giving the first indication in Antarctica that there had been isostasy following shrinkage of a previously much thicker ice-cap. At that time it was the greatest single contribution to science in Antarctica.
Nordenskjöld never paid off the debts he incurred. The government refused to bail him out because they had already spent money sending the _Fridtjof_ south. In 1905 he was promoted to chair of geography at University of Gothenburg, and carried out expeditions to Greenland in 1909 to study geology and glaciology, and to Chile in 1920–21. His only attempt to return to Antarctica was cancelled as war loomed in Europe. He was one of the rounded Antarctic explorers. Travel was a means to an end, not an end in itself. They tended to have longer and happier careers than those for whom a geographical trophy was all, and the science merely a necessary part of the funding package. He also worked on peace between nations, the colonisation of America to relieve unemployment in Sweden, and he founded and became director of the Gothenburg School of Economics. Having survived everything remote regions could throw at him he died in a traffic accident, aged fifty-nine. Friends and family bailed out his estate's debts.
Nordenskjöld deserves better remembrance. There are several reasons for the amnesia. He did not want fame, nor did his funders. There was no professional photographer: no Frank Hurley or Herbert Ponting to build a myth. The pictures are often of poor quality, and record everyday events without dramatising them. The account of the expedition, published in Swedish, and not translated into more widely spoken languages for some time, was a compilation of the several authors. Carl Skottsberg's writing sometimes goes to the heart of events and makes it come alive, but much of the others' prose, including Nordenskjöld's, is bland. He generalises his feelings so much they feel second-hand. He seldom puts his finger on the telling detail that fixes an event in the mind or makes the people breathe and live. He was the kind of man who could organise a fine polar conference, but if he were the after-dinner speaker, you'd sneak out to the pub with Shackleton.
Otto Nordenskjöld
The reconnaissance Larsen made in his winter cruising round South Georgia in 1901 – 1902 was the foundation of his fortune as a whaler, despite it being received wisdom that the Southern Ocean was too remote from world markets to make a profitable industry. Grytviken began life as a whaling station on 16 November 1904 with two sailing ships, the _Louise_ and _Rolf_ , and a steam catcher, the _Fortuna_. The first whale they took was a humpback. On shore were three prefabricated houses for sixty Norwegians. They built the essentials in a month, including the shallow slip, known as a plan, onto which the whales were hauled for processing, and twelve cookers to extract oil from blubber. Larsen became a rich man through whaling, but no mansion would mean more to him than the hut he shared with nineteen others on the lonely shore at Paulet Island. Storms have reduced it; a few walls still reach shoulder height. Penguins have colonised it; the dividing wall between kitchen and dormitory is a favourite nesting spot for Adélies. Some door and window frames survive; you can touch the few timbers of the _Antarctic_ which are not now resting on the sea floor in the jaws of the Weddell Sea. The _Uruguay_ still sails from her home port of Buenos Aires, a proud part of national naval history.
# Antarctica with a Butler:
Jean-Baptiste Charcot's (1867–1936) Expeditions
_You secretly find out that your keel had been badly damaged by hitting a rock. Do you accept defeat and turn for home, or do you conceal the truth from friends and colleagues and sail back into the ice and unchartered waters that holed her?_
## THE FRANÇAIS 1903 – 05
On 26 June 1905 the President of the Royal Geographical Society in London welcomed his guest speaker back from the dead: 'You will remember the reports that were brought back to South America as to the probable loss of all concerned.' Jean-Baptiste Charcot had returned just two weeks before. He began by admitting it was 'not without a certain trepidation that I venture to address you.' This was not just formal modesty. Scott's _Discovery_ and Bruce's _Scotia_ had also just returned from Antarctica, and the audience included Captain Robert Falcon Scott. Charcot said that in France 'polar expeditions had been neglected' but he hoped France would take its place 'in the peaceful struggle against the unknown.' Because of limited resources 'it would be better to employ all our resources in thoroughly exploring a narrow corner, and thus securing trustworthy documents with accurate observations, than wandering listlessly up and down the seas' in a way 'more satisfactory to our vanity' but 'far less useful to science.'
Charcot was born on 15 July 1867 at Neuilly-sur-Seine, near Paris. His father Jean-Martin was a successful and wealthy pioneer neuro-psychologist who influenced Freud. The dancer Jane Avril, who featured as the flame-haired can-can artist in Toulouse-Lautrec's most famous poster, was one of his patients. His son also studied medicine, and as is common, found it hard to follow a famous father in the same field. Yet even after Papa died in 1893, leaving him 400,000 gold Francs, he continued to play the dutiful son, marrying a grand-daughter of Victor Hugo, and becoming Head of Clinic at the Faculty of Medicine in 1896. Only in 1902, aged thirty-five, did he throw over his old life and plan an expedition to Greenland. He went in at the deep end, contracting a purpose-built polar vessel from shipwright Père Gautier. Charcot consulted Adrien de Gerlache on the design, who recommended he incorporate the modifications developed by Arctic sealers and whalers, employing a heavier frame and bracing it with additional diagonal beams in the parts that took the direct hits from floating ice: the bow and the waterline. The propeller was to be retractable to avoid damage among dense ice. The forty-six-metre-long three-masted 245-ton schooner _Français_ took shape in St Malo shipyard. Singleton designs usually run-up singular bills, and Charcot's ate through his inheritance. To equip the on-board laboratories, he sold a Fragonard painting. Even so, there was too little money for a new engine and the 125HP second-hand one which was eventually secured was known to be short on power and reliability.
Early in 1903, as they were readying to go, concern was mounting over the failure of Captain Larsen to return from the Weddell Sea in the Antarctic. Nothing at all had been heard of Nordenskjöld. One of Charcot's sponsors, who had quickly become a close friend, was the industrial engineer Paul Pléneau, who had signed on as photographer. Charcot needed his agreement for a dramatic change of plan. Abandoning Greenland they would go south to search for Nordenskjöld. Would Pléneau come? He breezily replied, 'Where you like. When you like. For as long as you like.'
Public bodies contributed and the Paris newspaper _Le_ _Matin_ launched a public appeal for 150,000 Francs. When donations stalled at F90,000 the editor personally donated F60,000, and the Government conferred F90,000 more plus the status of being the first French National Antarctic expedition. Even so, his budget was still only half Bruce's, itself a thrifty enterprise. Aside from seeking Nordenskjöld, Charcot hastily assembled some aims. He made his core project discovering whether Antarctica was a continent or an archipelago: a secret its ice mantle hid well. Only modern remote sensing would solve it conclusively. In particular he planned to explore the north-west coast of the Palmer Archipelago around Liège, Brabant and Antwerp Islands, to study the south-west entrance to the Gerlache Strait, and to establish a wintering station as far south as the pack allowed. In the following spring and summer he would explore Graham Land especially the Bismarck Strait area, and push as far south as Alexander I Island. 'In a word, we had to continue and complete the labours of the de Gerlache and Nordenskjöld expeditions,' gathering botanical, zoological, hydrographic and meteorological data as they went.
De Gerlache took an advisory role but let himself be drawn into the project, and signed on as an expedition member, despite having recently become engaged to be married. There was a team of scientists and someone Charcot found indispensible: Monsieur Paurnelle, his butler. All were French with the exception of the Italian Alpine guide: Dayné. The first leave-taking of Le Havre on 15 August 1903 ended tragically when a stern cable parted, killing seaman Maignan, his only hand with Antarctic experience.
They re-started, and made Madeira and then Pernambuco in Brazil. Here de Gerlache told Charcot he had changed his mind and he felt honour-bound to return to his fiancée when they reached Buenos Aires. It may have been that de Gerlache had been given time to observe Charcot's leadership and thought he lacked the focus and determination to drive through a project under the stresses Antarctica could impose. Perhaps the aristocrat didn't enjoy being under another's command.
Reaching Buenos Aires on 16 November, Charcot learned that Nordenskjöld and Larsen had been rescued, so he waited there and met the returning men in December. Nordenskjöld studied his plans with approval, according to Charcot, who might have wanted to delicately counter de Gerlache's implied rebuke. He decided to avoid the fate of the _Belgica_ at all costs. De Gerlache's own account of the first Antarctic night glossed over its horrors, but it rang false alongside Dr. Cook's sober record of depression and breakdown. The Argentine Republic was cultivating a keen interest in Antarctica and Charcot found 'Nothing could exceed its generous welcome.' Buoyed up by the achievements of the _Uruguay_ in rescuing Nordenskjöld, Argentina committed itself to going down to the Peninsula to check for messages Charcot might need to leave. Charcot left on 23 December, bunkered coal in Ushuaia, and began acclimatising to the Southern Ocean with two gales in the Drake Passage.
Despite eighty years of Antarctic experience and over three hundred years of Arctic sailing, explorers were still allowing too little time for preparation, and arriving late in the short Antarctic summer. They did not make landfall until 1 February. For four days they coasted the north-west side of the Palmer Archipelago making a rough hydro graphic survey, hampered in turn by snow and fog. The ship's engine, so vital for close manoeuvring in ice, suffered serious failures as one boiler pipe burst and others became blocked. Looking for shelter to make repairs, they were driven from Flandres Bay, and ice and contrary winds blocked their attempted passage through the Lemaire Strait. Returning to Flandres Bay they anchored the ship to the ice, and stayed eleven days, making boat excursions in the area while engineer Goudier and his team laboured. Six years before, de Gerlache had found the area largely ice-free, it was now choked enough to suggest there would be little progress farther south. On 19 February they sailed and erected a cairn on Wiencke Island, in the beautiful Neumayer Channel, to act as a post box in case of emergencies. He had almost completed a circumnavigation of the island when he found a small bay at the south-west end of Wiencke Island which formed one of the best anchorages in the whole Peninsula area, naming it Port Lockroy after the French Minister of Marine. But Charcot wanted to overwinter farther south, and they pressed on, making a successful passage of the Lemaire Channel where Charcot vividly confessed the tension of navigating unchartered channels choked with ice. As they were forced right under one of the black walls of rock rising nearly vertically and disappearing into the clouds two thousand feet above, there was 'an agonizing moment in which one's whole body contracts, as though trying to make the movement' for the ship. The depth of the channel is highly variable. They were lucky; although close to the shore they were in one of the deep areas. When he emerged into the more open waters of the Penola Strait he compared the surveys of de Gerlache and Dallman a quarter of a century apart, and realised that the Belgian had unwittingly rediscovered and re-named Dallman's Booth and Krogmann Islands. Sometimes good exploration is just ironing out past errors. On the seaward side of Booth Island he found a bay he named Port Charcot: after his father, he was keen to stress, not himself. Within it was a tight cove which he named for the ship. With the hydrographer Matha making soundings from a dinghy ahead of the ship, the _Français_ edged in. If the coast farther south offered nothing better, they could retreat here.
They pushed on south hoping to make Cape Tuxen at the mouth of the mythical Bismarck Strait, but were stopped by ice in the Biscoe Islands at 65°59´S 066°22´W. Unwilling to retreat all the way back to Port Lockroy, they picked their way through waters spiked with rocks and reefs, and put in at the cove behind Booth Island, which, in their absence, had filled with wind-blown ice. They pushed their way in on 4 March, the very same day the Bruce was discovering Coats Land in the Weddell Sea. They were now at a similar latitude to Nordenskjöld's Snow Hill Base on the opposite side of the Peninsula, which would add interest to their meteorological and other observations.
The ship was anchored fast to the shore with seven feet of water at the bow and twenty astern. To counter the threat of ice driving in and crushing the ship, anchor chains were strung across the mouth of the bay, and a jam of middling-sized ice soon blocked out the biggest bergs. Huts were constructed ashore, including some fitted with sandstone pillars and marble slabs to provide solid bases for magnetic observations. A large whaleboat covered the petrol store. A selection of supplies were off-loaded as insurance against a ship's fire. They even managed to build igloos for meat including freshly caught seal which they killed in considerable numbers, for both men and dogs. In contrast to de Gerlache's loathing of seal meat, Charcot recorded: 'The meat is excellent' and the fat was used to melt ice for water, saving precious coal. The only two men who couldn't stomach seal enjoyed penguin meat. The signs for a healthy overwintering looked good. He also noted the kindliness of another creature: 'The cormorants, which were equally good eating, never left us.' The men were successful at ice-fishing.
They were well-supplied with food, drink, books and music. On Sundays there were concerts, poetry readings and lectures. Charcot himself gave English lessons. The bunks were designed with sliding doors to create a little private space for the men, but the atmosphere could be claustrophobic. Tensions developed, not concealed by Charcot's insistence that 'perfect harmony prevailed, and the best of spirits and good humour were maintained throughout the expedition.' That's too good to be true.
When choosing the harbour, Charcot had been satisfied that it was protected against winds from any direction except the north-east. The first thing his meteorological observations told him was that the prevailing wind, and the only one that brought gales and foul weather, was from the north-east. Such gales often followed dramatic rises in temperature, from –30°C to –3°C in a few hours. During the winter, the sea away to the west and south-west was always frozen as far as the eye could see, but the area a few miles to the south regularly broke up, though the ice never dispersed enough to use boats. This was unhelpful, as they were denied a solid reliable ice-platform for winter journeys, which would have kept men busy and active. The story of the winter months is of frustration which could only be relieved by the temptation to take risks.
To boost morale, on 30 May Charcot led a picnic excursion to Krogmann Island where they cut up the meat and butter with axes and ate while dancing up and down to keep warm. In mid-July, the First Officer and hydrographer Matha presented himself to the expedition leader in Charcot's role as medical officer. His symptoms were swollen limbs, palpitations, and exhaustion. Dr Cook would have known all too well what it was. Hanson on Borchgrevink's expedition had reported the same symptoms on the same day in July and been dead by October. Despite the fresh meat, Matha had scurvy. Charcot repeated Dr Cook's stove treatment, and augmented the diet with more fish and tinned milk. By late September Matha was fit for duty.
Champagne and cigars alongside the _Pourquois-Pas_?
They began planning a forty-day trip to begin on 15 August but on 2 August the ice broke. They set out for Krogmann a month later but fog came down and trapped them on an islet only twenty metres across. When it lifted they struggled over to Krogmann, found shelter under rock, and dumped a month's provisions before the pack closed up on them. They barely got back alive.
At long last spring came. On 24 November the whaleboat was loaded with twenty days' rations amounting to nine thousand kilos, scientific instruments and a sledge. The plan was to go to Petermann Island then cross the Penola Channel to the Graham Land coast where they thought they could see a slender open lead. The sea between Petermann Island and the mainland was an interesting mix of bergy bits too dense to sail through, separated by sea-ice too thin to be sure of standing on it. They spent a lot of time knee-deep in melt-water dragging the boat and taking what Charcot called frequent baths. Twelve days working up to eighteen hours a day were made doubly miserable when poor eye-protection caused snow-blindness. When they got close to shore the lead was not there; it had either closed over or been an illusion. They finally got ashore on 29 November in good weather south of Cape Tuxen, and were rewarded by long views into the interior and south down the coast. They surveyed and photographed Graham Land from Booth Island to the Biscoe Islands, from where they could see thirty miles south, proving the strait crossing the Peninsula, supposed by de Gerlache to lie south of Cape Tuxen, did not exist. Had de Gerlache got four miles farther south he would have discovered that himself. But the fictional Bismarck Strait still played tricks. 'I can almost say for certain [it] exists in pretty much the spot where he [Dallman] placed it.'
In December the Alpinist Pierre Dayné was able to make it to the highest peak of Booth Island's southern end and survey the road south, but the ship was still held fast, despite the attentions of spades, saws, crowbars, and even explosives. A lead opened to the south and by 17 December it had reached the ship, and a southerly wind took the ice away. They prepared busily, pausing on Christmas Day to celebrate with a Christmas tree brought from home. They sailed on the high water at 17:30 that evening, heading north first to update the message in the Wiencke Island cairn. Ironically, having decided to commend Port Lockroy to the whalers, Charcot was now iced in there. Dayné and Jabet climbed an adjacent peak and Gourdon ascended to a mountain pass which revealed that the de Gerlache Strait was open. A survey of Anvers Island found a new high point, the beautiful and imposing pyramidal mass which they named Français after their vessel. On 3 January they managed after four days to force clear of the ice and coast down the north side of Antwerp Island. By 7 January a fair wind enabled them to put out the boilers and sail south, heading for the Biscoe Islands at 66° south. On the 8 January, in squalls of snow, the visibility improved enough to reveal the _Français_ was about to be squeezed between two large bergs, but they slipped between them before they ground together. Charcot pressed farther south until late on 11 January he made a rash decision. He was in relatively open water with large swells. They were on the edge of close pack ice studded with fortress-like bergs and large plate-like ice-floes. He drove into the pack hoping to get close enough to the shore to land, but it quickly tightened until they could barely move. They forced their way out with great difficulty.
They drove through pack ice and successfully completed the channel between Adelaide Island and the mainland. They crossed the Antarctic Circle and on 11 January 1905 gazed on the dramatic landscape of Alexander Island. The profile was exactly as de Gerlache had described it, but the charted position was wildly inaccurate, putting it forty miles east and seventy-five miles south of its true position. The whole crew came on deck to enjoy the scenery. On 15 January Charcot handed over the bridge to Matha at 08:00 but stayed on deck as they entered a two-mile channel in which large bergs were slowly waltzing. The ship stopped dead, throwing them forwards, the foremast bent with the impact. Charcot wrote: 'We were about a mile from land when, passing approximately a cable's length from a large tabular iceberg more than 150 feet high, the ship received a terrible shock, the bow rearing almost vertically.' The ship fell backwards, and they saw ahead of her a single rock; the extent of the damage was unknown, but water was flooding in. The engine was running so poorly that the pumps had to be operated by hand. The engineer, Libois, lowered himself into the water in the bow and spent several hours working on the damaged hull. Officers joined the men on the hand-pumps. The coast offered no safe landing, and they had seen no wildlife on which to subsist. They did not know what other rocks lay in wait, and the pack was tightening up on them, hampering navigation of the crippled ship. A fault in Gautier's design of the watertight compartments now became apparent. The hole had been punched in a compartment ahead of their main bunkers which had no pumps, and no connection to any pumpable area. Men had to break open the front wall of the bunker to release the water. Charcot climbed on to the yards to direct her course. Out of the pack, but still fighting the leak, he asked his men to make one final sally south to join up, in his concept of the geography, two stretches of mainland coast. They agreed, though what the carpenter thought of this, below decks, waist-deep in freezing water, is not recorded. Charcot did not realise that one of these stretches of 'mainland' was in fact an island – Adelaide.
A north-west wind came, helping them exploit small channels and re-trace their route towards open water. They judged the leak was under control but the damage made it inadvisable to take risks among the ice. They set course north for the Biscoes and sat out a gale there on 19 January. Heavy seas and a three-day fog followed. Charcot stuck to his surveying plans and tried to complete survey work on the Schollaert Channel between Brabant and Anvers Islands, but in storms the propeller key, which locks the propeller to the drive shaft, sheared off. They limped into the shelter of Port Lockroy with relief: _'Oh! Etre au calme, savoir le bateau en securité et pouvoir dormir!'_ (Oh to be in calm water, to know the ship is safe, and to be able to sleep!)
The men were showing the strain. Charcot's comment that 'nearly all were enfeebled and ate little; but none complained, and they kept in good spirits' sounds like wishful thinking. From 8 to 11 February, they enjoyed the first good weather they could remember. The sunshine brightened spirits and refreshed tired bodies, but the clear water round their anchorage revealed that the damage to their bows was greater than anyone had thought. On 11 February, when the engineers had done all they could, they tried the engines, and they 'worked almost satisfactorily', which must have been almost reassuring. They went to Biscoe Bay, and completed their survey of the Schollaert Channel, but the summer was ending. It was time to go. They saw their last iceberg by Low Island, the southerly outpost of the South Shetlands, and on 4 March they made the windy open spaces of Puerto Madryn, in northern Patagonia, where they were greeted as if back from the dead. The _Uruguay_ had come looking for them but failed to find the cairns, and been stopped by ice short of 65°S before turning home, presuming them lost. Charcot found his wife had made efforts to organise relief, before divorcing him for abandonment. De Gerlache's return to his fiancée looked like a good call. At Buenos Aires the _Français_ could at last be inspected in a dock; she had lost seven metres of keel. She had been a brave vessel, the smallest of the era to go south, but she needed some tender loving care. The Argentine Government made a generous offer to purchase her as a supply vessel for her new bases, which Charcot accepted. He returned home a hero, met by the Minister of Marine bearing the _Légion d'Honneur_.
Charcot lost no one, and the expedition recorded successes in all areas of study. They drew new or improved maps of one thousand miles of coast, and made the first good chart of islands along the west coast of the Peninsula. From Booth Island southwards they achieved accurate latitude and longitude fixes for the outside of the Palmer Archipelago and the Schollaert Channel, for the Biscoe Islands, and two thirty-miles stretches of Graham Land that Biscoe had thought to be ice-joined islands. When Charcot delivered his preliminary paper at the RGS, one of the first comments from the floor came from one of Scott's companions, Dr H. R. Mill, who thanked him for nailing down 'the Biscoe Islands [which] have been drifting about on the chart.' In zoology they took a hundred bird specimens and fifteen species of fish, mostly from the top two hundred feet of the water column, which complemented the deeper water samples of the _Belgica_ expedition and others. One scientist who received specimens noted that the sea urchins, _crinoides_ , and starfish were not only all new, but highly interesting, and there was a 'superb collection of sponges.' In botany they had discovered both of Antarctic's only two flowering plants: _Deschampsia antarctica_ on Booth and in Biscoe Bay, and _Colobanthus crassifolius brevifolius_ on Antwerp Island.
The _Pourquois-Pas?_
## THE POURQUOIS-PAS? 1908–10
There is a small island called Petermann just south of the Lemaire Channel. For some cruise ships it is their most southerly point, as they pass through the Lemaire then continue just far enough south to see Adélie penguins. Petermann has the most northerly colony of this cold-loving penguin in the Peninsula islands. There is a small Argentine hut and on the shore just south of it, a cross remembering three young British men from the old Faraday Base who lost their lives returning from a late-winter sled journey when the ice broke up under them.
The smooth limbs of silver-grey andesite rocks offer a choice of landing places, the more popular being just north of the hut, at the painfully named Port Circumcision, christened for its discovery on the day of the Feast of the Circumcision, 1 January 1909. But the more humorous part of the name is really the word 'Port'; it is a very small rocky inlet.
When the President of the RGS introduced the returning hero to the membership for a second time, with Shackleton in the audience, he emphasised that his French guest was no less successful for having left the most glorious and high-profile destination to others. 'The fact that Dr Charcot has not reached the South Pole is perhaps sufficiently accounted for by his never having tried to do so.'
In his paper, given that night of 19 December 1911 Charcot set the scene:
The exploration of the Antarctic has been set going, and the movement ought not to slacken until the conquest – still only just begun – is complete. Captain Scott has just set out once more for the final attack on the South Pole itself [how Shackleton, who had fallen out bitterly with Scott, would have striven to appear impassive; none of them knew Amundsen had made the South Pole a few days before] and great expeditions are being talked of in Germany, Norway and Japan, while the Argentine Republic is maintaining a permanent observatory in the South Orkneys, and means to establish another on the west coast of Graham Land – the site of our winter quarters.
It was perfectly natural that Sir E Shackleton should return to this region, with the exploration of which his country had become so intimately connected, and it was equally natural, since he had announced his attention of so doing, that I should refrain from entering the same field, however tempting it might appear. The Antarctic is, besides, vast enough to permit many expeditions to work in it side by side with good results, and I decided to return to the region which I had begun to study in the _Français_ , in 1903– 5, continuing the labours of de Gerlache. My object – a thoroughly definite one – was to study from all points of view the greatest possible expanse of ground in the Antarctic without consideration of latitude. I had no hope of getting anywhere near the pole.
Specifically, was Alexander I Land really mainland or was it an island, and, unproven from his first expedition, was there an Antarctic continent?
Charcot led a crew of twenty-two, including eight former shipmates from the _Français_ , and a talented scientific team. Second-in-command Maurice Bongrain was in charge of astronomical, pendulum and seismographic observations, and hydrographic work. Third Officer Jules Rouch directed meteorological measurements, atmospheric electricity, and physical oceanography. Godfroy covered tidal observations, coastal hydrography, and the chemistry of the atmosphere; Gourdon, geology and glaciology; Senouque would conduct magnetic and actinometric studies (the science of rays) and the scientific photography. Liouville and Gain were zoologists.
He took advice from all the experienced Antarctic hands of the day: Bruce, de Gerlache, Shackleton, Nordenskjöld and Drygalski. He met Scott to trial experimental motor sledges and was impressed enough to accept three as sponsorship, though he would never find terrain suitable to even land them. For his vessel, Charcot first hoped to buy back the purpose-built _Français_ , now the _Austral_ , but Argentina was too pleased with its investment to part with her. Charcot went back to Père Gautier's St Malo shipyard and with an expedition budget of F800,000, swollen by public and private pride in his success, commissioned a larger vessel of eight hundred tons, forty metres long at the waterline (ten more than her predecessor) and with a 9.2 metre beam. He gave it the same name as his boyhood model yacht: the _Pourquois-Pas?_ , launching her on 18 May 1908. Her frame was even heavier than that of the _Français_. It was reinforced by diagonal braces below the waterline, and the wooden hull was sheathed in iron, then zinc. She had three laboratories fitted out courtesy of the Duke of Monaco, and an electrical generator would add to daily comfort and convenience, and assist greatly in the scientific pursuits. Her brand new engine was four times as powerful as the makeshift antique in the _Français_.
They left Le Havre on 15 August 1908. Poignantly, as Charcot passed the Banco Chico shallows in the River Plate in spring 1907, he saw the _Français_ stranded on it. He reached Punta Arenas on 10 December, and departed for Antarctica on 16 December. They made good use of their contacts to collect coal at the Magellan Whaling Company at Deception. Entering the caldera through Neptune's Bellows they saw three whale catchers, one with a whale in tow. Two turned to pilot them through and into Whalers Bay where they surprised Charcot by heading out to sea again; they had turned back in the breakneck high season of whaling purely as a courtesy to him.
It was a strange meeting. The shore was a slaughterhouse. They were invited aboard the largest factory ship, the _Gobernador Bories_ , but to make room for them to tie alongside it, the whalers had to move six waiting carcases, some of which exploded like cannon. 'The smell,' wailed Charcot, 'is unbearable.' The cabins of the _Pourquois-Pas?_ were cold, to save coal no stoves were lit, but they were greeted in the _Bories's_ ward-room by a parrot preening itself in front of a roaring coal fire. With satisfaction, he found they were using charts he had surveyed on his previous expedition.
Formalities and bunkering done, they removed themselves from the shambles of Whalers Bay and retreated up to Pendulum Cove, but found the tidelines hedged with whale skeletons still with enough meat on them to stink out the air. They climbed to a cairn and found a memento inside: a broken bottle containing a message from Captain Galindez of the _Uruguay_ dated 9 January 1905, when she had ventured south seeking news of them, and supposed them lost. Galindez had stated his intention to press on to Wiencke to look for them. They read it like Lazarus reading his own death certificate. They studied the layout of Pendulum Cove with interest. Foster's _Chanticleer_ illustrations from 1829 showed a little creek running away to the right as you approach the shore. Seventy-six years later this had gone, as the _Uruguay's_ 1905 sketch-charts showed, the cliffs apparently having collapsed and filled it. The cove was now an open bay.
The scientists spent the week before Christmas running loose over the island: 'the zoologists Liouville and Gain are scouring the neighbouring area, collecting and classifying all they can.' Late on the evening of 23 December Charcot realised he had not seen Bongrain and Boland for some time. He found them in a field tent using the midnight light to continue their pendulum observations. Charcot returned to ship and brought them cakes and Mariani wine, a patent tonic in which Bordeaux wine was fortified with cocaine. Rouch took soundings in the harbour and compared them with Foster's. In one spot where Foster had found 582 feet, Rouch found only 378 feet. The caldera floor had risen up around two hundred feet. Their medical expertise was now called into use when a whaler's steam slicer cut off four fingers from the operator's hand. The machine was surrounded by blood and guts so gangrene was likely; they operated to tidy and clean up the wounds. He recovered well. The whaling manager Andresen promised to keep tabs on the _Pourquois-Pas?_ and visit the Port Lockroy cairn, if possible in their first season, or failing that, a year thence, in January 1910. Charcot admitted past arrangements had not been perfect: 'I was keenly reproached over the last expedition for not having made sure of a shelter in an emergency.' Before leaving, Charcot recommended, as the whalers went farther south, that they use Port Lockroy as the best harbour in that area. They left Deception on Christmas Day 1908, heading for Port Charcot.
As they departed down the Gerlache Strait Charcot remarked 'About us an innumerable quantity of whales are plunging.' The contrast with modern ships, where passengers and staff scan for whale blows for fruitless hours, is heightened in irony by Charcot remarking: 'I personally claim to have done my small part' in reviving Southern Ocean whaling. They first went to Port Lockroy where they opened the message cairn to find it contained just one message: their own from 1905. On 27 December they neared their old overwintering site at Booth Island, and sent ahead their motor launch to see if their Port Charcot anchorage was clear. Its engine failed when they were in the northern mouth of the Lemaire Channel. Bitterly cold, they were forced to row most of the way back before it finally started.
Reaching Booth Island on 29 December, they manoeuvred the ship into their old anchorage in Français Cove at Port Charcot with some difficulty, since she was ten metres longer than her predecessor. They lodged the bows onto the beach before anchoring her to shore. They strung the steel hawsers from their drag nets behind the ship to try to protect her against wind-blown bergs, but this time the cove first filled with small ice which did not stop large bergs driving through them right up against the ship. They rigged up a hose running meltwater directly to the ship, a fuel – and time-saving luxury – and killed penguins, Charcot rather hypocritically musing: 'Why is man bound to do evil as soon as he visits any place?' before passing them to the cook.
The whaleboat from their previous visit was full of ice but otherwise intact; the magnetic hut was as good as new. Their temporary house had slid off down the slope and then been bombed to destruction by falling ice. They soon saw that the hawsers were not keeping out the bergs the way the chains had: they drove over them or under them. On New Year's Eve the swell rose during breakfast and they heard an ominous grinding noise above their heads; a large berg had broken through and was working against the rudder. With poles they forced it away. At 20:00 the same evening another large berg armed all round with ugly angles and edges made its way straight towards them. The men's nerves were frazzled, fearing the rudder would be destroyed. At the last minute, the wind died and the current bore it away. Following a Spanish custom, continued in South America, they took out a surprise luxury with which to welcome the midnight chimes and the New Year: a small supply of Malaga grapes: 'they tasted as if they had just been picked.'
They realised that Français Cove was a hazardous anchorage, and they had been extremely lucky on the previous expedition that the ice had come to shore in a way that fortified their own protective measures. The luck did not recur. They had to relocate.
On New Year's Day they moved a short distance to Petermann Island where, on the shore facing the Penola Strait, there was another inlet where they could lodge the ship which would not be so vulnerable to large bergs. They had seen it on their previous expedition, but that year it had been iced up for eleven months. They named it Port Circumcision. If you veer to starboard before landing and look to the right, you can see, except at high spring tides, a horizontal line incised roughly into the rock. In a crisp light you can also see two initials: PP, the abbreviation for a small ship whose name reflected the character of the man: the _Pourquois-Pas?_ – the _Why Not?_ Charcot would note that: 'Throughout the whole remaining ground covered by our expedition we did not discover a single spot at which one could think of wintering with a ship.' Antarctica was a tight-fisted host.
As on the first expedition, they found that the Grandidier Channel between the Biscoes and the mainland was always fraught with dense ice crammed with large bergs. On 4 January 1909, Charcot, Gourdon and Godfroy took the launch to Cape Tuxen, surveyed that coast and the Berthelot Islands. They struggled to return to ship, and had to be rescued by the _Pourquois-Pas?_ The launch party had no time to recover from that danger before the ship itself was imperilled. They struck a rock near Cape Tuxen. The stern settled lower in the water. The hull had been torn open, and she was sinking. Charcot ordered cargo to be moved forward, the pumps began to gain on the in-rushing water, but she did not lift free. They had to hope the next high tide would provide that last nudge to raise them clear, but tides in Antarctica have a small range, often no more than a metre and a half. The tide crept up and the ship just rose up enough to free itself. Investigations back at Port Circumcision found that although the hull was leaking, Gautier's super-strength specifications for her hull had left them structurally sound.
Charcot's response was to push south and drive his ship south of the Antarctic Circle and into the narrows where Hansen Island lies in Hanusse Bay tucked in the south of Crystal Sound. In this freezing alley between Adelaide Island and the Arrowsmith Peninsula, where I have sailed my farthest south, it was not, they found, too cold for an Adélie rookery. The elegant birds stood with the line of their chic black headscarves running below the plump cheek and down through the front arc of their flippers, their white eye-rings giving them a mad stare.
They went back eastwards then south, finding Adelaide island was much larger than John Biscoe had supposed. Despite good weather, Biscoe had wrongly guessed he was close to its shore, estimating there was an ice-edge ten feet high when it was actually one hundred-foot cliffs. Its area had been similarly underestimated; it was seventy miles long not eight. They discovered that in this locality the ice was much less than in 1905, when they had met dense pack seventy miles north of Alexander I Island. Now they passed south of it before meeting the pack. Charcot thought Jenny Island in Marguerite Bay, jointly named for Bongrain's wife and his own, might prove a good site for a base, and the modern British Rothera Base lies just to the north-east of the site he had in mind.
From 73°W to 80°W they sailed to south of the track of the _Belgica_ , at times following the path she had drifted when gripped in the pack. From 82°W to 87°W they recrossed to the north of it as far as Peter I Island, and were also to the north of Bellingshausen's route in this area. They correctly assessed that Peter I seemed to be an isolated island with no other land close. From Peter I they were able to hold a course to the south of Bellingshausen and to north of _Belgica's_ ice-drift as far as 103°W, where the latter escaped. In the area of 73° to 80°W they sailed with caution since Knox, sailing on the Wilkes expedition, had reported seeing an ice barrier here. There was nothing. Charcot supposed it to be another one of the mirages, caused by refraction through thermal layers in the air. Around 71°S and 107°W they crossed Cook's path, and met pack thirty miles north of his farthest south. It would have been possible to steam a little farther south and better his latitude but Charcot thought it was pointless to burn coal to pursue an egotistical goal of no likely geographical importance, and went on his way. Dwindling supplies of coal and an outbreak of scurvy forced Charcot to regroup.
Looking round at the shelterless shores and the huge bergs that seemed to reassure them of deepwater whereas in reality they were snagged on reefs, he pondered: 'A ship overtaken by a storm in this region would find itself in an almost desperate situation.' In this area they surveyed 120 miles of new lands, and saw decisively that Bellingshausen's Alexander I Land was an island. Attempts to land on it were all foiled by ice. Having decided it was impossible to overwinter in this area they returned to Petermann Island and spent the autumn making excursions in that area. They formed the first Antarctic Sports Club running sled and ski races. To occupy minds, there were back copies of _Le Matin_ from their sponsor, and fifteen hundred books, with another on the way: _L'Amant de la Dactylograph_ , or the _Typist's Lover_ , which Third Officer Jules Rouch wrote and read out in instalments. They set up four huts ashore, filled with scientific instruments and powered from the ship's electrical generator.
The winter was trying, with gale force winds for most of the nine months they were iced in. Despite the stores of fresh seal and penguin meat, several members developed severe scurvy. At the end of the winter, from August onwards, the sea ice became heavier and heavier. In spring, on 18 September 1909, a party attempted to cross Graham Land to the Weddell Sea, to join up the west Peninsula observations with the eastern ones of Larsen and Nordenskjöld. But it proved 'impossible to surmount the perpendicular wall of granite and ice which seems to bound the whole region.' Charcot was not well enough to travel, and stayed on board until mid-October, suffering from 'Polar anaemia.' This diagnosis was derived from Dr Cook's reports on the de Gerlache expedition, when reports of anaemia, a side-effect of scurvy, were frequent. On 31 October they decided to break out of the cove and take the _Pourquois-Pas?_ north to Deception for coal. With great difficulty they reached Deception in late November, where they learned of two historic journeys. Ten months before Shackleton had got within a hundred miles of the South Pole, and seven months before, Peary claimed, he had reached the North Pole. His claim would soon be seen to require an elastic view of the truth.
Charcot asked a Norwegian diver at the whaling station to look at the damage to his ship and report privately to him; the diver reported that it was unsafe, and they should get north while his luck held. Charcot swore him to secrecy, obtained one hundred tons of coal, and sailed on 7 January 1910. South.
In their injured ship, they returned to the south-west of Alexander I island and surpassed all previous farthest souths in that zone. He called the new land Charcot, again after his father, and his officers and crew might have wondered why he did not try harder to get close to shore and land; only he knew their hull could take no more shocks. They followed the edge of the pack, running along the latitude of 70°S as far as 124°W. 'The ice now became more abundant, the coal was exhausted, and the general health unsatisfactory, so that it was necessary to turn once more to the north, and after ten days of somewhat eventful navigation.' They reached Punta Arenas on 12 February, and were fêted by France on their arrival at Rouen on 4 June 1910.
His main account of the voyage was published in 1910. They had surveyed 1250 miles (2000 kilometres) of unknown or partly known coast, and drawn the most accurate map for decades to come. They returned with three thousand photos and would publish twenty-eight volumes of reports. In eight months of observations at the winter station, his team recorded two earthquakes and five seismic storms. From Deception, on 23 December 1908, they had observed the last contact of the moon departing the sun during a solar eclipse, valuable in measuring astronomical distances. The geologists had shown that the same diorite and granite formations known from Graham Land continued farther south. Collating their results with those of _Belgica_ , they established the existence of continental shelf, defined its western limits, and linked up Nordenskjöld's work to north and de Gerlache's to the south. They had made some surveys inland from the heights of the Graham Land glaciers, and throughout the trip the timing of their magnetic observations had been co-ordinated with inde pendent observations on Staten Island and South Orkney for future comparison. The zoologists had collected embryos of all penguins at all stages, and by ringing birds had established that penguins do not breed until they are two years old, after which pairs return to same rookery and the same nest. They collected specimens of the embryos of all the species of seal, including the first ever embryos of the leopard seal.
When Charcot finished giving his paper to the Royal Geographical Society in London, the President said, 'In opening the discussion, it is natural I should first call on those Englishmen who are best capable of expressing the thought-out opinions on the value of this work.'
Sir Ernest Shackleton took the floor: 'When I heard the accounts of the gales, and saw the efforts that had been made to keep that ship safe and quiet during those long months of storm, I came to the conclusion it is better to have a wintering party and no ship in such quarters; and keep the ship for navigation during the summer. What Dr Charcot has gone through to my mind must have been one of the hardest experiences possible.' The French Ambassador Cambon diplomatically complimented both countries in a phrase: 'You have Shackleton, we have Charcot.'
A great friend and fellow-traveller of Shackleton and Scott on the _Discovery_ was there: Hugh Mill, born near John O'Groats, as he said of himself, where 'no land lies between the shore of my place of birth and the North Pole.' Mill observed, 'Now Dr Charcot has, in the most interesting manner, connected together all the earlier related discoveries, and showed how much greater they were than the discoverers themselves had supposed.'
Louis Bernacchi, physicist on the _Discovery_ , said 'this quiet, unostentatious, and really solid scientific work of Dr Charcot compels our deep admiration and recognition,' and added, 'I think probably the climatic conditions and the general conditions are more severe in the latitude in which Dr Charcot wintered than in the corresponding latitude in the Ross Quadrant.'
Outside France, Charcot is now little known to the general public, but contemporaries often rated him the greatest explorer of his day. Amundsen summed up Charcot as 'the French savant and yachtsman', 'opening up a large extent of the unknown continent.' He thought 'the scientific results were extraordinarily rich' and 'the point that compels our special attention in Charcot's voyages is that he chose one of the most difficult fields of the Antarctic zone to work in.'
Charcot never returned to the Antarctic. During World War One, despite poor health, he worked in the antisubmarine force and was awarded the DSC. On 15 September 1936 he was sailing on the _Pourquois-Pas?_ when it was wrecked in a gale on the coast of Iceland. There was only one survivor; Charcot, and forty-two other men died.
#### CAPE HORN
People who have never seen the sea know Cape Horn's reputation as a destroyer of ships and men. Why, among all the seas, has this cape come to represent all that is wild and deadly dangerous about the sea?
The Horn was not discovered until 1616, when two small ships under the commands of Willem Schouten and Le Maire came down the east coast of Patagonia, bound for the Pacific. It was nearly a century since Magellan had first found and sailed through his strait, but since then, it had proved a graveyard for shipping. It was nearly sixty years before Francis Drake became the second person to force a route through and get home westward. These two Dutchmen from an insignificant port called Hoorn missed the eastern entrance to the strait, where the coast is sandy and backed by low hills, and continued south along an increasingly rocky and wild coast, where islands send out ridges into the surf, long-toothed like a log-cutter's saw, which the sea breaks into stacks, then rocks, then spume-smashed reefs. They kept far enough out to sea to stay safe on their southward course until they saw the coast begin to curve away towards the west. It was impossible to tell whether the land to starboard was a land mass or a constellation of islands. When, in the 1820s and 1830s, it was eventually surveyed properly by Captain Stokes, an earlier master of the _Beagle_ , it drove him to such despair that he shot himself in the head. His successor Captain Fitzroy gazed at the charts he had made and said the myriad rocks and islets looked less like the sea and more like the stars of the sky. One high headland, brutal, simple as a stone axe, rising nearly thirteen hundred feet straight out of the water was rounded, and the coast began to run north. They named it, like one of their little craft, after their home port: Kap Hoorn; it became Cape Horn.
The cape is dangerous because south of it, below 56°S, there is no land to obstruct storms, or check the waves. It is open ocean, all around the world. There is a west-to-east ocean current running through the Drake Passage, and the prevailing winds are westerly. A square-rigged ship was poor at sailing into the wind, so the passage from east to west had both wind and wave stacked against it. It was nothing to take a month to round it. At that distance from Europe, scurvy would now start to sicken the crew unless their diet had been carefully managed. Men were washed overboard, ripped from the rigging, or left helpless in the hammocks with broken bones. A sailor who fell from the top mast hit the deck with the same impact as a motorcyclist coming off at 50mph and hitting a house. Others simply slipped off into the night when feet became too cold to feel a rope and fingers too numb to grasp one. Captain Bligh, who would later suffer mutiny on the _Bounty_ , began the practice of sailing east to get to the Pacific: a much longer route, but with the wind behind them, driving them on. Many skippers also sailed with poor charts and limited navigation equipment; they saw land here only when they were too close to escape from being dashed upon it.
If it was this bad, why sail round the Horn at all? Why not stay in the shelter of the Strait of Magellan? Firstly, the strait is a winding S-shape round which clippers, not very manoeuvrable ships, had to thread a path, tacking in limited space, always looking for stray rocks and reefs. Although much of the channel is very deep indeed, there are isolated peaks rising up from the cold of the lightless depths to prick the hull of passing ships. There are local katabatic winds powerful enough to flatten a sailing ship, putting her on her beam ends, in other words, her masts horizontal. These go by the misleadingly cute native name for them: willi-waws. Seamen and owners preferred to fight the open sea, rather than the sea and the land together.
# Red Water: Whaling
_Herman Melville, writing in_ Moby Dick, _argued that whales would never be exterminated by hunting. How wrong he was._
Whaling was the only commercial activity successfully pursued for any time in Antarctica. Other chapters, particularly chapter six on Deception, have dealt with some aspects of whaling. This short chapter provides context for those sections and describes some sites you may visit where whaling was significant.
In _Moby Dick_ Melville dismisses fears that the great whales could be hunted to destruction. His final reason was, 'They have two firm fortresses, which, in all human probability, will forever remain impregnable... whales can at last resort to their Polar citadels, and diving under the ultimate glassy barriers and walls there, come up among icy fields and floes; and in a charmed circle of everlasting December, bid defiance to all pursuit from man.' Melville was quite wrong.
The small but very sheltered harbour of Neko, tucked deep in Andvord Bay, is a favoured place to set foot on the mainland. The shingle beach only becomes difficult to land on when ice blows hard against the shore. The presence from 1911 –16 and 1918 – 24 of Salvesen's factory ship the _Neko_ in the area became notorious. She was not, as is often assumed from name, part of the Japanese fleet. She was a 3,576-ton former passenger liner in the _Kosmos_ fleet, and was converted to a factory ship at Leith, before commencing Antarctic whaling in 1911.
The British feared the 'whaling industry may come to a natural end by the reduction of the three species of whales on which the industry is based.' In the June preceding the season 1921–22 they issued a ban on hunting sperm, right and humpback whales. There was outright mistrust between the Norwegian whalers and the British officials. That season the _Neko_ reported no humpbacks caught, but two of its labourers transferred the following season to rivals A/S _Hektor_ and personally confirmed to the British Magistrate and Whaling Officer, Arthur Bennett, that the _Neko_ had taken forty-three humpbacks. It had chosen Andvord Bay because it was remote from the officials in Deception Island, and used the isolation to flout new regulations requiring more efficient use of carcases. The _Neko's_ English master, Thomas Sinclair, was renowned, Bennett reported, for continually 'shifting ports and selecting ports for anchorage remote from, and most difficult of access.' Even among the grossly inefficient Norwegian whaling ships she was one of the worst. Her condition was filthy and her catch reports unreliable. She continued at this game until 1923 when she spent the season in South Georgia, before foundering on the Brazilian coast, returning at the end of the season in 1924. Beneath her decks a still worse practice went on. Whalers coming south called in at the Cape Verde Islands for water and fruit. The islands were, and still are, poor compared with other Atlantic islands. Misguided young men would stow away, perhaps hoping to get to America, or maybe simply to try to force a master to employ them. Salvesen's response, and doubtless some other companies' too, was to put them to work poorly fed and clothed, at shovelling coal. If they only worked ordinary shifts they were not paid at all. Opportunities to transfer them to company ships going north were spurned. Bennett believed Salvesen ships called at Cape Verde intending to entrap men. Whaling Officer Hardy's end of year report concluded: 'The company's intention was without doubt organized slavery, and the same could be said for Salvesen's South Georgia station at which large numbers of stowaways are stated to be working without pay.'
Today at Neko Harbour the only noticeable remains on shore are of a much later Argentine refuge, which fell into disrepair following severe fire damage to the Argentine navy's principal support vessel the _Irízar_ , named after Julian Irízar who captained the corvette _Uruguay_ when it rescued Otto Nordenskjöld from Snow Hill. In the winter of 2009 the hut was destroyed by storms.
Another whaling centre was Wilhelmina Bay between Emma Island and the Peninsula. It is usually visited only by ships which bring passengers to dive the wreck of the _Guvernøren_ , a state-of-the-art whale factory ship which caught fire and was beached there. But a Zodiac cruise along the shore reveals many industrial relics, beginning with the wooden navigation marks used to guide vessels through the reef-filled bay. Along the shore you see metal posts and chains used to anchor ships and, in the days before factory ships, hold whale carcases securely in the shallows to be flensed. When snow melts towards the end of the season, some of the small islands reveal abandoned boats and barrels, and the foundations of huts and storehouses.
On the south shore of Trinity Island is Mikkelsen Harbour, skirted by steep, thunderously grumbling glaciers. In the middle of it is a small island, unnamed on marine charts but christened D'Hainaut Island in 1952 after Chilean lieutenant Ladislao D'Hainaut. You can stun your expedition staff with this unimportant information.
It's a horrible landing for Zodiac drivers because the shore is guarded all the way in by rocks deep enough to be out of sight and shallow enough to wreck a propeller. The shore has several old water boats on it, and a sea of whalebones dating from 1910 –1917. The factory ships _Bombay_ and the _Ørn_ and _Ørn II_ were regulars here.
Port Lockroy off Wiencke Island was an early refuge harbour for whalers and a few remains are still visible. If you step ashore at the landing site nearer the hut, rather than below the boathouse, there are heavy chains stapled to the rocks. At the head of the bay are more, and the remnants of painted graffiti on the rocks above. Hubert Wilkins was responsible for some of the Lockroy graffiti when he persuaded the master of the _Solstreif_ to take him and John Lachlan Cope home from their disastrous expedition. The chains were fixed there by the factory ship _Solstreif_ , which means the _Sunbeam_. She used the mooring regularly until the crash of 1931, but her name, painted on the rocks, and the date 1912, are now fading. She was owned by A/S _Norge_ and at 4637 tons was the third largest factory ship in the Falkland Island Dependencies before World War One. She was one of the first ships to successfully run a guano drier at sea. In whaler's jargon, guano is the dried meat and bone ground up for fertiliser. British policy was to press for more efficient use of carcases but owners and captains didn't like the cookers on board. Guano was tricky to process but not highly priced, and it took up storage space that could otherwise be used for much more valuable whale oil. This didn't stop the _Solstreif_ doing so well in its first season that its owners paid dividends of 50%. One of its captains, Thorvald Andersen, personally harpooned over five thousand Antarctic whales. The whalers also bequeathed another gift, stored in the whaleboats whose outline can still be seen below the front door: dynamite. They carried it for blasting their way out of ice.
You might have thought that the twentieth century was a period of change towards more responsible management. It wasn't. Whaling is a perfect example of the problem of the commons, first stated by the British economist William Lloyd in 1833. It shows how individuals exploiting a shared, unowned resource will maximise their own short-term self-interest and destroy the resource.
To simplify matters, I'll look at the twentieth century kills of the three largest southern whales: the blue, and their less massive cousins, the fin whales, and sei whales. Humpback whales had already been slaughtered out of the equation.
Before the First World War, the British had talks with the whaling companies from the many nations who were using as bases Deception Island, and adjacent areas claimed and adminstered by Britain. Whales were so abundant only the blubber was cut out and the rest of the body, rich with oil and other products like bonemeal, was cast away. It made commercial sense to fill their holds easily with the highest value product, oil, and maximise profits. If they did not, rival companies would out-perform them and shareholders would rebel. In the early days returns for shareholders over a single season could exceed 50% or even 100%.
The First World War virtually stopped whaling, and from 1920-24 catches of the preferred species, the blue, were at about 4,500, and fin catches were around 4,000. Hardly any of the less profitable sei whales were taken. In the following five years catches of blue and fin doubled or more. To the technology of the explosive harpoon, were added stronger and faster hunter ships, feeding a factory ship. The cumbersome business of cutting up a whale in the water alongside the ship or at the beach had been replaced by a stern slipway allowing the carcase to be hauled up a ramp for processing on deck before being dropped into the tryworks, now situated in the hold. Cutting a large whale now took just a few hours. The invention of the reverse osmosis technique to convert seawater to freshwater meant that ships could stay out at sea all season, outside the jurisdiction of the fledgling attempts by the British to impose more responsible practices.
The combined impact of these advances was that the Antarctic whale catch rose from 30,655 whales in the 1929-30 season to 40,201 the following year. The market was glutted, prices collapsed and the 1931– 32 season saw catches fall below ten thousand. In the five-year period to 1934, blue whale catches were still double fin catches, with sei whales insignificant. Whaling picked up after the Depression but the five years to 1939 show a shocking turnaround in the species being taken, and it is driven by necessity. The take of blue whales falls, and the take of fins more than doubles, becoming the principal species.
The Second World War gave another respite but whales breed slowly. The blue whale catch fell steeply and declined steadily until 1965 after which none could be found. Catches of fin whales soared to a level far above any past takes of blue whales, totally dominating the catch from 1955–59. By this time sei whales began to form a small proportion of the catch, perhaps 5%. Within a decade little else was being caught. One species after another was annihilated.
Attempts were made to regulate the industry. The International Whaling Commission (IWC) was set up in 1946. In its own words, its purpose 'is to provide for the proper conservation of whale stocks and thus make possible the orderly development of the whaling industry.' Like any authority attempting to resolve the problem of the commons, it faced opposition from operators who had owners and shareholders at their backs. Stocks of whales continued to fall and the IWC decided at its meeting in 1982 that there should be a moratorium on commercial whaling on all whale stocks from 1985/86. Exceptions have been made for scientific research and for indigenous societies with a tradition of whale hunting. Both rationales have been questioned. The Japanese press for quotas which far exceed the maximum numbers needed to inform science, and some native societies like the Inuit take whales like the bowhead whose survival is in doubt. They argue that their hunting did not cause the problem.
Three main countries now wage the political battle to ease the moratorium: Iceland, which manages its own fishery waters independently and effectively, Norway, and Japan. Part of the problem is a sense of nationalism. Norway and Iceland do not like to be told what to do by nations who have failed to manage their own fisheries. Japan claims a cultural tradition of eating whales, but in truth it largely dates from expediency, in the fat and protein-starved years after their military defeat in 1945. In both Japan and Norway the traditional whaling areas are economically depressed areas with marginal constituencies. Political parties rebuff the whaling companies at their peril.
The integrity of the IWC is now widely seen as damaged. Japan pays the expenses of other delegates in return for helpful votes. Of its eighty-eight member states, all with full voting rights, ten are land-locked, two more border only the Black Sea, and others have only a corridor of territory leading to a one-port coast. Its members include the Saharan wastes of Mali, the Asian steppes of Mongolia, and the intrepid Leviathan chasers of the tax-free haven of San Marino. There is also illegal whaling; South Korea is suspected of tacitly permitting vessels to hunt.
But the real danger in the long term, biologists agree, is pollution. Most whale species recover once hunting stops or is restricted, but if the seas are not healthy, their food and their fate are unsure.
#### THE GUVERNØREN
Before World War One the _Guvernøren_ was the largest factory ship in the dependencies, at 5459 gross tons. She was built as the Europa to carry cattle between New York and London, before being converted to a factory ship and renamed the _Thøger_ , then the _Guvernøren_. She was managed by veteran Leif Bryde, and was the most sophisticated processing ship of her day. She had the new Hartmann rotary cookers available in 1912 which broke up the meat and bone as it cooked them, improving the quality of oil, making it lighter and less acidic. She also had a guano plant, making fertiliser from bones and other by-products. Except on shore stations, it was difficult to make this pay, and few ship owners tried. After unsuccessfully hunting off South Africa she came down to the Antarctic Peninsula in 1913/14 and produced 3500 tons of oil and 2500 sacks guano from 544 whales. She made good profits and seemed to be the future of the industry. Next season, on 27 January 1915, she caught fire; it probably started in her processing machines. All of the eighty-five men on board were saved, but the 2600 tons oil on board were lost; over four hundred whales had died for nothing.
Her bow and bridge are above water, and have been colonised by nesting Antarctic terns. You can look into the portholes, and divers report unused and used harpoon heads are still in the hold. The stern lies under water and you can take Zodiacs over it and, on a calm day, see the outline of its hull diving into the deeper water, where a near-complete whale skeleton lies on the sea bed next to her.
# The Boss: Ernest Shackleton (1877–1922)
_The tale of an expedition which barely got under way at all, which never reached the mainland, whose ship was destroyed on its maiden voyage, and whose country, engulfed in war, left them to die. Out of this shambles Shackleton forged the legend of_ Endurance.
Headmasters seldom recognise the abilities of children with unconventional talent. They run a system. People who do not conform to the system do not have other talents; they are wrong. In Dulwich College, green and genteel in the leafy London suburbs, a young Irish boy from Kildare was appraised for his school report. The verdict: lacking in motivation. The college was founded in 1619 by the Shakespearian-era actor and theatre manager, Edmund Alleyn, but its greatest treasure is not one of Will's First Editions, but an old wooden boat called the _James Caird_ which their unmotivated old boy once took on a two-week cruise from Antarctica to South Georgia, in the greatest open boat journey ever made. The boy was Ernest Henry Shackleton. Aged sixteen he left school for the Merchant Navy. His first captain had a sharper eye for character: he was the 'most pig-headed, obstinate boy I have ever come across.'
Like many of the British explorers, including his great rival Scott, he did not begin with a great passion for polar exploration. They simply saw it as the best opportunity of the time for a young officer to make a name. Polar expeditions put you in the headlines in a way that being second officer on a cargo vessel or administering an African backwater did not. Shackleton also wanted to impress a Mr and Mrs Dorman, whose daughter Emily he wanted to marry. Mr Dorman was a prosperous solicitor and probably had bigger ambitions for Emily than an officer fresh from his first voyage, bringing guano from Chile. Meanwhile Ernest wrote to her misquoting Robert Browning: 'I hold that a man should strive to his uttermost for his life's set prize.'
Shackleton first went south in 1901– 03, serving under Scott on the _Discovery_ Expedition to the Ross Sea; in fact Shackleton had been pressed on Scott by a sponsor the Irishman had charmed. Whilst resenting Shackleton's charisma and natural leadership, Scott recognised his physical power and took him on the final three-man push south with the doctor, Edward Wilson. Shackleton did not respect Scott's leadership and the two fell out. They suffered badly from scurvy; Shackleton was coughing up blood, and was a passenger for part of the journey back. Wilson told Scott that Shackleton would not survive the journey. The patient overheard and muttered, 'I'll outlive the pair of you.' He was sent home early on the relief ship. As it sailed away, he stood on deck sobbing his heart out at the humiliation. From now on, he would run his own expeditions. He would define himself by being everything that Scott was not. But one thing he could not blame Scott for was the scurvy. The man in charge of diet was Ernest Shackleton.
He returned in 1907 in the _Nimrod_ leading his own expedition funded by sponsors. With two companions he made it to 88°23´, just ninety-eight nautical miles from the Pole. Then he turned back. He knew he could reach the pole, but they did not have enough fuel or food to return to base. Wracked by starvation and lacerated by cold, they got back by the skin of their teeth. Had they gone on south for even one more day, they would have perished. He wrote to Emily, now his wife, 'I thought you would rather have a live donkey than a dead lion.' At his side for this and every future expedition was Frank Wild. Wild was not an immediate fan of his leader, but an incident on the return journey bound him to 'The Boss' forever. They had sat down to a meagre supper of stew and a few biscuits. Wild soon finished and sat suffering still from cold and hunger. Shackleton had one biscuit left. He forced it on Wild. In his journal, Wild wrote in his diary, underlining every single word with emotional emphasis: 'I do not suppose that anyone else in the world can thoroughly realise how much generosity and sympathy was shown by this; I DO by GOD I shall never forget it. Thousands of pounds could not have bought that one biscuit.' On return, Shackleton was knighted. He would never set foot on the Antarctic continent again.
That was Shackleton's last chance to be first to the Pole. Others would be ready to build on his achievements before he had time to launch another expedition. December 1911 saw Amundsen triumph. In January 1912 Scott reached the Pole but died on the return. His party was scurvy-ridden, and out of fuel and food. Most of all Scott was paralysed by his failure to get there first. For nine days they lay dying in their tent eleven miles from a huge supply depot. The weather was normal for that area. Scott showed enormous courage but he and his followers died of poor planning and bad leadership. He is an easy target, and to some degree a fair one. He persisted, to what now seems an absurd degree, in believing polar travel was a test of character where Englishmen would pull through. For ten years British expeditions persevered with man-hauling sleds. Amundsen called it stupid after one day. He was a polar fanatic and had learned and applied every hard-won lesson from Arctic exploration and Native peoples. On his way back from the Pole he left half a ton of excess food on the ice and fed his dogs chocolate for fun. Amundsen put on weight during the trip. Scott starved and froze to death.
Shackleton too had his faults, and these are now glossed over as part of re-writing history and its heroes to suit our times. When news of Scott's death reached Britain, his incompetence in the sanitised white wilderness was refashioned to make him a saint to self-sacrifice, a perfect message for a nation going to war. Shackleton's character was flawed in ways we now find it easier to relate to, even if we do not condone it. He was a heavy smoker and drinker, was unfaithful to his wife, and a poor provider for his family. His men would learn to trust him with their lives, but not their money or their wives. While Scott was a man who relied on rank and uniform for authority, Shackleton wielded it by force of personality, including moments of breathtaking generosity. He never asked a man to do more than he would do himself. Although he could be bad-tempered and moody, the worse things got the better his temperament became: a priceless virtue in an expedition leader. For precisely this reason, Sir Edmund Hillary said he was the explorer he would most like to have met.
This background is necessary to know how much Shackleton had staked on the Imperial Trans-Antarctic Expedition of 1914 –17, now usually referred to by the ship whose name summed up the ordeal: _Endurance_. The plan was to cross the entire continent, via the Pole. One party would land on the Australian side of Antarctica and follow Shackleton's 1907 route to the point where it left the Ross Ice Shelf at over 83°S. Their task was to lay depots for Shackleton to pick up. The other, under the Boss himself, would begin on the South American side and land in the south-east corner of the Weddell Sea. They would pioneer a totally new route to the Pole, then descend (the South Pole is two miles above sea level), picking up those depots.
The expedition was funded by sponsors, and sailed in debt. The photographs, cine film and every man's journals were Shackleton's property to be sold to pay the bills when he returned. By 1914 enough had been scraped together, but war was looming. If it was declared, they would be morally obliged to offer both ship and men for service. The decision would fall to the Admiralty, and a man with known scepticism for Antarctic adventures: Winston Churchill. When affairs couldn't get worse, Sir Ernest's brother Frank Shackleton was accused of stealing the Irish Crown Jewels. Frank was a homosexual Guards officer whose confidants included the King's brother-in-law, the Duke of Argyll. The potential for scandal affecting the Crown was enormous. Sponsors and money melted away. Ernest lost £23,000 of £30,000 committed. He managed to coax £29,000 from a Dundee jute magnate called James Caird. That was just enough to sail. He accepted a present of a Bible from Queen Alexandra and with an almighty sigh of relief gave orders to depart. That very day, war was declared. Shackleton cabled Churchill offering his ship and men for war service. He received a one-word reply: Proceed. Churchill probably thought the best place for a maverick during a war was out of the way. At Buenos Aires, slack discipline under the maverick Captain Frank Worsley led to dismissals, hirings and, they found out after sailing, a stowaway, a teenager from Newport, south Wales: Perce Blackborrow. Shackleton admired his pluck, and made him steward.
At South Georgia, they received news that the spring ice was the worst any of the old whaling captains had ever seen. He respected their judgement and delayed a month. But when they did sail on 5 December they met pack ice just three days later at 56°S, hundreds of miles north of its usual limit. They already had eleven thousand miles of sailing behind them. They were still a thousand miles away from the planned landing in Vahsel Bay. From now on, every yard would be fought for.
They had auxiliary steam engines and the ship was strong enough to batter through ice: it had been built as the first purpose-built polar cruise vessel, with Adrien de Gerlache as one of the partners, but others withdrew and the project foundered. This was the _Endurance's_ maiden voyage, and she was filmed and photographed in her progress by Frank Hurley, an ambitious Australian who, with Herbert Ponting, was one of the two greats of early Antarctic photography. He sat hatless and gloveless astride spars, exposed in the rigging, toting his forty-pound camera, to expose the glass plates. The sailors admired his toughness, but his high opinion of himself earned him a nickname: The Prince.
But ice-breaking gobbled up precious coal, and it was vital to retain stocks for manoeuvring in shifting ice, in those times when the winds were low or from the wrong quarter. On 18 January 1915 exceptionally cold weather fell on them and the sea froze them in tight. When it relented, they fought free but could not get closer than sixty miles from Vahsel Bay. They stared south over ice ruptured into badlands by colossal pressures created by the slow clockwise rotation of the ice in the vast bay that encloses the Weddell Sea. There was no way of moving their stores and equipment across such terrain. The men watched Shackleton closely to see how he would cope with seeing all his preparation, done under constant financial strain, mocked by the unforgiving continent before he could even step ashore. The sole member released by the Government, the eccentric marine Thomas Orde-Lees, recorded that it was 'a catastrophe which hardly bears thinking of from either a sentimental or financial point of view.' The surgeon Alexander Macklin had a professional interest in Shackleton's morale and wrote: 'He did not rage at all, or show outwardly the slightest sign of disappointment; he told us simply and calmly that we must winter in the Pack; (he) never lost his optimism.'
One special problem faced Shackleton that only a few others like de Gerlache and Nordenskjöld had faced: the ship's officers and men, who formed the greater part of his companions, didn't want to be there. They had never signed on to stay in the Antarctic. Shackleton's Merchant Navy experience put him in a good position to manage this. He understood the regular sailors. He was not much interested in science and scientists, they were necessary to justify and promote a polar expedition and attract funds, but he would have gone without them if he could. No one on board needed to feel that one group was privileged over any other. And while he famously appointed scientists on a whim about their appearance or character, he made sure his officers and sailors were technically competent, including one man who would first stage a one-man mutiny, then use his skills to make escape possible. From the outset Shackleton saw him as a risky appointment, but the man was so exceptional at his craft, Shackleton over-rode his own fears, saying, when recruitment was finished, he was 'the only man I am not dead certain of.' He was the carpenter McNeish.
It is important to remember that although the ship is trapped, during this time life is still as comfortable as it can be in Antarctica. They had all the amenities of a ship at sea and few of the discomforts. Boredom is the main enemy; it leads to discontent. The scientists at least have work to do, but some of the seamen spent all their leisure hours sleeping or smoking in their bunks. This apathy aroused contempt in the officers and men. Nor were the sailors impressed by the scientists, many just out of university. Seeing the gentlemen swabbing floors as part of Shackleton's democratic allocations of chores did not make the seamen feel any camaraderie. For their part, the boffins doubling as housemaids posed dutifully for photographs but recorded their resentments in private diaries.
There was one group everyone loved. Shackleton's previous trips had cured him of believing ponies were any use; he brought only dogs. Hugging a big cuddly dog or playing with a puppy was some substitute for the loss of the company and affections of loved ones. There was plenty of time to go out on the ice and train men and dogs in the arts of sledding and skiing. Igloos were built on the ice to shelter the dogs: they were soon named dogloos. Puppies were born and Tom Crean, the hard-as-nails Irish second officer, was more protective about them than their mother was. In May, the days became short; they contemplated the time coming, it proved to be seventy-nine days, when the sun, the source of hope for spring, as well as heat and light, would not show its face. Would they all be alive when it returned? Both the two previous parties to overwinter unplanned had suffered deaths and the crushing of the spirit that went with disposing of a body in that bitter wasteland far from home. Would they all be sane when light returned?
The two previous enforced overwinterings had very different outcomes for the ships. De Gerlache's _Belgica_ had been swept round at random then released into the ocean at the very end of the season. Nordenskjöld's _Antarctic_ had been crushed and destroyed. Shackleton found they were in a giant clockwise rotation. Where the ice was hindered by dragging along the shore, or reefs and islands, or ran up against large icebergs stranded on the sea floor, huge stresses were created. They would be helpless to save the ship if she were carried into those compression zones. The winter darkness passed but the ship had suffered. She had been tilted and squeezed until the timbers groaned and emitted thin screams. No ship on earth was built to withstand such pressure; Nansen's _Fram_ had successfully been designed to slip up above it. The ice itself moaned and growled relentlessly. Captain Worsley wrote at best it was like 'heavy distant surf', at worst like 'an enormous train with squeaky axles' or 'steamer whistles... underfoot' or even the 'moans and groans of damned souls in torment.'
All through the winter Shackleton talked to all the officers and men, listening to concerns, patting them on the back, disciplining where necessary, lifting spirits. He thought optimism was the greatest of virtues and he had the worst relationships with the pessimists like Orde-Lees and McNeish. Orde-Lees worried so much about protecting his supply of stores that it was hard to get him to issue anything, even food. He became known as the 'Belly-Burglar'. In July the bosun John Vincent was reported for bullying with blows and language, and demoted to seaman.
On 15 October, early spring in Antarctica, the ice opens into a lead and the ship floats upright. Just two days later it closes again. That is the last time they will see the _Endurance_ afloat. She is now leaking and by 24 October they are pumping twenty-four hours a day. They begin offloading stores onto the ice, but continue to live on the ship. Two days later the pressure flexes the ship so hard that her planking starts to gape apart. Worsley wrote that the wind in the rigging was 'making just the sort of sound you would expect a human being to utter if they were in fear of being murdered.' The dogs stare at it, howling. Nearly all sailors are superstitious and some more so than others. Tom McLeod refused to eat penguins because the souls of dead fishermen went to live in them. Now eight emperor penguins come to them across the ice, walking with their dignified gait, their flippers to their sides. They pause before the ship and begin singing to it.
The men ate their last meal on board on 27 October and moved hurriedly to the ice alongside. They called it Dump Camp. It soon became clear that the dogs were not going to be part of any solution. Needing fresh meat to augment the pemmican, they ate three times as much seal meat as all the men put together. That same day those animals too young or unfit for work were shot. The remaining dog pemmican could double as emergency rations. They even shot the cat, Mr Chippy, a favourite of the carpenter McNeish. It was a rare tactless gesture. What it ate was neither here nor there, compared with the dogs' ravenous appetites. The cat was worth its weight in morale.
With the ruins of the ship in the background, Shackleton summoned the men and addressed them simply. 'Ship and stores have gone – so now we'll go home.' But how? They had no radio transmitter; they couldn't afford one. The war begun in August 1914 had been predicted to end by Christmas but they could not rely on anyone looking for them. Shackleton had told friends not to start worrying until February 1916. If he waited for Churchill to start any relief, it would be a long time coming. Sir Winston was writing to his wife 'when all their impoverished & brokenhearted homes have been restored, when every hospital is gorged with money, & every charitable subscription is closed, then & not till then w[oul]d I concern myself with those penguins.'
There are words that after ten years lecturing on Shackleton I can barely speak without tears coming to my eyes: 'For scientific discovery, give me Scott; for speed and efficiency of travel give me Amundsen; but when disaster strikes and all hope is gone, get down on your knees and pray for Shackleton.' It is usually credited to Sir Edmund Hillary. While working with his son, Peter, I asked if he would mind asking his father if he'd actually said it. Peter came back: 'No he didn't, but he said "It was such a damn fine quote that if anyone ever asked me, I never denied it!"' It was actually written by Raymond Priestley, a geologist who worked with both Scott and Shackleton. In the days that followed, Shackleton would show why he deserved such emotional praise.
They had three small ship's boats, wholly open to the elements. They would need freakishly good weather to sail back to South America across the worst stretch of water in the world. I have crossed the Drake Passage over eighty times and spent six months of my life in it. Maybe six times I have seen conditions fit for a boat such as the _James_ _Caird_ , and never for the ten successive days it would have taken them to sail it. Their only hope seemed to be in reaching areas on the other side of the Peninsula, or in the South Shetlands, that were frequented by whalers. When Nordenskjöld had been trapped on Snow Hill Island, Shackleton had been consulted on what stores might best be dropped off for him, so he knew that at Snow Hill, Hope Bay and Paulet Island there were supplies. They could haul across the ice, or wait for it to melt and sail there. Worsley wanted to find a big stable iceberg and sit on it until the ice melted. Waiting for fate wasn't Shackleton's style. McNeish proposed taking the three boats apart and building a small sloop they could all sail in. Shackleton refused. He wanted to retain the mobility of small boats that could be lifted as the sea re-froze or the pack closed.
While these choices exercised Shackleton and Wild, the rest of them put up their tents and organised stores. The plan was soon announced. They would head for Nordenskjöld's old base at Snow Hill, 312 miles away. From there, a 130-mile crossing of the Peninsula would lead to Wilhelmina Bay, a popular anchorage for whalers. To emphasise the need to leave behind all excess weight, Shackleton led by example, throwing onto the ice the gold coins kept as emergency funding, his gold watch and the heavy Bible donated by the Queen. First he tore from it the dedication in her own hand, and the page bearing Job Chapter 38 Verses 29–30: 'Out of whose womb came the Ice? and the hoary frost of heaven, who hath gendered it? The waters are hid as with a stone, and the face of the deep is frozen.' One possession was saved: Hussey's banjo. The Boss thought music was good for morale. Three days later the sledges were loaded with provisions and fifteen men began hauling the first boat over the ice. With the stores in it, the load was around one ton. Orde-Lees, the only experienced skier, was deputed to move up and down the line of march, trouble-shooting. Shackleton watched him a while then remarked: 'Do you know I had no idea how quickly it was possible for a man to get about.' Orde-Lees reserved his dismay for his diary: it 'set me wondering why he had not come to this conclusion long before... Amundsen's rapid journey to the pole was enough to convince one of the value of skis.' But that first day's futile hauling taught them a brutal truth. The boats were too heavy; in two days they covered two miles. To ease the loss of face, Shackleton took Wild, Hurley and Worsley to a vantage point, surveyed the ice and announced the abandonment of the plan. Even the curmudgeonly carpenter McNeish felt part of the team decision, confiding in one of the better-spelled parts of his diary: 'We have made up our minds to stop on this floe until the ice opens.' They camped where they were, and called it Ocean Camp.
After their hasty evacuation, they did not have all the equipment and stores they wanted, but the ship's hatches and companionways had collapsed, blocking re-entry to the hull. The carpenter had only an adze, a chisel, a hammer and whatever nails he could re-use. Hurley proposed breaking in, and more was salvaged, including his glass photographic plates, stored incongruously in the ship's refrigerator, which served to keep them cool but not cold, sealed in soldered tins.
The _Endurance_ lingered another month, sinking head-first at five in the evening on 21 November. You might have thought it would be inexpressibly painful to see the ship which was your home and transport disappear, but when her wreckage slid into the depths, reactions varied. Hurley wrote, 'We are not sorry to see the last of the wreck – an object of depression to all who turned their eyes in that direction.' Shackleton felt the hurt deeply and tried to confide his feelings to his journal but could not: 'I cannot write about it.'
Because the shore party stores were being consumed by the crew as well as the expedition staff, they would not last without killing seals and birds. Shackleton's biggest error in leadership was to give disproportionate weight to the demoralising effect that stockpiling seal and bird meat would have on the sailors. He said they must not be seen to be planning for a long stay. First officer Lionel Greenstreet received the news while reading Nordenskjöld's account of the crew of the _Antarctic_ setting up camp on Paulet Island after the ship sank. The first thing they had done was stockpile meat.
As Christmas approached, Shackleton's feelers told him that the fickle men were getting restless. Although conditions for hauling were little different, he decided to make work for idle hands. He announced they would celebrate Christmas on 22 December and begin hauling the day after. When they tried, McNeish was missing his cat and suffering from piles. He felt the boat hauling was pointless, and he was frustrated that his suggestion to build a sloop had been rejected. On 23 December they laboured for eight hours and made just 1.25 miles. There was twenty-four-hour daylight but they worked during the night hours when the surface was a little colder and easier to travel. On 24 December leads opened through the ice and they did no hauling. On 25 December they covered 3 miles, and a miserly 1.75 miles the following day. On the 27 December McNeish threw down the harness in disgust and said he would do no more of this stupid work. The ship had sunk and he was no longer subject to orders. This was technically true. However Antarctica is a poor place to found a republic of one. Shackleton knew that McNeish's words would linger in the men's minds. Everyone knew McNeish was good at this job, and hard-working when he saw the point of the task. On another voyage a fellow sailor wrote of him: 'He could work seemingly endless hours, and if he ever knew fear, not a suspicion of it showed on the solid, graven brickwork of his face.' He was also one of the oldest and most experienced men in the party.
Shackleton felt he had to respond to the truth at the heart of the carpenter's argument. He gathered everyone and read out the articles they had signed, but almost certainly not read. He made up clauses that they would be paid after any sinking, while still serving the expedition. He had no funds to cover this promise. McNeish was now isolated, without support, but Shackleton never forgave him. He took his revenge much later, and many of the officers and crew were shocked and angry when they learned what he had done. McNeish's own diary has no word of his mutiny.
Most of the ice was now rotten: too wet to walk on, too dry to sail. They were forced to retreat half a mile to a more solid floe, and stay there three months. Shackleton signalled his revised intentions with the name: Patience Camp. That patience had to last until the ice broke up close enough to launch the boats. McNeish had been right, and they were now doing what Worsley had advocated in the beginning.
In mid-January most of the remaining dogs were shot, leaving just two teams. Hussey was complaining of severe lassitude and stopped his nightly round of visiting the tents playing his banjo. Not everyone missed it. McNeish recorded his relief while enjoying the evening silence once again: 'his 6 tunes is heart-breaking.' On 21 January the drifting pack took them north across the Antarctic Circle. By mid-February, Orde-Lees was judged to be taking too many risks hunting seals on thin ice, and was banned from hunting. Worsley muttered that the letter of the law was becoming more important than whether there was blubber to burn and meat to eat. Perhaps Shackleton's need to define himself against Scott made him obsess about his record of never losing a man. The men were eating around 1800 calories a day, barely enough for a sedentary life, never mind physical exertion in great cold. Short rations demanded risks be taken. The ice-floe that was the limit of their secure world was reduced by splitting and progressive erosion at the margins. Orders required everyone to sleep fully clothed, including boots, for evacuation within thirty seconds.
In early March Orde-Lees received another round of the abuse that he often invited. He announced he was seasick. He was right, the ice was breaking up. So were some of the men. One wound copper wire round his belly and announced he was going to walk home.
The slow currents of the pack had taken them away from where they needed to be if they were to strike west for Snow Hill, tucked close into the eastern coast of the Peninsula. Likewise it was bearing them away from Hope Bay. Paulet was their last chance to reach anywhere with even primitive resources: Larsen's rough stone hut and the unused stores. On 11 March the ice opened. They were eighty-five miles from Paulet. Two days later it closed. By 22 March the prospect of being locked in for another winter, this time without a ship, was the fear muttered only quietly in corners. The next day tantalised them with a distant glimpse of the Danger Islands at the entrance to the Antarctic Strait: the first land they had seen in sixteen months. Once more there was too little ice to sledge on and too little water to sail in. Soon, regardless of their position, they would have to take to the boats. There was no more use for the remaining dogs; they were shot on 30 March and eaten in a gloomy feast. Wild led the awful work. He remarked there were a great many men he would rather shoot before any of the dogs.
Sailing back west against the prevailing winds would be slow and dangerous. But there was little land left to the north, just two remote outposts of the South Shetlands: Clarence Island and Elephant Island, and they were a hundred miles away across open ocean. They had two sextants, and a chronometer they did not trust. There was no useful information about currents to help them assess their longitude, but they were always borne east of north: away from all known refuges.
By 9 April the ice floe was little bigger than their camp, and was crumbling under their feet, yet the surrounding ice closed up, denying them any water to launch into. Their floe cracked in two, then cracked again. At 1pm they entered the boats. They began the first leg of a journey which de Gerlache had contemplated, when trapped in winter ice, and concluded there would be no more than a 1 in 100 chance of success.
The first night they camped on a new floe, cooking up the pemmican in a stew they called hoosh. They were now eating the dog pemmican. The lack of carbohydrate in it soon gave them diarrhoea. At 11pm Shackleton felt what he described as an 'intangible feeling of uneasiness' and he left his tent to prowl. Their floe was lifted by a large swell and cracked beneath his feet and right through the sailors' tent. Seaman Holness was in the water in his sleeping bag. Shackleton took hold of the bag and single-handedly hauled him from the water. A few seconds later the two halves of the floe, many hundreds of tons of ice, came back together in a terrible thud. The next night they retreated into the pack for shelter and camped on a floe which was eroding rapidly, but in the morning they could not leave it because the surrounding ice had pressed too close to allow launching the boats. Shackleton swore there would be no more floe camps, and the next night only Green the cook was put ashore while the rest rowed all night to keep station. The next day was sunny and clear, and Worsley could take an accurate sextant sighting. They absorbed the shocking news; the current had taken them ten miles backwards and thirty miles east. The wind moved to the north-west making it impossible to head for the South Shetlands, so they ran instead for Hope Bay on the tip of the Peninsula, and the Nordenskjöld three's old camp. On the following night, 12 April, the wind changed again and they turned around and set course for their last chance of land: Elephant Island. The three boats were tied in a line. All this time Shackleton stood up, to be seen by the exhausted men to be caring for the cockleshell flotilla every hour of the day and night. Hurley lost his mittens and although Shackleton was in the most exposed condition, he offered his own mittens. When the Australian refused, Shackleton said he would throw them overboard if he did not accept. The high exposed mass of Elephant Island was now in sight. Wild glimpsed it briefly during the night but no one else could see it. Shackleton later admitted that he wondered whether his number two had started hallucinating. Worsley, after fifty hours on watch, finally took a break to sleep. At 5am third officer Alfred Cheetham thought he heard the _Dudley Docker_ crack in the centre. The men all moved aft to reduce the strain on it, putting themselves under the spray breaking over the stern.
Twenty-three nautical miles long, Elephant Island's shores are mostly ice-cliffs and rocks that repel any attempt at landing. Freezing katabatic winds rip down without warning from the frost-fractured mountain peaks. Offshore reefs snag icebergs which foul the course in to the shore. The large ones may stay there for years; it was here, in November 2007, that I sailed along a twenty-nine miles long tabular berg lying against Clarence Island.
In the dark they almost missed the landing. They managed to beach the boats and stagger ashore at the eastern end of the island at Cape Valentine. It was the first time in almost five hundred days they had set foot on land. They dragged the boats up the beach. In the photos Hurley somehow took amid all this, one man can be seen, sitting apart on the beach. It is the stowaway Perce Blackborrow, whose feet and fingers were so frostbitten he could not assist. Shackleton's bare hands suffered a badly frost-bitten finger. Some men stood shaking helplessly. Many knelt at the edge of the stream running down the beach and guzzled water. Others let the stones of the beach trickle through their fingers like pirates scooping treasure from a chest. The surgeon Macklin said Shackleton 'looked gaunt and haggard.' Another said he looked aged and bowed down. Macklin thought only Frank Wild looked something like himself. He did not, like Shackleton, feel himself morally responsible for the wellbeing of twenty-seven men. He sent the two climbers Orde-Lees and Wordie to scale the land behind. They reported no way up for men in their condition. Shackleton saw the vulnerability of the site to storm and rock-fall and immediately began planning to leave. Next day Wild reconnoitred the coast and found a spit seven miles off that looked more promising. It is now Point Wild. The glacier that provided them with fresh water has retreated, exposing the shingle to erosion. It now looks less inviting than Cape Valentine, and I visited it for ten years before I found conditions safe enough to land. But there were penguins and many other nesting birds, as well as seals.
Two boats were upended side by side and made into accommodation. The _James Caird_ , Shackleton announced, shouting above the gale that had accompanied them to Cape Wild, would sail with himself and five others north and east to South Georgia to get help. On that course the currents and winds would assist them; they would not if they headed north and west to South America. The whalers would help them return to pick up the other twenty-two. Standing on the narrow strip of shingle with an ice-choked sea for a back-drop, he asked for volunteers to get back into a small boat on their third day on land in six months, and sail off to their likely deaths. There were some volunteers, but his selection epitomises the soundness of Shackleton's man-management. He did not take the six bravest and best. He took Worsley for his navigating skills and uncanny instinct when working blind. He left Wild to be a firm fair hand in charge of the camp. He left both surgeons because it was likely Blackborrow would require his toes amputated to quell incipient gangrene. One would act as anaesthetist for the other's surgery. He was not expected to survive, nor was the first engineer Louis Rickinson who had suffered a heart attack. He took Crean because he needed someone he trusted absolutely, and another Irishman, seaman Timothy McCarthy, who was not only strong and resolute; Worsley thought him the most irrepressible optimist he had ever met. The other two are fascinating choices. He took McNeish for his carpentry skills and to keep the dissident away from a camp where there would be much idle time. Finally he took the demoted bosun John Vincent, another man much happier when active. McNeish now had to improvise a deck in a land without trees. He dismantled additions he had made to the _Dudley Docker_ , and cannibalised a sledge and some packing cases, creating a frame over which canvas could be stretched. The canvas was frozen so hard Cheetham and McCarthy had to work the needle with pliers. McNeish completed the work in three working days, sitting out one day of blizzards. The twenty-two-foot boat set sail just after midday on 24 April heading her blunt nose at twenty-two-foot miles of the worst ocean in the world. An albatross picked them out and kept them company. If successful, they expected to be able to return within two weeks. Orde-Lees argued without success that the shore party should kill all the game available in case they were there for the long haul. Wild ridiculed him, though Orde-Lees knew Speirs Bruce had seen a great reduction in South Orkneys' penguins during the winter. Orde-Lees was their Cassandra, his advice and predictions often true, but usually ignored. He also said the pack could delay relief, and the day after the _James Caird_ sailed, it closed up the shore and stayed put.
For the men left behind on the shore, the longest walk ended one hundred yards from camp; they were trapped on the beach by mountains steep as cliffs and a glacier to the west, their only view was out to the sea that had nearly killed them, but was now their only hope.
Out in the _James Caird_ , the reindeer sleeping bags were moulting and the Irish were keeping their spirits up.
Shackleton: 'Go to sleep Crean and don't be clucking like an old hen.'
Crean: 'Boss, I can't eat those reindeer hairs. I'll have an inside on me like a billy-goat's neck. Let's give 'em to the Skipper and McCarthy. They never know what they're eating.'
The _James Caird_ sailed poorly, it was around 25% over-ballasted because Shackleton had overruled Worsley, despite knowing less about boat sailing. They made forty-five miles on the first day as a Force 9 gale blew up creating twenty-foot waves and making all but McCarthy and Worsley seasick. The tiller-men were soaked by breaking waves, the sleeping men were doused by leaks through the canvas decking. When off-duty, three men slept in a space seven feet by five. Worsley sometimes started awake with a dream he had been buried alive. On 26 April Worsley got his first navigational fix. He had both sextants and the chronometer, but he thought the timepiece was not running true. On 29 April he got another fix and they made ninetytwo miles, but the next day a confused sea forced them to hove to on a sea-anchor. The leaky deck was cured by colder weather which made the spray freeze on impact; they took turns to chop it off. Whenever anyone began shivering Shackleton ordered hot drinks for all, never singling out anyone as weaker. On 2 May the sea anchor parted; a lump of ice had chafed through the painter. The third of May was fine enough to dry out their gear. Worsley got sightings this day and the next, which was very pleasant sailing. Spirits rose; they were two thirds of the way there. The good mood was dampened the next day. Firstly they had finished the first cask of water, and when they opened the second, they found it had leaked and was tainted with saltwater. From now on they were reduced to a quarter-pint of foul water each day. Then Shackleton thought the sky was clearing and remarked on it. Then he shouted 'For God's sake, hold on! It's got us!' as in panic he realised the light was the surf on top of a wave bigger than any he had ever seen in all his life at sea. They were swamped and only after ten minutes bailing did they start to believe they would survive.
One man did not care. A few days into the voyage, John Vincent, who began the expedition as the most physically powerful man of all, had collapsed mentally, becoming a passenger and patient. On the tiller, he did not register the wind shifting north and hitting them almost sideways on, so they were forced to stop sailing and hove to, shipping water. Two days' cloud reduced Worsley to calculating by dead reckoning that they were 115 miles from South Georgia. South Georgia is only one hundred miles long; staying on course was vital. The price of missing it and drifting too far north was discussed secretly by Worsley and Shackleton. The skipper recalled the verdict. Soon 'our water would be finished... Our food would have lasted a fortnight, but that didn't alter the problem if we had no water, so we dropped the discussion as it was futile.' They had set their course for the Willis Islands at the west end of South Georgia, intending to then slip round to the north-east coast where the whaling stations lay in the sheltered fjords. On 7 May Worsley made sextant sightings guided only by shooting the brightest part of the fog obscuring the sun's disc. He admitted to Shackleton that he could not be sure of his position to better than ten miles. Shackleton reset the course to ensure they would make landfall at some part of the south-west coast. This made their target bigger but less convenient; there was no habitation anywhere on that coast. At dusk they saw kelp and dreamed of land and fresh water. At eight the next morning a South Georgia shag appeared. Worsley knew they never flew more than fifteen miles from land. It was also the fifteenth day at sea: 8 May. At 12:30 McCarthy called out the single word: 'Land!' After a brief glimpse, the black crag sank back into the murk, a strange emblem of salvation.
Worsley found his latitude was accurate to within two miles, which would be excellent for readings taken from the stable bridge of a large ship, but was plain miraculous in their circumstances. However his longitude, calculated from the faulty chronometer, was twenty miles out. They lay off a large natural harbour called King Haakon Bay. The entrance is guarded by reefs. Even today, only a single safe route in is known to the small cruise ships that on rare occasions visit this site to pay homage to Shackleton's landfall. The only ship, the _Polar Star_ , to deviate from the line, to avoid an iceberg, holed itself on a rock. Parched, cold and wet as they were, Shackleton ordered them to stand off for the night. The reward for such brave caution came at 6pm. The wind came at Storm Force out of the WNW, harder than anything they had suffered on the voyage. They bailed all night; even Vincent roused himself to help with the drudgery. Dawn showed them what the night's sound and fury had only hinted at. They were trapped in a triangle of sea between South Georgia and the off-lying Annekov Island. The sea and air united in an unholy marriage, the wind scourging through the waves making the air so heavy with spray a man could drown in it, and the wind so laden with hard drops of ocean it was like raising your face to a sandstorm of small, cruel stones. Down-wind, where the Force 10 gales were blowing them, the uncharted coast of central South Georgia was a wall of surf. The _James Caird_ was strained to her limits as Worsley dared to add reefed sails to the main and mizzen mast to help her claw her way off the shore. The planks had been chafed thin by prolonged sailing in ice back on the Peninsula. She hit the waves so hard the planks in the bow flexed open with the shock and shot spurts of water into the boat. When the storm dropped, the wind shifted to the north-west and they could make no progress to rounding the west end of the island and continuing to a whaling station. As they relaxed and changed shift, Crean caught his shoulder and heard a small tinkle. It was the pin which held the foot of the mast in place. Had it come out during the storm the mast would have snapped and dragged them into the sea.
While Shackleton was asleep, Worsley, obsessed with finding an outpost of civilisation, steered for Willis Island. He had taken no account of the fact that they had no water and Vincent and McNeish were barely alive. When Shackleton woke he ordered him to make for King Haakon Bay. They found a cove. It was 10 May, the seventeenth day out. A small stream ran down the beach and they fell to their knees and drank in new life. Next morning they were still too weak to haul the boat clear. They had to dismantle McNeish's additions to drag her up the beach. At some point, her rudder had fallen off. She was now too strained to sail round to the north-east coast. So were Vincent and McNeish. Help had to come from the land. The coast of South Georgia was sketchily surveyed; the interior was unknown. They must cross it or die where they were.
They fed on albatross chicks, as big and fat as turkeys, fattened up for the coming winter. Worsley said the first time he killed one he felt like a murderer; after that all he could think of was how good it had tasted. On the third night their rest and recuperation was interrupted in the middle of the night by Shackleton calling out 'Look out, boys, look out! Hold on!' He was pointing at the cliff opposite, black capped with snow, and hallucinating the reappearance of that one giant wave that had brought them to the brink of destruction. To bring back his optimism, the boat's rudder bobbed ashore. With all the Atlantic to sail over, the rudder came back to the Boss's feet. On 15 May they sailed to the head of the fjord where they had seen a saddle above that offered access to the interior.
They dragged the _James Caird_ out of the water and turned her upside down. She was now Peggotty Camp, after that character's boat-house in Charles Dickens's _David_ _Copperfield_. By them on the beach was a pile of timber, all from ships wrecked on the coast or out to sea. This is what happened if you weren't successful. Ever on the lookout for easy money, Shackleton mused to Worsley: 'Some day, Skipper, you and I will come and dig here for old treasure or perhaps sleep quietly with all the other old seamen.' They waited for a moonlit night. McCarthy was judged too weak for major exertion. If they were to cross the island it would be Tom Crean, the Boss and the Skipper who must do it, and do it quickly. Good weather was short-lived, and in bad weather in South Georgia, people were short-lived. At three on the morning of 19 May a bright moon guided them up. From the breaking of the _Endurance_ to here had been a journey of fifteen hundred miles. Husvik whaling station and safety for some, if not all, of the party was twenty miles away. They carried a stove and three days' food, but no sleeping bags or tent. The ground was too broken for a sled. McNeish had taken brass screws from the boat and put them in their boot soles to give grip. Only two of them had the best cold-weather boots, Shackleton had given his to one of the men left behind. His adze was their ice-pick. Their one frayed rope was just fifty feet long.
Their weakness, even when starting, may be measured by the fact that they estimated most of the altitudes along the way at two to three times their actual height. When they reached the saddle now known as the Shackleton Gap they were the first people to set eyes on the island's interior. They saw their task spelled out: saw-toothed ridges, cloud-covered pinnacles, near-vertical cliffs, and steep glaciers tumbling in confusion beneath. Snow smothered everything, and they now walked onto a glacier. No one spoke of the condition of the rope holding them together. They saw a smooth even surface below and altered course towards it, looking for easier walking on what must be a frozen lake. Dawn showed them it was the ocean: Cook's Possession Bay, accurately shown on their charts. They re-traced hard-won steps. Their true route took them towards five peaks lying right across their path. They headed to the southernmost of the four passes. Soft snow gave to their knees every step. At nine they ate hot food and before noon they took the last steps that showed them they had laboured, breathless, to a cliff top fifteen hundred feet high, and impossible to descend. Trying to maintain their altitude, they traversed round to the next col and found another version of the same scene. The third pass showed them snow lying over ice. Worsley wanted to try it. Shackleton thought it too risky. It was late in the day and a cold fog was rising from the lower land. When they reached the final pass it was dusk. The fog obscured their footsteps. They began descending but agonisingly slowly. They risked dying from cold and caution. They could slide down but their landing was lost in the mist; it might be another precipice. Crean had been in a similar position once before, on the other side of Antarctica. Short of time and supplies, and faced with a three-day detour or tobogganing an ice-fall, they had taken the risk and lived. Shackleton said, 'It's a devil of a risk, but we've got to take it.' They coiled the ragged rope and sat down, the Boss in the dangerous front seat. Worsley again; 'We seemed to shoot into space. For a moment my hair fairly stood on end. The quite suddenly I was grinning! I was actually enjoying it... I yelled with excitement, and found that Shackleton and Crean were yelling too.' They slowed and stopped in a bank of snow, standing up in disbelief that they were still in one piece. Shackleton's cautious side reasserted itself: 'It's not good to do that kind of thing too often,' but 'the risk was justified this time.' They had descended fifteen hundred feet and much reduced the risk of exposure except on Shackleton's and Worsley's legs, as the ride had ripped their trousers. At 6pm they ate. Fifteen hours had taken them half-way. Though the moon had not yet risen they anxiously began walking into the darkness. The temperature sank, freezing the surface and improving the walking. They rested briefly each fifteen minutes and ate at four-hour intervals through the night. They found themselves descending and were convinced they were above Stromness. Then they saw they were on a glacier and remembered there was no glacier at Stromness. They were on the Fortuna Glacier, one bay to the west. Back up they climbed: Worsley suffered a 'hopeless feeling'. One blessing was the weather continued kind.
I cannot know what it was like to make that walk. They began after nine days' rest, following that terrible boat journey. The kind of rest you get on a poor diet, sleeping in old clothes under an upturned boat as winter closes its fist. The lives of twenty-five men depended on those three. In Shackleton's mind, they depended on him, and his mental moral contract to bring them all home. Without knowing it, they were severely dehydrated, weakening the body, befuddling the mind. I have walked for several days in a row, over thirty miles a day, at very high altitude on a poor diet. At the border of exhaustion is a place where the raw physical pain of each moment mingles with distracted thoughts which chatter away in the back of your mind. It is kinder to listen to the mindless chatter, than acknowledge the pain and exhaustion. They passed through that place to somewhere I never want to go, where thoughts impose themselves on your senses.
Worsley's account has no affectation, he says what he saw: 'Each step of that journey comes back clearly, and even now I find myself counting our party – Shackleton, Crean and I and – who was the other? Of course we were only three, but it is strange that in mentally reviewing the crossing we should always think of a fourth, and then correct ourselves.'
Shackleton needed to make it mean something, though he was dismissive of conventional religion. One of the attractions of the polar wastes to him was to experience thoughts and feelings beyond the everyday senses. He might have been interested in Edmund Burke's writing on the sublime: 'When we go but one step beyond the immediate sensible quality of things, we go out of our depth', or as Captain Ahab says to Starbuck in _Moby Dick_ : 'All visible objects, man, are but as pasteboard masks.' Shackleton's take was: 'I know that during that long march of thirty-six hours over the unnamed mountains and glaciers of Georgia it often seemed to me that we were four, not three. And Worsley and Crean had the same idea. One feels "the dearth of human words, the roughness of mortal speech," in trying to describe intangible things.'
I have three times walked the final section from Fortuna Bay. It is not demanding if you are fit, rested and well-clothed. They were deficient in all of these: moreover their clothes were held together with safety pins, and their bodies were unwashed in three months. At five in the morning Shackleton broke twenty-two hours without sleep and let them doze. He himself stayed awake, fearing they might sleep so long the cold would take them. After five minutes, he told them they had slept half an hour, and they tramped on. The moon paled. Dawn soon came. It showed them a sea-cliff a few miles ahead in which the rock strata were bent into a huge Z-shape. It was the headland off Stromness. At 06:30, as breakfast was being prepared, Shackleton thought he heard a whistle. Worsley took his chronometer from under his ragged clothes where he was keeping it warm. He knew two whistles sounded at the whaling stations each morning, half an hour apart. The first called them to breakfast, the second to work. At seven they all heard it and Shackleton realised, 'It was the first sound made by outside human agency that had come to our ears since we left in 1914.' They abandoned the empty Primus and began to descend below the major glacier of Fortuna Bay. They saw two signs of man's presence: a seal shot with bullets, and hoof prints in the sand and mud of the glacial outwash. The beach is still trodden by reindeer introduced for food by the Norwegian whalers. Because they live in the southern hemisphere and grow their antlers for the summer breeding season, they are the only reindeer in the world which have antlers at Christmas.
Their luck with the weather still held. Tom Crean found the tarn that bears his name by falling through its ice. Shackleton chose the most direct route: Stromness. Soon they stood by a small outcrop of fine-grained rock shattered by frost into paper-thin sheets. A glaciated valley ran fifteen hundred feet beneath, hidden as it went inland to the left. An open vista showed the route to the right where half of Stromness whaling station is visible beyond the shoulder of the hill, less than two miles away. Against advice, Shackleton insisted on walking down the line of a small stream despite the fact that its line further down was hidden by a fall in slope. The easy route to the right is longer but much faster. They came to a low waterfall and were forced to climb down on the worn rope while icy water cascaded over them. At the foot they picked up the objects they had thrown down before them: the adze and the log. They had sailed into Antarctica with a fully equipped ship. At four in the afternoon on 20 May they walked back into civilisation's meanest outpost, a sub-Antarctic whaling station with the clothes they stood up in and those two items. And one more: a saucepan. They were so tired they all forgot that they had jettisoned the stove.
The different accounts of what happened when three strangers walked off the mountain into a busy factory reflect the exhaustion of the walkers and the stunned disbelief of the whalers, who thought the _Endurance_ men long dead. The first two boys who saw them ran away. Shackleton found the foreman who took them to the officer of the manager, a man Shackleton knew. 'Is Mr Sørlle in?' Sørlle had sailed with Larsen to the Weddell Sea in 1892 and had predicted the _Endurance_ would never leave the ice.
The foreman told them to wait outside while he asked. 'There are three funny-looking men outside, who say they have come over the island and they know you. I have left them outside.'
The conversation that ensued must have been strange. Men who have undergone stressful isolation tend to be far less lucid and normal than they imagine when they first contact outsiders.
Mr Sørlle came out and said, 'Well?'
'Don't you know me?' said Shackleton.
'I know your voice.'
'My name is Shackleton.'
Sørlle put his hand out. 'Come in. Come in.' Some accounts say he cried.
Shackleton asked, 'Tell me, when was the war over?'
'The war is not over. Millions are being killed. Europe is mad. The world is mad.' No one had a camera with film handy. There are pictures of them later, bathed and shaved. The expression on Shackleton's face is not exultation, or even relief. It is simple suffering, a deep, dark inscription. They lay down in clean sheets on comfortable beds and were so unaccustomed to comfort they slept poorly. In the morning they looked up at the mountains. The night had brought a snowstorm which would have killed anyone on the open mountain. On the other side of the mountain there were still three men, two of them half-dead, and who knew how many still alive on Elephant Island. Worsley went the same evening on the _Samson_ to guide the whaler to Peggotty Camp, where McCarthy complained that no expedition member had returned with the rescue ship. After two years in his constant company, they did not recognise their clean and shaven captain. The whalers insisted on picking up a fourth hero: the _James Caird._
By Tuesday 23 May, Shackleton was already on the whaler _Southern Sky_ wondering what he would find on Elephant Island's lonely shore. It took him four voyages to get there. What had seemed like the easy last leg was frustrated by another year of dense pack-ice and then, when he was forced to go to South America to launch the third and fourth bids, unsuitable vessels and the wartime shortage and expensive of coal. Uruguay and Chile were both hugely generous with ships and coal, and it was in a converted tug, working as a lighthouse supply vessel, the _Yelcho_ , that an obscure pilot named Luis Pardo brought Shackleton to the savage coast of Point Wild. Or rather Worsley and Shackleton did: Pardo was quite happy to take a back seat. With his usual uncanny skill, Worsley had brought them through ice-clear waters to the north shore of the island when fog closed. Shackleton took them in blind, very slowly. A fourth failure was not countenanced. It was no longer May, when he had promised to return, or even June or July. As they found their way to the jagged stack which lies off the beach, it was the next to last day of August. It was over three months since they had left Wild in charge of a camp mostly comprising invalids, with a heart-attack victim and a case of gangrene.
Marston and Hurley were collecting limpets when an odd ship came into view and sailed past; Worsley had not recognised Point Wild coming at it from the west for the first time. The tough, irrepressible Hurley believed 'by this time we could not have held out for more than a few days.' They tried to halt the vessel by raising a shirt on a pole, but it jammed half way. Then they lit a dirty fire with the last paraffin can. It exploded producing little smoke. In the famous Hurley picture of a smoky fire on the flank of the rocks, the black pall was added in the developing room. But the strange little ship with the Chilean Navy ensign turned around and a boat was lowered. A thick-set man stood in the bow. The camp commander Wild stared to see who it was. Shackleton had seen the shirt at half-mast and assumed it signalled that some had died.
Wild had attracted unstinting praise for his leadership ashore from the prickliest of judges, Orde-Lees. When Wild saw it was Shackleton, he let his larrikin tough-boy image slip: 'I felt jolly near blubbing for a bit and could not speak for several minutes.' Someone on shore cried out: 'Thank God, the Boss is safe.' 'Are you well?' called Shackleton. 'All safe, all well.' 'Thank God.'
Each day Wild had made the men stow and lash their gear ready for departure: for 126 days. Now they were able to board and go in forty-five minutes. The surf was rough and Shackleton did not even walk up the beach to see the camp. Two hours earlier, said Wild, the ice would have blocked them landing. As they left the wind shifted and it again began to drive against the shore. The men began to read the newspapers Shackleton had brought to prepare them for the state of the world. Three million were dead. Orde-Lees noted casualty lists had been renamed Rolls of Honour.
Shackleton went to the Ross Sea and found that his other party had laboured under huge difficulties to lay depots that would never be used, three men dying in the task.
Neither Shackleton nor the world could understand each other for a long time. He could not absorb that the grotesque war had changed how people thought and felt. His saving himself and twenty-seven men was not a triumph; their playlet was an insulting irrelevance to the men laying down their lives in the mud and to the women and children who had lost their husbands and fathers. Many of Shackleton's men signed up immediately and one was killed within the month. Others died before the Armistice. He dedicated his book of the expedition 'To my comrades who fell in the white warfare of the south and on the red fields of France and Flanders.' He knew the two were equal in courage and endeavour.
The Naval establishment implacably opposed his efforts to serve a meaningful wartime role; Scott was their man. The Boss did force recognition of his men's service with the award of the Polar Medal to most members of the expedition. Those who had not lived up to his standards were snubbed, including John Vincent, the demoted bosun, and most controversially, the dour Dundee carpenter McNeish whose brilliantly adapted boat saved all their lives. One moment's challenge to the whole order of discipline would not be forgiven.
Shackleton suffered his own kind of social exile. He had become a poor resident of the non-polar life, and the post-war world. He launched one last ill-focused expedition in 1921. He whistled and most of the old crew came from all corners of the world. As they left Plymouth Harbour with the harbourmaster on board, Shackleton heard a bell-buoy toll and said to him, 'That's my death-knell.' Despite suffering a severe heart attack in Buenos Aires on the way down, he sailed on to South Georgia entering the harbour of Grytviken in improving mood. Grytviken Bay has a desolate beauty to the old whaling station and the small, rusting seal and whale-catching ships add poignant memory. That evening he wrote, 'In the darkening twilight I saw a lone star hover, gem-like above the bay.'
In the early hours of the next morning the surgeon, Macklin once again, was called to his cabin. Shackleton had suffered another heart attack. He was trying to be humorously philosophical about it: 'You're always wanting me to give up things, what is it I ought to give up?'
Macklin replied, 'Chiefly alcohol, Boss, I don't think it agrees with you.'
In a few minutes the great heart died. The men sent his body back home, but when the ship was at Montevideo they received a telegram from his wife, the sufferer of longtime infidelity from her husband. She told them to take his remains back south where he had been truly happy. As you enter Grytviken harbour, the promontory on the right at King Edward Point has a small cross where his men built a cairn and held their own ceremony while the Boss's body was making its way north. The first feature on the left shore is the cemetery, bounded by a low wooden fence painted white. It keeps out the fur seals and elephant seals that loll among the tussock and rough grass below.
Above the grave stands a roughly cut block of granite, taller than a man. On the reverse is written the words from Browning with which the youthful Shackleton, the obstinate boy, had declared his ambition to his fiancée:
I hold that a man should strive to his uttermost for his life's set prize.
There he lies among the old sailors, just as he had daydreamed when inspecting the timbers of the ships which had found their way to the head of King Haakon Bay. All the graves in the cemetery face east towards the dawn and the resurrection. For nearly ninety years he alone faced South. In 2011, Frank Wild's ashes, long lost in a crematorium storage facility, were found and returned to lie alongside his great friend.
# Living in Antarctica: The Bases
_The first generation of explorers brought little, the second brought flat-pack huts. It takes a special person to endure overwintering. Sometimes the selectors get it wrong, badly wrong._
Before tourism, most visitors to Antarctica were scientists, and the bases where they lived and worked were typically small, with scientists trained to look after themselves as much as possible, to minimise support staff. Early expeditions had taken scientists with them, but essentially they were ship-based, and even if temporary structures were erected ashore, the scientists left when everyone else did. That began to change at the beginning of the twentieth century, and de Gerlache's _Belgica_ Expedition of 1897–99 was a turning point. The leaders had taken with them working huts for observing and measuring but no shore accommodation. However they were prepared to overwinter in the ship, and weren't unduly disappointed when the _Belgica_ became trapped in the ice. She became the base for the winter and the next spring and summer. Nordenskjöld's 1901–03 expedition took it a step further, their huts included sleeping quarters and the scientists were intended to be dropped off in the summer and picked up after one winter. The contemporaneous Bruce expedition planned to push south, retreat to the edge of the Antarctic for the winter, and set up a base. He always wanted the base to continue in use, by someone else if not him, and their hut was surrounded by stone walls and named Omond House. Argentina were keen to take it over, and this was the beginning of planned continuity, of having a base to which men would return year after year.
For a long time this was an all-male society. Despite women being physiologically better adapted to withstand exposure, traditional views of gender in universities and institutions were slow to change. Wives accompanied husbands long before anyone employed them in their own right. It is usually said that the first women in Antarctica were the Americans Jenni Darlington and Edith, known as Jackie, Ronne, in 1946, who accompanied their husbands on the expedition of 1946 – 48 led by Finn Ronne which overwintered on Stonington Island, Marguerite Bay. The Filchner-Ronne Ice Shelf and Edith Ronne Land are bequests of this trip. But in the 1930s, the wives of two whaling captains, Caroline Mikkelsen and Ingrid Christensen, accompanied their husbands and had a mountain and a length of coast named after them. Sadly the first woman here was probably a sex worker, the Native woman whose bones were found in a sealer's hut on Desolation Island. Australia began employing women in 1976, when a female doctor overwintered at Macquarie Island, and Britain soon followed. Wendy Pyper of the Australian Antarctic Division kindly provided me the following statement about their staffing. 'Although the Australian Antarctic Division makes a conscious effort to encourage more women to apply for jobs in Antarctica, the ratio of wintering men to women remains at about eight to one and the profile of an average Australian expeditioner is still male, white and aged between 29 and 32. Many occupations, particularly the trade and technical positions, are fields traditionally dominated by men in the Australian community, and so it seems likely that the participation rate of women, particularly in wintering groups, will not increase dramatically unless the jobs change and women can take their children.'
Antarctic visitors soon realise how few places there are to go safely ashore; the wildlife worked this out a long time ago. Most bases are in or next to penguin colonies because people have the same shore needs as penguins: a beach seldom pestered by ice where you can come safely ashore, get clear of the surf, and find ground free of snow and ice. They are usually on islands. This is because the snow and ice on the mainland piles up thousands of feet deep, and where it meets the sea it usually forms ice cliffs or glaciers: no use to man or penguin. Large islands can also produce enough snow and ice to create a hard high barrier to anything trying to come ashore. Small islands give you a chance. In the early days, as the John Lackland Cope fiasco demonstrated so clearly, the price small parties paid for basing themselves on small islands or ledges of land was isolation. Cliffs and glaciers may prevent you from accessing the interior, and small wooden boats are extremely vulnerable to the ice and squalls of Antarctica. Research is then limited to what is in your immediate grasp.
The most popular place to build bases has been in the South Shetland Islands, especially the largest: King George Island. Despite being big enough to support glaciers it is relatively far north and has areas of flat sheltered land, especially in King George Bay facing the Bransfield Strait, which are clear each summer. It is also close to South America and these two factors make it economical and reliable to service and support a base. By Antarctic standards the weather is mild and the seas normally open. I asked a Russian working at Bellingshausen Base on King George Island if he minded working through the winter. He replied, 'Not at all, it's usually around –25° centigrade, maybe –40° in a cold spell. My home is in Siberia, there it's –50° every winter.' It's so comfy, the workers on the Ross Sea side and at the interior bases refer to the Peninsula and South Shetlands as the Banana Belt. Bases are also close enough to share the cost of bringing down fuel and materials: no need for expensive heroics if an accident to your support ship leaves a base short of supplies. There's a high-quality air strip built and operated by Chile but available to other countries. If you have an accident or illness that requires an emergency evacuation during an Antarctic Peninsula visit, the chances are you will be flown out of King George Island.
I shall cover only those bases which are commonly visited and have historic importance. Deception Base has its own section in the chapter on that island, so I'll begin with the most visited place in Antarctica; in 2007–08 it received over 17,500 visitors.
## PORT LOCKROY
On the west side of Wiencke Island lies a small bay called Port Lockroy, discovered by Charcot's first expedition and named after Édoard Lockroy, Vice-President of the French Chamber of Deputies, who had helped Charcot obtain Government backing. Its southern shore forms a low rocky tongue beneath a skirt of mountains rising to the 4735 feet (1435 metres) of the Luigi Peak. This is Jougla Point and just off it lies Goudier Island, named by Charcot after his Chief Engineer. The island is no more than a small stone boss; you can walk across it in two minutes. Once planed flat by ice from the glacier at the back of the bay it is now polished by the patient feet of the gentoo penguins that nest all over it; but in 1944 men scrambled ashore to begin building Base A of Operation Tabarin.
They were not the first men to stay here; it was a popular harbour for whalers up until the industry's collapse through over-fishing in 1931. The Tabarin team landed with their own supplies and some timber salvaged from the whalers' factory at Deception Island. They built Bransfield House, named for the British naval captain who accompanied William Smith back to the South Shetlands. Over time, the building was extended and altered, but it has now been restored to the layout and furnishings it would have had around 1960: more on the restoration later.
After the war, it was used as a scientific base studying the upper atmosphere. Men came down, usually for a stay of two and a half years, learning the ropes from outgoing staff over the summer, and then completing two years, spending the last summer passing on their know-how to the incoming rookies. The great block of valves and dials which sits in the middle of the laboratory room was used for this, and inevitably nicknamed 'Beastie'. The upper atmosphere conducts electrical charges towards the poles and lightning strikes in the southern hemisphere would send signals which were picked up as 'whistlers' enabling this tiny island to monitor electrical activity over huge areas of the earth.
##### INSIDE PORT LOCKROY BASE A
The centre is exceptionally well interpreted and manned by knowledgeable staff, so a detailed account of the contents is unnecessary. Do make sure you see these little gems:
□ photo of Prince Philip meeting Raymond Priestley, author of the much quoted 'when disaster strikes and all hope is gone, get down on your knees and pray for Shackleton'
□ poster advertising the Tabarin night club, Paris on wall opposite the door to the sleeping quarters
□ the bath-house on the left as you enter, with rules for collecting water and bathing
□ the sunshine recorder in the laboratory
□ the British Crown Property sign in the Post Office and shop
□ the period foodstuffs in the kitchen
In a small wooden box, on the right hand side as you enter this room, is a piece of meteorological equipment which is elegant in its simplicity. It is a sphere of glass which fits into a light steel cradle. To one side are curved pieces of thin card printed with a grid. This measures hours of sunlight. The crystal ball is placed in a north-facing window (in the southern hemisphere) and each morning a fresh piece of card is fitted beneath it. The ball acts as a lens and whenever the sun shines it burns a line in the paper. At the end of the day the hours of sunshine can be read off against the grid. It has no moving parts and needs about twenty seconds of attention a day.
## THE UNITED KINGDOM ANTARCTIC HERITAGE TRUST
Governments don't usually spend money preserving artefacts and buildings which only their better-off taxpayers have any chance of seeing. The rescue of Port Lockroy is down to a dedicated group of volunteers including former base staff. They formed a charitable trust and rebuilt the base as it was around 1960. The shop and Post Office pay for its operation and maintenance, and enough money is being made to contemplate other projects. Work has begun on the Wordie Hut at Vernadsky Base, and plans are being prepared to restore Detaille Base, and Damoy Refuge hut just north of Port Lockroy. They also have care of Stonington (Tabarin Base E) at 68°11´S in Marguerite Bay, once run by Sir Vivian Fuchs, and nearby Horseshoe (Base Y).
The highest part of the island is roped off from visitors to allow the Base Commander, currently Rick Atkinson, and his team of three, to compare the breeding success of penguins in the undisturbed area with that around the base where people have free access close to the nests. The results mirror those obtained in similar studies elsewhere in Antarctica. The birds in the visited area do as well or slightly better than those in the protected area. This probably reflects two things. Firstly birds become accustomed to the presence of people, in part because their only land predators are other birds, so bulky mammals are not seen as a threat. Secondly humans deter those predators, especially skuas, which in the penguin breeding season feed heavily on eggs and chicks. Of course skuas deserve the same freedom from human interference as penguins, so this is not a neutral outcome. Taking the Peninsula and its islands as a whole, human impact on the predators is very small, and may well be balanced by personnel at other bases bending the rules a little and feeding skuas. These intelligent, curious and fearless birds provide a little animal comfort to people who see only the same human faces for months at a time.
Mail is taken, mostly by cruise ships, to Stanley in the Falkland Islands where it joins Britain's regular postal network. There are many ways of marking nationality and saying, 'This is ours.' For the British, it's having a local Post Office. One Christmas Eve I picked up their last mail sacks before the holiday and ferried them back to our vessel. I felt like Santa Claus.
#### FOUR-LEGGED FRIENDS: BAS SLED DOGS
Arctic peoples have used dogs as draught animals for two thousand years, but it was 1899 before Carsten Borchgrevink took the first sled dogs to Antarctica. Roald Amundsen's efficient, if ruthless, use of dogs to reach the South Pole proved they were the best means of travel until mechanical sleds improved in the 1950s. Dogs could haul three times their own weight for thirty miles a day. The best dogs were bred from the four main breeds of Arctic huskies: Samoyed and Siberian from the Russian Arctic, and Malamute and Eskimo from the Americas and Greenland. In 1945 Britain brought twenty-five Labrador dogs to Hope Bay to start a permanent breeding population. They remained an important part of both the logistical and emotional support of Antarctic work.
In August 1963 base men were amused and annoyed to receive a fax from BAS in London advising that with the increasing use of mechanical transport 'no more breeding is to take place on bases except at Halley Bay who must plan to maintain three teams indefinitely.' It seemed premature to men kept busy excavating Eliason motor toboggans from crevasses they had dug themselves into, or repairing their engines and tracks. But in 1957 – 58 Vivian Fuchs and Edmund Hillary had crossed Antarctica mostly using motors-leds, proving the technology was now reliable.
As scientists became more mindful of dogs as alien species, their management became more cautious. Dogs had been fed partly on seals and penguins to augment the dog food brought in: a direct impact on native animals. They also defecated wherever they went, introducing new organisms and minerals. The import of new dogs was halted in the late 1970s to prevent new parasites or bugs entering the ecosystem. Dogs were bred only from resident Antarctic populations. From 1972 to 1992 BAS dogs were reduced from two hundred to thirty. There were other factors.
In 1991 a new clause was inserted in the Antarctic Treaty the international agreement which regulates activities in the continent: 'Dogs shall not be introduced onto land or ice shelves and a dogs currently in those areas shall be removed by April 1, 1994.' A particular concern was that dogs had introduced, or could introduce diseases like distemper to the seals. 'Had' is unlikely, 'could' is always possible. Perhaps the ban was too cautious, but the Antarctic ecosystem is brutally simple with few checks and balances; one epidemic could be catastrophic. It's too late to be cautious afterwards. By 1993 only two BAS teams remained: named the Huns and the Admirals. One final project was saved for them: to run across the Peninsula to Alexander Island recording the weather on each side. The teams were flown out on 22 February 1994. Most base men would have them back tomorrow, at their own expense. As Antarctic veteran Bernard Stonehouse told me, 'even the friendliest tractor is no substitute for cuddling a dog.'
For an entertaining and informative account of sledge dogs by two men who ran them, read _Of Dogs and Men_ by Kevin Walton and Rick Atkinson, a regular Port Lockroy Base Commander. Full details in Acknowledgements.
## DETAILLE, BRITISH BASE W
Just across the Antarctic Circle is a small island rising to a snow dome. On its flank, facing a knoll of eye-catching igneous rock embedded with angular boulders, is a long wooden hut surrounded by the daily junk of Antarctic life. There are cases of tins, rusted almost through, things used and discarded, gadgets superseded and the broken utensils of everyday life. It looks as if it was abandoned in a hurry with hopes to return, never realised. The skuas which nest in the rocks above do not approve of visitors, or indeed other life forms, which they see as just a way of keeping meat fresh until they can eat it. They strafe anyone wandering into an area they think of as their territory, and their ideas of territory are expansive.
This was Base W, built in 1956, and it was large and comfortable compared with the British bases of the 1940s. Detaille was named for the director of the Magellan Whaling Company in Punta Arenas who helped Charcot in his 1908 – 10 expedition by holding stores for him at Deception Island. Workshops, laboratories, offices, a kitchen and dormitories led off a long corridor. A large radio mast stood on the hill behind; its substantial base is still visible. But it was used for only three years because it could not fulfil its planned role. It was intended as a base for dog-sledging parties, and the short documentary _Dog-_ _Sledding_ _in_ _Antarctica_ was filmed at Detaille Island. However, they found the sea-ice between the island and the Peninsula was an unstable and treacherous surface to walk or sledge on. Plans were made to decommission it in 1958–59, removing as much of the equipment and garbage as possible. That summer they finally got thick ice, so strong that their support vessel the _Biscoe_ , supported by two US ice-breakers, could not get closer than thirty miles to the north. The good intentions of decommissioning base rather than just abandoning it were frustrated. They were forced to load only their essentials onto their sleds and sledge to the ship, leaving on 31 March 1959. They reached there without incident, but as they undid the dog traces to board, one animal called Steve broke loose and they could not catch him. The dog turned back south. They left, unable to tell him that this parting was final. Three months later staff at the British Horseshoe Base over sixty miles south saw a strange dog come bounding into camp; it was Steve.
Because of the way it was abandoned, the camp has a Marie Celeste feel to it. The attic space is full of stores, tools lie on work-benches, magazines on tables, and rusty tins of porridge moulder in the cupboards next to Stephens Navy Pickles (Piccalilli with a picture of a battleship on the label) and Pearce, Duff's lemonade powder in ten-pound tins. The sense of isolation is still strong. I travelled there with John, a man in his early seventies, who as a young graduate had carried out meteorology there. Now retired in Falmouth, in south-west England, he guided us round the rooms, and the half-century's absence had melted away. In the final room he found the blank weather report forms stacked exactly where he had left them.
## ESPERANZA
The Argentine Esperanza Base occupies the pleasant, sheltered site on the mainland shore of the Antarctic Sound first seen by Nordenskjöld on his way to the Weddell Sea, and later explored by Gunnar Andersson when he was set down there to seek Nordenskjöld over land. In February 1944 sea ice thwarted the British attempt to set up Hope Bay as Operation Tabarin's Base D. They later came back and set up a station which suffered a fire on 8 November 1948. The Argentines set up five hundred metres away in 1951 and Hispanicised Hope to Esperanza. When the British returned to rebuild Hope Bay a short distance from its original site, it sparked an incident when Argentine troops shot at and expelled British scientists. No one was hurt and when the affair was reported to Buenos Aires the Argentine commander was disciplined and removed. Hope was rebuilt. When Britain rationalised its bases, Hope Bay was transferred to Uruguay on 8 December 1997.
The site's special attraction to science is Mount Flora, first studied by Andersson and his team, and found to be rich in fossils, including ones from warm climates which would later be used as evidence for continents moving between climate zones, helping to confirm the theory which became plate tectonics, governing the fundamental processes that shape the land and sea masses of the earth.
It is one of the largest bases which can be visited. Like the bases of a number of nations, it is not technically a military base, which the Antarctic Treaty forbids, but whenever you go all the staff there happen to be from the military. It has a small dock and jetty, and a heli-pad. Small sections remain of the railway which used to help move heavy supplies from dock to base. The path to the base passes the remains of the hut where Andersson and his two companions overwintered. The Argentines restored the hut in 1966–67. The structure is pitifully small: a very cogent reminder of what they uncomplainingly endured.
The base appears more like a miniature settlement than a workplace, and this is deliberate. Whole families come down here; there is a school, chapel, postal service, and medical centre. Children have been born here including the first Antarctican: Emilio Marcos Palma. Over fifty people commonly overwinter and the carbon footprint of the base is large. The generators alone consume more than 180,000 litres of fuel and year and it has forty-three different buildings. The site is susceptible to high winds. A few years ago the roof was ripped off a building during a storm whose winds exceeded 140mph at which point the anemometer was torn apart.
The huge Adélie colony behind the base is the most northerly continental one.
## VERNADSKY BASE (FORMERLY FARADAY BASE) AND WORDIE HUT (BASE F)
The yellow and light blue flag of the Ukraine flies over Vernadsky Base, a range of buildings on the low-lying Argentine Islands. The archipelago is fringed with rocks and poorly charted reefs, so ships normally stick to safe routes they have sailed before, and anchor a long Zodiac ride away. It's a good place to look for leopard seals along the way.
In a snug channel at the side of Vernadsky is a little wooden hut built in 1947, and the predecessor of Faraday/Vernadsky. It is named the Wordie Hut after the young geologist James Mann Wordie who sailed with Shackleton on the _Endurance_ , and went on to enjoy a distinguished career in academic and public life. One of his students recalls asking if the student baths might have hot water, and having his ears blistered by a response that included a reference to the lack of hot baths on Elephant Island. The 1947 expedition had planned to head much farther south to Adelaide Island and build a new base, but was frustrated by poor weather and dense ice. They retreated back north and erected their hut here on the site which had been used by the 1935– 36 British Grahamland Expedition. That hut had lasted about ten years before being flattened by a wave from a calving glacier or possibly a tsunami. The new hut originally comprised a single eighteen-feet square building, which was the size of the longest timber conveniently available in the shortages following the war. That building is now the living space, organised around the stove. The base was extended in 1951, adding the workshop, laboratory and extra accommodation.
The men at Wordie carried out meteorological observations; it still has a crystal ball sunshine recorder on the window ledge and a variety of other instruments on the walls. The outdoor meteorological instruments stood on the concrete base a short distance to the right of the front door. A short scramble to the left of the front door is a bare wooden sign announcing British Crown Property, dating from the era before the Antarctic Treaty, when territorial claims were still actively pursued. The little hut closed as a base in 1954 and you may discover why: the channel in which it stands is early to freeze over and late to thaw, making efficient provisioning difficult. There were also fears of further catastrophic calving. The work was transferred to a new base half a mile away on the Argentine Islands, which was later named Faraday, after the English physicist Michael Faraday, then Vernadsky. Wordie was maintained for use as an emergency hut, and as a refuge for Faraday staff needing to escape from base for a night or two. Ironically it was used again as a base in 1960 by another British expedition which had hoped to overwinter at Adelaide Island, but been unable to get there. Wordie finally closed in 1996 when Faraday Base was sold to the Ukraine for one pound and renamed Vernadsky. The price tag is explained by the fact that Britain wanted to reduce the number of their bases, and decommissioning a base is more expensive than building it in the first place. Ukraine wanted a base following its emergence as a nation after the break-up of the Soviet Union; the coin which paid for it is set into the bar between the handpumps.
The United Kingdom Antarctic Heritage Trust began restoring Wordie Hut in 2009. The small office and bunkroom near the front door still has a filing box in which individual records of sled dogs were kept. The cards have now been removed for safe-keeping. The characteristics of each dog were noted: how well they worked, how quarrelsome they were, their fertility, and for the bitches, their nursing and mothering skills. The cards were used to select mates and improve the breed. The living accommodation includes books, a radio still bearing the old FIDS signature, clothing, and packs of pemmican. The food in the kitchen is mostly modern stores, from Vernadsky.
As you enter Vernadsky Base you take off your boots and heavy clothing in a changing area where there are portraits of past base teams and of Vladimir Vernadsky (1863 – 1945) after whom the base is named. He was a polymath whose greatest discovery was that the Earth had not always had an oxygen-rich atmosphere. Early life had developed at a time when oxygen was just a trace element in the atmosphere, along with many others. He showed that it was not a physical or chemical change that had to happen before life could evolve, it was the other way around. Primitive organisms did not need oxygen, but they did create it as a by-product. This was the first great environmental change created by life itself. Later life found it so useful it developed, through evolution, new forms which used and depended on oxygen.
The main corridor which leads away from the cloakroom has offices and labs on either side and even a gymnasium. Notice how the calendars, magazine photos and computer screen-savers, when they are not pin-up girls, are green fields and countryside. Remember Nordenskjöld, and Bruce's men, pining for colour for his starved eyes. Near the end of the corridor the word Ozone is written above your head and a narrow ladder leads up into a low attic. The original Dobson's spectrophotometer is still up there next to the modern replacement: an instrument that showed us how quickly and perilously we can change a vital part of the environment.
The second most famous room after that housing the Dobson's spectrophometer is the bar, which dates from its British days. Hardwood was sent down for the carpenter to make a new observation hut. He made the hut out of spare wood and built a bar with the hardwood. It cost him his job, but the bar's still there. Look out for the pound coin with which the base was bought; it's embedded between the handles of the beer pumps. The base is said to order more potatoes per head of staff than any other base, and the result is a very good home-made vodka. I thought the year they made it in the fire-extinguishers was a particularly good vintage, though you wouldn't have wanted to be there when a fire broke out.
Women were introduced here for one season and when there was friction, they decided it was due to tensions created by their presence. It might have been better to root out the troublemakers whatever their sex.
Hanging behind the bar is a collection of bras, collected simply by offering a free shot of vodka in return for taking off your bra in the bar. It worries me how many women who can afford an expensive cruise think this is a good deal.
#### THE OZONE LAYER
The ozone layer used to be like a car tyre; no one thought about it much until it had a hole in it. Ozone molecules are a rare kind of oxygen. Unlike regular oxygen molecules, they are made from the bonding together of three atoms (O3) not two (O2). In every ten million air molecules, only three are ozone, and they are mostly found high in the upper atmosphere, around twenty kilometres up.
Each element absorbs radiation, including the narrow band of wavelengths we see as light, in a unique way. Ozone is best at absorbing ultra-violet rays. It plays a vital role in reducing the amount of ultra-violet light that reaches the surface of the earth, and is crucial to maintaining the balance of life on earth; however, this defence is extremely fragile. If O3 occurred in a pure band, unmixed with normal oxygen, it would be only 3mm thick: the thickness of a British one-pound coin. Yet we depend on it to survive.
The ozone layer occurs because solar light of a wavelength less than 2.4 nanometres can break up regular O2 oxygen into O-radicals, single O atoms which are very active in combining with remaining O2 molecules to make ozone. At 20 kilometres conditions are ideal to create and maintain O3. Above this, there are a great many energetic ultra-violet rays, but not many oxygen molecules to react with. Lower down, there is a rich supply of O2 molecules to react with, but few rays make it that far.
Ozone is not stable; in time it reverts back to O2, so the ozone layer is not a fixed supply of the same molecules, but more like a pond through which a stream flows: it stays the same size, even though the water in it is constantly replaced.
In the 1930s an English scientist called Gordon Dobson pioneered Ozone Layer studies and created a unit to describe the density of ozone and a machine to measure it. When he was first able to measure it, the Earth's atmosphere at 0°C had the equivalent of approximately 3mm of pure ozone; he designated this as 300 Dobson Units or DUs. During subsequent research this was regarded as the norm. Hand-built measuring instruments were made and, to enjoy unpolluted conditions, put to work in Antarctica at two British bases: Faraday and Halley, which stands on the ice of the Weddell Sea.
They discovered an annual cycle with low levels each spring. Then, in the late 1970s, something funny began happening at Faraday. Every year they detected lower and lower levels of ozone. They did not know why, but wondered if their ageing instrument was becoming less sensitive. Halley's readings also showed the same decline year on year. The problem was, hardly anyone else was measuring ozone levels, so they only had each other to compare results with. If ozone levels were really declining it was dangerous for everyone. Fearing needless controversy they mulled over it for some years before deciding to publish, admitting it might be instrument error. New machines were built and their readings showed the same trend; the changes were in the environment, not the kit. Levels in the Arctic were falling too, though not so fast, and the timing of the Arctic change lagged behind the Antarctic. They pieced together a picture of ozone in the atmosphere. The fastest rate of ozone formation was in the tropics which maintained levels of 400-450 DUs. The stratospheric winds common in those latitudes carry it around the globe and mix it up. Levels at the Poles are lower, averaging 300DUs.
The speed of change had been so rapid that consequences were already evident. In areas where the thinning of the ozone, or the 'hole in the ozone layer' as it soon became known, was greatest, skin cancers in livestock and people increased in frequency and severity. It seemed likely that some human activity was seriously affecting the system. The villains turned out to be two groups of chemicals called CFCs (chlorofluorinated carbons) and hydrochlorofluorocarbons (HCFCs). These gases are designer chemicals used in the compressors of refrigerators, and in air-conditioning systems and aerosols. Ironically, CFCs had been selected because at sea level they are inert, showing no interest in reacting with other chemicals. Twenty kilometres up, that all changes. The chlorides and bromides they contain become catalysts; when they meet ozone, they rip off the third oxygen atom. While ozone continued to be made at the same rate, it was now being broken down more quickly. Each summer the rising levels of radiation created ozone faster, and levels recovered, but because CFCs created an ever more rapid breakdown of ozone, levels couldn't bounce back to last year's levels. There was therefore less ozone to block incoming radiation. To return to the analogy of the pond, the same flow of water entered the pond, but each spring it flowed out more quickly, and the pond became shallower. Ozone levels are lowest in September and October before mixing of air masses begins to restores levels from mid-November onwards.
This theory asked industry and governments to accept that there was a link between current human activity and measurable and hazardous environmental change. There were other chemicals which could be substituted for CFCs and HCFCs. Governments signed the Montreal Protocol for future use in 1987, and most big manufacturers ratified it in 1989. Lows in the ozone layer began to fill, endorsing the theory and the remedy, though some countries, notably China, at first argued that they had only just begun to manufacture such goods and had some credit at the environmental bank where the First World had already exhausted its credit. Since the lucrative western markets for their goods prohibited CFCs and HCFCs, market forces brought the ethics into line. Most atmospheric models predict the atmosphere will by 2020–25 return to ozone levels equal to those first observed.
The differences between the behaviour of the two polar zones seems to be caused by the Antarctic atmosphere being more self-contained. Air circulations called Polar stratospheric clouds carry more nitrite acid and water vapour than most air systems. When that stronger mix meets CFCs it helps switch on the chlorides and bromides, which then strip O3 back to O2. This action is stimulated by light and so it increases in intensity each spring. In winter, Antarctica has an atmospheric current which greatly reduces mixing, allowing Polar Stratospheric Clouds to form in great cold and isolation, creating a stronger brew of nitrite acid and water vapour, freeing more chlorides and bromides to break down ozone.
#### MEASURING OZONE: SPECTROPHOTOMETERS
The light that our eyes see is a tiny part of the spectrum of electro-magnetic waves. The violet end of the visible spectrum is the shortest wavelength our eyes can see. Some animals can see even shorter waves: ultra-violet ones. We can't, but they can damage our bodies, causing burns and cancers.
In the 1920s, Gordon Dobson designed a machine which measured the amount of sunlight in different wavelengths within the ultra-violet range. It is useful to measure light levels at two wavelengths: 305 nanometres and 325 nanometres (abbreviated to nm: there are 1000 million nanometres in a metre.) These are emitted in the same quantities by the sun, but while 305 nm light is strongly absorbed by ozone, 325 nm is not absorbed. If you measure the difference in light received at the ground between the two wavelengths, you can calculate how much ozone must have been there to block it.
## GONZÁLEZ VIDELA BASE/WATERBOAT POINT
González Videla is a Chilean base named after a former president. It sits on a small low promontory which is separated at high tide from the mainland rising behind it. However, it is regarded as a landing on the mainland for the purpose of completing a continental landing. The site is home to a small group of leucistic gentoo penguins, a genetic variation which results in muted colouring, so the familiar patter of plumage is still there, but in variations of a pinkish biscuit colour. Eye colour is normal, because the faulty gene does not control the nervous system, of which the eyes are part.
It is often said that history has unjustly neglected someone. This is not true of the man who led the British Imperial Antarctic Expedition of 1920, Dr John Lackland Cope; anonymity is too good for him. He began by proposing to spearhead three teams of 120 men supported by twelve airplanes on a circumnavigation lasting five years. If it had been completed as projected, Cope's expedition would have colonised three-quarters of the continent for Britain by setting up permanent stations. He ended up leading eight dogs with nothing to do and four men who all hated each other. One of them was Hubert Wilkins who afterwards said the only thing to be learned from it was how not to organise a polar expedition. The Royal Geographical Society had examined the man and his plans then declined to support either. He bragged of having $750,000 worth of backing from three governments, yet he soon declared that all the money raised had been spent buying the dogs, leaving none to pay the whaler that was now to take them down instead of the twelve airplanes. Embezzlement was suspected. Thomas Bagshawe was nominally the geologist, and Maxime Lester, formerly second mate on a tramp, was designated surveyor. The Norwegian whaling magnate Lars Christensen gave them free passage, and on Christmas Eve 1920, they entered Port Foster, Deception Island. Then they begged passage on the Norwegian steamer _Solstreif_ to Paradise Harbour, landing 12 January 1921. Even this would not have happened had not Wilkins pledged his expensive photo equipment as security against their expenses.
The base staff are proud of the rather eccentric history of their location: Waterboat Point. They struggled to find a route up from the coast to access the interior, and nearly died trying. They also explored in a five-metre-long lifeboat, utterly vulnerable to the weather. When Wilkins, Cope and Lester were caught in a storm in Andvord Bay the boat took in water until they were nearly swamped. Only Wilkins was well enough to move, and he had barely enough strength to cut down their sail. Eventually Wilkins found a Norwegian whaler to take them home, but Bagshawe and Lester wanted to stay. Cope, hoping their adventure might help him raise more funds, not only encouraged them, but discouraged any whalers from checking on them. There was an abandoned waterboat left by Salvesen's factory ship _Neko_ in 1913, which they upturned for living quarters. Their food comprised only biscuits, baked beans, pemmican, sweets and alcohol. When they settled down to their first meal they found they had forgotten to bring forks. They had a phonograph and some records. Lester and Bagshawe quarrelled regularly, but the anger was as quick to die down as it was to flare up. Wilkins wrote: 'Two men who can have an angry row in the morning and forget about it by afternoon can get along anywhere for any length of time.' Frank Debenham, the Australian founder of Cambridge's Scott Polar Research Institute, said they 'possessed that quality so annoying to the great Napoleon of not knowing when they were defeated.' Cope never returned. When he reached the Falkland Islands, the authorities were so keen to move him on they got him a job potato peeling on a Scottish steamer. It was the end of his career as explorer. Next year Captain Andersen sent the whale catcher _Graham_ with the Deception Island Magistrate-cum-Whaling Officer Arthur Bennett and a prayer book. He expected to read the Burial Service for the Dead over the corpses. He found them alive and reasonably well on 18 December 1921. Indeed Bennett may have suffered more on that day, for in a year, Bagshawe and Lester had taken only one bath.
The site where they stayed is now on the edge of the Chilean González Videla Base. They upturned their boat on a rocky knoll on a rocky shore, just yards from the water. It is a perfect place for gentoo penguins to nest, which is what they have done for ninety years since. Their guano has helped rot down the remains of the wooden boat, but the outline of the gunnels is still visible among the nests, though much fainter than ten years ago.
## ARCTOWSKI BASE
Arctowski was founded in 1977 and named after the geologist, meteorologist and oceanographer Henry Arctowski who accompanied de Gerlache. It is located on a triangular shingle foreshore where two steep storm beaches meet at a cluster of rocks. Your arrival is watched over by a shrine to the Virgin Mary, and more practically, by a navigation light. It is a location where, from 1905, the first factory whaling ship in Antarctica, the _Admiralen_ , used to anchor. Colossal bones from the head are arranged on the shore, probably from blue whales or fin whales.
The base conducts research into glaciology, meteorology, geology, birds and sea-life, especially invertebrates and parasites of fish. It is a station where staff overwinter, but with a personnel reduced from summer levels of around twenty, as many of the projects are seasonal, studying creatures which only visit to nest, or which are locked beneath the sea-ice all winter.
## BELLINGSHAUSEN AND FREI
Two bases sit side by side above the same beach in Admiralty Bay, King George Island. The former is Russian, named for Admiral Bellingshausen. The latter is Chilean, named for a recent President who visited Antarctica: Eduardo Frei. Frei Base plays a similar political role to the Argentine base, Esperanza. A mother gave birth there in 1984, but it was not until 1995 that they built their first science building, a rather modest affair.
The Russian base suffered from poor housekeeping during the break-up of the Soviet Union, but was assisted in a big clean-up operation by volunteers from other nations. It is not hard to tell which side of the bay is Russian; on a small hill above it is a Russian Orthodox church, built of Siberian pine, smelling richly of resin, and home.
## ALMIRANTE BROWN
In Paradise harbour, within sight of the Chilean González Videla Base, lies a small base, named for the Irish-born William Brown (1777 – 1857), founder of the Argentine Navy. Work on meteorology, marine water temperatures and salinity have been carried out there. Since the support vessel, the ice-breaker _Irízar_ , was damaged by fire, there seems to have been no activity. The original base burned down in 1984. There are a few stories about it; my best source, an Argentine I won't name, said it was arson, by a station doctor who received mail telling him his sweetheart had finished with him. He determined to rush home and mend the relationship, and to avoid completing his contract, he made sure there was nowhere to work by burning down the base.
# Who Owns Antarctica?
_Visitors undoubtedly visit another continent, for some it's the motivation for coming, but have they been to another country? And why, in 1982, did two modern industrial countries go to war over possession of sub-Antarctic and adjacent islands?_
This chapter looks at who claims Antarctica, how it is run, and why Britain and Argentina went to war over the Falklands and South Georgia.
When you go to Antarctica you visit a new continent, but not a new country. Many bases will stamp your passport, and they never forget to include their country's name, but it's a souvenir. None of these stamps has any legal validity. Predictably, penguins feature heavily on them, though not necessarily with the anatomical fidelity you might expect given the leisure they have to study them. I like those on the Ukrainian Vernadsky Station, which look like Cold War veterans standing to attention, gazing up at a fly-past during a May Day parade.
In the era of exploration, there were various loose tests of whether a sovereignty claim was valid; there were procedures for claiming sovereignty which you followed as best you could. Going ashore was important, although drive-by claims were made when the weather was bad, or time pressing. Cook's 1775 claim on South Georgia was a model. He made a rough survey of the north coast, before going ashore at a place he smartly christened Possession Bay, in case anyone who followed was slow on the uptake. It was claimed for Britain in the name of His Majesty King George III and a volley of small arms discharged into the air. The event was recorded not just in the ship's log, but through materials left behind. Trees, where available, would be carved with due particulars, or have plaques nailed to them bearing the relevant boasts. Caches of coins showing the reigning monarch would be sealed in caskets or bottles: constitutional voodoo dolls to warn off unbelievers in the British values of fair play, in so far as fair play was compatible with world domination.
How seriously other countries took this depended on a number of factors. An important consideration was whether they were able politically or militarily to stand up to Britain. In Cook's time few could stand up to British arms; in the century that followed, virtually none.
When the British began probing around the South Shetlands they did what they always did and claimed it for Britain. In the case of Antarctica claims were simplified by the lack of indigenous peoples. Of course staking a claim was just the prelude to coming back to occupy the territory and make money. Antarctica was different. Whales and seals were two obvious sources of income, but much money could be made without owning territory, and the climate was too hostile to occupy all year round. Most countries never made formal claims. Those who did tended to be the early arrivals, plus two who were geographically closest, that is, Australia, France, New Zealand, Norway and the United Kingdom, plus Argentina and Chile. Notably the USA and Russia were as well qualified as these to lodge claims but chose not to.
These claims and counterclaims were mostly expressed in rhetoric and paperwork, though shots were fired on a few occasions, notably in 1952 by the maverick Argentine officer at Esperanza Base. But usually conflict was expressed in symbols. When the Second World War broke out the Argentines went down to a vacant Deception and painted over the Union Flag with the Argentine. The British came back in the austral summer of 1943 – 44 with their own paint pots.
It seemed likely that as the Cold War escalated in the 1950s, it might also be fought on the coldest front of all: Antarctica. That it wasn't was the remarkable result of applying scientific co-operation to politics, and it was an idea with a long pedigree. When Clements Markham was pitching a proposal for a new Antarctic expedition to the British Government, F. Sidney Parry, the Private Secretary to the Treasury, advised him to emphasise the practical value of magnetism, and minimise the role of pure science. But he had mis-read his Prime Minister. Arthur Balfour said, 'I take a different view – a view based on the scientific experience of the past. If our predecessors in the last two centuries had taken any narrow utilitarian view of their work, it is manifest that our ignorance of the planet on which we live would be much more profound than it is at present. There cannot be any territorial rivalry between any of the countries engaged in Antarctic exploration; and such rivalry as there may be, must be of a purely scientific character.'
## THE INTERNATIONAL GEOPHYSICAL YEAR (IGY)
The modern scientific venture which guided future politics was the initiative begun in 1950, when US and UK scientists revived ideas of the International Polar Years of 1882 – 83 and 1932 – 33. They had been chosen as years of either unusually high or low sunspot activity, when certain atmospheric effects are exaggerated. 1957–58 was expected to be a maximum sunspot year, so in 1952 the International Council of Scientific Unions set up a committee to organise an International Geophysical Year, which was a long one, running from 1 July 1957 to 1 January 1959.
The focus was to be on Antarctica and space. In both spheres, it was recognised that the expense and danger of research in those areas made co-operation vital, if scientific investment was to be efficient and effective. It galvanised unprecedented activity. Twelve nations established fifty new stations in Antarctica, staffed with over five thousand personnel. They co-ordinated studies on glaciology, meteorology, geology, geomagnetism and the upper atmosphere.
In one harmless sense, the Cold War continued; naturally the biggest contributors were the USA and USSR who built seven new stations each. The USSR had a base at Pole of Inaccessibility. To see the significance of the American choice, you must remember that many Antarctic claims had been made after exploring a length of coastline, then claiming all the land in the sector between those two latitude lines, until they converged at the South Pole. The USA set up Scott-Amundsen, a large shelter right over the Pole, with a toe in every sector. But still it made no territorial claims. Ironically, the Pole itself moves with time, and the ice on which the base is built flows at ten metres a year, so no building can stay over the true Pole for long.
Despite wider political tensions the IGY was a great success, and it was decided that the principles could be extended to future management of the continent. The result was the Antarctic Treaty, signed on 1 December 1959 and coming into force on 23 June 1961. The original signatories were: Argentina, Australia, Belgium, Chile, France, Japan, New Zealand, Norway, South Africa, USSR, UK and USA.
Diplomats and civil servants are often criticised, but the people who drafted the Treaty deserve the utmost praise. Faced with a large and delicate problem in a tense political climate, they devised a solution that has survived intact for fifty years. They began by setting out what was valuable and worthy of protection: it was in the interest of all mankind that Antarctica be used for peaceful purposes, not discord, and to recognise the contribution made to science by international co-operation in Antarctica. The main body of the treaty is expressed in a series of articles which can be summarised as follows. They cleverly avoid requiring anyone to lose face or renounce past claims.
###### Article I
It was for peaceful use only, with no military or weapon testing
But military personnel may be deployed for science or other peaceful purposes.
###### Article II
Freedom of scientific investigation and co-operation pioneered in IGY shall continue, subject to the provisions of this Treaty.
###### Article III
To promote co-operation
a) future plans should be shared
b) personnel exchanged
c) results freely available.
###### Article IV
Nothing in this Treaty is to be interpreted as
a) renouncing any asserted sovereignty claims
b) renouncing any claims consequent on that country's activities
c) prejudicing views on any other country's claims
d) no acts undertaken while Treaty in force can support or detract from sovereignty claims
e) no new or enhanced claim may be made while Treaty in force.
###### Article V
Antarctica to be a non-nuclear area, and nuclear agreements signed by signatories to Treaty also apply in Antarctica.
###### Article VI
Provisions of Treaty apply south of 60°S including ice shelves but without prejudice to the International Law of the Sea.
###### Article VII
All signatories have absolute freedom of access and inspection. All signatories must inform other signatories of all expeditions, manned stations and military movements.
###### Article VIII
Clarification of responsibilities for actions of observers and exchanged staff.
###### Article IX
Future actions required by signatories to further aims of the Treaty.
###### Article X
Each signatory monitors others' compliance.
###### Article XI
Resolution of disputes through negotiation and mediation, or, failing that, The International Court of Justice.
###### Article XII
Means of modifying Treaty is by unanimous agreement. After 30 years, any signatory may call a conference to review the Treaty's operation.
###### Article XIII
Treaty subject to ratification by signatory states. Open to succession by any UN member state.
###### Article XIV
Deposition of copies of Treaty in USA for transmission to members.
A number of protocols have been added, of which the most important for visitors is the Environmental Protocol signed 4 October 1991. It designates Antarctica as a 'natural reserve, devoted to peace and science', and prohibits activities related to mineral resources except for science, though the distinction may be hard to make in practice.
The Treaty was put in force for fifty years, and may only be changed unanimously. The current signatories represent 80% of the world's population. It includes those who first came to the table: Argentina, Australia, Belgium, Chile, France, Japan, New Zealand, Norway, South Africa, USSR, UK, and USA, plus thirty-three others, including major powers of the coming era like China and India, plus some countries whose polar credentials seem a bit thin, such as Papua New Guinea.
It has been a huge success. Some political manoeuvring has continued, thankfully without gunfire. But some countries, including Chile and Argentina, have invested heavily in developments which seem to overlook some of the articles they have signed up to, particularly Article IV d) and e), that you can't reinforce your claim after signing the treaty. Both have turned bases into miniature settlements with spouses and children living there. On 7 January 1978, baby Emilio Marcos de Palma was born to Silvia and her husband Argentine Army Captain Jorge de Palma at Esperanza Base. She was flown in heavily pregnant with doctors, gave birth, and took a photo-call before being flown home. In the next five years seven more births followed. Not to be outdone, in 1984 a baby was born at the Chilean Eduardo Frei Base. These are not the priorities of the Treaty, but childbirth is a healthier form of competition than gunfire.
## GOVERNING TOURISM
IAATO is a trade organisation founded in 1991 to advocate, promote and practice safe and environmentally responsible private-sector travel to the Antarctic. It was founded in 1991 by seven private tour operators already operating there and now has over a hundred members. It was a smart move satisfying several important needs:
□ it demonstrated responsible practice to governments with a stake in Antarctica and
□ has allowed the industry to operate and develop without government interference
□ has promoted good practice
□ has promoted management and
□ monitoring to inform future policy
## WHO OWNS THE FALKLAND ISLANDS (ISLAS MALVINAS)?
The Falklands are the principal islands, and the only ones with a significant population, in a number of territories, known as the Falkland Islands Dependencies, that share related issues of sovereignty and administration.
British sovereignty claims over the Falkland Islands and South Georgia are different from British claims in Antarctica proper, because both those territories lie north of the limit of the Antarctic Treaty.
Beginning with South Georgia, in 1775 Cook landed, made a model claim, and drew a chart, but no one came to colonise. When the Norwegian Carl Anton Larsen wanted to set up a whaling station in Grytviken, South Georgia in the first years of the twentieth century, Britain was almost embarrassed to find she had the best claim on the island, and felt obliged to administer it. Since then she has been firm in maintaining that claim, and has been actively governing the whaling and fishing industries, and more recently, tourism.
Two countries now claim sovereignty over the Falklands, remote islands with a total population the same as the crew of HMS _Hermes_ , the flagship of the British Falklands War Task Force of 1982. Before relating the British and Argentine claims, it's helpful to give a brief history of the discovery and government of the islands. It may be helpful to read these in the light of three core questions to which the debate returns, put by D. George Boyce in his excellent and concise book _The Falklands War_ , which has greatly informed this chapter.
□ What right has Britain to remote islands 8000 miles from its shores?
□ What right has Argentina to govern a people of wholly different racial, linguistic and cultural background?
□ What right does a small population with a dependent economy have to self-determination?
## HISTORY
Several navigators have been credited with the first sighting of the Falklands. Like many remote islands they were chanced upon because a vessel had been blown off course by a storm, as was the fine English seaman John Davis when he saw land on 14 August 1592. Of course the circumstances in which early mariners were least sure of their position was when tossed around in a storm, but it usually taken that he saw the Falklands, though he did not land. The earliest sighting that has been conclusively authenticated was by Dutchman Sebald van Weert in 1600. The first known landing was not made until 1690, by the British naval captain John Strong in the thirty-eight-gun privateer _Welfare_. Strong named the islands after his patron Viscount Falkland, who soon afterwards became First Lord of the Admiralty. A few years later, passing French sailors called the Islands _Les Îles Malouines_ after their home port of St Malo. The Spanish translated this as _Las Islas Malvinas_. It remained uninhabited.
In 1764, a small French colony, Port Louis, was established on East Falkland, claiming possession for France. Comically, the following year, a British expedition, not knowing the French were established there, reached Saunders Island, West Falkland, and anchored in a harbour which it named Port Egmont. It took formal possession of it and of 'all the neighbouring islands' for King George III. The following year, another British expedition established a settlement of about 100 people at Port Egmont, and the year after, 1767, France sold Port Louis to Spain for £24,000, who renamed it Puerto de la Soledad. The dual occupation was short-lived. In 1770, the Spanish drove the British out of Port Egmont. Britain negotiated a return but almost as soon as they got back, they evacuated the islands on economic grounds in 1774, but without relinquishing British sovereignty.
In 1810 Argentina declared independence from Spain but it was not yet a functioning state. The Spanish settlement on East Falkland was itself withdrawn in 1811, forty-four years after acquiring it, leaving the Islands without inhabitants or any form of government. The islands continued to be used by sealers, principally British and American, until in November 1820, Colonel David Jewett, an American national, visited the Islands and claimed formal possession of them in the name of the Government of Buenos Aires, but stayed on the islands for only a few weeks.
In 1820 the Government of the Province of Buenos Aires, trying to clear debts, sold the rights to exploit fish and the plentiful wild cattle running free on the Falklands to the French-born, US-trained businessman Louis Vernet. He also sought permission from the British. By 1828 the business was thriving, under the management of Matthew Brisbane, who had been master of the tiny cutter _Beaufoy_ when James Weddell discovered the Weddell Sea. They sold fresh and salted beef, vegetables and dairy products to ships, but felt threatened by the visits of numbers of US sealers, returning from Antarctica, who used the islands as their own. When Vernet asked for the protection of a naval ship, Buenos Aires refused, but appointed him Governor. In 1831, after warning the US _Harrier_ for unauthorised exploitation of the islands he confiscated the vessel and sent its master to Buenos Aires for trial. In reprisal the United States sent the warship _Lexington_ , which trashed Louis Vernet's settlement and declared the Falklands free from all government.
The islands remained without authority until September 1832, when the Government of Buenos Aires appointed Juan Mestivier as Civil and Political Governor on an interim basis. The British Government protested to the Buenos Aires Government that this appointment infringed British sovereignty over the Islands, but Mestivier sailed to the Falklands at the end of 1832 on the schooner _Sarandi_ with some convicts and twenty-five soldiers, most of whom had been deported as part-punishment for offences. Four days after arrival he was murdered by his own soldiers. In January 1833, the British warship HMS _Clio_ arrived at Puerto de la Soledad with instructions to exercise British rights of sovereignty, and give the remaining Argentines the opportunity to stay or return to Buenos Aires. They left Matthew Brisbane in command. He was murdered by the convicts.
The British naval officer First Lieutenant Henry Smith arrived on HMS _Challenger_ the following year to administer the Islands. With four seamen and Royal Marines, he spent four years restoring Vernet's work. In 1841, a civil Lieutenant Governor was appointed and, in 1843, the civil administration was put on a permanent footing by an Act of the British Parliament. The Lieutenant Governor's title was changed to Governor and, in 1845, the first Executive and Legislative Councils were set up. There was a majority of appointed officials in the Legislative Council until 1951, but from 1949, members elected by universal adult suffrage were introduced into the Council.
British rule continued until the Falklands were invaded and occupied by Argentine military forces on 2 April 1982. A British task force was despatched and, following a conflict in which 255 British and 625 Argentine lives were lost, the Argentine forces surrendered on 14 June 1982. British records show that suicides from post-traumatic stress have now exceeded the fatalities suffered in direct combat. Argentine figures are similar. Therefore total casualties now approach the 1982 population of the islands, which was eighteen hundred souls.
## THE BRITISH CASE FOR THE FALKLAND ISLANDS
There was no indigenous or settled population before British settlement. With the exception of two months' illegal occupation in 1982, the Falklands have been continuously, peacefully and effectively inhabited and administered by Britain since 1833. Over many years, international postal agreements have been made confirming British adminis tration with little or no challenge.
Not until 1946 did the Fascist dictator Peron claim the Falklands. In 1947 Britain invited Argentina to subject sovereignty to arbitration at the International Court of Justice at the UN, binding itself to the judgement; Argentina refused. Three further similar invitations were refused.
Many people who live in the Falklands trace their origins in the Islands back to the early nineteenth century. The islands have been listed by name among British dependencies since 1887. Enquiries for leases have been directed to The British Government since the nineteenth century. Argentina was formally notified of Letters of Patent (a legal instrument granting title) claiming the Falklands in 1908; they made no objection, and took no action to contest it.
#### THE LINDBLAD EXPLORER: LARS-ERIC LINDBLAD INVENTS ANTARCTIC TOURISM
In the late 1950s, both Chile and Argentina took over five hundred paying passengers on various voyages to the Antarctic Peninsula using naval transport ships. Systematic tourism began in 1966, when Lars-Eric Lindblad wanted to visit Antarctica without working on a base: the usual route for the curious. He found enough like-minded souls to charter a ship and visit for pleasure. The voyage was highly successful and his one-off trip became a business. In 1969 he launched a purpose built ship, the _Lindblad Explorer_ ; its bright red hull and slightly oversized funnel were familiar to Antarctic travellers for nearly forty years, during which she made over 250 voyages there.
Lindblad devised the model for Antarctic tourism: sail from South America across the Drake, visit the Peninsula and South Shetlands, and use the ship as the hotel. Landings were made using inflatable boats, Zodiacs, which were tough, versatile and stable.
Many other companies followed the model. The big growth came with the break-up of the former Soviet Union during 1990 – 91, which brought many small ice-strengthened vessels into the market for sale or charter. With them came officers and crews experienced in ice navigation.
There was rapid growth and visitor numbers reached nearly fifteen thousand by the 1999 – 2000 season, and were close to thirty thousand as the end of that decade approached. The slump in many developed nations' economies produced a check. At the same time, several incidents provoked discussion about the appropriateness of tourism in a hazardous and pristine environment. In January 2007, the MS _Nordkapp_ hit a rock off Deception Island and spilled a small quantity of light fuel oil. Scientists were quick to ask why risks should be taken to allow tourists to come sightseeing in a fragile and unspoiled environment. They had not made the equivalent response when the only severe pollution incident happened on 28 January 1989. The Argentine vessel _Bahia Paraiso_ ran aground off the American Palmer Station, after ignoring advice and entering the bay by a channel known to the base personnel to be full of rocks. It was carrying a quarter of a million gallons of fuel for Argentine bases. The men were safely evacuated; the ship sank.
But the most notable incident occurred in the first minutes of 23 November 2007 when the _Lindblad Explorer_ , renamed the _Explorer_ , made world headlines. She was sailing down the Bransfield Strait east of Penguin Island, and sailed into the edge of an ice field which contained glacier ice as well as sea ice. The captain was experienced in Baltic waters, which have sea ice but not glacier ice. He was not experienced in Antarctic ice. The Inquiry would find he had been overconfident in approaching at speed. The ship came to a halt: not unusual. But at 00:05 an alarm sounded on the bridge indicating flooding in cabin 312. An Able Seaman was sent to look at the problem and confirmed at 00:08 that it was seawater and there was a hole in the hull. The captain went to look for himself, returned to the bridge, and in the words of the official report said, 'Oh [expletive deleted]... This is serious.' At 00:12 the general emergency alarm was sounded calling all passengers to the assembly point. Pumping could not keep pace with the flooding, which could not be isolated in the compartment that was first breached. At 02:15 the ship suffered a total power blackout. She had a list of 15°-18°, a large iceberg was approaching, and the ship could not move. As a precaution, at 02:35 the order was given to abandon ship. There were several other cruise ships in the area; I was on the _Polar Star_ approaching the Peninsula but we were too distant to assist.
The engineers laboured by flashlight below decks to restore power long enough to take the ship into clearer, safer water. All one hundred passengers and fifty-four crew were successfully evacuated to lifeboats, though unlike modern specialist polar vessels, she had open lifeboats with canvas covers, and if the weather deteriorated and rescue was delayed, they would be exposed to the elements. Inexplicably, the crew forgot to put up the covers. There was a Force 6 wind making enough spray to wet everyone in the boats. The lifeboat engines were hand-cranked, cold, and cussed. Three out of the four failed to start, and Zodiacs had to be used to pull them away from the ship.
The Norwegian vessel _Nordnorge_ arrived with the _Endeavour_ at 06:25 and the _Nordnorge_ began to pick up passengers. Two hours after the last person was picked up, the wind had risen to gale force. The ship sank at 19:00 in deep water at twenty-five miles SE of Penguin Island. Passengers were transferred to nearby King George Island and put up at the Chilean and Uruguayan bases until they could be flown out.
The marketing name of the cruise was _The Spirit of Shackleton_ , which was more realistic than anyone would have liked. But the captain's precautionary early evacuation, and the engineer's bravery in restoring power, were judged by the inquiry to have saved lives. No one died, no one was seriously injured.
Tourism will always be under scrutiny from unsympathetic eyes, but it has a value far beyond the pleasure of the moment. One of Lindblad's motives for running trips to Antarctica was 'You can't protect what you don't know.' If only scientists visit, the continent has few advocates, and they would be ones with a vested interest, easy for politicians to sideline. Every year thirty thousand new advocates for Antarctica return to their home countries. Without tourism, Antarctica would not just be the White Continent, but also the silent one.
The Falkland Islanders have repeatedly made known their wish to remain British. A poll, conducted in 1994 revealed that 87% of them would be against any form of discussion with Argentina over sovereignty, under any circumstances.
Although the United Nations General Assembly has not debated the question of the Falklands since 1988, the Committee of Twenty-Four has continued to adopt resolutions calling for negotiations between Britain and Argentina. These resolutions are flawed because they make no reference to the Islanders' right to choose their own future. Several members of the Committee have acknowledged this omission. The principle of self-determination is included in every other resolution considered by the Committee. The British position that sovereignty is not for negotiation remains unaltered. There will be no change in the status of the Falklands without the Islanders' consent. Britain notes that Argentina's claim to the Falklands principally depends firstly on a Spanish title granted by Pope Alexander VI in the treaty of Tordesillas in 1493, and secondly on the conquest of the adjacent continental territories. But that treaty was a direct deal between one petitioning country, Spain, and a Papacy whose authority was tainted by Alexander VI being Spanish and having become Pontiff by bribery. At the time, the Catholic King François of France thought the deal preposterous and demanded to 'see the clause in Adam's will which entitled the Kings of Castile and Portugal to divide the earth between them.' The islands were seen and claimed for Britain before Spain had any knowledge of them, and lie over three hundred miles from the coast.
When Jewett claimed the islands for the Government of Buenos Aires, in 1820, that government of Buenos Aires was not recognised by any other foreign power. No act of occupation followed Jewett's visit and the islands remained without effective government. Five years before Buenos Aires had declared independence from Spain, in 1816, the Spanish settlement had been evacuated, leaving the Islands without inhabitants or any form of government. Argentina's subsequent attempts at settlement were sporadic and ineffectual. As for territorial contiguity, says the FCO, it 'has never been a determinant for title to islands (otherwise the Canary Islands, for example, might be Moroccan) and should not be used to overrule the right of self-determination.'
## THE ARGENTINE CASE FOR LAS ISLAS MALVINAS
By the Treaty of Tordesillas in 1493, Pope Alexander VI apportioned discoveries in the New World between Portugal and Spain. The Falklands lie in the area given to Spain. In 1820 the Government of Buenos Aires was the only functional government of that area and had inherited the rights exercised by Spain. Argentina claims sovereignty over the Falklands through her inheritance, on independence, of Spain's possessory title and through her attempts to settle the Islands between 1826 and 1833, and through the concept of territorial contiguity, since she has undisputed sovereignty over the adjacent mainland. On 10 June 1829, the Buenos Aires Government issued a decree setting forth its rights, derived from the Spanish Viceroyalty of La Plata, placing the Islands under the control of a political and military commandant, Louis Vernet. Its remaining occupants were expelled by force by the British.
In March 1967 and again in August 1968 Britain made statements and signed a memorandum of understanding that it would concede and recognise sovereignty subject to certain conditions, and respecting the wishes of the Falklanders. In the early 1970s, the Edward Heath Government made economic agreements that resulted in many Falkland Island services being provided by Argentina at costs subsidised by the Argentine exchequer, a _de facto_ acceptance of an emerging Argentine role in administration. In 1994, the Argentine Constitution was amended to include a clause asserting sovereignty over the Islands, which would be pursued 'in accordance with international law.'
The possession of territories remote from the mother country, with populations planted by colonial settlement, is an anachronism in the twenty-first century. (This view is shared by most of Latin America, notwithstanding that a strand of Argentina's case is that it inherited rights from Spain, a colonial power which acquired them by violence.) Foregrounding the rights of a very small number of inhabitants of British descent skews the debate, moving the focus from wider issues of principle to the self-interest of a tiny community. The United Nations has not given precedence to their rights of self-determination in its resolutions about future governance.
The British Government has, under both Labour and Conservative administrations, given strong signals that Britain did not desire governance in perpetuity, and its policy was to manage the transfer of sovereignty in a manner that respected the future rights of Falklanders. From the mid-1970s, British foreign policy was that the best solution was to cede to Argentina's claims, but leaseback the islands for seventy years, so few adult islanders would live to undergo the transfer. In 1981 they withdrew full British citizenship from Falklands passport holders (the status was only restored after the 1982 war). Whenever talks approached meaningful progress towards transfer, British Governments have retreated, and negotiations have been unproductive because they were insincere on the British side.
## THE UNITED NATIONS' POSITION
In 1960 the United Nations General Assembly adopted its Declaration of the Granting of Independence to Colonial Countries and Peoples. A committee was set up to oversee implementation of this resolution. This Committee, which became known as the Committee of Twenty-Four, considered the question of the Falklands for the first time in 1964. Following its recommendations, the General Assembly adopted Resolution 2065 in 1965, inviting the British and Argentine Governments to begin negotiations with a view to finding a peaceful solution to the problem, bearing in mind the interests of the population of the Falkland Islands. During 1967 and 1968 Britain entered into negotiations with Argentina based on a willingness to transfer sovereignty. Britain maintained throughout that any transfer of sovereignty must be subject to the wishes of the Islanders. It was on this issue that negotiations foundered.
## THE FUTURE
The dispute raises interesting questions about historical rights and commitments in the modern world. No part of the British mainland is as fiercely patriotic as the Falkland Islands. Their Britishness and relative autonomy, fortified by their isolation and self-sufficiency, is at the heart of their culture and identity. Living in some future _Malvinas_ would not be life. The 1982 war hardened attitudes in the Falklands and the United Kingdom. On 1 April 1982 not one in a hundred British people at home, including me, could have found the Falklands on a map. Many thought they were north of the British Isles, somewhere near the Shetlands, perhaps. But when Britain went to war it was not for economic interest, but for principles of defending citizens and rights; the Argentine government of the day was a brutal, undemocratic, military regime. Now that many hundreds of lives, mostly military, a few civilian, have been given for the Falklands, how could any Government quietly hand them over to the country which began the war?
In Argentina, the war is perceived as a just cause badly handled. Likewise the loss, directly and indirectly, of over a thousand lives means abandoning their ambitions is unthinkable. No single political ambition in Argentina is as unavoidable to tourists as the desire to 'reclaim' the Malvinas. Slogans such as _The Malvinas are ours, and one day we will return_ , stand outside every military institution. In March 2009 British Prime Minister Gordon Brown told Argentine President Cristina Fernández (Kirchner) he would not even talk about future sovereignty.
Argentina's greatest writer Jorge Luis Borges said that the Falklands War was two bald men fighting over a comb. It would be beyond farce or tragedy if they fought over it twice. Let us hope another generation of young men and women will not pay the price.
# The Future of Antarctica
Within a year of its discovery, Antarctica's wildlife was being butchered. The Antarctic fur seal barely survived the onslaught. The last expedition to South Georgia by Captain Budington in 1892 yielded only 135 skins, all taken at small rookeries previously unknown. But once the killing stopped, the seals showed their resilience. They are superbly adapted to their environment, and they fed and bred their way back to prosperity. There are beaches in South Georgia, such as Elsehul Bay, Cooper Bay and Stromness, where in recent years I have had to abandon or cut short tourist landings when territorial males refused to concede passage across the beach. It's a nice problem, coping with their success. There is even talk of culling them in areas like the Bay of Isles in South Georgia where rarer species, like albatrosses, suffer the erosion of their nesting areas in the tussock grass by the crush of seals.
Whales were hunted right into the 1980s. The forests of spouts which welcomed the first explorers may never be seen again. Some species are recovering much better than others, and may now be close to their pre-hunting levels; others species struggle, and we can only guess why. Blue whales have been reduced to such low numbers that recovery seems difficult. Their social structures, migration habits, and the behavioural savvy passed down by animals over two centuries old, have all been damaged. Scavenging birds probably flourished on the waste of whaling and will have fallen back in recent years. Penguins and the krill-specialist crabeater seals may have stayed at much higher levels than before whaling, exploiting the plankton and small fish no longer scooped up by whales. They may be inhibiting some whales' recovery by harvesting their food. In the future, man may add to that threat. Fishing fleets have eyed the huge tonnage of protein represented by krill, so vast the annual spring bloom can be monitored from space. The moment they die, fluorine from their exoskeleton migrates into the flesh, and the body is useful only for animal feed or fertiliser. So far this has protected them from gross exploitation. One discovery can change all that.
As to current human impacts, the only ones of any importance come from outside Antarctica. Biologists have been shocked at the industrial chemicals which are showing up in animals like seals, penguins and whales which live in the earth's remotest corners. When populations vary wildly, we seldom understand why, because we learned little about the ecology of the animals before killing them or their food. Only when albatross numbers began to crash did studies of their feeding flights revealed that Falkland Island black-browed albatross parents were going to the southern Indian Ocean to feed, and being drowned when they took the hooks baited with squid lowered in their tens of thousands from long-line fishing vessels. For each drowning, a chick starved to death. Techniques have been developed to reduce this tragic by-catch to virtually nil.
People actually working in Antarctica generally have a small environmental footprint, visitors even less. But we must remember that in a simple fragile ecosystem (the two go together) small inputs can have radical impacts. The Antarctic Treaty has steered the continent through difficult times and protected it well, though cynics might argue that Antarctica's climate continues to offer the best line of defence against commercial development and pollution. Antarctica has its share of mineral reserves, and techniques for exploiting them will only improve. If significant warming occurred on the continent exploitation would become easier and more economical as more land was exposed. Current indications are that it is not experiencing any overall change in temperatures, despite some regional trends, such as the warming of the northern and central parts of the Peninsula and adjacent islands, and the cooling of the south. This has been caused by a strengthening of a wind gyre which has drawn in more warm oceanic air to the north, and which is returning from the continent more strongly chilled farther south. As the world's resources become scarcer and more expensive, it will become harder to argue and enforce a view that economic growth should ignore Antarctica's untouched minerals. Then we will find out if a genuine will exists to protect it as wilderness, or if governments have simply been making a virtue out of necessity.
Polar exploration has been consciously used as a theatre to promote individual fame and national status. The stage was huge and the cast small, but many stars were made. The same stories are revisited by each new generation. This book is a step in redressing the balance between the romantic stars and the unsung heroes.
Polar struggle always has an audience. The expedition becomes a metaphor for the struggle for life, and failure becomes death, and death itself becomes some transcendent delivery beyond worldly care. Death is not simply the price paid for ambition; it is a sacrifice, and they are transfigured. Their every moment might have been driven by a selfish hunger for fame, but in dying they are redeemed, they die for their country, for all of us. Even if their bodies are lost, devoured by white winds, the man is resurrected, in stone and bronze and oils, poetry and song, in books, newspapers and magazines; medals are struck, a cenotaph raised, and garlands laid.
A continent cannot be discovered in a moment. Like Columbus, the first Antarctic visitors saw islands, then larger coasts. They made contradictory interpretations of what they had found. Ice stopped them answering with certainty even simple questions of geography, keeping them from the shore, capturing their vessels and carrying them off in a slow-motion fairground ride to madness. The weather could kill in summer, and for many years was unendurable all winter. Slowly their guesses got better, the scale of the discovery was revealed in piecemeal portions, then joined up, until there could be no doubt that this was a continent.
The men who succeeded often fade from view. They did their job, there is no redemption in that. If you have offended no one, a promotion is expected, maybe a mention in an honours list. If the politics changed while you were away, and there is a new King or President, perhaps just a different Admiral in high office, you may be sidelined, made absolute lord of a verandahed bungalow in a pestilential clime. Sometimes it may simply be easier to ignore you.
Maritime historians will always remember the dashing Captain James Clark Ross. Anyone who has looked at a map of Antarctica cannot miss James Weddell's name, written on a frozen sea. Cook's later career has been reassessed but his reputation is worldwide and goes beyond his own sphere, which is true fame. Shackleton seems to have gained an uneclipsable position in the firmament, which probably means he is due for a fall. But two men, Palmer and Bellingshausen, who vie for the honour of first seeing continental land, are famous only in their home towns. De Gerlache, who endured the first Antarctic overwintering, on the edge of madness, is barely known outside his native Belgium.
In the book's opening, I said it was like a conspiracy, the way names have been forgotten. Politics is part of it. The nations who were there at the beginning trumpet their heroes and camouflage the role of rival nations. Heroes who suited the times, like Scott, were embraced, and his despair reinvented as self-sacrifice, while it took different times and different values to canonise Shackleton. Wilkes, the supreme hankerer after recognition and status, was brutally sacrificed for the politics of the moment.
There is a possibility that the first men to see the last continent were the sick and broken sailors of the Spanish warship the _San Telmo_ , if their vessel survived the Drake Passage only to be smashed to scantling on Livingston Island. Perhaps only their 644 ghosts looked on.
The name of the American sealer whose boot first trod on the continent may never be known, unless a stray ship's log or paper puts a name to the man who led the boat ashore. Most of all, Captain William Smith deserves to be a household name. He was not an explorer or a scientist, he was a working seaman, earning a tough living, who took time out to be curious, although curiosity didn't pay. He believed in the worth of the land on the edge of the world, which he saw once in a storm. He returned three times and brought the world in his wake. Smith died thinking that wild shore was worth a pension: something to save him from the poorhouse, or from walking the streets of Whitechapel in thin-soled boots. His petition for recognition is a sad paper chain that runs out, leaves him standing cap in hand, ignored by men sitting each day before coal fires in Admiralty offices. His adventure was worth a pension. It was worth immortality. I close with a toast, a hot rum toddy for the real hero: Bill Smith.
# A Glossary of Specialist Polar and Maritime Words
Binnacle: deck mounting holding a compass.
Brig: a vessel with two masts, both square-rigged.
Brigantine: a two-masted vessel with the foremast square-rigged and the other fore-and-aft.
Collier: a ship for carrying coal.
Confused sea: a sea in which waves come from more than one direction, producing a lumpy unpredictable motion.
Corvette: a patrol vessel smaller than a frigate.
Dead reckoning: the craft, rather than science, of estimating your progress and position from sailing speed, wind, currents and other factors when instruments are lacking or can't be used.
Finnesko: Norwegian for Lapp shoes, soft reindeer-skin boots with the fur worn outside, traditionally lined with an Arctic plant called _sennegras_ which insulates and absorbs moisture.
Föhn: a strong wind from mountains, warming as it descends, which can cause rapid snow melt.
Heave to: to turn a vessel into the wind and hold the same position.
Ice blink: a brightness on cloud low above the horizon, occurring when there is extensive ice just beyond what can be directly seen.
Isostasy: when a large ice-cap forms, its weight depresses the land. The rebound when it melts is called isostasy.
Jury rig: a temporary repair to major items of sailing gear, like spars masts and rudders, until proper repairs can be carried out.
Katabatic wind: a violent local wind caused by very cold, dense air flowing off the land, typically from ice-caps.
Lahar: a flow of volcanic ash and water which can travel much faster than lava.
Landfall: confusingly, this means sighting land after a sea voyage, not going ashore.
Lead: a temporary linear opening in the ice.
Nunatak: a mountain peak showing above an ice-cap, usually as a slender pyramid.
Pack ice: dense ice from various sources which surrounds the continent completely each winter and melts back each summer. It is now defined as seven-tenths or more coverage of the sea surface.
Pemmican: a meat product adapted from a high-energy hunting food made by Cree and other Natives of the northern tribes. It mostly comprises meat and fat and needs carbohydrates to be added to make a basic diet. Lacks vitamin C.
Phytoplankton: unicellular plants which are the base level of the Antarctic food chain.
Polynya: a Russian term for a permanent, non-linear opening in sea ice.
Reefed sails: sails reduced in size to cope with high winds by rolling up an edge and tying it down.
Rorqual whales: large filter-feeding whales with pleats in their throats to help gulp and strain krill from the ocean.
Schooner: ship with two or more masts with the aft equal or larger to those in front of it.
Sea anchor: a floating device on a length of rope. In rough weather it is thrown overboard, attached to the bow. It is designed to drag in the water and keep the bow to wind and so present the most buoyant form to the seas.
Sea ice: ice formed when the sea freezes.
Sennegras: boot insulation, see finnesko.
Shallop: a small ship's boat, stored on deck or davits and powered by a sail or oars.
Ship's Officers: captain has three related meanings: a rank in the Navy, the holder of a Master Mariner's certificate, the man in charge of a vessel. In the eighteenth century, discipline problems became serious when officers of the army or marines were on board ship and refused to take command from Navy men or gentlemen civilian officers. The doctrine of a captain's word being absolute clarified this. Under sail, with no engineers on board, the next most senior officer was a man variously called, First Mate, Chief Mate, Chief Officer or sometimes First Officer. The officer below was therefore Second Mate or Second Officer, and so on. Midshipmen were gentlemen apprentices training to become full officers. In navies, there was usually a glass floor between these gentlemen and the men. It was exceptional to cross it, however able you were. Private vessels were more socially fluid. Above this ceiling the English names, like admiral and captain, are Latin in origin via French, because such work fell to the upper classes. Below the ceiling, starting with the bosun (Anglo-Saxon 'boatswain' meaning boatmaster) the men and their titles are all from the underclass.
Sloop: single-masted vessel with fore-and-aft rig.
Starboard: the right side of the ship when you look forward, a corruption of steerboard when most ships were steered by a trailing oar which right-handed people could most conveniently handle on that side. In port, to stop it being caught against the dock or bank, you manoeuvred the other side close in: the port side.
Tabular ice bergs: named for their flat white tops and vertical sides, like table-tops. They break off the huge ice shelves attached to sheltered parts of the continent.
Tonite: an explosive used in the late 1800s and early 1900s consisting of equal weights of barium nitrate and guncotton. The name derives from the Latin verb _tonare_ , to thunder.
Zooplankton: small crustacea which drift in the water. The 'plan' part of the word is the same base as 'planet': which means wanderer. The best known is krill.
# Acknowledgements
My deepest thanks to all the friends and colleagues who helped me through the many obscure corners of knowledge touched on in this book. Charles Aithie drew the fine, clear maps. Special thanks to my patient editor Kathryn Gray, who added creativity and imagination as well as controlling my text. More thanks to Rick Atkinson, and all the Port Lockroy staff, and Laura Baker, Robert Burton, James and Sharon Lowen, Tan Pearson, Richard and the Monday Drinks Club, Wendy Pyper, the Foyle Reading Room staff at the RGS, John Sparks, Callum and Jane Thompson, Josef Verlinden, Nancy Watson, Richard Williams and all my friends on the _Polar Star._
I met Celia Ansdell in Antarctica and I knew she understood me when she suggested that we meet up in the UK, and coaxed me by saying, 'I have a little place ten minutes walk from the Royal Geographical Society and twenty minutes from the British Library and the British Museum.' Her version is that despite knowing souvenirs of Antarctica are forbidden she brought me back and it's too much trouble to return me. Thanks for everything.
A reference and reading list is on my website: www.cloudroad.co.uk but the following sources were especially useful.
BOOKS
_The Falklands War_ by D. George Boyce (Palgrave Macmillan, 2005). I derived some of my core arguments on sovereignty from this book.
_From Venus to Antarctica: The Life of Dumont d'Urville_ by John Dunmore (Exisle Publishing, 2007) is a very readable biography.
_Sea of Glory_ by Nathaniel Philbrick (Viking, 2003) is an entertaining account of the Wilkes Expedition.
_A Visitor's Guide to South Georgia_ by Sally Poncet and Kim Crosbie (WildGuides Ltd, 2005)
Kevin Walton's and Rick Atkinson's _Of Dogs and Men_ (Images Publishing, Malvern 1996) tells you what it was like to run dogs. Great photos and text.
WEBSITES
Antarctic Treaty Secretariat www.ats.aq
British Antarctic Survey www.bas.ac.uk
Falkland Islands Government www.falklands.gov.fk
Scott Polar Research Institute www.spri.cam.ac.uk
South Georgia Government website www.sgisland.gs
South Georgia Museum www.sgmuseum.gs
www.south-pole.com is a quirky site with twin interests in polar discovery and philately.
# About the Author
John Harrison's award-winning travel writing with _Cloud Road_ and _Where the Earth Ends_ has featured journeys in South America and Antarctica. He has won the Alexander Cordell Prize twice and the 2011 Wales Book of the Year. His next project is _1519: A Journey to the End of Time_ , following overland the route of the Cortés expedition which destroyed the Aztecs. When not guiding and driving powerboats in polar regions, or travelling for his own interests, he lives in London and Cardiff.
# Copyright
First published in 2012
by Parthian
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This ebook edition first published in 2012.
All rights Reserved
© John Harrison
The publisher acknowledges the financial support of the Welsh Books Council.
Cover design by www.theundercard.co.uk
Cover photograph by John Harrison
Typesetting by books@lloydrobson.com
The right of Bryony Rheam to be identified as author of this work has been asserted in accordance with Section 77 of the Copyright, Designs and Patents Act 1988
This ebook is copyright material and must not be copied, reproduced, transferred, distributed, leased, licensed or publicly performed or used in any way except as specifically permitted in writing by the publishers, as allowed under the terms and conditions under which it was purchased or as strictly permitted by applicable copyright law. Any unauthorised distribution or use of this text may be a direct infringement of the author's and publisher's rights, and those responsible may be liable in law accordingly.
ISBN 978–1–908946–21–8
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<introduced datetime="2017-03-17"/>
<titles>
<title type="short" as="introduced">Physician Assistant Higher Education Modernization Act of 2017</title>
<title type="short" as="introduced">Physician Assistant Higher Education Modernization Act of 2017</title>
<title type="official" as="introduced">To amend the Higher Education Act of 1965 to improve education opportunities for physician assistants, and for other purposes.</title>
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<summary date="2017-03-17T04:00:00Z" status="Introduced in House">Physician Assistant Higher Education Modernization Act of 2017
This bill amends title IV (Student Assistance) of the Higher Education Act of 1965 to assist qualified individuals in receiving the education and training needed to become a physician assistant.
Graduate students enrolled in physician assistant education programs are eligible for: (1) increased Unsubsidized Stafford Loan limits, and (2) loan forgiveness programs for those who serve in areas of national need upon graduation.
The Department of Education (ED) must:
provide Historically Black Colleges and Universities and Predominantly Black Institutions with grants to fund physician assistant studies at the masters level, prioritize the awarding of grants for physician assistant education programs at the postbaccalaureate level that will benefit Hispanics and low-income students, and provide grants to prepare certified physician assistants to become faculty members in accredited graduate-level physician assistant education programs. In addition, ED is authorized to award grants to rural-serving institutions of higher education for: (1) increasing enrollment and graduation rates of students from rural areas in postbaccalaureate physician assistant education programs, and (2) training physician assistants in rural areas through the creation and expansion of physician assistant education programs in rural areas.
ED may also award grants to physician assistant education programs for enhancing the programs and improving patient safety by supporting the purchase of new infrastructure and technology.</summary>
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Scalable Information Systems
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Editors in Chief:
Xiaohua Jia (City University of Hong Kong)
Hua Wang (Victoria University)+ BioHua is a full time Professor at the Centre for Applied Informatics (CAI), Victoria University (VU). Before joining VU, he was a Professor at the University of Southern Queensland (USQ) during 2011-2013. He obtained his PhD in Computer Science from USQ in 2004. Hua has over 200 peer-reviewed research papers mainly in data security, data mining, access control, privacy and web services, as well as their applications in the fields of e-health and e-environment. Recent publications K. Cheng, L. Wang, Y. Shen, H. Wang, Y. Wang, X. Jiang. Secure k-NN Query on Encrypted Cloud Data with Multiple Keys. Accepted by Transactions on Big Data. 2017. Y. Shen, T. Zhang, Y. Wang, H. Wang and X. Jiang. MicroThings: A generic IoT Architecture for flexible data aggregation and scalable service cooperation. Accepted by IEEE Communications Magazine, May, 2017. M. Peng, W. Gao, H. Wang, Y. Zhang, G. Tiang, Parallelization of Massive Text Stream Compression Based on Compressed Sensing. Accepted by ACM Transactions on Information Systems. 2017. Y. Zhang, Y. Shen, H. Wang, J. Yong, X. Jiang, On Secure Wireless Communications for IoT under Eavesdropper Collusion. IEEE Trans. Automation Science and Engineering 13(3) (external link): 1281-1293 (2016). J. Zhang, H. Li, X. Liu, Y. Luo, F. Chen, H. Wang, L. Chang. On Efficient and Robust Anonymization for Privacy Protection on Massive Streaming Categorical Information. IEEE Transactions on Dependable and Secure Computing, no.1, pp. 1, 01/2015; DOI: 10.1109/TDSC.2015.2483503. Y. Zhang, Y. Shen, H. Wang, Y. Zhang, X. Jiang, On Secure Wireless Communications for Service Oriented Computing, IEEE Transactions on Services Computing, no. 1, pp. 1, doi:10.1109/TSC.2015.2478453 J. Ma, L. Sun, H. Wang, Y. Zhang and U. Aickelin, Supervised Anomaly Detection in Uncertain Sensor Data Streams, ACM Trans. Internet Techn. 16(1) (external link): 4:1-4:20 (2016).
Dr Manik Sharma (DAV University Jalandhar, India)
Area Editors:
Hamed Taherdoost (Research Club / Research and Development Department | Hamta Group / Hamta Business Solution Sdn Bhd)+ BioHamed Taherdoost holds a PhD in computer science from Universiti Teknologi Malaysia, MSc in information security, and BSc in the science of power electricity. With over 17 years of experience in both industry and academia, Hamed has established himself as an industry leader in the fields of Management and Information System. Hamed specializes in business management and information technology management. He started his career in 2001 as an electrical designer and programmer in Tablokar, a leading switchgear panel manufacturer in the Middle East. He has subsequently worked in IT transformation, systems analysis, and user experience analysis, and has held positions as varied as Business Manager, Project Manager, Executive Manager, CITO, CBO, CTO and CEO at international companies from Cyprus, the UK, Belize, Iran, and Malaysia such as at HAMTA GROUP | Hamta Business Solution Sdn Bhd, at an IT company, Beshnas Technology Sdn Bhd, at Ahoora Ltd, a management consultation group, and at Simurgh Pvt, an international trade company. Aside from his experience in industry, he also has numerous outstanding achievements in academia. Hamed has authored over 120 scientific articles in peer-reviewed international journals and conference proceedings, as well as six books. His academic employment started in 2009 when he became a lecturer at IAU and Payam Noor University. Moreover, he serves as an advisory board member and reviewer for many international journals and has organized and chaired numerous conferences and conference sessions respectively. Currently, he is the Team Leader & Supervisor of Research Club and Hamta Academy, Malaysia, Advisory Board of Cambridge Scholars Publishing, UK and Scientific Board of Nan Yang Academy of Sciences (NASS), Singapore. He is a Senior Member of the IEEE, Member of British Science Association, Sales Management Association, Production and Operations Management Society, American Association for Science and Technology, Scientific and Technical Research Association, IEEE Cybersecurity, Cyber Ethics and Cyber Peace, IEEE Blockchain Community, IFIP TC 11 Working Group and many other professional bodies. Particularly, he has completed Associate in Project Management (CAPM), Information Systems Auditor (CISA), Certified Ethical Hacker (CEH), Information Security Manager (CISM), PMI Risk Management Professional, Project Management Professional (PMP), Computer Hacking Forensic Investigator (CHFI) and Certified Information Systems (CIS), courses. His research and industry knowledge areas are Management of Information System, Technology Acceptance Models and Frameworks, Information Security, Information Technology Management, Web Service Quality and Security, Performance Evaluation, Project Management, Business Management, Leadership, Manufacturing Strategies, Business and Research Proposal Development, International Business, IT/IS Management, Multi-Criteria Decision Analysis (TOPSIS, Analytic Network Process, Analytical Hierarchy Process and SMART), Business Analysis and Market Research, User Acceptance Testing and Web Project Management.
Phan Cong Vinh (Nguyen Tat Thanh University (NTTU))+ BioPhan Cong Vinh received a PhD in computer science from London South Bank University (LSBU) in the United Kingdom. He finished his PhD dissertation with the title of "Formal Aspects of Dynamic Reconfigurability in Reconfigurable Computing Systems" at LSBU where he was affiliated with the Center for Applied Formal Methods (CAFM) at the Institute for Computing Research (ICR). At present, he is an Associate Professor of Nguyen Tat Thanh University (NTTU) to take on the responsibility of a senior research scientist. He has been author or co-author of many refereed contributions published in prestigious journals, conference proceedings or edited books. He is editor of three books titled, "Autonomic Networking-on-Chip: Bio-Inspired Specification, Development and Verification" (CRC Press, 2012), "Formal and Practical Aspects of Autonomic Computing and Networking: Specification, Development and Verification" (IGI Global, 2011) and "Nature-Inspired Networking: Theory and Applications" (CRC Press, 2018). He has served on many conference program committees and has been general or technical (co)chair and (co)organizer of several international conferences such as ICCASA and ICTCC series. His research interests center on all aspects of formal methods in computing, context-awareness, nature of computation and communication, and applied categorical structures in computer science.
Jose Machado (Uniiversity of Minho)
Giancarlo Fortino (University of Calabria)
Ajay Kattepur (Tata Consultancy Services)
Tzung-Pei Hong (National University of Kaohsiung, Kaohsing City, Taiwan)
José Merseguer (Universidad de Zaragoza)
Houbing Song (Embry-Riddle Aeronautical University)+ BioHoubing Song (M'12–SM'14) received the Ph.D. degree in electrical engineering from the University of Virginia, Charlottesville, VA, in August 2012. In August 2017, he joined the Department of Electrical, Computer, Software, and Systems Engineering, Embry-Riddle Aeronautical University, Daytona Beach, FL, where he is currently an Assistant Professor and the Director of the Security and Optimization for Networked Globe Laboratory (SONG Lab, www.SONGLab.us). He served on the faculty of West Virginia University from August 2012 to August 2017. In 2007 he was an Engineering Research Associate with the Texas A&M Transportation Institute. He serves as an Associate Technical Editor for IEEE Communications Magazine. He is the editor of four books, including Smart Cities: Foundations, Principles and Applications, Hoboken, NJ: Wiley, 2017, Security and Privacy in Cyber-Physical Systems: Foundations, Principles and Applications, Chichester, UK: Wiley-IEEE Press, 2017, Cyber-Physical Systems: Foundations, Principles and Applications, Boston, MA: Academic Press, 2016, and Industrial Internet of Things: Cybermanufacturing Systems, Cham, Switzerland: Springer, 2016. He is the author of more than 100 articles. His research interests include cyber-physical systems, cybersecurity and privacy, internet of things, edge computing, big data analytics, unmanned aircraft systems, connected vehicle, smart and connected health, and wireless communications and networking. Dr. Song is a senior member of ACM. Dr. Song was the very first recipient of the Golden Bear Scholar Award, the highest campus-wide recognition for research excellence at West Virginia University Institute of Technology (WVU Tech), in 2016.
Tuan Phung-Duc (University of Tsukuba, JP)
Chang Choi (Chosun Univ.)
Sherman S.M. Chow (The Chinese University of Hong Kong)
Mirjana Ivanovic (University of Novi Sad)
Manik Sharma (DAV University, India)+ BioDr Manik Sharma is MCA and UGC-NET qualified, and received his PhD degree in Distributed Query Optimization. He has more than 12 years of teaching experience, having previously worked as head of the Department of Computer Science and Application and officiating principal at Sewa Devi S.D. College, India. Currently, he is working as an Assistant Professor in the Department of Computer Science and Applications at DAV University, Jalandhar, India. His main research work focuses on query optimization, data mining, soft computing and machine learning. Currently, he is working in the area of data mining, machine learning and soft computing, with a specialization in the diagnosis of diseases and modeling of DSS query optimizers. The thrust area of his research is to improve predictive accuracy in early diagnosis of several fatal human disorders. He has published four books related to computer networks, internet applications, management information systems and database theory. In addition, he has published 25 research papers in various journals of international repute including some indexed by Scopus and Thomson Reuters. He also presented more than 10 research articles in various national and international UGC and IEEE conferences, and has given several talks on query optimization, data mining and health informatics. He is an active member of the international program committees, technical program committees, and advisory committees of several academic conferences.
Aniello Castiglione (University of Naples Parthenope)
Dr.VIJAYAKUMAR PONNUSAMY (ASSOCIATE PROFESSOR,SRM UNIVERSITY)+ BioPublications: • International Journal Published: 58 (SCI:6;Scopus indexed-23; others 29) • Conferences paper presented: International: 30; National: 23 Few Selected Publications: 1. Vijayakumar Ponnusamy,abajieet,balaji,sangeetha,"A Palm Vein Recognition System based on support vector machine", IEIE Transaction on smart processing and computing ,accepted ,Dec 2018.(Scopus indexed). 2. Vijayakumar KP; Pradeep Mohan Kumar K; Kottilingam K; Karthick T; Ganeskumar P; Vijayakumar P, "An Adaptive Neuro Fuzzy Logic based Jamming Detection System in WSN", Soft Computing(ISSN: 1432-7643), Science Citation Index, IF= 2.367),nov 2018. 3. Arumbhu, S. Malarvizhi, P.Vijayakumar, "Design of TAS-STBC-ESM(F) transceiver and Performance analysis for 20 bpcu", IEEE access ,March 2018. (Science Citation Index, IF=3.244) 4. Arumbhu, S. Malarvizhi, P.Vijayakumar," Performance Analysis of SPSK with Dual Polarized Transmit Antennas over Rayleigh fading channel", selected in the Journal of Telecommunications and Information Technology, no. 1/2018. (Scopus indexed). 5. P.Vijayakumar, Ranjan R., Malarvizhi S, "Waveform Generation and Reception of IEEE 802.11p Standard for CR-VANET Application", Smart Computing and Informatics, Smart Innovation, Systems and Technologies, vol 77.,pp 583-591, 2018, Springer (Scopus indexed, book series). 6. P.Vijayakumar, George J, Malarvizhi S, Sriram A, "Analysis and Implementation of Reliable Spectrum Sensing in OFDM Based Cognitive Radio", Smart Computing and Informatics, Smart Innovation, Systems and Technologies, vol 77, pp 565-572 ,2018, Springer(Scopus indexed, book series). 7. P. Vijayakumar, S. Malarvizhi, "Self-Diagnosis of Cognitive Relay on the Joint Impact of Hardware Impairment and Channel Estimation Error", Int. J. of Systems, Control and Communications, 2017, Vol.8, No.4, pp.335 - 347 (Scopus indexed). 8. P. Vijayakumar, S. Malarvizhi, "Wide Band Full Duplex Spectrum Sensing with SelfInterference Cancellation–an Efficient SDR Implementation "Mobile Networks & Applications, 22(4), 702-711,2017 (SCI indexed ;IMP=1.538). 9. P.Vijayakumar, S.Malarvizhi, "Fuzzy Logic Based Decision System for Context Aware Cognitive Waveform Generation", Wireless Personal Communications (ISSN No.0929- 6212), 94(4), pp. 2681–2703, June 2017 (Science Citation Index, IF=0.951).10. P. Vijayakumar,S. Malarvizhi , "Hardware Impairment Detection and Prewhitening on MIMO Precoder for Spectrum Sharing", Wireless Personal Communication(ISSN No.0929- 6212) , Springer Journal, 96(1), pp.1557–1576 ,September 2017 (Science Citation Index, IF=0.951). 11. P.Vijayakumar, S. Malarvihi," Green Spectrum Sharing: Genetic Algorithm Based SDR Implementation", Wireless Personal Communication (ISSN No.0929-6212), Springer Journal, 94(4), pp 2303–2324, June 2017 (Science Citation Index, IF=0.951). 12. P.Vijayakumar , Bahraini Jayapandian Kasthuri and Mayuri.B.R, "Game Theory based Hybrid Cognitive Radio Transmission", Indian Journal of Science and Technology, Vol 9(20), May 2016(Scopus indexed) 13. P.Vijayakumar, S.Malarvizhi , LTE-U Link Adaptation for Next Generation Wireless Networks", International Journal of Control Theory and Applications, Vol.9, No.34, (2016),pp.no 391-399 (Scopus indexed) 14. P. Vijayakumar, S. Malarvizhi "MIMO Cognitive Radio Spectrum Sharing Using spatial coding and user Scheduling for Fading Channels" , International Journal Of Multimedia And Ubiquitous Engineering , Vol.11, No.3 (2016), pp.103-114 (Scopus indexed) 15. P. Vijayakumar, S. Malarvizhi, "Reconfigurable Filter Bank Multicarrier Modulation for Cognitive Radio Spectrum Sharing - A SDR Implementation", Indian journal of science and technology, vol 9.issue 6, Feb 2016. (Scopus indexed) Education: Completed Data scientist using R specialization in datacamp -Online (23 courses ) Completed PhD, Wireless Communication (Cognitive Radio-applied machine learning in wireless communication) in SRM IST Tamil Nadu, India, on Feb, 21, 2018. Dissertation Topic: "Design of Fuzzy Decisive Cognitive Engine and Context Aware Waveform Generation" Certified "IOT Specialist" through online course , University of California, Irvine M.E (Applied Electronics) - College Of Engineering, Guindy, Anna University, India, May 2006 with First class. B.E (Electronics and Communication Engineering) From University of Madras, India on April 2000 with First class Research Experience: (along with teaching) Research Projects ongoing: BRNS grant: Rs 23,95,650.00, Project titled: "Real time hardware based raw data processing for dual energy X-ray baggage Inspection system XBIS" Investigators: Dr S.Malarvizhi and Vijayakumar.P; dated 1/8/17; sanction number 34/14/07/2017-BRNS/34282. Methods involved: Machine learning algorithms of neural network, convolutional neural network, FPGA implementation and computational complexity optimization for real time object detection. Completed: o NI sponsored AWR software tool of two licenses (worth 2x $57,600 ) for one year , for case study making under two Projects of 1. "Full Duplex Cognitive Hybrid Multiband System for V2V Communication" - Methods involved: SDR implementation of 802.11P with Game theory. 2. "Chaotic Communication System Simulation " o DST-FIST MIMO OFDM project –PhD Research Scholar o Mentor for SRM Nano SAT Project in Onboard Computing and Communication Subsystem o Pilot Project-Vehicle Tracking System ,1 Lakh Sponsored by SRM University Award Received: o Received best paper award in international conference of internet of things ICIOT- 2019 for the paper "performance analysis of millimeter wave radar for connected vehicles" held in SRM IST, from 11 to 15 March,2019. o Nominated and certified as Best Reviewer of ICVISP 18 international conference 12 Oct 2018 o I2OR-Outstanding Researcher Award 2018 by international institute of organized research, Dec 2018. o Dr .Abdul Kalam Young Scientist Award 13 Oct 2018 from Marina Lab,Chennai. o Awarded "Adarsh Vidya Saraswati Rashtriya Puraskar" by Global Management Council on 15th August 2018 o IEAE Research Excellence Award, March, 2018. o Selected for Prestigious IRDP Award 2018"Teaching and Research Excellence" for remarkable achievements in Teaching, Research and Publications at Chennai on 30th May 2018 o NI Engineering impact award November 19th 2015-acadamic category for innovative application development for full duplex CR transmission using NI USRP RIO 2943R ((XILINK Kintext7 FPGA programming and signal processing optimization) http://india.ni.com/nieia/winners/2015 Patent: o An Indian Patent filled: A System and Method for Self Diagnosis and Mitigation of Hardware Impairments In Wireless Transceivers, Application#: 201841025130 and Date of Filing: July 05, 2018 ( PhD work) With Machine learning algorithm of SVM and Neural network.o An Indian Patent filled: a system and method for achieving higher spectral and energy efficiency in index modulation transceiver, Application#: 201841035475 and Date of Filing: September 20, 2018 Skill Set: Hardware and Software Tools: Machine Learning and Deep Learning Programming on MATLAB, Python on Anaconda Navigator: Scikit-learn, Tensorflow, PyTorch and R programming. RTOS -DSP-BIOS Languages- C, C++, JAVA, MATLAB, VHDL, Labview, Python Embedded Processor -89C51RD2, Adriano, AVR, Raspberry Pi & it is IDEs. DSP -TMS 320C5X, TMS320C6713 programming. FPGA - Altera CYCLONE-III, Kinext 7 SDR-USRP 2943R;PXie platform based VSG and VGA Wireless Protocol Development on SDR-LTE, 802.11 Mathematical and algorithmic skill: Machine learning :SVM,K-means clustering, Neural Network ,Game theory ,Genetic algorithm, Fuzzy logic ,regression models, correlation analysis ,Decision Tree, Deep neural network, auto encoder, reinforcement learning, Bayesian network. Probability Theory and Random Processing Statistical Signal Processing Linear algebra Linear Optimization techniques Teaching Experience: Working as Associate Professor in Dept. of ECE, SRM IST, Kattankulathur Campus, and Chennai from July 2018 to till date Worked as Assistant Professor (Selection Grade) in Dept. of ECE, SRM IST, Kattankulathur Campus, Chennai from June 2017 to July 2018. Worked Assistant Professor (Senior Grade) in Dept. of ECE, SRM IST, Kattankulathur Campus, and Chennai from June 2007 June 2017. Worked as lecturer in Dept. of ECE, SRM IST, Ramapuram Campus, and Chennai from August 2005 to June 2007. Worked as Senior Lecturer\HOD in charge in Dept. of ECE, Danish College of Engineering, padappai, Chennai from June 2004 to June 2005. Worked as a lecturer\HOD in charge in the Dept. of ECE SMK Fomra Institute of Tech, kelambakkam, Chennai from July 2002 to June 2004. Worked as lecturer in the Dept. of ECE, A.M.S College of engineering, Namakkal from June 2000 to July 2002. Handled courses of communication theory, microwave and RF , Mobile computing and digital image processing in NOKIA Chennai as visiting faculty from 2009 to 2011 Handled DSP level II course & embedded projects as part time faculty in AU-PERSembedded technologies-Anna university from 2003-2004.Online Courses Completed: 57 Online courses completed with certificate of accomplishment in Coursera, Edx, NPTEL, udemy platform namely: Artificial Intelligence : Learn How To Build An AI , Python, Introduction to R ,Deep Learning : Hands-On Artificial Neural Networks, game theory ,machine learning, IOT,MIMO,OFDM,CDMA, a System View of Communication ,image and video processing, Linear and Integer Programming, Outcome Based Pedagogic Principles for Effective Teaching ,Computer Networks, Elements of Learner-Centric MOOCs , Introduction to Cloud Computing ,Probability and Random Variables/ Processes for Wireless Communications, writing in the Sciences and IEEE Certified Course on Innovation Management and Intellectual Properties Workshop / Training Program Attended: 29 workshop and training programs attended from 2010 to till date Professional Experience: Reviewers/Editorial member: Technical conference committee Member for the 3rd International Conference on Vision, Image and Signal Processing (ICVISP 2019) held in , Vancouver, Canada , 2019 Reviewer in 9th international conference on Electronics, Communication and Networking (CECNet 2019), Japan. Reviewer in IEEE Access journal Reviewer in IEEE Internet of Things Journal Reviewer in Acta Polytechnica Reviewer in Wireless personal communication, springer journal Reviewer in WSEAS Transactions on Communications Reviewer in applied soft computing – Elsevier journal Reviewer in digital communication and network – Elsevier journal Reviewer in journal of electrical engineering and technology Reviewer in COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Reviewer in Biomedical and Pharmacology Journal Reviewer in CDVE2018: The 15th International Conference on Cooperative Design, Visualization and Engineering, Hangzhou, China Editorial member in EnPress Publisher, USA for journal of Architecture and Design Review Editorial member in Int. J. of Vehicle Information and Communication Systems (Inderscience Publishers) Area Editor for the EAI Transactions on Journal e-Learning Technical conference committee Member for the 2nd International Conference on Vision, Image and Signal Processing (ICVISP 2018) will be held in Las Vegas, USA on August 27-29, 2018. Technical program committee member in ICCASA 2018 - 7th EAI International Conference on Context-Aware Systems and Applications, November 22-23, 2018,Viet Tri, Vietnam(http://iccasa.org/technical-program-committee/) Reviewer in the 8th International Conference on Digital Information and Communication Technology and its Applications (DICTAP2018), Lodz, Poland. Reviewer in the Fifth International Conference on Artificial Intelligence and Pattern Recognition (AIPR2018) September 17-19, 2018, Lodz, Poland. Reviewer in EAI Transactions on The Energy Green, Intelligent in Computing & Communication Technologies Guest editor for Special Issue of Information systems enabling digitalized advances in society- (ItAIS2017, the 14th Conf. of the Italian Chapter of AIS (Association for Information Systems)
J Amudhavel (VIT Bhopal University)
As the data volumes continue to increase and the ways of information dispersion across the globe continue to diversify, new scalable methods and structures are needed for efficiently processing those distributed and autonomous data. Grid computing, P2P technology, distributed information retrieval technology and networking technology all must be merged to address the scalability concern.
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\section{Cvitanovic's conjecture for g-2 in QED}
In their pioneering calculation of the $g-2$ factor of the electron to sixth
order in 1974, Cvitanovic and Kinoshita \cite{cvikin74} found a coefficient which was much
smaller numerically than had been expected by a naive estimate based on
the number of Feynman diagrams involved. A detailed analysis revealed
extensive cancellations inside gauge invariant classes of diagrams. This led
Cvitanovic \cite{cvitanovic1977} to conjecture that, at least in the quenched
approximation (i.e. excluding diagrams involving virtual fermions) these cancellations
would be important enough numerically to render this series convergent
for the $g-2$ factor.
Although nowadays there exist a multitude of good arguments against convergence of
the QED perturbation series (see, e.g., Ref. ~\refcite{dunnerev}), all of them are based
on the presence of an unlimited number of virtual fermions, so that Cvitanovic's conjecture
is still open today. Moreover, should it hold true for the case of the $g-2$ factor,
it is natural to assume that it extends to arbitrary QED amplitudes in this quenched approximation.
In previous work \cite{dunschSD,colima} the QED effective
Lagrangian in a constant field was used for analyzing the
$N$ -- photon amplitudes in the low-energy limit. Based on
existing high--order estimates for the imaginary part of this Lagrangian,
Borel dispersion relations, and a number of two--loop consistency checks,
this very different line of reasoning makes ``quenched convergence'' appear
quite plausible for the case of the $N$ -- photon amplitudes.
Its central point is an all-order conjecture for the imaginary
part of the constant-field effective Lagrangian for Scalar QED in the weak field
limit due to Affleck, Alvarez, and Manton \cite{afalma} (AAM).
Here we present ongoing work towards a first three-loop check of this
conjecture \cite{wip}.
\section{The AAM conjecture}
Let us start with recalling the representation obtained by Euler and Heisenberg \cite{eulhei}
for the one-loop QED effective Lagrangian in a constant field,
\begin{eqnarray}
{\cal L}^{(1)}_{\rm spin}(F) &=& - \frac{1}{8\pi^2}
\int_0^{\infty}{dT\over T^3}
\,\,{\rm e}^{-m^2T}
\biggl[
{(eaT)(ebT)\over {\rm tanh} (eaT){\rm tan} (ebT)}
- {1\over 3}(a^2-b^2)T^2 -1
\biggr]
\nonumber\\
\label{eulhei}
\end{eqnarray}\noindent
Here $T$ is the proper-time of the loop particle and $a,b$ are defined by
$a^2-b^2 = B^2-E^2,\quad ab = {\bf E}\cdot {\bf B}$.
The analogous formula for Scalar QED was obtained by Weisskopf \cite{weisskopf}
but will also be called ``Euler-Heisenberg Lagrangian'' (EHL) in the following.
Except for the magnetic case,
these effective Lagrangians have an imaginary part.
Schwinger \cite{schwinger} found the following representation for the
imaginary parts in the purely electric case,
\begin{eqnarray}
{\rm Im} {\cal L}_{\rm spin}^{(1)}(E) &=& \frac{m^4}{8\pi^3}
\beta^2\, \sum_{k=1}^\infty \frac{1}{k^2}
\,\exp\left[-\frac{\pi k}{\beta}\right] \nonumber\\
{\rm Im}{\cal L}_{\rm scal}^{(1)}(E)
&=&
-\frac{m^4}{16\pi^3}
\beta^2\, \sum_{k=1}^\infty \frac{(-1)^{k}}{k^2}
\,\exp\left[-\frac{\pi k}{\beta}\right]\nonumber\\
\label{schwinger}
\end{eqnarray}
($\beta = eE/m^2$). These formulas imply that any constant electric field
will lead to a certain probability for electron-positron pair creation from
vacuum. The inverse exponential dependence on the field
suggests to think of this as a tunneling process in which virtual pairs
draw enough energy from the field to turn real.
In the following we will be interested only in the
weak field limit $\beta \ll 1$, which allows us to truncate the
series in (\ref{schwinger})
to the then dominant first ``Schwinger exponential''.
For the Scalar QED case, Affleck et al. \cite{afalma} proposed in 1982 the
following all-loop generalization of (\ref{schwinger}),
\begin{eqnarray}
{\rm Im}{\cal L}^{({\rm all-loop})}_{\rm scal}(E)
\,\,
&{\stackrel{\beta\to 0}{\sim}}&
\,\,
\frac{m^4\beta^2}{16\pi^3}
\,{\rm exp}\Bigl[ -{\pi\over\beta}+\alpha\pi \Bigr]
\label{ImLallloop}
\end{eqnarray}\noindent
This formula
is highly remarkable for various reasons.
Despite of its simplicity it is a true all-loop result; the rhs
receives contributions from an infinite set of
Feynman diagrams of arbitrary loop order,
including also mass renormalization counterdiagrams.
Moreover, the derivation given in Ref. \refcite{afalma} is very
simple, if formal. Based on a stationary path approximation of
Feynman's worldline path integral representation \cite{feynman} of
${\cal L}_{\rm scal}(E)$, it actually uses only
a one-loop semiclassical trajectory, and arguments that this
trajectory remains valid in the presence of virtual photon
insertions.
An independent derivation of (\ref{ImLallloop}), as well as extension to
the spinor QED case, was given by Lebedev
and Ritus \cite{lebrit} through the consideration of higher-order corrections
to the pair creation energy in the vacuum tunneling picture.
At the two-loop level, (\ref{ImLallloop}) has
also been verified by a direct calculation of the EHL \cite{dunsch1} (for the
spinor QED case), as well as been extended to the case of a
self-dual field \cite{dunschSD}.
\section{Connection between the AAM and Cvitanovic conjectures}
Writing the AAM formula (\ref{ImLallloop}) as
\begin{eqnarray}
{\rm Im}{\cal L}^{({\rm all-loop})}_{\rm scal}(E)\,\,
=
\,\,\sum_{l=1}^{\infty}{\rm Im}{\cal L}^{(l)}_{\rm scal}(E)
\,\,
&{\stackrel{\beta\to 0}{\sim}}&
\,\,
{\rm Im}{\cal L}^{(1)}_{\rm scal}(E)\,\,{\rm e}^{\alpha\pi}
\label{LalltoL1}
\end{eqnarray}\noindent
it states that an all-loop summation has produced the convergent
factor $\,{\rm e}^{\alpha\pi}$, clearly an observation similar in vein to Cvitanovic's.
Moreover, at least at a formal level it is not
difficult to transfer this loop summation factor from
${\rm Im}\Gamma(E)$ to the QED photon amplitudes \cite{dunschSD,colima} .
Consider the weak field expansion of the $l$-loop contribution to the electric
EHL:
\begin{eqnarray}
{\cal L}^{(l)}(E) &=& \sum_{n=2}^{\infty} c^{(l)}(n) \Bigl(\frac{eE}{m^2}\Bigr)^{2n}
\label{wfe}
\end{eqnarray}\noindent
Using Borel dispersion relations, (\ref{ImLallloop}) can be shown \cite{dunsch1,dunschSD}
to imply that, at any fixed loop order $l$, the weak field expansion coefficients
have the same asymptotic growth,
\begin{eqnarray}
&& c^{(l)}(n)\quad {\stackrel{n\to \infty}{\sim}} \quad c^{(l)}_{\infty}\, \pi^{-2n}\Gamma[2n - 2]
\label{asymp}
\end{eqnarray}\noindent
where the constant $c^{(l)}_{\infty}$ relates directly to the prefactor of the
corresponding leading Schwinger exponential in the weak field limit:
\begin{eqnarray}
{\rm Im}{\cal L}^{(l)}(E)\,\, &{\stackrel{\beta\to 0}{\sim}} &\,\, c^{(l)}_{\infty}\,\,{\rm e}^{-\frac{\pi}{\beta}}
\label{borel}
\end{eqnarray}\noindent
As is well-known, the $n$th term in the weak field expansion of the $l$ - loop EHL
carries information on the corresponding $N=2n$ - photon amplitudes in the low energy limit.
Let us assume that the asymptotic behaviour should not depend on the choice of photon polarizations
$\varepsilon_i$ (this is plausible and supported by two-loop results \cite{colima}).
Since the kinematical
structure of the $N$ - photon amplitudes in this limit reduces to a prefactor which
is the same at any loop order
\cite{mavisc}, one can eliminate it by dividing the $l$ - loop amplitude by the one-loop one.
Expanding (\ref{ImLallloop}) in $\alpha$ and combining it with (\ref{borel}) and (\ref{asymp})
one then arrives at a formula for the ratio of amplitudes in the limit of large photon number,
\begin{eqnarray}
{\rm lim}_{N\to\infty}
\frac
{\Gamma^{(l)}[k_1,\varepsilon_1;\ldots ;k_N,\varepsilon_N]}
{\Gamma^{(1)} [k_1,\varepsilon_1;\ldots ;k_N,\varepsilon_N]}
&=&
\frac{({\alpha\pi})^{l-1}}{(l-1)!}
\label{ratio}
\end{eqnarray}\noindent
If we could now sum both sides over $l$ and interchange the sum and
limit, we could reconstruct the $\,{\rm e}^{\alpha\pi}$ factor, and conclude
that the perturbation series for the $N$ - photon amplitudes,
at least in this low energy limit, is perfectly convergent!
But this is too good to be true, since so far we have nowhere made a
distinction between quenched and unquenched contributions to
the photon amplitudes, and convergence of the whole
perturbation series can certainly be excluded.
However, as was noted in Ref. \refcite{colima} this distinction comes in
naturally if one takes into account that in the path integral derivation of
(\ref{ImLallloop}) in Ref. \refcite{afalma} the rhs comes entirely
from the quenched sector; all non-quenched contributions are suppressed
in the weak field limit. And since (switching back to the usual Feynman diagram
picture) the importance of non-quenched diagrams is growing with
increasing loop order, it is natural to assume that their inclusion will slow
down the convergence towards the asymptotic limit with increasing $l$,
sufficiently to invalidate the above naive interchange of limits.
On the other hand, there is no obvious reason to expect such a slowing
down of convergence inside the quenched sector, which led to the
prediction \cite{colima} that Cvitanovic's ``quenched convergence''
will indeed be found to hold true for the photon amplitudes.
As a further step in this line of reasoning,
one should now check that the convergence of
(\ref{asymp}) does not show a slowing down when
going from two to three loops if one keeps only quenched diagrams.
However, a calculation of any three-loop EHL,
be it in Scalar or Spinor QED, for an electric or self-dual field,
poses an enormous computational challenge.
Now, in 2006 M. Krasnansky \cite{krasnansky} calculated the two-loop EHL
in 1+1 dimensional Scalar QED and found it, surprisingly, to have a structure
almost identical to the one of the corresponding self-dual EHL in the
four-dimensional case:
\begin{eqnarray}
{\cal L}_{\rm scal}^{(2)(4D)}(\kappa)
&=&
\alpha \,{m^4\over (4\pi)^3}\frac{1}{\kappa^2}\left[
{3\over 2}\xi^2
-\xi'\right],\quad
\xi(\kappa):= -\kappa\Bigl(\psi(\kappa)-\ln(\kappa)+\frac{1}{2\kappa}\Bigr)
\nonumber\\
{\cal L}_{\rm scal}^{(2)(2D)}(\kappa)
&=&
-\frac{e^2}{32\pi^2}\left[
\xi^2_{2D}
-4\kappa \xi_{2D}'\right] ,\quad
\xi_{2D}:= -\Bigl(\psi(\kappa+{1\over 2})-\ln (\kappa)\Bigr)\nonumber\\
\label{compscal2D4D}
\end{eqnarray}\noindent
($\psi(x)=\Gamma^\prime(x)/\Gamma(x)$, $\kappa := m^2/(2ef)$,
$f^2={1\over4} F_{\mu\nu}F^{\mu\nu}$).
This led us to consider 2D QED as a toy model for studying the above asymptotic predictions.
\section{Extension of the AAM conjecture to 1+1 QED}
Of course, this will make sense only if the AAM formula (\ref{ImLallloop}) can be extended to the
2D case. The worldline instanton approach of \cite{afalma} can be
extended to the 2D case straightforwardly \cite{wip}, yielding the
following analogue of (\ref{ImLallloop}):
\begin{eqnarray}
{\rm Im}{\cal L}(E)
&\sim&
\,{\rm e}^{-\frac{m^2\pi}{eE} + \tilde\alpha \pi^2 \kappa^2}
\label{ImLallloop2D}
\end{eqnarray}\noindent
($\tilde\alpha := 2e^2/\pi m^2$).
Note that, contrary to the 4D case, the second term in the exponent also involves the
external field. This leads also to a somewhat more complicated form of the
corresponding asymptotic limit statement:
\begin{eqnarray}
{{\rm lim}_{n\to\infty}} {c^{(l)}(n)\over c^{(1)}(n+l-1)}
&=& {(\tilde\alpha\pi^2)^{l-1}\over (l-1)!}
\label{asymp2D}
\end{eqnarray}\noindent
\section{Three loop Euler-Heisenberg Lagrangian in 1+1 QED}
At the one and two-loop level, we have obtained the
EHL in 2D Spinor QED explicitly in terms of the gamma and
digamma functions \cite{wip}:
\begin{eqnarray}
{\cal L}^{(1)}(\kappa) &=&
-{m^2\over 4\pi} {1\over\kappa}
\Bigl[{\rm ln}\Gamma(\kappa) - \kappa(\ln \kappa -1) +
{1\over 2} \ln \bigl({\kappa\over 2\pi}\bigr)\Bigr]
\label{L2D1}\\
{\cal L}^{(2)}(\kappa) &=& {m^2\over 4\pi}\frac{\tilde\alpha}{4}
\Bigl[ \tilde\psi(\kappa) + \kappa \tilde\psi'(\kappa)
+\ln(\lambda_0 m^2) + \gamma + 2 \Bigr]
\label{L2D2}
\end{eqnarray}\noindent
Here $\tilde\psi(\kappa):=-\xi(\kappa)/\kappa$, and
$\lambda_0$ is an IR cutoff for the photon propagator which becomes
necessary at two loops in 2D.
Curiously, in the 2D case the two-loop Spinor QED result (\ref{L2D2})
is simpler (just linear in the digamma function) than the corresponding
Scalar QED one (\ref{compscal2D4D}).
Using the well-known large - $x$ expansion of ${\rm ln}\Gamma(x)$
in terms of the Bernoulli numbers $B_n$ one
can then easily verify that (\ref{asymp2D}) does indeed hold true for
$l=2$.
\begin{figure}
\begin{center}
\epsfig{file=graphs.eps,width=4.5in}
\end{center}
\caption{Diagrams contributing to the three-loop EHL}
\label{aba:fig1}
\end{figure}
At three loops our results are rather preliminary.
There are three diagrams contributing to the
EHL, depicted in fig. 1 (the solid line represents the electron propagator
in a constant field).
For all three we have obtained
representation in terms of fourfold proper-time integrals.
The first six coefficients $c^{(3)}(n)$ for the quenched part
(diagrams A and B) were then
obtained in part analytically, in part by numerical integration. As it turns out,
at three loops all coefficients of the weak field expansion except the first one depend on the
IR cutoff $\lambda_0$. Introducing the modified cutoff
$\Lambda := {\rm ln}(\lambda_0m^2)+\gamma$ the coefficients can
be written in the form
\begin{eqnarray}
c^{(3)}(n) &=& \tilde\alpha^2
\Bigl(
c^{(3)}_2 (n)\Lambda^2+ c^{(3)}_1(n)\Lambda + c^{(3)}_0(n)
\Bigr)
\label{c3split}
\end{eqnarray}\noindent
where the coefficients $c^{(3)}_{1,2} (n)$ are rational
numbers, while $c^{(3)}_0(n)$ contains a $\zeta(3)$ already for $n=0$.
Since the prediction (\ref{asymp2D}) is cutoff-independent,
it can involve only the $c^{(3)}_0(n)$'s, so that the $c^{(3)}_{1,2}(n)$'s
must be subdominant. For the series $c^{(3)}_2(n)$ we have been able to
compute a sufficient number of coefficients to verify that this is the case. Showing that the series
$c^{(3)}_0(n)$ indeed satisfies (\ref{asymp2D}) is, however, not possible
with the coefficients obtained.
\section{Summary}
Extending the worldline instanton method of \cite{afalma} to 2D QED
we have obtained a prediction for the asymptotic growth of the weak field
expansion coefficients of the 2D EHL at any loop order. At two loops
we have verified this prediction by an analytic calculation
of the EHL. At three loops we have obtained an integral representation
of the EHL suitable for a numerical calculation of the expansion coefficients,
and we expect to be able shortly to verify (or refute) the three main facts relevant for
the AAM conjecture, namely that (\ref{asymp2D}) holds at the $\tilde\alpha^2$ level,
independence of spin, and asymptotic suppression of the non-quenched diagram C.
On the slowing down issue, relevant for Cvitanovic's conjecture, it unfortunately
seems not to be possible to get information from the 2D QED case, due to the
dependence of the three-loop expansion coefficients on the IR cutoff $\Lambda$;
although its numerical value does not affect the asymptotic limit, it
does have an influence on the rate of convergence towards it, which thus remains
ambiguous. Thus further progress in this line of attack on Cvitanovic's conjecture
presumably has to await the calculation of the three-loop EHL in 4D.
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Q: Why the textfield shouldChangeCharactersInRange method is not executing? Hi in my application I have a registration form in that I have few textfields. Along with these textfields I am using few delegate methods. The delegate methods which I am using are
*
*Did Begin Editing
*Did End Editing
*Should Change characters in range methods
Out of these three methods only 1st and 2nd methods are executing while editing the text-fields but 3rd method is not executing.
Please let me know the reason for this behaviour. Even I set the delegate to the textfield. If is there any problem with delegate link even 1st and 2nd methods are also won't be execute right? So please help me to resolve this issue.
A: Finally i found the reason for this behavior.That is Actually i am using category concept in my project.In that category class i have this Should Change characters in range method and the same method i have in main class also.Due to this that category class method is executing but actually required behavior code is available in main class method due to this i did not get output. Thanks.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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One of the most powerful sounds we hear at the Family Wellness Center is the much anticipated "Bell Ring." When a child or teen rings the Bell, it means they have worked hard to complete their therapy with us.
Last month, 10 year old Nick* was our Bell Ringer… and what an emotional day it was! When Nick first came to us, he was going through some significant family changes which left him confused, hurt, and getting into lots of trouble at school.
Nick – and his parents – worked hard through private and group therapy sessions for the past year. And when they all completed their therapy, it was time to ring that Bell!
At the end of his final session, all of the other patients and therapists in the building stopped what they were doing to join Nick, his family and his therapist in the hallway to cheer him on. As everyone congratulated him on his wonderful accomplishment, Nick proudly rang the Bell. Today, Nick's mom says he is confident, proud, doing great in school and has plenty of friends. Congratulations Nick – we are so proud of you!
For over 13 years, the San Diego Center for Children has partnered with the County of San Diego Behavioral Health Services and four East County school districts to provide extensive outpatient therapeutic services for children and families that may not otherwise receive the counseling and skill-building they so vitally need.
At any point, between 100-150 children, ages 4-20, are enrolled in this comprehensive outpatient program that targets individual coping, communication and social skills, improved emotional health and overall well-being at home, school and in the community. Learn more about the Family Wellness Center.
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{
"redpajama_set_name": "RedPajamaC4"
}
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namespace {
// Thread used to run the test suite.
class CefTestThread : public base::Thread {
public:
explicit CefTestThread(CefTestSuite* test_suite)
: base::Thread("test_thread"),
test_suite_(test_suite) {
}
void RunTests() {
// Run the test suite.
retval_ = test_suite_->Run();
// Wait for all browsers to exit.
while (TestHandler::HasBrowser())
base::PlatformThread::Sleep(base::TimeDelta::FromMilliseconds(100));
// Quit the CEF message loop.
CefPostTask(TID_UI, base::Bind(&CefQuitMessageLoop));
}
int retval() { return retval_; }
protected:
CefTestSuite* test_suite_;
int retval_;
};
// Called on the UI thread.
void RunTests(CefTestThread* thread) {
// Run the test suite on the test thread.
thread->message_loop()->PostTask(FROM_HERE,
base::Bind(&CefTestThread::RunTests, base::Unretained(thread)));
}
#if defined(OS_LINUX)
int XErrorHandlerImpl(Display *display, XErrorEvent *event) {
LOG(WARNING)
<< "X error received: "
<< "type " << event->type << ", "
<< "serial " << event->serial << ", "
<< "error_code " << static_cast<int>(event->error_code) << ", "
<< "request_code " << static_cast<int>(event->request_code) << ", "
<< "minor_code " << static_cast<int>(event->minor_code);
return 0;
}
int XIOErrorHandlerImpl(Display *display) {
return 0;
}
#endif // defined(OS_LINUX)
} // namespace
int main(int argc, char* argv[]) {
#if defined(OS_LINUX)
// Create a copy of |argv| on Linux because Chromium mangles the value
// internally (see issue #620).
CefScopedArgArray scoped_arg_array(argc, argv);
char** argv_copy = scoped_arg_array.array();
#else
char** argv_copy = argv;
#endif
#if defined(OS_WIN)
CefMainArgs main_args(::GetModuleHandle(NULL));
#else
CefMainArgs main_args(argc, argv);
#endif
void* windows_sandbox_info = NULL;
#if defined(OS_WIN)
// Manages the life span of the sandbox information object.
CefScopedSandboxInfo scoped_sandbox;
windows_sandbox_info = scoped_sandbox.sandbox_info();
#endif
// Parse command-line arguments.
CefRefPtr<CefCommandLine> command_line = CefCommandLine::CreateCommandLine();
#if defined(OS_WIN)
command_line->InitFromString(::GetCommandLineW());
#else
command_line->InitFromArgv(argc, argv);
#endif
// Create a ClientApp of the correct type.
CefRefPtr<CefApp> app;
client::ClientApp::ProcessType process_type =
client::ClientApp::GetProcessType(command_line);
if (process_type == client::ClientApp::BrowserProcess) {
app = new client::ClientAppBrowser();
} else if (process_type == client::ClientApp::RendererProcess ||
process_type == client::ClientApp::ZygoteProcess) {
app = new client::ClientAppRenderer();
} else if (process_type == client::ClientApp::OtherProcess) {
app = new client::ClientAppOther();
}
// Execute the secondary process, if any.
int exit_code = CefExecuteProcess(main_args, app, windows_sandbox_info);
if (exit_code >= 0)
return exit_code;
// Initialize the CommandLine object.
CefTestSuite::InitCommandLine(argc, argv_copy);
CefSettings settings;
CefTestSuite::GetSettings(settings);
#if defined(OS_MACOSX)
// Platform-specific initialization.
extern void PlatformInit();
PlatformInit();
#endif
#if defined(OS_LINUX)
// Install xlib error handlers so that the application won't be terminated
// on non-fatal errors.
XSetErrorHandler(XErrorHandlerImpl);
XSetIOErrorHandler(XIOErrorHandlerImpl);
#endif
// Initialize CEF.
CefInitialize(main_args, settings, app, windows_sandbox_info);
// Create the test suite object. TestSuite will modify |argv_copy|.
CefTestSuite test_suite(argc, argv_copy);
int retval;
if (settings.multi_threaded_message_loop) {
// Run the test suite on the main thread.
retval = test_suite.Run();
} else {
// Create the test thread.
scoped_ptr<CefTestThread> thread;
thread.reset(new CefTestThread(&test_suite));
if (!thread->Start())
return 1;
// Start the tests from the UI thread so that any pending UI tasks get a
// chance to execute first.
CefPostTask(TID_UI, base::Bind(&RunTests, thread.get()));
// Run the CEF message loop.
CefRunMessageLoop();
// The test suite has completed.
retval = thread->retval();
// Terminate the test thread.
thread.reset();
}
// Shut down CEF.
CefShutdown();
#if defined(OS_MACOSX)
// Platform-specific cleanup.
extern void PlatformCleanup();
PlatformCleanup();
#endif
return retval;
}
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{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,149
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\section{INTRODUCTION}
The X-Ray Imaging and Spectroscopy Mission (XRISM) Resolve instrument is expected to collect high energy X-ray spectra of unprecedented quality\cite{Tashiro18}. Rigorous analyses of these spectra require iterative forward-fitting, in which a model spectrum is specified and convolved with the instrument response to produce predicted photon counts, which are compared with the observed photon counts, and followed by adjustment of the model parameters to minimize the difference between the predicted and observed counts\cite{Smith11}. This procedure can be overly complicated and time-consuming for one who wants to quickly examine a spectrum, as it requires the user to specify and then refine a model spectrum.
As a precursor to this rigorous analysis, the XRISM Science Data Center (SDC) has developed XSLIDE (X-Ray Spectral Line IDentifier and Explorer), which allows scientific investigators to rapidly browse and survey available Resolve spectra. This will allow for easy access to XRISM data for astronomers from outside the field of high-resolution X-ray spectroscopy, quick browsing/surveying of available XRISM data prior to a decision to focus on a given spectrum, and simple extraction and characterization of XRISM spectra for non-publishing use such as writing proposals or giving presentations.
XSLIDE has been designed as a simple graphical user interface available as a desktop or web application (see Fig.~\ref{fig:xslide-desktop-and-web}) which allows users to interactively plot X-ray spectra. It walks the user through loading and modifying a spectrum, detecting lines, identifying those lines with known lines from atomic databases, performing diagnostics, and exporting the results. Following exploratory work performed with XSLIDE, more rigorous tools such as XSPEC\cite{Arnaud96} can be used to do further analyses on spectra of interest.
In this paper, we give a brief overview of XSLIDE. We discuss the methodology by which it allows users to view a spectrum with no selection of models for forward-fitting required and we assess the validity of the assumptions required for this process. We then outline the features of XSLIDE and show how the user is walked through the application step-by-step. Finally, we discuss the software architecture of XSLIDE to give an idea of how the program is built.
\begin{figure} [ht!]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=12.5cm]{desktop-and-web.png}
\end{tabular}
\end{center}
\caption
{\label{fig:xslide-desktop-and-web}
Screenshots of the XSLIDE user interfaces, demonstrating the similarity between the desktop (above) and web (below) versions.}
\end{figure}
\section{Methodology}
A spectrometer obtains counts ($C$) within particular instrument channels ($I$). This is related to the source spectrum ($S$, in $\frac {\text{photon}} {\text{cm}^2 \text{ second keV}}$) by
\begin{equation}
\label{eq:specmaster}
C(I) = T \int R_{\text{RMF}}(I, E) R_{\text{ARF}}(E) S(E) dE \, ,
\end{equation}
where $T$ is the total observation time, $R_{\text{RMF}}$ is the unitless redistribution matrix file (RMF) which gives the probability of a photon of a given energy ending up as a count in a particular channel, and $R_{\text{ARF}}$ is the ancillary response file (ARF) which contains the effective area in $\frac {\text{cm}^2 \text{ count}} {\text{photon}}$\cite{Smith11}.
As Equation~(\ref{eq:specmaster}) cannot be analytically inverted to solve for $S$, it is typically solved by forward-fitting, whereby a model spectrum $S$ is specified, and the resulting predicted counts are compared to the observed counts, adjusting the parameters of the model to minimize some statistical measure of the error between the predicted counts and the observed
counts\cite{Smith11}. This procedure can be quite complicated and time-consuming as it requires the user to specify and then refine the model.
XSLIDE instead assumes that the RMF is a diagonal matrix (providing an ideal one-to-one mapping between incident photon energy and detector channel) and that the ARF is slowly varying such that it is approximately constant between neighboring instrument channels. This allows for $S$ to be discretely solved for as
\begin{equation}
\label{eq:specsolved}
S(E) = \frac{C(I)}{R_{\text{ARF}}(E) T \Delta E } \, ,
\end{equation}
which can be computed using the ARF to map the instrument channel to the spectrum energy. This allows a spectrum to be viewed directly, requiring the user to provide only the spectrum and ARF files.
\begin{figure} [ht]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=8cm]{test_notsparse_rmf_log2_cbar_edited.png}
\end{tabular}
\end{center}
\caption
{\label{fig:hitomi-rmf}
A sample RMF from the Hitomi SXS instrument, showing a sharp peak along the diagonal. The RMF shown was generated using the Hitomi ``sxsrmf'' tool for a single SXS pixel and for HighRes events, using the ``x-large'' format to include full input model resolution with all response components. For testing and display purposes, the level of the electron continuum component was increased by a factor of ten over the CalDB value.}
\end{figure}
\subsection{Validity}
As the RMF is not truly diagonal, the assumption of its being diagonal will result in an inexact spectrum; hence the need for the use of XSLIDE to be followed up with more rigorous tools such as XSPEC\cite{Arnaud96}. However, if the RMF is strongly peaked along the diagonal without significant off-diagonal contributions, then the locations of the spectral peaks will be correct, while only the peak widths will be incorrect. Thus, the approximate spectra gained using this assumption will be valid for the purpose of line identification and certain analyses. Although RMFs are not yet available for the XRISM mission, it is expected that the XRISM Resolve instrument's RMF will be similar to that of the Hitomi SXS instrument\cite{Tashiro18}, for which the RMF has been observed to be strongly peaked along the diagonal (see Fig.~\ref{fig:hitomi-rmf}).
The assumption of the ARF being locally constant can bring about spurious lines or hide lines at energies where the ARF changes quickly. The user can verify that the ARF is not changing quickly by overlaying it on the spectrum. In addition, if any detected lines are found in regions where the ARF is changing quickly, a warning is shown to the user that these lines may be artifacts.
As a check on whether these assumptions are sufficiently valid for XSLIDE to be used reliably with spectra taken by the XRISM Resolve instrument, we compared the spectrum of the Perseus cluster taken by the Hitomi SXS instrument as produced rigorously in Tashiro et al.\cite{Hitomi16} with a spectrum generated by XSLIDE. As Fig.~\ref{fig:spectrum-comparison-perseus} shows, the spectrum produced by XSLIDE well-reproduces the more rigorously produced spectrum.
\begin{figure} [ht]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=13.5cm]{spectrumComparisonPerseus.pdf}
\end{tabular}
\end{center}
\caption
{\label{fig:spectrum-comparison-perseus}
Comparison between spectra of the Perseus cluster taken by the Hitomi SXS instrument as shown in Tashiro et al.\cite{Hitomi16} (above) and as produced by XSLIDE (below). The XSLIDE image used the spectrum in ObsID 100040020, rebinned to 4.0 eV. See Tashiro et al.\cite{Hitomi16} for full details of their methods used. Note that some differences between the spectra are expected due to Tashiro et al.\cite{Hitomi16} having combined multiple observations for a longer cumulative exposure time and having performed additional corrections, which likely explain the small peaks at 5.6, 6.1, and 6.3 keV that were not observed with XSLIDE.}
\end{figure}
\section{Running XSLIDE}
XSLIDE is designed to be simple and easy to use. The user is guided through ordered steps and substeps, with little need for actions to be taken outside this guided procedure. In this section, we give a brief overview of these steps to illustrate XSLIDE's capabilities.
\subsection{Step 1: Spectrum}
In this step, the spectrum is loaded and modified.
The user can either load the spectrum and ARF files from the local desktop or perform an online query of NASA's High Energy Astrophysics Science Archive Research Center (HEASARC)\cite{heasarc}. After the spectrum is loaded, the user has the option to clip the edges of the spectrum, which can be used for such purposes as removing artifacts due to artificially low ARF. The user can then rebin the spectrum to improve the signal-to-noise ratio.
Fig.~\ref{fig:step1} shows a sample screenshot of XSLIDE after Step 1 is complete.
\begin{figure} [ht]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=12.5cm]{step1.png}
\end{tabular}
\end{center}
\caption
{\label{fig:step1}
Screenshot of the desktop version of XSLIDE after Step 1 is complete.}
\end{figure}
\subsection{Step 2: Detect}
In this step, spectral lines are detected.
The first substep in accomplishing this is fitting a continuum, which can help with the next step of automatically detecting lines when the lines are placed over a non-flat continuum, such as can occur with bremsstrahlung radiation. Two methods exist for continuum fitting. First, XSLIDE can automatically fit a power law function in a piecewise manner to segments of the spectrum. Second, the user can manually specify points, which XSLIDE will then interpolate to form the continuum.
XSLIDE then provides two methods for automatic line detection. The piecewise statistics method works by first calculating the average and standard deviation statistics of piecewise segments of the spectrum, and then setting aside points with flux outside a user-specified number of standard deviations of the segment's average as outliers. This process is then repeated iteratively several times, calculating the average and standard deviation statistics for the segments using only points that are not considered outliers until the statistics converge. Sets of consecutive outliers are considered detected lines. The second method works using the continuous wavelet transform (CWT), convolving the spectrum with Ricker wavelets of user-defined widths to effectively search the spectrum for regions which match the shape of the wavelets. The parts of the spectrum which better match the shape yield higher CWT coefficients, and points with local maximums in these CWT coefficients and which are above a user-defined signal-to-noise ratio are labeled as peaks. Each of the two methods has strengths and weaknesses. The piecewise statistics method can search for lines of varying widths, and it finds both emission and absorption lines, but it can fail to identify closely-spaced peaks as being separate lines if the valleys are sufficiently far away from the continuum, and it requires appropriate binning and continuum fitting to work. The CWT method is better at separately identifying closely-spaced peaks, and proper binning and continuum fitting are not critical to its success, but the user needs to specify the peak widths to search for, and it is mainly only capable of searching for either emission or absorption lines (but not both) in a given spectrum.
After automatically detecting lines, the user can then make manual adjustments as required. Undetected lines can be added, spurious lines can be removed, and existing lines can be narrowed or broadened as required.
We note that automatic line detection is far from a solved problem; neither of the methods provided in XSLIDE will be able to find all the peaks and discard all of the noise that can be readily discerned with a human eye. Rather, the point of XSLIDE's automatic line detection is to quickly find a spectrum's dominant features to determine if further analysis with a rigorous tool such as XSPEC\cite{Arnaud96} is warranted.
\subsection{Step 3: Identify}
In this step, the detected lines are identified using an atomic line database.
The user must first choose a set of atomic lines to use. XSLIDE comes packaged with AtomDB\cite{Smith01} version 3.0.9, several atomic databases custom-generated with XSTAR\cite{Mendoza21, Kallman21, Kallman01} version 2.58, and a neutral species database that was created using the data in Table 1-3 of the X-Ray Data Booklet\cite{Kortright09}. These built-in databases are expected to meet the requirements of the majority of XSLIDE users, and XSLIDE also allows users with niche needs to load their own atomic databases. Atomic line filters are implemented to allow the user to focus on particular ions or ``stronger'' lines with cutoffs for oscillator strength or emissivity.
Before the detected lines can be matched with the atomic lines, the spectrum's velocity shift (generally referred to simply as "redshift") needs to be accounted for so that the lines are energetically aligned. Several methods are available to specify the velocity shift. The manual method works by having the user simply enter a value. With the single line alignment method, the user selects a detected line and an atomic line, and the velocity shift gets set such that the energies of these two lines will be equal. This method of velocity shift determination is often done for intracluster medium spectra using the prominent Fe24+ line at 6.7 keV, for example. A more experimental method is the full line search, where a user-specified range of velocity shifts are generated, and for each velocity shift, a loss function is determined by calculating the deviation between each detected line and its nearest atomic line, summing the absolute values of these deviations. Velocity shifts with lower loss functions are worth further examination and can be applied by the user. This method is shown in Fig.~\ref{fig:xslide-desktop-and-web}, where it can be observed that a velocity shift of 0.0178 was found for the Perseus cluster observation by the Hitomi SXS instrument, quite close to the value of 0.01756 calculated in Tashiro et al.\cite{Hitomi16}.
Once the detected lines and atomic lines are aligned with each other, it is simple to identify the detected lines. XSLIDE simply matches each detected line with all atomic lines that are within the energy range determined by the edge points of the detected line. To help the user identify detected lines for which multiple atomic lines are possible, XSLIDE shows the oscillator strength and emissivities of the atomic lines to inform the user of the strength of the various lines. The user can choose to display labels next to the lines on the graph to indicate the identified atomic transitions.
Fig.~\ref{fig:xslide-desktop-and-web} shows sample screenshots of XSLIDE after Step 3 is complete.
\subsection{Step 4: Diagnostics}
In this step, the user views diagnostics that are computed using the identified lines. These diagnostics consist of ratios of line fluxes which have been found to correlate with system properties that may be of interest, such as temperature or ionization parameter.
The purpose of the XSLIDE diagnostics is only to provide a preliminary analysis, alerting the user to the possibility of the spectrum's diagnostic potential. The diagnostics should not be considered quantitatively reliable, as the detected line fluxes used for these calculations are only approximations which are highly dependent on the continuum and the number of points that define the detected lines. Any overlap between detected lines can drastically distort these calculated fluxes. To actually compute reliable diagnostics, a more rigorous tool such as XSPEC\cite{Arnaud96} is required.
Three diagnostics are available. In the hydrogen diagnostics (see Fig.~\ref{fig:step4} for a sample screenshot), ions for which the Lyman lines have been identified in the spectrum will have the Ly$\beta$/Ly$\alpha$ ratio calculated. This ratio will also be computed for the selected ion using the data in the atomic database as a function of temperature or ionization parameter and shown to the user in a graph; by seeing where the calculated ratio of the detected line fluxes intersect this graph, the temperature or ionization parameter may be approximated (as long as the assumptions that were used to create the selected atomic database are valid).
In the helium diagnostics (see Fig.~\ref{fig:step4}), ions for which the common helium-like lines (w, x, y, and z; also known as resonance (R), intercombination (I), intercombination (I), and forbidden (F)) have been identified in the spectrum will have the G ($\frac{x+y+z}{w}$) and R ($\frac{z}{x+y}$) ratios calculated. Similarly to the hydrogen diagnostics, the G ratio will also be computed for the selected ion using the data in the atomic database as a function of temperature or ionization parameter and shown to the user in a graph for the purpose of approximating the temperature or ionization parameter. The R ratio is commonly used as a diagnostic to indicate the density of the observed source, but assisting the user in approximating this value is beyond the scope of XSLIDE.
\begin{figure} [ht]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=14.5cm]{step4.png}
\end{tabular}
\end{center}
\caption
{\label{fig:step4}
Partial screenshots of the hydrogen (left) and helium (right) diagnostics from Step 4 of XSLIDE. The hydrogen diagnostics shown indicate a temperature less than 30 million K (emissivities are not provided in AtomDB for temperatures low enough to give a Ly$\beta$/Ly$\alpha$ ratio of 0.0619 for Fe25+), and the helium diagnostics shown indicate a temperature of 16.2 million K.}
\end{figure}
In the K diagnostics, ions for which K$\beta$ or K$\alpha$ lines have been identified in the spectrum will have the K$\beta$/K$\alpha$ ratio of the detected line fluxes calculated. This ratio is also commonly used as a diagnostic to indicate the temperature or ionization parameter of the observed source.
\subsection{Step 5: Export}
In this step, the results can be exported to various formats. This allows the results to be presented or utilized in other programs.
Output formats include a PDF report, PNG or SVG figures, and CSV or Excel tables of data. In addition, XSLIDE can export a model spectrum in a format that can be read directly into XSPEC\cite{Arnaud96}. This model spectrum will consist of the lines and continuum that have been detected in XSLIDE, giving the XSPEC user a starting point that can be further modified as desired.
\subsection{Other Features}
A key feature of XSLIDE is the interactive plot of the spectrum. The user can pan and zoom with intuitive controls to focus on particular features of the plot as desired. The user can also customize the plot by setting line colors, styles, widths, and symbols. In addition to being able to view the spectrum and the fitted continuum, the user can also choose to display the spectrum minus the continuum, the raw counts, and the ARF.
XSLIDE is available in both English and Japanese. The user can select the language as a menu option.
An extensive user manual is provided with XSLIDE which documents the various methods and parameters available in each step. It also provides a walkthrough of the software, demonstrating how XSLIDE can be used to approximately reproduce Figure 1 of Tashiro et al.\cite{Hitomi16}.
\section{Software Architecture}
XSLIDE has a Model-View-Controller (MVC) software design, which keeps a level of separation between the data and the GUI. "Data" refer to the physical quantities that XSLIDE is operating on, such as spectra, ARFs, continua, theoretical line lists, emissivity models, etc. The Model performs operations on the data elements, the View displays the current state of the Model to the user, and the Control part of the GUI manipulates data in response to user events by sending commands to operate on the Model.
This separation of programming elements allows for a central underlying Model to be displayed to multiple Views, allowing for separate desktop and web applications to be written with minimal reproduction of effort. It further simplifies development by providing a guided process by which to add a new feature, where first the Model is updated such that the new feature is available in the command line, then the View is updated such that new GUI elements showing the state of the Model are displayed, and finally the Control is updated so that the Model appropriately responds to user events. The separation helps with testing as well, as unit tests are written for the Model independently from functional tests written for the final GUI applications.
There are ultimately three XSLIDE products: a Python command-line-interface (XSLIDEpy), the desktop GUI (XSLIDE for desktop), and the web GUI (XSLIDE for web). The Model elements of the MVC are largely contained within XSLIDEpy, whereas the View and Controller elements are separately written for the desktop and web GUIs. Most users will only access either the desktop and/or web GUIs, with XSLIDEpy largely reserved for use in the development process.
XSLIDEpy is a Python package that leverages the standard scientific Python ecosystem of NumPy\cite{Harris2020array}, SciPy\cite{2020SciPy-NMeth}, and Astropy\cite{astropy:2013, astropy:2018} to quickly conduct common astronomical operations. XSLIDE uses PyVO, an Astropy-affiliated package, to query NASA's HEASARC\cite{heasarc} archive for spectrum and ARF files from the XRISM and Hitomi missions. Unit tests cover a large portion of the source code.
XSLIDE for desktop is built using Qt\cite{qt}, which allows XSLIDE to be run cross-platform on Mac and multiple Linux distributions. The Qt for Python (PySide6) bindings are used to develop the Control, and PyQtGraph\cite{pyqtgraph} is used for the interactive plot. PyInstaller\cite{pyinstaller} is used to bundle the application into a single package that can be installed via simple drag-and-click. Squish\cite{squish} is used to run automated functional tests that cover most of the functionality available to the end user. Squish runs the application through scripts of user actions, recognizing GUI objects and interacting with them by performing mouse and keyboard clicks, and then verifying that these actions produced the expected results by confirming that GUI objects contain predetermined values and that screenshots are visually equivalent to prepared standards. A video of Squish running one of the functional tests can be seen in the link provided in Fig.~\ref{fig:video1}.
\begin{figure} [ht]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=6cm]{video1.png}
\end{tabular}
\end{center}
\caption
{\label{fig:video1}
A video of Squish\cite{squish} performing one of XSLIDE's functional tests can be accessed as an ancillary file.}
\end{figure}
XSLIDE for web is built using HTML, CSS, and Bokeh\cite{bokeh} to design the View and interactive plot. JavaScript and a Bokeh server are used to develop the Control. Selenium\cite{selenium} is used to run functional tests in a manner similar to what is done by Squish for the desktop version.
For both the desktop and web versions of XSLIDE, Japanese language translations are provided using Qt's QTranslator class and user documentation is prepared with Sphinx\cite{sphinx}.
\acknowledgments
We thank Lorella Angelini for laying the foundational groundwork of XSLIDE, Tim Kallman for generating the XSTAR atomic database files, Megumi Shidatsu for assistance in creating Japanese translations, and Edmund Hodges-Kluck and Kenji Hamaguchi for providing invaluable user feedback that helped guide the development process.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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Q: How to create facebook application for a group I am a php web programmer. I don't have much experience in facebook application. I have a facebook group of 42 people. all are my friends. I want to build an application for this group only. Only the group member will have permission to access the application. When a member visits the group page, he will see the application and use it. How can I do this?
I do know a little about facebook application. I have a personal hosting which I will use to host program files and mysql database. But I want a basic guideline. Please tell me from scratch. If anyone know a online documentation or guideline, please mention the link. Thanks.
A: Unfortunately, I don't have the time right now to help you with a complete guide, but here's the main idea:
Make a facebook app that asks the user for the user_groups permissions.
With this permission, you can see the groups the user is a part of. Then, you can simply check to see if YOUR group is among them. If it is... super! show him the content. If not, show him a blank page, an error page or whatever, it's totally up to you.
Some other helpful links are these: Facebook PHP references and Facebook JavaScript references.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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Habeck in Sweden: warming up for the subsidy race?
Munich: tree felling on the Isar – Munich
Germany's help for Ukraine: If you want to lead, you also have something to explain
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Storming of the US Capitol: Oath Keepers leader Stewart Rhodes sentenced
The leader of the extreme right-wing militia "Oath Keepers" has been sentenced almost two years after the storming of the US Capitol. As US media reported from the court in the US capital Washington on Tuesday, a jury found Stewart Rhodes guilty of "seditious conspiracy" – a crime rarely recognized in the country's judicial history.
Rhodes was accused, along with co-defendants, of plotting to use force to prevent the transfer of power after the 2020 presidential election. After three days of deliberations, the jury found Rhodes, 56, of Texas, guilty of three counts, including obstruction of an official process and tampering with documents. Another defendant, Kelly Meggs, was found guilty of sedition charges. However, three other defendants were acquitted of these charges.
Maximum penalty of up to 20 years imprisonment
Earlier this year, the US Department of Justice filed charges against Rhodes and other participants in the Capitol attack. Among other things, they planned to travel to Washington on January 6, 2021 and organized weapons, paramilitary equipment and training in combat techniques in advance, it said. Several of the defendants had entered the Capitol themselves, while others had taken care of further coordination outside the seat of Congress and partly outside the city. The Justice Department said the maximum penalty could be up to 20 years in prison for "seditious conspiracy." Rhodes' sentence will be determined at a later date.
The guilty verdicts are seen as a major victory for the Justice Department in its effort to hold individuals accountable for the storming of the Capitol. Supporters of then President Donald Trump stormed the Houses of Parliament in Washington on January 6, 2021 to prevent Democrat Joe Biden's November 2020 election victory from being confirmed. Five people died in connection with this. 140 police officers were attacked and more than $2 million in damage was caused to the building.
Attack on the heart of US democracy
The attack on the heart of US democracy shook the country. Trump had previously incited his supporters in a speech. Rhodes claimed during the trial that he had no plans to attack the US Capitol. "On January 6, our democracy was attacked," Assistant US Attorney Kathryn Rakoczy said in her closing argument. "For these defendants, it was 'everything we trained for,'" Bloomberg said.
Open detailed view
Supporters of US President Donald Trump stormed the US Capitol in Washington DC on January 6, 2021
(Photo: Essdras M. Suarez/dpa)
During the trial, the defendants testified that they had no malicious intent and had traveled to Washington to protect prominent figures at pro-Trump events. They also claimed that the so-called rapid reaction force, which was armed with firearms, was only for emergencies, such as an attack by left-wing activists or if Trump invoked a law they thought he would give them the power to act as a militia, according to Bloomberg.
Two other seditious conspiracy trials are pending, involving the Oath Keepers and another far-right group called the Proud Boys.
source site
Tagged CapitolDonald TrumpJoe BidenkeepersleaderoathpoliticsRhodesRight-wing extremismsentencedSouthgerman newspaperStewartstormingStorming the US CapitolUnited States
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{"url":"http:\/\/arxitics.com\/articles\/1709.03974","text":"arXiv Analytics\n\narXiv:1709.03974 [math.CO]AbstractReferencesReviewsResources\n\nCombinatorics of cyclic shifts in plactic, hypoplactic, sylvester, Baxter, and related monoids\n\nPublished 2017-09-12Version 1\n\nThe cyclic shift graph of a monoid is the graph whose vertices are elements of the monoid and whose edges link elements that differ by a cyclic shift. This paper examines the cyclic shift graphs of `plactic-like' monoids, whose elements can be viewed as combinatorial objects of some type: aside from the plactic monoid itself (the monoid of Young tableaux), examples include the hypoplactic monoid (quasi-ribbon tableaux), the sylvester monoid (binary search trees), the stalactic monoid (stalactic tableaux), the taiga monoid (binary search trees with multiplicities), and the Baxter monoid (pairs of twin binary search trees). It was already known that for many of these monoids, connected components of the cyclic shift graph consist of elements that have the same evaluation (that is, contain the same number of each generating symbol). This paper focusses on the maximum diameter of a connected component of the cyclic shift graph of these monoids in the rank-$n$ case. For the hypoplactic monoid, this is $n-1$; for the sylvester and taiga monoids, at least $n-1$ and at most $n$; for the stalactic monoid, $3$ (except for ranks $1$ and $2$, when it is respectively $0$ and $1$); for the plactic monoid, at least $n-1$ and at most $2n-3$. The current state of knowledge, including new and previously-known results, is summarized in a table.\n\nComments: 61 pages. Complete proofs of results previously announced in arXiv:1611.04152\nCategories: math.CO, math.GR\nSubjects: 05E99, 05C12, 20M05\nRelated articles: Most relevant\u2002|\u2002Search more\narXiv:math\/0310082 [math.CO] (Published 2003-10-06)\nThe Algebra and Combinatorics of Shuffles and Multiple Zeta Values\narXiv:math\/0606346 [math.CO] (Published 2006-06-14, updated 2007-05-18)\nOn the combinatorics of hypergeometric functions\narXiv:0704.2518 [math.CO] (Published 2007-04-19)\nCombinatorics Of RNA Structures With Pseudoknots","date":"2017-09-22 11:51:44","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\": 0.7011728286743164, \"perplexity\": 1729.2118430272994}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-39\/segments\/1505818688940.72\/warc\/CC-MAIN-20170922112142-20170922132142-00450.warc.gz\"}"}
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\section{Background}
\label{sec:background}
Image description, a task to generate sentences to describe given
images, has been a active topic recently.
Many state-of-the-art frameworks~\cite{zhou2016image,you2016image,yang2016review,lu2016knowing,xu2015show,vinyals2015show} for this task adopt
the \emph{maximum likelihood} principle for learning.
Such a framework usually works as follows.
Given an image $I$, it first derives a feature representation $\vf(I)$,
and then generates the words $w_1, \ldots, w_T$ sequentially, following
a Markov process conditioned on $\vf(I)$.
The model parameters are learned via maximum likelihood estimation (MLE),
\ie~maximizing the conditional log-likelihood
of the training samples, as:
\begin{equation}
\sum_{(I_i, S_i) \sim \cD}
\sum_{t=0}^{T_i}
\log p \left(w_i^{(t)} | \vf(I), w_i^{(t-1)}, \ldots, w_i^{(t-n)}\right).
\end{equation}
Here, $I_i$ and $S_i = (w_i^{(0)}, \ldots, w_i^{(T_i)})$ are the image and
the corresponding descriptive sentence of the $i$-th sample,
and $n$ is the order of the Markov chain
-- the distribution of the current work depends on $n$ preceding words.
Along with the popularity of deep neural networks, latest studies often
adopt neural networks for both image representation and language modeling.
For example, \cite{xu2015show} uses a CNN for deriving the visual features
$\vf(I)$, and an LSTM~\cite{hochreiter1997long}~to express the sequential relations among words.
Despite the evolution of the modeling choices, the maximum likelihood principle
remains the predominant learning principle.
As illustrated in Figure \ref{fig:mle_limit}, when similar images are presented,
the sentences generated by such a model often contain \emph{repeated} patterns.
This is not surprising -- under the MLE principle, the joint probability of a sentence
is, to a large extent, determined by whether it contains those n-grams that appear
frequently in the training set. Therefore, the model trained in this way will tend
to produce such n-grams. In particular, when the generator yield a couple of words
that match the prefix of a frequent n-grams, the remaining words of that
n-gram are very likely to be generated following the Markov chain.
The quality of the generated sentences are usually assessed using
metrics originating from the machine translation area, such as
BLEU~\cite{papineni2002bleu}, ROUGE~\cite{lin2004rouge}, METEOR~\cite{lavie2014meteor}, and CIDEr~\cite{vedantam2015cider}.
Such metrics mostly rely on matching n-grams with the \emph{``ground-truths''}.
For example, the modified precision score $p$, which is the core
statistics in BLEU~\cite{papineni2002bleu}, is defined as
\begin{equation}
p = \frac
{\sum_{\omega \in \Omega} h_\text{clip}(\omega)}
{\sum_{\omega^\prime \in \Omega} h(\omega^\prime)}.
\end{equation}
which basically counts the appearance of all n-grams in \emph{``ground-truths''} and generated sentences,
and computes their ratio.
As a result of the reliance of n-gram matching, sentences that contain frequent
n-grams will get good scores as compared to those using variant expressions,
as shown in Figure~\ref{fig:oldmetrics}.
Note that whereas image captioning is sometimes
considered as \emph{translating an image to a sentence}, it differs signficantly
from a conventional translation task, \eg~from English to Spanish.
When presented a photo, different people would probably give quite different
descriptions that do not overlap much in the wording patterns.
This diversity in expression is an essential property of human languages,
which, however, is often overlooked in previous work (both generation
and evaluation).
This study explores an alternative approach --
instead of emphasizing n-gram matching, we focus on improving the
\emph{naturalness}, \ie~generating sentences that feel like
what real people would say.
\section{Conclusion}
\label{sec:concls}
This paper presented an alternative approach to generating image descriptions.
Compared to existing methods, which are mostly focused on the match of detailed wording,
our approach, instead, aims to improve the overall quality,
which involves \emph{semantic relevance}, \emph{naturalness}, and \emph{diversity}.
Some of these properties are often overlooked in previous efforts.
We proposed a formulation based on conditional GAN that jointly
trains a generator $G$ and an evaluator $E$, and applied Policy Gradient and
early feedbacks to tackle the technical challenges in end-to-end training.
On both MSCOCO and Flickr30k, the proposed method produced descriptions
that are more natural, diverse, and semantically relevant as compared
to a state-of-the-art MLE-based model. This is clearly
demonstrated in our user studies, qualitative examples, and retrieval applications.
Our framework also provides an evaluator that is more consistent with human's evaluation.
\vspace{-13pt}
\paragraph{Acknowledgment}
This work is partially supported by
the Big Data Collaboration Research grant from SenseTime Group (CUHK Agreement No.TS1610626),
the General Research Fund (GRF) of Hong Kong (No.14236516)
and the Early Career Scheme (ECS) of Hong Kong (No.24204215).
\section{Experiment}
\label{sec:experiment}
\input{tab_sentgen.tex}
\input{fig_userstudy.tex}
\input{tab_imgrank.tex}
\input{fig_noise.tex}
\input{fig_compare.tex}
\input{fig_paragen.tex}
\paragraph{Datasets}
We conducted experiments to test the proposed framework
on two datasets:
(1) \emph{MSCOCO} \cite{lin2014microsoft},
which contains $82,081$ training images and $40,137$ validation images.
(2) \emph{Flickr30k} \cite{young2014image}, which contains $31,783$ images in total.
We followed the split in \cite{karpathy2015deep}, which has $1,000$ images for validation,
$1,000$ for testing, and the rest for training.
In both datasets, each image has at least $5$ ground truth sentences.
Note that our experiments involve comparison between human descriptions and
model-generated ones.
As we have no access to the ground-truth annotations of the testing images
in MSCOCO, for this dataset, we use the training set for both training and
validation, and the validation set for testing the performance.
\vspace{-11pt}
\paragraph{Experimental settings}
To process the annotations in each dataset,
we follow \cite{karpathy2015deep}~to remove non-alphabet characters,
convert all remaining characters to lower-case,
and replace all the words that appeared less than $5$ times with a special word \emph{UNK}.
As a result, we get a vocabulary of size $9,567$ on MSCOCO,
and a vocabulary of size $7,000$ on Flickr30k.
All sentences are truncated to contain at most $16$ words during training.
We respectively pretrain $G$ using standard MLE~\cite{vinyals2015show}, for $20$ epoches,
and $E$ with supervised training based on Eq~\eqref{eq:e_loss}, for $5$ epoches.
Subsequently, $G$ and $E$ are jointly trained, where
each iteration consists of one step of G-update followed by one step of E-update.
We set the mini-batch size to $64$, the learning rate to $0.0001$,
and $n=16$ in Monte Carlo rollouts.
When testing, we use beam search based on the expected rewards from E-GAN,
instead of the log-likelihoods,
which we found empirically leads to better results.
\vspace{-11pt}
\paragraph{Models}
We compare three methods for sentence generation:
(1)\textbf{Human}: a sentence randomly sampled from ground-truth annotations of each image
is used as the output of this method.
Other human-provided sentences will be used as the references for
metric evaluation.
This baseline is tested for the purpose of comparing human-provided
and model-generated descriptions.
(2)\textbf{G-MLE}: a generator trained based on MLE~\cite{vinyals2015show} is used
to produce the descriptions.
This baseline represents the state-of-the-art of mainstream methods.
(3)\textbf{G-GAN}: the same generator trained by our framework proposed in this paper,
which is based on the conditional GAN formulations.
For both \emph{G-MLE} and \emph{G-GAN},
\emph{VGG16}~\cite{simonyan2014very} is used as the image encoders.
Activations at the \texttt{fc7} layer, which are of dimension $4096$,
are used as the image features and fed to the description generators.
Note that \emph{G-GAN} also takes a random vector $\vz$ as input.
Here, $\vz$ is a $1024$-dimensional vector, whose entries are
sampled from a standard normal distribution.
\vspace{-11pt}
\paragraph{Evaluation metrics}
We consider multiple evaluation metrics, including six conventional metrics
BLEU-3 and BLEU-4\cite{papineni2002bleu},
METEOR\cite{lavie2014meteor},
ROUGE\_L\cite{lin2004rouge},
CIDEr\cite{vedantam2015cider},
SPICE\cite{anderson2016spice},
and two additional metrics relevant to our formulation:
\emph{E-NGAN} and \emph{E-GAN}.
Particularly,
\emph{E-GAN} refers to the evaluator trained using our framework,
\emph{E-NGAN} refers to the evaluator trained according to Eq~\eqref{eq:e_loss}
without updating the generator alternatively.
In other words, it is trained to distinguish between human-provided sentences
and those generated by an MLE-based model.
Table \ref{tab:sentgen} lists the performances of different generators under these metrics.
On both datasets, the sentences produced by \emph{G-MLE} receive considerably higher
scores than those provided by human, on nearly all conventional metrics.
This is not surprising. As discussed earlier, such metrics primarily focus on
n-gram matching \wrt~the references, while ignoring other important properties,
\eg~naturalness and diversity.
These results also clearly suggest that these metrics may not be particularly suited
when evaluating the overall quality of the generated sentences.
On the contrary, \emph{E-GAN} regards \emph{Human} as the best generator,
while \emph{E-NGAN} regards \emph{G-GAN} as the best one. These two metrics obviously take into account
more than just n-gram matching.
\vspace{-11pt}
\paragraph{User study \& qualitative comparison}
To fairly evaluate the quality of the generated sentences as well as
how \emph{consistent} the metrics are with human's perspective,
we conducted a user study.
Specifically, we invited $30$ human evaluators to compare the outputs
of different generators. Each time, a human evaluator would be
presented an image with two sentences from different methods
and asked to choose the better one. Totally, we collected
about $3,000$ responses.
The comparative results are shown in Figure \ref{fig:userstudy}:
From human's views,
\emph{G-GAN} is better than \emph{G-MLE} in $61\%$ of all cases.
In the comparison between human and models,
\emph{G-MLE} only won in $9\%$ of the cases, while
\emph{G-GAN} won in over $24\%$.
These results clearly suggest that the sentences produced
by \emph{G-GAN} are of considerably higher quality, \ie~being
more natural and semantically relevant.
The examples in Figure~\ref{fig:compare} also confirm
this assessment.
Particularly, we can see
when \emph{G-MLE} is presented with similar images,
it tends to generate descriptions that are almost the same.
On the contrary, \emph{G-GAN} describes them with more distinctive and diverse ones.
We also varied $\vz$ to study the capability of \emph{G-GAN}
in giving \emph{diverse} descriptions while maintaining the semantical relatedness.
The qualitative results are listed in Figure~\ref{fig:noise}.
For the evaluation metrics, the assessments provided by \emph{E-GAN}
are the most consistent with human's evaluation,
where the Kendall's rank correlation coefficient between \emph{E-GAN} and \emph{HE} is 0.14,
while that for CIDEr and SPICE are -0.30 and -0.25.
Also note that \emph{E-GAN} yields a larger numerical gap
between scores of human and those of other generators as compared to \emph{E-NGAN},
which suggests that
adversarial training can improve the discriminative power of the evaluator.
\vspace{-15pt}
\paragraph{Evaluation by retrieval}
To compare the \emph{semantic relevance}, we conducted an experiment
using generated descriptions for retrieval.
Specifically, we randomly select $5,000$ images from the MSCOCO validation set;
and for each image, we use the generated description as a query,
ranking all $5,000$ images according to the similarities between the images and the descriptions,
computed by \emph{E-GAN},
as well as the log-likelihoods.
Finally, we compute the recall of the original image that appeared in the top-$k$ ranks.
The results for $k=1, 3, 5, 10$ are listed in Table \ref{tab:imgrank},
where \emph{G-GAN} is shown to provide more discriminative descriptions,
outperforming \emph{G-MLE} by a large margin across all cases.
\vspace{-15pt}
\paragraph{Failure Analysis}
We analyzed failure cases and found that a major kind of errors is
the inclusion of incorrect details. \eg colors (red/yellow hat),
and counts (three/four people).
A possible cause is that there are only a few samples for each particular detail,
and they are not enough to make the generator capture these details reliably.
Also, the focus on diversity and overall quality may also encourage the generator to include more details,
with the risk of some details being incorrect.
\vspace{-11pt}
\paragraph{Paragraph Generation}
We also tested our framework on paragraph generation (See Sec~\ref{sec:paragen}).
We use the dataset provided by \cite{krause2016paragraphs},
which contains $14,575$ training images, $2,487$ validation images, and $2,489$ testing images.
Example results are shown in Figure \ref{fig:paragen}.
Again, we found that G-GAN can produce diverse and more natural descriptions as compared
to G-MLE, which tends to follow similar patterns across sentences.
\section{Framework}
\label{sec:frmwork}
\input{fig_structure.tex}
We propose a new framework for generating image descriptions based on
the conditional GAN~\cite{mirza2014conditional}~method, which consists
of a generator $G$, and an evaluator $E$.
Given an image $I$,
the former is for generating \emph{natural} and \emph{semantically relevant}
descriptions; while the latter is for evaluating how well a sentence or
paragraph describes $I$.
We start with generating single sentences as descriptions,
and then extend our framework to paragraph generation.
\subsection{Overall Formulation}
Our framework contains a \emph{generator} $G$ and a \emph{evaluator} $E$,
whose structures are respectively shown in Figure~\ref{fig:structure} (a) and (b).
It is worth noting that our framework is orthogonal to works
that focus on architectural designs of the $G$ and the $E$.
Their structures are not restricted to the ones introduced in this paper.
In our framework, given an image $I$, the generator $G$ takes two inputs:
an image feature $\vf(I)$ derived from a convolutional neural network
(CNN) and a random vector $\vz$.
In particular, we follow the setting in NeuralTalk2\footnote{\url{https://github.com/karpathy/neuraltalk2}},
adopting \emph{VGG16}~\cite{simonyan2014very} as the CNN architecture.
The random vector $\vz$ allows the generator to produce
different descriptions given an image. One can control the
\emph{diversity} by tuning the variance of $\vz$.
With both $\vf(I)$ and $\vz$ as the initial conditions,
the generator relies on an LSTM~\cite{hochreiter1997long}~net as a decoder,
which generates a sentence, word by word.
Particularly, the LSTM net assumes a sequence of latent states
$(s_0, s_1, \ldots)$. At each step $t$, a word $w_t$ is
drawn from the conditional distribution $p(w | s_t)$.
The evaluator $E$ is also a neural network, with an architecture
similar to $G$ but operating in a different way.
Given an image $I$ and a descriptive sentence $S = (w_0, w_1, \ldots)$,
it embeds them into vectors $\vf(I)$ and $\vh(S)$ of the same dimension,
respectively via a CNN and an LSTM net.
Then the \emph{quality} of the description, \ie~how well it describes $I$,
is measured by the dot product of the embedded vectors, as
\begin{equation}
r_{\veta}(I, S) = \sigma \left(
\langle \vf(I, \veta_I), \vh(S, \veta_S) \rangle
\right).
\end{equation}
Here, $\veta = (\veta_I, \veta_S)$ denotes the evaluator parameters,
and $\sigma$ is a logistic function that turns the dot product
into a probability value in $[0, 1]$.
Note that while the CNN and the LSTM net in $E$ have the same structure
as those in $G$, their parameters are not tied with each other.
For this framework, the learning objective of $G$ is to
generate descriptions that are \emph{natural}, \ie~indistinguishable
from what humans would say when presented with the same image;
while the objective of $E$ is to distinguish between artifical
descriptions (\ie~those from $G$) and the real ones (\ie~those from
the training set).
This can be formalized into a minimax problem as follows:
\begin{equation}
\min_{\vtheta} \max_{\veta} \cL(G_{\vtheta}, E_{\veta}).
\end{equation}
Here, $G_{\vtheta}$ and $E_{\veta}$ are a generator with parameter $\vtheta$
and an evaluator with parameter $\veta$. The objective function $\cL$ is:
\begin{equation}\label{eq:minmax}
\Ebb_{S \sim \cP_I} \left[
\log r_{\veta}(I, S)
\right]
+
\Ebb_{\vz \sim \cN_0} \left[
\log(1 - r_{\veta}(I, G_{\vtheta}(I, \vz)))
\right].
\end{equation}
Here, $\cP_I$ denotes the descriptive sentences for $I$
provided in the training set, $\cN_0$ denotes a standard
normal distribution, and $G_{\vtheta}(I, \vz)$ denotes
the sentence generated with $I$ and $\vz$.
The overall learning procedure alternates between the
updating of $G$ and $E$, until they reach an equilibrium.
This formulation reflects an essentially different philosophy
in \emph{how to train a description generator} as opposed to
those based on MLE. As mentioned, our approach aims at the
\emph{semantical relevance} and \emph{naturalness},
\ie~whether the generated descriptions feel like what human would say,
while the latter focuses more on word-by-word patterns.
\subsection{Training $G$: Policy Gradient \& Early Feedback}
\label{sec:g_train}
As mentioned, unlike in conventional GAN settings, the production
of sentences is a discrete sampling process, which is \emph{nondifferentiable}.
A question thus naturally arises - how can we \emph{back-propagate the
feedback} from $E$ under such a formulation?
We tackle this issue via \emph{Policy Gradient}~\cite{sutton1999policy}, a technique
originating from reinforcement learning.
The basic idea is to consider a sentence as a sequence of \emph{actions},
where each word $w_t$ is an action. The choices of such ``actions'' are
governed by a \emph{policy} $\vpi_{\vtheta}$.
With this interpretation, the generative procedure works as follows.
It begins with an empty sentence, denoted by $S_{1:0}$, as the initial state.
At each step $t$, the \emph{policy} $\vpi_{\vtheta}$ takes the conditions
$\vf(I)$, $\vz$, and the preceding words $S_{1:t-1}$ as inputs,
and yields a conditional distribution $\vpi_{\vtheta}(w_t | \vf(I), \vz, S_{1:t-1})$
over the extended vocabulary, namely all words plus an indicator of
sentence end, denoted by $e$. This computation is done by moving forward
along the LSTM net by one step.
From this conditional distribution, an action $w_t$ will be sampled.
If $w_t = e$, the sentence will be terminated, otherwise $w_t$ will
be appended to the end.
The \emph{reward} of this sequence of actions $S$ is $r_{\veta}(I, S)$,
the score given by the evaluator $E$.
Now, we have defined an action space, a policy, and a reward function,
and it seems that we are ready to apply the reinforcement learning method.
However, there is a serious issue here -- a sentence can only be evaluated
when it is \emph{completely} generated. In other words, we can only see the
reward at the end. We found empirically that this would lead to a number
of practical difficulties, \eg~gradients vanishing along a long chain
and overly slow convergence in training.
We address this issue through \emph{early feedback}. To be more specific,
we evaluate an \emph{expected future reward} as defined below
when the sentence is \emph{partially} generated:
\begin{equation}\label{eq:policy_value}
V_{\vtheta, \veta}(I, \vz, S_{1:t}) =
\Ebb_{S_{t+1:T} \sim G_{\vtheta}(I, \vz)}
[r_{\veta}(I, S_{1:t}\oplus S_{t+1:T})].
\end{equation}
where $\oplus$ represents the concatenation operation.
Here, the expectation can be approximated using
Monte Carlo rollouts~\cite{yu2016seqgan}.
Particularly, when we have a part of the sentence $S_{1:t}$,
we can continue to sample the remaining words by simulating
the LSTM net until it sees an end indicator $e$. Conducting this
conditional simulation for $n$ times would result in $n$ sentences.
We can use the evaluation score averaged over these simulated sentences
to approximate the \emph{expected future reward}.
To learn the generator $G_{\vtheta}$, we use maximizing this expected reward
$V_{\vtheta,\veta}$ as the learning objective.
Following the argument in~\cite{sutton1999policy}, we can derive
the gradient of this objective \wrt~$\vtheta$ as:
\begin{equation} \label{eq:policy_grad}
\tilde{\Ebb}\left[
\sum_{t = 1}^{T_\text{max}}\sum_{w_t \in \cV}
\nabla_{\vtheta} \pi_{\vtheta}(w_t|I, \vz, S_{1:t-1}) \cdot
V_{\vtheta^\prime, \vpsi}(I, \vz, S_{1:t} \oplus w_t)
\right].
\end{equation}
Here, $\cV$ is the vocabulary, $T_\text{max}$ is the max length of a description,
and $\tilde{\Ebb}$ is the mean over all simulated sentences
within a mini-batch.
$\vtheta'$ is a copy of the generator parameter $\vtheta$
at the begining of the update procedure of the generator.
During the procedure, the generator will be updated multiple times,
and each update will use the same set of parameters ($\vtheta'$) to compute Eq (\ref{eq:policy_value}).
Overall, using policy gradients, we make the generator trainable
with gradient descent.
Using expected future reward, we can provide early feedback
to the generator along the way, thus substantially improving
the effectiveness of the training process.
Note that policy gradients have also been used in image
description generation in \cite{rennie2016self,liu2016optimization}.
These works, however, adopt conventional metrics, \eg~BLEU and CIDEr
as rewards, instead of relying on GAN.
Hence, their technical frameworks are fundamentally different.
\subsection{Training $E$: Naturalness \& Relevance}
The primary purpose of $E$ is to determine how well a description $S$
describes a given image $I$. A good description needs to satisfy
two criteria: \emph{natural} and \emph{semantically relevant}.
To enforce both criteria, inspired by \cite{reed2016generative} we extend Eq (\ref{eq:minmax}) to consider three types of descriptions
for each training image $I$:
(1) $\cS_I$: the set of descriptions for $I$ provided by human,
(2) $\cS_G$: those from the generator $G_{\vtheta}$, and
(3) $\cS_{\backslash I}$: the human descriptions for different images,
which is uniformly sampled from all descriptions that are not associated with the given image $I$.
To increase the scores for the descriptions in $\cS_I$
while suppressing those in the others, we use a joint objective
formulated as:
\begin{equation}
\max_{\veta} \quad \cL_E(\veta) = \frac{1}{N} \sum_{i=1}^N \cL_E(I_i;\veta).
\end{equation}
Here, $N$ is the number of training images. The term for each image $I_i$
is given by:
\begin{align}
\cL_E (I;\veta)
&= \Ebb_{S \in \cS_I} \log r_{\veta}(I, S) \notag \\
&+ \alpha \cdot \Ebb_{S \in \cS_G} \log (1 - r_{\veta}(I, S)) \notag \\
&+ \beta \cdot \Ebb_{S \in \cS_{\backslash I}} \log (1 - r_{\veta}(I, S)).
\label{eq:e_loss}
\end{align}
The second term forces the evaluator to distinguish between the human descriptions
and the generated ones, which would in turn provide useful feedbacks to $G_{\vtheta}$,
pushing it to generate more \emph{natural} descriptions.
The third term, on the other hand, ensures the \emph{semantic relevance},
by explicitly suppressing mismatched descriptions.
The coefficients $\alpha$ and $\beta$ are to balance the contributions of these terms,
whose values are empirically determined on the validation set.
\subsection{Extensions for Generating Paragraphs}
\label{sec:paragen}
We also extend our framework to generate \emph{descriptive paragraphs}
by adopting a \emph{Hierarchical LSTM} design.
Specifically, our extended design is inspired by \cite{krause2016paragraphs}.
As shown in part (c) of Figure \ref{fig:structure}, it comprises two LSTM levels --
a \emph{sentence-level} LSTM net and a \emph{word-level} LSTM net.
Given the conditions $\vf(I)$ and $\vz$, to produce a paragraph,
it first generates a sequence of vectors based on $\vf(I)$, each encoding the topics of a sentence.
Then for each sentence, it generates the words conditioned on the
corresponding topic and the random vector $\vz$.
For evaluating a paragraph, the evaluator $E$ also adopts a hierarchical
design, but reversing the steps. Given an image $I$ and a paragraph $P$,
it first embeds each sentence into a vector via a word-level LSTM net, and
then embeds the entire paragraph by combining the sentence embeddings
via a sentence-level LSTM net.
Finally, it computes the score by taking the dot product between the paragraph
embedding $\vp$ and the image representation $\vf(I)$, and turning it into a
probability as $\sigma(\vp^T \vf(I))$, where
$\sigma$ is the logistic function.
After pretraining, we fix the sentence-level LSTM net of $G$ and only update the
word-level LSTM net of $G$ during the CGAN learning procedure.
This can effectively reduce the cost of Monte Carlo rollouts.
With a fixed sentence-level LSTM net, the policy gradients for each sentence
will be computed separately, following the steps in Sec~\ref{sec:g_train}.
Other parts of the training procedure remain the same.
\section{Introduction}
\label{sec:intro}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{imgs/teaser.pdf}\\[-1mm]
\caption{\small
This figure shows two images with descriptions generated by humans,
an LSTM net trained with our GAN-based framework (\emph{G-GAN}),
and an LSTM net trained with MLE (\emph{G-MLE}).
The last two columns compare the metric values of BLEU-3 and \emph{E-GAN},
the evaluator trained using our method.
As we can see, the sentences generated by \emph{G-GAN} are more natural
and demonstrate higher variability, as compared to those by \emph{G-MLE}.
Also, the \emph{E-GAN} metrics are more consistent with human's evaluations,
while BLEU only favors those that significantly overlap
with the training samples in detailed wording.}
\label{fig:teaser_g}
\vspace{-3.0mm}
\end{figure}
Generating descriptions of images has been an important task in
computer vision. Compared to other forms of semantic summary, \eg~object
tagging, linguistic descriptions are often richer, more comprehensive,
and a more natural way to convey image content. Along with the recent surge of deep learning technologies,
there has been remarkable progress in image captioning
over the past few years~\cite{vinyals2015show,xu2015show,you2016image,yang2016review,krause2016paragraphs}. Latest studies on this topic often
adopt a combination of an LSTM or its variant and a CNN. The former is
to produce the word sequences while the latter is
to capture the visual features of the images.
The advance in image captioning has been marked as a prominent
success of AI\footnote{ARTIFICIAL INTELLIGENCE AND LIFE IN 2030, \url{https://ai100.stanford.edu/2016-report}}.
It has been reported~\cite{vinyals2015show,xu2015show} that with certain
metrics, like BLEU \cite{papineni2002bleu}~or CIDEr \cite{vedantam2015cider}, state-of-the-art techniques
have already surpassed human's performance.
A natural question to ask is then: {\em has the problem of generating image
descriptions been solved?}
Let us take a step back, and look at a sample of the current results.
Figure~\ref{fig:teaser_g} shows
two vivid scenes together with three sentences produced by
the Encoder-and-Decoder model~\cite{vinyals2015show} (marked as ``G-MLE''), a state-of-the-art caption generator.
Though faithfully describing the content of the images,
these sentences feel rigid, dry, and lacking in vitality.
This is not surprising. Our brief survey (see Section~\ref{sec:relwork})
shows that existing efforts primarily focus on \emph{fidelity},
while other essential qualities of human languages,
\eg~\emph{naturalness} and \emph{diversity}, have received less attention.
More specifically, mainstream captioning models, including those
based on LSTMs~\cite{hochreiter1997long}, are mostly trained with the
(conditional) maximum likelihood objective. This objective
encourages the use of the \emph{n-grams} that appeared in the
training samples. Consequently, the generated sentences will
bear high resemblance to training sentences in \emph{detailed wording},
with very limited variability in expression~\cite{devlin2015exploring}.
Moreover, conventional evaluation metrics,
such as
BLEU \cite{papineni2002bleu},
METEOR \cite{lavie2014meteor},
ROUGE \cite{lin2004rouge}, and
CIDEr \cite{vedantam2015cider},
tend to favor this \emph{``safe''} but restricted way.
Under these metrics, sentences that contain matched n-grams
would get substantially higher scores than those using
variant expressions~\cite{anderson2016spice}.
This issue is manifested by the fact that human descriptions
get considerably lower scores.
Motivated to move beyond these limitations, we explore an alternative
approach in this work. We wish to produce sentences that possess
three properties:
(1) \textbf{Fidelity}: the generated descriptions should reflect the visual
content faithfully. Note that we desire the fidelity in \emph{semantics}
instead of \emph{wording}.
(2) \textbf{Naturalness}: the sentences should \emph{feel} like what real
people would say when presented with the image. In other words, when these
sentences are shown to a real person, she/he would ideally not be able to tell
that they are machine-generated.
(3) \textbf{Diversity}: the generator should be able to produce notably
different expressions given an image -- just like human beings, different
people would describe an image in different ways.
Towards this goal, we develop a new framework on top of the
Conditional GAN~\cite{mirza2014conditional}.
GAN has been successfully used in image generation.
As reported in previous works~\cite{reed2016generative,isola2016image}, they can produce \emph{natural}
images nearly indistinguishable from real photos,
freely or constrained by conditions.
This work studies a different task for the GAN method, namely,
generating \emph{natural} descriptions conditioned on a given image.
To our best knowledge, this is the first time the GAN method is
used for image description.
Applying GANs to text generation is nontrivial.
It comes with two significant challenges due to the special
nature of linguistic representation.
First, in contrast to image generation, where the transformation
from the input random vector to the produced image is a deterministic continuous
mapping, the process of generating a linguistic description is a
\emph{sequential sampling} procedure, which samples a \emph{discrete}
token at each step. Such operations are \emph{non-differentiable},
making it difficult to apply back-propagation directly.
We tackle this issue via \emph{Policy Gradient},
a classical method originating from reinforcement learning~\cite{sutton1999policy}.
The basic idea is to consider the production of each word
as an \emph{action}, for which the reward comes from
the evaluator. By approximating the stochastic policy
with a parametric function approximator, we allow gradients
to be back-propagated.
Second, in the conventional GAN setting, the generator
would receive feedback from the evaluator when an entire sample
is produced. For sequence generation, this would lead to several
difficulties in training, including \emph{vanishing gradients} and
\emph{error propagation}.
To mitigate such difficulties, we devise a mechanism that allows the
generator to get early feedback. Particularly, when a description
is \emph{partly} generated, our framework would calculate an approximated
\emph{expected future reward} through Monte Carlo rollouts~\cite{yu2016seqgan}.
Empirically, we found that this significantly improves the efficiency
and stability of the training process.
Overall, our contributions can be briefly summarized as follows:
(1) We explore an alternative approach to generate image descriptions,
which, unlike most of the previous work, encourages not only \emph{fidelity}
but also \emph{naturalness} and \emph{diversity}.
(2) From a technical standpoint, our approach relies on the conditional GAN
method to learn the generator, instead of using MLE, a paradigm
widely adopted in state-of-the-art methods.
(3) Our framework not only results in a generator that can produce
natural and diverse expressions, but also yields a description evaluator
at the same time, which, as we will show in our experiments, is
substantially more consistent with human evaluation.
\section{Related Work}
\label{sec:relwork}
\paragraph{Generation.}
Generating descriptions for images has been a long standing topic in computer vision.
Early studies mostly adopted \emph{detection-based} approaches.
Such methods first detect visual concepts (\eg~object categories, relationships, and attributes)
using CRFs~\cite{farhadi2010every, kulkarni2013babytalk, dai2017detecting},
SVMs~\cite{li2011composing}, or CNNs~\cite{fang2015captions, lisce2017},
then generate descriptions thereon using simple methods,
such as sentence templates~\cite{kulkarni2013babytalk,li2011composing},
or by retrieving relevant sentences from existing
data~\cite{farhadi2010every,fang2015captions,lebret2014simple,kuznetsova2012collective}.
In recent years,
the Encoder-and-Decoder paradigm proposed in~\cite{vinyals2015show}
became increasingly popular.
Many state-of-the-art frameworks~\cite{zhou2016image,you2016image,yang2016review,lu2016knowing,xu2015show,vinyals2015show} for this task adopt
the \emph{maximum likelihood} principle for learning.
Such a framework usually works as follows.
Given an image $I$, it first derives a feature representation $\vf(I)$,
and then generates the words $w_1, \ldots, w_T$ sequentially, following
a Markov process conditioned on $\vf(I)$.
The model parameters are learned via maximum likelihood estimation (MLE),
\ie~maximizing the conditional log-likelihood
of the training samples, as:
\begin{equation}
\sum_{(I_i, S_i) \sim \cD}
\sum_{t=0}^{T_i}
\log p \left(w_i^{(t)} | \vf(I), w_i^{(t-1)}, \ldots, w_i^{(t-n)}\right)
\end{equation}
Here, $I_i$ and $S_i = (w_i^{(0)}, \ldots, w_i^{(T_i)})$ are the image and
the corresponding descriptive sentence of the $i$-th sample,
and $n$ is the order of the Markov chain
-- the distribution of the current word depends on $n$ preceding words.
Along with the popularity of deep neural networks, latest studies often
adopt neural networks for both image representation and language modeling.
For example, \cite{xu2015show} uses a CNN for deriving the visual features
$\vf(I)$, and an LSTM~\cite{hochreiter1997long}~net to express the sequential relations among words.
Despite the evolution of the modeling choices, the maximum likelihood principle
remains the predominant learning principle.
\begin{figure}
\centering
\includegraphics[height=0.22\textheight]{imgs/mle_limit.pdf}\\[-1.5mm]
\caption{\small
We illustrate the procedures of image description generation
and evaluation for state-of-the-art approaches.
While the generation procedure tends to follow observed patterns,
the evaluation procedure also favors this point.
Best viewed in color.}
\label{fig:mle_limit}
\vspace{-1mm}
\end{figure}
\begin{figure}
\centering
\includegraphics[height=0.24\textheight]{imgs/teaser_d.pdf}\\[-1.5mm]
\caption{\small
Examples of images with two semantically similar descriptions,
selected from ground-truth annotations.
While existing metrics assign higher scores to those with more matched n-grams,
\emph{E-GAN} gives scores
consistent with human evaluation.}
\label{fig:oldmetrics}
\vspace{-4mm}
\end{figure}
As illustrated in Figure~\ref{fig:mle_limit}, when similar images are presented,
the sentences generated by such a model often contain \emph{repeated} patterns \cite{devlin2015language}.
This is not surprising -- under the MLE principle, the joint probability of a sentence
is, to a large extent, determined by whether it contains the frequent n-grams from the training set. Therefore, the model trained in this way will tend
to produce such n-grams. In particular, when the generator yields a few of words
that match the prefix of a frequent n-gram, the remaining words of that
n-gram will likely be produced following the Markov chain.
\vspace{-12pt}
\paragraph{Evaluation.}
Along with the development of the generation methods,
various evaluation metrics have been proposed to assess
the quality of the generated sentences.
Classical metrics include BLEU~\cite{papineni2002bleu} and
ROUGE~\cite{lin2004rouge}, which respectively focuses on
the precision and recall of n-grams.
Beyond them, METEOR~\cite{lavie2014meteor} uses a combination of both the
precision and the recall of n-grams.
CIDEr\cite{vedantam2015cider} uses weighted statistics over n-grams.
As we can see, such metrics mostly rely on matching n-grams with
the \emph{``ground-truths''}.
As a result, sentences that contain frequent n-grams will get higher
scores as compared to those using variant expressions,
as shown in Figure~\ref{fig:oldmetrics}.
Recently, a new metric SPICE~\cite{anderson2016spice} was proposed.
Instead of matching between n-grams, it focuses on those linguistic
entities that reflect visual concepts (\eg~objects and relationships).
However, other qualities, \eg~the naturalness of the expressions,
are not considered in this metric.
\vspace{-12pt}
\paragraph{Our Alternative Way.}
Previous approaches, including both generation methods and evaluation metrics,
primarily focus on the \emph{resemblance} to the training samples.
While this is a \emph{safe} way to generate plausible descriptions,
it is \emph{limited}.
For example, when presented an image, different people would probably
give different descriptions that do not overlap much in the wording patterns.
This diversity in expression is an essential property of human languages,
which, however, is often overlooked in previous works (both generation
and evaluation).
In this work, we explore an alternative approach --
instead of emphasizing n-gram matching, we aim to improve the
\emph{naturalness} and \emph{diversity}, \ie~generating sentences that
feel like what real people would say, rather than focusing on word-by-word matching.
Specifically, our approach jointly trains a generator $G$ and an evaluator $E$
in an adversarial way, where $G$ is to produce natural descriptions,
while $E$ is to distinguish irrelevant or artificial descriptions
from natural ones.
From a technical standpoint, our approach is based on the conditional GAN approach.
GANs~\cite{goodfellow2014generative} and conditional GANs~\cite{mirza2014conditional} are
popular formulations for learning generators.
For computer vision, GAN was originally introduced to generate images~\cite{reed2016generative}.
In a recent work~\cite{yu2016seqgan}, a text generator based on the GAN method was proposed.
Note that this is an unconstrained generator that does not take into account any conditions.
Hence, it can not be directly used for generating descriptions for images
-- in this task, the relevance of the generated text to the given image is essential.
To our best knowledge, this is the first study that explores
the use of \emph{conditional} GAN in generating image descriptions.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,630
|
\section{Introduction}
Modern communication networks have moved beyond traditional point-to-point communication scenarios into multiterminal communication. Many communication scenarios in modern networks also involve delivery of large content \cite{Cis}, with low latency requirements. Owing to the reduction in the cost of storage media, a reasonable option to reduce the network load is to cache a fraction of the total content near the receivers, and deliver the rest of the requested files through the main network. This technique was formally presented for the single transmitter noiseless broadcast network in \cite{MaN}. It was shown in \cite{MaN} that huge gains can be achieved by a properly jointly designed caching and a delivery scheme which involves sending coded file-parts corresponding to demands of multiple users at the same time. The rate achieved by the scheme in \cite{MaN} was in fact shown to be optimal for uncoded caching in \cite{WTP}. The gains shown using the \textit{coded caching} paradigm established by \cite{MaN} were extended to a number of other scenarios, for instance, D2D communication \cite{D2D}, combination networks \cite{combinationnetworks}, and also multisource-multisink settings as in \cite{interferencemanagement}.
In \cite{interferencemanagement}, the authors consider the problem of managing interference in a multi-source ($K_T$ of them) multi-sink ($K_R$ of them) setting connected through an AWGN channel with complex channel gains between the transmitters and the receivers. Each of the receivers can demand one of the $N$ files present in the library. Each transmitter has some local storage (cache) which can store a $\frac{M_T}{N}$ fraction of the library, while each receiver has a cache that can store a $\frac{M_R}{N}$ fraction. A centralized caching scheme and a one-shot delivery scheme that employs linear combinations of subfiles, developed based on the scheme in \cite{MaN}, is designed in \cite{interferencemanagement}, which serves $t_T+t_R$ (where $t_T= \frac{{M_T}{K_T}}{N}$ and $t_R=\frac{{M_R}{K_R}}{N}$) receivers in each transmission round, which is known as the \textit{achievable one-shot linear sum-$\mathsf{DoF}$} (degrees of freedom). The scheme achieves this $\mathsf{DoF}$ using two tools, (a) using the transmitter cache content to zero-force interference at the receivers, and (b) by designing a coded one-shot delivery scheme based on \cite{MaN} exploiting the receiver caches. This achievable $\mathsf{DoF}$ is then shown to be within a multiplicative factor of two of the optimal one-shot $\mathsf{DoF}$ for the interference channel setting of \cite{interferencemanagement}.
However, this near-optimality comes at a price. The implementation of the scheme in \cite{interferencemanagement} requires that the number of subfiles of any file, called the \textit{subpacketization}, should be at least $\binom{K_T}{t_T}\binom{K_R}{t_R}\binom{K_R-t_R-1}{t_T-1}t_R!(t_T-1)!$. As $K_R$ grows large (as to be expected in practical scenarios) for constant cache fraction $\frac{M_R}{N}$, the subpacketization grows exponentially in $K_R$ As the subpacketization parameter associated with a coded caching scheme gives a lower bound on the size of the files in order to implement the scheme, this presents a practical difficulty that has to be overcome to reap the benefits of coded caching. The root of this problem lies in the scheme of \cite{MaN} for broadcast settings, and all schemes which utilize this scheme of \cite{MaN} also inherit this problem.
\subsection{Other related Work}
The first work which considered applying the strategy of coded caching to interference channels was \cite{CacheAidedInterferenceChannels}. The authors of \cite{CacheAidedInterferenceChannels} focused on the case with only transmitter caches, which were exploited to get both interference cancellation and interference alignment gains. This was further expanded upon by \cite{DoFHachem} where the DoF region of the interference channel with both transmitter and receiver channels was characterized and a near-optimal scheme (optimal except for a multiplicative gap) was obtained using general delivery schemes (beyond one-shot schemes). Coded caching for multi-antenna interference channels was considered in \cite{MultiAntennaInterference}. A one-shot delivery based coded caching scheme for the setup of \cite{interferencemanagement} was presented in \cite{hypercube_interference}, which achieves smaller subpacketization than the scheme in \cite{interferencemanagement}. However the subpacketization still remains exponential in the number of receivers $K_R$.
\subsection{Contributions and Organization}
The present work involves low subpacketization constructions of coded caching schemes for the interference management problem under the communication scenario in \cite{interferencemanagement}. The contributions and organization of this work are as follows.
\begin{itemize}
\item
We first review the system model from \cite{interferencemanagement} and also present the conditions for valid transmission rounds of a one-shot delivery scheme in Section \ref{systemmodel}. By doing this, we setup the strategy for designing our delivery schemes.
\item
In Section \ref{modified1}, we show that we can achieve the same $\mathsf{DoF}$ as \cite{interferencemanagement}, with a smaller subpacketization of $\binom{K_T}{t_T}\binom{K_R}{t_R}\binom{K_R-t_R-1}{t_T-1}$. However this subpacketization continues to be exponential in $K_R$ for constant $\frac{M_R}{N}.$ We do this by giving a modified delivery scheme of \cite{interferencemanagement} for the caching scheme of \cite{interferencemanagement}.
\item Inspired from subexponential subpacketization schemes for the broadcast coded caching setting developed in \cite{haribhavanaprasad}, in Section \ref{projective geometry based scheme}, we develop a coded caching scheme based on projective geometry over finite fields, which has subpacketization subexponential ($q^{O(K_T+(log_qK_R)^2)}$, where $q$ is some prime power) in the number of receivers $K_R$ for receiver cache fraction $\frac{M_R}{N}$ upper bounded by a small constant (these asymptotics are shown in Section \ref{analysis}). We give a numerical comparison of our projective geometry based scheme with the scheme proposed in Section \ref{modified1} (which is improved from \cite{interferencemanagement}), and also with \cite{hypercube_interference} in Section \ref{projective geometry based scheme} (Table \ref{table 1}). The results show that we outperform all known schemes for the interference channel setup in terms of the subpacketization, however trading it off with a lower achievable $\mathsf{DoF}$ which is $\Theta(log_qK_R+K_T)$.
\end{itemize}
\textit{Notations and Terminology:} $\mathbb{Z}^{+}$ denotes the set of positive integers.
We denote the set $\{1,\hdots,n\}$ by $[n]$ for some $n \in \mathbb{Z}^{+}$. For sets $A,B$, the set of elements in $A$ but not in $B$ is denoted by $A\backslash B$. For some element $x$, we denote $A\backslash\{x\}$ by $A\backslash x$ also. The set of $r$ sized subsets of a set $A$ is denoted by $\binom{A}{r}$.
For $i,j,m\in \mathbb{Z}^{+}$, we define $i\boxplus_{m}j \triangleq 1+((i+j-1) \textit{ mod } m)$. The finite field with $q$ elements is ${\mathbb F}_q$. The dimension of a vector space $V$ over ${\mathbb F}_q$ is given as $dim(V)$. For two subspaces $V,W$, their subspace sum is denoted by $V+W$. Note that $V+W=V\oplus W$ (the direct sum) if $V\cap W=\phi$. The span of two vectors $\mathbf{v_1},\mathbf{v_2}\in V$, is represented as $span(\mathbf{v_1},\mathbf{v_2})$.
\section{System Model, Basic Terminologies and a Technical Lemma}
\label{systemmodel}
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{sys_fig.jpg}
\caption{Wireless interference network}
\label{fig system}
\end{figure}
We follow the model as in \cite{interferencemanagement}. We consider a discrete time AWGN channel with $K_T$ transmitters, denoted by a set ${\cal K}_T$, and $K_R$ receivers, denoted by a set ${\cal K}_R$, as shown in Fig. \ref{fig system}. A collection of $N$ files exist in the system denoted as $W_{n}, n\in[N]$. Each transmitter has the capability of caching a fraction $\frac{M_T}{N}$ of the library, while each receiver can cache a fraction $\frac{M_R}{N}$ of the library.
The system operates in two phases. In the \textit{caching phase}, each file is divided in $F_C$ subfiles (equal-sized) and prefetched into the caches of the transmitters and the receivers. The subfiles of a file $W_n$ are indexed by a set ${\cal F}_C$ as $W_{n,f}:f\in {\cal F}_C$. In this work, we discuss caching schemes in which the transmitters cache $M_T F_C$ subfiles, the receivers cache $M_R F_C$ subfiles, and also each subfile is cached at $t_T$ transmitters and $t_R$ receivers. Such \textit{symmetric} caching schemes are commonly used in literature. We assume that $\frac{min(M_TK_T,M_RK_R)}{N}\geq 1$, i.e., the cumulative caches at the transmitters (equivalently, the receivers) can hold the entire library. During the \textit{delivery phase}, each receiver demands one of the files in the library, and the transmitters must cooperatively deliver the missing subfiles of each file to the receivers. In the delivery phase, we allow for further splitting of the subfiles into (equal-sized) \textit{packets} to reduce the delivery time. In this paper, we assume that the number of packets within each subfile of each file is some constant, denoted by $F_P$.
Designing the coded caching scheme for the interference channel, also known as an \textit{cache-aided interference management scheme}, consists of designing the caching phase (what to place in the cache) and the delivery phase as well. We consider delivery schemes which are \textit{one-shot linear schemes}, as in \cite{interferencemanagement},
described generally as follows. The delivery phase consists of several \textit{rounds} of transmissions. In each round, each transmitter picks some packets corresponding to the subfiles missing in the receiver caches, maps them to complex vectors, and then transmits a linear combination of them. Specifically, let $\boldsymbol{w}_j:j\in [n]$ be some $n$ packets missing at some receivers, and to be transmitted by some particular transmitter $t\in{\cal T}$. Then the packet $\boldsymbol{w}_j$ is mapped to a complex vector $\boldsymbol{\overline{w}}_j$. Then the transmitter transmits $\boldsymbol{x}_t=\sum_{j=1}^n v_j\boldsymbol{\overline{w}}_j$, where $v_j$ is the complex beamforming coefficients chosen by the transmitter $t$. We assume a power constraint $\frac{1}{b}||\boldsymbol{x}_t||^2\leq \mathsf{SNR}$ on the transmissions, where $b$ denotes the length of $\boldsymbol{x}_t$.
The output at the receiver $r\in{\cal R}$ is given as
\[
\boldsymbol{y}_r=\sum_{t\in{\cal K}_T}h_{r,t}\boldsymbol{x}_t+\boldsymbol{z}_r,
\]
where $h_{r,t}$ represents the complex channel gain from the transmitter $t$ to the receiver $r$ (assumed to be chosen i.i.d from ${\cal CN}(0,1)$ but remaining constant for a round of transmission), and $\boldsymbol{z}_r$ represents the additive noise vector at receiver $r$, with each component of $\boldsymbol{z}_r$ being i.i.d with distribution ${\cal CN}(0,1)$.
To precisely define the ability of a transmission round to deliver the packets involved in that round, we define the idea of a \textit{valid transmission round}.
\begin{definition}
For some $n\in{\mathbb Z}^+$, a collection of $n$ packets $P=\{\boldsymbol{w}_r:r=1,..,n\}$ intended to $n$ corresponding distinct receivers $R=\{1,..,n\}$ is said to participate in a \underline{valid transmission round} if there exists a choice of the beamforming coefficients at the transmitters containing $P$ for transmitting a linear combination of $P$ such that each receiver $r$ in $R$ can obtain (after certain interference terms being zero-forced by choice of the beamforming coefficients, and cancelling other interference terms present in cache) a quantity $\overline{\boldsymbol{w}}_r+\boldsymbol{z}_r$ (therefore enabling the decoding of $\boldsymbol{w}_r$ with vanishing error probability as $\mathsf{SNR}$ increases), where $\overline{\boldsymbol{w}}_r$ is the complex vector corresponding to demanded packet $\boldsymbol{w}_r$ and $\boldsymbol{z}_r$ denotes the complex AWGN noise.
\end{definition}
We then have the definition of \textit{a delivery scheme} as follows.
\begin{definition}
A \underline{(one-shot) delivery scheme} is one in which all the missing packets of the demanded file of each receiver are successfully delivered (i.e., decoded with vanishing error probability with increasing $\mathsf{SNR}$) in $S$ valid transmission rounds (for some finite $S$). The \textit{rate} of the delivery scheme is then defined as
\begin{align}
\label{rate}
\text{Rate}=\frac{S}{F},
\end{align}
where $F=F_CF_P$ denotes the effective subpacketization (simply, the subpacketization), for $F_C$ being the number of subfiles of each file during the caching phase, and $F_P$ is the number of packets of each subfile during the delivery phase.
\end{definition}
A rate $R$ is said to be \textit{achievable} if there exists a caching scheme and valid delivery scheme with rate $R$. Note that a rate of $K_R(1-\frac{M_R}{N})$ is the naively achievable, with a caching scheme in which $\frac{M_R}{N}$ fraction of each file is cached in the library of each receiver, and a delivery scheme in which the uncached fractions of the receiver demands are delivered on a time-sharing basis. Based on the rate, we now define $\mathsf{DoF}$.
\begin{definition}
[DoF]For an achievable scheme with rate $R,$ we refer to the fraction $\frac{K_R(1-\frac{M_R}{N})}{R}$ as the corresponding \textit{achievable} \textit{one-shot linear $\mathsf{DoF}$ (degrees of freedom)}. The \textit{one-shot linear $\mathsf{DoF}$} of the interference channel is defined as the fraction $\frac{K_R(1-\frac{M_R}{N})}{R^*}$, where $R^*$ is the optimal coded caching rate, i.e., the infimum of all achievable rates given the system parameters $K_T,M_T,K_R,M_R$ and $N$.
\end{definition}
\begin{remark}
Note that if the delivery scheme serves some $\gamma$ packets to $\gamma$ receivers in each round, then as the total number of missing subfiles is $K_R(1-\frac{M_R}{N})F$, we have from (\ref{rate}) that achievable $\mathsf{DoF}=\gamma$. Thus for such schemes we think of the achievable $\mathsf{DoF}$ as the number of receivers served in each round.
\end{remark}
This system model is developed in \cite{interferencemanagement}, after which the authors show a construction of a coded caching scheme which achieves a $\mathsf{DoF}$ of $t_T+t_R$ (when $t_T+t_R\leq K_R$), where $t_T=\frac{M_TK_T}{N}, t_R=\frac{M_RK_R}{N}$, by obtaining a caching scheme and delivery scheme inspired from the broadcast coded caching scheme of \cite{MaN}.
In this work, we give new cache-aided interference management schemes with reduced subpacketizations. We now give a small technical lemma, which we shall use in the rest of the paper for this purpose. The proof of the lemma can actually be inferred from \cite{interferencemanagement} (Lemma 3 of \cite{interferencemanagement}), however we give it here in an explicit fashion for our convenience.
\begin{lemma}
\label{ZF}
Consider a set of $n$ packets $P=\{\boldsymbol{{w}}_{d_{r_1},f_1},...,\boldsymbol{{w}}_{d_{r_n},f_n}\}$, where $\boldsymbol{{w}}_{d_{r_i},f_i}$ is a packet demanded by receiver $r_i$, and each packet in $P$ is cached at atleast $n-t_T$ receivers among the $n$ receivers $\{r_1,..,r_n\}$ (where $t_T$ refers to number of transmitters in which each packet is cached). Then the set of packets $P$ can participate in a valid transmission round.
\end{lemma}
\begin{IEEEproof}
On the transmitter side we have,
\begin{align}
\label{trans}
\begin{bmatrix}
\boldsymbol{x}_1\\ \vdots \\\boldsymbol{x}_{K_T}
\end{bmatrix}=\begin{bmatrix}
\overline{M}_1 & \cdots & \overline{M}_n
\end{bmatrix}\begin{bmatrix}
\boldsymbol{\overline{w}}_{d_{r_1},f_1}\\ \vdots \\\boldsymbol{\overline{w}}_{d_{r_n},f_n}
\end{bmatrix}
\end{align}
where $\boldsymbol{x}_i \in \mathbb{C}^{1 \times b}: i \in [K_T]$ denotes the signal transmitted by $i^{th}$ transmitter and $\overline{M}_i \in \mathbb{C}^{K_T \times 1} : i \in [n]$ is a column vector of size $K_T \times1$ which denotes the beamforming vector of (complexified) packet $\overline{\boldsymbol{{w}}}_{d_{r_i},f_i}: i \in [n]$. Note that the rows of $\overline{M}_i$ corresponding to the indices of transmitters where packet $\boldsymbol{{w}}_{d_{r_i},f_i}$ is not cached contain zeros. Thus $\overline{M}_i$ can have at most $t_T$ non-zeros for each $i \in [N]$, which are to be chosen by the respective transmitters. We want to show that there exists a choice of beamforming coefficients in $\overline{M}_i, \forall i$ such that packets $\boldsymbol{{w}}_{d_{r_1},f_1},...,\boldsymbol{{w}}_{d_{r_n},f_n}$ can be decoded at corresponding receivers $r_1,...,r_n$.
On the receiver side we have,
\begin{align}
\label{recv}
\begin{bmatrix}
\boldsymbol{y}_{r_1}\\ \vdots \\\boldsymbol{y}_{r_n}
\end{bmatrix}=H\begin{bmatrix}
\boldsymbol{x}_1\\ \vdots \\\boldsymbol{x}_{K_T}
\end{bmatrix}+Z
\end{align}
where $\boldsymbol{y}_{r_i}: i \in [n]$ denotes the signal received by receiver $r_i$, $H_{n\times K_T}$ denotes the channel state matrix containing the complex channel gain coefficients and $Z_{n \times b}$ denotes the additive Gaussian noise.
From (\ref{trans}) and (\ref{recv}) we have,
\begin{align*}
\begin{bmatrix}
\boldsymbol{y}_{r_1}\\ \vdots \\\boldsymbol{y}_{r_n}
\end{bmatrix}&=H\begin{bmatrix}
\overline{M}_1 & \cdots & \overline{M}_n
\end{bmatrix}\begin{bmatrix}
\overline{\boldsymbol{{w}}}_{d_{r_1},f_1}\\ \vdots \\\overline{\boldsymbol{{w}}}_{d_{r_n},f_n}
\end{bmatrix}+Z\\
&=C\begin{bmatrix}
\overline{\boldsymbol{{w}}}_{d_{r_1},f_1}\\ \vdots \\\overline{\boldsymbol{{w}}}_{d_{r_n},f_n},
\end{bmatrix}+Z
\end{align*}
where $C=[C_1 \hdots C_n],$ such that $C_i=H \overline{M}_i: i \in [n]$. Let $[v_{k_1} \hdots v_{k_n}]$ denote the $k^{th}$ row of $C$. For successful decoding of $\boldsymbol{{w}}_{d_{r_k},f_k}$ at receiver $r_k$, we should have
\[
\boldsymbol{y}_{r_k}=\sum_{i=1}^{n}v_{k_i} \overline{\boldsymbol{{w}}}_{d_{r_i},f_i}+\boldsymbol{z}_{r_i},
\]
such that $v_{k_i}=0$, for all $i$ such that $\boldsymbol{{w}}_{d_{r_i},f_i}$ is not present in cache of receiver $r_k$ and $v_{k_i} \neq 0$ for $i=k$ (w.l.o.g, we may assume $v_{k_k}=1$). The beamforming vectors $\overline{M}_i$s have to be chosen such that the above condition holds. Since it is given that any packet $\boldsymbol{{w}}_{d_{r_i},f_i}$ is cached at atleast $n-t_T$ receivers and demanded by $r_i$, the vector $\overline{M}_i$ must be chosen so that the $i^{th}$ column $C_i$ must have precisely $1$ at the $i^{th}$ position and $0$'s in $e \leq (t_T-1)$ positions (corresponding to the row indices of those receivers which have not cached $\boldsymbol{{w}}_{d_{r_i},f_i}$). Let $C'_i$ be the subvector of $C_i$ restricted only to these $e+1$ positions. Then we have,
\[
C'_i = H' \overline{M}_i
\]
where $H'_{(e+1)\times K_T}$ is some appropriate submatrix of $H$. Further since any packet is available at $t_T$ transmitters by the caching design, the packet $\boldsymbol{{w}}_{d_{r_i},f_i}$ is cached in $t_T$ transmitters, which means $K_T-t_T$ entries of $\overline{M}_i$ are zero. Let ${\overline{M^{''}_i}}$ be the subvector of $\overline{M}_i$ after removing all these $K_T-t_T$ zero positions of $\overline{M}_i$. Then
\[
C'_i = {H^{''}} {\overline{M^{''}_i}},
\]
for some appropriate submatrix $H^{''}_{(e+1)\times t_T}$ of $H'$. Since $e+1 \leq t_T$ and as the channel coefficients are generated i.i.d from ${\cal CN}(0,1)$, the matrix $H^{''}$ is of rank $e+1$. Thus there exists some solution to the vector ${\overline{M^{''}_i}}$, which gives us the beamforming vector $\overline{M}_i$ (by appending zero to ${\overline{M^{''}_i}}$).
\end{IEEEproof}
\begin{remark}
Though we have considered $K_T$ transmitters participating in each valid round of transmission in Lemma \ref{ZF}, it is sufficient to have $t_T$ transmitters participating in each valid round of transmission as used in the constructions in the remainder of the paper.
\end{remark}
\section{A Cache-Aided Interference Management Scheme with Reduced Subpacketization achieving $\mathsf{DoF}=t_T+t_R$}
\label{modified1}
The work \cite{interferencemanagement} presents a construction of a coded caching scheme for the interference channel achieving $\textsf{DoF}=t_T+t_R=\frac{K_TM_T+K_RM_R}{N}$ (we restrict ourselves to the case $K_R\geq t_T+t_R$) with the subpacketization $F=\binom{K_T}{t_T}\binom{K_R}{t_R}\binom{K_R-t_R-1}{t_T-1}t_R!(t_T-1)!$. In this section, we obtain a new delivery scheme for the same caching scheme which obtains the same $\mathsf{DoF}=t_T+t_R$ with a smaller subpacketization $F=\binom{K_T}{t_T}\binom{K_R}{t_R}\binom{K_R-t_R-1}{t_T-1}$. Note that as a result, the subpacketization of this new scheme can be exponentially smaller than that of \cite{interferencemanagement}, for large $K_R, K_T$. A numerical comparison of these is provided in Table \ref{table 1}, in the end of Section \ref{projective geometry based scheme}.
\subsubsection{Caching Scheme}
\label{modified_caching}
We use the caching technique given in \cite{interferencemanagement}. To describe the caching strategy, we first breakdown the files present in the library into subfiles. We consider $t_T = \frac{{K_T}{M_T}}{N}$ and $t_R = \frac{{K_R}{M_R}}{N}$. Each subfile $W_i:i \in [N]$ in the library is divided into disjoint subfiles denoted as $\{W_{i,{\cal T},{\cal R}}: {\cal T} \in \binom{[K_T]}{t_T}, {\cal R} \in \binom{[K_R]}{t_R}\}$. Hence the subpacketization during the caching phase is $F_C=\binom{K_T}{t_T}\binom{K_R}{t_R}$. Each transmitter and receiver caches a set of subfiles from the library. The caching strategy on the transmitter side is such that transmitter $t_j \in [K_T]$ caches subfiles $\{W_{i,{\cal T},{\cal R}}: t_j \in {\cal T}\}$. Similarly, the caching strategy on the receiver side is such that receiver $r_j \in [K_R]$ caches subfiles $\{W_{i,{\cal T},{\cal R}}: r_j \in {\cal R}\}$.
\subsubsection{Transmission Scheme}
\label{modified_transmission}
Let $W_{d_{r_j}}$ denote the file demanded by $r_j\in [K_R]$ in the delivery phase. Thus, each receiver $r_j \in [K_R]$ is to be served the subfiles $\{W_{d_{r_j},{\cal T},{\cal R}}: r_j \notin {\cal R},d_{r_j} \in [N]\}$ which are not available in its cache.
We seek to construct valid transmission rounds following Lemma \ref{ZF}. Towards that end, we further divide the demanded subfiles into packets and transmit $t_T+t_R$ packets in each round of transmission. Every subfile $W_{d_{r_j},{\cal T},{\cal R}}$ demanded by receiver $r_j$ is divided into $\binom{K_R-t_R-1}{t_T-1}$ packets denoted as
\[
W_{d_{r_j},{\cal T},{\cal R}}= \bigg\{W_{d_{r_j},{\cal T},{\cal R},{\cal R'}}: {\cal R'} \in \binom{[K_R]\setminus{({\cal R} \cup \{r_j\})}}{t_T-1}\bigg\}
\]
The index ${\cal R'}$ essentially denotes the indices of receivers at which the packet $W_{d_{r_j},{\cal T},{\cal R},{\cal R'}}$ is zero-forced.
We now describe the set of packets that can be sent in a valid round of transmission. For a set ${\cal U} \in \binom{[K_R]}{t_T+t_R}$ denoted as $\{u_1,...,u_{t_T+t_R}\} $, consider some set $U \subset {\cal U}$ denoted as $\{u_{i_1},\cdots,u_{i_{t_R}}\}$, then we define $U\boxplus_{|\cal U|} l \triangleq \{u_{(i_1\boxplus_{|\cal U|} l)},\cdots,u_{(i_n\boxplus_{|\cal U|} l)}\}$ where $l \in \{0,..,t_T+t_R-1\}$.
\begin{lemma}
\label{transmission lemma}
For a set ${\cal T} \in \binom{[K_T]}{t_T}$, consider a set ${\cal U} \in \binom{[K_R]}{t_T+t_R}$ denoted as $\{u_1,...,u_{t_T+t_R}\}$. Pick some $u_j \in {\cal U}$. Let $\mathfrak{U}=\binom{{\cal U}\setminus u_j}{t_R}$. Now choose $U \in \mathfrak{U}$. Consider a set of packets ${\cal P}({\cal T},u_j,{\cal U},U)=\{W_{d_{u_{(j {\boxplus}_{|{\cal U}|} l)}},{\cal T},{({U {\boxplus}_{|{\cal U}|} l}),{\cal U}\setminus \big({(U {\boxplus}_{|{\cal U}|} l ) \cup \{u_{(j {\boxplus}_{|{\cal U}|} l)}\} }\big)}}: l \in \{0,..,t_T+t_R-1\}\}$. Then there is a valid round of transmissions in which ${\cal P}({\cal T},u_j,{\cal U},U)$ can participate in.
\end{lemma}
\begin{IEEEproof}
The set ${\cal P}({\cal T},u_j,{\cal U},U)$ contains $t_T+t_R$ packets. Consider a packet $W_{d_{u_{(j {\boxplus}_{|{\cal U}|} l)}},{\cal T},{({U {\boxplus}_{|{\cal U}|} l}),{\cal U}\setminus \big({(U {\boxplus}_{|{\cal U}|} l ) \cup \{u_{(j {\boxplus}_{|{\cal U}|} l)}\} }\big)}}$ demanded by receiver $u_{(j {\boxplus}_{|{\cal U}|} l)}$. Following the defined caching scheme, it is easy to check that the packet is cached at the set of $t_R$ receivers $({U {\boxplus}_{|{\cal U}|} l})$. Since each packet in the set ${\cal P}$ is cached at $t_R$ receivers, there exists a valid round of transmission which delivers these packets $\{W_{d_{u_{(j {\boxplus}_{|{\cal U}|} l)}},{\cal T},{({U {\boxplus}_{|{\cal U}|} l}),{\cal U}\setminus \big({(U {\boxplus}_{|{\cal U}|} l ) \cup \{u_{(j {\boxplus}_{|{\cal U}|} l)}\} }\big)}}: l \in \{0,..,t_T+t_R-1\}\}$ to their respective receivers $\{u_{(j {\boxplus}_{|{\cal U}|} l)} : l \in \{0,..,t_T+t_R-1\} \}$ as in Lemma \ref{ZF}.
\end{IEEEproof}
From the above lemma, it is easy to see that any arbitrary packet $W_{d_{u_j},{\cal T},{\cal R},{\cal R'}}$ can be decoded from the unique round of transmission consisting of the set of packets ${\cal P}\big({\cal T},u_j,(u_j\cup{\cal R} \cup {\cal R'}),{\cal R}\big)$. We thus have Algorithm \ref{trans algorithm} which describes the transmission algorithm.
\begin{algorithm}[htbp]
\caption{Transmission Scheme}
\label{trans algorithm}
\begin{algorithmic}[1]
\Procedure{Transmissions}{}
\For{each ${\cal T} \in \binom{[K_T]}{t_T}$}
\For {each ${\cal U} \in \binom{[K_R]}{t_T+t_R}$}
\State pick $u_j \in {\cal U}$ and let $\mathfrak{U}=\binom{{\cal U}\setminus{u_j}}{t_R}$
\For {each $U \in \mathfrak{U}$}
\State Obtain a valid round of transmission (as in Lemma \ref{ZF}) using the packets $\{W_{d_{u_{(j {\boxplus}_{|{\cal U}|} l)}},{\cal T},{({U {\boxplus}_{|{\cal U}|} l}),{\cal U}\setminus \big({(U {\boxplus}_{|{\cal U}|} l ) \cup \{u_{(j {\boxplus}_{|{\cal U}|} l)}\} }\big)}}: l \in \{0,..,t_T+t_R-1\}\}$
\EndFor
\EndFor
\EndFor
\EndProcedure
\end{algorithmic}
\end{algorithm}
We thus have the theorem which summarizes our results.
\begin{theorem}
Given a caching scheme as described in Section \ref{modified_caching}, we get a transmission scheme with subpacketization level $F=\binom{K_T}{t_T}\binom{K_R}{t_R}\binom{K_R-t_R-1}{t_T-1}$ and $\mathsf{DoF}= t_T+t_R$.
\end{theorem}
\section{A new projective geometry based scheme with low subpacketization}
\label{projective geometry based scheme}
In this section, we obtain a projective geometry based scheme which achieves subexponential subpacketization, however at the cost of lower $\mathsf{DoF}$. In particular, we achieve a $\mathsf{DoF}$ of $\Theta(log_qK_R+K_T)$, with a subpacketization $F=q^{O(K_T+(log_qK_R)^2)}$ for constant cache fractions at transmitters and receivers ($\frac{M_T}{N}\leq \frac{1}{q^{\alpha-1}},\frac{M_R}{N}\leq \frac{1}{q^{\beta-1}}$, for $\alpha,\beta\in{\mathbb Z}^+$) . Towards presenting our new scheme, we first review some basic concepts from projective geometry.
\vspace{-0.3cm}
\subsection{Review of projective geometries over finite fields \cite{hirschfeld1998projective}}
Let $a,b,q\in \mathbb{Z}^+$ such that $q$ is a prime power. Let ${\mathbb F}_q^{a}$ be $a$-dim vector spaces over a finite field ${\mathbb F}_q$. Consider an equivalence relation on ${\mathbb F}_q^a\setminus\{\boldsymbol{0}\}$ (where $\boldsymbol{0}$ represents the zero vector) whose equivalence classes are $1$-dim subspaces (without $\boldsymbol{0}
$) of ${\mathbb F}_q^a$. The $(a-1)$-dim \textit{projective space} over ${\mathbb F}_q$ is denoted by $PG_q(a-1)$ and is defined as the set of these equivalence classes.
For $b\in [a]$, let $PG_q(a-1,b-1)$ denote the set of all $b$-dim subspaces of ${\mathbb F}_q^{a}$.
From Chapter $3$ in \cite{hirschfeld1998projective}. it is known that $|PG_q(a-1,b-1)|$ is equal to the \textit{Gaussian-binomial coefficient} $\gbinom{a}{b}$, where
$
\begin{bmatrix}a\\b\end{bmatrix}_q
=\frac{(q^{a}-1)\hdots(q^{a-b+1}-1)}{(q^{b}-1)\hdots(q-1)}
$ (where $a\geq b$).
In fact, $\gbinom{a}{b}$ gives the number of $b$-dim subspaces of any $a$-dim vector space over ${\mathbb F}_q$. Further, by definition, $\gbinom{a}{0}=1.$
Let $\mathbb{C} \triangleq \{C : C \in PG_q(a-1,0) \}$. Let $\theta(a)$ denotes the number of distinct $1$-dim subspaces of $\mathbb{F}_q^a$. Therefore $\theta (a) = |\mathbb{C}|=\gbinom{a}{1}= \frac{q^a -1}{q-1}$.
The following lemma and corollary of \cite{hari_SPAWC} will be used to prove the statements in forth coming subsections.
\begin{lemma} \cite{hari_SPAWC} \label{no of sets of LI 1D spaces}
Let $k,a,b \in \mathbb{Z}^+$ such that $1\leq a+b\leq k$. Consider a $k$-dim vector space $V$ over ${\mathbb F}_q$ and a fixed $a$-dim subspace $A$ of $V$. The number of distinct (un-ordered) $b$-sized sets $\{C_1,C_2,\cdots,C_b\}$ such that $C_i\in \mathbb{C}, \forall i\in[b]$ and $(A\oplus C_1 \oplus C_2 \oplus \cdots \oplus C_b) \in PG_q(k-1,a+b-1)$ is
$\frac{\prod\limits_{i=0}^{b-1}(\theta(k)-\theta(a+i))}{b!}$.
\end{lemma}
\begin{corollary} \cite{hari_SPAWC} \label{no of 1D spaces outside a hyper space}
Consider two subspaces $A,A'$ of a $k$-dim vector space $V$ over ${\mathbb F}_q$ such that $A'\subseteq A, dim(A)=a, dim(A')=a-1$. The number of distinct $C\in \mathbb{C}$ such that $A'\oplus C=A$ is $q^{a-1}$.
\end{corollary}
We now proceed to construct a new coded caching scheme for interference channel using projective geometry.
\subsection{Projective geometry based caching scheme} \label{our scheme B subsection}
Consider $k_t,m_t,l_t,k_r,m_r,l_r,q \in \mathbb{Z}^+$ such that $q$ is a prime power, $k_t\geq m_t+l_t$ and $k_r\geq m_r+l_r+\gbinom{m_t+1}{1}$. Consider $k_t$-dim vector space ${\mathbb F}_q^{k_t}$ and $k_r$-dim vector space ${\mathbb F}_q^{k_r}$. Let
\begin{align*}
\mathbb{C}_t \triangleq \{C : C \in PG_q(k_t-1,0) \}. \\
\mathbb{C}_r \triangleq \{C : C \in PG_q(k_r-1,0) \}.
\end{align*}
Therefore $\mathbb{C}_t,\mathbb{C}_r$ represents set of all distinct $1$-dim subspaces of ${\mathbb F}_q^{k_t}$ and ${\mathbb F}_q^{k_r}$ respectively.
Let $L_t$ be a fixed $(l_t-1)$-dim subspace of ${\mathbb F}_q^{k_t}$. Consider the following sets of subspaces of ${\mathbb F}_q^{k_t}$, where each such subspace contains $L_t$.
\begin{align*}
\mathbb{U} &\triangleq \{U \in PG_q(k_t-1,l_t-1): L_t\subseteq U \}. \text{(indexes Tx's)}.\\
\mathbb{P}_t &\triangleq \{P_t \in PG_q(k_t-1,m_t+l_t-1): L_t\subseteq P_t \}.\\
\mathbb{X}_t & \triangleq \left\{\{U_1,U_2,\cdots,U_{m_t+1}\}: \forall U_i \in \mathbb{U}, \sum\limits_{i=1}^{m_t+1}U_i \in \mathbb{P}_t \right\}.
\end{align*}
Let $L_r$ be a fixed $(l_r-1)$-dim subspace of ${\mathbb F}_q^{k_r}$. Consider the following sets of subspaces of ${\mathbb F}_q^{k_r}$, where each such subspace contains $L_r$.
\begin{align*}
\mathbb{V} &\triangleq \{V \in PG_q(k_r-1,l_r-1): L_r\subseteq V \}.\text{(indexes Rx's)}. \\
\mathbb{P}_r &\triangleq \{P_r \in PG_q(k_r-1,m_r+l_r-1): L_r\subseteq P_r \}.\\
\mathbb{X}_r & \triangleq \left\{\{V_1,V_2,\cdots,V_{m_r+1}\}: \forall V_i \in \mathbb{V}, \sum\limits_{i=1}^{m_r+1}V_i \in \mathbb{P}_r \right\}.
\end{align*}
Let $\{W_i, \forall i\in [N]\}$ be the set of $N$ files available at the library. Each transmitter has capacity to store $M_T$ number of files and each receiver has capacity to store $M_R$ number of files.
Let $\mathbb{U},\mathbb{V}$ denote the set of transmitters and receivers respectively. Therefore $K_T=|\mathbb{U}|$ and $K_R=|\mathbb{V}|$. During the caching phase files are divided into subfiles which are represented as $W_{i,X_t,X_r}: i\in [N], X_t \in \mathbb{X}_t, X_r\in \mathbb{X}_r$.
Let $F_T=|\mathbb{X}_t|$, $F_R=|\mathbb{X}_r|$. Thus $F_C=F_T F_R$ is the subpacketization in the caching phase.
We now present an algorithm for our projective geometry based caching scheme in Algorithm \ref{caching algorithm}.
\begin{algorithm}
\caption{Caching scheme}
\label{caching algorithm}
\begin{algorithmic}[1]
\Procedure{Placement Phase}{}
\For{each $i\in [N]$}
\State Split $W_i$ into $\{W_{i,X_t,X_r}:X_t\in\mathbb{X}_t,X_r\in\mathbb{X}_r\}$.
\EndFor
\For {each $U \in \mathbb{U}$}
\State Transmitter $U$ caches the subfiles $W_{i,X_t,X_r}, \forall i\in [N], \forall X_r \in \mathbb{X}_r$ and $ \forall X_t\in \mathbb{X}_t$ such that $U\subseteq \sum\limits_{U_i \in X_t}U_i$.
\EndFor
\For {each $V \in \mathbb{V}$}
\State Receiver $V$ caches the subfiles $W_{i,X_t,X_r}, \forall i\in [N], \forall X_t \in \mathbb{X}_t$ and $ \forall X_r\in \mathbb{X}_r$ such that $V\subseteq \sum\limits_{V_i \in X_r}V_i$.
\EndFor
\EndProcedure
\end{algorithmic}
\end{algorithm}
We now find the values of $K_T,K_R,F_T,F_R,t_T,t_R$.
Consider an arbitrary subfile $W_{i,X_t,X_r}$.
Let $t_T,t_R$ represents the number of transmitters and receivers this subfile is cached respectively.
It is easy to see that,
$t_T=|\{U:U \subseteq \sum\limits_{U_i\in X_t}U_i \}|$ and $t_R=|\{V:V \subseteq \sum\limits_{V_i\in X_r}V_i \}|$.
\begin{lemma}
\label{K_T,K_R,t_T,t_R,F expressions}
\begin{itemize}
\item $K_T =|\mathbb{U}| = \gbinom{k_t-l_t+1}{1}.$
\item $K_R =|\mathbb{V}| = \gbinom{k_r-l_r+1}{1}.$
\item $F_C=F_T F_R$ \text{, where}
\begin{itemize}
\item $F_T=|\mathbb{X}_t| = \frac{q^{\frac{m_t(m_t+1)}{2}}}{(m_t+1)!}\prod\limits_{i=0}^{m_t}\gbinom{k_t-l_t+1-i}{1}.$
\item $F_R=|\mathbb{X}_r| = \frac{q^{\frac{m_r(m_r+1)}{2}}}{(m_r+1)!}\prod\limits_{i=0}^{m_r}\gbinom{k_r-l_r+1-i}{1}.$
\end{itemize}
\item $t_T = \gbinom{m_t+1}{1}.$
\item $t_R = \gbinom{m_r+1}{1}.$
\end{itemize}
\end{lemma}
\begin{IEEEproof}
Finding $K_T=|\mathbb{U}|$:
Finding $|\mathbb{U}|$ is equivalent to counting the number of distinct $ C \in \mathbb{C}_t$ which gives distinct $L_t \oplus C \in \mathbb{U}$.
By Lemma \ref{no of sets of LI 1D spaces} and Corollary \ref{no of 1D spaces outside a hyper space} we have,
$K_T=\frac{\theta(k_t)-\theta(l_t-1)}{q^{l_t-1}} = \frac{q^{k_t}-q^{l_t-1}}{q^{l_t-1}(q-1)}=\frac{q^{k_t-l_t+1}-1}{q-1}= \gbinom{k_t-l_t+1}{1}$.
Similarly we can prove $K_R$.
Finding $|\mathbb{X}_t|$:
Finding $|\mathbb{X}_t|$ is equivalent to counting the number of distinct sets $ \{C_1,C_2,\cdots, C_{m_t+1}\}$ (such that $ C_i\in \mathbb{C}_t~\forall i\in [m_t+1]$ and $L_t\oplus C_1 \oplus C_2 \oplus \cdots \oplus C_{m_t+1} \in \mathbb{P}_t $) which gives distinct $\{L_t\oplus C_1,L_t\oplus C_2, \cdots ,L_t\oplus C_{m_t+1}\} \in \mathbb{X}_t$.
By Lemma \ref{no of sets of LI 1D spaces} we have, the number of distinct sets $ \{C_1,C_2,\cdots,C_{m+1}\}$, such that $ C_i\in \mathbb{C}_t ~(\forall i\in [m_t+1])$ and $L_t\oplus C_1 \oplus C_2 \cdots \oplus C_{m_t+1} \in \mathbb{P}_t $, is
$\frac{\prod\limits_{i=0}^{m_t}(\theta(k_t)-\theta(l_t-1+i))}{(m_t+1)!}$. It is easy to check that $\{L_t\oplus C_1,L_t\oplus C_2, \cdots ,L_t\oplus C_{m_t+1}\} \in \mathbb{X}_t$. By Corollary \ref{no of 1D spaces outside a hyper space} we have, the number of distinct $C\in \mathbb{C}_t$ such that $L_t\oplus C= U$ for some fixed $U\in \mathbb{U}$ is $q^{l_t-1}$. Therefore for each $\{L_t\oplus C_1,L_t\oplus C_2, \cdots ,L_t\oplus C_{m_t+1}\} \in \mathbb{X}_t$ there exist $(q^{l_t-1})^{m_t+1}=q^{(m_t+1)(l_t-1)}$ distinct $\{C_1',C_2',\cdots, C_{m_t+1}'\}$ (where $C_i'\in \mathbb{C}_t~ \forall i\in [m_t+1]$) such that $L_t\oplus C_i=L_t\oplus C_i', \forall i\in [m_t+1]$.
Therefore we can write
\begin{align*}
|\mathbb{X}_t|&=\frac{\prod\limits_{i=0}^{m_t}(\theta(k_t)-\theta(l_t-1+i))}{(m_t+1)!~ q^{(m_t+1)(l_t-1)}}\\
&= \frac{\prod\limits_{i=0}^{m_t}(q^{k_t}-q^{l_t-1+i})}{(m_t+1)!~ q^{(m_t+1)(l_t-1)}~(q-1)^{(m_t+1)}}\\
&= \frac{q^{(m_t+1)(l_t-1)}\left(\prod\limits_{i=0}^{m_t}q^i\right)\left(\prod\limits_{i=0}^{m_t}(q^{k_t-l_t+1-i}-1)\right)}{(m_t+1)!~ q^{(m_t+1)(l_t-1)}~(q-1)^{(m_t+1)}} \\
&= \frac{q^{\frac{m_t(m_t+1)}{2}}}{(m_t+1)!}\prod\limits_{i=0}^{m_t}\gbinom{k_t-l_t+1-i}{1}.
\end{align*}
Similarly we can prove $|\mathbb{X}_r|$.
Finding $t_T$: Consider an arbitrary subfile $W_{i,X_t,X_r}$. We have
$t_T=|\{U:U \subseteq \sum\limits_{U_i\in X_t}U_i \}|$.
We know that $\sum\limits_{U_i\in X_t}U_i =P_t$ for some $P_t\in \mathbb{P}_t$. Now finding $t_T$ is equivalent to counting the number of distinct $C\in \mathbb{C}_t$, which gives distinct $L_t \oplus C \in \mathbb{U}$ such that $L_t \oplus C\subseteq P_t$. By Lemma \ref{no of sets of LI 1D spaces} and Corollary \ref{no of 1D spaces outside a hyper space} we have, $t_T=\frac{\theta(m_t+l_t)-\theta(l_t-1)}{q^{(l_t-1)}} = \frac{q^{(m_t+l_t)}-q^{(l_t-1)}}{q^{(l_t-1)}(q-1)}= \frac{q^{(m_t+1)}-1}{q-1}=\gbinom{m_t+1}{1} $. Similarly we can prove $t_R$.
\end{IEEEproof}
\subsection{Projective geometry based delivery scheme}
We now present our delivery scheme for the caching scheme presented above. This delivery scheme is essentially inspired from Section \ref{modified1}. We first split the demanded subfiles into packets, and then transmit the missing packets by coding them across multiple rounds of transmissions, each of which are obtained according to Lemma \ref{ZF}. Towards this end, we consider the following sets.
$\mathbb{D} \triangleq \{D \in PG_q(k_r-1,l_r+t_T-3): L_r\subseteq D \}.$
$\mathbb{Y} \triangleq \left\{\{V_1,V_2,\cdots,V_{t_T-1}\}: \forall V_i \in \mathbb{V}, \sum\limits_{i=1}^{t_T-1}V_i \in \mathbb{D} \right\}.$
$\mathbb{E} \triangleq \{E \in PG_q(k_r-1,m_r+l_r+t_T-1): L_r\subseteq E \}.$
$\mathbb{Z} \triangleq \left\{\{V_1,\cdots,V_{m_r+t_T+1}\}: \forall V_i \in \mathbb{V}, \sum\limits_{i=1}^{m_r+t_T+1}V_i \in \mathbb{E} \right\}.$
Note that for an arbitrary demanded subfile $W_{d_{V_a},X_t,X_r}$ for some $X_r\in \mathbb{X}_r$, we have that $(V_a+\sum\limits_{V_i\in X_r}V_i)$ is a $(m_r+l_r+1)$-dim subspace containing $L_r$. We now present the subfile-splitting technique and the delivery scheme in Algorithm \ref{transmission algorithm}. For the purpose of the key step (Step 11) of the Algorithm, we need to define a notation. For some $Z=\{V_1,\hdots,V_{m_r+t_T+1}\}\in{\mathbb Z}$, let $S=\{V_{i_1},...,V_{i_{m_r+1}}\}\subset Z$. Then, for any $l\in\{0,1,\hdots,m_r+t_T\}$, define
\[
S\boxplus_{|Z|}l\triangleq\{V_{(i_1\boxplus_{|Z|}l)},...,V_{(i_{m_r+1}\boxplus_{|Z|}l)}\}.
\]
\begin{algorithm}
\caption{Transmission scheme}
\label{transmission algorithm}
\begin{algorithmic}[1]
\Procedure{Splitting of demanded subfiles}{demand of receiver $V$ is represented as $W_{d_{V}}$}
\For{each $V\in{\mathbb V}$, each $X_t\in \mathbb{X}_t$ and each $X_r\in{\mathbb X}_r$}
\State Split $W_{d_V,X_t,X_r}$ into $\{W_{d_V,X_t,X_r,Y}:Y\in\mathbb{Y}$ and $\{V\}\cup X_r\cup Y\in \mathbb{Z}\}$.
\EndFor
\EndProcedure
\Procedure{Transmissions}{}
\For{each $X_t \in \mathbb{X}_t$}
\For {each $Z \in \mathbb{Z}$}
\State pick $V_j \in \mathbb{Z}$ and let $\mathfrak{S}=\binom{Z\setminus{V_j}}{m_r+1}$
\For {each $S \in \mathfrak{S}$}
\State Obtain a valid round of transmission (as in Lemma \ref{ZF}) using the packets ${\cal P}(X_t,V_j,Z,S)=\{W_{d_{V_{j {\boxplus}_{|Z|} l}},X_t,{({S {\boxplus}_{|Z|} l}),Z\setminus \big({(S {\boxplus}_{|Z|} l ) \cup \{V_{j {\boxplus}_{|Z|} l}\} }\big)}}: l \in \{0,..,m_r+t_T\}\}$
\EndFor
\EndFor
\EndFor
\EndProcedure
\end{algorithmic}
\end{algorithm}
The idea of the splitting of a subfile $W_{d_V,X_t,X_r}$ into packets $\{W_{d_V,X_t,X_r,Y}\}$ is that $Y$ denotes the receivers where the packet $W_{d_V,X_t,X_r,Y}$ is to be zero-forced. We now verify that the delivery scheme indeed delivers all the missing packets to all receivers. We do this in two steps.
\begin{itemize}
\item \textit{Verification of the Step 11 of Algorithm:} Note that the the packet $W_{d_{V_{j {\boxplus}_{|Z|} l}},X_t,{({S {\boxplus}_{|Z|} l}),Z\setminus \big({(S {\boxplus}_{|Z|} l ) \cup \{V_{j {\boxplus}_{|Z|} l}\} }\big)}}$ is cached at the $(m_r+1)$ of the $(m_r+t_T+1)$ receivers indexed by $Z$ given by the indices $S\boxplus_{|Z|}l.$ Thus Lemma \ref{ZF} can apply here, and the collection of packets in Step 11 can participate in a valid transmission.
\item \textit{Ensuring all missing packets are delivered:} Consider a demanded packet $W_{d_{V_a},X_t,X_r,Y}$. This packet is delivered in the unique transmission round consisting of the packets ${\cal P}(X_t,V_a,V_a\cup Y\cup X_r,X_r).$ Thus all missing packets of the demanded files are delivered to the corresponding receivers.
\end{itemize}
We now obtain the parameter $F_P$ which is the number of packets into which each demanded subfile is divided.
\begin{lemma}
\label{F_P expression}
$F_P=\frac{q^{\frac{(t_T+2m_r+2)(t_T-1)}{2}}}{(t_T-1)!}\prod\limits_{i=1}^{t_T-1}\gbinom{k_r-m_r-l_r-i}{1}$.
\end{lemma}
\begin{IEEEproof}
Consider an arbitrary demanded subfile $W_{d_{V_a},X_t,X_r}$ for some $X_r\in \mathbb{X}_r$. Finding $F_P$ is equivalent to counting the number of distinct sets $Y\in \mathbb{Y}$ such that $\{V_a\} \cup X_r \cup Y \in \mathbb{Z}$. This is equivalent to counting number distinct sets $ \{C_1,C_2,\cdots, C_{t_T-1}\}$ (such that $ C_i\in \mathbb{C}_r~\forall i\in [t_T-1]$ and $V_a+\sum\limits_{V_i\in X_r}V_i + \sum\limits_{i=1}^{t_T-1}C_i \in \mathbb{E} $) which gives distinct $\{V_a\}\cup X_r \cup \{L_r\oplus C_1,L_r\oplus C_2, \cdots ,L_r\oplus C_{t_T-1}\} \in \mathbb{Z}$.
By following the similar proof technique, as employed in Lemma \ref{K_T,K_R,t_T,t_R,F expressions} we can write,
\begin{align*}
F_P&=\frac{\prod\limits_{i=0}^{t_T-2}(\theta(k_r)-\theta(m_r+l_r+1+i))}{(t_T-1)!~ q^{(t_T-1)(l_r-1)}}\\
&=\frac{\prod\limits_{i=1}^{t_T-1}(\theta(k_r)-\theta(m_r+l_r+i))}{(t_T-1)!~ q^{(t_T-1)(l_r-1)}}\\
&= \frac{1}{(t_T-1)!~ q^{(t_T-1)(l_r-1)}}\prod\limits_{i=1}^{t_T-1}\frac{q^{k_r}-q^{m_r+l_r+i}}{q-1}\\
&= \frac{q^{(m_r+l_r)(t_T-1)}\prod\limits_{i=1}^{t_T-1}q^{i}}{(t_T-1)!~ q^{(t_T-1)(l_r-1)}}\prod\limits_{i=1}^{t_T-1}\frac{q^{k_r-m_r-l_r-i}-1}{q-1}\\
&= \frac{q^{(m_r+1)(t_T-1)}q^{\frac{(t_T-1)(t_T)}{2}}}{(t_T-1)!}\prod\limits_{i=1}^{t_T-1}\gbinom{k_r-m_r-l_r-i}{1}\\
&= \frac{q^{\frac{(t_T+2m_r+2)(t_T-1)}{2}}}{(t_T-1)!}\prod\limits_{i=1}^{t_T-1}\gbinom{k_r-m_r-l_r-i}{1}.
\end{align*}
\end{IEEEproof}
\begin{remark}
The subpacketization of the proposed scheme is $F= F_C F_P=F_T F_R F_P$.
\end{remark}
We now summarize our scheme parameters in the following theorem. The proof of the theorem follows from the earlier lemmas and discussions in this section. The $\mathsf{DoF}$ follows from the observation that in Algorithm \ref{transmission algorithm}, each round serves $m_r+t_T+1$ packets to the same number of distinct receivers.
\begin{theorem}
Given a coded caching scheme as described in Section \ref{projective geometry based scheme}, with parameters $K_T,K_R,\frac{M_T}{N}=\frac{t_T}{K_T}, \frac{M_R}{N}=\frac{t_R}{K_R}, F_C,F_P$ (as per Lemma \ref{K_T,K_R,t_T,t_R,F expressions}, \ref{F_P expression}) we get a transmission scheme with subpacketization level $F=F_C F_P$ and $\mathsf{DoF}= m_r+t_T+1$.
\end{theorem}
Finally in Table \ref{table 1}, we give a numerical comparison of the schemes proposed in Sections \ref{modified1}, \ref{projective geometry based scheme} with the scheme given in \cite{hypercube_interference} (for the exact expressions for the parameters of \cite{hypercube_interference}, please refer to \cite{hypercube_interference}).
Note that we don't compare with \cite{interferencemanagement} because our scheme in Section \ref{modified1} achieves same $\mathsf{DoF}$ as \cite{interferencemanagement} with smaller subpacketization. The $-$ term indicates that there is no scheme in \cite{hypercube_interference} with the specific choice of parameters.
From the table it is clear that our scheme presented in Section \ref{projective geometry based scheme} outperforms the state of the art schemes in terms of the subpacketization with reduced $\mathsf{DoF}$.
\newcolumntype{g}{>{\columncolor{Gray}}c}
\begin{table}
\setlength{\tabcolsep}{4pt}
\renewcommand{\arraystretch}{1.5}
\centering
\resizebox{8.85cm}{!} {
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
$K_T$ & $K_R$ & $\frac{M_T}{N}$ & $\frac{M_R}{N}$ & DoF & DoF & $F$ & $F$ & $F$ \\
& & & & \cite{hypercube_interference}, \ref{modified1} & \ref{projective geometry based scheme} & \cite{hypercube_interference}& \ref{modified1} & \ref{projective geometry based scheme} \\
\hline
7 & 31 & .428 & .097 & 6 & 5 & $3\times 10^{6}$ & $10^{7}$ &
$2\times 10^{6}$\\
\hline
7 & 63 & .428 & .111 & 10 & 6 & - & $10^{13}$ & $7\times 10^{8}$
\\
\hline
7 & 127 & .428 & .055 & 10 & 6 & - & $10^{16}$ & $4\times 10^{10}$
\\
\hline
13 & 364 & .308 & .011 & 8 & 6 & $3\times 10^{16}$ & $ 10^{18}$ &
$2\times 10^{13}$
\\
\hline
40 & 364 & .10 & .011 & 8 & 6 & $3\times 10^{18}$ & $10^{20}$ &
$3\times 10^{14}$
\\
\hline
40 & 1093 & .10 & .004 & 8 & 6 & $7\times 10^{21}$ & $ 10^{24}$ &
$8\times 10^{16}$
\\
\hline
\end{tabular}
}
\caption{For some specific values of $K_T,K_R,\frac{M_T}{N}, \frac{M_R}{N}$, we compare the results of \cite{hypercube_interference} with the schemes presented in Sections \ref{modified1}, \ref{projective geometry based scheme} }
\label{table 1}
\end{table}
\section{Analysis of the scheme}
\label{analysis}
In this section, we analyse the asymptotic behaviour of subpacketization$(F)$ and $\mathsf{DoF}$ of the scheme presented in Section \ref{projective geometry based scheme} for large $K_R$ (this represents a practical regime of interest) by upper bounding $\frac{M_T}{N}, \frac{M_R}{N}$ by some constant.
We use the following lemma.
\begin{lemma}\cite{PK}
\label{approximations}
Let $a,b,f \in \mathbb{Z}^+$ and $q$ is some prime power. Then,
\begin{align}
\label{eqn qbinom inequality 1}
&q^{(a-b)b}&\leq &\gbinom{a}{b} \leq & q^{(a-b+1)b}\\
\label{eqn qbinom inequality 2}
&q^{(a-f-1)b} &\leq &\frac{\gbinom{a}{b}}{\gbinom{f}{b}}\leq & q^{(a-f+1)b}
\end{align}
\end{lemma}
From Lemma \ref{K_T,K_R,t_T,t_R,F expressions} we have,
$K_T = \gbinom{k_t-l_t+1}{1}$ and $K_R = \gbinom{k_r-l_r+1}{1}$.
We first upper bound $\frac{M_T}{N}$ and $\frac{M_R}{N}$ by some constants. Consider,
\[\frac{M_T}{N} =\frac{t_T}{K_T}=\frac{\gbinom{m_t+1}{1}}{\gbinom{k_t-l_t+1}{1}}\stackrel{(\ref{eqn qbinom inequality 2})}{\leq} q^{m_t-k_t+l_t+1}.\]
To upper bound $\frac{M_T}{N}$ by a constant, let $k_t-m_t-l_t=\alpha$, for some constant $\alpha \in \mathbb{Z}^{+}$. We thus have $\frac{M_T}{N} \leq \frac{1}{q^{\alpha-1}}$.
Similarly to upper bound $\frac{M_R}{N}$ by a constant, let $k_r-m_r-l_r=\beta$, for some constant $\beta \in \mathbb{Z}^{+}$.
We thus have $\frac{M_R}{N} \leq \frac{1}{q^{\beta-1}}$.
Now by using $(\ref{eqn qbinom inequality 1})$ we have,
$q^{k_t-l_t} \leq K_T \leq q^{k_t-l_t+1}$.
So we have,
\begin{align}
\label{eqn inequality k_t-l_t}
{\log_{q}{K_T}}-1 \leq (k_t-l_t)\leq {\log_{q}{K_T}}
\end{align}
Similarly,
\begin{align}
\label{eqn inequality k_r-l_r}
{\log_{q}{K_R}}-1 \leq (k_r-l_r)\leq {\log_{q}{K_R}}
\end{align}
The $F_T,K_T$ expressions are same that of $F,K$ in \cite{haribhavanaprasad} respectively. Therefore from \cite{haribhavanaprasad} (Section V) we have $F_T=q^{O\left((\log_{q}{K_T})^2\right)}$. Similarly $F_R=q^{O\left((\log_{q}{K_R})^2\right)}$.
Now we will analyze the asymptotics of $F_P$.
From Lemma \ref{F_P expression} we have,
\begin{align*}
F_P &= \frac{q^{(m_r+1)(t_T-1)}\left(\prod\limits_{i=1}^{t_T-1}q^{i}\right)}{(t_T-1)!}\prod\limits_{i=1}^{t_T-1}\gbinom{k_r-m_r-l_r-i}{1} \\
& \stackrel{(\ref{eqn qbinom inequality 1})}{\leq} \frac{q^{(m_r+1)(t_T-1)}\left(\prod\limits_{i=1}^{t_T-1}q^{i}\right)}{(t_T-1)!}\prod\limits_{i=1}^{t_T-1} q^{k_r-m_r-l_r-i} \\
&= \frac{q^{(m_r+1)(t_T-1)}q^{(k_r-m_r-l_r)(t_T-1)}}{(t_T-1)!}.
\end{align*}
Hence,
\begin{align}
\label{eqn F_P ineuqlity}
F_P &\leq \frac{q^{(k_r-l_r+1)(t_T-1)}}{(t_T-1)!}.
\end{align}
From Lemma \ref{K_T,K_R,t_T,t_R,F expressions} we have,
$t_T = \gbinom{m_t+1}{1}$. Now we can write,
$q^{m_t}\stackrel{(\ref{eqn qbinom inequality 1})}{\leq} t_T$.
From this we can write, $\frac{1}{(t_T-1)!}\leq \frac{1}{(q^{m_t}-1)!}$.
Also from (\ref{eqn inequality k_t-l_t}) we have,
${\log_{q}{K_T}}-1 \leq m_t+\alpha$.
which can be written as
$q^{{\log_{q}{K_T}}-1-\alpha}-1 \leq q^{m_t}-1$.
Therefore we have,
\begin{align*}
\frac{1}{(t_T-1)!} \leq \frac{1}{( q^{m_t}-1)!}\leq \frac{1}{(q^{{\log_{q}{K_T}}-1-\alpha}-1)!}.
\end{align*}
Consider,
$t_T -1 \stackrel{(\ref{eqn qbinom inequality 1})}{\leq} q^{m_t+1}-1$.
But
$m_t \stackrel{(\ref{eqn inequality k_t-l_t})}{\leq} {\log_{q}{K_T}} -\alpha$.
Therefore we have,
$t_T -1 \leq q^{{\log_{q}{K_T}} -\alpha +1} -1$.
Also we have ,
\begin{align*}
(k_r-l_r+1) \stackrel{(\ref{eqn inequality k_r-l_r})}{\leq} {\log_{q}{K_R}}+1.
\end{align*}
By using these inequalities in (\ref{eqn F_P ineuqlity}) we have,
\[F_P\leq \frac{q^{({\log_{q}{K_R}}+1)(q^{{\log_{q}{K_T}} -\alpha +1} -1)}}{(q^{{\log_{q}{K_T}}-1-\alpha}-1)!}.\]
Now by using Stirling's approximation and simple manipulations we have,
$F_P=q^{O\left(K_T\log_{q}{K_R}\right)}$.
Therefore,
\begin{align*}
F&=F_T F_R F_P \\ &=q^{O\left((\log_{q}{K_T})^2\right)}q^{O\left((\log_{q}{K_R})^2\right)}q^{O\left(K_T\log_{q}{K_R}\right)} \\
&=q^{O\left(K_T+(\log_{q}{K_R})^2\right)}.
\end{align*}
Now we will analyse the asymptotics of $\mathsf{DoF}=m_r+t_T+1$ .
From \ref{eqn inequality k_r-l_r} we have
\begin{align}
\label{eqn m_r+1 ineuqality}
{\log_{q}{K_R}}-\beta \leq (m_r+1)\leq {\log_{q}{K_R}}-\beta +1
\end{align}
By using \ref{eqn qbinom inequality 1} we have,
$q^{m_t}\leq t_T \leq q^{m_t+1}$.
By using \ref{eqn inequality k_t-l_t} we have, \[{\log_{q}{K_T}}-1-\alpha \leq m_t \leq {\log_{q}{K_T}}-\alpha.\]
From this we can write
$q^{{\log_{q}{K_T}}-1-\alpha}\leq q^{m_t}$ and $q^{m_t+1} \leq q^{{\log_{q}{K_T}}+1-\alpha}$.
So we have,
\begin{align}
\label{eqn t_T inequality}
q^{{\log_{q}{K_T}}-1-\alpha}\leq t_T \leq q^{{\log_{q}{K_T}}+1-\alpha}.
\end{align}
Hence by using (\ref{eqn m_r+1 ineuqality}) and (\ref{eqn t_T inequality}) we have
${\log_{q}{K_R}}-\beta +q^{{\log_{q}{K_T}}-1-\alpha}\leq m_r+1+t_T \leq {\log_{q}{K_R}}-\beta +1+q^{{\log_{q}{K_T}}+1-\alpha}.$
Therefore
$\mathsf{DoF} = \Theta(log_qK_R+K_T)$.
\bibliographystyle{IEEEtran}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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.. _apiref:
*************
API Reference
*************
.. highlight:: c
configuration.h
===============
The configuration module reponsible for managing the configuration of xy.
Any change to xy's runtime configuration (RC) file will cause the configuration
to be reloaded (see :func:`config_reinit`).
.. type:: CONFIG
The type made available after a call to :func:`config_init`. It contains all
the necessary configuration options xy uses while running.
.. function:: CONFIG * config_init()
Initializes xy's configuration. Returns the configuration or NULL on failure.
This function makes :type:`CONFIG` available for use.
.. function:: void config_terminate()
The configuration module's shutdown hook.
.. function:: void write_default_config()
Writes the default xy configuration to the xy runtime configuration path.
.. function:: void config_reinit()
Reinitializes the configuration from the xy runtime configuration path.
util.h
======
A collection of utility functions meant to stand alone from other xy modules.
.. function:: char * rc_path()
Returns the path of xy's runtime configuration. The returned string should be
freed by the caller.
.. function:: bool streq(const char *, const char *)
Returns ``true`` if the strings are equal or null, ``false`` otherwise.
.. function:: char * trim(char *)
Removes leading and trailing whitespace from the string and returns a
pointer starting from the first non-whitespace character. The returned
string is contained in the same memory as the argument.
Example::
char *conststr = " test string ";
char *dup = strdup(conststr);
char *trimmed = trim(dup); // trimmed: "test string"
free(dup);
.. function:: void dump_stack(uint num_frames)
Dumps *num_frames* stack frames to stderr.
.. function:: void parse_command(char *cmd, char **argv)
Tokenizes *cmd* by spaces into tokens placed in *argv*.
Example::
char *command = "some string here";
char *dup = strdup(command);
char *argv[3];
parse_command(dup, argv);
// argv[0]: "some"
// argv[1]: "string"
// argv[2]: "here"
free(dup);
.. function:: void exec(const char * cmd)
Executes the supplied command by calling *execvp*. The command will be
parsed before the call is made (see `parse_command`).
inotify.h
=========
The inotify module. XY uses the inotify module to react to changes occurring on
the filesystem.
types
-----
.. type:: in_fd
The inotify file descriptor made available after a call to
:func:`xy_inotify_init`. This file descriptor is suitable for system calls
like *select* and *epoll*.
functions
---------
.. function:: void xy_inotify_init()
Initializes xy's inotify module. This function makes :type:`in_fd`
available for use, which may be -1 on failure.
.. function:: void xy_inotify_reinit()
Reinitializes the inotify module. This function makes a new :type:`in_fd`
available for use.
.. function:: void xy_inotify_terminate()
The inotify module's shutdown hook.
.. function:: void xy_inotify_read()
Drains the inotify event queue of all events.
|
{
"redpajama_set_name": "RedPajamaGithub"
}
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Thomas Grey, 15:e baron Grey de Wilton, född ett okänt år, död 9 juli 1614, var en engelsk aristokrat och soldat. Han var son till Arthur Grey, 14:e baron Grey de Wilton och kämpade mot den spanska armadan 1588. Grey tog över titeln som lord efter sin far 1593 och ansåg sig själv tillhöra puritanismen. Han var angelägen att få leda ett regemente och när Robert Devereux, 2:e earl av Essex åkte till Irland i mars 1599 följde Grey med honom. Essex bad Grey att svära sin trohet till honom och avsäga sig all vänskap med Robert Cecil, 1:e earl av Salisbury, något Grey inte kunde göra då han stod i tacksamhetsskuld till Salisbury. Efter detta behandlade Essex (och hans vän Henry Wriothesley, 3:e earl av Southampton) Grey som en fiende. Grey kände sig sviken och begav sig från Irland i maj 1600.
Grey stred under Moritz av Oranien och deltog i slaget vid Nieuwpoort i Belgien den 2 juli 1600, men var tillbaka i London igen tidigt året därpå. Elisabet I av England bad att Grey skulle skipa fred med Southampton, men istället attackerade Grey Southampton och skickades till Fleetfängelset. Han släpptes dock kort därefter och när Essex försökte göra uppror mot drottning Elisabet var Grey med och kämpade mot honom. Upproret misslyckades och vid rättegången mot Essex den 19 februari 1601 satt Grey med bland de som dömde honom till döden; när Greys namn lästes upp i domstolen ska Essex ha skrattat föraktfullt åt honom.
Efter Elisabets död den 24 mars 1603 blev Jakob I av England kung. Jakob lovade att upphöra med förföljelserna av katolikerna, men dessa löften infriades inte och de besvikna katolikerna började nu fundera på att genomföra ett antal sammansvärjningar mot honom. Grey var delaktig i en sådan, där deras planer gick ut på att fånga in Jakob och konvertera honom till katolicismen samt ta kontroll över Towern och göra William Watson till Lord Keeper. De inblandade bestämde sig för att träffas i Greenwich den 24 juni 1603, men sammansvärjningen var ett totalt misslyckande. Några jesuiter hade nämligen avslöjat deras planer för myndigheterna och flera av de inblandade arresterades; Grey fångades in i juli samma år i Sluis. Under rättegången mot dem den 15 november 1603 dömdes de till döden, men Grey benådades kort innan sin avrättning. Han fängslades i Towern och efter 11 år i fångenskap avled han där den 9 juli 1614. Grey blev den sista baron Grey de Wilton att ha existerat.
Referenser
Tryckta källor
Brittiska brottslingar
Personer i England under 1500-talet
Personer i England under 1600-talet
Avlidna 1614
Män
Personer under Tudoreran
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"We explored thousands of scenarios in our simulation model, and we did not find even one plausible circumstance under which that strategy would be sufficient to contain an outbreak. You simply cannot catch up with this virus," Paltiel noted.
Most universities are considering less extensive COVID-control measures than those called for by Paltiel and his colleagues.
For example, University of California, San Diego plans to test students once a month, coupled with other monitoring methods designed to catch the coronavirus before an outbreak occurs, said Natasha Martin, an infectious disease modeler with the university.
"That was based on our simulations indicating that level of testing would allow us to identify outbreaks early. We want to detect outbreaks early, and we feel like we have the test capacity and the situation in place to do that," Martin said during the same HD Live! interview.
Despite their differences, both Paltiel and Martin agreed that some level of proactive testing is needed to stop the virus in its tracks and prevent a campus shutdown.
"If you're trying to understand why that might be, a lot of it has to do with the asymptomatic spread of the disease and the fact that it can be transmitted by asymptomatic so-called 'silent spreaders,'" Paltiel said. "It's very hard to play catch-up with this virus. You want to get out ahead of it, and the best way to do that is to screen with high frequency."
One innovative approach UC San Diego plans to use is to test the wastewater flowing from college dorms and residences for signs of COVID-19 coronavirus, Martin said.
The university hopes that wastewater monitoring "can give us an early signal in terms of if there's viral activity or infections within a specific dorm, and then we can go in and do more widespread testing to identify those infections and act on them," Martin explained.
Colleges need to plan for outbreaks, closures
Paltiel said his approach overlaps with Martin's in the "view that one has to hit this early, hit it hard, recognize the silent spreaders, test frequently, stop an outbreak in its tracks before it starts, recognize that nothing you do is going to be perfect and therefore you're going to have to both set aside sufficient isolation capacity and have an exit strategy.
"My difficulty is that many schools are planning as if everything that could possibly go wrong is going to go right," Paltiel said. "They aren't giving themselves enough wiggle room."
Both experts said it is inevitable that there will be COVID-19 infections on campus, so colleges have to be equipped with plans to control infections and, in the worst case, efficiently and safely shut down.
"I think every university needs to plan an exit strategy and have a criterion that says this is when we are out of control and we really need to move to a plan where we shut the school down," Paltiel said.
One potential positive is that an early pilot run of UC San Diego's plan found solid student buy-in for testing and COVID control, Martin said.
"One of the very strong messages we learned from them is they wanted to know this testing was part of a broader public health program, that it was something everybody was engaged with to keep the campus as safe as possible and to enable some amount of in-person activities on campus," Martin said. "They were less interested in the idea they would know their status or that they would be testing for themselves. They really wanted the message to be about the broader public good, which it is."
Given that student support, it's now the job of universities that are reopening to promote strategies and options that will keep students engaged on-campus in safe ways, Paltiel said.
"The schools can't just hand out a mask and a bottle of hand sanitizer and maybe a leaflet and wag their fingers at the students and say 'behave,'" Paltiel said. "That's just setting them up to fail. It's also setting them up to be scapegoated at the end if something does go awry.
"We're the grownups in the room," Paltiel continued. "We have a responsibility to provide them with resources and with imaginative, compassionate, realistic low-risk options to stay socially connected, to get the human interaction that they richly deserve and that they need, and to make the on-campus social event sufficiently attractive that it discourages them from wanting to head down the road to do Jell-O shots in the basement of the frat house, which is what I really want to avoid."
Colleges also need to pay attention to the surrounding community when managing their COVID-control strategies, Martin added.
Colleges "need to be thinking about what their capacity is on campus, but also what the situation is in their local community and really assessing it week by week to ensure they feel they are in the right position to reopen," Martin said.
UC San Diego has more about its campus reopening strategy.
SOURCES: David Paltiel, Ph.D., M.B.A., professor, health policy, Yale School of Public Health, New Haven, Conn.; Natasha Martin, D.Phil./Ph.D., infectious disease modeler, UC San Diego
Amenorrhea in Teens
Helping Your Teen Manage Asthma
Teen Health Quiz
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\section{Introduction}
\label{section:introduction}
This work is devoted to the linear wave equation in a heterogeneous medium with multiple highly varying length-scales. We are looking for an unknown wave function $u^\eps$ that fulfills the equation
\begin{align}
\nonumber\label{wave-equation-strong} \partial_{tt} u^\eps(x,t) - \nabla \cdot \left( a^\eps(x) \nabla u^\eps(x,t) \right) &= F(x,t) \qquad \mbox{in } \Omega \times(0,T], \\
u^\eps(x,t) &= 0 \hspace{48pt} \mbox{on } \partial \Omega \times [0,T], \\
\nonumberu^\eps(x,0) = f(x) \quad \mbox{and} \quad \partial_t u^\eps(x,0) &= g(x) \hspace{35pt} \mbox{in } \Omega.
\end{align}
Here, $\Omega$ denotes the medium, $[0,T] \subset \mathbb{R}^+$ the relevant time interval, $a^\eps$ the wave speed, $F$ a source term and $f$ and $g$ the initial conditions for the wave and its time derivative respectively. The parameter $\varepsilon$ is an abstract parameter which simply indicates that a certain quantity is subject to rapid variations on a very fine scale (relative to the extension of $\Omega$). The parameter $\varepsilon$ can be seen as a measure for the minimum wave length of these variations. Precisely, we assume that for a function $z^{\varepsilon}$, $\| z^\varepsilon\|_{H^s(\Omega)}$ is large for $s>1$ and cannot be approximated with a FE function on a coarse grid while the typical fine mesh needed to approximate such functions is computationally too expensive.
However, we stress that we do not assign a particular value or meaning to $\varepsilon$ in this work. Due to the fast variations in the data functions, which take place at a scale that is very small compared to the total size of the medium, these problems are typically referred to as {\it multiscale problems}. Equations such as (\ref{wave-equation-strong}) with a multiscale character arise in various fields such as material sciences, geophysics, seismology or oceanography. For instance, the propagation and reflection of seismic waves can be used to determine the structure and constitution of subsurface formations. In particular, it is necessary in order to locate petroleum reservoirs in earth's crust.
Trying to solve the multiscale wave equation (\ref{wave-equation-strong}) with a
direct computation, using e.g. finite elements or finite differences, exceeds typically the possibilities of today's super computers. The reason is that the computational mesh needs to resolve all variations of the coefficient matrix $a^\eps$, which leads to extremely high dimensional solution spaces and hence linear systems of tremendous size that need to be solved at every time step.
In order to tackle this issue, numerical homogenization can be applied. The term numerical homogenization refers to a wide set of numerical methods that are based on replacing the multiscale problem (\ref{wave-equation-strong}) by an effective/upscaled/homogenized equation which is of the same type as the original equation, but which has no longer multiscale properties (the fine scale is \quotes{averaged out}). Hence, it can be solved in lower dimensional spaces with reduced computational costs. The obtained approximations yield the effective macroscopic properties of $u^\eps$ (i.e. they are good $L^2$-approximations of $u^\eps$). Multiscale methods that were specifically designed for the wave equation, can be e.g. found in \cite{AbG11,EHR11,KoM06,OwZ08,VMK05}. In Section \ref{subsection-surbey-multiscale-methods} we give a detailed overview on these approaches.
In this paper, we will present a multiscale method for the wave equation which does neither require structural assumptions such as a scale separation nor does it require regularity assumptions on $a^\eps$. We will not exploit any higher space regularity than $H^1$. Furthermore, it is not necessary to solve expensive global elliptic fine scale problems in a pre-process (sometimes referred to as the 'one-time-overhead', cf. \cite{JEG10,JiE12,OwZ08}). Our method is based on the following consideration: the $L^2$-projection $P_{L^2}$ of the (unknown) exact solution $u^\eps$ into a coarse finite element space is assumed to be a good approximation to an (unknown) homogenized solution. Furthermore, the $L^2$-projection $P_{L^2}(u^\eps)$ can be well approximated in a low dimensional finite element space. If we can derive an equation for $P_{L^2}(u^\eps)$, all computations can be performed in the low dimensional space and are hence cheap. This approach fits into the framework of the Localized Orthogonal Decomposition (LOD) initially proposed in \cite{MaP14} and further developed in \cite{HeP13,HeM14}. The idea of the framework is to decrease the dimension of a high dimensional finite element space by splitting it into the direct sum of a low dimensional space with high $H^1$-approximation properties and a high dimensional remainder space with negligible information. The splitting is based on an orthogonal decomposition with respect to an energy scalar product. In this work we will pick up this concept, since the remainder space in the splitting is nothing but the kernel of the $L^2$-projection.
The general setting of this paper is established in Section \ref{section:motivation}, where we also motivate the method. In Section \ref{main-section-LOD} we introduce the space discretization that is required for formulating the method in a rigorous way. In Section \ref{section-main-results-and-misc} we state our main results and we give a survey on other multiscale strategies.
These main results
are proved in Section \ref{section:proofs}. Finally, numerical experiments confirming our theoretical results are presented in Section \ref{section-numerics}.
\section{Motivation - Numerical homogenization by $L^2$-projection}
\label{section:motivation}
In the following, we consider the wave equation (\ref{wave-equation-strong}) in weak formulation, i.e. we seek $u^\eps \in L^2(0,T;H^1_0(\Omega))$ and $\partial_{tt} u^\eps \in L^2(0,T;H^{-1}(\Omega))$ such that for all $v \in H^1_0(\Omega)$ and a.e. $t > 0$
\begin{align}
\label{wave-equation-weak}
\nonumber\langle \partial_{tt} u^\eps (\cdot,t), v \rangle + \left( a^\eps \nabla u^\eps (\cdot,t), \nabla v \right)_{L^2(\Omega)} &= \left( F (\cdot,t), v \right)_{L^2(\Omega)},\\
\left( u^\eps( \cdot , 0 ) , v \right)_{L^2(\Omega)} &= \left( f , v \right)_{L^2(\Omega)},\\
\nonumber \left( \partial_t u^\eps( \cdot , 0 ) , v \right)_{L^2(\Omega)} &= \left( g , v \right)_{L^2(\Omega)}.
\end{align}
Here, the dual pairing is understood as $\langle L, v \rangle=L(v)$ for $L\in H^{-1}(\Omega)$ and $v \in H^1_0(\Omega)$. In the following, we make use of the shorthand notation $W^{m,p}(H^s_0):=W^{m,p}(0,T;H^s_0(\Omega))$ for $0\le m < \infty$, $1\le p \le \infty$ and $0\le s \le 1$. We denote $H^0_0(\Omega):=L^2(\Omega)$ and $W^{0,p}:=L^p$.
In order to guarantee the existence of a unique solution of the system (\ref{wave-equation-strong}), we make the following assumptions:
\begin{itemize}
\item[(H0) $\bullet$] $\Omega\subset\mathbb{R}^{d}$, for $d=1,2,3$, denotes a bounded Lipschitz domain with a piecewise polygonal boundary;
\item the data functions fulfill $F \in L^2(0,T;L^2(\Omega))$, $f \in H^1_0(\Omega)$ and $g \in L^2(\Omega)$;
\item the matrix-valued function $a^\eps\in [L^\infty(\Omega)]^{d\times d}_{ \mbox{\tiny \rm sym} }$ that describes the propagation field is symmetric and it is uniformly bounded and positive definite, i.e. $a^\eps \in \mathcal{M}(\alpha,\beta,\Omega)$ for $\beta\geq\alpha>0$. Here, we denote
\begin{eqnarray}\label{e:spectralbound}
\lefteqn{\mathcal{M}(\alpha,\beta,\Omega) :=} \\
\nonumber&\hspace{0pt}&\{ a \in [L^\infty(\Omega)]^{d\times d}_{ \mbox{\tiny \rm sym} }| \hspace{5pt}
\alpha |\xi|^2 \le a(x) \xi \cdot \xi \le \beta |\xi|^2 \enspace\text{for all } \xi \in \mathbb{R}^d \mbox{ and almost all }x\in \Omega\}.
\end{eqnarray}
\end{itemize}
Under the assumptions listed in (H0) there exists a unique weak solution $u^\eps$ of the wave equation (\ref{wave-equation-strong}) with $\partial_t u^\eps \in L^2(0,T;L^2(\Omega))$. Furthermore, $u^\eps$ is regular in time, in the sense that $u^\eps \in C^0(0,T;H^1_0(\Omega))$ and $\partial_t u^\eps \in C^0(0,T;L^2(\Omega))$. This result can be e.g. found in \cite[Chapter 3]{LiM72a}.
In addition to the above assumptions, we also implicitly assume that the wave speed $a^\eps$ has rapid variations on a very fine scale which need to be resolved by an underlying fine grid. The dimension of the resulting finite element space (for the spatial discretization) is hence very large. The method proposed in the subsequent sections aims to reduce the computational cost that is associated with solving the discretized wave equation in this high dimensional finite element space.
In order to simplify the notation, we define
\begin{align}
\label{definition:problem:bilinearform}b^{\eps}(v,w):=\int_{\Omega} a^\eps \nabla v \cdot \nabla w \qquad \mbox{for } v,w \in H^1_0(\Omega).
\end{align}
We next motivate a multiscale method for the wave equation and discuss the framework of our approach. All the subsequent discussion will be later rigorously justified by a general convergence proof. We are interested in finding a {\it homogenized} or {\it upscaled} approximation of $u^\eps$. In engineering applications this can be a function describing the macroscopically measurable properties of $u^\eps$ and from an analytical perspective it can be defined as a suitable limit of $u^\eps$ for $\varepsilon \rightarrow 0$ (see Section \ref{subsection:homogenization:setting} below for more details).
Since $u^\eps$ is a continuous function in $t$, we restrict our considerations
to a fixed time $t$. Hence, we leave out the time dependency in the notation and denote e.g. $u^\eps = u^\eps (\cdot,t)$.
Let $\mathcal{T}_H$ denote a given coarse mesh and let $V_H \subset H^1_0(\Omega)$ denote a corresponding coarse finite dimensional subspace of $H^1_0(\Omega)$ that is sufficiently accurate to obtain good $L^2$-approximations. To quantify what we mean by \quotes{sufficiently accurate}, let $P_H$ denote the $L^2$-projection of $H^1_0(\Omega)$ on $V_H$, i.e. for $v \in H^1_0(\Omega)$ the projection $P_H(v)\in V_H$ fulfills
\begin{align}
\label{L2-projection}\int_{\Omega} P_H (v) w_H = \int_{\Omega} v w_H \qquad \mbox{for all } w_H \in V_H.
\end{align}
We assume that
$$\| u^\eps - P_H(u^\eps) \|_{L^2(\Omega)} \le \delta_H ,$$
where $\delta_H$ is a given small tolerance. Let us denote $u_H :=P_{H}(u^\eps) \in V_H$.
Obviously, the $L^2$-projection will average out all small oscillations that cannot be seen on the coarse grid $\mathcal{T}_H$ (in this sense the projection homogenizes $u^\eps$). By definition, $u_H$ is the best approximation of $u^\eps$ in $V_H$ with respect to the $L^2$-norm. Next, we want to find a macroscopic equation that is fulfilled by $u_H$.
Since $\nabla u_H$ does not approximate $\nabla u^\eps$, we are interested in a corrector $Q(u_H)$, such that
\begin{align*}
\int_{\Omega} a^\eps (\nabla u_H + \nabla Q(u_H) ) \cdot \nabla v_H = \int_{\Omega} a^\eps \nabla u^\eps \cdot \nabla v_H \qquad \mbox{for all } v_H \in V_H,
\end{align*}
or in a symmetric formulation
\begin{align}
\label{preliminary-method} \int_{\Omega} a^\eps (\nabla u_H + \nabla Q(u_H) ) \cdot (\nabla v_H + \nabla Q(v_H) ) = \int_{\Omega} a^\eps \nabla u^\eps \cdot (\nabla v_H + \nabla Q(v_H) )
\end{align}
for all $v_H \in V_H$. A suitable corrector operator $Q$ needs to fulfill two properties:\\
1. $Q(u_H)$ must be in the kernel of the $L^2$-projection $P_H$ in order to preserve the $L^2$-best-approximation property
\begin{align}
\label{trax-L2} \int_{\Omega} u^\eps v_H = \int_{\Omega} u_H \hspace{2pt} v_H = \int_{\Omega} (u_H + Q(u_H)) v_H \qquad \mbox{for all } v_H \in V_H.
\end{align}
2. It must incorporate the oscillations of $a^\eps$. A natural way to achieve this is to make the ansatz
\begin{align*}
\int_{\Omega} a^\eps (\nabla u_H + \nabla Q(u_H) ) \cdot \nabla v_h = 0,
\end{align*}
where the test function $v_h$ should be in $H^1_0(\Omega)$, but with the constraint $v_h \in \mbox{kern}(P_H)$. The constraint is sufficient to make the problem well posed (solution space and test function space are identical).
In summary, we have the following strategy if $u^\eps$ is a known function: find $u_H \in V_H$ that fulfills equation (\ref{preliminary-method}) and where for a given $v_H \in V_H$ the corrector $Q(v_H) \in \mbox{kern}(P_H)$ solves $\int_{\Omega} a^\eps (\nabla v_H + \nabla Q(v_H) ) \cdot \nabla v_h = 0$ for all $v_h \in \mbox{kern}(P_H)$. Observe that this $u_H$ fulfills indeed $u_H = P_H(u^\eps)$ as desired, because (by equation (\ref{trax-L2})) the function $e:=u^\eps - u_H - Q(u_H)$ is in the kernel of $P_H$. Hence for all $v_H \in V_H$
\begin{align*}
\int_{\Omega} u_H v_H = \int_{\Omega} (u_H + Q(u_H)) v_H = \int_{\Omega} (u_H + Q(u_H) + e) v_H = \int_{\Omega} u^\eps v_H,
\end{align*}
which means just $u_H=P_{H}(u^\eps)$. Consequently, we also have the estimate
\begin{align}
\label{abstract-delta-est}\| u^\eps - u_H \|_{L^2(\Omega)} \le \delta_H.
\end{align}
The only remaining problem is that we do not know the term $\int_{\Omega} a^\eps \nabla u^\eps \cdot (\nabla v_H + \nabla Q(v_H))$ on the right hand side of (\ref{preliminary-method}). However, we know that
\begin{align*}
\int_{\Omega} a^\eps \nabla u^\eps \nabla v = \int_{\Omega} F v - \int_{\Omega} \partial_{tt} u^\eps \hspace{2pt} v.
\end{align*}
If the solution $u^\eps$ is sufficiently regular then $\partial_{tt} u^\eps$ is well approximated by $\partial_{tt} u_H = P_{H}(\partial_{tt} u^\eps)$ in the sense of (\ref{abstract-delta-est}).
This suggests to replace $\partial_{tt} u^\eps$ by $\partial_{tt} u_H$ and to solve the approximate problem to find $\bar{u}_H \in V_H$ with
\begin{eqnarray*}
\int_{\Omega} a^\eps (\nabla \bar{u}_H + \nabla Q(\bar{u}_H)) \cdot (\nabla v_H + \nabla Q(v_H)) &=&\int_{\Omega}(F-\partial_{tt} \bar{u}_H)
(\nabla v_H + \nabla Q(v_H))\\
&\approx& \int_{\Omega} a^\eps \nabla u^\eps (\nabla v_H + \nabla Q(v_H))
\end{eqnarray*}
for all $v_H \in V_H$.
Note that the above presented strategy is not yet a ready-to-use method, since the exact computation of the corrector $Q$ involves global fine scale problems. In order to overcome this difficulty, a localization of $Q$ is required together with a suitable fine scale discretization. The final method is presented in the Section \ref{section:multiscale:methods}. Before we can formulate the method, we introduce a suitable fully discrete space-discretization.
\section{The LOD method for the wave equation}
\label{main-section-LOD}
In this section we propose a space discretization for localizing the fine scale computations in the previously described ansatz. For that purpose, we make use of the tools of the Localized Orthogonal Decomposition (LOD) that was introduced in \cite{MaP14} (see also \cite{EGM13,HeM14,HeP13,HMP14,MaP15} for related works) and that originated from the Variational Multiscale Method (VMM, cf. \cite{HFM98,LaM07,Mal11}). We then derive our multiscale method for the wave equation with a continuum of scales.
\subsection{Spatial discretization}
\label{subsection-LOD}
The spatial discretization involves two discretization levels. On the one hand, we have a quasi-uniform coarse mesh on $\Omega$ that is denoted by $\mathcal{T}_H$. $\mathcal{T}_H$ consists either of conforming shape regular simplicial elements or of conforming shape regular quadrilateral elements. The elements are denoted by $K \in \mathcal{T}_H$ and the coarse mesh size $H$ is defined as the maximum diameter of an element of $\mathcal{T}_H$. On the other hand we have a fine mesh that is denoted by $\mathcal{T}_h$. It also consists of conforming and shape regular elements. Furthermore, we assume that $\mathcal{T}_h$ is obtained from an arbitrary refinement of $\mathcal{T}_H$,
with the additional requirement that
$h \le (H/2)$, where $h$ denotes the maximum diameter of an element of $\mathcal{T}_h$.
In practice we usually have $h \ll H$.
In particular, $\mathcal{T}_h$ needs to be fine enough to capture all the oscillations of $a^\eps$.
In contrast, the coarse mesh is only required to provide accurate $L^2$-approximations.
For $\mathcal{T}=\mathcal{T}_H,\mathcal{T}_h$ we denote
\begin{align}
\label{definition:finite:element:spaces}P_1(\mathcal{T}) &:= \{v \in C^0(\omega) \;\vert \;\forall K\in\mathcal{T},v\vert_K \text{ is a polynomial of total degree}\leq 1\}, \\
\nonumber Q_1(\mathcal{T}) &:= \{v \in C^0(\omega) \;\vert \;\forall K\in\mathcal{T},v\vert_K \text{ is a polynomial of partial degree}\leq 1\}.
\end{align}
With this, we define the classical coarse Lagrange finite element space $V_H$ by $V_H:=P_1(\mathcal{T}_h)\cap H^1_0(\Omega)$ for a simplicial mesh and by $V_H:=Q_1(\mathcal{T}_h)\cap H^1_0(\Omega)$ for a quadrilateral mesh. The fine scale space $V_h$ is defined in the same way.
Subsequently, we will make use of the notation $a \lesssim b$ that abbreviates $a\leq Cb$, where $C$ is a constant that can dependent on $d$, $\Omega$, $\alpha$, $\beta$ and interior angles of the coarse mesh, but not on the mesh sizes $H$ and $h$. In particular it does not depend on the possibly rapid oscillations in $a^\eps$. We write $a\lesssim_T b$ if $C$ is allowed to further depend on $T$ and the data functions $F$, $f$ and $g$.
The set of the interior Lagrange points (interior vertices) of the coarse grid $\mathcal{T}_H$ is denoted by $\mathcal{N}_H$. For each node $z \in \mathcal{N}_H$ we let $\Phi_z \in V_H$ denote the corresponding nodal basis function that fulfills $\Phi_z(z)=1$ and $\Phi_z(y)=y$ for all $y \in \mathcal{N}_H \setminus \{ z \}$.
In the next step, we define the kernel of the $L^2$-projection (\ref{L2-projection}) restricted to $V_h$ in a slightly alternative way. Recall that this kernel was required as the solution space for the corrector problems discussed in Section \ref{section:motivation}. However, from the computational point of view it is more suitable to not work with the $L^2$-projection directly, since it involves to solve a system of equations in order to verify if an element is in the kernel. For that reason, we subsequently express kern$(P_H\vert_{V_h})$ equivalently by means of a weighted Cl\'ement-type quasi-interpolation operator $I_H$ (cf. \cite{Car99}) that is defined by
\begin{align}
\label{def-weighted-clement} I_H : H^1_0(\Omega) \rightarrow V_H,\quad v\mapsto I_H(v):=
\sum_{z \in \mathcal{N}_{H}}
v_z \Phi_z \quad \text{with }v_z := \frac{(v,\Phi_z)_{L^2(\Omega)}}{(1,\Phi_z)_{L^2(\Omega)}}.
\end{align}
With that, we define
$$W_h := \mbox{\rm {Ker}}(I_H\vert_{V_h}).$$
Indeed the space $W_h$ is the previously discussed kernel of the $L^2$-projection. This claim can be easily verified: if $I_H(v_h)=0$ for an element $v_h \in V_h$, then we have by the definition of $I_H$ that $(v_h,\Phi_z)_{L^2(\Omega)}=0$ for all $z \in \mathcal{N}_H$. Since $\Phi_z$ is just the nodal basis of $V_H$ we have $(v_h,\Phi_H)_{L^2(\Omega)}=0$ for any $\Phi_H \in V_H$. Hence $W_h=\mbox{Ker}(I_H\vert_{V_h}) \subset \mbox{Ker}(P_H\vert_{V_h})$. The reverse inclusion $\mbox{Ker}(P_H\vert_{V_h}) \subset W_h$ is straightforward. In particular, we have the splitting $V_h = \mbox{Im}(P_H\vert_{V_h}) \oplus \mbox{Ker}(P_H\vert_{V_h}) = {\mbox{Im}(I_H\vert_{V_h}) \oplus \mbox{Ker}(I_H\vert_{V_h})}= V_H \oplus W_h$.
We can now define the optimal corrector $Q_{h,{\Omega}}:V_H \rightarrow W_h$ (in the sense of Section \ref{section:motivation}) as the solution $Q_{h,{\Omega}}(v_H) \in W_h$ of
\begin{align}
\label{projection-orthogonal}
b^{\eps}( v_H + Q_{h,{\Omega}}(v_H) , w_h ) = 0 \qquad \mbox{for all } w_h \in W_h,
\end{align}
where $b^{\eps}(\cdot,\cdot)$ is defined in (\ref{definition:problem:bilinearform}). However, finding $Q_{h,{\Omega}}(v_H)$ involves a problem in the whole fine space $V_h$ and is therefore very expensive. For this purpose, we wish to localize the corrector $Q_{h,{\Omega}}$ by element patches.
For $k\in \mathbb{N}$, we define patches $U_k(K)$ that consist of a coarse element $K \in \mathcal{T}_H$ and $k$-layers of coarse elements around it. More precisely $U_k(K)$ is defined iteratively by
\begin{equation}\label{def-patch-U-k}
\begin{aligned}
U_0(K) & := K, \\
U_k(K) & := \cup\{T\in \mathcal{T}_H\;\vert\; T\cap U_{k-1}(K)\neq\emptyset\}\quad k=1,2,\ldots .
\end{aligned}
\end{equation}
Practically, we will later see that we only require small values of $k$ (typically $k=1,2,3$).
With that, we define the localized corrector operator in the following way:
\begin{definition}[Localized Correctors]
\label{definition:localized:ms:space}
For $k\in \mathbb{N}$, $K\in \mathcal{T}_H$ and $U_k(K)$ defined according to (\ref{def-patch-U-k}), we define the localized version of $\mbox{Ker}(P_H\vert_{V_h})$ by
$$W_h(U_k(K)):=\{ w_h \in W_h| \hspace{2pt} w_h=0 \enspace \mbox{in } \Omega \setminus U_k(K) \}.$$
The localized version of the operator (\ref{projection-orthogonal}) can be constructed in the following way. First, for $v_H \in V_H$ find $Q_{h,k}^{K}(v_H) \in {W}_h(U_k(K))$ with
\begin{align}
\label{local-corrector-problem}\int_{U_k(K)} a^\eps \nabla Q_{h,k}^{K}(v_H)\cdot \nabla w_h = - \int_K a^\eps \nabla v_H \cdot \nabla w_h \qquad \mbox{for all } w_h \in W_h(U_k(K)).
\end{align}
Then, the global approximation of $Q_{h,k}$ is defined by
\begin{align}
\label{global-corrector}Q_{h,k}(v_H):=\sum_{K\in \mathcal{T}_H} Q_{h,k}^{K}(v_H).
\end{align}
Observe that if $k$ is large enough so that $U_k(K)=\Omega$ for all $K \in \mathcal{T}_H$ (a case that is only useful for the analysis), we have $Q_{h,k}=Q_{h,{\Omega}}$, where $Q_{h,{\Omega}}$ is the corrector operator introduced in (\ref{projection-orthogonal}).
\end{definition}
\begin{remark}[Splittings of $V_h$]
The space that is spanned by the image of $(I+Q_{h,k})_{\vert V_H}$ is given by
\begin{align}
\label{localized-ms-space}
V_{H,k}^{\ms}:=\{ v_H + Q_{h,k}(v_H)| \hspace{2pt} v_H \in V_H\}.
\end{align}
Furthermore, we denote $V_{H,\Omega}^{\ms}:=\{ v_H + Q_{h,\Omega}(v_H)| \hspace{2pt} v_H \in V_H\}$ for the optimal corrector. This gives us the following splittings of $V_h$:
\begin{align*}
V_h &= V_H \oplus W_h, \qquad \hspace{7pt} \mbox{where } \quad V_H \perp W_h \quad \hspace{7pt} \mbox{w.r.t. } (\cdot , \cdot)_{L^2(\Omega)},\\
V_h &= V_{H,\Omega}^{\ms} \oplus W_h, \qquad \mbox{where } \quad V_{H,\Omega}^{\ms} \perp W_h \quad \mbox{w.r.t. } b^{\eps} (\cdot , \cdot),\\
V_h &= V_{H,k}^{\ms} \oplus W_h.
\end{align*}
\end{remark}
Beside the operator $I_H$ that we defined in (\ref{def-weighted-clement}) other choices of interpolation operators (such as the classical Cl\'ement interpolation) are possible to construct splittings $V_h = V_{H,k}^{\ms} \oplus W_h$ with $V_{H,k}^{\ms} \perp_{b^{\eps} (\cdot , \cdot)} W_h$. If the operator fulfills various standard properties (like interpolation error estimates, $H^1$-stability, etc.; cf. \cite{HMP14} for an axiomatic list) the space $V_{H,k}^{\ms}$ will have similar approximation properties as the multiscale space that we use in this contribution. However, we note that the particular Cl\'ement-type interpolation operator from (\ref{def-weighted-clement}) yields the $L^2$-orthogonality $V_H \perp W_h$ which is typically not the case for other operators. This is central in our approach. In this paper we particularly exploit this feature to show that we obtain higher order convergence rates under the assumption of additional regularity. We also note that the Lagrange-interpolation fails to yield good approximations (cf. \cite{HeP13}).
\begin{remark}
Observe that the solutions $Q_{h,k}(v_H)$ of (\ref{local-corrector-problem}) are well defined by the Lax-Milgram theorem. Furthermore, it is was shown that solutions such as $Q_{h,k}(v_H)$ (with localized source term) decay with exponential speed to zero outside of the support of the source term (cf. \cite{MaP14,HeM14}). More precisely, we will later see that we have an estimate of the type $\| \nabla (Q_{h,k} - Q_{h,\Omega})(v_H) \|_{L^2(\Omega)} \lesssim k^{d/2} \theta^{k} \| \nabla v_H \|_{L^2(\Omega)}$ for a generic constant $0<\theta <1$. Hence, we have exponential convergence in $k$ and small values for $k$ (typically $k=2,3$) can be used to get accurate approximations of $Q_{h,\Omega}$. For small values of $k$, the local problems (\ref{local-corrector-problem}) are affordable to solve, they can be solved in parallel and $Q_{h,k}(\Phi_z)$ is only locally supported for every nodal basis function $\Phi_z \in V_H$.
\end{remark}
\subsection{Semi and fully-discrete multiscale method}
\label{section:multiscale:methods}
Based on the discretization and the correctors defined in Section \ref{subsection-LOD}, we present a semi-discrete multiscale method for the wave equation and state a corresponding a priori error estimate. As an example of a time-discretization, we also present a Crank-Nicolson realization of the method and state a fully discrete space-time error estimate. In order to abbreviate the notation, we define effective/macroscopic bilinear forms for $v_H,w_H \in V_H$ by
\begin{align*}
b_{H,k} (v_H, w_H) &:= b^{\eps}( v_H + Q_{h,k}(v_H), w_H + Q_{h,k}(w_H)) \quad \mbox{and}\\
(v_H, w_H)_{H,k} &:= ( v_H + Q_{h,k}(v_H), w_H + Q_{h,k}(w_H))_{L^2(\Omega)},
\end{align*}
where $b^{\eps}$ is defined in (\ref{definition:problem:bilinearform}).
\subsubsection{Semi-discrete multiscale method}
We can now formulate the method.
\begin{definition}[Semi-discrete multiscale method]
Let $k \in \mathbb{N}$ denote the localization parameter that determines the patch size $U_k(K)$ for $K\in \mathcal{T}_H$ (according to (\ref{def-patch-U-k})) and hence also determines the localized corrector operator $Q_{h,k}$. The semi-discrete approximation $u_{H,k} \in H^2(0,T;V_H)$ (of $u^\eps$ in $L^2$) solves the following system for all $v_H \in V_H$ and $t > 0$
\begin{align}
\label{semi-discrete-lod-equation}
\nonumber ( \partial_{tt} u_{H,k} (\cdot,t), v_H )_{H,k} + b_{H,k}( u_{H,k} (\cdot,t), v_H ) &= \left( F (\cdot,t), v_H + Q_{h,k}(v_H)\right)_{L^2(\Omega)},\\
(u_{H,k}+Q_{h,k}(u_{H,k}))( \cdot , 0 ) &= \pi_{H,k}^{\mbox{\rm \tiny ms}}(f),\\
\nonumber \partial_t (u_{H,k}+Q_{h,k}(u_{H,k}))( \cdot , 0 ) &= P_{H,k}^{\mbox{\rm \tiny ms}}(g),
\end{align}
with the projections $\pi_{H,k}^{\mbox{\rm \tiny ms}}$ and $P_{H,k}^{\mbox{\rm \tiny ms}}$ defined in (\ref{elliptic-projection-pi_Hk}) and (\ref{L2-projection-P_Hk}) below.
\end{definition}
Recall $V_{H,k}^{\ms}$ be the space defined in (\ref{localized-ms-space}). We subsequently define two elliptic projections for $v \in L^{2}(0,T;H^1_0(\Omega))$ and one $L^2$-projection.
\begin{enumerate}
\item The projection $\pi_h : L^{2}(0,T;H^1_0(\Omega)) \rightarrow L^{2}(0,T;V_h)$ is given by:\\
find $\pi_h(v) \in L^{2}(0,T;V_h)$ with
\begin{align}
\label{elliptic-projection-pi_h}b^{\eps} ( \pi_{h}(v)(\cdot,t) , w ) = b^{\eps} ( v(\cdot,t) , w ) \qquad \mbox{for all } w \in V_h, \quad \mbox{for almost every } t \in (0,T).
\end{align}
\item The projection $\pi_{H,k}^{\mbox{\rm \tiny ms}} : L^{2}(0,T;H^1_0(\Omega)) \rightarrow L^{2}(0,T;V_{H,k}^{\ms})$ is given by:\\
find $\pi_{H,k}^{\mbox{\rm \tiny ms}}(v)(\cdot,t) \in V_{H,k}^{\ms}$ with
\begin{align}
\label{elliptic-projection-pi_Hk}b^{\eps}( \pi_{H,k}^{\mbox{\rm \tiny ms}}(v)(\cdot,t) , w ) = b^{\eps} ( v(\cdot,t) , w ) \qquad \mbox{for all } w \in V_{H,k}^{\ms}, \quad \mbox{for almost every } t \in (0,T).
\end{align}
For $U_k(K)=\Omega$, we denote by $\pi_{H,\Omega}^{\mbox{\rm \tiny ms}}$ the above projection mapping from $L^{2}(0,T;H^1_0(\Omega))$ to $L^{2}(0,T;V_{H,\Omega}^{\ms})$.
\item The $L^2$-projection $P_{H,k}^{\mbox{\rm \tiny ms}} : L^{2}(0,T;H^1_0(\Omega)) \rightarrow L^{2}(0,T;V_{H,k}^{\ms})$ is given by:\\
find $P_{H,k}^{\mbox{\rm \tiny ms}}(v)(\cdot,t) \in V_{H,k}^{\ms}$ with
\begin{align}
\label{L2-projection-P_Hk}( P_{H,k}^{\mbox{\rm \tiny ms}}(v)(\cdot,t) , w )_{L^2(\Omega)} = ( v(\cdot,t) , w )_{L^2(\Omega)} \qquad \mbox{for all } w \in V_{H,k}^{\ms}, \quad \mbox{for a.e. } t \in (0,T).
\end{align}
\end{enumerate}
\begin{remark}[Existence and uniqueness]
If assumption (H0) is fulfilled, the system (\ref{semi-discrete-lod-equation}) has a unique solution. This result directly follows from standard ODE theory, after a reformulation of (\ref{semi-discrete-lod-equation}) into a (finite) system of first order ODE's with constant coefficients and applying Duhamel's formula. Due to the Sobolev embedding theorems, we have $u_{H,k} \in C^1([0,T];V_H)$. If additionally $f \in C^0(0,T;L^2(\Omega))$, we even get $u_{H,k} \in C^2([0,T];V_H)$. The corrector $Q_{h,k}(u_{H,k})$ inherits this time regularity.
\end{remark}
\subsubsection{Fully discrete multiscale method}
In this section, for $J\in \mathbb{N}$, we let $\triangle t:=\frac{T}{J}>0$ denote the time step size and we define $t^n:=n \triangle t$ for $n \in \mathbb{N}$.
In order to propose a time
discretization of (\ref{semi-discrete-lod-equation}), we introduce some simplifying notation. First, recall that for every coarse interior node $z \in \mathcal{N}_H$, we denote the corresponding nodal basis function by $\Phi_z$. The total number of these coarse nodes shall be denoted by $N_H$ and we assume that they are ordered by some index set, i.e. $\mathcal{N}_H= \{ z_1, \dots, z_N \}$. With that we define the corresponding stiffness matrix $S_{k} \in \mathbb{R}^{N \times N}$ by the entries
\begin{align*}
(S_{k})_{ij} := b_{H,k}( \Phi_{z_j}, \Phi_{z_i} )
\end{align*}
and the entries of the corrected mass matrix $M_{k} \in \mathbb{R}^{N \times N}$ by
\begin{align*}
(M_{k})_{ij} := ( \Phi_{z_j}, \Phi_{z_i} )_{H,k}.
\end{align*}
The load vectors $G_{k},\bar{f}_{k},\bar{g}_{k} \in \mathbb{R}^{N}$ arising in (\ref{semi-discrete-lod-equation}) are defined by
\begin{align}
\label{def-data-vectors-cn}\nonumber(G_{k})_i(t) &:= \left( F (\cdot,t), \Phi_{z_i} + Q_{h,k}(\Phi_{z_i}) \right)_{L^2(\Omega)},\\
(\bar{f}_{k})_i \mbox{ is such that } \pi_{H,k}^{\mbox{\rm \tiny ms}}(f) &= \sum_{i=1}^N (\bar{f}_{k})_i (\Phi_{z_i} + Q_{h,k}(\Phi_{z_i})),\\
(\bar{g}_{k})_i \mbox{ is such that } P_{H,k}^{\mbox{\rm \tiny ms}}(g) &= \sum_{i=1}^N (\bar{g}_{k})_i (\Phi_{z_i} + Q_{h,k}(\Phi_{z_i})),
\end{align}
with $\pi_{H,k}^{\mbox{\rm \tiny ms}}$ being the elliptic projection on $V_{H,k}^{\ms}$ (see (\ref{elliptic-projection-pi_Hk})) and
$P_{H,k}^{\mbox{\rm \tiny ms}}$ being the $L^2$-projection on $V_{H,k}^{\ms}$ (see (\ref{elliptic-projection-pi_Hk})).
Hence, we can write (\ref{semi-discrete-lod-equation}) as the system: find $\xi_k(t)\in \mathbb{R}^{N}$ with
\begin{align*}
M_k \ddt\xi_k(t) + S_k \xi_k(t) &= G_k(t), \quad \hspace{10pt} \mbox{for } 0<t<T
\end{align*}
and $\xi_k(0)=\bar{f}_{k}$ and $\dt \xi_k(0)=\bar{g}_{k}$. This yields $u_{H,k} (\cdot,t) = \sum_{i=1}^N (\xi_k(t))_i \Phi_{z_i}$.
In order to solve this system we can apply the Newmark scheme.
\begin{definition}[Newmark scheme]
For $n\ge 1$, given initial values $\xi_k^{(0)} \in \mathbb{R}^N$ and $\xi_k^{(1)} \in \mathbb{R}^N$, and given load vectors $G_k^{(n)} \in \mathbb{R}^N$, we
define the Newmark approximation $\xi_k^{(n+1)}$ of $\xi_k(t^{(n+1)})$ iteratively as the solution $\xi_k^{(n+1)} \in \mathbb{R}^{N}$ of
\begin{eqnarray*}
\lefteqn{(\triangle t)^{-2} M_k \left( \xi_k^{(n+1)} -2 \xi_k^{(n)} + \xi_k^{(n-1)} \right)}\\
&+& \frac{1}{2} S_k \left( 2 \hat{\beta} \xi_k^{(n+1)} + (1 - 4 \hat{\beta} + 2 \hat{\gamma} ) \xi_k^{(n)} + (1 + 2\hat{\beta} -2\hat{\gamma}) \xi_k^{(n-1)} \right) = G_k^{(n)}.
\end{eqnarray*}
Here, $\hat{\beta}$ and $\hat{\gamma}$ are given parameters.
An example for an implicit method is given by the choice $\hat{\beta}=1/4$ and $\hat{\gamma}=1/2$, which leads to the classical Crank-Nicolson scheme. Another example is the leap-frog scheme that is obtained for $\hat{\beta}=0$ and $\hat{\gamma}=1/2$. The leap-frog scheme is explicit (up to a diagonal mass matrix which can be obtained by mass lumping).
\end{definition}
As one possible realization, we subsequently consider the case $\hat{\beta}=1/4$ and $\hat{\gamma}=1/2$, i.e. the Crank-Nicolson scheme (see Definition \ref{definition-fully-discrete-method} below). We state a corresponding a priori error estimate and the numerical experiments in Section \ref{section-numerics} are also performed with this method. Before we present the main theorems, let us detail the method by specifying the initial values and the load vectors for the Crank-Nicolson method.
\begin{definition}[Fully-discrete Crank-Nicolson multiscale method]
\label{definition-fully-discrete-method}
As before, let $k \in \mathbb{N}$ denote the localization parameter that determines the patch size $U_k(K)$ for $K\in \mathcal{T}_H$ (according to (\ref{def-patch-U-k})). The load functions $G_{k},\bar{f}_{k},\bar{g}_{k} \in \mathbb{R}^{N}$ are defined according to (\ref{def-data-vectors-cn}) and we denote
$G_k^{(n)}:=\frac{1}{2}( G_k{t^n} + G_k(t^{n-1}) )$ for $t^n=n \triangle t$, $n\ge 1$. Defining $\xi_k^{(0)}:=\bar{f}_{k}$ and $\eta_k^{(0)}:=\bar{g}_{k}$, the approximation $(\xi_k^{(n)},\eta_k^{(n)}) \in \mathbb{R}^N \times \mathbb{R}^N$ in the $n$'th time step is given as the solution of the linear system
\begin{align*}
(\frac{\triangle t^2}{4} S_k + M_k) \eta_k^{(n)} &= (M_k - \frac{\triangle t^2}{4} S_k ) \eta_k^{(n-1)} - \triangle t S_k \xi_k^{(n-1)} + \triangle t G_k^{(n)}
\end{align*}
and $\xi_k^{(n)}:= \frac{\triangle t}{2} \eta_k^{(n)} +\frac{\triangle t}{2} \eta_k^{(n-1)} + \xi_k^{(n-1)}$.
With that, the Crank-Nicolson approximation of (\ref{semi-discrete-lod-equation}) is defined as the piecewise linear function $u_{H,\triangle t, k}$ with
\begin{align}
\label{fully-discrete-lod-approximation}u_{H,\triangle t, k}(\cdot,t) := \sum_{i=1}^N \left( \frac{t^{n+1}-t}{\triangle t} (\xi_k^{(n)})_i + \frac{t-t^{n}}{\triangle t} (\xi_k^{(n+1)})_i \right)\Phi_{z_i} \qquad \mbox{for } t \in [t^{n},t^{n+1}].
\end{align}
\end{definition}
\begin{remark}
Existence and uniqueness of $(\xi_k^{(n)},\eta_k^{(n)})$ in Definition \ref{definition-fully-discrete-method} is obvious
since the system matrix $(\frac{\triangle t^2}{4} S_k + M_k)$ has only positive eigenvalues.
\end{remark}
\section{Main results and survey of other multiscale strategies}
\label{section-main-results-and-misc}
\subsection{Convergence results under minimal assumptions on the initial data}
\label{subsection:homogenization:setting}
In this section we recall some fundamental results concerning the homogenization of the wave equation and relate it to our multiscale method.
In this framework, we are able to give a convergence result between the homogenized solution and our multiscale approximation with the weakest possible assumptions: no scale separations, no assumptions on the initial data except the one that guaranties existence and uniqueness of the original problem \eqref{wave-equation-strong}.
The essential question of classical homogenization is the following: if $(a^\eps)_{\varepsilon>0}$ represents a sequence of coefficients and if we consider the corresponding sequence of solutions $(u^\eps)_{\varepsilon>0}$ of (\ref{wave-equation-weak}), does $u^\eps$ converge in some sense to a $u^0$ that we can characterize in a simple way? The hope is that $u^0$ fulfills some equation (the homogenized equation) which is cheap to solve since it does no longer involve multiscale features (which were averaged out in the limit process for $\varepsilon \rightarrow 0$). With the abstract tool of $G$-convergence it is possible to answer this question:
\begin{definition}[$G$-convergence]\label{def-G-convergence}
A sequence $(a^\eps)_{\varepsilon >0}\subset \mathcal{M}(\alpha,\beta,\Omega)$ (i.e. with uniform spectral bounds in $\varepsilon$) is said to be $G$-convergent to $a^0 \in \mathcal{M}(\alpha,\beta,\Omega)$ if for all $F \in H^{-1}(\Omega)$ the sequence of solutions $v^{\varepsilon} \in H^1_0(\Omega)$ of
\begin{align*}
\int_{\Omega} a^\eps \nabla v^{\varepsilon} \cdot \nabla v = F(v) \qquad \mbox{for all } v \in H^1_0(\Omega)
\end{align*}
satisfies $v^{\varepsilon} \rightharpoonup v^0$ weakly in $H^1_0(\Omega)$, where $v^0 \in H^1_0(\Omega)$ solves
\begin{align*}
\int_{\Omega} a^0 \nabla v^0 \cdot \nabla v = F(v) \qquad \mbox{for all } v \in H^1_0(\Omega).
\end{align*}
\end{definition}
One of the main properties of $G-$convergence is the following compactness result \cite{Spa68,Tar77}: let $(a^\eps)_{\varepsilon >0}$ be a sequence of matrices in $\mathcal{M}(\alpha,\beta,\Omega)$, then there exists a subsequence $(a^{\varepsilon '})_{\varepsilon '>0}$ and a matrix $a^0\in \mathcal{M}(\alpha,\beta,\Omega)$ such that $(a^{\varepsilon '})_{\varepsilon '>0}$ $G-$converges to $a^0$. For the wave equation, we have the following result obtained in \cite[Theorem 3.2]{BFM92}:
\begin{theorem}[Homogenization of the wave equation]
\label{theorem:homogenization:wave:equation} Let assumptions (H0) be fulfilled and let the sequence of symmetric matrices $(a^\eps)_{\varepsilon >0}\subset \mathcal{M}(\alpha,\beta,\Omega)$ be $G$-convergent to some $a^0 \in \mathcal{M}(\alpha,\beta,\Omega)$. Let $u^\eps \in L^{\infty}(0,T;H^1_0(\Omega))$ denote the solution of the wave equation (\ref{wave-equation-weak}). Then it holds
\begin{align*}
u^\eps &\rightharpoonup u^0 \quad \mbox{weak-}\ast \mbox{ in } L^{\infty}(0,T,H^1_0(\Omega)),\\
\partial_t u^\eps &\rightharpoonup \partial_t u^0 \quad \mbox{weak-}\ast \mbox{ in } L^{\infty}(0,T,L^2(\Omega))
\end{align*}
and where $u^0 \in L^2(0,T;H^1_0(\Omega))$ is the unique weak solution of the homogenized problem
\begin{align}
\label{homogenized-equation-weak}
\nonumber\langle \partial_{tt} u^0 (\cdot,t), v \rangle + \left( a^0 \nabla u^0 (\cdot,t), \nabla v \right)_{L^2(\Omega)} &= \left( F (\cdot,t), v \right)_{L^2(\Omega)} \qquad \mbox{for all } v \in H^1_0(\Omega) \mbox{ and } t > 0,\\
\left( u^0( \cdot , 0 ) , v \right)_{L^2(\Omega)} &= \left( f , v \right)_{L^2(\Omega)}
\qquad \hspace{21pt} \mbox{for all } v \in H^1_0(\Omega),\\
\nonumber \left( \partial_t u^0( \cdot , 0 ) , v \right)_{L^2(\Omega)} &= \left( g , v \right)_{L^2(\Omega)}
\quad \hspace{32pt} \mbox{for all } v \in H^1_0(\Omega).
\end{align}
\end{theorem}
This Theorem and the compactness result stated above show that for {\it any} problem \eqref{wave-equation-strong} based on a sequence of matrices with $a^\eps\in\mathcal{M}(\alpha,\beta,\Omega)$, we can extract a subsequence such that the corresponding solution of the wave problem convergence to a homogenized solution. Except for special situations, e.g., locally periodic coefficients $a^\eps$, i.e. tensor $a^\eps(x)=a(x,\frac{x}{\varepsilon})$ that are $\varepsilon$-periodic on a fine scale or for random stationary tensors, it is not possible to construct $a^0$ explicitly. The next theorem states that the solution
$u_{H,K}$ of our multiscale method \eqref{semi-discrete-lod-equation} converges to the homogenized solution also in the case where no explicit solution of
$a^0$ is known. This is a convergence result for the multiscale method to the effective solution of problem \eqref{wave-equation-strong}
under the weakest possible assumptions. Let $a^0$ denote the $G$-limit of $a^\eps$ and let $f^{\varepsilon} \in H^1_0(\Omega)$ be the solution to
\begin{align}\label{def-f-eps}
\int_{\Omega} a^\eps \nabla f^{\varepsilon} \cdot \nabla v = \int_{\Omega} a^0 \nabla f \cdot \nabla v \qquad \mbox{for all } v \in H^1_0(\Omega),
\end{align}
where $f$ is the initial value in problem \eqref{wave-equation-strong}.
By the definition of $G$-convergence, we have that $f^{\varepsilon} \rightharpoonup f$ weakly in $H^1_0(\Omega)$ and hence $f^{\varepsilon} \rightarrow f$ strongly in $L^2(\Omega)$. Define $e_{\mbox{\tiny \rm hom}}=\| u^\eps - u^0 \|_{L^{\infty}(L^2)} +\| f^\varepsilon - f \|_{(L^2)}$.
Note that under the assumptions of Theorem \ref{theorem:homogenization:wave:equation}, we have $\lim_{\varepsilon \rightarrow 0} e_{\mbox{\tiny \rm hom}}(\varepsilon) = 0.$
\begin{theorem}[A priori error estimate for the homogenized solution]\label{apriori-lod-homogenization}$\\$
Consider the setting of Theorem \ref{theorem:homogenization:wave:equation}, let $h<\varepsilon <H$ and assume (H0), $g\in H^1_0(\Omega)$, $\partial_t F \in L^2(0,T,L^2(\Omega))$ and $\nabla \cdot (a^0 \nabla f) + F(\cdot,0)\in L^2(\Omega)$.
By $u^0$ we denote the homogenized solution given by (\ref{homogenized-equation-weak}) and by $u_{H,k}$ we denote the solution of
\eqref{semi-discrete-lod-equation}.
Under these assumptions there exists a generic constant $C_\theta$ (i.e. independent of $H$, $h$ and $\varepsilon$) such that if $k \ge C_{\theta} |\ln(H)|$
it holds
\begin{eqnarray}
\label{main-result-est-1}
\lim_{\varepsilon\rightarrow 0} \lim_{h\rightarrow 0}
\| u^0 - u_{H,k} \|_{L^{\infty}(L^2)} &\lesssim_T& H \\
\label{main-result-est-2}
\lim_{h\rightarrow 0}\| u^\eps - u_{H,k} \|_{L^{\infty}(L^2)} &\lesssim_T& H + e_{hom}(\varepsilon).
\end{eqnarray}
If we replace the elliptic projection $\pi_{H,k}^{\mbox{\rm \tiny ms}}(f)$ in (\ref{semi-discrete-lod-equation}) by the $L^2$-projection $P_{H,k}^{\mbox{\rm \tiny ms}}(f)$,
the estimate
still remains valid.
\end{theorem}
The theorem is proved in Section \ref{section:proofs}, where the dependencies on $\varepsilon$ and $h$ are elaborated.
In particular, the following sharpened result can be extracted from the proof of Theorem \ref{apriori-lod-homogenization}:
Let $g\in H^1_0(\Omega)$ and $\partial_t F \in L^2(0,T,L^2(\Omega))$. If $f=0$ the estimate in Theorem \ref{apriori-lod-homogenization} can be improved to
\begin{align}
\label{main-result-est-1-sharp}
\lim_{h\rightarrow 0}\| u^\eps - u_{H,k} \|_{L^{\infty}(L^2)} \lesssim_T H.
\end{align}
Estimate \eqref{main-result-est-1}
guarantees convergence of our method
with respect to the $L^{\infty}(L^2)$-error under the weakest possible assumptions in the general setting of $G$-convergence without any restrictions on the initial values. However, for some choices of the initial values, these estimates can be still improved significantly. This case is discussed in the next section.
\subsection{Convergence results for well-prepared initial data}
\label{subsection-well-prepared-initial-values}
In the previous section, we showed convergence of our method in the setting of $G$-convergence. We obtained a linear rate in $H$ for the $L^{\infty}(L^2)$-error. In this section, we show that this convergence can be improved for {\it well-prepared} initial values by using correctors from the kernel of the $L^2$-projection. In particular, we obtain $L^{\infty}(H^1)$- and $W^{1,\infty}(L^2)$-error estimates with respect to the exact solution $u^\eps$. In a first step, we define what we mean by {\it well-prepared initial values} and why they are crucial for improved estimates. Consequently, we need to assume that $f$, $g$ and $F(\cdot,0)$ are $\varepsilon$-dependent such that they interact constructively with $a^\eps$ (in the sense specified in Proposition \ref{proposition-time-regularity} below). We note that in this section, $\varepsilon$ is an abstract parameter and functions $z$ with superscript $\varepsilon$
are assumed to have a large $\| z^\varepsilon \|_{H^s(\Omega)}$ norm for $s>1$.
For the wave equation, this kind of blow-up cannot only be triggered by the spatial derivatives, but also by the time derivatives.
Typically we have a large $\| u^\eps \|_{W^{m,2}(0,T;H^s(\Omega))}$ norm when $m+s>1$ (in a homogenization context
$\| u^\eps \|_{W^{m,2}(0,T;H^s(\Omega))} \rightarrow \infty$ for $\varepsilon \rightarrow 0$ when $m+s>1$, see e.g., \cite{CiD99}). However, this statement can be relaxed. More precisely, under certain assumptions on the initial data, it is possible to show that $\| u^\eps \|_{W^{m,2}(0,T;H^1(\Omega))}$ remains bounded independent of $\varepsilon$. To make this statement precise, we state the following regularity result.
\begin{proposition}[Time-regularity and regularity estimates]\label{proposition-time-regularity}
Let assumption (H0) be fulfilled and let $F \in W^{m,2}(0,T;L^2(\Omega))$ for some $m\in \mathbb{N}$. Furthermore we define iteratively
\begin{align}
\label{higher-order-initial-values}w_0^{\varepsilon}:=f, \qquad w_1^{\varepsilon}:=g, \qquad w_j^{\varepsilon} := \partial_t^{j-2} F(\cdot , 0) + \nabla \cdot ( a^\eps \nabla w_{j-2}^{\varepsilon}) \quad \mbox{for } j=2,3,\cdots,m+1.
\end{align}
If $w_j^{\varepsilon} \in H^1_0(\Omega)$ for $0\le j \le m$ and $w_{m+1}^{\varepsilon}\in L^2(\Omega)$, we have
$$
\partial_t^{m} u^\eps \in L^{\infty}(0,T;H^1_0(\Omega)); \enspace \partial_t^{m+1} u^\eps \in L^{\infty}(0,T;L^2(\Omega))
\enspace \mbox{and} \enspace \partial_t^{m+2} u^\eps \in L^{2}(0,T;H^{-1}(\Omega))
$$
and the regularity estimate
\begin{eqnarray}
\label{reg-estimates}
\nonumber\lefteqn{\| \partial_t^{m} u^\eps \|_{L^{\infty}(0,T;H^1(\Omega))} + \| \partial_t^{m+1} u^\eps \|_{L^{\infty}(0,T;L^2(\Omega))}}\\
&\lesssim_T& \| F \|_{W^{m,2}(0,T;L^2(\Omega))} + \sum_{j=0}^m \| w_j^{\varepsilon} \|_{H^1(\Omega)} + \| w_{m+1}^{\varepsilon} \|_{L^2(\Omega)}.
\end{eqnarray}
\end{proposition}
A proof of this proposition can be extracted from the results presented in \cite[Chapter 7.2]{Eva10}. From the regularity estimate \eqref{reg-estimates} we see that the higher order time derivatives of $u^\eps$ might not be independent of $\varepsilon$ (i.e. the data oscillations) if the initial values $f$, $g$ and $\partial_t^{m} F(\cdot,0)$ are not {\it well-prepared}, i.e. if the functions $w_j^{\varepsilon}$ cannot be bounded independent of $\varepsilon$. In the previous section, we learned that the method shows a convergence of linear order for the $L^{\infty}(L^2)$-error, even if $\| w_2^{\varepsilon}\|_{L^2(\Omega)}$ grows with $\varepsilon$ (not well-prepared). This should be seen as a worst-case result. In this section, we want to derive improved estimates for the case of {\it well-prepared} initial values. In the light of the regularity estimate, we hence formulate the following compatibility condition.
\begin{definition}[Compatibility condition and well-preparedness]
\label{def-of-comp-condition}
Let $k\in \mathbb{N}$ and let $w_0^{\varepsilon},\ldots,w_{k+1}^{\varepsilon}$ be the initial data defined in \eqref{higher-order-initial-values}. We say that the data is {\it well-prepared and compatible of order $k$} if $F \in W^{k,2}(0,T;L^2(\Omega))$, $w_j^{\varepsilon} \in H^1_0(\Omega)$ for $0\le j \le k$ and $w_{k+1}^{\varepsilon}\in L^2(\Omega)$ and if
\begin{align}
\label{bound-well-prep}\sum_{j=0}^k \| w_j^{\varepsilon} \|_{H^1(\Omega)} + \| w_{k+1}^{\varepsilon} \|_{L^2(\Omega)} \le C_w,
\end{align}
where $C_w$ denotes a (generic) constant that can depend on $F,f,g,\Omega,\alpha$ and $\beta$, but not on $\varepsilon$.
\end{definition}
\begin{remark}[Fulfillment of compatibility and well-preparedness]
Observe that the initial data is always
well-prepared and compatible of order $0$. Furthermore, if $f=0$, $g=0$ and $\partial_t^{j} F(\cdot , 0)=0$ for $0\le j \le k-2$, the compatibility condition of order $k$ is trivially fulfilled. For any other case, we note that $C_w$ in \eqref{bound-well-prep} is a computable constant, since $w_j^{\varepsilon}$ are known functions. Consequently, we can check a priori if the initial value is compatible and well-prepared.
\end{remark}
Well-prepared initial data is often crucial in homogenization settings. For instance, for deriving a homogenized model that captures long-time dispersion in the multiscale wave equation, well-prepared initial values are essential \cite{Lam11b}. Similar observations can be made for parabolic problems with a large drift (cf. \cite{AlO07}) which also can be seen as having a hyperbolic character.
The next main result of this paper are high order convergence rates in $L^{\infty}(L^2)$, provided that we use the correctors $Q_{h,k}( u_{H,k} )$ and that the data is {\it well-prepared and compatible}. Furthermore, we also show that the method yields convergence in $W^{1,\infty}(L^2)$ and $L^{\infty}(H^1)$.
For arbitrary initial values, we can only guarantee a convergence in $L^{\infty}(L^2)$ according to Theorem \ref{apriori-lod-homogenization}. In the following theorem, $u^\eps$ denotes the exact solution of the wave equation (\ref{wave-equation-weak}) and $u_{H,k}$ the numerically homogenized solution defined by (\ref{semi-discrete-lod-equation}).
\begin{theorem}[A priori corrector error estimates for the semi-discrete method]\label{apriori-semidiscrete}$\\$
Assume (H0) and that the data is {\rm well-prepared and compatible of order $2$} in the sense of Definition \ref{def-of-comp-condition}. Then
there exists a generic constant $C_\theta$ (i.e. independent of $H$, $h$ and $\varepsilon$) such that if
$k \ge C_{\theta} |\ln(H)|$
the following a priori error estimates hold:
\begin{align}
\label{main-result-est-2a}
\| u^\eps - (u_{H,k} + Q_{h,k}(u_{H,k})) \|_{L^{\infty}(L^2)} \lesssim_T H^2 + e_{\mbox{\tiny \rm disc}}^{\mbox{\tiny$(1)$}}(h);
\end{align}
if $F \in L^{\infty}(H^1_0)\cap W^{1,1}(H^1_0)$ and if the data is {\rm well-prepared and compatible of order $3$}:
\begin{align}
\label{main-result-est-3a}
\| u^\eps - ( u_{H,k} + Q_{h,k}( u_{H,k} )) \|_{L^{\infty}(L^2)} \lesssim_T H^3 + e_{\mbox{\tiny \rm disc}}^{\mbox{\tiny$(1)$}}(h);
\end{align}
if the data is {\rm well-prepared and compatible of order $3$}:
\begin{align}
\label{main-result-est-2b}
\| \partial_tu^\eps - \partial_t (u_{H,k} + Q_{h,k}(u_{H,k})) \|_{L^{\infty}(L^2)}
+ \| u^\eps - (u_{H,k} + Q_{h,k}( u_{H,k} )) \|_{L^{\infty}(H^1)} \lesssim_T H + e_{\mbox{\tiny \rm disc}}^{\mbox{\tiny$(2)$}}(h);
\end{align}
if $F \in L^{\infty}(H^1_0)$, if the data is {\rm well-prepared and compatible of order $3$} and
if the initial value in (\ref{semi-discrete-lod-equation}) is picked such that we have $\partial_t (u_{H,k}+Q_{h,k}(u_{H,k}))( \cdot , 0 ) = \pi_{H,k}^{\mbox{\rm \tiny ms}}(g)$:
\begin{align}
\label{main-result-est-3b}
\| \partial_tu^\eps - \partial_t ( u_{H,k} + Q_{h,k}( u_{H,k} )) \|_{L^{\infty}(L^2)}
+ \| u^\eps - ( u_{H,k} + Q_{h,k}( u_{H,k} )) \|_{L^{\infty}(H^1)} \lesssim_T H^2 + e_{\mbox{\tiny \rm disc}}^{\mbox{\tiny$(2)$}}(h).
\end{align}
Here, the fine scale discretization errors $e_{\mbox{\tiny \rm disc}}^{\mbox{\tiny$(1)$}}(h)$ and $e_{\mbox{\tiny \rm disc}}^{\mbox{\tiny$(2)$}}(h)$ are given by
$$e_{\mbox{\tiny \rm disc}}^{\mbox{\tiny$(1)$}}(h) := \| u^\eps - \pi_h(u^\eps) \|_{L^{\infty}(L^2)} + \| \partial_t u^\eps - \pi_h(\partial_t u^\eps) \|_{L^1(L^2)}$$
and
$$
e_{\mbox{\tiny \rm disc}}^{\mbox{\tiny$(2)$}}(h) := \| \partial_t u^\eps - \pi_h(\partial_t u^\eps) \|_{L^{\infty}(L^2)}
+ \| \partial_{tt} u^\eps - \pi_h(\partial_{tt} u^\eps) \|_{L^1(L^2)} + \|u^\eps - \pi_h(u^\eps) \|_{L^{\infty}(H^1)},
$$
where $\pi_h$ is the elliptic-projection on $V_h$ (cf. (\ref{elliptic-projection-pi_h})). Note that $e_{\mbox{\tiny \rm disc}}^{\mbox{\tiny$(i)$}}(h)$ will only yield optimal orders in $h$, if $u^\eps$ is sufficiently regular with respect to the spatial variable.
\end{theorem}
As for the $G-$convergence setting we have that if $g\in H^1_0(\Omega)$, $\partial_t F \in L^2(0,T,L^2(\Omega))$ and $f=0$, the estimate in Theorem \ref{apriori-lod-homogenization} can be improved to
\begin{align}
\label{main-result-est-1-sharp_bis}
\| u^\eps - u_{H,k} \|_{L^{\infty}(L^2)} \lesssim_T H + e_{\mbox{\tiny \rm disc}}^{\mbox{\tiny$(1)$}}(h).
\end{align}
The proof is analogous to the proof of Lemma \ref{lemma-apriori-hom} presented later.
A proof of Theorem \ref{apriori-semidiscrete} including refined estimates (i.e. estimates where all dependencies on $u^\eps$ and $k$ are worked out in detail) is given in Section \ref{section:proofs}.
Our last main result is an optimal $L^{\infty}(L^2)$ error estimate for the Crank-Nicholson version of the multiscale method. In order to obtain the optimal convergence rates with regard to the time step size, we observe that we need slightly higher regularity assumptions than in the semi-discrete case.
\begin{theorem}[A priori error estimates for the Crank-Nicolson fully-discrete method]\label{apriori-fullydiscrete}$\\$
Assume (H0) and that the data is {\rm well-prepared and compatible of order $3$} in the sense of Definition \ref{def-of-comp-condition}. Beside this, let the notation from Theorem \ref{apriori-semidiscrete} hold true and let $u_{H,\triangle t}:=u_{H,\triangle t, k}$ be the fully discrete numerically homogenized approximation as in Definition \ref{definition-fully-discrete-method}. Then there exists a generic constant $C_\theta$ (i.e. independent of $H$, $h$ and $\varepsilon$) such that if $k \ge C_{\theta} |\ln(H)|$ it holds
\begin{align}
\label{main-result-est-2-fully-discrete}\max_{0 \le n \le J} \| (u^\eps - (u_{H,\triangle t} + Q_{h,k}(u_{H,\triangle t})))(\cdot, t^n ) \|_{L^2(\Omega)} \lesssim_T H^2 + \triangle t^2 + e_{\mbox{\tiny \rm disc}}(h).
\end{align}
If furthermore $F \in L^{\infty}(H^1_0)$ and $\partial_t F \in L^2(H^1_0)$, we obtain the improved corrector estimate
\begin{align}
\label{main-result-est-3-fully-discrete} \max_{0 \le n \le J} \| (u^\eps - (u_{H,\triangle t} + Q_{h,k}(u_{H,\triangle t})))(\cdot, t^n ) \|_{L^2(\Omega)} \lesssim_T H^3 + \triangle t^2 + e_{\mbox{\tiny \rm disc}}(h).
\end{align}
Here, the fine scale discretization error $e_{\mbox{\tiny \rm disc}}(h)$ is given by
$$e_{\mbox{\tiny \rm disc}}(h) := \| u^\eps - \pi_h(u^\eps) \|_{L^{\infty}(L^2)} + \| \partial_t u^\eps - \pi_h(\partial_t u^\eps) \|_{L^2(L^2)}.$$
The convergence rates in $h$ hence dependent on a higher order space regularity of $u^\eps$.
\end{theorem}
The theorem is proved at the end of Section \ref{section:proofs}.
Note that if we are in a situation, where the data is not well-prepared (i.e. \eqref{bound-well-prep} does not hold) not only the mesh size needs to be small enough, but also the time step size $\triangle t$ requires a resolution condition such as $\triangle t \lesssim \varepsilon$. This will become obvious in the proof of Theorem \ref{apriori-fullydiscrete}.
\subsection{Survey on other multiscale methods for the wave equation}
\label{subsection-surbey-multiscale-methods}
The number of existing multiscale methods for the wave equation is rather small, compared to the number of multiscale methods that exist for other types of equations. Subsequently we give a short survey on existing strategies to put our method into perspective.
One way of realizing numerical homogenization is to use the framework of the Heterogeneous Multiscale Method (HMM) (cf. \cite{Abd05b,Abd06a,Abd11b,EE03,HeO09}). The method is based on the idea to predict an effective limit problem of (\ref{wave-equation-strong}) for $\varepsilon \rightarrow 0$. This can be achieved by solving local problems in sampling cells (typically called cell problems) and to extract effective macroscopic properties from the corresponding cell solutions. In some cases it can be explicitly shown that this strategy in fact yields the correct limit problem for $\varepsilon \rightarrow 0$. The central point of the method is that the cell problems are very small and systematically distributed in $\Omega$, but do not cover $\Omega$. This makes the method very cheap. For the wave equation, an HMM based on Finite Elements was proposed and analyzed in \cite{AbG11}. An HMM based on Finite Differences can be found in \cite{EHR11}. Since the classical homogenized model is known to fail to capture long time dispersive effects (cf. \cite{Lam11b}) another effective model is needed for longer times. Solutions for this problem by a suitable model adaptation in the HMM context can be found in \cite{AGS13,AGS14,EHR12}. The advantage of the HMM framework is that it allows to construct methods that do not have to resolve the fine scale globally, allowing for a computational cost proportional to the degrees of freedom of the macroscopic mesh.
But it requires scale separation and the cell problems must sample the microstructure sufficiently well.
In many applications, especially in material sciences, these assumptions are typically well justified, in geophysical applications on the other hand, they might be often problematic. In this work, we hence focus on the latter case, where the HMM might not be applicable.
Beside the multiscale character of the problem, one of the biggest issues is the typically missing space regularity of the solution. In realistic applications, the propagation field $a^\eps$ is discontinuous. For instance in geophysics or seismology, the waves propagate through a medium that consists of different, heterogeneously distributed types of material (e.g. different soil or rock types). Hence, the properties of the propagation field cannot change continuously. This typically also involves a high contrast. The missing smoothness of $a^\eps$ directly influences the space regularity of the solution $u^\eps$ which is often not higher than $L^{\infty}(0,T;H^1(\Omega))$. As a consequence, the convergence rates of standard Finite Element methods deteriorate besides being very costly.
To overcome these issues (multiscale character and missing regularity of $u^\eps$), Owhadi and Zhang \cite{OwZ08} proposed an interesting multiscale method based on a harmonic coordinate transformation $G$. The method is only analyzed for $d=2$, but it is also applicable for higher dimensions. The components of $G=(G_1,\dots,G_d)$ are defined as the weak solutions of an elliptic boundary value problem $\nabla ( a^\eps \nabla G_i ) = 0$ in $\Omega$ and $G_i(x)=x_i$ on $\partial \Omega$. Under a so called Cordes-type condition (cf. \cite[Condition 2.1]{OwZ08}) the authors managed to prove a compensation theorem saying that the solution in harmonic coordinates yields in fact the desired space-regularity. More precisely, they could show that $(u^\eps \circ G) \in L^{\infty}(0,T;H^2(\Omega))$ and furthermore that
$$ \| u^\eps \circ G \|_{L^{\infty}(0,T;H^2(\Omega))} \le
C(F,g) + C \| \partial_{tt} u^\eps(\cdot ,0) \|_{L^2(\Omega)},
$$
where $C(F,g)$ and $C$ are constants depending on the data functions, but not on the variations of $a^\eps$. Consequently, by using the equality $\partial_{tt} u^\eps(\cdot ,0) = \nabla \cdot ( a^\eps \nabla f ) - F(\cdot ,0)$, the $L^{\infty}(H^2)$-norm of $u^\eps \circ G$ can be bounded {\it independently of the oscillations} of $a^\eps$ if the choice of the initial value is such that $\| \nabla \cdot ( a^\eps \nabla f ) \|_{L^2(\Omega)}$ can be bounded independent of $\varepsilon$, i.e. if the initial value is well-prepared. Note that $\| u^\eps \|_{L^{\infty}(0,T;H^2(\Omega))}$ (if it even exists) is normally proportional to the $W^{1,\infty}$-norm of $a^\eps$ (if it exists), which is the reason why classical finite elements cannot converge unless this frequency is resolved by the mesh. The harmonically transformed solution of the wave equation does not suffer from this anymore. With this key feature, an adequate analysis (and corresponding numerics) can be performed in an harmonically transformed finite element space, allowing optimal orders of convergence. The method has only two drawbacks: the approximation of the harmonic coordinate transformation $G$ and the validity of the Cordes-type condition. Even though the Cordes-type condition can be hard to verify in practice, the numerical experiments given in \cite{OwZ08} indicate that the condition might not be necessary for a good behavior of the method. The approximation of the harmonic coordinate transformation $G$ on the other hand can become a real issue, since it involves the solution of $d$ global fine-scale problems. This is an expensive one-time overhead. Furthermore, spline spaces are needed and it is not clear how the analytically predicted results change, when $G$ is replaced by a numerical approximation $G_h$. Compared to \cite{OwZ08}, our method has therefore the advantage that it does not involve to solve global fine scale problems and relies on localized classical P1-finite element spaces.
Another multiscale method applicable to the wave equation was also presented by Owhadi and Zhang in \cite{OwZ11}. Here a multiscale basis is assembled by localizing a certain transfer property (which can be seen as an alternative to the aforementioned harmonic coordinate transformation). In this approach, the number of local problems to solve is basically the same as for our method. However, the local problems require finite element spaces consisting of certain $C^1$-continuous functions. Furthermore, the diameter of the localization patches must at least be of order $\sqrt{H} |\ln(H)|$ to guarantee an optimal linear convergence rate for the $H^1$-error, whereas our approach only requires $H |\ln(H)|$.
The Multiscale Finite Element Method using Limited Global Information
by Jiang et al. \cite{JEG10,JiE12} can be seen as a general framework that also covers the harmonic coordinate transformation approach by Owhadi and Zhang. The central assumption for this method is the existence of a number of {\it known} global fields $G_1, \dots, G_N$ and an unknown smooth function $H=H(G_1,\dots,G_N,t)$ such that the error $e=u^\eps - H(G_1,\dots,G_N,t)$ has a small energy. Based on the size of this energy, an a priori error analysis can be performed. The components of the harmonic coordinate transformation $G$ are an example for global fields that fit into the framework. Other (more heuristic) choices are possible (cf. \cite{JEG10,JiE12}), but equally expensive as computing the harmonic transformation $G$. The drawback of the method is hence the same as for the Owhadi-Zhang approach: the basic assumption on the existence of global fields can be hard to verify and even if it is known to be valid, there is an expensive one-time overhead in computing them with a global fine scale computation.
Excluding the HMM approach, we want to stress that each of the above multiscale methods is only guaranteed to converge in the regime $H>\varepsilon$ if the data is well-prepared in the sense of Definition \ref{def-of-comp-condition}. There are no available results with respect to arbitrary initial data.
With regard to the previous discussions, our multiscale method proposed in Definition \ref{definition-fully-discrete-method} has the following benefits. The method does not require additional assumptions on scale separation or regularity of $a^\eps$ and it does not involve one-time-overhead computations on the full fine scale. Furthermore, the method is guaranteed to converge even for not well-prepared initial values. On the other hand, if the initial-values are well-prepared, the method is independent of the homogenization setting and yields significantly improved convergence rates, even in $W^{1,\infty}(L^2)$ and $L^{\infty}(H^1)$, without exploiting any higher space regularity than $H^1$.
\section{Proofs of the main results}
\label{section:proofs}
This section is devoted to the proof of Theorem \ref{apriori-semidiscrete}. First, we derive some general error estimates in Subsection \ref{subsection-abstract-estimates}. In Subsection \ref{subsection-well-prepared-estimates} we analyze the case of well-prepared initial values,
and finally prove in Subsection \ref{subsection-G-conv-estimates} the convergence results without any assumption on the initial data but the one needed for the well-posedness of the wave equation.
\subsection{Abstract error estimates}\label{subsection-abstract-estimates}
Before we can start with proving the a priori error estimates, we present two lemmata. The first result can be found in \cite{Car99,MaP14}:
\begin{lemma}[Properties of the interpolation operator]
\label{lemma-properties-clement-op}
The interpolation operator $I_H : H^1_0(\Omega) \rightarrow V_H$ from (\ref{def-weighted-clement}) has the following properties:
\begin{align}
\label{lemma-properties-clement-op-est}\| v - I_H (v) \|_{L^2(\Omega)} + H \| v - I_H (v) \|_{H^1(\Omega)} \le C_{I_H} H \| v \|_{H^1(\Omega)},
\end{align}
for all $v \in H^1_0(\Omega)$. Here, $C_{I_H}$ denotes a generic constant, that only depends on the shape regularity of the elements of $\mathcal{T}_H$. Furthermore, the restriction $I_H\vert_{V_H} : V_H \rightarrow V_H$ is an isomorphism on $V_H$, with $(I_H\vert_{V_H})^{-1}$ being $H^1$-stable.
\end{lemma}
Observe that $((I_H\vert_{V_H})^{-1} \circ I_H) \vert_{V_H} = \mbox{\rm Id}$. On the other hand $((I_H\vert_{V_H})^{-1} \circ I_H) \vert_{W_h} = 0$. Hence, for any $v_h=v_H +w_h \in V_h=V_H \oplus W_h$ with $v_H \in V_H$ and $w_h \in W_h$ we have $((I_H\vert_{V_H})^{-1} \circ I_H)(v_h)=v_H=P_H(v_h)$ and therefore
\begin{align}
\label{L2-projection-by-interpolation} ((I_H\vert_{V_H})^{-1} \circ I_H) \vert_{V_h} = P_H \vert_{V_h}.
\end{align}
Furthermore we have the equation
\begin{align}
\label{equation-for-projection}\pi_{H,k}^{\mbox{\rm \tiny ms}}(v) = (P_H \circ \pi_{H,k}^{\mbox{\rm \tiny ms}})(v) + (Q_{h,k} \circ P_H \circ \pi_{H,k}^{\mbox{\rm \tiny ms}})(v) \qquad \mbox{for all } v \in H^1_0(\Omega).
\end{align}
The next proposition quantifies the decay of the local correctors.
\begin{proposition}
\label{proposition-stability-estimates}
Let assumptions (H0) be fulfilled and the corrector operators defined according to Definition \ref{definition:localized:ms:space} for some $k \in \mathbb{N}_{>0}$. Then there exists a generic constant $0<\theta<1$ (independent of $H$, $h$ or $\varepsilon$) such that
we have the estimate
\begin{align}
\label{decay-of-correctors}\| \nabla (Q_{h,k} - Q_{h,\Omega})(v_H) \|_{L^2(\Omega)} \lesssim k^{d/2}
\theta^{k} \| \nabla v_H \|_{L^2(\Omega)}.
\end{align}
for all $v_H \in V_H$.
Furthermore, the operator $Q_{h,k}$ is $H^1$-stable on $V_H$ and the operator $(P_{H} \circ \pi_{H,k}^{\mbox{\rm \tiny ms}})$ is $H^1$-stable on $H^1_0(\Omega)$, i.e.
\begin{align}
\label{stability-estimates-for-various-ops}\forall v_H \in V_H: \quad \hspace{29pt} \| Q_{h,k}(v_H) \|_{H^1(\Omega)} &\lesssim \| v_H \|_{H^1(\Omega)}
\quad \mbox{and}\\
\nonumber \forall v \in H^1_0(\Omega): \quad \| (P_{H} \circ \pi_{H,k}^{\mbox{\rm \tiny ms}})(v) \|_{H^1(\Omega)} &\lesssim \| v \|_{H^1(\Omega)}.
\end{align}
\end{proposition}
\begin{proof}
Estimate (\ref{decay-of-correctors}) was proved in \cite{HeM14}. It is hence sufficient to show \eqref{stability-estimates-for-various-ops}. Since $Q_{h,\Omega}$ is obviously $H^1$-stable on $V_H$ and since $k^{d/2}\theta^k$ is monotonically decreasing for growing $k$, the $H^1$-stability of $Q_{h,k}$ follows directly from (\ref{decay-of-correctors}). The elliptic projection $\pi_{H,k}^{\mbox{\rm \tiny ms}}$ is also obviously $H^1$-stable.
Finally, the $H^1$-stability of the $L^2$-projection $P_H$ on quasi-uniform meshes (as assumed for $\mathcal{T}_H$) was e.g. proved in \cite{BaD81,BaY14}. Combining these results gives us the desired $H^1$-stability of $P_{H} \circ \pi_{H,k}^{\mbox{\rm \tiny ms}}$ on $H^1_0(\Omega)$.
\end{proof}
The next lemma gives explicit error estimates for the elliptic projections on $V_{H,k}^{\ms}$.
\begin{lemma}
\label{lemma-multiscale-projection}
Let $u^\eps$ be the solution of (\ref{wave-equation-weak}) and let the corrector operator $Q_{h,k}$ be given as in Definition \ref{definition:localized:ms:space} for some $k \in \mathbb{N}_{>0}$. Furthermore, let $\pi_{H,k}^{\mbox{\rm \tiny ms}}$ and $\pi_h$ denote the elliptic projections according to (\ref{elliptic-projection-pi_Hk}) and (\ref{elliptic-projection-pi_h}). We further denote the $L^2$-projection of $V_h$ on $V_H$ by $P_H$. The following estimates hold for almost every $t \in [0,T]$.\\
If $\partial_t^i u^\eps \in L^1(0,T;H^1(\Omega))$ for $i \in \{0,1,2\}$, then it holds
\begin{eqnarray}
\label{projection-error-estimate-1}\lefteqn{\| (P_H \circ \pi_{H,k}^{\mbox{\rm \tiny ms}})(\partial_t^i u^\eps(\cdot,t)) - \partial_t^i u^\eps(\cdot,t) \|_{L^2(\Omega)}} \\
\nonumber&\lesssim& \| \partial_t^i u^\eps(\cdot,t) - \pi_h(\partial_t^i u^\eps(\cdot,t)) \|_{L^2(\Omega)} + (H + \theta^k k^{d/2}) \| \partial_t^i u^\eps(\cdot,t) \|_{H^1(\Omega)}.
\end{eqnarray}
Assume that $i \in \{0,1,2\}$ and $s,m \in \{0,1\}$. If $\partial_t^i u^\eps \in L^1(0,T;H^1(\Omega))$ and $\partial_t^{2+i} u^\eps, \partial_t^{i} F \in L^1(0,T;H^s(\Omega))$ it holds
\begin{eqnarray}
\label{projection-error-estimate-3-new-1}\lefteqn{\| \pi_{H,k}^{\mbox{\rm \tiny ms}}(\partial_t^i u^\eps(\cdot,t)) - \partial_t^i u^\eps(\cdot,t) \|_{H^m(\Omega)} \lesssim \| \partial_t^i u^\eps (\cdot,t)- \pi_h(\partial_t^i u^\eps(\cdot,t)) \|_{H^m(\Omega)} }\\
\nonumber&\enspace& \enspace + \left(H^{2+s-m} +
\left(k^{d/2} \theta^{k}\right)^{p(s,m)}
\right) \left( \| \partial_t^{2+i} u^\eps(\cdot,t) - \partial_t^{i} F(\cdot,t) \|_{H^s(\Omega)} + \| \partial_t^{i} u^\eps(\cdot,t) \|_{H^1(\Omega)} \right),
\end{eqnarray}
where
$p(s,m):= \left(\frac{s+2}{s+1}\right)^{1-m}\ge 1$.
\end{lemma}
\begin{proof}
{\it Error estimate under low regularity assumptions - (\ref{projection-error-estimate-1}).} We use an Aubin-Nitsche duality argument for some arbitrary $v \in L^1(0,T;H^1_0(\Omega))$. Let us define $e_{H,k}:= \pi_{H,k}^{\mbox{\rm \tiny ms}}(v) - \pi_h(v)$. We regard the dual problem: find $z_h \in L^1(0,T;V_h)$ with
\begin{align}
\label{dual-prob-1}b^{\eps}(w_h ,z_h(\cdot,t) ) = (e_{H,k}(\cdot,t) ,w_h)_{L^2(\Omega)} \qquad \mbox{for all } w_h \in V_h, \quad \mbox{for a.e. } t \in (0,T)
\end{align}
and the dual problem in the multiscale space: find $z_{H,k}^{\mbox{\rm \tiny ms}} \in V_{H,k}^{\ms}$ with
\begin{align}
\label{dual-prob-2}b^{\eps}(w^{\mbox{\rm \tiny ms}},z_{H,k}^{\mbox{\rm \tiny ms}}(\cdot,t) ) = (e_{H,k} (\cdot,t),w^{\mbox{\rm \tiny ms}})_{L^2(\Omega)} \qquad \mbox{for all } w^{\mbox{\rm \tiny ms}} \in V_{H,k}^{\ms}, \quad \mbox{for a.e. } t \in (0,T).
\end{align}
Obviously we have $b^{\eps}(w^{\mbox{\rm \tiny ms}},(z_h-z_{H,k}^{\mbox{\rm \tiny ms}})(\cdot,t))=0$ for all $w^{\mbox{\rm \tiny ms}} \in V_{H,k}^{\ms}$ and for almost every $t \in [0,T]$. This implies that $(z_h-z_{H,\Omega}^{\mbox{\rm \tiny ms}})(\cdot,t)$ is in the $b^{\eps}(\cdot,\cdot)$-orthogonal complement of $V_{H,\Omega}^{\ms}$ (for almost every $t$), hence it is in the kernel of the quasi-interpolation operator $I_H$. Omitting the $t$-dependency, we obtain
\begin{eqnarray}
\label{dual-estimate-1}\nonumber b^{\eps}( z_h-z_{H,\Omega}^{\mbox{\rm \tiny ms}}, z_h-z_{H,\Omega}^{\mbox{\rm \tiny ms}} ) &=& (e_{H,k} ,z_h-z_{H,\Omega}^{\mbox{\rm \tiny ms}} )_{L^2(\Omega)} \\
\nonumber &=& (e_{H,k} , (z_h-z_{H,\Omega}^{\mbox{\rm \tiny ms}}) - I_H( z_h-z_{H,\Omega}^{\mbox{\rm \tiny ms}}) )_{L^2(\Omega)} \\
&\overset{(\ref{lemma-properties-clement-op-est})}{\lesssim}& H \| e_{H,k} \|_{L^2(\Omega)} \| z_h-z_{H,\Omega}^{\mbox{\rm \tiny ms}} \|_{H^1(\Omega)}.
\end{eqnarray}
Next, let us define the energy
$$E(v_H):=b^{\eps}(z_h - v_H - Q_{h,k}(v_H), z_h - v_H - Q_{h,k}(v_H)) \quad \mbox{for } v_H \in V_H$$
and let
us write $z_{H,\Omega}^{\mbox{\rm \tiny ms}}=z_{H,\Omega} + Q_{h,\Omega}(z_{H,\Omega})$ and $z_{H,k}^{\mbox{\rm \tiny ms}}=z_{H,k} + Q_{h,k}(z_{H,k})$ with $z_{H,\Omega},z_{H,k} \in V_H$. Since we have
$$b^{\eps}( z_h - z_{H,k} - Q_{h,k}(z_{H,k}), v_H + Q_{h,k}(v_H))=0 \qquad \mbox{for all } v_H \in V_H,$$
we know that this is equivalent to the fact that $z_{H,k} \in V_H$ must minimize the energy $E(\cdot)$ on $V_H$.
Hence
\begin{eqnarray}
\nonumber\| z_h - z_{H,k} - Q_{h,k}(z_{H,k}) \|_{H^1(\Omega)} &\lesssim& \| z_h - z_{H,\Omega} - Q_{h,k}(z_{H,\Omega}) \|_{H^1(\Omega)}\\
\nonumber
&\le& \| z_h - z_{H,\Omega}^{\mbox{\rm \tiny ms}} \|_{H^1(\Omega)} + \| (Q_{h,\Omega} - Q_{h,k})(z_{H,\Omega}) \|_{H^1(\Omega)} \\
\nonumber
&\overset{(\ref{dual-estimate-1}),(\ref{decay-of-correctors})}{\lesssim}& H \| e_{H,k} \|_{L^2(\Omega)} + \theta^k k^{d/2} \| Q_{h,\Omega}(z_{H,\Omega})
\|_{H^1(\Omega)}\\
\label{newstep-trolu}&\overset{(\ref{stability-estimates-for-various-ops})}{\lesssim}& (H + \theta^k k^{d/2}) \| e_{H,k} \|_{L^2(\Omega)}.
\end{eqnarray}
Note that in the last step, we used $\| z_{H,\Omega}\|_{H^1} =
\| P_{H} (z_{H,\Omega})\|_{H^1}
= \| P_{H} ( z_{H,\Omega} + Q_{h,\Omega}(z_{H,\Omega}) )\|_{H^1}
= \| P_{H} ( z_{H,\Omega}^{\mbox{\rm \tiny ms}} )\|_{H^1}$, which together with
\eqref{stability-estimates-for-various-ops} (i.e. the $H^1$-stability of $P_H$ on quasi-uniform grids) yields
$$
\| Q_{h,\Omega}(z_{H,\Omega})
\|_{H^1(\Omega)} \lesssim \| z_{H,\Omega}\|_{H^1}
= \| P_{H} ( z_{H,\Omega}^{\mbox{\rm \tiny ms}} )\|_{H^1}
\lesssim \|z_{H,\Omega}^{\mbox{\rm \tiny ms}}\|_{H^1}
\lesssim \| e_{H,k} \|_{L^2(\Omega)}.
$$
As a direct consequence of \eqref{newstep-trolu}, using $b^{\eps}(e_{H,k}, z_{H,k}^{\mbox{\rm \tiny ms}})=0$ (combining (\ref{elliptic-projection-pi_h}) and (\ref{elliptic-projection-pi_Hk}) for test functions in $V_{H,k}^{\ms}$)
\begin{align}
\label{L2-estimate-for-general-v-prestage}\| e_{H,k} \|_{L^2(\Omega)}^2 = b^{\eps}( e_{H,k} , z_h ) = b^{\eps}( e_{H,k} , z_h - z_{H,k}^{\mbox{\rm \tiny ms}}) \lesssim \| e_{H,k} \|_{H^1(\Omega)} (H + \theta^k k^{d/2}) \| e_{H,k} \|_{L^2(\Omega)}.
\end{align}
The bound $\| e_{H,k} \|_{H^1(\Omega)} \lesssim \| v \|_{H^1(\Omega)}$ and $\pi_{H,k}^{\mbox{\rm \tiny ms}}(v) = (P_H \circ \pi_{H,k}^{\mbox{\rm \tiny ms}})(v) + (Q_{h,k} \circ P_H \circ \pi_{H,k}^{\mbox{\rm \tiny ms}})(v)$ conclude the estimate
\begin{eqnarray*}
\| (P_H \circ \pi_{H,k}^{\mbox{\rm \tiny ms}})(v) - v \|_{L^2(\Omega)} &\le&
\| \pi_{H,k}^{\mbox{\rm \tiny ms}}(v) - v \|_{L^2(\Omega)} + \| (Q_{h,k} \circ P_H \circ \pi_{H,k}^{\mbox{\rm \tiny ms}})(v) \|_{L^2(\Omega)}\\
&\overset{(\ref{lemma-properties-clement-op-est}),(\ref{stability-estimates-for-various-ops}),{(\ref{L2-estimate-for-general-v-prestage})}}{\lesssim}& \| v - \pi_h(v) \|_{L^2(\Omega)} + (H + \theta^k k^{d/2}) \| v \|_{H^1(\Omega)} + H \| v \|_{H^1(\Omega)}.
\end{eqnarray*}
Hence for all $v \in H^1_0(\Omega)$
\begin{align}
\label{L2-estimate-for-general-v}
\| (P_H \circ \pi_{H,k}^{\mbox{\rm \tiny ms}})(v) - v \|_{L^2(\Omega)} + \| \pi_{H,k}^{\mbox{\rm \tiny ms}}(v) - v \|_{L^2(\Omega)}
\lesssim \| v - \pi_h(v) \|_{L^2(\Omega)} + (H + \theta^k k^{d/2}) \| v \|_{H^1(\Omega)}.
\end{align}
The results follows with $v=\partial_t^i u^\eps(\cdot,t)$.\\
{\it Error estimate under high regularity assumptions - (\ref{projection-error-estimate-3-new-1}).} For the next estimate, we restrict our considerations to the solution $u^\eps$ of (\ref{wave-equation-weak}). Let the regularity assumptions of the lemma hold true and let us introduce the simplifying notation
\begin{align*}
v^\eps := \partial_t^i u^\eps \quad \mbox{and} \quad \bar{F}:= \partial_t^i F.
\end{align*}
We observe that $v^\eps$ solves the equation
\begin{align*}
( \partial_{tt} v^\eps(\cdot,t), w )_{L^2(\Omega)} + b^{\eps}(v^\eps(\cdot,t), w ) = (\bar{F}(\cdot,t) ,w )_{L^2(\Omega)}
\end{align*}
for all $w \in H^1_0(\Omega)$, for almost every $t \in (0,T)$. By the definition of projections, we have
\begin{align*}
b^{\eps}( (\pi_{H,\Omega}^{\mbox{\rm \tiny ms}}(v^\eps) - \pi_h(v^\eps))(\cdot,t) , w ) = 0 \qquad \mbox{for all } w \in V_{H,k}^{\ms}, \quad \mbox{for almost every } t \in (0,T).
\end{align*}
We conclude $(\pi_{H,\Omega}^{\mbox{\rm \tiny ms}}(v^\eps) - \pi_h(v^\eps))(\cdot,t) \in W_h$ for almost every $t$ and in particular
\begin{align}
I_H((\pi_{H,\Omega}^{\mbox{\rm \tiny ms}}(v^\eps) - \pi_h(v^\eps))(\cdot,t)) = 0 \quad \mbox{for almost every } t \in (0,T).
\end{align}
Furthermore, with the notation $\pi_{H,k}^{\mbox{\rm \tiny ms}}( v^\eps )=v_{H,k}+ Q_{h,k}(v_{H,k})$ and $\pi_{H,\Omega}^{\mbox{\rm \tiny ms}}(v^\eps)=v_{H,\Omega}+ Q_{h,\Omega}(v_{H,\Omega})$ we have again that $v_{H,k}(\cdot,t)\in V_H$ minimizes the
energy
$$E(\Phi_H):=b^{\eps}(\pi_h(v^\eps(\cdot,t) - \Phi_H(\cdot,t) - Q_{h,k}(\Phi_H)(\cdot,t),\pi_h(v^\eps(\cdot,t) - \Phi_H(\cdot,t) - Q_{h,k}(\Phi_H)(\cdot,t))$$
for $\Phi_H \in V_H$ and therefore
\begin{align*}
\| \pi_{H,k}^{\mbox{\rm \tiny ms}}(v^\eps(\cdot,t)) - \pi_h(v^\eps(\cdot,t)) \|_{H^1(\Omega)}
&= \| v_{H,k}(\cdot,t)+ Q_{h,k}(v_{H,k})(\cdot,t) - \pi_h(v^\eps(\cdot,t)) \|_{H^1(\Omega)} \\
&\lesssim \| v_{H,\Omega}(\cdot,t)+ Q_{h,k}(v_{H,\Omega})(\cdot,t) - \pi_h(v^\eps(\cdot,t)) \|_{H^1(\Omega)}.
\end{align*}
For brevity, let us from now on leave out the $t$-dependency in the functions for the rest of the proof.
Hence, we obtain in the same way as for the low regularity estimate
\begin{eqnarray}
\label{general-estimate-pi-H-k}\nonumber\lefteqn{\| \pi_{H,k}^{\mbox{\rm \tiny ms}}(v^\eps) - \pi_h(v^\eps) \|_{H^1(\Omega)}}\\
\nonumber&\le& \| u_{H,\Omega}+ Q_{h,\Omega}(v_{H,\Omega}) - \pi_h(v^\eps) \|_{H^1(\Omega)} + \| ( Q_{h,\Omega} - Q_{h,k})(v_{H,\Omega}) \|_{H^1(\Omega)} \\
\nonumber&\lesssim& \| v_{H,\Omega}+ Q_{h,\Omega}(v_{H,\Omega}) - \pi_h(v^\eps) \|_{H^1(\Omega)} + k^{d/2} \theta^k \| \pi_{H,\Omega}^{\mbox{\rm \tiny ms}}(v^\eps) \|_{H^1(\Omega)}\\
&\lesssim& \| \pi_{H,\Omega}^{\mbox{\rm \tiny ms}}(v^\eps) - \pi_h(v^\eps) \|_{H^1(\Omega)} + k^{d/2} \theta^k \| v^\eps \|_{H^1(\Omega)},
\end{eqnarray}
where we used again the $H^1$-stability of $P_H$ via the equation $v_{H,\Omega}= P_H(v_{H,\Omega}+ Q_{h,\Omega}(v_{H,\Omega})) = (P_H \circ \pi_{H,\Omega}^{\mbox{\rm \tiny ms}})(v^\eps)$.
We next estimate the term $\| \pi_{H,\Omega}^{\mbox{\rm \tiny ms}}(v^\eps) - \pi_h(v^\eps) \|_{H^1(\Omega)}$ in this estimate. For this, we use the equality
\begin{align}
\label{L2-quasi-orthogonality}( v , w_h )_{L^2(\Omega)} = ( v - I_H(v) , w_h - I_H(w_h) )_{L^2(\Omega)} \qquad \mbox{for all } v \in L^2(\Omega), w_h \in W_h.
\end{align}
This equation holds because of $I_H(w_h)=0$ for all $w_h \in W_h$ and $(v_H,w_h)_{L^2(\Omega)}=0$ for all $v_H \in V_H$ (because $w_h$ is in the kernel of the $L^2$-projection). With that we obtain
\begin{eqnarray*}
\lefteqn{b^{\eps}( \pi_{H,\Omega}^{\mbox{\rm \tiny ms}}(v^\eps) - \pi_h(v^\eps) \hspace{2pt},\hspace{2pt} \pi_{H,\Omega}^{\mbox{\rm \tiny ms}}(v^\eps) - \pi_h(v^\eps) )}\\
&=& b^{\eps}( \pi_h(v^\eps) \hspace{2pt},\hspace{2pt} \pi_h(v^\eps)-\pi_{H,\Omega}^{\mbox{\rm \tiny ms}}(v^\eps) ) \\
&=& b^{\eps}( v^\eps \hspace{2pt},\hspace{2pt} \pi_h(v^\eps)-\pi_{H,\Omega}^{\mbox{\rm \tiny ms}}(v^\eps) )\\
&=& ( \bar{F} - \partial_{tt} v^\eps \hspace{2pt},\hspace{2pt} \pi_h(v^\eps)-\pi_{H,\Omega}^{\mbox{\rm \tiny ms}}(v^\eps) )_{L^2(\Omega)} \\
&\overset{(\ref{L2-quasi-orthogonality})}{=}& \left( \hspace{2pt} (\bar{F} \hspace{-2pt} - \hspace{-2pt} \partial_{tt} v^\eps)-I_H(F \hspace{-2pt} - \hspace{-2pt} \partial_{tt} v^\eps) \hspace{2pt} , \hspace{2pt} (\pi_h(v^\eps) \hspace{-2pt} - \hspace{-2pt} \pi_{H,\Omega}^{\mbox{\rm \tiny ms}}(v^\eps)) - I_H(\pi_h(v^\eps) \hspace{-2pt} - \hspace{-2pt} \pi_{H,\Omega}^{\mbox{\rm \tiny ms}}(v^\eps)) \hspace{2pt} \right)_{L^2(\Omega)} \\
&\overset{(\ref{lemma-properties-clement-op-est})}{\lesssim}&
H^{s+1} \| \bar{F} - \partial_{tt} v^\eps \|_{H^s(\Omega)} \| \pi_h(v^\eps)-\pi_{H,\Omega}^{\mbox{\rm \tiny ms}}(v^\eps)) \|_{H^1(\Omega)}.
\end{eqnarray*}
Combining this with (\ref{general-estimate-pi-H-k}) we get
\begin{eqnarray}
\label{estimate-for-H1-error-pi-Hk}
\| \pi_{H,k}^{\mbox{\rm \tiny ms}}(v^\eps) - \pi_h(v^\eps) \|_{H^1(\Omega)} \lesssim H^{s+1} \| \partial_{tt} v^\eps - \bar{F} \|_{H^s(\Omega)} + k^{d/2} \theta^k \| v^\eps \|_{H^1(\Omega)},
\end{eqnarray}
which proves the estimate (\ref{projection-error-estimate-3-new-1}) for the case $m=1$. Now, we prove the estimate for the case $m=0$ by applying the same Aubin-Nitsche argument as above. Defining $e_{H,k}:= \pi_{H,k}^{\mbox{\rm \tiny ms}}(v^\eps) - \pi_h(v^\eps)$ we are looking for $z_h \in L^2(0,T;V_h)$ and $z_{H,k}^{\mbox{\rm \tiny ms}} \in V_{H,k}^{\ms}$ that are defined analogously to (\ref{dual-prob-1}) and (\ref{dual-prob-2}).
Hence we get with same strategy as before
\begin{align*}
\| z_{H,k}^{\mbox{\rm \tiny ms}} - z_h \|_{H^1(\Omega)} \lesssim (H + k^{d/2} \theta^k) \| e_{H,k} \|_{L^2(\Omega)}
\end{align*}
and hence together with (\ref{estimate-for-H1-error-pi-Hk}) and Youngs inequality
\begin{align*}
&\| e_{H,k} \|_{L^2(\Omega)}^2 = |b^{\eps}(e_{H,k}, z_h - z_{H,k}^{\mbox{\rm \tiny ms}})| \\
&\hspace{3pt}\lesssim
(H^{s+2} + k^{d{(s+2)/(2s+2)}} \theta^{k{(s+2)/(s+1)}}) \left( \| \partial_{tt} v^\eps - \bar{F} \|_{H^s(\Omega)} + \| v^\eps \|_{H^1(\Omega)} \right) \| e_{H,k} \|_{L^2(\Omega)}.
\end{align*}
In total we proved (\ref{projection-error-estimate-3-new-1}) for $m=0$, i.e.
\begin{eqnarray*}
\lefteqn{\| \pi_{H,k}^{\mbox{\rm \tiny ms}}(v^\eps) - v^\eps \|_{L^2(\Omega)}}\\
&\lesssim& \| v^\eps - \pi_h(v^\eps) \|_{L^2(\Omega)} + (H^{s+2} + k^{d{(s+2)/(2s+2)}} \theta^{k{(s+2)/(s+1)}}) \left( \| \partial_{tt} v^\eps - \bar{F} \|_{H^s(\Omega)} + \| v^\eps \|_{H^1(\Omega)} \right).
\end{eqnarray*}
\end{proof}
\subsection{Estimates for well-prepared initial values}\label{subsection-well-prepared-estimates}
In the next step, we exploit the estimates derived for the projections $\pi_{H,k}^{\mbox{\rm \tiny ms}}$ to bound the error for the numerically homogenized solutions $u_{H,k}$. The lemma is a data-explicit (in particular $\varepsilon$-explicit and $T$-explicit) version of the estimates (\ref{main-result-est-2a}) and (\ref{main-result-est-3a}) in Theorem \ref{apriori-semidiscrete}.
\begin{lemma}
\label{lemma-apriori-semidiscrete}
Assume that (H0) holds and let $s \in \{0,1\}$. If
$\partial_t u^\eps \in L^1(H^1_0)$; $\partial_{tt} u^\eps , F \in L^{\infty}(H^s_0)$ and $\partial_{ttt} u^\eps , \partial_t F \in L^1(H^s_0)$ it holds
\begin{eqnarray}
\label{lemma-apriori-semidiscrete-est-2}\lefteqn{\| u^\eps - (u_{H,k} + Q_{h,k}(u_{H,k})) \|_{L^{\infty}(L^2)} }\\
\nonumber
&\lesssim& \| u^\eps - \pi_h(u^\eps) \|_{L^{\infty}(L^2)} + \| \partial_t u^\eps - \pi_h(\partial_t u^\eps) \|_{L^1(L^2)} \\
\nonumber
&\enspace& + (H^{2+s} + k^{d} \theta^{k{(s+2)/(s+1)}}) \left(\| u^\eps \|_{L^{\infty}(H^1)} + \| \partial_t u^\eps \|_{L^1(H^1)} + \| \partial_{tt} u^\eps \|_{L^{\infty}(H^s)} + \| \partial_{ttt} u^\eps \|_{L^1(H^s)} \right. \\
\nonumber
&\enspace& \qquad \left. + \| F \|_{L^{\infty}(H^s)} + \| \partial_t F \|_{L^1(H^s)} \right).
\end{eqnarray}
Recall that the $\lesssim$-notation only contains dependencies on $\Omega$, $d$, $\alpha$, $\beta$ and the shape regularity of $\mathcal{T}_H$, but not on $T$ and $\varepsilon$.
\end{lemma}
The above Lemma proves the first part of Theorem \ref{apriori-semidiscrete}, i.e. estimates (\ref{main-result-est-2a}) and (\ref{main-result-est-3a}), as explained next.\\
\noindent[Proof of estimates (\ref{main-result-est-2a}) and (\ref{main-result-est-3a}) in Theorem \ref{apriori-semidiscrete}] \label{conclusion-3}
Exploiting the time-regularity result presented in Proposition \ref{proposition-time-regularity}, we observe that if $F \in L^{\infty}(H^s_0)\cap W^{1,1}(H^s_0)$ and if the data is {\it well-prepared and compatible of order $2+s$} in the sense of Definition \ref{def-of-comp-condition}, we obtain that $u^\eps$ is sufficiently regular for Lemma \ref{lemma-apriori-semidiscrete} to hold. Furthermore we have
$$\| u^\eps \|_{L^{\infty}(H^1)} + \| \partial_t u^\eps \|_{L^1(H^1)} + \| \partial_{tt} u^\eps \|_{L^{\infty}(H^s)} + \| \partial_{ttt} u^\eps \|_{L^1(H^s)} \le C_w$$
independent of $\varepsilon$.
In order to treat the $\theta$-terms in \eqref{lemma-apriori-semidiscrete-est-2}, we choose
$k:= \frac{(s+1)\ln(H)}{(s+2)\ln(\theta)}(s+2+\delta)$
for some $\delta>0$. This gives us
$$\theta^{k (s+2)/(s+1)} = \theta^{(s+2+\delta) \ln(H) / \ln(\theta)} =e^{ \ln(\theta) (s+2+\delta) \ln(H) / \ln(\theta)}
= e^{ (s+2+\delta) \ln(H) } = H^{s+2+\delta}.
$$
For $k$ as above and $\delta>0$ we hence have $k^{d} \theta^{k(s+2)/(s+1)} \lesssim H^{s+2}$. The constant $C_{\theta}$ in Theorem \ref{apriori-semidiscrete} can hence be chosen as $C_{\theta}=\frac{8}{3 |\ln(\theta)|}$ in the worst case.
This ends the proof of (\ref{main-result-est-2a}) and (\ref{main-result-est-3a}) in Theorem \ref{apriori-semidiscrete}.
\hfill$\square$
\begin{proof}[Proof of Lemma \ref{lemma-apriori-semidiscrete}]
To prove the result, we can follow the arguments of Baker \cite{Bak76}. For the numerically homogenized solution $u_{H,k}$ of (\ref{semi-discrete-lod-equation}), we define $u_{H,k}^{\ms}:=u_{H,k} + Q_{h,k}(u_{H,k})$. For brevity, we denote $(\cdot,\cdot):=(\cdot,\cdot)_{L^2(\Omega)}$. Furthermore, we use the notation from Lemma \ref{lemma-multiscale-projection} and define the errors
\begin{align*}
e^{\mbox{\rm \tiny ms}}
:= u^\eps - u_{H,k}^{\ms}, \quad
e^{\pi}
:= u^\eps - \pi_{H,k}^{\mbox{\rm \tiny ms}}(u^\eps) \quad \mbox{and} \quad
\psi^{\pi}
:= u_{H,k}^{\ms} - \pi_{H,k}^{\mbox{\rm \tiny ms}}(u^\eps).
\end{align*}
Observe that we have for $v \in L^2(0,T;V_{H,k}^{\ms})$ and almost every $t \ge 0$:
\begin{eqnarray*}
\lefteqn{0 = ( \hspace{2pt} \partial_{tt} \psi^{\pi}(\cdot,t) - \partial_{tt} e^{\pi}(\cdot,t) , v (\cdot,t) \hspace{2pt} ) + b^{\eps}( \hspace{2pt} \psi^{\pi}(\cdot,t), v (\cdot,t) \hspace{2pt} )}\\
&=& \partial_{t} ( \hspace{2pt} \partial_{t} \psi^{\pi}(\cdot,t) - \partial_{t} e^{\pi}(\cdot,t) , v (\cdot,t) \hspace{2pt} )
- ( \hspace{2pt} \partial_{t} \psi^{\pi}(\cdot,t) - \partial_{t} e^{\pi}(\cdot,t) , \partial_{t} v (\cdot,t) \hspace{2pt} )
+ b^{\eps}( \hspace{2pt} \psi^{\pi}(\cdot,t), v (\cdot,t) \hspace{2pt} )\\
&=& - \partial_{t} ( \hspace{2pt} \partial_{t} e^{\mbox{\rm \tiny ms}}(\cdot,t) , v (\cdot,t) \hspace{2pt} ) - ( \hspace{2pt} \partial_{t} \psi^{\pi}(\cdot,t) - \partial_{t} e^{\pi}(\cdot,t) , \partial_{t} v (\cdot,t) \hspace{2pt} )
+ b^{\eps}( \hspace{2pt} \psi^{\pi}(\cdot,t), v (\cdot,t) \hspace{2pt} ).
\end{eqnarray*}
For some arbitrary $0<t_0\le T$ we use the function $v (\cdot,t) = \int_t^{t_0} \psi^{\pi}(\cdot,s) \hspace{2pt} ds$ in the above equation (and the fact that $\partial_t v = - \psi^{\pi}$) to obtain
\begin{eqnarray*}
\lefteqn{ \frac{1}{2} \frac{d}{dt} \| \psi^{\pi}(\cdot,t) \|_{L^2(\Omega)}^2
- \frac{1}{2} \frac{d}{dt} b^{\eps} \left( \hspace{2pt} \int_t^{t_0} \psi^{\pi}(\cdot,s) \hspace{2pt} ds , \int_t^{t_0} \psi^{\pi}(\cdot,s) \hspace{2pt} ds \hspace{2pt} \right) } \\
&=& \partial_{t} \left( \hspace{2pt} \partial_{t} e^{\mbox{\rm \tiny ms}}(\cdot,t) , \int_t^{t_0} \psi^{\pi}(\cdot,s) \hspace{2pt} ds \hspace{2pt} \right) + ( \partial_{t} e^{\pi}(\cdot,t) , \psi^{\pi} (\cdot,t) \hspace{2pt} ).
\end{eqnarray*}
Integration from $0$ to ${t_0}$ yields
\begin{eqnarray*}
\lefteqn{ \frac{1}{2} \| \psi^{\pi}(\cdot,t_0) \|_{L^2(\Omega)}^2 - \frac{1}{2} \| \psi^{\pi}(\cdot,0) \|_{L^2(\Omega)}^2
+ \frac{1}{2} b^{\eps} \left( \hspace{2pt} \int_0^{t_0} \psi^{\pi}(\cdot,s) \hspace{2pt} ds , \int_0^{t_0} \psi^{\pi}(\cdot,s) \hspace{2pt} ds \hspace{2pt} \right) } \\
&=& - \left( \hspace{2pt} \partial_{t} e^{\mbox{\rm \tiny ms}}(\cdot,0) , \int_0^{t_0} \psi^{\pi}(\cdot,s) \hspace{2pt} ds \hspace{2pt} \right) + \int_0^{t_0} ( \partial_{t} e^{\pi}(\cdot,t) , \psi^{\pi} (\cdot,t) \hspace{2pt} ) \hspace{2pt} dt.
\end{eqnarray*}
Hence
\begin{eqnarray*}
\lefteqn{\| \psi^{\pi}(\cdot,t_0) \|_{L^2(\Omega)}^2}\\
&\le& \| \psi^{\pi}(\cdot,0) \|_{L^2(\Omega)}^2 - 2 \left( \hspace{2pt} \partial_{t} e^{\mbox{\rm \tiny ms}}(\cdot,0) , \int_0^{t_0} \psi^{\pi}(\cdot,s) \hspace{2pt} ds \hspace{2pt} \right) + 2 \int_0^{t_0} ( \partial_{t} e^{\pi}(\cdot,t) , \psi^{\pi} (\cdot,t) \hspace{2pt} ) \hspace{2pt} dt \\
&\le& \| \psi^{\pi}(\cdot,0) \|_{L^2(\Omega)}^2 + 2 \int_0^{t_0} ( \partial_{t} e^{\pi}(\cdot,t) , \psi^{\pi} (\cdot,t) \hspace{2pt} ) \hspace{2pt} dt \\
&\le& \| \psi^{\pi}(\cdot,0) \|_{L^2(\Omega)}^2 +
2 \| \partial_{t} e^{\pi} \|_{L^1(0,T;L^2(\Omega))}^2
+ \frac{1}{2} \| \psi^{\pi} \|_{L^{\infty}(0,T;L^2(\Omega))}^2.
\end{eqnarray*}
By moving the term $\| \psi^{\pi} \|_{L^{\infty}(0,T;L^2(\Omega))}^2$ to the left hand side, we get
\begin{align}
\label{lemma-first-est-befor-tri-ineq}\| \psi^{\pi} \|_{L^{\infty}(0,T;L^2(\Omega))} \lesssim \| \psi^{\pi}(\cdot,0) \|_{L^2(\Omega)} +
\| \partial_{t} e^{\pi} \|_{L^1(0,T;L^2(\Omega))}.
\end{align}
However, since $u_{H,k}^{\ms}(\cdot,0)=\pi_{H,k}^{\mbox{\rm \tiny ms}}(f)$, we get $\psi^{\pi}(\cdot,0)=\pi_{H,k}^{\mbox{\rm \tiny ms}}(f)-\pi_{H,k}^{\mbox{\rm \tiny ms}}(u^\eps(\cdot,0))=0$. Hence, together with the triangle inequality for $\psi^{\pi}=(u_{H,k}^{\ms} - u^\eps) + (u^\eps - \pi_{H,k}^{\mbox{\rm \tiny ms}}(u^\eps))$, equation (\ref{lemma-first-est-befor-tri-ineq}) implies
\begin{align*}
\| u_{H,k}^{\ms} - u^\eps \|_{L^{\infty}(0,T;L^2(\Omega))} \lesssim \| u^\eps - \pi_{H,k}^{\mbox{\rm \tiny ms}}(u^\eps) \|_{L^{\infty}(0,T;L^2(\Omega))} +
\| \partial_{t} u^\eps - \pi_{H,k}^{\mbox{\rm \tiny ms}}(\partial_{t} u^\eps) \|_{L^1(0,T;L^2(\Omega))}.
\end{align*}
Together with Lemma \ref{lemma-multiscale-projection} this finishes the proof of (\ref{lemma-apriori-semidiscrete-est-2}).
\end{proof}
Next, we prove an $\varepsilon$-explicit and $T$-explicit version of the estimates
from the second part of Theorem \ref{apriori-semidiscrete}.
\noindent[Proof of estimates (\ref{main-result-est-2b}) and (\ref{main-result-est-3b}) in Theorem \ref{apriori-semidiscrete}]
Similarly as for the first part of Theorem \ref{apriori-semidiscrete}, these
are obtained by combining Lemma \ref{lemma-apriori-semidiscrete-H1} below with the regularity statement in Proposition \ref{proposition-time-regularity}.
\hfill$\square$
\begin{lemma}
\label{lemma-apriori-semidiscrete-H1}
Let (H0) be fulfilled, let $s \in \{0,1\}$ and assume
$\partial_t u^\eps \in L^{\infty}(H^1_0)$; $\partial_{ttt} u^\eps, \partial_t F \in L^{\infty}(L^2)$; $\partial_{tt} u^\eps, F \in L^{\infty}(H^s_0)$; $\partial_{tt} u^\eps \in L^1(H^1_0)$; $\partial_{t}^4 u^\eps , \partial_{tt} F \in L^1(L^2)$ and $g \in H^1_0(\Omega)$.\\
If $s=0$, it holds
\begin{eqnarray}
\label{lemma-apriori-semidiscrete-est-H1-2}\lefteqn{\| \partial_t u^\eps - \partial_t (u_{H,k} + Q_{h,k}(u_{H,k})) \|_{L^{\infty}(L^2)}
+ \| u^\eps - (u_{H,k} + Q_{h,k}(u_{H,k})) \|_{L^{\infty}(H^1)}}\\
\nonumber
&\lesssim& \| \partial_t u^\eps - \pi_h(\partial_t u^\eps) \|_{L^{\infty}(L^2)}
+ \| \partial_{tt} u^\eps - \pi_h(\partial_{tt} u^\eps) \|_{L^1(L^2)} + \|u^\eps - \pi_h(u^\eps) \|_{L^{\infty}(H^1)} \\
\nonumber
&\enspace& + (H + k^{d/2} \theta^{k}) \left(
{\sum_{i=0}^1} \| \partial^i_t u^\eps \|_{L^{\infty}(H^1)}
+ \| \partial_{tt} u^\eps \|_{L^{\infty}(L^2)}
+ \| \partial_{tt} u^\eps \|_{L^{1}(H^1)}
+ \| \partial_{ttt} u^\eps \|_{L^{\infty}(L^2)} \right)\\
\nonumber&\enspace& + (H + k^{d/2} \theta^{k}) \left(
\| \partial_{t}^4 u^\eps \|_{L^{1}(L^2)}
+ {\sum_{i=0}^1} \| \partial_t^i F \|_{L^{\infty}(L^2)}
+ \| \partial_{tt} F \|_{L^1(L^2)}
+ \| g \|_{H^1(\Omega)} \right).
\end{eqnarray}
If $s=1$ and if the initial value in (\ref{semi-discrete-lod-equation}) is picked such that $\partial_t (u_{H,k}+Q_{h,k}(u_{H,k}))( \cdot , 0 ) = \pi_{H,k}^{\mbox{\rm \tiny ms}}(g)$, then we obtain the improved estimate
\begin{eqnarray}
\label{lemma-apriori-semidiscrete-est-H1-3}\lefteqn{\| \partial_t u^\eps - \partial_t (u_{H,k} + Q_{h,k}(u_{H,k})) \|_{L^{\infty}(L^2)}
+ \| u^\eps - (u_{H,k} + Q_{h,k}(u_{H,k})) \|_{L^{\infty}(H^1)}}\\
\nonumber
&\lesssim& \| \partial_t u^\eps - \pi_h(\partial_t u^\eps) \|_{L^{\infty}(L^2)}
+ \| \partial_{tt} u^\eps - \pi_h(\partial_{tt} u^\eps) \|_{L^1(L^2)} + \|u^\eps - \pi_h(u^\eps) \|_{L^{\infty}(H^1)} \\
\nonumber
&\enspace& + (H^2 + k^{d/2} \theta^{k}) \left(
{\sum_{i=0}^2} \| \partial^i_t u^\eps \|_{L^{\infty}(H^1)}
+ \| \partial_{tt} u^\eps \|_{L^{1}(H^1)}
+ \| \partial_{ttt} u^\eps \|_{L^{\infty}(L^2)} \right)\\
\nonumber&\enspace& + (H^2 + k^{d/2} \theta^{k}) \left(
\| \partial_{t}^4 u^\eps \|_{L^{1}(L^2)}
+ \| F \|_{L^{\infty}(H^1)}
+ \| \partial_t F \|_{L^{\infty}(L^2)}
+ \| \partial_{tt} F \|_{L^1(L^2)} \right).
\end{eqnarray}
Again, recall that the $\lesssim$-notation only contains dependencies on $\Omega$, $d$, $\alpha$, $\beta$ and the shape regularity of $\mathcal{T}_H$, but not on $T$ and $\varepsilon$.
\end{lemma}
\begin{proof}
Again, we define the errors
$
e^{\mbox{\rm \tiny ms}}
:= u^\eps - u_{H,k}^{\ms}$, $e^{\pi}:= u^\eps - \pi_{H,k}^{\mbox{\rm \tiny ms}}(u^\eps)$ and $ \psi^{\pi}
:= u_{H,k}^{\ms} - \pi_{H,k}^{\mbox{\rm \tiny ms}}(u^\eps)$.
We only consider the case $\partial_t u_{H,k}^{\ms}( \cdot , 0 ) = P_{H,k}^{\mbox{\rm \tiny ms}}(g)$ (i.e. estimate (\ref{lemma-apriori-semidiscrete-est-H1-2})), the case $\partial_t u_{H,k}^{\ms}( \cdot , 0 ) = \pi_{H,k}^{\mbox{\rm \tiny ms}}(g)$ (i.e. estimate (\ref{lemma-apriori-semidiscrete-est-H1-3})) follows analogously with $\partial_t \psi^{\pi}(\cdot,0)=0$.
By Galerkin orthogonality we obtain for a.e. $t\in[0,T]$
\begin{align*}
(\partial_{tt} e^{\mbox{\rm \tiny ms}}(\cdot,t) , v^{\mbox{\rm \tiny ms}} )_{L^2(\Omega)} + b^{\eps}( e^{\mbox{\rm \tiny ms}} (\cdot,t), v^{\mbox{\rm \tiny ms}} ) = 0 \qquad \mbox{for all } v^{\mbox{\rm \tiny ms}} \in V_{H,k}^{\ms}
\end{align*}
and hence
\begin{align*}
(\partial_{tt} \psi^{\pi}(\cdot,t) , v^{\mbox{\rm \tiny ms}} )_{L^2(\Omega)} + b^{\eps}( \psi^{\pi} (\cdot,t), v^{\mbox{\rm \tiny ms}} ) =
(\partial_{tt} e^{\pi}(\cdot,t) , v^{\mbox{\rm \tiny ms}} )_{L^2(\Omega)}
\qquad \mbox{for all } v^{\mbox{\rm \tiny ms}} \in V_{H,k}^{\ms}.
\end{align*}
Testing with $v^{\mbox{\rm \tiny ms}} = \partial_t \psi^{\pi}$ yields for a.e. $t \in [0,T]$
\begin{eqnarray*}
\frac{1}{2} \frac{\mbox{d}}{\mbox{d}t} \left(
\| \partial_t \psi^{\pi}(\cdot,t) \|^2_{L^2(\Omega)} + b^{\eps}( \psi^{\pi}(\cdot,t),\psi^{\pi}(\cdot,t))
\right) = (\partial_{tt} e^{\pi}(\cdot,t) , \partial_t \psi^{\pi}(\cdot,t) )_{L^2(\Omega)}.
\end{eqnarray*}
By integration from $0$ to $t_0 \le T$ we obtain
\begin{eqnarray*}
\lefteqn{\frac{1}{2} \| \partial_t \psi^{\pi}(\cdot,t_0) \|^2_{L^2(\Omega)} + \frac{\alpha}{2} \| \psi^{\pi}(\cdot,t_0) \|_{H^1(\Omega)}^2}\\
&\le& \frac{1}{2} \| \partial_t \psi^{\pi}(\cdot,0) \|^2_{L^2(\Omega)} + \frac{\beta}{2} \| \psi^{\pi}(\cdot,0) \|_{H^1(\Omega)}^2
+ \int_{0}^{t_0} |(\partial_{tt} e^{\pi}(\cdot,t) , \partial_t \psi^{\pi}(\cdot,t) )_{L^2(\Omega)}| \hspace{2pt} dt.
\end{eqnarray*}
Since we have $\psi^{\pi}(\cdot,0) = \pi_{H,k}^{\mbox{\rm \tiny ms}}(f) -\pi_{H,k}^{\mbox{\rm \tiny ms}}(f) =0$ we get with the Young and the Cauchy-Schwarz inequality
\begin{eqnarray*}
\lefteqn{\frac{1}{2} \| \partial_t \psi^{\pi}(\cdot,t_0) \|^2_{L^2(\Omega)} + \frac{\alpha}{2} \| \psi^{\pi}(\cdot,t_0) \|_{H^1(\Omega)}^2}\\
&\le& \frac{1}{2} \| \pi_{H,k}^{\mbox{\rm \tiny ms}}(g) - P_{H,k}^{\mbox{\rm \tiny ms}}(g) \|^2_{L^2(\Omega)}
+ \| \partial_{tt} e^{\pi} \|_{L^1(0,T;L^2(\Omega))}^2 + \frac{1}{4} \| \partial_t \psi^{\pi} \|_{L^{\infty}(0,T;L^2(\Omega))}.
\end{eqnarray*}
By taking the supremum over all $0\le t_0 \le T$ we obtain
\begin{eqnarray*}
{\| \partial_t \psi^{\pi} \|^2_{L^{\infty}(L^2)} + 2 \alpha \| \psi^{\pi} \|_{L^{\infty}(H^1)}^2}
\le2 \| \pi_{H,k}^{\mbox{\rm \tiny ms}}(g) - P_{H,k}^{\mbox{\rm \tiny ms}}(g) \|^2_{L^2(\Omega)}
+ 4 \| \partial_{tt} e^{\pi} \|_{L^1(L^2)}^2.
\end{eqnarray*}
The term $\| \partial_{tt} e^{\pi} \|_{L^1(L^2)}$ can be treated with Lemma \ref{lemma-multiscale-projection}, equation (\ref{projection-error-estimate-3-new-1}). Hence it only remains to estimate the term $ \| \pi_{H,k}^{\mbox{\rm \tiny ms}}(g) - P_{H,k}^{\mbox{\rm \tiny ms}}(g) \|^2_{L^2(\Omega)}$. Observe that this term vanishes in \eqref{lemma-apriori-semidiscrete-est-H1-3} as $\partial_t \psi^{\pi}(\cdot,0)=0$. For this term, we have
\begin{align*}
\| \pi_{H,k}^{\mbox{\rm \tiny ms}}(g) - P_{H,k}^{\mbox{\rm \tiny ms}}(g) \|_{L^2(\Omega)} &\le \| \pi_{H,k}^{\mbox{\rm \tiny ms}}(g) - g \|_{L^2(\Omega)} + \| P_{H,k}^{\mbox{\rm \tiny ms}}(g) - g \|_{L^2(\Omega)}\\
&\le 2 \| \pi_{H,k}^{\mbox{\rm \tiny ms}}(g) - g \|_{L^2(\Omega)} = 2 \| \pi_{H,k}^{\mbox{\rm \tiny ms}}(g) - g - I_H( \pi_{H,k}^{\mbox{\rm \tiny ms}}(g) - g) \|_{L^2(\Omega)} \\
&\lesssim H \| \pi_{H,k}^{\mbox{\rm \tiny ms}}(g) - g \|_{H^1(\Omega)} \lesssim H \| g \|_{H^1(\Omega)}.
\end{align*}
Hence, the triangle inequality yields
\begin{eqnarray*}
\lefteqn{\| \partial_t e^{\mbox{\rm \tiny ms}} \|_{L^{\infty}(L^2)} + \| e^{\mbox{\rm \tiny ms}} \|_{L^{\infty}(H^1)}}\\
&\lesssim& \| \partial_t u^\eps- \pi_{H,k}^{\mbox{\rm \tiny ms}}(\partial_t u^\eps) \|_{L^{\infty}(L^2)} + \| u^\eps- \pi_{H,k}^{\mbox{\rm \tiny ms}}( u^\eps) \|_{L^{\infty}(H^1)}\\
&\enspace& + \| \partial_{tt} u^\eps- \pi_{H,k}^{\mbox{\rm \tiny ms}}(\partial_{tt} u^\eps) \|_{L^1(L^2)} + H \| g \|_{H^1(\Omega)}.
\end{eqnarray*}
Lemma \ref{lemma-multiscale-projection} finishes the proof.
\end{proof}
The proof of Lemmas \ref{lemma-apriori-semidiscrete}
and \ref{lemma-apriori-semidiscrete-H1} hence finish as explained the
proof of Theorem \ref{apriori-semidiscrete}.
We conclude this section by proving the fully-discrete estimate stated in Theorem \ref{apriori-fullydiscrete}.
\begin{proof}[Proof of Theorem \ref{apriori-fullydiscrete}]
The first part of the proof is completely analogous to the one presented by Baker \cite[Section 4]{Bak76} for the classical finite element method. With the same arguments, we can show that
\begin{eqnarray*}
\lefteqn{\max_{0 \le n \le J} \| (u^\eps - u_{H,\triangle t, k} - Q_{h,k}(u_{H,\triangle t, k}) )(\cdot, t^n ) \|_{L^2(\Omega)}}\\
&\lesssim& \max_{0 \le n \le J} \| (u^\eps - \pi_{H,k}^{\mbox{\rm \tiny ms}}(u^\eps)) (\cdot, t^n ) \|_{L^2(\Omega)} \\
&\enspace& + \| (u_{H,\triangle t, k} + Q_{h,k}(u_{H,\triangle t, k}))(\cdot,0) - \pi_{H,k}^{\mbox{\rm \tiny ms}}(u^\eps(\cdot,0)) \|_{L^2(\Omega)} \\
&\enspace& + \| (\partial_t u^\eps - \pi_{H,k}^{\mbox{\rm \tiny ms}}(\partial_t u^\eps)) \|_{L^2(0,T;L^2(\Omega))} + \triangle t^2 \left( \| \partial_{t}^3 u^\eps \|_{L^2(0,T;L^2(\Omega))} + \| \partial_{t}^4 u^\eps \|_{L^2(0,T;L^2(\Omega))} \right),
\end{eqnarray*}
where we note that the term $ \| (u_{H,\triangle t, k} + Q_{h,k}(u_{H,\triangle t, k}))(\cdot,0) - \pi_{H,k}^{\mbox{\rm \tiny ms}}(u^\eps(\cdot,0)) \|_{L^2(\Omega)}$ is equal to zero, since obviously $(u_{H,\triangle t, k} + Q_{h,k}(u_{H,\triangle t, k}))(\cdot,0) = \pi_{H,k}^{\mbox{\rm \tiny ms}}(f)$ by definition of the method.
The last two terms are already readily estimated. It only remains to bound the two terms $\| (u^\eps - \pi_{H,k}^{\mbox{\rm \tiny ms}}(u^\eps)) (\cdot, t^n ) \|_{L^2(\Omega)}$ and $\| (\partial_t u^\eps - \pi_{H,k}^{\mbox{\rm \tiny ms}}(\partial_t u^\eps)) \|_{L^2(L^2)}$ using (\ref{projection-error-estimate-3-new-1}). That finishes the proof.
\end{proof}
\subsection{Estimates in the setting of $G$-convergence}\label{subsection-G-conv-estimates}
In this subsection we prove the homogenization result stated in Theorem \ref{apriori-lod-homogenization}. Consequently, we assume that we are in the homogenization setting of $G$-convergence as established in Definition \ref{def-G-convergence} and Theorem \ref{theorem:homogenization:wave:equation}.
Before we can start the proof, we need to introduce an auxiliary problem. Define
$\hat{u}^\eps \in L^2(0,T;H^1_0(\Omega))$
as the solution to the wave equation
\begin{align}
\label{wave-equation-weak-f-eps}
\nonumber\langle \partial_{tt} \hat{u}^\eps (\cdot,t), v \rangle + \left( a^\eps \nabla \hat{u}^\eps (\cdot,t), \nabla v \right)_{L^2(\Omega)} &= \left( F (\cdot,t), v \right)_{L^2(\Omega)},\\
\left( \hat{u}^\eps( \cdot , 0 ) , v \right)_{L^2(\Omega)} = \left( f^{\varepsilon} , v \right)_{L^2(\Omega)},\quad
\left( \partial_t \hat{u}^\eps( \cdot , 0 ) , v \right)_{L^2(\Omega)} &= \left( g , v \right)_{L^2(\Omega)}
\end{align}
for all $v \in H^1_0(\Omega)$ and a.e. $t > 0$,
where the $f^{\varepsilon}$ are defined from the initial value of the original wave equation by \eqref{def-f-eps}.
We recall the definition of the homogenization error
\begin{align}
\label{def-hom-error}
e_{\mbox{\tiny \rm hom}}(\varepsilon)
:= \| u^0 - u^\eps \|_{L^{\infty}(L^2)} + \| f - f^{\varepsilon} \|_{L^2(\Omega)}
\end{align}
and define the following fine scale discretization error
\begin{align}
\label{def-fine-disc-error}
&\hat{e}_{\mbox{\tiny \rm disc}}(h) := \\
\nonumber &\quad
\| \hat{u}^\eps \hspace{-2pt}-\hspace{-2pt} \pi_h(\hat{u}^\eps) \|_{L^{\infty}(L^2)}
+ \| \partial_t \hat{u}^\eps \hspace{-2pt}-\hspace{-2pt} \pi_h(\partial_t \hat{u}^\eps) \|_{L^2(L^2)}
+ \| f^{\varepsilon} \hspace{-2pt}-\hspace{-2pt} \pi_h(f^{\varepsilon}) \|_{L^2(\Omega)}
+ \| f \hspace{-2pt}-\hspace{-2pt} \pi_h(f) \|_{L^2(\Omega)}.
\end{align}
The following lemma is the main ingredient to prove Theorem \ref{apriori-lod-homogenization}.
\begin{lemma}
\label{lemma-apriori-hom}
Assume that (H0) holds. Let furthermore $g\in H^1_0(\Omega)$, $\partial_t F \in L^2(0,T,L^2(\Omega))$ and $\nabla \cdot (a^0 \nabla f) + F(\cdot,0)\in L^2(\Omega)$.
Then it holds
\begin{eqnarray}
\label{lemma-apriori-semidiscrete-est-1}\lefteqn{\| u^0 - u_{H,k} \|_{L^{\infty}(L^2)} \lesssim_T e_{\mbox{\tiny \rm hom}}(\varepsilon) + \hat{e}_{\mbox{\tiny \rm disc}}(h)} \\
\nonumber
&+& ( H + \theta^k k^{d/2}) \left( \| F \|_{W^{1,2}(L^2)} + \| f \|_{H^1(\Omega)} + \| g \|_{H^1(\Omega)}
+ \| \nabla \cdot (a^0 \nabla f)
\|_{L^2(\Omega)} \right).
\end{eqnarray}
If we replace the elliptic projection $\pi_{H,k}^{\mbox{\rm \tiny ms}}(f)$ in (\ref{semi-discrete-lod-equation}) by the $L^2$-projection $P_{H,k}^{\mbox{\rm \tiny ms}}(f)$, the term $\| f \hspace{-2pt}-\hspace{-2pt} \pi_h(f) \|_{L^2(\Omega)}$ in the definition of $\hat{e}_{\mbox{\tiny \rm disc}}(h)$ can be dropped.
\end{lemma}
\noindent[Proof of estimates \eqref{main-result-est-1} and \eqref{main-result-est-2} in Theorem \ref{apriori-lod-homogenization}.]
We observe that under the assumptions of Theorem \ref{apriori-lod-homogenization} we obviously have
\begin{align*}
\lim_{\varepsilon \rightarrow 0} e_{\mbox{\tiny \rm hom}}(\varepsilon) = 0 \qquad \mbox{and} \qquad \lim_{h \rightarrow 0} \hat{e}_{\mbox{\tiny \rm disc}}(h) = 0.
\end{align*}
Combining this observation with \eqref{lemma-apriori-semidiscrete-est-1} gives \eqref{main-result-est-1}. Next using a triangle inequality
for the term $(u^\varepsilon- u_{H,k})=(u^\varepsilon- u^0)+(u^0-u_{H,k})$ together with the estimate \eqref{main-result-est-1} yields \eqref{main-result-est-2}.
\hfill$\square$
\begin{remark}
If we are in the homogenization setting of $G$-convergence and if $h<\varepsilon$, we can assume $e_{\mbox{\tiny \rm hom}}(\varepsilon) + \hat{e}_{\mbox{\tiny \rm disc}}(h) \lesssim H$. This bound resembles the fact the fine grid resolves the microstructures and that the coarse grid is still coarse compared to the speed of the data oscillations in $a^\eps$. In this case estimate \eqref{lemma-apriori-semidiscrete-est-1} simplifies to
$$
\| u^0 - u_{H,k}\|_{L^{\infty}(L^2)} \lesssim_T H + \theta^k k^{d/2}.
$$
\end{remark}
\begin{proof}
First, recall that $P_{H,k}^{\mbox{\rm \tiny ms}}$ denotes the $L^2$-projection on $V_{H,k}^{\ms}$ and $\pi_{H,k}^{\mbox{\rm \tiny ms}}$ the elliptic projection. In this proof we treat both cases, i.e. $(u_{H,k}+Q_{h,k}(u_{H,k}))( \cdot , 0 ) = \pi_{H,k}^{\mbox{\rm \tiny ms}}(f)$ and $(u_{H,k}+Q_{h,k}(u_{H,k}))( \cdot , 0 ) = P_{H,k}^{\mbox{\rm \tiny ms}}(f)$, at the same time. The main proof consists of six steps. In the first step, we state some properties for the auxiliary problem defined via \eqref{wave-equation-weak-f-eps}. In the second step we split the error $\| u^0 - u_{H} \|_{L^{\infty}(L^2)}$ into several contributions. In the last four steps, these contributions are estimated and combined.
$\\$
{\it Step 1 (auxiliary problem and properties).} Let $f^{\varepsilon} \in H^1_0(\Omega)$ denote the solution of the auxiliary problem \eqref{def-f-eps}. Hence, we have
\begin{align}
\label{energy-f-eps}\| \nabla f^{\varepsilon} \|_{L^2(\Omega)}\le \frac{\beta}{\alpha} \| \nabla f \|_{L^2(\Omega)}.
\end{align}
With this, we use the following notation. By $u^\eps$
we denote the solution to the original wave equation \eqref{wave-equation-weak} and by
$\hat{u}^\eps$ the solution to the wave equation \eqref{wave-equation-weak-f-eps}, i.e. the wave equation with modified initial value $f^{\varepsilon}$. First, exploiting \eqref{energy-f-eps}, we observe that $\hat{u}^\eps$ fulfills the classical energy estimate
\begin{align}
\label{data-est-1}
\| \partial_t \hat{u}^\eps \|_{L^{\infty}(0,T;L^2(\Omega))}
+ \| \hat{u}^\eps \|_{L^{\infty}(0,T;H^1(\Omega))}
\lesssim_{T} \| F \|_{L^2(0,T;L^2(\Omega))} + \| g \|_{L^2(\Omega)} + \| \nabla f \|_{L^2(\Omega)}.
\end{align}
Furthermore, since by construction $\nabla \cdot (a^\eps \nabla f^{\varepsilon}) = \nabla \cdot (a^0 \nabla f) \in L^2(\Omega)$, we can use Proposition \ref{proposition-time-regularity} to verify that
$$
\partial_t \hat{u}^\eps \in L^{\infty}(0,T;H^1_0(\Omega)) \qquad \mbox{and} \qquad \partial_{tt} \hat{u}^\eps \in L^{\infty}(0,T;L^2(\Omega)).
$$
Consequently we have for all $v\in H^1_0(\Omega)$
\begin{eqnarray*}
\lefteqn{(\partial_{tt} \hat{u}^\eps(\cdot, 0) , v )_{L^2(\Omega)} = - ( a^\eps \nabla \hat{u}^\eps(\cdot, 0) , \nabla v)_{L^2(\Omega)} + (F(\cdot, 0) , v )_{L^2(\Omega)}}\\
&=& - ( a^\eps \nabla f^{\varepsilon} , \nabla v)_{L^2(\Omega)} + (F(\cdot, 0) , v )_{L^2(\Omega)}
= (F(\cdot, 0) + \nabla \cdot (a^0 \nabla f ) , v )_{L^2(\Omega)}
\end{eqnarray*}
and therefore $\| \partial_{tt} \hat{u}^\eps(\cdot, 0) \|_{L^2(\Omega)} = \| F(\cdot, 0) + \nabla \cdot (a^0 \nabla f ) \|_{L^2(\Omega)}$.
This yields the second energy estimate
\begin{eqnarray}
\label{data-est-2}
\nonumber\lefteqn{\| \partial_{tt} \hat{u}^\eps \|_{L^{\infty}(0,T;L^2(\Omega))}
+ \| \partial_{t} \hat{u}^\eps \|_{L^{\infty}(0,T;H^1(\Omega))}}\\
\nonumber&\lesssim_T& \| \partial_{t} F \|_{{L^2}(0,T;L^2(\Omega))} + \| \nabla g \|_{L^2(\Omega)} +\| \partial_{tt} \hat{u}^\eps(\cdot, 0) \|_{L^2(\Omega)}\\
\nonumber&=& \| \partial_{t} F \|_{{L^2}(0,T;L^2(\Omega))} + \| \nabla g \|_{L^2(\Omega)} +\| F(\cdot, 0) + \nabla \cdot (a^0 \nabla f ) \|_{L^2(\Omega)} \\
&\lesssim_T& \| F \|_{W^{1,2}(0,T;L^2(\Omega))} + \| \nabla g \|_{L^2(\Omega)} +\| \nabla \cdot (a^0 \nabla f ) \|_{L^2(\Omega)},
\end{eqnarray}
where we used $\| F \|_{L^{\infty}(L^2)} \lesssim_T \| F \|_{W^{1,2}(L^2)}$.
$\\$
{\it Step 2 (error splitting).} Next, we let $u_{H,k} \in H^2(0,T;V_H)$ denote the solution of the multiscale method (\ref{semi-discrete-lod-equation}) and we let $\hat{u}_{H,k} \in H^2(0,T;V_H)$ denote the solution of (\ref{semi-discrete-lod-equation}) with $f$ replaced by $f^{\varepsilon}$.
For simplicity, we also define
$$
u_{H,k}^{\ms}:=u_{H,k} + Q_{h,k}(u_{H,k}) \qquad \mbox{and} \qquad \hat{u}_{H,k}^{\ms} := \hat{u}_{H,k} + Q_{h,k}(\hat{u}_{H,k}).
$$
With that, we split the total error in the following contributions
\begin{eqnarray*}
\lefteqn{\| u^0 - u_{H,k} \|_{L^{\infty}(L^2)} \le
}\\
&\enspace&
\underset{=:\mbox{I}}{\underbrace{\| u^0 - \hat{u}^\eps \|_{L^{\infty}(L^2)}}} +
\underset{=:\mbox{II}}{\underbrace{\| \hat{u}_{H,k}^{\ms} - u_{H,k}^{\ms} \|_{L^{\infty}(L^2)}}} +
\underset{=:\mbox{III}}{\underbrace{\| \hat{u}^\eps - \hat{u}_{H,k}^{\ms} \|_{L^{\infty}(L^2)}}} +
\underset{=:\mbox{IV}}{\underbrace{\| Q_{h,k}(u_{H,k}) \|_{L^{\infty}(L^2)}}}.
\end{eqnarray*}
{\it Step 3.} We start with estimating $I$. Let us denote $\hat{e}^{\varepsilon}:=u^\eps-\hat{u}^\eps$. Exploiting the definitions of $u^\eps$ and $\hat{u}^\eps$ we obtain via \eqref{wave-equation-weak} that
\begin{align*}
\langle \partial_{tt} \hat{e}^{\varepsilon} (\cdot,t), v \rangle + \left( a^\eps \nabla \hat{e}^{\varepsilon} (\cdot,t), \nabla v \right)_{L^2(\Omega)} &= 0 \qquad \mbox{for all } v \in H^1_0(\Omega).
\end{align*}
Furthermore, we have $\hat{e}^{\varepsilon}( \cdot , 0 ) = f-f^{\varepsilon}$ and
$\partial_t \hat{e}^{\varepsilon}( \cdot , 0 ) = 0$.
Testing with $v(\cdot,t)=\int_{t}^{t_0} \hat{e}^{\varepsilon} ( \cdot, s ) \hspace{2pt} ds$ for any $t_0 \in [0,T]$ we obtain
\begin{eqnarray*}
\lefteqn{\frac{d}{dt} \langle \partial_{t} \hat{e}^{\varepsilon} (\cdot,t), \int_{t}^{t_0} \hat{e}^{\varepsilon} ( \cdot, s ) \hspace{2pt} ds \rangle
- \frac{1}{2} \frac{d}{dt}\left( a^\eps \nabla \int_{t}^{t_0} \hat{e}^{\varepsilon}( \cdot, s ) \hspace{2pt} ds, \nabla \int_{t}^{t_0} \hat{e}^{\varepsilon} ( \cdot, s ) \hspace{2pt} ds \right)_{L^2(\Omega)}
}\\
&=& - \frac{1}{2} \frac{d}{dt}
\left( \hat{e}^{\varepsilon} (\cdot,t), \hat{e}^{\varepsilon} (\cdot,t) \right)_{L^2(\Omega)}. \hspace{250pt}
\end{eqnarray*}
Consequently by integration over the interval $[0,t_0]$ and using $\partial_{t} \hat{e}^{\varepsilon} ( \cdot , 0)=0$ we get
\begin{eqnarray*}
\lefteqn{\| \hat{e}^{\varepsilon}( \cdot , t_0 ) \|_{L^2(\Omega)}^2}\\
&=& \| \hat{e}^{\varepsilon}( \cdot , 0 ) \|_{L^2(\Omega)}^2
- b^{\eps}( \int_{0}^{t_0} \hat{e}^{\varepsilon}( \cdot, s ) \hspace{2pt} ds, \int_{0}^{t_0} \hat{e}^{\varepsilon}( \cdot, s ) \hspace{2pt} ds)
+ 2 \langle \partial_{t} \hat{e}^{\varepsilon} ( \cdot , 0), \int_{0}^{t_0} \hat{e}^{\varepsilon} ( \cdot, s ) \hspace{2pt} ds \rangle\\
&\le& \| \hat{e}^{\varepsilon}( \cdot , 0 ) \|_{L^2(\Omega)}^2 = \| f - f^{\varepsilon} \|_{L^2(\Omega)}^2.
\end{eqnarray*}
Consequently, $I \le \| u^0 - u^\eps \|_{L^{\infty}(L^2)} + \| f - f^{\varepsilon} \|_{L^2(\Omega)} = e_{\mbox{\tiny \rm hom}}(\varepsilon)$.
$\\$
{\it Step 4.} We can estimate term $II$ in a similar way as term $I$. Hence, proceeding as before (and exploiting definition \eqref{semi-discrete-lod-equation}) gives us
\begin{align*}
\mbox{II} = \| \hat{u}_{H,k}^{\ms} - u_{H,k}^{\ms} \|_{L^{\infty}(L^2)} \le \| (u_{H,k}^{\ms} - \hat{u}_{H,k}^{\ms})( \cdot , 0 ) \|_{L^2(\Omega)}.
\end{align*}
Now we need to distinguish two cases. Case 1: If $(u_{H,k}^{\ms} - \hat{u}_{H,k}^{\ms})( \cdot , 0 )=P_{H,k}^{\mbox{\rm \tiny ms}}(f -f^{\varepsilon})$ we can exploit the $L^2$-stability of $P_{H,k}^{\mbox{\rm \tiny ms}}$ to directly get
\begin{align*}
\mbox{II} \le \| P_{H,k}^{\mbox{\rm \tiny ms}}(f -f^{\varepsilon}) \|_{L^2(\Omega)} \lesssim \| f -f^{\varepsilon} \|_{L^2(\Omega)} \le e_{\mbox{\tiny \rm hom}}(\varepsilon).
\end{align*}
Case 2: $(u_{H,k}^{\ms} - \hat{u}_{H,k}^{\ms})( \cdot , 0 )=\pi_{H,k}^{\mbox{\rm \tiny ms}}(f -f^{\varepsilon})$. In this case, we do not have $L^2$-stability and get the following estimate
\begin{eqnarray*}
\lefteqn{\mbox{II} = \| \hat{u}_{H,k}^{\ms} - u_{H,k}^{\ms} \|_{L^{\infty}(L^2)} \le \| \pi_{H,k}^{\mbox{\rm \tiny ms}}(f -f^{\varepsilon}) \|_{L^2(\Omega)}}\\
&\le& \| f -f^{\varepsilon} \|_{L^2(\Omega)} + \| f -f^{\varepsilon} - \pi_{H,k}^{\mbox{\rm \tiny ms}}(f -f^{\varepsilon}) \|_{L^2(\Omega)}\\
&\le& \| f -f^{\varepsilon} \|_{L^2(\Omega)} + \| f -f^{\varepsilon} - \pi_{h}(f -f^{\varepsilon}) \|_{L^2(\Omega)}
+ ( H + \theta^k k^{d/2}) \| \nabla \pi_{h}( f -f^{\varepsilon}) \|_{L^2(\Omega)}\\
&\lesssim& \| f -f^{\varepsilon} \|_{L^2(\Omega)} + \| f -f^{\varepsilon} - \pi_{h}(f -f^{\varepsilon}) \|_{L^2(\Omega)}
+ ( H + \theta^k k^{d/2}) \| \nabla f \|_{L^2(\Omega)},
\end{eqnarray*}
where we exploited the $H^1$-stability of $\pi_{h}$, the energy estimate $\| \nabla f^{\varepsilon} \| \le \frac{\beta}{\alpha} \| \nabla f \|_{L^2(\Omega)}$
and used the identity \eqref{L2-estimate-for-general-v-prestage}. Consequently, in both cases we obtain
\begin{align*}
\mbox{II} \lesssim e_{\mbox{\tiny \rm hom}}(\varepsilon) + \hat{e}_{\mbox{\tiny \rm disc}}(h) + ( H + \theta^k k^{d/2}) \| \nabla f \|_{L^2(\Omega)}.
\end{align*}
{\it Step 5.} Next, we treat the third term $\mbox{III}= \| \hat{u}^\eps - \hat{u}_{H,k}^{\ms} \|_{L^{\infty}(L^2)}$. We start with the triangle inequality to obtain $\| \hat{u}^\eps - \hat{u}_{H,k}^{\ms} \|_{L^{\infty}(L^2)} \le \| \hat{u}^\eps - \pi_{H,k}^{\mbox{\rm \tiny ms}}(\hat{u}^\eps) \|_{L^{\infty}(L^2)} + \| \psi^{\pi}\|_{L^{\infty}(L^2)}$, where $\psi^{\pi}:= \hat{u}_{H,k}^{\ms} - \pi_{H,k}^{\mbox{\rm \tiny ms}}(\hat{u}^\eps)$. For $\| \psi^{\pi}\|_{L^{\infty}(L^2)}$ we can use estimate (\ref{lemma-first-est-befor-tri-ineq}) that we obtained in the proof of Lemma \ref{lemma-apriori-semidiscrete}, i.e. we have
\begin{align}
\label{lemma-first-est-befor-tri-ineq-2}\| \psi^{\pi} \|_{L^{\infty}(0,T;L^2(\Omega))} \lesssim \| \psi^{\pi}(\cdot,0) \|_{L^2(\Omega)} + \| \partial_{t} \hat{u}^\eps - \pi_{H,k}^{\mbox{\rm \tiny ms}}(\partial_{t} \hat{u}^\eps) \|_{L^1(0,T;L^2(\Omega))}.
\end{align}
Again, we need to distinguish between $\hat{u}_{H,k}^{\ms}(\cdot,0)=\pi_{H,k}^{\mbox{\rm \tiny ms}}(f^{\varepsilon})$ and $\hat{u}_{H,k}^{\ms}(\cdot,0)=P_{H,k}^{\mbox{\rm \tiny ms}}(f^{\varepsilon})$. In the first case, we observe $\psi^{\pi}(\cdot,0)=0$ and the remaining estimate can be established using Lemma \ref{lemma-multiscale-projection}.
Consequently, we consider the non-trivial $L^2$-projection case, i.e. $\hat{u}_{H,k}^{\ms}(\cdot,0)=P_{H,k}^{\mbox{\rm \tiny ms}}(f^{\varepsilon})$. In this case, we can use the triangle inequality in (\ref{lemma-first-est-befor-tri-ineq-2}) for $\psi^{\pi}=(\hat{u}_{H,k}^{\ms} - \hat{u}^\eps) + (\hat{u}^\eps - \pi_{H,k}^{\mbox{\rm \tiny ms}}(\hat{u}^\eps))$ to obtain
\begin{eqnarray}
\label{estimate-ems}\nonumber\lefteqn{\mbox{III}=\| \hat{u}_{H,k}^{\ms} - \hat{u}^\eps \|_{L^{\infty}(0,T;L^2(\Omega))} \lesssim \underset{=:\mbox{III}_1}{\underbrace{\| (\hat{u}_{H,k}^{\ms} - \hat{u}^\eps)(\cdot,0) \|_{L^2(\Omega)} + \| (\hat{u}^\eps - \pi_{H,k}^{\mbox{\rm \tiny ms}}(\hat{u}^\eps))(\cdot,0) \|_{L^2(\Omega)}}}}\\
&+& \| \hat{u}^\eps - \pi_{H,k}^{\mbox{\rm \tiny ms}}(\hat{u}^\eps) \|_{L^{\infty}(0,T;L^2(\Omega))} + \| \partial_{t} \hat{u}^\eps - \pi_{H,k}^{\mbox{\rm \tiny ms}}(\partial_{t} \hat{u}^\eps) \|_{L^1(0,T;L^2(\Omega))}. \qquad \hspace{60pt}
\end{eqnarray}
The first term on the right hand side \eqref{estimate-ems} can be estimated as follows
\begin{eqnarray}
\label{estimate-initial-value-f}\nonumber\mbox{III}_1 &=& \| f^{\varepsilon} - P_{H,k}^{\mbox{\rm \tiny ms}}(f^{\varepsilon}) \|_{L^2(\Omega)} + \| f^{\varepsilon} - \pi_{H,k}^{\mbox{\rm \tiny ms}}(f^{\varepsilon})\|_{L^2(\Omega)}\\
\nonumber&\le& \inf_{v \in V_{H,k}^{\ms}} \| f^{\varepsilon} - v \|_{L^2(\Omega)} + \| f^{\varepsilon} - \pi_{H,k}^{\mbox{\rm \tiny ms}}(f^{\varepsilon}) \|_{L^2(\Omega)} \\
&\le& 2 \| f^{\varepsilon} - \pi_{H,k}^{\mbox{\rm \tiny ms}}(f^{\varepsilon})\|_{L^2(\Omega)}
\overset{(\ref{L2-estimate-for-general-v-prestage})}{\lesssim} \| f^{\varepsilon} -\pi_{h}(f^{\varepsilon}) \|_{L^2(\Omega)} + (H + \theta^k k^{d/2} ) \| f \|_{H^1(\Omega)},
\end{eqnarray}
where we again used $\| f^{\varepsilon} \|_{H^1(\Omega)}\lesssim \| f \|_{H^1(\Omega)}$. Consequently with (\ref{equation-for-projection}), we have
\begin{eqnarray}
\label{estimate-for-term-II}
\lefteqn{\lefteqn{\mbox{III} \lesssim
\| f^{\varepsilon} -\pi_{h}(f^{\varepsilon}) \|_{L^2(\Omega)} + (H + \theta^k k^{d/2} ) \| f \|_{H^1(\Omega)}}} \\
\nonumber &+& \| \hat{u}^\eps - P_H(\pi_{H,k}^{\mbox{\rm \tiny ms}}(\hat{u}^\eps)) \|_{L^{\infty}(L^2)} + \| \partial_{t} (\hat{u}^\eps - P_H(\pi_{H,k}^{\mbox{\rm \tiny ms}}(\hat{u}^\eps))) \|_{L^1(L^2)}\\
\nonumber &+& \|(Q_{H,k}^{\mbox{\rm \tiny ms}} \circ P_H \circ \pi_{H,k}^{\mbox{\rm \tiny ms}})(\hat{u}^\eps) \|_{L^{\infty}(L^2)} + \| (Q_{H,k}^{\mbox{\rm \tiny ms}} \circ P_H \circ \pi_{H,k}^{\mbox{\rm \tiny ms}})(\partial_{t} \hat{u}^\eps ) \|_{L^1(L^2)}.
\end{eqnarray}
The terms $\| \hat{u}^\eps - P_H(\pi_{H,k}^{\mbox{\rm \tiny ms}}(\hat{u}^\eps)) \|_{L^{\infty}(L^2)}$ and $\| \partial_{t} (\hat{u}^\eps - P_H(\pi_{H,k}^{\mbox{\rm \tiny ms}}(\hat{u}^\eps))) \|_{L^1(L^2)}$ can be estimated with Lemma \ref{lemma-multiscale-projection}, inequality (\ref{projection-error-estimate-1}).
For the last two terms involving $Q_{H,k}(\cdot)$, we can subtract the Cl\'ement-type interpolation $I_H(Q_{H,k}(v))=0$ and use the interpolation error estimate (\ref{lemma-properties-clement-op-est}). This gives us $\mathcal{O}(H)$-terms. Then, we can then use the $H^1$-stability estimates in (\ref{stability-estimates-for-various-ops}) to obtain
\begin{eqnarray*}
\lefteqn{\mbox{III} \lesssim \| f^{\varepsilon} -\pi_{h}(f^{\varepsilon}) \|_{L^2(\Omega)} + \| \hat{u}^\eps - \pi_h(\hat{u}^\eps) \|_{L^{\infty}(0,T;L^2(\Omega))} + \| \partial_t \hat{u}^\eps - \pi_h(\partial_t \hat{u}^\eps) \|_{L^1(0,T;L^2(\Omega))}}\\
\nonumber
&+& ( H + \theta^k k^{d/2}) \left( \| f \|_{H^1(\Omega)} + \| \hat{u}^\eps \|_{L^{\infty}(0,T;H^1(\Omega))} + \| \partial_t \hat{u}^\eps \|_{L^1(0,T;H^1(\Omega))} \right).\hspace{80pt}
\end{eqnarray*}
Using the energy estimates \eqref{data-est-1} and \eqref{data-est-2} and the definition of
$\hat{e}_{\mbox{\tiny \rm disc}}(h)$, we get
\begin{align*}
\mbox{III} \lesssim_T \hat{e}_{\mbox{\tiny \rm disc}}(h) + ( H + \theta^k k^{d/2}) \left( \| F \|_{W^{1,2}(L^2)} + \| f \|_{H^1(\Omega)} + \| g \|_{H^1(\Omega)} +\| \nabla \cdot (a^0 \nabla f ) \|_{L^2(\Omega)} \right).
\end{align*}
$\\$
{\it Step 6.} The term $\mbox{IV}=\| Q_{h,k}(u_{H,k}) \|_{L^{\infty}(L^2)}$ can be also estimated in the same way, using (\ref{lemma-properties-clement-op-est}) and (\ref{stability-estimates-for-various-ops}). Finally, the energy estimate
$\| u_{H,k}^{\ms} \|_{L^{\infty}(H^1)} \lesssim_T \|F\|_{L^2(L^2)} + \|f\|_{H^1(\Omega)} + \| g\|_{L^2(\Omega)}$ is required to bound $\mbox{IV}$ independent of $H$ and $h$. We obtain the same type of estimate as for $\mbox{III}$. Combining the estimates for $\mbox{I-IV}$ finishes the proof.
\end{proof}
\section{Numerical experiments}
\label{section-numerics}
In this section we present the results for four different model problems. The first model problem is taken from \cite{OwZ08} and involves a microstructure without scale separation, which however can be described by a smooth coefficient. In the second model problem we abandon the smoothness and consider a problem which involves a highly heterogenous discontinuous coefficient. The third model problem is also inspired by a problem presented in \cite{OwZ08}. Here, we add an additional conductivity channel to the heterogenous structure of model problem 2, which results in a high contrast of order $10^4$. Finally, in the last experiment, we investigate
the behavior of the method for smooth, but not well-prepared initial values.
In all computations, we fix the considered time interval to be $[0,T]:=[0,1]$ and the time step size to be $\triangle t:=0.05$. In order to compute the errors for the obtained multiscale approximations, we use a discrete reference solution $u_{h,\triangle t}$ as an approximation to the exact solution of problem (\ref{wave-equation-strong}). This reference solution is determined with the Crank-Nicolson scheme for the time discretization (using equidistant time steps with time step size $\triangle t=0.05$) and a Finite Element method on the fine mesh $\mathcal{T}_h$ for the space discretization. We use a linear interpolation between the solutions obtained for each time step. Hence, $\partial_t u_{h,\triangle t}$ is well defined on each time interval $[t^n,t^{n+1}]$. By $u_{H,\triangle t, k}$ we denote the multiscale approximation defined according to (\ref{fully-discrete-lod-approximation}).
In this section, we use the following notation for the errors:
\begin{align}
\nonumber e^{0,n} &:= u_{H,\triangle t, k}(\cdot,t^n) - u_{h,\triangle t}(\cdot,t^n)\\
\label{errors-definition} e^{\mbox{\rm \tiny ms},n} &:= (u_{H,\triangle t, k}+Q_{h,k}(u_{H,\triangle t, k}))(\cdot,t^n) - u_{h,\triangle t}(\cdot,t^n),\\
\nonumber \partial_t e^{\mbox{\rm \tiny ms},n} &:= \lim_{t \nearrow t^n} \partial_t(u_{H,\triangle t, k}+Q_{h,k}(u_{H,\triangle t, k}))(\cdot,t) - \partial_t u_{h,\triangle t}(\cdot,t).
\end{align}
By $\| \cdot \|_{L^2(\Omega)}^{\mbox{\tiny rel}}$ (respectively $\| \cdot \|_{H^1(\Omega)}^{\mbox{\tiny rel}}$) we denote the relative error norms, i.e. the absolute errors divided by the associated norm of the reference solution $u_{h,\triangle t}$. Furthermore, for an error $\| e_H \|$ on a coarse grid $\mathcal{T}_H$ and an error $\| e_{H/2} \|$ on a coarse grid $\mathcal{T}_{H/2}$, the EOC (experimental order of convergence) is given by $\mbox{EOC}_{H}:=\log_2(\| e_H \| / \| e_{H/2} \|)/\log_2(2)$.
\subsection{Model problem 1}
The first model problem is extracted from \cite{OwZ08}. As pointed out in \cite{OwZ08}, a sufficiently accurate reference solution $u_{h,\triangle t}$ is obtained for a uniform fine grid with resolution $h=2^{-7}$. Hence, we fix $\mathcal{T}_h$ to be a uniformly refined triangulation of $\Omega$ with $66.049$ DOFs.
\begin{problem}
Let $\Omega:= ]-1,1[^2$ and $T:=1$. Find $u^\eps \in L^{\infty}(0,T;H^1_0(\Omega))$ such that
\begin{align}
\nonumber\label{eq:model} \partial_{tt} u^\eps(x,t) - \nabla \cdot \left( a^\eps(x) \nabla u^\eps(x,t) \right) &= F(x) \qquad \mbox{in } \Omega \times(0,T], \\
u^\eps(x,t) &= 0 \hspace{35pt} \mbox{on } \partial \Omega \times [0,T], \\
\nonumberu^\eps(x,0) = 0 \quad \mbox{and} \quad \partial_t u^\eps(x,0) &= 0 \hspace{39pt} \mbox{in } \Omega,
\end{align}
where $F$ is a Gaussian source term given by $F(x_1,x_2)=(2 \pi \sigma^2)^{-1/2} e^{-(x_1^2 + (x_2-0.15)^2)/(2 \sigma^2)}$ for $\sigma=0.05$ and
\begin{align}
\label{diff-coefficient-1}a^\eps(x_1,x_2)&:=\frac{1}{6} \left( 1 + \sin(4 x_1^2 x_2^2) + \frac{1.1 + \sin(2 \pi x_1 / \varepsilon_1)}{1.1 + \sin(2 \pi x_2 / \varepsilon_1)} + \frac{1.1 + \sin(2 \pi x_1 / \varepsilon_2)}{1.1 + \cos(2 \pi x_2 / \varepsilon_2)} \right.\\
\nonumber&\left. + \frac{1.1 + \cos(2 \pi x_1 / \varepsilon_3)}{1.1 + \sin(2 \pi x_2 / \varepsilon_3)} + \frac{1.1 + \sin(2 \pi x_1 / \varepsilon_4)}{1.1 + \cos(2 \pi x_2 / \varepsilon_4)} + \frac{1.1 + \cos(2 \pi x_1 / \varepsilon_5)}{1.1 + \sin(2 \pi x_2 / \varepsilon_5)} \right).
\end{align}
with $\varepsilon_1=1/5$, $\varepsilon_2=1/13$, $\varepsilon_3=1/17$, $\varepsilon_4=1/31$ and $\varepsilon_5=1/65$. The coefficient $a^\eps$ is plotted in Figure \ref{diffusion-plot-1}, together with the reference solution $u_{h,\triangle t}$ for $t=1$.
\end{problem}
Note that the Gaussian source term will become singular for $\sigma \rightarrow 0$. Hence it influences the regularity of the solution and we expect the multiscale approximation to be less accurate than for a more regular source term. In particular, $F$ has already a very large $H^1$-norm, which is why we cannot expect to see the third order convergence O$(H^3)$ in (\ref{main-result-est-3-fully-discrete}), unless $H/\|F\|_{H^1(\Omega)} \ll 1$.
{\small
\begin{figure}[h!]
\centering
\includegraphics[scale=0.22]{diffusion_1_fem.png}
\caption{\it Model Problem 1. Left Picture: Plot of the coefficient $a^\eps$ given by (\ref{diff-coefficient-1}). Right Picture: reference solution $u_{h,\triangle t}$ at $t=1$ for $h=2^{-7}$.}
\label{diffusion-plot-1}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[scale=0.22]{comparison_isolines_and_LOD_8_16_2.png}
\caption{\it Model Problem 1, results for $t^n=1$. Left Picture: Comparison of the isolines of the reference solution $u_{h,\triangle t}$ for $h=2^{-7}$ (black isolines) with the multiscale approximation $u_{H,\triangle t, k}+Q_{h,k}(u_{H,\triangle t, k})$ for $(H,h,k)=(2^{-3},2^{-7},2)$ (colored isolines). Right Picture: Plot of the multiscale approximation $u_{H,\triangle t, k}+Q_{h,k}(u_{H,\triangle t, k})$ for $(H,h,k)=(2^{-3},2^{-7},2)$.}
\label{model-problem-1-isolines}
\end{figure}
}
{\small
\begin{table}[h!]
\caption{\it Model Problem 1, results for $t^n=1$. The table depicts relative $L^2$- and $H^1$-errors for the obtained multiscale approximations with respect to the reference solution. The errors are defined in (\ref{errors-definition}).}
\label{table-layers-results-1}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline $H$ & $k$
& $\| e^{0,n} \|_{L^2(\Omega)}^{\mbox{\tiny rel}}$
& $\| e^{\mbox{\rm \tiny ms},n} \|_{L^2(\Omega)}^{\mbox{\tiny rel}}$
& $\| e^{\mbox{\rm \tiny ms},n} \|_{H^1(\Omega)}^{\mbox{\tiny rel}}$
& $\| \partial_t e^{\mbox{\rm \tiny ms},n} \|_{L^2(\Omega)}^{\mbox{\tiny rel}}$
& $\| \partial_t e^{\mbox{\rm \tiny ms},n} \|_{H^1(\Omega)}^{\mbox{\tiny rel}}$\\
\hline
\hline $2^{-1}$ & 1 & 0.1448 & 0.1341 & 0.4532 & 0.8718 & 0.9957 \\
\hline $2^{-1}$ & 2 & 0.1394 & 0.1334 & 0.4627 & 0.8312 & 0.9822 \\
\hline
\hline $2^{-2}$ & 1 & 0.0780 & 0.0688 & 0.3517 & 0.6464 & 0.9424 \\
\hline $2^{-2}$ & 2 & 0.0687 & 0.0521 & 0.2919 & 0.5439 & 0.8949 \\
\hline $2^{-2}$ & 3 & 0.0675 & 0.0499 & 0.2835 & 0.5362 & 0.8929 \\
\hline
\hline $2^{-3}$ & 1 & 0.0368 & 0.0328 & 0.2279 & 0.5824 & 1.1262 \\
\hline $2^{-3}$ & 2 & 0.0242 & 0.0130 & 0.1212 & 0.3285 & 0.7769 \\
\hline $2^{-3}$ & 3 & 0.0234 & 0.0105 & 0.1036 & 0.2846 & 0.6998 \\
\hline
\end{tabular}\end{center}
\end{table}
}
\begin{table}[h!]
\caption{\it Model Problem 1, results for $t^n=1$. Overview on the EOCs associated with errors from Table \ref{table-layers-results-1}. We couple $k$ and $H$ by $k=k(H):=\lfloor |\ln(H)| + 1 \rfloor$. For each of the errors $\|e_H\|$ below (for $H=2^{-i}$), we define the average EOC by EOC$:= \frac{1}{2} \sum_{i=1}^2 \log_2(\| e_{2^{-i}} \| / \| e_{2^{-(i+1)}} \|)/\log_2(2)$.}
\label{table-EOC-results-1}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline $H$ & $k(H)$
& $\| e^{0,n} \|_{L^2(\Omega)}^{\mbox{\tiny rel}}$
& $\| e^{\mbox{\rm \tiny ms},n} \|_{L^2(\Omega)}^{\mbox{\tiny rel}}$
& $\| e^{\mbox{\rm \tiny ms},n} \|_{H^1(\Omega)}^{\mbox{\tiny rel}}$
& $\| \partial_t e^{\mbox{\rm \tiny ms},n} \|_{L^2(\Omega)}^{\mbox{\tiny rel}}$
& $\| \partial_t e^{\mbox{\rm \tiny ms},n} \|_{H^1(\Omega)}^{\mbox{\tiny rel}}$\\
\hline
\hline $2^{-1}$ & 1 & 0.1448 & 0.1341 & 0.4532 & 0.8718 & 0.9957 \\
\hline $2^{-2}$ & 2 & 0.0687 & 0.0521 & 0.2919 & 0.5439 & 0.8949 \\
\hline $2^{-3}$ & 3 & 0.0234 & 0.0105 & 0.1036 & 0.2846 & 0.6998 \\
\hline
\hline \multicolumn{2}{|c|}{EOC} & 1.31 & 1.84 & 1.06 & 0.81 & 0.25 \\
\hline
\end{tabular}\end{center}
\end{table}
In Table \ref{table-layers-results-1} the relative errors are depicted for various combinations of $H$ and $k$ (recall that $k$ denotes the truncation parameter defined in (\ref{def-patch-U-k})). The errors are qualitatively comparable to the errors obtained in \cite{OwZ08} for similar computations. Furthermore, we observe that the error evolution is consistent with the theoretically predicated rates. EOCs are given in Table \ref{table-EOC-results-1}.
For $k\approx |\ln(H)|+1$, we observe roughly a convergence rate of $1.3$ in $H$ for the $L^2$-error of the numerically homogenized solution $u_{H,\triangle t, k}$. Adding the corresponding corrector $Q_{h,k}(u_{H,\triangle t, k})$, the rate is close to $2$ in average. In Figure \ref{model-problem-1-isolines}, a visual comparison between the reference solution and the multiscale approximation is shown. We observe that for $(H,h,k)=(2^{-3},2^{-7},2)$, the solution $u_{H,\triangle t, k}+Q_{h,k}(u_{H,\triangle t, k})$ looks the same as the reference solution $u_{h,\triangle t}$ depicted in (\ref{diffusion-plot-1}). This is also stressed by the comparison of isolines in Figure \ref{model-problem-1-isolines}.
\subsection{Model problem 2}
\begin{figure}
\centering
\includegraphics[scale=0.22]{diffusion_2_fem.png}
\caption{\it Model Problem 2, plots for $t^n=1$. Left Picture: Plot of the coefficient $a^\eps$ given by (\ref{diff-coefficient-2}). Right Picture: reference solution $u_{h,\triangle t}$ at $t=1$ for $h=2^{-8}$.}
\label{diffusion-plot-2}
\end{figure}
In Model Problem 2 we investigate the influence of a discontinuous coefficient $a^\eps$ in our multiscale method. According to the theoretical results, it should not influence the convergence rates. The fine grid $\mathcal{T}_h$ is a uniformly refined triangulation with resolution $h=2^{-8}$.
\begin{problem}
Let $\Omega:= ]0,1[^2$ and $T:=1$. Find $u^\eps \in L^{\infty}(0,T;H^1_0(\Omega))$ such that
\begin{align}
\nonumber\label{eq:model} \partial_{tt} u^\eps(x,t) - \nabla \cdot \left( a^\eps(x) \nabla u^\eps(x,t) \right) &= 1 \qquad \hspace{17pt} \mbox{in } \Omega \times(0,T], \\
u^\eps(x,t) &= 0 \hspace{37pt} \mbox{on } \partial \Omega \times [0,T], \\
\nonumberu^\eps(x,0) = 0 \quad \mbox{and} \quad \partial_t u^\eps(x,0) &= 0 \hspace{38pt} \mbox{in } \Omega.
\end{align}
Here, we have
\begin{align}
\label{diff-coefficient-2}a^\eps(x)&:= (h \circ c_\varepsilon)(x)
\qquad \text{with} \enspace h(t):=\begin{cases}
t^4 &\text{for} \enspace \frac12 < t < 1 \\
t^{\frac{3}{2}} &\text{for} \enspace 1 < t < \frac{3}{2} \\
t &\text{else}
\end{cases}
\end{align}
and where
\begin{displaymath}
c_\varepsilon(x_1,x_2):=1 + \frac{1}{10} \sum_{j=0}^4 \sum_{i=0}^{j} \left( \frac{2}{j+1} \cos \left( \bigl\lfloor i x_2 - \tfrac{x_1}{1+i} \bigr\rfloor + \left\lfloor \tfrac{i x_1}\varepsilon \right\rfloor + \left\lfloor \tfrac{ x_2}\varepsilon \right\rfloor \right) \right).
\end{displaymath}
The coefficient $a^\eps$ is plotted in Figure \ref{diffusion-plot-2} together with the reference solution on $\mathcal{T}_h$ for $t=1$.
\end{problem}
\begin{table}[h!]
\caption{\it Model Problem 2. Overview on relative $L^2$- and $H^1$-errors for Model Problem 2 for $t^n=1$. The errors are defined in (\ref{errors-definition}).}
\label{table-layers-results-2}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline $H$ & $k$
& $\| e^{0,n} \|_{L^2(\Omega)}^{\mbox{\tiny rel}}$
& $\| e^{\mbox{\rm \tiny ms},n} \|_{L^2(\Omega)}^{\mbox{\tiny rel}}$
& $\| e^{\mbox{\rm \tiny ms},n} \|_{H^1(\Omega)}^{\mbox{\tiny rel}}$
& $\| \partial_t e^{\mbox{\rm \tiny ms},n} \|_{L^2(\Omega)}^{\mbox{\tiny rel}}$
& $\| \partial_t e^{\mbox{\rm \tiny ms},n} \|_{H^1(\Omega)}^{\mbox{\tiny rel}}$\\
\hline
\hline $2^{-2}$ & 1 & 0.1299 & 0.0613 & 0.1802 & 0.1762 & 0.6615 \\
\hline $2^{-2}$ & 2 & 0.1223 & 0.0245 & 0.0800 & 0.1298 & 0.6323 \\
\hline
\hline $2^{-3}$ & 1 & 0.0914 & 0.0616 & 0.1926 & 0.2194 & 0.7255 \\
\hline $2^{-3}$ & 2 & 0.0753 & 0.0191 & 0.0841 & 0.1049 & 0.5902 \\
\hline $2^{-3}$ & 3 & 0.0741 & 0.0085 & 0.0563 & 0.0870 & 0.5688 \\
\hline
\hline $2^{-4}$ & 1 & 0.0327 & 0.0243 & 0.1401 & 0.1197 & 0.6710 \\
\hline $2^{-4}$ & 2 & 0.0240 & 0.0047 & 0.0505 & 0.0600 & 0.5109 \\
\hline $2^{-4}$ & 3 & 0.0239 & 0.0029 & 0.0347 & 0.0562 & 0.5004 \\
\hline
\end{tabular}\end{center}
\end{table}
\begin{table}[t]
\caption{\it Model Problem 2, results for $t^n=1$. Overview on the EOCs associated with errors from Table \ref{table-layers-results-2}. We couple $k$ and $H$ by $k=k(H):=\lfloor |\ln(H)| + 0.5 \rfloor$. The average EOCs are computed according to (\ref{eoc-def}).}
\label{table-EOC-results-2}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline $H$ & $k(H)$
& $\| e^{0,n} \|_{L^2(\Omega)}^{\mbox{\tiny rel}}$
& $\| e^{\mbox{\rm \tiny ms},n} \|_{L^2(\Omega)}^{\mbox{\tiny rel}}$
& $\| e^{\mbox{\rm \tiny ms},n} \|_{H^1(\Omega)}^{\mbox{\tiny rel}}$
& $\| \partial_t e^{\mbox{\rm \tiny ms},n} \|_{L^2(\Omega)}^{\mbox{\tiny rel}}$
& $\| \partial_t e^{\mbox{\rm \tiny ms},n} \|_{H^1(\Omega)}^{\mbox{\tiny rel}}$\\
\hline
\hline $2^{-2}$ & 1 & 0.1299 & 0.0613 & 0.1802 & 0.1762 & 0.6615 \\
\hline $2^{-3}$ & 2 & 0.0753 & 0.0191 & 0.0841 & 0.1049 & 0.5902 \\
\hline $2^{-4}$ & 3 & 0.0239 & 0.0029 & 0.0347 & 0.0562 & 0.5004 \\
\hline
\hline \multicolumn{2}{|c|}{EOC} & 1.22 & 2.20 & 1.19 & 0.82 & 0.20 \\
\hline
\end{tabular}\end{center}
\end{table}
In Table \ref{table-layers-results-2} we depict various relative $L^2$- and $H^1$-errors for $t^n=1$. We observe that the numerically homogenized solution $u_{H,\triangle t, k}$ already yields good $L^2$-approximation properties with respect to the fine scale reference solution. These approximation properties can still be significantly improved by adding the corrector $Q_{h,k}(u_{H,\triangle t, k})$. We can see that the total approximation $u_{H,\triangle t, k}+Q_{h,k}(u_{H,\triangle t, k})$ is also accurate in the norms $\| \nabla \cdot \|_{L^2(\Omega)}$ and $\| \partial_t \cdot \|_{L^2(\Omega)}$. The errors in the norm $\| \partial_t \cdot \|_{H^1(\Omega)}$ are noticeably larger than the errors in all other norms. Hence, the multiscale solution does not necessarily yield a highly accurate $H^1(0,T,H^1(\Omega))$ approximation, even though the discrepancy is still tolerable.
All errors for Model Problem 2 are of the same order as the errors for Model Problem 1 depicted in Table \ref{table-layers-results-1}. In particular, we do not see any error deterioration caused by the discontinuity of the coefficient $a^\eps$. The method behaves nicely in both cases. This is stressed by the experimental orders of convergence (EOCs) shown in Table \ref{table-EOC-results-2}. For the EOCs in Table \ref{table-EOC-results-2}, we use the average
\begin{align}
\label{eoc-def} \mbox{EOC} := (\mbox{EOC}_{2^{-2}} + \mbox{EOC}_{2^{-3}})/2.
\end{align}
Motivated by Theorem \ref{apriori-fullydiscrete}, we couple the coarse mesh $H$ and the truncation parameter $k$ to be the closest integer to $|\ln(H)|$, i.e. we pick $k=k(H):=\lfloor |\ln(H)| + 0.5 \rfloor$ for the computation of the EOCs. This gives us $k=1$ for $H=2^{-2}$, $k=2$ for $H=2^{-3}$ and $k=3$ for $H=2^{-4}$. The corresponding results are stated in Table \ref{table-EOC-results-2}. We observe a close to linear convergence for the $L^2$-error for the numerically homogenized solution $u_{H,\triangle t, k}$ (as predicted by the theory). Adding the corrector $Q_{h,k}(u_{H,\triangle t, k})$, the convergence rate increases to $2.2$ (slightly worse than the optimal rate of $3$). For the $H^1$-error, we observe linear convergence. The convergence rate of the $L^2$-error for time derivatives is slightly below linear convergence, but still very satisfying. Only the convergence rate for the $H^1$-error for the time derivatives is close to stagnation.
Note that the deviation of these rates from the perfect rates comes from that fact that we do not know the generic constant $C_{\theta}$ in Theorem \ref{apriori-fullydiscrete}. We picked $k=|\ln(H)|$, instead of $k=C_{\theta} |\ln(H)|$. Still we observe that approximating $C_{\theta}$ by $1$ yields highly accurate results that are close to the optimal rates. Practically, this justifies the use of small localization patches $U_k(K)$.
\subsection{Model problem 3}
\begin{figure}
\centering
\includegraphics[scale=0.22]{diffusion_3_fem.png}
\caption{\it Model Problem 3. Left Picture: Plot of the coefficient $a^\eps$. The basis structure of $a^\eps$ is given by (\ref{diff-coefficient-2}), but this structure is perturbed by an arc-like conductivity channel (pink). In this arc $a^\eps$ takes the value $100$. Right Picture: reference solution $u_{h,\triangle t}$ at $t=1$ for $h=2^{-8}$.}
\label{diffusion-plot-3}
\end{figure}
\begin{problem}
Let $\Omega:= ]0,1[^2$ and $T:=1$. Find $u^\eps \in L^{\infty}(0,T;H^1_0(\Omega))$ such that
\begin{align}
\nonumber\label{eq:model} \partial_{tt} u^\eps(x,t) - \nabla \cdot \left( a^\eps(x) \nabla u^\eps(x,t) \right) &= F(x,t) \qquad \hspace{17pt} \mbox{in } \Omega \times(0,T], \\
u^\eps(x,t) &= 0 \hspace{66pt} \mbox{on } \partial \Omega \times [0,T], \\
\nonumberu^\eps(x,0) = 0 \quad \mbox{and} \quad \partial_t u^\eps(x,0) &= 0 \hspace{66pt} \mbox{in } \Omega.
\end{align}
Here, we have $F(x_1,x_2,t) = \sin( 2.4 x_1 - 1.8 x_2 + 2 \pi t )$ and $a^\eps$ is given by equation (\ref{diff-coefficient-2}) but additionally it is disturbed by a high conductivity channel of thickness 0.05. The precise structure of $a^\eps$ is depicted in Figure \ref{diffusion-plot-3}, together with the reference solution on $\mathcal{T}_h$ for $t=1$.
\end{problem}
\begin{table}[t]
\caption{\it Model Problem 3, results for $t^n=1$. Overview on relative $L^2$- and $H^1$-errors defined as in (\ref{errors-definition}).}
\label{table-layers-results-3}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline $H$ & $k$
& $\| e^{0,n} \|_{L^2(\Omega)}^{\mbox{\tiny rel}}$
& $\| e^{\mbox{\rm \tiny ms},n} \|_{L^2(\Omega)}^{\mbox{\tiny rel}}$
& $\| e^{\mbox{\rm \tiny ms},n} \|_{H^1(\Omega)}^{\mbox{\tiny rel}}$
& $\| \partial_t e^{\mbox{\rm \tiny ms},n} \|_{L^2(\Omega)}^{\mbox{\tiny rel}}$
& $\| \partial_t e^{\mbox{\rm \tiny ms},n} \|_{H^1(\Omega)}^{\mbox{\tiny rel}}$\\
\hline
\hline $2^{-2}$ & 1 & 0.2468 & 0.1564 & 0.3321 & 0.2066 & 0.4486 \\
\hline $2^{-2}$ & 2 & 0.2270 & 0.0782 & 0.1992 & 0.1168 & 0.3269 \\
\hline
\hline $2^{-3}$ & 1 & 0.1451 & 0.1046 & 0.3305 & 0.1639 & 0.4588 \\
\hline $2^{-3}$ & 2 & 0.1184 & 0.0329 & 0.1535 & 0.0607 & 0.2724 \\
\hline $2^{-3}$ & 3 & 0.1174 & 0.0202 & 0.1024 & 0.0468 & 0.2333 \\
\hline
\hline $2^{-4}$ & 1 & 0.0550 & 0.0433 & 0.2186 & 0.0667 & 0.3349 \\
\hline $2^{-4}$ & 2 & 0.0390 & 0.0095 & 0.0803 & 0.0250 & 0.1896 \\
\hline $2^{-4}$ & 3 & 0.0385 & 0.0046 & 0.0464 & 0.0198 & 0.1758 \\
\hline
\end{tabular}\end{center}
\end{table}
\begin{table}[t]
\caption{\it Model Problem 3, results for $t^n=1$. Overview on the EOCs associated with errors from Table \ref{table-layers-results-3}. We couple $k$ and $H$ by $k=k(H):=\lfloor |\ln(H)| + 0.5 \rfloor$. The average EOCs are computed according to (\ref{eoc-def}).}
\label{table-EOC-results-3}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline $H$ & $k(H)$
& $\| e^{0,n} \|_{L^2(\Omega)}^{\mbox{\tiny rel}}$
& $\| e^{\mbox{\rm \tiny ms},n} \|_{L^2(\Omega)}^{\mbox{\tiny rel}}$
& $\| e^{\mbox{\rm \tiny ms},n} \|_{H^1(\Omega)}^{\mbox{\tiny rel}}$
& $\| \partial_t e^{\mbox{\rm \tiny ms},n} \|_{L^2(\Omega)}^{\mbox{\tiny rel}}$
& $\| \partial_t e^{\mbox{\rm \tiny ms},n} \|_{H^1(\Omega)}^{\mbox{\tiny rel}}$\\
\hline
\hline $2^{-2}$ & 1 & 0.2468 & 0.1564 & 0.3321 & 0.2066 & 0.4486 \\
\hline $2^{-3}$ & 2 & 0.1184 & 0.0329 & 0.1535 & 0.0607 & 0.2724 \\
\hline $2^{-4}$ & 3 & 0.0385 & 0.0046 & 0.0464 & 0.0198 & 0.1758 \\
\hline
\hline \multicolumn{2}{|c|}{EOC} & 1.34 & 2.54 & 1.42 & 1.69 & 0.68 \\
\hline
\end{tabular}\end{center}
\end{table}
\begin{figure}
\centering
\includegraphics[scale=0.22]{comparison_isolines_and_LOD_8_16_2_prob3.png}
\caption{\it Model Problem 3, results at $t=1$. Left Picture: Comparison of the isolines of the reference solution $u_{h,\triangle t}$ for $h=2^{-8}$ (black isolines) with the multiscale approximation $u_{H,\triangle t, k}+Q_{h,k}(u_{H,\triangle t, k})$ for $(H,h,k)=(2^{-4},2^{-8},2)$ (colored isolines). Right Picture: Plot of the multiscale approximation $u_{H,\triangle t, k}+Q_{h,k}(u_{H,\triangle t, k})$ for $(H,h,k)=(2^{-4},2^{-8},2)$.}
\label{model-problem-3-isolines}
\end{figure}
This model problem is inspired by a model problem in \cite{OwZ08}. The source term as in \cite{OwZ08} is given by $F(x_1,x_2,t) = \sin( 2.4 x_1 - 1.8 x_2 + 2 \pi t )$ and
as in \cite{OwZ08} $a^\eps$ contains a conductivity channel that perturbs the original structure. As we could not access the data for the channel given in this reference we model a new one in this paper.
This model problem is set to investigate the approximation quality of our multiscale approximations for problems with channels (which do not have to be resolved by the coarse grid) and a high contrast $\beta/\alpha \approx 10^4$. As in the previous model problem, we chose $\mathcal{T}_h$ as a uniformly refined triangulation with resolution $h=2^{-8}$.
We see in Table \ref{table-layers-results-3} that the additional channel in the problem does not deteriorate the convergence rates compared to Model Problem 2.
Again, close to optimal convergence rates are obtained for the choice $k=k(H):=\lfloor |\ln(H)| + 0.5 \rfloor$ (which slightly underestimates the optimal truncation parameter $k$). The corresponding results are given in Table \ref{table-layers-results-3}. The method yields accurate results even in the case of a conductivity channel and despite that the coarse grid does not resolve the channel. Furthermore the high contrast of order $10^4$ does not significantly influence the size of the optimal truncation parameter $k$. In the model problem we can still work with small localization patches $U_k(K)$, independent of the conductivity channel. This is further stressed by Figure \ref{model-problem-3-isolines} which depicts the multiscale approximation (i.e. $u_{H,\triangle t, k}+Q_{h,k}(u_{H,\triangle t, k})$) for the case $(H,k)=(2^{-4},2)$. The solution looks almost identical to the reference solution $u_{h,\triangle t}$ and the corresponding isolines match almost perfectly.
\subsection{Model problem 4}
In this model problem we investigate the performance of the multiscale method in the case of smooth, but not well-prepared initial values. In the light of Theorem \ref{apriori-lod-homogenization}, we can only expect a linear convergence rate with respect to $H$ and in the $L^{\infty}(L^2)$-norm and we cannot expect any rates in the $W^{1,\infty}(L^2)$- and $L^{\infty}(L^2)$-norms, since Theorem \ref{apriori-semidiscrete} is not valid under general assumptions. As a model problem, we consider the setting of Model Problem 2, but with different source and initial values. As in the previous model problem, we chose $\mathcal{T}_h$ as a uniformly refined triangulation with resolution $h=2^{-8}$. We consider the following problem.
\begin{problem}
Let $\Omega:= ]0,1[^2$ and $T:=1$. Find $u^\eps \in L^{\infty}(0,T;H^1_0(\Omega))$ such that
\begin{align}
\nonumber\label{eq:model} \partial_{tt} u^\eps(x,t) - \nabla \cdot \left( a^\eps(x) \nabla u^\eps(x,t) \right) &= F(x,t) \qquad \hspace{17pt} \mbox{in } \Omega \times(0,T], \\
u^\eps(x,t) &= 0 \hspace{66pt} \mbox{on } \partial \Omega \times [0,T], \\
\nonumberu^\eps(x,0) = f \quad \mbox{and} \quad \partial_t u^\eps(x,0) &= g \hspace{66pt} \mbox{in } \Omega.
\end{align}
Here, we have $F(x_1,x_2) = \sin( 2 \pi x_1 ) \sin( 2 \pi x_2 )$, $f(x_1,x_2)=x_1 (1-x_1)x_2 (1-x_2)$, $g(x_1,x_2)=\sin( 2 \pi x_1 ) x_2 (1 - x_2)$ and $a^\eps$ is given by equation (\ref{diff-coefficient-2}).
\end{problem}
\begin{table}[t]{
\caption{\it Model Problem 4, results for $t^n=1$. Overview on relative $L^2$- and $H^1$-errors defined as in (\ref{errors-definition}).}
\label{table-layers-results-4}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline $H$ & $k$
& $\| e^{0,n} \|_{L^2(\Omega)}^{\mbox{\tiny rel}}$
& $\| e^{\mbox{\rm \tiny ms},n} \|_{L^2(\Omega)}^{\mbox{\tiny rel}}$
& $\| e^{\mbox{\rm \tiny ms},n} \|_{H^1(\Omega)}^{\mbox{\tiny rel}}$
& $\| \partial_t e^{\mbox{\rm \tiny ms},n} \|_{L^2(\Omega)}^{\mbox{\tiny rel}}$
& $\| \partial_t e^{\mbox{\rm \tiny ms},n} \|_{H^1(\Omega)}^{\mbox{\tiny rel}}$\\
\hline
\hline $2^{-2}$ & 1 & 0.2809 & 0.2598 & 0.6180 & 0.3522 & 0.9735 \\
\hline $2^{-2}$ & 2 & 0.1865 & 0.1229 & 0.4954 & 0.3147 & 0.9692 \\
\hline
\hline $2^{-3}$ & 1 & 0.1579 & 0.1420 & 0.5680 & 0.3243 & 0.9691 \\
\hline $2^{-3}$ & 2 & 0.1188 & 0.0894 & 0.4869 & 0.2473 & 0.9593 \\
\hline $2^{-3}$ & 3 & 0.1145 & 0.0820 & 0.4741 & 0.2372 & 0.9573 \\
\hline
\hline $2^{-4}$ & 1 & 0.0885 & 0.0857 & 0.4774 & 0.2891 & 0.9850 \\
\hline $2^{-4}$ & 2 & 0.0466 & 0.0361 & 0.3249 & 0.1925 & 0.9506 \\
\hline $2^{-4}$ & 3 & 0.0423 & 0.0289 & 0.3042 & 0.1823 & 0.9479 \\
\hline
\hline $2^{-5}$ & 1 & 0.0499 & 0.0489 & 0.3481 & 0.1849 & 0.9593 \\
\hline $2^{-5}$ & 2 & 0.0198 & 0.0149 & 0.2256 & 0.1277 & 0.9204 \\
\hline $2^{-5}$ & 3 & 0.0173 & 0.0109 & 0.2059 & 0.1229 & 0.9171 \\
\hline
\end{tabular}\end{center}}
\end{table}
\begin{table}[t]{
\caption{\it Model Problem 4, results for $t^n=1$. Overview on the EOCs associated with errors from Table \ref{table-layers-results-4}. We couple $k$ and $H$ by $k=k(H):=\lfloor |\ln(H)| + 0.5 \rfloor$. The average EOCs are computed with $\mbox{EOC} := (\mbox{EOC}_{2^{-2}} + \mbox{EOC}_{2^{-3}} + \mbox{EOC}_{2^{-4}})/3$.}
\label{table-EOC-results-4}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline $H$ & $k(H)$
& $\| e^{0,n} \|_{L^2(\Omega)}^{\mbox{\tiny rel}}$
& $\| e^{\mbox{\rm \tiny ms},n} \|_{L^2(\Omega)}^{\mbox{\tiny rel}}$
& $\| e^{\mbox{\rm \tiny ms},n} \|_{H^1(\Omega)}^{\mbox{\tiny rel}}$
& $\| \partial_t e^{\mbox{\rm \tiny ms},n} \|_{L^2(\Omega)}^{\mbox{\tiny rel}}$
& $\| \partial_t e^{\mbox{\rm \tiny ms},n} \|_{H^1(\Omega)}^{\mbox{\tiny rel}}$\\
\hline
\hline $2^{-2}$ & 1 & 0.2809 & 0.2598 & 0.6180 & 0.3522 & 0.9735 \\
\hline $2^{-3}$ & 2 & 0.1188 & 0.0894 & 0.4869 & 0.2473 & 0.9593 \\
\hline $2^{-4}$ & 3 & 0.0423 & 0.0289 & 0.3042 & 0.1823 & 0.9479 \\
\hline $2^{-5}$ & 3 & 0.0173 & 0.0109 & 0.2059 & 0.1229 & 0.9171 \\
\hline
\hline \multicolumn{2}{|c|}{EOC}
& 1.34 & 1.53 & 0.53 & 0.51 & 0.03 \\
\hline
\end{tabular}\end{center}}
\end{table}
The corresponding numerical results are depicted in Table \ref{table-layers-results-4}. In order to compute the experimental orders of convergence, we pick as before the relation $k=k(H):=\lfloor |\ln(H)| + 0.5 \rfloor$ to compute a suitable truncation parameter. The results are shown in Table \ref{table-EOC-results-4}. The crucial observation that we make is that Theorem \ref{apriori-lod-homogenization} seems to be indeed optimal and that Theorem \ref{apriori-semidiscrete} does not generalize to {\it not well-prepared} initial values. We observe that the $L^{\infty}(L^2)$-error converges with an order between $1$ and $1.5$, which is just what we predicted. We also note that even though the error becomes slightly smaller when we use the corrector $Q_{h,k}(u_{H,k})$, the rates of convergence remain basically the same (this is in contrast to the case of well-prepared data, where we could improve the rates with the corrector). For the convergence rates for the error in the $W^{1,\infty}(L^2)$- and $L^{\infty}(L^2)$-norms we still observe a rate of order $0.5$, which might be an indication that a relaxed version of Theorem \ref{apriori-semidiscrete} might still hold for arbitrary initial values. However, from the analytical point a corresponding argument is still missing. The main conclusion that we can draw from Model Problem 4 is that we can also numerically confirm that
the proposed multiscale method still works in the general setting of not well-prepared data. This is a significant observation that distinguishes our multiscale method from other multiscale methods for the wave equation.
\medskip
$\\$
{\bf Acknowledgements.}
This work is partially supported by the Swiss National Foundation, Grant: No $200021\_134716/1$ and by the Deutsche Forschungsgemeinschaft, DFG-Grant: OH 98/6-1. We would also like to thank the anonymous reviewers for their constructive feedback that helped us to improve the contents of this paper.
\def$'${$'$}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,788
|
\section{Introduction}
\label{sec1}
\hspace{6mm}With the recent operation of the Large Hadron Collider (LHC) at
CERN the opportunity to study the Bose\,--\,Einstein Correlations (BEC)
of identical bosons at very high hadron-hadron collision energy
has been opened \cite{cms1,cms2,cms3,alice1,alice2,alice4}. In
particular the energy dependence of the BEC dimension $R$ has
recently been investigated and was found to increase with
log$(\sqrt{s_{NN}})$ both in proton-proton ($pp$) and in heavy ion
($AA$) collisions \cite{alexander,gaitai,okorokov}. In the present
work we investigate the energy behavior of the chaoticity
parameter $\lambda$ in $pp$ collisions which determines the
strength of the measured BEC effect. To this end we utilize the
relation between the BEC dimension $R$, the number of pion source clusters
and $\lambda$. We further stipulate that the pion
source clusters are proportional to the average charged particle
multiplicity produced in the hadron reactions.\\
The BEC is measured in terms of the two identical particle
correlation function
\begin{equation}
C(p_1,p_2)\ =\ \frac{\rho(p_1,p_2)}{\rho_0(p_1,p_2)}\ ,
\label{bec}
\end{equation}
where $p_1$ and $p_2$ are the 4-momenta of the two hadrons, $\rho$
is the two particle density function and $\rho_0$ is the two
particle density function in the absence of the BEC effect. This
$\rho_0$ is often referred to as the reference sample against
which the BEC is measured. There are several ways to construct
$\rho_0$ which were adopted by the different BEC experiments
\cite{review}. These and the different background
conditions and the variety of BEC analysis methods should be appraised
when their physics implications are determined.
Throughout this work we assume that the BEC
background is well accounted for and that one can safely ignore
the influence of the long range correlations on the $\lambda$
properties.
\section{The one dimension BEC analysis}
\hspace{6mm}Among the various BEC representations one of the frequently used
in the one dimension (1D) analysis of hadrons emerging from a
sphere volume, is the Goldhaber parametrization of a static
Gaussian source in the plane-wave approach \cite{goldhaber},
namely
\begin{equation}
C_{Gauss}(Q)\ =\ 1+\lambda_{Gauss}e^{-Q^2R^2_{Gauss}}\ ,
\label{gauss}
\end{equation}
which assumes for the particles emitter a
spherical volume with a radial Gaussian
distribution. The second often used
parametrization, which assumes a radial Lorentzian distribution of
the source, is given by
\begin{equation}
C_{Expo}(Q)\ =\ 1+\lambda_{Expo}e^{-QR_{Expo}}\ ,
\label{expo}
\end {equation}
which generally was found at low $Q$ values, e.g.
$Q \leq 0.1$ GeV, to fit beter the measured
BEC distribution than the
the Gaussian parametrization \cite{atlasbec}.
In both representations ${Q^2=-(p_1-p_2)^2}$ is the difference
squared of the 4-momentum vectors of the two correlated identical
bosons. The $\lambda$ factor, also known as the chaoticity
parameter, lies in the range between 0 and 1.\\
In the 1D analysis the relation between $R_{Gauss}$ and $R_{Expo}$
dimensions can be evaluated from the requirement that the first
$Q$ moment in a given BEC distribution will be equal whether it is
treated by a fit to a Gaussian distribution or to an Exponential one,
namely
\begin{equation}
\frac{\displaystyle
\int_{Q_1}^{\infty}{Qe^{-R^2_{Gauss}Q^2}dQ}}{\displaystyle
\int_{Q_1}^{\infty}{e^{-R^2_{Gauss}Q^2} dQ}}= \frac{\displaystyle
\int_{Q_1}^{\infty}{Qe^{-R_{Expo}Q}dQ}}{\displaystyle
\int_{Q_1}^{\infty}{e^{-R_{Expo}Q} dQ}}\ .
\label{rcrg1}
\end{equation}
This relation remains essentially the same as long as the upper
integration value is higher than 2 GeV. The dependence of
$R_{Expo}/R_{Gauss}$ on the lower integration limit
$Q_1$ is shown in Fig. \ref{qmin}. In the case that
$Q_1=0$ GeV one obtains the known relation
\begin{equation}
R_{Gauss}=\frac{R_{Expo}}{\sqrt{\pi}}\ .
\label{rr}
\end{equation}
\begin{figure}[ht]
\centering{\psfig{file=fig-1.eps,height=8.0cm,
bbllx=46pt,bblly=150pt,bburx=552pt,bbury=682pt}} \caption{\small
$R_{Expo}/R_{Gauss}$ as a function of the lower integration limit
$Q_1$ in Eq. (\ref{rcrg1}) and the upper integration value is
$\geq$ 2 GeV.}
\label{qmin}
\end{figure}
A relation between $\lambda_{Gauss}$ and $\lambda_{Expo}$
can in principle be estimated by considering the relations
\begin{equation}
\int^{Q_2}_{Q_1}\lambda_{Expo}e^{-QR_{Expo}}dQ\simeq
\int^{Q_2}_{Q_1}\lambda_{Expo}e^{-QR_{Gauss}\sqrt{\pi}}dQ\simeq
\int^{Q_2}_{Q_1}\lambda_{Gauss}e^{-Q^2R_{Gauss}^2}dQ\ ,
\label{good}
\end{equation}
where in the range between $Q_1$ and $Q_2$ GeV the Gaussian and
the Exponential BEC parametrizations fit equally well the measured
$Q$ distribution. In this case for the values of $Q_1=0$ and $Q_2=\infty$ GeV
one finds that
\begin{equation}
\lambda_{Gauss}\ =\ \frac{2\lambda_{Expo}}{\pi}\ .
\label{lamratio}
\end{equation}
Experimentally it has been found that the chaoticity
parameter depends on several of the data properties such as
the average particle pair transverse momentum, $\langle
k_{T}\rangle$, and in particular on the outgoing
charged particle multiplicity $N_{ch}$.
Hence the additional division of the experimental measured
$\lambda$ data into different categories is advisable
before attempting a meaningful comparison with a model prediction.
For this reason we have here, unlike in a former preliminary study
\cite{ICPPA-2015}, considered two distinct $pp$ collision data
sample, the first without any multiplicity restrictions labeled by
MB (Minimum-Bias events) and the second the High Multiplicity
events, labeled by HM.
The measured values of
$\lambda_{Gauss}^{MB}$ and $\lambda_{Gauss}^{HM}$ are given in
Table \ref{table1}.
\vspace{2mm}
The measured ratios $\lambda_{Expo}/\lambda_{Gauss}$, evaluated from Table \ref{table1},
are significantly different from that of $\pi/2$ given by Eq.
(\ref{lamratio}) as they are lying in the neighborhood of
the value 2. The relation between the parameter pairs
$(R_{Gauss}, \lambda_{Gauss})$ and $(R_{Expo}, \lambda_{Expo})$
obtained in BEC fits to the measured $Q$
distributions are studied in some details in Ref. \cite{note}
where the $\lambda_{Expo}/\lambda_{Gauss}$ obtained ratios are
matched much better to the measured ones inferred from Table \ref{table1}.
\section{The $R$ and $\lambda$ dependence on the $pp$ energy}
\hspace{6mm}It has been shown that in the 1D BEC there exists a
relation between the dimensions $R_{n=1}$ and $R_{n>1}$, where $n$
is the number of independent similar emitting pion source clusters
\cite{rlamn}, namely
\begin{equation}
R_n = n\frac{\lambda_n}{\lambda_1}R_{n=1}\ .
\label{rs}
\end{equation}
which relates the one sources dimension $R_1$ at low
energies with its value $R_n$ for $n$ identical cluster of sources.
The $\lambda_1$ and $\lambda_n$ parameters are the chaoticity
values respectively for $n=1$ and $n$ clusters. Since we limit
ourselves to the study of
the behavior of the chaoticity parameter with energy
it is sufficient to evaluate the
energy dependence of the ratio $\lambda=\lambda_n/\lambda_1$ .
\vspace{2mm}
Recently the one dimension $R$ dependence on the $pp$ collision
center of mass energy energy was observed to increase
from low energies of less than 100 GeV to those reached by the
LHC at 7 TeV. This rise was fitted
\cite{alexander} to yield
\begin{equation}
R(s)\ =\ \left\{(1.64\pm 0.11)+(0.14\pm
0.02)\ln(\sqrt{s/s_0})\right\}R_{n=1}\ ,
\label{r1d}
\end{equation}
where $s$ is in TeV$^2$ units and $s_0=1$ TeV$^2$.
\vspace{2mm}
\renewcommand{\arraystretch}{1.2}
\begin{table}[h]
\caption{\small Measured $\lambda$
values obtained from the 1D Bose$-$Einstein correlations of two
identical pion pairs produced in $pp$ collisions using the
Gaussian and/or the Exponential parametrizations according to Eqs.
(\ref{gauss}) and (\ref{expo}). The superscript MB and HM
refer respectively to results from all events and from only high
charged multiplicity events. Whenever the statistical and
systematic errors were reported separately they were added in
quadrature in the table.}
\begin{center}
\begin{tabular}{||cl|c||c|c||c||}
\hline\hline \multicolumn{3}{||c||}{1D BEC analyzes}
& \multicolumn{3}{c||}{The chaoticity $\lambda$ parameter}\cr
\cline{1-6} &Reference & $\sqrt{s}$ (GeV) &
$\lambda_{Gauss}^{MB}$ & $\lambda_{Gauss}^{HM}$ & $\lambda_{Expo}$
\cr \hline \hline &\cite{uribe}& 7.21 & 0.466 $\pm$ 0.015\ & 0.532
$\pm$ 0.013\ &
--- \cr
&\cite{bailly}&26.0&
0.32 $\pm$ 0.08\ & 0.43 $\pm$ 0.13\ & --- \cr
&\cite{break}& 31.0 &
0.41 $\pm$ 0.02\ & 0.35 $\pm$ 0.04\ & --- \cr
&\cite{break}&44.0& 0.40 $\pm$ 0.02\ & 0.42 $\pm$ 0.04\ & --- \cr
&\cite{akesson-1}&58.1& 0.34 $\pm$ 0.04\
& --- & --- \cr
&\cite{break}&62.0& 0.43 $\pm$ 0.02\ & 0.42 $\pm$ 0.08\ & --- \cr
&\cite{akesson-2}&63.0 & 0.39 $\pm$ 0.07\ & --- & 0.77$\pm$
0.07 \cr
&\cite{breakstone}&63.0&0.45 $\pm$ 0.03\ & --- & --- \cr
&\cite{agg}&200$^{*}$& 0.35 $\pm$ 0.04\
& 0.36 $\pm$ 0.04\ & --- \cr
&\cite{alice1}&900& 0.35 $\pm$ 0.03\
& 0.31 $\pm$ 0.03\ & 0.55 $\pm$ 0.05 \cr
&\cite{atlasbec,sykora}&900& 0.34 $\pm$ 0.03\ & --- & 0.74 $\pm$ 0.11 \cr
&\cite{cms1,cms2}&900& 0.315 $\pm$
0.014\ & --- & 0.63 $\pm$ 0.03 \cr
&\cite{cms1}&2360& 0.32 $\pm$ 0.01\
& --- & 0.66 $\pm$ 0.09 \cr
&\cite{alice3}&7000& 0.65 $\pm$ 0.05\ & 0.66 $\pm$ 0.07\ & --- \cr
&\cite{atlasbec,sykora}&7000
& 0.33 $\pm$ 0.02\ & 0.25 $\pm$ 0.02\ & 0.53 $\pm$ 0.05 \cr
&\cite{cms2}&7000 &
---
& --- & 0.62 $\pm$ 0.04\cr
\hline\hline
\multicolumn{6}{l}{$^{*}$\rule{0pt}{10pt}\footnotesize The
relative systematic uncertainty for $\lambda$ is taken to be
equal to the corresponding error associated}\vspace{-2mm} \cr
\multicolumn{6}{l}{\footnotesize with the BEC radius.} \cr
\end{tabular}
\end{center}
\vspace{-3mm} \label{table1}
\end{table}
\renewcommand{\arraystretch}{1.0}
It has further been shown that the experimental BEC results are
depending only slightly, if at all, on the
rapidity\footnote{Throughout this work we refer to pseudorapidity
by rapidity.}
extent used in the accumulation of
pion-pairs data sample \cite{rapidity}. Thus $R$
is essentially
independent of the rapidity domain used in the experimental BEC
analyzes and as such should also be valid for the results obtained
from particle tracks at the mid rapidity region.
\vspace{2mm}
The energy behavior of $\lambda$ in terms of Eqs. (\ref{rs})
and (\ref{r1d}) thus requires a solution of the equation
\begin{equation}
\left\{(1.64\pm 0.11)+(0.14\pm
0.02)\ln(\sqrt{s/s_0})\right\}R_{n=1}\ =\ n(s)\lambda R_{n=1}\ ,
\label{extraction}
\end{equation}
where $n(s)$ is the number of source clusters which depends
on energy in the chosen rapidity domain. As has
been found that $R$ increases with the average charged
multiplicity $\langle N_{ch}\rangle$ of the colliding hadrons
and since our aim is to estimate the $\lambda$ dependence on
energy but not on its absolute value, it is sufficient
to require that the number of source clusters is proportional to the
average charged multiplicity.
\subsection{The $\lambda_{1D}$ energy dependence}
\hspace{6mm}A compilation of $\lambda_{Gauss}$ and $\lambda_{Expo}$ deduced
from the 1D BEC of pion-pairs produced in $pp$ collisions is given
in Table \ref{table1} ordered according to their $pp$ energy. The
measured 1D $\lambda_{Gauss}$ are also plotted in Fig.
\ref{data1d} where the chaoticity values in the energy region of 20
to 60 GeV are seen to be scattered somewhat, most probably due to the
different adopted experimental procedures as pointed out in
Section \ref{sec1}. In spite of this, a general decrease with
energy of the $\lambda_{Gauss}$ values is apparent. Here it should
be noted that the ALICE results at $\sqrt{s}=7$ TeV \cite{alice3},
which are outside the boundary of the figures are quite different
from those of the ATLAS experiment \cite{atlasbec} and also are
far from being part of the general pattern of the $\lambda(s)$
energy dependence.
\vspace{2mm}
To evaluate the $\lambda_{Gauss}$ dependence on energy we follow
the formalism outlined in Ref. \cite{nardi} where the
hadron-hadron collisions is contributed by two components. The
first is the ``hard'' component, with a contribution fraction $x$,
which is due to the number of binary collisions $N_{coll}$, and
the remaining $1-x$ fraction originates from the number of
participants $N_{part}$ referred to as the ``soft '' processes. In
the case of $pp$ collisions one has $N_{part}=2$ and
$N_{coll}=1$, so that the number of outgoing charged particles per
rapidity unit in $pp$ can be noted as
$\left.(dN_{ch}/{d\eta})\right|_{\eta=0}= n_{pp}$ in accordance
with \cite{nardi}. \vspace{2mm}
In general in hadron collisions the BEC is analyzed in a rapidity
range symmetric to its central value of $\eta=0$ and it is only
slightly dependent, if at all, on the extent of the rapidity
domain used in the analysis (see, e.g. Refs. \cite{abe,alicerap}).
Thus one should expect that the charge particle multiplicity
utilized in a BEC analysis, is approximately proportional to
$\left.(dN_{ch}/{d\eta})\right|_{\eta=0}$. \vspace{2mm}
For the energy dependence of the charged multiplicity
mid-rapidity density we have
considered three log and power series expressions
\cite{abe,EPJPlus-128-45-2013,alice}
given by
\begin{equation}\label{eq:11}
\left.\biggl(\frac{\textstyle dN_{ch}}{\textstyle d\eta}\biggr)\right|_{\eta=0}=\left\{
\begin{array}{lr}
\vspace*{0.35cm} \sum_{i=0}^2a_i\ln^{i}(s/s_{0}),&
\hspace*{0.0cm}{\rm{(a)}}\\ \vspace*{0.35cm}
\sum_{i=0}^2a_i\ln^{i}(s_{a}/s_{0}),&
\hspace*{0.0cm}{\rm{(b)}}\\
a_{0}(s/s_{0})^{a_{1}}.& {\rm{(c)}}
\end{array}
\right.
\label{eta}
\end{equation}
where $s_{a} \equiv (\sqrt{s}-2m_{p})^{2}$ and $m_{p}$ is the proton
mass. The $a_i$ are the free parameters which were determined from the
data to yield the values given in Table \ref{table1.add}. To note is that in Eq.
(\ref{eq:11}) and in the subsequent formulas $s$ is given in
units of GeV$^2$ and $s_0=1$ GeV$^{2}$, unless otherwise
specifically indicated.
\vspace{2mm}
\renewcommand{\arraystretch}{1.2}
\begin{table}[h]
\caption{\small The free parameter values
obtained for the
rapidity density of charged multiplicity in $pp$ collisions.}
\begin{center}
\begin{tabular}{||l|c||c|c|c||}
\hline\hline Reference & Eq. & $a_{0}$ & $a_{1}$ &$a_{2}$ \cr
\hline \hline \cite{abe} & (11a) & 2.5 $\pm$ 1.0\ & -0.25 $\pm$
0.19\ & 0.023 $\pm$ 0.008\cr \cite{EPJPlus-128-45-2013} & (11b) &
0.39 \ & 0.09 & 0.011 \cr \cite{alice} & (11c) & 0.75 $\pm$ 0.06 \
& 0.114 $\pm$ 0.003& -- \cr \hline\hline
\end{tabular}
\end{center}
\vspace{-3mm} \label{table1.add}
\end{table}
\renewcommand{\arraystretch}{1.0}
In Fig. \ref{dNch-pp} are shown the energy dependence of the
mid-rapidity charge particle densities according to Eq. (\ref{eq:11})
using their parameter values given in Table \ref{table1.add}.
As can be seen, the three Eq. (\ref{eq:11}) expressions
agree among themselves in
the energy range from $\sqrt{s} \sim 30$ GeV up to
of 8 TeV and as such do follow well the measured charge multiplicity
density $(dN_{ch}/d\eta)|_{\eta=0}$ in the range
$\sqrt{s}\sim 200$ GeV to 8 TeV. For our analysis we have chosen
the parametrization given by Eq. (\ref{eq:11}c)
which agrees well with the measured data
further up to 13 TeV and quotes the smallest relative
errors for its components.
\begin{figure}[ht]
\centering{\psfig{file=fig-2.eps,height=8.0cm,
bbllx=0pt,bblly=10pt,bburx=542pt,bbury=552pt}}
\caption{\small
Energy dependence of pseudorapidity density of secondary charged
particles produced in proton-proton and anti-proton-proton
collisions at mid-rapidity ($\eta=0$). The dashed curve
corresponds to the analytic form (\ref{eq:11}a) with parameters
from \cite{abe}, the solid line represents the modified log-series parametrization
(\ref{eq:11}b) from \cite{EPJPlus-128-45-2013} and the dotted curve is
the power function (\ref{eq:11}c) from \cite{alice} with its uncertainty band.
\label{dNch-pp}}.
\end{figure}
Thus we have
\begin{equation}
\left.(dN_{ch}/{d\eta})\right|_{\eta=0}(s)=(0.75 \pm
0.06)(s/s_0)^{0.114 \pm 0.003}\ .
\end{equation}
Inserting $n(s)=\left.(dN_{ch}/{d\eta})\right|_{\eta=0}(s)$ into
Eq. (\ref{extraction}) one obtains the following relation:
\begin{equation}
C\left\{(0.67\pm 0.18)+(0.14 \pm
0.02)\ln(\sqrt{s/s_0})\right\}\ =\ \lambda\left\{(0.75 \pm 0.06)(s/s_{0})^{0.114 \pm 0.003}\right\},
\label{extraction.1}
\end{equation}
where $C$ is a normalization factor.
Solving $\lambda$ from
Eq. (\ref{extraction.1}) one obtains
\begin{equation}
\lambda_{Gauss}(s) \simeq C\frac{(0.89\pm 0.25)+(0.19 \pm
0.03)\ln(\sqrt{s/s_0})}{(s/s_0)^{0.114 \pm 0.003}}.
\label{lamvsen}
\end{equation}
In the present work the normalization of Eq. (\ref{extraction.1})
was determined by requiring that our calculated $\lambda$ will be
equal to the experimentally well measured $\lambda_{Gauss}$ values
at 200 GeV \cite{agg} given in Table \ref{table1}. This yielded
for the MB and HM data samples respectively nearly the equal values of
$C_{MB}=0.63 \pm 0.11$ and $C_{HM}=0.63 \pm 0.13$.
\vspace{2mm}
The experimentally determined $\lambda$ values
are shown in Fig. \ref{data1d}
as a function of $\sqrt{s}$ for $pp$ collision data
free of charged particle multiplicity limitation (left)
and for only high multiplicity
events (right). The data are compared in both
figures
with our normalized calculated estimations
accompanied by a
$\pm$1 s.d. band limits drawn by the dotted curves.
As can be seen, our calculated $\lambda_{Gauss}$ behavior
with the $pp$ energy is within 1 s.d in good agreement
with the general decrease with energy of the measured chaoticity
values obtained
from the HM data sample. For the BEC deduced $\lambda_{gauss}$
from the MB data sample
our approach seem somewhat to deviate from the data
at $pp$ energies above $\sim$1 TeV.
From this
follows that in $pp$ collisions at $\sqrt{s}=13$
TeV and in the current highest planned LHC energy of $\sqrt{s}=14$
TeV, the expected 1D $\lambda_{Gauss}$ values for HM events should approach
$\sim 0.20$.
\begin{figure}[ht]
\centering{\psfig{file=fig-3a.eps,height=8.0cm,
bbllx=0pt,bblly=10pt,bburx=542pt,bbury=552pt}\
{\psfig{file=fig-3b.eps,height=8.0cm,
bbllx=0pt,bblly=10pt,bburx=542pt,bbury=552pt}}}
\caption{\small
The 1D two-pion BEC results for $\lambda_{Gauss}$ as a function
of $\sqrt{s}$.
The expected $\lambda_{Gauss}$ dependence on energy calculated in this work
is shown by the continuous lines normalized to the measured
$\lambda_{Gauss}$ at 200 GeV. The dotted lines represent its
$\pm$1 s.d. limits. Left: Events without a charged multiplicity cut.
The multiple
$\lambda_{Gauss}$ values at 900 was averaged
as well as an average was taken of the $\lambda_{Gauss}$ values
at $\sqrt{s}=62 - 63$ GeV.
Right: Measured $\lambda_{Gauss}$ in high charged
multiplicity events.}
\label{data1d}
\end{figure}
\subsection{The $\lambda_{3D}$ energy dependence}
\hspace{6mm}The BEC analysis in three dimensions (3D) is
frequently represented in its Gaussian form by
\begin{equation}
C_{3D}(Q_{long},Q_{out},Q_{side})\ =
\ 1\ +\ \lambda_{3D}
e^{-(R_{long}^2Q_{long}^2+R_{out}^2Q_{out}^2+R_{side}^2Q_{side}^2)}\ ,
\label{threed}
\end{equation}
where the directions $long$, $out$ and $side$ are defined in the
Longitudinal Center of Mass System (LCMS). (see e.g. Ref.
\cite{review}). The $\lambda_{3D}$ measured values deduced from
BEC carried out in $pp$ collisions
at center of mass energies of 200, 900 and 7000 GeV
are listed in Table \ref{table2}.
At the LHC energies as well as at $\sqrt{s}=200$ GeV
there are only a qualitative
indication for a smooth decrease of $\lambda_{3D}$ with multiplicity
together with a relative small dependence on $k_{T}$ \cite{alice2}.
\vspace{2mm}
This $k_{T}$-dependence of the chaoticity parameter
is studied in 3D BEC analyzes but not in the Gaussian representation of the
1D BEC studies \cite{agg}.
As only $\lambda_{1D}$ and $\lambda_{3D}$ values derived
at similar, or approximately, experimental conditions can be used
for a meaningful comparison,
the $\lambda_{3D}$ values
given in Table \ref{table2} were those obtained at the lowest $k_{T}$.
Furthermore for a comparison purpose the chaoticity data shown
in Table \ref{table2} are those deduced from
the BEC analyzes of the high multiplicity $pp$ collision data.
As can be seen, the $\lambda_{3D}$ values are
higher at the LHC than the corresponding 1D $\lambda_{Gauss}$
obtained at the same $pp$ center of mass energy\footnote{It should
be noted that this conclusion is also valid if the estimation
$\lambda_{3D} \sim 0.49$
is used for the minimum bias events at the
LHC energies based on the qualitative information
from \cite{alice2} together with corresponding values for $\lambda_{1D}$
given in Table \ref{table1}.} and the ratio
${\lambda_{3D}}/{\lambda_{1D}}$ indicates
some growth with energy within their uncertainties.
\vspace{2mm}
\renewcommand{\arraystretch}{1.2}
\begin{table}[h]
\caption{\small A comparison between the $\lambda_{3D}$ and
$\lambda_{1D}$ measured in the high multiplicity $pp$ collision events.}
\begin{center}
\begin{tabular}{||cl|c||c|c|c||}
\hline\hline \multicolumn{3}{||c||}{3D BEC in $pp$
collisions} & \multicolumn{3}{c||}{The measured
$\lambda_{Gauss}$ in HM events}\cr
\cline{1-6} &Reference & $\sqrt{s}$
(GeV) & $\lambda_{1D}$ & $\lambda_{3D}$
&${\lambda_{3D}}/{\lambda_{1D}}$ \cr \hline \hline
&\cite{agg}&200$^{*}$& 0.35 $\pm$ 0.04\
& 0.42 $\pm$ 0.04\ & 1.20 $\pm$ 0.18 \cr
&\cite{alice1,alice2}&900& 0.31 $\pm$ 0.03 \
& 0.42 $\pm$ 0.04 & 1.35 $\pm$ 0.18 \cr
&\cite{sykora,alice2}&7000& 0.25 $\pm$ 0.02 \ & 0.42 $\pm$ 0.04&
1.68 $\pm$ 0.21 \cr \hline\hline
\multicolumn{6}{l}{$^{*}$\rule{0pt}{10pt}\footnotesize The
relative $\lambda$ systematic uncertainty is taken to be equal to
the corresponding error of the}\vspace{-2mm} \cr
\multicolumn{6}{l}{\footnotesize BEC dimension.} \cr
\end{tabular}
\end{center}
\vspace{-3mm}
\label{table2}
\end{table}
\renewcommand{\arraystretch}{1.0}
\section{Summary}
\hspace{6mm}The 1D BEC measured values of $\lambda_{Gauss}$ and with them also
the values of $\lambda_{Expo}$, show a general decrease with
the $pp$ collision energy in particular in the high charged
multiplicity events which does point to
an increase in the coherent pion production.
\vspace{2mm}
The approach adopted here, in which the dimension
$R$ and the multiplicity increase with the $pp$ collision energy
are correlated to the number of source clusters, yield a
decrease of the 1D $\lambda_{Gauss}$ with energy.
The results of this approach agree well
with experimentally 1D BEC deduced $\lambda_{Gauss}$ values obtained
from the high charged
particle multiplicity data up to the $pp$ multi-TeV energy region.
As for the $\lambda_{Gauss}$ obtained
from event samples without any cut on the
outgoing charged particle multiplicity, there is some
discrepancy in the multi-TeV energy region
between the data
and the calculated model expectation
\vspace{2mm}
The chaoticity values extracted from the 3D Bose\,--\,Einstein
correlations are significantly higher than those
obtained in the 1D analyzes, and they seem to remain
essentially constant at the high energy end of the
currently available data.
\subsection*{Acknowledgments}
We would like to thank T. Cs{\"{o}}rg{\'{o}}, C. Pajares and E.K.G. Sarkisyan for helpful suggestions and comments.
|
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{"url":"https:\/\/kar.kent.ac.uk\/33288\/","text":"# Rings of invariants for modular representations of elementary abelian p-groups\n\nCampbell, Eddy, Shank, R. James, Wehlau, David L. (2013) Rings of invariants for modular representations of elementary abelian p-groups. Transformation Groups, 18 (1). pp. 1-22. ISSN 1083-4362 (online 1531-586X). (doi:10.1007\/s00031-013-9207-z) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:33288)\n\n The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided. (Contact us about this Publication) Official URLhttp:\/\/dx.doi.org\/10.1007\/s00031-013-9207-z\n\n## Abstract\n\nWe initiate a study of the rings of invariants of modular representations of elementary abelian $p$-groups. With a few notable exceptions, the modular representation theory of an elementary abelian $p$-group is wild. However, for a given dimension, it is possible to parameterise the representations. We describe parameterisations for modular representations of dimension two and of dimension three. We compute the ring of invariants for all two dimensional representations; these\n\nrings are generated by two algebraically independent elements. We compute the ring of invariants of the symmetric square of a two dimensional representation; these rings are hypersurfaces. We compute the ring of invariants for all three dimensional representations of rank at most three; these rings are complete intersections with embedding dimension at most five. We conjecture that the ring of invariants for any three dimensional representation of an elementary abelian $p$-group is a complete intersection.\n\nItem Type: Article 10.1007\/s00031-013-9207-z modular invariant theory Q Science > QA Mathematics (inc Computing science) > QA150 Algebra Faculties > Sciences > School of Mathematics Statistics and Actuarial Science > Pure Mathematics James Shank 01 Mar 2013 12:46 UTC 06 May 2020 03:07 UTC https:\/\/kar.kent.ac.uk\/id\/eprint\/33288 (The current URI for this page, for reference purposes) https:\/\/orcid.org\/0000-0002-3317-4088","date":"2020-10-29 08:51:55","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\": 0.8282111883163452, \"perplexity\": 1060.3015360639624}, \"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-2020-45\/segments\/1603107903419.77\/warc\/CC-MAIN-20201029065424-20201029095424-00470.warc.gz\"}"}
| null | null |
Tadeusz Nawrot (ur. 21 lutego 1921 w Szypowicach, zm. 25 lutego 1943 w Auschwitz) – polski działacz niepodległościowy.
Syn leśniczego ze wsi Sierbowice – Feliksa Nawrota, brat Feliksa Nawrota, zamieszkały w gajówce w Sierbowicach, uczeń-gimnazjalista, współredaktor gazetki konspiracyjnej "Wyzwolenie" a później "Przedświt". Aresztowany jako zakładnik 4 sierpnia 1942 roku za innego mieszkańca tej miejscowości Bronisława Wałka. Przebywał do stycznia 1943 r. w więzieniu Montelupich w Krakowie, skąd później przewieziono go do niemieckiego obozu koncentracyjnego Auschwitz, gdzie zginął.
Przypisy
Bibliografia
Adam Cyra, Mieszkańcy Ziemi Olkuskiej w hitlerowskich więzieniach i obozach koncentracyjnych, Państwowe Muzeum Auschwitz-Birkenau w Oświęcimiu, Oświęcim-Olkusz, 2005,
Henryk Błażkiewicz OFM, Pilica Zarys dziejów miejscowości
Ludzie związani z polską prasą podziemną 1939–1945
Polskie ofiary niemieckich obozów koncentracyjnych
Więźniowie KL Auschwitz
Więźniowie więzienia Gestapo na Montelupich w Krakowie
Żołnierze Armii Krajowej
Urodzeni w 1921
Zmarli w 1943
|
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## Eloisa James
## A Wild Pursuit
For my wonderful critique partner, Jessica Benson,
who lends me her intelligent language
and her biting wit.
Readers should be aware that
all the funniest lines are hers.
## Contents
In Which Scandal Brews in Wiltshire
A Hen Party...Plus One
So Young and Yet So Diabolic
The Garden of Eden
Anticipation
The Contrariness of Men Hardly Bears Repeating
A Saint, a Sinner, and a Goat
The Sewing Circle
Prudishness...That Coveted Quality
The Heights of Pleasure
The Delights of Poetry
Beds, Baths, and Night Rails
In Which Countess Godwin Learns Salutary Lessons about Desire
Because the Library is Not Yet Emptied of Books
The Imprint of a Man's Skin
The Unexpected Pleasure of Your Company
Playing at Billiards
In Which Curiosity Runs Rampant
Yours to Woo
Twenty Minutes Later...Privacy at Last
In Which a Marquess Pays a Call on a Lady
The Infernal Circle
Various Forms of Advertisement
Waltzing on One's Deathbed
A Taste for Seduction
The Experience That Divides the Ladies from the...Women
Sweet William
In the Library
Spousal Relations
In the Midst of the Night
A Proposal
And Motherly Love, Part Two
In Which the Goat Eats a Notable Piece of Clothing
Yours Till Dawn
Lady Beatrix Entertains
Because It Takes Courage to Admit a Mistake
Nights of Ecstatic Union
The Poetry Reading
The First Epilogue:
Plump as a Porker
The Second Epilogue:
In Which a Puritan Loses His Reputation
A Note on Shakespeare and his Wilder Brethren
About the Author
Praise
Books by Eloisa James
Copyright
About the Publisher
##
## In Which Scandal Brews in Wiltshire
Shantill House
Limpley-Stoke, Wiltshire
It is a truth universally acknowledged by women that it is far easier to dress when the point is to cover one's body, than when one desires to leave expanses of flesh delectably uncovered.
In the days of Esme Rawlings's reign over London society, it took her hours to clothe herself. She would emerge as a caterpillar from its coccoon: silky black curls gleaming over pearly shoulders, bodice miraculously suspended in air at the very moment of dropping to her waist, delectable curves swathed in a fabric so light and revealing that many gentlemen weakened at the knees at her very sight. Other gentlemen stiffened. It was all a matter of constitution.
These days it took precisely twenty minutes to throw on enough clothing to cover herself, and gentlemen in her vicinity never showed reaction beyond a sharpish discomfort at the apparition of a woman with a stomach the size of a large cannonball.
"I am plump as a pork pudding," Esme said, peering at herself in the mirror over her dressing table.
"I wouldn't say that," her aunt said with her characteristic drawl. Viscountess Withers was seated in a small chair, riffling through her reticule. "Drat, I cannot find my handkerchief."
"Stupendously stout," Esme said disconsolately.
"You are carrying a babe," Arabella said, looking up and narrowing her eyes. Clearly a pair of pince-nez would have come in handy, but spectacles were inconceivable, given the dictates of fashion. "I never liked the look of it. But you, my dear, might go far to changing my mind. How dare you look so delightful? Perhaps your example will finish the ridiculous habit of women confining themselves. Such a punitive word, confinement."
"Oh pooh," Esme said, rather rudely. "I am reaching elephantine proportions. No one would wish to see me on the streets of London."
"I believe that your size is normal, not that I've had much to do with childbearing. In fact, this is the first time I have seen a woman so close to her time. So when do you expect it, my dear? Tomorrow?"
"Babies aren't like house guests, Aunt Arabella. They choose their own moment, or so I gather. The midwife seems to think it might be a matter of a few weeks." Privately, Esme thought the midwife had to be mistaken. If she grew any larger, she'd be confined to a bath chair, like the Prince of Wales when he had the gout.
"Well! Here I am, ready to help in every way!" Arabella threw out her hands as if she expected to catch the baby in midair.
Esme had to grin at that. Arabella was her very favorite relative, and not only because her reputation was as scandalous as Esme's own. "It's very kind of you to visit me, Aunt Arabella. Not to mention positively self-sacrificing in the midst of the season."
"Nonsense! One can have just as much pleasure outside of London. Even in Wiltshire, if one applies oneself. I knew that you would be quite dreary in the country all by yourself. Always struck me as a foolish habit, women rusticating themselves in the wilderness merely because they're carrying a babe. The French are much more sensible. Marie Antoinette was dancing up to the moment she gave birth."
"I suppose so," Esme said, wondering whether a black gown would diminish the look of her waist. She was no longer in full mourning, and the idea of returning to blacks was dispiriting. But then, so was her girth.
"I took the liberty of asking just a few persons to follow me tomorrow," her aunt went on briskly. "We shall dine alone tonight, unless Stephen Fairfax-Lacy joins us in time. I suppose you know that your friend the Duchess of Girton is enceinte? If she births a male, obviously Fairfax-Lacy will lose his title. Mind you, it was only an honorary one, but having had it for eight years at least, the man will probably feel as if he's lost his hair. We'll have to cheer him up, won't we, darling?"
Esme looked up, startled. "Fairfax-Lacy? I am not in a position to entertain a house party, particularly one which includes a man I have only the slimmest acquaintance with!"
Arabella ignored her. "And of course I've brought my dame de compagnie with me. Why be on our lonesome when we needn't? It is the season, but I fancy that my invitation outweighs any tedious little parties that might be occuring in London."
"But Aunt Arabella, this is not entirely suitable—"
"Nonsense! I shall take care of everything. In fact, I already have. I brought some of my staff with me, dearest, because there are such terrible difficulties with people hired in the country, are there not?"
"Oh," Esme said, wondering how her butler, Slope, had taken this news. The extra footmen might come in handy if she was reduced to being hoisted about in a chair.
"As I said, a very few persons will follow tomorrow, just to enliven dinner, if nothing else. Of course, we won't hold any public gatherings, or perhaps only a very, very small one, because of your condition."
"But—"
"Now darling," Arabella said, patting her hand, "I've brought you a basket absolutely full of the latest creams and soaps made by that Italian man, the one with the funny little shop in the Blackfriars. They are all absolutely efficacious. You must try them immediately! Your mother's skin was disastrous when she was carrying you." She peered at Esme's face. "But yours appears to be remarkable. Ah well, you always did take after me. Now, I shan't expect you downstairs until dinner. You do remember that Fairfax-Lacy is a Member of Parliament?"
Esme was starting to have an odd feeling about the presence of Stephen Fairfax-Lacy.
"Aunt Arabella," she said, "you wouldn't be thinking of matchmaking, would you? My husband died only eight months ago."
Her aunt raised her exquisitely shaped and dyed eyebrows. "If you call me aunt again, my dearest, I shall scream! It makes me feel quite ancient. Arabella to you, thank you very much. We are family, after all."
"I would be delighted," Esme said, "but—"
Arabella was never one to respect another's sentence. "It's a dreary business, being a widow. I know, as I've been one three times over." She lost her train of thought for a second and then continued. "Now, I'm not saying that I couldn't be married if I chose, because I could."
"Lord Winnamore would marry you in a heartbeat," Esme agreed.
"Precisely," Arabella said, waving her hand. "I've invited Winnamore as well; he should arrive tomorrow. But my point is, darling, that being a widow is rather—daunting. Fatiguing, really."
"Oh, dear," Esme said, thinking that her aunt did look rather more tired than she had seen her in the past. "You must make a long visit."
"Nonsense," Arabella said smartly. "I shall stay with you for the time being. But where's the excitement in living with a woman, hmmm?"
Her wicked smile made her lose at least twenty years.
Esme grinned back. "I'll take your word for it. Miles and I only lived together for a year, and that was years ago, so I can hardly speak from experience."
"All the more reason to marry again," Arabella observed. "Now I've been thinking about Stephen Fairfax-Lacy. He's just the man for you. Lovely laugh lines around his eyes. That's important. And he's strong too. Apparently he boxes regularly, so he won't keel over in the act the way your late husband did."
"It wasn't in the act," Esme protested. Her husband had suffered an attack in their bedchamber. The fact that it had occurred during the first night they had spent together in years was not relevant.
"Close enough. Not that we can fault poor Miles too much. After all, he got the deed done, didn't he?" She waved vaguely in the direction of Esme's belly.
"Yes," Esme said, dismissing the thought of another possible contributor to her situation.
"Fairfax-Lacy is not a man to leave you in the breech, so to speak." Arabella almost choked on her smirk.
"I'm glad you're enjoying this discussion," Esme said pointedly. "It's nice to know that my husband's demise affords someone pleasure."
"For goodness' sake, Esme, don't start taking on airs like your mother. The way Fanny wept over your father could hardly be believed. And yet she couldn't stand the fellow. Well, who could?"
Arabella began opening the jars on Esme's dressing table and sniffing each of them delicately. "This is the best of the lot," she said, holding up a small jar. "Almond paste, straight from Italy and ground by nuns. Has a glorious perfume to it. Rub it on your chest every night and it will keep your skin as white as snow!" The viscountess had never been acclaimed as a beauty, but she didn't let that fact get in her way, any more than she was allowing age to dampen her flair. Her hair had faded slightly from a fiery mass to a gingery pink, but it was swept up in an exuberant mass of curls. Her face paint could not have been more exquisitely applied: it alone took some ten years from her true age.
She put the jar down with a little thump. "Now, let's see. Fairfax-Lacy has a good strong leg, and I like his buttocks too." She rubbed a little of the miraculous almond cream into her neck. "He has plenty of blunt, not that you'll need it, since Rawlings left you well established. The point is, Fairfax-Lacy is a good man and he won't give out in the long term. Stamina, that's what you want in a man. Look at me: married three times, and not a single one of them survived past a few years."
Esme sighed. Clearly poor Mr. Fairfax-Lacy was about to be thrown in her direction until his head spun.
"We're dreadfully awkward numbers tonight, with so few men," Arabella said, patting the almond cream into her cheeks. "Yourself and I, of course, and your friend Lady Godwin, and my dame de compagnie."
"Who is she?" Esme asked without much interest.
"Well, poor duck, she's really my goddaughter. I don't suppose you know her. She debuted four years ago."
"But what's her name?"
Arabella fidgeted with the glass jar for a moment, looking uncharacteristically hesitant. "I shouldn't want you to—well, I can trust you to be kind to the gel. It's not as if you've a puritanical past yourself."
Esme looked up at her aunt. "Her name?"
"Lady Beatrix Lennox."
One of the most irritating things about pregnancy, to Esme's mind, was that she seemed to have no firm grasp on her memory anymore. "I'm afraid I know nothing of her," she said finally.
"Yes, you do," her aunt said rather brusquely. "Beatrix is one of the daughters of the Duke of Wintersall. In her first season, unfortunately—"
"That daughter?" Now Esme remembered. She raised an eyebrow at her aunt. "I suppose you consider her your protegée, so to speak?"
"You're hardly one to talk, my girl," Arabella observed, patting her curls in the mirror. "You've made quite a few scandals yourself in the past ten years, and I'll have you know many a person considers you my protegée. Including your mama. Lord knows, Fanny has complained of my influence enough."
Esme was trying to remember the scandal. "Wasn't Lady Beatrix actually caught in flagrante delicto at a ball? I never did that."
"Naturally, I would never inquire about such a delicate subject," Arabella said, raising an eyebrow, "but perhaps you were merely never caught?"
Esme suddenly remembered a certain drawing room at Lady Troubridge's house, and kept a prudent silence.
"I'm never one to approve of the pot calling the kettle black," Arabella said, favoring her niece with a smug smile. "Poor Bea was only a baby, after all, and no mama to take charge. The duke had found some doddering old cousin to act as chaperone, and naturally Bea was lured into a closed room by Sandhurst. Happens to girls all the time, but the father generally hushes it up. Instead Wintersall decided to make her into an example for his other five daughters, or so he had the impudence to tell me. Apparently, he told Bea she was fit for nothing more than a hothouse and gave her the address of one!"
"Oh, the poor girl," Esme said. "I had no idea." At least she herself had been safely married when she'd embarked on a life that had earned her the sobriquet Infamous Esme.
"Well, don't go thinking that she's a wilting lily. Bea can hold her own among the best of them. I'm glad that I took her after her father disowned her. She keeps me young."
Esme had a sudden thought. "You didn't do this simply to irritate Mama, did you?"
"It did have a miraculous effect on your mother's temper," Arabella said with a smirk. "Fanny wouldn't have me in her house for at least six months. Lately, I have contemplated a major renovation of my town house, if only because I could insist on staying with my sister for a time, and naturally I would bring my dame de compagnie with me."
Esme couldn't help laughing. "Poor Mama."
" 'Twould do your mother good to be around Bea for a time. The gel has a backbone of steel, and she enjoys putting people in a stir. Thinks it's good for them. Wait till you meet her, my dear. She'll go far, mark my words!"
"Oh my," Esme said, suddenly remembering her Sewing Circle and their likely reaction to Beatrix Lennox. "I forgot to tell you, Aunt Arabella, that I've become respectable."
Arabella blinked and then snorted. "You? Why on earth would you wish to do such an odd thing?"
"I promised Miles before he died that I would avoid any sort of notoriety. He wished to live in Wiltshire, you know. I've been establishing myself among the local people, and—"
"I thought it was deuced odd that you, Esme Rawlings, were whiling out your confinement in Wiltshire like some sort of milk-and-water country miss! So you've decided to change your wicked ways, have you?"
"I have," Esme said, ignoring her aunt's smirk. Arabella could smirk all she liked; Esme was determined to live the life of a respectable widow and mother from now on.
"So how are you effecting this miraculous transformation, then?" Arabella asked, having the impudence to appear utterly disbelieving. "The information might be useful in the unlikely event that I'm...I'm..." Arabella clearly couldn't think of any circumstances that would drive her into respectability.
Esme shrugged. "It's not so difficult. I joined the local Sewing Circle and—"
But when Arabella was in the room, it was always difficult to finish a sentence. "You? You've joined a Sewing Circle?"
She needn't hoot with laughter like that. They could probably hear her in the next county.
"I have," Esme said with dignity. "It's a very worthy cause, Aunt Arabella. We sew sheets for the poor."
"Far be it from me to get in your way! Just let me know when the ladies are descending on the house, and I'll make myself scarce," Arabella chortled. "I'll have a word with Bea as well. I daresay she'll flee into the village rather than be trapped with a bunch of seamstresses."
Esme scowled at her. "There's no need to mock me."
"I'm not mocking you, love...well, not entirely. Would you prefer that I return to London and left you to the worthy matrons?"
"No!" And Esme found she really meant it. "Please don't go, Aunt Arabella. It is truly wonderful having someone here, at the moment. Not that I wish Mama could, but—"
"There's nothing wrong with wishing your mama wasn't such a stiff-rumped old chicken," Arabella put in. "My sister has always been a fool. Docile as a sheep. Let you be married off to Miles Rawlings without a by-your-leave, although anyone could tell that the two of you would never suit. Fanny never did learn to say no to your father, but what's her excuse now? Your father went his way these two years, and has she come out from his shadow? No. Just as prissy as he was. The only thing that woman thinks about is her reputation."
"That's quite harsh," Esme protested. "Mama has had a most difficult life. I know she has never recovered from the death of my little brother."
"That was a grievous sorrow, to be sure. He was an enchanting lad."
"Sometimes I'm terrified for my babe," Esme confessed. "What if—what if—" But she couldn't finish the sentence.
"That will not happen," Arabella stated. "I won't allow it. I do wish to point out, Esme, that while your mother has experienced tragedy, she needn't have responded by becoming so highty-tighty.
"Just don't turn into her, with all your plans for propriety. Promise me that. Poor Fanny hasn't had a day in years in which she didn't find some impropriety to turn her mouth sour. That's the problem with caring overmuch about your reputation: it leads to caring overmuch about other people's reputations as well."
"I would never do that," Esme said. "I merely promised Miles that I wouldn't be a scandalous mother to our child."
"Deathbed promise, eh? I've made a few of those myself." Arabella was silent for a second.
"It wasn't exactly a deathbed promise. We had discussed how we would raise our child a few days before he died."
Arabella nodded. "It's difficult to ignore the wishes of a dead man. I agree with you." She seemed to shake off a melancholy thought. "Hey-ho for the Proper Life! Your mother will be pleased, I suppose. Actually, your ambitions are all the more reason to consider Fairfax-Lacy as a husband. He's proper enough to suit your mama, and yet he's not tiresome. Which reminds me. Such a hen party tonight. The only man in the lot will be Fairfax-Lacy, if he arrives, and even I can't see the point of dressing for a man half my age."
"He's not half your age," Esme pointed out. "He's just slightly younger. You're only fifty, and he must be in his forties."
"Too young," Arabella said firmly. "Do you know, I once took a lover who was ten years younger, and it was altogether an exhausting experience. I had to dismiss him after a few days. Too, too fatiguing. The truth is, darling, I'm getting old!"
Esme gathered her wandering thoughts just in time to answer properly. "No!"
"Surprising, but true." Arabella looked at her reflection, but without a trace of melancholy. "I find that I don't mind very much. In fact, I rather like it. But your mother complains endlessly about her aches and pains." She turned around and fixed Esme with a fierce eye. "You are my favorite niece—"
"I'm your only niece," Esme put in.
"Just so. And what I mean to say is, I want you to take life on the hilt, rather than sit back and complain about it. Not that I don't love your mama, because I do. But you've more of my blood, and you always have done."
She turned back to the mirror. "The only thing I regret about age are the wrinkles. But I have high hopes for this new almond cream! Do you know, that Italian apothecary promises the cream will make one's skin as soft as a baby's cheek? Once your child arrives, we'll have a viable comparison. Not having seen a baby in years, how would I know what its skin looks like?"
"I'm glad my condition will prove to be of use," Esme said rather tartly.
##
## A Hen Party...Plus One
Stephen Fairfax-Lacy straightened his cravat and wondered, for the hundredth time, what in the world he was doing attending a house party while Parliament was in session. It wasn't just any house party either. No, in joining a flock of degenerates at Infamous Esme Rawlings's house he was missing indubitably important speeches on the Corn Laws. Castlereagh was expecting him to keep an eye on things in Parliament while the foreign secretary was off in Vienna, carving up Europe like slices of woodcock pie. There were problems brewing between the Canadian frontier and those cursed American colonies—correction, former colonies—not to mention the ongoing irritation of possible riots due to the Corn Laws. He had a strong feeling there would be deadly riots soon, protesting the increase in food prices.
But he just couldn't bring himself to care. He'd spent the last ten years fighting for the good of the common man. He had never used the honorary title he held as heir to his cousin Camden, the Duke of Girton. No, he'd been elected to the Commons on his own merits. Due to his own convictions.
And where were those convictions? Ten years of battling back and forth about Corn Laws and Enclosure Acts had drained the passion out of him. Years of trying to convince his own party to reconsider its position on Enclosures. Six years ago he'd put up fevered opposition to a proposed Enclosure Act. Now another such act was presented every week. He could barely bring himself to vote. No matter what he did, more and more families were being forcibly evicted from their farmland so that rich men could build fences and graze herds of sheep. He was a failure.
He threw away the cravat, which was hopelessly creased. Generally he could make a simple trone d'amour in under eight minutes, but this evening he had ruined two cloths already. "Sorry, Winchett," he said, as his valet handed him yet another starched cloth.
He stared at his reflection for a moment as he deftly tied the neck cloth. Even if this go at the trone d'amour finally seemed to be going well, nothing else in his life was. He felt old, for one thing. Old at forty-three years old. And, damn it all—he felt lonely. He knew exactly why that was. It was the visit to Cam. His cousin and wife had just returned from a visit to Greece. The duchess was radiant, intelligent, and expecting a child. And Cam—Cam who had been forced into a marriage, and then spent ten years hiding in Greece rather than acknowledge it—Cam was bursting with pride.
It was Gina and Cam's sense of companionship that underscored Stephen's loneliness. He'd distinctly seen Gina, the Duchess of Girton, tell her husband to close his mouth, and without saying a word! And Cam had done it. Amazing. Cam was friends with his wife.
Stephen's mouth took on a grim line as he folded the last piece of linen. There were no women like Gina wandering around London. Not as intelligent and yet untouched, with that bone-deep innocence. Naturally, one had to have that quality in a wife. But he was—just to repeat—all of forty-three. Too old for a debutante.
Finally Stephen shrugged on a coat and walked down the stairs. Perhaps he would plead work and return to London first thing in the morning. He might even attend a ball at Almack's and acquire some fresh young thing who didn't mind what an old man he was. After all, he was a good catch, to put it vulgarly. He had a resoundingly good estate.
Of course, he hardly remembered what the property looked like, given that his work in Parliament had taken virtually all his time in the past ten years. He had a wash of longing for the lazy days of his youth, sitting around with Cam, whittling boats and fishing for trout they rarely caught. These days he fished for votes.
What I need, he thought suddenly, is a mistress. It's a lengthy business, fishing for a wife, and likely tedious as well. But a mistress would offer an immediate solution to his malaise. No doubt life had a plodding sensation because he hadn't had a mistress in a donkey's age.
He paused for a moment and thought. Could it really have been a year since he entered a woman's bedchamber? How could that be? Too many smoky late nights, talking votes with whiskey-soaked men. Had it truly been a year since Maribell had kissed him good-bye and walked off with Lord Pinkerton? Over a year ago. Damn.
No wonder he was always in a foul mood. Still, Esme Rawlings's house would be an excellent hunting ground for a mistress. He walked into the salon with a surge of enthusiasm and bowed over his hostess's hand. "I must beg your forgiveness for my importunate arrival, my lady. Lady Withers assures me that she treats your house as her own. I trust she didn't prevaricate?"
Lady Rawlings chuckled, that deep, rich laugh that had entranced half the men in London. Of course, she was great with child and had presumably curtailed her seductive activities. Beautiful woman, though. She was even more lush than he remembered, with breasts that gave a man an instant ache in the groin. In fact...Stephen caught himself sharply before he formed the image. I must be getting desperate, he thought, kissing her hand.
There was something about the way Lady Rawlings's eyes met his that made him think she could read his thoughts, so he turned quickly to the lady next to her. It was shameful to be entertaining such thoughts about a woman on the verge of giving birth.
"This is Lady Beatrix Lennox," Lady Rawlings said. There was an odd tone in her voice, as if he were expected to recognize the girl. "Lady Beatrix, Stephen Fairfax-Lacy, the Earl of Spade."
"I do not use the title," he said, bowing. Lady Beatrix was clearly unmarried but equally clearly not eligible to be his wife. A wife had to have an angelic air, a sense of fragility and purity, whereas Lady Beatrix looked like a high-flying courtesan. Her lips were like a pouting rosebud, and that rosebud never grew in nature. Given that her skin was as pale as cream and red curls tumbled down her back, those velvety black eyelashes were obviously false too.
A beauty, a seductively false beauty. He almost laughed. Wasn't she exactly what he hoped for? A woman the precise opposite of his future bride. A woman who would likely be unrecognizable in the morning, were he ever foolish enough to spend a night in her bed. Too bad she was both well-bred and unmarried, and thus ineligible for an affair.
"Mr. Fairfax-Lacy," she said, and her voice had the practiced, husky promise of a coquette. "What a pleasure to meet you."
He brushed a kiss on the back of her hand. Sure enough, she wore French perfume, the sort that a certain kind of woman considers akin to a night rail.
"The pleasure is all mine," he said. She had high, delicate eyebrows, and the fact that she'd colored them black somehow suited her face.
Lady Arabella appeared at his side. "Ah, I see you've met my dame de compagnie," she said. "Bea, Mr. Fairfax-Lacy is quite a paragon of good works. Just imagine—he's a Member of Parliament! Commons, you know."
"At the moment," Stephen heard himself say and then wondered what on earth had led him to say such a thing.
Lady Beatrix looked bored by this revelation, so he bowed again and left. He'd just caught sight of Countess Godwin on the other side of the room. Now she was a distinct possibility, given that she hadn't lived with her husband for years. Moreover, she was beautiful, in a pale, well-bred way. He liked the way she wore her hair in a nest of braids. It showed a flagrant disregard for the current fashion of dangling frizzled curls around the ears.
Unfortunately, Lady Godwin's reputation was damn near irreproachable. She would be a challenge. But wasn't that what he needed? A challenge? He strode across the room toward her.
In the kind of serendipity that happens only too infrequently betwixt the sexes, his companion was thinking along precisely the same lines.
Helene, Countess Godwin, had watched Stephen enter the room and had been instantly struck by how remarkably good-looking Mr. Fairfax-Lacy was. He had the long, narrow face and high cheekbones of an English aristocrat. Moreover, he was immaculately dressed, a quality she considered to be of the highest importance, since her point of reference was her husband. He was bowing over Esme's hand and smiling at her. He couldn't be interested in a flirtation with Esme, could he? Under the circumstances? Men were always flirting with Esme, Helene thought with a dispirited pang. But the next moment he walked across the room directly toward her.
Helene felt a blush rising in her neck. Of course she shouldn't have been caught staring at the man, for all the world like a debutante. But it would be a pleasure to make his further acquaintance, if only because he was such a conscientious member of Parliament. Her own father said he was the best man in London on grain. More to the point, he was remarkably good-looking. His hair just brushed his neck, whereas her husband let his hair wave around his shoulders like some sort of wild animal. Oh, if only she'd married someone like Mr. Fairfax-Lacy instead of Rees, all those years ago!
But of course Stephen Fairfax-Lacy would never in a million years have eloped with someone as young and foolish as she had been. In fact, it seemed unlikely he'd ever marry. The man must be in his forties.
She sank into a curtsy before him. "I am delighted to meet you again, Mr. Fairfax-Lacy. What are you doing in the country? I thought Parliament was in session. And you, sir, are known to be the House's most determined whipper-in!" She allowed him to place her on a settee and sit beside her.
He smiled at her, but the smile fell short of his eyes. "They can spare me for a week or so," he said lightly.
"It must be rather difficult to keep up with all those issues," Helene said. He truly had lovely blue eyes. They were so respectable and clear, as opposed to her husband's muddy, scowling look.
"I don't find it difficult to follow the issues. But I'm finding it difficult to care about them as much as I used to." Stephen was feeling more cheerful by the moment. What he needed was a woman to purge this sense that the world was gray and fruitless. Lady Godwin's bashful charm was a perfect antidote.
"Oh dear," Helene said, touching his fingers with her gloved hand. "I am sorry to hear that. I sometimes think that you are a remarkable voice of clarity in the midst of the Tories. I, sir, seem naturally inclined to Whiggery."
"I find that alarming. What is it that draws you to the enemy side?"
His eyes crinkled in the corners when he smiled at her. Helene almost lost track of their conversation. He had very long, lean fingers. Didn't that...hadn't Esme told her something about a man's hands? She snatched her mind away from that vastly improper subject. "I have not found the past years of Tory government satisfactory," she said hastily.
"Oh?"
His eyes actually looked interested in what she was saying. Helene made a conscious effort to sound intelligent. "To tell the truth," she said, "I believe the government is making a huge mistake by ignoring the numbers of unemployed men in the country. The homeless, jobless soldiers wandering the roads are a reproach to all of us."
Stephen nodded and made a conscious effort to sound like a scrupulous and sympathetic politician. "I know. I wish I were convinced that a change of government would shift people's perception of the discharged soldiers." She was so slender that one really had to wonder whether she bothered to wear a corset. He'd never liked those garments, although women seemed to find them obligatory.
"I should not berate you of all men," Helene said. "Didn't I just read a speech of yours on the subject, transcribed in the Times? And you were quite eloquent on the state of the hungry laborers."
It was appalling, how tired he was of thinking of the plight of the poor. "Thank you," Stephen said, "but I am afraid my speeches are like water on stone; they seem to have little effect."
She leaned forward. "Never say that! If good men such as yourself did not stand up for the poor and downhearted, well then, what should become of them?"
"I've told myself the same, time out of mind, but I must admit, Lady Godwin, that I find the counsel far more engaging when spoken by such an intelligent woman." She was wearing a corset. He could tell by the way she moved toward him rather stiffly, like a marionette. It wasn't as if she had extra flesh to confine, so why on earth was she wearing that garment?
Helene pinked and realized that in her excitement she had picked up Mr. Fairfax-Lacy's hand. Blinking, she made to draw hers away, but he held it for a moment.
"It is a great pleasure to meet a woman interested in the political life of the nation."
He had a lovely voice, she thought. No wonder his speeches were so closely attended! Luckily (because she really had no idea what to say), Slope brought them both a sherry, which broke the oddly intimate feeling of the moment.
But they sat together for a moment in silence, and any reasonable observer could have noticed that Lady Godwin was slightly pink in the cheeks. The same observer might have glimpsed Mr. Fairfax-Lacy stealing a look at Lady Godwin's face, while she examined her sherry with rapt attention.
An acute observer—the kind who can see into human hearts—could have perceived speculation. Rampant speculation that led to a few conclusions.
Countess Godwin decided that Mr. Fairfax-Lacy had beautiful lean cheeks. She rather liked his thighs too, although she would never have formulated that thought into words. She was also still trying rather desperately to remember what it was that Esme had told her about men with long fingers.
As it happened, Mr. Fairfax-Lacy's attention was also caught by the question of fingers. Countess Godwin's fingers were slender, pink tipped and strikingly feminine. Being male, that formulation turned directly to self-interest. He liked the little flush the countess got every time she looked him in the eyes. And those fingers...
One thought prevailed: how would those slender fingers feel on my body? The image brought neglected parts of his anatomy to attention. Perhaps corsets weren't such an impediment, a thought supplanted by an image of a Norse goddess, pale hair swirling over her slender shoulders, unlacing her corset with delicate fingers....
##
## So Young and Yet So Diabolic
Lady Beatrix Lennox was inclined to think that she had wasted her efforts dressing. She had expected more excitement from a house party being given by the scandalous Lady Rawlings. But Countess Godwin was the only guest other than those Arabella had brought with her, and the countess didn't interest Bea. First of all, she was female. Secondly, she was prudish, proper and a strange choice of friend for the infamous Lady Rawlings. Thirdly, Bea had little patience for the martyred wife role.
Were I foolish enough to marry, Bea thought, wandering toward the windows, and were my husband as flagrantly unfaithful as is Earl Godwin, I'd take a fork to him. Outside there was nothing to see but a few stone walls with rusty ferns growing from them. She took a sip of sherry. It had a smoky sharpness that went with the gray afternoon.
A husband who invited an opera singer to reside in his wife's bedchamber obviously deserved violence. Shattered china came to mind. She would have quickly taught the man better manners.
When someone tapped on her shoulder, Bea was far away, imagining a confrontation with her imaginary husband's imaginary mistress. She spun around with a suppressed gasp. The countess herself stood before her.
They curtsied and exchanged the usual trivialities, and then the countess turned and stared at the same rusty ferns Bea had been looking at. After a second, she said, "You looked so absorbed by the view that I thought it must be magnificent. I forgot that this window looks only into the back courtyard."
Bea was feeling that pulse of wicked boredom, the one that always got her in trouble. "I was meditating on unfaithful husbands," she said, looking at the ferns and not at her companion.
"Oh?" the countess sounded startled, but not appalled. "I have one of those. I hope you're not planning to follow my example."
Bea laughed. "I have no plans to marry, and so hopefully I shall avoid that conundrum."
"I eloped," the countess said rather dreamily. "That was the problem, I do believe. Elopement is about the intoxication of acquaintance. And acquaintance is hardly a solid basis for marriage."
"I always thought elopement was rather romantic," Bea said curiously. It was hard to imagine anyone wishing to elope with Lady Godwin, to be honest. The countess was a slender woman with stark cheekbones and a good deal of braided hair, not a look that Bea admired much. It made her look positively medieval. Plus, she was hideously flat-chested. Bea's own undergarments were cleverly designed to enhance every inch of flesh she had, as well as suggesting many inches that she didn't have, and she maintained a lively scorn for any woman who didn't avail herself of such garments.
"I must have thought elopement was romantic as well," the countess said, sitting down. "I can hardly credit it now. Of course, that was years ago, and I was a foolish girl."
Bea's mind had jumped back to her bloodythirsty fantasies. "Do you ever think of taking your husband in hand?" she asked.
"Taking him in hand?" The countess looked up at her, one eyebrow raised.
Bea's streak of mischief grew larger. Surely, listening to the countess's marital woes would be more fun than examining rusty ferns out the window. She sat down as well. "Why haven't you evicted the opera singer from your bedchamber?" she asked, precisely as if she were inquiring the time of day. This was a deliciously improper conversation, even given that Bea rather specialized in unsuitable topics. Surprisingly, Countess Godwin didn't turn a hair at her impropriety.
"Absolutely not," she said, gazing into her glass of sherry.
"I would never allow another woman to sleep in my bedchamber."
"To evict the woman in question would imply that I had an interest in entering that bedchamber."
Bea waited. She had discovered that silence sometimes inspired interesting confidences.
"If she weren't in my bed," the countess continued, "who would be there? I think of her as a necessary evil. A nuisance because everyone is so aware of her presence. Along the lines of a bed warmer."
Bea choked. She had just discovered why the notoriously proper Countess Godwin was friends with the equally notoriously improper Lady Rawlings. "A bed warmer?"
The countess nodded, looking as serene as a dowager discussing a baptism.
Bea could see her point. If Lady Godwin didn't want to bed her husband, the opera singer might as well do the chore for her. But all the world knew that Lady Godwin lived in her mother's house, rather than in her husband's house on Rothsfeld Square.
"That's not equitable," she pointed out. "You should be able to sleep in your own house. You are married to the man."
The countess cast her a sardonic glance. "Have you found that life is fair to females, then, Lady Beatrix? I think we would both sum it up as deplorable."
Until then, Bea hadn't been quite sure whether the countess remembered her scandalous past. "I don't consider my situation a deplorable one."
"If my memory serves, you were caught in an indiscretion with Sandhurst. His reputation was untouched by the scandal; yours was ruined. You were forced out of your childhood home, and"—she paused, looking for the right word—"ostracized by a great many people you once knew."
"But I didn't want to marry Sandhurst," Bea pointed out. "Had I married the man, I suppose it would have all blown over. I refused him."
"I admit, I thought the offer had not been made," the countess admitted. Then, after a moment, she added, "Why didn't you wish to marry him?"
"I didn't like him very much."
The countess swirled her sherry, then drank it in one gulp. "You are wiser by far than I, Lady Beatrix. I didn't discover a similar dislike until I was already married."
Bea smiled at her. "They should outlaw Gretna Green weddings, perhaps."
"Perhaps. Do you really think that you'll never marry?"
"Yes."
"And did you always feel that way?"
Presumably the countess knew as well as Bea did that no respectable man would wish to marry a person like her. Bea didn't say anything.
"Of course you thought to marry," the countess said to herself. "Otherwise, you never would have refused Sandhurst's offer. I'm sorry."
Bea shrugged. "This is a case where dreams have been supplanted by reality. I could not tolerate a husband such as yours, my lady. I'd probably take to him with a blunt instrument. Truly, I am better off in my position."
Lady Godwin was grinning. Bea was surprised to find how enlivened her face was by humor. She didn't look boringly medieval anymore, but sparkling and quite lovely, in a slender kind of way.
"And just what would you do to my husband?" she asked with some curiosity. "And by the way, you must call me Helene. This is one of the most intimate conversations I've ever had with a complete stranger, after all." In fact, Helene was surprised at herself. There was something about Beatrix Lennox, some sort of mischievous sparkle, that reminded her of Esme. Which must explain why she, Helene, was being so uncharacteristically indiscreet.
"I would love to, as long as you call me Bea. I gather that you do not wish for your husband to...play an active role in your life," Bea said, trying for a delicate tone. Subtlety wasn't exactly her strong point.
Helene laughed, a short, rather bristly laugh. "No."
"I would make him sorry, then. I would make him very, very sorry that he ever thought to leave my bed. At the same time that I made it clear he hadn't the faintest hope of returning."
"Revenge is mine?" Helene asked, eyebrow raised again. She rather liked the idea of revenge. There were whole days—such as the one when Rees appeared in the Godwin opera box, doxy in tow—when she thought of nothing but doing Rees serious injury.
"Precisely," Bea nodded. "Besides, revenge is not only sweet in itself, but enjoyable. You, Lady Godwin—"
"Helene."
"Helene," Bea repeated obediently. "You have the kind of reputation that the three other women in this room could only dream of. That is, if we had the desire for such dreams."
Helene looked around. True enough, Bea, Lady Arabella and Esme herself could hardly be called champions of propriety. "Esme is turning over a new leaf," she pointed out. "I believe she does indeed dream of being a proper matron, or widow, rather."
Bea shrugged. "Lady Rawlings may be aspiring to a chaste reputation, but I certainly am not. And I've seen no signs of such ambition on Arabella's part either. The point is, though, that you are the one of us who has been most flagrantly slighted by a man, and yet you are the most prudent of all of us. If I were you, I would be flaunting my affairs before my husband."
"Perhaps if he cared, I would. But Rees wouldn't give a hang, to be honest."
"Nonsense. Men are like dogs: they want the whole manger, even though they don't eat hay themselves. If you have an affair, especially one in the public eye, it will curdle his liver." Bea said it with a certain relish. It was gratifying to see how closely the countess was listening to her. "Not to mention the fact that you will enjoy yourself."
"My goodness," Helene said. Then she smiled again. "Naturally, I like the idea of curdling his liver."
"Your husband has the best of all worlds," Bea insisted. "He has that opera singer, and he has you. The world and all knows that you're faithful to him."
Helene chewed her lip for a moment. "The problem is that I'd have to have an affair in order to flaunt one," she pointed out.
"Precisely!" Bea said, grinning at her. "You have nothing to lose but reputation, and what has that got you?"
"Respectability?"
But Bea knew she had her. She paused and looked at Helene from the top of her tightly coiled braid to the tips of her slippers. Her gaze spoke for herself.
"I think they warned me about women like you when I was in the schoolroom," Helene observed.
Bea fluttered her eyelashes. "So young and yet so diabolic?"
"Something of the sort." But Helene had come down to earth with a thump. She looked back into the depths of her sherry. "It hardly signifies, because I haven't the faintest hope of attracting a man with whom to have an affair, if you must know. No one has made me an indecent proposal in years. In fact, I think my husband may have been the first, and the last, to do so." She felt a crawling mortification at the admission.
"Nonsense. Available men are everywhere," Bea said, giving her an encouraging smile.
From Bea's point of view, Helene thought glumly. She was likely propositioned every other day.
"Men do seem a bit thin on the ground at this particular party," Bea continued. "What about that—that politician Arabella dragged out here? I've forgotten his name." She nodded toward him.
"Mr. Fairfax-Lacy?" Helene asked. "I'm not sure that—"
"I know, I know. I thought just the same: Church fathers, propriety, honor, Old Testament...A boring old Puritan!" Puritan was Bea's worst insult.
"I didn't mean that! I actually find Mr. Fairfax-Lacy quite attractive, but he is unlikely to make imprudent love to me. Let alone in front of my husband. Men simply do not think of me in those terms."
Bea hesitated. She could hardly inform a woman whom she had just met that she needed a new wardrobe. "Sometimes those Old Testament types are longing for a diversion," she said. "If not, why on earth did the man take up Arabella's invitation? This is not the house party for a prudent public servant. Arabella is not interested in him for herself; she would have told me. Besides, she dislikes younger men."
They both stared across the room at Mr. Fairfax-Lacy, who was talking to their hostess.
"Do you think he knows anything of music?" Helene asked dubiously.
"What's that got to do with the price of oranges?"
"I couldn't—I'm very fond of—that is, I couldn't spend my time with someone who didn't like music."
At that very moment, Mr. Fairfax-Lacy turned to the pianoforte in the corner of the room, sat down with a twinkling smile at Esme, and began to play a lively tune.
"Does he pass muster?" Bea asked. She herself had been trained on the harp, since her father considered tinkling little tunes to be indicative of ladylike thoughts.
"Not in terms of taste," Helene said a bit sourly. "He's playing one of my husband's arias. You do know that my husband writes comic operas, don't you?"
Bea nodded, even though she hadn't had the faintest idea. Helene was married to an earl. Did earls write comic operas?
"The piece he's playing comes from an opera called The White Elephant. Drrread-ful," Helene said. "Overall, the opera wasn't bad. But that particular song was absolutely dreadful."
"What's the matter with it?"
"The soprano has to sing an F in alt. The poor girl nearly strangled herself trying to reach it, and the audience thought her stays were pinching," Helene said, gazing across the room. "And the overture had so many dissonances that the orchestra sounded as if it were sight-reading the piece. Disaster. It was an utter disaster. The fact that Mr. Fairfax-Lacy liked it enough to memorize the piece doesn't say much for his taste."
But Bea had already made up her mind that Helene and the politician were a possible match, and she wasn't going to allow his inadequate musical judgment to influence Helene. "I'll walk you across the room, and you can improve Mr. Puritan's musical taste," Bea said encouragingly. "Men love it when a beautiful woman corrects them. Meanwhile we can assess whether he is worth your time and effort. He's old enough to be going soggy at the waistline, which is far worse than a lack of musical ability. Trust me on this."
"It hasn't been my experience that men enjoy correction," Helene said, "and I'm hardly—" but Bea was pulling her across the room like a determined little towboat.
Stephen looked up to find the glorious bit of disrepute, Lady Beatrix, and the graceful Lady Godwin peering over the pianoforte. His fingers almost stumbled when he realized what a mistake he'd made in choosing a piece of music, and he leaped to his feet.
But the countess was smiling at him, and there was amusement in her eyes. He gave her a wry grin.
Lady Beatrix also smiled at him, but damned if she didn't turn a normal greeting into a shamelessly wanton invitation. It was something about her eyes, the way they melted into a sultry little examination of his body and lingered around his middle. Luckily his stomach was as flat as the day he left Oxford—or was she looking lower? But the last thing he needed was a flagrant affair with an unmarried lass who already had the reputation of a highflier.
He wrenched his eyes away and looked to the countess. "Lady Godwin, I had the pleasure of hearing a canzone of yours at a musicale some years ago. Will you honor us with a composition?"
Lady Godwin gave him a reserved but genuinely friendly smile and took his place at the keyboard. "I'd be happy to play something else for you, but I rarely play my own compositions in public."
To Stephen's surprise, Beatrix Lennox didn't seem to have realized that he had snubbed her; perhaps she was so ready with her invitations that they weren't even personal. She leaned over the pianoforte, looking like a schoolgirl, an absurd comparison given that her bodice was so low that her breasts almost touched the glossy surface of the pianoforte.
"I didn't know you wrote music, Helene!" she said. "What a wonderful gift. Will you play us something you have written yourself?" And then, when Lady Godwin hesitated, "Please?"
Stephen had to admit that Lady Beatrix was pretty damn near irresistible when she pleaded. Lady Godwin blushed and nodded.
"Would you like to hear something polished or something quite new?"
"Oh, something new!" Lady Beatrix exclaimed.
Naturally, Stephen thought to himself. That sort of flippery young woman would always be looking for the very newest attraction.
Lady Godwin smiled. "All right. But I have to ask a favor of mine own, then."
He bowed. "For the pleasure of your music, my lady, anything."
"I'm working on a waltz at the moment, and it is so difficult to maintain the rhythm during the transitions. Would you and Lady Beatrix dance while I play?"
Stephen blinked. "I'm afraid that I haven't had much practice in waltzing."
Lady Beatrix was looking at him with one slim black eyebrow raised. "One Christmas I taught my grandfather, who is quite unsteady on his feet, to waltz," she put in, with a sweet smile that didn't deceive him for a moment.
She thought he was akin to her grandfather. Stephen felt a stab of pure rage.
"It's not a question of skill," Helene said earnestly. "I'm quite certain that you will be nimbler than my music, Mr. Fairfax-Lacy." She called to their hostess. "Esme, may I employ your guests for a practical purpose? Mr. Fairfax-Lacy and Lady Beatrix are kind enough to attempt one of my waltzes."
"I only wish I were capable of dancing myself," Lady Rawlings said cheerfully, hoisting herself from a chair and waving at her butler. A moment later the footmen had cleared a long, polished expanse down the center of the Rose Salon.
Stephen eyed it with distrust. Holding a seat in the House of Commons hadn't left him a great deal of time to spin women around the dance floor, especially in this newfangled German dance. Damn it, he'd probably only waltzed three or four times in his life. And now he had to try it before an audience. He stalked to the floor. She flitted out before him, the better to display that round little body of hers. Well, she wasn't so very little. He was a quite tall man, and yet she wasn't dwarfed by his height, as so many women were.
He glanced back at Lady Godwin. Truly, she was very attractive. She looked like a cool drink of water.
"This is so kind of you," she called. "You must tell me precisely what you think."
Stephen snapped a bow in the direction of Lady Beatrix. "May I have this dance?"
"My pleasure," she said demurely.
If demure was the correct word. That sleepy, sensual smile of hers ought to be outlawed. It said everything, without saying anything. And yet it was more a matter of her eyes than her mouth. Why on earth was she bothering to give him, a man her grandfather's age, apparently, such an invitation? Naturally his body didn't understand that it wasn't personal.
"There's a small introduction before the waltz proper starts," Lady Godwin said. She nodded, lowered her hands and the music splashed around them.
The waltz had none of the ceremonial pacing that Stephen vaguely remembered from the waltzes he'd encountered in the past. No, it leaped from the keyboard.
For a moment he was frozen in place, already behind in the beat. Then he literally grabbed Lady Beatrix's waist, pulled her hand into the air, and plunged into the cleared space.
They galloped down the center of the room. Stephen didn't attempt a twirl; it was all he could do to keep them on time when the music suddenly broke off.
"I'm so sorry!" Lady Godwin called from the pianoforte. "I've set it far too fast. I see that now. One minute—"
His companion was giggling. "You were far more agile than my grandfather." Her face was pink and her chest was heaving.
There was always the chance that her dress would fall to her waist, Stephen thought with a flash of interest. She had glorious breasts for a schoolgirl. Not that she was a schoolgirl, except in relation to his years.
"You don't seem at all out of breath," she observed.
"We'll start again, please," Lady Godwin called.
Stephen settled his hand more firmly on his partner's waist. This time the music began more slowly, so Stephen ventured a turn. He suddenly remembered that he had once considered dancing a delight, but that was long ago, before he'd discovered politics. Now he had no time for such frivolity. The melody drove them on. It was beginning to speed up again. One Two Three! One Two Three! Faster and faster they circled and spun. Lady Beatrix was grinning like the schoolgirl she wasn't, her eyes shining with delight.
"May I offer you my compliments?" she said, obviously rather out of breath. "You keep to this rather rapid pace extremely well."
Was her compliment in respect to his age? "I should say the same to you," he said stiffly. It was annoying to realize that his hand on her waist was tingling. That he was taking huge pleasure in holding such a ripe piece of womanhood in his arms...and all the time she was thinking that he was fit for the knacker's yard. It was repugnant.
Yet any man would feel a pang of interest. For one thing, he could tell from his hand on her back that Lady Beatrix did not wear a corset. His leg brushed hers as he turned her again. If this dance had been in vogue when I was young, Stephen thought suddenly, I'd be married by now. It was intoxicating to hold a woman in one's arms. No wonder all the old biddies thought the waltz was too scandalous for Almack's. This was the closest he'd ever come to lovemaking by music.
The waltz reached out and pulled them forward. It suddenly grew slower and rather melancholy, shifting to a minor key. They floated down the room on the sadness of it. That deep curve to her bottom lip was not something that could be enhanced by art, he thought absentmindedly.
"She must be putting her marriage into the music," Lady Beatrix said, meeting his eyes. "The music is so sorrowful now."
It was extraordinarily imprudent to remark to a perfect stranger about the countess's marriage! She spoke as if they were acquaintances of old, as if he were her uncle, or her infernal grandfather. And she was waiting for a response. "I would disagree," Stephen said, rather stiffly. "I'm not sure the music is sad as much as resigned."
"That's even sadder," Lady Beatrix observed.
Stephen dropped his hand from her waist the instant the music stopped. He didn't want her to think that she'd enticed him, with all her uncorseted beauty. "That was indeed a pleasure, Lady Beatrix," he said, with just the faintest touch of irony.
She caught it. Her eyelids flickered, and she gave him a langorous look that drifted down his front and made his private parts shoot to attention. "The pleasure," she said, "was entirely mine."
Damn it, she was worse than a courtesan!
Lady Godwin was rising from the pianoforte. The countess would never be so indecorous. Stephen felt his blood cool to a steady beat just watching her. The fact was, he had neglected that part of his life for too long. Now he seemed to have the unruly enthusiasm of an adolescent, lusting after every woman who crossed his path. Steady, he told himself. Steady ahead.
He strode over to Lady Godwin, took her hand and raised it to his mouth for a kiss. "That was a delightful performance," he told her softly. "Your waltz is exquisite."
"No, it isn't!" she protested. "It is far too fast. You must be quite fatigued." But she was smiling.
Stephen decided to take a chance. He turned her hand over and pressed a kiss into her palm. "Nothing you could do would ever make me tired," he told her, looking straight into her eyes.
She truly had a delightful blush.
##
## The Garden of Eden
Regular reading of the Tatler would convince anyone that English gentlewomen seduced their butlers and their footmen on a regular basis.
"This journal is a disgrace!" Mrs. Cable said, dropping the offending paper to the table. "If Lady Syndenham were indeed foolish enough to run away with her footman—and I see no reason to disbelieve the report—the information ought to be suppressed, so others don't follow her lead!"
Her companion's response was as frivolous as her nature. "Reading of Lady Syndenham's adventures is not likely to prompt one to cast a lascivious eye at a footman," Esme Rawlings pointed out. "At least, not unless one's footmen were better looking than those in my household."
"There'll be no end to it," Mrs. Cable snapped. "Before we know it, impressionable young ladies will be marrying footmen—nay, even gardeners! You may laugh, Lady Rawlings, but 'tis a serious concern." She stood up and gathered her reticule and shawl. "I myself am starting a campaign to weed out incorrigible sinners from my staff, and I sincerely hope you will do the same."
Mrs. Cable made a point of visiting Lady Rawlings, since the poor woman was widowed with a child on the way, but she often found her efforts unrewarding. Lady Rawlings's inclination to levity was disturbing. Mrs. Cable found herself all too often reminded that Esme Rawlings was considered something of a fast woman. Infamous Esme, that's what they used to call her in London.
All the more reason for Mrs. Cable to make frequent visits and impress the wisdom of the Bible on Lady Rawlings. Even looking at her now made Mrs. Cable uneasy. Lady Rawlings was entirely too beautiful, despite carrying a child. There was something about the color of her cheeks that looked feverish, as if she were ill. And that smile curving her lips...Mrs. Cable could only hope the woman wasn't thinking about one of her footmen. Surely not! Even Esme Rawlings would never smile at such a sin.
Mrs. Cable couldn't quite articulate her thoughts, but she knew what she saw, and if Lady Rawlings were one of her maids, she'd turn her off without a reference. She herself had never smiled like that in her life. She must remember to drop off some improving tracts on the morrow.
Mrs. Cable was right.
Esme had not been thinking of her butler, a worthy man by the name of Slope. Nor had she thought of her footmen, a callow group of country lads who suffered mightily under Slope's tutelage. It was worse. She had lost track of the conversation for a moment because she was thinking about her gardener.
Esme bid farewell to Mrs. Cable. Then she sat down in her sitting room and tried to remember all the good reasons she had to be respectable. Mrs. Cable wasn't one of them. She had a sharp nose, the beady, inquisitive eyes of a swallow, and a flock of acquaintances that rivaled that of the Regent himself. Mrs. Cable considered propriety next to godliness, and if she ever discovered the truth, Esme's reputation would be blackened the length and breadth of England.
Normally, Esme wouldn't be caught within ten yards of such a woman. But these days, she didn't have that luxury. Mrs. Cable led the Sewing Circle, an inner sanctum of ladies dedicated to the virtuous and charitable life. When the Sewing Circle was not hemming acres of coarse sheets for the deserving poor, it monitored the reputations of everyone within five counties. Manuevering her way into the circle had taken the diplomacy of a reformed rake aspiring to a bishopric in the Church of England, and Esme found the idea of forfeiting her newly acquired virtue galling.
Yet what was she to do? The gardener refused to leave her employ. Presumably, he was roaming around her garden at this very moment, although it was noon. He had likely retreated to the hut at the bottom of the apple orchard and was sitting there without a care in the world, reading Homer and not even considering the deleterious effect his presence might have on her reputation.
Of course she wouldn't visit him. This was her new life, a principled life, a life in which she would conduct herself in a respectable fashion. She had promised her husband, Miles, as much. Before he died, they agreed that he was going to give up his mistress, Lady Childe, and she was going to become the sort of woman who wore little lace caps and sewed sheets for the poor. And never, ever, thought about gardeners.
She bundled herself into a pelisse two minutes later, explaining to her maid that she wished fresh air. It wasn't as if her child was born yet, she told herself as she headed down the slope into the apple orchard. Once the child was born she would never see the gardener again. In fact, she would have her butler terminate his employment. Esme's pace quickened.
The hut was a small, roughly built structure at the bottom of the garden. It had one of everything: one chair, one bench, one table, one fireplace. One bed. And one gardener.
He was standing by the fireplace with his back to her when she pushed open the door. He didn't turn until she closed the heavy wood door with a thump. Then he whirled around so suddenly that the pot over the fire tipped and its contents cascaded across the wood floor. What appeared to be lumps of carrot and beef dripped into the cracks between the boards. Esme's stomach growled. Pregnancy had the unfortunate effect of making her always hungry.
He looked at her without greeting, so she tried a jaunty smile. "Never tell me that you're learning to cook?"
He still didn't say anything, just took a step toward her. Her gardener was big, with a rider's body, tousled blonde curls, and eyes the blue of a patch of sky in summer. His features were as regular as if they were chiseled from marble. No man had a right to be so beautiful. He was a danger to all womankind, perhaps even to Mrs. Cable. "Did you cook that stew yourself?" she insisted, waving at the pot.
"Rosalie, in the village, brought it to me."
Esme narrowed her eyes. "Rosalie? Who is she?"
"The baker's daughter," he said, shrugging. He took another long step toward her. "Is this a social call, my lady?" Something had sparked in his eyes, something that made her heart skip and her knees feel weak.
She opened her mouth to inform him that he was shortly to be discharged from his position, and found herself saying something entirely different. "How old is this Rosalie?"
"Rosalie is a mere lass," he said negligently.
"Ah," Esme said, realizing that there was nothing she could say to that. She herself was no lass. No, she was all of twenty-seven years old, and huge with child in the bargain.
He was just in front of her now, all golden and beautiful in his rough workman's shirt. He'd rolled it to the elbow, and his forearms swelled with muscle. He was everything the smooth, delicate gentlemen of her acquaintance were not: There was something wild and untamed about him. Esme felt a shock of shyness and couldn't meet his eyes.
"My lady," he said, and his voice was as smooth and deep as that of any marquess. "What are you doing in my humble abode?"
She bit her lip and said nothing. Embarrassment was creeping up her spine. Hadn't she told him last time that she would never visit again?
"You are responsible for the loss of my meal," he said, and his hand pushed up her chin so she had to meet his eyes. He loomed in front of her, the sort of man all young girls are warned to stay away from. The kind who knows no laws and no propriety, who sees what he wants and takes it.
"It was purely an accident," Esme pointed out.
"Then you must provide me with another." She barely caught a glimpse of the hunger in his eyes before his mouth closed on hers.
It was always the same with them. There were no words for it, really. Esme had been married. She'd had lovers. But she clung to Baring, her gardener, as if he were the first man on earth, and she the first woman. As if a smoky little hut smelling of charred stew were the famous Garden itself and she, Eve shaking in Adam's arms. And he held her with the same desperate hunger and the same deep craving.
It was a good ten minutes later when Esme remembered why she'd come to the hut. By then she was tucked in his arms and they were sitting on the bed, albeit fully clothed. "You're sacked," she said against his shoulder. He smelled of woodsmoke, and Rosalie's stew and more, strongly, of a clean, outdoors smell that no nobleman had.
"Indeed?" His voice had a husky, sleepy tone that made her breasts tingle.
"Mrs. Cable is beginning a campaign to stamp out all incorrigible sinners in the village, and surely you qualify."
"Is she a little woman who wears her hair scraped into a bun?"
Esme nodded.
"She's already tried," he said with a chuckle. "Came around to The Trout and handed out a lot of pamphlets to the lads last week. They were all about God's opinion of the Ways of the Wicked. I gather she forgot that reading is not a strong point in the village."
"Wait until she discovers that my aunt Arabella has arrived and brought a houseful of guests with her. Not a one of them has a decent reputation. Are you listening?"
"Of course." He was dropping small kisses on her neck.
"It's not a laughing matter," Esme said crossly. "You of all people should understand how important it is to be respectable. Why, only last year you were thought of as the most proper man in all the ton."
At that, he did grin. "Yes, and you can see how much that affected me. Here I am, living in disgrace on the Continent, and a very small Continent it is," he added, glancing around his hut.
"Entirely your fault!" she snapped. Esme was starting to feel a wicked temper. "If you hadn't lurched into my bedchamber in the middle of the night, you'd still be in the judgement seat, pronouncing verdicts on all the poor dis-repectable souls like myself." She brooded over that for a moment. "I used to feel as if you were always watching me."
She glanced up and found he was indeed watching her. His eyes were a darkish blue form of periwinkle.
"I was."
"Not just watching. Judging."
"I had to," he said cheerfully. "I was so utterly miserable about your married state that it drove me mad."
Esme felt a slight cheer in her heart. No woman in the world would dislike hearing that. "Truly, Sebastian, what am I to do? I know you think it's foolish, but I did promise Miles that I would become a respectable wife once we had a child. I can't have one of Arabella's scandalous parties in my house. I'm in confinement! All Arabella will say is that Marie Antoinette was dancing a minuet up to the moment she gave birth."
"Why don't you just accept my proposal? I'll make an honest woman of you, and we'll turn up our noses at the gossips."
Esme's heart skipped a beat and then steadied. She scowled at him. "To begin with, I can't marry you because you are even more scandalous than I am. Half the world believes you seduced your fiancée."
"Former fiancée," he put in.
"But that is nothing to the scandal if they discovered your current whereabouts. Arabella, for one, would instantly recognize you, and she's invited any manner of persons, all of whom could also identify you."
"Mmmmm."
He wasn't paying attention. "I don't understand why you consider my wishes to be so insignificant!" she said sharply, pushing his hand off her breast.
He just grinned down at her, all thick golden hair and laughing eyes. "Because I've given up all that respectability you want so much, Esme. I don't have it anymore. And I don't give a damn. Do you know that I once actually scolded Gina for trying to kiss me in public?"
Esme pursed her mouth. She didn't like to think about Sebastian kissing his former fiancée, for all Gina was one of her closest friends. "That sounds just like you," she observed. "Holy Willy, always standing on your consequence."
"I'd still have my Sir Sanctimonious credentials if I hadn't gotten mixed up with you," he observed. "My mother will likely faint when she hears of my new position."
"You didn't tell your mother!"
He grinned. "No. But I'm going to visit her tomorrow, and I shall."
"Noooo," Esme wailed. "You can't. You absolutely cannot do that!" She tended to keep well away from the more stiff-rumped members of the ton, such as Marchioness Bonnington. Sebastian's mother was one of those women who prided themselves on the fact that they needn't be magnanimous to lesser mortals. And her son, at least before he'd become a gardener, had been an unexceptional successor to her manifold virtues.
He shrugged. His hand was stealing up toward her breast, and his eyes had that look again.
"It will be a terrible shock for her," Esme said, trying to find a shred of sympathy and instead finding an evil ray of pleasure in her heart. "Aren't you rather old to be growing rebellious? I sowed my wild oats a good ten years ago."
Sebastian snorted. "And your mother still hasn't recovered. She's a bosom beau of my mother's, you know."
"I wasn't aware of their friendship." Esme didn't feel it necessary to add that she and her mother hadn't spoken except in passing for three years. She had no idea who Fanny's friends were. Her mother communicated only by letter, and that infrequently. "My mother has decided not to attend my confinement," she admitted. Why on earth was she relating that pitiful fact? She hadn't even told Helene.
"Your mother is as foolish as mine, then," he said, dropping a kiss on her nose.
"Fanny is not foolish," Esme felt compelled to defend her. "She simply cares a great deal for her reputation. And I've—well, obviously, I've been a great disappointment to her. I am her only child."
"So you are," Sebastian said. "All the more fool she, not to be here when her grandchild is born."
"I'm afraid that my mother has...has quite discarded the idea of our further acquaintance." It was absurd to find that she had a lump in her throat. She hadn't even had a cup of tea with her mother for some three years. Why should she miss her now?
"Is that why you have such a fierce wish to become respectable?" Sebastian inquired. "So that your mother will accept you again?"
"Of course not! It's only because of Miles, as I told you."
"Hmmm." But he wasn't really listening. He was kissing her ear.
"I don't think my mother likes me very much," Esme said dolefully.
To Sebastian's mind, her mother's behavior had made that clear for years, but it didn't seem politic to say so. "I expect she has some affection for you," he said in as comforting a manner as he could manage, given that he had Esme's delicious body on his lap. He felt like a starving man at a feast. "I am almost certain that my mother has some affection for me, although she would never acknowledge such a thing."
"You were a perfect son to her. And you will be again. Once you return from the Continent, everyone will forget the scandal, and you can return to being the very proper Marquess Bonnington. Snobby old sobersides."
"Never again. Never."
"Why not?"
"I shall never again believe that it matters a bean whether I kiss the woman I love in a garden or my own bedchamber. All that propriety, respectability, it's nothing but a trap, Esme, don't you see?"
"No," she said. Secretly she was a bit shaken by the vehemence in his voice. "I wish—oh, I do wish—that I hadn't been unfaithful to Miles in the first year of our marriage. Perhaps if I'd been more respectable, we could have found a way to be married again. To live together and raise a family."
She was startled by the look in his eyes. "Why? Why, Esme? Why Miles?"
"Because he was my husband," Esme said earnestly. This was at the heart of all their arguments. "I should have honored our vows," she explained.
"You vowed to love him forever. Yet you didn't even know him when you married him. He was weak, charming but weak. Why on earth are you harboring the idea that the two of you could ever have been happy together?"
"Because it would have been the right thing to do." She knew she sounded like a stubborn little girl, but he had to understand.
"Ah, the right thing," he said, and there was a dark tiredness in his voice. "I can't fight with that. But if you, Esme, were able to fall in love with your husband because it was the right thing to do, you would have been a very unusual woman indeed."
"I could have tried!" she said with a flare of anger. "Instead I flaunted my affairs before him and the rest of London."
Esme was missing the point. The trouble was that Sebastian wasn't sure how to make himself clear without risking her stamping out of his hut in a rage. He tried to put it delicately. "Your husband, Miles, didn't seem to take much notice of those affairs."
"Yes, he did."
My God, she was a stubborn woman. "You began flirting with other men in an attempt to get Miles's attention," Sebastian said. "Fool that he was, he simply concluded that the marriage was not successful. And to be honest, I don't think he cared very much. He was in love with Lady Childe, these many years before he died." His voice was calm but merciless.
Esme was silent for a moment. "We could have tried," she said finally.
"You did reconcile just before Miles died," Sebastian pointed out. "To my knowledge, you had one night together." He drew her even closer against his chest. "Did it pass in a blaze of passion, then?"
Esme turned her face into his rough shirt. "Don't laugh at Miles," she warned. "He was my husband, and I was very fond of him."
"I would never laugh at Miles. But I would never make the mistake of thinking that the two of you could have had a successful marriage, either."
"Perhaps not. I suppose not. It's just that I'm so...so ashamed of myself!" It burst out of her. "I wish I hadn't done all those things. I just wish I hadn't."
Sebastian was beginning to kiss her again, and his kisses were drifting toward her mouth. Suddenly Esme was tired of whimpering about her miserable marriage and her reputation. "You know when you used to watch me so crossly?" she said huskily. Sebastian's large hands were leaving tingling paths in their wake. He was a beautiful man, with his honey skin and tumbling hair. She couldn't tear her eyes away from him. Why was she even thinking about Miles?
"Of course," he drawled. He was watching her now too, except his eyes were below her chin. He was watching his hand on her breast.
"You had the most arrogant, sulky look," she said. "You used to lean against the wall and frown at me, and I knew you were thinking that I was an absolute tart."
The corner of his mouth curled up. "Something like that, I suppose."
She was getting breathless because of what he was doing, but she wanted to make herself clear. "I used to do some of it for you," she said, pushing his chin up so he met her eyes.
"Do what?"
"Flirting." She smiled and put all the seductive joy she felt into that smile. "You would be frowning at me from the side of the ballroom, with that gloriously sulky mouth of yours, and I'd play for you."
"Play for me?"
She nodded, giggling. "Be even more wanton. Do you remember when I kissed Bernie Burdett on the ballroom floor at Lady Troubridge's house party?"
"Of course," he growled, and he nipped her bottom lip with his teeth. He used to feel half mad, watching Esme Rawlings flirting with her latest conquest, allowing that intolerable Burdett to partner her in dance after dance. While he—he'd rarely danced with her. She'd been married, and he'd been engaged to her best friend. The very memory made him take her mouth with a growl of desire.
"Even as I kissed Bernie, I was wondering what you would do if I simply waltzed up to you and kissed you," she said after a little while, and with a catch in her voice. "I decided you'd probably be up in arms about it, prig that you are, so I kissed Bernie instead."
He raised his head for a moment. "You deliberately—"
"Exactly," she said smugly. Then she ran her lips along the strong, sun-browned column of his neck. "You were so disdainful of me and yet—something—I thought I saw something in your eyes."
He growled again, that deep male sound that made her thighs tremble. "So you were longing to kiss me, were you?"
It was frightening to hear it aloud. Esme chose to keep silent, turning her cheek against his shoulder so he couldn't see her eyes.
"So kiss me now, then," he said. And his voice had that dark, insistent throb that she couldn't disobey. It made her feel ravishing rather than pregnant. She didn't know why she'd ever thought he was priggish: He kissed her like a wild man. With one last gasp of rational thought, she said, "But Sebastian, I meant it when I said you have to leave. Tomorrow. It's too dangerous now that a house party is arriving."
"And what shall I do for a living, eh?"
"You'll have to go back to what you did before."
"Before..." His voice was dark now, velvet dark, muffled against her skin. "I spent all my time before arguing with a certain lady."
"You were extremely vexing," Esme said. "You were always scolding me because I was brazen, and—"
He bent and kissed her shoulder. "Brazen," he agreed. "Improper." He dropped another kiss on the little juncture between her neck and her collarbone. "Strumpet. I'll have to lend you that pamphlet on the Ways of the Wicked."
"And all because I was having a wee flirtation with Bernie Burdett," she said, grinning up at him. "Ravishing man that he was. How I miss—"
"That Bertie," he said against her mouth.
"Bernie!"
"Whatever," he growled. "The pain he caused me!"
She reached up and put her hand to his cheek. "Bernie and I never had an affair. It was a mere flirtation."
"I know that." He smiled down at her then, a lazy, dangerous smile. "Bertie would have made a tedious lover." He dragged his lips over the sweetness of her cheek and the long delicate stretch of her neck. "And you, my darling Esme, are not a woman to tolerate tedium in your bedchamber."
"And how would you know, sir?" she said, sounding a little breathless. "You have something of a lack of experience in these matters, wouldn't you say?" It was one of the most joyous memories of her life when the beautiful Marquess Bonnington threw off his cravat in Lady Troubridge's drawing room, announced he was a virgin, and proceeded to lose that virginity.
"It would be no different if I were Adam himself, and you Eve," he said. His eyes were burning again. "No one can make love to you the way I do." His hands slipped from her shoulders to her breasts, shaped their exuberance in his hands. She arched up with a gasp. His knee nudged her legs apart, and with one swift motion, he pulled her to the end of the bed, where he would put no weight on her belly.
Then he was there, bending over her, and she was laughing, and to him, it felt as if there were only the two of them in the world. He and his intoxicating, ravishing mistress, his very own Esme, his infamous lover...
As if his garden were the first garden itself.
As if his Esme, with her plump mouth and her seductive wit, were the very first woman in the world. She moaned, and he shook with desire. Took up a rhythm that he knew drove her to distraction, made her whimper and grow incoherent. Standing there, making sweet, slow love, he was the only man in the world...or the first...it didn't matter.
Marquess Bonnington was well and truly ravished.
##
## Anticipation
Stephen had made up his mind to approach—not seduce—Lady Godwin. One couldn't use a disreputable word of that sort in respect to such a delightfully ladylike woman. He organized his campaign in the same orderly fashion with which he approached all important arguments undertaken in Parliament.
First, Helene Godwin had eloped at age seventeen, which surely indicated a certain unconventionality, even if she showed no signs of it now. Second, the lady's husband proved to be a reprobate, tossing his wife out the front door and establishing a changing show of young women in her bedchamber. Nonetheless, third, the lady had maintained an irreproachable reputation. She would not be an easy woman to win. But, finally, he fancied that he did have a chance of winning. A long shot, perhaps, but that blush.... She blushed whenever she saw him.
Stephen grinned to himself. He was used to assessing the odds of any given victory in the House. He gave himself a forty percent chance of victory over Helene. Sufficient odds to make it a challenge. Already he felt much more himself than he had in the last few months. Enclosure Acts just weren't enough to keep a man's interest. He had been suffering from a healthy dose of lust.
A deliciously bashful countess, intelligent, musical and neglected by her husband, would solve all his problems.
He strode into Lady Rawlings's Rose Salon and paused for a moment. The house party had apparently been augmented by neighbors of Lady Rawlings; country gentlefolk drifted around the room in little groups. The countess was sitting next to the fireplace, talking to their host. Her skin was so pale that it looked translucent. Frosty, almost. Like snow or ice. Stephen loved ices, sweet and cool to the tongue.
He was far too adept a campaigner to approach Lady Godwin immediately. Instead he walked over to greet an old friend, Lord Winnamore, whom he knew well from various skirmishes between the Houses of Lords and Commons.
Winnamore was as amiable as ever. "Another escapee from matters of business, I see," he said, greeting him.
"I should be in London," Stephen admitted. Come to think of it, what was Winnamore doing in the deeps of Wiltshire?
"Life has a way of creating distractions," Winnamore said. He was watching Lady Arabella.
"Thank goodness!" Stephen was startled by the vehemence of his own exclamation. It certainly wasn't as if he ever would consider deserting the House before his term was up. Or even at that point. There was no threat to his reelection, after all.
"This isn't the sort of party where I'd have thought to meet you," Winnamore said, giving him a shrewd glance over his spectacles.
"I am finding it quite enjoyable," Stephen said, checking to make certain that Lady Godwin was still in the corner. In another moment, he would stroll in that direction.
"Enjoyable, yes. Respectable, no. Have you met Lady Beatrix yet?" Winnamore said cheerfully, looking at the door to the salon. Stephen looked as well. Lady Beatrix was making what she clearly considered a spectacular entrance. Apparently the curls of yesterday had been compliments of a curling iron; today her shining copper hair was straight as a pin. Yesterday, her skin had been sunkissed; tonight it was pale as snow. Yesterday her lips had been ripe as a cherry; tonight they were a pale, languid pink. Even her pert expression of the previous night had been replaced by a faintly melancholy gaze—except if one looked very, very closely, mischief brewed.
"That young woman is a work of art," Stephen said, not without admiration.
"A lovely child, in fact," Winnamore said. "She is a great comfort to Lady Arabella."
Stephen could think of no reason why Lady Arabella, known far and wide for her three marriages and various other dalliances, would have need of comfort, but he kept prudently silent. Besides, Lady Arabella herself swept up to them that very moment.
"Mr. Fairfax-Lacy," she cried, taking a grip on his elbow, "I must insist that you greet my niece. Dear Esme is not as nimble as she is normally, and so I have appointed myself the duty of bringing sufficient conversationalists to her side."
It was suddenly quite clear to Stephen why he had been invited to this particular house party. Lady Arabella had selected him as a prospective husband to her niece. Well, there was nothing new in that. Matchmaking mamas had been chasing him for years.
He bowed to Lady Rawlings but sought Lady Godwin's eyes as he did so. She was just as lovely as he remembered, pure and delicate as a—he couldn't think. Poetry was hardly his forte. She was blushing again and looking rather adorably shy.
Too shy. A moment later she jumped to her feet like a startled gazelle and fled across the room. He'd have to go even slower than he had planned. He didn't look over his shoulder at the countess, but sat down next to Lady Rawlings.
For her part, Esme was watching Stephen Fairfax-Lacy with a good deal of interest. Unless she was mistaken (and she was never mistaken when it came to men), the man was attracted to Helene. Marvelous. Poor Helene had suffered so much from the cruelties of her careless husband. A kindly, handsome, respectable man such as Mr. Fairfax-Lacy would do wonders to restore her sense of confidence and allow her to hold her head high before that reprobrate of a husband.
"Are you enjoying yourself?" she asked, remembering rather belatedly that she was nominally, at least, a hostess. Arabella had taken over all the duties of running the house, the better for Esme to concentrate on her supposed confinement. "Is your chamber acceptable?"
"Truly, it has been all that is comfortable," he said. And then changed the subject. "I much enjoyed Countess Godwin's waltz. Her husband is not invited to this gathering, I presume?"
Yes! Esme felt all the exuberance of an old friend. Helene appeared to have made a remarkable impression on Fairfax-Lacy. "Absolutely not," she hastened to say. "Helene and Rees have had little to do with each other for years. He has other interests. She and her husband have an entirely amiable friendship," she added. One wouldn't want the M.P. to be frightened off by the notion of an irate husband.
Stephen was watching Helene talk to Bea on the other side of the room. Esme didn't quite like the contrast that conversation presented: Bea was such a vividly colored young woman that she made Helene look pale and washed out. "If you'll excuse me," she said brightly, "I must confer with my butler." She allowed Fairfax-Lacy to haul her to her feet and then trundled off toward the door, stopping next to Helene and Bea.
"He was just asking for you!" she whispered to Helene.
Helene looked adorably confused. "Who was?"
"Fairfax-Lacy, of course! Go talk to him!"
Helene looked across the room, and there was Stephen Fairfax-Lacy smiling at her. But she felt a strange reluctance; it was all she could do to hover next to the door and not flee to her bedchamber. Her life, to this point, had not been easy. In fact, although she only admitted it to herself in the middle of the night, sometimes she felt as if she must have been cursed at birth. It had only taken one foolish decision—the foolish, foolish decision to elope with an intoxicating man by the name of Rees—to ruin her entire life. But in the last year she had realized that if she didn't do something about it now, the rest of her life would follow the pattern of the past seven years. The years hadn't been unpleasant: She lived with her mother and she was welcome everywhere. But she had no life, no life that mattered. No child.
She glanced again at Fairfax-Lacy. He looked like a gentleman, not like that savage she had married. Perhaps, just perhaps, she would even like having intimacies with him. It wouldn't be terrifyingly messy and embarrassing as it had been with Rees. It would be...proper. Acceptable. He was quite lovely: all rangy, lean, English gentleman. And without a doubt it would curdle Rees's liver to see her with such a man. If anything could curdle her husband's liver, given the qualities of brandy he drank. So why wasn't she walking straight into Mr. Fairfax-Lacy's arms?
Suddenly a pert voice spoke just at her left elbow. "Shall I walk you across the room again?"
Helene blinked. Bea's eyes were sparkling with mischief. She repeated, "Shall I walk you across the room, Helene? Because I believe you are expected."
"Ah—"
"This way," Bea said efficiently, taking her elbow and strolling toward the far end of the room, where Stephen waited. "He is quite lovely, isn't he?"
Helene was so nonplussed that she couldn't quite bring out an answer. "Who?" she finally said lamely.
"Mr. Fairfax-Lacy, naturally!"
"I thought you found him Old Testament."
"That too. But it seems obvious to me that the two of you are perfectly suited," Bea said in a coaxing voice, as if she were taking a mare over a high jump. "There he is, a perfect specimen of the English gentleman, and here you are, exactly the same in a female form. Both impeccably virtuous too, which must add luster to your friendship. And I think he's quite, quite interested in you," Bea said confidentially. "He looked straight in your direction when he entered the room. Whenever I speak to him, he simply glances around the room. Normally"—her smile grew—"I am used to complete attention."
Bea had on a dinner dress that had neither a front nor a back. One could only guess how it stayed above her waist, given that her plump little breasts threatened to escape her scrap of a bodice. Men must simply slaver over her, Helene thought enviously. She herself was wearing a gown of Egyptian net over a dark blue silk. She had felt very a la mode in her chamber, but now she felt dismally overdressed, like a dog wearing a sweater.
But Bea seemed to follow her train of thought perfectly. "I'm certain that he doesn't like my gown," she said. "Last night at dinner he kept looking at me as if I had something stuck between my teeth. Come along!" She jiggled Helene's arm. "You don't want to wait too long, do you? What if Arabella manages to convince the man that he should wed Lady Rawlings? You could hardly have a liaison with your friend's husband!"
Helene thought about that as they moved across the room.
"You see," Bea said, not quite as softly as Helene would have liked, "he's looking at you right now!"
But when Helene looked up, it seemed to her that Stephen was watching her companion, although with an expression of deep annoyance. She swallowed and curtsied before Stephen Fairfax-Lacy. "Sir," she said. Bea had glided away without even greeting Mr. Fairfax-Lacy.
He smiled down at her, and Helene realized again what a good-looking man he was. There wasn't a whisker on his face, not like her husband, who always had a shadowed jaw by evening.
"How are you?" he asked.
"I'm quite well."
There was a moment's silence while Helene thought desperately of a conversational tactic. "Did you read this morning's paper?" she finally asked. "Napoleon has escaped from Elba and is in France again! Surely the French army will not support him."
"I believe you are quite correct, Lady Godwin," Stephen said, looking away. He had decided to play this game very, very slowly, so as not to startle her.
Helene felt a crawling embarrassment. How on earth could she have ever thought to seduce a man? She couldn't even carry on a simple conversation.
"What do you think of the fact that Catholics cannot sit in Parliament?" she asked.
He blinked, not prepared for philosophical reasoning. "I have long felt that the prohibition should be rethought," he said finally.
"I believe it has to do with the wordings of the oaths they would have to take. Wouldn't it violate their religious vows to take Parliamentary oaths?"
"Most of the men I know don't give a fig for those oaths," Stephen said.
Helene heard a faint bitterness in his voice and wondered about it. Why was Mr. Fairfax-Lacy in Wiltshire rather than sitting in the House of Commons?
"Why should we expect Catholics or Jews to be more circumspect than Anglicans?" he continued.
"Surely to establish oneself as a Catholic in this country, given its Anglican past, implies a deeper fidelity to religion than one might expect from an ordinary gentleman," Helene said. She was quite enjoying herself now. He wasn't regarding her in a lustful fashion, just with the sort of normal engagement one might expect during a conversation.
But she waited in vain for a reply. He appeared to be looking over her shoulder.
"Mr. Fairfax-Lacy," she said, with a bit of sharpness to her voice.
He snapped to attention. "Yes, Lady Godwin? Do forgive me."
"Is there something interesting that I should see as well?" Helene said, deciding on the basis of his really quite charming smile that she wasn't insulted after all.
"It is merely that impudent little chit, Lady Beatrix," Stephen said. "I truly can't imagine what Lady Withers is thinking, allowing the girl to dress in that unseemly fashion."
Helene turned as well. Bea was sauntering across the room toward them.
Stephen felt as if the girl were some sort of irritating gnat. Here he was, having a remarkably informed and intelligent conversation with the woman who might well become his future mistress, and there she was again. About to interrupt their fascinating discussion of religious oaths. Lady Beatrix seemed to have dropped the melancholy pose with which she had originally entered the room. She looked strikingly exotic and utterly unnatural. And potent. Too potent.
"Do you know, I don't think that is the true color of her hair?" Stephen said. He could hear the rancor in his own voice. Why on earth did the girl get under his skin in such a fashion? "Look at that bronze. Have you ever seen such a color in nature?"
"But why would she color her hair?" Helene asked with some fascination. "She can hardly be showing gray."
"Of course not!" he agreed. "She's barely out of the schoolroom."
Helene didn't agree with that pronouncement. Beatrix Lennox was obviously far too ripe for a schoolgirl, and besides, hadn't she debuted some three years ago? That would put her at about twenty years old.
"I expect she colored her hair merely to shock people," Stephen said with a shrug. "She's obviously artificial." He turned back to her. "Not like you...a true English gentlewoman, bred to the bone."
Helene felt a pang of envy toward Bea. It wasn't high on her list of wishes to be described as a well-bred filly at Tattersall's. Naturally, she ought to be pleased by the compliment. But it would be fun if just once, she were considered dangerously attractive. Able to shock someone. Helene had never shocked anyone in her life. Well, perhaps her husband. There was that time with the chamber pot...Helene wrenched her thoughts away from the unsavory topic.
"Thank you for the compliment," she said, opening her fan. Esme always flirted with her fan to great effect. Unfortunately, Helene hadn't the faintest idea how to do the same thing. She waved it gently, but the only result was that she was unable to see Stephen at all. She snapped it closed.
At that moment Bea joined them. "We have been discussing poetry," she said with a twinkle. "And I am sent to discover each person's favorite poem. Arabella has had the splendid idea that we shall have a poetry reading on Friday evening."
"I haven't read any poetry in years," Stephen observed.
Bea looked up at him from under her lashes. "We'll have to do something about that. Perhaps I'll lend you a book from my private library."
To Helene's amazement, a ruddy tone appeared in Stephen's lean face. "That won't be necessary," he said brusquely. "I was quite fond of poetry as a boy. I'm certain I can remember something."
"Have you a favorite poem?" Bea asked Helene.
"I am acquainted with Shakespeare's sonnets," Helene said uncertainly. "But some of them are hardly suitable for reading aloud."
"I'm sure you will find something you deem appropriate," Bea said, and Helene was unable to dismiss the idea that the girl was laughing at her.
"And your favorite poem?" Stephen asked her.
"A love poem by Lord Byron," Bea said, drifting away. "It's quite, quite beautiful."
"That girl is trouble," Stephen said, rather unoriginally.
But Helene had had enough of this torturous flirtation. She was exhausted. "If you'll excuse me, Mr. Fairfax-Lacy," she said with a curtsy, "I will join Lady Rawlings."
Helene had hardly sat down next to Esme when Bea plumped herself on Esme's other side. "Disastrous!" Bea announced.
"What?" Helene asked, but Esme seemed to know precisely what she was speaking about and responded with a choked giggle. Helene narrowed her eyes. "What are you discussing?"
"You, darling," Esme said, with such fondness in her voice that it removed the sting. "Bea and I have been conspiring to bring you together with that estimable gentleman on the other side of the room, but you're not doing your share."
Helene already felt tired; now she felt obstinate as well. "While I much dislike the idea of my affairs being discussed in public," she said, "I also resent the imputation that I have not attempted to...to sway Mr. Fairfax-Lacy's attentions. I am wearing a new dress, and I allowed myself to be walked over to him, like a lamb to slaughter. It is not my fault that the man has no conversation."
"You must have discussed something," Esme said.
"Topics I introduced," Helene snapped. "First I brought up Napoleon's escape and then the position of Catholics in the government. He had nothing to say to either issue. Really, if this is what he's like in Commons, it's no wonder the government never gets anything done!"
Bea sighed. "He doesn't want to talk about legalities, Helene. The man is bored with the House. He wants to talk about frivolous things. Men always pretend that they want intelligence in their mates, but it's not really the case."
"What sort of frivolous things?" Helene asked.
"I don't agree," Esme put in. "I think Bea has the wrong end of the stick. In my experience, it doesn't even matter what you talk about. The man is burnt to the socket. Look at those circles under his eyes. Unless I miss my guess, he's rather desperately hoping to find a warm body to curl up with. All you have to do is indicate that interest, Helene."
"You make it sound easy," Helene muttered.
"It is easy," Bea said. "You watch, and I'll do it right now. He's utterly uninterested in me, so there's no threat to your future."
Helene grabbed her arm. "I can't let you do that!"
"Why on earth not? I do it enormously well," Bea said with some satisfaction. "In fact, I think one could fairly say that I am an expert." She sauntered off, and sure enough, even the very sway of her hips was a promise.
"I do believe that girl is more outrageous than I ever was," Esme said thoughtfully. "She must be quite unhappy."
"Nonsense. She's having the time of her life," Helene said. "Look at her now!"
Bea was laughing up at Stephen, waving her fan gently before her face. Her piquant little face was glowing, her eyes sending the man a speaking invitation. Her bosom brushed against his arm, and even from the other side of the room, Helene could see him start.
"I can't possibly do that sort of thing," Helene said flatly. "I just couldn't." She felt positively riddled with embarrassment at the very thought.
"Bea is not doing much," Esme said. "There's only one important thing, and that's to let your eyes tell Stephen that you're available. That's all. It's easy."
"Easy?" Helene said in an appalled voice. "That's not easy! Available? How on earth does one indicate such an unseemly thing?"
Across the room, Bea was laughing up at Stephen. She seemed to be vibrating with desire. Then she turned around for the merest moment and grinned at them. The desire wiped from her face and was replaced by pure mischief. She looked like a girl just out of the school room. The next second she turned around and threw Stephen another languishing look.
"Ah," Esme said with some satisfaction, "she can still be herself."
"I have no idea what you're talking about," Helene said, feeling just on the edge of tears. "I can't do this. I must be missing the ability. Rees always said—" She snapped her mouth shut. She didn't want even her own best friend to know that she was a frigid woman who would never enjoy bedding a man. Her own husband had said so, and she was fairly certain he was right.
"Don't despair, darling. Mr. Fairfax-Lacy doesn't like what Bea is doing. See?"
Sure enough, Fairfax-Lacy was frowning at Bea and clearly growling some sort of reproach. "He's just the man for you," Esme said with satisfaction. "Not Bea's type at all."
A fact which Bea exuberantly seconded a moment later. "He told me to go wash my face," she reported with some glee. "I do believe that Mr. Higher Than Thou M.P. doesn't like my maquillage, even though it is imported all the way from Paris."
Helene felt a little steadier. She had never worn rouge in her life and couldn't imagine why she ever would. Perhaps she and Stephen were suited after all.
Just look available, she told herself. "So, I simply look...look—"
"As if you want to bed him," Bea said.
"I'll try," Helene muttered. Never mind the fact that she didn't wish to bed anyone, and couldn't believe that any woman would wish to do so voluntarily. Except for reasons of revenge.
"Or you could just tell him," Bea suggested with a wicked grin.
"I most certainly could not!"
"I have an idea! The poetry! We'll use the poetry."
"What do you mean?" Esme asked her.
"We are each supposed to read a favorite poem on Friday, remember? If Helene reads the right kind of poem, and looks at Fairfax-Lacy while she does it, it won't fail! That way you need not embarrass yourself," she told Helene. "The poem will do it all. And I'll warrant he'll visit your chamber that very night."
"An excellent idea," Esme said, nodding.
"But I don't know any love poetry," Helene pointed out. "Besides that of Shakespeare."
"Good," Bea interjected, "because we don't want love poetry, silly!"
"We don't?"
"Do you love him?" she asked.
"Well, no."
"Precisely my point. This is an altogether different type of poetry. And not to worry, I never travel without my favorite authors."
"You are remarkable. You travel with...with this sort of poetry all the time?" Helene asked Bea.
"Naturally," Bea said, opening her fan.
Helene watched with fascination as Bea shook the delicate, lacy confection slightly. She held it just below the level of her eyes, and somehow she looked ten times more delectable. I shall practice with my fan tonight, Helene thought. In front of the mirror. If I read the poem with a fan covering my face, no one can see me blush. Helene loathed the fact that she blushed constantly, like some sort of green girl.
"Don't forget that your friendship with Mr. Fairfax-Lacy will curdle your husband's liver," Bea said with relish.
"Of course I haven't forgotten that!" Helene said. Why on earth would she even consider doing such an immoral act otherwise?
"Just remember to look at Stephen while you read," Esme advised. "I shall put the two of you next to each other at supper so you can practice giving him desiring looks. Naturally I'll have to be on his other side, since Arabella is determined that we should marry."
"I rather agree with Arabella," Helene said. "He would undoubtedly make a good husband, Esme. I was just thinking how very much I wish that I had married someone like him, rather than Rees."
"He's not for me," Esme said, shrugging.
"Nor me," Bea said, with the little yawn of a cat. "He's all yours, Helene. If you can stomach all that virtue and pomposity, that is."
"He's not pompous!" Helene protested, and then realized that her two friends were laughing at her.
"Not pompous—perfect for you. We'll confer over poetry tomorrow, shall we?" Bea said, a twinkle in her eye.
"Better not," Helene said, biting her lip. "If I have to read something shocking, and"—she narrowed her eyes at Bea—"I have the feeling that your choice will be along those lines, I'd much rather not know the worst before the moment arrives."
Esme put an affectionate arm around her shoulder. "I'll be there, cheering you on."
"As will I!" Bea put in brightly.
Helene looked at Stephen Fairfax-Lacy again. He was leaning against the mantelpiece, deep in conversation with a stout lady from a neighboring estate. He was the very picture of a timeless kind of elegance. The kind of elegance that her husband didn't even dream of. Rees didn't give a toss what coat he drew on in the morning. He'd never tied a cravat in such intricate, snowy folds in his life. And since no decent servants would stay in his employ, he didn't have a gentleman's gentleman to tie it for him.
Stephen Fairfax-Lacy was just what she needed: an antidote to her loathed husband. Rees's antithesis. Helene's hands curled into fists at her sides. She would do it. She would do it, and then she would tell Rees that she had. And when he looked stricken with jealousy...
The smile on Lady Helene Godwin's face reflected pure feminine glee.
When Rees was stricken with jealousy—and suffering from a curdled liver—she would just laugh and walk away.
##
## The Contrariness of Men Hardly Bears Repeating
Bonnington Manor
Malmesbury, Wiltshire
Marchioness Bonnington was not accustomed to opposition from the male sex. She had ruled—and survived—two husbands and fourteen male lapdogs. To her mind, there was no question as to which group had provided the better companionship. And as for logic...her own son was an excellent example of the worthiness of lapdogs over humans.
"Did I understand you to say that you are living in a garden hut, Bonnington? A garden hut?"
Her son nodded. The marchioness let silence fall between them. She had not invited him to sit, since she considered sons to be inferiors, along the lines of a butler: willing to take advantage, and needing to be continually reminded of their place. Not that her only son Sebastian had ever shown much proclivity for rebellion. He was a quite appropriate example of his sex, if she said so herself. Never caused her a moment of worry, until she had heard he had been courting the Duchess of Girton and persuading her to seek an annulment of her marriage.
That had ended in disaster, as she had known it would. In the end, her only son had been exiled to Europe, labeled unmarriageable, tarred as a liar and deceiver. The only thing that had sustained her in the past eight months had been a lifetime's knowledge that the sins of young, very wealthy men seemed to dissolve after a year or so. She had fully intended to recall him to England in the summer and rehabilitate him in the eyes of the ton by marrying him to an upright young woman, perhaps someone who reminded her of herself at an earlier age.
Except here he was. Back in England without her permission.
She placed her hands carefully on top of her walking stick, which was planted in front of her chair. "May I ask why you have chosen such an insalubrious location in which to lodge?" she asked gently. Neither of them was deceived by her tone. The marchioness tolerated insubordination in no one.
"I am living in a garden hut, Mother," her son said now, smiling at her for all the world as if he were a natural rather than a marquess, "I am living in a garden hut because I am working as a gardener on the estate of—"
She raised her hand. "I do not wish to hear her name spoken out loud."
He looked at her and said, "On the estate of Lady Rawlings, Mother, the woman whom I shall marry."
Of all possible outcomes to her son's disastrous impudence, this was the worst.
"I cannot fathom it," she said, punctuating each word with vigorous disapproval. "I understood when you were courting the Duchess of Girton last year. I was as aware as anyone that Ambrogina Camden's marriage was not consummated. She was a respectable woman, an excellent choice for marchioness, if one could disregard the unfortunate annulment that would have had to occur." She paused and gripped the carved top of her walking stick even harder.
"As I say, I understood your wish to marry her. Marriage to a duchess, even one who has annuled her previous marriage, can never be seen as a mistake. But marriage to Esme Rawlings is—is beyond my—I cannot fathom it. The woman took lovers under her husband's nose. Everyone in London knew what she was up to. Her own mother has publicly expressed horror at her behavior. I was never so surprised as when I heard that Lady Rawlings was actually entertaining her husband in that bed of hers; Lord knows all of London had been there at some point or other."
"If you repeat that comment one more time, you'll never see me again." His voice was calm, but the fury there made the marchioness blink.
She rallied quickly. "Don't be a fool!" she said sharply. "In my estimation, the gossip probably didn't cover half of what she did. I know for a fact—" Her eyes widened, and Sebastian saw that she had only just grasped the full ramifications of the situation.
"You to marry her! You, who killed her husband?"
"I did not kill her husband," Sebastian said, standing taller. "Rawlings's heart failed him on my unexpected entry to the chamber."
"You killed her husband," his mother said. "You entered that room looking for the bed of your duchess—oh, don't give me that folderol about a false wedding certificate. I don't believe common gossip. You had been bedding the duchess, but you crawled into the wrong bedchamber and encountered a husband. I call that killing the man! In my day"—she said it with grim triumph—"a man ascertained whose door he was entering before he did so."
Sebastian suppressed a grimace. "I mistook the room," he said stolidly, "and it had an unfortunate effect."
"Then why in the name of blazes should you marry the woman? A mistaken notion of paying for your crimes? If so, I shall have the vicar speak to you. Because one can overemphasize the doctrine of reconciliation, and marrying a doxy simply because one killed her husband is Going Too Far."
Sebastian sighed and looked about him. He was tired of standing like a schoolboy before his mother. She was perched on a thronelike chair in which the Regent would have felt comfortable, fitted out with claw feet and serpentine arms. He spotted a reasonably comfortable chair in the corner and strode over to fetch it.
"What are you doing?" his mother barked. "I didn't give you permission to sit down, Bonnington!"
"My name is Sebastian," he said, putting down the chair with a decisive thump and seating himself directly before her. "My name is Sebastian, and I am your son. Your only son. It would make me feel a great deal more comfortable if you did not refer to me as having killed Lord Rawlings. He had a weak heart, and the doctor had given him until the end of the summer. It was truly unfortunate that I was the cause of his seizure—and I would give anything to have not instigated that episode. But I did not kill him."
The marchioness blinked. Her ever-courteous, ever-proper, almost boring son appeared to be showing a little backbone for the first time in his life. She didn't know whether to be pleased or horrified.
She chose horrified.
"The only man with whom I have ever been on a first-name basis was your father," she said with some distaste, "and that only in the most intimate of situations. You, Bonnington, are my son, and as such should offer me only the greatest respect."
He inclined his head. "And that I do, Mother." But he stayed seated. He had her looks, that son of hers. When she was young, men wore their hair powered and women wore patches. But it would be a pity if Sebastian powdered his hair. He had her hair, the color of sunshine, that's what Graham called it. Of course, Graham hadn't been bad-looking either. Those were his deep-set eyes looking at her. After her first husband died, she had married the most handsome man in London, and if Graham Bonnington wasn't a lively conversationalist, he knew his place. He listened to her. She said enough for both of them.
She thumped her stick on the floor. The stick made some of the younger servants quite ill with anxiety, but her son merely glanced at the floor, as if checking for scuff marks. She decided to stay with the most crucial point.
"You cannot marry a doxy out of some misplaced sense of obligation. The Bonningtons are an ancient and respected family. Make Lady Rawlings an allowance, if you must. The estate can certainly bear the cost."
"I intend to marry her," Sebastian said. "But not out of obligation."
"No?" She invested the word with as much scorn as she could.
"No. I love her."
The marchioness closed her eyes for a moment. The day had begun with the unpleasant shock of seeing her son in England, and it was rapidly turning into something truly odious.
"We don't marry out of love," she sat flatly. "Marry a decent woman, and you can always see about Lady Rawlings later."
"I love her, and I will marry her."
"I believe I have fallen into a comic opera. And I detest musical theater. Are you planning to break into song?"
"Not at this moment."
"Let me see if I understand you: you feel yourself to be in love with a doxy who has shared her bedchamber with half the men of London, and whose husband you didn't kill, but certainly helped to his grave?"
"This is your last warning, Mother." He said it through clenched teeth. "You speak of the woman I intend to marry, who will be marchioness after you. Speak so again, and you will have no part in our life."
The marchioness rose with some difficulty—the gout in her left foot was growing worse by the moment—and thumped her stick for good emphasis, although it seemed to have little effect. She was pleased to note that her son rose when she did. At least he hadn't discarded all manners.
"The day you marry that doxy, I shall disown you," she said, as if she were commenting on the weather. "But I am quite certain that you knew that would be the outcome. I may remind you that my portion is not inconsiderable. Any child you—"
Sebastian groaned inwardly. The other shoe had dropped.
"By God, the woman is enceinte! I'd forgotten that trollop is carrying a child. Tell me you are not planning to marry Esme Rawlings before that child is born!"
Sebastian toyed with the idea of threatening to marry Esme tomorrow, an action that would make her unborn child his heir. But he didn't want to be responsible for his mother having heart palpitations. Miles Rawlings's death already weighed heavily on his conscience. More to the point, Esme still refused to marry him at all.
"Lady Rawlings has not accepted me," he admitted.
A look of grim satisfaction crossed his mother's face. "Well, at least someone is showing intelligence. Of course she won't accept you. You killed her husband." She began to stump her way toward the door. "I don't know where you got this devilishly self-sacrificing side to you. Your father didn't show any penchant for that sort of nonsense."
Suddenly Sebastian felt his temper, which had been growing at a steady rate, flare into life. He walked around his mother and stopped before the door.
"Move aside!" she said.
"I will make Esme Rawlings marry me. She will accept me because she loves me as well. Moreover, I shall expect you to attend the wedding and behave in a respectable fashion."
"There won't be a wedding," his mother replied calmly. "I felt a momentary anxiety, true. But from what I know of her, Esme Rawlings is as intelligent as she is dissolute. She won't marry you. She won't even think of it. I've no doubt but what Rawlings left her warm enough in the pocket, and a woman like that doesn't need a protector, or yet a husband either. Now if you'll excuse me, I will return to my chamber."
And she walked past him.
Sebastian spun on his heel and walked over to the other side of the room. He looked down at his clenched fist, pulling it back on the point of putting it through the window. His mother had said no more than Esme herself had done, although she had never said he wasn't the one to father her child. But she probably thought it. How could a man serve as father to a babe when the whole world—his mother included—thought he'd killed the child's true father?
Sebastian Bonnington had faced few obstacles in his life. Thanks to his mother, he was both remarkably beautiful for a man and rigidly aware of proprieties. When other men strayed to mistresses and gambling, losing their estates and their minds in dissolute activities, he had watched and not partaken. Before he'd met Esme, in fact, he had never even felt the urge to commit an indecorous act.
He shook his head, staring blindly at the garden. Oh, he loved Esme's delicious curves and her beauty, but it was her eyes that he found irresistible. There was no other woman in the world with eyes at once seductively enchanting and secretly sad. They had taken his head, robbed his heart, and stolen his senses. Something about her made him love her, willy-nilly.
And if to love her and to marry her was indecorous or foolish, he had no choice in the matter. All he had to do was convince her of the same.
##
## A Saint, a Sinner, and a Goat
Lady Beatrix Lennox was bored. There wasn't a man to flirt with in the entire house. Lord Winnamore was eligible, but he was hopelessly besotted with Arabella. Too old, naturally, although he was curiously attractive in a ponderous kind of way. But Bea would never, ever take a man from her godmother. She wasn't proud of many of her characteristics, but she had always been loyal.
Bea drifted over to the mirror and practiced a seductive pout. She had dressed herself for a walk, but she didn't know why: there was nothing she found more tedious than the countryside. In fact, the very idea of traipsing through a meadow, gazing at cows, filled her with boredom.
Yet here she was, dressed up like a trussed turkey. In fact, distinctly like a turkey, given that she was wearing a walking dress of Austrian green, exuberantly adorned with ribbons. Little bows marched all the way up her bodice, the better to emphasize her bosom (amply padded with cotton). But there was no one in the house to enjoy it.
Except, of course, Mr. Fairfax-Lacy.
Mr. Fairfax-Lacy had one of those lean, well-bred faces that would have looked as attractive in an Elizabethan ruff as it did in fashionable garb. His grandfather probably wore one of those huge collars. Still, Elizabethans in portraits always seemed to have slightly piggish, avaricious eyes, whereas Fairfax-Lacy had—
A curt voice made her jump. "Lady Beatrix, your godmother is going to the village for a brief visit. Would you like to join her?"
Talk of the devil. She turned around slowly and gave Mr. Fairfax-Lacy a smouldering look, just for practice. The one that began just at the edge of her eyes and then turned into a promise.
He looked unmoved. Indifferent, as a matter of fact. "Lady Beatrix?"
A pox on his well-bred nature! He really was a Puritan. Or perhaps he was simply too old to play. He had to be forty. Still, the combination of her reputation and personal assets had made Bea widely admired by the male gender, irrespective of age.
She sauntered over to him and put her hand on his arm. His eyes didn't even flicker in the direction of her bosom, something she found quite disappointing, given the amount of cotton she had bundled under her chemise. "I would rather take a walk," she said. He was much better looking than a cow, after all; his presence might make a country stroll palatable.
"It has been raining on and off all day. Perhaps tomorrow would be a more pleasant experience for you."
"Oh, but I love rain!" she said, giving her sweetest smile, the one that always accompanied outrageous fibs.
Sure enough, he responded like a parrot: "In that case, I would be enchanted to accompany you." But was there a trace of irony in that enchanted? Did the Boring Puritan have a little bit of depth to him after all?
Bea thought about that while the footman fetched her spencer. Luckily her walking costume came with a matching parasol, because the idea of allowing even a drop of rain to disorder her face or hair made her shiver.
It was appalling to see how wet it was outside. Bea could hardly say that she didn't want her little jean half-boots to touch the ground, given as she'd squealed about loving rain. So she picked her way over the cobblestones in front of the house, hanging onto Mr. Fairfax-Lacy's arm so that she didn't topple over and spoil her spencer with rain-water.
At least he seemed to be enjoying himself. She sneaked a look, and he was smiling as they started down a country lane—a messy, dirty little path guaranteed to ruin her boots. Oh well. Bea had had lots of practice saying good-bye to people and things—her sisters, her father—what was a pair of boots? She let go of Fairfax-Lacy's arm and tramped along on her own. The path was lined with sooty-looking, thorny bushes with nary a flower to be seen.
He wasn't exactly the best conversationalist in the world. In fact, he didn't say a word. Bea had to admit that the landscape was rather pretty, with all those sparkling drops hanging off branches (waiting to destroy one's clothing, but one mustn't be squeamish about it). And the birds were singing, and so forth. She even saw a yellow flower that was rather nice, although mud-splattered.
"Look!" she said, trying to be friendly. "A daffodil."
"Yellow celandine," her companion said curtly.
After that, Bea gave up the effort of conversation and just tramped along. Helene was welcome to the Puritan. In the city there was always someone to look at: an old woman peddling lavender, a dandy wearing three watch fobs, a young buck trying to catch his whip. Bea found the street endlessly amusing.
But here! This lane had only one inhabitant.
"Hello, there," Fairfax-Lacy said, and he had a gentle smile on his face that she'd never seen before. He had nice creases around his eyes when he smiled like that. Of course, it would all be rather more attractive if he weren't scratching a goat.
The man ignored her cotton-enhanced bosom and saved his smiles for a goat! Still, the goat seemed to be the only object of interest, so Bea poked her way across the lane. The animal stuck its wicked-looking face over the gate and rolled an eye in her direction.
"He looks quite satanic," she said. She'd seen that face before, in the grandest ballrooms in London. "Evil, really."
"He's just an old billy goat," Fairfax-Lacy said, scratching the goat under his chin. The goat had a nasty-looking beard, as if it had been partially eaten while he wasn't watching.
"Aren't you worried that you will catch fleas?"
"Not particularly, given that goats don't carry fleas."
Well, that was an exciting exchange. Bea was just standing there, thinking about how hairy the goat's ears were, when the beast suddenly turned its head and clamped its yellowing teeth on the sleeve of her spencer. Luckily it was belled, in the Russian style, and he didn't manage to chomp her arm, although that was undoubtedly his intention.
"Help!" she shrieked, tugging at her spencer. The goat rolled its eyes at her and bared its teeth but didn't let go of her sleeve.
Instead, he began to back up, and a second later Bea found herself plastered against an extremely wet fence, desperately trying to pull her sleeve away from the monster's mouth as it tried to back into the field.
"Do something!" she bellowed at Fairfax-Lacy. She was shocked to see that he was trying to conceal the fact that he was laughing. Quite overcome by laughter, in fact.
"You bloody beast!"
"Me or the animal?"
"Either! Get—this—animal—now!"
"At your service!" He hopped over the fence and approached the billy goat. But for all the fact that Fairfax-Lacy had been on the very best of terms with the animal a moment before, it wasn't very loyal. As soon as Fairfax-Lacy got close, the goat's rear leg shot out, caught him on the hip, and tossed him into a mud puddle.
Bea was trying to get her left arm out of her spencer. It was difficult trying to squirm out of the garment while hanging onto a fencepost. But even with such pressing business at hand, she stopped to have a laugh at Fairfax-Lacy's expense.
He shot her a level look and got up. He was plastered with mud from his shoulders to his knees. Even his hair was flecked with brown.
Bea was laughing so hard that her stomach hurt. "What sort of mud is it?" she called out, breaking into a fresh storm of giggles.
"The kind women slap on their faces to improve their complexions," he growled over his shoulder. "May I bring you a handful?" This time he managed to avoid the goat's kick, but he couldn't get close enough to grab her spencer. Every time he approached the animal, it bared its ugly yellow teeth and kicked at him again.
Finally Fairfax-Lacy turned back to her. "Take it off."
"What do you think I'm trying to do?" Bea cried, all laughter disappearing from her voice.
"He's eaten the sleeve already."
"Bloody hell!"
"You swear far too much," the Puritan said.
"I swear just as much as I wish to," Bea retorted, starting to unbutton. The goat hadn't given an inch; it just stood there chewing on her sleeve as if he was making a supper of it.
"You're going to have to help me," she finally said sourly. "I can't unbutton the rest without letting go of the fencepost. And if I do that he'll undoubtedly drag me straight over the fence." She eyed Fairfax-Lacy. "Not that I want you anywhere near me. Does that mud smell as potent as it looks?"
"Yes," he said, sauntering over to her.
He was the most infuriating man. This was literally—literally!—the first look he'd given her that acknowledged her as a woman. In fact, it was as if he were seeing her for the first time. He didn't look Elizabethan at all. He looked...
Bea's stomach took a funny little hop, and she felt a wave of unaccountable shyness. So she kept her eyes down as he unbuttoned the rest of her spencer. It was all very romantic, what with the odoriferousness of his person and the grinding sound of a goat munching her extravagantly expensive garment.
Once it was unbuttoned, she managed to squirm the rest of the way out of her left sleeve, and then quickly shed the right. One could have sworn that the goat had been waiting for that moment. The very second her body was free of the spencer he took a bigger bite and then bared his teeth in a smile.
Bea felt a wave of anger. "Go get him!" she ordered the Puritan.
He laughed. He was still looking at her as if she were a person, rather than an annoying insect, but Bea didn't let that distract her.
"Then I shall do so myself," she said, unlatching the gate and pushing it open. There was a ghastly squishing noise as her boot sank into brown muck. Bea ignored it.
He closed the gate behind her and leaned on it with a huge grin on his face. She thought about sticking her tongue out at him and rethought it. She was twenty-three, after all.
"Goat," she said, in the low, threatening tone she had perfected on her four smaller sisters. "Goat, give me that garment."
The goat stopped chewing for a second and looked at her, and Bea knew she had him.
She walked over, ignoring the Puritan's shouts. Apparently Fairfax-Lacy had realized she was serious and seemed perturbed that she might get injured.
"Don't even think about kicking me," she told the goat. "I'll tie your ears in a bow and you'll look so stupid that no lady goat will ever look at you again."
He stopped chewing. Bea took another step and then held out her hand. "Drop that coat!" she said sharply.
The goat just stared at her, so she used the meanest tone she had, the one she reserved for little sisters who were caught painting their cheeks with her Liquid Bloom of Roses. "Drop it!"
He did, naturally.
Bea cast a triumphant look over her shoulder and bent to pick up her coat. Fairfax-Lacy was tramping across the field after her, no doubt impressed by her magnetic effect on animals.
Time has a way of softening memories. Yes, her meanest tone had been successful. But how could she have forgotten that her wicked little sisters often found retribution?
The kick landed squarely on her bottom and actually picked her off her feet. She landed with a tremendous splash, just at the feet of Mr. Stephen Fairfax-Lacy.
"Ow!"
At least he didn't laugh at her. He squatted next to her, and his blue eyes were so compassionate that they made her feel a little teary. Or perhaps that was due to the throbbing in her bottom.
"You've still got your spencer," he said reassuringly.
Bea looked down at her hand, and sure enough, she was clutching a muddy, chewed-up garment. The goat may have got his revenge, but she'd kept his supper. She started to giggle.
A smile was biting at the corners of the Puritan's mouth too. A splatter of warm rain fell on Bea's cheeks, the kind that falls through sunshine. Water slid behind her ears and pattered on the leaves of a little birch. Bea licked her lips. Then, as suddenly as it started, the shower stopped.
"I didn't realize how much you treasure your clothing," he said, touching her cheek. For a moment Bea didn't know what he was doing, and then she realized he was wiping mud from her face.
Without even thinking, she leaned against the Puritan and just let laughter pour out of her. She howled with laughter, the way she used to, back when she and her sisters would lark around in the nursery. The way she did when the world was bright and fresh and new.
She laughed so hard that she almost cried, so she stopped.
He wasn't laughing with her. Damned if the Puritan didn't have the sweetest eyes in the whole world. He scooped her off the ground and then strode over to the birch and sat down, back against its spindly trunk. Bea found it very interesting that when he sat down he didn't put her on the grass, but on his lap.
"You have triumphed," he told her. Sunlight filtered through the birch leaves in a curiously pale, watery sort of way. It made his eyes look dark blue, an azure bottom-of-the-sea type of blue.
She raised an eyebrow. Actually, now that she thought of it, all the color she'd put into her eyebrows and lashes had probably made its way down her cheek. Oh well, he likely thought it was just mud.
"A goat conqueror."
"One of my many skills," she said, feeling a little uncomfortable.
"I just want to suggest that you rest on your laurels," he said, and his eyes had a touch of amusement that made Bea feel almost...almost weak. She never felt weak. So she leaned against him and thought about how good that felt. Except she wasn't quite following the conversation.
"What do you mean?" she finally asked.
There was a definite current of laughter in his voice. "Your bonnet."
Bea shrieked and clapped a hand to her head, only just realizing that she had felt rain falling on her head as well.
"There." He pointed to the right. The damned goat was chewing up her very best hat. The green plume hung drunkenly from his mouth, and he seemed to be grinning at her.
Bea started up with a shriek of rage.
"I think not!" The Puritan had arms like steel. He didn't pay a bit of attention to her wiggling, just picked her up and turned her around. When she looked up at his face, she suddenly stopped protesting.
He didn't kiss like a Puritan. Or an old man either.
He kissed like a hungry man. Bea's first sensation was triumph. So the Puritan had pretended that he didn't notice her charms. Ha! That was all an act. He was just...he was just like...but then somehow, insidiously, she lost her train of thought.
He was kissing her so sweetly, as if she were the merest babe in arms. He didn't even seem to wish to push his tongue into her mouth. Instead he rubbed his lips against hers, danced on her mouth, his hands cupping her head so tenderly that she almost shivered. She quite liked this.
Oh, she felt his tongue. It sung on her lips, patient and tasting like raspberries. Without thinking, her own tongue tangled with his for a second. Then she realized what she was doing and clamped her mouth shut. There was nothing she hated more than a man pushing his great tongue where it didn't belong.
But he didn't. His lips drifted across her face and pressed her eyes shut, and then closed back on her lips with a ravenous hunger that made her soften, ache deep inside.
He probably thinks I'm a virgin, Bea thought in a foggy sort of way.
His mouth was leaving little trails of fire. He was nibbling her ear, and she was tingling all over. In fact, she wanted—she wanted him to try again. Come back, she coaxed silently, turning her face toward his lips. Try to kiss me, really kiss me. But he didn't. Instead, his tongue curled around the delicate whorls of her ear, and Bea made a hoarse sound in her throat. He answered it by nipping her earlobe, which sent another twinge deep between her legs.
He tugged her hair and she obediently tipped her face back, eyes closed, and allowed him to taste her throat, all the time begging silently that he return, return, kiss her again...But he seemed to be feasting on her throat. She opened her mouth to say something, but at that moment he apparently decided he had tormented her enough, and his mouth closed over hers.
She could no more fight that masculine strength than she could rise to her feet. He didn't coax this time; he took, and she gave. And it wasn't like all the other times, when she'd tolerated a moment or two of this kind of kissing. The Puritan's kiss was dark and sweet and savage all at once. It sent quivers through her legs and made her strain to be closer. His hands moved down her back, assured, possessive. In a moment he would bring them around to her front, and her breasts were aching for...
That was the thought that woke Bea. She hadn't been thinking of grappling in the field when she'd dressed in the morning. These particular breasts weren't meant to withstand a man's hand. There was more cotton than flesh. She tore her mouth away, gasping, and stared at him. She didn't even think about giving him a seductive glance. She was too stunned.
"I like you when you're like this," he said, and there was that sweetness to his eyes again. He reached out and rubbed a splatter of mud from her cheek. "You look rain-washed and very young. Also rather startled. It seemed to me that you've been inviting kisses. Was I wrong?"
"No," she said, trying hard to think what to say next. All her practiced seductive lines seemed to have fled from her head.
"Alas," he said, even more gently, tucking her hair behind her ear. "I can hardly offer marriage to a woman half my age. So I'm afraid that I shall have to leave your kisses, sweet though they are, to some younger man."
Bea's mouth almost fell open. Marriage? Didn't he know who she was? "I don't want—" she began, but her voice was hoarse. She stopped. "As it happens, I am not interested in marriage either," she said quite sedately. "I find that I am, however, very interested in you." She twisted forward and kissed his lips, a promise of pleasure. And she was absolutely honest about that. With him, there would be no boundaries.
But it was he who pulled back. She had been so sure he would lunge at her that she'd smiled—but the smile faded.
He was a Puritan. His eyes had gone cold, dark, condemning. "I thought you played the lusty trollop for fun."
She raised her chin. "Actually, no," she said, and she was very pleased to find her tone utterly calm and with just a hint of sarcasm. "I play myself."
"Yourself? Do you even know who you are, under all that face paint?"
"I assure you that I do."
"You play a part you needn't," he said, eyes fixed on hers. "You are young and beautiful, Beatrix. You should marry and have children."
"I think not."
"Why?"
"You simply want to make me like everyone else," she said sharply. "I like wearing macquillage. I would rather not look like myself, as you put it. And I find it incalculably difficult to imagine myself sitting by the fire wearing a lace cap and chattering about my brood of children."
"I think yourself is beautiful. All your paints have washed away at the moment. You never needed them."
"I didn't say I needed them. I enjoy them," she retorted, and then added, deliberately, "just as I occasionally enjoy the company of a man in my bedchamber."
For a moment they just looked at each other, Puritan to trollop. "Am I to understand that you are not interested in taking a mistress?" she asked, meeting his eyes. She was no child to be whipped by his condemnation.
"Actually, I am," he said. "But I have little interest in one so...practiced."
Bea got to her feet, shaking out her skirts. Then she bent over and picked up her mangled spencer, shaking it out and folding it over her arm, taking a moment to make absolutely certain that her face wouldn't reveal even for a second what she felt.
"I have often noticed that men of your years seem to overprize naiveté," she replied calmly.
He showed no reaction, but her quip was so untrue that she gained no joy from saying it. He wasn't old. Suddenly, she decided to be honest. Looking him in the eye, she said, "That was cruel, and quite shabby, Mr. Fairfax-Lacy. I would not have expected it of you."
"I'm sorry."
She nodded and began to turn toward the gate. After all, she'd had much worse things said to her, mostly by women, but then there was her dear father. So when he caught her arm, she turned toward him with a little smile that was almost genuine.
"Don't you think we should take our bedraggled selves home?"
There was real anguish in his eyes. "I feel like the worst sort of bastard. Kissing you in a field and then insulting you."
At that, she grinned. "I gather you wish I were an innocent, Mr. Fairfax-Lacy. But I am not. I truly enjoyed that kiss." The smile she gave him was as wicked and lazy as any she'd ever bestowed on a man. "And I would very much have enjoyed your company in my bedchamber as well. But I have never forced myself on a man. I fully understand that you are looking for a far more respectable mistress." Helene was an altogether perfect alternative.
At that moment, Bea made up her mind. Helene would never be able to lure the Puritan on her own. She, Beatrix, would have to help, if only to prove that she didn't hold grudges, even when rejected. She would give him to Helene as a present.
She turned and made her way across the field, and when the goat rolled his wicked eyes and snapped his lips over a Pomona green satin ribbon, all that remained of her bonnet, she just smiled at him.
Which startled the animal so much that he galloped off to the other end of the field, leaving her hat behind.
##
## The Sewing Circle
To Esme's great relief, Mrs. Cable swept into her morning parlor on the very strike of ten o'clock. Esme had been putting crooked stitches into a sheet for at least five, perhaps even ten, minutes and hadn't got further than two hands' lengths. She hastily bundled the sheet to the side to greet her guest.
"My goodness, Lady Rawlings!" Mrs. Cable said. "How very becoming that cap looks on you! You are verily an illustration of the good book of Timothy, which says that women should adorn themselves in modest apparel, with shamefacedness and sobriety rather than gold and pearls."
Esme touched her head self-consciously. It was the very first time that she had ever worn a cap, and she felt like a fool. Like one of those Renaissance fools, with bells hanging off their caps. It felt like rank hypocrisy, as if wearing a trifling bit of lace on top of one's head would make up for the fact that two days ago she'd reveled in indecencies with her gardener. One could only imagine what would happen if her guest knew the truth!
Esme pushed away that thought and offered Mrs. Cable some tea.
"I would be grateful," Mrs. Cable said, plumping herself onto the settee next to Esme, and showing no inclination whatsoever to pick up an unhemmed piece of cotton. "For a body must have sustenance, and that's a fact!"
"I quite agree," Esme said, pouring tea into a cup and ruthlessly repressing visions of other kinds of bodily sustenance, types of which she doubted Mrs. Cable would approve quite so heartily.
Mrs. Cable sipped and raised her eyebrows. "She is like the merchants' ships; she bringeth her food from afar."
Esme was not someone with a facility in biblical verses. Oddly enough, contact with Mrs. Cable seemed to be increasing her irritation rather than her piety. "Indeed?"
"Proverbs," Mrs. Cable said briskly. "This is India tea, is it not? An expense, a dear expense, but quite delicious. I have brought with me six sheets, which I managed to hem in my spare time this week."
"How marvelously industrious you are!" Esme gushed. She herself couldn't seem to sew anything except under the direct supervision of the Circle itself, so she never participated in the weekly count of completed sheets.
"You must have a great deal of time on your hands these days, Lady Rawlings."
Esme resisted the temptation to tell Mrs. Cable that having a houseful of dissolute guests made for rather a lot of work. "So one would think."
Luckily Slope opened the door. "Lady Winifred," he announced, "and Mrs. Barret-Ducrorq."
"What a pleasure to see you, Mrs. Barret-Ducrorq," Esme exclaimed. "And here we thought you were enjoying yourself in London and we wouldn't see you until the season ended!"
"We are all assembled," Mrs. Cable put in, "as when the good book says that the elders were assembled."
"I'd take it as a personal compliment if you'd not refer to me as an elder, Mrs. Cable," Mrs. Barret-Ducrorq snapped. "Lucy and I have fled London for a week or so. The poor girl is quite, quite worn out by all the festivities. As am I," she added, looking remarkably robust. "Sponsoring a debut is a quite exhausting business." Mrs. Barret-Ducrorq's sister had recently died, leaving her to administer her niece's debut.
"And by all accounts, Lucy is having a particularly exciting time," Lady Winifred said with a good-natured chuckle. Lady Winifred had three grown daughters living in London; while she no longer traveled to the city for the season, she seemed to know of even the tiniest event.
Mrs. Barret-Ducrorq leveled a glare at Lady Winifred, who was demurely threading a needle. "I expect that, as always, accounts of the incident have been grossly exaggerated."
Mrs. Cable's eyes were bulging out with pure excitement. "Never tell me that something happened to sweet Miss Aiken! Your niece could not create a scandal. There must be some mistake!"
Mrs. Barret-Ducrorq's mouth twisted. She was a rather corpulent woman, whose body seemed to have focused itself in her bosom; it jutted below her chin like the white cliffs of Dover. Generally, she had an air of victory, but today she looked rather deflated.
Esme put down her sheet. "What on earth has happened to Miss Aiken?" she asked. Lucy Aiken had always seemed a pallidly unimaginative girl and certainly not one to achieve notoriety.
"It's her father's blood coming out," Mrs. Barret-Ducrorq said heavily.
Mrs. Cable gasped. "Never say so!"
"I do say so! If my sister hadn't married beneath her, none of this would have happened!"
"It didn't sound particularly outrageous to me," Lady Winifred observed, turning the corner on her hem. "After all, many girls do foolish things in their first season. It's almost expected. And it's not as if she created some sort of true scandal!"
Aha, Esme thought to herself. That would have been my role...in the old days. She was astounded that neither Mrs. Barret-Ducrorq nor Lady Winifred had mentioned her cap. Did they really think she was old enough, stodgy enough, widowed enough, to wear one of these? Even Arabella didn't wear a cap!
"My niece insulted the great Brummell himself," Mrs. Barret-Ducrorq said heavily.
"What on earth did Miss Aiken say to him?" Esme asked, fascinated despite herself. She'd often wanted to insult Brummell.
"He did her the inestimable honor of complimenting her complexion, and then asked what preparation she used on her freckles." Mrs. Barret-Ducrorq shuddered. "Lucy was rather tired, and apparently she did not entirely understand the breadth of Mr. Brummell's importance in the ton. Or so she tells me."
"And?" Mrs. Cable said.
"She snapped at the man," Mrs. Barret-Ducrorq admitted. "She informed him that any preparations she chose to use on her complexion were her business, and no one else's."
"The snare of vanity," Mrs. Cable said darkly.
"The vanity is all Mr. Brummell's," Esme pointed out. "The man takes a spiteful delight in pointing out the faults that one most wants to hide."
"She loathes her freckles," Mrs. Barret-Ducrorq said. "I blame them on her father's side of the family. We have nothing of the sort in our family, and so I have told Lucy, time out of mind."
"Vanity—," Mrs. Cable put in.
Everyone ignored her. "You were right to bring poor Lucy to the country for a week," Lady Winifred said. "Everyone will have forgotten by next Monday."
"True enough. More importantly, has she met any gentlemen whom she finds acceptable?" Esme put in.
Mrs. Barret-Ducrorq looked slightly more cheerful. "Several gentlemen have paid her marked attention. I am hopeful that they will overlook both her slip of the tongue and the freckles."
"Poor Lucy just didn't understand that we fairly beg Mr. Brummell to be discourteous to us," Esme said. "He's a horrid little beast, and so I shall tell Lucy when I see her."
"Lady Rawlings!" Mrs. Cable said with a gasp. "Mr. Brummell is a leader of the ton! It would never do for Miss Aiken to insult him yet again."
Esme bit her lip before she retorted that she too was a leader of the ton, and knew better than Mrs. Cable what a song and dance one was supposed to make before the great Brummell. Or the penniless Brummell, as was rumored.
At that moment the door opened, and Arabella swept in. "Ah, this must be my niece's group of virtuous laborers," she said, laughing. "I thought I'd join you and bring a little frivolity to lighten your exertions!"
"How kind of you," Esme said, giving Arabella a pointed look. If she undermined Esme's new respectability, Esme would have to flay her, relative or no. She had sewn too many sheets to give up her place in the Circle now. "Ladies, may I present my aunt, the Dowager Viscountess Withers? Aunt, this is Mrs. Cable, Mrs. Barret-Ducrorq, and—"
"Winifred!" Arabella crowed. "How are you, dear girl?"
Esme watched, rather stupefied, as Lady Winifred came to her feet with a great creaking of stays and Arabella bounded into her embrace. Lady Winifred was a florid woman with a bewildering range of acquaintances. Still, Esme wouldn't have put her aunt among them, given that Lady Winifred spent a great deal of her time impugning the reputations of women with far fewer sins than had Arabella. Perhaps Lady Winifred was losing her memory.
"I haven't seen you in an age!" Lady Winifred boomed. "It's all my fault, of course. I've grown as large as a horse, and as lazy as one too. Nowadays I loathe London."
"I know just what you mean," Arabella said, patting her hand. "There are days when I feel every bone in my aged body and I can't think of a single activity that might please me."
Esme just stopped herself from rolling her eyes. Arabella was wearing an utterly charming and provocative morning gown made of a cotton so light it floated on the breeze. If she didn't look precisely youthful, she did appear to have a good twenty years before she'd feel even a touch of rheumatism.
The look on Mrs. Cable's face made it clear that she, at least, was having no trouble remembering the kind of activities for which Arabella was famed. "How unusual to find such a distinguished personage in Limpley-Stoke," she said with a titter. "I'm afraid that you'll find our little village quite drab!"
Esme suddenly saw Mrs. Cable through her aunt's eyes. Mrs. Cable's small, dark eyes were glistening with dislike. Her mouth was thinned with contempt. The worst thing of all, from Arabella's point of view, would be the fact that Mrs. Cable was wearing a dress of pomona green poplin, just the color to emphasize the sallow color of her cheeks.
"No place that contained my niece could be tedious!" Arabella replied, whisking herself into a chair. "I do believe I would even travel to America to see her. And that's a profound compliment, as I'm sure you all know how sea air can ruin one's complexion."
"I am honored," Esme said, pouring Arabella a cup of tea. "Thank goodness you needn't go to such lengths, dear aunt. At your age," she added.
Arabella narrowed her eyes at her. "I see you've taken up wearing a cap, dearest niece. At your age."
Lady Winifred had settled herself back with a length of cotton. "I won't offer you a piece of this, Arabella," she said with a booming laugh. "I don't think of you as a needle-mistress!"
"But of course, you're right," Arabella agreed. "I can't sew to save my life."
"Sometimes these sheets are all that come between the poor and the cold floor," Mrs. Cable said pointedly. "Whoso stoppeth his ear at the cry of the poor, she also shall cry herself and not be heard."
Rag-mannered, Esme thought to herself. Could Miles truly have wished her to spend time with the likes of Mrs. Cable?
Apparently Lady Winifred agreed with Esme's assessment. "I have been meaning to mention to you, Mrs. Cable, that there is something just slightly vulgar about quoting the Bible, unless, of course, it is the vicar himself who ventures to recite."
Mrs. Cable thrust back her head, rather like a rooster preparing to battle an impudent hen, and said, "I fear not, but testify unto every man."
Arabella raised one eyebrow and said pleasantly, "My goodness, you do seem to have the Bible at your fingertips. I do congratulate you. It is such an unusual skill to find among the gently bred."
Mrs. Cable turned a deepish puce color. Arabella turned to Mrs. Barret-Ducrorq with her charming smile. "I don't believe we've met. But as it happens, I did meet your delightful ward, Miss Aiken, just two weeks ago, at Almack's. Sally Jersey introduced me. We both thought her manners were remarkably engaging, with very little of that strident awkwardness that seems rampant this season, and I certainly applauded Sally's decision to give her a voucher to Almack's."
Mrs. Barret-Ducrorq had silently watched the skirmish between Arabella and Mrs. Cable to this point, but she was instantly wooed and won.
"That is tremendously kind of you, Lady Withers," she said, putting her sewing to the side, "and I must ask you a question. I have been longing to know the truth behind the Countess of Castignan's extraordinary marriage, and I expect you know all about it."
Arabella laughed. "Well, as to that, Petronella is one of my dearest friends...."
Esme risked a look at Mrs. Cable. She was sitting like a dour crow, stitching so quickly that her needle was a blur. Even for the sake of Miles, her departed—if not terribly dear—husband, could she contemplate a lifetime in Mrs. Cable's company?
##
## Prudishness...That Coveted Quality
Bea woke in the morning feeling rather ashamed of herself. Of course, there was nothing new in that sensation. Her father had often bellowed his amazement that he'd never managed to teach her a single thing, but she secretly thought he had had no difficulty imparting shame. She'd simply refused to reveal it, to his everlasting fury.
But she should never have kissed Stephen Fairfax-Lacy in the goat pasture. Never. He was singled out for Helene, and if there was one thing that Bea did not do, it was steal men from other women.
I'll dress in such a way as to make it absolutely clear to Mr. Puritan that he's not to kiss me again, Bea thought. Then she remembered that the Puritan didn't want to kiss her, now he knew of her experience. If that pang in the region of her stomach was shame, Bea refused to acknowledge it.
"I'll wear the new morning gown," she told her maid, Sylvie. "The one with blonde lace."
"But, my lady, I thought you had decided that gown was entirely too prudish," Sylvie lisped in her French accent.
"It is rather prudish, isn't it? Wonderful. I'm in a Puritanical mood."
"If you say so," Sylvie said resignedly. She was rather hoping that her mistress had taken a permanent dislike to the gown and would hand it on to her.
Sometime later Bea looked at herself in the glass with some satisfaction. She looked—as her grandmother might have said—as if butter wouldn't melt in her mouth. The dress was made of the finest jaconet muslin in a pale amber, trimmed with deep layers of pointed blonde lace. It had long sleeves, and while the bodice clung to every inch of her bosom (and several inches that weren't hers at all), it was so high-necked that it practically touched her ears.
"No Spanish papers," Sylvie suggested, as Bea sat herself at the dressing table. Once she'd gotten over the disappointment of having her mistress actually wear the coveted gown, she'd started enjoying the dressing, as always. Truly, she was lucky. Her mistress was lovely, invariably cheerful, and, most importantly, took clothing very, very seriously.
"You're absolutely right," Bea said, nodding at her in the mirror. "The papers are far too red. My cheeks should be just the palest pink. Didn't I buy something called Maiden's Blush at that shop in Bedford Square?"
Sylvie was rummaging through a smallish trunk that stood open to the right of Bea's dressing table. "Here it is!" she said, holding up a small bottle. "Although you may wish to consider the Royal Tincture of Peach," she added, handing over another bottle as well.
Bea tipped both colors onto a bit of cotton and considered them carefully. "Maiden's Blush, I think," she decided. "The Peach is lovely, though. Perhaps I'll use it on my lips."
"Don't you think it will be rather pale?" Sylvie asked doubtfully.
"No, no," Bea said, deftly applying a translucent layer of rouge. "I'm nothing more than a seedling today. Utterly missish." She ignored the little voice in the back of her mind that kept insisting on the contradictory nature of her actions. Why shouldn't such an experienced trollop as herself dress any way that she pleased? Illogical or no.
"Ahh," Sylvie said. She loved a challenge. "In that case, I shall change your hair, my lady. Perhaps if I twisted a simple bandeau through it? These beads are entirely too knowing."
"You are a blessing," Bea told her with satisfaction. "What on earth would I do without you?" A few moments later, she grinned at herself in the mirror. Her hair had the simplicity of a fourteen-year-old. She looked utterly milk-and-water. A mere infant!
She refused to think about the perverse impulse that was driving her to demonstrate to Stephen Fairfax-Lacy that she was not as experienced as—well, as she was. For a moment she almost deflated. Why on earth was she pretending to a virtue she didn't possess and had never before aspired to, either?
There was a knock on the door, and Sylvie trotted away. Bea delicately applied kohl to her lashes. Not even for the sake of innocence would she emerge from her room without coloring her lashes.
"May Lady Rawlings visit for a moment?" Sylvie called from the door.
Bea hopped up, slipping her feet into delicate white kid slippers. "Esme! Do come in, please!"
Sylvie opened the door, but Esme just stood there for a moment, blinking. "Bea?" she said weakly, "is that you?"
"Do you like it?" Bea said, laughing.
Esme dropped her considerable girth into a chair by the fireplace. "You look like a green girl, which I gather must be your aspiration."
"Precisely," Bea replied triumphantly.
"I do like the color you're wearing on your lips, although I could never wear something so pale myself. Where did you buy it?"
"It was the perfumer on St. James Street, wasn't it, Sylvie?" Bea said.
"Indeed it was, my lady," Sylvie replied.
"I haven't been to London in over six months," Esme said, wiggling her toes in front of the fire. "I hardly remember what the inside of a perfumer looks like!"
"How appalling," Bea said, tucking herself into the chair opposite. "I suppose that carrying a child does limit one's activities." She felt very pleased at the idea that she herself would never be banned from London for that many months. Being unmarried had definite advantages.
"Actually, it's this respectability business," Esme answered.
"Lady Godwin did mention that you are—" Bea stopped, unable to find a tactful way to phrase Esme's ambitions.
"Aspiring to be above reproach," Esme said.
"We all aspire to something, I suppose," Bea said, rather doubtfully.
"Did you buy those slippers from Mrs. Bell?" Esme inquired. "I adore the daisy clocks on the ankle."
"Mrs. Bell tried to convince me to buy a shawl with the same daisy pattern. But I thought that might be too kittenish."
"You're risking kittenish now, if you don't mind my saying so, but you somehow manage to look delightful instead. At any rate," Esme said with a sigh, "I came to warn you that although my Sewing Circle has finally departed, I was maneuvered into asking them to return for a late luncheon. So please, feel free to eat in your chambers unless you wish to be showered in Bible verses."
"Sewing Circle?" Bea repeated rather blankly.
"Did Arabella forget to tell you?" Esme said, standing up and shaking out her skirts. "I've joined a local Sewing Circle. We meet every week, at my house, due to my delicate condition. Arabella joined us this morning, which caused great excitement and led to the luncheon invitation."
"Never tell me that Arabella is able to sew!" Bea said with fascination.
"Absolutely not. But her tales of the Countess of Castignan certainly kept everyone awake. The problem is that the most repellent of the seamstresses, Mrs. Cable, and my aunt have taken a fervent dislike to each other. So there is more than a slim possibility that lunch will be a demonstration of gently bred fury."
Esme paused at the door. "I have been trying to come up with a seating arrangement that will keep my aunt and Mrs. Cable apart, and I have decided to scatter small tables in the Rose Salon." She gave Bea an alluring smile. "If you feel sufficiently brave, I would love to put you at a table with Mrs. Cable. She has a marked tendency to punctuate her conversation with ill-chosen Bible verses. Given your current appearance, she will deem you among the saved and be cordial."
Bea managed to simper. "Actually, I am quite well versed in the Bible myself."
"Oh goodness, how wonderful! I shall seat you just beside Mrs. Cable, if you don't mind. You can quote at each other in perfect bliss."
"Mr. Fairfax-Lacy seems quite sanctimonious," Bea put in before she could stop herself. "Mrs. Cable would likely approve of him. All those good works."
"Do you think so?" Esme asked with some doubt. "I believe that the man is undergoing some sort of internal upset. He doesn't appear to be interested in Parliamentary doings at all. And that is his reputation, you know."
"All work and no play?"
"Precisely." Bea thought about Stephen's activities in the goat pasture and rather agreed with Esme. The man was not thinking about Parliament. No: he was on the hunt for a mistress. Or perhaps a wife.
"But he must be accustomed to tedious speeches, so I shall put him at your table," Esme continued. "Helene can sit there as well and rehearse dallying with Mr. Fairfax-Lacy. You must prompt her if she neglects her practice. Although I must tell you, Bea, it's my opinion that your poetry will have to do the trick if Helene and Fairfax-Lacy are to become intimate. I've known Helene for years, and it simply isn't in her nature to play the coquette."
"But she did elope," Bea said, wondering how on earth that had happened. Who would elope with a woman who had all the sensual appeal of a matron of sixty? Yet when Helene laughed, she was surprisingly captivating.
Esme shrugged and opened the door. "Her husband, Rees, effected that miracle somehow, and they've both spent the last ten years regretting it. I do believe the marriage ended before they even returned from Gretna Green, although they resided together for quite some time.
"I am counting on your bravery at luncheon, then." She paused for a moment and looked at Bea. "Amazing! I would hardly have recognized you. I suppose you are revisiting the artless Lady Beatrix Lennox of age sixteen or thereabouts."
Bea gave her a rather crooked smile. "I hate to disillusion you, Esme, but I was fourteen when my father discovered that I was coloring my lashes with burnt cork. He never recovered from that initial shock."
"Oh, parents!" Esme said, laughing. "You should hear my mother on the subject of my innocence! Or the lack thereof. According to my mother, I sprang from the womb a fully fledged coquette—shaped in my aunt's image, as it were."
Bea grinned. "You could do worse."
"Much," Esme said with an answering grin. "At luncheon, then!"
When Bea slipped into a chair next to the redoubtable Mrs. Cable, her mind was not on the meal. She was wondering precisely how a Puritan gentleman greets a woman he vigorously kissed in the goat pasture the previous afternoon. Would Stephen pretend that they had never grappled with each other? That his tongue hadn't slipped between her lips? That she hadn't—
Bea could feel that rare thing, a real flush, rising in her cheeks, so she hurriedly pushed the memory away. She hadn't spent a good twenty minutes painting herself with sheer layers of Maiden's Blush only to find herself blushing.
The gentleman in question was rather exquisitely dressed himself, if the truth be told. Bea watched under her lashes as he strode into the room. He was wearing a costume of the palest fawn, with a severely cut-away jacket. For a man who spent his time roaming about the House of Commons, he seemed to have unaccountably powerful thighs.
"Oh Lord, there he is," Helene moaned, sitting down next to her. "This is such a foolish idea."
"You've no reason to worry," Bea said to her encouragingly. "The poem can do the work for you."
"Countess Godwin," Mrs. Cable announced, snapping her napkin into her lap, "we have met, although I expect you have no memory of the event."
"I remember perfectly," Helene said. "And how pleasant to see you again."
"It was a dinner that Lady Rawlings gave some few months ago," Mrs. Cable told Bea.
"How lovely that must have been!" Bea said breathlessly. She was rather enjoying playing the role of a virtuous maiden. It was a new experience, after all, since she'd spent her youth trying to infuriate her father with less-than-virtuous antics.
"It was not lovely," Mrs. Cable said darkly, "not at all. Countess Godwin, I daresay you have formed the same aversion as I to even thinking of the occasion. Quite scandalous."
Bea clasped her hands and widened her eyes. Stephen was on his way to their table, and she wanted him to see her in the midst of full-blown girlishness. "Oh, what could have happened!" she cried, just as Stephen arrived.
Helene, who had just noticed Bea's transformation, gave her a sardonic look. "Nothing you couldn't have topped, Bea."
The Puritan created a diversion by bowing and introducing himself to Mrs. Cable, who seemed enraptured at the idea of sharing a table with a Member of Parliament.
Somewhat to Bea's disappointment, he didn't even blink when she gave him a girlish smile and a giggle. Instead he bowed just as one would to a damsel still in the schoolroom, then turned readily to Countess Godwin and kissed her hand.
"Earl Godwin was there, of course," Mrs. Cable said in her sharp voice, returning directly to the subject. "Mr. Fairfax-Lacy, we are discussing an unfortunate dinner that I and Countess Godwin attended in this very room, some months ago. I won't go into the details in present company." She cast a motherly look at Bea, who bit her lip before she could grin, and then looked modestly at her hands.
Stephen caught a glimpse of Bea's downcast eyes and felt like bellowing with laughter. She was a minx. It wasn't only that she was dressed as primly as an escapee from a nunnery. Somehow she had managed to make her whole face look as guileless as a babe in arms. Gone was the mischievous twinkle and the lustful glances. She had the aura and the innocence of a saint, and only that one dimple in her cheek betrayed the fact that she was enjoying herself mightily. Other than that dimple, she was the picture of a naive duke's daughter. If there was such a thing in England.
"I daresay your husband told you," Mrs. Cable was saying to Lady Godwin, "that he and I exchanged some pointed words on the subject of matrimony. Not harsh, not at all. But I think I made my point." She smiled triumphantly.
Helene smiled weakly and took a sip of wine. "It would appear to have slipped Rees's mind."
Bea felt a surge of admiration. She herself would likely have lost her temper by now and started screeching at that harpy.
Mrs. Cable shook her head. "A man shall leave his father and mother, and cleave unto his wife, and so it says in the Bible."
"Alas, Rees is notorious for his defiance of authority," Helene replied.
Bea watched Helene trying to defend herself and felt a surge of fury. Who was this old harridan, and what right had she to say such an unaccountably insulting thing to Helene?
Mrs. Cable looked at Stephen. "I'm certain you won't mind if I act as if we are all old friends," she informed him. "I have given much thought to Lady Godwin's situation in the ensuing months since I dined with her husband." She paused for a drink of water.
Bea saw that Helene's slender hand was clenched so tightly on her napkin that her knuckles were white. "Were you not quoting Genesis just now, Mrs. Cable?" Bea cooed.
Mrs. Cable gave her the approving look of a headmistress with a promising student. "Precisely, Lady Beatrix. It's a true pleasure to meet a young lady with a proper education. Now, Lady Godwin, if I might offer a few—"
"My father puts great faith in religious instruction," Bea interrupted.
"Quite right," Mrs. Cable rejoined. "Now I think that I can bring some wisdom to bear on the situation." And she turned back to Helene.
That old snake can see that Helene is defenseless, Bea thought in a fury. "Why, when I fell in love with one of my father's footmen," she said in a high, ringing voice, "my father made me memorize the entire Book of Maccabees in punishment."
"Indeed," Mrs. Cable said, obviously taken aback at this information.
"Yes," Bea said, favoring her with a dulcet smile, "I offered myself to the footman in question, you see, and my father truly did think that I should not have done so."
Mrs. Cable's eyes widened.
"But I don't agree," Bea continued blithely. "Because, of course, the Gospel of John counsels us to love one another. That's chapter thirteen," she told Mrs. Cable. "But I expect you know that." Stephen was shaking with suppressed laughter. Helene's hand had relaxed, and she was biting back a smile as well.
"Yes, I—"
"Even if my love was unconventional," Bea said with a soulful tremor in her voice, "I'm quite certain that it was ripe with virtue."
"Ripe would be the word," Stephen said dryly.
Bea ignored him. "After all, while it is true that a footman would have had difficulty supporting me as a wife"—she glanced modestly at her gown, which cost more than a footman earned in six months—"Proverbs does say that where love is, a dinner of herbs is better than a stalled ox. Although I always wondered what a 'stalled ox' is? Mr. Fairfax-Lacy, perhaps you have come across the term in your many years in Parliament?"
Alas, Stephen didn't have a chance to deliver his opinion because Mrs. Cable sputtered into life again, like a candle that found itself briefly in the path of a rainshower. Now she was viewing Bea with the acute horror of someone who has discovered that an exquisite pastry is rotten in the center.
"Lady Beatrix," she said on an indrawn breath, "I am quite certain that you do not realize the impression your little story might create on the assembled company." She swept a glance around the table.
Helene met her eyes blandly. "Lady Beatrix never fails to surprise me, for one," she said. "A footman, did you say, Bea? How very adventuresome of you!"
"I don't know that I agree," Stephen drawled. He felt a thrill of danger when Bea's eyes met his, especially since the thrill went right between his legs. She was a glorious, impudent piece of womanhood, and he liked her defense of Lady Godwin. If only she knew, she had utterly ceased to look sixteen years old. Her face was too alive for all this nonsense she affected. "I, for one, would like to know how the footman answered Lady Beatrix's overtures," he put in. "Didn't you notice, Mrs. Cable, that while Lady Beatrix apparently offered herself to the footman, she said nothing of his response. Can it be that the man in question refused her?"
Mrs. Cable huffed. "I cannot fathom why we would even discuss such a repellent subject! Surely Lady Beatrix is merely seeking to shock us, for—"
"Not at all," Bea said. "I would never do that, Mrs. Cable!"
Mrs. Cable narrowed her eyes. "And where is your father, my lady?"
"At his house," Bea replied, suddenly reverting to her maidenlike docility. "I'm a sad disappointment to him,
Mrs. Cable. In fact I make my home with Lady Withers now."
Mrs. Cable huffed. "And the footman—"
"Oh, it wasn't due to the footman," Bea said blithely. "Father moved the footman to a house in the country. It was—"
"I'll not listen!" Mrs. Cable said shrilly. "You're making a May game of me, my lady, and it's not kind of you. I could take one look at you and know that you aren't one of those scandalous women you're pretending to be."
Helene threw Bea a warning look and put a gentle hand on Mrs. Cable's wrist. "You're absolutely right, of course," she said. "I do keep begging Lady Beatrix to be less frivolous, but I'm afraid that she's quite a romp. But, naturally, it's all in fun, Mrs. Cable."
"I knew that," Mrs. Cable said, blinking rapidly. "I'm a fair judge of character, to which Mr. Cable agrees. Now Lady Beatrix, you may attempt to shock us, but your true purity of character shines through. It's written all over your face. What did you say that was?" she asked the footman. "A regalia of cowcumbers? Indeed, I'll try some."
Stephen looked at Bea for a moment, and she had no trouble deciphering his thoughts. He was thinking of the goat pasture, and the true purity of her character.
##
## The Heights of Pleasure
By the time that Esme finished luncheon, she was resigned to the fact that the house was gradually filling with her aunt's friends, not one of whom was precisely respectable. Her Sewing Circle was doubtless scandalized by her guests, since the said guests substituted cynical wit for gentility. And since they delighted above all in displaying that wit, the house rang with laughter.
Or perhaps it was more accurate to say that the house simply rang with noise. Lady Arabella had taken charge of the housekeeping from top to bottom and seemed to be bent on proving her mothering ability by cleaning from the attics to the cellars. Mind you, it wasn't as if she touched dirt herself.
"I've instructed the maids that we want this house to shine from top to bottom," she announced to Esme. "This is what a mother would do. Remove all worries! You have enough to think about. When will that child come?"
Never mind the fact that Esme had no interest in the attics at all. She was hardly the matronly sort, even in her current respectability. But Arabella didn't stop with the attics. "And I've sent men up on the roof to fix the slate. I've no tolerance for gardeners simply sitting about, and there's nothing to be done outside at any rate." March rain was taking fitful turns with sunshine.
Esme had been listening rather absentmindedly, but she snapped awake at that one. "You sent the gardeners up on the roof?"
Arabella blinked at her. "Haven't you heard the hammering? They started first thing this morning. I noticed that the slate had practically evaporated from your roof in several parts. Without repair, we shall have leaks in short order. No doubt the task will take a few weeks or perhaps a month. But it needs doing."
"It's not safe!" Esme said. Panic surged into her stomach, and she suddenly felt a little dizzy.
"Of course it's safe," Arabella said. "They won't drop slate off the roof. Most of the work's being done in the back of the house. But perhaps I'll station a footman at the front door so he can check that the coast is clear before anyone leaves the house. In fact, darling, that is an excellent idea. We have far too many footmen as it is. I seem to have overestimated the difficulty of hiring staff in the country."
"It's not safe for the gardeners," Esme said, trying to calm her racing heart. Sebastian was up there. Up on a slippery roof, likely on the verge of falling to his death. She could not bear it if that happened. Not—not after Miles.
"Gardeners? Gardeners? They're likely ecstatic to be up in the air," Arabella said, waving her beringed hands. "Much more engaging than digging up weeds, believe me."
She left before Esme could say another word. Perhaps she should tell Arabella the truth about Sebastian. There was probably no one in the world who would be more receptive to the idea that Esme had her former lover installed in the bottom of her garden. Panic beat in her throat. Sebastian had to come down from the roof this very moment.
She went downstairs, bundled herself into a mantle, and slipped out the side door. The sound of hammering bounced off the neighboring hills. Starlings were converging on the elms at the side of the house, pirouetting against the grayish sky. Every blade of grass bent with rain. Now and then she heard the echo of male voices, but she walked all the way to the back of the house without seeing a soul.
And then, as she rounded the house to the west, there he was. Sitting with his back to a chimney, eating a hunk of bread as if he hadn't a care in the world. Marquess Bonnington wasn't hanging from the gutter by one fingernail. He wasn't spread-eagled in the rainy grass, face drained of color. He was—he was fine!
In fact, Esme could hardly believe that Sebastian was a marquess. Not this great muscled man, wearing a rough white shirt and sleeves rolled up to show great muscled forearms. No gentleman had muscles like that. Nor thighs, either.
She pulled herself together. What was the point of staring at Sebastian like a lovesick cat? The man would probably roll off the roof in a moment. He wasn't trained for this sort of activity.
"You!" she shouted. Her voice evaporated into the air. He tipped his head back against the chimney, turning his face up to the sun. It turned his neck to honey, kissed his hair with gold...that hadn't changed. He was just bigger...stronger. There was more of him.
What was he calling himself these days? She couldn't remember. But she could hardly shout "Bonnington!" either. If any of her guests discovered that Marquess Bonnington was snugly living on her estate, they'd dine out on it for days. Her name—and her child's name—would be mud. The thought gave her backbone.
She picked up a rock and threw it at the roof as hard as she could. It skittered across the sandstone. She tried again and managed to get up to the level of the slate roof, but all the rock did was ping gently and fall to the gutter.
"Drat!" Esme muttered, eyeing the ladders that were braced against the house. Of course she couldn't climb a ladder.
At that moment a voice spoke nearly in her ear. "May I be of service, madam?"
Esme jumped into the air. "Slope!" she gasped.
Her butler bowed. "I noticed your progress from the Rose Salon, my lady, and I ventured forth in the hopes of being of service."
Esme's cheeks burned. What was she to say? What the devil was she doing out here, anyway?
But Slope didn't wait for a reply. "Baring!" he bellowed at the roof. "Her ladyship wishes to speak with you. Be quick, man!"
Baring—or Marquess Bonnington, however one wished to think of it—looked down the roof with such a sweet smile that Esme felt her stomach turn over. He pulled on a cap and descended the ladder. Esme watched for a moment as he climbed down, but she found her eyes lingering on muscled thighs, so she turned to Slope.
"I simply wished to ascertain if the gardener—" she began.
But Slope raised a finger. "If you and the marquess were to retire to the rose arbor, my lady, you would be less likely to be seen from the house."
And with that astounding statement, he bowed and retreated.
Esme stood staring after him, mouth open. But here was her gardener, doffing his hat and fingering the brim, for all the world as if he were indeed an outdoorsman, planning to give his account to the lady of the house.
"How dare you climb my roof!" she snapped, turning her back on him and walking toward the rose arbor, which had so many ancient rose trees growing up its latticed sides that it was impossible to see in or out.
"I wish I could take your arm," Sebastian said, his voice so low that she could hardly hear it.
She didn't bother to turn around. It was quite difficult to pick her way down a slope slick with rain. The last thing she wanted to do was slip off her feet; Sebastian would likely strain his back heaving her back up.
"What the devil are you doing up on that roof?" Esme snapped, turning around the moment they entered the arbor.
Sebastian smiled, that easy smile that never failed to make her feel—greedy. The very thought made her indignation rise. "You have no right to risk your life on my roof! I want you off my property, Sebastian. Today!"
He strolled toward her. The rain had dampened his shirt and it clung to his shoulders, outlining a swell of muscles.
"What do you have to say to that?" she demanded, feeling her advantage weaken. Damn him for being so beautiful.
"I say," and his voice was as slow and deep as the rest of him, "first I say hello to your babe, here." He walked just before her and cupped his great hand over her belly.
"Hello," he whispered, looking straight into her eyes, not at her stomach. As if he could hear, the child stirred under his hand.
Sebastian laughed. "He must be rather cramped in there these days." He dropped to his knees and cupped her stomach with both hands. "Hello!" he said against the cloth of her gown. "Time to greet the world."
He looked up at Esme, and there was such wild joy in his eyes that she shivered all over. Then he stood up, and his hands slid around her body to her back.
"First I say hello to the babe," he said, and his voice was as slow and wicked as molasses, "and then I saw hello to his mother."
There wasn't even a thought in her head of avoiding that kiss. He bent his head and his hands pulled her against him, lips settling to hers as gently as the kiss of the sun. "Oh God, Esme, I've missed you," he groaned against her lips. And when she opened her mouth to reply, he plundered.
His tongue was rough and warm and God help her—a woman with child, a widow, a mature, respectable widow—Esme leaned into his kiss and wound her arm around his neck. He tasted like farmer's bread and he smelled like rain. He didn't move his hands. They stayed, huge and powerful, on her back, making her feel as delicate as a bird. He didn't even twitch a finger toward her breasts, and yet they melted toward him and longed, and other parts of her too...
That wave of longing brought her hands from his neck to his shoulders. It was more than longing: it was exquisite relief. He was whole. He hadn't fallen from the roof. The very thought brought her a measure of rationality.
"What were you doing up on the roof?" she said, frowning.
He ignored her. His warm, rough tongue plunged into her mouth, stole her words, brought that melting weakness to her knees. Willy-nilly, she curled her fingers into his hair, returned his kiss fiercely, until—
"You could have broken your neck!" Her voice sounded weak, a thread away from silence.
"No," he said. His hands were starting to roam now. He cupped her stomach again, kissed her so sweetly that tears came to her eyes. "Hello there," he whispered, "mama-to-be."
He scooped her up without even seeming to notice what an elephant she'd become and sat on the wrought-iron bench, holding her on his lap. She could feel his welcome stiffly, right through her pelisse.
It had never seemed to matter to Sebastian that her breasts were now so large that she couldn't wear the delicate gowns in fashion. His hands ranged, not roughly but possessively, over the front of her gown. It was almost embarrassing. Her nipples were so tender these days that he merely drew a thumb across her gown and a low moan hung in the air between them.
He looked at Esme's eyes then. They had lost all the fierceness. They didn't snap at him like a mother lion. He pulled her head against his shoulder, brushed a silky black curl from her ear and whispered, "What's the most beautiful mother in the world doing out in the rain, then?"
Her head popped off his shoulder before he'd had a chance to kiss her ear. "Rescuing you!" she said, and her eyes were snapping fire again. "What in the bloody hell were you doing up on the roof?"
He couldn't help it; a smile curled the corner of his mouth. She wouldn't be so fierce if she didn't care for him.
"Mending the slate," he said, knowing it would drive her mad. But he liked her furious, those gorgeous eyes blazing at him, breasts heaving, focused on him.
She jerked her head away from him. But she didn't move to stand up, so he kept his hands exactly where they were. One on her narrow back and the other cupping the swell of her breast. His fingers longed to move, to caress, nay to take her breast and—
He pulled himself back to a state in which it was possible to listen. She was scolding him for being reckless, heedless, brash, daring, inconsiderate...His fingers trembled, and so he allowed himself to take her breast more snugly in his hand. He imagined its glorious weight on his chest.
He was imprudent, unwise and altogether foolish....
He was maddened by the desire to push off her pellisse, sweep a hand under her gown and claim her as his. Again. Every time. Those few times she had visited his hut, he'd found that his sense of ownership, of a primitive she-is-mine feeling had lasted only an hour or so after she'd left. She'd returned to the house, and he'd stayed in his hut and dreamed of her.
His hand closed on her breast, and his thumb rubbed over her nipple again. The flow of words stopped and there was a tiny gasp. He did it again, and again, and then bent to her mouth. Those lips, so dark, cherry dark, were his. She whimpered and trembled against his chest. He memorized each quiver.
"You are mine," he said, and the growl of it surprised him.
She leaned back against his shoulder, silky curls falling over his shirt, her eyes closed. Her breathing grew shallow, and she clutched his shirt as his thumb rubbed again and again with the roughness of desire, with the roughness with which he wanted to plunge between her legs.
But he couldn't. They were in a rose arbor, after all. Slowly he eased her back against his shoulder and let his hand cup her breast, sending a silent apology to the nipple that begged against his palm.
He knew instantly when she returned to herself. It wasn't that she sprang to her feet. It was an imperceptible change in the air, in the very air they were breathing.
"No," she said, and the anguish in her voice struck him in the heart. "I don't want this!"
"I know," he said, as soothingly as he could, tracing with one finger the graceful curve of her neck. "I know you don't."
"Obviously, you don't care for my wishes! Otherwise you would have returned to the Continent by now. What if one of my guests decides to take a breath of fresh air?"
"I do care for your wishes. You wish to be respectable. You wish to remain a widow. You also"—he dropped a kiss onto the sweet cream of her neck—"you also wish to bed me."
"I can live without the latter."
"I don't know that I can," he said, his mouth glazing her neck. Her perfume was surprisingly innocent for such a worldly lady. She didn't smell like some sort of exotic inhabitant of the East Indies, but like an almond tree in flower.
"I admit that I find you—enticing," she said, and he spared a moment to admire the steadiness of her voice. "But the game is over. Slope, my butler, knows who you are. In fact, he must have known from the moment you applied for a position. While he is unlikely to gossip about the matter, it is a matter of time before one of my house guests finds out your identity. The house is full of people who know you, Sebastian. I'll be ruined. And I can't bear that, not when so much is at stake.
"And I don't want you to fall off my roof either!" she said, her hand gripping his shoulder. "I cannot bear it if something happens to you, Sebastian. Not after Miles. Not after—Don't you understand?" Esme felt as if her breath caught in her chest at even saying it aloud.
Oh, he understood all right. He'd probably have five little marks on his shoulder, one love mark for each of her fingers. The smile that grew on his face came from his heart, and if she didn't recognize that..."You want me to leave?" he said, and he had to steady his voice because she might recognize the rough exaltation there.
She nodded fiercely. "No more Baring the Gardener," she said. "You must go."
Much to his regret, he rather agreed with her. It was time to say good-bye to his disguise, much though he loved the simple life. "Do you really, truly wish me to return to the Continent—or, to be specific, France?"
She nodded again. But Sebastian noted the way she swallowed, and he had to bite back another growl of triumph.
"If you truly wish me to go," he said into her hair, "you'll have to grant me a wish."
"A wish?"
Another curl of her perfume caught him, and he had to stop himself from licking her face, simply drinking her. She was so beautiful, in all her silken, sulky anger and fear for him. "One wish." His voice sounded drunken.
"I wish for you to go," Esme said primly. "It is certainly—"
He cut her off. "One night," he said. "I want one night."
Her backbone straightened. "What?"
"I'll come to you tonight. I'll come to your bedroom," he said into her ear, and his tongue lingered there for a second. "I'll take you in my arms, and put you on the bed—"
"You certainly will not!"
He smiled into her curls. "Do you truly wish me to leave your property?"
"Immediately!" she snapped.
"Then I demand compensation." He let his hand spread on her breast again, warm and possessive on the curve. He felt the quiver that rolled through her body as acutely as she did. "One night," he said hoarsely, and he couldn't keep all the lust and love from tangling together in his voice. "One night and I'll leave your employ and retire as a gardener."
She was silent, likely worrying about whether they'd be discovered, fretting about her respectability. Only he, whose reputation was absolutely ruined, seemed to understand how very little respectability mattered in life.
His hand trailed over the fabric of her dress, touching the roundness of her thigh. "Oh God, Esme, give me this." But she was holding something back. He could tell.
"Are you sure that you would want to make love to me in this state?" Her eyes met his, direct as ever. "I've grown even more ungainly and—"
He caught her silliness in his mouth. "I want to devour you." That seemed to silence her; her cheeks turned pink. "In fact, you should take a nap this afternoon, because there won't be much sleep tonight. I mean to have you every way I can. I mean to intoxicate you and torment you so that you know precisely how I feel about you." His finger trailed down her cheek and tipped up her chin.
"Don't mistake what is going to happen tonight." His voice was sinful, dark and hoarse. "You will never forget the imprint of my skin after tonight, Esme. Waste your life chitchatting with ladies in lace caps. Raise your child with the help of your precious Sewing Circle. But in the middle of all those lonely nights, you will never, ever, forget the night that lies ahead of us."
Esme's heart was beating so fast that she could hardly speak.
"Tonight." He held her gaze. "And then I'll leave for France because...because that's what you want, true?"
At the moment she couldn't quite remember what it was she wanted. Besides the one thing, of course. That thing was pressing against her backside as they spoke.
And the Sewing Circle. She mustn't forget the Sewing Circle.
##
## The Delights of Poetry
Tonight Helene was going to seduce Stephen Fairfax-Lacy, otherwise known as the Puritan, and Bea was perfectly reconciled to that fact. In fact, she was the instigator. She herself had selected an exquisitely desirous bit of verse for Helene to read. Not only that, but she, Helene and Esme had had an uproarious time trying to teach Helene to use a fan and various other flirtatious tricks.
The only reason I feel a bit disconsolate, Bea thought to herself, is that I have no one to play with. If only Arabella had invited sufficient gentlemen to this house party, she wouldn't have had the slightest qualm while assisting Helene to use the stodgy M.P. in order to curdle her husband's liver. If there was a dog in the manger here, it was Bea herself. Because of course she would never want Mr. Fairfax-Lacy, not really.
I merely feel, Bea instructed herself, a mild anxiety at the upcoming performance of my protégée. For it was her poem Helene was reading and her idea to use Mr. Fairfax-Lacy to make Helene's husband jealous. Thus Helene's success or failure reflected on Bea. And why she didn't just keep her mouth shut when she had the impulse to meddle in the lives of perfect strangers, she didn't know.
Lord Winnamore had elected to be the first to read. He was standing before the fireplace, droning on and on from Virgil's Second Eclogue. Whatever that was. Bea didn't care if it had been translated into English by Shakespeare himself; it was as boring as dirt.
"Well, Winnamore," Arabella said briskly, the very moment he fell silent, "that certainly was educational! You've managed to put my niece to sleep."
Esme sat up with a start, trying hard to look as if she hadn't been daydreaming about the way Sebastian would—might—"I'm not asleep," she said brightly. "The eclogue was utterly fascinating."
Arabella snorted. "Tell it to the birds. I was asleep, if no one else was."
But Lord Winnamore just grinned. "Do you good to hear a bit of the classics," he told her mildly.
"Not if they're that dreary. I've no need for them. Am I right in thinking that the whole thing was praise for a dead man?"
When Lord Winnamore nodded, Arabella rolled her eyes. "Cheerful." Then she turned to the company at large. "Let's see, we'll just put that painful experience behind us, shall we? Who wants to go next?"
Esme shot Helene an encouraging look. She was sitting bolt upright on a wing chair, looking desperately uneasy. As Esme watched, Bea handed Helene a small leather book, open in the middle.
Helene turned even paler, if that were possible. She seemed terrified. "Helene!" Esme called across the room. "Would you like to read a poem, or shall we save your performance for tomorrow?" But Esme saw in Helene's eyes terror mixed with something else: a steely, fierce determination.
"I am quite ready," she answered. She stood up and walked over to the fireplace to stand where Lord Winnamore had been. Then she turned and smiled at Stephen Fairfax-Lacy. Esme almost applauded. No one could call that a lascivious smile, but it was certainly cordial.
"I shall read a poem entitled 'The Shepherdess's Complaint,'" she said.
"Lord, not another bloody shepherd!" Arabella muttered.
Lord Winnamore sent her an amused look. "Lady Godwin did say shepherdess, not shepherd."
Helene was starting to feel reckless. It was too late for second thoughts. Fairfax-Lacy would come to her bed, and then she would flaunt—yes, flaunt—him in front of Rees.
She threw Stephen another smile, and this one truly was warm. He was going to make it happen. What a lovely man!
"Well, do go on," Arabella said rather impatiently. "Let's kill off this shepherdess, shall we? Lord, who ever thought that poetry was so tedious?"
Helene looked again at Stephen Fairfax-Lacy, just to make certain that he realized he was the benefactor of her poetry reading, and began:
If it be sin to love a sweet-faced Lad,
Whose amber locks trussed up in golden trammels
Dangle down his lovely cheeks with joy—
"Trammels?" Arabella interrupted. "Trammels? What the devil is the poet talking about?"
"The man in question has his hair caught up in a net," Winnamore told her. "Trammels were used by fishermen—" Helene cast him a look as well, and he fell silent. She felt rather like a schoolmistress. One kind of look for Stephen, a look that said Come to my room! Another kind of look for Lord Winnamore—Hush in the back, there! "I shall continue," she announced.
When pearl and flower his fair hair enamels
If it be sin to love a lovely Lad,
Oh then sin I, for whom my soul is sad.
Helene had to grin. This was perfect! She looked down at Bea with thanks, but Bea jerked her head almost imperceptibly at Stephen. Obediently, Helene looked at Stephen again. It was getting easier to smile at the man. And all this talk about sin had to make it clear what she had in mind.
O would to God (so I might have my fee)
My lips were Honey, and thy mouth a Bee.
Then shouldst thou sucke my sweete and—
and my—
Helene stopped. She could feel crimson flooding up her neck. She couldn't read this—this stuff!
"That's a bit of all right!" Arabella called. "Lady Godwin, you are showing unexpected depths!"
But Esme was crossing the room and taking the book from Helene, who seemed to be frozen in place. "It's too deep for me," she said, giving Helene a gentle push toward her chair. "I am a respectable widow, after all." She glanced down at Bea and then decided not to ask that minx to read. "I think we have time for only one more poem tonight." It wasn't that she was particularly anxious to retire to her room...except that Sebastian might be waiting for her. A lady never kept a gentleman waiting.
"Mr. Fairfax-Lacy," she said, turning to him, "did you find a poem that you liked in my library?"
"I did. And I'd be most pleased to read it," he said, getting up. "Now as it happens, mine is also supposedly written by a shepherd."
"Who would have thought," Arabella said in a jaundiced voice, "that sheepherders were quite so literary?"
Helene's heart was racing with humiliation. How could she have read those words aloud? Why—why—hadn't she read the poem before accepting it from Bea? She should have known that any poem Bea chose would be unacceptable. Finally she drew a deep breath and looked up at Stephen.
She met his eyes. They were utterly kind, and she felt imperceptibly better. In fact, he grinned at her.
"I'm afraid that my poem is far less interesting than was Lady Godwin's," he said with a little bow in her direction, "but then, so am I."
That's a compliment! Helene thought. Mr. Fairfax-Lacy had a lovely voice. It was deep and rolled forth, quite as if he were addressing the entire House.
Ah beauteous Siren, fair enchanting good,
Sweet silent retoric of persuading eyes...
He paused and looked directly at Helene. She felt an unmistakable pang of triumph. He'd understood her! She stopped listening for a moment and wondered which of her night rails she should wear. It wasn't as if she had any luscious French confections such as Esme presumably wore when she embarked on an affair.
But slowly she was drawn back into listening, if only because Stephen's voice was truly so beautiful. He made each word sound as if it were of marvelous meaning.
Such one was I, my beauty was mine own,
No borrowed blush, which bank-rot
beauties seek,
The newfound shame, a sin to us unknown,
The adulterate beauty of a falsified cheek...
"I'm not sure I like this one any more than the first poem," Arabella said grumpily to Lord Winnamore. "Feel as if I'm being scolded. Falsified cheek indeed! And what's a bank-rot beauty? We've none of those in this room."
"It was not so intended at all, Lady Arabella, I assure you," Stephen said, glancing down at Bea to make sure she was listening. She was curled up on a stool like a little cat. He could see a bewitching expanse of breast. Naturally her bodice was the size of his handkerchief.
The adulterate beauty of a falsified cheek,
Vile stain to honor and to woman also,
Seeing that time our fading must detect,
Thus with defect to cover our defects.
"Enough of that!" Arabella said briskly. "Last thing I need is a lecture about what time is doing to my face, and you'll be lucky, Mr. Fairfax-Lacy, if I don't take it amiss that you've even mentioned fading in my presence!"
"I'm truly sorry," Stephen said. "Naturally I viewed this poem as utterly inapplicable to anyone in this room." He bowed and kissed Arabella's hand. "I certainly detect no fading in your beauty, my lady." He gave her a look of candid repentance, the one he used when his own party was furious because he had voted for the opposition.
"Humph," Arabella said, somewhat mollified.
He'd made his point; he was fairly certain that he saw a gleam of fury in Lady Beatrix's eyes. Now he intended to pursue the more important goal of the evening.
Helene found with a start that Stephen Fairfax-Lacy was drawing her to her feet. "May I show you a volume of poetry that I discovered while searching for a suitable text for this evening?" he said, nodding toward the far end of the room.
Helene rose lightly to her feet. "I'd be most pleased," she said, steadying her voice. She put her fingers on his arm. It was muscled, as large as Rees's. Were all gentlemen so muscled under their fine coats?
They walked across the room until they reached the great arched bookshelves at the far end. Once there, Helene looked up at Stephen inquiringly, but he didn't pull a volume from the shelf.
"It was merely an excuse to speak with you," he said with an engaging smile. "You seemed startled by the content of your poem, and I thought you might like to escape the company for a moment."
Helene felt that traitorous blush washing up her neck again. "Well, who wouldn't be?" she said.
"Lady Beatrix Lennox?" he said, and the note of conspiratorial laughter in his voice eased Helene's humiliation.
"She did give me the poem to read," Helene admitted.
"I thought so." He took her hand in his. "You have lovely fingers, Lady Godwin. Musician's hands."
Her hands looked rather frail in his large ones. Helene quite liked it. She never felt frail.
"And I thought your waltz was truly exquisite." He was stroking her fingers with his thumb. "You have an amazing talent, as I'm sure you know."
Helene's heart melted. No one ever praised her music. Well, she rarely allowed her music to be played in public, so no one had the opportunity. But she melted all the same. "It's rather a daring piece," she murmured, watching his fingers on her hands.
"How so?"
"Because it's a waltz," she explained. He truly didn't seem to understand, so she elaborated. "The waltz is considered unforgivably fast, Mr. Fairfax-Lacy. You do know that it hasn't been introduced to Almack's yet, don't you?"
He shrugged. "I haven't been to Almack's in years, and I count myself lucky."
"Respectable women seldom dance the waltz, and they certainly don't write them."
"I enjoyed it." He was smiling down at her, and she felt a little thrill all the way to her toes. "Was that the very first waltz that you have written?"
"No." She hesitated. "But it is the first to receive a public airing."
"Then the fact that I danced it is truly one of the greatest honors of my life," he said, with another elegant bow.
Mr. Fairfax-Lacy was truly...truly all that was admirable. "Would you consider," she asked impulsively, "coming to my chamber tonight?"
He blinked, and for one dreadful moment Helene had an icy sense of error. Horror swept up her spine.
But he was smiling and bowing. "You anticipated my own question," he said, kissing the very tips of her fingers. "May I pay a visit to your chambers later this evening?"
"I'd like that very much," Helene managed. His smile deepened. He really is handsome, she told herself.
"I believe it is time to retire, Lady Godwin. Our hostess appears to be taking her leave."
"Yes, lovely," Helene said breathlessly. So this is how it was done! How simple, really. She invited; he accepted. She almost pranced back across the library on his arm. Esme twinkled at her. Bea kissed her cheek and whispered something Helene couldn't hear. Probably advice. Arabella frowned a little; she had probably only just realized that her scheme to marry Esme to Mr. Fairfax-Lacy was in danger.
Helene felt a surge of triumph. She had just taken the most eligible man in the house and summoned him to her room! She was not a frigid, cold woman as her husband had told her.
She had a lover!
##
## Beds, Baths, and Night Rails
He wasn't in her chamber when Esme opened the door. Of course, she was glad of that. What would her maid think, to find the gardener in her bedchamber? It sounded like a tidbit from a gossip rag: "A certain lady widow, in the absence of a husband, seems to be relying on her staff." Tomorrow she'd start her new life as a respectable mother. Of course she wouldn't take lovers once her baby was born—for one thing, she could never risk having another child, since she had no husband.
But she couldn't seem to concentrate on her future respectability. Her whole body was humming, talking of the night to come. She felt almost dizzy. She and Sebastian had never had an assignation before. They'd made love once in a drawing room last summer. Then she had visited him in his gardener's hut a few times, but always on the spur of the moment. He had never come to her. Well, how could he?
She had never known beforehand that he would enter her room at night. That she would watch him undress. That he would lean over her bed with that smoldering look of his. Her inner thighs pricked at the thought.
"I feel particularly tired," Esme told her maid, Jeannie. "A bath immediately, please, with apricot oil." Jeannie chattered on about the household while Esme tried to ignore the fact that merely washing her body was making her feel ripe...aware....
Suddenly a tiny movement caught her eye. Her windows were hung with long drapes of a rich pale yellow. And under one of them poked the toe of a black boot. Not a gentleman's boot either. A gardener's boot.
A great surge of desire sank right to the tips of Esme's toes. He was watching. Her whole body sung with awareness of those hidden eyes. Jeannie had bundled her hair atop her head to keep it dry; Esme reached up as if to ascertain that no hairpins fell. Her breasts rose from the bath, drops of water sliding over her sleek skin. The curtain moved again, just the faintest twitch.
Esme smothered a grin and lay back against the edge of the bath. "My skin is so dry these days," she said to Jeannie, hoping that the maid didn't notice that her voice seemed deeper. "May I have the oil, please?"
Jeannie poured some into her hand and slowly, very slowly, Esme opened her hand and let the sweet-smelling oil trickle first down her neck, and then down the slick curve of her breast. Jeannie was darting around the room, folding up clothes and talking a constant stream as she did so. Esme spread a hand across the swell of one breast. The oil sank into her moistened skin, turned it satiny smooth. The curtain moved again, and Esme smiled, a smile for him. For the man waiting for her. Those unseen eyes made a simple bath feel scandalous, forbidden...made her feel sensuous and erotic. She raised her arms to her hair again, a ballet of tantalization.
The curtains swayed. He was watching...
"Now, that's odd," Jeannie said, starting toward the windows. "I could have sworn I shut that window, but there must be a draft."
"There's no draft!" Esme croaked.
"I'll just make certain, my lady."
"No!"
Jeannie stopped short of the windows. "My lady?"
"I think that I shall take a longer bath than I expected. Why don't you go downstairs and"—her mind was utterly blank—"help Mrs. Myrtle with something?"
Jeannie looked utterly astonished, but at least she turned away from the windows. "But my lady, Mrs. Myrtle doesn't need my help! She's far too grand to ask me to do aught for her!"
That was likely true enough. Esme's housekeeper was a formidable sort of woman. "I would like to be alone," Esme said bluntly.
"Of course, my lady! I'll just return in ten minutes and—"
"No! That is, I shall put myself to bed tonight."
Jeannie's mouth actually fell open. "But my lady, how will you rise from the bath? And if you fall? And—-"
Jeannie had a point, but Esme could hardly say that she had an assistant at hand. "Help me up," she said, reaching out a hand. Jeannie brought her to her feet, and Esme stepped onto the warm rug next to the bath, grabbing the toweling cloth Jeannie held out. The last thing she wanted to do was give Sebastian a good look at the enormous expanse of her belly. He'd probably run for the entrance. Then she waved toward the door in dismissal.
Jeannie was obviously bewildered. "Shall I just return in—"
"I will be quite all right," Esme said firmly. "Good night."
Jeannie knew a command when she heard one. She stood blinking for a second and then curtsied. She ran down the back stairs, utterly confused. She was that distracted that she actually told Mrs. Myrtle what had happened, although in the normal course of things she would never share an intimacy with that dragon of a housekeeper.
Mrs. Myrtle raised her eyebrows. In the old days, of course, such behavior would have meant that the missus had other plans for the evening. But obviously that wasn't the case. "Pregnant women are like that," she advised Jeannie. "Irrational as the day is long. My own sister ate nothing but carrots for an entire week. We all thought she'd turn orange. Never mind, Lady Rawlings will be as right as rain in the morning."
If Jeannie had but known, the very experienced maid, Meddle, who attended Helene, Countess Godwin, was just as bewildered. Her mistress had also ordered a bath. And then she had tried on all four night rails she'd brought with her, seeming to find fault with each. One wasn't ironed correctly, another had a pulled seam...Obviously her mistress had an assignation that evening. But with whom?
"It's plain as a pig's snout," Mr. Andrews said, waving his fork about. "She must have an assignation with my gentleman, Lord Winnamore. He's had no success with Lady Withers, and he's decided to cultivate greener fields." Andrews was a boisterous Londoner who had only served Winnamore for a matter of days.
"I do not agree," Mr. Slope said magisterially. As the butler, he never allowed even the mildest discussion of their mistress, but he had been known to lend the benefit of his expertise when it came to the foibles of other gentlepersons. And his expertise was considerable; everyone had to agree to that. After all, as butler to one of the most notorious couples in London for some ten years, he'd seen every sort of depravity the peerage got up to.
Mrs. Myrtle raised an eyebrow. "You're not suggesting Mr. Fairfax-Lacy, my dear Mr. Slope? And do have a bit more of this pickled rarebit. I think Cook has outdone herself."
Mr. Slope chewed and swallowed before replying; his manners were an example to the understaff. "I am indeed suggesting Mr. Fairfax-Lacy."
"My gentleman a partner to adultery? Never!" Mr. Fairfax-Lacy's valet was on the naive and elderly side. Mr. Fairfax-Lacy had rescued him from near destitution, when the poorhouse had been staring him in the very face.
"It would be an act of kindness," Meddle pointed out. "What's poor Lady Godwin to do, then? Her husband left her ten years ago. If the stories be true—it was before my time—he put the missus out on the street. Made her take a common hackney to her mother's house. Didn't even allow her to take the carriage with her when she left! That's evil, that is."
"Ah, if it's an act of kindness you want, then Mr. Fairfax-Lacy is the one to do it," his valet said, leaning back satisfied.
"I think Lord Winnamore is an excellent choice," Andrews said stubbornly. "My master is known the breadth and length of London. And he's rich as well."
"He's known for faithful courtship of Lady Withers," Mr. Slope pointed out. "Now you, Mr. Andrews, have admitted to all of us that you are still green in service." A few of the younger footmen looked blank, so he explained, "Mr. Andrews has not served as a gentleman's gentleman for a great period of time."
"That's so," Andrews said. "Came to the business from tailoring," he explained. "I finished my apprenticeship and found I couldn't stomach the idea of twenty years of sewing. So I found this position."
"When you are further along, you will learn to read the signs. Now Mr. Winnamore...where is he at this very moment?"
"Why, he's in bed, I suppose," Andrews said. "In bed with the countess!"
"You undressed him?"
"In a manner of speaking." Andrews had found to his great relief that his master didn't need any personal assistance. He didn't think he was up to pulling down another man's smalls, even for the sake of a steady wage.
"That proves it," Mr. Slope said with satisfaction.
"Why?"
"A gentleman doesn't undress before he visits a lady's chamber. What if he were seen in the corridor? He makes pretense that he is fetching a book from the library or some such." He chuckled. "There have been nights in this house when the library would have been empty of books, if all the stories were true!"
Andrews had to accept that. It sounded like the voice of experience. And Mr. Winnamore certainly hadn't looked as if he were planning an excursion down the hallway when Andrews left him. He'd been reading in bed, the same as he always did.
"Mr. Fairfax-Lacy, eh? He's a Member of Parliament, isn't he?" Andrews said, throwing in the towel.
"That's right," Mr. Slope nodded. "An esteemed one at that. Lady Godwin couldn't have chosen better. I wouldn't mind another bite of that shepherd's pie, Mrs. Myrtle, if you would be so good. Now, perhaps we should all discuss the proper manner of exiting a room during dinner, because I happened to notice this evening that young Liddin barged through that door as if a herd of elephants were after him."
##
## In Which Countess Godwin Learns
Salutary Lessons about Desire
Helene was battling pure terror. If she could have thought of a way to send Mr. Fairfax-Lacy a note without creating a scandal in the household, she would have done so in a second. The note would have said that she had come down with scarlet fever and couldn't possibly entertain callers in her chambers.
She felt like...like a bride! Which was incredibly ironic. She remembered distinctly waiting for Rees to walk into their room at the inn. They hadn't even married yet; they'd still been on their way to Gretna Green. But Rees had guessed, correctly, that her papa wouldn't bother to follow them, and so they'd stopped at an inn the very first night.
If only she had had enough character to walk out of that inn the very next morning and return, unmarried, to her father's house. She had waited in that chamber just like any other giddy virgin, her eyes shining. Because she'd been in love—in love! What a stupid, wretched concept. When Rees had appeared, it had been immediately clear that he'd been drinking. He'd swayed in the doorway, and then caught himself. And she—fool that she'd been!—had giggled, thinking it was romantic. What was romantic about a drunken man?
Nothing.
The very thought steadied her. Stephen Fairfax-Lacy would no sooner appear in his bride's doorway the worse for liquor than he would appear in Parliament dressed in a nightshirt. Which made her wonder whether he would come to her room in a nightshirt. If she simply pretended that he was no more unexpected than Esme paying her a visit...
There was a knock on the door, and Helene almost screamed. Instead she tottered over to the door, opened it, and said rather hoarsely, "Do come in." He was fully dressed, which she found daunting. She was wearing nothing more than a cotton night rail. Helene straightened her backbone. She had survived marriage with Rees; she could survive anything.
He seemed to see nothing amiss, though. With a little flourish, he held up a flagon and two small glasses.
"How thoughtful of you, Mr. Fairfax-Lacy," she said.
He put the glasses on the table and walked over to her. "I think you might call me Stephen?"
His voice had that rich, dark chocolate sound that he must use to mesmerize the House of Commons.
"And may I call you Helene?"
Helene on his lips sounded French and almost exotic. She nodded and took a seat by the fireplace. He sat down next to her and poured a little glass of golden liquor. Helene tried to picture what would happen next. Would he simply disrobe? Should she turn to the wall and allow him some privacy? How was she to take off her night rail? Luckily Mr. Fairfax-Lacy—Stephen—seemed perfectly content to sit in silence.
"I've got no practice with this sort of thing," she finally said, taking a gulp of the liquor. It burned and glowed to the pit of her stomach.
He reached out and took her hand in a comforting sort of way. "There's nothing very arduous to it, Helene. You and your husband have not lived together for years, have you not?"
"Almost nine years," Helene said, feeling that inextricable pang again. It was just that she hated to admit to such a failure.
"You cannot be expected to go to your grave without companionship," he said. His thumb was running gently over the back of her hand, and it felt remarkably soothing. "As it happens, I have never met a woman whom I wished to marry. So I am also free to take my pleasure where I may, and I would very much like to take it with you."
Helene could feel a little smile trembling on her lips. "I'm just worried about...about..." But how did one ask bluntly when he was going to leave? If he spent the night with her and her maid found out, she would die of shame.
"I am perfectly able to prevent conception," he said. He moved his hand, and her fingers slipped between his.
Helene's heart skipped a beat. She wanted a child—desperately, in fact. But not this way. "Thank you, that would be very kind," she said, feeling the ridiculousness of it. Oh for goodness' sake, perhaps they should just get it over with, and then she could begin the process of curdling Rees's liver. "Would you like to go to bed now?" she asked.
He stood looking at her for a moment and nodded. "It would be a pleasure, my dear."
Helene crawled into her bed and pulled the covers up. "I shall close my eyes so as to give you some privacy," she said. Surely he would be grateful for that small kindness. After all, there was no reason why they had to behave like wild animals simply because they were embarking on an affair.
A moment later she felt the bed tilt slightly as he got under the covers. She opened her eyes and hastily shut them again. He was leaning over her, and he hadn't any shirt on. "You forgot to snuff the candles," she said in a stifled voice.
"I shall do so immediately," he replied.
Stephen was so different from Rees. His voice was always calm and helpful, ever the gentleman. Would Rees have snuffed the candles on her request? Never. And Rees's chest was all covered with black hair, whereas Stephen's was smooth. Almost—almost feminine, except that was such a disloyal thought that she choked it back.
He returned to bed, and she made herself turn toward him. Thank goodness, the room had fallen into a kind of twilight, lit only by the fireplace off to the side. She took a deep breath. Whatever happened, she was ready.
Except that nothing happened for a few moments.
If the truth be known, Stephen was rather perplexed. Helene clearly wanted to have an affair. But she wasn't exactly welcoming. That's because she's a true English lady, and not a trollop, he told himself, dismissing the image of Bea's creamy breasts that popped up, willy-nilly, to his mind. He had doubts about those breasts anyway. They'd seemed slightly skewed to the left after she'd wriggled herself out of her spencer in the goat pasture.
With a start he realized that he was in quite a different bed and should be thinking very different thoughts. He bent over and kissed Helene. Her lips were cool and not unwelcoming. He slipped his hands around her shoulders. Her husband must have been something of a boor; the poor woman was trembling, and not with passion.
But Stephen was nothing if not patient. He kissed her slowly and delicately, each touch promising that he would be a gentleman, that he would be slow, that she could take her pleasure as she would. And slowly, slowly, Helene stopped trembling. She didn't exactly participate, though. He kept having to push away fugitive thoughts about the way Bea had made little sounds in her throat when he kissed her in the goat pasture.
Twenty minutes later, he judged that they had reached a point at which she wouldn't mind being touched. He ran a hand down her shoulder and edged toward her breasts. Helene gasped and went rigid again.
"May I touch your breast?" he whispered. A small voice in the back of his head was saying quite obstinately that this was all extraordinarily unexciting. The last woman he bedded who'd shown as little initiative as Lady Godwin had been his very first. And she was all of fifteen, as was he. But Lady Godwin—Helene—was clearly trying.
"Of course you may," she whispered.
It was the whisper that did it. The very small trickle of desire that had crept into his veins died on the spot. She was being polite, and she was being brave. Neither emotion did much for Stephen's desire. The desire he had wilted, in all senses of the word. He slid his hand very carefully around her back and pulled her close. She felt rather like a fragile bird, nestled in his arms. Then he rested his chin on her hair and said, "I thought I knew why I was here, but now I'm not sure that I do."
There was silence. Then: "Because we are beginning an affair?"
He couldn't even tell where all that desperation was coming from. From the idea of bedding him? In that case, why on earth was she putting herself through such an ordeal?
He chose his words very carefully. "Generally, when a couple embarks on such a...a relationship, it is because they feel a mutual attraction. I certainly think you are a beautiful woman—"
Helene chimed in with the exquisite manners that accompanied everything she said or did. "You are extremely handsome as well."
"But do you really wish to sleep with me?" He ran a coaxing hand down her arm.
When she finally spoke, it sounded as if she were near tears. "Of course!"
"I've never been so attractive that a woman felt she must bed me," Stephen said teasingly, trying to lighten the atmosphere. It didn't work. He could feel his chest growing damp from her tears. Damn it, the whole day was a fiasco, from beginning to end.
"I should never have done it," Helene said shakily, wiping tears away as fast as she could. "I simply thought..." Her voice trailed off.
Stephen was struck by a sudden thought. "Did you think to use me to prove adultery?" That would destroy his career in two seconds flat—a notion that didn't seem to bother him as much as it should.
"No," Helene sobbed. "I would never have used you in such a way. I thought we might—we might enjoy—and then I could tell my husband, and—" Her voice trailed off.
They lay there for a while, a lanky English gentleman and a sniffling countess. "I'm sure my face is quite red," Helene said finally.
Her wry tone told Stephen that she had regained control. Her face was indeed all red, and her hair was starting to fall out of its braid into wisps around her face. For some reason, he found it very sweet that she hadn't even known enough about an assignation with a man to loosen her hair.
"Helene," he said gently, "this isn't going to work."
"Why not?"
He blinked and realized her surprise was genuine. "Because you don't truly wish to make love to me," he pointed out.
Helene could have screamed with vexation. How stupid could the man be! If she didn't want to make love to him, would he be in her room? Would she have humiliated herself by appearing in dishabille before a man? Would she have allowed an unclothed man anywhere in her vicinity? "I do wish to make love to you," she managed.
He reached out and rubbed away a tear. "I don't think you do," he said, and there was a sweet look on his face.
A sweet, condescending look.
A whole flood of naughty words—the words she had been taught not to use, and indeed, had never even used in thought—came to Helene's mind. "That's rot," she said. "You're a man. Men always want to make love to women under any circumstances. Everyone knows that."
Stephen bit his lip, and Helene had a terrible feeling he was trying to keep back a smile. "They generally like to feel that the lady they are with is willing."
"I am willing!" Helene said, hearing her voice rising. "How much more willing can I be?"
He looked embarrassed now. "Perhaps I'm not phrasing this correctly."
"I'm willing!" Helene said. She reached up and wrenched open the buttons that ran down the front of her night rail. "Go ahead. Do whatever you wish."
For a moment they both just stared at Helene's breasts. They were small in comparison to Esme's, but they had a nice jaunty air to them. At least that's what Helene thought until she forced herself to look at Stephen. He looked absolutely mortified. But Helene was starting to enjoy herself. It seemed that she was capable of shocking people!
She wriggled her night rail back over her shoulders so it pooled around her waist. "Now if I remember this whole procedure correctly from my marriage," she said, feeling a slightly hysterical giggle coming, "you should be overcome by lust at this point. At least, my husband always was."
Stephen looked almost goggle-eyed. "He was? I mean, of course he was!"
At this point one obviously laughed or cried. Helene chose to laugh. There was only so much humiliation a woman could take in one evening. She folded her hands over the top of the sheet and grinned at Stephen, quite as if they were at a tea party. "I suppose we could have an old-fashioned game of 'you show me, I show you,'" she said. "Or we could simply give it up."
His eyes flew to hers, and the relief in them was palpable.
"I gather that I need more practice before I can induce a man to actually stay in my bed," Helene said. "I have to tell you, Stephen, that it is quite a personal triumph that I lured you into my bedchamber at all."
He reached over and pulled the sheet up above her breasts, tucking it about her, quite as if he were tucking a child in at night. Then he said, "Now you'll have to explain to me, Helene, exactly why you lured me up here. After all, your husband is not a member of this house party."
Helene swallowed. But he deserved a real explanation. "I want a divorce. But when I asked my husband whether we could simply manufacture the evidence of my adultery, he laughed and said no one in the world would believe that I was adulterous. It has to be a woman's adultery that dissolves a marriage, you know. It's grotesquely unfair, but the letter of the law."
"I agree with you," Stephen said, nodding. "Especially in cases such as yours, there ought to be other provisions. And I'm sure there will be changes to the law, in time. So..."
"So I thought perhaps you and I...we...might..." Helene trailed off and then stiffened her backbone. For goodness' sake! She was half naked in bed with the man; she might as well be frank. "I like you very much," she said, looking into his eyes. "I thought perhaps we might have an affair, but I see now that I was mistaken. There's a great deal I don't understand about bedroom matters."
Stephen pulled her snug against his side again. "There's time."
Helene couldn't help grinning. Here she was in bed, half naked, and snuggled up to a naked man! If only Esme could see her now! Or Rees, for that matter! "It was a lowly impulse," she said, feeling more generous now that the acute sense of humiliation was gone. "I just wanted revenge. Rees laughed when I asked him for the divorce. He says I'm frigid, and no man would ever want me." Her tone had a bitterness that she couldn't hide.
Stephen's arm tightened. "That's cruel nonsense," he said firmly. They sat for a moment, Helene tucked against Stephen's shoulder while he thought about beating Rees Godwin into small pieces.
"Are you absolutely certain that it wouldn't work between us?" she asked.
Stephen looked down at her. "Are you trembling with desire because my arm is around you? Are you secretly wishing that I would push down your sheet and take your breast—"
"No! No, I'm not," she said hastily, tucking the sheet more firmly under her arm. "All right. I accept that it won't work between the two of us. It's just such a shame, because you are quite perfect, and I'm not sure I have enough...enough bravery to go through this again."
"Ah, but if you truly desired the man in question, it wouldn't take that much bravery."
Helene didn't agree at all, but she bit her tongue.
"It seems to me," Stephen said slowly, "that you're not quite certain that you wish for the affair itself, Helene. You are more interested in the appearance of an affair."
"True. At the heart I'm terribly prudish about marriage. I am married. Or perhaps," she added, rather sadly, "I'm just prudish. That's what Rees would say."
"If only your husband could see you now," Stephen said, a mischievous glint in his eye.
"Yes, wouldn't that be wonderful? Because I do like you more every moment."
"The feeling is entirely mutual." He gave her a little squeeze.
"And there's no one else at this party whom I could even consider inviting to my bedchamber," she continued. "There's no hope for it. I shall have to wait until I can return to London, and that won't be for quite a while. I just wish that Rees knew where I am, right now!"
"Invite him," Stephen said, a wicked lilt in his voice.
"Invite him? Invite him where?"
"Here. Invite him to the house. We can make certain that he sees you in a compromising situation."
Helene gasped. "With you?"
"Exactly."
She started to giggle. "It would never work."
"I don't see why not. I've never met your husband. But I don't like what you've told me about him. So why not fashion a comeuppance for the man?"
"It would be wonderful," Helene breathed, imagining it. All the revenge without having to go through with the unpleasant bits. Could there be anything better?
"Unless there's a chance he might grow violent," Stephen said, thinking of various nasty stories he'd read over the years about irate husbands.
"Rees wouldn't bother. Truly. He lives with an opera singer, you know."
"I have heard that," Stephen admitted.
Helene clutched his arm. "Would you do it, Stephen, really? Would you do it for me? I would be so grateful; I can't even tell you how much."
He looked down at her and laughed, and the joy of it came right from the heart. "Do you know what I do with my days? I try to win votes. I count votes. I bargain for votes. I beg for votes."
"That is very important work."
"It doesn't feel important. This feels more important. So, summon the philandering husband!" Stephen said magisterially. "I always wanted to play a part in a romantic comedy. Sheridan, Congreve—here I come!"
Helene broke into laughter and he joined her, two proper, half-clothed members of the English peerage.
##
## Because the Library is
Not Yet Emptied of Books
Bea was creeping down the corridor toward the main stairs and library when the laugh rumbled through a door just at her shoulder. She would know that laughter anywhere. There wasn't another man in London with such a lovely, deep voice as Mr. Puritan Fairfax-Lacy himself.
It wasn't that she didn't want Helene and Mr. Fairfax-Lacy to find pleasure in each other. Of course she did. Why, she was instrumental in bringing them together, wasn't she? She headed directly down the stairs, trying to erase all thoughts of what might have brought on the Puritan's delighted peal of laughter. What had Helene done? Did she know as much as Bea did about pleasuring a man? It seemed unlikely.
Probably that was the sort of laughter shared by people who don't know everything, who discover new pleasures together. She couldn't remember laughing while in bed with a man. She mentally revisited the three occasions in question. There had been a good deal of panting and general carrying on...but laughter? No.
The thought made her a little sick, so she walked downstairs even faster. Once in the library she wandered around the shelves, holding her candle up high so she could read the titles. But it was no use.
The idea of returning to her cold bed was miserable. The idea of pretending to read one of these foolish books was enough to make a woman deranged. Instead she plunked down on a chilly little settee, drew up her feet under her night rail (a delicious, frothy concoction of Belgian lace that was far more beautiful than warm) and tried to think where things had gone wrong in her life.
The world would have said, without hesitation, that it was the moment when Lady Ditcher walked into a drawing room and was paralyzed with horror to see one of the Duke of Wintersall's daughters prostrate in the arms of a gentleman. Not that her arms were a problem, Bea thought moodily to herself. It was the sight of long white thighs and violet silk stockings. That's what had done the trick for Lady Ditcher.
But the truth was that the trouble started long before. Back when she was fifteen and fell in love with the head footman. Never mind the fact that Ned the Footman must have been thirty. She adored him. Alas, she wasn't very subtle about it. Her entire family knew the truth within a day or so. Finally her father sent the overly handsome footman to one of his distant country estates. He didn't really get angry, though, until he discovered she had been writing Ned the Footman letters, one a day, passionate, long letters...
That's where she went wrong. With Ned the Footman.
Because Ned rejected her. She offered herself to him, all budding girlhood and thrilled with love, and he said no. And it wasn't to preserve his position, either. Ned the Footman wasn't interested. She could read it on his face. After her father transferred him to the country, he never answered a single letter. With the wisdom of time, she realized Ned may not have been able to read, but honesty compelled her to admit that he wouldn't have wanted to write back. He thought she was tiresome.
Ever since then, she seemed to be chasing one Ned after another...except all the Neds she found were endlessly willing, and therefore endlessly tiresome.
She curled up her toes and rocked back and forth a little. She was certain that she wasn't merely a lusty trollop, as her father characterized her. She truly did want all those things other women wanted: a husband, a baby, two babies, love.... Real love, not the kind based on breasts propped up by cotton pads.
You've gone about the wrong way of finding that sort of love, she thought sourly. And it was too late now. It wasn't as if she could let her hair down and put away her rouge, and swear to never utter another profanity. She liked being herself; she truly did. It was just...it was just that being herself was rather lonely sometimes.
"Oh damn it all," she said out loud, rubbing her nose hard to stop the tears from coming. "Damn it all! And damn Ned too!"
A slight noise made her look up, and there in the doorway was Mr. Laughing Lover himself, looking tall and broad-shouldered and altogether aristocratic. He could never be a footman. Not even Ned had looked at her with that distant disapproval, that sort of well-bred dismay. Ofcourse, the man was sated by his midnight excursion. That alone would make him invulnerable to her charms, such as they were.
"Ned?" he said, eyebrow raised. "I gather the gentleman has not joined us but remains in your thoughts?"
"Precisely," she said, putting her chin on her knees and pretending very hard that she didn't mind that he had been with Helene. "And how are you, Mr. Fairfax-Lacy? Unable to sleep?"
"Something of the sort," he said, looking as if butter wouldn't melt in his mouth.
Why on earth was he in the library instead of snuggling beside the skinny body of his mistress? Uncharitable thought, she reminded herself. You're the one with a padded bosom. The reminder made her irritable.
"So why are you in the library?" she asked. "I thought you had other fish to fry."
"A vulgar phrase," he said, wandering forward and turning the wick on the Argand lamp. "In fact, I came to see if I could find the book of poetry you gave to Lady Godwin."
"Why, are you having a private reading?" she asked silkily.
The minx was nestled on the settee, little pointed chin resting on her knees. She was curled up like a child, and with her hair down her back, she should have looked like a schoolgirl. It must be the dimple that gave her such a knowing look. That and the way her lips curled up, as if they were inviting kisses.
He walked over to her. "Why on earth did you give Lady Godwin that particular poem to read?"
"Didn't you like it?"
Close up, she didn't look like a child. Her hair was the color of burning coals. It tumbled down her back, looking as delectable and warm as the rest of her. "You've washed your face," he said. Ignoring the danger signals sent by his rational mind, he crouched down before her so their eyes were on the same level. "Look at that," he said mockingly. "I do believe that your eyebrows are as yellow as a daisy."
"Pinkish, actually," she said. "I absolutely loathe them. And in case you're planning to comment on it, my eyelashes are precisely the same color."
"It is rather odd. Why aren't they the same red as your hair?"
She hugged her knees tighter and wrinkled her nose. "Who knows? One of my sisters has red hair, and she has lovely eyelashes. But mine fade into my skin unless I color them."
"They're very long, though." He just stopped himself from touching them.
"And they curl. I should be grateful that I have material to work on. They look quite acceptable after I blacken them. Naturally, I never allow a man to see me in this condition."
"And what am I?" Stephen said. She was actually far more seductive like this than when she was being seductive, if only she knew. She smelled like lemons rather than a thick French perfume. Her lips were a gloriously pale pink, the color of posies in a spring garden.
"I suppose you are a man. But sated men have never interested me."
"What an extraordinarily rude person you are. And how unaccountably vulgar."
"I can't think why it surprises you so much," she said, seemingly unmoved by his criticism. "Surely you must have talked once or twice in your life to a woman who wasn't as respectable as yourself."
"Actually, brothels have never interested me. I have found ready companionship in other places."
Bea shrugged. He wasn't the first to imply that she belonged in such an establishment, although to her mind, that signaled his stupidity. There was a vast difference between taking occasional pleasure in a man's company and doing the same thing for money, and if he couldn't see the difference, he was as stupid as the rest.
"Where did you find that poem, anyway?" he said, getting up and walking toward the bookshelves.
"I brought it with me."
He swung around. "You travel about the country with a collection of libidinous poetry?"
"I have only just discovered Stephen Barnfield, and I like his poetry a great deal. The piece Helene read is by far his most sensuous. And it worked, didn't it? Called you to her side like a barnyard dog!"
"Not just a dog, but a barnyard dog?" he said, wandering back and sitting down next to her. His rational mind told him to stop acting like the said barnyard dog. And every cell in his body was howling to move closer to her.
"If you'd like to read the poetry yourself, I believe Lady Godwin left it on the table."
He grabbed the book and then returned to the settee and sat down again. He didn't want to look at Bea anymore. Her thick gold eyelashes were catching the firelight. "I shall borrow it, if I may," he said, leafing through the pages.
"I was surprised to find that you knew Spenser's poetry, for all you chose an unpleasantly vituperative bit to read aloud. You should have known that Lady Arabella would take it amiss if you read aloud poetry criticizing women for growing old."
"I wasn't criticizing age," he said, reaching out despite himself and picking up a lock of her hair. It was silky smooth and wrapped around his finger. "That poetry was directed at you and all your face painting."
"I gathered that." Bea felt as if little tendrils of fire were tugging at her legs, tugging at her arms, telling her to fall into his arms. She lay her head sideways on her knees and looked at him. He had dropped her hair and was reading the book of poetry. Who would have thought he liked poetry? He looked such a perfect English gentleman, with that strong jaw and elegant cheek. Even after (presumably) shaking the sheets with Helene, he was as irreproachably neat and well dressed as ever. Only the fact that he wore no cravat betrayed his earlier activities.
"Where's your cravat?" she asked, cursing her own directness. She didn't want to know the answer, so why ask?
"I've found the poem." His eyebrows rose as he read. "Goodness, Bea, you are a surprising young woman."
"Only in my better moments. So where is your cravat?" He'd undoubtedly left it on the floor of Lady Godwin's room, lost as he'd wrenched it from his throat in his urgency to leap into bed with the chaste—or not so chaste—countess. "Did you leave it on Helene's floor?" she asked, jealousy flooding her veins.
"No, I didn't," he said, looking at her over the book. The somber look in his eyes, that disapproval again, told her she was being vulgar.
She threw him a smoldering invitation, just to make him angry. It worked.
"I hate it when you practice on me," he observed, his eyes snapping. "You don't really want me, Bea, so don't make pretenses."
She threw him another look, and if he weren't so stupid, he would know—he would see that it was real. That the shimmer of pure desire racing through her veins was stronger than she'd ever felt.
But he didn't see it, of course. He merely frowned again and then reached into his pocket and pulled out his cravat.
"Oh, there it is," she said, rather foolishly.
"A gentleman is never without a cravat," he said, moving suddenly toward her.
Bea raised her head, thinking he was finally, finally going to kiss her. A moment later he had tied the cravat neatly over her eyes. She felt him draw away and then heard the crackle of pages.
"Let me know when you wish to return to your chambers," he said politely, although any idiot could hear the amusement in his voice. "I think we'll both be more comfortable this way."
For a moment Bea sat in stunned silence. She didn't even move. She still had her arms around her knees. But she couldn't see a thing. Her senses burst into life. His leg was a mere few inches from hers, and her memory painted it exactly, since she couldn't see it: the way his muscles pulled the fine wool of his trousers tight when he sat. The way his shirt tucked into those trousers without the slightest plumpness. Even—and no good woman would have noticed this, obviously—the rounded bulge between his legs that promised pleasure.
Bea wiggled a little. It was worse than when she could see him. Sensation prickled along every vein, pooled between her legs. Perhaps if she leaned back against the couch and pretended to stretch? Her night rail was fashioned by Parisian exiles and made of the finest lace. Perhaps it could do what she could not seem to do: seduce him. At least make him feel a portion of the yearning desire she felt.
But she'd tried all of that before. It was a little embarrassing to realize how much she had tried to create a spark of lust in his eyes. She had rubbed against him like a cat, leaned forward and showed her cleavage so often he must worry she had a backache. None of it had created the slightest spark of interest in the man. Only when she'd been rained on and covered with mud had he kissed her.
Bea chewed her lip. Maybe she should just return to her room. Except honesty told her that she would no more leave his presence than she would stop breathing. Not when he might kiss her, when he might change his mind, when he might—
Oh, please let the poem excite him, since I don't seem to be able to, she prayed to any heathen goddess who happened to be listening. Please let it work for me as well as it worked for Helene.
"If it be sin to love a sweet-faced Boy," he read.
His voice was so dark, so chocolate deep that it sent shivers down her spine. The poem certainly worked on her. Bea felt him lean toward her. She didn't let herself move.
"Whose amber locks trussed up in golden trammels, dangle down his lovely cheeks..."
A wild shiver ran down Bea's back as his hand rested on her head.
"Your hair is darker than amber, Lady Beatrix. Your hair is more the color of—"
"Wine?" Bea said with a nervous giggle. She felt utterly unbalanced by her inability to see. She was used to directing the conversation.
"Not rust. Beetroot, perhaps?"
"How very poetic of you. I prefer comparisons to red roses or flame."
"Beetroot has this precise blending of deep red and an almost orange undertone."
"Marvelous. Bea the Beetroot."
"Mellifluous," he agreed. "Of course, it might look less beetlike if you had pearls and flowers entangled in it like the boy in this poem. After all, his amber locks are—let me see if I've got this right—enameled with pearl and flowers."
"Flowers are not in style," Bea said dismissively. "A feather, perhaps. Pearls are so antiquated."
"If it be sin to love a lovely Lady," he read, "Oh then sin I, for whom my soul is sad."
Bea almost couldn't breath. She wanted to drink that voice; she wanted that voice to drink her. She wanted that voice to tell her—"You've changed the poem," she said rather shakily. "The line reads, 'If it be sin to love a lovely lad.'"
"There's no lad in my life whom I love," Stephen said. He couldn't not touch her for another moment. He closed the book and put it to the side. She was still curled like a kitten, strangely defenseless without those flashing eyes that seemed to send invitations in every direction. He rather missed them.
Dimly he noticed that his fingers were trembling as they reached toward her. He lifted her head and just rubbed his lips across hers. She sighed—could it be that she wanted his kiss as he longed to give it? Her arms slipped around his neck.
But he didn't like the fact that those magnificent eyes of hers were covered. So what if they sent him a message they'd sent a hundred other men? He pulled the cravat off her head in one swift movement and then, before she could even open her eyes, he cupped that delicate face in his hands and kissed her again, hard this time, demanding a true response: one that she hadn't given another man.
Her lips didn't taste of the worldly smiles that so often sat there. They tasted sweet and wild, and they opened to him with a gasp of pleasure. He invaded her mouth, only meaning to tell her that he felt desire when he wanted to, not when she willed it.
But she tasted like lemons, sweet and tart, and her mouth met his with a gladness that couldn't be feigned. Nor could the shiver in her body when he pulled her against him, nor could the tightness with which she wound her arms around his neck. Oh, she was—she was glorious, every soft, yielding inch of her. He longed to lick her whole body, to see if she tasted as tart and sweet behind her knees, and on her belly, and between her legs...aye, there too. Because she would let him: he knew that without a doubt. All the respectable women he'd slept with, wives and widows, none of them had even dreamed of such a thing.
He had never even tried, knowing one can only take such liberties with a courtesan, a woman paid to accept the indignities of sensual activity. But Bea...sweet, unmarried Bea...
God, what was he doing?
He tore his mouth away and she leaned back toward him, her mouth bee-stung and her eyes closed. He went back for one last taste, just licked her mouth, except she opened to him and then her lips drew his tongue into her mouth. His hands turned to steel on her shoulders, even as his lower body involuntarily jerked toward her. Lust exploded in his loins at the precise moment that rage turned his vision dark.
"Where the devil did you learn that trick?" he said, pulling back.
She opened her eyes, and for a moment Stephen was bewildered: her eyes were so velvety soft, innocent seeming, dazzled looking. She must have looked dazzled for many a man. Even as he watched her eyes focus. But she didn't lose her languorous, desirous look.
"Do this?" she said softly, leaning forward. She almost took him by surprise, but he jerked back.
Bea sighed. Obviously the rake had turned back into a Puritan. She might as well infuriate him since he clearly wasn't planning on further kisses—or anything else, for that matter. "I believe that was Billy Laslett," she said. Now she really wanted to return to her bedchamber. How excruciatingly embarrassing this was. "Lord Laslett now, since his father died a few months ago."
"Laslett taught you that kiss and didn't marry you?" Stephen asked, feeling as if he'd been pole-axed.
"Oh, he asked," Bea said, standing up. Her knees were still weak. "He asked and asked, if that makes you feel any better."
Stephen felt sour and enraged. He stood up and towered over Bea. "At least you remember his name," he said with deliberate crudity.
Bea rolled her eyes. "There haven't been that many, Mr. Fairfax-Lacy. I'm only twenty-three. Ask me again when I'm fifty. But may I say that I am quite impressed with your stamina? After all, it's not every man in his forties who could frolick so gaily with a countess and then prepare such an...impressive welcome for me." She let her eyes drift to his crotch.
Then she smiled gently at the outrage in his face and walked from the room, leaving Stephen Fairfax-Lacy alone in the library with a libidinous book of poetry.
And an impressive welcome.
##
## The Imprint of a Man's Skin
Esme was having second thoughts. Her heart was still pounding from the pure terror of seeing Jeannie move toward the windows. She wrapped the towel tighter around her shoulders. "This is a foolish idea. The babe is due any day now."
"Oh, I know that," he said with some amusement. "I can count as well as any, you know. Last July, when you and I met in Lady Troubridge's drawing room, is almost precisely eight and a half months ago."
"This is Miles's baby," she said, fully aware of just how obstinate she sounded. But it was terribly important to her that Miles have the child he wanted so much.
"Surely it has occurred to you that your certitude that this child is Miles's may be an error? After all, you had not yet reconciled with your husband when you and I enjoyed each other's company."
"Miles and I reconciled the very next night," Esme said hastily. But she knew it was of no use.
"This may well be my babe. Mine and yours. In fact, mathematically speaking, I believe that Miles and I are in a dead heat, given that we each had the pleasure of one night, and one night only."
"It is surely Miles's. He wanted a child so much!"
"Unfortunately, wishes have never influenced paternity in the past."
Esme had to acknowledge the truth of that statement. "Do you remember when I told you about my mother's letter? The one in which she told me that she didn't feel she could attend my confinement?"
"Of course." He began unbuttoning his shirt.
"She added a postscript saying that she hoped I knew who the father was. That was the worst of it. Because I don't know. If only we hadn't slept together, I could have written my mother an indignant letter, and perhaps she would have attended my confinement. Perhaps she would be here at this very moment."
"I trust not in this particular room," Sebastian said, pulling her into his arms. He smelled gloriously male and windswept. "I wish your mother felt differently."
His hand was so comforting on her back. It was no wonder that Esme kept blurting out every humiliating secret she had.
"I love you, do you know that?" he said.
She chose her words carefully. "I believe that you think you love me, Sebastian. But I know the guilt you feel because of Miles's death as well. There is no need to compensate for what happened, truly."
"Compensation has nothing to do with what I feel for you."
"How could it not?" she asked, looking into his clear blue eyes.
"Because when I fell in love with you, Miles was alive and sporting with his mistress," Sebastian said, watching her just as steadily. "I loved you all the time I was engaged to your friend Gina. I watched you dance; I watched you flirt; I watched you think about an affair with that abominable idiot, Bernie Burdett."
Esme turned away, wrapping her towel tighter around her shoulders. She would cry in a moment, and she needed to keep her head.
"Esme," he said.
She sat down heavily in a chair, heedless of the fact that her damp towel was likely to spot the pale silk. "I know you think you love me. But there's lust—and then love. And I don't think you know the difference."
He watched her for a second. She bit back tears. Why couldn't he just see that it was impossible? She couldn't marry the man who had caused her husband's death. The scandal would never die, and she couldn't revisit that scandal on her child.
He walked over and picked her up.
"I must be straining all your gardener's muscles," she whispered, turning her face against his shirt.
"Not you," he said, carrying her to the bed. "I think we've talked enough, Esme mine. The night is long."
She felt the breath catch in her chest. But Sebastian was as methodical in seduction as he was in every other part of his life. He turned down the wicks and poked the fire before he returned to the bed. She watched the long line of his thigh and tried to remember if his legs had been so muscled last summer when he'd been a mere earl. Before he was a gardener. She didn't think so. His legs had been muscled, but not with the swelling sense of power they had now.
"Oh, Sebastian," she said, and the aching desire was there in her voice for anyone to hear.
He strode to the other side of the room to snuff yet another light. Firelight danced on his back. He must have done some work without a shirt because his skin was golden to his waist, and then it turned a dark honey cream over his buttocks. There were two dimples just there...Esme found herself moving her legs restlessly, and she almost blushed. He had snuffed the candle and seemed to be inspecting the wick.
Finally he turned around. Esme's mouth went dry. He stood there with just a whisper of a smile on his face. He knew what he did to her. Firelight flickered over his thighs, over large hands, over golden skin, over...
And still he smiled, that wicked, slow smile that promised everything.
"Was there something you wished?" he asked, mischief dancing in his eyes.
Esme felt nothing more than liquid invitation between her thighs. How had she survived even a single night without this—without him? How could she survive another moment? "You are beautiful," she said, and the hoarseness in her own voice surprised her into silence. He sauntered over toward the bed, looking like Adonis and Jove all rolled into one: golden boy and arrogant king, sensual devil and English aristocrat.
It was no time to worry about what she looked like. If a woman is lucky enough to lure such a man into her bedchamber, it would be a true waste to let an enormous expanse of belly get in the way. So she sat up and reached for him. When he stood just before her, she wrapped her legs around his so he couldn't escape.
"I have you trapped," she said, smiling a little.
"And what will you do with me?" he said, and he wasn't smiling at all.
She reached up and ran her fingers over his nipples, felt the tiny tremor that rippled through his body, right down through his legs. Her fingers drifted south, touching muscled ridges, skin kissed by the sun, drifted around his bottom and pulled him even closer. He seemed to be holding his breath, silenced.
Whereas she...she felt greedy and loving all at once. She wanted him never to forget her. In fact, had Esme admitted it, the thoughts she was having were hardly generous. Distilled, they ran like this: she couldn't marry him herself, but she could make it very, very hard for him to marry someone else.
And besides, she wanted him, every sun-ripened inch of him. There was no better place to start than the hard length of him, straining toward her even as he stood still. She bent forward and he said something, strangled in his throat, drowned by her warm mouth. She pulled him closer, hands on his muscled rump, and he arched, not pushing forward, simply a body exalting in pleasure.
The pleasure she was giving him. A shiver of delight pulsed down Esme's body, and she leaned even closer, torturing him, loving him. He arched his back again and groaned, a deep pulse of need that made Esme's heart pound.
But then he reached down and pushed her back onto the bed. She resisted for a second and then melted under the pressure of his powerful hands. She felt like the merest wisp of a girl, lying back on the bed with Sebastian towering over her. "I can't wait," she said, her voice revealingly hoarse. But there was no room for embarrassment between them.
Powerful hands pulled her to the edge of the bed. He leaned over her, cupping her face in his hands, kissing her until she was senseless, delirious, but not so lost that she didn't feel him there.
Asking.
"You do remember," he said some time later, and now that wicked grin was back, "that I'm a virgin, don't you?"
She couldn't help laughing.
"Not anymore."
Sebastian's voice was an amused, dark whisper against her skin. "Do you remember the night when you took my virginity, Esme?" His hand was on her womb. "This child might well be mine," he said into her hair.
"Or Miles's," she said, but the shrillness of her tone was wearisome even to her. She closed her eyes and leaned back against his shoulder, letting him continue his gentle caress.
"It makes me very happy." She could hear the joy in his voice. "The very thought of the child."
"And what if we married, and the child was Miles's? You would never know."
"I would love him or her as my own," he said. "I would never do otherwise, Esme."
"I know," she said, humbled by the look on his face.
"If you allow me to have a place in this child's life," he said, cupping her belly in his warm hand, "propriety will not be foremost on my mind. I'm not criticizing Miles's wish that you become a respectable woman. But I don't think it's the most important aspect of raising a child either." She couldn't see his face because a lock of hair had fallen over his eyes.
He leaned forward and dropped butterfly kisses on her stomach. "You have to understand that I don't want to imitate my father, for all you wish to imitate your mother. He was quite respectable. I have trouble remembering his first name."
She reached out and pushed the hair back so she could see his eyes.
"You'll be a wonderful mother, Esme."
She bit her lip hard. It was that or cry, and she had firmly resolved not to cry. "I worry," she said, and her voice cracked.
"Nothing to worry about. The child is lucky to have you."
"I couldn't...I didn't..." The tears were coming anyway. They blinded her.
"Why on earth are you worried, sweetheart?"
"Benjamin," she said, "just my brother Benjamin. You do remember that he died as a baby? I'm afraid. I'm...I'm just afraid."
"Of course I remember that you told me of Benjamin." He folded her in his arms then, and rocked her back and forth. "Nothing will happen to your baby. I promise you."
They fell asleep together, she curled in his arms as if he could protect her from all the evils that life could offer. When she woke, hours later, Sebastian was still holding her against his chest. The fire had burned out, and the room had taken on a pearly, luminous light. He was sleeping, lashes thick against his cheeks. His hair gleamed as if it were gilt. All her fear seemed to have been burned away.
"Sebastian," she said, and his eyes opened immediately. They looked black in this light. She licked her lips and tasted salt tears and desire.
"How are you?" His voice was deep with sleep.
It set off a quiver between her thighs. "I don't think that I have the imprint of your body on mine yet," she whispered.
"Oh?" He raised one eyebrow. How had she ever thought he was a priggish Holy Willy? She must have been blind.
"Not at all." She shook her head sadly. "I'm sorry. All your efforts don't seem to have succeeded."
"You'll have to excuse my failures." His voice purred with seductive power. "I am practically a virgin." One hand brushed over her nipple, returned, returned again.
A strangled little sigh came from Esme's throat.
"I need practice." His voice was dark, gutteral, possessive. A shiver of ecstasy jolted Esme's spine. "You will have to give me another chance."
She couldn't answer. His lips had replaced his hands, and his hands had drifted lower. He was fierce and possessive, and he left no space for words. All Esme could do was try to stop the broken moans that came from her chest. But she had his smooth skin to put her mouth to, all those muscles to shape with her tongue.
It was around an hour later that he asked her a question. "Did you ever read Romeo and Juliet?"
"Well, of course. I only read it once. She was daft to kill herself for the sake of that lovelorn boy, that I remember."
"My hardheaded Esme," he said, dropping a kiss on her nose. "That's the sound of the lark outside your bedchamber window. I must leave soon."
The light filtering in her window was a watery yellow, filtered through spring leaves. Esme didn't want to acknowledge what those greenish ribbons of light meant. "Would you massage my back?" she asked, ignoring the whole exchange.
Sebastian pushed his thumbs into the very base of Esme's spine. She seemed to have forgotten that mornings always come. That she had told him to leave at the very first light. The sun was pouring under the curtains, and her maid would arrive any moment. She moaned like a woman in ecstasy. Her gorgeous hip rose from her waist like a creamy wave.
"My back hurts more than normally this morning," Esme said in a fretful voice. "You don't suppose we did it any injury, do you?"
Sebastian rolled her over on her back and grinned down at the huge mound of belly that reared between them. "Not the slightest," he said, rubbing a little hello to the babe. His babe.
"I suppose you should make your way out of here," Esme said, eyeing him. She had a distinctly jaundiced and irritable air. "Where are you planning to travel, anyway?"
"I've always enjoyed France," he said rather evasively.
If he didn't wish to give her his direction, that was quite all right. "Well, drink some champagne for me," Esme managed.
"Don't you wish to give me a weeping farewell?"
"I'm not up to hysterical farewells at the best of times," Esme snapped. She struggled up on her elbows and then Sebastian helped her to her feet. "You'll have to leave, because Jeannie will appear soon."
Sebastian smiled to himself. Esme was protecting that vulnerability of hers, the heart she hid amidst all her seductions and flirtations. The heart she had never given to anyone—but him, he thought. Although she didn't seem to know it.
He bundled a dressing gown around her and pushed her glorious tumbling locks back over her shoulders. "You're beautiful in the morning," he said, cupping her face in his hands.
"I am not," Esme said, pulling away. "I have a perfectly foul taste in my mouth and my back hurts like the devil. I am not in the mood for sentiment, Bonnington, and so I'll thank you to find your way out before the household awakes."
Sebastian obediently pulled on his trousers and shirt as she watched. He was buttoning the last button on his placket when he realized that tears were sliding down her face. "Sweetheart." He pulled her into his arms. "Don't cry."
"I can't help it," Esme said, sobbing. "I know you have to go—you have to go!—but I'm so lonely without you. I'm a fool, a weak, silly fool. I'm just...I'm just—"
"I love you, Esme," he said, finally. "If you need me, just ask. I'll always come to you."
"I need you to leave! I can't have an earl hiding as a gardener on my estate. Everyone will know in a matter of minutes, and my reputation will be blackened, more than it is already."
He handed her a handkerchief.
"Thank you. And I wouldn't even mind my reputation being ruined," she wailed, "except for the baby. But you know all this, Sebastian, you know it, and I know it, and—and there's nothing to be done. So please, go."
He didn't move.
"Go!" She looked at him, face shiny with tears, eyes red, a handkerchief balled in her hands, and Sebastian knew that he would never love anyone so much as he loved her.
He leaned forward and kissed her quite simply on the lips. "Good-bye," he said. Then he put his hands over her belly. "And good-bye to you, little one."
"Oh God, I can't bear it!" Esme said, her breath caught on a sob. "You have to leave, or I'll lose my resolve. Please go."
He slipped through the door and looked to the right and left. He'd entered Esme's chamber by means of one of the ladders being used to fix the roof; he had never actually been in the upper reaches of her house before.
Suddenly there was a polite cough, almost at his shoulder. "If I might help you, my lord?"
He swung about to find Esme's butler bowing before him. "Slope, isn't it?" Sebastian asked.
"Just so, my lord."
"I know your mistress trusts you implicitly. I trust her loyalty is not misplaced."
"Absolutely not," Slope replied, with just a tinge of offense in his voice.
"A workman, Rogers, is stealing slate and selling it in the village," Sebastian told him. "You might want to tell the foreman such. And I am leaving Lady Rawlings's employ, so you'll have to find a new gardener."
Slope bowed again. "I am most grateful for the information, my lord. May I direct you toward the side door, under the circumstances?"
"Thank you," Sebastian said, walking into the light of dawn.
##
## The Unexpected Pleasure of Your Company
Esme placed compresses on her eyes for a full hour, but it didn't reduce the swelling in her eyes. When Arabella entered her room, she advised a cucumber mask, but that didn't help, either. Esme suspected there wasn't much one could do to plaster over a broken heart. I sent him away because I had to, she told herself fiercely. The only problem was that he'd actually left. That was the worst of it: the petty, mean, screaming little voice in the back of her head that kept saying, He wouldn't have left if he really loved you! If he really—
And then the tears would well up again, because why should Sebastian be any different from the other men she'd known? Miles never really loved her. Sebastian said he loved her, and perhaps he did. But it felt like a stab in the heart. If he loved her, really loved her, he wouldn't have left, no matter how many times she commanded. Didn't he know how many women died in childbirth? Didn't he care?
The ache in her chest answered that. He did care. He just didn't care as much as she wanted him to care. You chased away Miles by creating a scandal, Esme thought dully. And then you chased away Sebastian in order to avoid a scandal. But it was all the same, really. If either man had truly loved her, he wouldn't have left. He would have fought for her. But Miles had just smiled politely and slipped away to other pursuits, other bedchambers.... Sebastian smiled painfully and slipped away to the Continent to protect her reputation. It was exactly the same situation. Apparently she was the kind of woman whom men found easy to leave.
The tears welled up so fast and furious that Esme felt she would never stop crying. But she did, finally. More cucumber compresses and an hour later, she even thought drearily about going downstairs. The only reason she would consider it was to talk to Helene. She was faintly curious about the outcome of the previous evening.
Any questions were answered when she entered the sitting room. Helene looked happier than Esme had ever seen her, sitting across from Stephen Fairfax-Lacy and playing chess. An utterly suitable game for such an intelligent couple. She herself didn't even know how to play.
"Hello," she said, standing at Helene's shoulder. Stephen immediately jumped to his feet and gave Esme his chair. She sank gratefully into it as Helene waved Stephen away with a smile. He bowed and pressed a kiss into her palm before strolling off. They certainly seemed to have got themselves on intimate terms in a hurry. Well, a shared bed could do that.
"Esme!" Helene said with a huge smile, "would you mind terribly if we invited Rees to make a brief visit?"
Esme blinked. Had she heard correctly? "Rees? Rees, your husband, Rees?"
Helene laughed. "Of course, that Rees."
"Naturally you may invite anyone you wish," Esme said. She looked around a bit sourly. Winnamore and Arabella were practicing a duet on the harpsichord. Bea seemed, rather surprisingly, to be embroidering something. "No one else is taking notice of the fact that I am in confinement, so why should you?"
For her part, Helene suddenly brought Esme's face into focus. For goodness' sake, what was wrong with Esme? She looked utterly haggard and was obviously out of sorts. "I am being utterly thoughtless," she said repentantly. "Of course I won't invite Rees to the house. Esme, what's the matter?"
Esme ground her teeth. Her nerves were on the edge of total distraction. "I didn't say you shouldn't invite him!" she snapped. "Clearly there is a house party occurring, so why not invite one more? At least it would go toward evening out our numbers, and that will make Arabella happy."
Helene hesitated. "I don't know if he will come."
"More to the point, why on earth would you wish him to? I can assure you that these friendships are better conducted away from one's husband." Lord knows, she was an expert on that subject.
"Not in my case," Helene whispered. "Esme, we are going to flaunt ourselves in front of Rees!"
"Flaunt yourself?" Esme repeated. Her back felt as if a carriage had driven over it. That was Sebastian's fault. Last night had obviously been far too energetic for a woman in her condition. Perhaps she would be permanently crippled.
"He's agreed to it," Helene whispered.
"Agreed to what?" Esme asked.
"Flaunting!"
"Oh for God's sake," Esme snapped. "What on earth are you talking about?"
"Stephen and I are going to demonstrate to Rees that I am not undesirable and frigid," Helene said. There was a high, startled color in her cheeks, but she looked straight at Esme.
"Rees never said that!" Esme narrowed her eyes. "That reprobate dared to say such a thing to you!"
Helene nodded.
"It's a good thing he's not here," Esme said through clenched teeth. "I'd tear him limb from limb. Men are all the same. Lechers and knaves, all of them!"
"You seem to be in a less than excellent humor," Helene said, examining her closely. "Didn't you sleep last night, Esme? You have marked circles under your eyes. How do you feel? Is the baby on the way?"
"No. Once a day the midwife emerges from the kitchen, prods me, and announces that nature must take its course. I'm so tired of that phrase!" Esme put her head on the back of the chair. On the ceiling of the salon, overindulgent-looking gods and goddesses shaped from plaster were eating grapes dangled in the air by cupids. The goddesses were all slim. Very slim. Probably Esme would never be able to shed the extra weight she'd gained.
"What do you think of my plan?" Helene asked.
Esme blinked at her. "Plan? What plan?"
"Esme," Helene said firmly, "you are not yourself. Would you like me to accompany you to your chamber?"
Esme was trying to think about whether that would make her feel worse—indeed, whether anything could make her feel worse—when Slope appeared in the doorway.
"My lady," he said, and there was a warning note to his voice that made every head in the room turn. "The Marchioness Bonnington has arrived to pay a brief visit."
Esme straightened up as if she'd heard the last trumpet itself. She clutched Helene's arm. "It can't be!"
Helene obviously sensed danger. "What on earth is Lady Bonnington doing here? Her son is on the Continent. I certainly hope she is not planning to call you to account for his actions last summer! I shall rout her instantly, if that's the case," Helene said, bristling like a mother goose sensing danger.
All the blood was draining from Esme's head, and she felt a curious airiness in her knees. "I'm going to faint," she whispered.
But there was no time to faint. Lady Bonnington herself was standing in the doorway and surveying the room. Esme forced away her dizziness and stood up. "My lady," she said weakly, "what a pleasure to welcome you to Shantill House."
The marchioness was wearing a carriage costume of straw-colored sarsenet lined in white satin. Her gown was trimmed with black and finished with two of the French ruffs that had just come into fashion. She looked formidable and, to Esme's eyes, utterly terrifying.
"The pleasure is mutual," she said, surveying Esme from head to foot through a pair of pince-nez with an air of vigorous and personal condemnation. That seemed to be the extent of her polite conversation. "Lady Rawlings, I daresay you're within a day or so of giving birth. And yet by all appearances you are hosting a house party. How very peculiar."
"That would be my doing," Arabella drawled, drifting over. "And what a surprise to see you here, Honoratia. My goodness, how long it has been since we were in school together. And yet when I see you, the years melt away!"
"I suppose that's a compliment," Lady Bonnington replied acidly. "One can so rarely tell exactly what you mean, Arabella."
"Such a failing," Esme's aunt replied, smiling. "Whereas one always knows precisely what point you wish to make. So kind of you to clarify your every thought. Now why on earth are you here? Not that your presence isn't a remarkable pleasure."
Lady Bonnington humphed and banged her stick for emphasis. "I merely wish to speak to your niece for a moment." She gave Esme a pointed glance. "In private, if you would be so kind."
"Of course," Esme said, leading the way to the door. "If you would accompany me to the library?" She desperately wanted to remove Sebastian's mother from the vicinity of her closest friend and aunt, both of whom looked likely to burst from curiosity. It was just her luck that Arabella had been at school with Lady Bonnington. Be brave, she counseled herself, walking into the library.
"You'd best sit down," Lady Bonnington said, waving her stick at the couch. "Good lord, you look as if you're about to birth a water buffalo."
"One assumes not," Esme managed. What an extraordinarily rude old woman. She sat down without waiting for the marchioness to do so.
"I've come for my son," Lady Bonnington said, lowering herself into a chair.
"Am I to assume that you hope to find him here?" Esme said, with an air of disinterest.
"To my vast regret, yes."
"I am sorry to disappoint you. He is not here. To the best of my knowledge, he is on the Continent."
"I have information to the contrary. He told me himself that he was working in a menial capacity in your household. I don't approve, Lady Rawlings. I cannot approve. You may have led a rather imprudent life before this date, but I assure you that this current escapade will result in complete exile from the ton."
"Escapade?" Esme cried. "He took the position without my knowledge. And then he refused to leave!"
"I thought as much," Lady Bonnington said, with an odd tone of satisfaction. "I've been thinking of nothing else for the past few days. It's the blood coming out."
"Indeed? To what blood do you refer, madam?"
"My father's blood. My father was not a man to be crossed. He had a streak of obstinacy that ran a mile wide. I never thought my son had the least touch of him, but I see it now. Of course he won't leave. My father wouldn't have either."
"Be that as it may, your son is no longer in my employ," Esme pointed out.
"He tried to pull the wool over my eyes," Lady Bonnington said. Now her satisfaction was unmistakable. "Gave me fluff and such-and-such about love. I didn't raise him to pay attention to that kind of nonsense. Naturally, I paid that no mind. I stayed up half the night wondering whether he'd gone mad as a March hare due to guilt over killing your husband. But it didn't ring true."
She leaned forward, gray eyes as piercing as an eagle sighting a rabbit. "You're carrying his child, aren't you?"
Esme opened her mouth, but nothing came out.
"Aren't you!" the marchioness thundered, stamping her stick for emphasis.
Esme narrowed her eyes. "No, I am not," she said coolly.
"Poppycock," Lady Bonnington replied, and there wasn't even a speck of hesitation in her voice. "My son is no fool. The more I thought about it, the more I knew that he would never have entered the wrong bedchamber. He entered yours because you were carrying on an affair with him. Your husband was likely just paying you a courtesy visit. All the world knows they could have found him in Lady Childe's bedchamber, if they wished."
"I am carrying my husband Miles's child!"
"I've no doubt but that you wish you were. I expect we all wish you had kept that bedchamber door a bit more securely fastened."
Flaming circles crept into Esme's cheeks. "I beg your pardon?"
"The important point is that my foolish son has taken the quixotic notion that it's his child about to be born, hasn't he? And he wants to marry you on account of that."
"The child is Miles's, and his birth as a Rawlings is utterly appropriate." Esme's words bristled with rage.
"Think clearly, girl," Lady Bonnington snapped back. "Even if you managed to lure Rawlings into your bed, that child was more than likely fathered by my son. Miles Rawlings was as weak as a cricket; everyone knew that. I expect you know as well as I do that Rawlings's doctor gave him naught more than a few weeks to live. How could he have fathered a child? It takes strong red blood, you know."
"Miles's blood was quite red enough for the task," Esme retorted. "It is unfortunate that Miles died before his son or daughter was born, but this babe will not be the first nor the last posthumous child. May I remind you, Lady Bonnington, that for Miles's child to be born a Bonnington would be just as much an affront to Miles's name?"
"So you admit that the child could be my grandchild," the marchioness said with grim satisfaction.
Esme opened her mouth to reply, but Lady Bonnington thumped her stick.
"In my day, we didn't spend as much time as your generation does worrying about whose bedroom door was open. I'll put my cards on the table. I would greatly dislike my son to marry a woman with your reputation. And I want my son out of the woodshed, or wherever he is, and back in the drawing room where he belongs." She pursed her lips. "Surely I need not articulate my position further?"
Esme felt as if rage were bursting under her skin. "Your son is not here," she said, punctuating each word with deliberation. "I sent him away. This is my husband's child, and under no circumstances would I marry your son. You do realize, don't you, that Marquess Bonnington is responsible for taking my husband's life?"
"You know as well as I do that Rawlings could have popped off at any time."
But Esme could see just the faintest hesitation in her face. "If your son hadn't entered my bedchamber uninvited and grappled with Miles in the dark, my husband might still be alive," she said flatly. "I cannot marry a man under those circumstances. I could never make that man into my child's father."
"It always surprises me that the most flamboyant hussies are the most missish at the core," Lady Bonnington observed. The whole encounter didn't seem to have ruffled her sensibilities at all. "Mind you, your own mother is one of the most punctilious women in the ton." She stood up, leaning heavily on her stick. "But it's all well and good that you refuse to marry him, no matter your reasons. My only remaining concern is the parentage of that child. Don't underestimate my son, Lady Rawlings. If he feels the child is his, he'll likely take you to Gretna Green without permission or delay. It's my father's blood coming out."
"I won't marry him," Esme said. "Neither in Gretna Green nor in St. Paul's Cathedral. And may I point out again: he did leave my estate, Lady Bonnington. He shows rather less resolution than you give him credit for."
"Miss him, do you?" Lady Bonnington asked.
Esme colored. She was hideously observant, this awful old woman. "Not at all!"
"In his absence, I shall remain until the birth and ascertain whether the child is indeed a member of my family," the marchioness said. "If the child is Rawlings's, this whole debacle will be quickly forgotten by all of us."
"How on earth are you going to know that? Newborn children look remarkably similar, you know," Esme said, nettled beyond all patience. "From what I've been told, they're all equally red and wrinkled."
"If he's a Bonnington, he'll have a spangled mark at the base of his spine."
"No!" Esme gasped. Sebastian did have a small brown mark at the base of his spine.
Lady Bonnington gave a little cackle of laughter. "Don't be a fool! My son has a blemish, but it's his alone. What do you think this is, a fairy story? I'll look at the child and see whether it resembles our side of the family or your huband's. And then I will inform my son of my observations. Since you do not wish to marry him, you might hope for red hair. We have no redheads in our family."
She stumped to the door and then turned. "You're not the daughter-in-law I would have chosen, as I think I've made clear."
"The feeling is mutual," Esme said with scathing precision. "I would prefer that you put the notion from your mind immediately."
"But you're surprisingly acceptable," the marchioness said, not heeding Esme at all. "Mind you, you're related to Arabella Withers, and she and I have been at loggerheads since we were at school together. And that was a donkey's age ago, for all she tries to act as if she's got no more than thirty years to her. You may have the reputation of a coal scuttle, but you seem to have some backbone too."
Esme literally saw red. She dropped into a faint excuse for a curtsy. "If you'll forgive me, I shall retire to my room in order to recover from that compliment. I can occupy myself by praying for red hair."
The corners of Lady Bonnington's mouth curled upwards. "I am rather reminded of myself, as a matter of fact."
And that comment, as Esme later thought to herself, was the cruelest stroke of all.
##
## Playing at Billiards
There are certain times in a man's life when the only thing he wants is the company of other men. After a dinner marked by an incomprehensible female subtext and a ballet of barbed comments, Stephen wanted nothing more than an evening of hard drinking, cards, and bawdy jokes. Alas, the sole male in the household other than himself, Winnamore, rambled off to his bed directly after the meal. Still, there were two places in the house likely free from women: Stephen's own bedroom and the billiards room.
But when he pulled open the door to the billiards room, he saw a trim little bottom lean over the woolen cloth covering the table as Beatrix Lennox stretched to make a shot. Stephen decided on the spot that perhaps one female was acceptable company.
"Good evening, Mr. Fairfax-Lacy," she said, glancing over her shoulder as the shot caromed one of the red balls off two walls and directly into a pocket.
He paused as if transfixed. The oil lamps suspended above the table turned her hair into a flaming gold. She straightened with exquisite grace and deliberation, as if she were conscious of precisely what that little movement had done to his loins.
"Do you play billiards?" she asked, pulling balls from the corner pockets.
Stephen nodded. Blood seemed to be thundering through his body, every beat speaking to the sultry rhythm of her body.
She pulled the fifteen balls together. "Pyramids?"
He nodded. "Where did you learn to play?" Stephen said, walking over to pick up a cue stick and trying to appear utterly natural.
Bea shrugged. "I found one of the footmen secretly playing on our billiards table when I was no more than twelve. He would have been instantly dismissed had anyone found out. I'm afraid that I coerced him into giving me lessons."
"Do take the first turn," Stephen said, wanting her to bend over again.
She looked at him, and there was a little smile playing around her mouth that made his face burn. Then she slowly, slowly bent over the table. She was wearing an evening dress so slim that it reminded him of a chemise. It was a faint pink that should have looked awful with her hair but didn't. Around it billowed an overdress of transparent washed silk, embroidered with fleur-de-lis. All that transparent cloth emphasized the trim curves of her body every time she moved.
She broke the triangle, and balls scattered in all directions like drops of water falling on a plate. Three caromed into corner pockets.
Stephen looked at the table. "That footman must have been a remarkable player."
"Why do you say so?" she asked.
"Because you are obviously an excellent player," he said, trying to decide which ball to take down.
"The implication is that a woman can only reflect the skill of her teacher. As it happens, Ned was a mediocre player. I could beat him within four months."
"There," Stephen said, indicating the far right pocket.
He bent down and chose a ball. With casual precision he sent the ball on a voyage from one side to the other, into a collision with another ball, and finally into the pocket he designated.
"You would seem to be a much more formidable opponent than my footman," Bea observed.
He straightened. "I apologize for the inference regarding female skill. You are, as a matter of fact, the first female player I have encountered."
She shrugged, and a few of the loose red curls that tumbled down her back fell forward onto her creamy shoulder. "I might point out that it is difficult for women to demonstrate a skill that no one offers to teach them. I'll take that ball." She whipped the ball off another ball and into a corner, sending it directly into its pocket.
"Le coup sec," Stephen said, admiration leaking into his tone. He walked over to stand just next to her. Her French perfume reached him, a promise, a smoky promise of reckless sensuality.
Bea smiled at him over her shoulder, and he wanted to bend her backwards on the table. Push the balls to the side and take her there. Anywhere.
"I thought I'd take that ball," he said, pointing. His voice was a husky question.
She moved slightly to the side and then peered down at the ball. "Were you planning a low stroke?"
He nodded. He had just noticed that for all her calm, there was a pulse beating madly in her throat. In her beautiful creamy throat that he longed to lap, to kiss, to taste. "If I may," he said, and even to his ears, his voice was deeper, slower, lazy. He put a hand to her back and moved her oh so slightly to the side. Then he bent over, just as slowly and deliberately as she had. He could feel her eyes on his body, on his legs.
He straightened. "This is a difficult shot," he said, looking down at her. There was a faint, faint crimson stain in her cheeks that didn't owe its color to art. "I'll remove my jacket, if it wouldn't offend you, Lady Beatrix."
"Bea," she said. "Please call me Bea."
She watched as he wrenched the jacket from his shoulders and rolled up his sleeves. He knew he had a muscled body, a body a woman would admire, even a woman who had presumably enjoyed more than a few male bodies. The only way he could dispel the tension of hours spent in Parliament was to visit Gentleman Jackson's boxing salon. He'd never resorted to deliberately exhibiting it before, but for Bea—
He bent over again, lining up the shot with elaborate care, his hip nearly touching hers. By some miracle his fingers were steady. The shot went into a gentle reverse spin, glanced at another ball, danced by a third, spun sedately into the designated pocket.
"Your turn," he said, straightening.
"Hmmm. You do have skill."
He threw caution to the winds and let a reckless grin spread across his face. "In many areas, Lady Bea."
"Just Bea," she said. But there was a sparkle to her eyes.
She walked away from him, and it took all his strength not to pull her back to his side. "I believe that I shall take...that ball." Her lips pursed. It was torture. Would she—How experienced was she? Would she do things that ladies never did? Already she had kissed him like a wanton. Would she—Images danced through his mind, tormenting him.
She was on the opposite side of the table now. She bent down, focusing on her stick, and Stephen could see directly inside her bodice. Her gown was low, and her breasts were cradled against the hard pad lining the table, resting as they might in the palm of his hand.
Stephen made a hoarse sound in his throat, and she glanced up for a moment. Her cheeks were flaming now. But, "I shall try a jenny in the middle pocket," she said.
"You could better your grip," Stephen said, just as she was lengthening her arm to take a shot.
She straightened, and he saw amusement in her eyes. "And I gather you know a better posture?"
"A better grip," he corrected.
She looked at him through her lashes, a smile playing on her lips. "Mr. Fairfax-Lacy, naturally I would be quite pleased to learn a new grip. I'm not a woman who chooses ignorance over knowledge. But I must point out to you that you presumably have a busy night before you."
He raised his eyebrows. Something about her, about the way she looked at him, made him feel recklessly gorgeous, decadent, lustful, wild—all the things that a thoughtful man of words never felt. "I would never be too busy for you," he said. "And my name is Stephen."
She perched a rounded hip on the edge of the table. Stephen watched her, feeling another surge of animal lust. He felt in his skin, in his body, in a way he hadn't since he was a restless, lustful adolescent. He put down his stick deliberately, and then stretched, letting his chest draw the fine linen of his shirt tight against his chest muscles.
Her eyes darkened. "Alas, I would guess that the duties of a man with a brand-new mistress leave no time for lessons."
"I can be the judge of that," he said easily, coming around the table to her. He felt like a tiger, stalking his prey. She stood absolutely still and let him come up to her. So he moved to stand behind her, just as if they were about to make love, as if he were going to bend her over the billiards table. Then he brought her body into the curve of his, tucking her sweet little bottom against his groin, and leaned down.
"If you straightened your right shoulder, your aim would improve." It was quite a triumph that his voice sounded much as usual. He tucked her fingers back against her stick.
But Bea was no tender lamb, to be driven by a tiger. She slowly straightened, and his body moved with her. Then she turned within the circle of his arms, reaching back and bracing herself on the table.
"Mr. Fairfax-Lacy," she said softly, "I assume that's not your pool cue at my backside? What precisely are you playing?"
He didn't look a proper Englishman now. There was an open male swagger about him, a masculine vigor that she had never seen before.
"Seducing you."
"And if I don't choose to be seduced?"
"Don't you?" He bent his head and brushed her lips. "Don't you, Bea? Because I thought you told me that you were—seduceable."
"I don't invite married men into my bed," Bea said gently, but there was steel in her voice.
"But I'm not married!"
She shrugged. "You are Helene's. I do not betray other women."
Stephen picked her up and seated her on the horsehair pad lining the pool table. Her lips were pale cherry again. The color had worn off. As soon as she allowed him, he would run his tongue along her mouth, bite her round lower lip. "As of yet, I belong to no woman," he drawled. Then he lowered his head, finally, finally, burying his mouth against her, raking his lips against her rosy mouth.
For a second she relaxed against him and her mouth opened slightly, just barely yielding to his hunger. And then she pushed him away with all the determination of a pure-as-the-driven-snow duke's daughter.
"Behave yourself!"
"Bea," he said, and the word had all the hunger he felt in it. "Loyalty in matters of marriage is an entirely commendable emotion. But Helene and I have taken no vows. We are merely friends." He looked directly at her. Her eyes were a warm brown, with just the faintest tinge of exotic green, just enough to make them tempting beyond all resistance.
"Friends?" There was an edge to her voice. "You offer euphemisms with practiced ease, Mr. Fairfax-Lacy."
"I am a politician," he said with a sardonic grin.
"I thought you didn't care to take mistresses from women with experience. Too much experience," she clarified.
He looked at her, cursing his own stupidity. "That was cruel, and rather shabby," he said, echoing her own comment. "My excuse is that I want you so much that I—"
"I'll take it into consideration," she said, standing up.
Longing spread through him, coursed down his legs and made him tremble from head to foot. Dimly, he wondered what in the hell was happening to him. Why would this woman—this small, impudent, less-than-chaste woman—drive him into a fever of lust?
"We haven't finished our game," he said hoarsely.
She grinned at that, and the way her rosy lips curled sent his heart dancing. She had a way of smiling that made it look as if her whole body was dancing with joy. "There's no need to finish." She nodded toward the table. "You cannot win after my last shot."
He jerked her against his chest and swallowed her laughter, taking her mouth again and again, driving his tongue in a rhythm his whole body longed to repeat. "You," he said hoarsely, "I want you, Bea."
Her eyes slowly opened, and now they had that slumbrous interest he remembered. She melted against him and silenced him with her mouth, with a trembling sweetness, a speaking silence.
"Might I seduce you with poetry? I gather it is a method that you recommend." His voice was dark and slow, and his hands ran down her back with unsteady promise. She looked at him, and her eyes seemed more green than brown now, all exotic beauty and one dimple. But there was something in her face.... She had expected him to react this way. What he glimpsed now was not an aching lust akin to his but the faintest hint of satisfaction.
Men no doubt wooed Lady Beatrix all the time. Her beauty and her reputation would bring them like moths to a flame. She dressed to please, to attract; she made up her face so that she looked even more exotic—and approachable. She dared them all to come to her, and Stephen had no doubt but that they came.
Yet he sensed that Bea didn't succumb herself. She found pleasure, but not delirium. He wanted to bring her delirium, or nothing. "On second thought, perhaps I won't seduce you after all," he said, dropping his arms from about her and rolling down one sleeve. He watched her through her lashes.
She looked surprised but not particularly heartbroken.
"I shall wait for you to woo me. After all, I shall be quite busy in the next few days, as you kindly pointed out."
"I don't woo," Bea said, her small nose in the air.
He leaned back against the billiard table and looked at her. He had never, ever, felt as if his body were so valuable. Deliberately he spread his legs and watched her glance catch for a second and then fly away. "Did you never see a man whom you wanted rather desperately?"
"I have been fortunate in that—" and she stopped. Clearly something—or someone—had occurred to her.
He let his eyes glide over her breasts, linger where she was most sensitive. "It will depend, of course, on whether you think that I am worth competing for."
A corner of her mouth turned up wryly. She was no green girl to be brought directly to heel, that was clear. "I shall have to consider the matter," she said gravely. "You see, I am not altogether certain why Helene desired to summon you to her side. You, a sober party official, seem an unusual choice."
"Can you think of nothing?" The question hung in the air between them.
"I suppose there's your voice," she said.
Mentally, Stephen cheered. She liked his voice! He walked over to her, and his words came out in the dark, liquid language he had used to convince reluctant politicians but never a woman before. "I shall have to hope that this voice is potent enough so that you enter the fray."
She stared at him, eyes dark. He tipped up her chin and saw in her eyes the expectation of a kiss. So he bent and kissed her hand instead. "Lady Bea," he said. "I wish you good night."
She was surprised, he could see that. He doubted any man had ever left her company without begging for greater liberties. He hooked his coat with a finger and slung it over his shoulder. Then he walked to the door, feeling his body in an unfamiliar masculine swagger, in a walk so unlike him that he almost laughed.
"Stephen?" Her voice was so soft that it was no more than a whisper in the night air.
But of course he stopped. Whether she knew it or not, she was a siren, and he would follow her anywhere.
"Are you certain you're worth it? Two women vying for your attentions?"
His smile was as proud as a sultan's. "I've no doubt of that, Bea. To my mind, the only real question is—which of you will win me?"
She shrugged. "Not me. I don't woo."
"A pity, that," he said, and turned on his heel to go.
Bea stared at the closed door in blank astonishment. No man since Ned had ever walked away from her. In fact, she saw her role in society as a fairly simple one. She adorned herself; they came.
He was infuriating, if intriguing. But she'd be damned before she would chase a man, she who already had the reputation of a demimonde. That was one thing that was quite clear in her mind. She might have taken lovers—although far fewer than Stephen appeared to believe—but she had never, since Ned, allowed one of those men to believe that she was desperate for their company. Because she never was. She enjoyed male company. That was all.
And if Mr. Fairfax-Lacy wanted some sort of vulgar exhibition of interest, he was bound to be disappointed.
##
## In Which Curiosity Runs Rampant
Rees Holland, Earl Godwin, was in a pisser of a mood, as his butler put it belowstairs. "Got some sort of note from his wife, he did," Leke confirmed.
Rosy, the downstairs maid and Leke's niece, gasped. "I saw a pantomime on my last half-day where the husband poisoned a love letter and when his wife kissed it, she died. Maybe the countess saw the pantomime as well and she's poisoned him!"
"He deserves it then," grunted Leke. He found Earl Godwin difficult to work for, and he didn't like the irregularity of the household. On the one hand, his master was an earl, and that was good. On the other hand, the man had a dastardly temper, not to mention the fact that his fancy piece was living in the countess's quarters.
"And there's something to clean up there as well, so you'd better get to it."
"Don't tell me he spilled coffee on all them papers again," Rosy said, scowling. "I'm finding another position if he doesn't pick up those papers. How can I clean with that much muck about my ankles?"
"Don't you touch his papers," Leke said. "It's worth your life. Anyhow, it's not coffee this time. 'Twas a vase of flowers the strumpet was foolish enough to put on his piano."
"It's a wicked temper he has," Rosy said with relish. "How the strumpet puts up with it, I don't know."
The strumpet was Alina McKenna, erstwhile opera singer and inamorata of the bad-tempered earl. The term strumpet wasn't truly pejorative; both Leke and his niece rather liked Lina, as she called herself. Not that one could truly like a woman of that type, of course. But she wasn't as hard to work for as a great many more virtuous ladies, and Leke in particular knew that well enough.
He shrugged. "Thank the Lord, the master's taken himself off, at least."
"Where'd he go?"
"How could I know? Something in response to that letter from his wife, I've no doubt. Time for you to go about your duties, Rosy, before the strumpet makes her way home." The only reason Rosy's mum allowed her to work in such a house of ill repute was due to her uncle's presence. He took his responsibilities seriously and did his best to arrange her duties so that she rarely encountered one of the inhabitants of the house.
"I'd best go clean the sitting room then," Rosy said. It was a rare moment when the master wasn't in there pounding on one of them three pianos he had. And now there was likely water all over the floor.
A moment later she flew back downstairs, finding her uncle polishing silver. "I found the note," she said. "The note from his wife. He'd crumpled it up and left it right there, on the piano." She stuck out her hand.
Leke hesitated.
"Go on, Uncle John! You've simply got to read it—you know you do!"
"I oughtn't to."
"Mum will just murder you if you don't," Rosy said with relish. And that was true enough. Rosy's poor mum, Leke's only sister, was stuck in the house caring for Rosy's little sisters. She lived for stories about the goings-on at the earl's house that Leke and Rosy brought from the great house. That and the discarded gossip papers that the strumpet read and threw to the side.
Leke pursed his lips to indicate disapproval and then flattened the piece of paper. "It's from the countess all right," he confirmed. "Looks like she's staying in Wiltshire somewhere." He peered at the direction. "Can't really make it out. Perhaps Shambly House? That can't be right."
"Never mind where she is!" Rosy said, dancing with impatience. "What did she say? Where's he gone to, then?"
"'Rees,'" Leke read, "'I've contracted pleurisy. If you wish to see me alive, please come at your earliest convenience.'"
Rosy gasped. "No!"
Leke was reading it again. "That's what it says, all right. I'm thinking it's a bit odd—what is pleurisy, anyhow?"
"Likely some awful, awful disease," Rosy said, clasping her hands. "Oh, the poor countess! I only hope she's not deformed by it."
"You've never met her. Are you crying?"
For Rosy was wiping away tears. "It's just so sad! Here she's probably been pining away for her husband, and longing for him to come back to her, and now it's too, too late!"
"Use your head, girl. If you were the earl's wife, would you be pining for him to return?"
Rosy hesitated. "He's very handsome."
Her uncle snorted. "Like a wild boar is handsome, maybe. Face facts, Rosy. You wouldn't like to be married to the man, would you?"
"Well, of course not! He's awfully old, and so messy, too."
"The countess was better off without him. Funny, though, about that pleurisy. Pleurisy. What is pleurisy?"
"Mum would know," Rosy said.
"Neither of us has a half-day for another fortnight," her uncle said dismissively.
"But you could go over this afternoon, Uncle," Rosy pleaded. "You know you could. The master's gone to Wiltshire, to his wife's deathbed!" Her eyes were huge with excitement.
Leke hesitated and looked at the paper.
"That's our own mistress dying. We must needs know why. What if people ask?"
"I don't see what difference it makes. If she dies, the only thing we need are blacks. That is, if the master even sees fit to go into blacks for her death. Mayhap he and the strumpet will carry on just as usual."
"Oh no, they wouldn't!" Rosy clasped her hands again. "Perhaps this will be enough to reform him. He'll—"
"You're dreaming, lass. Now up you go to the sitting room, and I'll see how I do with the polishing. If I can finish this lot, I'll nip over to your mother's."
It wasn't until that evening that Rosy and her uncle met up again. Theirs was a small household, due to a combination of the earl's unconventional habits and the reluctance of decent servants to stay in a house of iniquity. Supper in the servants' dining room was merely Cook, Rosy, Uncle, and three footmen, not one of whom was quite as intelligent as he might have been. The scullery girl and shoeblack ate in the kitchen.
Rosy had filled Cook in on all the details of the afternoon before Leke made his way to the head of the table.
Rosy waited while he said a brief grace, and then burst out, "What is it, Uncle? What is pleurisy? Did Mum know?"
"Your mum is a keen woman," Leke said, taking some roast beef from the plate handed him by James, the third footman. "Tuck your hand under the plate, James. You don't want us to have to stare at your fingers, do you? Put us right off our food, it would."
James curled his fingers under the plate, and Leke nodded at him.
"She did know what pleurisy was, and that's a fact."
"I thought pleurisy was some sort of thing children caught," Cook said. Cook was a sturdy woman with bright red cheeks and a generous smile who had once cooked for the Prince of Wales and never forgot it. She was a genius in the kitchen, or so the prince had said. Earl Godwin had to pay her one hundred guineas a year to keep her in his house.
"That's right," Leke said, nodding. "You're another shrewd one, just like Rosy's mum. It's a disease children catch. In fact, my sister had never heard of an adult with it."
"But the countess isn't a child," Rosy said, perplexed.
"I do know someone who caught measles and it killed him," Cook said. "Mr. Leke, what do you think of this lamb pie?" Frustrated by the complete lack of visitors to the house, Cook had taken to serving up dishes for the staff as if Prinny himself were expected to sit down with them. "Have to keep my hand in, don't I?" or so she justified it. And it wasn't as if the master noticed anything wrong with the household bills. Rich as Croesus, he was.
"I'm liking it," Leke said, chewing with proper gravity. "There's just a touch of allspice, is it?"
"Correct," said the Cook. "I like a man with a knowing mouth, that I do." She beamed at him and then turned to Rosy. "People die in the strangest ways. There's no telling what might happen to a soul. Why, I just heard the other day that a man was riding his horse across the moors, right in the daylight, mind, and..."
##
## Yours to Woo
It took two days—two whole days—for Esme's heart to form a hard little shell that stopped her from thinking about the marquess. He was gone. That story was finished. True, his mother was still in the house, sparring with Arabella and occasionally flinging an insult at Esme, but her presence was irrelevant. Sebastian was gone, as Miles had gone, and as men always went. She decided to stop thinking about him. Forever. Of course, that didn't stop her from waking at the first light of dawn and brooding. It's a very good thing that Sebastian took himself off to France, Esme told herself, because I was in danger of believing his protestations and vows of love. More fool I. He didn't love me enough to defy me when I told him to leave. He just left. Probably thinking she'd be waiting when he returned from a leisurely exploration of French vineyards.
Why on earth cry for such a man? A potent, useful rage was filling the empty spaces in her heart. It was his fault that she was forced to entertain his mother. And it was his fault that she was carrying an elephantine child (never mind the irrationality of that). And it was his fault that she was husbandless and in the awkward position of not knowing who'd fathered her own child.
All together, her situation was all his fault, and the only pity was that he was no longer there so she could blister his ears with the truth of it. And if Sebastian were standing in my bedroom at this very moment, Esme thought, I would tell him that his attempt to imprint himself on my skin didn't work. That the only result of his exertions was an aching back and a desire to never see him again. She set her jaw to stop hot tears from running down her cheeks.
Of course, if his memories of that night were anything like hers, Sebastian might have trouble believing her. The solution would be to flirt madly in front of him. Perhaps do more than flirt. Why should he think—as he clearly did—that she was some sort of light-skirt who would allow him to waltz in and out of her bedchamber at will? Marriage would be the perfect solution. Especially if she married long before he wandered back from France and thought to pick up where he left off.
Perhaps she would marry Fairfax-Lacy, since her aunt had been kind enough to bring him to the house for precisely that purpose. Helene wasn't acting at all loverlike toward Fairfax-Lacy, and Esme had seen enough surreptitious lovers to recognize the signs. Or the lack thereof. So there was nothing—nothing!—to prevent her from taking such an eligible husband. Moreover, her mother would appreciate her marriage. Esme suspected that the only way on earth Fanny would receive her in public again would be if Esme remarried a man of the highest character. Sebastian certainly wasn't in that category. Not that she ever considered marrying him.
Fairfax-Lacy had a reputation for high moral fibre. And he was handsome too, in a sort of well-bred fashion. He didn't have Sebastian's raw beauty. But Fairfax-Lacy would make a perfect husband. A perfect, respectable husband whom her mother would adore. He would never leave her on the verge of giving birth.
That was the crux of it: Sebastian didn't seem to realize how frightening it was to give birth. He just didn't care enough to be frightened for her. Esme cried over that for a while and then, infuriatingly, found herself crying over her mother's similar lack of concern. Nobody cares, Esme thought savagely, conveniently forgetting Arabella and Helene. Not Sebastian, not Miles, not her own mother.
She didn't make it downstairs for luncheon, having dissolved into a humiliating, childish pit of despair. But by late afternoon, the hard little shell was back in place. Of course she wouldn't die in childbirth. She would be just fine. There was nothing she could do about the fact that Sebastian didn't love her as much as she wished. Better to forget it, push that fact away, not think about it. She rang her bell and asked Jeannie to prepare yet another cucumber poultice for her eyes.
By the time Esme descended the stairs in the evening, she had managed to channel an ocean's worth of rage and grief into one question: was Stephen Fairfax-Lacy indeed appropriate husband material? She didn't think he had Sebastian's ability to overlook her belly. He was unlikely to be attracted to her in her condition. But she could certainly make up her mind whether he was suitable for a life's worth of dinner conversations.
And so it was that Stephen Fairfax-Lacy, who strode into the dining room hoping against all hopes that a certain lady had decided to woo him, found, to his utter surprise, that his hostess appeared to have made that decision instead. And Lady Rawlings, nine months with child or not, was a formidable wooer.
Naturally, she was seated at the head of the table, but she placed him to her right. And Stephen had no sooner seated himself than Lady Rawlings leaned toward him with a very marked kind of attention. There was a sleepy smile in her eyes that would make any man under the age of seventy think of bed—nay, dream of bed. Yet it wasn't until Lady Beatrix Lennox was ushered into a seat across from him that Stephen began enjoying himself. As Bea sat down, Esme—as she'd asked him to call her—was showing him the intricate figures on the back of her fan. And he glimpsed something in Bea's face. Just enough to make him draw closer to Esme and bend his head over her fan.
He was, after all, an old hand at campaigning.
"Romeo and Juliet, are they?" he asked Esme, peering at the little figures painted with exquisite detail on the folds of her fan.
"Exactly. You see"—one of Esme's curls brushed his cheek—"there's Romeo below the balcony, looking up at Juliet. Bea, would you like to see it? The workmanship is quite elegant."
The Marchioness Bonnington was sitting at Stephen's right. "Goodness, what a hen party!" she said briskly. "Why on earth didn't Arabella even out the numbers when she issued her invitations?"
Esme looked up, and her tone evened to a polite disinterest. "I can't say, Lady Bonnington. I believe that Earl Godwin will arrive tomorrow. His presence should ameliorate the situation."
"Humph," Lady Bonnington said. "Least said of that reprobate, the better. So what's on that fan you are regarding so closely, Lady Beatrix?"
Bea blinked down at the fan. "Romeo and Juliet," she murmured. There was something odd happening here. She glanced across the table while pretending to examine the fan. Esme's impending child was hidden beneath the tablecloth, which meant that she looked like any other gloriously beautiful woman in London—except there were very few women who could match Esme. And to all appearances Esme had decided to seduce Stephen Fairfax-Lacy. Her Stephen. In fact, Esme presumably had decided to follow her aunt's advice and marry, not seduce, Stephen. Of course she wasn't thinking of seduction, given her delicate condition.
The realization gave Bea a most peculiar sinking feeling. Esme's hair was caught up in a loose topknot; fat, silky curls caressed her shoulders and cheeks. She wore a gown of French violet silk cut very low in the bosom and very short in the sleeves. But more importantly, she was burning with a kind of incandescent sensual beauty.
"Romeo and Juliet, did you say?" Lady Bonnington barked.
"The balcony scene," Bea explained, pulling herself together and handing over the fan. She didn't want to woo Stephen. Therefore, it hardly mattered if Esme decided to do so. "I've always thought it was an absurd scene."
"How so?" Stephen asked, one dark eyebrow raised.
Bea blinked, trying to see what it was about the man that drove all the women in his vicinity to hanker after him. He was handsome, but she'd seen better. Somewhere. He was waiting for a reply, so she shrugged. "Romeo stands below, wailing up at Juliet like a pining adolescent."
"That seems a bit harsh. He is in love."
"He only met the woman twenty minutes earlier. But you're right, he thinks he's in love. The funny part, to my mind, is when Juliet suddenly says: do you plan to marry me, and if so, where?"
Esme grinned. "How extraordinary. I read the play, of course, but I never realized that Juliet proposed to him."
"'If that thy bent of love be honorable,'" Bea quoted, "'thy purpose marriage, send me word tomorrow.'" Juliet bluntly asks him to marry her, although he hasn't said a word on the subject previously."
Esme's eyes flicked to Stephen with a meaningful laughter that made Bea's stomach twist. She was so beautiful! It was almost too much to bear. Bea could paint her cheeks the color of the rainbow, but she could never reproduce that flair of raw sensuality that Esme had just tossed in Stephen's direction.
"I saw a hilarious parody of the balcony scene once," Esme was saying, her voice a glorious, husky alto.
"Oh?" Stephen bent toward her, his eyes bold and appreciative.
Naturally, Bea thought. Given the pick of the three women in the house, Helene, herself and Esme, what man wouldn't choose Esme?
"This Juliet almost threw herself off the balcony in her eagerness to join Romeo," Esme remarked. Her eyes seemed to be speaking volumes. Bea considered pleading a sick stomach and leaving the table.
The Marchioness Bonnington had been examining the painted fan; she put it down with a little rap. "That sounds very unlike Shakespeare."
"Do share it with us," Stephen said.
If he got any closer to her shoulder, he could start chewing on her curls, Bea thought. Just like the goat.
"I only remember a line or two," Esme said, and her crimson lips curled into a private smile for Stephen, so seductively potent that Bea felt it like a blow.
"Romeo stands below the balcony, bellowing at Juliet," Esme continued. "And she says 'Who's there?'"
Stephen had just caught a tantalizing glimpse of Bea's eyes. She looked...pained. Stricken? That was too strong. He deliberately returned Esme's smoldering gaze with one of his own. "And what does Romeo reply?" He pitched his voice to a deep purr.
Esme flashed a smile around the table. "I do hope this won't embarrass any of you."
"I doubt it," Lady Bonnington said sourly. "After the astonishments of the last month, I consider myself fairly unshockable."
"The scene takes place in the early morning, if you remember. Juliet says, 'Who, Romeo? O, you're an early cock in truth! Who would have thought you to be so rare a stirrer?'" Esme said it with dulcet satisfaction.
There was a moment of silence and then Stephen roared with laughter. "I'll warrant you Romeo clambered up the vine as fast as he was able!"
"She wouldn't allow him to do so," Esme said. Her eyes were sparkling with mischief, and she had a slim hand on Stephen's arm. "The next line was something like this: 'Nay, by my faith, I'll keep you down, for you knights are very dangerous if once you get above.'"
Stephen laughed again, and then tilted his head toward Esme and murmured something in her ear. Obviously, it was a comment meant for her alone. Likely something about getting above. Bea chewed very precisely and swallowed her beef. Perhaps Arabella would allow her to return to London on the morrow. It wasn't that she was jealous, because she wasn't. It was just that no man could resist Esme, and certainly not Stephen, who had frankly told her that he hoped to marry. Slope was bending down at Esme's shoulder, interrupting her tête-à-tête with Stephen. Bea looked back at her beef. She liked Esme. She really did.
"My lady," Slope said quietly into Esme's ear. "We have an unexpected guest."
"All right," Esme said, only half listening. She'd forgotten how much fun flirting was. She was actually enjoying herself. She hadn't thought about wretched, wretched Sebastian for at least a half hour. Arabella was right. Stephen Fairfax-Lacy was charming, and he had a ready wit. He was fairly handsome. She had almost decided to marry him. Of course, first she had to make certain that Helene didn't want him for herself.
Slope, seeing that the unexpected guest in question had followed him into the dining room, although his mistress hadn't yet noticed, straightened and announced, "The Marquess Bonnington."
Esme's head jerked up. There he was.
No gardener ever wore a pearl gray coat of the finest broadcloth, with an elaborately tied cravat of a pale, icy blue. He looked every inch a nobleman, from the top of his elegantly tousled hair to the tips of his shining Hessians.
There were murmurs all down the table. The scandalous marquess had returned from the Continent! Or from the garden, if only they'd known.
She met his eyes, and there was a flare of amusement in them that made her smoldering rage burst into flame. No doubt he thought to simply return to her bedchamber. Without giving a thought for her reputation, for her child's reputation, for her future.
"Ah, Bonnington," his mother said. "There you are." She sounded as if he'd been to a horse race rather than exiled to the Continent.
But he waited, as polite as ever, for his hostess's acknowledgement. Esme's hands clenched into fists. How dare he think he could simply come and go in her house, just as he had walked into her bedchamber at Lady Troubridge's house?
"Lord Bonnington," she said, inclining her head. "How can it be anything other than a pleasure to see you, after so many months." She reached over and put a hand on Stephen Fairfax-Lacy's shoulder. He had broad shoulders. She was almost certain that he would be as good a lover as Sebastian. He certainly would be less exhausting.
Fairfax-Lacy looked up, and Esme smiled down at him brilliantly. "Marquess Bonnington has joined us just at the very moment I was to make an important announcement. May I introduce my fiancé, Mr. Fairfax-Lacy?"
There was a moment of utter silence in the dining room.
Then Sebastian went into a low bow, the kind with a flourish and a good deal of gloved violence. His eyes were pitch black in the candlelight, but Esme wouldn't have been surprised if they'd burned straight through her. "I seem to have arrived just in time for a celebration," he said, and the sardonic note in his voice was clear for all to hear.
Esme swallowed and tightened her hand on her new fiancé's shoulder. She had always been impetuous, but this was without a doubt her wildest moment yet.
"What a delightful surprise!" the Marchioness Bonnington crowed. Obviously, she saw her son's freedom within reach.
"Yes, indeed," Helene chimed in, giving Esme a darkling look that said, clear as day: I have use for that man, remember?
Even little Bea seemed shaken, although she said nothing. And to Esme's endless relief, her brand-new fiancé also refrained from expressing his surprise.
##
## Twenty Minutes Later...Privacy at Last
"You needn't really marry me. After all, it's not as if you asked me."
"My thought precisely."
"In fact, no one need even know that we were engaged—"
"We are not engaged!"
"Would you mind terribly if we just pretended that we are?"
Stephen Fairfax-Lacy was perplexed. Even after some twenty years of being an eligible bachelor, he seemed to have reached an unexpected peak of desirability. "Lady Rawlings—"
"Oh, please, you must call me Esme! After all, we're—"
"Engaged," he put in. He couldn't help smiling a little. "In that case, you must call me Stephen."
"Thank you," Esme said, with evident relief.
"But I insist, Esme, that you tell me why we are engaged."
Esme fidgeted and rearranged her fingers. Stephen had seen that look before, many times. It was the look that a Member of Parliament wore who had been courted away by the other party, who had to disclose that he'd already given a crucial vote away.
"Esme?"
"Perhaps you are aware that Marquess Bonnington and I—uh—" She looked agonized, so Stephen came to her rescue.
"Of course, I am aware that you had an unpleasant experience at Lady Troubridge's house party last year, during which your husband unfortunately suffered a spasm and died."
Esme nodded. "You put it remarkably concisely."
Stephen waited. Esme looked at him and then away again. "I was having an affair with him. With the marquess," she clarified.
Stephen thought for a second. "In that case, I believe I understand why the marquess has returned from the Continent. He has just discovered that you are carrying a child?"
"He wishes to compensate for what happened last summer. Marquess Bonnington believes that marrying me will ameliorate his guilt."
"Guilt is an interesting concept," Stephen said. "I wish I could induce guilt in more of the men I deal with on a regular basis."
"But I don't wish to marry a man who seeks to assuage his guilt. And when I saw him, I panicked."
Stephen was beginning to enjoy himself. While he had never begged for any woman's attentions, they had never stood in line and begged for his either. "I gather I appeared to be a useful solution to your problem?" he suggested.
"I'm truly sorry to have used you so. But would you greatly dislike pretending to be my fiancé, merely until Marquess Bonnington returns to the Continent? I'm certain we can arrange it so that no one outside this small party discovers our brief engagement. His mother is, naturally enough, anxious to turn his thoughts in another direction; perhaps she will manage to convince him to leave by tomorrow morning. He need feel no further guilt when he thinks I am marrying such an estimable man as yourself."
"I bow to your greater knowledge of Marquess Bonnington. I must say that I would not have judged him as one to easily give up. I would describe him along the lines of a terrier with a bone."
"I don't want to be that bone," Esme said despairingly. "I know I'm not looking my best, and I'm not a very appealing fiancée under the circumstances, but if you would play a devoted future spouse in front of the marquess, I would be endlessly grateful."
His laughter echoed around the room. Stephen stood up and kissed Esme's hand, and then helped her to her feet. "Since you are my future wife, perhaps I could take the liberty of telling you that you look exhausted. May I escort you upstairs?"
"Oh, thank you!" Esme said, taking his arm. They encountered no one, and Stephen saw his presumed wife into her chamber with an unmistakable sense of relief.
In fact, he actually leaned his head back against the corridor wall, closed his eyes, and wondered if he'd been caught in a dream. It seemed impossible that he—a staid, proper, boring member of the House of Commons—was pretending not only to be carrying on a flagrant affair for the benefit of one woman's husband but also to be passionately in love with another woman, a drama to be played out before her lover.
He heard a rustle of silk. Of course it was Bea. She seemed to be everywhere, with her painted eyebrows and her red mouth. And the rest of her: those far too intelligent eyes, curved little body, and sultry looks.
"Time for bed?" he said, and he let a deliberately suggestive tone slide into his voice like cream.
"Good night, Mr. Fairfax-Lacy." She appeared to be walking toward her chamber. He stretched his leg out so that she would have to step ungracefully across him to continue down the corridor.
"Sir?" she asked. The very tone of her voice had changed. Where was the impudent suggestion? Where were the smoldering looks that she practiced on him so regularly? (Because he knew quite well that she didn't feel desire; she issued such invitations as a matter of course.)
"Will you let me pass, please?" She was getting nettled now.
But Stephen was surrounded by women begging him to pretend to be their lover. What he wanted was just one truthful request. And the fact she had refused to woo him for two days bothered him more than it should. "I should like to read more of that poetry you brought with you," he said.
"I can lend you the book, if you wish. Or you can find it yourself. I left it in the library, since it seems to have become an object of curiosity for all." Her eyes were shadowed, and he couldn't read them.
He reached out and slipped his hand under her elbow. God, but he was consumed with lust. Even the sleekness of the bare skin of her arm made him leap to attention.
She shook her head, frowning. "I think not, Mr. Fairfax-Lacy."
"A further introduction to poetry," he said, his voice as persuasive as he could make it.
"I gather you wish me to accompany you to the library?"
He nodded. Not that he had actually thought it out.
"Why?" She stared at him, and for once her eyes were neither sultry, nor inviting, nor even particularly friendly. "You, a newly engaged man, must have many places to be."
"Because," he said, through clenched teeth. "Because of this." He folded her into his arms, and his whole body throbbed with gratitude. She smelled like an exotic perfume tonight, some thick heavy flower of the Nile.
He spread his fingers through her hair, cupping the back of her head and pulling it gently back so he could reach her lips. He could see the perfect oval of her cheek in the dim light. He could see the darker glowing red of her lips. Black lashes fringed her eyes. But none of it mattered, because he couldn't see those eyes well enough to read them.
Did she feel even a fraction of the desire that pulsed through his body? Was she almost trembling? Or was this all the fantasy of an aging man, caught off-guard by a young woman's seductive beauty? Believing—
He refused to think too hard. Instead he pulled her head closer to his, closed his mouth on hers, plunging inside. He never kissed like this. He prided himself on consummate expertise, on dancing over a woman's lips, coaxing her to give him her inner sweetness, to reward him with her lips, her mouth. It was all a foretaste of his future treatment of her body. He was a thoughtful lover, cherishing his partner's pleasure as his own.
Not with Bea. His heartbeat pounded with the same rhythm as his tongue. As for technique...what technique? It was all he could do to stay upright, to control his hunger.
Yet she melted in his arms with a fervor he had never awoken in a lover before. If he was rudely plundering her mouth, she certainly wasn't fainting at the intrusion. Her arms were around his neck, and she was—she was offering herself. Yet after a second she stepped back in a swish of silk, and he released her. "Where are you going?"
She smiled over her shoulder, and it was the same smile Cleopatra gave Antony. Antony had no hope of escaping; why should he?
"I'm not interested in wooing you, Mr. Fairfax-Lacy, as I think I've made clear. And I might add that my lack of interest is all the greater since you are now engaged to marry."
"I'm—" But he stopped before he said not. Instead he smiled at her, an imitation of all those smiles she gave him, a sexy dance in his eyes. "Too much competition?" he asked softly.
Bea paused and turned her nose in the air. "I don't compete."
He leaned back against the wall, and it was happening again: around Bea, and only around her, he felt in his body, if only in fragile control of it. He deliberately spread his legs. They felt muscle hard...as did other parts of his body. Her eyes widened slightly. In one stride he had her pinned against the opposite wall. God, he loved the fact that she was tall enough for him. So many women felt like fragile little dolls in his arms.
"Bea," he growled, looking down at her.
"Mr. Fairfax-Lacy?" she said pertly. But she didn't try to move away. Not even when he brought his mouth down on hers, without apology, without warning, without pleading. Instead she just gasped and shuddered in his arms as his mouth drank from hers, came to her again and again with savage tenderness.
He kissed her until he knew she couldn't pull away and give him that Cleopatra smile. Gone was the seasoned beauty, wise in the ways of the world and quick with her seductive invitations. If he didn't know better, he'd have said she was a pure innocent. It was in her eyes, in the way she trembled in his arms, in the way she clutched his shoulders.
"I do wish you'd change your mind," he said. His voice didn't come out that siren call of the polished politician. No, it sounded deep, dangerous. The voice of a man who would seduce a young unmarried girl. Who instructed her to woo him. The kind of man who had a mistress, and a fiancée, and wanted yet a third woman.
Stephen reveled in it. He ran a slow hand down her side, and then swiftly, before he or she could think better, slid that hand around her sweet little bottom and pulled her hard against his legs. She gasped, and her arms spun tighter around his neck.
For one blissful second he pressed her into the wall, letting her know just how primitive their joining would be.
Then he snapped back and dropped his arms. "Because should you decide to compete," he said, "I think you would find it worth your while."
His smile was wild and tender and utterly unpolished. It was all Bea could do not to gasp yes, plead, beg...woo. Whatever he wanted. Her body was throbbing, liquid with desire, beating through her legs. Even her toes tingled. He wasn't like the gentlemen she'd toyed with in the past. He was a man. More: he was a dangerous man, the sort of man who didn't think twice about taking on a fiancée and a mistress in the same week. What would she be? The third woman?
She couldn't drag her eyes off him though, off his broad shoulders, and off those wicked, wicked laughing eyes. How had she ever thought he was proper? He was some sort of a satyr! She licked her lips and watched his eyes narrow. If he reached for her again, she would do—whatever it was he wanted. The wooing he demanded.
How humiliating. If she did that, begged him in so many words, there would be no escape from all the words her father flung at her. No escape in her own mind. They all crowded together: wanton, short-heeled, soiled, doxy.
No. Bea swallowed hard, pushed herself from the wall, and started down the corridor without a backward glance. She couldn't look back.
##
## In Which a Marquess Pays a Call on a Lady
As Esme prepared for bed, she wondered exactly how much time she had to herself before Sebastian Bonnington joined her. Because he would, not matter how many future husbands she pretended to have.
She didn't have to wait long. She was barely tucked into bed, with Jeannie sent back to the nether regions of the house, when her door opened. Esme was propped up against the pillows, wide awake. She was unable to sleep very much these days; her back and her belly seemed to be competing to make her uncomfortable.
Sure enough, he had that disapproving look that he always used to have, back when she was married to Miles and flirting with Bernie Burdett. Esme frowned. She never liked it when he played the Holy Willy then, and she didn't appreciate it now, either.
"What are you doing in my room?" she demanded.
He walked slowly over to the bed. "Thinking about corporal punishment," he said, staring down at her. "Hell-born brat. I can't leave you alone for two days without finding you've attached a male to your skirts."
Esme held on to her anger. She was the angry one. He had left her when she was on the verge of having a child (but he did come back, a little voice reminded her).
"I could have died while you were gone," she said. Her voice sounded petulent and childish. "In childbirth," she qualified.
"I talked to your midwife before I left, and she had no expectations that you would give birth before a week at least," he said, still staring down at her. There was something in his eyes that made her feel uncomfortable. As if she'd disappointed him.
"Midwives don't know everything!" she said shrilly.
He folded his arms across his chest. "I sent my mother to look after you."
"Your mother!" she gasped. "Your mother is here to make certain that I don't marry you!"
"I told her I was a gardener here because I knew she wouldn't be able to resist calling on you. I had to visit my estate, Esme. I've done as much work as I could from afar, but I needed to be there, if only for a day." He ran a hand through his hair. "I stayed up two nights so I could return to you as soon as possible. But it seems you had no trouble occupying yourself."
Esme shot him a swift glance. Sure enough, there were weary circles under his eyes. And a bleak note in his voice that clutched her heart.
"I thought you'd left me," she said, pleating the linen sheet with her fingers. "That you—"
"That I'd obeyed you?" he lifted her chin. "Because you did tell me in no uncertain terms that you never wanted to see me again, Esme. That I would ruin your reputation."
"And so you will!" Esme managed.
"Not with my mother here," he said.
There was nothing she could say to that. Of course, he was right. The very presence of the formidable Marchioness Bonnington would stop all gossip about his presence on her estate.
"But I see I didn't need to worry," he said ironically. "It seems you made other plans for the protection of your name."
"I can't marry you, Sebastian," she said in a low voice. "I want respectability. Our marriage would be the greatest scandal to reach the ton in years. Your mother said that, and she's right. I don't want to be Infamous Esme anymore. Please understand!"
"I understand all right," he said.
That was definitely disappointment in his voice. Esme swallowed hard. Her back was aching, and he was angry at her. And he was right. She shouldn't have made that pretense of an engagement with Fairfax-Lacy.
Suddenly he pushed her over a few inches and sat down on the bed. "Back hurting?" And at her nod, he said, "Roll over."
Esme rolled to the right, and those big hands started rubbing her neck and shoulders. The relief was so great that she literally forgot everything else for a few moments. Sebastian had miraculous hands. Somehow he was ironing away all the pain that crouched in her spine.
A half hour later she rolled back, propped herself up against the pillow, and eyed Sebastian. He had to leave her bedchamber. Women in confinement did not entertain gentlemen to whom they were not married. But she had to try to explain her own stupidities first.
"I thought to marry Mr. Fairfax-Lacy because—"
He interrupted her. "Are you sure that you remembered to warn poor Mr. Fairfax-Lacy of his impending marriage? Of course, I would never suggest that he looked disagreeably surprised, but he seemed to me...disagreeably surprised."
Esme raised her chin. "He was merely startled by my public announcement. We had thought to wait until after the child's birth."
Sebastian didn't seem angry anymore. "I haven't even said hello to that child yet." He spread his hands on the soft cambric of her night rail. "He's all lumps and bumps. I don't think there's any room in there."
"The midwife told me today that he is...well, ready," Esme said. She felt a pulse of worry. The child seemed impossibly large to her.
But Sebastian looked up and grinned. "Don't worry. He'll slip out like a greased pig."
"That is so vulgar!" Esme scolded.
"Look at this!" he said, disregarding her. "If I push on his little foot, he pushes back!"
They watched for a second and then burst into laughter.
"Oh, no!" Esme said, clapping a hand over her mouth. "My goodness, I hope no one heard that."
"They'll think you're entertaining your future husband," Sebastian said, shrugging a bit. "Although no one here would give a bean who you were entertaining. I have to say, Esme, I've heard about your aunt's house parties for years, but this one takes the cake. Who's that extraordinarily luxurious-looking girl with all the paint on her face?"
"Lady Beatrix Lennox," Esme said, "and don't say anything cruel about her, because I like her hugely."
"The scandalous one? Daughter of the Duke of Wintersall?"
"Exactly."
Sebastian gave a little whistle. "Quite a gathering. You were certainly right when you thought it might endanger your reputation."
"My aunt invited a few of her friends without my knowledge, and one thing led to another. And what about you? If anyone finds out you've returned from France and are attending this gathering, the ton will dine out on it for days."
"Not with my mother here. And I don't give a hang if they do," Sebastian said, rubbing her tummy all over. "Face it, sweetheart. You're not made for the respectable life. You collect scandals the way other women collect china. I have some trouble envisioning you as a dutiful wife of a party member."
He leaned over, his face just an inch from hers. A dark blonde curl fell over his forehead. She could smell him...all that potent, clean-smelling male body.
"What are you doing in my bedroom?" Esme asked, quite annoyed to find that her voice was breathless.
"Paying a respectable visit to my future wife." His eyes were the blue of a mountain lake. Except no mountain lake ever had that smoky look way down deep that made her want to squirm. "Surely you aren't expecting your estimable fiancé to visit your chambers this evening. Since I intend to be your next fiancé, I have every right to be here. Besides, I feel a certain discontent with my performance. I must not have imprinted myself on your skin, given that you leaped directly into another man's embrace."
She couldn't squirm because he was hovering over her. "Certainly not," she said, pulling herself together. "Return to your own room, if you would be so kind. I'm sure you did a very good job of—of imprinting yourself on my skin. More than adequate. Now I'd like to ask you to leave."
She put her palms on his chest to push him away. He was warm and big, and somehow her palms just stuck and forgot to shove him off the bed.
He lowered his head and just kissed the top of her ear. "I'd rather stay with you." His lips slid to her mouth. He tasted of cognac and Sebastian.
Just a kiss, Esme told herself as his tongue touched hers. She couldn't help it; her mouth opened with a gasp. He tasted so good, so male, so comforting and intoxicating, all at once. He moved so they were lying side by side.
"We're not going to make love again," she managed to say. "My back hurt all day after you left."
"I'm sorry about that," and he actually sounded sorry. Except he had his hands under her night rail, and that wasn't her back he was stroking.
Esme gave up. Her body melted the moment his fingers slid up her thighs. So she buried her hands in his hair and let herself stroke circles down his neck with her tongue.
He pushed her leg up to give him better access, and she didn't protest, just jerked at his shirt so that he reared back and stripped it off, giving her all that honey-sweet skin to kiss and lick and touch.
They didn't say much for a while, there being no need for speech. Esme was gasping and moaning, and when she absolutely had to make a point, her voice came out in a husky mixture of a moan and a squeak. "Sebastian...please!"
"We can't," he said. "Your back." His voice sounded strangled, deep and hungry. He repeated what he was doing, and Esme clutched him feverishly.
"I don't care about my back!"
But he knew her, he knew her body, he knew everything...she couldn't stop now, not when he was stroking her like that, hands so smooth and rough at once. It took his mouth to stop the scream that tore from her chest.
The shame of it, Esme realized in the early dawn, was that she'd promptly fallen asleep in his arms, having given no thought to his pleasure. When was the last time she'd slept straight through the night, without waking over and over because her back hurt?
His tousled hair was the color of guinea coins. He was lying on his stomach, and the sheet was pulled to his mid-back. All Esme could see was the flare of his shoulders.
The babe seemed to be asleep. Sebastian was definitely asleep. As she watched, he gave a little humph, almost a snore, and lapsed into deep breathing. He'd stayed up at night so that he could come back to her...She had to push down the fierce joy she felt. Respectable widows didn't feel this sort of thing.
It was too much temptation for any woman to endure, even a widow bent on the respectable life.
She scooted the linen sheet down onto his legs. His back curved down to a sweet spot with two dimples and that little brown mark that wasn't hereditary, according to his mother. It looked like a small star. She would have leaned down to kiss it, but an awfully large stomach was in the way. So she contented herself with finger kisses, walking her way over all those muscles, circling his dimples, climbing back up that taut pair of buttocks.
He shifted under her fingers and groaned a little in his sleep. Sebastian made her feel more sensual than she ever had in bed with a man. As if her mere touch were enough. Before, it always seemed that men were interested in her breasts, in her legs—in all the parts of her that she'd been born with. Not in the way she touched, or kissed. Not in what she thought they ought to do next.
The very thought had her heart racing. She spread her hand and cupped one of his muscled buttocks.
Suddenly he made a noise in his throat and turned onto his back. Her fingers slid away and ended up on his stomach. He was still sleeping, lashes dark against his cheek. It was almost frightening how much she desired him. A lady shouldn't feel such a dark pounding wave of lust. That was certain.
What she should have done was wake this slumbering god and sent him on his way. Because she needed him to scoop up his terrible mama and leave her house, so that she could have her baby and begin her life again. Despite herself, her fingers trailed downwards. He was magnificent.
When she looked up, he was looking at her. And he didn't seem to be sleepy anymore.
##
## The Infernal Circle
When Dante was writing The Inferno, making up all those circles of hellish occupants—the gluttons, the adulterers, the...the whatevers—he should have included the Sewing Circle. To Esme's mind, they deserved a circle all their own. Admittedly, her memory of The Inferno was rather foggy, but weren't the gluttons sitting around eating and eating as punishment for a life of overly rich dining? In Esme's version of hell, overly righteous women would have to sit on small, upright chairs and sew seam after seam in coarse white cotton while Mrs. Cable read improving literature aloud.
They had been sewing for about fifteen minutes when Mrs. Barret-Ducrorq smiled genially at Esme and said, "That child of yours won't want to wait much longer."
Esme looked down at her vast expanse of stomach, suppressing a wince as a foot made its presence known just under her ribs. "The midwife has suggested that it's only a matter of a few days."
"They don't know everything," Lady Winifred said comfortably, putting down her sheet.
Esme had noticed that everyone except Mrs. Cable took every opportunity to stop sewing.
"The midwife for my first child told me every day for a month that today was the day," Lady Winifred continued. "Consequently I refused to actually believe that I was in labor when the time came. Is Lady Withers going to join us today, my dear? Arabella is such an amusing woman. And so brave. I know the loss of three husbands has been a true source of grief in her life, but she never seems disheartened."
Mrs. Cable said, very frostily, "I doubt that Lady Withers has risen at this hour."
But Arabella pranced through the door at that very moment, blowing kisses in every direction. "Ladies!" she announced. "I come to you on an errand of mercy."
Arabella took a few moments to seat herself. She was wearing a morning dress of celestial blue muslin, which opened down the front and pulled back to reveal an underskirt of sprigged muslin. She looked charming, effortlessly elegant and, to Esme's eyes, unmistakably mischievous.
"Surely you heard who arrived at this house last evening!" she announced, once she had arranged her gown to her satisfaction.
Even Mrs. Cable looked up from her seam.
"The most disreputable man in all England!" Arabella trumpeted.
Esme groaned inwardly.
"The Duke of York!" Lady Henrietta exclaimed.
"No, no, slightly lower in rank," Arabella said, obviously enjoying herself hugely. "It seems quite overheated in the morning room, Esme my dear. Perhaps that fire is too high for the season." She took out a small blue fan and began fanning herself.
"I'm having trouble keeping myself on the cool side as well," Lady Winifred said, eyeing the fan. "We've entered that time of life, I suppose."
Arabella dropped the fan as if it had bitten her.
"Who is it?" Mrs. Barret-Ducrorq said eagerly. "Who arrived last night?"
"Bonnington," Arabella said after a magnificent pause, "has returned."
It was a good line. And if it weren't for the fact that Esme's own life was being paraded before the gossips, she would have applauded Arabella's dramatic turn of phrase.
There was a collective intake of breath. Lady Winifred was obviously amused; Mrs. Barret-Ducrorq was shocked; Mrs. Cable was so horrified that she covered her face with her hands, as if she'd been faced with the devil himself.
"He's reformed," Arabella dropped into the silence that followed.
"I doubt that very much!" snapped Mrs. Cable, seemingly unable to contain herself.
"Astounding, yet true." Arabella picked up her fan again and glanced significantly at the ladies. "He's come back to England to prostrate himself at my niece's feet!"
"As well he might," Mrs. Barret-Ducrorq said rather sourly. "After all, he did..." But her voice trailed off when she realized that mentioning the fact that Esme's husband had died grappling with her latest guest wasn't entirely well mannered.
Esme looked down at her sheet and very precisely put another crooked stitch into the hem. That foot was still in her ribs. Oddly enough, she didn't feel the pinched sensation that she usually got at the very mention of Miles. Poor Miles. She placed another stitch. Dear Miles.
"Prrrrostrate himself," Arabella said with pleasure. "As you say, Mrs. Barret-Ducrorq, Bonnington is at least partially responsible for the death of poor Lord Rawlings. Although his doctors had said that Esme's late husband was liable to die at any moment. I lost a husband to a weak heart myself; it's a terrible circumstance. At any rate, Marquess Bonnington is overcome with contrition. Quite beside himself."
Everyone looked at Esme, so she tried to look like a grieving widow. Far be it from her to diminish Arabella's performance. Why was it that whenever she was supposed to look miserable, she felt cheerful? "The marquess has certainly expressed his repentance," she agreed, placing another stitch so as to avoid Mrs. Cable's piercing glance. Really, sewing had its uses.
"How can Bonnington possibly think to alleviate Lady Rawlings's situation?" Mrs. Cable demanded. "What's done is done. The man should stay on the Continent, where he is less likely to corrupt others."
"Unless he's asked for her hand in marriage?" Lady Winifred said, giving Esme a shrewd look.
"A revolting proposition," Mrs. Cable said tightly. "Lady Rawlings is not even out of a full year of mourning! Only think of the scandal!"
"Oh, one can always think of the scandal," Lady Winifred said. "But it's so seldom worth the effort. The marquess, after all, has a very fine estate."
"My thought precisely," Arabella said, beaming. "I do believe the man is genuinely overcome by penitence. He wishes to mitigate the evils he visited on her in any way possible."
"What makes you believe that his intentions are honorable?" Mrs. Cable wanted to know. "After his behavior last summer!"
Esme felt a pang of guilt. She was hardly innocent when it came to the loss of Sebastian's sterling reputation, since he had fabricated a story of depravity in order to protect her reputation. "His mother accompanied him to this house, which seems to bode well for his sincerity," she noted. "The Marchioness Bonnington is also staying with us."
"My goodness!" Lady Winifred exclaimed. "If Bonnington persuaded his mother to accompany him, the man must indeed be serious. Lady Bonnington is as stiff-rumped a lady as I've ever met!"
"I sincerely hope that you informed him that marriage was impossible," Mrs. Cable told Esme.
Esme suddenly remembered her supposed engagement to Fairfax-Lacy. There was more than one reason why marriage to Sebastian was impossible. Rather than answer, she started sewing again.
"After all, the man forged a marriage certificate in order to take a lady's chastity!" Mrs. Cable continued. "The poor Duchess of Girton might well have been taken in by his depravity, if it hadn't been for the happenstance of his stumbling into your bedchamber rather than hers. And that's not to mention his hand in your husband's death."
Arabella leaned forward. From the look of pure pleasure in her eyes, Esme could see her aunt had prepared herself for just this moment.
"A woman of mercy does not spurn a geninely remorseful soul," Arabella intoned. "By doing so, she would be responsible for any lapses in judgement that followed. No, Esme's path is clear. She must aid and succor the poor unfortunate sinner in his moment of contrition."
"The devil is full of all subtlety and all mischief," Mrs. Cable snapped. "Acts."
"By mercy and truth, iniquity is purged," Arabella retorted, without even pausing for breath. "Proverbs."
Esme bit her lip so she wouldn't ruin the moment by laughing. Mrs. Cable was flattened, trapped between the Bible and her abhorrence of iniquitous behavior.
Lady Winifred jumped in at this point. "I quite agree with you, Arabella dearest. It takes a truly charitable heart to recognize where the path of goodness lies."
Arabella was obviously trying to look as if she had a charitable heart. To Esme, it looked as if she had wind.
"I don't support it," Mrs. Cable snapped. "The man is a poisonous influence. You'll have to watch the young women in the house very carefully, Lady Rawlings. He may besmirch them, corrupt them, deprave them!"
No, Esme thought ruefully, he's only interested in besmirching me. Although she wouldn't argue with the idea that Sebastian was depraved. He had no sense of propriety in bed. Esme's cheeks grew hot at the very thought of the liberties he had taken the previous night. She wrenched her attention back to Mrs. Cable.
"A man like that is more than likely to seduce the maids," she was saying. "There'll be no woman in the house safe from him."
Too tired, Esme thought. He's definitely too tired for the maids.
Arabella giggled. "It's a pity I'm too old for the man."
Mrs. Cable gasped, but Lady Winifred chuckled. "Handsome, isn't he? I remember seeing Bonnington riding to the hounds, last year it was, before all the scandal broke. He looked as regal as a prince. A prince in a fairy story," she clarified, "not one of our own." Everyone accepted that. The royal dukes were more easily described as fat and friendly than regal.
With pressed lips, Mrs. Cable backed down. "Well, you won't accept Bonnington's proposal, of course," she instructed Esme. "But I do acknowledge Lady Withers's point about improving his soul. It is not ours to question why the Lord places a sinner at our doorstep. We must simply endure while we aid in the cultivation of a better life."
"I must try saying that to my husband," Lady Winifred murmured to Arabella. "I endure, and he never seems to cultivate. Perhaps I could bore him into virtue by reading the Bible aloud."
But Mrs. Cable heard her, and the Sewing Circle disbanded on an acrimonious note.
##
## Various Forms of Advertisement
Lady Rawlings's Rose Salon
"I suppose your mother felt she couldn't attend you," Lady Bonnington said to Esme with her usual lack of finesse. "Fanny does have strict notions of propriety."
"My dear sister is very preoccupied by the fate of the poor," Arabella said, with a little snap of her teeth. "She cannot be in as many places as she would wish."
"She wrote me as much," Esme put in. Though why on earth she always defended her mother, she didn't know.
The marchioness's expression showed exactly what she thought about Arabella's fib. "Yet during the confinement of her only daughter!" Lady Bonnington said. "Quite dismaying. You must find her absence painful," she said to Esme.
Esme smiled tightly. "Naturally I am proud of Mama's unfailing attention to those less fortunate than ourselves."
To her surprise, Lady Bonnington's eyes were not scornful; Esme could see a gleam of sympathy there. "As you undoubtedly know," she announced, "I am close friends with your mother. Perhaps the combination of my presence and your entirely acceptable engagement will be enough to change her mind. I fancy I do have some small authority in society, you know." She bent toward Esme with the fanged smile of a leopard about to spring. "If I champion your reentry into society upon your marriage to Mr. Fairfax-Lacy, I feel quite certain that the ton will quickly dismiss the foibles of your youth."
Esme gave her a weak smile. Obviously Lady Bonnington was offering her a pact. Marry Fairfax-Lacy instead of her son, and the marchioness would reinstate Esme in the good graces of her mother and society. She nodded, meeting Lady Bonnington's eyes. "That would be most kind."
At that moment the rest of the party entered the room. Sebastian strolled over to Esme. "How are you?" he said, leaning over her sofa and speaking in her ear with unmistakable intimacy.
"Stop that!" she scolded, trying to avoid Lady Bonnington's glare.
Sebastian followed her glance. "Ah, my dear mother is here. Now where's your inamorato? Mr. Fairfellow. What is his name? I loathe double-barreled names, don't you?"
"Hush, you monster!" she said, pinching his arm. Under his laughter she caught a spark of something—jealousy, perhaps? She decided that her plan wasn't a failure after all. So she held out a languid hand to Fairfax-Lacy. "Ah, there you are!" she cried. "It seemed ages since the men retired for port!"
A few moments later, Bea entered the salon to find that Stephen Fairfax-Lacy was dancing attendance on Esme in a manner that could only be called lavish. They were snugly tucked into a small couch together, and as Bea watched, Stephen tenderly rearranged the cushion behind Esme's back. She felt a prick of jealousy. Apparently Esme and Stephen had discovered a shared affinity for bawdy jokes; Stephen kept murmuring things into Esme's ear that made them both roar with laughter.
They certainly looked like an affianced couple. But Bea couldn't work out what exactly had happened the previous evening. Why had Esme announced that she and Stephen were marrying? Presumably because they had agreed to marry, a sensible little voice in the back of her head insisted. But—and this seemed the crucial question to Bea—what was Marquess Bonnington doing in the house, and what was his relation to Esme? As Bea watched, the marquess strode over to join the lovebirds. Esme began sparkling like a tree decorated with candles, and laughing (Bea thought uncharitably) like a hyena.
Bea herself was dressed for attention, and she wasn't going to get that if she kept hugging the fireplace like a debutante wearing too many ruffles to dance. So she drifted over to the group and paused for a second until they looked up.
Esme's face lit with pleasure. "Bea, darling! Do join us. Mr. Fairfax-Lacy is telling me abominable jests about codpieces."
"Codpieces?" Bea inquired, walking toward her. She was wearing a gown of slate-gray silk. Slate-gray was the kind of color governesses wore, but this gown was cut with cunning precision to make it appear that she was a governess hiding the soul of a Jezebel. The bosom was as low as an evening gown's, but the addition of a trifling bit of lace gave the bodice a faint claim to respectability. "What is a codpiece?"
Naturally, the gentlemen stood at her arrival, so Bea nimbly slipped next to Esme, taking Stephen's seat.
Stephen himself answered her question, one dark eyebrow raised. "Have you not heard of codpieces, Lady Beatrix? Gentlemen wore them in the sixteenth century. Rounded pieces of leather sometimes decorated with ribbons."
"Wore them? Where did they—" Bea broke off, suddenly guessing where they wore them. Now she thought of it, she had seen portaits of men wearing codpieces over their tights. It was wicked of him to laugh at her in such a fashion, though.
"Life must have been so much easier for women in those days," Esme said, her voice spiced with mischief. "One could presumably choose a man by the number of ribbons he wore. Bea, we must sit together all evening. Our gowns suit each other extraordinarily."
Esme was dressed in a dark silvery crimson gown whose bosom was as low as Bea's but didn't include any disguising lace. Given the fact that Esme was approximately twice as endowed in the chest area, Bea figured that the contrast was personally unfortunate. But it was better than watching Stephen nestle up to Esme's curves.
"So, would you insist your husband match his daily ribbons to your gown?" Bonnington asked Esme. There was a sardonic twist to his lips. To Bea's mind, something smoldered in the marquess's eyes when he looked at Esme. And the same could be said for the way her lips curved up at his question. If she laughed a great deal while talking to Stephen, she got a husky undertone when she spoke to the marquess that was utterly suggestive.
"Ah, what a dilemma!" Esme said. "I doubt my fiancé would agree to wear rosy ribbons, were I to wear a pink costume." She threw Stephen a languishing look.
Stephen sat down in a chair beside the settee. He was suffering from awareness of the fact that if he were indeed an Elizabethan gentleman, wearing little more on his legs than some thin stockings, he'd be grateful for a codpiece, because his body's reaction to Bea's outrageous gown would have been all too obvious. "For you, Lady Rawlings, I would wear the colors of the rainbow," he said, pitching his voice to a velvety earnestness.
"How fortunate that you, rather than I, are marrying Lady Rawlings," the marquess drawled, leaning back in his chair and crossing his legs. "Lady Beatrix, would you demand that a man make an ass of himself?"
Bea could feel Stephen watching her. She gave the marquess a look of liquid promise. Bea had a distinct preference for dark hair, but the marquess's tawny golden-brown hair could well nigh change her mind. "I do believe I would insist on the removal of all ribbons."
"Oh?" he asked. He had lovely blue eyes. If only she weren't so fond of dark eyes. "You prefer a naked codpiece, Lady Beatrix?"
"I would prefer that my husband not advertise," she said. "Don't you agree, Esme? If a man wore too many ribbons, he might become the target of many women's attentions." Bea looked at Stephen, her face as innocent as she could manage. "And the next thing one knew, one's husband would have virtually turned into a peacock, thinking that every woman within eyesight is longing for his attentions."
Vixen, Stephen thought. "Do you mean his eyesight or theirs?" he asked.
"I shall have to take the idea of naked codpieces into consideration," Esme put in. "Perhaps we should have a game of charades. There must be some Elizabethan clothing up in the attics."
She turned to Stephen and said, with a simper, "But, darling, wouldn't you mind dreadfully if I stripped you of ornamentation?"
Bea thought Esme was playing a dangerous game. There was something wild about the marquess, something ungentleman-like, that made Bea a trifle nervous. And yet Esme was toying with him as if he were a mouse and she a kitten. But it was closer to the truth to see him as a tiger, and Esme a mouse.
For his part, Stephen was fairly certain that his courtship of Esme was piquing Bea's jealousy. There was a stormy something in her eyes that he liked. So he picked up Esme's hand and told her, "I would strip myself naked, if you wished."
"Even in this state?" Esme said, gesturing toward her nonexistent middle.
"If carrying a child made every woman as beautiful as you, Lady Rawlings, England's population would be growing by leaps and bounds." Stephen kissed Esme's hand as he watched Bea out of the corner of his eye. Her hands were clenched into fists. Stephen felt a burst of cheer. As long as he wasn't knocked into a corner by Bonnington, his plan was a success.
"I do believe that most women would faint at the idea of gaining such a waistline," Esme was saying.
"The most beautiful things in nature are those about to burst into flower: a bud on the verge of becoming a rose, a tree dripping with ripe apples. And you are more beautiful than a rose, Lady Rawlings."
"Quite the dandy, aren't you?" Marquess Bonnington said to Stephen. There was a dangerous gleam in his eyes. "I wouldn't have thought a politician would have so much address. You could do much worse for a husband, Lady Rawlings."
"I merely speak the truth when I feel pressed," Stephen said promptly, hoping that Bonnington wouldn't lose control and floor him. Clearly the man had a prior claim. "Lady Rawlings is so beautiful that one can hardly stop oneself from singing her praises. It was the most surprising moment of my life when she agreed to marry me." He sighed, a languishing expulsion of breath. "The keen pleasure of that moment will never leave my memory."
Esme blushed faintly, and Bea realized that Esme had, indeed, decided to marry Stephen, no matter what her previous relationship with Marquess Bonnington might have been. Who could possibly choose to raise a child alone when she might have Stephen as a father? To Bea's annoyance, Stephen began kissing Esme's every fingertip. Now her stomach was churning with jealousy.
"Your eyes are the color of sapphires," Stephen said, his voice a low croon. "And your lips are finer than rubies."
Bea cleared her throat. Stephen looked around in a faintly irritated fashion and then said, "Forgive me, Lady Beatrix, Marquess Bonnington. You must forgive the flush of early love, the delight with which one greets his bride-to-be...."
"I've never met a woman whom I wanted to compare to sapphires," the Marquess Bonnington said with an easy shrug of his shoulders. "What appeals to me is a kind of willowy grace...an elegance of form."
Esme stiffened slightly.
"Isn't it the poet Petrarch who compares his lady to a slender willow, swaying in the breeze? That appeals to me much more than comparing my lady to flinty little gems."
"Petrarch loved a woman who was only twelve years old," Stephen said dismissively. "I leave the younger set to you, Lord Bonnington. I find young women tiresome. A woman who is a woman is the most appealing." He carefully didn't even glance at Bea. Unless he was much mistaken, a pale pink nipple was just visible through the lace of her bodice. One more look at her chest and he would pick her up and stride right out of the room, and it wouldn't be his decoration that was stripped.
Bea was having trouble biting back an unpleasant comment. Clearly she was a member of the younger set whom Stephen professed to find tiresome. And presumably Stephen expected her to compete with Esme, though how she was supposed to do that, short of stuffing her corset with all the cotton in Wiltshire, she had no idea. The least she could do was to help the cause of true love.
"Lord Bonnington," she said rather jerkily, "I brought the most exquisite book of poetry with me. And you had not yet joined the house party when we read some of it aloud. Would you like me to introduce you to the work?"
"I would be more than pleased," he said, rising and giving her an elegant bow.
Bea didn't look to see what Stephen thought. He was probably grateful. After all, if she took Bonnington off of Esme's hands, he had no competition to worry about.
They walked into the corridor together. She took a deep breath and gave Lord Bonnington the full benefit of one of her smoldering looks. There must be something wrong with her. He looked no more impressed than had Sebastian. Bea blinked to hold back sudden tears. Was she...was she losing her attractiveness to men? That was inconceivable. It was all she had.
The library was just down the corridor from the Rose Salon. Esme's library was a snug nutshell of a room, all lined with books that gave off a sleepy, satisfied smell. Bea felt better immediately. The library had been one of the few places in her father's house where she'd felt happy.
Lord Bonnington walked away from her and looked out one of the arching windows that faced into the garden, so Bea followed. She still could hardly believe that he hadn't shown her the faintest interest. Perhaps—perhaps it had been too dim in the corridor. Perhaps he hadn't seen the expression in her eyes.
It had rained all day. A silver layer of mist crept over the garden, drifting down to a blocky structure that Bea knew was the rose arbor.
"I gather you think that Lady Rawlings should marry Mr. Fairfax-Lacy," Lord Bonnington said abruptly, looking at the garden, and not at her.
"I—"
"And you brought me here to give them breathing space."
Bea swallowed. She could hardly say that she'd brought him to the library in a weak effort to make Stephen Fairfax-Lacy jealous. Or to prove that she was still desirable.
"I do think that Lady Rawlings would be happier if she were married," she said, steadying her voice.
"Married to him?"
The scorn in his voice lashed her into speech. "Esme would be extremely fortunate to marry Mr. Fairfax-Lacy!"
"He's a stick," Bonnington said, still gazing out into the garden.
"No, he's not. He's quite handsome, and he's funny, and kind. And he...he seems to care for her," Bea said.
"So do I."
What could she say to that? She stood next to him, feeling the chill that breathed off the leaded window panes.
"Did she tell you to take me away? Did she send you some sort of signal?"
"No, no," Bea said. "It wasn't like that at all! I merely...I merely..."
He turned and looked down at her. After a moment, he said, "We're in the same boat, then."
She couldn't ask what boat that was because she was afraid that she knew. "Absolutely not," she replied stiffly.
"Are you saying that you don't wish to marry that proper M.P. in there?" The touch of disbelief in his voice made her raise her head.
"I do not."
There was a skeptical curl to his lip.
"I don't wish to marry anyone." She walked over to the couch and sat down, not bothering to tilt her hips from side to side in the walk she had perfected at age fifteen. The man was not interested in her. That slow fire she saw in his eyes was for Esme, not for her.
But he did follow her, throwing himself down on the couch. "If I thought jealousy would help, I would have a go at pretending to be in love with you. But it wouldn't make any difference," Bonnington said flatly. "I'm sorry to say that the man appears enamored of Esme Rawlings. And once she draws you in, it's damn hard to look at another woman."
"I am not interested in Mr. Fairfax-Lacy," Bea insisted, more for the sake of her pride than anything else.
He didn't even answer her. "I expect he thinks you're too young."
"Too scandalous," Bea put in, unable to pretend any longer.
"Scandalous, hmmm?"
She nodded. She knew Marquess Bonnington by reputation; well, who didn't? He used to be considered one of the most upright men in the ton. There'd never been a whisper of scandal about the man until last summer. Not even a shred. If he knew her past, he would spit at her and leave the room immediately. But he didn't seem to be reacting with condemnation.
"Didn't you side-step with Sandhurst? Why on earth did you choose that odious mushroom?" he asked, and she couldn't hear any censure in his voice. Just a kind of lazy curiosity.
She shrugged. "He had a lovely bow. He complimented me."
He looked at her without saying anything.
"And my father loathed him," she added.
"I expect the noble public servant holds it against you, though." The marquess's eyes were kind, too. As kind as Stephen's. What was it with these men? They didn't react to her best overtures, and then they made her feel like crying.
"Actually, Mr. Fairfax-Lacy said that he wanted a mistress with less experience," she said, her wry grin crooked.
He stared at her. "Fairfax-Lacy said that?"
She nodded.
"You're better off without him. Why on earth would you wish to be a mistress to such an intolerable lout? Or a mistress to any man, for that matter?" He was looking at her so intently that Bea wondered whether he'd suddenly noticed she was a woman. Was he going to offer her a consolatory kiss? For all she'd drawn him to the library, she didn't want him to touch her.
"I suppose I don't wish to be a mistress," she said, dismissing the memory of Stephen's kisses. "Nor a wife either."
"Humph," he said, looking unconvinced. "Well, then, where's that poetry you brought me in here for? I shouldn't like to go back to the salon without having read some of it. Lord knows what they'll think we were doing."
Bea smiled back, feeling an unwilling pulse of friendship. He got up and threw another log into the fireplace and then walked back to the couch.
"Here it is," she said, plucking the book off the end table.
He started reading and his eyebrows rose. "I suppose this is from Esme's personal library?"
"No." She blushed. "I brought it with me. Truly, some of the poetry is quite...quite unexceptional."
Bea liked his chuckle. She drew up her legs and curled into her favorite position—the one she would never assume before a man because it didn't emphasize how slender her limbs were.
"I like this," he said. ' "O faire Boy, trust not to thy beauty's wings.'"
She nodded.
He looked over at her with a wry grin. "I spent a great deal of my life trusting the wrong things. My title, for example."
"Your beauty?" she said daringly.
"Not so much...I was convinced that I had to live up to the dignity of my title. I suppose I trusted my reputation too much."
Bea's smile mirrored his. "Whereas I simply threw mine away."
"Then perhaps you are the one who trusts your beauty overmuch." He put the book to the side. "Shall we return to the salon, Lady Beatrix?"
She put her feet down and rose. He looked down at her, and Bea felt a faint blush rise in her cheeks. "If I hadn't met Esme first, you likely would have been the making of me, Lady Beatrix."
"I'm not suitable for someone who honors their reputation," she observed, starting toward the door.
A large hand curled around her hand, drawing it under his arm. "Ah, but it wouldn't have taken long for you to convince me of the worthlessness of reputation. Esme didn't even try, and I was ready to throw it away as soon as I met her."
She looked a question as they walked through the corridor.
"She was married at that point."
"Now she isn't," Bea observed.
"And therein lies my trouble. I am of the fixed opinion that Esme should marry me and no other." He glanced down at Bea. "I am telling you this merely because I wouldn't want you to worry if I have to take out your darling Fairfax-Lacy."
"Take out?" Bea said sharply. "What on earth do you mean by that, sir?"
He shrugged. "I doubt it will come to violence. But no one is going to marry Esme but myself."
##
## Waltzing on One's Deathbed
Trying to not feel guilty because one's wife is dying is a difficult task. Damn near impossible, Rees finally decided. After all, they'd been married for years—nine or ten, he estimated. He'd married Helene when she was barely out of the schoolroom. They were both too young to know better. Yet it wasn't entirely his fault the marriage failed, no matter what she said about it.
But he never, ever thought of her as not being there. Not there to send him nagging letters, or curl her lip at him as they passed. Not there to send him horrid little notes after he debuted a new piece of music, putting her finger directly on the weakest spot, and not saying a word about the best of it.
Damn it, she couldn't die.
He'd been to Lady Rawlings's house a mere few months ago, and Helene had seemed perfectly healthy then. A little too thin, perhaps. But she was always thin. Not like Lina's overflowing little body, all curves and fleshy parts. Rees frowned. Surely it wasn't correct to think about one's mistress while riding in a carriage to greet one's dying wife. And was greet the right word?
It was with a great sense of relief that Rees realized his carriage was finally pulling up in front of Shantill House. It wasn't that he cared for his wife, of course. He didn't. Hadn't the faintest feelings for her of that nature. It was merely natural anxiety that had his chest feeling as if it were clamped in a vise. His fists kept curling, and he could bellow with rage. At what? Helene, for growing ill? No!
He had to be sweet, calm, tell her loving things. Because she was dying. His bitter-tongued, frigid little wife was dying.
God knows, that should probably have given him a sense of relief. Instead he couldn't seem to swallow, and he actually had to support himself on the side of the carriage when he descended, because his knees felt weak for a moment.
He could tell by the butler's minatory gaze—Slope, wasn't that his name?—that he probably should have changed his garments before leaping into a carriage. Instead he ran a hand through his hair, doubtless disheveling it more than before. "I've come to see my wife," he said brusquely, heading past Slope and up the stairs. He knew where Helene stayed when she was at Lady Rawlings's house. Not that he visited her bedchamber, naturally, but he'd noted the room.
Dimly he realized that Slope was calling after him. Impatient, he stopped and glared down the stairs. "What is it, man?"
"The countess is not in her chamber. She can be found in the Rose Salon."
Rees blinked. Seemed an odd place to stage a dying scene, but who was he to cavil? Perhaps she wouldn't die until tomorrow. He all but galloped down the stairs, brushed by Slope—and stopped.
A typical scene of English country life greeted him. A stout peer was dozing in a low chair by the fire. A beautiful little tart of a girl was leaning over her embroidery, her lips painted a fantastic red. And there were a few other remnants of English nobility strewn around the room.
But it was the piano that held his attention.
He'd know her playing anywhere, of course. She was seated at the pianoforte, and not by herself, either. They were playing one of Beethoven's sonatas in E-flat major. And she was laughing. As he watched, her companion leaned over and kissed her on the cheek. Kissed Helene! True, it was just a brush of a kiss. But Helene blushed.
Rees's body went from cold to burning hot and back to cold again, in the mere moment he stood in the doorway. Suddenly he was aware of the butler standing just at his shoulder, of the wintery morning sunlight making Helene's pale hair look like strands of silver. Of the very—aliveness of her. They started playing again and she was swaying slightly, her shoulder just bumping her partner's arm. Her face was glowing with joy, as it always did when she played. Always. Helene and he had only lived in the same house for a matter of months, but he'd never forgotten the way she looked when she played the piano.
It was that joy that had made him fall in love with her. The very thought shocked his senses back into movement. Fall in love? Ha!
"I see that the report of your demise was overhasty," he drawled in the nastiest tone he could summon. And Earl Godwin was pretty much an expert at giving offense.
Helene looked up, and he saw her mouth form a little Oh. But the next moment she turned to her partner and said saucily, "I'm so sorry; I almost lost my place, Stephen." And her fingers flew over the keys again, just as if he weren't there.
Stephen? Who the hell was Stephen?
Rees had a vague sense he'd seen the man before. He was handsome, in a pallid, English sort of way. Damn it, he'd been rooked. Although it wasn't clear to him why he had been called as audience. Why in the hell had his wife wanted him to jump to her bidding? He wasn't going to stand around and give her the satisfaction of gloating over his presence. For tuppence he'd turn around and head straight back for London. But he'd been on the road for two days, and his horses were exhausted.
"Excuse me," an amused voice said, just at his elbow. Lady Withers smiled at him. She was a quite lovely woman of a certain age and Esme Rawlings's aunt, if he weren't mistaken.
"Lord Godwin," she said. "How splendid to have you join us. The countess did mention that you might make a brief visit." For a moment her eyes danced over to his wife, cozily tucked against her piano partner.
"Who the hell is that?" he snarled, jerking his head backwards, dismissing the fleeting thought that he might actually greet Lady Withers.
She blinked as if the room were so filled with gentry that she might have trouble identifying the pallid Englishman. "Mr. Fairfax-Lacy is the Member of Parliament for Oxfordshire, and such an intelligent man. He also holds the honorary title of the Earl of Spade, although he chooses not to use it. We are all enjoying his company."
Rees was pulling himself together. He'd be damned if he showed any sort of husbandly emotion before a smirking viscountess. And since he wasn't feeling any of those husbandly feelings, that should be simple. Unless murderous was considered a husbandly emotion.
Then Helene was before him, holding out her hand and sinking into a curtsy. "Rees. I must apologize for my letter," she said, as tranquil as ever. "While the midwife in the village did suggest I had pleurisy, it turned out to be something far more innocent."
"Oh?"
"Well, you see pleurisy starts with a red rash. But I had beard burn, as it turned out," she said, laughing slightly. "Aren't I the naive one, then? I suppose you were so young when we married that I never encountered this problem."
Her laugh was breathy, perhaps a sign of nerves. But Rees wasn't going to give her the satisfaction. Any satisfaction. He just looked at her, and the giggle died on her lips. "You are still my wife—" he began.
She put a hand on his arm. This was not the naive girl he'd married. Not the Helene he woke up with the day after they returned from Gretna Green, a girl who veered madly between shrieking tantrums and sullen tears. She was poised, cool, and utterly unapproachable.
"Only in name, Rees. Another woman shares your bedchamber now."
He looked over her shoulder. Fairfax-Lacy was practicing chords. He played well. Presumably she wasn't sleeping alone in her bedchamber. "A gentleman who planned to be at your side during divorce proceedings wouldn't sit at the piano while you face an irate husband," he said, his tone polished steel.
"You are hardly an irate husband," she said, shrugging. "I asked Stephen to remain where he is. I hardly think you are interested in making his acquaintance. And who said anything about divorce?"
"So you've taken a lover," Rees sneered, on the verge of crashing his fist into that sleek bastard's face. "What is it all in aid of, Helene?"
"Pleasure," she said, and the smile on her face burned down his spine. "My pleasure, Rees."
He turned on his heel and then back at the last second. "Who did that arrangement of Beethoven for four hands?"
"I did. I've been rearranging all of them."
He should have known that. The sonata sounded half like Beethoven and half like Helene, an odd mixture.
"Now we have that little discussion out of the way," Lady Withers said brightly, coming up from somewhere, "why don't I show you to your room, Lord Godwin? I do hope you'll make a long stay with us."
Rees turned like a cornered lion and snarled at her, then strode out of the room. As Arabella described it later to Esme, who hadn't been in the Rose Salon at the time, Earl Godwin acted precisely like the Wild Man of Deepest Africa whom Arabella had seen once in a traveling circus.
"All hair, and such a snarl, Esme!" Arabella paused, thinking about it. "Honestly, Helene, your husband is quite—quite impressive." There was reluctant respect in her voice.
"Oh, Rees is very good at snarling," Helene said. She, Arabella and Esme were cozily seated in Esme's chamber, drinking tea and eating gingerbread cakes.
Esme looked up from her plate, her eyes sparkling with laughter. "The important thing is he snarled because you managed to tangle his liver—or whatever that phrase is that you keep using, Helene."
"Curdle his liver," Helene repeated, and there was a growing spark of happiness in her eyes. "He did seem chagrined by our conversation, didn't he, Lady Withers?"
"Chagrined is not the word," Arabella replied, stirring a little sugar into her tea. "He was incensed. Absolutely incensed. Purple with rage."
"I hope he's not feeling too violent," Esme said. "I can hardly have my future husband mangled by your present husband, Helene. It would all be such fodder for gossip if the servants shared what they know."
Helene thought about the difference between what the servants thought they knew about her and Stephen, and what the truth was. "I do think you could have left Stephen to me," she told Esme somewhat peevishly. "What if Rees discovers that you have claimed my lover as a future husband?"
"I doubt very much that your husband will raise the subject with Stephen," Esme replied. "Rees already announced that he will stay one day at most, so Stephen only has to briefly juggle a fiancée and a mistress. He'll not be the first to do so. I can't tell you how many times I found myself at a table that included Miles and Lady Randolph Childe. Miles always acted with the greatest finesse, and if my husband could do it, so can Stephen."
Arabella chortled. "Supper will be an interesting meal. Mr. Fairfax-Lacy will face quite a difficult task. You, Helene, wish him to impress your husband with his devotion, and you, Esme, wish to impress the marquess with Mr. Fairfax-Lacy's devotion. Hmmm, perhaps I could ask Bea to create a diversion by flirting with Earl Godwin?"
"There's no need to go as far as that," Helene said hastily. "And do you know, I have the strangest feeling that Bea might be having some feelings for Mr. Fairfax-Lacy herself? There's something odd about the way she looks at him."
Esme laughed. "That would make three of us chasing the poor man. Arabella, are you certain that you have no use for Mr. Fairfax-Lacy?"
"Quite sure, thank you, darling," Arabella said, carefully choosing a perfectly browned gingerbread. "It seems to me that the poor man must be growing tired. I dislike fatigued men. Still, it seems to be quite enlivening for him," she continued rather absently. "The man was getting hidebound. He looked so cheerful this morning. And that, of course, is your doing," she said, beaming at Helene.
Helene hid a pulse of guilt. She was hardly enlivening poor Stephen's nights, even though the whole house believed she was. Now Esme was smiling at her too. Her sense of guilt grew larger.
"I'm very proud of Helene," Esme said. "Arabella, you can't imagine how impossibly rude Rees has been to poor Helene over the years, and she's never staged even the slightest rebellion until now."
"Now that you've rebelled," Arabella asked Helene with some curiosity, "what will be the outcome? Are you wishing to continue your relationship with Mr. Fairfax-Lacy? That is, if Esme gives up her rather dubious claim to him?"
"I wouldn't call it dubious," Esme put in. "Merely unexpected."
"No," Helene admitted. "I don't wish to remain his friend."
"I knew that," Esme said. "I watched the two of you. Otherwise, I would not have claimed him as my own future husband, I assure you, Helene."
"Stephen Fairfax-Lacy is good marriage material," Arabella said. "I am never wrong about that sort of thing. All three of my husbands were excellent spouses." She finished her gingerbread and added, meditatively, "Barring their short life spans, of course."
"I have to tell you something," Helene said rather desperately.
"I do hope you are going to tell us intimate details," Arabella said. "There's nothing more pleasurable than dissecting a man's performance in bed. I believe it's my favorite activity, perhaps even more fun than actually being in that bed." She looked faintly appalled. "I surprise myself," she said, picking up another cake. "Ah, well, that's the benefit of being an elderly person."
"You're not elderly, Aunt Arabella!" Esme said. "You're barely out of your forties."
"I'm not really bedding Mr. Fairfax-Lacy," Helene blurted out.
Arabella's mouth fell open for a second before she snapped it shut.
"I thought so," Esme said with some satisfaction. "You don't have the air of a couple besotted with each other."
Helene could feel her face reddening. "We didn't suit."
"I had that happen to me once," Arabella said. "I won't bore you with the details, my dears, but after his third attempt, I called for a truce. A laying down of arms," she clarified with a naughty smirk. "Well, who would have thought? Fairfax-Lacy looked so—"
"No!" Helene cried, horrified by the conclusion Arabella had drawn. "It truly was all my fault. I'm just not—" She stopped.
To her horror, she felt tears rising to her eyes. How could she possibly admit to a failure in bedroom activity when she was seated with two of the most desirable women in the ton?
"You know, I find him quite uninteresting as well," Esme said quickly. "It's something about the Englishness of his face. And his chest is quite narrow, isn't it? Moreover, I never liked a man with a long chin."
Helene threw Esme a watery smile. "It's not that I don't like the way he looks. I do. It's just that I found myself unable to countenance the prospect of bedding him." Her voice dropped. "He was very kind about the whole thing."
Arabella nodded. "There are always men whom one simply cannot imagine engaging in intimate circumstances. Unfortunately, I felt that way about my second husband. But what really interests me," she said, turning a piercing eye on Esme, "is what exactly you are doing, announcing your engagement to a man with an overly long chin? Or, to put the same question another way, what is Marquess Bonnington doing in this house, Esme?"
Esme almost choked on the gingerbread she was eating. "Expressing his repentance?" she said hopefully.
"Don't repeat that poppycock story that I fed your Sewing Circle!" Arabella cried. "You managed to evade my every effort at a confidential talk last night by clinging to your new fiancé's arm, but now I would like to know the truth. Why has the marquess arrived in your house?"
Helene leaned forward. "I would like to know that as well, Esme. I accepted that his mother was likely here due to the circumstances of last summer, although it seemed extremely odd—"
Arabella interrupted, naturally. "Odd? There's something deuced smoky about Honoratia Bonnington's arrival."
Esme sighed.
"You sound like a bellows," her aunt observed. "Out with it!"
Esme looked up at her aunt for a moment. Arabella looked as delicate and sensual as if a wisp of wind would blow her away, and yet she and the formidable Lady Bonnington were certainly forged of the same steel. So Esme told. "But I don't wish to marry anyone," she finished. "Least of all Lord Bonnington. It wouldn't be fair to Miles, or to the babe."
After a moment of stupefied silence, Arabella burst into a cackle of laughter. "Want to just keep Bonnington around for those lonely nights, do you? And you had him working in the garden during the day? Here I thought you were bent on a sober widowhood! Lord, Esme, even I never created a scandal akin to this one!"
"What scandal?" Esme demanded. "You stopped the Sewing Circle from even considering the possibility with all your quotations."
"Which took me a good hour of poring over a Bible, I'll have you know!" Arabella said.
"Esme, do you think it might be time to give up the Sewing Circle?" Helene asked tentatively. "Things are a trifle...complicated in your life. Perhaps it would be best if you weren't under quite so close surveillance."
"It's part of my new respectability," Esme said stubbornly. "I rather enjoy it."
"Not that I noticed," Arabella commented. "You sewed a miserable seam. Some people are simply not gifted in that department."
"You know, Mama makes shirts for the poor," Esme said. "The whole shirt, even the collar and cuffs."
Arabella was silent for a moment. "Lord, Esme, I never like to think of myself as someone who could say ill of my own sister, but Fanny is dim-witted. She's spending all that time making up collars for people she doesn't even know, and her own daughter is alone in the country. She's got her priorities in a tangle." She reached over and gave Esme's hand a little squeeze. "Don't go changing yourself into your mother. You have always had a merry soul. But Fanny has grown into a rather dreary adult, if I say so myself."
"That's not fair," Esme objected. "Mama has had a great many disappointments in life." Obviously her daughter was foremost on the list.
"She's dispirited," Arabella said firmly, "although it's good of you to defend her. Fanny spends all her time gazing at the world and pursing her lips. I've always been glad to have one relation with a grain of sense in her head. I can't afford to lose you to the ranks of straitlaced matrons."
"Your aunt is right," Helene put in. "I have only the slimmest acquaintance with your mother. But the idea of you growing as prim and prissy as that Mrs. Cable is simply dispiriting. She's not a very nice woman, Esme."
"I know," Esme said. "Believe me, I know."
Arabella took a look at her niece and judged it time to change the subject. But while she and Helene chatted about the Venetian lace points that adorned Helene's sleeves, Esme sat in silence. She had promised Miles that she would be a respectable mother, yet Miles was gone. She thought never to make another scandal, and she could think of no scandal with the explosive force that her marriage to Marquess Bonnington would have.
##
## A Taste for Seduction
The next morning Bea stamped down the lane to visit the goat. She'd taken to visiting the devilish creature every morning, from the pure boredom of life. Of course, she could have spent her time flirting with the Puritan. But annoyingly, irritatingly, hatefully, he and Helene were learning to play a four-handed piece. The sight of Helene's pale braids bent close to Stephen's dark head as they whacked away on that pianoforte gave Bea a strange kind of longing, the kind that pinches your heart. It wasn't an emotion that Bea was familiar with at all.
The one time they had been alone together after breakfast, for the merest moment, he had looked down at her with a rather wintery smile and said, "I gather that you have decided not to woo me?"
She had answered, "I never woo," hoping that he would kiss her or smile at her the way he did at Esme and Helene. But all he'd done was bow and walk away. Bea had realized in that very moment, watching his back, that there was nothing in the world she wanted more than to woo the man. But Stephen showed little desire to secure any sort of relationship with her. How could he, in truth? He had no time. When he wasn't playing an instrument with his mistress, he was exchanging bawdy jests with his fiancée. Lord knows where he was at night. Bea ground her teeth together. She was making a regular occupation out of thinking about Stephen Fairfax-Lacy and then admonishing herself for doing it. She held out a branch she'd brought the goat and watched him chew it into kindling.
In fact, Lady Beatrix Lennox was suffering from a mighty loss of confidence. First Mr. Fairfax-Lacy had refused her as a mistress and taken Helene instead. And Marquess Bonnington hadn't shown the faintest interest in her from the very beginning. Bea had to blink very hard to hold back tears.
The goat was chewing so loudly that it was no surprise that Bea didn't hear anyone approaching her. "Aren't you afraid to approach that spencer-eating beast?" said a voice at her ear.
She was like some sort of trained dog, Bea thought miserably. All she had to do was hear his voice and her knees weakened.
"The goat doesn't bother me," she said, not turning to look at him. What was the point? He was leaning on the stile next to her, seemingly unperturbed by her graceless welcome.
"We should introduce the rest of the party to this fascinating creature," he said idly. "I don't believe that Esme even knows of his existence. Whereas I find myself compulsively visiting the beast every day."
Bea's heart hardened. "I thought you and Lady Godwin were spending your time together," she said, being deliberately rude. "Or is it Lady Rawlings who occupies more of your time?"
"Not every moment. And never tell me that you're jealous." His voice took on that dark, sweet note that drove Bea to distraction.
"Absolutely not!" she said, turning and facing him for the first time. He was—He wasn't so fabulously handsome. He had wrinkles on the edges of his eyes. And his chin was too long. God, how she hated a long chin!
"I'm glad," Stephen said. She couldn't read his eyes. Was he making fun of her? No, that was a look of concern. Damn it all.
"Because Esme and I..." He hesitated.
"You needn't tell me," Bea put in. "I can see the truth for myself. And I assure you that I haven't the slightest feeling about it except happiness for the two of you."
"I'm glad to hear that." It was so unfair that his smile could make her stomach clench. Long chin, long chin, long chin, she thought to herself.
"Esme and I seem to have so many interests in common." Apparently he was feeling quite chatty now that he'd cleared away any misconceptions Bea might have had. "I had forgotten how much I enjoy word play and jests."
"Lovely," Bea said listlessly. She had been routed by a fleshy woman nine months with child. The fact that Bea genuinely liked Esme (and Helene, for that matter) didn't help.
Stephen looked aside at his little Bea. Unless he was quite mistaken, his campaign was working. She was lurid with jealousy. "Do you enjoy jests?" he asked.
Apparently she was supposed to engage in a contest of bawdy jests in order to obtain the great honor of being yet another woman in Mr. Fairfax-Lacy's life. Of course she wouldn't do such an ignominious thing. "I know a few," she said, despite her best intentions. "Do you know the ballad that begins: 'He's lain like a log of wood, in bed for a year or two, and won't afford me any good, he nothing at all would do?' There are quite a few verses."
He laughed. "Perhaps men don't care to repeat that particular ballad amongst themselves." His eyes warmed her to her stomach, sent pangs of warning to her heart.
"I am thinking of returning to London, Mr. Fairfax-Lacy," Bea said, making up her mind on the spot. "I must visit my mantua-maker. After all, my favorite garment was eaten by this animal." The goat rolled his eyes at her.
"Oh," he said. And then, "Are you then determined not to woo?"
"How many times must you ask me?" Bea snapped. The arrogance of the man was incredible. Incredible! Bea peeked a look at him from under her lashes. He looked almost—well—anxious.
"My besetting sin is arrogance, it would seem," he said. "Although I had not realized it until recently. I truly apologize if I misinterpreted your interest in me when we played billiards together."
"No, you didn't!" she wanted to shriek. Why wasn't he wooing her? Why wasn't he trying to seduce her?
She peeked another look. It was no use. He had the longest chin in Christendom, perhaps, but she wanted to kiss him desperately. Or rather, she wanted to be kissed by him. And it seemed that there was still a chance, before Esme scooped him into a forty-year waltz. But she couldn't quite bring herself to give him one of her seductive looks. She was feeling paralyzingly shy, and there they were in front of the goat, and—
"I'll think about it," she mumbled.
"What? I'm sorry, I didn't quite catch what you said." He was leaning slightly against the fence. He looked like the most respectable, prudish, Puritan in the world. Not her sort at all. Too old, for one thing. And too opinionated. And too—too desirable.
"I said, I'll decide today whether I wish to woo you," she said painstakingly.
"Oh, good."
The infuriating man acted as if they were discussing a trip to see a Roman monument. Bea couldn't think of anything else they had to say to each other, so she made her farewell and then walked listlessly up the lane, swinging her parasol at a rock with the misfortune to be in her way. It was only in front of him that she pretended there was a decision to be made: and that was merely because of an instinctive feminine wish to protect herself.
Tonight she would spent an hour bathing, two hours dressing, and even longer painting her face, and she would seduce that man, by God, if he were seducable.
##
## The Experience That Divides
the Ladies from the...Women
Esme stared out the window of the drawing room. They were having a late spring flurry of snow. The white flakes were making the yellow crocuses on the side of the house look pale and betrayed. Or perhaps it was she who was betrayed. Or was it she who was betraying?
The comedy of errors that made up this particular house party was astonishing. She and Mr. Fairfax-Lacy, to all eyes, were apparently planning to marry. Equally well known to all was the fact that Helene was having an affair with the said Mr. Fairfax-Lacy, although it didn't seem to have given Helene's husband even a qualm. The earl was leaving the next morning, but as far as Esme could determine, he was thoroughly enjoying bickering with Helene over her reformulations of Beethoven and had paid no attention whatsoever to Stephen Fairfax-Lacy's lavish compliments to his wife.
Today the pain in her lower back was even worse than usual. She could hardly stand up, it hurt so much. The door opened behind her.
"Hello," she said, not bothering to look around. It was amazing how closely her ears were attuned to the sound of his step, rather than those of the other two dozen persons thronging her house. He stood just behind her and, without even being asked, pressed his thumbs sharply into the base of her spine. It felt so good that Esme's knees almost collapsed.
"Steady there," he said. "How is that babe this morning?"
"I received a letter from my mother," Esme said, turning around and looking up at him. "Fanny is coming to visit, thanks to your mother's persuasive powers. Much though I loath it, I am going to have to express gratitude to Marchioness Bonnington."
Sebastian narrowed his eyes. Didn't Esme have any idea why his mother would have done such an act of benevolence? "My mother didn't do it out of the kindness of her heart," he pointed out.
"I know, I know." The smile that spread across her face was genuine. "But I am glad that Mama is coming. It must be because I'm having a child myself. And because Miles is dead, of course."
Of course, Sebastian thought cynically. He was getting sick of Esme referring to her husband as if he'd ever played a significant role in her life.
"Don't you see that your mother is coming here solely to ensure that you do indeed marry Fairfax-Lacy?" he asked harshly. "Once you disappoint her again, she'll drop you like a hot potato."
"There's always the small chance that I won't disappoint her," Esme replied icily.
Sebastian snorted. "Your mother is the sort of woman who would find something to criticize if you had taken on a nun's habit."
"I mean to be respectable, and I shall be," Esme said. But her heart wasn't in the argument: her back hurt too much.
"You are pretending not to be in love with me. You're a hypocrite, Esme, and you're making a terrible mistake."
"I don't feel very well," Esme mumbled. It wasn't only because she didn't want to think about Sebastian's offensive comment. Her back hurt so much that she seemed to be hearing his voice through a fog, as if filtered through cotton wool. "Perhaps I ought to go to my chambers."
At that moment the door opened and a flood of chatting houseguests swept in. Lady Bonnington took one look at Esme and announced, "I do believe Lady Rawlings is having that baby now."
"Well, you've done this sort of thing before," Arabella said to her with a tone of mild panic. "Tell the poor girl what to do."
"Don't be more of an idiot than nature made you!" Lady Bonnington snapped. "Obviously she needs to retire to her bedchamber."
"I see no occasion for rudeness," Arabella replied, bristling.
Esme took a deep breath. She was surrounded by a ring of faces. A second later Arabella was pushed to the side, and Sebastian bent over Esme.
"Up you go," he said to her, with a tone of unmistakable intimacy. Before she could protest, he picked Esme up in his arms and started carrying her up the stairs, looking for all the world as if he knew directly where he was going.
"Oh!" Esme gripped his arm as her entire body shuddered and seemingly attempted to turn itself inside out. She dug her fingernails into his arm.
"Call the midwife!" Sebastian yelled over his shoulder. A moment later he had her in one of the spare bedrooms, on a bed specially prepared for just this occasion. But Esme didn't let him put her down.
"Wait!" she gasped. He started to lower her to the bed. "Wait, damn it!" She hung on for dear life as another wave swept through her body. Just then the door popped open, and in streamed Arabella, Helene, Marchioness Bonnington, and three maids.
"All right, Bonnington," Arabella said importantly. "If you could just put my niece on the bed, we'll carry on from here. The midwife will be here directly; the silly woman had taken a walk to the village. Just try to keep that baby where it is until she arrives, all right, Esme?"
"Don't be a widgeon!" Lady Bonnington said, marching over to the side of the bed. "The babe will not arrive for hours."
"God, I hope that's not the case," Esme gasped.
"That's the way of it," the marchioness said. Her tone was not unsympathetic.
Esme let go of Sebastian's hand. He bent over her for a moment, pressed a kiss on her forehead, and then he was gone. She felt a bit like crying, except another pain rushed up from her toes and stole her attention away. "Bloody hell," Esme said in a near shout, reaching out and grabbing Arabella's hand. The pain receded, and she flopped back on the bed, drained.
"Profanity will not ease the pain," Lady Bonnington observed. "My own mother told me that what distinguishes a lady from a lower being is that a lady accepts pain without rebuke."
Esme ignored her. "How many of these pains will there be?" she demanded of the midwife as she entered the room.
Mrs. Pluck was a thick-set woman who was cheerfully confident about the "course of nature," as she called it. "I expect you're in some discomfort," she said, bustling about with a stack of towels. "But you've got the hips for a quick one." She chuckled in a wheezing sort of way. "We must let nature take its course, that's what I say."
"My niece will dispatch this business with...with dispatch," Arabella announced, surreptitiously examining the red patches on her hand where Esme had squeezed her. "Bring me a wet cloth," she snapped at one of the maids. "Esme, darling, you're rather unbecomingly flushed. I'll just bathe your forehead."
"Took me all of six hours," Lady Bonnington trumpeted.
Esme immediately decided that she was going to birth her baby in less than six hours. She'd never survive an ordeal as long as that. "Oh no," she moaned. "Here it comes again."
Arabella hastily dropped her wet cloth, and Esme grabbed her hand. The tidal wave came, swept her down and under, cast her up gasping for air. "I don't like this," Esme managed to say in a husky whisper.
"Never knew a woman who did!" the marchioness said cheerfully from the side of the bed. "All a lady can do is endure with fortitude, showing her well-bred nature in every moment."
Esme responded with flat profanity.
As the marchioness thought later, if she hadn't already known that Esme Rawlings was an appallingly ill-bred woman, she would have known from that moment on. The gel just had no idea how a lady behaved under duress.
##
## Sweet William
Giving birth in the presence of two elderly ladies of the ton was without a doubt the most uncomfortable experience of Esme's life. Arabella stood at her right, bathing her forehead every time one of the pains ended. Esme emerged from a swooping black wave of pain to find that Lady Bonnington, standing to her left, was exhorting her to greater efforts, and Arabella, not to be outdone, was instructing the midwife to hurry things along.
"There's no need to hurry things along," Mrs. Pluck, the midwife, responded with a glimmer of irritation. "The course of nature will do it. And Lady Rawlings has the hips for it, that she does."
"A little less conversation about my niece's hips, if you please," Arabella snapped. "There's no need to be vulgar."
"Arabella, you're a fool," Lady Bonnington announced with her usual politeness.
Esme took a breath, feeling the pain coming again. It was worse than she had ever imagined, rather like being scalded from the toes up. She struggled her way back out of the pain a moment later, dimly hearing Arabella's congratulations. Her aunt seemed to have decided that Esme needed applause after every contraction. And Esme definitely agreed with her. "Where...where's Helene?" she gasped at one point.
Lady Bonnington looked shocked. "Naturally, we sent her out of the room. The poor girl hasn't had a babe of her own, you know. This is enough to put her off for life."
"Oh no," Esme moaned. The next contraction was coming, sweeping up from her toes—
"Fortitude, darling, fortitude!" Arabella said, taking her hand even more firmly. Esme clutched her hand.
"You've got the hips for it," Mrs. Pluck said from the bottom of the bed. And then, "We're almost there, my lady. I told you this would be a ride in the park, didn't I?"
A ride in the park it wasn't. But Esme couldn't summon up the breath to argue the point. Instead she let the pain wrench her bones from their sockets, or so it felt. Arabella was alternating between putting a cool cloth on Esme's head and wrapping it around her own hand.
"All right, my lady," Mrs. Pluck said loudly. "Time to bring the little master into the world."
Or daughter, Esme thought, although she couldn't summon up the wherewithal to say so. But Mrs. Pluck was right.
Squealing, indignant, fat and belligerent, William Rawlings entered the world in a burst of rage. Esme propped herself on her elbows. There he was: dark red from pure anger, kicking jerkily, waving his fists in the air. Her heart turned over with a thump. "Oh, give him to me," she cried, pushing herself into a half seated position and reaching out.
"He'll need a good bath, and after that I will check all his toes and make certain that he is presentable," Mrs. Pluck replied, handing the baby to the waiting nursemaid.
"He seems to be a boy," Arabella said, ogling the baby. "My goodness, Esme. He's remarkably well endowed!" She giggled. "It looks as if he has two turnips between his legs."
"They're all like that," Lady Bonnington said with a tinge of nostalgia in her voice. "My son was just the same. I thought he was going to be a satyr."
"Just a minute, my lady," Mrs. Pluck said. "Just one little push now."
A few minutes later, Esme hoisted herself into a sitting position. "I'd like to hold my son, please," she said hoarsely. "Please—now!"
Mrs. Pluck looked up. "Everything in good time, my lady. After we—"
Arabella reached over and snatched the baby out of the nursemaid's arms. "Lady Rawlings wants to hold her son." She put him, rather awkwardly, in Esme's arms. He was still howling, fat little legs jerking out of the blanket.
"This isn't wise," Mrs. Pluck scolded. "It's best if the baby is washed within the first five minutes of its birth. Cleanliness is essential to good health."
"There's time for many a bath in his future," Arabella said, bending over the bed. "He's so plumpy, isn't he, Esme? And look at his gorgeous little toes!"
Esme had never felt anything quite like it. It was as if the world had narrowed to a pinhole, the size of herself and the baby. He was so beautiful that her heart sang with it. And yet he was remarkably homely as well. "Why is his face so red?" she asked. "And why is his head this peculiar shape?"
"The course of nature," Mrs. Pluck answered importantly. "They all look like that. Now you'll have to give up the baby, my lady. We have just a few more things to do here."
But the baby had decided to open his eyes. Esme clutched him closer. "Hello there," she whispered. "Hello, love." He blinked and closed his mouth. His eyes were the pale blue of the sky on a very early morning, and he looked up at her, quite as if he were memorizing her face. "I know you think you're smiling at me," Esme told him, kissing his nose and his forehead and his fat little cheeks. "You just forgot to smile, didn't you, sweet William?"
"Are you naming him William?" Lady Bonnington asked. "I suppose that must be an old name in Lord Rawlings's family—his father, you know," she told the nursemaid, who looked blank.
William's eyes were sweet and solemn, trusting Esme to keep him safe, trusting her to nourish and protect him. For years and years to come. A chill of fear fell on Esme's heart. Benjamin, her own little baby brother, had died. Of course, it wasn't the same, but sweet William was so enchantingly dear. He frowned a little as a drop fell on his cheek.
"What is it, darling?" Arabella asked. "Oh no, the baby's getting wet. Shall I take him?"
"He's the most beautiful baby I ever saw," Esme said, hiccuping. "I lo-love him so much. But he's so little! What if something happens to him? I couldn't bear it!"
"She's having some sort of reaction to the birth," Lady Bonnington observed. "Takes some women that way. My second cousin twice removed went into a decline after her daughter was born. Mind you, that husband of hers was enough to send anyone into a decline."
Esme swallowed and dried her eyes on the sheet. "He has Miles's eyes," she said to Arabella, ignoring the marchioness. "See?" She turned William toward Arabella. "They have just that sweetness that Miles had. Miles's eyes were that same blue. And Miles is dead."
"But his son isn't," Arabella said, smiling down at her. "William is a fine, sturdy baby, with nothing fragile about him."
"I agree," Lady Bonnington said promptly. "I knew immediately that the baby was the image of your husband."
Arabella threw her a look of potent dislike. "Why don't you go transmit this happy news to your son, Honoratia?"
"I shall," the marchioness said, "I shall. And may I say, Lady Rawlings, that I am impressed by your handling of this entire delicate matter?"
Whether Lady Bonnington meant to refer to the process of giving birth, or that of identifying Esme's child's father, no one could tell. Mrs. Pluck took the baby, who promptly started crying again.
"He wants to be with me," Esme said, struggling to sit upright.
"He has a good healthy voice," Mrs. Pluck said, handing him to the wet nurse. "But the course of nature must take its course, my lady," she said, rather obscurely.
Helene had come in and was peering at the baby as the wet nurse wrapped him in a warm cloth. "Oh, Esme, he's absolutely lovely," she said.
"Does he look healthy to you?" Esme asked the nursemaid.
"Fat as a suckling pig," the wet nurse said promptly. "Now shall we see if he'd like some breakfast?"
She sat down and pulled open the neck of her gown. Esme watched as William turned to the nurse and made a little grunting noise. He had those great blue eyes open, and he was looking up at the nursemaid. From Esme's point of view, it looked as if he were giving the woman the very same blinking, thoughtful glance he had given her.
White-hot jealousy stabbed her in the chest. That was her baby, her own sweet William. "Give him to me!" she said sharply.
The wet nurse looked up, confused. She had William's head in position and was about to offer him her breast.
"Don't you dare nurse my baby!" Esme said, her hands instinctively clenched into fists. "Give me William this instant!"
"Well, my goodness," the wet nurse said. "You hired me, my lady."
"I changed my mind," Esme snapped. She was not going to have William think that anyone else was his mother. She would do everything for him that needed to be done, including feeding.
The wet nurse pursed her lips, but she brought over the child. "It doesn't come easily, nursing a child," she said. "It's quite painful at the beginning, and there's many a woman who can't master the art of it. And ladies don't have the breasts for it, to tell the truth."
"I have the breasts for it," Esme said with all the authority she could command. "Now, if you'll just tell us how to do this, I'd like to give William his breakfast."
"If you don't mind my being blunt," Lady Bonnington announced, "that very idea makes me feel squeamish. A lady is more than a milch cow, Lady Rawlings!"
"Oh do go on, Honoratia," Arabella said impatiently. "Don't you have something important to tell your son?"
Lady Bonnington left the bedchamber feeling a bit piqued. After all, Lady Rawlings was no relative of hers, and yet there she'd stayed, for all of three hours, counseling her to greater efforts. It was quite likely due to her efforts that Lady Rawlings had managed to get through the birth so quickly. But, on the other hand, she couldn't have hoped for a better outcome. Lady Rawlings herself had identified the child's father, and that was all there was to it. Sebastian would have to acknowledge himself free of responsibility now.
"He doesn't look a bit like you," the marchioness told her son with more than a twinge of satisfaction. "He's bald as a belfry, just like his father."
"Miles Rawlings had hair until a few years ago," Sebastian pointed out.
"You wait until you see that child," his mother said, rather gleefully. "He's the image of his father. You needn't feel a moment's anxiety about whether you have a responsibility to him. You haven't. Lady Rawlings started weeping the moment she saw him, because the child has her husband's eyes. There's no doubt about it. Miles Rawlings has a posthumous son."
Lady Bonnington paused and looked at her son. He seemed a little pale. "You're free," she said, rather more gently.
He looked at her, and the expression in his eyes shocked her to the toes. "I don't suppose she asked for me?"
"No," his mother said, shaken. "No, she didn't."
She bit her lip as Sebastian turned about without another word and walked from the room. Could it be that he was more entangled with this woman than she thought? No. But it seemed that she had underestimated the amount to which Sebastian had hoped the child would be his own. I'll have to get the boy married off as soon as possible, the marchioness decided. To a girl from a large family, a woman who wouldn't be loathe to have more than one child herself. Although if my future daughter-in-law shows the faintest interest in turning herself into a milk cow, I'll have to set her straight.
There are some things that would never happen in the Bonnington family. And that sort of ludicrously ill bred behavior was one of them.
##
## In the Library
Beatrix Lennox had made up her mind. She had dillydallied enough over the question of Stephen Fairfax-Lacy. In fact, she had given him far too much importance in her life. She had never had the faintest wish to invite a man to her bed twice; actually inviting one into her chamber was the best way she knew of to utterly blot out any future desire for his company.
Dressing to seduce Stephen took her all afternoon. At the end, she was certain that she was utterly delectable. She was scented and polished and curled and colored...every inch of her. She wore no corset, and no cotton padding; instead she chose a gown that offered everything she had to the world in a burst of pagan enthusiasm. It was of French silk, shaded in a subdued blue-green color that turned her hair to flame. It was daringly low, and ornamented with ribbons of a slightly darker shade.
There were very few covers at the table for dinner, of course. Esme apparently would not even rise from her bed for a matter of days or weeks. Bea paid Stephen almost no attention during the meal, allowing him to flirt as he wished with Helene while Earl Godwin watched with a sardonic expression. She had no wish to engage in a noticeable competition with Helene. After all, just the previous evening Helene had lavishly thanked Bea for her help in gaining Stephen's friendship. She would likely be somewhat startled if Bea snatched him back before her very eyes.
When she sauntered into the salon after supper, the earl and countess were, naturally enough, already hammering away at the piano. Stephen's eyes darkened when he registered her gown. No matter how censorious the man might be about her face painting, he liked that dress. There wasn't a man alive who wouldn't like it.
"You look delicious!" Arabella cried, holding out her hands. "Trust my own Bea to keep us from falling into country doldrums. If we spend too much time here, we're likely to stop dressing for dinner at all!"
Bea gave a faint shudder. The idea of wearing the same clothing through an entire day was intolerable.
"Bea," Helene called, looking up from the piano, "would you very much mind trying my waltz again? I would like to demonstrate it to Rees."
Perfect.
Bea turned around to find that Stephen was already at her side. His eyes were almost black, and she felt a surge of female triumph. Why shouldn't she woo if she wanted? Men had their way far too often in life, as it was. She dropped into a low curtsy, putting out her hand to Stephen. He bowed and straightened, kissing her hand. Then he paused for one second, gazing at her arm. Bea looked down. There was nothing odd about her arm. "Are you quite all right?" she asked.
"I had a sudden recollection of dancing with a young lady who shall remain unnamed," he said. "She left marks of white powder all over my coat."
Bea raised an eyebrow. "This white is all my own."
Their eyes caught for a second, and she let her smile tell him that the rest of her was just as white, and just as unpowdered.
Then the music began. Helene had curbed the waltz's frenzied pace somewhat. It still rollicked, though. Bea was shivering with excitement. Now that she'd finally got herself to this point, she couldn't imagine why she'd ever wasted the previous fortnight thinking about it. Wooing—wooing was like breathing, for her. Why hadn't she seen that? She smiled at him and let just a hint of the desire that was pounding through her body show. Just a hint.
He didn't respond, which was a little disappointing. All he did was swirl her into another series of long circles that carried them down the long length of the Rose Salon. Bea couldn't help it; the very feeling of his hand on her waist made her feel greedy. She edged closer. He seemed to push her away. Her heart was beating so fast that she could hardly hear the music.
"Have you joined the adoring hordes in the nursery?" Stephen asked.
And what would be the point of that? Bea wanted to shriek. Ladies such as herself never had children! They had men...not babies. She didn't want one anyway. William looked like a buttery little blob to her. The moment she peered at him, he started to cry, and the very sound put her teeth on edge. "I'm not very maternal," she said.
Stephen drew her into a sweeping circle. "I don't seem to be developing a paternal side either," he said, once they were straight again. "Helene is agog over that child."
Bea didn't want to talk about Helene. She had to get this done. "Mr. Fairfax-Lacy," she said, and stopped.
He bent his head, that dark head that was so beautifully shaped. "Yes?"
"If you would join me in the library, I should like to discuss a poem with you."
His eyes were inscrutable. Yet surely he knew what she was saying! He'd told her to woo, after all. She managed to smile, but it was hardly a seductive triumph. Then she just waited, heart in her throat.
"I would very much like to finish reading you a poem," she said steadily, when he didn't respond.
He raised an eyebrow. He looked ages older and more worldly than she. Perhaps he didn't feel the same sort of ravishing longing that she did. "The Barnfield poem," she clarified.
"Ah."
So when the waltz was over, she bid everyone good night and left the Rose Salon. She didn't look to see whether he followed. Because if he didn't follow, she was going to cry, and then she was going to pretend that none of this had ever happened. In fact, she'd probably go to London the next morning and stay with friends.
But he did follow.
She walked into the library, Stephen behind her. He was mesmerized by the way Bea's hips swayed. It seemed to promise everything, that little swing and sway. "Did you practice that walk?" he asked, lighting the lamp with a candle from the mantelpiece. He had the oddest feeling of disappointment. She had invited him to an impromptu seduction. What else had he expected when he'd told her to woo? She was, after all, just what she appeared to be: remarkably available.
He turned around and she was smiling, nestled into the arm of a high-backed settee like a wanton. "What do you think?"
"I think you're too damned practiced," he said bluntly.
Her smile disappeared, and there was something uncertain in her eyes. Almost diffident. He walked over. "You needn't look like a little girl denied a sweet. You can have any man you please."
"At the moment, I would like you." Trust Bea to go straight to the point.
Her hair had the sheen of a feverish rose. Stephen had never felt anything like the lust he had at the moment. And yet every civilized bone he had in his body fought it. She was a young, unmarried woman. He didn't succumb to such wiles. In fact, he realized with an almost visible start, he'd never been seduced. He had only seduced. It was a great deal more uncomfortable the other way around.
She turned from him and picked up the small leather book on the table. "Shall I start with the poem which gave everyone so much excitement?" she asked. There was a satin thread in her voice that made Stephen's entire body stiffen.
O would to God (so I might have my fee)
My lips were Honey, and thy mouth a Bee.
He couldn't stop himself. He drifted over to the settee. His will was strong enough that he didn't sit down, but he found himself leaning over the tall back, standing just at her shoulder. She looked up at him, a sparkling glance, and he found to his torment that this position merely gave him an excellent view of her breasts. They were a perfect white that had nothing to do with powder, not that snowy perfection.
Then shouldst thou sucke my sweete and my faire flower
That now is ripe, and full of honey-berries...
Stephen could just make out the outline of Bea's nipples, puckered under the frail silk of her bodice. He gave in, reached a hand over her shoulder, and wantonly, deliberately, took one of her breasts in his hand. There was a gasp, and she stopped reading.
But she didn't jump away or protest. That was disappointing too. What a fool I am, he thought. Why not just enjoy what is being offered?
Her breast was perfect. Somehow he'd thought it would be larger, fleshier. But it was flawlessly tender, an unsteady weight in his fingers.
"'Full of honey-berries,'" he prompted. His voice was rough and unsteady. He supposed dimly that her other lovers had been more debonair, probably, less—
He couldn't pretend this was normal behavior for him. Or normal desire, for that matter.
"'Then would I leade thee to my pleasant Bower,'" she said, and the quaver had moved from his to her voice. "'Filled full of grapes, of mulberries and cherries. Then shouldst thou be my Wasp, or else my Bee, I would be thy hive, and thou my honey bee.'"
He brought his other arm around her neck and took both breasts in his hands. She moaned, a throaty little sound, and dropped the book, arching her head back into the curve of his neck. He let his mouth play along her cheek. She smelled like lemons, clean and sweet, an English smell. Her ear was small and neatly placed against her head. In fact, her ear was like the rest of her: small, perfectly shaped, rounded, beautiful. He nipped it in anguish. Why did she have to be so—so beautiful and so available?
Her arms were tangling in his hair, pulling his head closer to her mouth. The small gasps that fell from her lips didn't seem practiced. They sounded wrenched from her throat. God knows the hoarse sounds he kept swallowing were wrenched from his own chest.
Her breasts seemed to swell in his hands, and he hadn't even allowed himself to move his hands. "Bea." His voice was hoarse and embarrassingly gruff. It sounded like an old man's voice.
This time he managed to speak clearly. "Bea, we cannot do this."
Her eyes closed, and her arms fell from his hair. He lifted his hands from her breasts—what if someone walked into the library? He waited a second, but she didn't open her eyes.
"Bea?" he asked. He was standing straight now, as straight as possible given the strain in his pantaloons.
"You may leave," she said. She didn't open her eyes.
"What?"
"I'm going to sit here and pretend that you aren't a stick-in-the-mud Puritan," she said. "I'm going to pretend that you actually had the courtesy to go through with the invitation that you ordered me to issue, if I remember correctly. Or is it a lack of guts that's the issue?"
"That's incredibly vulgar," he said slowly.
She opened her eyes. "Mr. Fairfax-Lacy, listen to me carefully."
She seemed to be waiting for a response, so he nodded curtly.
"I can be far more vulgar than this. I am a vulgar woman, Mr. Fairfax-Lacy." Her eyes were flashing, for all her voice was even. She was in a rage, and Stephen didn't know why that would make him feel better, but it did.
"Look at this, Mr. Puritan-Lacy!" she said, grabbing her bodice and pulling it down. Two perfectly shaped breasts, satin smooth, white velvet, fell free of her bodice. "I-am-a-vulgar-woman," she said, emphasizing every word. "I am the sort of woman who allows herself to be handled in the library by—"
He was at her side. "No, you are not." His voice was dry, authoritative and utterly commanding. In one split second he hauled her bodice so high over her breasts that it almost touched her collarbone.
She narrowed her eyes. "How dare you say what I am or am not?"
"I know you," he said calmly, although his hands were shaking. "You are no vulgar woman, Bea."
"Well—," she said, obviously about to rush into a hundred examples, but he stopped her with a kiss. They drank each other as if manna had fallen between them, as if kisses were the bread of life.
"You're worse," he said against her lips, a moment later. He felt them curve beneath his mouth, and he wanted her so fiercely that his entire body throbbed. "It must be tiring being so much worse than vulgar day and night."
She could not answer because his lips were crushing hers. And somehow his hands were back at her breasts. He brushed her nipple through the silk of her bodice, and she gasped.
"These must be your honey-berries," he said in her ear.
"That's so vulgar," she said, a hint of laughter in her voice.
He yanked down her bodice, the mere inch that kept her nipple from the evening air, and flicked his tongue over it. She stiffened and clutched his shoulders so hard that he would likely have bruises. He did it again. And again.
"Stephen," she whispered. Her voice didn't sound so practiced now. It was ragged and hoarse.
Finally his mouth closed over her breast. She arched against him, shaking all over. He felt a stab of pure arrogance. She may have slept with other men, but he couldn't believe that she reacted to them like this.
Of course, that's exactly what every other man thought.
"I want to be courted," he said fiercely.
"What's the difference?" she said. She sounded genuinely perplexed.
"I am not wooing you at this moment," Stephen said. "I am seducing you." He ran a hand up her leg, past the sleek silk stocking and the slight bump of her garter. "You need to learn the difference, Bea." His voice was rough with lust. His fingers trembled as they danced over the skin of her inner thighs, closer, closer—
She reached forward, pulling his hair toward her. "Kiss me!" she said, and her voice had an unsteadiness that sent his blood in a dizzying swirl.
So he kissed her, took her mouth with an untamed exuberance, at the very moment his fingers slipped into her warmth, pressed up and in with a strength that made her arch helplessly against him. She was ripe and plump, and it took every bit of strength he had to let his fingers drift where his body longed to be. To drive her mad, make her shudder under his hand even as he drank her cries with his mouth.
"This is seduction," he said to her, and his voice was raw with it.
He could feel the coil in her, feel the tension growing. She was so beautiful, trembling in his arms, coming closer and closer...
"Would you do this for me?" he said fiercely.
Her eyes opened. They were magnificent, drenched, beautiful..."Of course!" she choked. She reached out for him. "Please..."
"It's seduction."
"It's glorious."
He made his fingers still, just stay there, in the melting warmth. Then just as she was about to stir, he moved again. She gasped, and her body jerked against his. He stopped. And then pressed hard again.
"Stephen, don't!" she cried.
"Don't? Don't?" He let his fingers take a rhythm then. And allowed himself, finally, to return to her lips, beautiful dark and swollen, not with false colors but with kisses.
She was writhing against him now, panting little bursts of air, a scream building in her—he could feel it, could feel an answering shout in his own chest, a desperate longing—
She shuddered all over and clutched his shoulders so hard that he could feel her fingernails bite into his flesh, even through his coat.
And then she was pliant in his arms, a sweet, curved womanly body. He whispered into her hair. "That was a seduction, Bea."
There was silence in the library, and then she said, "I think I guessed that. At some point." The thread of laughter in her voice would always be part of living with Bea.
She didn't pull away from him, though. She stayed, nestled into his arm like a dove. He had to leave the room or he'd lose his resolve. Stephen had the sense he was fighting the greatest battle of his life: his own Enclosure Act. He had to enclose her, keep her, marry her.
And he had to make her understand that.
"I want more from you," he said into her ear.
She opened her eyes rather drowsily and smiled at him. His blood licked like fire at the look in them. "I'm amenable," she said sleekly.
"You don't know what I want," he pointed out.
She blinked. "Couldn't you teach me?"
"I want to be courted, Bea." He watched her carefully. "Not seduced, wooed."
"Do I have to consult a dictionary?"
"I hope not. May I escort you to your chamber?"
She was the most beautiful thing he had ever seen in his life, with her hair tumbling over her shoulders and a faint rosy color high in her cheeks. It took every bit of self-control he had in his body to leave her at her bedchamber door. But he was playing for keeps.
##
## Spousal Relations
Candles were being snuffed in bedchambers all over the house. Rooms were sinking into darkness, into the pleasant intimacy that welcomes a lover's step, a silent kiss, a whispered invitation. But Rees Holland, Earl Godwin, was hardly in a loverlike mood. He stared at the door to his bedchamber, grimly awaiting—
His wife.
And wasn't that an irony? That he should feel such a revulsion of feeling, such a disinclination to even speak to the woman, that he felt like dashing out of the house and saddling a horse on the moment? But there it was. She was a viper, Helene was. She could say the merest thing to him, and it would sting to the bottom of his soul for days.
And yet—he told himself again—he only wanted the best for her. Fairfax-Lacy wasn't a man to stand by her during a divorce. 'Twould ruin his career, for one thing. She was infatuated with the man, he could tell that. But it wouldn't last. Fairfax-Lacy was naught more than a smooth-talking politician, a silver-tongued devil, as his grandmother would have said. He didn't look at her with much desire, either. Rees had caught Fairfax-Lacy looking at Beatrix Lennox with real interest in his eyes.
That was the crux of it. He himself had been a damned failure as a husband to Helene. Not that she'd been any good as a wife. But presumably she was bedding Fairfax-Lacy, so perhaps it was only with him that she felt so—revolted. It was amazing to find that it still stung, years later. Even now, when he saw her, he had the impulse to put on a cravat, to cover any stray chest hair that might show. Because it disgusted her. She had said that again and again. Hairy beast, that he was.
Rees grimaced. What the hell was he even worrying about her for? She was a sharp-tongued little devil. Except he couldn't let her make the same mistake again. She needed to find a husband who'd be true to her this time. And Fairfax-Lacy wasn't the one. Not with the liquorish way he glanced at Lady Beatrix when no one was looking. He never looked at Helene that way. Oh, he was wooing her: teasing her with extravagant compliments about her moonlit hair and other such blather. But he didn't look at her with that smoky longing that a man looks at a woman he wants to bed. Can't bear not to bed, in fact.
And yet she was obviously planning to ask for a divorce. Presumably Mr. M.P. thought he could get an Act of Parliament allowing her to remarry. But if Helene married Fairfax-Lacy, she'd find herself with yet another unfaithful husband. He, Rees, had allowed her to go her own way and find a consort of her own. He'd given her her own life back. But Mr. Proper Fairfax-Lacy would never do that. No, he would dally with strumpets on the side, embarrassing Helene in public and private, but he'd never give her freedom to do the same.
There was a scratching on the door, and it silently swung open. Rees marveled for a moment: the doors in Lady Rawlings's house seemed to have been greased, they moved so quietly.
Helene looked rather like a silvery ghost. She was muffled up in a thick dressing gown, looking as drearily proper as any matron in all England. Rees had to admit that he was rather glad she had found a consort. The burden of being the only adulterous one in their marriage was exausting for his conscience.
"Forgive me for my informal attire," she told him. Her voice was cool, with just the faintest edge that told him that she expected him to be rude. Vulgar, even. She always thought he was vulgar.
So he bowed and settled her in a chair with all the manners he could summon to mind.
"I've come to ask for a divorce," she said abruptly, "but I'm sure you've guessed that."
"Has Mr. Fairfax-Lacy agreed to exposure as your consort?" Perhaps the skepticism in his tone was audible. "He will allow me to sue him for adultery?"
But she was shaking her head, perfectly calmly. "Oh, no, that might infringe upon his career. Stephen has a very important role in the government, in the life of the nation. We'll simply have to hire some man to stand in his place."
Rees didn't have to think hard to know that a writer of comic operas wasn't considered important to the life of the nation. "Shouldn't Fairfax-Lacy be in session at this very moment, if he's so vital?" he asked.
"Stephen is quite, quite exhausted by the ordeals of a recent parliamentary debate," Helene said, waving her hand in the air.
Rees thought sourly about exhausted men and their proclivities to entertain themselves with other people's wives. "Ah, exhausted. I see."
"You wouldn't understand, Rees. Stephen has a critical role in the House. He just finished orchestrating a tremendous battle against an Enclosure Act. That's when a rich man fences in land that was originally openly used for grazing by villagers. Stephen had to go against his own party!"
"I know what an Enclosure Act is," Rees said irritably. "And I fully understand that he is a worthy man."
"So it would be better for all concerned if we simply created evidence of my adultery."
"I don't see any reason for us to go to the tremendous expense of effecting a divorce," Rees said. Despite all his caution, he was starting to get angry. It was something about that martyr role that she played so well: as if he had ruined her life. Whereas it was more the opposite—she had ruined his life!
Her jaw set. "I don't wish to be married to you any longer, Rees."
"We can't all have what we want. And now you seem to have the best of all worlds, if you'll excuse a little plain speaking. You have the proper politician for a bit of kiss and tumble on the side, as well as the title of countess and the very generous allowance I make you."
"I don't give a fig for the allowance," she said. Her eyes were glacial.
"No, I don't suppose you do." He was losing his temper again. Damn it, but she had a way of getting under his skin. "Because if you did, you might actually buy some clothing designed to appeal to a man. How the hell does Fairfax-Lacy fight his way through that thing you're wearing?" He eyed her thick woolen dressing gown.
She raised her chin and squared her shoulders. She could have been wearing the mantle of a queen. "The allowance, the title—they're nothing. It's a baby that I want," she said. And to Rees's horror, her voice wobbled. Helene and he never, ever showed vulnerability to each other. It was beyond possibility that he should comfort her.
"A baby. I believe you told me that before," he said, giving her time to gather herself together.
Helene took a deep breath and leaned forward. She had to convince Rees: she simply had to. Never mind the fact that she had no intention of marrying Stephen. It took ages to obtain a divorce, and she could find someone else to marry during the process. "Have you seen Esme's baby?" she asked.
"Of course not. Why on earth would I venture into the nursery to peer at a newborn?"
"William is the dearest little boy that you ever saw," Helene said, trying vainly to convey the stab of longing that overtook her at the very sight of the baby. "His eyes are a lovely clear blue. And he looks at Esme so sweetly. I think he already knows who she is."
Rees couldn't stand children. They mewled, spit and vomited on a regular basis. They also created any manner of revolting odors without the slightest consideration for others in the room. Moreover, there was something about the slavish adoration in her voice that set his teeth on edge.
"A baby is unlikely, in your situation," he said bluntly. "You would do better to avoid the nursery, if a mere visit sends you into this kind of transport of emotion."
Helene had been smiling a little, but the smile withered immediately. "Why not?" she demanded. "And what precisely do you mean by my situation?" Rees was grateful to hear that her voice was not trembling now; she sounded more likely to garrotte him on the spot.
"You would do better to simply accept the truth," he said. "I have done so, I assure you. I have no hope of having an heir." Never mind the fact that he'd never wanted one. "I think it is far better that we simply accept our situation."
"And that is?"
"We're married to each other and, obviously, our marriage isn't tenable. But no putative second husband has presented himself. Fairfax-Lacy won't even stand by you during the divorce. Therefore, he's extremely unlikely to marry you afterwards."
"He would!" Her voice was shrill now, but Rees far appreciated shrill over teary.
"I doubt it. And frankly, my dear, he's eyeing that wanton little friend of Lady Withers's. So even if he did obtain an Act of Parliament allowing him to marry you—and I suppose, given his position, he has more chance of success than most men—he'd be as unfaithful to you as I am." Rees rather liked the way he had summed up their situation. "If you find a braver consort, I'd be happy to reconsider the idea of divorce," he added.
"Bloody hell!" She exploded out of her chair like one of those Chinese firecrackers he'd seen in London. "How bloody generous of you! You are the most stubborn, disgusting man in all England!"
"I think I am being perfectly reasonable," Rees said, staying right where he was. Surely husbands didn't have to follow that nonsense about standing up every time a lady did.
"Reasonable!"
"You would be happier if you simply accepted the situation," he said.
"You bastard!"
That caught him on the raw. "Don't you think that I would like something more?" he roared. He bounded from his chair and grabbed her by the shoulders. "Don't you think that I would like a real marriage? A wife I could love, could talk with, laugh with?"
For a second she shrunk away from him, and then she raised her head and glared. "Am I responsible for that? No! You eloped with me when I was barely out of the schoolroom!"
"I was barely of age myself," he said. "We were fools together, Helene, don't you see?" He gave her a little shake. "God knows I'd love to take back the moment I asked you to elope. I wanted—want—more from life than I have! I see Darby and Henrietta together, and I wish—" He turned away. There was no point in continuing this subject. He sat down, plopping into the chair with a sense of exhaustion that took his whole being.
There was silence in the room for a moment and then, with a faint rustle of wool, she sat down opposite him.
"So," she said finally, "you see your friend Darby and his new wife, and you wish for a more appropriate spouse. Someone as charming and beautiful as Henrietta, I gather. Whereas I see Esme's baby, and I feel equally envious."
"My only point," he said, feeling as tired as he ever had in his life, "is that at some point one simply has to accept what's happened. I made a mistake, and God knows I've paid for it."
"You've paid for it," she whispered, incredulous. He could see her small fists clenched in her lap. "I'm the one who lives to be ridiculed, to have everyone tell me about your—your opera singer. I'm the one who wants a child and will never be able to have one. I'm the one who can't even attract a man willing to face the scandal of marrying me! Your life is perfect. You have your music and your opera singer. And I don't believe for a moment that you really want Henrietta: she's not at all musical."
"I don't want Henrietta. I just want—I want what Darby has with his wife." Rees leaned his head back against the carved oak of his chair. "Fool that I am, I want a woman's companionship."
After that, they just sat there.
Helene didn't say anything about the light his comment cast on his relationship with his opera singer, the woman living in Helene's own bedchamber.
And Rees didn't say anything about the fact that Helene herself had admitted that Fairfax-Lacy didn't have the nerve to stand by her during a divorce.
There had been precious few acts of kindness in this particular marriage, but sometimes silence can be the greatest kindness of all.
##
## In the Midst of the Night
It was two o'clock in the morning, and Sebastian had only just managed to wave his mother off to her bedchamber. She had spent the evening babbling of balls and Almack's and other locales at which he was sure to meet an eligible wife. A fertile one, as she kept emphasizing. But he didn't want a wife, not even the fruitful woman his mother had set her heart on. Naturally, he couldn't visit Esme's bedchamber during the day. But now...was it late enough? Surely the child would be snug in the nursery, and that would leave Esme able to receive visitors. She'd been through an ordeal. He had to see her.
He walked up the darkened stairs feeling as cautious as a man rendezvousing with a new mistress. Esme's room was quite light due to a roaring fire in the grate. Esme herself was sitting in the middle of her bed, rocking back and forth, the baby in her arms. Her hair had fallen forward over her face, and she apparently didn't hear him come in.
He closed the door softly. "Esme," he whispered.
Her whole body jerked, and she stared up at him. She looked so exhausted and drawn that Sebastian swallowed. He knew birth was difficult, but my God, she looked as if she'd been through the wars.
And then he heard a thin wail. Esme shot him a look of pure rage, and turned to the child. Miraculously, the anger dropped away, and her face turned to an expression of pure adoration. She started cooing at the baby, and smiling at him, and dusting his little face with kisses. Naturally the child stopped crying immediately. Even more so when Esme offered him her breast. Sebastian sat down next to the bed and watched as moonlight washed over Esme's tumbling hair, her breast, the baby's little hand holding her finger as he suckled. Surely it was wrong to feel such longing to be part of the group. But he did. He wanted to climb onto the bed, to help position William at her breast, to...
To be there.
"He seems to be a good-natured child," Sebastian ventured, once Esme had tucked William over her shoulder and was patting his back.
Her eyes flashed at him. "He's very delicate. I shall have to be quite careful that he doesn't take cold."
Sebastian watched William's chubby legs kicking. "He is delicate?"
Esme nodded as William gave a burp.
"He sounds like one of the lads down at the tavern," Sebastian said. William looked at him with a beery expression. "He even looks drunk."
"He does not!" Esme said indignantly. "But Sebastian, don't you think he's the image of Miles? I knew he would be. I just knew he would be."
To Sebastian, William looked like most of the babies he'd seen in his life: bald, round and red. Yes, he looked like Miles. But then all babies looked like Miles.
"I have blue eyes," he said, unable to stop himself.
"Not that azure blue," Esme said. "And it's not the color that matters anyway. It's the way he looks at one, with such a deep sweetness...just like Miles. He's the sweetest boy in the whole world, aren't you?" And she gathered little William up into her arms and kissed him all over his face again. "Now he has to go to sleep," Esme said, looking expectantly at Sebastian. So he left.
She likely thought he would leave the house as well. But no. He would stay. And his mother would have to stay as well, like it or not. If the ton discovered the whereabouts of the notorious Marquess Bonnington, the presence of his stodgy mother would likely dispel some suspicion.
Or perhaps it wouldn't. He didn't really give a damn.
William slept most of the following day while Esme and Arabella hovered over the crib, pointing out the pink perfection of his toes, and the sweetness of his round tummy. Esme was convinced he already knew who she was. "That's a loving look," she told her aunt when William finally opened his eyes.
"If you say so," Arabella said.
"I know it. Do you think William is warm enough? I think his cheek is a bit chilled." She felt it with the back of her hand and tucked his blankets around him even more securely.
"I'll ring the footman for another log," Arabella said, going to the other side of the room. That was one of the things that Esme loved about her aunt. Unlike the nursemaid, who was annoyingly contentious, and even her own nanny, Arabella never questioned Esme's judgement.
Esme picked William up and put his little cheek against her own. It felt like the softest silk in the world.
"So your mother writes that she will visit at her earliest convenience?" Arabella asked, coming back and fingering the delicate embroidery on William's coverlet.
"I'm so happy about it," Esme said. "It was disappointing when she didn't visit during my confinement. But she can't help but love William."
Arabella cast her a worried look. "Of course Fanny will love William. But I just..." Her voice trailed off.
"Don't worry. William will enchant her."
Arabella took another glance at her niece's hopeful face and decided that she had to speak. "I'm worried for you, Esme. Your mother has suffered many disappointments in her life. She's not always as agreeable as she could be."
"I know that," Esme said instantly. She had always been aware that she was her mother's primary trial. "But William will make up for all that, don't you see? And of course, I'm going to be just the sort of daughter she always wished for. She won't have to be ashamed of me anymore."
"Yee-es. I certainly hope that's the way of it."
"But you don't believe it?"
"I'm afraid that you may be disappointed," Arabella admitted. "I shall scream at Fanny if she hurts your feelings. But, I just—"
"You mustn't worry so much. Truly, Mama has always wished me to be respectable, and now I am. I'm living in Wiltshire like the virtuous widow I am. What more could she want?"
"Fanny has a difficult nature. She has spent a great deal of her adult life berating you for one thing or another, and I never approved of it, never."
Esme gave her a rueful grin. "It's not as if I didn't deserve it. I'm the first to say that my reputation was precisely as black as it deserved to be."
"But Fanny was disappointed even before you grew into such a convenient whipping boy. When I was growing up with her, she was always finding fault with me as well. In fact, Esme, my sister Fanny is a bit of a malcontent. Your grandfather used to call her Miss Tart, because of the way she pranced around the house with her mouth tight, finding fault with everyone."
"I know Mama has had an onerous life," Esme said. She was tickling William through his blankets, and he looked as if he might start smiling any moment. "But perhaps having a grandchild will transform some of her discouragement, especially if he starts smiling. Smile, William! Smile for Mama! And he is beautiful. Even Sebastian said last night that—" She stopped short and turned around to find Arabella looking at her and shaking her head.
"A woman after my own heart," Arabella said. "I told you that living with a man was far more fun than living alone, didn't I?"
Esme bit her lip. "Sebastian only—"
"Never mind the details. What about Fairfax-Lacy? When are you planning to drop that pretense?"
"Not yet! Not until after my mother visits. The only reason she is making a visit is due to Lady Bonnington telling her of my engagement."
"In that case, I would advise that you wait until nighttime to entertain Bonnington again. Lord knows, if Honoratia Bonnington found out that her son was making secret trips to your bedchamber, and you still in confinement, she'd likely shriek the roof down."
Esme smiled a bit at that. "I'm not worried about the marchioness. It's Mother, of course, whom I would prefer not to realize that Sebastian and I have a friendship."
"Naturally," Arabella agreed. "We definitely do not wish for sainted Fanny to discover that you have a man visiting your bedchamber in the wee hours of the night."
"There's nothing salacious about his visits," Esme hastened to say.
Arabella bent over the crib again, and Esme couldn't see her face. "My first husband, Robbie, used to look at me the way Marquess Bonnington looks at you."
"I don't think I remember him," Esme said. "And Sebastian doesn't look at me in any particular fashion."
"Robbie died when you were a young girl. You likely never would have met him anyway. How he loathed your father!"
"Were you very much in love?" Esme asked.
"Too much," Arabella answered flatly. She turned around, and her smile was bright. "Never fall in love, darling. It makes farewells utterly dreary."
Esme didn't say anything to that nonsense, but just gave her a kiss.
"I can't be as angry at your mother as I'd like to be," Arabella said, with one of her lightning changes of conversation, "because I was so lucky with my first marriage, and she so unlucky. Robbie was a sweet-hearted man. He died laughing, you know. We were riding in the country, and he was laughing at something I said. He wasn't paying attention, and his horse caught a rabbit-hole."
"Oh, Arabella," Esme said, putting her arms around her.
"I only tell you because your memories of your father are undoubtedly as clear as mine," Arabella said. "That man never laughed a day in his life. Being married to him was a terrible hardship, for all your mother won't acknowledge it. Terrible. I was allowed to choose my husband, as I was the homely sister. But Fanny went to the highest bidder."
"Don't you think that William will assuage some of her grief?"
"I hope so, darling. I hope so."
##
## A Proposal
There was no saying when, if ever, Arabella might decide to return to London for the rest of the season. As far as Bea could see, Arabella spent every spare moment fussing over the baby's clothing and counting his toes. She and Esme hovered over that fat little creature as if it were made of spun sugar. Earl Godwin left; Lord Winnamore finally left as well. There was no one in the house to talk to, given that Helene and Stephen pounded away at the keyboard for hours.
Not that it mattered much, because all Bea could think about was the Puritan, and his wish to be wooed rather than seduced. Wooed. Whatever that meant.
She headed off down the lane to visit the goat. It was very cold, and the wind was sprinkling small white flowers all over the lane, almost as if it were snowing. It could be a pear tree losing all its blossoms; Bea had almost made up her mind to try to find a book identifying plants. That was how boring she was becoming.
When she rounded the bend, he was there. Bea slowed down. Wooing, wooing. She was no good at wooing. She was only good at seduction. Why didn't he understand that? Why didn't he know that she had nothing else to offer? She tramped up beside him and leaned on the fence without bothering to say hello. A large hand curled around her neck.
"Bea," he said. Why did he have to have a voice like that?
"Doesn't the Parliament miss you?" she asked aimlessly, trying to take her attention away from that warm hand.
"It doesn't seem that way. According to the morning paper, they just passed a law giving poachers seven years' hard labor. I keep thinking of an old man named Maidstone, who lived on our estate when I was young. He was such a one, poached in my father's forest his entire life. It was an art to old Maidstone. My father sent me out with him to learn to shoot."
"I wish I could shoot," Bea said. "My father didn't consider it a ladylike pursuit."
"Perhaps I'll teach you."
The statement hung in the air between them. She finally risked a glance at him, and he was smiling at her.
"When you're my wife."
She suddenly felt the splintering wood of the gate under her fingers. "You are engaged already."
"You know as well as I do that the engagement is a temporary one, having naught to do with love nor even desire."
White petals had floated onto his dark hair. "I could never marry you. I thought you understood that."
"You must have misunderstood." He moved closer and looked down at her. His eyes were flames, telling her something. But the misery in her heart was beating in her ears.
"Men like you don't take wives like me," she cried.
"Am I too old?"
"Don't be a fool."
He smiled a little. "Too rigid?"
"Something like that. It would destroy your career."
"I don't care about my career."
"Who will save the poachers from seven years' hard labor?"
"Someone else," he said. "I will go home and take care of Maidstone's son, who is undoubtedly poaching every pheasant I have in the world."
"You can't marry me." It seemed imperative to make him understand. "I am—ruined, Stephen." Her face was wet with tears, and she didn't even know where they had come from. "Don't you understand that? And damn you for making me say it aloud! Why do you insist on being so cruel? I told you, you can have me!" Her voice broke with the humiliation of it. And the truth of it. He stared at her, eyes veiled. "You could have had me, anywhere," she said brokenly. "On the billiards table, in the library. Anywhere. You're some sort of debauchee, aren't you? You just enjoy tormenting me. You don't even really want me."
"No." His voice was hard. "That's not the case and you know it, Bea. I want you." He took her shoulders. "I want you more than any other man has wanted you. But I don't want only your delectable body. Or your mouth. Or even the direction to your bedroom door. I want more, Bea, and if you can't give it to me, I don't want any of it."
She stared ahead, and the goat's sharp little horns were blurred by her tears. "I wish things were different," she said. "I wish I weren't myself, or that myself was—"
"No! You don't understand. I want you with all your face paints, and all your sultry glances, and all your wicked poetry. I want you just as you are, Bea."
It was probably a tribute to her father that she didn't believe him for a moment.
She cleared her throat. "That is remarkably kind of you. I am honored, truly. Of course, I might be even more honored if you didn't already have a fiancée. But I appreciate your willingness to add me to the list."
"Stuff that," he said, and his voice was harsh and utterly unlike the smooth cadences that members were used to hearing in the House. "Don't be honored," he said fiercely. "Be my wife."
"I can't do that." She turned and faced him, head high. "I care for you too much. You may be exhausted by your position at the moment, but you will long for it after a few months. I don't see you spending your days fishing and befriending poachers, Stephen. After a month, even a year perhaps, you would yearn for your position back. And they'd never give it to you, never. Not after you married me."
"I disagree, Bea. I could marry you and stay in Parliament. But I want to resign. If I grow bored in the country, I'll find something to do. But not Parliament. I don't want to think about votes again. I'd rather think about you."
"Leave," she said tersely, hanging onto the splintered boards with all her might. "Just leave, Stephen."
The smile fell from his face.
"Please leave," she whispered.
##
## And Motherly Love, Part Two
Esme's mother arrived on a beautiful spring day, a week after her grandson's birth. Esme looked out her bedchamber window and there it was, rounding the bend before the road to Shantill House: a squat, ugly carriage that she remembered from her childhood. The family used to travel to and from London in it. The seats were made of slippery horsehair and sloped upward. As a child, Esme constantly slid to the carriage floor, earning a scolding for her fidgety nature.
William was sleeping in her arms, his long eyelashes curling against his cheek. "I'll never make you ride in a carriage for hours," she whispered to him. And then rethought that promise. "Well, perhaps only if we make many stops."
Then she turned aside and rang the bell. "My mother has arrived," she told Jeannie. "I must change my clothing. I'll wear the gray morning gown with the white lace trim, the one with the small tippet. And I shall wear a cap as well, perhaps with a silver ribbon so it matches."
Jeannie looked surprised. "But, madam, that gown is half-mourning, and so heavy for this weather. Wouldn't you prefer to wear something more cheerful? Surely your lady mother will wish to see you more lighthearted. We don't even have such a thing as a silver ribbon in the house!"
"No, the gray dress will be perfect." Fanny had worn full mourning for two years after Esme's father died. The least Esme could do was appear to have a virtue, even if she had it not.
"Shall I take Master William to the nursery?" Jeannie said, once Esme was dressed in gray, complete with a lace cap but no silver ribbon.
"I'll bring him downstairs with me. I'm certain my mother is quite eager to see her grandson."
"Of course she is! And he's the bonniest boy that's ever lived. She'll likely cry with pure joy. I know my mother would."
When Esme entered the morning parlor, she found her mother seated with Marchioness Bonnington and Arabella. To Esme's relief, Bea was nowhere to be seen. Esme had a secret fear that her mother would take affront at the idea of staying in the same house with her sister's dame de compagnie and leave without delay.
She could see on the instant that Fanny and Arabella were already twitting at each other. They were seated opposite each other, and Arabella had the look of someone who has just delivered a magnificent set-down. Fanny was shaking her head sadly and looking at her younger sister as if she were addled. Esme hurried across the room toward them.
Fanny looked like an exquisite watercolor rendition of Arabella. Arabella's hair was ginger; Fanny's was a pale rose. Arabella's complexion was a tribute to French face paints; Fanny's face had a delicate bloom all her own. Arabella's face somehow just missed being beautiful, but Fanny had been acknowledged as flawless from the moment she'd toddled into her papa's arms.
"Mama, it is such a pleasure to see you!" Esme cried. "I've brought William, who is longing to meet his grandmother."
All three ladies looked up. Her mother gave her the melancholy smile with which she always greeted her daughter, a perfect blend of responsibility and disappointment. Impulsively Esme went to her knees beside her mother and folded the blanket back from around William's face so Fanny could see him. He was still sleeping peacefully, as beautiful a child as she'd ever seen. William was the one thing in life that Esme had done perfectly.
But her mother looked at her rather than at William. "Esme," she said, "I must ask you to seat yourself properly. We are not en famille here. There is no need for such boisterous manners."
Lady Bonnington leaned forward. "Please don't insist on convention on my account, Fanny dear. I find your daughter's affection for her child quite refreshing."
Esme rose and seated herself next to her mother on the settee. Fanny raised her eyebrows slightly and then finally looked down at William. For a moment she stared at him in utter silence.
"Isn't he beautiful?" Esme said, unable to stop herself. "Isn't he the most darling baby you ever saw, Mama?"
Her mother closed her eyes and put out a wavering hand as if to push William away. "He looks just like your brother," she murmured, turning her face away and shading her eyes. Her hand stayed in the air, shaking slightly with the strength of her emotion.
Esme bit her lip. "William doesn't resemble Benjamin so much," she ventured. "Benjamin had such a lovely cap of black hair, do you remember? Even when he was—"
"Naturally I remember every moment of my son's short life!" her mother broke in. "You do me great disservice, daughter, to suggest that I could forget the smallest detail of my little angel's face." She sat with her face shaded by her hand, overcome by grief.
Esme was stricken into silence. She literally didn't know what to say.
"William is quite an adorable child," Arabella said. There was a crackling warning in her voice. "And I do think that he has the look of his father rather than Esme. In fact, I would say that William is the spitting image of Miles Rawlings. Why don't you look at William more closely, Fanny?"
Esme's mother visibly shuddered. "I couldn't...I just couldn't." She waved her slim white hand in the air. "Please, remove the child. I simply am not strong enough for this sort of blow. Not today. Perhaps when I am having a better day."
"Of course, Mama," Esme said quietly, tucking William's blanket around his face. "I'll take him back to the nursery."
"Give him to the footman," her mother instructed, sounding a bit stronger. "I didn't come all the way to this house merely to watch you act like a servant."
Esme had never given William to one of the servants, but she handed him over without a murmur. She should have realized how much pain the baby would cause her mother. No wonder Fanny hadn't attended her confinement. The whole event was undoubtedly too distressing to contemplate. As she returned to the parlor, Esme braced herself for the look of disapproval that always crossed her mother's face. But it was, miraculously, not there. Esme blinked and almost stumbled.
"Do come here, daughter," Fanny said, patting the seat next to her.
Esme sat down next to her, careful to not allow her back to touch the back of the settee.
"We were just discussing how much your cap suits you," Fanny said. "I think you will find that a cap truly eases one's life. It does the necessary work of informing lecherous men that you are a woman of propriety and virtue. They never, ever make indecent proposals to a woman in a cap."
Arabella looked at Esme with a faint smile. "I've just told your mother that she needn't lend me one of hers."
Fanny ignored that. "And Lady Bonnington has been regaling me with tales of your fiancé's devotion. I must say, he sounds like an estimable gentleman. What a shame that Mr. Fairfax-Lacy stands to lose his courtesy title if the Duke of Girton's wife gives birth to a son. The Earl of Spade, isn't he? Of course, the duchess may birth a girl. We shall have to hope for the best."
"Mr. Fairfax-Lacy doesn't use his title," Esme murmured.
But her mother swept on. "It would be even better were the earl to give up his seat in Parliament. The House of Commons is so very...common, is it not?"
"Mr. Fairfax-Lacy plans to resign his seat," Esme said. "He wishes to spend more time on his estate."
Her mother gave her a smile and patted her hand. "I'm certain that you can effect the earl's resignation without delay. I feel quite heartened by this news, dearest."
"I'm very glad to hear it, Mama."
"Perhaps you could marry by special license," her mother continued. "That would be by far the more respectable choice. No one to gawk, as would happen in a public ceremony."
"Choice? What choice does she have?" Arabella said, and there was a distinct jaundiced note in her voice.
"Whether to remain a widowed woman or marry Mr. Fairfax-Lacy immediately," Fanny said sharply. "Given our plans to rehabilitate dear Esme's position in society, I tend to think that immediate marriage would not be frowned upon. What do you think, Honoratia?" she asked the marchioness.
"While I am naturally eager to see Lady Rawlings settled in such a beneficial position," Lady Bonnington announced, "I do not approve of marriages within the first twelve months of mourning."
Esme breathed a sigh of relief.
Arabella gave her a wink. "You must be eager to find an appropriate spouse for your son," she said, turning to Lady Bonnington, "since he has returned from the Continent. I know there is no one of the slightest interest to him at this house party, but I am quite certain that you must have some thoughts on the subject."
Esme's mother stiffened. Clearly she had had no idea that her friend's disreputable son was even in the country, let alone in the very house in which she sat. "May I ask—" she said, her voice shrill.
But Lady Bonnington broke in. She was magnificently quelling, Esme had to admit. "Fanny, there is no one in the world who deprecates my son's behavior more than I do. But I decided he had been in exile long enough. He has naturally attended me here; as a dutiful son, he is engaged in accompanying his mother wherever I wish to be."
"But this particular household is surely not the appropriate place to be!" Fanny sputtered. "Given the events of last summer—"
"We do not speak of that," Lady Bonnington said with magnificent hauteur.
Fanny snapped her mouth shut.
Esme had to hide a smile. Perhaps she could learn something of Lady Bonnington's technique herself.
"The events of last summer were grievous for everyone in this room." Lady Bonnington gave Esme a little nod, and then turned back to Fanny. "You must understand, Fanny, that I have decided to keep that boy on a very tight rein. Where I go, he goes. I found London entirely too stuffy and tedious this season, and I decided to retreat to the country."
Fanny nodded. "I agree with you. It is far too early for the marquess to reenter London society. But must he be here, in my daughter's house?"
"No one could possibly question his presence, given that I am here," the dowager trumpeted.
"That is certainly true," Arabella put in merrily. "And now that you are here as well, Fanny, this party is positively taking on the air of a wake!"
"Your levity is repugnant," Fanny snapped. "My only pleasure in making this visit is finding that my daughter has changed so much." She patted Esme's hand. "You have become the daughter I always dreamed of."
"Yes, Esme has been remarkably silent, hasn't she?" Arabella put in.
"Silence is a virtue that few women understand. Believe me, a virtuous silence is a far greater blessing than the kind of impudent chatter that you consider conversation," Fanny retorted.
"You must ask Esme to tell you about her Sewing Circle," Arabella said, standing up and shaking out her skirts. "I am afraid that the very sanctity of this room is wearying to such a devout Jezebel as myself."
Esme felt an unhappy hiccup in the area of her heart. Fanny had leveled the same disapproving glare at her sister that she usually gave to her daughter. On the one hand, it was a pleasure not to be the target of her censure. But Esme didn't like to see Arabella slighted either.
"Aunt Arabella was a blessing to me during my confinement," she said after the door closed. "I don't know what I would have done without her."
"Really?" Fanny asked with languid disinterest. "I can't imagine what that light-heeled sister of mine could possibly do to help anyone. Except perhaps a womanizer. I doubt she would have any hesitation helping such a man."
Esme blinked. She had never before realized the amount of vitriol that her mother felt toward her sister. "In fact, Arabella was quite helpful during William's birth," she said cautiously.
"I knew you would see fit to reproach me for not attending you," Fanny said in a peevish voice. "When you see how much pain it caused me to merely look at a young child, I wonder that you would even bring it up!"
"I didn't mean to imply such a thing."
Lady Bonnington had been sitting silently, watching Fanny and Esme with a rather odd expression on her face. "I will do Lady Withers the credit of saying that she was a source of strength to Lady Rawlings during the birth. Much more so than I was."
Fanny shuddered. "You attended the birth, Honoratia? Why on earth would you put yourself through such an ordeal?"
" 'Twas your daughter who went through an ordeal," Lady Bonnington pointed out. "I merely counseled from the bedside."
"Yes, well," Fanny said in a fretting tone of voice. "Naturally I am ecstatic if Arabella actually managed to summon up an ounce of family feeling. When has she ever thought of me? She simply made one short-lived marriage after another, and never a thought for my wishes in the matter."
"Aunt Arabella can hardly be blamed for the deaths of her husbands," Esme pointed out, and then wished that she hadn't opened her mouth.
"She drove them into their graves," Fanny spat. "I grew up with the woman, and I've always known what she was like."
Esme rose and rang the bell. "Why don't I ask Slope to bring us some tea," she suggested. "You must be exhausted after your long carriage ride, Mama."
"As to that, I've been staying a mere hour or so from here, at dear Lady Pindlethorp's house," her mother said. "The season is just too tiring for someone my age, I find. Lady Pindlethorp and I have had a perfectly lovely time in the past fortnight. We have so many interests in common."
Esme turned around slowly. "You mean you have been living at a short distance? But—but you could have come for a visit at any time!"
Fanny blinked at her. "Not until I was quite certain that you had reformed, my dear. I would never risk my reputation merely on dear Honoratia's assurance, although of course I took her advice quite seriously. No, indeed. I will admit that I had quite given up hope of your reformation, as I believe I mentioned in my letter. I always thought you took after my sister, although naturally I am pleasantly surprised to find you so much changed."
Esme's jaw set. I will not scream, she thought. She felt her face growing red with the effort of not lashing out at her mother. Lady Bonnington seemed to guess, because she quickly turned to Fanny and asked her if she would like to stroll among the roses in the conservatory.
"Only if I need not step a foot outside," Fanny said. "I'm afraid that my poor departed angel, Benjamin, inherited his weak constitution from me. I take a chill at the slightest breath of wind. I am virtually housebound these days, if you can believe it."
Esme curtsied to her mother, walked up the stairs to her chambers, and jerked the cap off her head so harshly that hairpins spilled on the floor. Throwing the cap on the floor didn't help. Neither did stepping on it. Neither did ripping off that horrible gray dress with its foolish little lace tippet that worked so well to give the wearer a nunlike air. None of it helped. She stood in the middle of her bedchamber, chest heaving with tears and pure rage.
She had achieved it all: the Sewing Circle, the respectability, her mother's approval, Miles's wishes—why did success make her feel so terribly enraged? And so terribly, terribly afraid, at the same time?
##
## In Which the Goat Eats a Notable Piece of Clothing
The irritating man hadn't left Shantill House, even after Bea had begged him. He stopped opportuning her and made no seductive moves. Instead he played duets with Helene, which left Bea embroidering on the other side of the room and trying not to think about the Puritan. She stayed away from him. No more flirtatious glances. No more flirtation, period. Certainly no more failed seductions.
It was late morning, and they were gathered in Esme's morning parlor. Arabella and her sister were conducting a genteel squabble; Esme was presumably in the nursery. Naturally, Helene and Stephen were practicing the piano. Bea sat by herself, stitching away on her tapestry.
When Slope arrived with the morning post, Bea looked in the other direction. It was foolish of her to wish that one of her sisters would write. They had never answered her letters, and she was fairly certain that her father was intercepting them. Surely Rosalind would have written. They were only separated in age by a few years. Rosalind was to make her debut next year, and Bea wanted so much to tell her—
Well, to tell her not to make her mistake. Or did she mean to tell her to follow her example? Bea kept thinking and thinking about it. On the one hand, it was grievously hard to turn down Stephen's marriage proposal on the grounds that by accepting, she would ruin his career. On the other hand, had she married whomever her father had seen fit to select as her husband, she would still have fallen in love with Stephen at some point, she was sure of that.
So Bea bent over her tapestry and surreptitiously watched the way Stephen leaned toward Helene, the way their shoulders touched as they played. What would it mean to him, to no longer be the estimable Member of Parliament? Would he be happy? If he were married, would he give up his mistresses, not to mention his supposed fiancée, Esme?
Helene received a letter. "I'm going from pillar to post," she told Stephen. "This is from my friend Gina, asking me to visit her during her confinement."
"I gather you refer to the Duchess of Girton?" Stephen said. And at her nod, he added, "Cam, her husband, is my cousin."
Wonderful, Bea thought sourly. Splendidly cozy.
"She and the duke returned from Greece a few months ago," Helene was saying, "and now they are living on their estate. Apparently Gina will be having a child this summer." She made a funny, rueful face.
Bea bit her lip as Stephen put a comforting arm around Helene. They had the intimacy of an old married couple.
"I can't even bear to look at William. Although I love him." The agony in Helene's voice mirrored that in Bea's heart. Nothing more was said, and after a moment Helene and Stephen returned to playing a Turkish march for four hands. Bea was sick of pieces written for four hands. She was sick of everything that had to do with one prim countess and one proper politician.
Abruptly she got up and walked out of the room. She might as well visit the goat. She still kept a daily pilgrimage to the ungrateful beast, although she hadn't encountered Stephen again in the lane. He seemed to be avoiding the goat, as well as her.
As she tramped down the lane, regardless of the mud clinging to her boots, Bea was actually beginning to think that perhaps she could live in the country. Some sort of wild rose grew over the hedges in the lane. They were pale pink and hung down like faded curtains. For the first time in her life, she had a sense of what happened in spring. A scraggly tree next to the road had broken out all over in white buds. They stuck out from the branches like the knotted ribbons on debutantes' slippers.
And there were daisies growing all up and down the lane. Impulsively Bea started gathering them. Finally she took off her bonnet and filled it with daisies. It hardly mattered if her skin colored in the sun. She could powder it white, or powder it pink. The sun felt kind on her cheeks. Finally she reached the end of the lane and leaned on the pasture gate. He was there, of course, the old reprobate. He trotted over and accepted a branch Bea gave him to chew. Bea even walked in his pasture sometimes; he had never again tried to chew her clothing. She pushed open the gate and headed for the small twisted tree in the center. There were no daisies in the pasture, of course. The goat presumably ate them the moment they poked up their heads. But the tree was in the sun, and surrounded by a patch of grass.
It was when she was sitting against the tree that she realized what she had to do. She had to go home. Go home. Back to her irate father, who wouldn't throw her out again if she promised to be a model of proper behavior. And back to her sisters. She missed her sisters. She didn't want to play the voluptuary role anymore, not after meeting Stephen. He made her games seem rather shabby and hollow, rather than excitingly original.
Without really thinking about it, she picked all the daisies from her bonnet and braided a daisy chain, a rather drunken daisy chain that had a few stems sticking out at right angles. It was just the sort she used to make for her little sisters. Perhaps she would ask Arabella to send her home tomorrow morning.
He was there, in front of her, before she even noticed his arrival. "How you do sneak up on one!" she snapped.
"You are the very picture of spring," he said, staring down at her.
Bea allowed him a smile. She rather fancied that compliment, since she was wearing a horrendously expensive Marie Antoinette–styled shepherdess dress that laced up the front and had frothy bits at the sides. Suddenly he dropped onto his haunches in front of her, and she blinked at him. His eyes were dark and—
She reached out and touched his cheek. "What's the matter, Stephen? Are you all right?" She forgot they weren't on intimate terms and that, in fact, she had hardly spoken to him in virtually a week.
"No, I'm not," he said, rather jerkily. "I've made rather a mess of my life."
"Why do you say that?" Bea asked, taken aback.
"Because I asked a lady to woo me," he said, and the look in his eyes made her knees weak. "Because I asked a lady to woo me, and she very properly refused. I was unfathomably stupid to ask such a thing."
Bea bit her lip. "Why?" Don't say that you never wanted me, she prayed inside. But there was that something in his eyes that gave her hope.
"Because I should have said, 'Seduce me. Take me. Please.'"
Bea supposed that was her cue to leap on him like a starving animal, but she stayed where she was. Her heart was beating so fast that she almost couldn't feel her own disappointment. Wasn't this just what she wanted? Of course it was.
"You see, I need her any way she'll have me," Stephen said. His voice had lost all those liquid rolling tones he used so well. It was almost hoarse. "Any time she'll give me. I don't care. I won't make any demands."
Bea couldn't quite meet his eyes. She fidgeted with the ribbon on her parasol, tilting it slightly so that she couldn't see his face. "I've decided to return to my father's house," she said almost inaudibly. He was silent, and all she could hear was her own pulse beating in her throat and the goat ambling away to the other side of the pasture.
"Am I too late, then?" he said finally. There was a bleakness in his voice that wrenched her heart.
She took the parasol and neatly closed it. He would always have a patrician's face. It was the face of an English gentleman, long chin and lean cheeks, laughter wrinkles around his eyes, tall, muscled body. He would wear well. She raised her eyelashes and gave him the most smoldering look she had in her repertoire.
He made a hoarse sound in his voice and pulled her into his arms so fast that her parasol flew into the air.
"Will you, Bea, will you let me..." He was plundering her mouth, and he couldn't seem to finish the sentence. Finally he raised his mouth a fraction of an inch from hers, so close that she was almost touching his lips. His voice was husky. "Will you seduce me, Bea? Or let me seduce you?"
She strained forward, trying to catch his mouth with hers, but he held back.
"Please?" The urgency in his voice awed her. "I was a fool to refuse you. I'll take anything, any little bit you'll give me. Of course you don't wish to woo me, marry me. But I'll take whatever you give me, Bea. Please."
She closed her eyes. One of the proudest gentlemen in the kingdom was literally, as well as metaphorically, at her feet. "I didn't mean that," she whispered, clutching his shoulders as hard as she could. "It's not that I don't wish to marry you—"
"Hush," he said, rubbing his lips across hers. "I know you don't want to marry me. I was a conceited fool to think you'd even consider me. But I don't care, Bea. Just—just seduce me, Bea."
She could untangle this later. At the moment she unwrapped her arms from his neck and smiled at him with the slumberous smile of Cleopatra. "But what if I lead you to do things that are less than gentlemanly?"
"You already have," he said. "This is absolutely the first time in my life that I have begged a young unmarried woman to seduce me."
"Oh, well, in that case," she said, with a gurgle of laughter. Then she settled back against the tree trunk and, looking at him, very, very slowly raised the ruffled dimity of her skirt. She was wearing gossamer silk stockings, with clocks, and her slender ankes were crossed. She pulled her skirts up just past her knee, so that Stephen could see the pale blue stocking, and its darker garter, and then the pale cream of her thigh.
She saw him swallow. "Bea, what are you doing?" he said, and the rasp in his voice was a warning.
"Seducing you." Her smile was blinding. He didn't seem to be able to stop staring at her legs.
"What if someone comes?"
"No one ever comes down this lane," she said blissfully. "It leads nowhere except to the goat. And you and I, Stephen, are the only persons who have ever shown interest in the goat."
Just as deliberately she uncrossed her legs and drew them slightly higher. Her skirt fell back against her thighs.
"And where is the damned goat?" he said hoarsely.
"The other side of the field." Her knees came a little higher, and her skirts slid farther down, exposing smooth, milky thighs.
"If I touch you, Bea, there's no stopping this," Stephen said, meeting her eyes.
Her heart tumbled in her chest. "I wouldn't want to stop you. I never have."
He put his hands gently on her ankles. "Last chance, Bea. Are you sure you wish to make love in a goat's pasture?" But she was laughing, and her eyes were shining. There was desire there, so that was all right. And obviously, she didn't mind the goat's pasture. So Stephen let his fingers wrap around that delicate little ankle, slide up the faint softness of her stockings. He stopped at the garters and untied them. They left angry red marks on her skin.
She was watching him with a half smile, but there was something uncertain there too, for all she was such an accomplished seductress. He smoothed the red marks with his fingers. "Why so ruthless with your poor skin?" he said, as he lowered his head and ran his tongue along the groove in her leg.
She gasped and squirmed in his hands. "It's particularly difficult to keep stockings this flimsy from collapsing around my ankles."
"Ah." He had his hands on both her knees now, and he pulled them apart. She resisted for a moment and then gave in. She was wearing some sort of fluttering gown that obediently fell back, as if it had been designed for outdoor games. Stephen ran a finger down the inside of her thigh. He stopped at a burst of lacy cotton, then ran his finger over all the fabric.
She visibly shuddered and reached for him. But he pushed her back against the tree and knelt in front of her, between her raised knees, and pressed his lips there, on the inside of a quavering knee. And then let his lips drift down, down smooth, ivory flesh.
And all the time his finger was running inquisitively over the white cotton between her legs, dancing a little surface dance that made her hips jiggle a bit. He could hear her uneven little whoosh of breath, and it made him feel a steely wave of triumph, and then a wave of lust so pure that he almost wrenched that cotton down—
"What do you call this?" he asked, and his voice came out hoarse. He put his hand between her legs, firm, and rocked forward.
"Oh," she said, and her voice seemed very small.
He ran his thumb under the frilly border. "This?"
"Pantalettes," she said, quivering all over.
He leaned foward and put a leg over her left knee so he was straddling her, and then he let that thumb sink, fall into sleek, hot folds. She had been lying against the tree as if she were too shocked to move, but that shudder woke her up; she reached out and pulled his head toward her.
Her lips trembled under his, and opened, and Stephen let his thumb take on the same rhythm as his tongue, although his chest felt like bursting for lack of air, or for the thumping of his heart in his chest.
Her eyes fluttered open, and she was beautiful. This close, her eyes had the green of a rock glimpsed at the river bottom, greeny blue, with small specks of light. All the more beautiful for being slighty glazed.
Suddenly she focused on him. "You seem to have forgotten that this is my seduction," she said. Her voice was such a deep purr that he almost didn't catch her meaning. But with one flip of her hip, she pushed his hand away and came up on her knees. Alas, her skirt fell down and covered her legs again.
He reared up so he was facing her. Then he very, very deliberately took his thumb and rubbed it over his lips. She gasped in shock, and he felt a throb of pleasure. She wasn't so jaded then. He licked his lips, enjoying the faint taste of her.
"Stephen!" she said. He grinned. But she was pulling at his neck cloth. She seemed to have some trouble undoing it, so finally he tossed it to the side and undid the placket on his shirt.
It was her turn then to inch that shirt up his muscled abdomen. Her fingers were everywhere, delicate, admiring. The shirt billowed past his eyes and disappeared. Now her fingers were at his waist. But she couldn't seem to undo the buttons there either. She looked so serious.
"I thought you'd make my clothes fly off like greased lightning," he said teasingly. But she didn't look up, so he pushed up her chin. "That was only a jest, Bea. In poor taste, to be sure, but a jest."
"I—" Her eyes were larger, not so passionate now. Stephen felt a pang of pure fear. She'd changed her mind. She didn't want him. He was too old.
"I'm afraid I'll disappoint you," she said.
"Never."
"I don't—I don't have as much experience as you might think," she said, staring fixedly at his waistband as she tried to undo it. The very feeling of her fingers fumbling around his pantaloons was driving Stephen crazy.
But once he registered what she'd said, he laughed. "I don't care what kind of experience you've got, Bea. All I want is you. You." He pushed up her chin again. Her lips were swollen with his kisses. "Oh God, Bea, you're so beautiful."
But she wasn't really listening. "You see, I did—that is, there was Sandhurst, but it was only once, and I'm afraid I didn't learn very much, especially as we were interrupted by Lady Ditcher. And then I allowed Billy Laslett, but I didn't truly enjoy it towards the end, and so I told him to go."
Stephen laughed. "Are you trying to tell me that the bold seductress herself didn't find the experience pleasurable?"
Bea blushed. "No, I did. Although I wish I hadn't."
"Why?"
"Because it would make me almost like a virgin, wouldn't it?" Her eyes were shadowed. "But I did—did enjoy it, up to a point. I haven't liked—well, that's irrelevant. I took another lover once too." The last came out in a rush of admissions. "So you see, I've had three lovers. But I never gave anyone a second chance, and I'm not certain that I actually learned very much, if you see what I mean."
Stephen threw back his head and laughed, laughed so hard that four starlings and a wren flew out of the crooked tree and wheeled into the sunlight. When he looked back, she was still there, blinking at him, looking a little defensive, extraordinarily lovely, and far too young.
"Bea, you are over twenty-one, aren't you?" he said.
"I'm twenty-three."
"Good. Are you trying to tell me that you won't let me have a second round? That one time with lovely Bea is all any man could hope to achieve?" He let his hands settle on her waist.
She blushed faintly. "No." But he could hardly hear her.
"Because I want more, Bea." He lowered his head and brushed his mouth over hers. She opened to him, willing and shuddering. "I'm going to take more," he told her.
Her eyes closed, and she wrapped her arms around his neck. "Take me, Stephen."
An invitation no man could refuse. He took over the job of removing his pantaloons himself. And threw off his boots and every other stitch of clothing he had on as well. She sat on the ground in front of him, mouth open.
He laughed at her. The sun was warm on his shoulders, and under her eyes he had that sense of his body that he only seemed to have with her. A sense of powerful muscle and a lean stomach. He came down on his haunches. She watched him in fascination, her eyes looking either at the powerful muscles in his thighs—or between them. He wasn't quite sure. But she seemed to like what she saw. That faint blush in her cheeks had turned rosy.
"I can't believe you're quite naked in the outdoors!" she said. She had her hand over her mouth, but giggles escaped.
"Your turn," he said, and her eyes grew serious.
"Oh, Stephen, I don't know...I wasn't thinking..." She kept squealing. But Stephen was very good at removing ladies' clothing, and so he had her dress over her head in a moment, and her chemise followed. She wore no corset, to his great interest. He left her only that flimsy little garment she called her pantalettes, a foolish little trifle of white cotton and lace.
The sun threw dancing spots over her ivory skin, skipping shadows of dappled color. Her face was quite rosy. She sat on the ground with her hands covering her breasts, for all the world like a timid virgin. Though of course, even an experienced courtesan might never have made love outdoors.
He kneeled just before her and put his hands over hers. "It's all right, love," he whispered. "Truly, no one will come down the lane."
"It's not that!"
He peeled one of her hands away from the alluring curve of her breast. They were perfect, rosy-tipped, uptilted, just the size for a man's hand. He bent his head and drew her nipple into his mouth, roughly for such a sweet bit of flesh. One hand flew away from her breast and curled around his neck instead.
He couldn't play this game much longer. It had been too long, weeks of longing for her, watching her secretly, watching her openly, dreaming of her. He swept her up in one decisive movement and then put her down gently on top of his jacket. As he kissed her, he let one hand shape her breast so she strained into his hand, and he let his other hand pull down that bit of cotton she called a pantalette.
She wasn't sure about that. "What if someone?..." but her voice was melting. He moved down, kissed her breast in passing until she squeaked out loud, until she writhed upwards, kept going further down her body until he found her. Until he had all that sweet, lemony flesh in front of him, and she was moaning, all deep in her throat and begging him, and begging him, and—
She reached out, grabbed his hair and yanked it hard. Bea could hardly breathe, because her whole body was on fire, but she knew there was a remedy here. There had to be. And his tormenting her was not going to be the answer.
"I want you," she said fiercely, having got his face where she could see it.
"It's your seduction, darling," he said. His lopsided grin made her heart somersault, and she almost forgot and just started kissing him again. Instead, she reached down and wrapped her fingers around him, and that did give her a shred of sanity. He was a great deal larger than Billy Laslett, and a great deal, well, firmer than Sandhurst.
For a moment she froze. What if this wasn't possible? Billy had been difficult enough. It was embarrassing to have been a party to that encounter. She had been phenomenally pleased when he'd stopped bucking about on top of her and taken himself away.
But Stephen was smiling down at her, and he seemed to know exactly what she was thinking. He unwrapped her fingers and brought himself forward, nudging her knee out of the way. Bea couldn't help herself. She arched up to meet him. But he was just teasing her, bringing her that hardness and taking it away again.
She may not have learned much, but she had learned one thing, because Billy Laslett had asked her to.... She brought her hands down from his neck and deliberately brushed his flat nipples with her fingers. He jumped and arched forward for a moment, deliciously hard. How could she ever have thought that—but this wasn't the moment for comparisons.
Instead, she gave him the same lazy, mischievous grin he gave her, and leaned forward and nipped him with her teeth. He groaned and drove forward. The rush of feeling was so exquisite that she flopped backwards and clutched his shoulders. And this time their eyes were serious.
"All right?" he said, hardly able to recognize his own voice.
And she nodded, clutching him so hard that he was going to have ten small bruises on his shoulders. He drove forward again. She cried out, unintelligible, the sound swallowed into the bright air. But it didn't seem to be pain she was registering.
He bent to kiss her, and she made startled, gulping sounds, as if she thought he might lose his balance if he tried to do two things at once. He finally managed to coax her mouth open, but she kept trying to speak.
"What is it?" he finally said, huskily.
"Nothing—oh! Don't stop that!"
Stephen smiled to himself. He pulled himself even higher and listened to her squeals floating into the meadow.
After a bit, he came up on his knees and caught her slender hips in his hands. She gasped and said, "No!" and then said nothing. So he taught her that if she lifted her hips to meet him, that was very pleasant too.
At some point she really did seem to have something to say, so he stopped kissing her. "Do you...", she was panting. "Do you—could you just keep going a little longer?"
He grinned, a fiendish grin. "I'm better at this than I am at billiards," he said. His voice was guttural, deep with desire. She was coming to meet him now, matching him. Her skin was gleaming with sweat in the sunlight. Stephen knew at that exact moment that his Bea had experienced no real woman's pleasure with those other lovers of hers.
She was a virgin, in all real senses of the word.
He felt as if the raw joy burning in the back of his throat might explode, so he simply tucked back, concentrating on showing the woman he loved that she didn't know a thing about making love. Great waves of passion kept swamping the joy. Far off in the distant recesses of his mind not occupied by the sweet undulations of her body, with the way she panted with surprise and the way her eyes were squeezed tight now, as if she were going somewhere that couldn't be seen, he was conscious of two things. One was that his buttocks had never been exposed to an English summer, and they were definitely beginning to feel as if a sunburn might be in the offing. And the second was that that infernal goat had stolen Bea's dress and galloped to the other side of the field with yards of white lace falling from its mouth.
But then even those bits of rational thought flew from him. He dove higher into her body, and she cried out, cries that spiraled, falling away into the bright air. Stephen ground his teeth and said hoarsely, "Come on, Bea, come with me!"
And Bea opened her eyes and saw him poised above her, outlined in the indigo blue sky, her beautiful, proper Puritan.
He stopped for a moment, bent his head and crushed his mouth against her. "I love you," he said hoarsely. "My Bea."
She arched up to meet him, heard his groan, lost herself in the prism of sunshine and pleasure that rained on her, spiraling through her arms and legs, driving her against his chest, telling her without words the difference between wooing and seduction.
##
## Yours Till Dawn
"Esme, what's the matter?" She was even whiter than when he'd seen her last, her face pallid and drawn. There was a gleaming trail of tears down her cheek. "Is William all right?" Sebastian sat down on the bed and peered at the babe. William looked just as moon-faced as he had last week. Long lashes brushed his cheeks, and he was snoring a little bit. Sebastian felt a funny sensation around his chestbone. He was a sweet-looking child, as children went.
"He's caught a cold," Esme said, her voice strangling on a sob.
Sebastian could see that she had obviously been crying for a long time. He put an arm around her shoulder and peered down at William again.
His rosy little lips opened in a snore.
"There! Do you hear it?" Esme said.
"He's snoring," Sebastian said. "Did Miles snore?"
"That's not a snore. He's caught a cold...probably inflammation of the lung," Esme said, tears rolling down her face. "Now I'll only have him with me for a few days at most. I knew this would happen; I knew this would happen!" Her voice rose to a near shriek.
William stirred. He could hardly move, he was wrapped in so many blankets.
"I think he's hot," Esme continued, and the broken despair in her voice caught Sebastian's heart. She put a trembling hand to the baby's head. "I keep feeling his head and one moment I think he's caught a fever, and the next he seems to be perfectly all right. What do you think, Sebastian?"
"I'm hardly an expert." He cautiously felt William's forehead. It felt sweaty to him. "Do you think he might be wearing a few too many blankets? There's quite a fire in here, after all."
"No, no," Esme said, tucking his blankets around him even more securely.
"Why don't you ask your nanny?" Sebastian asked, inspired.
"I sent her to bed. She's too old to be awake at night."
"The nursemaid, then? Surely you have some help at night."
"I sent the woman away. She just didn't understand babies. She didn't understand William, not at all. She never forgave me for nursing him myself, and she was always trying to bathe him in the midst of a cold draft."
"Oh," Sebastian said. He fished in his pocket and pulled out a handkerchief.
Esme wiped her eyes. "She kept talking about strengthening him. But William is far too frail to be exposed to drafts, or to the fresh air. Why, she actually wanted to take him outdoors! She was being grossly imprudent, and I had to tell her so."
She sniffed, and a few more tears rolled down her cheeks. "And then—and then she said that William was as fat as a porkchop and didn't have a cold at all. It was as if she'd never been around babies at all! Any fool could hear that William was having trouble breathing when he's asleep."
William snored peacefully. Sebastian looked closely at Esme and was shocked. All the generous lushness in her face was gone, replaced by a drawn exhaustion and a brutal whiteness. "Poor darling," he said. "You're all topped out, aren't you?"
"It's just that it's so tiring! No one understands William, no one! Even nanny keeps saying he's a brawny boy and I should just leave him in the nursery at night. But I can't do that, Sebastian, you must see that. What if he needed me? What if he were hungry? What if his cold worsened, or his blankets slipped?"
Sebastian pushed himself back against the headboard and then gently pulled Esme into his arms. She leaned back with a great, racking sigh, her head falling on his shoulder.
"He's a bonny lad," he said.
"Yes." She was utterly exhausted. He could see violet shadows under her eyes. Slowly he curled an arm around her and eased her back more comfortably against his shoulder. "Rest," he said softly.
"You shouldn't be here!" she said, sitting up again. "My mother—well, surely you met my mother at dinner. She's come for a visit."
Sebastian had decided not to say a word about Esme's mother. "She can have no idea that I'm in your chamber. Rest, Esme."
William snored on. After a few moments, Esme's long eyelashes fluttered closed and her body relaxed against his. Sebastian waited for a few minutes more, eased her back against the pillows, and gently took William from her arms.
Esme's eyes popped open. "Make sure you hold his head up," she said blearily. "Tuck in his blankets."
"I will," Sebastian said soothingly. "Lie down."
"You mustn't forget to prop up his neck," she insisted, but she was already toppling to the side, her whole body a testament to acute exhaustion.
Sebastian experimented cautiously for a moment and discovered what she was talking about. William's head seemed to be too heavy for his body. "I hope you outgrow this problem," he told the baby, walking over to the rocking chair by the fire. Perhaps it was just because the child was sleeping.
In the light thrown by the firelight, he could see two things. One was that William was definitely overheated. His hair was damp with sweat and his cheeks were rosy. But it didn't look like a fever; it looked as if four blankets were too much. He gently loosened some of the blankets, and it seemed to him that the baby was a little more comfortable. The second thing he noticed was that William did indeed look like Miles Rawlings. His eyes were closed, of course, but surely those were Miles's plump cheeks and Miles's rounded chin? Even the fact that William had no hair seemed evocative of Rawlings's balding state.
So Sebastian, Marquess Bonnington, rocked the baby in front of the fire and thought hard about how much he wanted the child to be his, because he hoped that if the child was his, Esme couldn't deny him fatherhood. But fatherhood wouldn't be enough anyway. He looked over at the utterly silent mound of womanhood in the bed. He didn't want Esme as a wife merely because she felt it necessary to give his son a father.
He wanted Esme to love him for himself, love him so much that she braved scandal. It was almost comical. How on earth had it happened that he, an excruciatingly correct marquess whose ideas of propriety were so rigidly enforced, had ended up asking a lady to disregard social mores, cause a scandal of profound proportions, and marry him?
And more to the point, how was he to get her to that point? He knew instinctively that it was no use asking her to marry him again. She cared only for William at the moment. Somehow, he had to bring her around to see him as a man again. And herself as a woman, as well as a mother. Sebastian rocked and thought, and William snored.
##
## Lady Beatrix Entertains
Since Bea had never allowed a gentleman to repeat the experience of bedding her, she had no idea whether she was expected to articulate a further invitation, or whether Stephen would take it for granted that he could knock at her bedchamber door. He had given no sign of his intentions over dinner. But fairness led her to admit that there was little he could have done, since he was seated between Arabella and Fanny. The two ladies spent dinner hissing insults around his shoulders, and ignoring his attempts at polite conversation. Bea's own enjoyment in the meal was dimmed when she distinctly heard Esme's mother reproach Arabella for allowing Bea to live in the same house with the pure little soul in the nursery.
Bea clenched her fists at the memory. Could she possibly marry Stephen? She, with her tarnished reputation and a malevolent influence that apparently extended to babes in the nursery? She dismissed the thought for the four hundredth time. Tonight was just another seduction, not a wooing. And she had dressed for that seduction—or undressed, howsoever one wished to put it. After all, her flimsy negligée was, well, flimsy. And she was painted, and perfumed, and curled to within an inch of her life. The only thing that seemed to calm her was applying another layer of kohl to her eyelashes, or adjusting the candles so that they fell on the bed just so. For a while she lay on the bed in a posture that displayed her entire body to its best advantage, but her stomach was jumping so much that she had to hop off the bed and pace.
There was nothing to worry about. The candles were lit, and she was perfumed in every conceivable spot that he might wish to kiss. She'd even placed a glass of water next to the bed, as she'd felt appallingly thirsty after their encounter in the goat pasture. But should she have arranged two glasses of water there, offering him one? Or would that look too rehearsed?
By the time the knock came on her door, Bea was more overwrought than she'd ever been in her entire life. "One moment!" she croaked, flinging herself toward the center of the bed. To her horror, the edge of her trailing sleeves caught the glass of water. It arched through the air, splashing water as it flew, and ended up on the bed next to her hip.
"Damnation!" Bea cried, under her breath. There was another discreet knock on the door. Of course Stephen didn't want to stand about in the corridor: what if he were seen by Helene, Esme or—a rather more terrifying possibility—Esme's mother?
"Enter!" she called hoarsely, rolling on top of the wet spot and positioning herself on her side with a hand propping up her head. Her hair was falling in the right direction to be enhanced by the pearl blue of her negligée, but she was uncomfortably aware of dampness soaking through the said garment.
He walked through the door looking as urbane and composed as if he often conducted this sort of excursion. Which, of course, he did, Bea reminded herself. Stephen was the man with two mistresses and a fiancée, after all.
"Good evening, lovely Bea," he said, closing the door and walking over to the bed.
Bea cleared her throat. "Good evening," she managed, with reasonable serenity. She looked surreptitiously down her body and was horrified to see that the silk of her negligée was apparently soaking up the water from her coverlet. Just at her hip there was a spreading patch of dark greenish–looking silk. Quickly she pulled the silk behind her and rolled onto her back so that her bottom covered the spilled water.
"And how are you, sir?" she said, smiling up at Stephen. He had seated himself on the side of the bed and was looking at her with a rather quizzical expression.
"The better for seeing you," he said.
What was that in his eyes? Bea wiggled a little. Her bottom was growing distinctly damp. Who would have thought there could be that much water in one glass?
He leaned forward and dropped a kiss on her forehead. "My word, that's a very elegant perfume you're wearing," he whispered against her cheek.
He was hovering above her. Perhaps she should give him a kiss? She brushed her lips over his, but he pulled back suddenly and sneezed. Bea sat up, realizing as she did so that she was now damp all the way to the small of her back. If she didn't change clothing, she would be sneezing as well.
"Excuse me," he said, bracing a hand on the bed and reaching into his pocket, presumably for a handkerchief.
Bea shivered. His shoulders...and the way his neck rose out of his shirt. Who would have thought Stephen Fairfax-Lacy was a symphony of muscle under all that linen? She was trembling, literally trembling, to take off his clothes again. She leaned toward him. "I missed you during dinner," she said. The naked longing in her voice was rather embarrassing. Why hadn't he given her a proper kiss?
He frowned, held up his hand and said, "Bea, your coverlet appears to be rather damp."
Bea bit her lip. "I spilled a glass of water."
"Ah." He bent close to her again and—sneezed. "I'm sorry," he apologized. "I'm terribly sorry to say that I—achoo!"
"You caught a chill in the pasture," Bea said, her heart sinking.
"Not I." He looked at her and smiled. For the first time since he entered the room Bea felt a rush of confidence. His smile said volumes about the cut of her bodice. She shifted slightly, just enough so the neckline fell off her shoulder.
The look in his eyes was dark and seductive. Bea quivered all over. Her knees suddenly felt weak, and her breath disappeared. A strong hand rounded her ankle, and the melting sensation crept up to her middle. He was on the bed now, leaning over her; Bea raised her arms to pull that hard body down on hers and—
He sneezed again.
"You are ill!" Bea said with anguish as he pulled away again.
Stephen almost wished he were. But there was no way he was leaving the room without tasting Bea's perfect little body. "It's the perfume," he admitted.
Bea's eyes widened. "My perfume?"
He nodded.
"One moment. I shall—" She scrambled off the bed and headed toward her dressing table and the pitcher of water that stood there. She began pouring water into a bowl.
Stephen swallowed. The backside of her negligée was drenched. The wet silk clung to the middle of her back, clung to the round curve of her ass, to a secret curve that turned inward, drawing a man's eye. He was off the bed in a moment, splaying his hand across that sweet bottom, eyes meeting hers in the mirror.
"Stephen!" she cried, shocked.
"Yes, Bea?" he said with a grin, his fingers slipping over the wet silk, letting the cool fabric rumple against his fingers, against the smooth skin of her bottom as he curved his fingers in and under. Silk met silky flesh and her head fell back against his shoulder. Stephen reached around her with his free hand and scooped water from the bowl.
"This may be chilly," he murmured, opening his hand on the smooth column of her neck. Her eyes flew open and she began to protest, but he had her now, wet silk over one breast, and wet silk below, and both hands slipping and rubbing. Her head fell back again and she made that little throaty moan he loved. It sounded different in a bedchamber than it had in the pasture: less thin, more deep with womanly delight. She was liquid in his arms, and the chilly silk was taking heat from her burning skin.
She turned in his arms, and her curious eyes, always so vigilant, so watchful, so wicked, were dazed. He kissed her fiercely and she begged him without words, so he cupped her bottom and pulled her hard against him.
But he couldn't concentrate because of the damn perfume, so he pulled the negligée over her head in a moment, took more water, and used his fingers as a facecloth. He started at her neck, at the smooth skin just under her ears, water dripping from his fingers, shaping her body, singing over her skin, licking kisses from his fingers. Over her collarbone, down her arms, back to her breasts, further down.... He was on his knees, and the water came with him, cooling her burning skin until he worked his way up her legs and there, then and there, his control snapped.
Bea was throbbing so much that she felt unable to speak or move. She hardly noticed when he picked her up and put her down on the wet part of the bed. She scarcely realized that he had shed his clothing. She was too busy twisting toward him. But then he was pushing her legs apart, and that dark head was there, and she was quivering, crying, pleading....
Then he cupped her face in his hands and pressed his lips to hers, and she opened to him as gladly as she wound her legs around him, as joyously as she surged against him, with as much urgency as she shattered around him, waves of pleasure flooding to the very tips of her fingers.
##
## Because It Takes Courage to Admit a Mistake
The following afternoon
Marchioness Bonnington was having a most unusual sensation. It took Honoratia quite a while to identify precisely what it was: not an incipient warning of gout, not an attack of indigestion, not a premonition that rain would soon fall. It wasn't until the gentlemen had retired to take port and the ladies to take tea in Lady Rawlings's private sitting room that Sebastian's mother knew exactly why she had a queasy feeling in the back of her stomach. There was a chance—a slim chance, but a chance nonetheless—that she was Making a Mistake.
An odd sensation, Honoratia considered. One with which she, for obvious reasons, had very little familiarity.
Mistakes seemed to generate an oddly bilious sensation in her middle section. She had it every time she looked at Lady Rawlings, who had joined them for supper on the first occasion since her child was born. She was astonishingly beautiful, that girl. Her skin had a magnolia creaminess to it. The ripeness on those lips didn't come from a bottle. Overall, though, the marchioness thought that Esme Rawlings probably gained most of her appeal from her nature, from those clever, laughing remarks of hers. From the way her eyes lit up with pleasure when she mentioned her baby.
Fanny clearly did not approve of her daughter's nature. She visibly stiffened every time Lady Rawlings laughed. "Modulate your voice, my dear," Honoratia had heard her snap during dinner. "A lady finds little to laugh at in a strident fashion."
"I'm sorry, Mother," Lady Rawlings had said instantly. She was trying so hard to make this reconciliation a success. But Honoratia thought the chances were slim.
"I find that dress rather unappealingly low in the chest," Fanny announced as soon as the ladies seated themselves.
Lady Rawlings gave the bodice of her gown an uneasy little tug. "It's only because my bosom is enhanced by the situation."
"Yes, you have gained some flesh," Fanny said, eyeing her up and down. "Perhaps a brisk walk every morning. A diet of cucumbers and vinegar can be efficacious. Dear Mr. Brummell confided in me that even he has occasionally undertaken a slimming project."
"Oh, I couldn't do that," her daughter said with a smile. "Mama, may I give you a lemon tartlet?"
"Absolutely not. I never partake of sweets in the evening. And I certainly hope you won't take one yourself."
Honoratia swallowed a smile as Lady Rawlings quickly transferred the tartlet she was about to put on her own plate to that of Lady Godwin.
"Why should you not try a cucumber diet?" Fanny insisted. "I judge you to be in rather desperate need of a slimming plan."
"It's not advisable for nursing mothers to undertake such a drastic step."
Lady Bonnington had always counted herself dear friends with Fanny, but as it happened, this was the first time they had encountered each other at the same house party. It was a bit demoralizing to realize that after a mere two days, she already recognized the thin white lines that were appearing next to Fanny's mouth as a sign of temper.
"Helene, did I understand you to say that you are leaving us?" Lady Rawlings said, turning to Lady Godwin.
"I'm afraid I must," Lady Godwin said quickly, demonstrating that she too had come to understand the signs that indicated Fanny's impending attack of temper. "Gina, the Duchess of Girton, writes me that she is expecting a child and she would be grateful for companionship. I am planning to take a carriage in two days, if you have no immediate need for my presence."
"Nursing mother? That must be some sort of witticism you thought up to horrify me," Fanny said acidly, ignoring her daughter's diversionary tactics. "My stomach is positively turning at the very thought." And she looked it. Honoratia thought there was a fair chance that Fanny would lose her supper.
"Mama, perhaps we could discuss this at a later time," Lady Rawlings said pleadingly, putting her arm on her mother's sleeve.
She shook it off. "I shall not be fobbed off. And I am certain that these ladies are as repulsed by what you said as I am!"
Honoratia took a sip of her tea. When Lady Rawlings first demanded to nurse her baby, she had been repulsed, certainly. The very idea of allowing a child to munch from one's private parts was instinctively revolting. But then she had been in the nursery yesterday while Esme nursed William, and it was hard to reconcile that experience with her own repulsion.
"While I am quite glad to have utilized a nursemaid myself," she announced, "I do not find Lady Rawlings's actions distasteful."
Fanny flashed her a hostile look that had Honoratia stiffening. Didn't Fanny realize that she was of far lower rank than she, Marchioness Bonnington? Why, it was pure kindness on her part that kept the friendship intact.
"Be that as it may," Fanny said with frigid severity, "the majority of the polite world agrees with me. Are you telling me that the fleshy expanse of chest that you are exposing to the world is due to this unsavory practice, Esme?"
Lady Rawlings sipped her tea quietly. "Yes it is, Mama."
Honoratia had to admit, Esme Rawlings had backbone.
"Had I ever been blessed by a child, I hope I would have had the courage to be as excellent a mother as is Esme," Arabella put in.
Her sister turned to her with the lowering look of a striking serpent. "It was the will of God that you not be given children, and no more than you deserve!"
Arabella went pure white, rose from her chair and walked out. There was no sound other than a faint swish of silk and then the click of the door shutting behind her.
"That was most unkind," Lady Rawlings said, looking straight at her mother. "It was unworthy of you."
"I spoke the truth as I saw it."
"I would urge you to apologize to Aunt Arabella. She has a forgiving soul, and if you make haste, she may overlook your unkindness."
Fanny merely took a sip of tea. There was a suppressed air of triumph about her. "Now," she said brightly, "you must all forgive us for this unwarranted display of poor judgement. I assure you that our family is not generally so rag-mannered!"
But her daughter was standing up. "You will have to forgive me," she said to the company at large. "Mama, I know you will act as a hostess in my absence. I shall speak to my aunt." And she was gone.
Fanny turned to Lady Beatrix Lennox. "As my sister's dame de compagnie," she said with a sapient smile, "perhaps you would like to join her, given that my daughter seems to think that Lady Withers might be distressed?"
Lady Beatrix gave her a stony look and stood up, curtsying. "I can think of little that would give me greater pleasure."
"Now we can be cozy," Fanny said, once the door closed again. "I find the presence of impure women to be extremely trying on my nerves. One has such an impulse to help, and yet no help is ever enough. Once lost, a woman's reputation can never be recovered." She shook her head. "I fear it is all a question of nature. Clearly, my daughter inherited my sister's disposition."
That was the moment when Lady Bonnington discovered what it felt like to have Made a Mistake. She accepted a tart from Fanny while she thought about it.
Countess Godwin was a lovely, if rather pale, woman. Yet when she leaned forward, Honoratia caught her breath. In profile, the countess looked like an accusing angel, a stone statue of Saint Michael standing at the gates of Paradise with a sword. "I wish you to be the first to know," she said, speaking with great precision.
"Oh?" Fanny said, looking a bit uneasy.
"I am having an affair with your daughter's fiancé, Mr. Fairfax-Lacy. We enjoy each other in ecstastic union every night."
Fanny gasped. "What a thing to say to me!" she said shrilly.
"If it be sin to love Mr. Fairfax-Lacy...well, then sin I!" retorted Lady Godwin. She stood up. "I expect my presence will make you uncomfortable, so I shall leave."
Honoratia raised her eyebrows. There was something distinctly odd about the phrasing of Lady Godwin's parting shot. And as someone who'd watched many a marriage and many a sinful union, she doubted that Lady Godwin had ever experienced ecstatic union. Still, loyalty was an admirable quality, and Lady Godwin had it in spades.
Fanny had stopped looking horror-struck and was eating one of those lemon tartlets that she never consumed in the evening. They were left alone, two hardened old harridans with shining reputations and naught much else. Neither of them had had an illicit proposal in years.
Fanny patted her mouth delicately. "I wonder that you chose this house to retire from the season, dear Honoratia," she said. "I leave tomorrow at dawn to return to Lady Pindlethorp's house. I told Esme as much this morning, and now my mind is made up. You would be more than welcome to join me."
"Wouldn't you rather stay and make further acquaintance with your grandson?"
"It's far, far too painful. My daughter has no understanding of the grief I still bear every time I think of my dear departed son. And I am very much afraid that my initial qualms about my daughter's rehabilitation are entirely correct. I admire your generous nature, my dear, but you are far too optimistic. Are you aware that my daughter has no real idea whose child she birthed?"
"Certainly not!" Honoratia replied in her most quelling tone of voice. Surely—surely—Esme's own mother wouldn't repeat such a vicious piece of gossip about her own daughter.
Fanny took a bite of tartlet. "I queried her on the matter, most discreetly, you understand, through the post. She did not respond to my query, which speaks for itself, does it not? This tea is quite cold." She rang the bell. "As I said, I would be more than welcome for your company tomorrow morning."
Honoratia stood up. Fanny looked up, startled. Honoratia thumped her stick, and, sure enough, Fanny quailed with as much fear as any lazy housemaid. "You will not say a word to anyone about your grandson's patrimony," she ordered.
"Well, naturally, I—" Fanny said, flustered. "I only tell you as you are a very close friend!"
"From this moment, we are not close friends," Honoratia said, pulling herself even straighter. "In fact, we are not friends at all. If I ever hear a breath of scandal about your daughter or your grandson that has begun at your lips, Fanny, I shall ruin you."
Fanny stared up at her, faded eyes wide.
"Do I make myself clear?"
Fanny jumped but said nothing.
"Do I make myself clear?" Honoratia said, with the snap of a carnivorous turtle.
Fanny twittered. "I can't imagine why you would think that I would ever do something as ill-bred as gossip about my daughter's debased circumstances." Then she faltered, seeing Honoratia's expression. "I shall not!" she said shrilly.
Honoratia didn't bother with a reply. She just stumped over to the door and left Fanny there among the crumbs of lemon tarts and cooling cups of tea.
##
## Nights of Ecstatic Union
"And then I said that we spend every night in ecstatic union with each other!"
"Ecstatic what?" Esme asked.
"Ecstatic union. It was the only thing that came to mind. It is a rather odd phrase, is it not? And then I quoted a bit of the poetry Bea lent me, the part being a sin to love. Your mother was quite horrified, Esme." Helene looked triumphant.
Esme choked with laughter. She was sitting on her aunt's bed, arm wound around her aunt's neck. Helene was standing before them like a militant, raging angel. Bea was curled up on the little armchair to the side.
"You didn't have to do that," Arabella said damply, blotting a last few tears with a handkerchief. "Drat! I've taken off all my facepaint. I must look a veritable hag."
"You look beautiful," her niece said, giving her a squeeze.
"Fanny really doesn't mean to be so horrible," Arabella said. "She's had a most difficult life."
"Yes she does," Helene said firmly. "I'm sorry, Lady Withers, but your sister is a truly poisonous woman. And I'm sorry for you, too, Esme."
Esme looked up with a rueful smile. "And what a dreadful thing in a daughter to agree with you." But she didn't disagree either.
Arabella gave a last sniff. "I haven't cried for years," she said, "so I suppose I was due for a bout of tears. Fanny's comments generally don't distress me very much. But Robbie and I did so want children. I thought perhaps when he died...well, I didn't have my flux for months. And I thought that perhaps I carried a bit of Robbie with me." She gave another sniff. "But finally the doctor said that it must have been due to grief." She wiped away some tears. "What a wet blanket I've become!"
"You're not a wet blanket," Esme said. "You're one of the bravest people I know."
Arabella chuckled damply. "Well, that's a new compliment for me. Thank you, my dear."
Esme's own smile wavered. "And the dearest as well. No mother could have helped me more than you have, Arabella, nor a sister more than you, Helene." She met their eyes, and now they were all a little teary.
"I couldn't have loved a child more than I love you, dearest," Arabella said.
Helene sat down hard on Arabella's dressing table stool. "Do you still feel a great deal of grief due to not having a child, Lady Withers? If you don't mind my asking?"
Arabella gave her an unsteady smile. "It is not terrible, no. But it is a sadness to me, since I would have been delighted to be a mother. Yet just having the chance to be with William is very healing in that respect."
Helene pressed her lips together. "I want you all to know that I am going to have a child."
Unexpectedly, Bea, who'd been sitting silently to the side, yelped, "What?" And then clapped her hand over her mouth. "I'm sorry! It's none of my business."
"My dissipated husband returned to London still refusing to divorce me, and I have decided to have a child irrespective of my marital situation. If Rees wishes to divorce me after the fact, on the ground of adultery, I truly don't give a bean."
"Would you then marry Mr. Fairfax-Lacy?" Bea asked. The strain in her voice made all three women look at her.
"Stephen? No!" Helene said. "Stephen has no aspirations to my hand. Or bed, for that matter, although he was kind enough to pretend so before my husband." There was a pause. "Are you going to marry him?"
Bea swallowed and then looked to Esme. "Lady Rawlings has precedence."
Esme laughed. "I surrender my claim."
"Then I am," Bea said sedately. A smile was dawning on her face. "I am going to marry him."
"Bravo!" Arabella said, tossing her handkerchief onto her dressing table. "I knew the man was good marrying material. Didn't I tell you so, dear?" she said to Esme.
"I merely have to ask him," Bea put in.
Helene blinked at her. "Hasn't he asked you?"
"Not in so many words. He wishes to be wooed."
"What an extraordinary thing," Helene said slowly. "Do you know, I am coming to have an entirely different idea of how to behave around men?"
Arabella nodded. "If you wish to have a child, you will need to move decisively. That's why I married so quickly after Robbie died. I wasn't in love, wasn't even in my right head, I think now. But I wanted a child. Mind you, it didn't work for me, but it might well for you."
Helene nodded. "You may not wish to acknowledge me in the future," she said, looking at Esme. "I will create a tremendous scandal by having a child. Everyone in the polite world knows that I have no contact whatsoever with my husband."
Esme stood up and gave her a fierce hug. "You never deserted me, and I would never desert you. What would I have done without you and Arabella these past few months? Besides, I do believe I shall give up some of my aspirations to respectability."
"Thank goodness!" Arabella said, with a world of meaning in her voice.
Helene turned to Bea. "I trust you don't mind my saying that you are very inspiring. I mean to copy down that poem, if you don't mind. Perhaps I shall have use for it another day."
Bea grinned. "As long as you are not planning to direct your invitation to Mr. Fairfax-Lacy, you may use it as you please."
"How are you going to ask him to marry you?" Esme asked, fascinated.
Bea bit her lip. "I only just this moment decided to do so. I really don't know."
"Poetry," Helene said positively. "Obviously, you must use poetry."
Esme clapped her hands. "We'll have a small party tomorrow night, just amongst ourselves, and we shall complete the poetry reading that we began."
"That means I shall have to find an appropriate poem," Bea said. "I suppose I had better hie me to the library." She looked at Esme. "You didn't read a poem at our last such reading."
"I haven't such a pressing need as yourself," Esme said lightly.
"Humph," Arabella snorted. "That's one way of putting it."
Esme frowned at her.
"Well, you've got an eligible man visiting your chambers on the sly," Arabella said irrepressibly. "You might as well let him make an honest woman of you."
Bea's eyes grew round. "Which man?"
Arabella replied. "The marquess, naturally."
Helene laughed. "Oh Esme," she said, "you are truly Infamous Esme, are you not?"
"I most certainly am not," Esme said with dignity. But all her friends were laughing, so after a bit she gave in and laughed as well.
##
## The Poetry Reading
Mrs. Cable was rather scandalized to find that she was attending a poetry reading. But while inviting the Sewing Circle, Lady Rawlings had noted that she herself intended to read from the Bible, and Mrs. Cable had decided that encouragement of such a devout practice was a virtue. And if she was honest, she was finding the presence of the scandalous Marquess Bonnington rather enthralling. He was, well, wickedly attractive. Mrs. Cable secretly thought that she'd never seen anyone quite so mesmerizing: those dusky golden curls, and he had such a powerful body! Although she hardly put it to herself like that. In truth, Mrs. Cable had some difficulty dragging her eyes away.
There certainly was enough to see at this particular gathering. She was absolutely certain that Lady Beatrix, for example, had reddened her lips, if not worse. Naturally Lady Winifred was having the time of her life trundling around the room with her dear friend Arabella. It was quite a sorrow to see how susceptible Lady Winifred was to the lures of the fashionably impure. And Mr. Barret-Ducrorq was almost as bad. He seemed to be fascinated by Lady Withers, and Mrs. Barret-Ducrorq had had to call her husband to heel quite sharply. Mrs. Cable looked with satisfaction at her own husband. He was sitting next to her, nursing his brandy and looking stolidly bored. Mr. Cable had attended the reading only after bitter protest; he did not consider poetry to be palatable entertainment.
Lady Rawlings clapped her hands. "For those of you who have recently joined us, we have been entertaining ourselves in the evening by giving impromptu poetry readings. We shall have two readings this evening. First Lady Beatrix will read a piece from Shakespeare, and then I shall read a piece from the Bible."
Mrs. Cable felt cheered. She must have had an influence on the young widow. Shakespeare and the Bible: what could be more unexceptional than that? Lady Beatrix walked before the group and stood in front of the fireplace. She was wearing a dinner gown of moss silk, in a bright rose color. Of course, the bodice bared far more of her neck and bosom than Mrs. Cable considered acceptable. But Lady Beatrix looked nervous, which Mrs. Cable counted in her favor. A young lady entertaining a group of distinguished guests ought to be fairly shaking with fright.
And, indeed, had she but known, Bea was literally trembling. She kept sneaking glances at Stephen, but he hadn't even smiled at her. There was nothing in his demeanor to indicate that he had spent virtually the whole of last night in her bed. "I have chosen a dialogue," Bea told the assembled company, "from Romeo and Juliet."
"An excellent choice," Lady Bonnington commented. "I am very fond of Mr. Shakespeare's works. I don't hold with those who criticize him for frivolity."
"I suppose you need a man for your dialogue," Esme said. "Do choose a partner, Bea."
My goodness, but Esme's eyes had a wicked suggestiveness to them, Bea thought. It would serve her right if she chose Marquess Bonnington, if she stole Esme's supposedly unwanted suitor from under her nose. Naturally Esme was cushioned between the two most eligible men in the room. She had Stephen on her left and Marquess Bonnington on her right.
But Bea didn't chose Bonnington, of course. She turned to Stephen and gave him a melting smile. "Mr. Fairfax-Lacy, would you be so kind?"
His face gave nothing away. He came to his feet with easy grace and accepted the open book she handed him.
"We'll read from the balcony scene," she told him.
"Very good! Very good!" Lady Bonnington trumpeted. "I've always been fond of 'Wherefore art thou, Romeo'?" She turned to her son. "Do you remember when we saw Edmund Kean perform as Romeo last year, dear?"
Sebastian frowned at her. He had the feeling that something quite important was happening and—more important—it looked to be the kind of event that might derail Esme's patently artificial engagement to Fairfax-Lacy. Lady Beatrix seemed to be a handful, but the way Fairfax-Lacy was looking at her, he was ready to take on the task.
Meanwhile Stephen looked down at Bea and felt as if his heart would burst with pure exhilaration. She was wooing him, his own darling girl had decided to woo him. He glanced down at the book. "'But, soft! What light through yonder window breaks? It is the east, and Juliet is the sun.'" His eyes told her silently the same things he read: She was his east, his sun, his life. But she hardly glanced at him, the silly girl, just kept looking at her book as if she might lose courage.
Bea gripped her book as if holding its pages would force her fingers to stop trembling. She was doing it: she was stealing him, taking him, ruining him..."'Good night, good night!'" she said steadily, "'As sweet repose and rest come to thy heart as that within my breast!'" She risked a look at him. The tender smile in his eyes was all she ever wanted in life. She took a deep breath and kept reading until there it was before her. She glanced at the group watching: met Esme's laughing eyes, and Helene's steady gray ones, Sebastian Bonnington's sardonic, sympathetic gaze, and Lady Bonnington's look of dawning understanding. Then she turned back to Stephen.
She had no need of the book, so she closed it and put it to the side. "'If that thy bent of love be honorable,'" she said clearly, ' "thy purpose marriage, send me word tomorrow...'"
But his voice joined hers as he held out his hands. "'Where and what time thou wilt perform the rite, and all my fortunes at thy foot I'll lay, and follow thee my lord throughout the world.'"
"I will," Stephen said, smiling at her in a way that broke her heart and mended it again, all in one moment. "I will, Bea, I will."
"You will?" she asked with a wobbly smile, clinging to his hands. "You will?"
"What's that? Part of the play?" Mr. Barret-Ducrorq said. "Quite the actor, isn't he?"
"I will marry you," Stephen said. His voice rang in the room.
Bea's knees trembled with the shock of it. The smile on her lips was in her heart. She'd wooed a man. His mouth was hungry, violent, possessive, and she nestled into him like the very picture of—of a wife.
"Ladies and gentlemen," Stephen said a moment later. He turned, his arm snug around Bea. "May I present the future Mrs. Fairfax-Lacy?"
Esme was laughing. Marquess Bonnington bellowed, "Good man!" Even Lady Bonnington gave a sedate little nod of her head, although she quickly turned to Esme. "You would appear to have lost your fiancé," she observed. And then, "How fortuitous that your mother left this morning."
"Yes, isn't it lucky," Esme said, smiling at her.
Stephen pulled Bea away to sit next to him on the settee, where he could presumably whisper things in her ear not meant for public discussion. Esme straightened her shoulders. Her heart was hammering in her chest from nerves. "I shall read from the Bible," she said, picking up the book from the table and walking to the front of the room. It was Miles's Bible that she carried, the family Bible, into which she had written William's name. But she had the feeling that Miles approved, almost as if he were there in the room, with his blue eyes and sweet smile.
"It is a pleasure to see a young widow immerse herself in the Lord's words," Mrs. Cable said loudly. "I believe I have set an example in that respect."
"You're not a widow yet," her husband said sourly.
Sebastian was the picture of sardonic boredom. Obviously he thought that Esme was merely cultivating her Sewing Circle, quoting the Bible in the hopes of polishing her reputation. Esme swallowed. He was looking down at his drink, and all she could see was the dark gold of his hair. "I shall read from the Song of Solomon," she said. Sebastian's head swung up sharply.
"'The song of songs, which is Solomon's,'" she read, steadying her voice. "'Let him kiss me with the kisses of his mouth: for thy love is better than wine.'"
"Didn't she say that she was going to read from the Bible?" Mr. Barret-Ducrorq asked, in great confusion.
"Hush!" Lady Bonnington said. She was sitting bolt upright, her stick clutched in her hands. Her eyes were shining and—wonder of wonders—she was smiling.
Esme kept reading. "'Stay me with flagons, comfort me with apples: for I am sick of love.'"
Abruptly Sebastian stood up. Mrs. Cable was looking at him. Esme looked at him too, telling him the truth with every word she read. "'My beloved spake, and said unto me, Rise up, my love, my fair one, and come away.'"
He strode toward her, skirting his mother's chair, the settee, Mrs. Cable sitting in rigid horror.
"'For lo, the winter is past,'" Esme said softly, only for him. "'The rain is over and gone; The flowers appear on the earth.'"
He was there before her, taking the book away, taking her hands in his large ones. She looked up at him.
"'My beloved is mine, and I am his: he feedeth among the lilies.'"
His arms closed around her with hungry violence. A shudder ran through Esme's body as she lifted her mouth to his. How could she ever have thought that anything mattered more than Sebastian, her love, her deep center, her heart.
He tore his mouth from hers for a moment. "I love you," he said hoarsely.
Joy raced through Esme's body, sang between them.
"And 'I am sick with love for you,'" she said softly, repeating the beautiful old words of the ancient book.
Mrs. Cable's mouth snapped shut. She grabbed her husband by the arm and hauled him to his feet. "I am appalled!" she hissed. "Appalled!"
Lady Rawlings didn't heed her, crushed as she was into that degenerate marquess's arms. Mrs. Cable could see what had happened. She had lost the battle for the widow's soul, yes, and the devil had won. Lust and Lasciviousness ruled this house.
"We are leaving!"
She turned to go and found her way blocked by Marchioness Bonnington. "I pity you!" Mrs. Cable croaked, narrowing her eyes. "But perhaps your son is well matched by such a lightskirt."
"I daresay he is," the marchioness replied. There was something in her eyes that gave Mrs. Cable pause. "Surely you wish to give the happy couple your congratulations before you leave so precipitously?"
But Mrs. Cable had a backbone to match the marchioness's. "I do not," she said, fixing her beady eyes on Lady Bonnington. "And if you would inform your dissolute daughter-in-law that her services are no longer desired in the Sewing Circle, I would be most grateful."
The marchioness stepped back, something to Mr. Cable's relief. He was beginning to fear that his wife would actually pummel a peeress of the realm.
"I should be most happy to fulfill your request," Lady Bonnington said.
The smile that played around the marchioness's mouth so enraged Mrs. Cable that she didn't even realize for several hours that the rest of her Sewing Circle had not followed her from the room.
Alas, it was the demise of that excellent institution.
A month or so later, Mrs. Cable began a Knitting Circle drawn from women in the village, priding herself on bringing the Lord's words to illiterate laborers. Without her leadership, the Sewing Circle drifted into dissolute activities such as attending Lady Rawlings's wedding to the degenerate marquess. Society noted that Lady Rawlings's mother did not attend. But the smiling presence of Marchioness Bonnington, and the weight of her formidable power in the ton, established the marriage as the most fashionable event of the season.
Rather more quietly, Lady Beatrix Lennox married Mr. Fairfax-Lacy from her own house, with only her immediate family in attendance. It was rumored that her only attendants were her sisters, and that they wore daisy chains on their heads, which sounded odd indeed. The newly wed couple returned to London, and by the time that society really noticed what had happened, and with whom, the new Mrs. Fairfax-Lacy proved to have such powerful friends that hardly more than a murmur was heard of her blackened reputation. Besides, the Tory party quickly realized that she showed considerable potential as a political wife.
Helene, Countess Godwin, traveled to attend her friend the Duchess of Girton's confinement. Through the whole summer and fall she brooded on the child she was determined to have. By hook or by crook, with the help of her husband, or without him.
But that's a story for another day....
## The First Epilogue
## Plump as a Porker
Esme started awake, as always, with a bolt of fear. Where was William? Was he all right? A second later she realized that what had woken her was a chuckle, a baby's chuckle. The curtains were open and early sunlight was streaming into the room. Sebastian was standing in front of the window, wearing only pantaloons. His shoulders were a ravishing spread of muscles. And there, just peeking over his left shoulder, was a tiny curled fist, waving in the air.
A cascade of baby giggles erupted into the room.
Sebastian was dancing William up and down on his arm. The question of chilly drafts leaped into Esme's throat. She never let William go anywhere near a window. But then...it felt as if high summer had come. Sebastian spun around and William screamed with laughter. He was sitting on the crook of Sebastian's arm, and he wasn't even wearing a nappy.
Esme's heart skipped a beat. She never took all William's clothes off at once!
But the baby was clutching Sebastian's hair and squealing. Sebastian obligingly bounced him up into the air again. Esme found herself looking at a stunningly beautiful man, all muscles and smooth golden skin, and tumbling curls.
And then, suddenly, she looked at William. It was rather like looking sideways and suddenly catching sight of oneself in the mirror without recognizing who it is. Because the naked man in her bedchamber was holding one of the fattest, healthiest, happiest babies she'd ever seen.
That was William. Her sickly, fragile son?
Esme's mouth fell open.
Sebastian still didn't know she was watching. He was holding William in the air and laughing up at him. Pudgy little legs kicked with delight. "You love that, don't you, son," he said. And every time he jiggled William, the baby giggled and giggled. Until Sebastian nestled him back against his chest. It was when Sebastian was kissing William's curls that he caught sight of Esme's wide eyes.
He was clearly unsure of Esme's reaction to William's undressed state. "He loves it, Esme," he said quickly. "See?" And he tickled William's plump little tummy. Sure enough, William leaned back against his shoulder and giggled so hard that all his fat little bits shook with delight. And there were many parts jiggling.
"He is healthy, isn't he?" Esme said with awe.
"He's a porker," Sebastian said.
"Oh my goodness," Esme breathed. "I just—I didn't—"
Sebastian brought William over to the bed. "I promise you that he's not chilled, Esme. Not in the slightest. I never would have removed his clothes if I thought he might take a chill."
William lay on the coverlet kicking his legs and waving his arms, gleefully celebrating freedom from three layers of woolens.
"It's summer, Esme," Sebastian said gently. "Roses are blooming in the arbor. And I do believe some exercise will do him good." He rolled the baby over. William squealed with delight and then poked up his large head inquisitively. "He's gaining some control over his neck," Sebastian said, looking as pleased as if William had taken a top degree at Oxford University.
Esme opened her mouth—and stopped.
The sun was shining down on the sturdy little baby's body, on his brown hair that was so like his father Miles's hair. Onto his unsteady head, blue eyes blinking up at Sebastian with precisely the sweetness that Miles had given to him.
And there, at the very base of his spine, was a small spangled mark. A mark that hadn't been there at his birth, but was indubitably present now.
"Sebastian," she said quietly. There was something in her voice that made him turn to her immediately. "Look."
Sebastian stared at the bottom of his son's spine and didn't say a word.
"What do you think?"
"I think it looks very much like the mark I have at the base of my spine," he said slowly. He looked puzzled rather than joyous. Then, after a moment, he laughed. "I was right! He may have suddenly become my blood relation, but I already loved him with every bit of my heart."
Esme looked up at him, eyes brimming. "Oh, Sebastian, what would I ever do without you?"
He stared at her for a moment, and then a little crooked smile curled his mouth. "I won't answer that, because it will never happen."
William rolled over, his naked little arms waving in the air. His mama and papa weren't watching him wave at the dust fairies playing in a ray of sunshine. They were locked in each other's arms, and his papa was kissing his mama in that way he had: as if she were the most delectable, desirable, wonderful person in the world. And she was kissing him back, as if she would throw away the world and all its glories merely to be in his arms.
William giggled again and kicked the air, scattering dust fairies like golden stars in all directions.
## The Second Epilogue
## In Which a Puritan Loses His Reputation
It was high summer. The air was heavy with dust and smoke, and the streets smelled of ripe manure. The odor crept into the houses of the very rich, even into an occasion as grand as Lady Trundlebridge's yearly ball, where bunches of lavender could do nothing for the stench. "Paugh!" exclaimed the Honorable Gerard Bunge as he held a heavily scented handkerchief to his nose. "I cannot abide the end of the season. Even I must needs think of the country, and you know I loathe the very sight of sheep."
"I feel precisely the same way," his cousin, Lady Felicia Saville, sighed, fluttering her fan so quickly that it would have ruffled hair less severely tamed by a curling iron. "London is simply abominable at the end of the season." She straightened and snapped shut her fan, making up her mind on the moment. "I shall leave for the country tomorrow, Gerard. The season is over. This ball, for example, is unutterably tedious."
Gerard nodded. "Nothing left but the dregs of gossip, m'dear. Did you catch a glimpse of Fairfax-Lacy and his bride?"
"A doomed marriage," she said, with some satisfaction. Alas, Lady Felicia Saville was something of a personal expert on the subject. "A man of such reputation marrying the notorious Lady Beatrix!" Her high-pitched laughter said it all. "Do you know, I believe I saw Sandhurst earlier. Perhaps she will recommence her alliance now she is safely married. Given Lady Ditcher's interruption, I would say their encounter left, shall we say, something to be desired?"
Gerard tittered appreciatively. "You do have a way with words, Cousin. Look: Lady Beatrix is dancing with Lord Pilverton. She is rather exquisite; you can't fault Sandhurst for taste."
But Felicia had never been fond of musing over other women's attractions, particularly those of women like Lady Beatrix, who appeared to have a flair for fashion rivaling her own. "I should like to walk in the garden, Gerard," she commanded.
"My red heels!" he protested. "They're far too delicate for gravel paths."
"And far too out of fashion to protect. This year no one wears red heels other than yourself, although I haven't wanted to mention it." And she swept through the great double doors into the garden, her cousin reluctantly trailing behind her.
They weren't the only people to escape the stuffy ballroom. The narrow little paths of Lady Trundlebridge's garden were fairly heaving with sweaty members of the aristocracy, their starched neckcloths hanging limply around their necks. Stephen Fairfax-Lacy, for example, was striding down a path as if he could create a breath of fresh air just by moving quickly. Bea had talked him into giving up his pipe, and while he thought that it was a good idea on the whole, there were moments when he longed for nothing more than the smell of Virginia tobacco. Thinking of Bea, and pipes, he turned the corner and found himself face-to-face with—
Sandhurst.
Bea's Sandhurst. The man disreputable enough to seduce a young girl in a drawing room. The man who'd ruined Bea's reputation.
Sandhurst was a sleek-looking man, with his hair swept into ordered curls and a quizzing glass strung on his chest by a silver chain. He took one look at Fairfax-Lacy and didn't bother with prevarication. "I offered to marry her," he said, his voice squeaking upward.
Stephen didn't even hear him. He was stripping off his coat. There was a reason why he'd trained in Gentleman Jackson's boxing salon, day after day for the past ten years. True, he hadn't known what it was, but now he realized.
"Mr. Fairfax-Lacy!" Sandhurst squealed, backing up. "Couldn't we simply discuss this like gentlemen?"
"Like what?" Stephen asked, advancing on him with the slow, lethal tread of a wolf. "Like gentlemen?"
"Yes!" Sandhurst gulped.
"You forfeited that title a few years ago," Stephen said, coming in with a swift uppercut. There was a satisfying thunk of fist on bone. Sandhurst reeled back, hand to his jaw.
"Fight!" yelled an enthusiastic voice at Stephen's shoulder. He paid no mind. His arm shot out. A sledge-hammer, in Jackson's best manner. Sandhurst fell back, tripped, and landed on his ass. Stephen was conscious of a thrum of disappointment. Was the man simply going to stand there and play the part of a punching bag? He watched dispassionately as Sandhurst picked himself off the gravel.
There was a growing circle around them in the shadowy garden, calling to each other to discover who was in the fight, hushing to a whisper as the relation between the two men was explained. A voice bellowed from behind Sandhurst: "For God's sake, man, pull yourself together!" Others joined in, rather like a crowd at a cockfight. "Show yourself a man, Sandhurst! By God, you're nothing more than a nursling! A molly! A..." Stephen blanked the voices from his mind and watched his opponent, who was being goaded into a decent effort. He was pulling off his jacket with the air of a maddened bull.
I think, a nobber, Stephen thought. Yes, and then a left hook. And after that, he dodged a hit, feigned right, launched a chop at Sandhurst's jaw. Took one himself in the right eye—damn, now Bea would demand an explanation. The irritation he felt at that translated to his right arm: a leveller, and Sandhurst dropped to the ground like a fallen tree. Stephen nudged him with his foot to make sure he was completely out, looked up, and caught the eye of his hostess. She deliberately threw up her fan and said something Stephen couldn't hear to the lady beside her, who laughed shrilly and said, "It's what comes naturally after associating with the House of Commons!"
He was picking up his coat when he felt a hand on his arm. "Mr. Fairfax-Lacy," said Lady Felicia Saville, her voice sweet as honey. "Would you be so kind as to escort me to the house?"
Stephen bowed. Apparently barbarous—nay, common—behavior was the way to this gentlewoman's heart. "If you will allow me to replace my jacket," he said.
"Hardly the behavior of the prudent man of Parliament," Felicia laughed up at him as they strolled back toward the house, quite as if nothing had taken place at all. "You will be quite the man of the hour."
"I highly doubt that. I'm afraid Lady Trundlebridge did not appreciate my behavior." He didn't feel like a Member of Parliament. He felt damn near—exuberant.
Felicia shrugged. "You were defending your wife's honor. Any woman of sense must applaud you, sir!" There was a flutter of warmth in Felicia's stomach when he smiled at her compliment. Perhaps once Lady Beatrix returned to her wandering ways, she could comfort Beatrix's neglected husband.
Just inside the ballroom doors, Stephen bowed. "If you will excuse me, Lady Felicia, I shall locate my wife."
He walked away without a backward glance, leaving Felicia with her mouth all but hanging open. Why had she never noticed how muscled and attractive the man was? She turned to meet the curious eyes of one of her bosom friends.
"Did you see the fight itself?" Penelope squealed. "Is it true that he called Sandhurst a blathering blackguard?"
Felicia's eyes were still a little dreamy. "Now there's a man worth having," she whispered to Penelope. "He was like a medieval knight protecting his wife's honor. He flattened Sandhurst!"
"Do you think he means to keep it up?" Penelope giggled. "Unless marriage changes Lady Beatrix's nature, he's going to be a busy man."
Felicia was watching his dark head as he made his way to the other side of the room. "She'd be a fool to stray," she sighed.
Bea was growing a little tired. Her shoes pinched loathsomely, and thanks to an overly energetic waltz, Pilverton had left a damp patch from his hand on the back of her gown. She turned gratefully at the sound of her husband's voice, and then gasped. "Stephen! What on earth happened to you?"
But he was grinning. "Nothing important. Are you ready to leave, m'dear? It's damnably hot in here."
"Stephen!" Bea said, her voice rising. "You tell me this moment what you've been up to."
"Making a spectacle of myself," he told her obligingly. "Fistfight in public. Shouldn't wonder if my reputation for tolerant debate isn't ruined." He said it with distinct relish, towing her out of the ballroom as he spoke. "I think it's time to retire to the country."
"We can't go to the country yet," Bea said, stopping and looking up at him suspiciously. "The House isn't closing session for at least a week." His eye was growing darker by the moment. "Just who have you been tussling with? Don't tell me you actually resorted to blows over that Enclosure Act?"
He reached around behind her and opened the door to the library. When she was inside, he leaned against it and grinned at her. "Something of the kind," he drawled.
"Really!" Bea said, rather amused. "It's hard to believe that solid, respectable members of Parliament can bring themselves to violence." And then, "What on earth are you doing, Stephen?"
He had turned the key in the lock. "I'm not a solid, respectable member, Bea. I'm resigning tomorrow morning, and I won't stand for reelection either." There was a sound at his back.
"Someone wishes to enter," Bea observed. "Stephen!" For he was walking toward her with an unmistakably lustful glint in his eye. There was something tantalizing about the air of wild exuberance that hung around him. "Did you take a blow to the head?" Bea asked, her voice rising to a squeak.
"No," he said, and his voice was rich with laughter. There was a bang at the door. "It's Fairfax-Lacy," he bellowed. "I'm in here kissing my wife. Go make yourself useful by telling Lady Trundlebridge."
There was a sound of rapidly retreating footsteps, and then the room was quiet but for the faint hum of the ball continuing on the other side of the house.
"Stephen Fairfax-Lacy!" his wife gasped.
"I'm a madman in love with my wife." He had her now, cupping her face in his hands. "I do believe I shall make love to you at Lady Trundlebridge's ball, and ruin my reputation for once and for all." One hand slid to her breast, and that rush of melting pleasure that came at his slightest touch rushed down Bea's legs. He kissed her until she was limp, until he had backed her onto a couch, until she was gasping, pink in the cheeks, almost—almost lost.
"Stephen," she said huskily, removing his hand, which had somehow managed to get under her gown and was touching her in a flagrantly ungentlemanly fashion.
"Darling." But he was busy. The necklines of Bea's gowns were so useful that he didn't know why he'd ever thought they were too low. They were perfect.
She pushed at his shoulders. Something was prickling the back of her mind. "Stephen, with whom precisely did you fight?"
He raised his head and looked at her. His right eye was almost swollen shut, but the gleam of desire was there. He feathered his lips over hers.
"Stephen!"
"Sandhurst," he said obligingly.
Bea gasped.
"We were fighting over an Enclosure Act, just as you guessed. I'm like all those nasty sheep farmers, Bea. You're mine. I've enclosed you."
"But—but—"
"Hush," he said and kissed her again.
Bea looked up at him, and there were tears in her eyes. "Oh Stephen," she whispered. "I love you."
"Can we go home now, Bea? We've been in London for a month and have been received everywhere. I've tramped off to the House and listened to assinine debates. Our marriage didn't ruin my career. In fact, with the way Lord Liverpool looks at you, I stand to be named to the cabinet if I'm not smart enough to resign quickly."
She smiled at him mistily. "Are you saying I told you so?"
"With any luck, I just ruined my career," he said, kissing her. "Now may we leave London, please? Shall we go home and chase each other around the billiards table, and start a goat farm, and perhaps a baby, and make love in the pasture?"
Bea wanted to weep for the joy of it, for her luck in finding him, for the bliss of realizing he was right. He was right. She hadn't ruined his career. "Oh, Stephen," she said huskily, "I do love you."
"I made you woo me," he said, looking into her eyes. "I think it's time that I courted you, don't you think?" His arms closed around her, arms that would never abandon her, and never let go. "Flowers at dawn," he whispered into her ear, "daisy chains for lunch, champagne in your bath."
Bea swallowed hard so she wouldn't cry. "I love you," she said again.
"I think Romeo said it best," her husband said, brushing his lips over hers. "You are, indeed, my love, my wife."
## A Note on Shakespeare and his Wilder Brethren
The last words of A Wild Pursuit were written by Shakespeare, and spoken by Romeo. I decided to close the novel with Romeo's farewell to his bride because Renaissance poetry is so important to this book as a whole. Bea uses Romeo and Juliet to propose to Stephen Fairfax-Lacy; Esme uses the King James version of The Song of Solomon to propose to Sebastian Bonnington.
But the book is also punctuated by works far less known than these two famed pieces of love poetry. Richard Barnfield published only two books of verse, which appeared in 1594 and 1595, precisely when Romeo and Juliet was likely first performed. For their time, both Shakespeare's play and Barnfield's poetry were shockingly original. Juliet's proposal to Romeo, not to mention the speech in which she longs for their wedding night to begin, both startled and delighted London audiences. Romeo and Juliet was a howling success; ten years later, young courtiers were still quoting the play to each other on the street. Its popularity is attested to by the fact that in 1607 a company of boys put on the stage a play called The Puritan, which contains a riotous parody of Juliet's balcony scene. Some lines from that play are used by Esme to poke fun at Romeo and Juliet, precisely as the original boy actors did back in 1607.
Richard Barnfield's poetry was, in a different fashion, as shocking as Shakespeare's portrayal of Juliet. The book that Bea brings with her to Esme's house party was an odd amalgam of love poetry and narrative verse. Amongst the various odes and lyrics Barnfield wrote are some of the most beautiful, sensual, and explicit poems written before the twentieth century. As you can perhaps tell from the reaction Helene has to reading aloud a Barnfield poem, neither Renaissance nor Regency readers were accustomed to expressing in public a wish that My lips were honey, and thy mouth a bee. I sometimes receive letters from readers contending that aristocrats living in the Regency period would have acted with propriety at all times, even in the privacy of their own bedchambers. I thought it well to present some poetry written over two hundred years before the Regent took the throne. Barnfield may have been one of the first Englishmen to put this desire in print; he was neither the first, nor the last, to express it.
## About the Author
Author of seven award-winning romances, Eloisa James is a professor of English literature who lives with her family in New Jersey. All her books must have been written in her sleep, because her days are taken up by caring for two children with advanced degrees in whining, a demanding guinea pig, a smelly frog, and a tumbledown house. Letters from readers provide a great escape! Write Eloisa at eloisa@eloisajames.com or visit her website at www.eloisajames.com.
Visit www.AuthorTracker.com for exclusive information on your favorite HarperCollins authors
## Praise
## USA TODAY BESTSELLING AUTHOR
ELOISA JAMES
"Writes with a captivating blend of charm, style, and grace that never fails to leave the reader sighing and smiling and falling in love. Her style is exquisite, her prose pure magic. Nothing gets me to a bookstore faster than a new novel by Eloisa James."
New York Times bestselling author Julia Quinn
"Brings to mind the best of
Amanda Quick and Judith McNaught."
Booklist
"Offers romance readers what they love."
Columbia (S.C.) State
"Weaves a story as rich in plot as in character."
Publishers Weekly
"Has taken the Regency to a new level."
San Antonio Express-News
## [Books by
Eloisa James](9780061801952_contents.html#adp_01)
SEVEN MINUTES IN HEAVEN
FOUR NIGHTS WITH THE DUKE
THREE WEEKS WITH LADY X
WITH THIS KISS: THE COMPLETE COLLECTION
ONCE UPON A TOWER
AS YOU WISH
SEDUCED BY A PIRATE
THE UGLY DUCHESS
A FOOL AGAIN
THE DUKE IS MINE
WINNING THE WALLFLOWER
WHEN BEAUTY TAMED THE BEAST
STORMING THE CASTLE: AN ORIGINAL SHORT STORY WITH BONUS CONTENT
A KISS AT MIDNIGHT
A DUKE OF HER OWN
THIS DUCHESS OF MINE
WHEN THE DUKE RETURNS
DUCHESS BY NIGHT
AN AFFAIR BEFORE CHRISTMAS
DESPERATE DUCHESSES
PLEASURE FOR PLEASURE
THE TAMING OF THE DUKE
KISS ME, ANNABEL
MUCH ADO ABOUT YOU
YOUR WICKED WAYS
A WILD PURSUIT
FOOL FOR LOVE
DUCHESS IN LOVE
Copyright
This book is a work of fiction. The characters, incidents, and dialogue are drawn from the author's imagination and are not to be construed as real. Any resemblance to actual events or persons, living or dead, is entirely coincidental.
A WILD PURSUIT. Copyright © 2004 by Eloisa James. All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the non-exclusive, non-transferable right to access and read the text of this e-book on-screen. No part of this text may be reproduced, transmitted, down-loaded, decompiled, reverse engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of HarperCollins e-books.
EPub Edition © SEPTEMBER 2008 ISBN: 9780061801952
06 07 08 09 10
## About the Publisher
Australia
HarperCollins Publishers (Australia) Pty. Ltd.
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Pymble, NSW 2073, Australia
http://www.harpercollinsebooks.com.au
Canada
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http://www.harpercollinsebooks.co.uk
United States
HarperCollins Publishers Inc.
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http://www.harpercollinsebooks.com
|
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"redpajama_set_name": "RedPajamaBook"
}
| 8,518
|
#include "SVM_StringUtils.h"
namespace Shogun
{
namespace Util
{
String string_format(const String fmt_str, ...) {
int final_n, n = ((int)fmt_str.size()) * 2; /* reserve 2 times as much as the length of the fmt_str */
String str;
std::unique_ptr<char[]> formatted;
va_list ap;
while (1) {
formatted.reset(new char[n]); /* wrap the plain char array into the unique_ptr */
strcpy(&formatted[0], fmt_str.c_str());
va_start(ap, fmt_str);
final_n = vsnprintf(&formatted[0], n, fmt_str.c_str(), ap);
va_end(ap);
if (final_n < 0 || final_n >= n)
n += abs(final_n - n + 1);
else
break;
}
return String(formatted.get());
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,478
|
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{
"redpajama_set_name": "RedPajamaC4"
}
| 576
|
\section{Introduction}
In recent years, fractional derivatives (FDs) in the sense of Caputo type
have used to describe anomalous behaviors of diffusive phenomena in
mathematical physics involving different kernels, such as the power-law
\cite{1}, exponential \cite{2}, Mittag-Leffler \cite{3}, stretched exponential \cite{4}, and
stretched Mittag-Leffler \cite{5} functions. For example, the fractional
diffusion-wave, in the power-law function kernel was considered in \cite{6}. The
numerical solution for the space-fractional diffusion equation was presented
in \cite{7}. The Cauchy problem for the time-fractional diffusion equations was
investigated in \cite{8}. With the help of the FD involving the
exponential-function kernel, the heat-diffusion problem with respect to a
non-singular fading memory was proposed in \cite{9}. The heat transfer problem
within the non-singular second grade fluid was discussed in \cite{10}. The
non-singular unsteady flow of the ordinary couple stress fluid was studied
in \cite{11}. With the use of the FD involving the stretched
Mittag-Leffler-function kernel, the Irving--Mullineux oscillator \cite{12} and the
Allen-Cahn equation \cite{13} were also analyzed. For more details see \cite{13a,13b}.
FDs in the sense of Riemann--Liouville type were developed in \cite{14,15,16}. We can
mention the studies about not only the Fokker-Planck \cite{17} and the diffusion \cite{18,19,20,21}
equations, but also the wave propagation \cite{22}. Furthermore, the Chen's system of the
Riemann--Liouville type \cite{23}, the~monotone iterative method
for neutral fractional differential equations \cite{24} and the
time-fractional-order Harry-Dym equation \cite{25} were also discussed.
Readers can find the more details about the distinct versions of FDs in \cite{26}.
The normalized $\sinc$ function, structured by Whittaker in \cite{27}, and its
properties were considered in \cite{28}. Furthermore, the Fourier \cite{29}, Laplace
\cite{29} and Sumudu \cite{30} transforms of the NSF were formulated. However, the FD involving
the normalized $\sinc$ function without singular kernel has not proposed.
Motivated by the idea, the present article derives a
new FD with respect to the normalized $\sinc$ function without singular kernel.
Furthermore, based on the new concept it is considered the applications in
one-dimensional anomalous heat-transfer problems.
The structure of the present paper is as follows. In Section 2, a new FD
with respect to the normalized $\sinc$ function without singular kernel is
presented. In Section 3, the anomalous heat-diffusion models and their
solutions are analyzed by means of the Laplace transform. Finally, the conclusion is
outlined in Section 4.
\section{Preliminaries, definitions and integral transforms}
In this section, we derive the FD involving the
normalized $\sinc$ function without singular kernel.
\subsection{A new FD involving the normalized $\sinc$
function without singular kernel}
\begin{definition}
The normalized $\sinc$ function is defined by \cite{27,28}:
\begin{equation}
\label{eq1}
\sinc\left( x \right)=\frac{\sin \left( {\pi x} \right)}{\pi x},
\end{equation}
where $x\in {\rm R}$.
If $\varphi \left( x \right)$ is any~smooth function with~compact support,
where $x\in {\rm R}$, then \cite{28}
\begin{equation}
\label{eq2}
\mathop {\lim }\limits_{\varpi \to 0} \frac{1}{\varpi }\sinc\left(
{\frac{x}{\varpi }} \right)=\mathop {\lim }\limits_{\varpi \to 0} \frac{\sin
\left( {\frac{\pi x}{\varpi }} \right)}{\pi x}=\delta \left( x \right),
\end{equation}
\begin{equation}
\label{eq3}
\sin c\left( 0 \right)=1,
\end{equation}
where
\begin{equation}
\label{eq4}
\mathop {\lim }\limits_{\varpi \to 0} \int\limits_{-\infty }^\infty
{\frac{\varphi \left( x \right)\sinc\left( {\frac{x}{\varpi }}
\right)}{\varpi }dx} =\varphi \left( 0 \right).
\end{equation}
\end{definition}
\begin{definition}
Let $\Pi \left( x \right)\in H^1\left( {a,b} \right)$ and $\mbox{ }b>a$. A
new FD involving the normalized $\sinc$ kernel of the function $\Pi \left( \mu \right)$ of order $\varpi
\mbox{ }\left( {\varpi \in \left( {0,1} \right)} \right)$ is defined as:
\begin{equation}
\label{eq5}
{ }_aD_\mu ^{\left( \varpi \right)} \Pi \left( \mu \right)=\frac{\varpi \wp
\left( \varpi \right)}{1-\varpi }\int\limits_a^\mu {\sinc\left(
{-\frac{\varpi \left( {\mu -x} \right)}{1-\varpi }} \right)\Pi ^{\left( 1
\right)}\left( x \right)dx} ,
\end{equation}
where $a\in \left( {-\infty ,\mu } \right)$, and $\wp \left( \varpi \right)$ is
a normalization constant depending on $\varpi $ such that $\wp \left( 0
\right)=\wp \left( 1 \right)=1$.
\end{definition}
Following Eq.(\ref{eq1}), we obtain
\begin{equation}
\label{eq6}
\mathop {\lim }\limits_{\varpi \to 0} \frac{1}{\varpi }\sinc\left(
{\frac{\mu -x}{\varpi }} \right)=\mathop {\lim }\limits_{\varpi \to 0}
\frac{\sin \left( {\frac{\pi \left( {\mu -x} \right)}{\varpi }} \right)}{\pi
\left( {\mu -x} \right)}=\delta \left( {\mu -x} \right),
\end{equation}
where $\varphi \left( x \right)$ is any~smooth function with~compact support
where $x\in {\rm R}_{ }$ such that
\begin{equation}
\label{eq7}
\mathop {\lim }\limits_{\varpi \to 0} \int\limits_{-\infty }^\infty {\varphi
\left( x \right)\frac{1}{\varpi }\sinc\left( {\frac{x-\mu }{\varpi }}
\right)dx} =\varphi \left( \mu \right).
\end{equation}
Thus, we have
\begin{equation}
\label{eq8}
\begin{array}{l}
\mathop {\lim }\limits_{\varpi \to 0} { }_aD_\mu ^{\left( \varpi \right)}
\Pi \left( \mu \right)=\mathop {\lim }\limits_{\varpi \to 0} \frac{\varpi
\wp \left( \varpi \right)}{\left( {1-\varpi } \right)}\int\limits_a^\mu
{\sinc\left( {-\frac{\varpi \left( {\mu -x} \right)}{1-\varpi }} \right)\Pi
^{\left( 1 \right)}\left( x \right)dx} \\
=\left( {\mathop {\lim }\limits_{\varpi \to 0} \wp \left( \varpi \right)}
\right)\int\limits_a^\mu {\delta \left( {\mu -x} \right)\Pi ^{\left( 1
\right)}\left( x \right)dx} \\
=\Pi ^{\left( 1 \right)}\left( x \right). \\
\end{array}
\end{equation}
When
\begin{equation}
\label{eq9}
\mathop {\lim }\limits_{\varpi \to 1} \frac{\varpi \wp \left( \varpi
\right)}{1-\varpi }\left[ {\sinc\left( {-\frac{\varpi \left( {\mu -x}
\right)}{1-\varpi }} \right)} \right]=\mathop {\lim }\limits_{\varpi \to
\mbox{1}} \wp \left( \varpi \right)\frac{\varpi }{1-\varpi }\frac{\sin
\left( {\frac{\pi \left( {\mu -x} \right)}{\frac{1-\varpi }{\varpi }}}
\right)}{\pi \left( {\mu -x} \right)}=\mathop {\lim }\limits_{\varpi \to
\mbox{1}} \mbox{1,}
\end{equation}
we have
\begin{equation}
\label{eq10}
\begin{array}{l}
\mathop {\lim }\limits_{\varpi \to 1} { }_aD_\mu ^{\left( \varpi \right)}
\Pi \left( \mu \right)=\mathop {\lim }\limits_{\varpi \to 1} \frac{\varpi
\wp \left( \varpi \right)}{\left( {1-\varpi } \right)}\int\limits_a^\mu
{\sinc\left( {-\frac{\varpi \left( {\mu -x} \right)}{1-\varpi }} \right)\Pi
^{\left( 1 \right)}\left( x \right)dx} \\
=\mathop {\lim }\limits_{\varpi \to 1} \int\limits_a^\mu {\Pi ^{\left( 1
\right)}\left( x \right)dx} \\
=\Pi \left( \mu \right)-\Pi \left( a \right). \\
\end{array}
\end{equation}
For $n\ge 1$ and $\varpi \in \left( {0,1} \right)$, the FD $D_\mu ^{\left( {n+\varpi } \right)} \Pi \left( \mu \right)$ of
order $\left( {n+\omega } \right)$ is defined as:
\begin{equation}
\label{eq11}
{ }_aD_\mu ^{\left( {n+\varpi } \right)} \Pi \left( \mu \right):={ }_aD_\mu
^{\left( n \right)} \left( {{ }_aD_\mu ^{\left( \varpi \right)} \Pi \left(
\mu \right)} \right).
\end{equation}
\begin{property}
(T1) ${ }_0D_\mu ^{\left( \varpi \right)} \theta =0$, where $\theta $ is a
constant;
(T2) ${ }_0D_\mu ^{\left( \varpi \right)} \mu =\frac{\varpi \wp \left(
\varpi \right)}{1-\varpi }\int\limits_0^\mu {\sinc\left( {\frac{\varpi
x}{1-\varpi }} \right)dx} $.
\end{property}
\begin{proof}
We have from Eq.(\ref{eq5}) that
\begin{equation}
\label{eq12}
{ }_0D_\mu ^{\left( \varpi \right)} \theta =\frac{\varpi \wp \left( \varpi
\right)}{1-\varpi }\int\limits_0^\mu {\sinc\left( {-\frac{\varpi \left(
{\mu -x} \right)}{1-\varpi }} \right)\theta ^{\left( 1 \right)}dx} =0.
\end{equation}
We have, by using the definition Eq.(\ref{eq5}),
\begin{equation}
\label{eq13}
{ }_0D_\mu ^{\left( \varpi \right)} \mu =\frac{\varpi \wp \left( \varpi
\right)}{1-\varpi }\int\limits_0^\mu {\sinc\left( {-\frac{\varpi \left(
{\mu -x} \right)}{1-\varpi }} \right)dx} =\frac{\varpi \wp \left( \varpi
\right)}{1-\varpi }\int\limits_0^\mu {\sinc\left( {\frac{\varpi x}{1-\varpi
}} \right)dx} .
\end{equation}
\end{proof}
\subsection{Integral transforms of the new FD
involving the normalized $\sinc$ function without singular kernel}
Here, we have \cite{29}
\begin{equation}
\label{eq14}
\aleph \left\{ {\sinc\left( x \right)} \right\}=\aleph \left\{ {\frac{\sin
\left( {\pi x} \right)}{\pi x}} \right\}=\sqrt {\frac{1}{2\pi }} H\left(
{\pi -\left| \xi \right|} \right)
\end{equation}
such that
\begin{equation}
\label{eq15}
\begin{array}{l}
\aleph \left\{ {\sinc\left( {-\frac{\varpi x}{1-\varpi }} \right)}
\right\}=\aleph \left\{ {\frac{\sin \left( {-\frac{\varpi \pi x}{1-\varpi }}
\right)}{-\frac{\varpi \pi x}{1-\varpi }}} \right\} \\
=-\sqrt {\frac{1}{2\pi }} \frac{1-\varpi }{\varpi }H\left( {-\frac{\varpi
\pi }{1-\varpi }-\left| \xi \right|} \right) \\
=\sqrt {\frac{1}{2\pi }} \frac{1-\varpi }{\varpi }H\left( {\frac{\varpi \pi
}{1-\varpi }+\left| \xi \right|} \right), \\
\end{array}
\end{equation}
where $\aleph $ is the Fourier transform operator \cite{29}, and $H\left( x
\right)$ is the Heaviside function \cite{29}.
The Fourier transform of Eq.(\ref{eq5}) can be written as
\begin{equation}
\label{eq16}
\begin{array}{l}
\aleph \left\{ {{ }_0D_\mu ^{\left( \varpi \right)} \Pi \left( \mu \right)}
\right\} \\
=\aleph \left\{ {\frac{\varpi \wp \left( \varpi \right)}{1-\varpi
}\int\limits_0^\mu {\sinc\left( {-\frac{\varpi \left( {\mu -x}
\right)}{1-\varpi }} \right)\Pi ^{\left( 1 \right)}\left( x \right)dx} }
\right\} \\
=\frac{\varpi \wp \left( \varpi \right)}{1-\varpi }\aleph \left\{ {\sinc\left( {-\frac{\varpi x}{1-\varpi }} \right)} \right\}\aleph \left\{ {\Pi
^{\left( 1 \right)}\left( x \right)} \right\} \\
=\frac{\varpi \wp \left( \varpi \right)}{1-\varpi }\left[ {\sqrt
{\frac{1}{2\pi }} \frac{1-\varpi }{\varpi }H\left( {\frac{\varpi \pi
}{1-\varpi }+\left| \xi \right|} \right)} \right]\left[ {i\xi \Pi \left( \xi
\right)} \right] \\
=i\xi \sqrt {\frac{1}{2\pi }} \wp \left( \varpi \right)H\left(
{\frac{\varpi \pi }{1-\varpi }+\left| \xi \right|} \right)\Pi \left( \xi
\right), \\
\end{array}
\end{equation}
where $\aleph \left\{ {\Pi \left( \mu \right)} \right\}=\Pi \left( \xi
\right)$.
Similarly, we have \cite{29}
\begin{equation}
\label{eq17}
\Im \left\{ {\sinc\left( x \right)} \right\}=\Im \left\{ {\frac{\sin \left(
{\pi x} \right)}{\pi x}} \right\}=\frac{1}{\pi }\mbox{tan}^{-1}\left(
{\frac{\pi }{s}} \right)
\end{equation}
such that
\begin{equation}
\label{eq18}
\Im \left\{ {\sinc\left( {-\frac{\varpi }{1-\varpi }x} \right)}
\right\}=\Im \left\{ {\frac{\sin \left( {-\frac{\varpi }{1-\varpi }\pi x}
\right)}{-\frac{\varpi \pi x}{1-\varpi }}} \right\}=\frac{1}{\frac{\varpi
\pi }{1-\varpi }}\mbox{tan}^{-1}\left( {\frac{\frac{\varpi \pi }{1-\varpi
}}{s}} \right),
\end{equation}
where $\Im $ is the Laplace transform operator \cite{29}.
From Eq.(\ref{eq18}) the Laplace transform of Eq.(\ref{eq5}) can be given by:
\begin{equation}
\label{eq19}
\begin{array}{l}
\Im \left\{ {{ }_0D_\mu ^{\left( \varpi \right)} \Pi \left( \mu \right)}
\right\} \\
=\Im \left\{ {\frac{\varpi \wp \left( \varpi \right)}{1-\varpi
}\int\limits_0^\mu {\sinc\left( {-\frac{\varpi \left( {\mu -x}
\right)}{1-\varpi }} \right)\Pi ^{\left( 1 \right)}\left( x \right)dx} }
\right\} \\
=\frac{\varpi \wp \left( \varpi \right)}{1-\varpi }\Im \left\{ {\sinc\left( {-\frac{\varpi x}{1-\varpi }} \right)} \right\}\Im \left\{ {\Pi
^{\left( 1 \right)}\left( x \right)} \right\} \\
=\frac{\wp \left( \varpi \right)}{\pi }\mbox{tan}^{-1}\left(
{\frac{\frac{\varpi \pi }{1-\varpi }}{s}} \right)\left( {s\Pi \left( s
\right)-\Pi \left( 0 \right)} \right), \\
\end{array}
\end{equation}
where $\Im \left\{ {\Pi \left( \mu \right)} \right\}=\Pi \left( s \right)$.
As a direct result, we have \cite{30}
\begin{equation}
\label{eq20}
\Re \left\{ {\sinc\left( x \right)} \right\}=\Re \left\{ {\frac{\sin \left(
{\pi x} \right)}{\pi x}} \right\}=\frac{\mbox{tan}^{-1}\left( {\pi \zeta }
\right)}{\pi \zeta }
\end{equation}
such that
\begin{equation}
\label{eq21}
\Re \left\{ {\sinc\left( {-\frac{\varpi }{1-\varpi }x} \right)}
\right\}=\Im \left\{ {\frac{\sin \left( {-\frac{\varpi }{1-\varpi }\pi x}
\right)}{-\frac{\varpi \pi x}{1-\varpi }}}
\right\}=\frac{\mbox{tan}^{-1}\left( {\frac{\varpi \pi \zeta }{1-\varpi }}
\right)}{\frac{\varpi \pi \zeta }{1-\varpi }}_{,}
\end{equation}
where $\Re $ is the Sumudu transform operator \cite{30}.
Thus, we have from Eq.(\ref{eq13}) that
\begin{equation}
\label{eq22}
\begin{array}{l}
\Re \left\{ {{ }_0D_\mu ^{\left( \varpi \right)} \Pi \left( \mu \right)}
\right\} \\
=\Re \left\{ {\frac{\varpi \wp \left( \varpi \right)}{1-\varpi
}\int\limits_0^\mu {\sinc\left( {-\frac{\varpi \left( {\mu -x}
\right)}{1-\varpi }} \right)\Pi ^{\left( 1 \right)}\left( x \right)dx} }
\right\} \\
=\frac{\varpi \wp \left( \varpi \right)}{1-\varpi }\Re \left\{ {\sinc\left( {-\frac{\varpi x}{1-\varpi }} \right)} \right\}\Re \left\{ {\Pi
^{\left( 1 \right)}\left( x \right)} \right\} \\
=\frac{\wp \left( \varpi \right)}{\pi \zeta }\mbox{tan}^{-1}\left(
{\frac{\varpi \pi \zeta }{1-\varpi }} \right)\left( {\frac{\Pi \left( \zeta
\right)-\Pi \left( 0 \right)}{\zeta }} \right), \\
\end{array}
\end{equation}
where $\Re \left\{ {\Pi \left( \mu \right)} \right\}=\Pi \left( \zeta
\right)$.
\section{Modelling the anomalous heat-diffusion problems }
In this section, we model the anomalous heat-diffusion problems involving
fractional-time and -space derivatives of the normalized $\sinc$ function
without singular kernel.
\textbf{Example 1}
The anomalous heat-diffusion within the fractional-time derivative of the
normalized $\sinc$ function without singular kernel is written as:
\begin{equation}
\label{eq23}
{ }_0D_\tau ^{\left( \varpi \right)} \Pi \left( {\mu ,\tau } \right)=\kappa
\frac{\partial ^2\Pi \left( {\mu ,\tau } \right)}{\partial \mu ^2},\mbox{
}\mu >0,\mbox{ }\tau >0,
\end{equation}
subjected to the initial and boundary conditions:
\begin{equation}
\label{eq24}
\Pi \left( {\mu ,0} \right)=0,\mbox{ }\mu >0,
\end{equation}
\begin{equation}
\label{eq25}
\Pi \left( {0,\tau } \right)=\lambda \left( \tau \right)\mbox{, }\tau >0,
\end{equation}
\begin{equation}
\label{eq26}
\Pi \left( {\mu ,\tau } \right)\to 0,\mbox{ }as\mbox{ }\mu \to \infty
,\mbox{ }\tau >0,
\end{equation}
where $\kappa $ is the thermal diffusivity.
With the aid of Eq.(\ref{eq19}), Eq.(\ref{eq23}) can be transferred into
\begin{equation}
\label{eq27}
\frac{\wp \left( \varpi \right)}{\pi }\mbox{tan}^{-1}\left(
{\frac{\frac{\varpi \pi }{1-\varpi }}{s}} \right)\left( {s\Pi \left( {\mu
,s} \right)-\Pi \left( {\mu ,0} \right)} \right)=\kappa \frac{d^2\Pi \left(
{\mu ,s} \right)}{d\mu ^2}.
\end{equation}
From Eq.(\ref{eq24}) we have the following:
\begin{equation}
\label{eq28}
\frac{d^2\Pi \left( {\mu ,s} \right)}{d\mu ^2}=\frac{\wp \left( \varpi
\right)s}{\pi \kappa }\mbox{tan}^{-1}\left( {\frac{\frac{\varpi \pi
}{1-\varpi }}{s}} \right)\Pi \left( {\mu ,s} \right),
\end{equation}
which leads to
\begin{equation}
\label{eq29}
\Pi \left( {\mu ,s} \right)=\Omega _1 \exp \left( {-\mu \sqrt {\rm H} }
\right)+\Omega _2 \exp \left( {\mu \sqrt {\rm H} } \right),
\end{equation}
where $\Omega _1 $ and $\Omega _2 $ are two unknown constants and
\begin{equation}
\label{eq30}
{\rm H}=\frac{\wp \left( \varpi \right)s}{\pi \kappa }\mbox{tan}^{-1}\left(
{\frac{\frac{\varpi \pi }{1-\varpi }}{s}} \right).
\end{equation}
In view of Eq.(\ref{eq25}) and Eq.(\ref{eq26}), we have
\begin{equation}
\label{eq31}
\Omega _2 =0
\end{equation}
such that
\begin{equation}
\label{eq32}
\Pi \left( {\mu ,s} \right)=\lambda \left( s \right)\exp \left( {-\mu \sqrt
{\rm H} } \right),
\end{equation}
where $\Im \left\{ {\lambda \left( \mu \right)} \right\}=\lambda \left( s
\right)$.
Thus, the Laplace transform solution of Eq.(\ref{eq23}) is:
\begin{equation}
\label{eq33}
\Pi \left( {\mu ,s} \right)=\lambda \left( s \right)\exp \left( {-\sqrt
{\frac{\wp \left( \varpi \right)s}{\pi \kappa }\mbox{tan}^{-1}\left(
{\frac{\frac{\varpi \pi }{1-\varpi }}{s}} \right)} \mu } \right).
\end{equation}
\textbf{Example 2}
The anomalous heat-diffusion within the fractional-space derivative of the
normalized $\sinc$ function without singular kernel is
\begin{equation}
\label{eq34}
\frac{\partial \Pi \left( {\mu ,\tau } \right)}{\partial \tau }=\kappa {
}_0D_\mu ^{\left( 1 \right)} \left( {{ }_0D_\mu ^{\left( \varpi \right)} \Pi
\left( {\mu ,\tau } \right)} \right),\mbox{ }\mu >0,\mbox{ }\tau >0,
\end{equation}
with the initial and boundary conditions:
\begin{equation}
\label{eq35}
\Pi \left( {\mu ,0} \right)=0,\mbox{ }\mu >0,
\end{equation}
\begin{equation}
\label{eq36}
\Pi \left( {0,\tau } \right)=\lambda \left( \tau \right)\mbox{, }\tau >0,
\end{equation}
\begin{equation}
\label{eq37}
\Pi \left( {\mu ,0} \right)\to 0,\mbox{ }as\mbox{ }\mu \to \infty ,\mbox{
}\tau >0,
\end{equation}
where $\kappa $ is the thermal diffusivity, and
\begin{equation}
\label{eq38}
{ }_0D_\mu ^{\left( 1 \right)} \left( {{ }_0D_\mu ^{\left( \varpi \right)}
\Pi \left( {\mu ,\tau } \right)} \right)=\frac{\varpi \wp \left( \varpi
\right)}{1-\varpi }\frac{\partial }{\partial \mu }\int\limits_0^\mu {\sinc
\left( {-\frac{\varpi \left( {\mu -x} \right)}{1-\varpi }} \right)\Pi
^{\left( 1 \right)}\left( {x,\tau } \right)dx} .
\end{equation}
With the help of Eq.(\ref{eq19}) and Eq.(\ref{eq35}), Eq.(\ref{eq34}) can be written as:
\begin{equation}
\label{eq39}
{ }_0D_\mu ^{\left( 1 \right)} \left( {{ }_0D_\mu ^{\left( \varpi \right)}
\Pi \left( {\mu ,s} \right)} \right)=\frac{s}{\kappa }\Pi \left( {\mu ,s}
\right),
\end{equation}
where
\begin{equation}
\label{eq40}
{ }_0D_\mu ^{\left( 1 \right)} \left( {{ }_0D_\mu ^{\left( \varpi \right)}
\Pi \left( {\mu ,s} \right)} \right)=\frac{\varpi \wp \left( \varpi
\right)}{1-\varpi }\frac{\partial }{\partial \mu }\int\limits_0^\mu {\sinc
\left( {-\frac{\varpi \left( {\mu -x} \right)}{1-\varpi }} \right)\Pi
^{\left( 1 \right)}\left( {x,s} \right)dx} .
\end{equation}
By the integration of Eq.(\ref{eq39}) we have
\begin{equation}
\label{eq41}
{ }_0D_\mu ^{\left( \varpi \right)} \Pi \left( {\mu ,s} \right)=\frac{\varpi
\wp \left( \varpi \right)}{1-\varpi }\int\limits_0^\mu {\sinc\left(
{-\frac{\varpi \left( {\mu -x} \right)}{1-\varpi }} \right)\Pi ^{\left( 1
\right)}\left( {x,s} \right)dx} =\int\limits_0^\mu {\frac{s}{\kappa }\Pi
\left( {x,s} \right)dx} +\Theta ,
\end{equation}
where $\Theta $ is a constant.
By taking the Sumudu transform operator with $\mu $ and $\Theta =0$, we have
\begin{equation}
\label{eq42}
\frac{\varpi \wp \left( \varpi \right)}{1-\varpi }\int\limits_0^\mu {\sinc\left( {-\frac{\varpi \left( {\mu -x} \right)}{1-\varpi }} \right)\Pi
^{\left( 1 \right)}\left( {x,s} \right)dx} =\int\limits_0^\mu
{\frac{s}{\kappa }\Pi \left( {x,s} \right)dx} ,
\end{equation}
which implies that
\begin{equation}
\label{eq43}
\frac{\wp \left( \varpi \right)}{\pi \zeta ^2}\mbox{tan}^{-1}\left(
{\frac{\varpi \pi \zeta }{1-\varpi }} \right)\left( {\Pi \left( {\zeta ,s}
\right)-\Pi \left( {0,s} \right)} \right)=\frac{s\zeta }{\kappa }\Pi \left(
{\zeta ,s} \right).
\end{equation}
From Eq.(\ref{eq35}) and Eq.(\ref{eq43}), we have the following:
\begin{equation}
\label{eq44}
\frac{\wp \left( \varpi \right)}{\pi \zeta ^2}\mbox{tan}^{-1}\left(
{\frac{\varpi \pi \zeta }{1-\varpi }} \right)\left( {\Pi \left( {\zeta ,s}
\right)-\lambda \left( s \right)} \right)=\frac{s\zeta }{\kappa }\Pi \left(
{\zeta ,s} \right).
\end{equation}
Thus, we have
\begin{equation}
\label{eq45}
\Pi \left( {\zeta ,s} \right)=\frac{\frac{\wp \left( \varpi \right)\lambda
\left( s \right)}{\pi \zeta ^2}\mbox{tan}^{-1}\left( {\frac{\varpi \pi \zeta
}{1-\varpi }} \right)}{\frac{\wp \left( \varpi \right)}{\pi \zeta
^2}\mbox{tan}^{-1}\left( {\frac{\varpi \pi \zeta }{1-\varpi }}
\right)-\frac{s\zeta }{\kappa }}
\end{equation}
where $\Re \left\{ {\Pi \left( {\mu ,s} \right)} \right\}=\Pi \left( {\zeta
,s} \right)$ represents the Sumudu transform operator \cite{30}.
From Eq.(\ref{eq45}), the Laplace transform solution of Eq.(\ref{eq23}) is:
\begin{equation}
\label{eq46}
\Pi \left( {\mu ,s} \right)=\Re ^{-1}\left\{ {\frac{\frac{\wp \left( \varpi
\right)\lambda \left( s \right)}{\pi \zeta ^2}\mbox{tan}^{-1}\left(
{\frac{\varpi \pi \zeta }{1-\varpi }} \right)}{\frac{\wp \left( \varpi
\right)}{\pi \zeta ^2}\mbox{tan}^{-1}\left( {\frac{\varpi \pi \zeta
}{1-\varpi }} \right)-\frac{s\zeta }{\kappa }}} \right\},
\end{equation}
where $\Re ^{-1}\left\{ {\Pi \left( {\zeta ,s} \right)} \right\}=\Pi \left(
{\mu ,s} \right)$ represents the inverse Sumudu transform operator [30].
When $\varpi =0$, Eq.(\ref{eq33}) and Eq.(\ref{eq46}) become the Laplace transform solution
of the classical heat-diffusion equation \cite{31}:
\begin{equation}
\label{eq47}
\Pi \left( {\mu ,s} \right)=\lambda \left( s \right)\exp \left( {-\sqrt
{\frac{s}{\kappa }} \mu } \right).
\end{equation}
which is in agreement with the result in \cite{29}.
\section{Conclusions}
In the present study, we addressed a new FD in respect to
the normalized $\sinc$ function without singular kernel.
Moreover, the Fourier, Laplace and Sumudu transforms of the FD operator
and the Laplace--transform solutions of the anomalous heat-diffusion equations were
considered. The analytical solutions of the
classical and anomalous heat-diffusion equations in the form of the Laplace
transform were also compared. The new formulation may be used to support a new
perspective for describing the anomalous behaviors in mathematical
physics.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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{"url":"https:\/\/www.nature.com\/articles\/s42005-019-0271-0?error=cookies_not_supported&code=dab672ce-5da0-473f-aa59-0929d4ad98e4","text":"# Time-delay polaritonics\n\n## Abstract\n\nNon-linearity and finite signal propagation speeds are omnipresent in nature, technologies, and real-world problems, where efficient ways of describing and predicting the effects of these elements are in high demand. Advances in engineering condensed matter systems, such as lattices of trapped condensates, have enabled studies on non-linear effects in many-body systems where exchange of particles between lattice nodes is effectively instantaneous. Here, we demonstrate a regime of macroscopic matter-wave systems, in which ballistically expanding condensates of microcavity exciton-polaritons act as picosecond, microscale non-linear oscillators subject to time-delayed interaction. The ease of optical control and readout of polariton condensates enables us to explore the phase space of two interacting condensates up to macroscopic distances highlighting its potential in extended configurations. We demonstrate deterministic tuning of the coupled-condensate system between fixed point and limit cycle regimes, which is fully reproduced by time-delayed coupled equations of motion similar to the Lang-Kobayashi equation.\n\n## Introduction\n\nTime delay is widespread in nature and occurs when the constituents of a given system interact via signals with a finite propagation time1. If the characteristic timescale of the system, such as the period of a simple pendulum, is much longer than the signal propagation time, then one arrives at familiar examples such as Huygens clock synchronisation described by instantaneous interactions2. When propagation times are appreciably long, the role of the system\u2019s history is enhanced and the interactions are said to be time delayed. Such systems dictate the neurological function of our brains, affect traffic flow, influence economic activities, define population dynamics of biological species, regulate physiological systems, determine the stability of lasers, and have application in control engineering3,4. Besides their ubiquity in nature and science, coupled systems with continuous time-delayed interactions exhibit interesting mathematical properties such as an infinite dimensional state space, i.e. for a fixed time-delay $$\\tau$$ there are infinitely many initial conditions of the system in the time interval $$-\\tau \\le t\\le 0$$ needed to predict the dynamics for $$t \\, > \\, 0$$ (ref. 5). Time-delayed interaction or self-feedback is known to greatly increase the dynamical complexity of a system, giving rise to chaotic motion, chimera states6, as well as being able to both stabilise and destabilise fixed point and periodic orbit solutions4.\n\nTechnological applications of delay-coupled systems appear in diverse areas such as control engineering7, high speed random-bit generation8, secure chaos communication9, but have also recently emerged in machine learning, where the demand for neuro-inspired computing units has led to various hardware realisations of artificial neural networks10. The intrinsic high-dimensional state space of delay-coupled systems enables an efficient platform for tasks such as pattern recognition, speech recognition, and time series prediction as demonstrated in electronic11, photonic12,13,14, and optoelectronic15,16,17 systems.\n\nIn this work, we demonstrate the prospect of an ultrafast microscale platform for engineering time-delayed coupled oscillator networks based on condensates of microcavity exciton\u2013polaritons (from here on polaritons)18,19. Polaritons are bosonic quasi-particles that can undergo a power-driven quantum phase transition to a macroscopically occupied state with long-range phase coherence20. This phase transition is associated with the formation of a polariton condensate, a non-linear matter-wave quantum fluid, which differs from classical cold atom Bose\u2013Einstein condensates, and liquid light droplets21,22 due to its non-equilibrium nature. One of the greatest advantages of polariton condensates for optoelectronic applications is the easy implementation of arbitrary geometries, or graphs, using adaptive optical elements for the excitation laser beam23. We show controllable tuning between steady state (single-colour) and limit-cycle (two-colour) regimes of the two coupled condensate system and explain our observations through time-delayed equations of motion.\n\n## Results\n\n### Time-delayed coupled matter-wave condensates\n\nConventionally, networks and lattices of condensates consisting of cold atoms24, Cooper pairs25, photons26, or polaritons27,28 are studied in trap geometries, weakly coupled via tunnelling currents, that allow for the study of solid state physics phenomena such as superfluid-Mott phase transitions29, magnetic frustration30, $$PT$$-symmetric non-linear optics31, and Josephson physics28,32. Here we investigate the inverse case, wherein polariton condensates are freely expanding from small (point-like) sources experiencing dynamics reminiscent to macroscopic systems such as time-delayed coupled semiconductor lasers33. Ballistic expansion extends over two orders of magnitude beyond the pump beam waist, and occurs due to the repulsive potential formed by the uncondensed exciton reservoir injected by the non-resonant pump34. In the case of spatially separated polariton condensates, propagation from one condensate centre to another results in a substantial phase accumulation, interpreted as a retardation of information flow between the condensates. To date, coupled polariton condensates were restricted to distances $$d \\, \\lesssim \\, 40\\ \\upmu {\\rm{m}}$$23,35,36, wherein coupling was assumed to be instantaneous although propagation time was comparable or longer than the characteristic timescale of the systems. To unravel the role of time-delayed interactions, we demonstrate the synchronisation between two tightly pumped condensates separated by up to $$d\\approx 114\\, \\upmu {\\rm{m}}$$, as shown in Fig.\u00a01a. Figure\u00a01b, c compares schematically the conventional regime of coupled condensates separated by a potential barrier and described by a tunnelling current $$J$$, to the macroscopically coupled driven-dissipative matter-wave condensates interacting via radiative particle transfer subject to finite propagation time $$\\tau$$.\n\n### Dynamics of two interacting condensates\n\nWe utilise a strain compensated semiconductor microcavity to enable uninhibited ballistic expansion of polaritons over macroscopic distances37. We inject a polariton dyad with identical Gaussian spatial profiles, with full-width-at-half-maximum (FWHM) of $$\\approx \\!2\\, \\upmu {\\rm{m}}$$, using non-resonant optical excitation at the first Bragg minima of the reflectivity stop-band, and employ spatial light modulation to continuously vary the separation distance from $$6$$ to $$94\\, \\upmu {\\rm{m}}$$, while keeping constant the excitation density of each pump beam at $${P}_{1,2}\\approx 1.5\\times {P}_{{\\rm{thr}}}^{(1)}$$, where $${P}_{{\\rm{thr}}}^{(1)}$$ is the condensation threshold power of an isolated condensate. For each separation distance $$d$$ (more than 400 positions), we record simultaneously the spatial profile of the photoluminescence (real-space), the dispersion (energy vs in-plane momentum in the direction of propagation), and the photoluminescence in reciprocal (Fourier) space (see Supplementary Movie\u00a01). We observe that opposite to the twofold hybridisation of two evanescently coupled condensates28, the macroscopically coupled system is characterised by a multitude of accessible modes of even and odd parity (i.e., 0 and $$\\uppi$$ phase difference between condensate centres) that alternate continuously between opposite parity states with increasing dyad separation distance. For a range of separation distances only one resonant mode is present in the gain region of the dyad, wherein the polariton dyad is stationary, occupying a single energy level. Between the separation distances, wherein only one mode is present, we observe the coexistence of two resonant modes of opposite parity resulting in non-stationary periodic states.\n\nIn Fig.\u00a02 we present an example of the two different regimes, stationary and non-stationary states. Figure\u00a02a, b shows real-space photoluminescence, Fig.\u00a02c, d shows Fourier-space photoluminescence and Fig.\u00a02e, f shows the polariton dispersion along the axis of the dyad with separation distances of (a, c, e) $$d=12.7\\, \\upmu {\\rm{m}}$$ and (b, d, f) $$d=37.3\\ \\upmu {\\rm{m}}$$. Figure\u00a02g depicts the integrated spectra of Fig.\u00a02e, f using black dots and red squares, respectively. Absolute values of the energy levels are given as a blueshift with respect to the ground-state of the lower polariton branch. Although from the clear fringe pattern in Fig.\u00a02a one could infer that only one mode of odd parity is present, it is through the simultaneous recording of either the Fourier-space or the dispersion that the coexistence of an even parity mode becomes apparent. Each of the two modes has a well defined but opposite parity to the other. We note that as long as we maintain the mirror symmetry of the system by pumping both of the two condensate centres with the same power $${P}_{1}={P}_{2}$$, we do not observe formation of non-trivial phase configurations $${\\phi }_{1}-{\\phi }_{2} \\, \\ne \\, 0,\\pi$$.\n\nTo unravel the dynamics of the coupling on the separation distance between the two condensates, we obtain the spectral position, parity, and spectral weight of both energy levels for pump spot separation distances from $$d=5\\, \\upmu {\\rm{m}}$$ to $$d=66\\ \\upmu {\\rm{m}}$$. We have recorded configurations with more than two occupied energy levels for several distances $$d$$, but with the relative spectral weight of the third peak less than a few percent. In the following, we focus our analysis to the two brightest energy levels for each configuration. The measured normalised spectral weights and spectral positions of the two states versus pump spot separation distance $$d$$ are illustrated in Fig.\u00a03a, b using black dots for even parity states and red hollow squares for odd parity states, respectively. Figure\u00a03a shows that the system follows an oscillatory behaviour in the spectral weights of the two parity states giving rise to continuous transitions between even and odd parity states interleaved by configurations exhibiting both parity states. Interestingly, each period of these oscillations in relative intensity (starting and ending with a vanishing spectral weight) displays an \u2018energy branch\u2019 featuring a notable reduction in energy with increasing pump spot separation distance $$d$$ as is shown in Fig.\u00a03b. The energy of an isolated condensate is measured in the same sample space and its value $$\\approx {\\!}2.22\\ {\\rm{meV}}$$, above the ground state energy of the lower polariton dispersion, is illustrated with a blue dashed horizontal line. We observe that the phase-coupling of two spatially separated condensate centres is dominated by a spectral redshift with respect to the energy of an isolated condensate. The spectral size of each \u2018energy branch\u2019, i.e., the measurable redshift, is decaying from branch-to-branch with increasing pump spot separation distance $$d$$. The two dyad configurations exhibiting single-colour and two-colour states shown in Fig.\u00a02 are indicated with grey vertical dashed lines in Fig.\u00a03. Interestingly, it has been shown that the observation of phase-flip transitions, accompanied with changes in oscillation frequency to another mode is a universal characteristic of time-delayed coupled non-linear systems5,38. Such dynamics can be associated with neuronal systems39,40,41, and coupled semiconductor lasers33,42, but have also been demonstrated experimentally for other types of time-delayed coupled systems such as non-linear electronic circuits38,43, living organisms44, chemical oscillators45, and candle-flame oscillators46.\n\nTime averaged measurements over $$\\sim \\!1000$$ realisations of the system as presented in Fig.\u00a02a, c, e cannot reveal whether the system of two coupled polariton condensates exhibits periodic dynamics in the form of two-colour states or whether it stochastically picks one of the two parity states. However, it is well known that the power spectrum of a time-dependent signal is related to its auto-correlation function by means of Fourier-transform (Wiener\u2013Khinchin theorem). A two-colour solution, as shown in Fig.\u00a02g, results in the periodic disappearance and revival of the first-order temporal coherence function $${g}^{(1)}(\\bar{\\tau })$$. To characterise the temporal evolution of $${g}^{(1)}(\\bar{\\tau })$$, we perform interferometric measurements of the two coupled condensates and extract the fringe visibility $${\\mathcal{V}}$$ which\u2014up to a normalisation factor\u2014is proportional to the magnitude of the coherence function $$\\left|{g}^{(1)}\\right|$$. Assuming two coexisting states of opposite parity but equal spectral weight, one may simplify the complex amplitudes of the two coupled condensates $${\\Psi }_{1}$$ and $${\\Psi }_{2}$$ as\n\n$${\\Psi }_{1}(t)={\\psi }_{0}\\left({\\mathrm{e}}^{-i{\\mu }_{{\\rm{e}}}t\/\\hslash }+{\\mathrm{e}}^{-i{\\mu }_{{\\rm{o}}}t\/\\hslash }{\\mathrm{e}}^{i\\phi }\\right),$$\n(1)\n$${\\Psi}_{2}(t)={\\psi }_{0}\\left({\\mathrm{e}}^{-i{\\mu}_{{\\mathrm{e}}}t\/\\hslash }-{\\mathrm{e}}^{-i{\\mu }_{{\\rm{o}}}t\/\\hslash }{\\mathrm{e}}^{i\\phi }\\right),$$\n(2)\n\nwith time-independent complex amplitude $${\\psi }_{0}$$ and relative phase $$\\phi$$ between the two modes of opposite parity and energy $${\\mu}_{{\\rm{e}}},\\, {\\mu }_{{\\rm{o}}}$$ for the even and odd mode respectively. The mixture of two modes, with energy splitting $$\\hslash \\Delta =\\left|{\\mu }_{{\\rm{e}}}-{\\mu }_{{\\rm{o}}}\\right|$$, causes an antiphase temporal beating in the intensities $${\\left|{\\Psi }_{1,2}\\right|}^{2}$$ similar to oscillatory population transfer in coupled bosonic Josephson junctions. Therefore, while the occupation amplitude of both condensate centres oscillates with a period $$T=2\\pi\/\\Delta$$, they feature a relative phase shift of $$\\pi$$ due to mixture of even and odd parity states. We use a Michelson interferometer, comprising a retroreflector mounted on a translational stage, and measure both the time averaged interference of the photoluminescence of the same condensate $${I}_{11}(\\bar{\\tau })=\\langle{\\left|{\\Psi }_{1}(t)+{\\Psi }_{1}(t+\\bar{\\tau })\\right|}^{2}\\rangle$$ as well as the time averaged interference of the photoluminescence of opposite condensates $${I}_{12}(\\bar{\\tau })=\\langle{\\left|{\\Psi }_{1}(t)+{\\Psi }_{2}(t+\\bar{\\tau })\\right|}^{2}\\rangle$$, where $$\\bar{\\tau }$$ is the relative time delay controlled by the adjustable position of the retroreflector. Examples of the interferometric images are illustrated in Fig.\u00a04a for two condensates with a separation distance of $$d=10.3\\, \\upmu{\\rm{m}}$$, which demonstrates condensation in both an even and an odd parity state as shown in Fig.\u00a04b. Assuming the spectral composition as noted in Eqs. (1) and (2) the corresponding interferometric visibilities $${\\mathcal{V}}$$ can be written as\n\n$${\\mathcal{V}}_{11}=\\left|\\cos (\\Delta \\bar{\\tau }\/2)\\right|,$$\n(3)\n$${\\mathcal{V}}_{12}=\\left|\\sin (\\Delta \\bar{\\tau }\/2)\\right|.$$\n(4)\n\nThe experimentally extracted visibilities $${\\mathcal{V}}_{11}(\\bar{\\tau })$$ and $${\\mathcal{V}}_{12}(\\bar{\\tau })$$ versus relative time delay $$\\bar{\\tau }$$ are illustrated in Fig.\u00a04c with blue circles and red squares, respectively. In agreement with the predicted correlations [Eqs. (3) and (4)] we observe high fringe visibility $${\\mathcal{V}}_{11}(0) \\, > \\, 0.8$$ for the interference of one condensate with itself and almost vanishing fringe visibility $${\\mathcal{V}}_{12}(0) \\, < \\, 0.1$$ for the interference of opposite condensates at zero time delay. Furthermore, we find periodic disappearance and revival of fringe visibility with period $$T\\approx 15.3\\ {\\rm{ps}}$$, which is consistent with the observed energy splitting of the two modes $$\\hslash \\Delta =270\\, \\upmu {\\rm{eV}}$$ (see Fig.\u00a04b). The relative phase shift between the two visibilities $${\\mathcal{V}}_{11}(\\bar{\\tau })$$ and $${\\mathcal{V}}_{12}(\\bar{\\tau })$$ is a direct result of the periodic population transfer between the two condensates. For comparison the inset in Fig.\u00a04c shows the expected visibilities (Eqs. (3) and (4)) multiplied with an exponential decay accounting for the finite coherence time of the system.\n\nIn Fig.\u00a04d we show the decay of temporal coherence for two coupled condensates in a single-colour state ($$d=20\\, \\upmu {\\rm{m}}$$), a two-colour state ($$d=20.5\\, \\upmu {\\rm{m}}$$), and for an isolated condensate excited with the same pump power density. We note that the coherent exchange of particles in both regimes of the coupled condensate system results in an enhanced coherence time. For the single-colour state ($$d=20\\, \\upmu {\\rm{m}}$$) and the isolated condensate the coherence time $${\\bar{\\tau }}_{{\\rm{c}}}$$ can be extracted from exponential fits yielding $$25.5$$ and $$10.2\\ {\\rm{ps}}$$, respectively.\n\n### Numerical analysis\n\nThe dynamics of polariton condensates can be modelled via the mean field theory approach where the condensate order parameter $$\\Psi ({\\bf{r}},t)$$ is described by a two-dimensional (2D) semiclassical wave equation often referred as the generalised Gross\u2013Pitaevskii equation coupled with an excitonic reservoir which feeds non-condensed particles to the condensate47. In Supplementary Note\u00a01 we give numerical results of the spatiotemporal dynamics of the dyad reproducing semi-quantitatively the experimentally observed results. We also provide a supplemental animation showing the calculated evolution of the two-colour condensate (Supplementary Movie\u00a02).\n\nIn the following, we show that our experimental observations are described as a system of time-delayed coupled non-linear oscillators. For simplicity we consider the one-dimensional (1D) Schr\u00f6dinger equation corresponding to the problem of $$\\mathrm s$$-wave scattering of the condensate wavefunctions with $${\\rm{\\delta }}$$-shaped complex-valued, pump induced, potentials. We start by characterising the energies of the time-independent non-hermitian single particle problem:\n\n$$E\\Psi (x)=\\left(-\\frac{{\\hslash }^{2}{\\partial }_{x}^{2}}{2m}+V(x)-i\\frac{\\hslash {\\gamma }_{{\\rm{c}}}}{2}\\right)\\Psi (x),$$\n(5)\n\nwhere $$V(x)$$ describes complex-valued $$\\delta$$-shaped potentials separated by a distance $$d$$, $$V(x)={V}_{0}\\left(\\delta (x+d\/2)+\\delta (x-d\/2)\\right)$$, and $${V}_{0}$$ lies in the first quadrant of the complex plane, i.e. repulsive interactions and gain. The eigenfunctions of Eq. (5) describing normalisable solutions of outwards propagating waves from the potential (non-resonant pump) centres are written as\n\n$$\\Psi (x)=\\left\\{\\begin{array}{ll}{\\hskip -26pt}A{\\mathrm{e}}^{-{i}kx},& {\\hskip 12pt}x \\le -d\/2\\\\ B{\\mathrm{e}}^{ikx}+C{\\mathrm{e}}^{-{i}kx}, & |x| \\, < \\, d\/2 \\\\ {\\hskip -35pt}D{\\mathrm{e}}^{ikx}, & x \\ge d\/2\\end{array}\\right.$$\n(6)\n\nwhere $$k$$ also belongs to the first quadrant of the complex plane. This problem is well known for the case of lossless attractors ($$\\Re ({V}_{0}) \\, < \\, 0$$, $$\\Im ({V}_{0})=0$$) describing electron states in a 1D diatomic Hydrogen molecule ion. The resonance condition of the system is written as\n\n$${\\left[\\frac{{V}_{0}}{\\frac{i{\\hslash }^{2}k}{m}-{V}_{0}}{\\mathrm{e}}^{ikd}\\right]}^{2}=1.$$\n(7)\n\nThe solutions of Eq. (7) can explicitly be written as\n\n$${k}_{n,\\pm }=-i\\tilde{V}+\\frac{i}{d}{W}_{n}(\\mp {\\mathrm{d}}\\tilde{V}{\\mathrm{e}}^{{\\mathrm{d}}\\tilde{V}}),\\quad n\\in {\\mathbb{Z}},$$\n(8)\n\nwith $$\\tilde{V}=m{V}_{0}\/{\\hslash }^{2}$$. Equation (8) describes infinitely many solutions of the system of even $$(+)$$ and odd $$(-)$$ parity, where $${W}_{n}$$ are the branches of the Lambert $$W$$ function. The corresponding complex-valued eigenvalues,\n\n$${E}_{n,\\!\\pm }=\\frac{{\\hslash }^{2}{k}_{n,\\!\\pm }^{2}}{2m}-\\frac{i\\hslash {\\gamma }_{{\\rm{c}}}}{2},$$\n(9)\n\nare illustrated in Fig.\u00a05 and qualitatively reproduce the experimental findings of the multiple energy branches shown in Fig.\u00a03. We interpret the experimental occurrence of predominantly two lasing modes with the behaviour of the imaginary values of $${E}_{n}$$ for the simplistic 1D-toy model. While there are periodically alternating regions of even and odd-parity solutions dominating the gain, we expect distances at which two modes (of opposite parity) have equal gain and thus can operate with equal intensity. It is worth noting that the Lambert $$W$$ function naturally arises for problems involving delay differential equations48.\n\nThe time-dependent problem can be formulated as a superposition of two displaced, normalisable, and ballistically propagating waves $${\\psi }_{1,2}(x)$$, each emerging from one of the condensate centres,\n\n$$\\Psi (x,t)={c}_{1}(t){\\psi }_{1}(x)+{c}_{2}(t){\\psi }_{2}(x).$$\n(10)\n\nHere, in analogy with Eq. (6), the normalised ansatz is written as $${\\psi }_{1,2}(x)=\\sqrt{\\kappa }{\\mathrm{e}}^{ik| {x}\\!\\pm\\!{d\/2}|}$$, with a complex-valued wavevector $$k={k}_{{\\rm{c}}}+i\\kappa$$. When the coupling between condensates is weak, i.e. small $$\\xi =\\exp(-\\kappa{d})$$, one can omit all terms of order $${\\mathcal{O}}({\\xi }^{2})$$ and higher. Then plugging Eq. (10) into the time-dependent form of Eq. (5) and integrating out the spatial degrees of freedom, assuming that Eq. (7) is satisfied, one gets (see Supplementary Note\u00a02),\n\n$$i\\hslash {\\dot{c}}_{i}=\\left[\\frac{{\\hslash }^{2}{k}^{2}}{2m}-i\\frac{\\hslash {\\gamma }_{{\\rm{c}}}}{2}+\\kappa \\left({V}_{0}-\\frac{i{\\hslash }^{2}k}{m}\\right)\\right]{c}_{i}+{V}_{0}\\kappa {\\mathrm{e}}^{ikd}{c}_{j}.$$\n(11)\n\nwhere $$j=3-i$$ and $$i=1,2$$ are the condensate indices. When setting $${c}_{i}=\\pm \\!{c}_{j}$$, and solving for stationary states, the above equation recovers the exact resonant solutions dictated by Eq. (8). Equation (11) then shows that inter-condensate interaction is in the form of a coherent influx of particles from condensate $$j$$ onto the condensate centre of condensate $$i$$ (and vice versa), with a phase retardation of $${k}_{{\\rm{c}}}d$$. When $${c}_{i}$$ and $${c}_{j}$$ oscillate at a fixed frequency $$\\omega$$ we can transform the phase-shifting term $$\\exp (i{k}_{{\\rm{c}}}d)$$ into an effective time delay,\n\n$${\\rm{e}}^{ikd}{c}_{j}(t)={\\rm{e}}^{-\\kappa d}{c}_{j}(t-\\tau ).$$\n(12)\n\nThe time delay $$\\tau$$ corresponds to an interaction lag between condensate centres $$i$$ and $$j$$ caused by their spatial separation and is given by $$\\tau ={k}_{{\\rm{c}}}d\/\\omega$$. In the case of weak coupling, i.e. small changes in oscillation frequency $$\\omega$$ compared to the frequency $${\\omega }_{0}$$ of a single unperturbed condensate, the time-delay is approximately proportional to the dyad separation distance $$d$$, i.e. $$\\tau \\approx {k}_{{\\rm{c}},0}d\/{\\omega }_{0}$$ where we also use the notation $${k}_{0}={k}_{{\\rm{c}},0}+i{\\kappa }_{0}$$ for the wavevector of the single unperturbed condensate. The exponentially decaying term on the right-hand side of Eq. (12) accounts for the 1D spatial decay of particles propagating in between the two condensate centres. Introducing local non-linear interactions, reservoir gain, and blueshift one can write the full non-linear equation of motion for the two coupled condensates as47\n\n$$i{\\dot{c}}_{i}=\\left[\\Omega +\\left(g+i\\frac{R}{2}\\right){n}_{i}+\\alpha | {c}_{i}{| }^{2}\\right]{c}_{i}+J{\\mathrm{e}}^{i\\beta }{c}_{j}(t-d\/v),$$\n(13)\n$${\\dot{n}}_{i}=-({\\Gamma }_{{\\rm{A}}}+R| {c}_{i}{| }^{2}){n}_{i}+P.$$\n(14)\n\nHere, $${n}_{i}$$ correspond to the pump induced exciton reservoirs providing blushift and gain into their respective condensates, $$g$$ is the polariton-reservoir interaction strength, $$R$$ is the rate of stimulated scattering of polaritons into the condensate from the active reservoir, $$\\alpha$$ is the interaction strength of two polaritons in the condensate, $$v={\\omega }_{0}\/{k}_{{\\rm{c}},0}$$ is the phase velocity of the polariton wavefunction, and $${\\Gamma }_{{\\rm{A}}}$$ is the radiative decay rate of the bottleneck reservoir excitons. The parameter $$\\Omega ={\\Omega }_{0}-i\\Gamma$$ captures the self-energy of each condensate which will, in general, have a contribution from a background of optically inactive dark excitons generated at the pump spot, and $$\\Gamma$$ denotes the effective linewidth of the polaritons expanding away from the pump spot. The complex valued coupling is written $$J\\exp (i\\beta )={V}_{0}\\kappa \\exp (-\\kappa d)$$ for brevity. Equation (13) is then in the form of a discretised complex Gross\u2013Pitaevskii equation but with time-delayed interaction between the bosonic particle ensembles: which greatly increases the dimensionality of phase-space and complexity of the coupled system. Our system then has strong similarity with the famous Lang\u2013Kobayashi equation49,50 where in our case each condensate acts as a radiating antenna of symmetrically expanding waves. The two spatially separated antennas interfere and maximise their gain by adjusting both their common frequency and their relative phase difference $$\\phi =\\arg ({c}_{i}^{* }{c}_{j})$$. Similarities of the dynamics of coupled polariton condensates to equations of motion in the form of the Lang\u2013Kobayashi equation was discussed in theoretical works recently for instantaneously coupled condensates with complex-valued couplings51. Unlike trapped ground state bosonic systems, such as cold atoms, the ballistically expanding polariton matter-wave condensates necessarily experience time-delayed coupling since $${k}_{{\\rm{c}},0}d\\gg 1$$ similar to inter-cavity coupling of semiconductor lasers33.\n\nIn order to accurately reproduce experimentally observed spectra using Eqs. (13) and (14) we fit the distance dependence of the coupling amplitude $$J(d)$$ to the spatial envelope of a single condensate. It is known that in a 2D system the 0-order Hankel function of the first kind describes the cylindrically symmetric radial outflow of particles in the linear regime34, therefore we choose\n\n$$J(d)={J}_{0}\\left|{H}_{0}^{(1)}({k}_{0}d)\\right|,$$\n(15)\n\nwhere $${J}_{0}$$ is a parameter describing the coupling strength (see Supplementary Note\u00a03). Numerical integration of Eqs. (13) and (14) is computed for separation distances $$d \\, > \\, 10\\, \\upmu {\\rm{m}}$$ using Gaussian white noise as initial conditions which, in close analogy to the experimental findings, reproduces periodic parity-flip transitions accompanied by cyclic solutions in the transition region. Figure\u00a06a depicts the experimentally measured and normalised emission spectra of the polariton dyad versus pump spot separation distance $$d$$ on a grey-scale colour-map, for which the extracted two most dominant spectral peaks are depicted in Fig.\u00a03. The red dots represent the numerically calculated spectral peak from Eq. (13) when in single mode operation, or the two most dominant spectral peaks when multiple spectral components exist, showing excellent agreement with experiment. As an example, we illustrate a continuous transition of the system from an anti-phase to in-phase state described by the time-delayed coupled model in Fig.\u00a06b and c showing the corresponding spectral decomposition and phase-space diagrams of the system for an increasing set of distances $$d$$ from $$20$$ to $$21\\, \\upmu {\\rm{m}}$$. Similar stationary and oscillatory behaviour, for separation distances $$d=20$$ and $$20.5\\, \\upmu {\\rm{m}}$$ respectively, is shown in Fig.\u00a04d. The numerically simulated phase-space diagrams depict periodic orbits in the transition region, involving periodic oscillations of the phase difference $$\\phi =\\arg ({c}_{1}^{* }{c}_{2})$$ and population imbalance $$z=(| {c}_{1}{| }^{2}-| {c}_{2}{| }^{2})\/(| {c}_{1}{| }^{2}+| {c}_{2}{| }^{2})$$, which is also confirmed by 2D-simulations of the generalised Gross\u2013Pitaevskii equation (see Supplementary Note\u00a01). We have verified through numerics that time-delay physics are indeed an accurate representation of the coupled condensate dynamics (see Supplementary Note\u00a02).\n\nWe note that in the limit of fast active reservoir relaxation, $${\\Gamma }_{{\\rm{A}}}^{-1}\\ll {\\Gamma }^{-1}$$, one can adiabatically eliminate Eq. (14) and introduce an effective non-linear term to Eq. (13), $$({\\alpha }_{\\text{eff}}-i\\sigma )| {c}_{i}{| }^{2}$$, accounting for polariton\u2013polariton and polariton\u2013reservoir interactions, as well as condensate gain saturation. The dynamical equations of two coupled condensates are then described by time-delayed coupled Stuart\u2013Landau oscillators,\n\n$$i{\\dot{c}}_{i}=\\left(\\Omega +({\\alpha }_{\\text{eff}}-i\\sigma )| {c}_{i}{| }^{2}\\right){c}_{i}+J{\\mathrm{e}}^{i\\beta }{c}_{j}(t-d\/v).$$\n(16)\n\nNumerical analysis of Eq. (16) is given in Supplementary Note\u00a04 showing qualitative agreement with experiment.\n\n## Discussion\n\nWe present an extensive experimental and theoretical study of a system of two ballistically expanding (untrapped) interacting polariton condensates, the fundamental building block of polariton graphs with higher connectivity. We demonstrate a regime for coupled matter-wave condensates, wherein the coupling is not instantaneous but mediated by a particle flow inherently connected with time retardation effects. We observe deterministic selection of steady state (single-colour) or dynamical (two-colour) modes of the system by controlling the separation distance between the condensates. Time-delay polaritonics potentially offers an ultrafast platform for simulating the dynamics of real-world systems of time-delayed coupled non-linear oscillators that appear in photonics, electronics, and neural circuits. Given the high non-linearities of polaritons and the ease of optically imprinting multiple condensates of arbitrary geometries on planar microcavities, the system offers promising applications for neuromorphic devices based on lattices of history dependent, non-trapped, strongly interacting, polariton condensates with a wide range of coupling strengths, fast optical operations (input), dynamics (processing), and readout.\n\n## Methods\n\n### Microcavity sample and experimental methods\n\nThe microcavity sample used is a strain compensated $$2\\lambda$$ GaAs microcavity with three pairs of 6\u2009nm InGaAs quantum wells embedded at the anti-nodes of the electric field37. The intracavity layer contains a wedge and the position on the sample is chosen such that the cavity photonic mode is redshifted from the excitonic mode at zero inplane wavevector ($$| {\\bf{k}}| =0$$) by $$\\approx -5.5$$\u2009meV. For all experiments the microcavity is held in a cold finger cryostat (temperature $$T\\approx 6\\, {\\mathrm{K}}$$) and is optically pumped with a circularly polarised continuous wave monomode laser blue-detuned above the cavity stopband ($$\\lambda \\approx 785\\ {\\rm{nm}}$$). To prevent heating of the sample, an acousto-optic modulator is used to generate square wave packets at a frequency of 10\u2009kHz and duty cycle of $$5 \\%$$. A liquid crystal spatial light modulator (SLM) imprints a phase pattern such that when the beam is focused through the 0.4 numerical aperture microscope objective lens it excites the sample with the desired spatial geometry. The phase patterns are carefully designed so that when changing the pump separation distance $$d$$ of the two-spot excitation pattern, the diffraction efficiency of the SLM remains constant and both pump spots retain equal excitation power and width. The photoluminescence is collected in reflection geometry and spectrally resolved using an 1800 grooves\/mm grating in a $$750\\ {\\rm{mm}}$$ spectrometer, which is equipped with a charge-coupled device (CCD). Real-space, Fourier-space and dispersion images are acquired using exposure times in the order of milliseconds.\n\n### Interferometry\n\nA modified Michelson interferometer, where one mirror is replaced with a retroreflector mounted on a translational stage, is used for measuring the interference and temporal coherence of the emission. By tilting the angle of the emission entering the interferometer, we select to spatially overlap the emission of either opposing condensates or the same condensates onto a CCD camera. Interference fringe visibilities are extracted from the normalised $${{\\rm{first}}}$$ diffraction order of the computed discrete Fourier transform of each interference image.\n\n### Image processing\n\nImage displayed in Fig.\u00a01a has been digitally processed with a low-pass filter to increase the visibility of interference fringes.\n\n## Data availability\n\nThe data that support the findings of this study are openly available from the University of Southampton repository (https:\/\/doi.org\/10.5258\/SOTON\/D1149)52.\n\n## References\n\n1. 1.\n\nMackey, M. C. & Glass, L. Oscillation and chaos in physiological control systems. Science 197, 287\u2013289 (1977).\n\n2. 2.\n\nOliveira, H. M. & Melo, L. V. Huygens synchronization of two clocks. Sci. Rep. 5, 11548 (2015).\n\n3. 3.\n\nErneux, T. Applied Delay Differential Equations, Vol. 3 (Springer-Verlag, New York, 2009).\n\n4. 4.\n\nAtay, F. M. Complex Time-Delay Systems: Theory and Applications (Springer-Verlag, Berlin Heidelberg, 2010).\n\n5. 5.\n\nSchuster, H. G. & Wagner, P. Mutual entrainment of two limit cycle oscillators with time delayed coupling. Prog. Theor. Phys. 81, 939\u2013945 (1989).\n\n6. 6.\n\nLarger, L., Penkovsky, B. & Maistrenko, Y. Virtual chimera states for delayed-feedback systems. Phys. Rev. Lett. 111, 054103 (2013).\n\n7. 7.\n\nSch\u00f6ll, E. & Schuster, H. G. Handbook of Chaos Control, Vol. 2 (Wiley-VCH, Weinheim, 2008).\n\n8. 8.\n\nUchida, A. et al. Fast physical random bit generation with chaotic semiconductor lasers. Nat. Photonics 2, 728 (2008).\n\n9. 9.\n\nArgyris, A. et al. Chaos-based communications at high bit rates using commercial fibre-optic links. Nature 438, 343 (2005).\n\n10. 10.\n\nVan der Sande, G., Brunner, D. & Soriano, M. C. Advances in photonic reservoir computing. Nanophotonics 6, 561\u2013576 (2017).\n\n11. 11.\n\nAppeltant, L. et al. Information processing using a single dynamical node as complex system. Nat. Commun. 2, 468 (2011).\n\n12. 12.\n\nLarger, L. et al. Photonic information processing beyond turing: an optoelectronic implementation of reservoir computing. Opt. Express 20, 3241\u20133249 (2012).\n\n13. 13.\n\nBrunner, D., Soriano, M. C., Mirasso, C. R. & Fischer, I. Parallel photonic information processing at gigabyte per second data rates using transient states. Nat. Commun. 4, 1364 (2013).\n\n14. 14.\n\nVandoorne, K. et al. Experimental demonstration of reservoir computing on a silicon photonics chip. Nat. Commun. 5, 3541 (2014).\n\n15. 15.\n\nPaquot, Y. et al. Optoelectronic reservoir computing. Sci. Rep. 2, 287 (2012).\n\n16. 16.\n\nMartinenghi, R., Rybalko, S., Jacquot, M., Chembo, Y. K. & Larger, L. Photonic nonlinear transient computing with multiple-delay wavelength dynamics. Phys. Rev. Lett. 108, 244101 (2012).\n\n17. 17.\n\nLarger, L. et al. High-speed photonic reservoir computing using a time-delay-based architecture: million words per second classification. Phys. Rev. X 7, 011015 (2017).\n\n18. 18.\n\nKavokin, A., Baumberg, J. J., Malpuech, G. & Laussy, F. P. Microcavities (Oxford University Press, Oxford, 2007).\n\n19. 19.\n\nLi, F. et al. Tunable open-access microcavities for solid-state quantum photonics and polaritonics. Adv. Quantum Technol. 2, 1900060 (2019).\n\n20. 20.\n\nKasprzak, J. et al. Bose-Einstein condensation of exciton polaritons. Nature 443, 409\u2013414 (2006).\n\n21. 21.\n\nWu, Z. et al. Cubic-quintic condensate solitons in four-wave mixing. Phys. Rev. A 88, 063828 (2013).\n\n22. 22.\n\nZhang, Z. et al. Particlelike behavior of topological defects in linear wave packets in photonic graphene. Phys. Rev. Lett. 122, 233905 (2019).\n\n23. 23.\n\nBerloff, N. G. et al. Realizing the classical xy Hamiltonian in polariton simulators. Nat. Mater. 16, 1120\u20131126 (2017).\n\n24. 24.\n\nBloch, I. Ultracold quantum gases in optical lattices. Nat. Phys. 1, 23 (2005).\n\n25. 25.\n\nCataliotti, F. et al. Josephson junction arrays with Bose-Einstein condensates. Science 293, 843\u2013846 (2001).\n\n26. 26.\n\nDung, D. et al. Variable potentials for thermalized light and coupled condensates. Nat. Photonics 11, 565 (2017).\n\n27. 27.\n\nKim, N. Y. et al. Dynamical d-wave condensation of exciton-polaritons in a two-dimensional square-lattice potential. Nat. Phys. 7, 681 (2011).\n\n28. 28.\n\nAbbarchi, M. et al. Macroscopic quantum self-trapping and Josephson oscillations of exciton polaritons. Nat. Phys. 9, 275\u2013279 (2013).\n\n29. 29.\n\nGreiner, M., Mandel, O., Esslinger, T., H\u00e4nsch, T. W. & Bloch, I. Quantum phase transition from a superfluid to a mott insulator in a gas of ultracold atoms. Nature 415, 39 (2002).\n\n30. 30.\n\nStruck, J. et al. Quantum simulation of frustrated classical magnetism in triangular optical lattices. Science 333, 996\u2013999 (2011).\n\n31. 31.\n\nZhang, Z. et al. Observation of parity-time symmetry in optically induced atomic lattices. Phys. Rev. Lett. 117, 123601 (2016).\n\n32. 32.\n\nAlbiez, M. et al. Direct observation of tunneling and nonlinear self-trapping in a single bosonic josephson junction. Phys. Rev. Lett. 95, 010402 (2005).\n\n33. 33.\n\nSoriano, M. C., Garc\u00eda-Ojalvo, J., Mirasso, C. R. & Fischer, I. Complex photonics: dynamics and applications of delay-coupled semiconductors lasers. Rev. Mod. Phys. 85, 421\u2013470 (2013).\n\n34. 34.\n\nWouters, M., Carusotto, I. & Ciuti, C. Spatial and spectral shape of inhomogeneous nonequilibrium exciton-polariton condensates. Phys. Rev. B 77, 115340 (2008).\n\n35. 35.\n\nChristmann, G. et al. Oscillatory solitons and time-resolved phase locking of two polariton condensates. N. J. Phys. 16, 103039 (2014).\n\n36. 36.\n\nOhadi, H. et al. Nontrivial phase coupling in polariton multiplets. Phys. Rev. X 6, 031032 (2016).\n\n37. 37.\n\nCilibrizzi, P. et al. Polariton condensation in a strain-compensated planar microcavity with ingaas quantum wells. Appl. Phys. Lett. 105, 191118 (2014).\n\n38. 38.\n\nPrasad, A. et al. Universal occurrence of the phase-flip bifurcation in time-delay coupled systems. Chaos: an interdisciplinary. J. Nonlinear Sci. 18, 023111 (2008).\n\n39. 39.\n\nAdhikari, B. M., Prasad, A. & Dhamala, M. Time-delay-induced phase-transition to synchrony in coupled bursting neurons. Chaos: an interdisciplinary. J. Nonlinear Sci. 21, 023116 (2011).\n\n40. 40.\n\nBarardi, A., Sancrist\u00f3bal, B. & Garcia-Ojalvo, J. Phase-coherence transitions and communication in the gamma range between delay-coupled neuronal populations. PLoS Comput. Biol. 10, e1003723 (2014).\n\n41. 41.\n\nDotson, N. M. & Gray, C. M. Experimental observation of phase-flip transitions in the brain. Phys. Rev. E 94, 042420 (2016).\n\n42. 42.\n\nClerkin, E., O\u2019Brien, S. & Amann, A. Multistabilities and symmetry-broken one-color and two-color states in closely coupled single-mode lasers. Phys. Rev. E 89, 032919 (2014).\n\n43. 43.\n\nRamanaReddy, D. V., Sen, A. & Johnston, G. L. Experimental evidence of time-delay-induced death in coupled limit-cycle oscillators. Phys. Rev. Lett. 85, 3381\u20133384 (2000).\n\n44. 44.\n\nTakamatsu, A., Fujii, T. & Endo, I. Time delay effect in a living coupled oscillator system with the plasmodium of physarum polycephalum. Phys. Rev. Lett. 85, 2026\u20132029 (2000).\n\n45. 45.\n\nCruz, J. M. et al. Phase-flip transition in coupled electrochemical cells. Phys. Rev. E 81, 046213 (2010).\n\n46. 46.\n\nManoj, K., Pawar, S. A. & Sujith, R. I. Experimental evidence of amplitude death and phase-flip bifurcation between in-phase and anti-phase synchronization. Sci. Rep. 8, 11626 (2018).\n\n47. 47.\n\nWouters, M. & Carusotto, I. Excitations in a nonequilibrium Bose-Einstein condensate of exciton polaritons. Phys. Rev. Lett. 99, 140402 (2007).\n\n48. 48.\n\nAsl, F. M. & Ulsoy, A. G. Analysis of a system of linear delay differential equations. J. Dyn. Syst. Meas. Control 125, 215\u2013223 (2003).\n\n49. 49.\n\nLang, R. & Kobayashi, K. External optical feedback effects on semiconductor injection laser properties. IEEE J. Quant. Electronics 16, 347\u2013355 (1980).\n\n50. 50.\n\nKozyreff, G., Vladimirov, A. & Mandel, P. Global coupling with time delay in an array of semiconductor lasers. Phys. Rev. Lett. 85, 3809 (2000).\n\n51. 51.\n\nKalinin, K. P. & Berloff, N. G. Polaritonic network as a paradigm for dynamics of coupled oscillators. Preprint at https:\/\/arxiv.org\/abs\/1902.09142 (2019).\n\n52. 52.\n\nT\u00f6pfer, J. D., Sigurdsson, H., Pickup, L. & Lagoudakis, P. G. Data for Time-Delay Polaritonics (University of Southampton Repository, 2019).\n\n## Acknowledgements\n\nThe authors are grateful to N. G. Berloff for fruitful discussions and acknowledge the support of the UK\u2019s Engineering and Physical Sciences Research Council (grant EP\/M025330\/1 on Hybrid Polaritonics).\n\n## Author information\n\nP.G.L. led the research project. P.G.L., J.D.T., and L.P. designed the experiment. J.D.T. and L.P. carried out the experiments and analysed the data. J.D.T. and H.S. developed the theoretical modelling. H.S. performed numerical simulations. All authors contributed to the writing of the manuscript.\n\nCorrespondence to J. D. T\u00f6pfer or P. G. Lagoudakis.\n\n## Ethics declarations\n\n### Competing interests\n\nThe authors declare no competing interests.\n\nPublisher\u2019s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.\n\n## Supplementary information\n\n### 42005_2019_271_MOESM3_ESM.mp4\n\nSupplementary Movie 1\n\n### 42005_2019_271_MOESM4_ESM.mp4\n\nSupplementary Movie 2\n\n## Rights and permissions\n\nReprints and Permissions\n\nT\u00f6pfer, J.D., Sigurdsson, H., Pickup, L. et al. Time-delay polaritonics. Commun Phys 3, 2 (2020). https:\/\/doi.org\/10.1038\/s42005-019-0271-0","date":"2020-04-06 15:17:36","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 2, \"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\": 0.7131853699684143, \"perplexity\": 2195.166897713952}, \"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-2020-16\/segments\/1585371637684.76\/warc\/CC-MAIN-20200406133533-20200406164033-00164.warc.gz\"}"}
| null | null |
Anne Schillings Photography specializes custom high school senior, college & teen portrait sessions. Located in beautiful Northern California.
Nationally published & selected as a one of the top 100 senior portrait photographers in the nation.
Your session will be designed for you! From styling assistance, use of our accessories closet to professional hair & makeup services & posing guidance you will look your absolute best! Get the magazine style photos you have always dreamed of.
I was absolutely amazed by my senior pictures! Anne did such a good job and was so sweet! She was so easy to talk to and fun. Overall I had a great experience!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,730
|
var should = require("should");
var when = require("when");
var sinon = require("sinon");
var passport = require("passport");
var auth = require("../../../../red/api/auth");
var Users = require("../../../../red/api/auth/users");
var Tokens = require("../../../../red/api/auth/tokens");
var settings = require("../../../../red/settings");
describe("api auth middleware",function() {
describe("ensureClientSecret", function() {
it("leaves client_secret alone if not present",function(done) {
var req = {
body: {
client_secret: "test_value"
}
};
auth.ensureClientSecret(req,null,function() {
req.body.should.have.a.property("client_secret","test_value");
done();
})
});
it("applies a default client_secret if not present",function(done) {
var req = {
body: { }
};
auth.ensureClientSecret(req,null,function() {
req.body.should.have.a.property("client_secret","not_available");
done();
})
});
});
describe("revoke", function() {
it("revokes a token", function(done) {
var revokeToken = sinon.stub(Tokens,"revoke",function() {
return when.resolve();
});
var req = { body: { token: "abcdef" } };
var res = { status: function(resp) {
revokeToken.restore();
resp.should.equal(200);
return {
end: done
}
}};
auth.revoke(req,res);
});
});
describe("login", function() {
beforeEach(function() {
sinon.stub(Tokens,"init",function(){});
sinon.stub(Users,"init",function(){});
});
afterEach(function() {
Tokens.init.restore();
Users.init.restore();
});
it("returns login details - credentials", function(done) {
auth.init({adminAuth:{}},null);
auth.login(null,{json: function(resp) {
resp.should.have.a.property("type","credentials");
resp.should.have.a.property("prompts");
resp.prompts.should.have.a.lengthOf(2);
done();
}});
});
it("returns login details - none", function(done) {
auth.init({},null);
auth.login(null,{json: function(resp) {
resp.should.eql({});
done();
}});
});
});
});
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,485
|
package org.reldb.rel.tests.external;
import org.junit.AfterClass;
import org.junit.BeforeClass;
import org.junit.Test;
import org.reldb.rel.tests.helpers.BaseOfTest;
public class TestExternal01 extends BaseOfTest {
@BeforeClass
public static void testExternal01() {
String src =
"BEGIN;" +
"OPERATOR COMPARE_TO(s CHAR, anotherString CHAR) RETURNS INTEGER Java FOREIGN\n" +
" // Compares two strings lexicographically.\n" +
" return ValueInteger.select(context.getGenerator(), s.stringValue().compareTo(anotherString.stringValue()));\n" +
"END OPERATOR;" +
"END;" +
"true";
String expected = "true";
testEquals(expected, src);
}
@Test
public void testExternal02() {
String src =
"COMPARE_TO('string 1', 'string 2')";
String expected = "-1";
testEquals(expected, src);
}
@Test
public void testExternal03() {
String src =
"COMPARE_TO('string 1', 'string 1')";
String expected = "0";
testEquals(expected, src);
}
@AfterClass
public static void testExternal04() {
String src =
"BEGIN;" +
"DROP OPERATOR COMPARE_TO(CHAR, CHAR);" +
"END;" +
"true";
String expected = "true";
testEquals(expected, src);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,071
|
import threading
import time
import subprocess
class mpstat(threading.Thread):
def __init__(self, callback):
"""Form the monitor and register the monitored metrics."""
super(mpstat, self).__init__()
self.stop_flag = False
self.interval = None
self.callback = callback
self.callback.add('cpu', 'inuse')
self.callback.add('cpu', 'steal')
def stop(self):
"""Prepare monitor to stop."""
self.stop_flag = True
def set(self, argument, value):
"""Set a monitor argument.
Arguments:
argument -- arugment name
value -- value for the argument
"""
if argument == 'interval':
self.interval = value
def run(self):
"""Monitor the metrics."""
if not self.interval:
raise RuntimeError
# open mpstat as a subprocess
process_parameters = ['mpstat', str(self.interval)]
try:
mpstat_subprocess = subprocess.Popen(process_parameters,
stdout=subprocess.PIPE,
stderr=subprocess.PIPE)
except:
print('Could not create mpstat monitor, check if the sysstat package is installed in your system.')
return
# discard prelude
mpstat_subprocess.stdout.readline()
mpstat_subprocess.stdout.readline()
# get column numbers
output_line = mpstat_subprocess.stdout.readline().decode('ascii')
output_line_columns = output_line.split()
idle_col = output_line_columns.index('%idle')
steal_col = output_line_columns.index('%steal')
while mpstat_subprocess.poll() is None and not self.stop_flag:
# read one line from the output
output_line = mpstat_subprocess.stdout.readline().decode('ascii')
output_line_columns = output_line.split()
# in_use = 100.0 - %idle
# (%idle: 12th column in line)
idle = float(output_line_columns[idle_col].replace(',', '.'))
inuse = 100.0 - idle
self.callback.set_metric('cpu', 'inuse', inuse)
# steal = %steal
# (%steal: 9th column in line)
steal = float(output_line_columns[steal_col].replace(',', '.'))
self.callback.set_metric('cpu', 'steal', steal)
# sleep some time
time.sleep(self.interval)
#if mpstat_subprocess.poll() is not None:
# mpstat_subprocess.terminate()
self.callback.remove('cpu', 'inuse')
self.callback.remove('cpu', 'steal')
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,966
|
Zahuur h. Minnesota
Islamophobia by Zahuur H
My topic is islamophobia in America and the steps we can take to help resolve it
Dear President,
We should try to find resolutions for Islamophobia. Muslims suffer a lot of discrimination throughout their daily lives. At first it was mostly just verbal abuse and whispers while walking by but now it's also physical. Hate crimes against Muslims are at their highest rates ever in the United States. People can have their own opinions,but when it gets physical and mothers fathers and kids are getting hurt. Then I think something needs to happen.
The system of media and the social world has a lot to do with Islamophobia in America today. According to Al Jazeera statistics, the percentage of Americans who respect Muslims and their beliefs dropped from 47% to 25%. According to The Huffington Post's 2015 survey, nearly 52% of Americans do not like Muslims. The media has a big role in how situations and things that happen in the world are viewed. In the morning before people go to work the first thing they'll do is turn on the 9 o'clock news or grab a newspaper on their way to wherever they are heading. The news is not only influential to adults but it is influential to all ages. According to nydailynews.com, just recently 7 year old Ahmed Usmani was beaten up by his classmates for being Muslim. All because some parent seen the news and was scared of Muslims or was islamophobic and passed it onto their kids. The media has to stop having deep root islamophobic articles and broadcasts. The media also needs to stop saying saying bias and islamophobic things. As I was looking for more information and sources about this topic I stumbled across what CNN anchor John Vaus has said. "Why is it that no one within the Muslim community in France knew what these guys were up to?" These are the type of quotes that perpetuates Islamophobia. They allow the audience to think all Muslims are terrorists. This also feeds into xenophobia that's commonly against Muslim immigrants.
Xenophobia is one of the branches that lead to Islamophobia. When someone discriminates against someone because they are from another country it is hard for Muslims and immigrants to adjust to where they have migrated to. A lot of Muslims and non-Muslims migrate to other places because of forced or chained migration. Chained migration is when one person from a certain group migrates to a destination and other people from that group follow. when the immigrants migrate to another place to start a new life and find job opportunities and to take care of their loved ones Being scared of Muslim, immigrants can lead to them not getting the right resources. According to society of human resources management a man named Zaw oo was randomly picked to check if he was a terrorist just because he is Muslim. "One minute I was in my unit and then because I am a Muslim I was being investigated for being a terrorist supporter" Zaw oo said. Zaw oo was in the U.S army. And even after he was cleared he was still getting sent anonymous death threats. Zaw oo's co workers have disrespected and mocked the Quran. He even got sent a video from a coworker who was threatening him. "I can't afford to lose my job I have a wife and a child" Zaw oo said. A lot of people have to go through many hardships just to keep their job everyday. But a positive point is according to the society of human resources management (shrm), 89 % percent of Muslims are satisfied with their job. "I would be very cautious of hiring a Muslim we have no way of knowing how many will support America," Joseph Hepp, a retired executive of Battle Creek illustrations said.
Whenever news outlets or anywhere else is mentioning Muslim, they only mention Middle Eastern countries. Why? ⅔ of the Muslims population is from Asia and Africa. People are very uneducated about Muslims and our diversity. I feel like that's also one of many reasons why there is a lot of islamophobia happening throughout the world. The world's 1.6 billion Muslims are connected because we are all from one religion. But that doesn't mean we are all from the Middle East. Throughout my whole life I have seen the beauty in the diversity of Islam. I just wish others could see it too. 53% of Africa is Muslim. 62% of Asia is Muslim. Every ethnicity/race/country in this world has Muslims. So why is it always only the Middle East that's brought up. I would like people to see how much diversity we have in this world that makes us who we are as a whole. I have seen Caucasian Muslims, Black Muslims, Asian Muslims, Hispanic Muslims, etc. In Fact according to teachmideast.com, the country with the most Muslims in the world is Indonesia. But a lot of people will only have a stereotype that all Muslims are only from the Middle East. When we are from every part of this world. And I think if people knew this it could be one step into demolishing Islamophobia.
This problem that we call Islamophobia needs to be resolved. I'm not saying that it will take a day or a year or a month I know it will take time. But the more we don't do anything to fix it the more Islamophobia grows and the more it passes on to other generations. It hurts Muslims a lot when Islamophobic things are said. Just today a kid threw a backpack next to me in school and said " Allahu Akbar!" And that really hurt me deep. Now of course I defended myself and tried to educate him more. But islamophobia goes deeper then something only a couple of people in the world can solve. One way we can start resolving islamophobia and the violence of it is by education! If more people knew the positivity Islam has and its effects on the world I think there would be less Islamophobic people in the world today. If they knew what Muslims invented if they knew how diverse we are and if they knew we come from every single country and continent on the map. Another way we can resolve things is by using the media for a positive outlook. Instead of the media making Muslims seem mean and dangerous they can talk about how Muslim countries are having the same problem with terrorism. And how Muslims are getting killed raped and murdered by terrorists. How Muslims have to flee and sometimes leave their loved ones and maybe never see them again. We should stick together and educate everyone on what's going on. The man thing we can do to resolve Islamophobia is education. I think that is the only way we are going to make this world a more positive and beautiful place.
I wish the world didn't have Islamophobia and I wish we could have more possible positive coverage for every religion throughout the world. I hope that one day I can tell my kids or grandkids about our history in the world and about how we have conquered Islamophobia and how in my generation we focused on making the world a positive place for all religions and all races. But at the rate we are going I don't know if I will be able to. But I will persevere and try everyday to have a positive outlook on the world and educate the world on Muslim people. But until we can solve Islamophobia I will remember one of my favorite quotes that give me hope and remind me about positivity. "When everything seems to be going against you, remember that the airplane takes off against the wind not with it" - Henry Ford. I hope you will help America take steps into the right direction future president. I hope you will be one of the best.
Zahuur Hassan
👏 Applauded
media discrimination xenophobia islamophobia diversity
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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Lacrosse Film Review: Yale Defense vs. Duke In NCAA Lacrosse National Championship
by Ted Hust | Dec 4, 2018 | Lacrosse Film Review Series, Lacrosse News | 0 comments
Yale faced Duke in the NCAA Lacrosse National Title Game last Memorial Day (2018) and it was a heck of a game. We're going to take a look at Duke's first possession of the game as the Blue Devils poke and prod Yale defenders with dodges from the wing to the middle of the field, and down and up the alley's on both sides of the field.
Yale communicates well, only has to slide once (probably unnecessary) and by the end of the possession, their patience and communication pay off and they force a bad pass and cause a turnover.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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\section{Introduction}
To gain better understanding of string
theories and M-theory,
intersecting branes
have played an important role, and
worldvolume analyses have been powerful approaches to investigate
the intersecting
branes\cite{cal1}\cite{how3}\cite{gib1}\cite{tow6}\cite{gaun1}.
In
ref.\cite{cal1}\cite{how3}\cite{gib1}
solutions of
worldvolume field theories of branes
(in flat backgrounds)
with nontrivial
worldvolume gauge fields were obtained.
In the case of D-branes,
it was shown
that
an appropriate excitation of one of
the transverse scalar fields is needed
in order to obtain a
{\it supersymmetric} (i.e. BPS)
point charge solution of the worldvolume gauge field\cite{cal1}.
Each of the solutions was interpreted as
a fundamental string ending on the D-brane
on the basis of the fact that
the (regularized) energy of
each solution defined on the worldvolume
is proportional to the
(regularized) value of the scalar field,
which is considered to be the length of
the string.\footnote{Since the energies of the point charge solutions
are infinite,
(UV) regularization is needed to calculate them.}
This interpretation
is also consistent with the charge conservation suggested in
ref.\cite{openpbrane} (see also ref.\cite{surgery}).
These analyses are very important in that
they made clear
``how one of intersecting two branes is described''
from the viewpoint of another brane's worldvolume.
In fact, however,
{\it the energy
defined on the brane's worldvolume
does not always agree with
the target-space mass of the string},
though the agreement of the two holds for the case of ref.\cite{cal1}.
We can easily understand this by considering
the fact that
the worldvolume energy depends on the definition of the time
of the worldvolume,
while the mass of the string should not depend on it.
That is,
the derivation of the mass of the string
must be considered more carefully.
The main purpose of this paper is
to present
a generalized ``mass formula'' of a fundamental string
described as a BPS solution of a D-brane's worldvolume,
which holds independent of definition of its worldvolume time.
Our idea to obtain the mass formula
is as follows:
the mass of the string does not always agree with
the worldvolume energy, to be sure, but
it is also true
that the two are very close to each other, since
the two give the same results
at least in the cases discussed in ref.\cite{cal1}.
So, we construct
the mass formula in a heuristic way,
by using the Hamiltonian density defined on the worldvolume,
based on the invariance
of the mass under the
coordinate transformations of the worldvolume.
Moreover,
we also construct explicitly
point charge solutions
(with appropriate excitations of single transverse
scalar fields)
of branes' worldvolumes in a {\it curved} background.\footnote{Some
worldvolume solutions of branes in a curved (brane) backgrounds
are discussed for other purposes in
ref.\cite{sjrey}\cite{sato3}.}
There are two advantages to consider these solutions:
First, in each of the cases, the time component of
the induced worldvolume metric $\tilde{g}_{00}$
(as well as spatial ones) becomes nontrivial,
So, the differences
originating from the contribution of $\tilde{g}_{00}$ become
apparant
in discussing the mass formula.
Second,
in each case of the solutions,
{\it the mass of
the corresponding fundamental string should be
proportional to
the geodesic distance from the brane to the point
parameterized by the (regularized)
value of the scalar field},
and that the proportional coefficient should be
an appropriate tension of the string,
as discussed in ref.\cite{sen1}.
This requirement is very tight.
So, once we find out
the quantity which gives the mass stated above,
it is expected to be the correct mass formula,
even if it is constructed by hand.
We will construct it based on this idea.
(The discussion on the worldvolume interpretation of the string mass
will be given finally in section 3.)
The worldvolume theories we discuss here are
the two cases: those of
a test D-4-brane and a test D-8-brane both
embedded parallel to (a subspace of)
the worldvolume of
the D-8-brane background\cite{pol1}\cite{berg3}
(i.e. a massive IIA background).
First, we present the two reasons to choose this background:
One is that
this background has only one transverse coordinate,
leading to the fact that
the harmonic function depends linearly on the coordinate.
Only in this case,
we can obtain {\it explicitly}
the exact solutions of the worldvolumes
without any extra assumption
(as we will see later).
We note that obtaining the explicit form is
is crucial not only in order to find out the mass formula {\it by hand}
and but for other discussions.
Another reason is that
choosing this background,
we can see the supersymmetry preserved
in the solutions
by using superalgebras in a massive IIA background via brane probes''
in ref.\cite{sato2}.
\footnote{In the previous paper\cite{sato3}
we have confirmed that
the supersymmetry preserved in the solution of brane's
worldvolume can be derived via ``superalgebras in brane
backgrounds''
(see also \cite{sato1}).}
From the supersymmetry, we can confirm
that the solutions are BPS states
and that their target-space interpretation
is consistent.
Next, we explain why we choose the two worldvolume theories
embedded in those ways:
This is because
at least one overall transverse space is needed
{\it after} embedding (test) D-branes' worldvolumes
in order to obtain supersymmetric point charge solutions.
In the case of
a D-2-brane (a D-6-brane),
only the intersection with the background D-8-brane
on a string (a 5-brane) leads
to the preservation of supersymmetry,\footnote{
We do not discuss bound states like (6$|$D6,D8)
since
the D-8-brane solution is ``singular'' just on the D-8-brane
hyper-surface.}
but,
there is no overall transverse space.
A D-0-brane
is not adequate to this worldvolume description
since there is no world space.
So, we consider
the above two cases,
which preserve at this moment 1/4 and 1/2
supersymmetry, respectively.
Concrete procedures are in the following:
In each of the two cases we construct
explicitly an point charge solution
with an appropriate excitation of
the only overall transverse scalar field.
The consistency of
the interpretation of each solution as a fundamental string
is confirmed in two ways:
by discussing its behavior
in the flat background limit
and by checking its preserved supersymmetry.
Then, we propose the generalized mass formula,
and examine
whether it gives the geodesic distance
multiplied by an appropriate tension of the string.
Moreover, we also discuss
the condition that the mass of the string agrees with
the energy defined on the D-brane's worldvolume.
The organization of this paper is as follows:
In section 2 we construct the point charge solutions
and discuss their mass formula.
In section 3 we give the conclusion and some discussions
on the consistency and the interpretation
of the mass formula
The notations in this paper is as follows:
We use ``mostly plus'' metrics for both spacetime and
worldvolumes.
We denote coordinates of each p-brane's worldvolume as
$\xi^{i},\xi^{j},..$ (i,j =0,1,..,p),
those of 10D spacetime as $x^{m}, x^{n},..$ (m,n=0,..,9),
fermionic coordinates as $\theta^{\alpha},\theta^{\beta},..$,
and those of superspace
as $Z^{M}$.
We use {\it hatted letters}
($\hat{M},\hat{m},\hat{\alpha}..$)
for {\it all the local Lorentz frame indices}
and {\it under-barred} letters ($\underline{m},
\underline{i}$) for {\it spatial} indices
(but not time one), respectively.
We denote gamma matrices as
$\Gamma_{\hat{m}}$, which are all real and satisfy
$ \{ \Gamma_{\hat{m}} , \Gamma_{\hat{n}} \}
= 2\eta_{\hat{m}\hat{n}}$.
$\Gamma_{\hat{0}}$ is antisymmetric and others
symmetric. Charge Conjugation is ${\cal C}=\Gamma^{\hat{0}}$.
\section{Point charge solutions
of 10D IIA D-branes' worldvolumes and their mass formula
}
\setcounter{footnote}{0}
In this section we construct the point charge solutions corresponding
to fudamental strings and discuss their mass formula.
The D-p-brane action
in a general 10D massive
IIA background\cite{rom1}\cite{berg3}
takes the form\cite{green1}\cite{berg5}
\begin{eqnarray}
S_{{\rm Dp}} = S_{{\rm Dp}}^{BI}+S_{{\rm Dp}}^{WZ}
&=& -T_{p}\int d^{p+1}\xi e^{-\phi}
\sqrt{-{\rm det}(\tilde{g}_{ij}+{\cal F}_{ij})}\nonumber\\
& &+T_{p}\int [\tilde{C} e^{{\cal F}}|_{{\rm (p+1)-form}}
+\frac{m}{(p/2+1)!}V(dV)^{p/2}],
\label{Dpaction}
\end{eqnarray}
where $\tilde{g}_{ij}
=E_{i}^{\ \hat{m}}E_{j}^{\ \hat{n}}\eta_{\hat{m}\hat{n}}$
is
the induced worldvolume metric
where $E_{i}^{\ \hat{m}}=\partial_{i}Z^{M}
E_{M}^{\ \hat{m}}$ where $E_{M}^{\ \hat{N}}$ is the supervielbein.
$\phi$ is the dilaton field and
${\cal F}_{ij}$ are the components of a modified
worldvolume 2-form field strength
\begin{eqnarray}
{\cal F}=dV-\tilde{B}_{2}
\end{eqnarray}
of the worldvolume 1-form gauge field $V$ where
$\tilde{B}_{2}$ is the worldvolume 2-form induced by the superspace
NS-NS 2-form gauge potential $B_{2}$.
$\tilde{C}$ is a formal sum of worldvolume r-forms
$\tilde{C}^{(r)}$ (r=1,3,5,7,9)
induced by the superspace R-R r-form gauge potentials $C^{(r)}$.
$m$ is a mass parameter
which is the dual of the 10-form
field strength $F^{(10)}$
of a R-R 9-form $C^{(9)}$\cite{pol2}\cite{berg3}.
$T_{p}$ is the ``formal'' (but not physical) tension of the D-p-brane
which is given by\cite{green1}\footnote{
We choose $\alpha'=1$ in this paper.}
\begin{eqnarray}
T_{p}=\frac{1}{(2\pi)^{p}}.
\end{eqnarray}
We take the background of the action (\ref{Dpaction})
to be the D-8-brane solution
given by\cite{pol1}\cite{berg3}
\begin{eqnarray}
ds^{2} &=& H^{\epsilon/2}dx^{\mu}dx^{\nu}\eta_{\mu\nu}
+H^{-5\epsilon /2-2}dy^{2}
\nonumber\\
e^{\phi} &=& H^{5\epsilon /4}\nonumber\\
C^{(9)}_{01\cdots 8}&=&H^{\epsilon}\label{d8sol}
\end{eqnarray}
where $x^{\mu}$ and $x^{\nu}$ ($\mu,\nu$=0,..,8 )
are the spacetime coordinates parallel to the D-8-brane and
$y$ is a single transverse
coordinate. $\epsilon$ is
a nonzero parameter which cannot be determined by the equations
of motions of 10D massive IIA supergravity.
We note that the solution
(\ref{d8sol}) with $\epsilon=-1$ is the standard form of
the D-8-brane solution
since it is obtained via T-duality
from the other D-p-brane solutions\cite{berg3}.
$H=H(y)$ is a harmonic function on $y$.
In this paper we set
\begin{eqnarray}
H(y)=c_{1}+\frac{m}{|\epsilon|}|y|,\label{harm1}
\end{eqnarray}
which means that the D-8-brane lies at $y=0$.
We choose $c_{1}>0$ and $ m>0$ to avoid a singularity at
$y=0$ and to get a real dilaton.
We note that
the solution (\ref{d8sol}) becomes the flat spacetime metric
in the massless limit
\begin{eqnarray}
\left\{
\begin{array}{lll}
m &\rightarrow & 0 \\
c_{1} &\rightarrow &1 \ ({\rm \ via\ diffeomorphism}).
\end{array}
\right. \label{masslesslim}
\end{eqnarray}
The Killing spinor of (\ref{d8sol}) has the form
$\varepsilon = H^{\epsilon/8}\varepsilon_{0} $ where $\varepsilon_{0}$
has a definite chirality,
i.e. $ \Gamma_{\hat{y}}
\varepsilon_{0}=+\varepsilon_{0}$ for $y>0$ and $\Gamma_{\hat{y}}
\varepsilon_{0}=-\varepsilon_{0}$ for $y<0$.
We first consider
a point charge solution
of a D-4-brane worldvolume
parallel to the background D-8-brane.
Since the solutions we construct here is a bosonic one,
we set fermionic coordinates $\theta$ to be zero.
Moreover, we consider the ansatz
\begin{eqnarray}
\left\{
\begin{array}{ll}
x^{i}=\xi^{i}
& (i=0,1,\cdots,4 )\\
x^{5},x^{6},x^{7},x^{8}:&{\rm \ \ constants}\\
y=y(r)>0& \ \\
V_{0}= V_{0}(r),& V_{\underline{i}}=0\\
\end{array}
\right.\label{pointsolansatz}
\end{eqnarray}
where $r$ is defined as
$r\equiv\sqrt{\Sigma_{\underline{i}=1}^{4}
(\xi^{\underline{i}})^{2}}$.
We note that
the upper two columns of (\ref{pointsolansatz}) mean that
the D-4-brane is embedded in the 1234-hyper-plane.
That is, from target-space point of view,
we consider
the following intersection of three branes:
\begin{center}
\begin{tabular}{lllllcrrrrr}
{\rm background D8 (at $y=0$)}:& 0&1&2&3&4&5&6&7&8&-\\
{\rm worldvolume D4 (at $y=y_{0}$)}:& 0&1&2&3&4&-&-&-&-&-\\
{\rm fundamental string } :& 0&-&-&-&-&-&-&-&-&9
\end{tabular}
\end{center}
where $y_{0}$ is a positive constant (and $x^{9}=y$).
Then, since $V_{\underline{i}} =\theta=0$,
$ S_{{\rm D4}}^{WZ}$ does not contribute to the equations of motion,
and
the equations of motion to solve
are given by
\begin{eqnarray}
\frac{\delta{\cal L}^{BI}}{\delta x^{m}}=
\partial_{i}[\frac{\delta{\cal L}^{BI}}{\delta \partial_{i}x^{m}}],
\ \
\frac{\delta{\cal L}^{BI}}{\delta V_{i}}&=&
\partial_{j}[\frac{\delta{\cal L}^{BI}}
{\delta \partial_{j}V_{i}}].\label{eom1}
\end{eqnarray}
So, let us examine more about ${\cal L}^{BI}$
The induced worldvolume metric
$\tilde{g}_{ij}$ is given by
\begin{eqnarray}
\tilde{g}_{ij}=
\left(
\begin{array}{cc}
-H^{\epsilon/2}&0\\
0&H^{\epsilon/2}\cdot [\delta_{\underline{i}\underline{j}}
+H^{-3\epsilon-2}
\partial_{\underline{i}} y\partial_{\underline{j}} y]
\end{array}
\right), \label{wvmetric1}
\end{eqnarray}
whose determinant is
${\rm det}\tilde{g}_{ij}=H^{5\epsilon/2}
[1+H^{-3\epsilon-2}(\partial
y)^{2}]$.
At this moment,
det
$(\tilde{g}_{ij}+{\cal F}_{ij})$ arising in ${\cal L}^{BI}$
is very complicated.
However, setting the condition
\begin{eqnarray}
\partial_{\underline{i}}y=\partial_{\underline{i}}V_{0},
\label{yveual1}
\end{eqnarray}
results in
the simple form of the determinant:
${\rm det}(\tilde{g}_{ij}+{\cal F}_{ij})
=H^{5\epsilon/2}$.\footnote{
It is shown that
no supersymmetry is preserved in this case without (\ref{yveual1}),
by using a superalgebra via brane probe
appearing later (eq.(\ref{spalgd4})).
So, (\ref{yveual1}) is considered to
correspond to the BPS condition.
}
Then, the equations of motion
(\ref{eom1}) become
the following two simple equations
\begin{eqnarray}
\sum_{\underline{i}}
\partial_{\underline{i}}(H^{-3\epsilon-2}\partial_{\underline{i}}y)
&=&0\label{eq11}\\
\sum_{\underline{i}}
\partial_{\underline{i}}(H^{-2\epsilon-1}\partial_{\underline{i}}y)
&=&0.\label{eom2}
\end{eqnarray}
((\ref{eq11}) arises from the 9th component of
the former of (\ref{eom1}), (\ref{eom2}) from the time
component of the latter of (\ref{eom1}), and the others are solved.)
So, requiring the two equations to be
compatible,
it needs to hold $\epsilon=-1$,
and the two are combined into one equation
\begin{eqnarray}
\sum_{\underline{i}}
\partial_{\underline{i}}(H\partial_{\underline{i}}y)=0.\label{eom3}
\end{eqnarray}
We note that
the harmonic function
gives nontrivial contribution to (\ref{eom3}),
which means that
the equations of motion of the D-4-brane worldvolume
are affected by the background D-8-brane.
By using (\ref{harm1}) and $y>0$,
the equation (\ref{eom3}) is written as
\begin{eqnarray}
\sum_{\underline{i}}(\partial_{\underline{i}})^{2}
(y+\frac{c_{1}}{m})^{2}=0.
\end{eqnarray}
We choose the boundary condition
\begin{eqnarray}
\left\{
\begin{array}{lll}
y&\rightarrow & y_{0} \ (>0)\\
V_{0}&\rightarrow & 0
\end{array}
\right. \label{boundary1}
\end{eqnarray}
for $r \rightarrow \infty$, which means that
the D-4-brane lies at $y=y_{0}$.
Then, the solution
is obtained, with the following unusual form,
as
\begin{eqnarray}
y(r)&=&[(\frac{c_{1}}{m}+y_{0})^{2}+\frac{c_{2}}{r^{2}}]^{1/2}
-\frac{c_{1}}{m}\nonumber\\
V_{0}&=&[(\frac{c_{1}}{m}+y_{0})^{2}+\frac{c_{2}}{r^{2}}]^{1/2}
-(\frac{c_{1}}{m}+y_{0})\label{sol1}
\end{eqnarray}
where $c_{2}$ is a constant proportional to the electric
charge of the gauge field.
The electric charge $Q_{1}$ is defined as \cite{gib1}
\begin{eqnarray}
Q_{1}=\int_{S_{3}} \star D \label{charge0}
\end{eqnarray}
where $S_{n}$ is the n-sphere, $\star$ is the worldvolume Hodge
dual and $D$ is the 2-form defined by
$D^{ij}=-\frac{1}{T_{4}}
\frac{\delta {\cal L}_{D4}}{\delta F_{ij}}$.
Then, we have
\begin{eqnarray}
Q_{1}
=\int_{S_{3}} \star (HdV)=m c_{2}\Omega_{3}\label{charge1}
\end{eqnarray}
where $\Omega_{n}$ is the volume of the unit n-sphere.
We note that in this definition
the string coupling $g_{s}=e^{\phi}$
is included in
$Q_{1}$ since ${\cal L}_{D4}^{BI}$ is proportional to
the inverse of $g_{s}$.
Since the solution has been
obtained explicitly,
we next give some pieces of
evidence that the solution corresponds to the fundamental
string ending on the D-4-brane.
First, we discuss the massless limit of the solution (\ref{sol1}).
We assume here that the charge $Q_{1}$ is independent of $m$.
(The validity of the assumption is discussed later.)
Then, in the massless limit (\ref{masslesslim}),
the solution (\ref{sol1}) with
(\ref{charge1}) behaves as
\begin{eqnarray}
y&\rightarrow & y_{0}+\frac{Q_{1}}{2\Omega_{3}r^{2}}\nonumber\\
V_{0}&\rightarrow & \frac{Q_{1}}{2\Omega_{3}r^{2}}.
\label{masslesssol}
\end{eqnarray}
The right hand side of (\ref{masslesssol})
is exactly the solution of
a D-4-brane's worldvolume
in the 10D {\it flat} spacetime,
which corresponds to the fundamental string\cite{cal1}.
So, it is expected to correspond to a fundamental string.
Next,
we check the preserved supersymmetry of the solution
by using ``superalgebras in brane backgrounds via brane
probes''\cite{sato1}\cite{sato2}\cite{sato3}.
The superalgebra in a D-8 brane background via a D-4-brane probe
is given in ref. \cite{sato2} as
\begin{eqnarray}
\{ Q_{\alpha}^{+}, Q_{\beta}^{+}\}&=&
2\int_{{\cal M}_{4}}d^{4}\xi\ \Pi_{\mu}
({\cal C}\Gamma^{\mu})_{\alpha\beta}
+2\int_{{\cal M}_{4}}d^{4}\xi{\cal P}^{(0)\underline{i}}
\partial_{\underline{i}}y
({\cal C}\Gamma_{y}\Gamma_{11})_{\alpha\beta}\nonumber\\
& &+\frac{2T_{4}}{4!}
\int_{{\cal M}_{4}}H^{5/4}dx^{\mu_{1}}..dx^{\mu_{4}}
({\cal C}\Gamma_{\mu_{1}..\mu_{4}}\Gamma_{11})_{\alpha\beta}
+2T_{4}\int_{{\cal M}_{4}}H^{5/4}dx^{\mu}dydV
({\cal C}\Gamma_{\mu y})_{\alpha\beta}\nonumber\\
& &+2T_{4}\int_{{\cal M}_{4}}
H^{5/4}(dV)^{2}({\cal C}\Gamma_{11})_{\alpha\beta}
+{\cal O}(\theta^{2})\label{spalgd4}
\end{eqnarray}
where $Q_{\alpha}^{+}= \frac{1+\Gamma_{\hat{y}}}{2}Q_{\alpha}$ is the
supercharge preserved in (\ref{d8sol})
and ${\cal M}_{p}$ is the worldspace of the D-p-brane.
${\cal P}^{(0)\underline{i}}$ is almost equivalent to
the conjugate momentum of
$V_{\underline{i}}$. (The contributions of the Chern-Simons term
are subtracted.)
Substituting the solution for the right
hand side of (\ref{spalgd4}),
the superalgebra can be written as
\begin{eqnarray}
\{ Q_{\alpha}^{+}, Q_{\beta}^{+}\}&=&
4T_{4}\int_{{\cal M}_{4}}d^{4}\xi\
[ H^{1/4}(\frac{1+{\cal C}\Gamma_{\hat{1}\hat{2}\hat{3}\hat{4}}
\Gamma_{11}
}{2})_{\alpha\beta}
+H^{5/4}(\partial_{i}y)^{2}
(\frac{1+{\cal C}\Gamma_{\hat{y}}\Gamma_{11}}{2})_{\alpha\beta}]
\label{spalgd4'}.\ \
\end{eqnarray}
Since the three gamma matrix products $\Gamma_{\hat{y}},
{\cal C}\Gamma_{\hat{1}\hat{2}\hat{3}\hat{4}}\Gamma_{11}$ and
${\cal C}\Gamma_{\hat{y}}\Gamma_{11}$ (arising in (\ref{spalgd4'}))
commute with each other,
all of them can be simultaneously diagonalized.
Since the square of each matrix product is
equal to the identity and each is traceless,
both of
the matrices
$\frac{1+{\cal C}\Gamma_{\hat{1}\hat{2}\hat{3}\hat{4}}\Gamma_{11}}
{2}$
and $ \frac{1+{\cal C}\Gamma_{\hat{y}}\Gamma_{11}}{2}$
are projection operators.
So, we conclude that the solution
has 1/8 supersymmetry, hence
is consistent with the target-space interpretation.
This also shows that the solution is a BPS state.
Now, we discuss the mass of the string.
Let us first consider the energy of the solution.
For this purpose,
we pass to the Hamiltonian formalism
as done in ref.\cite{cal1}.
If we assume that V is purely electric and that
only the scalar $y$ is excited,
$S_{D4}$ reduces to
\begin{eqnarray}
S_{D4}= -T_{4}\int d^{5}\xi
\sqrt{\{1-H(F_{0\underline{i}})^{2}\}
\{1+H(\partial_{\underline{i}}y)^{2}\}
+H^{2}(F_{0\underline{i}}\partial_{\underline{i}}y)^{2}
-H\dot{y}^{2}}
\end{eqnarray}
(where $H$ is the harmonic function).
The canonical momenta of $y$ and $V_{\underline{i}}$,
are defined respectively as
\begin{eqnarray}
P&=&\frac{T_{4}H\dot{y}}{\sqrt{\{1-H(F_{0\underline{i}})^{2}\}
\{1+H(\partial_{\underline{i}}y)^{2}\}
+H^{2}(F_{0\underline{i}}\partial_{\underline{i}}y)^{2}
-H\dot{y}^{2}}}
\nonumber\\
\Pi_{\underline{i}}&=&\frac{T_{4}H[F_{0\underline{i}}\{
1+H(\partial_{\underline{i}}y)^{2}\}-H\partial_{\underline{i}}y
(F_{0\underline{j}}\partial_{\underline{j}}y)
]}
{\sqrt{\{1-H(F_{0\underline{i}})^{2}\}
\{1+H(\partial_{\underline{i}}y)^{2}\}
+H^{2}(F_{0\underline{i}}\partial_{\underline{i}}y)^{2}
-H\dot{y}^{2}}}\ .
\end{eqnarray}
The Hamiltonian $\bar{H}$ is constructed as
$\bar{H}\equiv\int_{{\cal M}_{4}}d^{4}\xi {\cal H}$
where ${\cal H}$ is the Hamiltonian
density given by
\begin{eqnarray}
{\cal H}=T_{4}
\sqrt{\{1+H(\partial_{\underline{i}}y)^{2}\}
(1+T_{4}^{-2}H^{-1}P^{2})+T_{4}^{-2}H^{-1}(\Pi_{\underline{i}})^{2}
+T_{4}^{-2}(\Pi_{\underline{i}}\partial_{\underline{i}}y)^{2}
}.\label{hami1}
\end{eqnarray}
We note that
$\Pi_{\underline{i}}$ is subject to the constraint
$\partial_{\underline{i}}\Pi_{\underline{i}}=0$.
Substituting the solution (\ref{sol1})
for $\bar{H}$,
we can obtain the energy of the solution $E$
defined on the worldvolume.
We note that for a BPS solution like this case,
it generically happens that
the square root of $\bar{H}$ becomes a perfect square
and that the energy becomes a sum of
the two parts:
the part originating from the D-p-brane and
that from the string.
so, we denote the first part of ${\cal H}$ as ${\cal H}_{1}$
and the second part as ${\cal H}_{2}$.
Concretely, the energy in this case takes the form
\begin{eqnarray}
E=T_{4}\int_{{\cal M}_{4}}d^{4}\xi[1+H(\partial_{\underline{i}}y)^{2}]
\equiv E_{1}+E_{2}.
\end{eqnarray}
The first term $E_{1}$
is the ``energy'' of the D-4-brane itself,
and the second term $E_{2}$
is the energy of the excitation (i.e. the string),
both defined on the worldvolume.
Since we are interested in the second part,
we compute only $E_{2}$ here.
$E_{2}$ is infinite in this case, but
if we regularize it by introducing a small parameter
$\delta$,
we can get the energy for $r\geq \delta$ as
\begin{eqnarray}
E_{2}&=&T_{4}\int d\Omega_{3}
\int^{\infty}_{\delta} r^{3}dr (c_{1}+my(r))
(\partial_{\underline{i}} y(r) )^{2}\nonumber\\
&=&T_{4}Q_{1}(y(\delta)-y_{0}).
\end{eqnarray}
That is, the energy
is (again) proportional to the difference of the coordinate.
Thus, we conclude that
{\it the energy defined on the brane's worldvolume
does not agree with the mass of the string
in the case of D-branes in curved backgrounds.}
We note that this result is rather reasonable, in a sense,
in the case of $\tilde{g}_{00}\ne 1$,
because the energy has the same transformation property
as $\partial_{0}$ under the reparametrization of $\xi^{0}$.
Now, we construct the mass formula.
Since
it should be invariant
under the reparametrization of $\xi^{0}$,
we propose the generalized mass formula $M$ for a string
described as a solution
of a D-p-brane's worldvolume,
as
\begin{eqnarray}
M
=\int_{{\cal M}_{p}}d^{p}\xi \sqrt{-\tilde{g}^{00}}{\cal H}_{2}
\label{stringmass}
\end{eqnarray}
where ${\cal H}_{2}$ is the second part of
the Hamiltonian density defined on the D-p-brane
(originating from the excitation corresponding to a string).
Let us calculate the mass of the string in this case,
based on the formula.
Substituting the solution for (\ref{stringmass}),
we find
\begin{eqnarray}
M(\delta)&=& T_{4}\int_{{\cal M}_{4}}d^{4}\xi
H^{5/4}(\partial_{\underline{i}}y)^{2}=
T_{4}\int d\Omega_{3}
\int^{\infty}_{\delta} r^{3}dr
(c_{1}+my(r))^{5/4}(\partial_{\underline{i}} y )^{2}\nonumber\\
&=&\frac{4T_{4}Q_{1}}{5m}[\{(c_{1}+my_{0})^{2}+\frac{c_{2}}{\delta^{2}}
\}^{5/8}
-(c_{1}+my_{0})^{5/4}].
\end{eqnarray}
On the other hand,
the geodesic distance $l$
from the
D-4-brane (lying at $y_{0}$) to the point parametrized by
$y(\delta)$
is given by
\begin{eqnarray}
l(y(\delta);y_{0})&\equiv& \int^{y(\delta)}_{y_{0}}
\sqrt{g_{yy}}dy
=\frac{4}{5}m^{1/4}[(y(\delta)+\frac{c_{1}}{m})^{5/4}
-(y_{0}+\frac{c_{1}}{m})^{5/4}].\label{geodesic}
\end{eqnarray}
So, we obtain the proportional relation:
\begin{eqnarray}
M=T_{4}Q_{1}\cdot l(y(\delta);y_{0}).\label{proporel}
\end{eqnarray}
Furthermore,
we can show that
the coefficient $ T_{4}Q_{1}$ reproduces
the tension of the fundamental string correctly.
To derive this, we discuss the unit electric
charge for a (1,0) (i.e. a fundamental) string.
First, we review the discussion
about the case of
point charge solutions of the D-p-brane in the flat
spacetime\cite{cal1} (and ref.\cite{sjrey2}).
Let us consider
a triple junction of strings:
a (0,1) string, a (n,0) string and a (n,1) string.
If the string coupling
$g_{s}$ is small,
it holds
\begin{eqnarray}
\frac{\Delta T}{T_{(0,1)}}
=\frac{(g_{s})^{2}n^{2}}{2}\label{tensionrel1}
\end{eqnarray}
where $T_{(p,q)}$ is the tension of a (p,q) string and
$\Delta T$ is
the additional tension $\Delta T \equiv T_{(n,1)}-T_{(0,1)}$.
On the other hand,
the solution of a D-1-brane worldvolume
corresponding to
the above string junction is
given in ref.\cite{sjrey2}.
In the flat background with the ansatz
that $x^{0}=\xi^{0}, x^{1}=\xi^{1}, x^{9}=y(\xi^{1}),
x^{m}=$ constant for m=2,..,8, and $\theta=0$,
the D-1-brane action is written as
\begin{eqnarray}
S_{D1}=-\frac{T_{1}}{g_{s}}\int d^{2}\xi
\sqrt{1+(\partial_{1}y)^{2}-(F_{01})^{2}
-(\partial_{0}y)^{2}}.
\end{eqnarray}
The solution of the D-1-brane's worldvolume as the triple string
junction with an electric charge $q_{1}$ is\cite{sjrey2}
\begin{eqnarray}
y(\xi_{1})=V_{0}(\xi_{1})=
\left\{
\begin{array}{ll}
-q_{1}\xi_{1}\ & {\rm for}\ \xi_{1}>0 \\
0 &{\rm for}\ \xi_{1}<0.
\end{array}
\right. \label{triplesol}
\end{eqnarray}
The energy of the solution can be computed by using the Hamiltonian,
and the additional tension $\Delta T$ is also derived from this
correctly.
By taking into account the bending of the (n,1) string\cite{sjrey2},
it is obtained as
$\Delta T /T_{(0,1)}=(1/2)(F_{01})^{2}$.
Comparing this with (\ref{tensionrel1}),
the ``charge quantization condition''
$F_{01}=q_{1}=g_{s}n$ is deduced for a point charge $q_{1}$ (for
an integer $n$).
By T-dualizing with respect to
the directions of $x^{m}$ for m=2,..,8,
the charge quantization condition
for a electric point charge $q_{1}$ of the D-p-brane's worldvolume
is shown to be
\begin{eqnarray}
q_{1}\equiv \frac{1}{(2\pi)^{p-1}}\int_{S_{p-1}} F_{0r}=g_{s}n.
\end{eqnarray}
Next, we discuss the case
which is related by T-duality to the case of (\ref{sol1}).
Let us suppose
a D-1-brane parallel to a subspace of
the worldvolume of a D-5-brane background,
and that some number $n$ of fundamental strings
are absorbed in the D-1-brane.
(This is also a BPS configuration
since 1/8 spacetime supersymmetry is preserved\cite{sen3}.)
If we consider a D-5-brane background solution,
the string coupling $g_{s}$
becomes a {\it local} function on
the transverse coordinates $y^{a}$ ($g_{s}=e^{\phi}=H^{-1/2}$
where $H$ is the harmonic function on $y^{a}$), and
so is the tention $T_{(n,1)}$.
If the ``test'' D-1-brane is put near the D-5-brane,
the string coupling $g_{s}$ is considered to be
sufficiently small
around the D-1-brane.
So, the equation
(\ref{tensionrel1}) with $g_{s}=H^{-1/2}$
holds on the basis of the same discussion.
On the other hand,
suppose that a electric point charge $q_{1}'$ is added to
the D-1-brane's worldvolume with an excitation of a scalar field
$y^{9}$.
Then,
the D-1-brane action is
\begin{eqnarray}
S_{D1
=-T_{1}\int d^{2}\xi \sqrt{1+H(\partial_{1}y^{9})^{2}-H(F_{01})^{2}
-H(\partial_{0}y^{9})^{2}}.\label{d1d51}
\end{eqnarray}
If we assume the existence of the correponding point charge
solution,
the additional tension due to the field strength is
derived in the same way, as
\begin{eqnarray}
\frac{\Delta T}{T_{(0,1)}}=
\frac{1}{2(T_{1})^{2}}\cdot H^{-1}(\Pi_{1})^{2}
=\frac{1}{2}\cdot H (F_{01})^{2}.\label{tensionrel5}
\end{eqnarray}
So, comparing this with (\ref{tensionrel1})
we have
the charge quantization condition
$H^{1/2}F_{01}=ng_{s}$ for an integer $n$.
By using the
the ``electric induction'' $D_{01}=-\frac{1}{T_{1}}\cdot
\frac{\delta {\cal L}_{D1}}{\delta F_{01}}$ ($=HF_{01}$
in this case)
it can be rewritten in a more generic form as
\begin{eqnarray}
D_{01}=n.\label{chargeqcond3}
\end{eqnarray}
The higher dimensional D-brane cases are related to
(\ref{chargeqcond3}) by T-dualities.\footnote{
We note that the background fields should also be transformed
by T-dualities simultaneously.}
So, we have
\begin{eqnarray}
\frac{1}{(2\pi)^{p-1}}\int_{S_{p-1}}D_{0r}=n.\label{chargeqcond4}
\end{eqnarray}
This is the ``generalized''
charge quantization condition
for the point charge of the D-p-brane parallel
to the worldvolume of
a D-(p+4)-brane background.
Let us return to
to the case of (\ref{sol1}).
the left hand side of (\ref{chargeqcond4}) with $p=4$
is equivalent to
the charge $Q_{1}$ of (\ref{sol1}) multiplied by
$1/(2\pi)^{3}$.
So,
the unit charge $Q_{1}$ for a (1,0) string in this case is
$Q_{1}=(2\pi)^{3}$.
That is,
the unit
charge
is the same as the one in the case of the flat background,
(This result is consistent from physical point of view,
since
the unit charge $Q_{1}$
is considered to be
independent of the background.)
Thus, the proportional coefficient of (\ref{proporel}) for
a (1,0) string is obtained as
$T_{4}Q_{1}=1/2\pi$, which is exactly
the tension of the string.
So, the quantity $M$ defined in (\ref{stringmass})
certainly gives the mass of the string correctly !
As another case,
we consider the case of a test D-8-brane parallel to the background
D-8-brane.
The worldvolume action is given in (\ref{Dpaction}) for p=8.
In this case, we consider the ansatz
\begin{eqnarray}
\left\{
\begin{array}{ll}
x^{i}=\xi^{i}
& (i=0,1,\cdots,8 )\nonumber\\
y=y(r)(>0)& \nonumber\\
V_{0}= V_{0}(r),& V_{\underline{i}}=0.
\end{array}
\right.\label{pointsolansatz2}
\end{eqnarray}
where $r$ is defined as
$r\equiv\sqrt{\sum_{\underline{i}=1}^{8}
(\xi^{\underline{i}})^{2}}$.
Then, combined with $\theta=0$,
only the term
including $\tilde{C}^{(9)}$ in $S_{D8}^{WZ}$
does contribute to the equations of motion
($\tilde{C}^{(9)}_{01..8}=H^{\epsilon}$).
The expression of the induced worldvolume
metric $\tilde{g}_{ij}$ is the same as (\ref{wvmetric1})
except for the range of the indices
(in this case $\underline{i},\underline{j}=1,2,..,8$).
Setting the same condition as (\ref{yveual1})
makes the determinant of $(\tilde{g}_{ij}+{\cal F}_{ij})$
simple, such as
${\rm det}(\tilde{g}_{ij}+{\cal F}_{ij})=H^{9\epsilon /2}$.
Then,
we find the equations of motion
\begin{eqnarray}
\sum_{\underline{i}}
\partial_{\underline{i}}(H^{-2\epsilon-2}\partial_{\underline{i}}y)
&=&0\\
\sum_{\underline{i}}
\partial_{\underline{i}}(H^{-\epsilon-1}\partial_{\underline{i}}y)
&=&0.
\end{eqnarray}
So, these two equations are again compatible only if
$\epsilon=-1$, and
the equations to solve become a single equation
\begin{eqnarray}
\sum_{\underline{i}}(\partial_{\underline{i}})^{2}y=0.\label{eom'3}
\end{eqnarray}
We note that unlike the D-4-brane case,
the harmonic function $H$ does not appear in
(\ref{eom'3}).
So, choosing the same boundary condition as (\ref{boundary1}),
the solution is obtained easily as
\begin{eqnarray}
y&=&y_{0}+\frac{c_{2}'}{r^{6}}\nonumber\\
V_{0}&=&\frac{c_{2}'}{r^{6}}\label{sol2}
\end{eqnarray}
where $c_{2}'$ is a constant proportional to the electric
charge of the solution.
By using the definition of the charge
similar to (\ref{charge0}),
we have the electric charge $Q_{1}'$:
\begin{eqnarray}
Q_{1}'\equiv -\frac{1}{T_{8}}
\int_{S_{7}}\star(\frac{\delta {\cal L}_{D4}}{\delta F_{ij}})
=\int_{S_{7}}\star(dV)
=6c_{2}'\Omega_{7}.
\end{eqnarray}
We note that the form
of this point charge
solution (\ref{sol2})
is completely the same as
that
of the D-8-brane worldvolume
in the {\it flat} background.
Here,
We derive the preserved supersymmetry of the solution
(\ref{sol2}) in the same way.
1/4 supersymmetry is expected to be preserved.
The superalgebra in a D-8 brane background via a D-8-brane probe
is\cite{sato2}
\begin{eqnarray}
\{ Q_{\alpha}^{+}, Q_{\beta}^{+}\}=
2\int_{{\cal M}_{8}}d^{8}\xi\ \Pi_{\mu}
({\cal C}\Gamma^{\mu})_{\alpha\beta}
+2\int_{{\cal M}_{8}}d^{8}\xi{\cal P}^{(0)\underline{i}}
\partial_{\underline{i}}y
({\cal C}\Gamma_{y}\Gamma_{11})_{\alpha\beta}\nonumber\\
\ \ \ \ \ \ \ \ \ \ \ \ \nonumber\\
+\frac{2T_{8}}{5!}\int_{{\cal M}_{8}}
H^{5/4}dx^{\mu_{1}}..dx^{\mu_{5}}dydV
({\cal C}\Gamma_{\mu_{1}..\mu_{5}y})_{\alpha\beta}
+\frac{2T_{8}}{4!}\int_{{\cal M}_{8}}
H^{5/4}dx^{\mu_{1}}..dx^{\mu_{4}}(dV)^{2}
({\cal C}\Gamma_{\mu_{1}..\mu_{4}}\Gamma_{11})_{\alpha
\beta} \nonumber\\
+ 2T_{8}\int_{{\cal M}_{8}}H^{5/4}dx^{\mu}dy(dV)^{3}
({\cal C}\Gamma_{\mu y})_{\alpha\beta}
+2T_{8}\int_{{\cal M}_{8}}
H^{5/4}(dV)^{4}({\cal C}\Gamma_{11})_{\alpha\beta}
+{\cal O}(\theta^{2}).
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\label{spalgd8}
\end{eqnarray}
The momentum $ \Pi_{\mu} $
includes the following two terms:
\begin{eqnarray}
\Pi_{\mu}=\Pi_{\mu}^{(0)} + \frac{T_{8}}{8!}H^{-2}my
\epsilon^{0i_{1}..i_{8}}\partial_{i_{1}}x^{\nu_{1}}..
\partial_{i_{8}}x^{\nu_{8}}\epsilon_{\mu\nu_{1}..\nu_{8}y}
\end{eqnarray}
where $\Pi_{\mu}^{(0)}$ is the contribution of $S^{BI}_{D8}$
and the second term is that of $S^{WZ}_{D8}$
(where $\epsilon^{01..8}=1$).
Substituting the solution for the right hand side of (\ref{spalgd8}),
the superalgebra can be written as
\begin{eqnarray}
\{ Q_{\alpha}^{+}, Q_{\beta}^{+}\}&=&
4T_{8}\int_{{\cal M}_{8}}d^{8}\xi\
H^{1/4}(\partial_{\underline{i}}y)^{2}
(\frac{1+{\cal C}\Gamma_{\hat{y}}\Gamma_{11}}{2})_{\alpha\beta}
\label{spalgd8'}.
\end{eqnarray}
By the same discussion as
done in the D-4-brane case,
it is shown
that 1/4 supersymmetry is preserved in this configuration,
which is consistent with the spacetime interpretation.
So, we interpret the solution (\ref{sol2}) as a fundamental string
again.
Applying the mass formula (\ref{stringmass})
to this case,
we obtain the result:
\begin{eqnarray}
M &\equiv& \int_{{\cal M}_{8}}\sqrt{-\tilde{g}^{00}}{\cal H}_{2}
=T_{8}\int_{{\cal M}_{8}}d^{8}\xi H^{1/4}
(\partial_{\underline{i}} y )^{2}\nonumber\\
&=&T_{8}\int d\Omega_{7}
\int^{\infty}_{\delta} r^{7}dr(c_{1}+my(r))^{1/4}
(\partial_{\underline{i}} y )^{2}\nonumber\\
&=& T_{8}Q_{1}'\cdot l(y(\delta);y_{0}).
\end{eqnarray}
Based on the same discussion as that done
in the D-4-brane case, the unit electric charge
$Q_{1}'$ for a (1,0) string ending on a D-8-brane can also be
derived as
$Q_{1}'= (2\pi)^{7}$.
So,
the tension of the string is reproduced correctly again,
and
(\ref{stringmass}) also gives
the mass of the string for the solution (\ref{sol2}) correctly.
Therefore,
we conclude that (\ref{sol2}) is the correctly
generalized mass formula,
which holds when the background of the D-brane is curved.
We note that
the energy defined on the worldvolume again
gives the difference of the coordinate $y$ multiplied by
$T_{8}Q_{1}'$.
That is, this case is another example that
the worldvolume energy does not agree with the mass of the
string.
\section{Summary and discussions}
In summary,
we have proposed (\ref{sol2}) as the generalized mass formula
which holds when the background of the D-brane is curved,
and have
shown explicitly using the two examples
that the formula certainly gives
the mass of a fundamental string
described as a BPS solution of a D-brane's worldvolume.
In addition, based on the obtained formula,
we can see that
the mass of the string agrees with the worldvolume energy
only in the cases $\tilde{g}_{00}=-1$
(where
$\tilde{g}_{ij}$ is the induced worldvolume metric).
which include
the case discussed in ref.\cite{cal1}.
Here, we
discuss
the consistency of the mass formula from another point of view,
especially
focusing on the factor $\sqrt{-\tilde{g}^{00}}$.
Suppose we consider the D-4-brane embedded parallel to the
D-8-brane background (\ref{d8sol}) {\it with no excitation of
worldvolume fields} (i.e. $x^{i}=\xi^{i}$ for i=0,..,4 and
$x^{5},..,x^{8},y$ : constants).
Then, on the analogy of the mass formula of a (p,q) string
given by Sen
in ref.\cite{sen1},
the target-space mass $m_{D4}$ of the D-4-brane
should be proportional to its spatial volume element
{\it measured by the geodesic distances} in the spacetime.
So, it should be given by
\begin{eqnarray}
m_{D4}=\int_{{\cal M}_{4}}d^{4}\xi T_{4} e^{-\phi}
\sqrt{{\rm det}\tilde{g}_{\underline{i}\underline{j}}'}
\end{eqnarray}
where $\tilde{g}_{\underline{i}\underline{j}}'$
is the induced {\it world-space} metric
of the D-4-brane.
In fact, the mass $m_{D4}$
obtained in this way can be shown to agree with
the quantity $M_{D4}$ defined as
\begin{eqnarray}
M_{D4}
=\int_{{\cal M}_{4}}d^{4}\xi \sqrt{-\tilde{g}^{00}}{\cal H}_{1}
(=\int_{{\cal M}_{4}}d^{4}\xi H^{1/4}\cdot 1)
\end{eqnarray}
where ${\cal H}_{1}$ is the first part (i.e.
originating from the D-4-brane) of
the Hamiltonian density of the solution (\ref{sol1})
defined on the D-4-brane.
This means that the information of the D-4-brane mass
can also be extracted from the solution (\ref{sol1})
by integrating the Hamiltonian density
multiplied by the factor $\sqrt{-\tilde{g}^{00}}$
with respect to world-space coordinates.
Thus, the factor $\sqrt{-\tilde{g}^{00}}$
arising in the formula is consistent in this sense, too.
Finally, let us discuss
the worldvolume interpretation of the mass formula (\ref{stringmass}).
If we define a worldvolume proper time $\tau$ as
$d\tau\equiv \sqrt{-\tilde{g}_{00}}d\xi^{0}$,
the Hamiltonian density multiplied by $\sqrt{-\tilde{g}^{00}}$
might be regarded as the energy density defined
with respect to $\tau$.
So, we might say that from worldvolume point
of view,
the mass of the string is interpreted as
``the energy defined
with respect to the worldvolume proper time''.
\par\bigskip\noindent
{\Large\bf Acknowledgement}
\par\bigskip\noindent
I would like to thank Taro Tani for useful discussions
and encouragement.
I would also like to thank Y. Imamura,
Tsunehide Kuroki and Shinya Tamura
for useful discussions
in computing the energy of the solution.
\par\bigskip\noindent
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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Bash completion is supported and can be activated as follows:
```bash
source ./cmd/swagger/completion/swagger.bash-completion
```
Note that this does require you already setup bash completion,
which can be done in 2 simple steps:
1) install `bash-completion` using your favorite package manager;
2) run `source /etc/bash_completion` in bash;
#### Zsh Completion
Zsh completion is supported and can be copied/soft-linked from:
```
./cmd/swagger/completion/swagger.zsh-completion
```
In case you're new to adding auto-completion to zsh completion,
here is how you could enable swagger's zsh completion step by step:
1) create a folder used to store your completions (eg. `$HOME/.zsh/completion`);
2) append the following to your `$HOME/.zshrc` file:
```zsh
# add auto-completion directory to zsh's fpath
fpath=($HOME/.zsh/completion $fpath)
# compsys initialization
autoload -U compinit
compinit
```
3) copy/soft-link `./cmd/swagger/completion/swagger.zsh-completion` to `$HOME/.zsh/completion/_swagger`;
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,284
|
\section{Introduction}
\label{sec:intro}
The LHC is a wonderful QCD machine. The large proton energy allows for probing
predictions of QCD at unprecedented energy scales in accelerator physics.
Due to the high luminosity of the machine, and large QCD cross sections,
especially for jet production, many analyses are limited by systematic
uncertainties right from the start.
This puts high demands on the understanding of the detectors, and also calls
for high performance jet algorithms to cope with the ever increasing
complications due to complex final states, and the occurrence of more than one
proton-proton interaction per bunch crossing (pileup) that potentially
deteriorates the jet energy resolution.
The results presented are all based on the LHC running at a proton-proton
centre-of-mass energy of $\rts=7$~\TeV.
For many of the topics discussed, results from the ATLAS and CMS experiments
exist, however, due to the limited space, for each topic only a single result
is shown, concentrating on published results, i.e.~additional preliminary
measurements are not included.
The paper is organised as follows: jet production for increasing jet
multiplicities is discussed in Section~\ref{sec:jetpro}. Adding a further hard
scale, heavy gauge boson plus jet production is detailed in
Section~\ref{sec:wzboson}.
A number of issues related to the production of top quarks are highlighted
in Section~\ref{sec:ttbar}.
Finally, the summary and conclusions are given in Section~\ref{sec:concl}.
\section{Jet production}
\label{sec:jetpro}
Due to the large proton energies, and the correspondingly large phase space for
jet production, very complex final states, with large jet transverse momenta
(\pt) occur.
An example of this, a six jet event observed in the ATLAS detector, is shown in
Figure~\ref{fig:sixjet}.
To properly reconstruct those final states, a high performance jet algorithm is
needed. The present choice of the LHC experiments for this is the anti-\kt\
algorithm~\cite{CAC-0801,CAC-0802}, which is a sequential clustering algorithm
that uses 1/\ptq\ as the weighting factor for the scaled distance, and the
$R$ parameter to define the jet resolution.
This algorithm exhibits a number of good features~\cite{CAC-0801}, which, on
top of its infra-red safety, makes it superior to other possible choices.
\begin{figure*}[tbph!]
\centering
\includegraphics[width=0.98\textwidth]{nisius_fig01}
\caption{Event display of a six-jet event in the ATLAS
detector~\protect\cite{ATL-2011-030}. Shown are: a view along, and a view
parallel to the beam axis, and the angular distribution of transverse energy
in pseudo rapidity ($\eta$) and azimuthal angle ($\Phi$). The measured jet
transverse momenta range from $\pt=84$~\GeV to $\pt=203$~\GeV.
\label{fig:sixjet}}
\end{figure*}
These features are: round and rigid jet shapes that lead to a clear and stable
spatial definition of jets; an almost \pt\ independent jet area, ensuring an
almost constant pileup correction as a function of \pt; and finally, very small
back reaction, i.e.~re-assignments of particles from the hard interaction to
jets for different pileup contributions, which guarantees stable definitions of
the jets stemming from the hard QCD process.
In search for optimised uncertainties of observables the $R$ parameter is
varied in the comparisons to the theoretical predictions, with typical values
in the range $R=0.4-0.7$. In this respect, jets with smaller R values are found
to be less dependent on pileup, and those with larger R values to be less
dependent on scale changes in the theoretical predictions~\cite{ATL-2011-030}.
For the ATLAS and CMS analyses, the observed jet distributions are corrected
for detector effects, and then, at the resulting stable particle level,
compared to the theoretical predictions which come in a large number of
flavours.
These comprise: leading-order (LO) $2\to2$ Matrix Elements~(ME) plus subsequent
Parton Shower~(PS) and underlying event~(UE) implemented in the programs
\Pythia~\cite{SJO-0601} and \Herwig~\cite{COR-0001} together with
\Jimmy~\cite{BUT-9601} (those will be referred to as LO $2\to2$ predictions);
LO $2\to n$ MEs provided by the \Sherpa~\cite{GLE-0901},
{\sc MadGraph}\xspace~\cite{STE-9401,ALW-0701} and \Alpgen~\cite{MAN-0301} programs, with
subsequent internal (\Sherpa) or external, i.e.~by other packages provided,
({\sc MadGraph}\xspace, \Alpgen) PS and UE (referred to as LO $2\to n$ predictions);
NLO ME calculations for $n\le 3$ outgoing partons featured by the
\Mcfm~\cite{CAM-0301} and \Nlojet~\cite{NAG-0301} programs;
NLO ME calculations matched to PS that are either provided by the
\Mcatnlo~\cite{FRI-0201,FRI-0301} together with the \Herwig\ software packages,
or by the \Powheg~\cite{FRI-0701} generator interfaced to either the
\Pythia\ or \Herwig\ programs;
and finally, an all order resummed calculation for wide angle emissions
implemented in the \Hej~\cite{AND-1001,AND-1101} program.
In addition, for the soft parts of the event simulation, a number of different
tunings of parameters that control those parts in the general purpose Monte
Carlo programs are in use, see for example~\cite{SKA-1001}.
This makes up for a very large variety of predictions for comparisons, a
number of them are shown below.
\begin{figure}[tbp!]
\centering
\includegraphics[width=0.48\textwidth]{nisius_fig02}
\caption{Inclusive jet production~\protect\cite{CMS-1104}. Shown is the double
differential inclusive jet cross section as a function of \pt, and in bins
of the absolute rapidity.
\label{fig:onejet}}
\end{figure}
In Figure~\ref{fig:onejet} the double differential inclusive jet
cross section~\cite{CMS-1104} as a function of \pt, for the range $18~\GeV< \pt
<1100~\GeV$, and in bins of the absolute value of the rapidity \arap, is shown.
Already now, the reach in \pt\ scales that are probed by the LHC experiments
extend those probed by the Tevatron experiments from about 700~\GeV to about
1100~\GeV.
The data are compared to the NLO prediction from the \Nlojet\ program, with
non-perturbative corrections estimated using the \Pythia\ and \Herwig\ models.
For CMS, jets are reconstructed using the so-called particle flow algorithm.
This algorithm combines information from a list of objects: leptons, photons,
and charged and neutral hadrons, into jets. For each of these objects, this
information is obtained from various components of the detector.
As for most of the inclusive jet cross section determinations, the experimental
uncertainty is dominated by the Jet Energy Scale (JES) uncertainty.
The description of the data by the NLO prediction over a large range in \pt\ is
fair. However, the prediction is systematically higher than the data,
especially so at large values of \arap, see Figure~\ref{fig:onejetrat}.
At the highest \pt\ the theoretical uncertainty (shown as solid lines above and
below unity) is dominated by the one from the Parton Density Functions (PDFs)
of the proton, parametrised as a function of the partons momentum \xk\ from the
proton, and consequently, the data start to constrain the PDFs, see
Figure~\ref{fig:onejetrat}.
\begin{figure}[tbp!]
\centering
\includegraphics[width=0.47\textwidth]{nisius_fig03}
\caption{Inclusive jet production~\protect\cite{CMS-1104}. Shown is the ratio of
the measured cross section and the NLO prediction as a function of \pt, and in
bins of the absolute rapidity.
\label{fig:onejetrat}}
\end{figure}
A similar conclusion holds for the double differential inclusive dijet
production cross section shown in Figure~\ref{fig:twojet} as a function of
\Mjj, the invariant mass of the two highest \pt\ jets in the event, and for
various bins in \arapm, the maximum of the absolute rapidity of these two jets.
In this analysis, the dijet production cross section probes the PDFs in the
range $0.0008<\xo\xt<0.25$.
The observable \Mjj\ can be reconstructed with a 7$\%$ (3$\%$) resolution at
$\Mjj=0.2$~\TeV (3~\TeV).
The low-mass region is mainly sensitive to the UE, whereas the high-mass region
potentially constrains the high-x region of the PDFs, see
Figure~\ref{fig:twojetrat}.
At present, the component of the cross section uncertainty caused by the JES
uncertainty amounts to 15$\%$ (60$\%$) at $\Mjj=0.2$~\TeV (3~\TeV).
\begin{figure}[tbp!]
\centering
\includegraphics[width=0.47\textwidth]{nisius_fig04}
\caption{Inclusive dijet production as a function of the dijet invariant mass
\Mjj, and in bins of maximum of the absolute rapidity of the two jets
\arapm~\protect\cite{CMS-1105}. The horizontal bars indicate the bin width,
the vertical bars the statistical uncertainty.
\label{fig:twojet}}
\end{figure}
The cross section uncertainty due to the non-perturbative corrections mentioned
above is comparably small, and amounts to 15$\%$ (2$\%$) at the same invariant
masses.
Finally, the present PDF uncertainty of the theoretical prediction is 5$\%$
(30$\%$), again at the same invariant masses.
Consequently, still some improvement in the JES uncertainty is needed to get a
precise constraint for the high-x region of the PDFs.
The next observable discussed is \RtW, the ratio of the 3-jet to 2-jet
cross sections.
For this analysis jets within $\arap=2.5$ and for $\pt>50$~\GeV\ are used.
Being a ratio, \RtW, shown in Figure~\ref{fig:ratjet} as a function of \HT, the
scalar sum of the \pt\ of all jets defined above, profits from cancellations of
many systematic uncertainties.
These are most notably the uncertainties from the imperfect knowledge of the
jet energy scale and the one from the jet selection efficiency.
\begin{figure}[tbp!]
\centering
\includegraphics[width=0.48\textwidth]{nisius_fig05}
\caption{Inclusive dijet production as a function of the dijet invariant mass
\Mjj, and in bins of maximum of the absolute rapidity of the two jets
\arapm~\protect\cite{CMS-1105}. Shown is the ratio of the observed and
predicted cross sections. The data are displayed as in
Figure~\ref{fig:twojet}. The uncertainties of the prediction are shown as
solid lines above and below unity.
\label{fig:twojetrat}}
\end{figure}
In this analysis the efficiency for 2-jet events is 100$\%$, whereas the
efficiency for 3-jet events increases from 72$\%$ at $\HT=0.2$~\TeV to 100$\%$
at $\HT=0.4$~\TeV.
The present resolution in \pt\ translates into a resolution in \HT\ of about
6$\%$ (3.5$\%$) at $\HT=0.05$~\TeV (1~\TeV).
The size of the correction to the particle level is small and only amounts to
about 4$\%$ (2$\%$) for the two regions $\HT<0.5$~\TeV~($>0.5$~\TeV).
\begin{figure}[tbp!]
\centering
\includegraphics[width=0.47\textwidth]{nisius_fig06}
\caption{The ratio of the 3-jet to 2-jet cross sections, \RtW, as a function of
the scalar transverse momentum sum \HT~\protect\cite{CMS-1106}. The vertical
bars denote the statistical uncertainties, the shaded area indicates the
systematic uncertainty. The data are compared to a number of predictions
explained in the text.
\label{fig:ratjet}}
\end{figure}
In this ratio, the total experimental uncertainty amounts to (4-10)$\%$ and is
mainly limited by the knowledge of the \pt\ dependence in the Monte Carlo
predictions entering the correction procedure. Consequently, the \RtW\ ratio is
considerably more precise than the inclusive jet cross sections discussed
above.
The ratio reaches a plateau of about 0.8, where the actual value depends on the
choice of jet algorithm and the jet selection criteria like the chosen rapidity
range.
The plateau region can be nicely described by a number of predictions, see
Figure~\ref{fig:ratjetrat}. However, the steep rise for low values of \HT,
originating from the increasing phase space for the emission of the third jet,
is only adequately followed by the {\sc MadGraph}\xspace\ prediction, i.e.~all other
predictions are too steep.
\begin{figure}[tbp!]
\centering
\includegraphics[width=0.48\textwidth]{nisius_fig07}
\caption{The ratio of the 3-jet to 2-jet cross sections as a function of the
scalar transverse momentum sum \HT~\protect\cite{CMS-1106}. Shown is the
ratio of the predicted and observed \RtW. The shaded area indicates the total
experimental uncertainty.
\label{fig:ratjetrat}}
\end{figure}
Finally, the cross section for multijet production for a minimum number of
jets \Njets\ with $\Njets\ge2, \ldots 6$ is shown in
Figure~\ref{fig:muljetincl}.
Jets within $\arap=2.8$ are used, where in a \pt\ ordered list the first jet is
required to fulfill $\pt>80$~\GeV, and all others to fulfill $\pt>60$~\GeV.
The present JES uncertainty is asymmetric. It amounts to 5$\%$ (2.5$\%$) at
$\pt=0.06$~\TeV (1~\TeV) and is larger than -3$\%$ everywhere.
Within uncertainties, the shape of the inclusive jet multiplicity can be
accounted for by all predictions~\cite{ATL-2011-030}, which however do show a
slightly steeper trend than the data.
There are very significant differences in the absolute predictions that result
in different overall scaling factors, ranging from 0.65 up to 1.22, which are
applied to individually normalise the predictions to the $\Njets\ge2$ bin, see
Figure~\ref{fig:muljetincl}.
The smallest scaling with only +6$\%$ is needed for the LO \Sherpa\ $2\to n$
prediction, the largest with -35$\%$ for the LO \Pythia\ $2\to 2$ prediction.
\begin{figure}[tbp!]
\centering
\includegraphics[width=0.50\textwidth]{nisius_fig08}
\caption{Inclusive multijet cross section as a function of the jet
multiplicity~\protect\cite{ATL-2011-030}. The darker shaded band corresponds
to the systematic uncertainty excluding the contribution from the
luminosity. The lighter shaded band corresponds to the systematic uncertainty
on the shape of the measured distribution. The theoretical predictions are
individually normalised to the $\Njets\ge2$ cross section.
\label{fig:muljetincl}}
\end{figure}
In addition to the multiplicity also the differential cross sections for
multijet production as a function of \HT, and for different jet
multiplicities is investigated.
In this analysis, the systematic uncertainty is about (10-20)$\%$ across \pt,
and increases to about 30$\%$ for the fourth leading jet differential
cross section.
The results for $\Njets\ge3$ and $\Njets\ge4$ are shown in
Figure~\ref{fig:muljetpttj} and Figure~\ref{fig:muljetptvj}.
\begin{figure}[tbp!]
\centering
\includegraphics[width=0.50\textwidth]{nisius_fig09}
\caption{Differential multijet cross section as a function of
\HT~\protect\cite{ATL-2011-030}, for $\Njets\ge3$. The predictions are
normalised as in Figure~\protect\ref{fig:muljetincl}.
\label{fig:muljetpttj}}
\end{figure}
Again, within uncertainties the data in both multiplicity bins can be described
by all shown predictions.
In these distributions the LO $2\to 2$ prediction from the \Pythia\ program is
steeper than the LO $2\to n$ predictions from either the \Alpgen\ or the
\Sherpa\ package, a trend that can generally be observed when comparing LO
$2\to 2$ to LO $2\to n$ predictions.
\begin{figure}[tbp!]
\centering
\includegraphics[width=0.50\textwidth]{nisius_fig10}
\caption{Differential multijet cross section as a function of
\HT~\protect\cite{ATL-2011-030}. Same as Figure~\protect\ref{fig:muljetpttj},
but for or $\Njets\ge4$.
\label{fig:muljetptvj}}
\end{figure}
The shape differences in the two predictions based on the \Alpgen\ software,
but using different programs and tunings for the soft part of the simulation
implemented in \Herwig\ or \Pythia\ are very small, demonstrating a low
sensitivity of the shape of these differential cross sections to soft effects.
In addition to inclusive jet production, also more detailed investigations in
quest for identifying BFKL signatures are performed~\cite{ATL-2011-029}.
In this analysis, starting from a dijet system defining a rapidity gap, the
properties of that gap are investigated.
The selection requirements for jets obtained with $R=0.6$ are: $\arap<4.4$,
$\pt>20$~\GeV, and an average transverse momentum of the two jets of the dijet
system of $\ptb>50$~\GeV.
\begin{figure}[tbp!]
\centering
\includegraphics[width=0.49\textwidth]{nisius_fig11}
\caption{The measured gap fraction as a function of \Dy\ for a bin in
\ptb~\protect\cite{ATL-2011-029}. The vertical bars represent the statistical
uncertainty, the band indicates the systematic uncertainty. The data are
compared to a number of predictions detailed in the text.
\label{fig:gapfrady}}
\end{figure}
\begin{figure}[tbp!]
\centering
\includegraphics[width=0.49\textwidth]{nisius_fig12}
\caption{The measured gap fraction as a function of \ptb\ for a bin in
\Dy~\protect\cite{ATL-2011-029}. See Figure~\ref{fig:gapfrady} for details.
\label{fig:gapfrapt}}
\end{figure}
From these jets, the dijet system is either formed from the two highest
\pt\ jets (leading \pt\ dijet selection), which typically have rather similar
\pt, or from the two jets with the largest rapidity gap, for which typically
their invariant mass is much larger than their \ptb.
The gap properties investigated are either the gap fraction, i.e.~the fraction
of events that do not contain any jet above a certain \pt\ threshold, chosen to
be $\qn=20$~\GeV, i.e.~this scale satisfies $\qn\gg\Lambda$, or the average jet
multiplicity of exactly those additional jets.
These observables probe wide angle soft gluon radiation for $\qn\ll \ptb$, BFKL
dynamics for large \Dy, and finally colour singlet exchange if both conditions
are fulfilled at the same time.
As an example, for the leading \pt\ dijet selection the corrected gap fraction
is shown in Figure~\ref{fig:gapfrady} as a function of the rapidity gap
\Dy\ for a given bin in \ptb, and in Figure~\ref{fig:gapfrapt} as a function of
\ptb\ for a given bin in \Dy. The corrections to the stable particle level
amount to about (2-4)$\%$.
The JES uncertainty is about (2-5)$\%$ for the central region and 13$\%$ for
the forward region, defined as $\arap>3.2$. The resulting uncertainty on the
gap fraction is about 3$\%$~(7$\%$) for the same rapidity ranges.
The comparison to the theoretical predictions reveals that the LO $2\to 2$
predictions from \Pythia\ and \Herwig\ follow the data, except for large values
of \Dy. In contrast, the \Alpgen\ $2\to n$ model predicts too many jets,
i.e.~a too small gap fraction, for both the \ptb\ and the \Dy\ dependence,
except for low scales.
In Figure~\ref{fig:gapfraho} the ratio of the predicted and the observed gap
fractions for various higher order predictions is displayed as a function of
\Dy, and for a number of narrow ranges in \ptb.
The NLO prediction from the \Powheg\ model generally has too much jet activity,
with the \Pythia\ fragmentation being closer to the data than the one from
\Herwig.
The partially large spread between \Powheg+\,\Pythia\ and
\Powheg+\,\Herwig\ indicates regions of phase space with sizeable contributions
from soft effects to this observable.
The difference of the \Powheg\ model to the data increases for increasing \Dy.
This can be attributed to the fact that the NLO plus PS prediction is lacking
the full QCD ME contributions that become important as \Dy\ increases.
The prediction from the \Hej\ program are shown at the parton level, i.e. they
will in addition be subject to soft effects that may be sizeable, see the
\Powheg\ discussion above.
This prediction, apart from the region of low \Dy, has too few jets in the gap,
especially so at large values of \Dy, and at large \ptb/\qn for all values of
\Dy.
Again, this deviation from the data is expected, since the theoretical QCD
prediction implemented in the \Hej\ program is only a valid approximation in
the limit where all jets have similar \pt.
\begin{figure}[tbp!]
\centering
\includegraphics[width=0.49\textwidth]{nisius_fig13}
\caption{The ratio of the predicted gap fraction to the one observed in the data
as a function of \Dy, and for various bins in
\ptb~\protect\cite{ATL-2011-029}. The data are displayed as in
Figure~\protect\ref{fig:gapfrady}. The band not centred around unity
represents the theoretical uncertainty in the \Hej\ calculation. For the two
\Powheg\ predictions only the central result is shown.
\label{fig:gapfraho}}
\end{figure}
\section{W/Z-Boson plus jet production}
\label{sec:wzboson}
Jet production, together with an additional hard scale provided by the mass of
a heavy boson, is investigated in the \WB\ plus 1-jet and \ZB\ plus 1-jet
production processes, using the leptonic decays of the heavy bosons
$W\to\ell\nu_\ell$ and $Z\to\ell^+\ell^-$ with $\ell=$ electron or
muon~\cite{ATL-2011-041}.
The driving idea in this analysis is to construct an observable with very small
experimental uncertainty to perform a precise QCD test. Therefore, firstly, not
individual cross sections, but the ratio of the \WB\ plus 1-jet and \ZB\ plus
1-jet production cross sections is utilised, and secondly, this ratio is
investigated as a function of the \pt\ threshold, and not in bins of \pt.
In this analysis, jets have to fulfill $\pt>30$~\GeV and $\aeta<2.8$, and
events with additional jets with $\pt>30$~\GeV are vetoed.
The background contribution is small, below 5$\%$ in all channels, but for the
QCD multijet background in the $W\to\mathrm{e}\nu_\mathrm{e}$ channel which is
19$\%$.
All background estimates are taken from Standard Model Monte Carlo samples,
except for the QCD multijet background. This is because this background can at
present not be reliably modelled in the Monte Carlo programs, and is therefore,
as for most analyses, taken from sideband regions in the data.
\begin{figure}[tbp!]
\centering
\includegraphics[width=0.48\textwidth]{nisius_fig14}
\caption{The ratio of \WB\ plus 1-jet to \ZB\ plus 1-jet production in the muon
decay channel as a function of the jet
\pt\ threshold~\protect\cite{ATL-2011-041}. The data are shown as points at
the respective threshold, together with their statistical uncertainty
(vertical bars), their systematic uncertainty (inner band), and their
total uncertainty (outer band). The dashed lines indicate the theoretical
uncertainty on the MCFM prediction dominated by the PDF and scale
uncertainties.
\label{fig:wbopromu}}
\end{figure}
The result in the muon decay channel, corrected to the stable particle level
with \Pythia, is shown in Figure~\ref{fig:wbopromu}.
As expected, the ratio decreases with increasing jet \pt\ threshold, because
the effective scale of the interaction becomes large with respect to the
difference in the heavy boson masses.
The systematic uncertainty (inner band) on the ratio is about (5-10)$\%$, and
in itself it has a large statistical component.
The largest contribution to the systematic uncertainty is due to imperfections
in the heavy boson reconstruction, including effects from the lepton trigger,
reconstruction and identification efficiencies, and scale uncertainties for the
lepton and missing transverse energy measurements.
For transverse momenta larger than about 50~\GeV, the total uncertainty is
dominated by the statistical uncertainty (vertical bars).
%
This uncertainty will soon decrease given the large statistics of the completed
2011 LHC run, which amounts to about 5/fb.
\begin{figure}[tbp!]
\centering
\includegraphics[width=0.48\textwidth]{nisius_fig15}
\caption{Same as Figure~\ref{fig:wbopromu} but for both lepton decay channels
combined~\protect\cite{ATL-2011-041}.
\label{fig:wbopro}}
\end{figure}
The measurements in both lepton decay channels are consistent and the combined
result, evaluated for a common phase space region for the leptons, and based on
data for an integrated luminosity of 33/pb, is shown in
Figure~\ref{fig:wbopro}.
The final value of the ratio, corrected to the phase space of the leptons
indicated, and for the lowest \pt\ threshold of 30~\GeV, is
\XZ{8.23}{0.18}{0.28}.
The data are compared to three predictions: a LO $2\to 2$ prediction based on
the \Pythia\ program; a LO $2\to n$ prediction from the \Alpgen\ software; and
finally, an NLO ME calculation for $2\to W/Z\,+\,2$ partons based on the
\Mcfm\ program.
All predictions fall within the still large experimental uncertainty band, but
clearly the deviations between data and predictions partly exceed the
systematic uncertainty of the data.
In particular, the data are well described by the NLO \Mcfm\ prediction, for
which the uncertainty (shown as dashed lines) is driven by the PDF uncertainty
and the one due to scale variations.
The experimental systematic uncertainties are smaller than those of these
predictions, especially at large \pt\ thresholds such that, after including the
2011 data, the experimental precision will challenge the NLO theoretical
prediction.
\section{Top-quark pair production}
\label{sec:ttbar}
The LHC is a top quark factory. At the present proton-proton centre-of-mass
energy of $\rts=7$~\TeV, the theoretical value of the \ttbar\ production
cross section, obtained from a computation approximating the NNLO prediction,
and for an assumed input top quark mass of $\mt=172.5$~\GeV\ is about
\sigttbar, with an uncertainty of about \dsigttbar~\cite{ALI-1101}, see below
for details.
This cross section is about 20 times larger than the corresponding
cross section at the Tevatron.
The LHC experiments have already analysed a wide spectrum of top quark
physics.
The two observables discussed are the \ttbar\ production
cross section~\cite{ATL-2010-016,CMS-1107} and the top quark mass
\mt~\cite{CMS-1101}.
The two most important quantities to be precisely evaluated for a cross section
determination are: the selection efficiency for signal events, and the amount
of background events present in the data.
To achieve a high precision estimate in the LHC analyses, not only the overall
normalisation of the background is used, but the shape of its contribution as
a function of one or more variables discriminating signal and background
processes is utilised.
An example of such a variable, the mass distribution of identified secondary
vertices, is shown in Figure~\ref{fig:ttbpro}~\cite{CMS-1107}. This figure
shows that the mass distributions originating from either light-quark jets or
\bquark\ jets are significantly different.
The mass distribution from jets stemming from charm quarks falls in between the
two.
Using this discriminant variable, a number of statistically independent
sub-sets of data with different signal to background compositions are
exploited.
\begin{figure}[tbp!]
\centering
\includegraphics[width=0.47\textwidth]{nisius_fig16}
\caption{The \ttbar\ production cross section~\protect\cite{CMS-1107}. Shown is
the vertex mass obtained from the charged particles assigned to a secondary
vertex as predicted in simulation.
\label{fig:ttbpro}}
\end{figure}
Figure~\ref{fig:ttbprofit} shows this set of distributions used in the fit for
the \ttbarlj\ decay in the \ejets\ and \mjets\ channels, for different jet
multiplicities, and depending on the number of \btagged\ jets.
The simulated signal events accumulate at large vertex masses, and their
fraction grows with the numbers of observed jets and \bjets.
The use of the profile likelihood method allows systematic uncertainties, which
are treated as nuisance parameters in the fit, to cancel each other within
bounds. Therefore, this method in general leads to smaller uncertainties than
are achieved when individually varying systematic effects to ascertain the
corresponding uncertainties.
This method requires a very good modelling of the correlation of the systematic
uncertainties, since otherwise fortuitous cancellations can happen.
\begin{figure}[tbp!]
\centering
\includegraphics[width=0.48\textwidth]{nisius_fig17}
\caption{The \ttbar\ production cross section~\protect\cite{CMS-1107}. Shown
are the mass distributions of identified secondary vertices for the two lepton
decay channels, and for different jet- and \bjet\ multiplicities. The data are
shown with their statistical uncertainty, together with the fitted
contributions of the predicted signal and background samples.
\label{fig:ttbprofit}}
\end{figure}
The variations of the systematic uncertainties are individually constrained in
the fit by Gaussian priors. The different sources of systematic uncertainty are
correlated or anti-correlated in the fit by up to absolute 70$\%$.
The determination of the cross section in the \ejets\ and \mjets\ channels
leads to consistent results.
The measured cross section, obtained from simultaneously fitting the
distributions of both channels using data corresponding to only 36/pb of
luminosity, Figure~\ref{fig:ttbprofit}, has an uncertainty of about 20~pb and
is already limited by systematic effects.
The measured cross section value is $\ensuremath{\sigma_\mathrm{t\bar{t}}}\xspace=(\XZ{154}{9}{17})$~pb with an
additional uncertainty of 6~pb due to the uncertainty on the
luminosity~\cite{CMS-1107}.
\begin{figure}[tbp!]
\centering
\includegraphics[width=0.48\textwidth]{nisius_fig18}
\caption{The predicted \ttbar\ cross section at the LHC, obtained from an NLO
(open markers) and an approximate NNLO (closed markers) calculation, as a
function of the strong coupling constant. The values are given for
$\rts=7$~\TeV\ and for a number of PDF sets~\cite{WAT-1101}. The markers are
placed at the predicted cross section and the \almzq\ value of the respective
PDF set. The horizontal bar span the \almzq\ uncertainty, and the vertical
bars indicate the PDF uncertainty of the cross section (inner bar), and the
PDF and \al\ uncertainty (outer bar). The lines indicate the cross section
variation with the \al\ dependent additional PDF sets.
\label{fig:sigtop}}
\end{figure}
This measured cross section can be used to further constrain the PDFs. Since,
at the LHC the \ttbar\ production process is largely dominated by gluon-gluon
processes, in contrast to the Tevatron, where it is dominated by
quark-antiquark processes, this mostly concerns the gluon distribution
function.
Figure~\ref{fig:sigtop} shows the present theoretical knowledge of this
cross section based on an NLO and an approximate NNLO calculation, and for
various PDF sets~\cite{WAT-1101}. The points are given at the respective value
of \almzq\ of the corresponding PDF set. The NNLO corrections are small.
The smallest cross section values are predicted when using the ABKM09 PDF
set. This is correlated to the smallest predicted gluon-gluon luminosity at
the \ttbar\ production threshold and above, as shown in
Figure~\ref{fig:gglum}~\cite{WAT-1101}.
\begin{figure}[tbp!]
\centering
\includegraphics[width=0.48\textwidth]{nisius_fig19}
\caption{The NNLO gluon-gluon luminosity at the LHC for $\rts=7$~\TeV\ as a
function of the gluon-gluon invariant mass scaled to the centre-of-mass
energy, as predicted by a number of PDF sets~\cite{THO-1101}. The
distributions are all normalised to the NNLO prediction from MSTW 2008. The
bands indicate the uncertainties obtained from the additional pdf sets.
\label{fig:gglum}}
\end{figure}
The difference in production cross section between the highest and lowest
prediction at NNLO is about 33~pb, i.e. about 1.5$\sigma$ of the above
described measurement.
With improved measurements that are underway, and by combining the results from
the ATLAS and CMS experiments, the gluon PDF can be significantly constrained.
The present average value of the top quark mass of $\mt=(\mttevo)$~\GeV is
obtained from direct measurements performed at the Tevatron~\cite{TEV-1101},
and has a total uncertainty of $0.6\%$.
The main methodology used to determine \mt\ at hadron colliders consists of
measuring the invariant mass of the decay products of the top quark candidates
and deducing \mt\ using sophisticated analysis methods.
The most precise measurements of this type use the decay \ttbarlj\ with
$\mathrm{lepton}=e,\mu$, where one \WB\ decays into a lepton and a neutrino and
the other into a pair of quarks.
These measurements rely on Monte Carlo programs to simulate the \ttbar\ final
state. The experimental observables are constructed such that they are unbiased
estimators of the top quark mass used as an input parameter in the Monte Carlo,
denoted with \mtMC, which is verified using pseudo-experiments performed on
large scale Monte Carlo simulated event samples.
Consequently, the top quark mass determined this way corresponds to \mtMC.
On the theoretical side, there are a number of definitions of the mass. The
definition of the pole mass, \mtpole, basically regards the quark as free and
long lived.
In contrast, for the \msb\ mass definition, \mtmsb, the mass is treated like a
coupling.
The masses expressed in the two renormalisation schemes are related, and
consequently can be converted into one another.
Their difference is sizable compared to the experimental precision:
$\mtpole=172$~\GeV\ leads to approximately $\mtmsb=162$~\GeV, a difference of
about 6$\%$, i.e.~ten times the experimental uncertainty.
Non of these definitions coincides with \mtMC\ defined above, which leads to a
problem in interpreting the experimental results.
There are theoretical arguments~\cite{BUC-1101} suggesting that \mtpole\ is
closer to \mtMC\ than \mtmsb, and that \mtpole\ is expected to be
\Ord{1~\mbox{GeV}} larger than \mtMC, but no proof of this relation from first
principles exists.
Theoretically, the \ttbar\ pair production cross section \ttbcrossv{\mtpole} is
known in a given renormalisation scheme. The calculations are performed at NLO,
NLO+(N)NLL or approximate NNLO precision, and have a strong dependence on the
top quark mass.
Consequently, it was suggested that by utilising this dependence, and
extracting the top quark mass from the cross section, the problem explained
above is absent, i.e.~the resulting \mt\ corresponds to a mass in a
theoretically well defined concept.
In addition, when the top quark mass is extracted from a comparison of the
measured production cross section with its prediction as a function of the
mass, one could profit from the fact that the relative uncertainty of the cross
section translates into an about five times smaller uncertainty on the top
quark mass, when neglecting the theoretical uncertainties.
\begin{figure}[tbp!]
\centering
\includegraphics[width=0.48\textwidth]{nisius_fig20}
\caption{Dependence of the \ttbar\ production cross section on the top quark
mass \mt\ for Tevatron conditions.
\label{fig:mtoptev}}
\end{figure}
This concept is only valid if the experimental determination of \ensuremath{\sigma_\mathrm{t\bar{t}}}\xspace\ does
not depend on the value of \mt\ itself, which unfortunately is not the case.
This is because also for the measurement of the \ttbar\ cross section,
\mtMC\ is needed, since the Monte Carlo models are indispensable for evaluating
the acceptance, efficiency and the systematic uncertainties in the
experimental determination of the cross section.
Figure~\ref{fig:mtoptev} shows the dependence of the measured production cross
section~\cite{D00-1101} and its experimental uncertainty as a function of
\mtpole\ for $\mtMC=\mtpole$ (flattest of the three bands), for the Tevatron
conditions.
The measured cross section value is $\ensuremath{\sigma_\mathrm{t\bar{t}}}\xspace=(\Y{8.13}{1.02}{0.90})$~pb,
with a symmetrised uncertainty of about 12$\%$.
\begin{figure}[tbp!]
\centering
\includegraphics[width=0.48\textwidth]{nisius_fig21}
\caption{Same as Figure~\protect\ref{fig:mtoptev}, but simulating the LHC
conditions at $\rts=7$~\TeV.
\label{fig:mtoplhc}}
\end{figure}
Since the measured cross section depends on the value of \mtMC, its relation to
\mtpole\ is important as can be seen from the second shifted band shown for the
assumption $\mtMC=\mtpole-10$~\GeV, which now means \mtMC\ approximately
coincides with \mtmsb.
Experimentally, the top quark mass is essentially extracted from the overlap of
the experimental band and the theoretical band (the steepest band shown), which
in this figure is based on~\cite{LAN-0901}. The result is indicated by the
vertical lines in the figure.
For a given assumption on \mtMC, this yields \mt\ with an uncertainty of about
3$\%$, showing the aforementioned reduction in relative uncertainty.
However, the uncertainty of where to put the experimental band leads to an
additional uncertainty on \mt.
For the example of the two extreme assumptions made in
Figure~\ref{fig:mtoptev}, the corresponding difference in the extracted top
quark mass is about 3~\GeV.
A similar situation is shown in Figure~\ref{fig:mtoplhc}, again using the
predictions from~\cite{LAN-0901}, but this time for LHC running conditions and
assuming a somewhat steeper dependence of the measured cross section on \mtMC.
For this situation, the difference in the extracted top quark mass is
correspondingly larger, and amounts to about 5~\GeV.
This investigation shows that to mitigate this uncertainty it is most important
to find an \mt\ independent selection, i.e.~to select the signal events while
depending as little as possible on absolute energy scales which directly relate
to the actual value of \mt.
In addition, to reach a precision on \mt\ of 0.6$\%$ as obtained in the direct
measurements, this indirect extraction needs to achieve about a 3$\%$ precision
on the measured cross section, which is a big challenge.
\section{Conclusions}
\label{sec:concl}
The analyses of the data from the first year of LHC running at a proton-proton
centre-of-mass energy of $\rts=7$~\TeV\ resulted in a large variety of physics
results concerning QCD observables, only a small part of which could be
discussed here.
The investigation of jet production for a number of jet multiplicities already
proved a helpful tool to better constrain QCD predictions of various types
implemented in a large number of programs.
The production of a heavy gauge bosons in conjunction with jets constitutes a
high precision QCD test, the potential of which has just been started to be
explored.
The LHC is a top quark factory with a twenty times larger production cross
section for pair production of top quarks than at the Tevatron. The first
measurements already give interesting hints on the size of the gluon PDF at
large values of x.
The determination of the top quark mass poses interesting challenges to the
experiments and also to the interpretation of the measured values.
All results presented were based on about 35/pb of data from the 2010 LHC
run. By now about 5/fb of data each have been collected by the ATLAS and CMS
experiments in 2011.
With this huge amount of data, the statistically limited analyses from 2010
data, can now be considerably expanded. For the analyses discussed in this
paper, this especially applies to the measurements of the ratio of the heavy
gauge bosons plus 1-jet cross sections.
The constantly increasing specific luminosity leads to more and more
proton-proton interactions per bunch crossing which pose an increasing
challenge to the proper treatment of the pileup.
With an even better understanding of the detectors, a large number of
interesting and precise measurements are ahead of us. The LHC experiments will
constantly extend the highest scales at which QCD has ever been probed at
accelerators.
\section*{Acknowledgements}
I like to thank the organisers not only for the stimulating spirit of this
meeting, but even more for their tireless enthusiasm during all the passed
years. I am glad I could participate in the last conference of this series.
\nocite{*}
\bibliographystyle{elsarticle-num}
|
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| 2,673
|
{"url":"https:\/\/www.physicsforums.com\/threads\/guass-law.710645\/","text":"# Guass Law\n\n## Homework Statement\n\nI. A non-conducting sphere of radius a has a spherically symmetric, but non-uniform charge distribution is placed on it, given by the volume density function: p(r) = C\u00b7r, where C is a positive constant, and 0 < r < a.\n\na. Find an algebraic expression for the total charge Q on the sphere, in terms of the parameters C and a.\n\nII. Use Gauss' Law to find an algebraic expression for the magnitude of the electric field at a distance R from the origin, in each of the following regions. Express your answer in terms of the following four parameters: the electrostatic constant k; the radius a of the sphere; the total charge Q on the sphere; and the radial distance R from the origin to the field point.\n\nb. Within the insulating sphere (i.e. for R <a):\n\nc. Outside the sphere (i.e. for R > a):\n\n## The Attempt at a Solution\n\nI got Q = Cpi(a^4) for part a which is correct.\n\nFor part 2 I am stuck both b and c.\n\nI got [(KQR^4)\/((a^4)(r^2))] which is incorrect.\n\nehild\nHomework Helper\nFor part 2 I am stuck both b and c.\n\nI got [(KQR^4)\/((a^4)(r^2))] which is incorrect.\n\nehild\n\nr is what I got with determining K. I got $$\\frac{1}{4\\pi \\epsilon_0 r^2},$$ but I replaced the 1\/4pi e_0 with K.\n\nvela\nStaff Emeritus","date":"2022-05-24 21:34:02","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\": 0.8192804455757141, \"perplexity\": 453.1571506900756}, \"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-2022-21\/segments\/1652662577259.70\/warc\/CC-MAIN-20220524203438-20220524233438-00120.warc.gz\"}"}
| null | null |
The Automatic Detective
From the award-winning author of Gil's All Fright Diner comes a fantastic sci-fi mystery read The Automatic Detective
Even in Empire City, a town where weird science is the hope for tomorrow, it's hard for a robot to make his way. It's even harder for a robot named Mack Megaton, a hulking machine designed to bring mankind to its knees. But Mack's not interested in world domination. He's just a bot trying to get by, trying to demonstrate that he isn't just an automated smashing machine, and to earn his citizenship in the process. It should be as easy as crushing a tank for Mack, but some bots just can't catch a break.
When Mack's neighbors are kidnapped, Mack sets off on a journey through the dark alleys and gleaming skyscrapers of Empire City. Along the way, he runs afoul of a talking gorilla, a brainy dame, a mutant lowlife, a little green mob boss, and the secret conspiracy at the heart of Empire's founders---not to mention more trouble than he bargained for. What started out as one missing family becomes a battle for the future of Empire and every citizen that calls her home.
Author Web Site
The Learned Council had an official name for Empire City.
Technotopia.
Yeah, it wasn't a real word, but that was kind of the point. The Council loved to reinvent things, improve them, make them new and snazzy. Of course...
Praise for The Automatic Detective
"Martinez tickles the funny bone in this delightful, fast-paced mishmash of SF and hard-boiled detective story. . . . A smart, rocket-fast read with a clever, twisty plot that comes to a satisfying conclusion." —Publishers Weekly (starred review) on The Automatic Detective
"Martinez crafts a private eye in the best tradition of hard-boiled futuristic detection, with plenty of beautiful babes and evil geniuses, and written in classic wise-cracking first-person narrative." —Booklist on The Automatic Detective
"Reformed warrior robot Mack Megaton searches for a missing family---and a moral purpose---in a city of exotic mutants. . . . Mack's smash-and-grab mission acquires some satisfying philosophical heft." —Entertainment Weekly on The Automatic Detective
"A terrific debut. The fast-paced plot is full of memorable incidents and wonderful observations." —Publishers Weekly (starred review) on Gil's All Fright Diner
A. Lee Martinez lives in Terrell, Texas.
Tweetsby @crimehq
|
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"redpajama_set_name": "RedPajamaCommonCrawl"
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| 333
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Le elezioni parlamentari a Malta del 1971 si tennero il 12 e 14 giugno e videro la vittoria del Partito Laburista.
Risultati
Collegamenti esterni
1971
Malta
|
{
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| 2,963
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Q: Why can't I purchase coins in Pokemon Go? I can't seem to buy any Poke Coins on my iPhone app. Has anyone else had this issue and how did you resolve it?
I usually go to the Shop and press on the coins that I want to purchase but it does nothing and we therefore can't buy any items in the shop.
A: If nothing happens when you tap on the coins in the shop, you probably have in-app purchases disabled on your phone, possibly through parental controls.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 2,129
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\begin{document}
{\title{\vskip-2truecm{\hfill {{\small \\
\hfill GUTPA/99/XX/X \\
}}\vskip 1truecm}
{\bf B-ball Baryogenesis and D-term Inflation$^*$}}
{\author{
{\sc John McDonald$^{1}$}\\
{\sl\small Department of Physics and Astronomy, University of Glasgow,
Glasgow G12 8QQ, SCOTLAND}
}
\maketitle
\begin{abstract}
\noindent
The MSSM has flat directions in its scalar potential, along which it is natural
for Bose condensates of squarks to form in the early Universe.
A baryon asymmetry can be induced in these condensates via Affleck-Dine baryogenesis. The condensates are unstable with respect to fragmentation to "B-balls", solitons made of squarks and carrying baryon number,
which, if they survive thermalization, fill the Universe down to low temperatures, much lower than that of the electroweak phase transition, with interesting cosmological consequences.
In particular, their decay implies a similar number density of baryons and dark matter neutralinos,
in accordance with observations. Evasion of thermalization and the
ability to account for the observed baryon asymmetry requires a very low reheating temperature,
which, it is argued, is a natural feature of presently favoured D-term inflation models.
\end{abstract}
\vfil
\footnoterule
{\small $^*$Presented at "Strong and Electroweak Matter '98",
NORDITA, Copenhagen}
\\
{\small $^1$mcdonald@physics.gla.ac.uk}
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The possibility of electroweak baryogenesis \cite{ewb}
in the Minimal SUSY Standard Model (MSSM) \cite{nilles} has become increasingly tightly constrained by
experiment. In particular, the requirement that the
electroweak phase transition is strong enough to prevent
subsequent wash-out of the
asymmetry imposes an upper bound on the Higgs
mass of 105-107 GeV \cite{Hupper} (the present 95 $\%$ CL LEP
lower bound is 77.5GeV \cite{Hlower},
with almost all of the remaining range to be tested by LEP200 \cite{Hupper}), and in addition requires that the right-hand
stop is light \cite{lights}, which in turn requires a negative SUSY breaking mass squared term for the r.h. stop
(but not for the other squarks and sleptons); a stiff challenge for supergravity (SUGRA) models.
However, there is another perfectly natural way to generate the baryon asymmetry
in the context of the MSSM, namely Affleck-Dine baryogenesis \cite{ad}. In this the
B asymmetry is induced in a squark Bose condensate which subsequently decays to (or is thermalized to) quarks.
This mechanism has recently been given a new spin \cite{kus,ks,bbb1} with the realization that in almost all realistic cases the squark condensate is unstable with respect to
fragmentation to B-balls (B carrying Q-balls \cite{Qballs,Qballs2}) made of squarks, which fill the Universe for a significant part of its history (until well after the electroweak phase transition in the case of gravity-mediated SUSY breaking
\cite{bbb1,bbb2} and right up to and including NOW, with interesting experimental and astrophysical consequences, in the case of gauge-mediated SUSY breaking \cite{ks,ks234}). These B-balls can have very significant consequences for cosmology;
in particular, in the case of
gravity-mediated SUSY breaking on which
we focus here, they can explain the
remarkable similarity of the {\it number density}
of baryons and dark matter particles for
the case where the dark matter
particles have masses $O(m_W)$ \cite{bbb2,bbbdm},
such as is typically the case for neutralinos.
To be precise, with currently favoured values compatible with distant supernovae observations \cite{sn},
$\Omega = 1$, $\Omega_{m} = 0.4$, and $\Omega_{\Lambda} = 0.6$, expansion rate $h$ in the range 0.6 to 0.8 (as determined by Hubble Space Telescope
\cite{hst,freedman} and limits on the age of the Universe \cite{ageu}) and
baryon number density $0.0048 \lae \Omega_{B}h^2 \lae 0.013$, as determined by primordial
nucleosynthesis \cite{sarkar}, we find
\be{n1} \frac{n_{B}}{n_{DM}} \approx (1.5-7.3) \; \frac{m_{DM}}{m_W} ~. \ee
Thus for $m_{DM} \sim m_W$, the baryon and dark matter particle number densities
are within an order of magnitude of each other; a remarkable result and a possible clue as to the
origin of baryons and dark matter. Although one might expect that the existence of
a significant mass density of baryons could be understood by appealing to anthropic selection, there
is still no reason for such a selection to
result in similar {\it number} densities
of baryons and dark matter. We refer to the scenario in which the B asymmetry at least partly comes from late-decaying B-balls as B-ball baryogenesis.
The details of this mechanism are intimately tied to the details of the full
supergravity (SUGRA) model and inflation, requiring a rather
global view of SUSY cosmology. This connection with SUSY cosmology as a whole, and the possibility of observable predictions \cite{bbbdm,bbbiso} (as discussed in
more detail in these Proceedings by Enqvist),
make this an exciting scenario, requiring, we emphasize, nothing more than the particles of the MSSM and a model for inflation.
B-ball baryogenesis is based on the
Affleck-Dine mechanism. This in turn depends on the
existence of renormalizable flat directions in the
scalar potential of the MSSM \cite{drt}. The scalar potential of the MSSM is complicated, with squarks and sleptons as well as Higgs fields. Renormalizable flat directions correspond to directions in the scalar potential with renormalizible F- and D-terms
vanishing, leaving only the soft SUSY-breaking terms and the Planck-suppressed
non-renormalizable terms which are expected in a SUGRA effective theory. These directions are characterized by the lowest dimension superpotential operators which are consistent with all symmetries and which which are non-vanishing when expressed as scalars along the flat directions.
We concentrate on the case of the MSSM with R-parity, which eliminates dangerous renormalizable B- and L-violating terms from
the superpotential and also allows for stable neutralino cold dark
matter \cite{nilles,susydm}. In this case
the flat directions are of even dimension \cite{drt} e.g.
\newline d=4 $H_{u}L$-direction
\be{e1} <\tilde{\nu}_L> = < \phi_{u}^{o}> \;\;\; ; \;\;\; (H_{u}L)^2 \neq 0 ~,\ee
\newline d=6 $u^c d^c d^c$-direction (the $d^cQL$ and $e^cLL$ directions are similar)
\be{e2} <\tilde{u}^{c\;1}> = <\tilde{d}^{c\;2}> = <\tilde{d}^{c'\;3}> \;\;\; ; \;\;\; (u^{c}d^{c}d^{c})^2 \neq 0 ~,\ee
where the indices denote colour and $\tilde{d}^{c}$ and $\tilde{d}^{c'}$ are orthogonal in flavour
space. We will be particularly interested in the d=6 $u^{c}d^{c}d^{c}$ direction, as d=4 directions are both incompatible with
B-balls which survive thermalization \cite{bbb2} and with the AD mechanism in the context of
D-term inflation \cite{kmr}.
The form of the soft SUSY-breaking terms (once Planck-suppressed SUGRA
couplings are taken into account) is different in the early Universe; the
energy density of the Universe breaks SUSY and results in O(H) corrections
to the SUSY-breaking terms \cite{h2O,h2}. As a result, it is natural for the
initial value of the AD squark field to take a large value during and/or soon
after inflation. This is true if the correction to the SUSY breaking mass squared term is
negative. The scalar potential then has the form
\be{e3} V = (m^2 - c H^2) |\Phi|^2 + \left( \frac{A_{\lambda} \lambda \Phi^6}{M^3} + h.c.\right) +
\frac{\lambda^2 |\Phi|^{10}}{M^6} ~,\ee
where $M = M_{Pl}/\sqrt{8 \pi}$ and $c \sim 1$.
The AD field will begin to oscillate coherently once $H \approx m$,
corresponding to a Bose condensate of squarks, with an initial amplitude
$\phi \approx (m M^3)^{1/4}\approx 10^{14} {\rm\ GeV}$ for the d=6 direction and
$\phi \approx (m M)^{1/2} \approx 10^{10} {\rm\ GeV}$ along
the d=4 direction. (It is the larger amplitude
along the d=6 direction which protects the
AD field from being thermalized
in D-term inflation models, by giving a large mass to the particles which couple to it, so
suppressing the scattering rate with light thermal particles \cite{kmr}).
In the absence of the A-terms, there is no B violation
or CP violation, and the real and imaginary parts of the AD scalar
oscillate in phase, corresponding to zero B asymmetry.
A B asymmetry is induced in the squark condensate if a phase shift is induced between the real and
imaginary parts of the AD field. The baryon asymmetry density is given by
\be{e4} n_{B} = i( \dot{\Phi}^{\dagger}\Phi - \Phi^{\dagger} \dot{\Phi}) ~.\ee
With $\Phi = (\phi_1 + i \phi_2)/\sqrt{2}$, $\phi_1 = \phi_o Cos(mt)$ and $\phi_2 = \phi_o Cos(mt + \delta)$,
the B asymmetry density is given by $n_B = (m \phi_o^2 Sin \delta )/2 $. The required
phase shift is provided by the A-term.
When the AD field starts oscillating, the A-term is of the same order of magnitude as the
mass term and non-renormalizable term in the potential, and distinguishes between the real and
imaginary directions (where the real direction is defined by the phase of the A-term). For the case of interest to us here,
D-term inflation \cite{dti}, in which there are no $O(H)$ corrections to the A-terms \cite{kmr}, the initial phase of the AD field is determined by its random initial value
during inflation and is therefore typically of the order of 1; thus within a few oscillations a phase difference of
order 1 will be induced in the AD field. Subsequently, as the Universe expands, the amplitude of oscillation
decreases; the A-term, which is proportional to $\phi^6$, rapidly decreases relative to the mass term,
effectively switching off B violation and leaving a fixed B asymmetry which we observe today. The final
B-asymmetry depends crucially on the reheating temperature after inflation; for the d=6 $u^{c}d^{c}d^{c}$ direction
the baryon to entropy ratio is \cite{bbbdti}
\be{e5} \eta_B \approx 3 \times 10^{-11} \left(\frac{T_R}{1 {\rm\ GeV}}\right) Sin \delta ~.\ee
Comparing with the observed B asymmetry, $\eta_{B\;obs} = (3-8) \times 10^{-11}$, we see that the reheating
temperature for d=6 AD baryogenesis must of the order of 1GeV in
D-term inflation models (or in any model in which the magnitude of the CP violating phase
is of the order of 1; a natural value); for higher
reheating temperatures there is {\it overproduction} of B.
Thus d=6 AD baryogenesis with O(1) CP violating phase (as in the D-term inflation scenario)
imposes the constraint on inflation models that the reheating temperature must be very low relative to the
energy scale of inflation. At first sight this seems
perhaps unfavourable; however, reheating in SUSY inflation models must already occur at a low temperature
relative to the inflation energy scale, $T_R \lae 10^9 GeV$, in order to avoid thermally overproducing
gravitinos \cite{sarkar,grav}. We will see that there is good reason to expect a very low reheating temperature in
D-term inflation models, resulting in an interesting self-consistency of the D-term inflation/AD baryogenesis scenario \cite{bbbdti}.
In the original AD mechanism it was assumed that the AD field would simply oscillate around the
minimum of its potential with decreasing amplitude until it either decayed or the AD field strength was small enough to allow it to thermalize.
However, it was recently realized that the AD condensate is generally unstable with respect to
fragmentation to B-balls \cite{ks,bbb1}.
In general, Q-ball solutions (charged non-topological solitons in a scalar theory with an approximate
$U(1)_{global}$, which in our case corresponds to baryon number) exist if $U(|\phi|)/|\phi|^2$ has a global minimum at $\phi \neq 0$ \cite{Qballs}.
This is likely to occur rather generally in SUSY models,
as a result of the soft SUSY breaking terms \cite{kus}.
The resulting solution has the form \cite{Qballs}
\be{e6} \Phi(r,t) = \frac{e^{i \omega t}}{\sqrt{2}} \phi(r) \;\;\; ; \;\;\; \phi(r) \approx \phi(0) e^{-r^2/R^2} ~,\ee
where $\omega$ is the effective mass of the scalars inside the Q-ball and
$\phi(r)$ here corresponds to the case of thick-walled
Q-balls \cite{bbb2}. For the particular case of flat directions
in models with gravity-mediated SUSY breaking, the mass squared
term rather generally becomes
smaller with increasing $\phi$ once radiative corrections are taken into account, as a result
of gauge loops \cite{bbb1} \begin{footnote}{Let us note some interesing work in connection with flat directions, proton stability, Q-balls and string models in this context in \cite{minos}.}\end{footnote} The RG equations generally have the form \cite{nilles}
\be{e7} \mu \frac{\partial m_{i}}{\partial \mu} = \alpha_{i} m_{i}^{2} - \beta_{\alpha} M_{\alpha}^{2}
~,\ee
where $m_{i}$ represents the scalar soft SUSY breaking terms and
$M_{\alpha}$ the gaugino masses.
These are typically
dominated by the gaugino masses,
causing $m_{i}$ to decrease with increasing scale.
Thus the AD scalar mass term decreases with increasing $\phi$. As a result,
the condition for the existence of Q-ball solutions is satisfied.
The homogeneous condensate
will naturally fragment to Q-balls, as the
Q-ball is the lowest energy solution for a given charge.
This can be also seen by noting that a homogeneous scalar field oscillating in a
potential that is flatter than
$\phi^2$ will behave as matter with a negative pressure \cite{turner}.
At 1-loop the correction to the mass term is
\be{e8} V(|\Phi|) = m^2 |\Phi|^2
+... \rightarrow m^2 \left(1 + K log \left(\frac{|\Phi|^2}{\mu^2}\right)\right) |\Phi|^2
+... ~,\ee
where typically $|K| \approx 0.01 -0.1$ and $ K < 0$.
For small $|K|$, $V \sim \phi^{2+K}$, which results in
an equation of state for the homogeneous field
\be{e9} p = \frac{K}{2} \rho ~.\ee
This negative pressure merely reflects the attractive interaction
between the scalars due to the $log$ term in the potential.
As a result, any spatial perturbations
will grow exponentially with time until they go non-linear and the
condensate will fragment into charged condensate lumps
which eventually condense into B-balls. The
exponential growth of the linear perturbations
is given by \cite{bbb2}
\be{e10} \delta \phi = \left(\frac{a_{o}}{a}\right)^{3/2}
\delta \phi_{o} exp\left(\frac{2}{H} \left(\frac{|K|}{2} \frac{\vec{k}^2}{a^2}
\right)^{1/2} \right) ~,\ee
where $\delta \phi_{o}$ is the
seed perturbation, expected to come from quantum fluctuations during
inflation \cite{bbb1}.
This is valid for perturbations with $|\vec{k}^2/a^2| \lae |2Km^2|$;
shorter wavelength perturbations have a positive
pressure due to their gradient energy which overcomes the negative pressure,
such that the perturbations merely oscillate.
The first perturbation to go non-linear has a diameter
$ l_{non-linear} \approx \pi / (2 |K| m^2)^{1/2}$.
The baryon number density at a given value of $H$
is given by $n_{B} = \eta_{B} H^2 M_{Pl}^2 / 2 \pi T_R$.
From these we find that the baryon number of
the condensate lumps (and so of the B-balls) is \cite{bbb2}
\be{e11} B \approx 10^{24} f_B |K|^{1/2}
\left( \frac{1 {\rm\ GeV}}{T_R} \right) ~,\ee
with the corresponding radius and field inside the thick-walled B-ball given by
$R \approx (|K|^{1/2} m)^{-1}$ and
$\phi(0) \approx 10^{14} \left(B/10^{26}\right)^{1/2} {\rm\ GeV}$.
Here $f_B$ is fraction of the B asymmetry which
ends up inside the B-balls; this requires
an analysis of the non-linear evolution of the condensate \cite{fb}
but we expect B-ball formation to be quite efficient; $f_B \sim 0.1-1$.
Due to their large charges and field strengths, the B-balls can survive thermalization
and decay at low temperatures. It may be shown that they survive thermalization
if $T_{R} \lae 10^{3-5}{\rm\ GeV}$ \cite{bbb2}. This is
easily satified for the d=6 D-term inflation case. (However, it rules out
survival of B-balls for the d=4 directions,
since in this case $T_{R} \gae 10^7 {\rm\ GeV}$ is required to
account for the observed B \cite{bbb2}).
Charge escapes from B-balls to light B-carrying fermions and
scalars a rate proportional to their area, $A$.
The decay rate of the B-balls is given by \cite{Qballs2,bbb2}
\be{e12} \frac{1}{B} \frac{dB}{dt}
\approx - \frac{\omega^3 f_{s}}{192 \pi^2} \frac{A}{B} \propto \frac{1}{B}
~,\ee
where $\omega \approx (1 + 3K/2)m \approx m$, with $m \sim 100GeV$ being the
mass of the AD scalar \cite{bbb2}.
($f_s$ is a factor accounting for the enhancement of the decay rate if
decay to scalar pairs is possible; we estimate $f_s \approx 10^3$ \cite{bbb2}).
Therefore larger charge B-balls decay later. The B-balls decay once
$ | \frac{1}{B} \frac{dB}{dt}| \gae H$; the resulting decay
temperature is
\be{e13} T_{d} \approx (0.01-1) \left(\frac{T_{R}}{1 {\rm\ GeV}}\right)^{1/2} {\rm\ GeV} ~.\ee
Thus, in general, B-balls which can evade thermalization will
decay after the electroweak phase transition.
This implies that the B-balls will have non-trivial
consequences for cosmology:
\newline $\bullet$ The baryon asymmetry inside the B-balls (where thermal particles cannot penetrate)
will be protected from washout due the combined effects of sphaleron B+L violation and additional L violating interactions. Such interactions commonly
arise in extensions of the MSSM; for example, in see-saw models of Majorana neutrino masses.
\newline $\bullet$ $n_{B} \sim n_{DM}$: If the reheating temperature is of the order of 1GeV,
then the B-balls will decay {\it after the dark matter neutralinos have frozen out of chemical equilibrium}. Neutralinos freeze out at $T_{f} \approx m_{\chi}/20 > 1 {\rm\ GeV}$, where $m_{\chi}$
is the neutralino mass \cite{susydm}.
The B-ball is made of squarks, with one unit of R-parity per 1/3 B. Thus when the B-ball decays we will obtain
3 neutralino LSPs per unit B.
This naturally results in $n_{B} \sim n_{DM}$, in accordance with
observations! Note that this produces {\it non-thermal}
neutralino dark matter, with a quite different dependence on the
MSSM parameter space than the conventional thermal relic dark matter,
a feature that can be tested experimentally
once SUSY is discovered in accelerators.
The actual number density ratio predicted by B-ball decay
(assuming no subsequent annihilations \cite{bbb2}) is crucially dependent
on $f_B$
\be{e15} \frac{n_{B}}{n_{\chi}} = \frac{1}{3 f_{B}} ~.\ee
Thus $f_{B}$ has to be less than 1 in order to be
compatible with observations (Eq.(1)) and the present
experimental lower bound on the LSP neutralino mass, $m_{\chi} \gae
25GeV$ \cite{mneut}.
However, $f_B$ in the range 0.1 to 1 can accomodate a
wide range of neutralino masses. In addition, if we can {\it calculate} $f_{B}$,
we can constrain the LSP mass. For example, for the $\Omega_{m} = 0.4$ case we find
\be{e15a} 0.046 f_B^{-1} \lae \frac{m_{\chi}}{m_{W}} \lae 0.22 f_B^{-1}
~,\ee
where the range is due to the uncertainties in the baryon asymmetry
estimate from nucleosynthesis and in the present expansion rate.
Since $\Omega_{m}$, $\Omega_{B}$ and $h$ will all be fixed to 1 $\%$ accuracy
by the MAP and PLANCK satellite missions \cite{mapp,freedman}, we will eventually be able to
{\it predict} $m_{\chi}$ if we can calculate $f_{B}$,
providing a "smoking-gun"
test of this scenario \cite{bbbdm}. It remains to be seen whether $f_{B}$ can be accurately
computed from the non-linear evolution of the condensate \cite{fb}.
\newline $\bullet$ Another observable consequence of late-decaying
B-balls, in the context of D-term inflation models, is that they
transfer the isocurvature fluctuations in the baryons
to the dark matter neutralinos, resulting in a significant
enhancement of the isocurvature fluctuations,
making them observable by MAP and PLANCK \cite{bbbiso}.
This is discussed in detail in these Proceedings by Enqvist.
So far we have shown that the d=6 Affleck-Dine
mechanism with O(1) CP violating
phase requires a reheating temperature of the order of 1GeV,
and we have noted that
O(1) phases are to be expected in D-term inflation
models. We now
address the question of whether such low reheating temperatures are a natural feature of these models.
SUSY inflation is characterized by the nature of the energy density driving inflation.
The SUSY potential has F- and D-term contributions,
$V_{susy} = |F|^2 + |D|^2$, with
D-term inflation driven by $|D|^2$. The reason D-term inflation
is favoured over F-term is that there are no O(H) corrections to the inflaton mass in this
case \cite{dti}. These terms would
otherwise ruin the flatness of the inflaton potential, preventing
slow-rolling and/or producing too large deviations from a scale-invariant
perturbation spectrum.
The minimal model for D-term inflation has a singlet $S$ and two fields with opposite
$U(1)_{FI}$ charges $\psi_{\pm}$, where $U(1)_{FI}$ is the Fayet-Illiopoulos (FI) gauge group with a FI
term $\xi^2$. The superpotential is
\be{e16} W = \lambda S \psi_{+} \psi_{-} ~\ee
and the scalar potential, including the $U(1)_{FI}$ D-term, is then
\be{e17} V = |\lambda|^2 (|\psi_{+} \psi_{-} |^2 + |S \psi_{+}|^2 + |S \psi_{-}|^2)
+ \frac{g^2}{2} ( |\psi_{+}|^2 - |\psi_{-}|^2 + \xi^2)^2 ~,\ee
where $g$ is the $U(1)_{FI}$ coupling. $\xi$ is fixed by cosmic microwave background measurements
to be $6.6 \times 10^{15} {\rm\ GeV}$ \cite{lr}. The inflation
model which results from this potential has the form of a
hybrid inflation model \cite{hybrid}; for
$S > S_{crit} \equiv g \xi/\lambda$ the minimum of the
potential corresponds to $ \psi_{\pm} = 0$,
resulting in a very flat, slow-roling potential
with $V = g^2 \xi^4 / 2 + \kappa \; log|S|$, where $\kappa$
comes from radiative corrections.
For sufficient inflation, $S > S_{55} \approx 0.9 M$
is necessary. However, this leads to a
generic problem for D-term inflation models \cite{kmr}.
In general, since we are considering SUSY models
to be the low energy effective theory of a SUGRA theory,
we expect all non-renormalizable
superpotential terms consistent with the
symmetries and suppressed by powers of $M$ to appear,
$\Delta W \sim S^m / M^{m-3}$.
However, the contribution to the scalar potential of such terms
will cause unacceptable deviation from
scale-invariant fluctuations at length scales corresponding to the observable Universe,
requiring the elimination of all such terms
for $ m \leq 9$ \cite{kmr}.
Thus we need to introduce a symmetry to eliminate such terms; we focus
on the case of an R-symmetry, which is
particularly effective (eliminating {\it all} purely $S$
superpotential terms if R(S) is negative,
for example), although we expect the same argument to apply
for more general discrete symmetries. The
connection between D-term inflation and low reheating
temperatures is then that the
symmetry which keeps the inflaton potential sufficiently flat
also tends to suppress the coupling of the
inflaton to light fields, so suppressing the decay rate of the
inflaton and resulting in
very low reheating temperatures \cite{bbbdti}
\begin{footnote}{ In D-term inflation models there are {\it two} stages
of reheating; that from decay of the $\psi_{-}$ oscillations, with
typically $T_{R}^{\psi} \approx 10^{15} {\rm\ GeV}$, and that from decay of the
$S$ oscillations \cite{kmr}. The first stage of reheating is important for
thermalizing the d=4 AD field, but otherwise the important reheating
temperature (for the gravitino bound and for B-balls) is that associated
with the $S$ field, whose oscillations dominate the energy
density throughout \cite{bbbdti}.}\end{footnote}.
This happens if there is a "mismatch" between the
transformation properties of the
inflaton sector $S, \psi_{\pm}$ and the MSSM sector fields.
As a simple example, consider the
case where the inflaton sector fields
have half-integral charges and the MSSM
sector fields have 1/3-integral charges.
With $R(S) = -n$, $R(\psi_{+}) = R(\psi_{-}) = (n+2)/2$
($n \in Z$) for the inflaton sector fields and
$R(Q) = R(L) = 5/3$, $R(u^c) = R(d^c) = R(e^c) = 1/3$
and $R(H_u) = R(H_d) = 0$ for the
MSSM fields (the MSSM charges have been chosen to allow
the $(u^{c}d^{c}d^{c})^2$ operator necessary for AD baryogenesis)
we find that, for n=1, the lowest dimensional superpotential term
allowing the inflaton to couple to the MSSM sector is of the form $W_{int}
= \kappa S(LH_{u})(u^c u^c d^c e^c)/M^{4}
\sim \kappa S \phi^r / M^{r-2}$ with r=6, where $\phi$ represents the light MSSM
fields \cite{bbbdti}. For n=2 a dimension r = 4 operator is
possible, but for larger n the dimension is $r \geq 6$.
The corresponding decay rate
is
\be{e18} \Gamma_{S} \approx \left( \frac{M_{S}}{M} \right)^{2(r-2)} \kappa^2 \beta_r M_S
~,\ee
where $\beta_r$ represents the phase factor for decay to $r$ light particles.
The corresponding reheating temperature is then
\be{e19} T_{R}^S \approx 4 \times 10^{15} (2.8 \times 10^-3)^{r-5/2} \kappa \lambda^{r-3/2} \beta_{r}^{1/2} {\rm\ GeV} ~.\ee
Thus for r=6 we find \cite{bbbdti}
\be{e19a} T_{R}^{S} \approx
150 \left( \frac{\lambda}{0.1} \right)^{9/2}
\kappa \beta_{6}^{1/2} {\rm\ GeV} ~,\ee
where $\beta_{6} \approx 10^{-6}$. Therefore low reheating temperatures,
as low as 1 GeV or less,
are a natural feature of D-term inflation models \cite{bbbdti}.
Of course, it is also necessary to check if reheating mechanisms other than single particle decay could be more efficient; in particular parametric resonance \cite{pr} via
Planck-suppressed operators. We find that parametric resonance is completely negligible if $ \lambda/g^{1/2} \gae 0.03$ \cite{bbbdti}. For values of $\lambda$ and $g$ not too small compared with 1 this condition will be easily satisfied.
In conclusion, we believe that Affleck-Dine baryogenesis with late
decaying B-balls (B-ball baryogenesis)
is a natural candidate for baryogenesis in the MSSM which has
the great advantage that it can explain why $n_B \sim n_{DM}$.
This model fits in very well with the currently favoured D-term inflation
scenario, with a natural consistency between the low
reheating temperature required for
d=6 AD baryogenesis and the suppression of
inflaton couplings required for a flat inflaton potential.
It is especially interesting
that this model can make definite predictions;
$n_B \sim n_{DM}$, non-thermal neutralino
dark matter, eventually (depending on the calculation of $f_B$ and results from MAP and PLANCK) the neutralino mass, and,
as discussed in these Proceedings by
Enqvist, isocurvature microwave fluctuations which should be observable by MAP and PLANCK.
The correlation between these various predictions should serve as a clear test of the B-ball baryogenesis/D-term inflation scenario.
\subsection*{Acknowledgements} Many thanks to Kari Enqvist
for a stimulating and continuing collaboration. Thanks also
to the European Union TMR programme, which supported most of the
work reported here, and to the Dept.
of Theoretical Physics, Helsinki
University, where it was carried out. This
report has been supported by the UK PPARC.
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{
"redpajama_set_name": "RedPajamaGithub"
}
| 80
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Despite the many innovations that make this project so fascinating, the Dawn team has not yet discovered how to travel backwards in time. (If it had, while writing this log, we would be able now to prevent the misspelling that occurred while writing the last log.) To see why Dawn seems to be reversing course, both heading toward the Sun and traveling more slowly now than at the beginning of its mission, we need to consider some of the principles that govern space travel.
Your correspondent offered some comments on these concepts in a log for a different interplanetary mission, Deep Space 1. If not for some unexpected legal issues with certain species in spiral galaxies capable of abstract thought, we would simply reprint that material here. Instead, we shall consider some of the same ideas but with different words.
The overwhelming majority of craft humans have sent into space have remained in the vicinity of Earth, accompanying that planet on its annual revolutions around the Sun. The satellites of Earth (including the moon) remain bound to it by its gravity. As fast as they seem to travel compared to residents of the planet, from a solar system perspective, their incessant circling of Earth means their paths through space are not very different from Earth's itself. Everything on the surface and in Earth orbit travels around the Sun at an average of around 30 kilometers/second (67,000 miles/hour), completing one full solar orbit every year. To undertake its interplanetary exploration and travel elsewhere in the solar system, Dawn needed to break free of Earth's grasp, and that was accomplished by the rocket that carried it to space last year. Dawn and its erstwhile home went their separate ways, and the Sun became the natural reference for the spacecraft's position and speed on its travels in deep space.
Whether it is for a spacecraft or moon orbiting a planet, a planet or Dawn orbiting the Sun, the Sun orbiting the Milky Way galaxy, or the Milky Way galaxy orbiting the Virgo supercluster of galaxies (home to a sizeable fraction of our readership), any orbit is the perfect balance between the inward tug of gravity and the inexorable tendency of objects to travel in a straight path. If you attach a weight to a string and swing it around in a circle, the force you use to pull on the string mimics the gravitational force the Sun exerts on the bodies that orbit it. The effort you expend in keeping the weight circling serves constantly to redirect its path; if you let go of the string, the weight's natural motion would carry it away in a straight line (ignoring the effect of Earth's gravity).
The force of gravity diminishes with distance, so the Sun's pull on a nearby body is greater than on a more distant one. Therefore, to remain in orbit, to balance the relentless tug of gravity, the closer object must travel faster, fighting the stronger pull. The same effect applies at Earth. Satellites that orbit very close (including, for example, the International Space Station, less than 400 kilometers from the surface) must streak around the planet at about 7.7 kilometers/second (more than 17,000 miles/hour) to keep from being pulled down. The moon, orbiting 1000 times farther above, needs only to travel at about 1.0 kilometers/second (less than 2300 miles/hour) to balance Earth's weaker grip at that distance.
The rocket that launched Dawn threw it hard enough to escape from Earth, sending it well beyond the International Space Station and the moon. Indeed, the spacecraft is now more than 1 million times farther away than the station. Dawn's maximum speed relative to Earth on launch day was so high that Earth could not pull it back. As mentioned in an earlier log, Dawn was propelled to 11.46 kilometers per second (25,600 miles per hour), well in excess of the space station's orbital speed given 3 paragraphs above. But it remains under the Sun's control.
On September 27, 2007, some 2 minutes after it had separated from its rocket, Dawn reached its highest speed relative to the Sun for the entire mission. At that time, it was traveling at about 38.95 kilometers/second (87,130 miles/hour). Earth (and its residents, including this writer) were moving around the Sun at the more leisurely pace of 29.70 kilometers/second (66,440 miles/hour).
So what has been the effect of Dawn's thrusting since then? By August 8 it had expended about 55.4 kilograms (122 pounds) of xenon propellant, some for tests during the initial checkout phase of the mission and most with the specific intent of altering its orbit around the Sun. If this were solely for the purpose of accelerating (as it is usually described, in these logs and elsewhere) without the complex patterns involved in orbital dances, the effect would have been to increase Dawn's speed by 1.38 kilometers/second (3090 miles/hour). But because of the way forces and velocities work in space travel, in pushing Dawn away from the Sun, allowing it to travel "higher" before the Sun pulled it back, the ion propulsion system helped Dawn continue away from the Sun until, on August 8, it was more than 1.68 AU from the master of the solar system. By then, its speed had fallen to 20.77 kilometers/second (46,460 miles/hour). At the same moment Dawn was orbiting the Sun at that rate, distant Earth was racing in its orbit at 29.38 kilometers/second (65,710 miles/hour).
Dawn will rendezvous with Vesta in about 3 years. To match that asteroid's orbit around the Sun, our robotic explorer will have to continue tuning its orbital parameters so that it will be almost 2.3 AU from the Sun while traveling at about 20 kilometers/second (45,000 miles/hour), farther and slower than its current orbit or that of its quondam planetary domicile.
Achieving the required speed and distance alone is not enough to ensure Dawn can slip into orbit around Vesta, but we will consider other aspects of this problem in a future log. In the meantime, we can think of the general problem of flying elsewhere in space as similar to climbing a hill. For terrestrial hikers, the rewards of ascent come only after doing the work of pushing against Earth's gravity to reach a higher elevation. Similarly, Dawn is climbing a solar system hill with the Sun at the bottom. It started from Earth, at 1 AU in elevation; and its first rewards await it higher up that hill at 2.3 AU, where Vesta, traveling at only about two thirds of Earth's speed, keeps its records of the dawn of the solar system. Ceres is still higher up the hill, moving even more slowly to balance the still-weaker pull of the Sun.
If this were only a climb, it would be easy to stop at the correct spot on the solar system hill. This simple analogy fails us here though, because everything is in orbital motion. With a big enough rocket, or gravitational boosts, it would not be difficult to throw Dawn hard enough that it would fly out to Vesta or beyond, and some other spacecraft have coasted past that distance from the Sun. But to enter orbit, Dawn must precisely match Vesta's path around the Sun, joining it for a portion of the asteroid's regular 3.6-year circuit around the Sun, just as Earth's natural and human-made satellites stay with it throughout its 1-year orbit. That is part of the reason the spacecraft needs ion propulsion. The ion propulsion system allows Dawn not only to carry its scientific instruments up that hill but also to "stop" on the slope, neither falling back toward the Sun nor coasting by the asteroid. When a subsequent log addresses what more is required than speed and distance, we will see why this is more difficult than it may appear. (And as we surely will have a link from that log to this one, on behalf of all present readers, we send greetings from the past to you future readers.) We are confident that in meeting this great challenge, should Dawn remain healthy, it will be a shoo-in in the next solar system Olympics, aiming not for a bronze medal, nor for one of silver or gold but rather for the most highly coveted: the xenon medal.
We promised near the beginning that for those who completed this arduous log (perhaps a challenge even greater than Dawn's interplanetary journey), the examinations at the prestigious Galactic Institute of Space Travel would not prove difficult. The reason is simple: there is no such organization; we made it up. Nevertheless, following Dawn's long and ambitious journey does not require mastery of the concepts touched upon here. All that really is needed is the desire to learn more about the cosmos, to share in one of humankind's bold adventures to explore the unknown as we set our sights on extraordinarily distant goals and aspire to something well beyond the confines of our humble home in the universe.
Dawn is 352 million kilometers (219 million miles) from Earth, or 955 times as far as the moon and 2.33 times as far as the Sun. Radio signals, traveling at the universal limit of the speed of light, take 39 minutes to make the round trip.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,859
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Digital Beat
Free, over-the-air television and radio; community-based, low-power FM radio stations; public radio and television; and the obligations of licensees to serve the public interest. A key principle of federal communications law is that in exchange for free use of the public airwaves broadcasters agree to take actions to benefit the public. These principles are enshrined in the Radio Act of 1927 and the Communications Act of 1934 in the mandate that "broadcasting serve the public interest, convenience and necessity."
Do We Still Care About Diversity?
On Wednesday, January 15, the House Commerce Committee's Subcommittee on Communications and Technology held a hearing on diversity in the media market.
While You Were Googling 'Impeachment'
Obviously, there's no bigger story this week than the possible impeachment of the 45th president of the United States. But if we still have your attention, here's some items of note we found this week.
Media, Government, Us
The media merger pot keeps boiling.
A Preview of the FCC's July Open Meeting: Taking the "E" Out of EBS and TV
Perhaps the biggest news of the week was the agenda for the Federal Communications Commission's July 10 Open Meeting, which FCC Chairman Ajit Pai laid out in a blog post o
Benton Joins Call for Public Scrutiny of Sinclair
The FCC ruled that there are substantial and material questions as to whether Sinclair is qualified to be a broadcast licensee. There is only one way to resolve these questions in a transparent manner that allows public participation: a hearing on
Bending the Arc Towards Media and Social Justice
I've spent just over 30 years working to ensure that all Americans benefit from accessible, affordable, and open communications networks that promote democratic values.
Tribune withdraws from Sinclair merger, saying it will sue for 'breach of contract'
Tribune Media will withdraw from its $3.9 billion merger with Sinclair Broadcast Group, saying it would sue Sinclair for "breach of contract" over its failed negotiations with regulators over the deal. "In light of the FCC's unanimous decision, re
FCC Issues Hearing Designation Order for Tribune Media Company and Sinclair Broadcast Group
On June 28, 2017, Sinclair Broadcast Group and Tribune Media Company filed applications seeking to transfer control of Tribune subsidiaries to Sinclair.
Statement of Chairman Ajit Pai On Sinclair/Tribune Transaction
Based on a thorough review of the record, I have serious concerns about the Sinclair/Tribune transaction.
FCC Opens Review of National Broadcast TV Ownership Cap
The Federal Communications Commission adopted a Notice of Proposed Rulemaking initiating a comprehensive review of the national television audience reach cap, including the so-called UHF discount used by broadcasters to determine compliance with t
Benton Institute
for Broadband & Society
727 Chicago Avenue
© 1994-2020 Benton Institute for Broadband & Society. All Rights Reserved.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,147
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Deep in the heart of Black Rock, Tim Herzog makes a living making beer. The former art teacher launched Flying Bison Brewing Co. in 1995, and he spends much of each day nurturing the business. At age 51, Herzog would like Buffalo to return to its halcyon days, when this proud city boasted 35 breweries. Herzog's a church guy, who describes himself as a situational drinker, marking 28 years of marriage.
Tim Herzog: Yeah, I like it. I like going to Mass. I was an altar boy. There was Latin Mass. Now I'm an usher, and there's this guy that I usher with on Sundays. He's 82 years old. I went to his wedding four years ago. He's robbing the cradle. The woman he married is 73.
>PT: You should be in the choir, your voice is so powerful.
TH: I went to an old-time Catholic school in downtown Rochester, and every day we had to stand up and read out loud. The rest is my dad. He has a legitimate singing voice. He sang in the choir at our church. He has a deep voice, even at 80.
>PT: What quality must a brewer have?
TH: You have to pay attention to color and smell and taste. I like paying attention. My college degree is in art education and graphic design. I look at a painting and really see it. People who make really good beer usually are people who have a refined palate. The guy we have at Flying Bison is a classically trained French chef.
>PT: Tell me your philosophy on beer.
TH: I approach it from the artistic side. What should it smell like? What should it taste like? Then I back into the start of the recipe from there.
>PT: Was the brewery a midlife crisis?
TH: No. I don't think I'm at midlife yet. I have a friend who bought a yellow sports car. Another guy I know bought a Harley. They admit these are toys, maybe a midlife crisis thing. I don't have to buy anything or do whatever this midlife crisis things is about.
>PT: What is the last thing you bought for yourself?
TH: A rugby jersey when I was in Scotland. I started playing rugby when I was in college. I still have really good friends I met through rugby. Once the kids came along, that was it for rugby, but I've always enjoyed the idea of it. It's not played for money anywhere in the world. It's played for beer and pride. Rugby is a lot about paying attention.
>PT: Do you work out?
TH: I used to run a lot, and I took ballet when I was in [Buffalo State] college. The way I pick up kegs has to do with what I learned in ballet class. I don't have a lot of unstructured time. My job chews up a lot. The brewery will be 10 years old in spring.
>PT: Would you ever return to teaching?
TH: Sometimes I kid my wife about that, but when I couldn't get a job as an art teacher, I didn't want to go back to school to broaden the degree. If I had stuck with the teaching thing, I could retire next year. Do I miss it? I don't think so. If the brewing industry suddenly dried up and went away, I guess I'd have to.
>PT: Where has beer taken you?
TH: Literally everywhere. Once a year I lead a beer tour. My wife comes, too. Last year we went to Germany, and I visited some obscure cities known for beer. I got to see where my relatives used to live. Somewhere way back in time, my relatives were the royal family of Bavaria. This year we went to Scotland. I got to have a beer in a pub that's been a pub since 1400-something, and it's been serving beer continuously.
>PT: What's better than a tall cold one?
TH: A tall cold one with some friends. It's not just about the glass of beer in front of you, but it's about all that history and tradition and art and culture that made it. The older I get, the more I enjoy people around me, whether it's my parents or my kids. Now that my kids are old enough to drink legally, it's fun.
>PT: You must know a lot of people.
TH: That's one of the great things about being in the beer industry. I've met the mayor. I've met Brian Higgins. Our garbage man buys our beer. He gets a growler filled every once in a while on Saturday.
>PT: What's its shelf life?
>PT: How are you like beer?
TH: Always developing. Beer always changes. There are beers that are meant to be aged, and beers that are meant to be served fresh. As it ages, there are some things about the beer that get worse, some things get better. My knees get worse. My appreciation gets better.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,822
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Cricotopus elegans est une espèce de diptères nématocères de la famille des Chironomidae et de la sous-famille des Orthocladiinae.
Elle est trouvée en Europe et aux États-Unis. Les larves minent les feuilles des plantes aquatiques du genre Potamogeton.
Références
Liens externes
Chironomidae
Espèce de Diptères (nom scientifique)
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,649
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Q: ErrorException Array to string conversion in Laravel form submission with repeater fields In my Laravel application, I have the following form to submit data:
{!! Form::open(array('route' => 'testresults.store','method'=>'POST')) !!}
<div class="row">
<div class="col-xs-12 col-sm-12 col-md-12">
<div class="form-group">
<strong>Test ID:</strong>
@foreach ($getTests as $object)
{!! Form::text('test_id', ''.$object->id .'', array('placeholder' => 'Test Name','class' => 'form-control','readonly')) !!}
@endforeach
</div>
</div>
<div class="col-xs-12 col-sm-12 col-md-12">
<table class="table table-bordered">
<tr>
<th>Test</th>
<th width="280px">Result</th>
</tr>
@foreach ($getTests as $key => $getTest)
@foreach (explode(',', $getTest->samp_list) as $samp_list)
<tr>
<td>
{!! Form::text('test_type[]', ''.$samp_list.'', array('placeholder' => 'Test Type','class' => 'form-control','readonly')) !!}
</td>
<td>{!! Form::text('test_result[]', null, array('placeholder' => 'Test Result','class' => 'form-control')) !!}</td>
</tr>
@endforeach
@endforeach
</table>
</div>
<div class="col-xs-12 col-sm-12 col-md-12">
<div class="form-group">
<strong>Test By:</strong>
{!! Form::text('test_by', ''.Auth::user()->name .'', array('placeholder' => 'Test Name','class' => 'form-control','readonly')) !!}
</div>
</div>
<div class="col-xs-12 col-sm-12 col-md-12 text-center">
<button type="submit" class="btn btn-primary">Submit</button>
</div>
</div>
{!! Form::close() !!}
This is my create.blade.php
And my controller looks like this:
public function store(Request $request)
{
try{
request()->validate([
'test_id' => 'required',
'test_type' => 'required',
'test_result' => 'required',
'test_by' => 'required',
]);
TestResult::create($request->all());
return redirect()->route('testresults.index')
->with('success','Test Type created successfully.');
}
catch(Exception $e){
return redirect()->route('testresults.create')
->with('failed','An error has been occured.');
}
}
Now the problem is, whenever I tried to submit the data, it gives me an error saying
ErrorException
Array to string conversion
test_type and test_result fields are repeaters. Due to that, I've used those field names as test_type[] and test_result[].
A: As you're trying to save an array in your database, you need to define a text or json column in your database and cast the field to one of array, json or collection types.
Define cast in TestResult model:
TestResult
{
protected $casts = [
'test_type' => 'array',
'test_result' => 'array',
];
}
Learn more about casting in laravel models from here
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,183
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Q: How to show County Codes using auto complete as [County Code], County Name, State Abberevation The county code should match only the starting characters not the characters in between. For example, If the user types "UN", it can only list counties starting with "UN"
thanks in advance
A: If its a TextBox you are using you want to use the TextChanged event
private void textBox_TextChanged(object sender, EventArgs e)
{
foreach(string County in MyCountyList)
{
if(County.StartsWith(textBox.Text))
{
//Do work (Add to list or ComboBox or whatever autocompletion you require)
}
}
}
If you are using a ComboBox, don't forget to clear the list each time the user changes the text entered. Hope this helps!
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,341
|
package org.scaleborn.linereg.calculation.statistics;
import org.elasticsearch.test.ESTestCase;
import org.junit.Test;
import org.scaleborn.linereg.TestModels;
import org.scaleborn.linereg.TestModels.TestModel;
import org.scaleborn.linereg.estimation.DerivationEquationSolver.EstimationException;
/**
* Tests for {@link StatsCalculator}.
* Created by mbok on 19.03.17.
*/
public class StatsBuilderTests extends ESTestCase {
@Test
public void testStats() throws EstimationException {
testStatsForModel(TestModels.SIMPLE_MODEL_1);
testStatsForModel(TestModels.MULTI_FEATURES_2_MODEL_1);
testStatsForModel(TestModels.MULTI_FEATURES_3_MODEL_1);
testStatsForModel(TestModels.MULTI_FEATURES_6_LONGLEY);
}
private void testStatsForModel(final TestModel testModel) throws EstimationException {
final StatsModel linearModel = testModel.evaluateModel();
final Statistics statistics = new StatsCalculator().calculate(linearModel);
testModel.assertStatistics(statistics);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,050
|
[via](http://auvertaveclili.fr/gateau-renverse-poires-avec-ou-sans-gluten-vegan/)
280 g de poires + 2 poires (soit 4 à 5 poires selon la taille)
60 g de sirop d'érable + 1 c. à soupe
2 c. à soupe de jus de citron
2 c. à soupe d'eau
60 g d'huile de coco fondue
40 g de sucre complet
2 c. à café bombées de poudre levante
1 petite c. à café de bicarbonate alimentaire
120 g de farine de blé T80 ou de farine de riz complet
1 c. à café de fève tonka râpée*
180° 30min
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,090
|
\section{Introduction}
Classification in high-dimensional data is a common problem which has created new challenges for traditional statistical methods. For instance, the classification of leukemia data \citep{golub1999} is a classic high-dimensional example in which there are 7129 genes and 72 samples coming from two classes. Due to the small sample size $n$ and large sample dimension $p$, which are often referred to as ``large $p$, small
$n$'' data, estimators of the sample mean and covariance matrix are usually unstable. In a seminal paper by \cite{bickel2004}, linear discriminant analysis (LDA) was proved to be no better than a random guess when $p/n \to \infty$. In the literature, researchers have proposed two classes of independent rules to deal with high-dimensional classification.
A natural method is to ignore the dependence among the variables and this leads to the so-called naive Bayes classifier, see \cite{dudoit2002} or \cite{bickel2004} for more details. This independent rule has also been well studied in many works such as \cite{dudoit2002a}, \cite{tib2002}, and \cite{barry2005}. However, the correlation ignored by the naive Bayes classifier may be very important for classification. This is partially evidenced by \cite{fan2010road}, who comment that the theoretical misclassification rate of the naive Bayes classifier is higher than that of Fisher's rule unless the true population covariance matrix is diagonal.
An alternative approach involves individual analysis. \cite{fan2008high} proposed using the two-sample $t$-statistic to select features. For every feature, a $t$-score is calculated and the features are chosen by their $t$-scores. Similar rules can also be found in \cite{zuber2009gene}, \cite{tibshirani2006correlation}, and \cite{lai2008genome}. In \cite{fan2008high}, the authors proved that the two-sample $t$-statistic could pick up all the differently expressed features. However, those differently expressed features may not be the best features for classification unless the true population covariance matrix is diagonal. For example, \cite{wu2009} pointed out that in gene analysis, most genes are not expressed sufficiently differently that they can be detected by the $t$-statistic.
\cite{fan2010road} and \cite{mai2012} found that the above rules could result in misleading feature selection and inferior classification based on feature selection by the $t$-statistic or the ignorance of correlations among features. As also pointed out in \cite{wu2009}, there is often a group of correlated genes in gene expression analysis in which correlations cannot be ignored, and the covariance information can help to reduce the misclassification rate. Assuming that the population covariance matrix and mean are sparse, a thresholding procedure is used in \cite{shao2011} to estimate parameters and plug these estimators into the LDA. A constrained $l_1$ minimization method is introduced in \cite{cai2011} to estimate the classification direction, and other methods include those of \cite{wu2009}, \cite{tong2011}, \cite{mai2012}, \cite{fan2010road}, \cite{li2001gene}, and \cite{goeman2004global}.
Just as \cite{fan2008high} commented, the difficulty of high- dimensional classification is intrinsically caused by the existence of many noise features that do not contribute to the reduction of the misclassification rate. Thus, if we can select a subset of important features, the high-dimensional classification will become manageable. In gene expression, especially in diagnostic tests, selecting signature genes for accurate classification is essential \citep{yeung2012predicting}. In this article, we study a theoretical rule to capture the discriminant features for classification. Generally, the best $s$ features for classification are those having the same (or almost the same) theoretical misclassification rate as all $p$ features. When the true linear discriminant
direction is sparse, we can select a subset of features having the same misclassification rate as all $p$ features. For the asymptotic sparsity situation, the misclassification rate based on our selected features is also close to the theoretical misclassification rate. Our results show that the main condition used in \cite{fan2010road}, \cite{cai2011}, \cite{mai2012}, and \cite{shao2011} ensures that such a small subset of important features which can be selected to derive a more stable and accurate classification result does exist.
In this work, a two-stage LDA (TLDA) is proposed to learn high- dimensional data. TLDA uses $l_1$ minimization, which is a linear program for selecting important features; LDA will then be constructed based on these selected features. Asymptotic results of the proposed TLDA are studied where the consistency and convergence results are given. Experiments show that, under the same regularity conditions as in \cite{fan2010road}, \cite{cai2011}, and \cite{mai2012}, TLDA achieves a better convergence rate. Simulation studies and experiments on real datasets support our theoretical results and demonstrate that TLDA outperforms existing methods.
The rest of the paper is organized as follows. In Section 2, we investigate the theoretical rule of choosing features and the asymptotic results. Evaluations in simulated data are included in Section 3. In Section 4, TLDA is applied to three real datasets to demonstrate its performance on real data. Finally, we conclude the article in Section 5. All the proofs are given in Appendix.
\section{Methods}
Let $X$ be a $p$-dimensional normal random vector belonging to class $k$ if $X \sim N_p(\mu_k,\Sigma),~k=1,2,$ where $\mu_1 \neq \mu_2$, and $\Sigma$ is a positive definite symmetric matrix.
If $\mu_1,\mu_2$, and $\Sigma$ are known, the optimal classification rule is Fisher's linear discriminant rule
\begin{eqnarray} \label{fish}
\delta_F(X)=I\{(X-\mu_a)^T \Sigma^{-1}\mu_d >0\},
\end{eqnarray}
where $\mu_a=(\mu_1+\mu_2)/2,~\mu_d=(\mu_1-\mu_2)/2$, and $I$ denotes the indicator function with value 1 corresponding to classifying $X$ to class 1 and 0 to class 2. Fisher's rule is equivalent to the Bayes rule with equal prior probabilities for two classes. The misclassification rate of the optimal rule is
\begin{eqnarray} \label{fishm}
R=1-\Phi(\Delta_p^{1/2}),~\Delta_p=\mu_d^T\Sigma^{-1}\mu_d,
\end{eqnarray}
where $\Phi$ is the standard normal distribution function.
In practice, Fisher's rule is typically not directly applicable because the parameters are usually unknown and need to be estimated from the samples. Let $\{X_{1,j},j=1,\cdots,n_1\}$ and $\{X_{2,j},j=1,\cdots,n_2\}$ be independent and identically distributed random samples from $N_p(\mu_1,\Sigma)$ and $N_p(\mu_2,\Sigma)$, respectively. The maximum likelihood estimators of $\mu_1,\mu_2,\Sigma$ are
\begin{eqnarray*}
&&\bar{X}_{k}=\frac{1}{n_k}\sum_{j=1}^{n_k} X_{1,j},~~k=1,2,\\
&&S_n=\frac{1}{n}\sum_{k=1}^{2}\sum_{j=1}^{n_k}(X_{k,j}-\bar{X}_k)(X_{k,j}-\bar{X}_k)^T,
\end{eqnarray*}
where $n=n_1+n_2$, and setting
\begin{eqnarray*}
\hat{\mu}_a=\frac{\bar{X}_1+\bar{X}_2}{2},~\hat{\mu}_d=\frac{\bar{X}_1-\bar{X}_2}{2},
\end{eqnarray*}
and $\Sigma^{-1}=S_n^{-1}$ (or generalized inverse $S_n^-$ when $S_n^{-1}$ does not exist), Fisher's rule becomes the classic LDA
\begin{eqnarray*}
\delta_{LDA}(X)=I\{(X-\hat{\mu}_a)^T S_n^{-1}\hat{\mu}_d>0 \},
\end{eqnarray*}
and the misclassification rate of LDA based on sample $\{X_{1,j},j=1,\cdots,n_1\}$ and $\{X_{2,j},j=1,\cdots,n_2\}$ is
\begin{eqnarray*}
R_{LDA}=\frac{1}{2}\Phi(\frac{(\hat{\mu}_a-\mu_1)S_n^{-1} \hat{\mu}_d}{(\hat{\mu}_d^TS_n^{-1}\Sigma S_n^{-1} \hat{\mu}_d)^{1/2}})+\frac{1}{2}\Phi(-\frac{(\hat{\mu}_a-\mu_2)S_n^{-1} \hat{\mu}_d}{(\hat{\mu}_d^TS_n^{-1}\Sigma S_n^{-1} \hat{\mu}_d)^{1/2}}),
\end{eqnarray*}
which has been well studied when $p$ is fixed; more details can be obtained from \cite{anderson2003}.
For classification, the best $s$ features are those with the largest $\Delta_s$, where $\Delta_s$ is the counterpart of $\Delta_p$. We begin with basic notation and definitions. For a vector $a=(a_1,\cdots,a_p)^T$, we define $|a|_0=\sum_{j=1}^p I(a_j \neq 0)$, $|a|_1=\sum_{j=1}^p |a_j|$, and $|a|_2=\sqrt{\sum_{j=1}^p a_j^2}$. For any index set $\mathcal{A} \subset \{1,\cdots,p\}$, $\mathcal{A}^c=\{j \in \{1,\cdots,p\}:j \not\in \mathcal{A}\}$ and $C$ is denoted as a constant which varies from place to place. For any two index sets $\mathcal{A}$ and $\mathcal{A}'$ and matrix $B$, we use $B_{\mathcal{A} \mathcal{A}' }$ to denote the matrix with rows and columns of $B$ indexed by $\mathcal{A}$ and $\mathcal{A}'$. For a vector $b$, $b_{\mathcal{A}}$ denotes a new vector with elements of $b$ indexed by $\mathcal{A}$. In particular, $\Delta_{\mathcal{A}}=(\mu_d)^T_{\mathcal{A}}(\Sigma^{-1})_{\mathcal{A} \mathcal{A}} (\mu_d)_{\mathcal{A}}$, which dominates the theoretical misclassification rate if we only use
features corresponding to index set $\mathcal{A}$.
The following propositions give solutions to the feature selection problem. Here and below we write $\beta_0=2 \Sigma^{-1}\mu_d$.
\begin{prop} \label{thm1}
Let $\mathcal{A}=\{k:(\beta_0)_k \neq 0\}$. We have
\begin{eqnarray} \label{eq1}
\Delta_{\mathcal{A}}=\mu_d^T\Sigma_p^{-1}\mu_d=\Delta_p.
\end{eqnarray}
\end{prop}
Proposition \ref{thm1} means that the best features are indexed by the support of $\beta_0$. If $\beta_0$ is approximately sparse, which means that many entries of $\beta_0$ are very small, we have the following result.
\begin{prop}\label{thm2}
Assuming that there is a constant $c_0$ (not dependent on $p$) such that $\frac{1}{c_0} \leq all~eigenvalues~of~\Sigma_p \leq c_0$ and there exists $\mathcal{A}_1 \subseteq \{1,2,\cdots,p\}$ satisfying $s_p=\sum_{ k \in \mathcal{A}^c_1} |(\beta_0)_k|^2 \to 0$, we have
\begin{eqnarray}
\Delta_p-\Delta_{\mathcal{A}_1}=O(s_p).
\end{eqnarray}
\end{prop}
Propositions \ref{thm1} and \ref{thm2} provide the theoretical foundations for choosing features, and next we will study how to recover the support of $\beta_0$ from the samples. In other fields, such as compressed sensing and high-dimensional linear regression, constrained $l_1$ minimization has been a common method for reconstructing a sparse signal \citep{donoho2006,candes2007}. In a recent work by \cite{cai2011}, the authors applied $l_1$ minimization to estimate $\beta_0$ directly. However, as \cite{candes2007} pointed out, a two-stage $l_1$ minimization procedure tends to outperform the practical results; more details can be found in the discussions in \cite{candes2007}. Motivated by this, we use $l_1$ minimization in our work to select features and construct LDA on
those selected features.
First, to ensure the identifiability of the important features, we assume that there exists $\mathcal{A} \subseteq \{1,2,\cdots,p\}$ satisfying $p_0=|\mathcal{A}|_0=o(\sqrt{n/\log{p}})$, $(\beta_0)_{\mathcal{A}^c}=0$, and $\min_{k \in \mathcal{A}}{|(\beta_0)_k|} \geq c_p$. Based on the samples, we first consider the $l_1$ minimization method,
\begin{eqnarray} \label{L1}
\hat{\beta} \in \arg\min_{\beta \in R^p}\{|\beta|_1 ~subject~to~|S_n \beta-(\bar{X}_1-\bar{X}_2)|_{\infty} \leq \la_n\},
\end{eqnarray}
where $\la_n$ is a tuning parameter. Second, important features will be selected as
\begin{eqnarray}
\mathcal{A}^{\ast}=\{j: |\hat{\beta}_j| is~among~the~first~largest~p_0~of~all\} .
\end{eqnarray}
Before introducing the asymptotic properties of TLDA, we specify the following regularity conditions
\begin{eqnarray} \label{maincd}
&&c_0^{-1} \leq n_1/n_2 \leq c_0,~c_0^{-1} \leq \la_{min}(\Sigma_p) \leq \la_{max}(\Sigma_p) \leq c_0, \nonumber \\
&&\log{p} \leq n,~\Delta_p \geq c_0^{-1} ~for~some~constant~ c_0>1, ~~~~~~~~~~~~~~
\end{eqnarray}
which are commonly used in high-dimensional settings. Our first result is the consistency of $\mathcal{A}^{\ast}=\mathcal{A}.$
\begin{thm} \label{thm3}
Let $\la_n=C \sqrt{\Delta_p \log{p}/n}$, with $C>0$ being a sufficiently large constant. Suppose that (\ref{maincd}) holds and that $c_p^2/(\Delta_p p_0 \sqrt{\log{p}/n}) \to \infty$. Then
\begin{eqnarray} \label{mthm3}
P(\mathcal{A}^{\ast}=\mathcal{A})=1-O(p^{-1}).
\end{eqnarray}
\end{thm}
From (\ref{mthm3}), we know that the truly important feature set $\mathcal{A}$ will be indexed by $\mathcal{A}^{\ast}$ with a high probability. If the LDA is constructed on those selected features, the following results demonstrate the explicit convergence rate of the misclassification rate based on features $\mathcal{A}^{\ast}$.
\begin{thm} \label{thm4}
Under the assumption of Theorem \ref{thm3}, and applying LDA to features $\mathcal{A}^{\ast}$, denoting the corresponding misclassification rate as $R_{\mathcal{A}^{\ast}}$, then the following hold.\\
(1) $R_{\mathcal{A}^{\ast}}-R \to 0$ in probability.\\
(2) If further assuming $\Delta_p p_0 \sqrt{\log{p_0}/n} \to 0$,
\begin{eqnarray} \label{crate}
\frac{R_{\mathcal{A}^{\ast}}}{R}-1=O(p_0 \Delta_p \sqrt{\log{p_0}/n}),
\end{eqnarray}
with probability greater than $1-O(p^{-1})$.
\end{thm}
\begin{remark}
According to Definition 1 of \cite{shao2011}, with probability greater than $1-O(p^{-1})$, TLDA is asymptotically optimal when $\Delta_p p_0 \sqrt{\log{p_0}/n} \to 0$. Furthermore, the conditions in Theorems \ref{thm3} and \ref{thm4} are similar to those in \cite{fan2010road}, \cite{mai2012}, and \cite{cai2011}, but our method has a better convergence rate. For example, Theorem 3 in \cite{cai2011} shows that $R_n/R-1=O(p_0 \Delta_p \sqrt{\log{p}/{n}})$. Noting that $p_0\ll p$, therefore our results outperform theirs in this case. This means that, compared with estimating $\beta_0$ directly, our two-stage method improves the results in theory.
\end{remark}
\section{Simulations}
In practice, the final LDA depends on parameters $\la_n$ which can be selected by maximizing the cross-validation (CV) as in \cite{cai2011} and $p_0$, which can also be selected by CV. Our algorithms are outlined below.
\begin{algorithm}
\caption{A Two-stage LDA based on $l_1$ minimization}
\begin{algorithmic}[1]
\State Calculating the sample covariance matrix $S_n$ and mean $\bar{X}_{k},k=1,2$;
\State $\hat{\beta}^{\la_n}= \arg\min_{\beta \in R^p} \sum_{k=1}^p |\beta_k| ~subject~to~|S_n \beta-(\bar{X}_1-\bar{X}_2)|_{\infty} \leq \la_n;$
\State Denoting the tuning parameters chosen by five-fold CV as $\hat{\la}_n$ and $\hat{p}_0$. Here we adjust $\hat{\la}_n$ as $\la=\sqrt{4/5} \hat{\la}_n$;
\State $\mathcal{A}^{\ast}=\{j: |\hat{\beta}^{\la}_j| is~among~the~first~largest~\hat{p}_0~of~all\}$;
\State $\beta^{\ast}=((S_n)_{\mathcal{A}^{\ast}\mathcal{A}^{\ast}})^{-1}((\bar{X}_1)_{\mathcal{A}^{\ast}}-(\bar{X}_2)_{\mathcal{A}^{\ast}})$;
\State If $(Y-(\bar{X}_1+\bar{X}_2)/2)_{\mathcal{A}^{\ast}}^T \beta^{\ast}>0$, classifying $Y$ to class 1, else class 2.
\end{algorithmic}
\end{algorithm}
The reason for adjusting $\hat{\la}_n$ as $\la=\sqrt{4/5} \hat{\la}_n$ is due to $\la_n=C \sqrt{\Delta_p \log{p}/n}$, and the fact that the sample size is $4n/5$ but not $n$ in five-fold CV. The simulations reported in Table 4 of \cite{cai2011} also support our adjustment here. Furthermore, the $l_1$ minimization is a linear program which is very attractive for high-dimensional data and can be implemented by many existing programs, such as the function $linprogPD$ included in the R package ``clime", which is available at \url{http://cran.r-project.org/web/packages/clime/index.html}.
We now present the results of simulation studies which were designed
to evaluate the performance of the proposed TLDA. For the purpose of comparison, we also apply several other methods to the data, specifically, linear programming discriminant (LPD) \citep{cai2011}, regularized optimal affine discriminant (ROAD) \citep{fan2010road,wu2009}, and the oracle Fisher's oracle rule (Oracle). The oracle rule is included as a benchmark. The LPD will be solved by the R package clime and the matlab code for ROAD is available at \url{http://www.mathworks.com/matlabcentral/fileexchange/40047}.
In simulations, we fix the sample size $n_1=n_2=100$ and without loss of generality we set $\mu_2=0$. For the true classification direction $\beta_0$, $(\beta_0)_{[(2k-1)/10]}=(-1)^{k+1}(k+1)/4,~k=1,\dots,5$ and all other elements are zero. Two kinds of population covariance matrix will be considered.
\begin{itemize}
\item Model~1. $\Sigma=(\sigma_{ij})_{p \times p}$, where $\sigma_{ij}=0.8^{|i-j|}$ for $1 \leq i,j \leq p$.
\item Model~2. $\Sigma=(\sigma_{ij})_{p \times p}$, where $\sigma_{ii}=1$ for $1 \leq i \leq p$ and $\sigma_{ij}=0.5$ for $i \neq j$.
\end{itemize}
The first simulation is to evaluate the performance of our proposed TLDA method and the two-sample $t$-statistic \citep{fan2008high}. The average misclassification rates based on 100 simulations are reported in Fig. \ref{fig1}, and here $p=100$.
\begin{figure}
\includegraphics[scale=0.70]{fig1.eps}
\caption{Plots for TLDA and the $t$-statistic. Upper: average misclassification rates versus number of selected features; Middle: average $\beta_0$ representing the signal of choosing features by TLDA; Lower: average $\mu_1-\mu_2$ representing the signal of choosing features by the $t$-statistic.}
\label{fig1}
\end{figure}
The figure shows that TLDA always selects more useful features than the two-sample $t$-statistic, which ignores the correlation between features. Specifically, due to correlations, features $30$ and $70$ cannot be detected by the two-sample $t$-statistic for Model 2.
In the second simulation, we study the misclassification rate of our TLDA method. In \cite{cai2011} and \cite{fan2010road}, the authors have conducted many numerical investigations to compare their methods with others, including the oracle features annealed independence rule (OFAIR) \citep{fan2008high} and nearest shrunken centroid (NSC) method \citep{tib2002}, and concluded that their methods perform better. We therefore compare TLDA only with LPD and ROAD and do not consider other classic methods. Table \ref{table1} shows the misclassification rates based on 100 replications for TLDA, LPD, ROAD, naive Bayes (NB) and Oracle.
\begin{table}[h!]
\caption{Average misclassification rates in percentage for sparse situations. Standard deviations are given in parentheses.}
\label{table1}
\begin{tabular}{cccccc}
\hline
$p$&TLDA&LPD&ROAD&NB&Oracle\\
\hline
\multicolumn{6}{c}{Model~1}\\
100&\textbf{13.41}(2.68)&13.58(2.48)&16.68(5.44)&16.94(2.64)&11.59(2.18)\\
200&\textbf{13.31}(2.45)&13.62(2.55)&16.19(5.05)&17.18(2.54)&11.66(2.38)\\
400&\textbf{13.99}(2.56)&14.06(2.69)&17.45(5.49)&18.86(2.67)&11.88(2.39)\\
800&\textbf{14.16}(2.94)&14.93(2.96)&18.22(5.08)&20.56(2.92)&11.74(2.30)\\
\hline
\multicolumn{6}{c}{Model~2}\\
100&\textbf{20.78}(3.01)&21.04(3.14)&25.01(4.47)&35.13(3.02)&18.41(2.66)\\
200&\textbf{20.91}(3.26)&21.58(3.27)&25.49(3.91)&35.92(2.76)&18.55(2.55)\\
400&\textbf{21.49}(3.50)&22.49(3.55)&26.04(3.88)&35.87(2.86)&18.60(2.76)\\
800&\textbf{21.99}(3.70)&23.31(3.75)&26.62(3.71)&36.04(3.03)&18.70(3.13)\\
\hline
\end{tabular}
\end{table}
From Table \ref{table1}, we can see that the performance of TLDA is similar to that of Oracle and is better than that of the other methods. Clearly, due to its fundamental drawback, the naive Bayes is the worst of all methods although it is better than random guess (whose misclassification rate is 50\%). Overall, compared with LPD and ROAD, TLDA has the smallest misclassification rate, and the standard deviation of TLDA is similar to that of LPD but smaller than that of ROAD. When the dimensionality $p$ increases from 100 to 800, TLDA is quite stable, whereas LPD and ROAD become increasingly worse. In particular, TLDA always has a smaller misclassification rate and standard deviation than ROAD. When $p$ is not large, TLDA and LPD have similar performance, while TLDA becomes better than LPD as $p$ increases; in particular when $p$ is sufficiently large (such as $p=800$), the difference between the misclassification rates of TLDA and LPD becomes bigger. In summary, simulations demonstrate that TLDA is a stable and superior classification method
compared to existing methods.
Next, we will study the estimators $\hat{\beta}_{TLDA},\hat{\beta}_{LPD}$, and $\hat{\beta}_{ROAD}$. Fig. \ref{fig2} plots the average estimators of 100 replications. Due to different assumptions, here we adjust $\hat{\beta}_{ROAD}$ to $|\beta_0|^2*\hat{\beta}_{ROAD}$ so that it fits the real situation.
\begin{figure}
\centering
\includegraphics[scale=0.6]{fig2.eps}
\caption{Average estimators of TLDA, LPD and ROAD for $p=100$. The true $\beta_0$ and the estimators are very sparse, which is why there is an almost solid line at zero.}
\label{fig2}
\end{figure}
From Fig. \ref{fig2}, we can see that TLDA correctly selects most of those five features but very few noise features. In particular, compared with LPD, which estimates the true $\beta_0$ directly, our two-stage estimators are much closer to $\beta_0$, which is consistent with the discussions in \cite{candes2007}.
The above simulations are conducted for scenarios where $\beta_0$ is sparse. In practice, it is quite common that there are many weak signals that are correlated with the main signals. It would be interesting to evaluate the performance of TLDA for these approximately sparse situations. Specifically, we will consider two scenarios with respect to $\mu_1$, as follows.
\begin{itemize}
\item Model 3. $\mu_1=(1_5,0_{p-5})$ in Model 1.
\item Model 4. $\beta_0=0.551*(3,1.7,-2.2,-2.1,2.55,(p-5)^{-1}1_{p-5})$ and $\mu_1=\Sigma*\beta_0$ in Model 2.
\end{itemize}
Here $n_1=n_2=100$ and $\mu_2=0$. Model 3 is similar to those in \cite{cai2011} and \cite{fan2010road}, and Model 4 comes from \cite{mai2012}. The average misclassification rates based on 100 replications are reported in Table \ref{tablen1}. It is again evident that TLDA performs favorably compared to existing
methods.
\begin{table}[h!]
\caption{Average misclassification rates in percentage for approximately sparse simulations. Standard deviations are given in parentheses.}
\label{tablen1}
\begin{tabular}{cccccc}
\hline
$p$&TLDA&LPD&ROAD&NB&Oracle\\
\hline
\multicolumn{6}{c}{Model 3}\\
100&\textbf{20.70}(3.12)&22.69(3.67)&26.85(5.91)&31.46(4.07)&18.56(2.54)\\
200&\textbf{20.89}(3.11)&24.03(3.83)&27.52(5.37)&33.74(3.68)&18.98(2.65)\\
400&\textbf{20.96}(3.18)&25.03(3.77)&28.03(5.36)&36.61(3.69)&18.65(2.59)\\
800&\textbf{21.75}(4.56)&26.77(4.60)&28.73(5.14)&40.71(3.63)&18.80(2.69)\\
\hline
\multicolumn{6}{c}{Model 4}\\
100&\textbf{11.99}(2.68)&12.30(2.59)&14.57(3.33)&21.87(2.68)&9.98(2.07)\\
200&\textbf{12.64}(2.58)&13.04(2.67)&15.15(3.19)&22.17(2.97)&10.60(2.06)\\
400&\textbf{12.70}(2.64)&13.52(2.40)&15.56(3.09)&22.28(2.79)&10.03(2.17)\\
800&\textbf{12.90}(3.01)&13.85(2.94)&15.35(3.75)&22.33(3.11)&10.08(2.21)\\
\hline
\end{tabular}
\end{table}
\section{Real data}
In this section, we apply the proposed TLDA to real datasets. Since real data usually has an ultra-high data dimension $p$, a sure independence screening (SIS) method \citep{fan2008sure} will be carried out before our proposed feature selection procedure to further improve the accuracy and control the computational cost. For brevity, we will apply the two-sample $t$-test statistic \citep{tib2002, fan2008high} to reduce the dimensionality from ultra-high to a moderate scale. Other screening steps such as that in \cite{fan2010road} can also be used, but we do not pursue them in detail.
First, TLDA is applied to study leukemia data, which is available at \url{http://www.broadinstitute.org/cgi-bin/cancer/datasets.cgi.} The dataset contains $p=7129$ genes for $n_1=27$ acute lymphoblastic leukemia (ALL) samples and $n_2=11$ acute myeloid leukemia (AML) samples in the training set; the test set consists of 20 ALL samples and 14 AML samples. More details can be found in \cite{golub1999}. By following similar pre-processing steps as \cite{dudoit2002} and \cite{fan2008high}, we standardize each sample to zero mean and
$S_n=\frac{1}{n}\sum_{k=1}^{2}\sum_{j=1}^{n_k}(X_{k,j}-\bar{X}_k)(X_{k,j}-\bar{X}_k)^T$ has unit diagonal elements.
For comparison with LPD in \cite{cai2011}, we use 2867 genes with the largest absolute values of the two-sample $t$-statistic ($|\mu_1-\mu_2|>0.5$). Fig. \ref{fig3} shows the mean difference and estimator $\hat{\beta}_0$ (tuning parameter $\la=1.2$), representing the feature selection signals of the two-sample $t$-statistic and TLDA, respectively.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.80]{fig3.eps}
\caption{True mean difference and estimator $\hat{\beta}_0$ of leukemia data.}
\label{fig3}
\end{figure}
Clearly, the signal for TLDA is sparse, while the signal for the two-sample $t$-statistic has no clear clues.
The classification results for TLDA, LPD, ROAD, OFAIR, NSC, and NB are shown in Table \ref{table3}.
\begin{table}
\caption{Classification errors of leukemia data by various methods}
\label{table3}
\begin{tabular}{ccccccc}
\hline
&TLDA&LPD&ROAD&OFAIR&NSC&NB\\
\hline
Training~Error&0/38&0/38&0/38&1/38&1/38&0/38\\
Test~Error&1/34&1/34&1/34&1/34&3/34&5/34\\
No. of Selected genes&8&151&40&11&24&7129\\
\hline
\end{tabular}
\end{table}
Table \ref{table3} shows that TLDA performs competitively in classification error with LPD and ROAD. However, TLDA only selects 8 genes, in contrast to 40 genes by ROAD and 151 genes by LPD. The 8 selected genes and their TLDA weights are given in Table \ref{table4}. For comparison, we also present their $t$-statistic rank in the 7129 genes.
\begin{table}
\centering
\caption{The eight genes of leukemia data selected by TLDA.}
\label{table4}
\begin{tabular}{cccccc}
\hline
Gene position & & TLDA weights& &Rank of $t$-statistic\\
\hline
461& &-3.203& &7\\
1779& &-4.455& &87\\
1834& & -5.039& &6\\
3320& & -0.960& &1\\
3525& &-3.876& &138\\
4847& &-6.389& &2\\
5039& &-1.187& & 4\\
6539& &-7.9933& &21\\
\hline
\end{tabular}
\end{table}
We further compare the methods on two more real datasets: the colon \citep{srivastava2007comparison} and breast cancer \citep{hess2006pharmacogenomic} datasets. A leave-one-out cross validation (LOOCV) is performed on the two datasets. For $i = 1, \cdots, n$ , the $p \times 1$ vector $x_i$ is treated as the testing set, while the remaining $n-1$ observations form the training set. A subset of 1000 genes is selected based on the two-sample t-statistic. The classification results for the TLDA, LPD, ROAD, and NB methods are shown in Table \ref{table5}. We can see that, on each dataset, the proposed TLDA has a competitive performance in terms of classification errors while using the fewest genes. Overall,
TLDA is also applicable in real datasets and performs favorably in comparison to existing methods.
\begin{table}[h!]
\centering
\caption{Classification error and number of genes selected by various methods for the colon and breast cancer datasets}
\label{table5}
\begin{tabular}{cccccc}
\hline
& & TLDA & LPD& ROAD&NB\\
\hline
Colon& Error(\%)&9.68&9.68&11.29&14.52\\
& No. of genes & 7.42(1.03)& 168.95(71.39)& 38.10(27.60)& 1000(0)\\
Breast & Error(\%)&21.80&25.56&31.58&34.59\\
& No. of genes & 14.61(2.40)& 332.45(103.56)& 44.14(47.26)& 1000(0)\\
\hline
\end{tabular}
\end{table}
\section{Discussions}
In this paper, we have proposed a solution for feature selection in high-dimensional data.
We have derived the optimal feature selection rule for LDA and proposed the selection of features based on the sparsity of $\Sigma^{-1}\mu_d$. An $l_1$ minimization method is used on the samples to select the important features and LDA is then applied to those selected features. Our proposed TLDA performs favorably compared to existing methods in theory and application. Our analysis shows that the independent rules such as the two-sample $t$-statistic and naive Bayes may not be efficient and may even lead to bad classifiers.
Suppose that there are $K>2$ classes (in this article we assume that $K=2$), our TLDA is also applicable. For this, $X$ will be classified to class $k$ if and only if
\begin{eqnarray}
(X-(\bar{X}_k+\bar{X}_l)/2)_{\mathcal{A}_{kl}^{\ast}}^T \beta_{kl}^{\ast}>0~for~all~k \neq l.
\end{eqnarray}
Moreover, the procedure can be extended to unequal prior probabilities $\pi_1$ and $\pi_2$ in which we classify $X$ to class 1 when
\begin{eqnarray}
(X-(\bar{X}_1+\bar{X}_2)/2)_{\mathcal{A}^{\ast}}^T \beta^{\ast}>\log{(\pi_2/\pi_1)},
\end{eqnarray}
where the parameters can also be estimated as $\hat{\pi}_1=n_1/n$ and $\hat{\pi}_2=n_2/n$. For non-Gaussian distributions, we can also derive similar results under the moment conditions, as in \cite{cai2011}.
Finally, we note that the number of selected features is $p_0=o(\sqrt{n /\log{p}})$ which is very small compared to $p$. Setting $n=O((\log{p})^\beta)$ for $\beta>1$, this means that only $o((\log{p})^{(\beta-1)/2})$ features can be selected from $p$ variables to apply LDA. This is due to the fact that LDA is stable only when $p_0 \sqrt{p_0/n} \to 0$, and a detailed result can be found in \cite{shao2011}. Our future research will focus on improving $p_0$.
\section*{Acknowledgments}
We are grateful for the valuable comments from the reviewers and editors. Cheng Wang's research was supported by NSF of China Grants (No. 11101397, 71001095 and 11271347). Longbing Cao's research was supported by Australian Research Council Discovery Grants (DP1096218 and DP1301691) and Australian Research Council Linkage Grant (LP100200774).
\section*{Appendix A: Proofs}
\subsection*{A.1. Proof of Theorem \ref{thm3}}
From the proofs of Theorem 2 in \cite{cai2011}, we know that
\begin{eqnarray} \label{ap1}
(\hat{\beta}-\beta_0)^T \Sigma (\hat{\beta}-\beta_0) \leq C |\beta_0|_1^2 \sqrt{\log{p}/n}+6 \la_n |\beta_0|_1,
\end{eqnarray}
with probability greater than $1-O(p^{-1})$. Using the Cauchy-Schwartz inequality,
\begin{eqnarray*}
&&|\beta_0|_1^2 \leq |\beta_0|_0 |\beta_0|_2^2 \leq c_0 p_0 (\beta_0^T \Sigma \beta_0)= 4 c_0 p_0 \Delta_p ,\\
&& (\hat{\beta}-\beta_0)^T \Sigma (\hat{\beta}-\beta_0) \geq c_0^{-1} (\hat{\beta}-\beta_0)^T (\hat{\beta}-\beta_0).
\end{eqnarray*}
Together with (\ref{ap1}), we have
\begin{eqnarray}
(\hat{\beta}-\beta_0)^T (\hat{\beta}-\beta_0) \leq C p_0\Delta_p \sqrt{\log{p}/n},
\end{eqnarray}
with probability greater than $1-O(p^{-1})$. For $j \in \mathcal{A}$,
\begin{eqnarray*}
|\hat{\beta}_j-(\beta_0)_j|^2 \leq C p_0\Delta_p \sqrt{\log{p}/n}.
\end{eqnarray*}
Then
\begin{eqnarray*}
|\hat{\beta}_j| &\geq& |(\beta_0)_j|-\sqrt{C p_0\Delta_p \sqrt{\log{p}/n}}\\
&\geq& c_p(1-\sqrt{C p_0\Delta_p \sqrt{\log{p}/n}}/c_p )\\
&>& c_p/2.
\end{eqnarray*}
Similarly, for $j \in \mathcal{A}^c$,
\begin{eqnarray*}
|\hat{\beta}_j| \leq \sqrt{C p_0\Delta_p \sqrt{\log{p}/n}} < c_p/2.
\end{eqnarray*}
Hence, we have proved that $P(\mathcal{A}^{\ast}=\mathcal{A})=1-O(p^{-1})$.
\subsection*{A.2. Proof of Theorem \ref{thm4}}
Applying the features selector $\mathcal{A}^{\ast}$ to the sample $\{X_{1,j},j=1,\cdots,n_1\}$ and $\{X_{2,j},j=1,\cdots,n_2\}$, we still denote the corresponding data as $X, \{X_{k,j},k=1,2\}$ for brevity. It is noted that here the dimension is $p_0$ not $p$. Setting
\begin{eqnarray*}
&&\bar{X}_{k}=\frac{1}{n_k}\sum_{j=1}^{n_k} X_{1,j},~~k=1,2,\\
&&S_n=\frac{1}{n}\sum_{k=1}^{2}\sum_{j=1}^{n_k}(X_{k,j}-\bar{X}_k)(X_{k,j}-\bar{X}_k)^T,
\end{eqnarray*}
and
\begin{eqnarray*}
\hat{\mu}_a=\frac{\bar{X}_1+\bar{X}_2}{2},~\hat{\mu}_d=\frac{\bar{X}_1-\bar{X}_2}{2}.
\end{eqnarray*}
The LDA procedure is
\begin{eqnarray*}
\delta_{LDA}(X)=I\{(X-\hat{\mu}_a)^T S_n^{-1}\hat{\mu}_d \},
\end{eqnarray*}
and the misclassification rate is
\begin{eqnarray*}
R_{\mathcal{A}^{\ast}}=\frac{1}{2}\Phi(\frac{(\hat{\mu}_a-\mu_1)S_n^{-1} \hat{\mu}_d}{(\hat{\mu}_d^TS_n^{-1}\Sigma S_n^{-1} \hat{\mu}_d)^{1/2}})+\frac{1}{2}\Phi(-\frac{(\hat{\mu}_a-\mu_2)S_n^{-1} \hat{\mu}_d}{(\hat{\mu}_d^TS_n^{-1}\Sigma S_n^{-1} \hat{\mu}_d)^{1/2}}).
\end{eqnarray*}
By the proofs of Theorem 1 in \cite{shao2011}, we know that
\begin{eqnarray*}
\frac{(\hat{\mu}_a-\mu_1)S_n^{-1} \hat{\mu}_d}{(\hat{\mu}_d^TS_n^{-1}\Sigma S_n^{-1} \hat{\mu}_d)^{1/2}}=-\Delta_p^{1/2}(1+O(p_0 \sqrt{\log{p_0}/n})),
\end{eqnarray*}
and a similar result also holds for $\Phi(\frac{(\hat{\mu}_a-\mu_1)S_n^{-1} \hat{\mu}_d}{(\hat{\mu}_d^TS_n^{-1}\Sigma S_n^{-1} \hat{\mu}_d)^{1/2}})$. Then
\begin{eqnarray}
R_{\mathcal{A}^{\ast}}=\Phi(-\Delta_p^{1/2}(1+O(p_0 \sqrt{\log{p_0}/n}))).
\end{eqnarray}
Noting that $p_0 \sqrt{\log{p_0}/n} \to 0$, therefore, in probability,
\begin{eqnarray}
R_{\mathcal{A}^{\ast}}-R \to 0.
\end{eqnarray}
From equation (12) of \cite{cai2011}, we know that
\begin{eqnarray*}
|\frac{\Phi(\frac{(\hat{\mu}_a-\mu_1)S_n^{-1} \hat{\mu}_d}{(\hat{\mu}_d^TS_n^{-1}\Sigma S_n^{-1} \hat{\mu}_d)^{1/2}})}{\Phi(-\Delta_p^{1/2})}-1|
\leq O(\Delta_p p_0 \sqrt{\log{p_0}/n})e^{O(\Delta_p p_0 \sqrt{\log{p_0}/n})}.
\end{eqnarray*}
Then
\begin{eqnarray*}
|\frac{R_{\mathcal{A}^{\ast}}}{R}-1|\leq O(\Delta_p p_0 \sqrt{\log{p_0}/n})e^{O(\Delta_p p_0 \sqrt{\log{p_0}/n})}.
\end{eqnarray*}
When $\Delta_p p_0 \sqrt{\log{p_0}/n} \to 0$, we get
\begin{eqnarray}
|\frac{R_{\mathcal{A}^{\ast}}}{R}-1|=O(\Delta_p p_0 \sqrt{\log{p_0}/n}).
\end{eqnarray}
The proof is completed.
\bibliographystyle{elsarticle-harv}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section{Introduction}\label{sec:prologo}
\setcounter{equation}{0}
%
EPW-sextics are defined as follows. Let $V$ be a $6$-dimensional complex vector space. Choose a volume-form
$\vol\colon\bigwedge^6 V\overset{\sim}{\lra}\CC$
and equip $\bigwedge^3 V$ with the symplectic form
\begin{equation}
(\alpha,\beta)_V:=\vol(\alpha\wedge\beta).
\end{equation}
Let $\lagr$ be the symplectic Grassmannian parametrizing Lagrangian subspaces of $\bigwedge^3 V$ - of course $\lagr$ does not depend on the choice of volume-form.
Let $F\subset\bigwedge^3 V\otimes\cO_{\PP(V)}$ be the sub vector-bundle
with fiber
\begin{equation}
F_v:=\{\alpha\in\bigwedge^3 V\mid v\wedge\alpha=0\}
\end{equation}
over $[v]\in\PP(V)$. Notice that $(,)_V$ is zero on $F_v$ and $2\dim(F_v)=20=\dim\bigwedge^3 V$; thus $F$ is a Lagrangian sub vector-bundle of the trivial symplectic vector-bundle on $\PP(V)$ with fiber $\bigwedge^3 V$.
Next choose $A\in\lagr$. Let
%
\begin{equation}\label{diecidieci}
F\overset{\lambda_A}{\lra}(\bigwedge^3 V/A)\otimes\cO_{\PP(V)}
\end{equation}
be the composition of the inclusion $F\subset\bigwedge^3 V\otimes\cO_{\PP(V)}$ followed by the quotient map.
Since $\rk F=\dim(V/A)$
the determinat of $\lambda_A$ makes sense.
Let
%
\begin{equation*}
Y_A:=V(\det\lambda_A).
\end{equation*}
A straightforward computation gives that $\det F\cong\cO_{\PP(V)}(-6)$ and hence
$\det\lambda_A\in H^0(\cO_{\PP(V)}(6))$.
It follows that if $\det\lambda_A\not=0$ then $Y_A$ is a sextic hypersurface. As is easily checked $\det\lambda_A\not=0$ for $A\in\lagr$ generic - notice that there exist \lq\lq pathological\rq\rq $A$'s such that $\lambda_A=0$ e.g.~$A=F_{v_0}$.
An {\it EPW-sextic} (after Eisenbud, Popescu and Walter~\cite{epw}) is a sextic hypersurface in $\PP^5$ which is projectively equivalent to $Y_A$ for some $A\in\lagr$.
Let $Y_A$ be an EPW-sextic: one constructs a double cover of $Y_A$ as follows. Since $A$ is Lagrangian the symplectic form defines a canonical isomorphism $\left(\bigwedge^3 V/A\right)\cong A^{\vee}$; thus~\eqref{diecidieci} defines a map of vector-bundles $\lambda_A\colon F\to A^{\vee}\otimes\cO_{\PP(V)}$.
Let $i\colon Y_A\hra\PP(V)$ be the inclusion map:
since a local generator of $\det\lambda_A$ annihilates $\coker (\lambda_A)$ there is a unique sheaf $\zeta_A$ on $Y_A$ such that we have an exact sequence
\begin{equation}\label{eccozeta}
0\lra F\overset{\lambda_A}{\lra} A^{\vee}\otimes\cO_{\PP(V)}\lra
i_{*}\zeta_A\lra 0.
\end{equation}
Let
\begin{equation}\label{patte}
\xi_A:=\zeta_A(-3).
\end{equation}
We will equip $\cO_{Y_A}\oplus\xi_A$ with a structure of commutative $\cO_{Y_A}$-algebra.
Choose $B\in\lagr$ transversal to $A$; thus we have a direct-sum decomposition
\begin{equation}\label{asommab}
\bigwedge^3 V=A\oplus B.
\end{equation}
Then $(,)_V$ defines an isomorphism $B\cong A^{\vee}$. Decomposition~(\ref{asommab}) defines a projection map $\bigwedge^3 V\to A$; thus we get a map $ \mu_{A,B}\colon F\to A\otimes\cO_{\PP(V)}$.
We claim that there is a commutative diagram with exact rows
%
\begin{equation}\label{spqr}
\begin{array}{ccccccccc}
0 & \to & F&\mapor{\lambda_A}& A^{\vee}\otimes\cO_{\PP(V)} & \lra & i_{*}\zeta_A
&
\to & 0\\
& & \mapver{\mu_{A,B}}& &\mapver{\mu^{t}_{A,B}} &
&
\mapver{\beta_{A}}& & \\
0 & \to & A\otimes\cO_{\PP(V)}& \mapor{\lambda_A^{t}}& F^{\vee} & \lra &
Ext^1(i_{*}\zeta_A,\cO_{\PP(V)}) & \to & 0
\end{array}
\end{equation}
In fact the second row is obtained by applying the $Hom(\,\cdot\, ,\cO_{\PP(V)})$-functor to~(\ref{eccozeta}) and the equality $\mu_{A,B}^t\circ\lambda_A=\lambda_A^t\circ\mu_{A,B}$ holds because
$F$ is a Lagrangian sub-bundle of $\bigwedge^3 V\otimes\cO_{\PP(V)}$.
Lastly $\beta_A$ is defined to be the unique map making the diagram commutative; it exists because the rows are exact. Notice that the map $\beta_A$ is independent of the choice of $B$ as suggested by the notation.
Next by applying the $Hom(i_{*}\zeta_A,\,\cdot\,)$-functor to the exact sequence
\begin{equation}
0\lra\cO_{\PP(V)}\lra\cO_{\PP(V)}(6)\lra\cO_{Y_A}(6)\lra 0
\end{equation}
we get the exact sequence
\begin{equation}\label{romaleone}
0\lra i_{*}Hom(\zeta_A,\cO_{Y_A}(6))\overset{\partial}{\lra}
Ext^1(i_{*}\zeta_A,\cO_{\PP(V)})\overset{n}{\lra} Ext^1(i_{*}\zeta_A,\cO_{\PP(V)}(6))
\end{equation}
where $n$ is locally equal to multiplication by $\det\lambda_A$. Since the second row of~(\ref{spqr}) is exact a local generator of $\det\lambda_A$ annihilates $Ext^1(i_{*}\zeta_A,\cO_{\PP(V)})$; thus $n=0$ and hence we get a canonical isomorphism
\begin{equation}\label{extugualehom}
\partial^{-1}\colon
Ext^1(i_{*}\zeta_A,\cO_{\PP(V)})\overset{\sim}{\lra} i_{*}Hom(\zeta_A,\cO_{Y_A}(6)).
\end{equation}
We define $\wt{m}_A$ by setting
\begin{equation}\label{emmea}
\begin{matrix}
\zeta_A\times\zeta_A & \overset{\wt{m}_A}\lra & \cO_{Y_A}(6) \\
(\sigma_1,\sigma_2) & \mapsto & (\partial^{-1}\circ\beta_A(\sigma_1))(\sigma_2).
\end{matrix}
\end{equation}
Let $\xi_A$ be given by~(\ref{patte}).
Tensorizing both sides of~(\ref{emmea}) by $\cO_{Y_A}(-6)$ we get a multiplication map
\begin{equation}\label{aridaje}
\xi_A\times\xi_A \overset{m_A}\lra \cO_{Y_A}.
\end{equation}
Thus we have defined a multiplication map on $\cO_{Y_A}\oplus\xi_A$.
\begin{prp}\label{prp:mappabeta}
Let $A\in\lagr$ and suppose that $Y_A\not=\PP(V)$.
Let notation be as above. Then:
\begin{itemize}
\item[(1)]
$\beta_A$ is an isomorphism.
\item[(2)]
The multiplication map $m_A$ is associative and commutative and hence it equips $\cO_{Y_A}\oplus\xi_A$ with a structure of commutative ring.
\end{itemize}
\end{prp}
\Ref{prp}{mappabeta} will be proved in~\Ref{subsec}{prodenne}. In fact we will give an explicit formula for~\eqref{aridaje} (known to experts, see~\cite{catanese}).
Granting~\Ref{prp}{mappabeta} we let
\begin{equation}
X_A:=\Spec(\cO_{Y_A}\oplus\xi_A),\qquad
f_A\colon X_A\to Y_A
\end{equation}
where $f_A$ is the structure morphism. Then $X_A$ is a {\it double EPW-sextic} and $f_A$ is its structure map.
The {\it covering involution} of $X_A$ is the automorphism $\phi_A\colon X_A\to X_A$ corresponding to the involution of
$\cO_{Y_A}\oplus\xi_A$ with $(-1)$-eigensheaf equal to $\xi_A$. Let
\begin{eqnarray}
Y_A(k)= & \{[v]\in\PP(V)\mid \dim(A\cap F_v)= k\}, \\
Y_A[k]= & \{[v]\in\PP(V)\mid \dim(A\cap F_v)\ge k\}.
\end{eqnarray}
Thus $Y_A(0)=(\PP(V)\setminus Y_A)$ and $Y_A=Y_A[1]$. Clearly $Y_A(1)$ is open in $Y_A$, it is dense in $Y_A$ if $A$ is generic.
\begin{clm}\label{clm:rivestimento}
The map
$f_A^{-1}(Y_A(1))\to Y_A(1)$ defined by restriction of $f_A$ is a topological covering of degree $2$ and its non-trivial deck transformation is equal to
the restriction of $\phi_A$ to $f_A^{-1}(Y_A(1))$.
\end{clm}
\begin{proof}
The restriction of $\xi_A$ to $Y_A(1)$ is an invertible sheaf. By Item~(1) of~\Ref{prp}{mappabeta} the restriction of $m_A$ to $Y_A(1)$ defines an isomorphism $\xi_A\otimes\xi_A\cong\cO_{Y_A}$; the claim follows.
\end{proof}
Double EPW-sextics come with a natural polarization; we let
\begin{equation}\label{ecceampio}
\cO_{X_A}(n):=f_A^{*}\cO_{Y_A}(n),\quad
H_A\in |\cO_{X_A}(1)|.
\end{equation}
Let
\begin{eqnarray}
\Sigma:= & \{A\in\lagr\mid \text{$\exists W\in{\mathbb G}r(3,V)$ s.~t.~$\bigwedge^3 W\subset A$}\},\\
\Delta:= & \{A\in\lagr\mid Y_A[3]\not=\es\}.
\end{eqnarray}
Then $\Sigma$ and $\Delta$ are closed subsets of $\lagr$. A straightforward computation, see~\cite{og5}, gives that $\Sigma$ is irreducible of codimension $1$. A similar computation, see~\Ref{prp}{codelta}, gives that $\Delta$ is irreducible of codimension $1$ and distinct from $\Sigma$.
Let
\begin{equation}\label{eccozero}
\lagr^0:= \lagr\setminus\Sigma\setminus\Delta\,.
\end{equation}
Thus $\lagr^0$ is open dense in $\lagr$.
In~\cite{og2} we proved that if $A\in\lagr^0$ then $X_A$ is a hyperk\"ahler (HK) $4$-fold which can be deformed to $(K3)^{[2]}$.
Moreover we showed that the family of polarized HK $4$-folds
%
\begin{equation*}
\{ (X_A,H_A)\}_{A\in \lagr^0}
\end{equation*}
is locally complete. Three other explicit locally complete families of projective HK's of dimension greater than $2$ are known - see~\cite{beaudon,debvoi,iliran1,iliran2}. In all of the examples the HK manifolds are deformations of the Hilbert square of a $K3$: they are distinguished by the value of the Beauville-Bogomolov form on the polarization class (it equals $2$ in the case of double EPW-sextics and $6$, $22$ and $38$ in the other cases).
In the present paper we will analyze $X_A$ for $A\in\Delta$, mainly under the hypothesis that $A\not\in\Sigma$.
Let $A\in(\Delta\setminus\Sigma)$. We will prove the following results
%
\begin{enumerate}
\item[(1)]
$Y_A[3]$ is a finite set and it equals $Y_A(3)$. If $A$ is generic in $(\Delta\setminus\Sigma)$ then $Y_A(3)$ is a singleton.
%
\item[(2)]
One may associate to $[v_0]\in Y_A(3)$ a $K3$ surface $S_A(v_0)\subset\PP^6$ of genus $6$, well-defined up to projectivities. Conversely the generic $K3$ of genus $6$ is projectively equivalent to $S_A(v_0)$ for some $A\in(\Delta\setminus\Sigma)$ and $[v_0]\in Y_A(3)$.
\item[(3)]
The singular set of $X_A$ is equal to $f_A^{-1}Y_A(3)$. There is a single $p_i\in X_A$ mapping to $[v_i]\in Y_A(3)$ and the cone of $X_A$ at $p_i$ is isomorphic to the cone over the set of incident couples $(x,r)\in\PP^2\times(\PP^2)^{\vee}$ (i.e.~$\PP(\Omega_{\PP^2})$). Thus we have two standard small resolutions of a neighborhood of $p_i$ in $X_A$, one with fiber $\PP^2$ over $p_i$, the other with fiber $(\PP^2)^{\vee}$. Making a choice $\epsilon$ of local small resolution at each $p_i$ we get
a resolution $X^{\epsilon}_A\to X_A$
with the following properties: There is a birational map $X^{\epsilon}_A\dashrightarrow S_A(v_i)^{[2]}$ such that the pull-back of a holomorphic symplectic form on $S_A(v_i)^{[2]}$ is a symplectic form on $X^{\epsilon}_A$. If $S_A(v_i)$ contains no lines (true for generic $A$ by Item~(2)) then there exists a choice of $\epsilon$ such that $X^{\epsilon}_A$ is isomorphic to $S_A(v_i)^{[2]}$.
\item[(4)]
Given a sufficiently small open (classical topology) $\cU\subset(\lagr\setminus\Sigma)$ containing $A$ the family of double EPW-sextics parametrized by $\cU$ has a simultaneous resolution of singularities (no base change) with fiber $X_A^{\epsilon}$ over $A$ (for an arbitrary choice of $\epsilon$).
\end{enumerate}
A remark: if $Y_A(3)$ has more than one point we do not expect all the small resolutions to be projective (i.e.~K\"ahler).
Items~(1)-(4) should be compared with known results on cubic $4$-folds - recall that if $Z\subset\PP^5$ is a smooth cubic hypersurface the variety $F(Z)$ parametrizing lines in $Z$ is a HK $4$-fold which can be deformed to $(K3)^{[2]}$ and moreover the primitive weight-$4$ Hodge structure of $Z$ is isomorphic (after a Tate twist) to the primitive weight-$2$ Hodge structure of $F(Z)$, see~\cite{beaudon}. Let $D\subset|\cO_{\PP^5}(3)|$ be the prime divisor parametrizing singular cubics. Let $Z\in D$ be generic: the following results are well-known.
%
\begin{enumerate}
\item[(1')]
$\sing Z$ is a finite set.
\item[(2')]
Given $p\in\sing Z$ the set $S_Z(p)\subset F(Z)$ of lines containing $p$ is a $K3$ surface of genus $4$ and viceversa the generic such $K3$ is isomorphic to $S_Z(p)$ for some $Z$ and $p\in\sing Z$.
\item[(3')]
$F(Z)$ is birational to $S_Z(p)^{[2]}$.
\item[(4')]
After a local base-change of order $2$ ramified along $D$ the period map extends across $Z$.
%
\end{enumerate}
Thus Items~(1')-(2')-(3') are analogous to Items~(1), (2) and~(3) above, Item~(4') is analogous to (4) but there is an important difference namely the need for a base-change of order $2$. (Actually the paper~\cite{og3} contains results showing that there is a statement valid for cubic hypersurfaces which is even closer to our result for double EPW-sextics,
the r\^ole of $\Sigma$ being played by the divisor parametrizing cubics containing a plane.)
We explain the relevance of Items~(1)-(4). Items~(3) and~(4) prove the theorem of ours mentioned above i.e.~that if $A\in\lagr^0$ then $X_A$ is a HK deformation of $(K3)^{[2]}$ (the family of polarized double EPW-sextics is locally complete by a straightforward parameter count). The proof in this paper is independent of the proof in~\cite{og2}. Beyond giving a new proof of an \lq\lq old\rq\rq theorem the above results show that away from $\Sigma$ the period map is regular, it lifts (locally) to the relevant classifying space and the value at $A\in(\Delta\setminus\Sigma)$ may be identified with the period point of the Hilbert square $S_A(v_0)^{[2]}$. We remark that in~\cite{og4} we had proved that the period map is as well-behaved as possible at the generic $A\in(\Delta\setminus\Sigma)$, however we did not have the exact statement about $X_A^{\epsilon}$ and we had no statement about an arbitrary $A\in(\Delta\setminus\Sigma)$.
The paper is organized as follows. In~\Ref{sec}{epwdoppie} we will give formulae that give the local structure of double EPW-sextics - the formulae are known to experts, see~\cite{catanese}, we will go through the proofs because we could not find a suitable reference. We will also perform the local computations needed to prove Item~(4) above.
In the very short~\Ref{sec}{famigliaepw} we will discuss (local) families of double EPW-sextics: we will determine the singular locus of the total space. In~\Ref{sec}{avoidelta} we will go through some standard computations involving $\Delta$.
In~\Ref{sec}{famdesing} we will prove Items~(1), (4) and the statements of Item~(3) which do not involve the $K3$ surface $S_A(v_0)$. In~\Ref{sec}{zitelle} we will prove Item~(2) and the remaining statement of Item~(3). In~\Ref{sec}{wedding} we will prove some auxiliary results on $3$-dimensional linear sections of $\mathrm{Gr}(3,\CC^5)$.
\vskip 3mm
\n
{\bf Notation and conventions:}
Throughout the paper $V$ is a $6$-dimensional complex vector space.
\vskip 2mm
\noindent
Let $W$ be a finite-dimensional complex vector-space. The span of a subset $S\subset W$ is denoted by $\la S\ra$. Let $S\subset\bigwedge^q W$. The {\it support of $S$} is the smallest subspace $U\subset W$ such that $S\subset\im(\bigwedge^{q}U\lra \bigwedge^q W)$: we denote it by $\supp(S)$, if $S=\{\alpha\}$ is a singleton we let $\supp(\alpha)=\supp(\{\alpha\})$ (thus if $q=1$ we have $\supp(\alpha)=\la\alpha\ra$). We define the support of a set of symmetric tensors analogously. If $\alpha\in\bigwedge^q W$ or $\alpha\in \Sym^d W$ the {\it rank of $\alpha$} is the dimension of $\supp(\alpha)$.
An element of $\Sym^2 W^{\vee}$ may be viewed either
as a symmetric map or as a quadratic form: we will denote the former by $\wt{q},\wt{r},\ldots$ and the latter by $q,r,\ldots$ respectively.
\vskip 2mm
\noindent
Let $M=(M_{ij})$ be a $d\times d$ matrix with entries in a commutative ring $R$. We let $M^c=(M^{ij})$ be the matrix of cofactors of $M$, i.e.~$M^{i,j}$ is $(-1)^{i+j}$ times the determinant of the matrix obtained from $M$
by deleting its $j$-th row and $i$-th column.
We recall the following interpretation of $M^c$. Suppose that $f\colon A\to B$ is a linear map between free $R$-modules of rank $d$ and that $M$ is the matrix associated to $f$ by the choice of bases $\{a_1,\ldots,a_d\}$ and $\{b_1,\ldots,b_d\}$ of $A$ and $B$ respectively. Then $\bigwedge^{d-1}f$ may be viewed as a map
%
\begin{equation}
\bigwedge^ {d-1}f\colon A^{\vee}\otimes\bigwedge^d A\cong
\bigwedge^{d-1} A \lra
\bigwedge^{d-1} B\cong B^{\vee}\otimes\bigwedge^d B.
\end{equation}
(Here $A^{\vee}:=\Hom(A,R)$ and similarly for $B^{\vee}$.)
The matrix associated to $\bigwedge^ {d-1}f$ by the choice of bases $\{a^{\vee}_1\otimes (a_1\wedge\ldots\wedge a_d),\ldots,a^{\vee}_d\otimes (a_1\wedge\ldots\wedge a_d)\}$ and $\{b^{\vee}_1\otimes (b_1\wedge\ldots\wedge b_d),\ldots,b^{\vee}_d\otimes (b_1\wedge\ldots\wedge b_d)\}$ is equal to $M^c$.
%
\vskip 2mm
\noindent
Let $W$ be a finite-dimensional complex vector-space. We will adhere to pre-Grothendieck conventions: $\PP(W)$ is the set of $1$-dimensional vector subspaces of $W$. Given a non-zero $w\in W$
we will denote the span of $w$ by $[w]$ rather than $\la w\ra$; this agrees with standard notation.
Suppose that $T\subset\PP(W)$. Then $\la T\ra\subset\PP(W)$ is the {\it projective span of $T$} i.e.~the intersection of all linear subspaces of $\PP(W)$ containing $T$.
\vskip 2mm
\noindent
Schemes are defined over $\CC$, the topology is the Zariski topology unless we state the contrary. Let $W$ be finite-dimensional complex vector-space: $\cO_{\PP(W)}(1)$ is the line-bundle on $\PP(W)$ with fiber $L^{\vee}$ on the point $L\in\PP(W)$. Let $F\in \Sym^d W^{\vee}$: we let $V(F)\subset\PP(W)$ be the subscheme defined by vanishing of $F$. If $E\to X$ is a vector-bundle we denote by $\PP(E)$ the projective fiber-bundle with fiber $\PP(E(x))$ over $x$ and we define $\cO_{\PP(W)}(1)$ accordingly.
If $Y$ is a subscheme of $X$ we let $Bl_Y X\lra X$ be the blow-up of $Y$.
\section{Symmetric resolutions and double covers}\label{sec:epwdoppie}
\setcounter{equation}{0}
%
We will examine local double covers. The proof of~\Ref{prp}{mappabeta} is given at the end of~\Ref{subsec}{prodenne}. \Ref{subsec}{desdop} contains the local construction needed to construct the simultaneous desingularization described in Item~(3) of~\Ref{sec}{prologo}.
\subsection{Product formula}\label{subsec:prodenne}
\setcounter{equation}{0}
Let $R$ be an integral Noetherian ring.
Let $N$ be a finitely generated $R$-module of rank zero and projective dimension $1$ i.e.
\begin{equation}\label{ipotenne}
\text{$\Ext^i(N,R)\not=0$ if and only if $i=1$.}
\end{equation}
Let $Fitt(N)$ be the $0$-th Fitting ideal of $N$; it is locally principal by~(\ref{ipotenne}). Given a homomorphism
\begin{equation}\label{morbeta}
\beta\colon N\to \Ext^1(N,Fitt(N))
\end{equation}
one defines a product $m_{\beta}\colon N\times N\to R/Fitt(N)$
as follows.
Apply the functor $\Hom(N,\,\cdot\,)$ to the exact sequence
\begin{equation}
0 \lra Fitt(N) \lra R \lra
R/ Fitt(N) \lra 0. \\
\end{equation}
Arguing as in the proof of Isomorphism~(\ref{extugualehom}) we get a coboundary isomorphism
\begin{equation}\label{isobordo}
\partial\colon \Hom\left(N,R/Fitt(N)\right)\overset{\sim}{\lra}
\Ext^1\left(N,Fitt(N)\right).
\end{equation}
%
We let
%
\begin{equation}\label{emmebeta}
\begin{matrix}
N\times N & \overset{m_{\beta}}{\lra} & R/Fitt(N)\\
(n,n') & \mapsto & (\partial^{-1}\beta(n))(n').
\end{matrix}
\end{equation}
The goal of this subsection is to give an explicit formula for $m_{\beta}$.
A local formula will be sufficient. Thus we may assume that
$N$ has a free resolution
\begin{equation}\label{risenne}
0\lra U_1\overset{\lambda}{\lra} U_0\overset{\pi}{\lra} N\lra 0.
\end{equation}
where
\begin{equation}\label{ranghi}
\rk\, U_1=\rk\, U_0=d>0.
\end{equation}
Let
$\{a_1,\ldots,a_d\}$ and $\{b_1,\ldots,b_d\}$ be bases of $U_0$ and $U_1$ repectively.
Let $M_{\lambda}$ be the matrix associated to $\lambda$ by our choice of bases. Then $Fitt(N)=(\det M_{\lambda})$. We choose the ismorphism
%
\begin{equation}\label{fittban}
\begin{matrix}
R & \overset{\sim}{\lra} & Fitt(N) \\
a & \mapsto & a \det M_{\lambda}
\end{matrix}
\end{equation}
From now on we view $\beta$ and $m_{\beta}$ as
\begin{equation}\label{emmeban}
\beta\colon N\to \Ext^1(N,R),\qquad
m_{\beta}\colon N\times N \lra R/(\det M_{\lambda}).
\end{equation}
Applying the $\Hom(\,\cdot\,,R)$-functor to~\eqref{risenne} we get the exact sequence
\begin{equation}\label{risext}
0\lra U^{\vee}_0\overset{\lambda^t}{\lra} U^{\vee}_1\overset{\rho}\lra \Ext^1(N,R)\lra 0.
\end{equation}
Thus we get that $\beta\circ\pi$ lifts to a homomorphisms $\mu^t\colon U_0\to U_1^{\vee}$ - the map is written as a transpose in order to conform to~\eqref{spqr}. It follows that there exists $\alpha\colon U_1\to U_0$ such that
%
%
\begin{equation}\label{unozeroenne}
\begin{array}{ccccccccc}
0 & \to & U_1 &\mapor{\lambda}& U_0 & \overset{\pi}{\lra} & N & \to & 0\\
& & \mapver{\alpha}& &\mapver{\mu^t} & &
\mapver{\beta}& & \\
0 & \to & U_0^{\vee} & \mapor{\lambda^{t}} & U_1^{\vee} & \overset{\rho}{\lra} & \Ext^1(N,R) & \to & 0
\end{array}
\end{equation}
is a commutative diagram.
Let $\{a^{\vee}_1,\ldots,a^{\vee}_d\}$ and $\{b^{\vee}_1,\ldots,b^{\vee}_d\}$ be the bases of $U_0^{\vee}$ and $U_1^{\vee}$ which are dual to the chosen bases of $U_0$ and $U_1$.
Let $M_{\mu^t}$ be the matrix associated to $\mu^t$ by our choice of bases.
\begin{prp}
Keeping notation as above we have
\begin{equation}\label{prodmin}
m_{\beta}(\pi(a_i),\pi(a_j))\equiv (M^c_{\lambda}\cdot M_{\mu^t})_{ji}\mod{(\det M_{\lambda})}
\end{equation}
where $M^c_{\lambda}$ is the matrix of cofactors of $M_{\lambda}$.
\end{prp}
%
\begin{proof}
%
Equation~\eqref{risext} gives an isomorphism
\begin{equation}\label{ecconu}
\nu\colon \Ext^1(N,R)\overset{\sim}{\to} U_1^{\vee}/\lambda^t(U_0^{\vee}).
\end{equation}
Let $\det(U_{\bullet}):=\bigwedge^d U_1^{\vee}\otimes\bigwedge^d U_0$.
We will define an isomorphism
\begin{equation}\label{tetaisom}
\theta\colon U_1^{\vee}/\lambda^t(U_0^{\vee})\overset{\sim}{\lra}
\Hom\left(N,\det(U_{\bullet})/(\det\lambda)\right).
\end{equation}
First let
\begin{equation}\label{mappazza}
\begin{matrix}
U_1^{\vee}=& \bigwedge^{d-1}U_1\otimes\bigwedge^d U_1^{\vee} &
\overset{\wh{\theta}}{\lra} &
\bigwedge^{d-1}U_0\otimes\bigwedge^d U_1^{\vee} & = &
\Hom(U_0,\det (U_{\bullet})) \\
&\zeta\otimes\xi & \mapsto & \bigwedge^ {d-1}(\lambda)(\zeta)\otimes\xi
\end{matrix}
\end{equation}
We claim that
\begin{equation}\label{jamesbiondi}
\im(\wh{\theta})=\{\phi\in \Hom(U_0,\det(U_{\bullet}))\mid
\phi\circ\lambda(U_1)\subset(\det\lambda)\}.
\end{equation}
In fact by Cramer's formula
\begin{equation}\label{cofattori}
M_{\lambda}^c\cdot M_{\lambda}^t=M_{\lambda}^t\cdot M_{\lambda}^c=\det M_{\lambda}\cdot 1
\end{equation}
and Equation~\eqref{jamesbiondi} follows.
Thus $\wh{\theta}$ induces a surjective homomorphism
\begin{equation}
\wt{\theta}\colon U_1^{\vee}\lra
\Hom\left(N, \det (U_{\bullet})/(\det\lambda)\right).
\end{equation}
One checks easily that
$\lambda^t(U_0^{\vee})= \ker\wt{\theta}$ - use Cramer again.
We define $\theta$ to be the homomorphism induced by $\wt{\theta}$; we have proved that it is an isomorphism.
We claim that
\begin{equation}\label{voila}
\theta\circ\nu=\partial^{-1},\qquad\text{$\partial$ as in~(\ref{isobordo}).}
\end{equation}
In fact let $K$ be the fraction field of $R$ and $0\to R\overset{\iota}{\to} I^0\to I^1\to\ldots$ be an injective resolution of $R$ with $I^0=\det(U_{\bullet})\otimes K$ and $\iota(1)=\det\lambda\otimes 1$.
Then $\Ext^{\bullet}(N,R)$ is the cohomology of the double complex $\Hom(U_{\bullet},I^{\bullet})$ and of course also of the single complexes $\Hom(U_{\bullet},R)$ and $\Hom(N,I^{\bullet})$.
One checks easily that the isomorphism $\partial$ of~(\ref{isobordo}) is equal to the isomorphism $H^1(\Hom(N,I^{\bullet}))\overset{\sim}{\to}H^1(\Hom(U_{\bullet},I^{\bullet}))$ i.e.
\begin{equation}\label{eccoeps}
\partial\colon
\Hom(N,\det (U_{\bullet})/(\det\lambda))=
\Hom(N,I^0/\iota(R))\overset{\sim}{\lra}H^1(\Hom(U_{\bullet},I^{\bullet})).
\end{equation}
Let
$f\in \Hom(N,\det (U_{\bullet})/(\det\lambda))$; a representative of $\partial(f)$ in the double complex $\Hom(U_{\bullet},I^{\bullet})$ is given by $g^{0,1}:=f\circ\pi\in \Hom(U_0,I^1)$. Let $g^{0,0}\in \Hom(U_0,\det( U_{\bullet}))$ be a lift of $g^{0,1}$ and $g^{1,0}\in \Hom(U_1,\det( U_{\bullet}))$ be defined by $g^{1,0}:=g^{0,0}\circ\lambda$. One checks that $\im(g^{1,0})\subset(\det\lambda)$ and hence there exists $g\in \Hom(U_1,R)$ such that $g^{1,0}=\iota\circ g$. By construction $g$ represents a class $[g]\in H^1(\Hom(U_{\bullet},R))=U_1^{\vee}/\lambda^t(U_0^{\vee})$ and $[g]=\nu\circ\partial(f)$. An explicit computation shows that $[g]=\theta^{-1}(f)$.
This proves~(\ref{voila}). Now we prove Equation~(\ref{prodmin}).
By~(\ref{voila}) we have
\begin{equation}
m_{\beta}(\pi(a_i),\pi(a_j))=(\partial^{-1}\beta\pi(a_i))(\pi(a_j))
=(\theta\nu\beta\pi(a_i))(\pi(a_j)).
\end{equation}
Unwinding the definition of $\theta$ one gets that the right-hand side of the above equation equals the right-hand side of~(\ref{prodmin}).
%
\end{proof}
%
Let $m_{\beta}$ be given by~\eqref{emmeban}: we define a product on $(R/(\det M_{\lambda})\oplus N)$ as follows. Let $(r,n),(r',n')\in (R/(\det M_{\lambda})\oplus N)$: we set
\begin{equation}\label{moltiplico}
(r,n)\cdot(r',n'):=(rr'+m_{\beta}(n,n'),rn'+r'n).
\end{equation}
In general the above product is not associative nor commutative. The following is an example in which the product is both associative and commutative.
Let $U$ be a free finitely generated $R$-module, and assume that we have an exact sequence
\begin{equation}\label{mappasimm}
0\lra U^{\vee} \overset{\gamma}{\lra} U \overset{\pi}{\lra} N \lra 0
\end{equation}
%
with $\gamma$ symmetric. Then we get a commutative diagram~\eqref{unozeroenne} by setting
%
\begin{equation}\label{scelte}
\alpha=\Id _U,\qquad
\mu^t=\Id _{U^{\vee}},
\end{equation}
and letting $\beta$ be the map induced by the above choices.
We denote by $m_{\gamma}$ the correponding map $N\times N\to R/(\det M_{\gamma})$.
\begin{prp}\label{prp:associo}
Suppose that we have Exact Sequence~(\ref{mappasimm}) with $\gamma$ symmetric. The product defined on $(R/(\det M_{\gamma})\oplus N)$ by $m_{\gamma}$ is associative and commutative.
\end{prp}
\begin{proof}
Let $d:=\rk\, U>0$.
Let $\{a_1,\ldots,a_d\}$ be a basis of $U$ and $\{a^{\vee}_1,\ldots,a^{\vee}_d\}$ be the dual basis of $U^{\vee}$. Let $M=M_{\gamma}$ i.e.~the matrix associated to $\gamma$ by our choice of bases.
Since $\gamma$ is a symmetric map $M$ is a symmetric matrix. By~\eqref{prodmin} we have
\begin{equation}\label{mucca}
m_{\gamma}(\pi(a_i),\pi(a_j))\equiv M^c_{ji} \mod{(\det M)}.
\end{equation}
Since $M$ is symmetric so is $M^c$ and hence we get that $m_{\gamma}$ is symmetric. It remains to prove that $m_{\gamma}$ is associative.
For $1\le i< k\le d$ and $1\le h\not= j\le d$ let $M^{i,k}_{h,j}$ be the $(d-2)\times(d-2)$-matrix obtained by deleting from $M$ rows $i,k$ and columns $h,j$.
Let $X_{ijk}=(X^h_{ijk})\in R^d$ be defined by
\begin{equation}
X^h_{ijk}:=
\begin{cases}
(-1)^{i+k+j+h}\det M^{i,k}_{j,h} & \text{if $h< j$}, \\
0 & \text{if $h=j$}. \\
(-1)^{i+k+j+h-1}\det M^{i,k}_{j,h} & \text{if $j<h$.}
\end{cases}
\end{equation}
%
A tedious but straightforward computation gives that
\begin{equation}\label{relazione}
M_{ij}^c a_k-M^c_{jk} a_i=\gamma(\sum_{h=1}^d X^h_{ijk}a_h^{\vee}).
\end{equation}
The above equation proves associativity of $m_{\gamma}$.
\end{proof}
\begin{hyp}\label{hyp:ipotizzo}
In Commutative Diagram~\eqref{unozeroenne} we have that $\mu^t$ is an isomorphism and $\alpha=\mu$.
\end{hyp}
\begin{prp}\label{prp:redux}
Assume that~\Ref{hyp}{ipotizzo} holds. Set
\begin{equation}\label{semplifico}
\gamma:=\lambda\circ\mu^{-1},
\qquad U:=U_1.
\end{equation}
Then~\eqref{mappasimm} is an exact sequence, $\gamma^t=\gamma$ and
\begin{equation}\label{redux}
m_{\beta}(\pi(a_i),\pi(a_j))\equiv
(M_{\gamma}^c)_{ij}\mod{(\det M_{\gamma})}.
\end{equation}
where $\{a_1,\ldots,a_d\}$ is the basis of $U^{\vee}=U_0$ introduced right after Equation~\eqref{ranghi}. In particular $(R/(\det M_{\lambda})\oplus N)$ equipped with the product given by~\eqref{moltiplico} is a commutative (associative) ring.
\end{prp}
\begin{proof}
The first statement is obvious. Equation~\eqref{redux} follows from~\Ref{prp}{associo}. The last statement follows from~\eqref{redux}.
\end{proof}
\begin{dfn}\label{dfn:simmetrizzo}
Suppose that~\Ref{hyp}{ipotizzo} holds: the {\it symmetrization of~\eqref{unozeroenne}} is Exact Sequence~\eqref{mappasimm} with $\gamma$ given by~\eqref{semplifico}.
\end{dfn}
\vskip 2mm
\noindent
{\it Proof of~\Ref{prp}{mappabeta}}
Let $[v_0]\in\PP(V)$. Choose $B\in\lagr$ transversal to $F_{v_0}$ (and to $A$ of course). Then $\mu_{A,B}$ is an isomorphism in an open neighborhood $U$ of $[v_0]$. It follows that $\beta_A$ is an isomorphism in a neighborhood of $[v_0]$. This proves Item~(1). Let's prove Item~(2). Let $B\in\lagr$ and $U$ be as above; we may assume that $U$ is affine. Let $N:=H^0(i_{*}\zeta_A|_U)$ and
$\beta:=H^0(\beta_A|_U)$. Thus $\beta\colon N\to {\rm Ext}^1(N,\CC[U])$.
By Commutativity of Diagram~(\ref{spqr}) and by~\Ref{prp}{redux} we get that the multiplication map $m_{\beta}$ is associative and commutative. On the other hand $m_{\beta}$ is the multiplication induced by $m_A$ on $N$; since $[v_0]$ is an arbitrary point of $\PP(V)$ it follows that $m_A$ is associative and commutative.
\qed
\subsection{Double covers}\label{subsec:rivdop}
\setcounter{equation}{0}
Suppose that we are given $\beta$ as in~\eqref{morbeta} and assume that~\Ref{hyp}{ipotizzo} holds. We let
\begin{equation}
X_{\beta}:= \Spec(R/(\det M_{\lambda})\oplus N),\qquad
Y_{\beta}:=\Spec(R/(\det M_{\lambda})).
\end{equation}
Let $f_{\beta}\colon X_{\beta}\to Y_{\beta}$ be the structure map.
The {\it covering involution} of $f_{\beta}$
is the automorphism $\phi_{\beta}$ of $X_{\beta}$ which corresponds to the isomorphism of the $R/(\det M_{\lambda})$-algebra
$(R/(\det M_{\lambda})\oplus N)$ acting as multiplication by $(-1)$ on $N$.
Let $Y_{\beta}[c]$ be the closed subscheme of $\Spec R$ defined by imposing that $\cork M_{\lambda}\ge c$ i.e.
\begin{equation}\label{eccozici}
Y_{\beta}[c]=\bigcap_{|I|=|J|=d+1-c}V(\det M_{\lambda}^{I,J})\,.
\end{equation}
($I,J,K,H$ are multi-indices, $M_{\lambda}^{I,J},M_{\lambda}^{K,H}$ the corresponding minors of $M_{\lambda}$.) Thus $Y_{\beta}[1]=Y_{\beta}$.
Let
\begin{equation}
Y_{\beta}(c):=Y_{\beta}[c]\setminus Y_{\beta}[c+1]
\end{equation}
Let $\gamma$ be the simmetrization of $\beta$ - see~\Ref{dfn}{simmetrizzo}. Then $f_{\beta}\colon X_{\beta}\to Y_{\beta}$ is identified with $f_{\gamma}\colon X_{\gamma}\to Y_{\gamma}$. Thus we may replace throughout $X_{\beta}$ by $X_{\gamma}$. We will denote $Y_{\beta}[c]$ by $Y_{\gamma}[c]$ etc.
We let $\phi_{\gamma}$ be the covering involution of $X_{\gamma}$.
We realize $X_{\gamma}$ as a subscheme of $\Spec(R[\xi_1,\ldots,\xi_d])$ as follows. Since the ring $(R/(\det M_{\gamma})\oplus N)$
is associative and commutative there is a well-defined surjective morphism of $R$-algebras
\begin{equation}
R[\xi_1,\ldots,\xi_d]\longrightarrow R/(\det M_{\gamma})\oplus N
\end{equation}
mapping $\xi_i$ to $a_i$. Thus we have an inclusion
\begin{equation}\label{esplicito}
X_{\gamma}\hra \Spec(R[\xi_1,\ldots,\xi_d]).
\end{equation}
\begin{clm}\label{clm:ideawu}
Referring to Inclusion~(\ref{esplicito}) the ideal of $X_{\gamma}$ is generated by the entries of the matrices
\begin{equation}
M_{\gamma}\cdot\xi\,,\ \ \xi\cdot\xi^t-M_{\gamma}^c\,.
\end{equation}
(We view $\xi$ as a column matrix.)
Furthermore the map $f_{\gamma}$ is induced by the natural projection $\Spec(R[\xi_1,\ldots,\xi_d])\to \Spec R$ and the covering involution $\phi_{\gamma}$ is induced by the automorphism of the $R$-algebra $R[\xi_1,\ldots,\xi_d]$ which multiplies each $\xi_i$ by $(-1)$.
\end{clm}
\begin{proof}
By Equation~(\ref{mucca}) the ideal of $X_{\gamma}$ is generated by $\det M_{\gamma}$ and the entries of the matrices
\begin{equation}
M_{\gamma}\cdot\xi,\qquad\xi\cdot\xi^t-M_{\gamma}^c\,.
\end{equation}
By Cramer's formula $\det M_{\gamma}$ belongs to the ideal generated by the entries of the first and second matrices above. This proves that the ideal of $X_{\gamma}$ is as claimed. The statements regarding $f_{\gamma}$ and $\phi_{\gamma}$ are obvious.
\end{proof}
Suppose that $R$ is a finitely generated $\CC$-algebra and that we have~\eqref{mappasimm} with $\gamma$ symmetric.
Let $p\in \Spec R$ be a closed point: we are interested in the localization of $X_{\gamma}$ at points in $f_{\gamma}^{-1}(p)$.
Let $J\subset U^{\vee}(p)$ be a subspace complementary to $\ker\gamma(p)$. Let ${\bf J}\subset U^{\vee}$ be a free submodule whose fiber over $p$ is equal to $J$. Let ${\bf K}\subset U^{\vee}$ be the submodule orthogonal to ${\bf J}$ i.e.
\begin{equation}
{\bf K}:=\{u\in U^{\vee}\mid \gamma(a)(u)=0\quad\forall a\in {\bf J}\}\,.
\end{equation}
The localization of ${\bf K}$ at $p$ is free. Let
$K:={\bf K}(p)$ be the fiber of ${\bf K}$ at $p$; clearly $K=\ker\gamma(p)$. Localizing at $p$ we have
%
\begin{equation}\label{ugeikappa}
U^{\vee}_p={\bf K}_p\oplus {\bf J}_p\,.
\end{equation}
Corresponding to~\eqref{ugeikappa} we may write
%
\begin{equation}\label{spettrale}
\gamma_p=\gamma_{\bf K}\oplus_{\bot} \gamma_{\bf J}
\end{equation}
where $\gamma_{\bf K}\colon {\bf K}_p\to {\bf K}^{\vee}_p$ and $\gamma_J\colon {\bf J}_p\to {\bf J}^{\vee}_p$ are symmetric maps. Notice that we have an equality of germs
%
\begin{equation}\label{yuguali}
(Y_{\gamma},p)= (Y_{\gamma_{\bf K}},p).
\end{equation}
Let $k:=\dim K$ and $d:=\rk\,U$. Choose bases of ${\bf K}_p$ and ${\bf J}_p$; by~\eqref{ugeikappa} we get a basis of $U^{\vee}_p$. The dual bases of ${\bf K}^{\vee}_p$, ${\bf J}^{\vee}_p$ and $U^{\vee}_p$ are compatible with respect to the decomposition dual to~\eqref{ugeikappa}. Corresponding to the chosen bases we have embeddings
%
\begin{equation}\label{inclusioni}
X_{\gamma_K}\hra Y_{\gamma_K}\times\CC^k,\quad
X_{\gamma}\hra Y_{\gamma}\times\CC^d\,.
\end{equation}
%
The decomposition dual to~\eqref{ugeikappa} gives an embedding
%
\begin{equation}
j\colon(Y_{\gamma_K}\times\CC^k)\hra
(Y_{\gamma}\times\CC^d)\,.
\end{equation}
%
\begin{clm}\label{clm:nucleo}
Keep notation as above. The composition
\begin{equation}
X_{\gamma_K}\hra (Y_{\gamma_K}\times\CC^k)\overset{j}{\lra}
(Y_{\gamma}\times\CC^d)
\end{equation}
%
defines an isomorphism of germs in the analytic topology
\begin{equation}
(X_{\gamma_{\bf K}},f_{\gamma_{\bf K}}^{-1}(p))\overset{\sim}{\lra}
(X_{\gamma},f_{\gamma}^{-1}(p))
\end{equation}
which commutes with the maps $f_{\gamma_{\bf K}}$ and $f_{\gamma}$.
\end{clm}
\begin{proof}
This follows from Decomposition~\eqref{spettrale} and Equation~\eqref{mucca}. We pass to the analytic topology in order to be able to extract the square root of a regular non-zero function.
\end{proof}
\begin{prp}\label{prp:gradodue}
Assume that $R$ is a finitely generated $\CC$-algebra. Suppose that~\Ref{hyp}{ipotizzo} holds. Then the following hold:
\begin{itemize}
\item[(1)]
$f_{\beta}^{-1}Y_{\beta}(1)\to Y_{\beta}(1)$ is a topological covering of degree $2$.
%
\item[(2)]
Let $p\in (Y_{\beta}\setminus Y_{\beta}(1))$ be a closed point. The fiber $f_{\beta}^{-1}(p)$ consists of a single point $q$. If $\beta$ is the isomorphism associated to a symmetric map $\gamma$ fitting into Exact Sequence~\eqref{mappasimm} then $\xi_i(q)=0$ for $i=1,\ldots,d$ where $\xi_i$ are the coordinates on $X_{\gamma}$ associated to Embedding~\eqref{esplicito}.
\end{itemize}
\end{prp}
\begin{proof}
(1): By~\Ref{prp}{redux} we may assume that $\beta$ is the map associated to a symmetric $\gamma$ fitting into Exact Sequence~\eqref{mappasimm}. Localizing at $p\in Y_{\beta}(1)$ and applying~\Ref{clm}{nucleo} we get Item~(1). (2): Since $\cork M_{\gamma}(p)\ge 2$ we have $M_{\gamma}^c(p)=0$ and thus Item~(2) follows from~\Ref{clm}{ideawu}.
\end{proof}
\subsection{Local models of double covers}\label{subsec:doppipiccoli}
\setcounter{equation}{0}
In the present subsection we assume that $R$ is a finitely generated $\CC$-algebra. Let $\cW$ be a finite-dimensional complex vector-space. We will suppose that we have an exact sequence
\begin{equation}\label{ancoramu}
0\lra R\otimes\cW^{\vee}\overset{\gamma}{\lra} R\otimes\cW\lra N\lra 0,\qquad
\gamma=\gamma^t.
\end{equation}
Thus we have a double cover $f_{\gamma}\colon X_{\gamma}\to Y_{\gamma}$.
Let $p\in Y_{\gamma}$ be a closed point.
We will examine $X_{\gamma}$ in a neighborhood of $f_{\gamma}^{-1}(p)$ when the corank of $\gamma(p)$ is small.
We may view $\gamma$ as a regular map $\Spec R\to \Sym^2 \cW$; thus it makes sense to consider the differential
\begin{equation}
d\gamma(p)\colon T_p \Spec R\to \Sym^2\cW.
\end{equation}
Let $K(p):=\ker\gamma(p)\subset\cW^{\vee}$; we will consider the linear map
%
\begin{equation}\label{difftre}
\begin{matrix}
T_p \Spec R & \overset{\delta_{\gamma}(p)}{\lra} &
\Sym^2 K(p)^{\vee} \\
\tau & \mapsto & d\gamma(p)(\tau)|_{K(p)}
\end{matrix}
\end{equation}
Let $d:=\dim\cW$; choosing a basis of $\cW$
we realize $X_{\gamma}$ as a subscheme of $\Spec R\times\CC^d$ with ideal given by~\Ref{clm}{ideawu}. Since $\cork\gamma(p)\ge 2$ \Ref{prp}{gradodue} gives that $f_{\gamma}^{-1}(p)$ consists of a single point $q$ - in fact the $\xi_i$-coordinates of $q$ are all zero. Throughout this subsection we let
\begin{equation}
f_{\gamma}^{-1}(p)=\{q\}\,.
\end{equation}
\begin{clm}\label{clm:pocheq}
Keep notation as above. Suppose that $d=\dim\cW=2$ and that $\gamma(p)=0$. Then $I(X_{\gamma})$ is generated by the entries of $\xi\cdot\xi^t-M^c_{\gamma}$.
\end{clm}
\begin{proof}
\Ref{clm}{ideawu} together with a straightforward computation.
\end{proof}
\begin{expl}\label{expl:rangodue}
{\rm Let $R=\CC[x,y,z]$, $\cW=\CC^2$. Suppose that the matrix associated to $\gamma$ is
\begin{equation}
M_{\gamma}=
\left(
\begin{matrix}
x & y \\
y & z
\end{matrix}
\right)
\end{equation}
Then $f_{\gamma}\colon X_{\gamma}\to Y_{\gamma}$ is identified with
\begin{equation}
\begin{matrix}
\CC^2 & \lra & V(xz-y^2) \\
(\xi_1,\xi_2) & \mapsto & (\xi_2^2,\,-\xi_1\xi_2,\,\xi_1^2)
\end{matrix}
\end{equation}
i.e.~the quotient map for the action of $\la -1\ra$ on $\CC^2$.}
\end{expl}
\begin{prp}\label{prp:biancabernie}
Keep notation as above. Suppose that
the following hold:
\begin{itemize}
\item[(a)]
$\cork\,\gamma(p)=2$,
\item[(b)]
the localization $R_p$ is regular.
\end{itemize}
Then $X_{\gamma}$ is smooth at $q$ if and only if $\delta_{\gamma}(p)$ is surjective.
\end{prp}
\begin{proof}
Applying~\Ref{clm}{nucleo} we get that we may assume that $d=2$. Let
\begin{equation}
M_{\gamma}=
\left(
\begin{matrix}
a & b \\
b & c
\end{matrix}
\right)
\end{equation}
By~\Ref{clm}{pocheq} the ideal of $X_{\gamma}$ in $\Spec R\times\CC^2$ is generated by the entries of $\xi\cdot\xi^t-M^c_{\gamma}$ i.e.
\begin{equation}\label{lassie}
I(X_{\gamma})=(\xi_1^2-c,\ \xi_1\xi_2+b,\ \xi_2^2-a)\,.
\end{equation}
Thus
\begin{equation}
\cod(T_q X_{\gamma},T_q (\Spec R\times\CC^2))=\dim\la da(p)\,,db(p)\,,dc(p)\ra\,.
\end{equation}
On the other hand $\cod_q(X_{\gamma}\,,\Spec R\times\CC^2)=3$ and hence we get that $X_{\gamma}$ is smooth at $q$ if and only if $\delta_{\gamma}(p)$ is surjective.
\end{proof}
\begin{clm}\label{clm:sempresing}
Keep notation and hypotheses as above. Suppose that $\cork\gamma(p)\ge 3$. Then $X_{\gamma}$ is singular at $q$.
\end{clm}
\begin{proof}
Let $I$ be the ideal of $X_{\gamma}$ in $\Spec R[\xi_1,\ldots,\xi_d]$. By~\Ref{clm}{ideawu} we get that $I$ is non-trivial but the differential at $q$ of an arbitrary $g\in I$ is zero.
\end{proof}
Next we will discuss in greater detail those $X_{\gamma}$ whose corank at $f^{-1}_\gamma(p)$ is equal to $3$.
First we will identify the \lq\lq universal\rq\rq example (the universal example for corank $2$ is~\Ref{expl}{rangodue}). Let $\cV$ be a $3$-dimensional complex vector space. We view $\Sym^2\cV$ as an affine ($6$-dimensional) space and we let $R:=\CC[\Sym^2\cV]$ be its ring of regular functions.
We identify $R\otimes_{\CC}\cV$ and $R\otimes_{\CC}\cV^{\vee}$ with the space of $\cV$-valued, respectively $\cV^{\vee}$-valued, regular maps on $\Sym^2\cV$.
Let
\begin{equation}
R\otimes_{\CC}\cV^{\vee} \overset{\gamma}{\lra}
R\otimes_{\CC}\cV
\end{equation}
be the map induced on the spaces of global sections by the tautological
map of vector-bundles $\Spec R\times\cV^{\vee}\lra \Spec R\times\cV$. The map $\gamma$ is symmetric. Let $N$ be the cokernel of $\gamma$: thus
\begin{equation}
0\lra R\otimes_{\CC}\cV^{\vee} \overset{\gamma}{\lra}
R\otimes_{\CC}\cV\lra N\lra 0
\end{equation}
is an exact sequence. Since $\gamma$ is symmetric it defines a double cover $f\colon X(\cV)\to Y(\cV)$ where
\begin{equation}
Y(\cV):=\{\alpha\in \Sym^2\cV\mid \rk\,\alpha<3 \}
\end{equation}
is the variety of degenerate quadratic forms. We let
\begin{equation}\label{invoix}
\phi\colon X(\cV)\to X(\cV)
\end{equation}
be the covering involution of $f$.
One describes explicitly $X(\cV)$ as follows. Let
\begin{equation}
(\cV\otimes\cV)_1:= \{\mu\in (\cV\otimes\cV)\mid \rk\,\mu \le 1\}.
\end{equation}
Thus $(\cV\otimes\cV)_1$ is the cone over the Segre variety $\PP(\cV)\times\PP(\cV)$. We have a finite degree-$2$ map
\begin{equation}\label{mappasigma}
\begin{matrix}
(\cV\otimes\cV)_1 & \overset{\sigma}{\lra} & Y(\cV) \\
\mu & \mapsto & \mu+\mu^t\,.
\end{matrix}
\end{equation}
\begin{prp}\label{prp:equarela}
Keep notation as above. There exists a commutative diagram
\begin{equation}\label{triangolo}
\xymatrix{ \\ (\cV\otimes\cV)_1\ar^{\sigma}[dr] \ar^{\tau}[rr] & & X(\cV)\ar_{f}[dl] \\
& Y(\cV) &}
\end{equation}
where $\tau$ is an isomorphism. Let $\phi$ be Involution~\eqref{invoix}: then
\begin{equation}\label{invtras}
\phi\circ\tau(\mu)=\tau(\mu^t),\qquad \forall\mu\in (\cV\otimes\cV)_1.
\end{equation}
\end{prp}
\begin{proof}
In order to define $\tau$ we will give a coordinate-free version of Inclusion~(\ref{esplicito}) in the case of $X(\cV)$.
Let
\begin{equation}\label{spigola}
\begin{matrix}
\Sym^2\cV\times(\cV^{\vee}\otimes\bigwedge^3\cV) &
\overset{\Psi}{\lra} & (\cV\otimes\bigwedge^3\cV)\times
(\cV^{\vee}\otimes\cV^{\vee}\otimes\bigwedge^3\cV\otimes\bigwedge^3\cV)
\\
(\alpha,\xi) & \mapsto & (\alpha\circ\xi,\ \xi^t\circ\xi - \bigwedge^ 2 \alpha)\,.
\end{matrix}
\end{equation}
A few words of explanation.
In the definition of the first component of $\Psi(\alpha,\xi)$ we view $\xi$ as belonging to $\Hom(\bigwedge^3\cV^{\vee},\cV^{\vee})$, in the definition of the second component of $\Psi(\alpha,\xi)$ we view $\xi$ as belonging to
$ \Hom(\cV\otimes\bigwedge^3\cV^{\vee},\CC)$. Moreover we make the obvious choice of isomorhpism $\CC\cong\CC^{\vee}$. Secondly
\begin{equation}
\bigwedge^ 2\alpha\in\Hom(\bigwedge^2\cV^{\vee},\,\bigwedge^2\cV)
=\Hom(\cV\otimes\bigwedge^3\cV^{\vee},\,
\cV^{\vee}\otimes\bigwedge^3\cV)=
\cV^{\vee}\otimes\cV^{\vee}\otimes
\bigwedge^3\cV\otimes\cV\,.
\end{equation}
Choosing a basis of $\cV$ we get an embedding $X(\cV)\subset \Sym^2\cV\times \CC^3$, see~\eqref{esplicito}. \Ref{clm}{ideawu} gives equality of pairs
\begin{equation}\label{coppie}
(\Sym^2\cV\times(\cV^{\vee}\otimes\bigwedge^3\cV),\,
\Psi^{-1}(0))= (\Sym^2\cV\times\CC^3,\,X(\cV))\,,
\end{equation}
where $\Psi^{-1}(0)$ is the scheme-theoretic fiber of $\Psi$.
Now notice that we have an isomorphism
\begin{equation}\label{cuffia}
\begin{matrix}
\cV\otimes\cV & \overset{\overset{T}{\sim}}{\lra} &
\Sym^2\cV\times(\cV^{\vee}\otimes\bigwedge^3\cV) \\
%
\epsilon & \mapsto & (\epsilon+\epsilon^t,\epsilon-\epsilon^t)
\end{matrix}
\end{equation}
Let $\tau:=T|_{(\cV\otimes\cV)_1}$: thus we have an embedding
\begin{equation}\label{eccotau}
\tau\colon(\cV\otimes\cV)_1\hra \Sym^2\cV\times(\cV^{\vee}\otimes\bigwedge^3\cV).
\end{equation}
We will show that we have
equality of schemes
\begin{equation}\label{ixgielle}
\im(\tau)=\Psi^{-1}(0)(=X(\cV)).
\end{equation}
First let
\begin{equation}
\begin{matrix}
\cV\oplus\cV &
\overset{\rho}{\lra} & (\cV\otimes\cV)_1 \\
(\eta,\beta) & \mapsto & \eta^t\circ\beta.
\end{matrix}
\end{equation}
Notice that $\rho$ is the quotient map for the $\CC^{\times}$-action on $\cV\oplus\cV$ defined by $t(\eta,\beta):=(t\eta,t^{-1}\beta)$. We have
\begin{equation}
\tau\circ\pi= (\eta^t\circ\beta+\beta^t\circ\eta,
\eta\wedge\beta).
\end{equation}
Let's prove that
\begin{equation}\label{tauinix}
\Psi^{-1}(0)\supset\im(\tau)\,.
\end{equation}
Notice that $\GL(\cV)$ acts on $(\cV\otimes\cV)_1$ with a unique dense orbit namely $\{\eta^t\circ \beta\mid \eta\wedge\beta\not=0\}$. An easy computation shows that $\tau(\eta^t\circ \beta)\in \Psi^{-1}(0)$ for a conveniently chosen $\eta^t\circ \beta$ in the dense orbit of $(\cV\otimes\cV)_1$; it follows that~\eqref{tauinix} holds. On the other hand $T$ defines an isomorphism of pairs
\begin{equation}\label{coppiedue}
(\cV\otimes\cV,\, (\cV\otimes\cV)_1)\cong
(\Sym^2\cV^{\vee}\times (\cV^{\vee}\otimes\bigwedge^3\cV),\,\im(\tau))\,.
\end{equation}
Since the ideal of $(\cV\otimes\cV)_1$ in $\cV\otimes\cV$ is generated by $9$ linearly independent quadrics we get that the ideal of $\im(\tau)$ in $\Sym^2\cV^{\vee}\times (\cV^{\vee}\otimes\bigwedge^3\cV)$ is generated by $9$ linearly independent quadrics. The ideal of $\Psi^{-1}(0)$ in $\Sym^2\cV\times(\cV^{\vee}\otimes\bigwedge^3\cV)$ is likewise generated by $9$ linearly independent quadrics - see~\eqref{spigola}. Since
$\Psi^{-1}(0)\supset\im(\tau)$ we get that the ideals of $\Psi^{-1}(0)$ and of $\im(\tau)$ are the same and hence~\eqref{ixgielle} holds.
This proves that $\tau$ is an isomorphism between $(\cV\otimes\cV)_1$ and $X(\cV)$. Diagram~\eqref{triangolo} is commutative by construction.
Equation~\eqref{invtras} is equivalent to the equality
\begin{equation}
\phi(\tau\circ\rho(\beta,\eta))=\tau\circ \rho(\eta,\beta)).
\end{equation}
The above equality holds because $\beta\wedge\eta=-\eta\wedge\beta$.
\end{proof}
The following result is an immediate consequence of~\Ref{prp}{equarela}.
\begin{crl}\label{crl:singixvu}
$\sing X(\cV)=\tau(0)=f^{-1}(0)$.
\end{crl}
Let
\begin{equation}\label{pimappa}
\pi\colon\wt{X}(\cV)\to X(\cV)
\end{equation}
be the blow-up of $\tau(0)$ and $\wt{\phi}\colon\wt{X}(\cV)\to\wt{X}(\cV)$ be the involution lifting $\phi$. The claim below follows immediatly from~\Ref{prp}{equarela}.
\begin{clm}\label{clm:antonioni}
Keep notation as above. Then
\begin{itemize}
\item[(1)]
$\wt{X}(\cV)$ is smooth,
%
\item[(2)]
the exceptional set $E$ of $\pi$ is canonically isomorphic to $\PP(\cV)\times\PP(\cV)$ - in particular it has pure codimension $1$,
\item[(3)]
the normal bundle $N_{E,\wt{X}(\cV)}$ is isomorphic to $\cO_{\PP(\cV)}(-1)\boxtimes\cO_{\PP(\cV)}(-1)$,
\item[(4)]
$\wt{\phi}$ acts on the exceptional set of $\pi$ (i.e.~$\PP(\cV)\times\PP(\cV)$) by exchanging the factors.
\end{itemize}
\end{clm}
By Item~(3) of the above claim $X(\cV)$ has two small resolutions $X^{\pm}(\cV)$: they are obtained by contracting $E$ along the two projections $E\to\PP(\cV)$. The small resolutions may be constructed explicitly as follows.
Let $X^{+}(\cV)\subset\PP(\cV)\times(\cV\otimes\cV)_1$ be defined by
\begin{equation}
X^{+}(\cV) :=
\{([\beta],\alpha\circ\beta)\mid [\beta]\in\PP(\cV),\ \alpha\in\cV\}.
\end{equation}
(Here and in what follows we view $\alpha,\beta\in\cV$ as elements of $\Hom(\CC,\cV)$ and of $\Hom(\cV^{\vee},\CC)$ respectively.) The map
\begin{equation}\label{fibroixpiu}
\begin{matrix}
X^{+}(\cV) & \lra & \PP(\cV) \\
([\beta],\alpha\circ\beta) & \mapsto & [\beta]
\end{matrix}
\end{equation}
is a $\CC^3$-fibration and hence
$X^{+}(\cV)$ is smooth.
One defines similarly
$X^{-}(\cV)\subset\PP(\cV)\times(\cV\otimes\cV)_1$ as
\begin{equation}
X^{-}(\cV) := \{([\alpha],\alpha\circ\beta)\mid [\alpha]\in\PP(\cV),\ \beta\in\cV\}.
\end{equation}
Let $\pi_{+}\colon X^{+}(\cV)\to X(\cV)$ be defined by
\begin{equation}
\begin{matrix}
X^{+}(\cV) & \overset{\pi_{+}}{\lra} & X(\cV) \\
([\beta],\alpha\circ\beta) & \mapsto & \tau(\alpha\circ\beta)
\end{matrix}
\end{equation}
Define $\pi_{-}\colon X^{-}(\cV)\to X(\cV)$ similarly.
Then $\pi_{\pm}$ is a small resolution of $X(\cV)$; the exceptional set
$Ex(\pi_{\pm})$ is isomorphic to $\PP(\cV)$.
We claim that $X^{+}(\cV)$, $X^{-}(\cV)$ are obtained by contracting $\wt{X}(\cV)$. In fact by~\Ref{prp}{equarela} we may identify $\wt{X}(\cV)$ with the blow-up of $(\cV\otimes\cV)_1$ at the origin. Thus we have a natural embedding $\wt{X}(\cV)\subset\PP(\cV)\times \PP(\cV)\times(\cV\otimes\cV)_1$ and a $\CC$-fibration
\begin{equation}
\xymatrix{ \wt{X}(\cV) \ar[d] & \supset &
\{([\beta],[\alpha],t\alpha\circ\beta)\mid t\in\CC\} \ar[d] \\
%
\PP(\cV)\times \PP(\cV) & \ni & ([\beta],[\alpha])}
\end{equation}
In fact this is the fibration that realizes
$\wt{X}(\cV)$ as the total space of the line-bundle $\cO_{\PP(\cV)}(-1)\boxtimes\cO_{\PP(\cV)}(-1)$ over $\PP(\cV)\times \PP(\cV)$. It follows that we have obvious forgetful maps $\rho_{\pm}\colon \wt{X}(\cV)\to X^{\pm}(\cV)$. As is easily checked $\rho_{+}$, $\rho_{-}$ are the maps contracting $E$ along the two projections $E\to\PP(\cV)$. Thus we get a commutative diagram
\begin{equation}\label{zigzag}
\xymatrix{ & \wt{X}(\cV) \ar_{\rho_{+}}[dl] \ar^{\pi}[dd] \ar^{\rho_{-}}[dr] & \\
%
X^{+}(\cV) \ar_{\pi_{+}}[dr] & &
X^{-}(\cV) \ar^{\pi_{-}}[dl] \\
%
& X(\cV) & \\}
\end{equation}
Next we notice that the rational map
\begin{equation}
\pi_{\mp}^{-1}\circ\pi_{\pm}\colon X^{\pm}(\cV)\dashrightarrow X^{\mp}(\cV)
\end{equation}
is identified with the flop of $Ex(\pi^{\pm})=\PP(\cV)$ (i.e.~the blow-up of $Ex(\pi^{\pm})$ followed by contraction of $\PP(\cV)^2$ along the second projection). On the other hand let $\phi_{\pm}$ be the rational involution of $ X^{\pm}(\cV)$ induced by the involution $\phi$ of $X(\cV)$, explicitly
\begin{equation}
\begin{matrix}
X^{+}(\cV) & \overset{\phi_{+}}{\lra} & X^{+}(\cV) \\
([\beta],\alpha\circ\beta) & \mapsto & ([\alpha^t],\beta^t\circ\alpha^t)
\end{matrix}
\end{equation}
and similarly for $\phi_{-}$. Then $\phi_{\pm}$ is identified with the flop
of $Ex(\pi^{\pm})=\PP(\cV)$ as well. It follows that the composition $\pi_{\mp}^{-1}\circ\pi_{\pm}\circ\phi_{\pm}$ (a priori a rational map) is an isomorphism $X^{\pm}(\cV)\cong X^{\mp}(\cV)$; explicitly
\begin{equation}\label{isopiccolo}
\begin{matrix}
X^{+}(\cV) & \overset{\pi_{-}^{-1}\circ\pi_{+}\circ\phi_{+}}{\lra} & X^{-}(\cV) \\
([\beta],\alpha\circ\beta) & \mapsto & ([\beta^t],\beta^t\circ\alpha^t)
\end{matrix}
\end{equation}
and $\pi_{+}^{-1}\circ\pi_{-}\circ\phi_{-}$ is the inverse of the above map.
\subsection{Desingularization of certain double covers}\label{subsec:desdop}
\setcounter{equation}{0}
Suppose that we are given $\beta$ as in~\eqref{morbeta} and that~\Ref{hyp}{ipotizzo} holds.
We will analyze $f_{\beta}\colon X_{\beta}\to Y_{\beta}$ in a neighborhood of $f_{\beta}^{-1}(p)$ where $p\in Y_{\beta}(3)$. Let $\gamma$ be the symmetrization of $\beta$ - see~\Ref{dfn}{simmetrizzo}. Then $f_{\beta}\colon X_{\beta}\to Y_{\beta}$ is identified with $f_{\gamma}\colon X_{\gamma}\to Y_{\gamma}$. We will replace throughout $X_{\beta}$ by $X_{\gamma}$.
Suppose that
$p\in Y_{\gamma}(3)$.
We may express $f_{\gamma}$ as the pull-back of $f\colon X(\cV)\to Y(\cV)$ via a suitable map $Y_{\gamma}\to Y(\cV)$. In fact it suffices to apply the reduction procedure of~\Ref{subsec}{rivdop}.
Let ${\bf K}$ be as in~\Ref{subsec}{rivdop}: by~\eqref{yuguali} we have $(Y_{\gamma_{\bf K}},p)=
(Y_{\gamma},p)$ and by~\Ref{clm}{nucleo} we have a natural isomorphism $(X_{\gamma_{\bf K}},f_{\gamma_{\bf K}}^{-1}(p))\overset{\sim}{\to}
(X_{\gamma},f_{\gamma}^{-1}(p))$ commuting with $f_{\gamma_{\bf K}}$ and $f_{\gamma}$. Shrinking $\Spec R$ if necessary we may assume that ${\bf K}$ is a trivial $R$-module; thus ${\bf K}=\cV\otimes R$ where $\cV$ is a complex vector-space and $\dim\cV=3$. Hence we may view $\gamma_{\bf K}$ as a map $\gamma_{\bf K}\colon \Spec R\to \Sym^2\cV^{\vee}$. Notice that we have equality of schemes $Y_{\gamma}=\gamma_{\bf K}^{-1}Y(\cV)$; thus the restriction of $\gamma_{\bf K}$ to $Y_{\gamma} $ defines a map $\mu\colon Y_{\gamma} \to Y(\cV)$. The following claim is immediate.
\begin{clm}\label{clm:tiroind}
Keep notation and hypotheses as above. There exists a map $\nu\colon X_{\gamma}\to X(\cV)$ such that the following diagram commutes
\begin{equation}\label{quadruniv}
\xymatrix{ X_{\gamma} \ar_{f_{\gamma}}[d] \ar^{\nu}[r] &
X(\cV) \ar^{f}[d] \\
%
Y_{\gamma} \ar^{\mu}[r] & Y(\cV)}
\end{equation}
Furthermore the above diagram identifies $X_{\gamma}$ with the fibered product $Y_{\gamma}\times_{Y(\cV)}X(\cV)$.
\end{clm}
Our next task is to give an explicit desingularization of $X_{\gamma}$ under suitable hypotheses.
Let
\begin{equation}\label{wuzeta}
W_{\gamma}:=(f_{\gamma}^{-1}Y_{\gamma}[3])\cap V(\xi_1,\ldots,\xi_d)\,.
\end{equation}
A few words of explanation. First $Y_{\gamma}[c]$ is given by~(\ref{eccozici}), secondly in defining $W_{\gamma}$ we we are refering to Embedding~(\ref{esplicito}) - with $\{\xi_1,\ldots,\xi_d\}$ a basis of $\cW$.
Notice that $f_{\gamma}$ defines an isomorphism of schemes
\begin{equation}\label{isodelta}
W_{\gamma}\overset{\sim}{\lra}Y_{\gamma}[3]\,.
\end{equation}
Let
%
\begin{equation}
\pi_{\gamma}\colon\wt{X}_{\gamma}\to X_{\gamma}
\end{equation}
be the blow-up of $W_{\gamma}$ and $\wt{f}_{\gamma}:=f_{\gamma}\circ\pi_{\gamma}$.
\begin{ass}\label{ass:realsteel}
Notation and hypotheses as above.
\begin{itemize}
\item[(a)]
$R$ is a regular finitely generated $\CC$-algebra.
%
\item[(b)]
$\cork\gamma(p)\le 3$ for all $p\in\Spec R$.
\item[(c)]
The map $\delta_{\gamma}(p)$ defined by~\eqref{difftre}
is surjective for all closed $p\in\Spec R$.
\end{itemize}
\end{ass}
\begin{prp}\label{prp:scoppiodelta}
Keep notation as above. Suppose that~\Ref{ass}{realsteel} holds.
Then
\begin{itemize}
\item[(1)]
$Y_{\gamma}[3]$ is smooth of dimension equal to $(\dim R-6)$.
\item[(2)]
Let $p\in Y_{\gamma}(3)$ be a closed point. Let $r:=(\dim_p X_{\gamma}-5)$. There exists an isomorphism between the analytic germ $(X_{\gamma},f_{\gamma}^{-1}(p))$ and the product $(\CC^r,0)\times(X(\cV),f^{-1}(0))$ such that the composition
\begin{equation}
(X_{\gamma},f_{\gamma}^{-1}(p))
\overset{\sim}{\lra}(\CC^r,0)\times(X(\cV),f^{-1}(0))\lra
(X(\cV),f^{-1}(0))
\end{equation}
is the germ of the map $\nu$ of~\eqref{quadruniv}.
\item[(3)]
$\wt{X}_{\gamma}$ is smooth.
\item[(4)]
Let $p\in Y_{\gamma}(3)$ be a closed point and $K(p):=\ker\gamma(p)$. There is a canonical isomorphism of schemes $\wt{f}_{\gamma}^{-1}(p)\cong\PP(K(p)^{\vee})\times\PP(K(p)^{\vee})$.
\end{itemize}
\end{prp}
\begin{proof}
Item~(1) follows at once from the hypotheses and the well-known description of the normal space to the locus of quadratic forms of corank at least $r$ at a quadratic form of corank $r$. Item~(2) follows from~\Ref{clm}{tiroind}.
Items~(3) and~(4) follow from Item~(2) above together with Items~(1) and~(2) of~\Ref{clm}{antonioni}.
\end{proof}
Let $E_{\gamma}\subset\wt{X}_{\gamma}$ be the exceptional set of $\pi_{\gamma}$. By~\Ref{prp}{scoppiodelta} the restriction of $\wt{f}_{\gamma}$ to $E_{\gamma}$ defines a fibration
\begin{equation}\label{brugola}
\xymatrix{ \PP^2\times\PP^2 \ar[r] & E_{\gamma} \ar[d]
\\
%
& Y_{\gamma}[3]}
\end{equation}
with fiber $\PP(K(p)^{\vee})\times\PP(K(p)^{\vee})$ over $p\in Y_{\gamma}[3]$. The following claim follows at once from Item~(4) of~\Ref{prp}{scoppiodelta}.
%
\begin{clm}\label{clm:contraggo}
Keep notation as above.
Let $p\in Y_{\gamma}[3]$. The restriction of the normal bundle $N_{E_{\gamma}/\wt{X}_{\gamma}}$ to the fiber over $p$ of~\eqref{brugola} is isomorphic to
\begin{equation}
\cO_{\PP(K(p)^{\vee})}(-1)\boxtimes
\cO_{\PP(K(p)^{\vee})}(-1).
\end{equation}
\end{clm}
Now assume that Fibration~(\ref{brugola}) is trivial - this may always be achieved by shrinking $\Spec R$ around a given point $p\in Y_{\gamma}(3)$. A trivialization of~\eqref{brugola} determines two distinct $\PP^2$-fibrations
%
\begin{equation}\label{fibiano}
\epsilon\colon E_{\gamma}\lra *
\end{equation}
where $*$ is itself a $\PP^2$-fibration over $Y_{\gamma}[3]$.
(The set of such structures of $\PP^2$-fibration on
$E_{\gamma}$ has cardinality $2^s$ where $s$ is the number of connected components of $Y_{\gamma}(3)$.)
By~\Ref{clm}{contraggo} we can blow down $\wt{X}_{\gamma}$ along the chosen fibration $\PP^2$-fibration $\epsilon$: let $c_{\gamma,\epsilon}\colon
\wt{X}_{\gamma}\lra X^{\epsilon}_{\gamma}$ be the contraction map. The complex space $X^{\epsilon}_{\gamma}$ (there is no guarantee that it is quasi-projective) will be smooth.
Every such structure $\epsilon$ of $\PP^2$-fibration on
$E_{\gamma}$ has a \lq\lq complementary\rq\rq $\PP^2$-fibration $\epsilon^c$ determined by choosing the \lq\lq other\rq\rq $\PP^2$-fibration on every connected component of $E_{\gamma}$.
We have a commutative diagram
\begin{equation}\label{diamante}
\xymatrix{ & \wt{X}_{\gamma} \ar_{c_{\gamma,\epsilon}}[dl] \ar^{\pi_{\gamma}}[dd] \ar^{c_{\gamma,\epsilon^c}}[dr] & \\
%
X^{\epsilon}_{\gamma} \ar_{\pi_{\gamma,\epsilon}}[dr] & &
X^{\epsilon^c}_{\gamma} \ar^{\pi_{\gamma,\epsilon^c}}[dl] \\
%
& X_{\gamma} & \\}
\end{equation}
\begin{prp}\label{prp:fattinostri}
Keep notation and assumptions as above.
Then the following hold:
\begin{itemize}
\item[(1)]
$\sing X_{\gamma}=W_{\gamma}$,
\item[(2)]
$\pi_{\gamma,g}$, $\pi_{\gamma,\epsilon}$ and $\pi_{\gamma,\epsilon^c}$ are resolutions of $X_{\gamma}$,
\item[(3)]
$\pi_{\gamma,\epsilon}$ and $\pi_{\gamma,\epsilon^c}$ are semi-small resolutions of $X_{\gamma}$,
\item[(4)]
the rational involution $\phi_{\epsilon}\colon X^{\epsilon}_{\gamma}\dashrightarrow X^{\epsilon}_{\gamma}$ induced by the involution $\phi_{\gamma}$ of $X_{\gamma}$ is the flop of the $\PP^2$-fibration~(\ref{fibiano}),
\item[(5)]
the rational map $\pi_{\gamma,\epsilon^c}^{-1}\circ\pi_{\gamma,\epsilon} \colon X^{\epsilon}_{\gamma}\dashrightarrow X^{\epsilon^c}_{\gamma}$ is the flop of the $\PP^2$-fibration~(\ref{fibiano}),
\item[(6)]
the (a priori rational) map $\pi_{\gamma,\epsilon^c}^{-1}\circ\pi_{\gamma,\epsilon}\circ\phi_{\epsilon}$ is actually an isomorphism
\begin{equation}
\pi_{\gamma,\epsilon^c}^{-1}\circ\pi_{\gamma,\epsilon}\circ\phi_{\epsilon}\colon
X^{\epsilon}_{\gamma}\overset{\sim}{\lra} X^{\epsilon^c}_{\gamma}\,.
\end{equation}
\end{itemize}
\end{prp}
\begin{proof}
Let $q\in X_{\gamma}$ and $p:=V$. By definition of $X_{\gamma}$ we have $1\le \cork\,\gamma(p)\le 3$. If $\cork\,\gamma(p)=1$ then $Y_{\gamma}$ is smooth at $p$ because by hypothesis the differential $\delta_{\gamma}(p)$ is surjective: thus
$X_{\gamma}$ is smooth at $q$ by~\Ref{prp}{gradodue}. If $\cork\,\gamma(p)=2$ then $X_{\gamma}$ is smooth at $q$ by~\Ref{prp}{biancabernie}. This proves that $\sing X_{\gamma}\subset W_{\gamma}$. On the other hand if $q\in W_{\gamma}$ then $X_{\gamma}$ is singular at $q$ by Item~(2) of~\Ref{prp}{scoppiodelta} and by~\Ref{crl}{singixvu}. This finishes the proof of Item~(1) of~\Ref{prp}{fattinostri}. The other Items follow from the corresponding statements for $\wt{X}(\cV)$, $X^{\pm}(\cV)$ etc.~together with Item~(2) of~\Ref{prp}{scoppiodelta}.
\end{proof}
\section{Families of double EPW-sextics}\label{sec:famigliaepw}
\setcounter{equation}{0}
Let
\begin{equation}\label{ennevu}
\NN(V):=\{A\in\lagr \mid Y_A=\PP(V)\}.
\end{equation}
Notice that $\NN(V)$ is closed.
Let $\cY$ be the tautological family of EPW-sextics
i.e.
\begin{equation}
\cY:=\{(A,[v])\in (\lagr\setminus\NN(V))\times\PP(V)\mid
\dim(A\cap F_v)>0\}\,.
\end{equation}
Of course $\cY$ has a description as a determinantal variety
and hence a
natural scheme structure.
Let $\cU\subset (\lagr\setminus\NN(V))$ be open and $\cY_{\cU}:=\cY\cap(\cU\times\PP(V))$. Suppose that there exist a scheme $\cX_{\cU}$ and a finite map
\begin{equation}\label{famdoppia}
f_{\cU}\colon \cX_{\cU}\to \cY_{\cU}
\end{equation}
such that for every $A\in\cU$ the induced map $f^{-1}Y_A\to Y_A$ is identified with $f_A\colon X_A\to Y_A$:
then we say that a {\it tautological family of double EPW-sextics parametrized by $\cU$ exists} - often we simply state that~\eqref{famdoppia} exists. Composing $f_{\cU}$ with the natural map $\cY_{\cU}\to\cU$ we get a map
%
\begin{equation}\label{mapparo}
\rho_{\cU}\colon \cX_{\cU}\to\cU
\end{equation}
such that $\rho_{\cU}^{-1}(A)\cong X_A$. Given $B\in \lagr$ we let
%
\begin{equation}\label{eccoua}
U_B:=\{A\in \lagr \mid A\pitchfork B\}\setminus\NN(V).
\end{equation}
\begin{prp}
Let $B\in\lagr$. A tautological family of double EPW-sextics parametrized by $U_B$ exists.
\end{prp}
\begin{proof}
Let $i_{U_B}\colon\cY_{U_B}\hra U_B\times\PP(V)$ be inclusion and let $\cA$ be the tautological rank-$10$ vector-bundle on $\lagr$ (the fiber of $\cA$ over $A$ is $A$ itself).
Going through the argument that
produced Commutative Diagram~\eqref{spqr} we get that there exists a commutative diagram
\begin{equation}\label{pazzaidea}
\begin{array}{ccccccccc}
0 & \to & \cO_{U_B}\boxtimes F&\mapor{\lambda_{U_B}}&
(\cA^{\vee}|_{U_B})\boxtimes\cO_{\PP(V)} &
\lra & i_{U_B,*}\zeta_{U_B} &
\to & 0\\
& & \mapver{\mu_{U_B}}& &\mapver{\mu^{t}_{U_B}} &
&
\mapver{\beta_{U_B}}& & \\
0 & \to & (\cA|_{U_B})\boxtimes\cO_{\PP(V)}&
\mapor{\lambda_{U_B}^{t}}&\cO_{U_B}\boxtimes F^{\vee} & \lra &
Ext^1(i_{U_B,*}\zeta_{U_B},\cO_{U_B\times\PP(V)}) & \to & 0
\end{array}
\end{equation}
Let $\xi_{U_B}:=\zeta_{U_B}\otimes\rho^{*}\cO_{\PP(V)}(-3)$ where $\rho\colon U_B\times\PP(V)\to\PP(V)$ is projection. Proceeding as in the definition of the multiplication on $\cO_{Y_A}\oplus\xi_A$ we get that $\beta_{U_B}$ defines a multiplication on $\cO_{\cY_{U_B}}\oplus\xi_{U_B}$. By~\Ref{prp}{redux} we get that $\cO_{\cY_{U_B}}\oplus\xi_{U_B}$ is an associative commutative ring.
Let $\cX_{U_B}:=\Spec(\cO_{\cY_{U_B}}\oplus\xi_{U_B})$ and $f_{U_B}\colon\cX_{U_B}\to\cY_{U_B}$ be the structure map.
\end{proof}
Of course $\lagr\setminus\NN(V)$ is covered by open sets as above and hence locally around any $A\in \lagr\setminus\NN(V)$ we do have a tautological family of double EPW-sextics. We leave the proof of the following result to the reader.
\begin{clm}
Let $\cU\subset (\lagr\setminus\NN(V))$ be open and suppose that $f_{\cU}\colon\cX_{\cU}\to \cY_{\cU}$ exists. Then $\rho_{\cU}\colon \cX_{\cU}\to\cU$ is flat over $\cU$.
\end{clm}
Let $\cU\subset (\lagr\setminus\NN(V))$ be open and such that $f_{\cU}\colon\cX_{\cU}\to \cY_{\cU}$ exists.
We will determine the singular locus of $\cX_{\cU}$.
Let
\begin{eqnarray}
\cY[d]:= & \{(A,[v])\in(\lagr\setminus\NN(V))\times\PP(V)
\mid \dim(A\cap F_v)\ge d\},\\
%
\cY(d):= & \{(A,[v])\in(\lagr\setminus\NN(V))\times\PP(V)
\mid \dim(A\cap F_v)= d\}.
\end{eqnarray}
Then $\cY[d]$ has a natural structure of closed subscheme of $\lagr\times\PP(V)$ given by its local description as a symmetric determinantal variety - see {\bf Subsection~2.2} of~\cite{og5}.
Let $\cU\in (\lagr\setminus\NN(V))$ be open. We let $\cY_{\cU}[d]:=\cY[d]\cap \cY_{\cU}$ and similarly for $\cY_{\cU}(d)$. Suppose that $f_{\cU}\colon\cX_{\cU}\to\cY_{\cU}$ is defined; we let
\begin{equation}
\cW_{\cU}:= f_{\cU}^{-1}\cY[3].
\end{equation}
Notice that the restriction of $f_{\cU}$ to $\cW_{\cU}$ defines an isomorphism
$\cW_{\cU}\overset{\sim}{\lra}\cY_{\cU}[3]$ - see~\eqref{isodelta}. We will prove the following result.
\begin{prp}\label{prp:singfam}
Let $\cU\subset (\lagr\setminus\NN(V))$ be open and suppose that $f_{\cU}\colon\cX_{\cU}\to\cY_{\cU}$ exists. Then $\sing\cX_{\cU}=\cW_{\cU}$.
\end{prp}
Let $(A,[v])\in\cY$. Choose $B\in\lagr$ such that $B\pitchfork A$ and $B\pitchfork F_v$. Let $\cN\subset\PP(V)$ be an open neighborhood of $[v]$ such that $B\pitchfork F_w$ for all $w\in\cN$. The restriction to $U_B$ of the tautological
rank-$10$ vector-bundle $\cA$ on $\lagr$ is trivial and the restriction to $\cN$ of $F$ is likewise trivial. Moreover the restriction of $\mu_{U_B}$ to $U_B\times\cN$ is an isomorphism. Let
\begin{equation}\label{gamfam}
\gamma:=\lambda_{U_B}|_{U_B\times\cN}\circ (\mu_{U_B}|_{U_B\times\cN})^{-1}.
\end{equation}
We have an exact sequence
\begin{equation}\label{pattypravo}
0\lra (\cA|_{U_B})\boxtimes\cO_{\cN}\overset{\gamma}{\lra} (\cA^{\vee}|_{U_B})\boxtimes\cO_{\cN}
\lra i_{U_B,*}\zeta_{U_B} |_{U_B\times\cN} \lra 0
\end{equation}
The map $\gamma$ is symmetric, in fact it is the symmetrization of the restriction of~\eqref{pazzaidea} to $U_B\times\cN$ - see~\Ref{dfn}{simmetrizzo}. The double cover $X_{\gamma}\lra \cY_{U_B}\cap(U_B\times\cN)$ is isomorphic to the inverse image of $U_B\times\cN$ for the double cover
$\cX_{U_B}\lra \cY_{U_B}$. Thus it suffices to examine $X_{\gamma}$.
Let
%
\begin{equation}\label{delav}
\delta_{\gamma}(A,[v])\colon T_{(A,[v])}\lagr
\times\PP(V)\lra \Sym^2(A\cap F_v)^{\vee}
\end{equation}
be as in~(\ref{difftre}).
\begin{lmm}\label{lmm:libero}
Keep notation as above. The map $\delta_{\gamma}(A,[v])$ of~\eqref{delav} is surjective.
\end{lmm}
\begin{proof}
The restriction of $\delta_{\gamma}(A,[v])$ to the tangent space to $\lagr$ at $A$ is surjective.
\end{proof}
\vskip 2mm
\noindent
{\it Proof of~\Ref{prp}{singfam}}
We may assume that $\cU$ is affine, thus $\cU\cong\Spec R$. Of course $R$ is a finitely generated regular $\CC$-algebra. Let $\gamma$ be as above: thus $\cX_{\cU}\cong\cX_{\gamma}$.
Let $q\in\cX_{\cU}$ and $f_{\cU}(q)=(A,[v])$.
Suppose that $q\notin\cW_{\cU}$ i.e.~that $\cork\gamma(p)\le 2$. If $\cork\gamma(p)=1$ then $Y_{\cU}=Y_{\gamma}$ is smooth because the differential $\delta_{\gamma}(A,[v])$ is surjective (\Ref{lmm}{libero}): by~\Ref{prp}{gradodue} we get that $\cX_{\cU}$ is smooth at $q$. If $\cork\gamma(p)=2$ then $\cX_{\cU}$ is smooth at $q$ by~\Ref{prp}{biancabernie} - recall that the differential $\delta_{\gamma}(A,[v])$ is surjective by~\Ref{lmm}{libero}.
This proves that $\sing\cX_{\cU}\subset\cW_{\cU}$. On the other hand $\cW_{\cU}\subset \sing\cX_{\cU}$ by~\Ref{clm}{sempresing}.
\qed
\vskip 2mm
\n
We will close the present section by proving a few results about the indivdual $X_A$'s. First a definition: for $A\in\lagr$ we let
\begin{equation}\label{eccoteta}
\Theta_A:=\{W\in\mathrm{Gr}(3,V) \mid \bigwedge^3 W\subset A\}.
\end{equation}
\begin{lmm}\label{lmm:critliscio}
Let $A\in(\lagr\setminus\NN(V))$ and $[v]\in Y_A$. Suppose that
$\dim(A\cap F_v)\le 2$ and that
there is no $W\in\Theta_A$ containing $v$.
Then $X_A$ is smooth at $f_A^{-1}([v])$.
\end{lmm}
\begin{proof}
Let $q\in f^{-1}_A([v])$. Suppose that $\dim(A\cap F_v)=1$. By Corollary~2.5 of~\cite{og5} we get that $Y_A$ is smooth at $[v]$, thus $X_A$ is smooth at $q$ by~\Ref{prp}{gradodue}.
Suppose that $\dim(A\cap F_v)=2$. Locally around $q$ the double cover $X_A\to Y_A$ is isomorphic to $X_{\ov{\gamma}}\to Y_{\ov{\gamma}}$ where $\ov{\gamma}$ is the symmetrization of the restriction of $\beta_A$ to an affine neighoborhood $\Spec R$ of $[v]$. Thus we may consider the differential $\delta_{\ov{\gamma}}([v])$ - see~\eqref{difftre}. The differential is surjective by Proposition~2.9 of~\cite{og5}, thus
$X_A$ is smooth at $q$ by~\Ref{prp}{biancabernie}.
\end{proof}
\begin{prp}\label{prp:buono}
Let $A\in(\lagr\setminus\NN(V))$. Then $X_A$ is smooth if and only if $A\in\lagr^0$.
\end{prp}
\begin{proof}
If $A\in\lagr^0$ then $X_A$ is smooth by~\cite{og2}. Suppose that $X_A$ is smooth. Then $A\notin\Delta$ by~\Ref{clm}{sempresing}. Assume that $A\in\Sigma$; we will reach a contradiction. Let $W\in\Theta_A$ and $[v]\in\PP(W)$ - notice that $\PP(W)\subset Y_A$. Let $q\in f_A^{-1}([v])$. Since $A\notin\Delta$ we have $1\le \dim(A\cap F_v)\le 2$. Suppose that $\dim(A\cap F_v)=1$. Then $Y_A$ is singular at $[v]$ by Corollary~2.5 of~\cite{og5}, thus $X_A$ is singular at $q$ by~\Ref{prp}{gradodue}. Suppose that $\dim(A\cap F_v)=2$. Let $\ov{\gamma}$ be as in the proof of~\Ref{lmm}{critliscio}. Then $\delta_{\ov{\gamma}}([v])$ is not surjective - see Proposition 2.3 of~\cite{og5} - and hence
$X_A$ is singular at $q$ by~\Ref{prp}{biancabernie}.
\end{proof}
\section{The divisor $\Delta$}\label{sec:avoidelta}
\setcounter{equation}{0}
\subsection{Parameter counts}
\setcounter{equation}{0}
Let $\Delta_{+}\subset\lagr$ and $\wt{\Delta}_{+},\wt{\Delta}_{+}(0)\subset\lagr\times\PP(V)^2$ be
\begin{eqnarray}
\Delta_{+}:= & \{A\in\lagr\mid | Y_A[3]|>1\}\,,
\index{$\Delta_{+}$}\\
\wt{\Delta}_{+}:=& \{(A,[v_1],[v_2])\mid [v_1]\not=[v_2],
\quad \dim(A\cap F_{v_i})\ge 3\}\,,\\
\wt{\Delta}_{+}(0):=& \{(A,[v_1],[v_2])\mid [v_1]\not=[v_2],
\quad \dim(A\cap F_{v_i})= 3\}\,.
\end{eqnarray}
Notice that $\wt{\Delta}_{+}$ and $\wt{\Delta}_{+}(0)$ are locally closed.
\begin{lmm}\label{lmm:codelta}
Keep notation as above. The following hold:
\begin{enumerate}
\item[(1)]
$\wt{\Delta}_{+}$ is irreducible of dimension $53$.
\item[(2)]
$\Delta_{+}$ is constructible and $\cod(\Delta_{+},\lagr)\ge 2$.
\end{enumerate}
\end{lmm}
\begin{proof}
(1): Let's prove that $\wt{\Delta}_{+}(0)$ is irreducible of dimension $53$.
Consider the map
\begin{equation}
\begin{matrix}
\wt{\Delta}_{+}(0) & \overset{\eta}{\lra} &
\mathrm{Gr}(3,\bigwedge^3 V)^2\times\PP(V)^2 \\
(A,[v_1],[v_2]) & \mapsto &
(A\cap F_{v_1},A\cap F_{v_2},[v_1],[v_2])
\end{matrix}
\end{equation}
We have
\begin{equation}
\im\eta=\{(K_1,K_2,[v_1],[v_2])\mid
K_i\in\mathrm{Gr}(3,F_{v_i}),\quad K_1\bot K_2,\quad [v_1]\not=[v_2]\}\,.
\end{equation}
We stratify $\im\eta$ according to $i:=\dim(K_1\cap F_{v_2})$ and to $j:=\dim(K_1\cap K_2)$; of course $j\le i$. Let $(\im\eta)_{i,j}\subset \im\eta$ be the stratum corresponding to $i,j$. A straightforward computation gives that
\begin{multline}
\dim\eta^{-1}(\im\eta)_{i,j}=10+7(3-i)+j(i-j)+(3-j)(4+i)+
\frac{1}{2}(j+5)(j+4)=\\
=53-4i-\frac{1}{2}j(j-1)\,.
\end{multline}
Since $0\le i, j$ one gets that the maximum is achieved for $i=j=0$ and that it equals $53$. It follows that $\wt{\Delta}_{+}(0)$ is irreducible of dimension $53$. On the other hand $\wt{\Delta}_{+}(0)$ is dense in $\wt{\Delta}_{+}$ (easy) and hence we get that Item~(1) holds.
(2): Let $\pi_{+}\colon \wt{\Delta}_{+}\to \lagr$ be the forgetful map: $\pi_{+}([v_1],[v_2],A)=A$. Then $\pi_{+} (\wt{\Delta}_{+})=\Delta_{+}$. By Item~(1) we get that $\dim\Delta_{+}\le 53$: since $\dim\lagr=55$ we get that Item~(2) holds.
\end{proof}
\begin{prp}\label{prp:codelta}
The following hold:
\begin{itemize}
\item[(1)]
$\Delta$ is closed irreducible of codimension $1$ in $\lagr$ and not equal to $\Sigma$.
\item[(2)]
If $A\in\Delta$ is generic then $Y_A[3]=Y_A(3)$ and it consists of a single point.
\end{itemize}
\end{prp}
\begin{proof}
(1): Let
\begin{equation}
\wt{\Delta}:=\{(A,[v])\mid \dim(F_v\cap A)\ge 3\},\qquad
\wt{\Delta}(0):=\{(A,[v])\mid \dim(F_v\cap A)= 3\}.
\end{equation}
Then $\wt{\Delta}$ is a closed subset of $\lagr\times\PP(V)$ and $\wt{\Delta}(0)$ is an open subset of $\wt{\Delta}$. Let
\begin{equation}\label{dimiss}
\pi\colon\wt{\Delta}\to\lagr
\end{equation}
be the forgetful map (restriction of projection). Thus
$ \pi(\wt{\Delta})=\Delta$: since $\pi$ is projective it follows that $\Delta$ is closed.
Projecting $\wt{\Delta}(0)$ to $\PP(V)$ we get that $\wt{\Delta}(0)$ is smooth irreducible of dimension $54$.
A standard dimension count shows that $\wt{\Delta}(0)$ is open dense in $\wt{\Delta}$; thus $\wt{\Delta}$ is irreducible of dimension $54$. It follows that $\Delta$ is irreducible.
By~\Ref{lmm}{codelta} we know that $\dim\wt{\Delta}_{+}\le 53$.
It follows that the generic fiber of $\wt{\Delta}\to\Delta$ is a single point, in particular $\dim\Delta=54$ and hence $\cod(\Delta,\lagr)=1$ because $\dim\lagr=55$. A dimension count shows that $\dim(\Delta\cap\Sigma)<54$ and hence $\Delta\not=\Sigma$. This finishes the proof of Item~(1). (2): Let $A\in\Delta$ be generic: we already noticed that there exists a unique $[v]\in\PP(V)$ such that $([v],A)\in \wt{\Delta}$, i.e.~$Y_A[3]$ consists of a single point. Since $\wt{\Delta}(0)$ is dense in $\wt{\Delta}$ and $\dim\wt{\Delta}=\dim\Delta$ we get that $([v],A)\in \wt{\Delta}(0)$, i.e.~$Y_A[3]=Y_A(3)$. This finishes the proof of Item~(2).
\end{proof}
\subsection{First order computations}
\setcounter{equation}{0}
Let $(A,[v_0])\in\wt{\Delta}(0)$. We will study the differential of $\pi$ - see~\eqref{dimiss} - at $(A,[v_0])$.
First some preliminaries. Given $v\in V$ we define
a quadratic form $\phi^{v_0}_v$ on $F_{v_0}$ as follows. Let $\alpha\in F_{v_0}$; then $\alpha=v_0\wedge\beta$ for some $\beta\in\bigwedge^2 V$. We set
\begin{equation}\label{quadricapluck}
\phi^{v_0}_v(\alpha):=\vol(v_0\wedge v\wedge\beta\wedge\beta).
\end{equation}
The above equation gives a well-defined quadratic form on $F_{v_0}$ because $\beta$ is determined up to addition by an element of $F_{v_0}$. Of course $\phi^{v_0}_v$ depends only on the class of $v$ in $V/[v_0]$. Choose a direct-sum decomposition
\begin{equation}\label{scelgodeco}
V=[v_0]\oplus V_0.
\end{equation}
We have the isomorphism
\begin{equation}\label{trezeguet}
\begin{matrix}
\lambda^{v_0}_{V_0}\colon\bigwedge^2 V_0 &
\overset{\sim}{\lra} & F_{v_0}\\
\beta & \mapsto & v_0\wedge\beta
\end{matrix}
\end{equation}
Under the above identification the Pl\"ucker quadratic forms on $\bigwedge^2 V_0$ correspond to the quadratic forms $\phi^{v_0}_v$ for $v$ varying in $V_0$.
Let
$K:=A\cap F_{v_0}$ and
\begin{equation}\label{botte}
\begin{matrix}
V_0 & \overset{\tau^{v_0}_K}{\lra} & \Sym^2 K^{\vee} \\
v & \mapsto & \phi^{v_0}_v|_K
\end{matrix}
\qquad
\begin{matrix}
\Sym^2 A^{\vee} & \overset{\theta^{A}_{K}}{\lra} & \Sym^2 K^{\vee} \\
q & \mapsto & q|_K
\end{matrix}
\end{equation}
The isomorphism
\begin{equation*}
\begin{matrix}
V_0 & \overset{\sim}{\lra} & \PP(V)\setminus\PP(V_0) \\
v & \mapsto & [v_0+v]
\end{matrix}
\end{equation*}
defines an isomorphism
\begin{equation*}
V_0\cong T_{[v_0]}\PP(V).
\end{equation*}
Recall that the tangent space to $\lagr$ at $A$ is canonically identified with $\Sym^2 A^{\vee}$.
\begin{prp}\label{prp:tandeltil}
Keep notation as above - in particular choose~\eqref{scelgodeco}. Then
\begin{equation}
T_{(A,[v_0])}\wt{\Delta}\subset T_{(A,[v_0])}\left(\lagr\times\PP(V)\right)=
\Sym^2 A^{\vee}\oplus V_0
\end{equation}
is given by
\begin{equation}
T_{([v_0],A)}\wt{\Delta}=
\{(q,v) \mid \theta^{A}_{K}(q)-\tau^{v_0}_K(v)=0\}.
\end{equation}
\end{prp}
\begin{proof}
Let $B\in\lagr$ be transversal both to $A$ and to $F_{v_0}$. Let $\cN\subset\PP(V)$ be a sufficiently small open neighborhood of $[v_0]$. There exists an exact sequence~\eqref{pattypravo} with $\gamma$ symmetric ($U_B$ is as in~\eqref{eccoua}). Thus $\wt{\Delta}\cap(U_B\times \cN)$ is a symmetric degeneration locus:
\begin{equation}\label{degsim}
\wt{\Delta}\cap(U_B\times \cN)=\{(A',[v])\in (U_B\times \cN) \mid \cork\gamma(A',[v])\ge 3\}.
\end{equation}
It follows that $(q,v)\in T_{([v_0],A)}\wt{\Delta}$ if and only if
\begin{equation*}
0=d\gamma(A,[v_0])(q,v)|_K=d\gamma(A,[v_0])(q,0)|_K+d\gamma(A,[v_0])(0,v)|_K.
\end{equation*}
It is clear that $d\gamma(A,[v_0])(q,0)|_K=\theta^{A}_{K}(q)$. On the other hand Equation~(2.26) of~\cite{og2} gives that
\begin{equation}\label{labambola}
d\gamma(A,[v_0])(0,v)|_K=-\tau^{v_0}_K(v).
\end{equation}
The proposition follows.
\end{proof}
\begin{crl}\label{crl:tanramdel}
$\wt{\Delta}(0)$ is smooth (of codimension $6$ in $\lagr\times\PP(V)$).
Let $(A,[v_0])\in\wt{\Delta}(0)$ and $K:=A\cap F_{v_0}$. The
differential $d\pi(A,[v_0])$ is injective if and only if $\tau^{v_0}_K$ is injective.
%
\end{crl}
\begin{proof}
Let $(A,[v_0])\in\wt{\Delta}(0)$ and $K:=A\cap F_{v_0}$. The map $\theta^A_K$ is surjective: by~\Ref{prp}{tandeltil} we get that $T_{(A,[v_0])}\wt{\Delta}(0)$ has codimension $6$ in $T_{(A,[v_0])}(\lagr\times\PP(V))$.
On the other hand the description of $\wt{\Delta}(0)$ as a symmetric degeneration locus - see~\eqref{degsim} - gives that $\wt{\Delta}(0)$ has codimension at most $6$ in $\lagr\times\PP(V)$: it follows that $\wt{\Delta}(0)$ is smooth of codimension $6$ in $\lagr\times\PP(V)$. The statement about injectivity of $d\pi(A,[v_0])$ follows at once form~\Ref{prp}{tandeltil}.
\end{proof}
A comment regarding~\Ref{crl}{tanramdel}. At a first glance one would think that the statement about smoothness of $\wt{\Delta}(0)$ is contained in the proof of~\Ref{prp}{codelta}: actually in that proof we consider $\wt{\Delta}(0)$ with its reduced structure, here it has the natural scheme structure given by its description as a symmetric degeneracy locus.
\begin{prp}\label{prp:aloha}
Let $(A,[v_0])\in\wt{\Delta}(0)$ and let $K:=A\cap F_{v_0}$. Then
$\tau^{v_0}_K$ is injective if and only if
\begin{itemize}
\item[(1)]
no $W\in\Theta_A$ (see~\eqref{eccoteta} for the definition of $\Theta_A$) contains $v_0$, or
\item[(2)]
there is exactly one $W\in\Theta_A$ containing $v_0$ and moreover
\begin{equation}
A\cap F_{v_0}\cap (\bigwedge^2 W\wedge V)=\bigwedge^3 W.
\end{equation}
\end{itemize}
If Item~(1) holds then $\im\tau^{v_0}_K$ belongs to the unique open $\PGL(K)$-orbit of $\mathrm{Gr}(5, \Sym^2 K^{\vee})$, if Item~(2) holds then $\im\tau^{v_0}_K$ belongs to the unique closed $\PGL(K)$-orbit of $\mathrm{Gr}(5, \Sym^2 K^{\vee})$.
\end{prp}
\begin{proof}
Let $V_0\subset V$ be a codimension-$1$ subspace transversal to $[v_0]$. Let
\begin{equation}\label{rocco}
\rho^{v_0}_{V_0}\colon F_{v_0}\overset{\sim}{\lra} \bigwedge^2 V_0
\end{equation}
be the inverse of Isomorphism~\eqref{trezeguet}.
Let ${\bf K}:=\PP(\rho^{v_0}_{V_0}(K))\subset\PP(\bigwedge^2 V_0)$; then ${\bf K}$ is a projective plane. Isomorphism $\rho^{v_0}_{V_0}$ identifies the space of quadratic forms $\phi^{v_0}_v$, for $v\in V_0$, with the space of Pl\"ucker quadratic forms on $\bigwedge^2 V_0$. Since the ideal of $\mathrm{Gr}(2,V_0)\subset\PP(\bigwedge^2 V_0)$ is generated by the Pl'ucker quadratic forms we get that $\tau^{v_0}_K$ is identified with the natural restriction map
\begin{equation}
V_0=H^0(\cI_{\mathrm{Gr}(2,V_0)}(2))\overset{\tau^{v_0}_K}{\lra} H^0(\cO_{{\bf K}}(2))=\Sym^2 K^{\vee}.
\end{equation}
It follows that if the scheme-theoretic intersection ${\bf K}\cap \mathrm{Gr}(2,V_0)$ is not empty nor a single reduced point then $\tau^{v_0}_K$ is not injective. Now suppose that ${\bf K}\cap \mathrm{Gr}(2,V_0)$ is
\begin{itemize}
\item[($1'$)]
empty i.e.~Item~(1) holds, or
\item[($2'$)]
a single reduced point, i.e.~Item~(2) holds.
\end{itemize}
Let
\begin{equation}
\PP(\bigwedge^2 V_0)\overset{\Phi}{\dashrightarrow} | H^0(\cI_{\mathrm{Gr}(2,V_0)}(2))|^{\vee} =\PP(V_0^{\vee})
\end{equation}
be the natural map: it associates to $[\alpha]\notin\mathrm{Gr}(2,V_0)$ the projectivization of $\supp\alpha$. We have a tautological identification
\begin{equation*}
{\bf K}\overset{\Phi|_{\bf K}}{\dashrightarrow}\PP(\im \tau^{v_0}_K)^{\vee}
\end{equation*}
and of course $\Phi|_{\bf K}$ is the Veronese embedding ${\bf K}\to |\cO_{\bf K}(2)|^{\vee}$ followed by
the projection with center $\PP(\Ann (\im \tau^{v_0}_K))$. Notice that $\tau^{v_0}_K$ is not injective if and only if $\dim\PP(\Ann (\im \tau^{v_0}_K))\ge 1$. Now suppose that~($1'$) holds. Then $\Phi|_{\bf K}$ is regular and in fact it is an isomorphism onto its image - see Lemma~2.7 of~\cite{og5}. Since the chordal variety of the Veronese surface in $|\cO_{\bf K}(2)|^{\vee}$ is a hypersurface it follows that $\dim\PP(\Ann (\im \tau^{v_0}_K))<1$ and hence $\tau^{v_0}_K$ is injective.We also get that $\Ann (\im \tau^{v_0}_K)$ is a point in $|\cO_{\bf K}(2)|^{\vee}$ which does not belong to the chordal variety of the Veronese surface and hence it
belongs to unique open $\PGL(K)$-orbit.
Now suppose that~($2'$) holds. Assume that $\tau^{v_0}_K$ is not injective. Then $\dim\PP(\Ann (\im \tau^{v_0}_K))\ge 1$. It follows that there exist $[x]\not=[y]\in{\bf K}$ in the regular locus of $\Phi|_{\bf K}$ (i.e.~neither $x$ nor $y$ is decomposable) such that $\Phi([x])=\Phi([y])$. By the description of $\Phi$ given above in terms of supports we get that $\supp(x)=\supp(y)=U$ where $\dim U=4$; since $\mathrm{Gr}(2,U)$ is a hypersurface in $\PP(\bigwedge^2 U)$ we get that the line $\la [x],[y]\ra\subset\PP(\bigwedge^2 V_0)$ intersects $\mathrm{Gr}(2,U)$ in a subscheme of length $2$. Since $\la [x],[y]\ra\subset{\bf K}$ it follows that ${\bf K}\cap \mathrm{Gr}(2,V_0)$ contains a scheme of length $2$, that contradicts Item~($2'$). This proves that if~($2'$) holds then $\tau^{v_0}_K$ is injective. It also follows that $\Ann(\tau^{v_0}_K)$ belongs to the Veronese surface in $ |\cO_{\bf K}(2)|^{\vee}$ i.e.~$\im(\tau^{v_0}_K)$ belongs to the unique closed $\PGL(K)$-orbit.
\end{proof}
\section{Simultaneous resolution}\label{sec:famdesing}
\setcounter{equation}{0}
\begin{dfn}\label{dfn:piazzata}
Let $\lagr^{*}\subset \lagr$ be the set of $A$ such that the following hold:
\begin{enumerate}
\item[(1)]
$A\notin\NN(V)$.
\item[(2)]
$Y_A[3]$ is finite.
%
\item[(3)]
$Y_A[3]=Y_A(3)$.
\end{enumerate}
\end{dfn}
\begin{rmk}\label{rmk:piazzata}
$\lagr^{*}$ is an open subset of $\lagr$.
\end{rmk}
\begin{clm}\label{clm:nodeco}
$(\lagr\setminus\Sigma)\subset\lagr^{*}$.
\end{clm}
\begin{proof}
Item~(1) of~\Ref{dfn}{piazzata} holds by Claim 2.11 of~\cite{og5}. Let's prove that Item~(2) of~\Ref{dfn}{piazzata} holds. Suppose that $Y_A[3]\not= Y_A(3)$ i.e.~there exists $[v_0]\in\PP(V)$ such that $\dim(A\cap F_{v_0})\ge 4$. Let $V_0\subset V$ be a codimension-$1$ subspace transversal to $[v_0]$ and let $\rho^{v_0}_{V_0}$ be as in~\eqref{rocco}. Let
${\bf K}:=\PP(\rho^{v_0}_{V_0}(A\cap F_{v_0}))$. Then $\dim{\bf K}\ge 3$; since $\mathrm{Gr}(2,V_0)$ has codimension $3$ in $\PP(\bigwedge^2 V_0)$ it follows that there exists $[\alpha]\in {\bf K}\cap\mathrm{Gr}(2,V_0)$. Let $\wt{\alpha}\in (A\cap F_{v_0})$ such that $\rho^{v_0}_{V_0}(\wt{\alpha})=\alpha$. Then $\wt{\alpha}$ is non-zero and decomposable, that is a contradiction because $A\notin\Sigma$. Lastly let's prove that Item~(3) of~\Ref{dfn}{piazzata} holds. Let $[v_0]\in Y_A[3]=Y_A(3)$. Then $(A,[v_0])\in \wt{\Delta}(0)$. Let $K:=A\cap F_{v_0}$ and $\tau^{v_0}_K$ be as in~\eqref{botte}. We have
\begin{equation*}
T_{[v_0]}Y_A[3]=T_{[v_0]}Y_A(3)=\ker\tau^{v_0}_K.
\end{equation*}
By~\Ref{prp}{aloha} the map $\tau^{v_0}_K$ is injective. Thus $[v_0]$ is an isolated point of $Y_A[3]$.
\end{proof}
Let $A\in\lagr^{*}$. Let
\begin{equation}\label{puntini}
Y_A[3]=\{[v_1],\ldots,[v_s]\}.
\end{equation}
%
Let
\begin{equation}\label{upiccolo}
A\in\cU\subset\lagr^{*}
\qquad\text{$\cU$ open and small.}
\end{equation}
In particular $\rho_{\cU}\colon \cX_{\cU}\to\cY_{\cU}$ exists.
We will construct $2^s$ simultaneous resolutions of
$\cX_{\cU}$.
Let $\pi_{\cU}\colon\wt{\cX}_{\cU}\to\cX_{\cU}$ be the blow-up of $\cW_{\cU}$ and $E_{\cU}$ be the exceptional set of $\pi_{\cU}$.
By~\Ref{lmm}{libero} and~\Ref{prp}{scoppiodelta} $\wt{\cX}_{\cU}$ is smooth and
we have a fibration
\begin{equation}\label{tilli}
\xymatrix{ \PP^2\times\PP^2 \ar[r] & E_{\cU} \ar[d]
\\
%
& Y_{\cU}(3)}
\end{equation}
with fiber over $(A,[v])$ canonically isomorphic to $\PP(A\cap F_v)^{\vee}\times\PP(A\cap F_v)^{\vee}$.
Furthermore by~\Ref{clm}{contraggo} the restriction of the normal bundle $N_{{E_{\cU}}/\wt{\cX}_{\cU}}$ to an arbitrary fiber of~\eqref{tilli} is isomorphic to $\cO_{\PP^2}(-1)\boxtimes\cO_{\PP^2}(-1)$.
We claim that Fibration~\eqref{tilli} is trivial.
In fact since $\cU$ is small and $\wt{\Delta}(0)$ is smooth we have that $\cY_{\cU}[3]$ has $s$ connected components - they are in one-to-one correspondence with $Y_A[3]$ - each isomorphic to a polydisk.
A choice of identification of $E_{\cU}([v_i])$ (the fiber of $E_{\cU}$ over $[v_i]$) with $\PP^2\times\PP^2$ for $i=1,\ldots,s$ determines a trivialization of~\eqref{tilli}.
\begin{rmk}\label{rmk:induista}
Let $\cU$ be as in~\eqref{upiccolo}. Let the $[v_i]$'s be given by~\eqref{puntini} and let
\begin{equation}\label{epsixa}
E_{\cU}([v_i])\lra \PP^2
\end{equation}
be a $\PP^2$-fibration for each $1\le i\le s$.
There exists a unique $\PP^2$-fibration
\begin{equation}\label{marchette}
\epsilon\colon E_{\cU}\lra \star
\end{equation}
where $\star$ is itself a $\PP^2$-fibration over $Y_{\cU}[3]$. We say that~\eqref{epsixa} is a {\it choice of $\PP^2$-fibration $\epsilon$ for $X_A$}.
\end{rmk}
Corresponding to $\epsilon$ we have a smooth contraction $c_{\cU,\epsilon}\colon\wt{X}_{\cU}\to
X^{\epsilon}_{\cU}$. Thus we
have a commutative diagram similar to~\eqref{diamante}.
\begin{equation}\label{scendo}
\xymatrix{ & \wt{X}_{\cU} \ar_{c_{\cU,\epsilon}}[dl] \ar^{\pi_{\cU}}[dd] \ar^{c_{\cU,\epsilon^c}}[dr] & \\
%
X^{\epsilon}_{\cU} \ar_{\pi_{\cU,\epsilon}}[dr] & &
X^{\epsilon^c}_{\cU} \ar^{\pi_{\cU,\epsilon^c}}[dl] \\
%
& X_{\cU} & \\}
\end{equation}
(Here $\epsilon^c$ is the $\PP^2$-fibration \lq\lq complementary\rq\rq to $\epsilon$ - see~\Ref{subsec}{doppipiccoli}.)
We let
\begin{equation}
f_{\cU}^{\epsilon}:=f_{\cU}\circ\pi_{\cU,\epsilon}\,.
\end{equation}
\begin{prp}\label{prp:piuomeno}
Let $\cU$ be as in~\eqref{upiccolo}. Then $\pi_{\cU,\epsilon}$ and $\pi_{\cU,\epsilon^c}$ are semi-small resolutions of $\cX_{\cU}$. Furthermore there exists an isomorphism $\cX_{\cU}^{\epsilon}\overset{\sim}{\lra}\cX_{\cU}^{\epsilon^c}$ fitting into the following commutative diagram
\begin{equation}\label{bulgari}
\xymatrix{ \\ \cX_{\cU}^{\epsilon}\ar_{f_{\cU,\epsilon}}[dr] \ar^{\sim}[rr] & & \cX_{\cU}^{\epsilon^c}\ar^{f_{\cU,\epsilon^c}}[dl] \\
& \cY_{\cU} &}
\end{equation}
\end{prp}
\begin{proof}
We may suppose that $\cU$ is affine, say $\cU=\Spec R$. We have $X_{\cU}\cong X_{\gamma}$ where $\gamma$ is given by~\eqref{gamfam} restricted to $\cU$ (we may assume that $\cU\subset U_B\times\cN$). We claim that~\Ref{ass}{realsteel} holds for $X_{\gamma}$. In fact Item~(a) of~\Ref{ass}{realsteel} holds is obvious, Item~(b) holds because $\cU\subset\lagr^{*}$ and Item~(c) holds by~\Ref{lmm}{libero}. Since~\Ref{ass}{realsteel} holds for $X_{\gamma}\cong \cX_{\cU}$ we get that $\pi_{\cU,\epsilon}$ and $\pi_{\cU,\epsilon^c}$ are semi-small resolutions by~\Ref{prp}{scoppiodelta}. The remaining assertion of the proposition follows from~\Ref{prp}{fattinostri}.
\end{proof}
Let $\cU\subset\lagr^{*}$ be a small open subset. Choose a $\PP^2$-fibration $\epsilon$ for $X_A$ - see~\Ref{rmk}{induista}. We let
\begin{equation}
\rho_{\cU}^{\epsilon}\colon \cX_{\cU}^{\epsilon}\to \cU
\end{equation}
be the map $f_{\cU,\epsilon}$ followed be the tautological map $\cY_{\cU}\to\cU$.
We let
\begin{equation*}
X^{\epsilon}_A:= (\rho^{\epsilon}_{\cU})^{-1}(A),\quad
f_A^{\epsilon}:= f_{\cU}^{\epsilon}|_{X_A^{\epsilon}},\quad
\cO_{X_A^{\epsilon}}(1):= (f_A^{\epsilon})^{*}\cO_{Y_A}(1),\quad
H_A^{\epsilon}\in |\cO_{X_A^{\epsilon}}(1)|.
\end{equation*}
Our notation does not make any reference to $\cU$ because the isomorphism class of the polarized couple $(X^{\epsilon}_A,\cO_{X_A^{\epsilon}}(1))$ does not depend on the open set $\cU$ containing $A$. Notice that if $A\in\Delta $ then $\cO_{X_A^{\epsilon}}(1)$ is not ample, in fact it is trivial on $s$ copies of $\PP^2$ where $s=|Y_A[3]|$.
\Ref{prp}{piuomeno} gives an isomorphism
\begin{equation}
(X^{\epsilon}_A,\cO_{X_A^{\epsilon}}(1))\cong
(X^{\epsilon^c}_A,\cO_{X_A^{\epsilon^c}}(1))\,.
\end{equation}
Of course
\begin{equation}
\text{$(X^{\epsilon}_A,\cO_{X_A^{\epsilon}}(1))\cong
(X_A,\cO_{X_A}(1))$ if $A\in(\lagr\setminus\Delta)$.}
\end{equation}
\begin{prp}\label{prp:fuorisig}
Let $A\in\lagr^{*}$ and let ${\epsilon}$ be a choice of $\PP^2$-fibration for $X_A$.
\begin{itemize}
\item[(1)]
$X_A^{\epsilon}$ is smooth away from $(f_A^{\epsilon})^{-1}(\bigcup_{W\in\Theta_A}\PP(W))$.
\item[(2)]
If $[v_i]\in Y_A[3]$ then
$(f_A^{\epsilon})^{-1}[v_i]\cong \PP(A\cap F_{v_i})^{\vee}$.
\item[(3)]
If $\epsilon'$ is another $\PP^2$-fibration~(\ref{fibiano}) there exists a commutative diagram
\begin{equation}
\xymatrix{ X_A^{\epsilon} \ar_{f_A^{\epsilon}}[dr] & \dra &
X_A^{\epsilon'} \ar^{f_A^{\epsilon'}}[dl] \\
%
& Y_A &}
\end{equation}
where the birational map is the flop of a collection of $(f_A^{\epsilon})^{-1}[v_i]$'s. Conversely every flop of a collection of $(f_A^{\epsilon})^{-1}[v_i]$'s is isomorphic to one $X_A^{\epsilon'}$.
\end{itemize}
\end{prp}
\begin{proof}
Let's prove Item~(1). $X_A^{\epsilon}$ is smooth away from $(f_A^{\epsilon})^{-1}(Y_A[3]\cup\bigcup_{W\in\Theta_A}\PP(W))$ by~\Ref{lmm}{critliscio}. It remains to prove that $X_A^{\epsilon}$ is smooth at every point of $(f_A^{\epsilon})^{-1}\{[v_1],\ldots,[v_s]\}$ where
\begin{equation}
\{[v_1],\ldots,[v_s]\}= Y_A[3]\setminus\bigcup_{W\in\Theta_A}\PP(W).
\end{equation}
Let $A\in\cU$ where $\cU$ be as in~(\ref{upiccolo}). Let $\wt{\rho}_{\cU}:=\rho_{\cU}\circ\pi_{\cU}$; thus $\wt{\rho}_{\cU}\colon\wt{X}_{\cU}\to\cU$.
For $1\le i\le s$ the fiber over $(A,[v_i])$ of Fibration~\eqref{tilli} is canonically isomorphic to $\PP(A\cap F_{v_i})^{\vee}\times\PP(A\cap F_{v_i})^{\vee}$. Let $\wh{X}_A\subset\wt{X}_{\cU}$ be the strict transform of $X_A$. Abusing notation we write
\begin{equation}
\wt{\rho}_{\cU}^{-1}(A)=\wh{X}_A\cup
\bigcup\limits_{i=1}^s \PP(A\cap F_{v_i})^{\vee}\times\PP(A\cap F_{v_i})^{\vee}\,.
\end{equation}
(Of course $\PP(A\cap F_{v_i})^{\vee}\times\PP(A\cap F_{v_i})^{\vee}$ denotes the fiber over $(A,[v_i])$ of Fibration~\eqref{tilli}.)
The components $\PP(A\cap F_{v_i})^{\vee}\times\PP(A\cap F_{v_i})^{\vee}$ are pairwise disjoint. We claim that for $i=1,\ldots,s$ the intersection
\begin{equation}
E_{A,i}:=\wh{X}_A\cap(\PP(A\cap F_{v_i})^{\vee}\times\PP(A\cap F_{v_i})^{\vee})
\end{equation}
is a smooth symmetric divisor in the linear system $|\cO_{\PP(A\cap F_{v_i})^{\vee}}(1)\boxtimes\cO_{\PP(A\cap F_{v_i})^{\vee}}(1)|$. In order to prove this we go back to Map~\eqref{mappasigma} - recall that $\cV$ is a $3$-dimensional complex vector space. Pull-back by $\sigma$ defines an isomorphism
\begin{equation}
\Sym^2\cV^{\vee}\overset{\sigma^{*}}{\lra}
(\cV^{\vee}\otimes\cV^{\vee})^{\ZZ/(2)}=:\Sym_2\cV^{\vee}
\end{equation}
which is $\GL(\cV)$-equivariant. Isomorphism $\sigma^{*}$ induces a $\PGL(\cV)$-equivariant isomorphism of projective spaces ${\bf p}\colon \PP(\Sym^2\cV^{\vee})\overset{\sim}{\lra}
\PP(\Sym_2\cV^{\vee})$.
Of course ${\bf p}$ maps a point in the unique open $\PGL(\cV)$-orbit
of $\PP(\Sym^2\cV^{\vee})$ to a point in the unique open $\PGL(\cV)$-orbit
of $\PP(\Sym_2\cV^{\vee})$. Now let $\cV=(A\cap F_{v_i})^{\vee}$. Let $K_i:=(A\cap F_{v_i})$ and $\tau^{v_i}_{K_i}$ be as in \eqref{botte}.
By~\Ref{prp}{aloha} we have that $\im(\tau^{v_i}_{K_i})$ belongs to the unique open $\PGL(K_i)$-orbit
of $\PP(\Sym^2(A\cap F_{v_i}))$. Commutative Diagram~\eqref{triangolo} gives that $E_{A,i}$ is a symmetric smooth divisor in $|\cO_{\PP(A\cap F_{v_i})^{\vee}}(1)\boxtimes\cO_{\PP(A\cap F_{v_i})^{\vee}}(1)|$. Thus we have described $\wt{\rho}_{\cU}^{-1}(A)$. Since $X^{\epsilon}_{\cU}$ is obtained from $\wt{X}_{\cU}$ by contracting $E_{\cU}$ along the $\PP^2$-fibration $\epsilon$ it follows that $X_A^{\epsilon}$ is smooth at every point of $(f_A^{\epsilon})^{-1}\{[v_1],\ldots,[v_s]\}$.
This proves Item~(1). Since $X^{\epsilon}_A$ is obtained from $\wh{X}_A$ by contracting each of the divisors $E_{A,i}$ along the fibration $\PP^1\to E_{A,i}\to \PP(A\cap F_{v_i})^{\vee}$ determined by $\epsilon$ (and similarly for $\epsilon'$) we also get Items~(2) and~(3).
\end{proof}
\begin{crl}
Let $A\in(\lagr\setminus\Sigma)$. Then we have a desingularization $\rho_A^{\epsilon}\colon X_A^{\epsilon}\to X_A$ for every choice of $\PP^2$-fibration $\epsilon$ for $X_A$.
\end{crl}
\begin{proof}
By~\Ref{clm}{nodeco} we know that $A\in\lagr^{*}$: thus~\Ref{prp}{fuorisig} applies to $X_A^{\epsilon}$. Since $A\notin\Sigma$ we get that $X_A^{\epsilon}$ is smooth by Item~(1) of~\Ref{prp}{fuorisig}.
\end{proof}
\begin{crl}\label{crl:unadef}
Let $A,A'\in(\lagr\setminus\Sigma)$ and ${\epsilon},\epsilon'$ be $\PP^2$-fibrations~\eqref{fibiano}.
The quasi-polarized $4$-folds $(X_{A}^{\epsilon},H_{A}^{\epsilon})$ and $(X_{A'}^{\epsilon},H_{A'}^{\epsilon})$ are deformation equivalent.
\end{crl}
\section{Double EPW-sextics parametrized by $\Delta$}\label{sec:zitelle}
\setcounter{equation}{0}
%
Let $A\in\Delta$ and $[v_0]\in Y_A(3)$. In the first subsection we will associate to $(A,[v_0])$ (under some hypotheses which are certainly satisfied if $A\notin\Sigma$) a $K3$ surface $S_A(v_0)$ of genus $6$, meaning that it comes equipped with a big and nef divisor class $D_A(v_0)$ of square $10$. We will also prove a converse: given a generic such pseudo-polarized $K3$ surface $S$ there exist $A\in\Delta$ and $[v_0]\in Y_A(3)$ such that the pseudo-polarized surfaces $S$ and $S_A(v_0)$ are isomorphic. In the second subsection we will assume that $A\in(\Delta\setminus\Sigma)$ - with this hypothesis $D_A(v_0)$ is very ample. We will prove that there exists a bimeromorphic map $\psi\colon S_A^{[2]}(v_0)\dashrightarrow X_A^{\epsilon}$ where $\epsilon$ is an arbitrary choice of $\PP^2$-fibration for $X_A$. That such a map exists for generic $A\in\Delta$ could be proved by invoking the results of~\cite{og4}. Here we will present a direct proof (we will not appeal to~\cite{og4} nor to~\cite{og2}).
Moreover we will prove that if $S_A(v_0)$ contains no lines (this will be the case for generic $A$) then there exists a choice of $\epsilon$ for which
$\psi$ is regular - in particular $X_A^{\epsilon}$ is projective for such $\epsilon$. Lastly we will notice that the above results show that a smooth double cover of an EPW-sextic is a deformation of the Hilbert square of a $K3$ (and that the family of double EPW-sextics is a locally versal family of projective Hyperk\"ahler manifolds): the proof is more direct than the proof of~\cite{og2}.
\subsection{EPW-sextics and $K3$ surfaces}\label{subsec:kapdel}
\setcounter{equation}{0}
\begin{ass}\label{ass:avuzero}
$A\in\lagr$, $[v_0]\in Y_A(3)$ and the following hold:
\begin{itemize}
\item[(a)]
There exists a codimension-$1$ subspace $V_0\subset V$ such that $\bigwedge^3 V_0\pitchfork A$.
\item[(b)]
There exists at most
one $W\in\Theta_A$ containing $v_0$.
%
\item[(c)]
If $W\in\Theta_A$ contains $v_0$ then $A\cap (\bigwedge^2 W\wedge V)=\bigwedge^3 W$.
%
\end{itemize}
\end{ass}
\begin{rmk}\label{rmk:pixmania}
Let $A\in(\Delta\setminus\Sigma)$. Let $[v_0]\in Y_A(3)$ ($=Y_A[3]$ by~\Ref{clm}{nodeco}). Then~\Ref{ass}{avuzero} holds. In fact Items~(b) and~(c) hold trivially while Item~(a) holds by Claim~2.11 and Equation~(2.81) of~\cite{og5}.
\end{rmk}
Let $(A,[v_0])$ be as in~\Ref{ass}{avuzero}: we will define a surface $S_A(v_0)$ of genus $6$.
The condition that $\bigwedge^3 V_0$ is transverse to $A$ is open: thus we may assume that we have a direct-sum decomposition
\begin{equation}\label{trasuno}
V=[v_0]\oplus V_0.
\end{equation}
We will denote by $\cD$ be the direct-sum decomposition of $V$ appearing in~\eqref{trasuno}.
Let
\begin{equation}\label{kappagrasso}
K^{\cD}_A:= \rho^{v_0}_{V_0}(A\cap F_{v_0}).
\end{equation}
where $\rho^{v_0}_{V_0}$ is given by~\eqref{rocco}.
Choose a volume-form on $V_0$. Wedge-product followed by the volume-form defines an isomorphism $\bigwedge^3 V_0\cong\bigwedge^2 V_0^{\vee}$ and hence it makes sense to let
\begin{equation}
F^{\cD}_A:=\PP(\Ann K^{\cD}_A)\cap\mathrm{Gr}(3,V_0).
\end{equation}
By~\Ref{prp}{fanoindue} and~\Ref{prp}{singfano} we know that $F^{\cD}_A$ is a Fano $3$-fold with at most one singular point. Next we will define a quadratic form on $\Ann K^{\cD}_A$. By Item~(a) of~\Ref{ass}{avuzero} the subspace $A$ is the graph of a map $\wt{q}_A^{\cD}\colon \bigwedge^2 V_0\to \bigwedge^3 V_0$: explicitly
\begin{equation}\label{giannibrera}
\wt{q}_A^{\cD}(\alpha)=\beta\iff
(v_0\wedge\alpha+\beta)\in A.
\end{equation}
%
The map $\wt{q}_A^{\cD}$ is symmetric because $A$, $\bigwedge^2 V_0$ and $\bigwedge^3 V_0$ are lagrangian subspaces of $\bigwedge^3 V$. Clearly $\ker\wt{q}_A^{\cD}=K^{\cD}_A$: it follows that
$\wt{q}_A^{\cD}$ induces an isomorphism
\begin{equation}
\wt{r}_A^{\cD}\colon \bigwedge^2 V_0/K^{\cD}_A\overset{\sim}{\lra}
\Ann K^{\cD}_A\subset \bigwedge^3 V_0.
\end{equation}
The inverse $(\wt{r}_A^{\cD})^{-1}$ defines a non-degenerate quadratic form $(r_A^{\cD})^{\vee}$ on $\Ann K^{\cD}_A$. For future reference we unwind the definition of $(\wt{r}_A^{\cD})^{-1}$ and $(r_A^{\cD})^{\vee}$.
Let $\beta\in \Ann K^{\cD}_A$ i.e.
\begin{equation}\label{killk}
v_0\wedge\alpha+\beta\in A,\qquad \alpha\in\bigwedge^2 V_0.
\end{equation}
Then
\begin{equation}\label{errepol}
(\wt{r}_A^{\cD})^{-1}(\beta)\equiv \alpha\pmod{K^{\cD}_A},\qquad
(r_A^{\cD})^{\vee}(\beta)= \vol(v_0\wedge\alpha\wedge\beta).
\end{equation}
%
Let $V((r_A^{\cD})^{\vee})\subset\PP(\Ann K^{\cD}_A)$ be the zero-scheme of $(r_A^{\cD})^{\vee}$: a smooth $5$-dimensional quadric. Let
%
\begin{equation}\label{essdec}
S_A^{\cD}:=V((r_A^{\cD})^{\vee})\cap F^{\cD}_A.
\end{equation}
Our first goal is to show that $S_A^{\cD}$ does not depend on the choice of the subspace $V_0\subset V$ complementary to $[v_0]$ i.e.~it depends only on $A$ and $[v_0]$. First we notice that $F^{\cD}_A$ is independent of $V_0$. In fact $\bigwedge^3 V_0$ is transversal to $F_{v_0}$; since both $\bigwedge^3 V_0$ and $F_{v_0}$ are Lagrangians the volume $\vol$ induces an isomorphism
\begin{equation}\label{elipattino}
g_{V_0}\colon \bigwedge^3 V_0\overset{\sim}{\lra} F_{v_0}^{\vee}\,.
\end{equation}
Thus $g_{V_0}$ defines an inclusion
\begin{equation}\label{osiris}
F^{\cD}_A\hra\PP(\Ann K_A)\,.
\end{equation}
\begin{rmk}\label{rmk:zoppas}
The image of Map~\eqref{osiris} does not depend on $V_0$ i.e.~it depends exclusively on $A$ and $[v_0]\in Y_A(3)$; we will denote it by $Z_A(v_0)$.
\end{rmk}
Similarly $g_{V_0}$ defines an inclusion
\begin{equation}\label{dovesta}
{\bf g}_{V_0}\colon S_A^{\cD}\hra\PP(\Ann K_A)\,.
\end{equation}
\begin{lmm}\label{lmm:intrinseco}
Keep notation and assumptions as above. Then ${\bf g}_{V_0}(S_A^{\cD})$ is independent of $V_0$, in other words it depends exclusively on $A$ and $[v_0]\in Y_A(3)$.
\end{lmm}
\begin{proof}
Let $V_0'\subset V$ be a codimension-$1$ subspace complementary to $[v_0]$ and transverse to $A$.
Let $\cD'$ denote the corresponding direct-sum decomposition of $V$; we must show that
\begin{equation}\label{stessosub}
{\bf g}_{V_0}(S_A^{\cD})={\bf g}_{V'_0}(S_A^{\cD'})\,.
\end{equation}
The subspace $V_0'$ is the graph of a linear function
\begin{equation}
\begin{matrix}
V_0 & \lra & [v_0] \\
v & \mapsto & f(v)v_0
\end{matrix}
\end{equation}
and hence we have an isomorphism
\begin{equation}
\begin{matrix}
V_0 & \overset{\psi}{\lra} & V_0' \\
v & \mapsto & v+f(v)v_0
\end{matrix}
\end{equation}
We notice that
\begin{equation}\label{contrazione}
\bigwedge^ 3\psi(\beta)=\beta+v_0\wedge (f{\mspace{1.5mu}\lrcorner\mspace{2mu}}\beta)
\end{equation}
where ${\mspace{1.5mu}\lrcorner\mspace{2mu}}$ denotes contraction.
In particular $g_{V'_0}\circ\bigwedge^ 3\psi=g_{V_0}$. Moreover $\phi:=\bigwedge^ 3\psi|_{\Ann K^{\cD}_A}$ is an isomorphism between $\Ann K^{\cD}_A\subset\bigwedge^3 V_0$ and $\Ann K^{\cD'}_{A'}\subset\bigwedge^3 V'_0$.
Thus it suffices to prove that
\begin{equation}\label{corrisponde}
\phi(S_A^{\cD})=S_A^{\cD'}\,.
\end{equation}
We claim that
\begin{equation}\label{amenodi}
\phi^{*}(r_A^{\cD'})^{\vee}-(r_A^{\cD})^{\vee}
\in H^0(\cI_{F^{\cD}_A}(2))\,.
\end{equation}
In fact let $\beta\in \Ann K^{\cD}_A\subset\bigwedge^3 V_0$; then~\eqref{killk} holds.
By~\eqref{contrazione} we get that
\begin{equation}
v_0\wedge(\alpha-(f{\mspace{1.5mu}\lrcorner\mspace{2mu}} \beta))+\phi(\beta)=
v_0\wedge\alpha+\beta\in A\,.
\end{equation}
By~\eqref{contrazione} we get that
\begin{multline}
\phi^{*}(r_A^{\cD'})^{\vee}(\beta)=
\vol(v_0\wedge(\alpha-(f{\mspace{1.5mu}\lrcorner\mspace{2mu}}\beta))\wedge\phi(\beta))=\\
\vol(v_0\wedge\alpha\wedge\phi(\beta))-
\vol(v_0\wedge (f{\mspace{1.5mu}\lrcorner\mspace{2mu}}\beta)\wedge\phi(\beta))=\\
\vol(v_0\wedge\alpha\wedge\beta)-
\vol(v_0\wedge (f{\mspace{1.5mu}\lrcorner\mspace{2mu}}\beta)\wedge\beta)=\\
(r_A^{\cD})^{\vee}(\beta)-
\vol(v_0\wedge (f{\mspace{1.5mu}\lrcorner\mspace{2mu}}\beta)\wedge\beta)\,.
\end{multline}
The second term in the last expression is the restriction to $\PP(\Ann K^{\cD}_A)$ of a Pl\"ucker quadratic form and hence it vanishes on $F^{\cD}_A$. This proves~(\ref{amenodi}) and hence~(\ref{corrisponde}) holds.
\end{proof}
By the above lemma we may give the following definition.
\begin{dfn}
Let $A\in\lagr$. Suppose that $[v_0]\in Y_A(3)$ and that~\Ref{ass}{avuzero} holds. Let $\cD$ be the direct-sum decomposition~\eqref{trasuno}. We set
\begin{equation}
S_A(v_0):={\bf g}_{V_0}(S_A^{\cD}).
\end{equation}
\end{dfn}
Keep assumptions and notation as above. We single out special points of $S_A(v_0)$ as follows. Suppose that $W\in\Theta_A$ (see~\eqref{eccoteta} for the definition of $\Theta_A$) and assume that $v_0\notin W$. Let $\gamma$ be a generator of $\bigwedge^3 W$ i.e.~$\gamma$ is decomposable with $\supp(\gamma)=W$. By hypothesis $\bigwedge^3 V_0\cap A=\{0\}$ and hence $W\not\subset V_0$; thus
\begin{equation}\label{mafalda}
\gamma=(v_0+u_1)\wedge u_2\wedge u_3,\qquad u_i\in V_0\,.
\end{equation}
Since $v_0\notin W$ we have $u_1\wedge u_2\wedge u_3\not=0$; thus $[u_1\wedge u_2\wedge u_3]\in F^{\cD}_A$. Moreover $[u_1\wedge u_2\wedge u_3]\in V((r_A^{\cD})^{\vee})$ by~\eqref{errepol} and hence $[u_1\wedge u_2\wedge u_3]\in S_A^{\cD}$. We let
\begin{equation}
\begin{matrix}
\Theta_A\setminus\{W\mid v_0\in W\} &
\overset{\theta_A^{\cD}}{\lra} & S_A^{\cD} \\
W & \mapsto & [u_1\wedge u_2\wedge u_3]
\end{matrix}
\end{equation}
The map
\begin{equation}
\theta_A(v_0):={\bf g}_{V_0}\circ \theta_A^{\cD}\colon (\Theta_A\setminus\{W\mid v_0\in W\})\to S_A(v_0)
\end{equation}
is independent of $\cD$, i.e.~it depends exclusively on $A$ and $[v_0]$. Notice that $\theta_A(v_0)$ is injective.
\begin{prp}\label{prp:transex}
Let $A\in\lagr$. Suppose that $[v_0]\in Y_A(3)$ and that~\Ref{ass}{avuzero} holds. Let $\cD$ be the direct-sum decomposition~\eqref{trasuno}.
The set of points at which the intersection $V((r_A^{\cD})^{\vee})\cap F^{\cD}_A$ is not transverse is equal to
\begin{equation}
\im\theta_A^{\cD}\coprod(S_A^{\cD}\cap \sing F^{\cD}_A).
\end{equation}
\end{prp}
\begin{proof}
Let $[\beta]\in S^{\cD}_A$. In particular $\beta$ is non-zero decomposable; let $U:=\supp \beta$.
Moreover since $[\beta]\in F^{\cD}_A$ we have that~\eqref{killk} holds; let $\alpha\in\bigwedge^2 V_0$ be as in~\eqref{killk}.
We claim that
\begin{equation}\label{vendette}
\text{$V((r^{\cD}_A)^{\vee})\pitchfork F^{\cD}_A$ at $[\beta]$ unless $\la\alpha,\,K^{\cD}_A\ra\cap\bigwedge^2 U\not=\es$.}
\end{equation}
In fact
the projective tangent space to $\mathrm{Gr}(3,V_0)$ at $[\beta]$ is given by
\begin{equation}\label{tangrass}
{\bf T}_{[\beta]}\mathrm{Gr}(3,V_0)=\PP(\Ann (\bigwedge^2 U))\,.
\end{equation}
On the other hand~\eqref{errepol} gives that
\begin{equation}\label{tanaka}
{\bf T}_{[\beta]}V((r^{\cD}_A)^{\vee})=\PP(\Ann \alpha)\cap\PP(\Ann K^{\cD}_A)\,.
\end{equation}
Statement~(\ref{vendette}) follows at once from~(\ref{tangrass}) and~(\ref{tanaka}).
Next we prove that
\begin{equation}\label{sinequa}
\text{$\la\alpha,\,K^{\cD}_A\ra\cap\bigwedge^2 U\not=\es$ if and only if
$[\beta]\in \sing F^{\cD}_A$ or $[\beta]\in \im\theta_A^{\cD}$.}
\end{equation}
Suppose that $[\beta]\in \sing F^{\cD}_A$; then Item~(1) of~\Ref{prp}{singfano} gives that $K^{\cD}_A\cap\bigwedge^2 U\not=\es$.
Next suppose that $[\beta]\in \im\theta_A^{\cD}$; then $\alpha\in\bigwedge^2 U$ by~(\ref{mafalda}).
This proves the \lq\lq if\rq\rq implication of~\eqref{sinequa}. Let us prove the \lq\lq only if\rq\rq implication. First assume that $K^{\cD}_A\cap\bigwedge^2 U\not=\{0\}$. Let $0\not=\kappa_0\in K^{\cD}_A\cap\bigwedge^2 U$. Then $\kappa_0$ is decomposable because $\dim U=3$ and hence $[\kappa_0]$ is the unique point belonging to $\PP(K^{\cD}_A)\cap \mathrm{Gr}(2,V_0)$. We get that $[\beta]$ is the unique singular point of $F^{\cD}_A$ by~\eqref{nelpiano}. Lastly assume that $K^{\cD}_A\cap\bigwedge^2 U=\{0\}$. Then
there exists $\kappa\in K^{\cD}_A$ such that $(\alpha+\kappa)\in\bigwedge^2 U$. Since $\kappa\in K^{\cD}_A$ we have $(v_0\wedge(\alpha+\kappa)+\beta)\in A$. The tensor $(v_0\wedge (\alpha+\kappa)+\beta)\in A$ is decomposable, let $W$ be its support. Then $v_0\notin W$ because $\beta\not=0$ and hence $[\beta]=\theta_A^{\cD}(W)$. This finishes the proof of~\eqref{sinequa} and of the proposition.
\end{proof}
\begin{crl}\label{crl:prekappa}
Let $A\in\lagr$. Suppose that $[v_0]\in Y_A(3)$ and that~\Ref{ass}{avuzero} holds. Asssume in addition that $\Theta_A$ is finite.
Then $S_A(v_0)$ is a reduced and irreducible surface with
\begin{equation}\label{puntising}
\sing S_A(v_0)=
\im\theta_A(v_0)\coprod(S_A(v_0)\cap \sing Z_A(v_0))\,.
\end{equation}
(See~\Ref{rmk}{zoppas} for the definition of $Z_A(v_0)$.)
\end{crl}
\begin{proof}
By~\Ref{prp}{transex} we know that $S_A^{\cD}$ is a smooth surface outside the right-hand side of~\eqref{puntising}. By hypothesis $\Theta_A$ is finite and hence the right-hand side of~\eqref{puntising} is finite. On the other hand by~\Ref{prp}{singfano} we know that $Z_A(v_0)$ is a $3$-fold with at most one singular point, necessarily an ordinary quadratic singularity, and $S_A^{\cD}$ is the complete intersection of $Z_A(v_0)$ and a quadric hypersurface.
It follows that $S_A^{\cD}$ is reduced and irreducible with singular set as claimed.
\end{proof}
\begin{crl}\label{crl:kappatre}
Let hypotheses be as in~\Ref{crl}{prekappa}. Suppose in addition that $S_A(v_0)$ has Du Val singularities. Let $\wh{S}_A(v_0)\to S_A(v_0)$ be the minimal desingularization. Then $\wh{S}_A(v_0)$ is a $K3$ surface.
\end{crl}
\begin{proof}
Let $\cO_{Z_A(v_0)}(1)$ be the pull-back by Map~(\ref{osiris}) of the hyperplane line-bundle on $\PP(\Ann (F_{v_0}\cap A))$. Then $S_A(v_0)\in |\cO_{Z_A(v_0)}(2)|$. By~\Ref{prp}{fanoindue} and~\Ref{prp}{singfano} there exist smooth divisors in $|\cO_{Z_A(v_0)}(2)|$ and they are $K3$ surfaces; by simultaneous resolution of Du Val singularities we get that $\wh{S}_A(v_0)$ is a $K3$ surface.
\end{proof}
\begin{crl}\label{crl:buonogrande}
Let $A\in(\Delta\setminus\Sigma)$. Let $[v_0]\in Y_A(3)$ (and hence~\Ref{ass}{avuzero} holds by~\Ref{rmk}{pixmania}). Then $S_A(v_0)$ is a (smooth) $K3$.
\end{crl}
\begin{proof}
Immediate consequence of~\Ref{crl}{kappatre}.
\end{proof}
Under the hypotheses of~\Ref{crl}{kappatre} let $\cO_{S_A(v_0)}(1)$ be the restriction to $S_A(v_0)$ of $\cO_{Z_A(v_0)}(1)$. Let $\cO_{\wh{S}_A(v_0)}(1)$ be the pull-back of $\cO_{S_A(v_0)}(1)$ to $\wh{S}_A(v_0)$. We set
\begin{equation}
D_A(v_0)\in | \cO_{S_A(v_0)}(1) | \qquad
\wh{D}_A(v_0)\in | \cO_{\wh{S}_A(v_0)}(1) |.
\end{equation}
\begin{rmk}\label{rmk:genersei}
Let hypotheses be as in~\Ref{crl}{kappatre}. Then $(\wh{S}_A(v_0),\wh{D}_A(v_0))$ is a quasi-polarized
$K3$ surface of genus $6$. Moreover the composition
\begin{equation}
\wh{S}_A(v_0) \lra S_A(v_0)\lra \PP(\Ann (F_{v_0}\cap A))
\end{equation}
is identified (up to projectivities) with the map associated to the complete linear system $| \wh{D}_A(v_0) |$.
\end{rmk}
\Ref{rmk}{genersei} has a converse; in order to formulate it
we identify $F_{v_0}\cong \bigwedge^2 (V/[v_0])$ (the identification is well-defined up to homothety).
%
\begin{ass}\label{ass:kappass}
$ K\in\mathrm{Gr}(3, F_{v_0})$ and
%
\begin{enumerate}
\item[(1)]
$\PP(K)\cap \mathrm{Gr}(2,V/[v_0])=\es$, or
\item[(2)]
the scheme-theoretic intersection $\PP(K)\cap \mathrm{Gr}(2,V/[v_0])$ is a single reduced point.
\end{enumerate}
\end{ass}
%
Let
\begin{equation}\label{voodoo}
W_K:=\PP(\Ann K)\cap \mathrm{Gr}(3,V/[v_0]).
\end{equation}
(This makes sense because we have an isomorphism $\bigwedge^2 (V/[v_0])\overset{\sim}{\lra}\bigwedge^3 (V/[v_0])^{\vee}$ well-defined up to homothety). Let
\begin{equation}\label{bombay}
S:=W_K\cap Q,\qquad \text{$Q\subset\PP(\Ann K)$ a quadric.}
\end{equation}
If $Q$ is generic then $S$ is a linearly normal $K3$ surface of genus $6$, see~\Ref{crl}{kappatre}. In fact the family of such $K3$ surfaces is locally versal. More generally suppose that~\Ref{ass}{kappass} holds, that $S$ is given by~\eqref{bombay} and that $S$
has DuVal singularities. Let $\wh{S}\to S$ be the minimal desingularization - thus $\wh{S}$ is a $K3$ surface. Let $D\in|\cO_S(1)|$ and $\wh{D}$ be the pull-back of $D$ to $\wh{S}$.
Consider the family $\cS\to B$ of deformations of $(S,D)$ obtained by deforming slightly $K$ and $Q$;
by Brieskorn and Tjurina there is a suitable base change $\wh{B}\to B$ such that the pull-back of $\cS$ to $\wh{B}$ admits a simultaneous resolution of singularities $\wh{S}\to\wh{B}$ with fiber $\wh{S}$ over the point corresponding to $S$. Of course there is a divisor class $\wh{\cD}$ on $\wh{\cS}$ whose restriction to $\wh{S}$ is $\wh{D}$ - thus $\wh{\cS}\to\wh{B}$ is a family of quasi-polarized $K3$ surfaces. The following result is well-known - we omit the (standard) proof.
\begin{prp}\label{prp:generikappa}
Keep notation and hypotheses as above.
The family $\wh{\cS}\to\wh{B}$ is a versal family of quasi-polarized $K3$ surfaces.
\end{prp}
\begin{lmm}\label{lmm:scabrosa}
Suppose that~\Ref{ass}{kappass} holds. Let $S$ be as in~\eqref{bombay} and assume that $Q$ is transversal to $W_K$ outside a finite set - thus $S$ is a surface with finite singular set. There exists a smooth quadric $Q'\subset\PP(\Ann K)$ such that $S=W_K\cap Q'$.
\end{lmm}
\begin{proof}
Since $W_K$ is cut out by quadrics Bertini gives that the generic quadric in $\PP(\Ann K)$ containing $S$ is smooth outside $\sing S$; let $Q_0=V(P_0)$ be such a quadric. Let $p\in \sing S$. The generic quadric $Q'=V(P')\in | \cI_{W_K}(2) |$ is smooth at $p$ and hence $V(P_0+P')$ is smooth at $p$. Since $\sing S$ is finite we get that the generic quadric $Q$ containing $S$ is smooth at all points of $\sing S$. It follows that the generic quadric $Q$ containing $S$ is smooth.
\end{proof}
The following corollary provides an inverse of the process which produces $S_A(v_0)$ out of $(A,[v_0])\in\wt{\Delta}(0)$ (with the extra hypotheses in~\Ref{ass}{avuzero}).
\begin{prp}\label{prp:pellegrini}
Suppose that~\Ref{ass}{kappass} holds. Let $S$ be as in~\eqref{bombay} and assume that
$Q$ is smooth and transversal to $W_K$ outside a finite set.
There exist $A\in\Delta$, $[v_0]\in\PP(V)$ and a codimension-$1$ subspace $V_0\subset V$ transversal to $[v_0]$ such that the following hold:
\begin{itemize}
\item[(1)]
$\bigwedge^3 V_0\cap A=\{0\}$,
\item[(2)]
Items~(c) and~(d) of~\Ref{ass}{avuzero} hold,
\item[(3)]
the natural isomorphism $\PP(\bigwedge^3(V/[v_0]))\overset{\sim}{\lra}\PP(\bigwedge^3 V_0)$ maps $S$ to $S_A^{\cD}$ where $\cD$ is the direct-sum decomposition of $V$ appearing in~\eqref{trasuno}.
\end{itemize}
If we replace the quadric $Q$ by a smooth quadric $Q'\subset\PP(\Ann K)$ such that $S=W_K\cap Q'$ and let $A'\in\Delta$ be the corresponding point, there exists a projectivity of $\PP(V)$ fixing $[v_0]$ which takes $A$ to $A'$.
\end{prp}
\begin{proof}
Let $Q=V(P)$. The dual of $\Ann K$ is $\bigwedge^2(V/[v_0])/K$; thus the polarization of $P$ defines a non-degenerate symmetric
map
\begin{equation}
\Ann K\overset{\sim}{\lra} \bigwedge^2(V/[v_0])/K.
\end{equation}
The inverse of the above map is
non-degenerate symmetric
map
\begin{equation}
\bigwedge^2(V/[v_0])/K\overset{\sim}{\lra}\Ann K.
\end{equation}
Composing on the right with $\bigwedge^2(V/[v_0])\overset{\sim}{\lra}\bigwedge^2(V/[v_0])$ and the quotient map $\bigwedge^2(V/[v_0])\to \bigwedge^2(V/[v_0])/K$ and on the left with $\Ann K\hra\bigwedge^3(V/[v_0])$ and $\bigwedge^3(V/[v_0])\overset{\sim}{\lra}\bigwedge^3(V/[v_0])$ we get a symmetric map
\begin{equation}
\bigwedge^2 V_0\lra \bigwedge^3 V_0
\end{equation}
with $3$-dimensional kernel corresponding to $K$. The graph of the above map is a Lagrangian $A\in\lagr$. One checks easily that~(1), (2) and~(3) hold. One gets that the projective equivalence of $A$ does not depend on $Q$ by going through the proof of~\Ref{lmm}{intrinseco}.
\end{proof}
\subsection{$X_A^{\epsilon}$ for $A\in(\Delta\setminus\Sigma)$}\label{subsec:hilbdue}
\setcounter{equation}{0}
Let $S$ be a $K3$.
Let $\Delta_S^{[2]}\subset S^{[2]}$ be the irreducible codimension $1$ subset parametrizing non-reduced subschemes. There exists a square root of the line bundle $\cO_{S^{[2]}}(\Delta_S^{[2]})$: we denote by $\xi$ its first Chern class.
There is a natural morphism of integral Hodge structures $\mu\colon H^2(S)\to H^2(S^{[2]})$ such that
$H^2(S^{[2]};\ZZ)=\mu(H^2(S;\ZZ))\oplus \ZZ\xi$, see~\cite{beau}.
Let $(\cdot,\cdot)$ be the Beauville-Bogomolov bilinear symmetric form on $H^2(S^{[2]})$. It is
known~\cite{beau} that
\begin{equation}\label{poldon}
(\mu(\eta),\mu(\eta))=\int_S c_1(\eta)^2,
\quad \mu(H^2(S;\ZZ))\bot \ZZ\xi,
\quad (\xi,\xi)=-2.
\end{equation}
Since $S$ and $S^{[2]}$ are regular varieties we may identify their Picard groups with $H^{1,1}_{\ZZ}(S)$ and $H^{1,1}_{\ZZ}(S^{[2]})$ respectively. Let $C\in \Pic(S)$; abusing notation we will denote by $\mu(C)$ the class in $\Pic(S^{[2]})$ corresponding to $\mu(\cO_S(C))\in H^{1,1}_{\ZZ}(S)$: if $C$ is an integral curve it is represented by subschemes whose support intersects $C$.
The following is the main result of the present subsection.
\begin{thm}\label{thm:ixhilb}
Let $A\in(\Delta\setminus\Sigma)$ and $[v_0]\in Y_A[3]$ ($=Y_A(3)$ by {Claim 3.11} of~\cite{og5}) - thus $S_A(v_0)$ is a $K3$ surface by~\Ref{crl}{buonogrande}. Then the following hold:
\begin{itemize}
\item[(1)]
If $S_A(v_0)$ does not contain lines (true for generic $A$ by~\Ref{prp}{generikappa}) then there exist a choice $\epsilon$ of $\PP^2$-fibration for $X_A$ and an isomorphism.
\begin{equation}\label{calispera}
\psi\colon {S}_A(v_0)^{[2]}\dashrightarrow X_A^{\epsilon}
\end{equation}
such that
%
\begin{equation}\label{tirodietro}
\psi^{*}H_A^{\epsilon}\sim
\mu(D_A(v_0))-\Delta^{[2]}_{{S}_A(v_0)}.
\end{equation}
\item[(2)]
Let $A$ and $\epsilon$ be arbitrary.
There exists a bimeromorphic map
\begin{equation}\label{calimero}
\psi\colon {S}_A(v_0)^{[2]}\dashrightarrow X_A^{\epsilon}
\end{equation}
such that~\eqref{tirodietro} holds.
%
\end{itemize}
\end{thm}
\begin{rmk}
Suppose that $S_A(v_0)$ contains a line $L$. The restriction of the right-hand side of~\eqref{tirodietro} to $L^{(2)}$ (embedded in $S_A(v_0)^{[2]}$) is $\cO_{L^{(2)}}(-1)$. Since $H_A^{\epsilon}$ is nef we get that in this case Map~\eqref{calimero} cannot be regular.
\end{rmk}
The proof of~\Ref{thm}{ixhilb} will be given after a series of auxiliary results.
Let $S\subset\PP^6$ be a linearly normal $K3$ surface of genus $6$ such that $\cI_{S/\PP^6}(2)$ is globally generated; then $S$ is projectively normal and hence Riemann-Roch gives that $\dim |\cI_S(2)|=5$. One defines a rational map ${S}^{[2]}\dashrightarrow
|\cI_S(2)|^{\vee}$ as follows. Given $[Z]\in{S}^{[2]}$ we let $\la Z\ra\subset \PP^5$ be the line spanned by $Z$. We let
%
\begin{equation}\label{bellaciao}
\begin{matrix}
({S}^{[2]}\setminus \bigcup\limits_
{\text{$L\subset S$
line}}
L^{(2)})
& \overset{g}{\lra} & |\cI_S(2)|^{\vee}\cong\PP^5 \\
& \\
[Z] & \mapsto &
\{Q\in |\cI_S(2)|\mid \text{s.t.}\ Q\supset\la Z\ra\}
\end{matrix}
\end{equation}
Let $D$ be a hyperplane divisor on $S$;
one shows (see Claim~(5.16) of~\cite{og1}) that
\begin{equation}\label{cacciatore}
g^{*}\cO_{\PP^5}(1)\cong \mu({D})-\Delta^{[2]}_{{S}}.
\end{equation}
(Notice that the set of lines on $S$ is finite and hence $\bigcup_
{L\subset S\text{line}} L^{(2)}$
has codimension $2$ in ${S}^{[2]}$.)
In fact $g$ can be identified with the map associated to the complete linear system $|(\mu({D})-\Delta^{[2]}_{{S}})|$. We will analyze $g$ under the assumption that $S$ is generic (in a precise sense).
\begin{ass}\label{ass:ipotesse}
Item~(1) of~\Ref{ass}{kappass} holds.
\begin{equation}
S:=W_K\cap Q
\end{equation}
where $Q\subset\PP(\Ann K)$ is a quadric intersecting transversely $W_K$.
\end{ass}
Let $S\subset\PP(\Ann K)$ be as in~\Ref{ass}{ipotesse}. Then $S$ is a linearly normal $K3$ surface of genus $6$ and $\cI_S(2)$ is globally generated. Thus the map $g$ of~\eqref{bellaciao} is defined.
Let $F(W_K)$ be the variety parametrizing lines in $W_K$. Since the set of lines in $S$ is finite (empty for generic $S$ by~\Ref{prp}{generikappa})
we have a map
\begin{equation}\label{corda}
\begin{matrix}
(F(W_K)\setminus\{L\mid L\subset S\})& \lra & S^{[2]} \\
L & \mapsto & L\cap Q
\end{matrix}
\end{equation}
\begin{dfn}\label{dfn:pianohilb}
Let $P^0_S\subset S^{[2]}$ be the image of Map~(\ref{corda}) and $P_S$ be its closure in $S^{[2]}$.
\end{dfn}
We recall that $F(W_K)\cong\PP^2$ by Iskovskih's ~\Ref{prp}{fanoindue}.
\begin{clm}
Let $S\subset\PP(\Ann K)$ be as in~\Ref{ass}{ipotesse}. Suppose moreover that $S$ contains no lines. Let $C_1,C_2,\ldots,C_s$ be the (smooth) conics contained in $S$ (of course the generic $S$ contains no conics).
Then $P_S,C_1^{(2)},\ldots,C_s^{[2]}$ are pairwise disjoint subset of $S^{[2]}$. Moreover there exists a biregular morphism
\begin{equation}\label{sgonfio}
c\colon S^{[2]}\lra N(S).
\end{equation}
%
contracting each of $P_S,C_1^{(2)},\ldots,C_s^{[2]}$.
Thus $N(S)$ is a compact complex normal space with
%
\begin{equation}\label{singenne}
\sing N(S)=
\{c(P_S),\ldots,c(C^{(2)}),\ldots\mid \text{$C\subset S$ a conic}\}
\end{equation}
and $c$ is an isomorphism of the complement of $P_S\cup C_1^{(2)}\cup\ldots\cup C_s^{[2]}$ onto the smooth locus of $N(S)$.
The map $g$ (regular on all of $S^{[2]}$ because $S$ contains no lines) descends to a regular map
\begin{equation}
\ov{g}\colon N(S)\to |\cI_S(2)|^{\vee},\qquad \ov{g}\circ c=g\,.
\end{equation}
\end{clm}
\begin{proof}
$P_S$ is isomorphic to $\PP^2$ by Iskovskih's ~\Ref{prp}{fanoindue} and each $C_i^{(2)}$ is isomorphic to $\PP^2$ because $C_i$ is a conic. Thus each of $P_S$, $C_i$ can be contracted individually. Let's show that $P_S,C_1^{(2)},\ldots,C_s^{[2]}$ are pairwise disjoint. Suppose that $[Z]\in P_S\cap C_i^{(2)}$. Let $\Lambda$ be the plane containing $C_i$. Then $\Lambda\cap W_K$ contains the line $\la Z\ra$ and the smooth conic $C_i$. Since $W_K$ is cut out by quadrics it follows that $\Lambda\subset W_K$, that is absurd because $W_K$ contains no planes. This proves that $P_S\cap C_i^{(2)}=\es$. On the other hand there does not exist $[Z]\in C_i^{(2)}\cap C_j^{(2)}$ by~\Ref{crl}{consufan}. that $P_S,C_1^{(2)},\ldots,C_s^{[2]}$ are pairwise disjoint. Thus the contraction~\eqref{sgonfio} exists. It remains to prove that $g$ is constant on each of $P_S,C_1^{(2)},\ldots,C_s^{[2]}$.
In fact if $[Z]\in P_S$ then $g([Z])=|\cI_{W_K}(2)|$, if $[Z]\in C_i^{(2)}$ then
\begin{equation*}
g([Z])=\{Q\in |\cI_S(2)| \mid Q\supset \la C_i\ra \}.
\end{equation*}
\end{proof}
Now we go back to the \lq\lq general\rq\rq case: we suppose that~\Ref{ass}{ipotesse} holds however $S$ may very well contain lines.
Let
\begin{equation}\label{hilbstella}
S^{[2]}_{\star}:=S^{[2]}\setminus P_S
\setminus\bigcup\limits_{\text{$R\subset S$ line or conic}}\Hilb^2 R\,.
\end{equation}
(Notice that if $R\subset S$ is a conic which is not smooth then we delete all $[Z]\in S^{[2]}$ such that $Z$ is contained in the scheme $R$.) The following result is essentially {\bf Lemma~3.7} of~\cite{og4}.
\begin{prp}\label{prp:biunivoca}
Suppose that~\Ref{ass}{ipotesse} holds.
\begin{itemize}
\item[(1)]
The fibers of $g|_{S^{[2]}_{\star}}$ are finite of cardinality at most $2$ and the generic fiber has cardinality $2$.
\item[(2)]
There exist an open dense subset $\cA\subset S^{[2]}_{\star}$
and an anti-symplectic (and hence non-trivial) involution $\phi\colon \cA\to \cA$ such that
\begin{equation}\label{invariante}
(g|_{\cA})\circ\phi=g|_{\cA}\,.
\end{equation}
The induced map
\begin{equation}\label{comequoz}
\cA/\la \phi\ra\lra
g(\cA)
\end{equation}
is a bijection.
%
\item[(3)]
If in addition $S$ does not contain lines $\phi$ descends to a regular involution $\ov{\phi}\colon N(S)\to N(S)$ such that $\ov{g}\circ\ov{\phi}=\ov{g}$ and
the induced map
\begin{equation}\label{quozdisc}
j\colon N(S)/\la \ov{\phi}\ra \lra
g(S^{[2]})
\end{equation}
is a bijection. Moreover
%
\begin{equation}\label{cofisso}
\cod(\Fix(\ov{\phi}),N(S))\ge 2
\end{equation}
where $\Fix(\ov{\phi})$ is the fixed-locus of $\ov{\phi}$.
\end{itemize}
\end{prp}
Let $A$ and $[v_0]$ be as in the statement of~\Ref{thm}{ixhilb}: we will perform the
key computation one needs to prove that theorem.
Let $V_0\subset V$ be a codimension-$1$ subspace transversal to $[v_0]$ and such that $\bigwedge^3 V_0\cap A=\{0\}$. Let $\cD$ be Decomposition $V=[v_0]\oplus V_0$ and $S_A^{\cD}$ be given by~(\ref{essdec}) - thus $S_A^{\cD}$ sits in $\PP(\Ann K^{\cD}_A)\cap\mathrm{Gr}(3,V_0)$ and is isomorphic to $S_A(v_0)$.
Let $f\in V_0^{\vee}$; we let $q_f$ be the quadratic form on $\bigwedge^3 V_0$ defined by setting
\begin{equation}
q_f(\omega):=\vol_0((f {\mspace{1.5mu}\lrcorner\mspace{2mu}}\omega)\wedge\omega)
\end{equation}
where $\vol_0$ is a volume-form on $V_0$.
Then $q_f$ is a Pl\"ucker quadric, in fact we have an isomorphism
\begin{equation}\label{puntura}
\begin{matrix}
V_0^{\vee} & \overset{\sim}{\lra} & H^0(\cI_{\mathrm{Gr}(3,V_0)}(2)) \\
f & \mapsto & q_f
\end{matrix}
\end{equation}
Let $V^{\vee}=[v_0^{\vee}]\oplus V_0^{\vee}$ be the dual decomposition of $\cD$; thus $v_0^{\vee}\in \Ann V_0$ and $v_0^{\vee}(v_0)=1$.
We have an isomorphism
\begin{equation}\label{padre}
\begin{matrix}
[v_0^{\vee}]\oplus V_0^{\vee} & \overset{\sim}{\lra} & H^0(\cI_{S_A^{\vee}}(2)) \\
x v_0^{\vee}+f & \mapsto & x (r_A^{\cD})^{\vee}+q_f
\end{matrix}
\end{equation}
We let
%
\begin{equation}\label{proiso}
\iota\colon |\cI_{S_A^{\cD}}(2)|^{\vee}\overset{\sim}{\lra}\PP(V)
\end{equation}
be the projectivization of the transpose of~(\ref{padre}).
\begin{prp}\label{prp:belcalcolo}
Let $A$ and $[v_0]$ be as in the statement of~\Ref{thm}{ixhilb} and keep notation as above. Let $g$ be Map~\eqref{bellaciao} for $S_A^{\cD}$ - this makes sense by~\Ref{crl}{buonogrande}.
Then $\iota(\im g)\subset Y_A$.
\end{prp}
\begin{proof}
Let
\begin{equation}\label{apertodenso}
[Z]\in ((S_A^{\cD})_{\star}^{[2]}\setminus\Delta_{S_A^{\cD}}^{[2]}\setminus
P_{S_A^{\cD}})\,.
\end{equation}
We will prove that
\begin{equation}
\iota(g([Z])\in Y_A\,.
\end{equation}
This will suffice to prove the lemma because the right-hand side of~(\ref{apertodenso}) is dense in $(S_A^{\cD})_{\star}^{[2]}$ and $Y_A$ is closed. By hypothesis $Z$ is reduced; thus $Z=\{[\beta],[\beta']\}$ where $\beta,\beta'\in\bigwedge^3 V_0$ are decomposable. The line $\la[\beta],\beta']\ra$ spanned by $[\beta]$ and $[\beta']$ is not contained in $F^{\cD}_A$ because $[Z]\notin P_{S_A^{\cD}}$. Thus $\la[\beta],\beta']\ra$ is not contained in
$\mathrm{Gr}(3,V_0)$ and it follows that the vector sub-spaces of $V_0$ supporting the decomposable vectors $\beta$ and $\beta'$ intersect in a $1$-dimensional subspace. Thus there exists a basis $\{v_1,\ldots,v_5\}$ of $V_0$ such that
\begin{equation}
\beta=v_1\wedge v_2\wedge v_3,\quad
\beta'=v_1\wedge v_4\wedge v_5\,.
\end{equation}
We may assume moreover that $\vol_0(v_1\wedge v_2\wedge v_3\wedge v_4\wedge v_5)=1$. By~\eqref{killk} and~\eqref{errepol} there exist $\alpha,\alpha'\in\bigwedge^2 V_0$ such that
\begin{equation}
v_0\wedge\alpha+\beta,\ v_0\wedge\alpha'+\beta'\in A,\qquad
\alpha\wedge\beta=\alpha'\wedge\beta'=0\,.
\end{equation}
Since $A$ is Lagrangian we get that
\begin{equation}
\vol_0(\alpha\wedge\beta')=\vol_0(\alpha'\wedge\beta)=:c\,.
\end{equation}
Let $t_0,\ldots,t_5\in\CC$; a straightforward computation gives that
\begin{equation}
(t_0 (r_A^{\cD})^{\vee}+\sum_{i=1}^5 t_i q_{v_i^{\vee}})(\beta+\beta')=
2ct_0+2t_1\,.
\end{equation}
Thus
\begin{equation}\label{imagogi}
\iota(g([Z]))=[cv_0+v_1]\,.
\end{equation}
It remains to prove that
\begin{equation}\label{appartiene}
[cv_0+v_1]\in Y_A\,.
\end{equation}
Let $K^{\cD}_A$ be as in~(\ref{kappagrasso}); we claim that it suffices to prove that there exist $(x,x')\in(\CC^2\setminus\{(0,0)\})$ and $\kappa\in K^{\cD}_A$ such that
\begin{equation}\label{desiderata}
(cv_0+v_1)\wedge
(x(v_0\wedge\alpha+\beta)+x'(v_0\wedge\alpha'+\beta')+
v_0\wedge\kappa)=0\,.
\end{equation}
In fact assume that~(\ref{desiderata}) holds. Then
\begin{equation}
0\not=(x(v_0\wedge\alpha+\beta)+x'(v_0\wedge\alpha'+\beta')+
v_0\wedge\kappa)\in A\cap F_{cv_0+v_1}.
\end{equation}
(The inequality holds because $\beta,\beta'$ are linearly independent.) A straightforward computation gives that~(\ref{desiderata}) is equivalent to
\begin{equation}\label{olimpia}
x(c\beta-v_1\wedge\alpha)+x'(c\beta'-v_1\wedge\alpha')=
v_1\wedge\kappa\,.
\end{equation}
As is easily checked we have
\begin{equation}\label{podio}
(c\beta-v_1\wedge\alpha),\ (c\beta'-v_1\wedge\alpha')\in
([v_1]\wedge(\bigwedge^ 2\la v_2,v_3,v_4,v_5\ra))\cap
\{v_2\wedge v_3,\ v_4\wedge v_5\}^{\bot}
\end{equation}
where perpendicularity is with respect to wedge-product followed by $\vol_0$.
Multiplication by $v_1$ gives an injection $K^{\cD}_A\hra
([v_1]\wedge(\bigwedge^ 2\la v_2,v_3,v_4,v_5\ra))$; in fact no non-zero element of $K^{\cD}_A$ is decomposable because $A\notin\Sigma$. Since the right-hand side of~(\ref{podio}) has dimension $4$ and $\dim K^{\cD}_A=3$ we get that there exists $(x,x')\in(\CC^2\setminus\{(0,0)\})$ such that~(\ref{olimpia}) holds.
\end{proof}
\begin{lmm}\label{lmm:rivcon}
Let $A\in(\lagr\setminus\Sigma)$. Then $Y_A(1)$ is not empty, the topological double cover $f_A^{-1}Y_A(1)\to Y_A(1)$ is not trivial and $Y_A$ is integral.
\end{lmm}
\begin{proof}
By~\Ref{clm}{nodeco} we know that $Y_A[3]$ is finite. On the other hand $(Y_A[2]\setminus Y_A[3])$ is a smooth surface - see Proposition~2.8 of~\cite{og2}. Since $\sing Y_A\subset Y_A[2]$ it follows that $Y_A$ is integral and $Y_A(1)$ is connected. Let $[v_0]\in (Y_A[2]\setminus Y_A[3])$.
By~\Ref{prp}{gradodue} we know that $f_A^{-1}([v_0])$ is a singleton $\{q\}$. Moreover $X_A$ is smooth at $q$ by~\Ref{lmm}{critliscio}. Thus there exists an open neighborhood $U$ of $[v_0]$ in $Y_A$ such that $f_A^{-1}U$ is smooth. Moreover $(f_A^{-1}Y_A[2])\cap f_A^{-1} U$ is nowhere dense in $f_A^{-1} U$. Since $f_A^{-1} U$ is smooth the complement $f_A^{-1}(Y_A(1)\cap U)$ is connected. Since $Y_A(1)$ is connected it follows that
$f_A^{-1}Y_A(1)$ is connected.
\end{proof}
\begin{prp}
Keep hypotheses and notation as in~\Ref{prp}{belcalcolo}. Then $\iota(\ov{\im g})=Y_A$.
\end{prp}
\begin{proof}
By Item~(1) of~\Ref{prp}{biunivoca} the map $g$ has finite generic fiber and hence $\dim\ov{\im g}=4$. By~\Ref{prp}{belcalcolo} we get that $\iota(\ov{\im g})$ is an irreducible component of $Y_A$. On the other hand $Y_A$ is irreducible by~\Ref{lmm}{rivcon}; it follows that $\iota(\ov{\im g})=Y_A$.
\end{proof}
\begin{rmk}\label{rmk:puntovuz}
Keep notation as in~\Ref{prp}{belcalcolo}; then
\begin{equation}
\iota\circ g(P^0_{S_A^{\cD}})=\iota(H^0(\cI_{F^{\cD}_A}(2)))=[v_0].
\end{equation}
\end{rmk}
\noindent
{\it Proof of~\Ref{thm}{ixhilb}.}
Let's prove that Item~(1) holds. Let $A$ and $[v_0]$ be as in the statement of~\Ref{thm}{ixhilb}. Let $V_0\subset V$ be a codimension-$1$ subspace transversal to $[v_0]$ and such that $\bigwedge^3 V_0\cap A=\{0\}$. Let $\cD$ be Decomposition $V=[v_0]\oplus V_0$. In order to simplify notation we set $S=S_A^{\cD}$; thus $S\cong S_A(v_0)$ and by hypothesis $S$ does not contain lines.
Let $j$ be the map of~\eqref{quozdisc}; by~\Ref{prp}{belcalcolo} the composition $\iota\circ j$ is a map
\begin{equation}\label{pettirosso}
\iota\circ j\colon N(S)/\la\ov{\phi}\ra \lra Y_A\,.
\end{equation}
%
We claim that $\iota\circ j$ is an isomorphism:
in fact it has finite fibers and is birational by~\Ref{prp}{biunivoca}, since $\dim \sing Y_A=2$ (because $A\notin\Sigma$) the hypersurface $Y_A$ is normal and hence $\iota\circ j$ is an isomorphism. Let $\pi\colon N(S)\to N(S)/\la\ov{\phi}\ra$ be the quotient map.
By~\eqref{cofisso} the singular locus of $N(S)/\la\ov{\phi}\ra$ is the image of $\Fix(\ov{\phi})$ (and thus isomorphic to $\Fix(\ov{\phi})$); since~(\ref{pettirosso}) is an isomorphism we get that
\begin{equation}\label{rivestodoppio}
\begin{matrix}
N(S)\setminus \Fix(\ov{\phi}) & \lra & Y_A^{sm} \\
x & \mapsto & \iota\circ j\circ\pi(x)
\end{matrix}
\end{equation}
%
is a topological covering of degree $2$.
We claim that
\begin{equation}\label{zetadue}
\pi_1(Y_A^{sm})\cong\ZZ/(2)\,.
\end{equation}
In fact $(N(S)\setminus \Fix(\ov{\phi}))\cong(S^{[2]}\setminus(P_S\cup \Fix(\phi|_{S^{[2]}\setminus P_S}))$. Since $(P_S\cup \Fix(\phi|_{S^{[2]}\setminus P_S}))$ is of codimension $2$ in the simply connected
manifold
$S^{[2]}$ we get that $(N(S)\setminus \Fix(\ov{\phi}))$ is simply connected. Thus~(\ref{rivestodoppio}) is the universal covering of $Y_A^{sm}$ and we get~(\ref{zetadue}). On the other hand $Y_A^{sm}\subset Y_A(1)$ by {Corollary~1.5} of~\cite{og5} and thus by~\Ref{lmm}{rivcon} we get that $f_A^{-1}Y_A^{sm}\to Y_A^{sm}$ is the universal covering of $Y_A^{sm}$ as well.
Hence both $X_A$ and $N(S)$ are normal completions of the universal cover of $Y_A^{sm}$ such that the extended maps to $Y_A$ are finite; it follows that they are isomorphic (over $Y_A$).
The singular locus of $N(S)$ is given by~\eqref{singenne}. On the other hand $\sing X_A=Y_A[3]$. By~\Ref{rmk}{puntovuz} we can order the set of (smooth) conics on $S$, say $C_1,\ldots,C_s$ and the set of points in $Y_A[3]$ different from $[v_0]$, say $[v_1],\ldots,[v_s]$ so that
\begin{equation}\label{buonatale}
\ov{\psi}(c(P_S))=[v_0],\qquad\ov{\psi}(c(C_i^{(2)}))=[v_i],\quad
1\le i\le s.
\end{equation}
(Recall~\Ref{rmk}{puntovuz}.) Let $\epsilon_0$ be a choice of $\PP^2$-fibration for $X_A$; then $\ov{\psi}$ defines a birational map $\psi_0\colon S^{[2]}\dra X_A^{\epsilon_0}$ such that
\begin{equation}\label{wurstel}
\psi_0^{*}H_A^{\epsilon_0}\cong\mu(D)-\Delta_S^{[2]}
\end{equation}
where $D$ is the hyperplane class of $S$ (thus $(S,D)$ is isomorphic to $(S_A(v_0),D_A(v_0))$). The birational map $\psi_0$ is an isomorphism away from
\begin{equation}\label{undisg}
P_S\cup C_1^{(2)}\cup\ldots\cup C_s^{(2)}.
\end{equation}
It follows that $\psi_0$ is the flop of a collection of irreducible components of~(\ref{undisg}). By~\Ref{prp}{fuorisig} we get that there exists a choice of $\PP^2$-fibration for $X_A$, call it $\epsilon$, such that the corresponding birational map $\psi\colon S^{[2]}\dra X_A^{\epsilon}$ is biregular. Equation~(\ref{tirodietro}) follows from~(\ref{wurstel}). This finishes the proof that Item~(1) holds.
Item~(2) follows from Item~(1) and a specialization argument - we leave the details to the reader.
%
\qed
\vskip 2mm
\n
We close the present subsection by reproving a result of ours. Let $h_A:=c_1(\cO_{X_A}(H_A))$.
\begin{thm}[{\rm O'Grady~\cite{og2}}]\label{thm:beatrice}
Let $A\in\lagr^0$. Then $X_A$ is a deformation of $(K3)^{[2]}$ and $(h_A,h_A)_{X_A}=2$. Any small deformation of $(X_A,H_A)$ (i.e.~a small deformation of $X_A$ keeping $h_A$ of type $(1,1)$) is isomorphic to $(X_B,H_B)$ for some $B\in\lagr^0$.
\end{thm}
\begin{proof}
Let $A_0\in (\Delta\setminus\Sigma)$ and $[v_0]\in Y_{A_0}[3]$. Suppose moreover that $S_{A_0}(v_0)$ does not contain lines. By~\Ref{thm}{ixhilb} there exists a choice $\epsilon$ of $\PP^2$-fibration for $X_{A_0}$ such that
we have an isomorphism
\begin{equation}
\psi\colon S^{[2]}\overset{\sim}{\lra} X_{A_0}^{\epsilon},
\qquad
\psi^{*}H_{A_0}^{\epsilon}\sim\mu(D_A(v_0))-\Delta_{S_{A_0}(v_0)}^{[2]}.
\end{equation}
On the other hand $(X_A,H_A)$ is a deformation of $(X_{A_0}^{\epsilon},H_{A_0}^{\epsilon})$ by~\Ref{crl}{unadef}.
This proves that $(X_A,H_A)$ is a deformation of $(S^{[2]},(\mu(D_A(v_0))-\Delta_{S_{A_0}(v_0)}^{[2]}))$.
By~\eqref{poldon} we get that $(h_A,h_A)_{X_A}=2$. Lastly we prove that an arbitrary small deformation of $(X_A,H_A)$ is isomorphic to $(X_{A'},H_{A'})$ for some $A'\in\lagr^0$. The deformation space of $(X_A,H_A)$ has dimension given by
\begin{equation}\label{defixacca}
\dim \Def(X_A,H_A)=h^{1,1}(X_A)-1=20\,.
\end{equation}
On the other hand $\lagr^0$ is contained in the locus of points in $\lag$ which are stable for the natural (linearized) $PL(V)$-action - this is proved in~\cite{og2}. Thus by varying $A\in\lagr$ we get
\begin{equation}\label{defepw}
\dim \lagr-\dim SL(V)=55-35=20
\end{equation}
moduli of double EPW-sextics. Since~(\ref{defixacca}) and~(\ref{defepw}) are equal we conclude that an arbitrary small deformation of $(X_A,H_A)$ is isomorphic to $(X_{B},H_{B})$ for some $B\in\lagr^0$.
\end{proof}
\section{Appendix: Three-dimensional sections of $\mathrm{Gr}(3,\CC^5)$}\label{sec:wedding}
\setcounter{equation}{0}
In the present section $V_0$ is a complex vector-space of dimension $5$.
Choose a volume form $\vol_0$ on $V_0$; it defines an isomorphism
\begin{equation}\label{zeppa}
\begin{matrix}
\bigwedge^ 2 V_0 & \overset{\sim}{\lra} & \bigwedge^ 3 V^{\vee}_0 \\
\alpha & \mapsto & \omega\mapsto \vol_0(\alpha\wedge\omega)
\end{matrix}
\end{equation}
Let $K\subset\bigwedge^ 2 V_0$ be a $3$-dimensional subspace
such that either
%
\begin{equation}\label{nointer}
\PP(K)\cap \mathrm{Gr}(2,V_0)=\es
\end{equation}
or else
\begin{equation}\label{puntosolo}
\PP(K)\cap \mathrm{Gr}(2,V_0)=\{[\kappa_0]\}=
\PP(K)\cap T_{[\kappa_0]}\mathrm{Gr}(2,V_0)\,.
\end{equation}
In other words either $\PP(K)$ does not intersects $\mathrm{Gr}(2,V_0)$ or else the scheme-theoretic intersection is a single reduced point.
We will describe
\begin{equation}
W_{K}:=\PP(\Ann K)\cap \mathrm{Gr}(3,V_0)
\end{equation}
First we recall that the dual of $\mathrm{Gr}(3,V_0)$ is $\mathrm{Gr}(2,V_0)$. More precisely let $[\alpha]\in\PP(\bigwedge^ 2 V_0)$: then
\begin{equation}\label{sezsing}
\sing(\PP(\Ann \alpha)\cap\mathrm{Gr}(3,V_0))=
\{U\in\mathrm{Gr}(3,V_0)\mid U\supset \supp \alpha\}.
\end{equation}
In particular $\PP(\Ann \alpha)$ is tangent to $\mathrm{Gr}(3,V_0)$ if and only if $[\alpha]\in\mathrm{Gr}(2,V_0)$ (and if that is the case it is tangent along a $\PP^2$).
Secondly we record the following observation (the proof is an easy exercise).
\begin{lmm}\label{lmm:senullo}
Let $U\subset V_0$ be a codimension-$1$ subspace. Let $\alpha\in\bigwedge^ 2 V_0$. Then
\begin{equation}
\alpha\wedge(\bigwedge^ 3 U)=0
\end{equation}
if and only if $ \supp \alpha\subset U$.
\end{lmm}
We recall the following result of Iskovskih.
\begin{prp}[{\rm Iskovskih~\cite{iskovskih}}]\label{prp:fanoindue}
Keep notation as above. Let $K\subset\bigwedge^ 2 V_0$ be a $3$-dimensional subspace
such that~(\ref{nointer}) holds. Then
%
\begin{itemize}
\item[(1)]
$W_{K}$ is a smooth Fano $3$-fold of degree $5$ with $\omega_{W_{K}}\cong\cO_{W_{K}}(-2)$,
\item[(2)]
the Fano variety $F(W_{K})$ parametrizing lines on $W_{K}$ (reduced structure) is isomorphic to $\PP^2$,
\item[(3)]
the projective equivalence class of $W_{K}$ does not depend on $K$.
\end{itemize}
\end{prp}
%
\begin{prp}\label{prp:singfano}
Keep notation as above. Let $K\subset\bigwedge^ 2 V_0$ be a sub vector-space
of dimension $3$ such that~\eqref{puntosolo} holds. Then
$W_{K}$ is a singular Fano $3$-fold of degree $5$ with $\omega_{W_{K}}\cong\cO_{W_{K}}(-2)$ and one singular point which is ordinary quadratic and belongs to
\begin{equation}
\{U\in\mathrm{Gr}(3,V_0)\mid U\supset \supp\kappa_0\}.
\end{equation}
\end{prp}
%
\begin{proof}
If $\kappa\in(K\setminus[\kappa_0])$ then $\kappa$ is not decomposable and hence $\PP(\Ann \kappa)$ is transverse to $\mathrm{Gr}(3,V_0)$; by~(\ref{sezsing}) we get that
\begin{equation}\label{nelpiano}
\sing W_{K}=
\{U\in\mathrm{Gr}(3,V_0)\mid U\supset \supp\kappa_0\}
\cap\PP(\Ann K)\,.
\end{equation}
We claim that the above intersection consists of one point. First notice that we have a natural identification
\begin{equation}\label{miomartello}
\{U\in\mathrm{Gr}(3,V_0)\mid U\supset \supp\kappa_0\}\cong
\PP(V_0/ \supp\kappa_0)
\end{equation}
and a linear map
\begin{equation}
\begin{matrix}
K & \overset{\nu}{\lra} & (V_0/ \supp\kappa_0)^{\vee} \\
\kappa & \mapsto & (\ov{v}\mapsto \vol_0(v\wedge\kappa_0\wedge\kappa))
\end{matrix}
\end{equation}
where $v\in V_0$ and $\ov{v}$ is its class in $V_0/ \supp\kappa_0$.
Given~(\ref{nelpiano}) and Identification~(\ref{miomartello}) we get that
\begin{equation}
\sing W_{K}=\PP(\Ann \im \nu)\,.
\end{equation}
Of course $\kappa_0\in\ker\nu$ and hence
in order to prove that $\sing W_{K}$ is a singleton it suffices to prove that $\ker\nu=[\kappa_0]$. If $\kappa\in(K\setminus[\kappa_0])$ then $\kappa_0\wedge\kappa\not=0$; in fact this follows from~(\ref{puntosolo}) together with the equality
\begin{equation}
\PP\{\kappa\in\bigwedge^ 2 V_0\mid
\kappa_0\wedge\kappa=0\}=T_{[\kappa_0]}\mathrm{Gr}(2,V_0)\,.
\end{equation}
Since $\kappa_0\wedge\kappa\not=0$ we have $\nu(\kappa)\not=0$.
This proves that $\sing W_{K}$ consists of a single point. The formula for the dualizing sheaf of $W_{K}$ follows at once from adjunction. It remains to prove that $W_{K}$ has a single singular point and that it is an ordinary quadratic point.
Let $\wt{W}_{K}\subset\PP(\supp\kappa_0)\times
\PP(V_0/ \supp\kappa_0)\times W_{K}$ be the closed subset defined by
\begin{equation}
\wt{W}_{K}:=\{([v],U,W)\mid
v\in W\subset U\}\,.
\end{equation}
The projection $ \wt{W}_{K}\to \PP(V_0/ \supp\kappa_0)$ is a $\PP^1$-fibration and hence $ \wt{W}_{K}$ is smooth. One shows that the projection $\pi\colon\wt{W}_{K}\to W_{K}$ is the blow-up of $\sing W_{K}$. Moreover $\pi^{-1}(\sing W_{K})\cong\PP^1\times\PP^1$ and one gets that the singularity of $W_{K}$ is ordinary quadratic.
\end{proof}
Our last result is about the base-locus of $3$-dimensional linear systems of quadrics containing $W_K$ for $K\subset\bigwedge^ 2 V_0$ a $3$-dimensional subspace such that~(\ref{nointer}) holds. First we consider the analogous question for the Grassmannian $\mathrm{Gr}(3,\bigwedge^ 3 V_0)$. Let's consider the rational map
\begin{equation}
\PP(\bigwedge^ 3 V_0)\overset{\Phi}{\dashrightarrow}
|\cI_{\mathrm{Gr}(3,V_0)}(2)|^{\vee}\cong\PP(V_0)
\end{equation}
where the last isomorphism is given by~(\ref{puntura}).
Let $Z\subset \PP(\bigwedge^ 3 V_0)\times\PP(V_0)$ be the incidence subvariety defined by
\begin{equation}
Z:=\{([\omega],[v])\mid v\wedge\omega=0\}\,.
\end{equation}
Then we have a commutative triangle
\begin{equation}\label{ranieri}
\xymatrix{
& Z \ar^{\wt{\Phi}}[dr] \ar_{\Psi}[dl] & \\
\PP(\bigwedge^ 3 V_0) &\overset{\Phi}{\dashrightarrow} & \PP( V_0)}
\end{equation}
where $\Psi$ and $\wt{\Phi}$ are the restrictions to $Z$ of the two projections of $\PP(\bigwedge^ 3 V_0)\times\PP(V_0)$. Moreover $\Psi$ is the blow-up of $\mathrm{Gr}(3,V_0)$. In particular the following holds: if $\omega\in\bigwedge^ 3 V_0$ is not decomposable then there exists a unique $[v]\in\PP(V_0)$ such that $v\wedge\omega=0$ and moreover $\Phi([\omega])=[v]$. Let $[v]\in\PP(V_0)$; by~(\ref{puntura}) we may view $\Ann (v)\subset V_0^{\vee}$ as a hyperplane in $|\cI_{\mathrm{Gr}(3,V_0)}(2)|$; by commutativity of~(\ref{ranieri}) we have
\begin{equation}\label{intspezzata}
\bigcap\limits_{f\in \Ann (v)}V(q_f)=\mathrm{Gr}(3,V_0)
\cup\{[\omega]\in\PP(\bigwedge^ 3 V_0)\mid v\wedge\omega=0\}.
\end{equation}
\begin{prp}\label{prp:baseweb}
Let $K\subset\bigwedge^ 2 V_0$ be a $3$-dimensional subspace such that~(\ref{nointer}) holds. Let $L\subset|\cI_{W_K}(2)|$ be a hyperplane (here $\cI_{W_K}$ is the ideal sheaf of $W_K$ in $\PP(\Ann K)$). Then
\begin{equation}
\bigcap\limits_{t\in L}Q_t=W_K\cup R_L
\end{equation}
where $R_L$ is a plane. Moreover $W_K\cap R_L$ is a conic.
\end{prp}
\begin{proof}
Restriction to $\PP(\Ann K)$ defines an isomorphism
\begin{equation}
|\cI_{\mathrm{Gr}(3,V_0)}(2)|\overset{\sim}{\lra} |\cI_{W_K}(2)|\,.
\end{equation}
By~(\ref{puntura}) we get that we may identify $L$ with $\PP(\Ann (v))$ for a well-defined $[v]\in\PP(V_0)$ and each quadric $Q_t$ for $t\in L$ with $\PP(\Ann K)\cap V(q_f)$ for a suitable $[f]\in \PP(\Ann (v))$. By~(\ref{intspezzata}) we have
\begin{equation}
\bigcap\limits_{f\in \Ann (v)}(\PP(\Ann K)\cap V(q_f))=
W_K\cup R_L
\end{equation}
where
\begin{equation}
R_L:=\PP(\Ann K)\cap
\{[\omega]\in\PP(\bigwedge^ 3 V_0)\mid v\wedge\omega=0\}.
\end{equation}
Thus $R_L$ is a linear space of dimension at least $2$. Now notice that we have an isomorphism
%
\begin{equation}
\begin{matrix}
\bigwedge^ 2 (V_0/[v]) & \overset{\sim}{\lra} & \{[\omega]\in\PP(\bigwedge^ 3 V_0)\mid v\wedge\omega=0\} \\
\ov{\alpha} & \mapsto & v\wedge\alpha
\end{matrix}
\end{equation}
where $\alpha\in \bigwedge^ 2 V_0$ is an element mapped to $\ov{\alpha}$ by the quotient map $\bigwedge^ 2 V_0\to \bigwedge^ 2 (V_0/[v])$. Since $\dim(V_0/[v])=4$ the Grassmannian $\mathrm{Gr}(2,V_0/[v])$ is a quadric hypersurface in $\PP(\bigwedge^ 2(V_0/[v]))$; it follows that either $R_L\subset W_K$ or $R_L\cap W_K$ is a quadric hypersurface in $R_L$. By Lefschetz $\Pic(W_K)$ is generated by the hyperplane class; it follows that $W_K$ contains no planes and no quadric surfaces. Thus necessarily $\dim R_L=2$, moreover $R_L\not\subset W_K$ and the intersection $R_L\cap W_K$ is a conic.
\end{proof}
\begin{crl}\label{crl:consufan}
Let $K\subset\bigwedge^ 2 V_0$ be a $3$-dimensional subspace such that~(\ref{nointer}) holds and $\cC(W_K)$ be the variety parametrizing conics on $W_K$ (reduced structure). Then we have an isomorphism
\begin{equation}
\begin{matrix}
|\cI_{W_K}(2)|^{\vee} & \overset{\sim}{\lra} & \cC(W_K) \\
L & \mapsto & R_L\cap W_K
\end{matrix}
\end{equation}
where $R_L$ is as in~\Ref{prp}{baseweb}. Moreover given $Z\in W_K^{[2]}$ there exists a unique conic containing $Z$ namely $R_L\cap W_K$ where $L\in |\cI_{W_K}(2)|^{\vee}$ is the hypeprlane of quadrics containing $\la Z\ra$.
\end{crl}
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"redpajama_set_name": "RedPajamaArXiv"
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\def\cms{\hbox{cm.s$^{-1}$}}
\def\cmss{\hbox{cm.s$^{-2}$}}
\def\gcm3{\hbox{g.cm$^{-3}$}}
\def\vsini{\hbox{$\upsilon \sin i_{\star}$}}
\def\sini{\hbox{sin\,$i$}}
\def\Msun{\hbox{M$_{\astrosun}$}}
\def\hbox{R$_{\astrosun}$}{\hbox{R$_{\astrosun}$}}
\def\hbox{M$_{\jupiter}$}{\hbox{M$_{\jupiter}$}}
\def\hbox{R$_{\jupiter}$}{\hbox{R$_{\jupiter}$}}
\def\hbox{M$_{\oplus}$}{\hbox{M$_{\oplus}$}}
\def\hbox{R$_{\oplus}$}{\hbox{R$_{\oplus}$}}
\def\hbox{$^\circ$}{\hbox{$^\circ$}}
\defT$_{\rm eff}${T$_{\rm eff}$}
\deflog~{\it g}{log~{\it g}}
\def[Fe/H]{[Fe/H]}
\def$\upsilon_\mathrm{micro}${$\upsilon_\mathrm{micro}$}
\def$\upsilon_\mathrm{macro}${$\upsilon_\mathrm{macro}$}
\title{An Earth-sized exoplanet with a Mercury-like composition}
\author{A. Santerne$^{\ref{LAM}}$,
B. Brugger$^{\ref{LAM}}$,
D. J. Armstrong$^{\ref{warwick}}$,
V. Adibekyan$^{\ref{IA}}$,
J. Lillo-Box$^{\ref{ESO}}$,
H. Gosselin$^{\ref{LAM}, \ref{UPS}}$,
A. Aguichine$^{\ref{LAM},\ref{ENS}}$,
J.-M. Almenara$^{\ref{geneva}}$,
D. Barrado$^{\ref{CAB}}$,
S. C. C. Barros$^{\ref{IA}}$,
D. Bayliss$^{\ref{geneva}}$,
I. Boisse$^{\ref{LAM}}$,
A. S. Bonomo$^{\ref{Torino}}$,
F. Bouchy$^{\ref{geneva}}$,
D. J. A. Brown$^{\ref{warwick}}$,
M. Deleuil$^{\ref{LAM}}$,
E. Delgado~Mena$^{\ref{IA}}$,
O. Demangeon$^{\ref{IA}}$,
R. F. D\'iaz$^{\ref{UBA},\ref{CONICET},\ref{geneva}}$,
A. Doyle$^{\ref{warwick}}$,
X. Dumusque$^{\ref{geneva}}$,
F. Faedi$^{\ref{warwick},\ref{catania}}$,
J. P. Faria$^{\ref{IA},\ref{UPorto}}$,
P. Figueira$^{\ref{ESO},\ref{IA}}$,
E. Foxell$^{\ref{warwick}}$,
H. Giles$^{\ref{geneva}}$,
G. H\'ebrard$^{\ref{IAP},\ref{OHP}}$,
S. Hojjatpanah$^{\ref{IA},\ref{UPorto}}$,
M. Hobson$^{\ref{LAM}}$,
J. Jackman$^{\ref{warwick}}$,
G. King$^{\ref{warwick}}$,
J. Kirk$^{\ref{warwick}}$,
K. W. F. Lam$^{\ref{warwick}}$,
R. Ligi$^{\ref{LAM}}$,
C. Lovis$^{\ref{geneva}}$,
T. Louden$^{\ref{warwick}}$,
J. McCormac$^{\ref{warwick}}$,
O. Mousis$^{\ref{LAM}}$,
J. J. Neal$^{\ref{IA},\ref{UPorto}}$,
H. P. Osborn$^{\ref{warwick},\ref{LAM}}$,
F. Pepe$^{\ref{geneva}}$,
D. Pollacco$^{\ref{warwick}}$,
N. C. Santos$^{\ref{IA},\ref{UPorto}}$,
S. G. Sousa$^{\ref{IA}}$,
S. Udry$^{\ref{geneva}}$
\& A. Vigan$^{\ref{LAM}}$}
\begin{document}
\maketitle
\begin{affiliations}
\item Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France\label{LAM}
\item Department of Physics, University of Warwick, Gibbet Hill Road, Coventry, CV4 7AL, UK\label{warwick}
\item Instituto de Astrof\'isica e Ci\^{e}ncias do Espa\c co, Universidade do Porto, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal\label{IA}
\item European Southern Observatory (ESO), Alonso de Cordova 3107, Vitacura, Casilla 19001, Santiago de Chile, Chile\label{ESO}
\item Universit\'e de Toulouse, UPS-OMP, IRAP, Toulouse, France\label{UPS}
\item Paris-Saclay Universit\'e, ENS Cachan, 61 av. du Pr\'esident Wilson, 94230 Cachan, France\label{ENS}
\item Observatoire Astronomique de l'Universit\'e de Gen\`eve, 51 Chemin des Maillettes, 1290 Versoix, Switzerland\label{geneva}
\item Depto. de Astrof\'isica, Centro de Astrobiolog\'ia (CSIC-INTA), ESAC campus 28692 Villanueva de la Ca\~nada (Madrid), Spain\label{CAB}
\item INAF -- Osservatorio Astrofisico di Torino, Strada Osservatorio 20, I-10025, Pino Torinese (TO), Italy\label{Torino}
\item Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales. Buenos Aires, Argentina\label{UBA}
\item CONICET - Universidad de Buenos Aires. Instituto de Astronom\'ia y F\'isica del Espacio (IAFE). Buenos Aires, Argentina\label{CONICET}
\item INAF -- Osservatorio Atrofisico di Catania, via S. Sofia 78, 95123, Catania, Italy\label{catania}
\item Departamento de F\'isica e Astronomia, Faculdade de Ciencias, Universidade do Porto, Rua Campo Alegre, 4169-007 Porto, Portugal\label{UPorto}
\item Institut d'Astrophysique de Paris, UMR7095 CNRS, Universite Pierre \& Marie Curie, 98bis boulevard Arago, 75014 Paris, France\label{IAP}
\item Aix Marseille Univ, CNRS, OHP, Observatoire de Haute Provence, Saint Michel l'Observatoire, France\label{OHP}\\
\end{affiliations}
\begin{bibunit}
\begin{abstract}
The Earth, Venus, Mars, and some extrasolar terrestrial planets\cite{dressing2015mass} have a mass and radius that is consistent with a mass fraction of about 30\% metallic core and 70\% silicate mantle\cite{stacey2005high}. At the inner frontier of the solar system, Mercury has a completely different composition, with a mass fraction of about 70\% metallic core and 30\% silicate mantle\cite{smith2012gravity}. Several formation or evolution scenarios are proposed to explain this metal-rich composition, such as a giant impact\cite{benz2008origin}, mantle evaporation\cite{cameron1985partial}, or the depletion of silicate at the inner-edge of the proto-planetary disk\cite{wurm2013photophoretic}. These scenarios are still strongly debated. Here we report the discovery of a multiple transiting planetary system (K2-229), in which the inner planet has a radius of 1.165$\pm$0.066~\hbox{R$_{\oplus}$}\ and a mass of 2.59$\pm$0.43~\hbox{M$_{\oplus}$}. This Earth-sized planet thus has a core-mass fraction that is compatible with that of Mercury, while it was expected to be similar to that of the Earth based on host-star chemistry\cite{thiabaud2015elemental}. This larger Mercury analogue either formed with a very peculiar composition or it has evolved since, e.g. by losing part of its mantle.
Further characterisation of Mercury-like exoplanets like K2-229~b will help putting the detailed in-situ observations of Mercury (with Messenger and BepiColombo\cite{benkhoff2010bepicolombo}) into the global context of the formation and evolution of solar and extrasolar terrestrial planets.
\end{abstract}
The star EPIC228801451 (TYC 4947-834-1 -- 2MASS J12272958-0643188 -- K2-229) was observed in photometry as part of campaign 10 of the \textit{K2} mission with the \textit{Kepler} space telescope, from 2016-07-06 to 2016-09-20 with a 30-min cadence. Analysis of the extracted and reduced light-curve using the POLAR pipeline\cite*{barros2016new} revealed two sets of periodic planetary transit-like events on periods of about 14~h and 8.3~d. A single transit-like event near the mid-campaign time was also detected (see Fig. 1 and methods). We refer to these planets as K2-229~b, c, and d, respectively. The light curve exhibits a large modulation with a 2\% peak-to-peak amplitude and 18-d period variability (see Fig. 1) that is caused by the presence of active regions (spots and/or faculae). The star is a bright (V=11) late-G / early-K dwarf\cite*{huber2016k2}, hence suitable for precise radial-velocity (RV) observations.
We observed EPIC228801451 with the HARPS spectrograph with the aim of confirming the planetary nature of the transiting candidates and measuring the mass through Doppler spectroscopy. We collected 120 RVs from 2017-01-26 to 2017-05-04 with up to four observations each night. We reduced the spectra using the online pipeline available at the telescope and derived the RV, the full width half maximum (FWHM) and bisector (BIS) of the averaged line profile as well as the spectroscopic indices of chromospheric activity in the core of five spectral lines (see methods and the Supplement Tables 3 and 4). The pipeline automatically rejected one poor-quality spectrum.
To assess the planetary nature of the detected transit signals and rule-out the presence of background stellar objects contaminating the light curve, we performed high-resolution imaging observations with the AstraLux lucky-imaging instrument. No background nor stellar companion is detected within the sensitivity limits of the data and within the photometric mask (see methods and Supplement Fig. 2).
We co-added the HARPS spectra and derived the spectral parameters of the host star\cite{santos2013sweet} (see methods). We find that the host star has an effective temperature of T$_{\rm eff}$\ = $5120\pm39$~K, a surface gravity of log~{\it g}\ = $4.51\pm0.12$~[cgs], an Iron abundance of [Fe/H]\ = $-0.06\pm0.02$~dex, and a micro-turbulence velocity of v$_{mic}$ = $0.74\pm0.08$~\kms. This identifies the host star as a K0 dwarf. We also derived the chemical abundance of the star (see the Supplement Table 2).
The 119 RV time series is displayed in Fig. 1 and exhibits a large variability at the level of about 50 \ms\ and a recurrence timescale of 18 days. This large modulation is not caused by planet reflex motions but is due to the presence of active regions on the surface of the star\cite{dumusque2014soap}. We performed a joint analysis of the photometric and spectroscopic data, together with the spectral energy distribution (SED) of the star with the \texttt{PASTIS} software\cite{diaz2014pastis} (see methods). We used three different approaches to correct the RV data for the activity signal of the star\cite{dumusque2017radial}: (I) a nightly RV offset, (II) a Gaussian process (GP) regression, and (III) diagnostics de-correlation with a moving average. For the inner transiting planet (K2-229~b), the three methods give consistent results (see Supplement Fig. 3). Its mass determination is therefore robust in regards to stellar activity. The reflex motion of planet c is only significantly detected with method III and thus requires further investigations to be fully secured. The outer transiting planet (planet d) is not significantly detected in the RV data with any of the methods. We finally adopted the physical parameters of the K2-229 system, that are reported in Table 1, using model II for the correction of the activity-induced RVs.
The inner transiting planet, K2-229~b, has a mass of $2.59\pm0.43$~\hbox{M$_{\oplus}$}\ and a radius of $1.165\pm0.066$~\hbox{R$_{\oplus}$}, hence a bulk density of $8.9\pm2.1$~\gcm3. The planets c and d have masses lower than 21.3~\hbox{M$_{\oplus}$}\ and 25.1~\hbox{M$_{\oplus}$}\ (95\% credible interval), and a radius of $2.12\pm0.11$~\hbox{R$_{\oplus}$}\ and $2.65\pm0.24$~\hbox{R$_{\oplus}$}, respectively. Note that for the outer transiting planet, K2-229~d, we find two different orbital solutions: with an orbital period close to 31 days or longer than about 50 days. This bi-modal distribution is due to the gap in the first half of the \textit{K2} light curve (see Fig. 1). In the first case, another transit should have occurred during this gap while in the latter case, there is only one transit event during the entire photometric campaign. We excluded the long-period scenario for stability reasons, as this requires large eccentricity to explain the relatively short transit duration ($2.65^{_{+0.15}}_{^{-0.20}}$~h, see Supplement Table 6), making planet d likely to cross the orbit of planet c.
In Fig. 2, we compare K2-229 b with other known Earth-sized planets\cite{pepe2013earth,berta2015rocky,gillon2017seven,2018arXiv180201377G}, together with theoretical compositions of terrestrial planets (assuming different fractions of metallic core, silicate mantle, and water layer)\cite{brugger2017constraints}. A comparison between K2-229~b and large rocky exoplanets as well as the rocky planets and major moons in the Solar system is also available in the Supplement Fig. 7. Given its high density of $8.9\pm2.1$~\gcm3, we can constrain the water-mass fraction to be less than a few percent at maximum and thus negligible in the planet-mass budget. Assuming K2-229~b is composed only of a metallic core (with a negligible fraction of Si) and a silicate mantle, its mass and radius are consistent with a core-mass fraction (CMF) of $68^{_{+17}}_{^{-25}}$\%\cite{brugger2017constraints}. The derived CMF differs from the one computed ($27^{_{+9}}_{^{-13}}$\% see Fig. 3) assuming the planet was formed with the same Fe/Si$_{c}$ ratio (see methods for a description of Fe/Si$_{c}$) as observed in the host star (Fe/Si$_{c}$=$0.71\pm0.33$). Therefore, the composition of this planet is likely (with an 88\% credible probability) different from the one expected based on the chemical composition of its host star. The reasons for this would be that the planet evolved since its formation, for instance by losing parts of its mantle\cite{cameron1985partial,benz2008origin}, or it was formed with a substantially different Fe/Si$_{c}$ ratio than its host star\cite{wurm2013photophoretic}. With such a potentially large core-mass fraction, K2-229~b stands out as a larger analogue of Mercury in the solar system, which has a CMF of 68\%\cite{stacey2005high,smith2012gravity}. Both planets might therefore share a common formation and evolution history.
Compared to Mercury, K2-229~b orbits much closer to its star, with an orbital period of about 14~h. Its day-side temperature can reach up to 2330~K, assuming synchronous rotation. These conditions make K2-229 b potentially more sensitive than Mercury to mantle evaporation. At this temperature, the mantle in the day-side of the planet is expected to volatilise into a saturated atmosphere of silicate vapour\cite{leger2011extreme}. Substantial evaporation of this atmosphere would result in planetary mass-loss. Current thermal escape and stellar X-ray irradiation are however not expected to substantially strip out this thin atmosphere (see methods). Other extremely hot rocky worlds have been reported with masses and radii consistent with an Earth-like composition\cite{dressing2015mass}, while a temperate, likely metal-rich, rocky planet has been recently detected: LHS 1140~b\cite{2017Natur.544..333D}. Therefore, stellar irradiation is not expected to contribute to the mass-loss rate (hence mantle evaporation) by more than a few percent of the total planet mass.
Since K2-229~b is orbiting extremely close to an active K dwarf (at 0.012AU), another hypothesis is that the thin layer of silicate vapour might escape from the planet through magnetic interaction with the host star\cite{lanza2013star, strugarek2016assessing}. Intense stellar wind and flares might also erode the planet's atmosphere\cite{2017arXiv170604617G}. However, these two mechanisms would be less efficient if the planet has its own magnetosphere, protecting the atmosphere from evaporation. More observations and modelling, which are outside the scope of this paper, are required to fully constrain this evolution scenario. The detection of a cometary-like tail\cite{mura2011comet} in the planet vicinity would be an evidence for these mechanisms evaporating the volatilised mantle of rocky planets. Searching for correlation between the CMF of rocky exoplanets and their magnetic environment\cite{moutou2016magnetic} would provide insights on the importance of the magnetic field in this possible mechanism.
Another scenario to explain the composition of Mercury is a Theia-like giant impact\cite{benz2008origin}. If this scenario is responsible for Mercury-like exoplanets, there might be some correlation between the multiplicity and architecture of the systems with the presence of such planets. In that case, these planets would also be prime targets to search for relatively large exo-Moons. However, more giant-impact modelling is needed to understand what would be the initial conditions, in terms of e.g. mass and velocity, for a Theia-like object to remove the mantle of rocky planets much more massive than Mercury such as K2-229~b.
Finally, if photophoresis is the mechanism that forms Mercury-like planets\cite{wurm2013photophoretic}, the comparison of their host star properties compared with those hosting Earth-like planets\cite{dressing2015mass} will provide important constraints on the conditions required for this process. However, given the properties of the Mercury- and Earth-like planets (the orbital separation versus the CMF), there is no clear correlation between the formation conditions and the composition of the system-innermost planet. One possibility would be that the Earth-like planets are formed further out in the disk than those with a Mercury-like composition. Further theoretical work are needed to fully understand the formation and migration of these planets under the photophoresis scenario.
Note that after submitting this paper, the planet K2-106~b, which was initially reported with an Earth-like composition\cite{sinukoff2017k2}, turns out to be much more rich in Iron\cite{guenther2017k2}. No clear comparison between the composition of K2-106~b and the one of its host star has been performed so far. Still, it shows that the existence of such Mercury-like planets is more common than previously thought. This is opening a new approach to understand the particular formation and evolution of these planets.
The planet K2-229~b appears to have abundances that are different from the ones of the the central star. This is the first time such situation is observed in an extrasolar system\cite{dorn2017generalized}. Its composition is similar with that of Mercury. However both planets do not share the same environmental conditions, the former being substantially hotter and closer to its star. Therefore, K2-229~b is an excellent laboratory to test the conditions to form Mercury analogues. The exploration of this new population of exoplanets can greatly complement Solar-system \textit{in-situ} missions like Messenger and BepiColombo\cite{benkhoff2010bepicolombo}, and will help to constrain the formation and evolution mechanisms of Mercury-like planets in various environments. It can then be used to refine the formation models of the Solar-system terrestrial planets. The increased precision in mass and radius measurements brought by upcoming dedicated missions like PLATO\cite{rauer2014plato} will bring an important advancement for this purpose.
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The Applied Mechanics Division (AMD) is a division in the American Society of Mechanical Engineers (ASME). The AMD was founded in 1927, with Stephen Timoshenko being the first chair. The current AMD membership is over 5000, out of about 90,000 members of the ASME. AMD is the largest of the six divisions in the ASME Basic Engineering Technical Group.
Mission
The mission of the Applied Mechanics Division is to foster fundamental research in, and intelligent application of, applied mechanics.
Summer Meeting
The Division participates annually in a Summer Meeting by programming Symposia and committee meetings. The principal organisers of the Summer Meetings rotate among several organizations, with a period of four years, as described below.
Year 4n (2020, 2024, etc.): International Union of Theoretical and Applied Mechanics (IUTAM).
Year 4n + 1 (2017, 2021, etc.): Materials Division of the ASME (joined with the Applied Mechanics Division of ASME, Engineering Mechanics of the American Society of Civil Engineers, and Society of Engineering Sciences).
Year 4n + 2 (2018, 2022, etc.): National Committee of Theoretical and Applied Mechanics.
Year 4n + 3 (2019, 2023, etc.): Applied Mechanics Division of the ASME (joined with Materials Division of ASME).
Publications
Newsletters of the Applied Mechanics Division
Journal of Applied Mechanics
Applied Mechanics Reviews
Awards
Timoshenko Medal
Koiter Medal
Drucker Medal
Thomas K. Caughey Dynamics Award
Ted Belytschko Applied Mechanics Award
Thomas J.R. Hughes Young Investigator Award
Journal of Applied Mechanics Award
These awards are conferred every year at the Applied Mechanics Division Banquet held during the annual ASME (IMECE) conference. Awards other than those mentioned above are also celebrated during this banquet, such as the Haythornthwaite Research Initiation Grant Award and the Eshelby Mechanics Award for Young Faculty.
Executive committee
The responsibility for guiding the Division, within the framework of the ASME, is vested in an Executive Committee of five members. The Executive Committee meets twice a year at the Summer Meeting and Winter Annual Meeting. Members correspond throughout the year by emails and conference calls. Three members shall constitute a quorum, and all action items must be approved by a majority of the Committee.
Each member serves a term of five years, beginning and ending at the conclusion of the Summer Meeting, spending one year in each of the following positions:
Secretary
Vice-Chair of the Program Committee
Chair of the Program Committee
Vice-Chair of the Division
Chair of the Division
New members of the Executive Committee are sought from the entire membership of the Division. Due considerations are given to leadership, technical accomplishment, as well as diversity in geographic locations, sub-disciplines, and genders. At the Winter Annual Meeting each year, the Executive Committee nominates one new member, who is subsequently appointed by the ASME Council.
The executive committee has an additional non-rotating position, the Recording Secretary. The responsibility of the Recording Secretary is to attend and record minutes for the Executive Committee Meeting at the Summer and Winter Annual Meeting and the General Committee Meeting at the Winter Annual Meeting. The Recording Secretary serves a term of two years and is selected from the junior members (i.e. young investigators) of the AMD.
Current members of the Executive Committee
Samantha Daly, University of California at Santa Barbara, Santa Barbara, CA, United States: Secretary
Kenji Takizawa, Waseda University, Tokyo, Japan: Vice-Chair of the Program Committee
Glaucio Paulino, Princeton University, Princeton NJ, United States: Chair of the Program Committee
Marco Amabili, McGill University, Montreal, Canada: Vice-Chair of the Division
Taher Saif, University of Illinois at Urbana-Champaign, Champaign IL, United States: Chair of the Division
Technical Committees
The mission of a Technical Committee is to promote a field in Applied Mechanics. The principal approach for a Technical Committee to accomplish this mission is to organize symposia at the Summer and Winter Meetings. Technical Committees generally meet at the Winter Annual Meeting and the Summer Meeting; they may also schedule special meetings.
There are 17 Technical Committees in the Applied Mechanics Division.
Technical Committees are established and dissolved by the Executive Committee.
Financial
History
See Naghdi's "A Brief History of the Applied Mechanics Division of ASME" for details of the history from 1927-77.
Past chairs of the Applied Mechanics Division
Pradeep Guduru (2022),
Yuri Bazilevs (2021),
Yonggang Huang (2020),
Balakumar Balachandran (2019),
Pradeep Sharma (2018),
Arun Shukla (2017),
Peter Wriggers (2016),
Huajian Gao (2015),
Lawrence A. Bergman (2014),
Ken Liechti (2013),
Ares Rosakis (2012),
Tayfun Tezduyar (2011),
Zhigang Suo (2010),
Dan Inman (2009),
K. Ravi-Chandar (2008),
Thomas N. Farris (2007),
Wing Kam Liu (2006),
Mary C. Boyce (2005),
Pol Spanos (2004),
Stelios Kyriakides (2003),
Dusan Krajcinovic (2002),
Thomas J.R. Hughes (2001),
Alan Needleman (2000),
Lallit Anand (1999),
Stanley A. Berger (1998),
Carl T. Herakovich (1997),
Thomas A. Cruse (1996),
John W. Hutchinson (1995),
L.B. Freund (1994),
David B. Bogy (1993),
William S. Saric (1992),
Ted Belytschko (1991),
Michael J. Forrestal (1990),
Sidney Leibovich (1989),
Thomas L. Geers (1988),
James R. Rice (1987),
Michael M. Carroll (1986),
Jan D. Achenbach (1985),
Charles R. Steele (1984),
William G. Gottenberg (1983),
R.C. DiPrima (1982),
R.M. Christensen (1981),
R.S. Rivlin (1980),
Richard Skalak (1979),
F. Essenburg (1978),
Yuan-Cheng Fung (1977),
J. Miklowitz (1976),
B.A. Boley (1975),
George Herrmann (1974),
J. Kestin (1973),
Paul M. Naghdi (1972),
S. Levy (1971),
H.N. Abramson (1970),
Stephen H. Crandall (1969),
P.G. Hodge Jr. (1968),
R. Plunkett (1967),
M.V. Barton (1966),
George F. Carrier (1965),
Daniel C. Drucker (1964),
E. Reissner (1963),
A.M. Wahl (1961, 1962),
S.B. Batdorf (1960),
William Prager (1959),
W. Ramberg (1958),
M. Hetenyl (1957),
Raymond D. Mindlin (1956),
Nicholas J. Hoff (1955),
N.M. Newmark (1954),
D. Young (1953),
R.E. Peterson (1952),
L.H. Donnell (1951),
R.P. Kroon (1950),
M. Golan (1949),
W.M. Murray (1948),
H.W. Emmons (1947),
H. Poritsky (1946),
J.N. Goodier (1945),
J.H. Keenan (1943, 1944),
H.L. Dryden (1942),
J.P. Den Hartog (1940, 1941),
C.R. Soderberg (1937,1938),
E.O. Waters (1936),
J.A. Goff (1935),
F.M. Lewis (1934),
J.M. Lessells (1933),
G.B. Pegram (1932),
A.L. Kimball (1931),
G.M. Eaton (1928, 1929),
Stephen P. Timoshenko (1927, 1930)
Relevant websites
Homepage of Applied Mechanics Division
iMechanica.org, a web of mechanics and mechanicians.
References
P.M. Naghdi, A brief history of the Applied Mechanics Division of ASME. Journal of Applied Mechanics 46, 723-794.
Bylaws of Applied Mechanics Division
Organizations established in 1927
American Society of Mechanical Engineers
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{"url":"https:\/\/www.physicsforums.com\/threads\/it-would-take-a-computer-bigger-than-the-universe-to-solve-11-electron.679825\/","text":"# It would take a computer bigger than the universe to solve 11 electron\n\n1. Mar 20, 2013\n\n### maverick_starstrider\n\nHi,\n\nI've seen quite a few textbooks and lectures start by saying something like \"to solve a quantum system of N particles would take the best computer in the world, to solve a quantum system of (N+a small number) of particles would take a computer bigger than the universe..\", where they give specific numbers. An example of this is Wen's book. My question is which SPECIFIC back of the envelope calculations are they using to get these numbers? It's not something as simple as the Hilbert space for spin growing like 2^N or something so what are the specific assumptions that are made when coming up with these numbers? Anyone know?\n\n2. Mar 20, 2013\n\n### bahamagreen\n\nProbably in reference to what are called P vs NP problems...\n\n3. Mar 20, 2013\n\n### maverick_starstrider\n\nI'm not asking why quantum systems are not computable on classical computers I'm asking what set of assumptions go in to a statement like \"To computer a system of blah electrons would require 2^blah bits\" or what have you. What is the Hilbert space, what are the assumptions? What is the EXACT calculation that produced those EXACT numbers.\n\n4. Mar 20, 2013\n\n### maverick_starstrider\n\nBy way of an example here is one of the opening paragraphs from Wen's book:\n\n\"However, in practice, the required computing power is immense. In the 1980s, a workstation with 32 Mbyte RAM could solve a system of eleven interacting electrons. After twenty years the computing power has increased by 100-fold, which allows us to solve a system with merely two more electrons. The computing power required to solve a typical system of 10^23 interacting electrons is beyond the imagination of the human brain. A classical computer made by all of the atoms in our universe would not be powerful enough to handle the problem\/' Such an impossible computer could only solve the Schrodinger equation for merely about 100\"\n\nWhere is he getting the numbers he's quoting? (he doesn't say)\n\n5. Mar 21, 2013\n\n### atyy\n\nBrian Swingle fills in some details in his essay. Basically, the size of the Hilbert space gets very large. He also points out that very good approximations can be obtained if you know which part of Hilbert space to hunt in for your particular application.\n\nhttp:\/\/fqxi.org\/community\/forum\/topic\/1559\n\n\"More generally, the ground state of the Hamiltonian HN(g1, ..., gk) depends on these k parameters, but the state is a 2N component vector and hence all 2N components are determined by just a few numbers! Even if N is merely 100 and k = 2 we are talking about a two dimensional subset of a 2100 dimensional space! ...\n\n\"So what, after all, are all those complex numbers really telling us? Do we need them to predict the results of physical measurements? If so, we\u2019re in trouble. Ignoring causality and the lack of materials, even if we \ufb01lled up our entire Hubble volume, the whole visible universe, with our best classical storage device, we could only store the quantum state of a few hundred spins using this huge classical memory. Suddenly the illusory nature of Hilbert space is brought into focus.\"\n\nLast edited: Mar 21, 2013\n6. Mar 21, 2013\n\n### cgk\n\nOP, these statements come from arguments which assume that you intent to represent the wave function in an exponential fashion, with N^M numbers, where N is the number of one-particle states and M the number of electrons. If you then go on and formulate a particularly bad representation of the 1-particle space (e.g., a uniform 3d grid of 100 points in 3 dimensions for a molecule) you get those estimates.\n\nWhat such estimates really show is the ignorance of the people providing them. For example, it is really not hard to find out that the electronic structure of molecules is routinely calculated by tens of thousands of theoretical chemists daily. You could do it on the computers your local super market sells. For friendly small molecules (say, <40 electrons) one can effectively get converged results to a few meV. These techniques of course require some information about the physical systems involved, *but this information is generally available* and can be checked in controlled approximations.\n\nIn short: If you read such a statement in a book, you can safely assume that you do not want to learn many-body theory from that book.\n\n7. Mar 21, 2013\n\n### Fightfish\n\n@cgk, those approximation methods require quite a number of assumptions though, in particular they do not work well for strongly-correlated systems, which are important in contemporary condensed matter.\nThis report provides some examples of the computing power required for classical simulation:\nhttp:\/\/sc05.supercomputing.org\/schedule\/pdf\/pap188.pdf\n\n8. Mar 21, 2013\n\n### f95toli\n\nA good example would be high-temperature superconductivity. We have a very good understanding of what goes on in metallic superconductors, including binary systems such as MgB2 and we can calculate all properties more or less from first principles; but no computer in the world is powerfull enough to fully simulate e.g a reasonably large YBCO lattice; which is what you would need in order to e.g. calculate Tc without any assumptions.\n\nNote that this hasn't stopped people from trying; there are a bunch of papers out there where people have tried various \"short-cuts\" but with limited success. Perhaps if we extended the lattice 10 times in each direction (x,y and z) it would work; but that would require 10^3=1000 faster computers than what we have now...\n\n9. Mar 22, 2013\n\n### cgk\n\nI recognize the fact that there are systems for which those approximation methods will not work well\/at all. But just saying that it is impossible to calculate wave functions for any meaningful systems, as the kinds of statements cited by the OP often imply, is still utterly wrong (this is particularly ridiculous if these statements are used to introduce density functional theory).\n\nEven just saying for strongly correlated systems it cannot be done'' is not enough: For example, a vast array of strongly correlated molecular systems are accessible to MCSCF+MRCI (multiconfiguration self-consistent field\/multiconfiguration configuration interaction). Also when the strong correlation extents only across one direction, there are methods like DMRG (density matrix renormalization group) which can in principle (and sometimes in practice) handle them. There are even infinite strongly correlated lattice models (like the 1D Hubbard model) for which some ground state wave functions can be calculated analytically with pen and paper with various variations of the Bethe Ansatz.\n\nI would be okay with those statements if they came with some restrictions in scope. For example, by saying that in the very worst case, for a random Hamiltonian with N-body terms and on which we have no further information, the wave function most likely cannot be represented without exponential cost (just like the Hamiltonian itself!). But that is often not the case; the authors make it sound like it is impossible to calculate wave functions of *any* interesting systems. And this is not true.","date":"2017-10-21 09:13:31","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\": 0.5359416007995605, \"perplexity\": 692.4851530877968}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-43\/segments\/1508187824675.67\/warc\/CC-MAIN-20171021081004-20171021101002-00036.warc.gz\"}"}
| null | null |
{"url":"https:\/\/www.gradesaver.com\/textbooks\/math\/calculus\/calculus-early-transcendentals-2nd-edition\/chapter-4-applications-of-the-derivative-4-2-what-derivatives-tell-us-4-2-exercises-page-257\/49","text":"## Calculus: Early Transcendentals (2nd Edition)\n\nThe local maximum of $1\/e$ at $x = 1$ is an absolute maximum. There is no absolute minimum, since the function is unbounded in the negative direction as $x\u2192\u2212\u221e$.\n$f'(x) = \u2212xe^{\u2212x} +e^{\u2212x} = e^{\u2212x}(1\u2212x)$, which is $0$ only for $x = 1$. Note that $f$ is continuous on $(\u2212\u221e,\u221e)$ and contains only one critical point. Note that $f' > 0$ for $x < 1$ and $f' < 0$ for $x > 1$. So there is a local maximum of $f(1) = 1\/e$ at $x = 1$. The local maximum of $1\/e$ at $x = 1$ is an absolute maximum. There is no absolute minimum, since the function is unbounded in the negative direction as $x\u2192\u2212\u221e$.","date":"2020-04-04 09:29:01","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.9302555322647095, \"perplexity\": 45.22732024939431}, \"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-2020-16\/segments\/1585370521574.59\/warc\/CC-MAIN-20200404073139-20200404103139-00360.warc.gz\"}"}
| null | null |
Bob Foulkrod has hunted from sweltering Africa to the frigid North. Bob is known as one of the premier hunters in the world and is a strong advocate of archery education. Inducted into the Anderson archery International Bowhunters Hall of Fame in 1993, as well as the International and National Hall of Fame and has had the opportunity to guide several key figures in the sport. Bob received "first issuances" for bowhunting licenses in the N.W.Territories and Labrador. Bob is the founder and instructor of the Bob Foulkrod Bowhunting School of Troy, PA, along with being certified for the PA Bowhunter Instructor program. He appears on many hunting videos and TV shows, including Bass Pro Shops and Winchester Legends. Bob writes for leading magazines as has his own online magazine at www.bobfoulkrod.com.
Bob completed over a five-year period, his Obsession Quest. He did what no other sportsman in history has ever accomplished. He took eighty species and was all captured on video. This included the list for North America, SCI's list for the World Slam on turkeys and Africa's legendary Big Six.
Stop at his booth for an autograph and chat session. Don't forget his seminar on how to hunt those mountains and stay successful.
Travel with Bob Foulkrod to Eastern Maryland on a sitka deer hunt in the swamp. Don't miss this unique bowhunting adventure.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,087
|
Q: Using d3j to select an element in an embedded SVG file <body>
<button type="button" id="changeButton" onClick="hide(this)">Hide</button><br>
<br>
<object data="graphdata.svg" id="myGraph">graph</object>
<script src="d3.js" charset="utf-8"></script>
<script>
state="visible";
function hide(button) {
console.log(d3.select("body").select("#myGraph").select("svg")); // ??
if (state=="visible") { value="none"; state="invisible"; }
else { value=""; state="visible"; }
button.firstChild.data = state=="visible" ? "Hide" : "Unhide";
}
</script>
</body>
The SVG in question is here.
The basic structure is
<svg>
<g id="edges"></g>
....
<g id="node-labels">
<text class="thing1"></text>
<text class="thing2"></text>
....
</g>
</svg>
I am trying to select the SVG element so I can set the attribute "display" to be "none", but every attempt I make returns a null element from .select().
A: The DOM selectors that D3 uses are, as their name suggests, confined to the particular DOM that you're in. By embedding an SVG like this, you end up with a new, separate DOM for it. The selector will not traverse DOM boundaries, but you can select the root element of the new DOM, change into it and run your selector there. The code would look something like this.
d3.select(document.getElementById("#myGraph").documentElement).select("svg");
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,880
|
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